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 |
|---|---|---|---|---|
3,088,503 | <p>For <span class="math-container">$r\in [0,1)$</span> define
<span class="math-container">$$
I(r) = \int_0^{2\pi}\frac{\cos(t) - r}{1 - 2r\cos t + r^2}\,dt.
$$</span>
Numerical experiments hint at <span class="math-container">$I(r) = 0$</span> for all <span class="math-container">$r\in [0,1)$</span> but I can't show ... | Myunghyun Song | 609,441 | <p>(Since it is not mentioned yet) The function
<span class="math-container">$$
P_r(t) = \sum_{n=-\infty}^\infty r^{|n|} e^{int} =\frac{1-r^2}{1-2r\cos t +t^2},\quad0\le r<1, 0\le t\le 2\pi
$$</span> is called the Poisson kernel. For any continuous function <span class="math-container">$f:[0,2\pi]\to \Bbb C$</span> ... |
1,009,996 | <p>Show that if $a,b$ are rational numbers and $x$ is a positive real number then $x^a$$x^b$ $=$ $x^{(a+b)}$</p>
<p>I honestly have no idea how to even do this. Anyone have any hints or a good explanation?
Thanks in advance!</p>
| Bumblebee | 156,886 | <p>We know that $$x^ax^b=x^{a+b}$$ is true for positive integer powers $a,b$ and also it can be easily extend to negative integers also using $$x^{-a}=\dfrac{1}{x^a}$$ where $x$ is a positive real number. Note that using usual rules for integer exponents we can prove that, $$\Big(x^{\frac{p_1}{q_1}}x^{\frac{p_2}{q_2}}\... |
2,089,980 | <p>Those are the mechanics:</p>
<ul>
<li>You always draw from 2 to 10 cards.</li>
<li>You only use from A to 10. So 40 cards.</li>
</ul>
<p>What I want to obtain:</p>
<ul>
<li>The % of times you gonna obtain a number repeated.</li>
</ul>
<p>I want to make a graphic with how much % of at least one repeat you obtain ... | b00n heT | 119,285 | <p>Hint: shuffle the boxes. In which case do you obtain new combinations, and in which you don't?</p>
|
1,080,833 | <p>There are several nontrivial powers containing every digit equally often, for example</p>
<p>$32043^2$ $2158479^3$ $69636^4$ $643905^5$ $3470187^6$</p>
<p>A necessary condition for a power with the desired property is that the base is
divisible by $3$ because the power must be a multiple of $9$.</p>
<p>My questi... | Robert Israel | 8,508 | <p>This is not a proof, but a heuristic argument.</p>
<p>Of the $10^{10m} - 10^{10m-1}$ positive integers with $10m$ digits,
$\dfrac{9}{10} \dfrac{(10m)!}{m!^{10}} \approx \dfrac{9}{32 \sqrt{5}} m^{-9/2} 10^{10m}$ have all digits equally often. Now the number of $k$'th powers with $10m$ digits is approximately
$(1 -... |
4,086,718 | <p>Prove <span class="math-container">$\lim\limits_{(x,y) \to (0,0)} \frac{xy(y^2-x^2)}{x^2+y^2}=0$</span></p>
<p>My attempt: For all <span class="math-container">$\epsilon > 0$</span> there is a <span class="math-container">$\delta > 0$</span> such that <span class="math-container">$\left | \frac{xy(y^2-x^2)}{x^... | Davide Trono | 494,745 | <p>You should try with polar coordinates, it gets way easier. Impose:
<span class="math-container">$$\begin{cases} x=\rho\cos\theta\\y=\rho\sin\theta
\end{cases}
$$</span>
and <span class="math-container">$(x,y)\longrightarrow(0,0)$</span> will turn into <span class="math-container">$\rho\longrightarrow0$</span>. As th... |
103,603 | <p>There is a lock with the password between 000-999. But we could use only two digits to open the lock. For example, if the password is 123, the lock is open when you try 1*3, or *23, or 12*.
questions:
a) compose a strategy that the try number of opening the lock is the smallest one under the worst situation
b) Prov... | AD - Stop Putin - | 1,154 | <p>Algorithm: </p>
<p>100 tests are sufficient <br>
1. 00* <br>
2. 01* <br>
3. 02* <br>
... <br>
9. 08* <br>
10. 09* <br>
11. 10* <br>
... <br>
90. 89* <br>
91. 90* <br>
... <br>
99. 98* <br>
100. 99* <br></p>
<p>My guess is that this can not be improved (if we are to be certain about finding the pw)... |
1,857,250 | <p>I am stuck on this problem - I really don't know how to set it up.</p>
<p>A soft drink can is $h$ centimeters tall and has a radius of $r$ cm. The cost of material in the can is $.0005$ cents per cm$^2$ and the soda itself costs $.001$ cents per cm$^3$. The cans are currently $9$ cm tall and have a radius of $4$ cm... | callculus42 | 144,421 | <p>You can estimate the changes of the surface and the volume by using the <a href="https://en.wikipedia.org/wiki/Total_derivative" rel="nofollow">total derivative</a>. </p>
<p>The formula for the surface S is $S= 2\cdot \pi r^2+2\cdot \pi r\cdot h$</p>
<p>And the change of the surface is approximately</p>
<p>$dS\a... |
761,578 | <p>I am looking at the Cantor Intersection Theorem from Apostol's Mathematical analysis.</p>
<p>Let {$Q_1, Q_2, $...} be a countable collection of nonempty sets in $R^n$ such that:</p>
<p>1) $Q_{k+1} \subset Q_k$</p>
<p>2) Each set $Q_k$ is closed and $Q_1$ is bounded.</p>
<p>Then the intersection $\bigcap_{k=1}^\i... | Prahlad Vaidyanathan | 89,789 | <p>$$
(34)(123)(456) = \begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 \\
1 & 2 & 4 & 3 & 5 & 6
\end{pmatrix} \begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 \\
2 & 3 & 1 & 4 & 5 & 6
\end{pmatrix} \begin{pmatrix}
1 & 2 & 3 & 4 & 5 & 6 \\
1... |
1,225,643 | <p>I was able to show that when $p ≥ 1$, the $L^p$ space on the interval $[0,1]$ has a countable dense set.</p>
<p>However, when $p$ is infinite, how to prove that $L^p$ space on the interval $[0,1]$ does not have a countable dense set? I can't find some way to approach.</p>
| Learnmore | 294,365 | <p>Consider all those elements $e_i$ whose terms are either $0$ or $1$ .They all belong to $L^\infty$ and they are uncountable having cardinality $c$</p>
<p>$||e_i-e_j||=1$</p>
<p>Now if we a countable dense set $D$ say then we should have for each $e_i$ an element $d_i$ such that $||e_i-d_i||<\epsilon $ for any $... |
1,225,643 | <p>I was able to show that when $p ≥ 1$, the $L^p$ space on the interval $[0,1]$ has a countable dense set.</p>
<p>However, when $p$ is infinite, how to prove that $L^p$ space on the interval $[0,1]$ does not have a countable dense set? I can't find some way to approach.</p>
| Umberto P. | 67,536 | <p>Let $f_s(x) = \chi_{[0,s]}(x)$. If $0 \le s < t \le 1$, what is $\|f_s - f_t\|_\infty$?</p>
|
276,264 | <p>Say we are given a system of differential equations </p>
<p>$$
\left[ \begin{array}{c} x' \\ y' \end{array} \right] = A\begin{bmatrix} x \\ y \end{bmatrix}
$$ </p>
<p>Where $A$ is a $2\times 2$ matrix. </p>
<p>How can I in general solve the system, and secondly sketch a solution $\left(x(t), y(t) \right)$... | Did | 6,179 | <p>Quite generally,
$$
\left[ \begin{array}{c} x(t) \\ y(t) \end{array} \right] = \mathrm e^{tA}\begin{bmatrix} x(0) \\ y(0) \end{bmatrix},
$$
where, by definition,
$$
\mathrm e^{tA}=\sum_{n=0}^{+\infty}\frac{t^n}{n!}A^n.
$$
To compute $\mathrm e^{tA}$ in the case at hand, note that $A^2=4A$, hence
$$
\mathrm e^{tA}=I+... |
3,318,166 | <p>This may be a silly question but I want to make sure my general intuition is correct. I don't think it is possible for an <span class="math-container">$n$</span>-dimensional manifold to live inside <span class="math-container">$\mathbb{R}^m$</span> for <span class="math-container">$m < n$</span> because in the "b... | Rybin Dmitry | 606,842 | <p>Your intuition is right. There are rigorous ways to show that. I've seen it done often the following way:</p>
<p>By definition <span class="math-container">$n$</span>-dimensional manifold is covered by open subset of <span class="math-container">$\mathbb{R}^{n}$</span>, e.g. open balls. Suppose there is an inclusio... |
2,076,379 | <p>I am asked:
<em>Determine whether the relation X on the set Z is reflexive, symmetric, antisymmetric, and/or transitive, where $(a,b) ∈ X$ if and only if a = 1</em>.</p>
<p>Apparently, the set is antisymmetric and transitive. I understand how it is not reflexive or symmetric, but I don't get why it is transitive. I... | Patrick Stevens | 259,262 | <p>You want to show that if $a \sim b$ and $b \sim c$ then $a \sim c$, where I've written $x \sim y$ for the statement that $(x,y) \in X$.</p>
<p>If $a \sim b$, then $a=1$.
Hence $a \sim c$ for all $c$.
We didn't actually need to assume that $b \sim c$.</p>
<hr>
<p>For completeness, I'll go over why it's antisymmetr... |
654,942 | <p>Im trying to use propositional logic to break this down but i have no clue. i know about the rule that if a wff ends in form <code>....implies (a implies b)</code>, the a can be added to the hypothesis in order to only get <code>b</code>. but in my problem ~A implies (A implies B) and my following hypothesis are <co... | amWhy | 9,003 | <p>$$\begin{align} \lnot A \rightarrow (A \rightarrow B) &\equiv (\lnot A \land A) \rightarrow B \\ \\ & \equiv \text{False} \rightarrow B \\ \\ & \equiv \text{True}\end{align}$$</p>
<p>Recall that in an implication, if the antecedent is false, the implication as a whole is true, regardless of the truth v... |
654,942 | <p>Im trying to use propositional logic to break this down but i have no clue. i know about the rule that if a wff ends in form <code>....implies (a implies b)</code>, the a can be added to the hypothesis in order to only get <code>b</code>. but in my problem ~A implies (A implies B) and my following hypothesis are <co... | Doug Spoonwood | 11,300 | <p>If you have $\lnot$A, and [$\lnot$A$\rightarrow$(A$\rightarrow$B)], then by modus ponendo ponens (or modus ponens or detachment) you then may infer (A$\rightarrow$B).</p>
<p>Since you have A also, and now you have (A$\rightarrow$B), then you may infer B by modus ponendo ponens. </p>
|
3,514,934 | <blockquote>
<p>If <span class="math-container">$x = 2 + \sqrt{3}$</span> then find <span class="math-container">$x^3 + \dfrac1{x^3}$</span>. </p>
</blockquote>
<p>My attempt: we know that the sum of a number and its reciprocal is always greater than or equal to <span class="math-container">$2$</span> but I don't kn... | an4s | 533,556 | <p><strong>Hint:</strong></p>
<p><span class="math-container">$$\begin{align}x &= 2 + \sqrt3\\x^3 &= (2 + \sqrt3)^3 =\,?_1 \\ \frac1{x^3} &= \dfrac1{2 + \sqrt3} = \frac1{2 + \sqrt3}\cdot\frac{2 - \sqrt3}{2 - \sqrt3} =\,?_2\\ x^3 + \frac1{x^3} &=\,?_1 + ?_2\end{align}$$</span></p>
<p>Also, what is <spa... |
418,880 | <p>Helmut Wielandt <a href="https://books.google.com/books/about/Finite_Permutation_Groups.html?id=npviBQAAQBAJ" rel="nofollow noreferrer">discussed</a> an old question (Chap. 2, Section 15, which can be dated back to Camille Jordan):</p>
<p>Let <span class="math-container">$g\neq 1$</span> be a permutation in some fin... | Seven9 | 78,173 | <p>I found the motivation in Luc Tartar's book <em>An Introduction to Sobolev Spaces and Interpolation Spaces</em>. It is in the first page of Chapter 24.</p>
<p>I quote:</p>
<blockquote>
<p>The K-method is the natural result of investigations which originated in questions of traces: if <span class="math-container">$u ... |
1,325,406 | <p>I'm having trouble grasping the proof of theorem E, section 5, chapter 1 in Halmos' <em>Measure Theory</em>.</p>
<p>Let $X$ be a nonempty set, and $\Omega$ a family of subsets of of $X$. Given $A\subset X$, denote by $\Omega \sqcap A$ the class of all sets of the form $E \cap A$ with $E \in \Omega$. Further, denote... | R.Findlayson | 564,489 | <p>I think the unstated part of this is that $S(\Omega\cap A)$ should be $S_A(\Omega\cap A)$ i.e. the $\sigma$-ring generated with respect to the set $A$ not the whole set $X$</p>
|
2,515,002 | <p>I was looking for an elementary proof for Sobolev's Inequality in one dimension and I found the proof in <a href="http://www-users.math.umn.edu/~garrett/m/fun/simplest_cptness.pdf" rel="nofollow noreferrer">this paper</a>, it's the proof of Sobolev imbedding. I would like an explanation about some steps of the proof... | David C. Ullrich | 248,223 | <p>Looking at the link, I actually don't think the first chain of inequalities is very clearly presented. Here's how I'd say the same thing: </p>
<p>Assume first that $f\in C^\infty$. Now for every $t$ it's clear that $$|f(x)|\le|f(t)|+|f(x)|+|f(x)-f(t)|.$$</p>
<p>Now since $|f'|=|f'|\cdot 1$, Cauchy-Schwarz shows th... |
288,097 | <p>For $x \in \mathbb{R}^n$ let us denote $x_{[i]}$ the $i$th largest component of $x$ s.t
$$
x_{[1]} \geq x_{[2]} \geq x_{[3]}\ge\cdots
$$
The function
$$
f(x)= \sum_{i=1}^r x_{[i]}
$$
is the sum of $r$ largest element of $x$. Here I want to show that for each $x \in \mathbb{R}^n$
$$
y^Tx \leq f(x)
$$
for $y \in \mat... | Arin Chaudhuri | 404 | <p>Assume wlog that $ x_1 \geq x_2 \geq \dots \geq x_n $.</p>
<p>Consider first the special case where the $x_i$'s are pairwise distinct,
i.e., $ x_1 > x_2 > \dots > x_n .$</p>
<p>The set $$ S = \{ (y_1,y_2,\dots,y_n) : y_i \in [0,1], \sum y_i = r \}$$ is compact, and hence there is a $y=(y_1,y_2,\dots,y_n) ... |
3,527,470 | <p>Compute <span class="math-container">$$\lim\limits_{n\to \infty} \int\limits_0^1 x^{2019} \{nx\} dx,$$</span> where <span class="math-container">$\{a\}$</span> denotes the fractional part of the real number <span class="math-container">$a$</span>.<br>
I firstly tried to apply the substitution <span class="math-conta... | Doug M | 317,176 | <p>For a finite value of <span class="math-container">$n$</span> our equation The graph of our function looks like a saw tooth, that touches the curve <span class="math-container">$x^{2019}$</span> when <span class="math-container">$x$</span> is a multiple of <span class="math-container">$\frac {1}{n}$</span></p>
<p>T... |
3,527,470 | <p>Compute <span class="math-container">$$\lim\limits_{n\to \infty} \int\limits_0^1 x^{2019} \{nx\} dx,$$</span> where <span class="math-container">$\{a\}$</span> denotes the fractional part of the real number <span class="math-container">$a$</span>.<br>
I firstly tried to apply the substitution <span class="math-conta... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
108,523 | <p>Is there, similar to the Mehler kernel, a closed formula for the heat kernel of the heat equation associated to the Laplacian
$$ -\sum_j \frac{d^2}{dx_j^2} + 2\sqrt{-1} \sum_j \lambda_j \frac{d}{dx_j} + \sum_{ij} a_{ij}x_ix_j$$
on $\mathbb{R}^n$? Here, the matrix $(a_{ij})$ is supposed to be symmetric and positive d... | Liviu Nicolaescu | 20,302 | <p>Yes there is. Here is how you do it. First find an orthogonal change in variables </p>
<p>$$ x_j=\sum_{jk} s_{jk}y_k $$</p>
<p>$(s_{jk})$ orthogonal matrix, so that in the new coordinates we have</p>
<p>$$ \sum_{i,j}a_{ij} x_ix_j = \sum_j \mu_j^2 y_j^2, $$</p>
<p>where $\mu_j^2$ are the eigenvalues of the sy... |
12,006 | <p>Recently I was asked to tutor some students individually on a basic course of Linear Programming, centered on all the "typical" introductory subjects like graphical methods, Simplex Algorithm and Transfer Pricing.</p>
<p>Now, this is not the first time I teach someone on a University level, but it is the first time... | Community | -1 | <blockquote>
<p>Should I teach a subject I don't like?</p>
</blockquote>
<p>It depends on other factors, but don't automatically <strong>not</strong> teach a subject just because you don't like it. I'm currently teaching a course that combines linear algebra with differential equations and I really can't stand line... |
2,072,294 | <p>Suppose I want a card A from a deck. I shuffle a deck and draw a card. The probability that this card is $A$ is $1/52$.</p>
<p>Now, suppose I reshuffle this deck n times and draw a card each time. I intend to find the mean number of times I would draw card $A$. Would I have to model this as a Binomial random variab... | Fernando Revilla | 401,424 | <p>$\displaystyle\lim_{h\to 0}\dfrac{\Gamma(h+\alpha) \;h}{\Gamma(h+1)\;\Gamma(\alpha)\; h}=\lim_{h\to 0}\dfrac{\Gamma(h+\alpha) }{\Gamma(h+1)\;\Gamma(\alpha)}\underbrace{=}_{\Gamma\text{ continuous}}\frac{\Gamma(\alpha)}{\Gamma(1)\Gamma(\alpha)}=1.$</p>
|
27,968 | <p>What is the "standard" way to denote all positive (or non-negative) real numbers? I'd think</p>
<p>$$ \mathbb R^+ $$</p>
<p>but I believe that that is usually used to denote "all real numbers including infinity".</p>
<p>So is there a standard way to denote the set</p>
<p>$$ \{x \in \mathbb R : x \geq 0\} \; ?$$ ... | Raphael | 3,330 | <p>Not that I knew of. There are many, e.g.</p>
<ul>
<li>$\mathbb{R^+_0}$,</li>
<li>$\mathbb{R^+}$ and</li>
<li>$[0, \infty)$.</li>
</ul>
|
268,525 | <p>I have heard stated the following</p>
<p><strong>Theorem.</strong> If $\Sigma$ is a (orientable) surface, then $\mathrm b_1(\Sigma)$ counts the maximum number of "circular cuts" (embedded circles $C_1,\ldots,C_m$) that you can make on $\Sigma$ without disconnecting it (i.e. with $\Sigma\smallsetminus (C_1\cup\ldots... | Bruno Martelli | 6,205 | <p><b> Edit. </b> I misunderstood the question, so what follows may not be relevant, but it may be useful anyway.</p>
<p>The <i>cut number</i> $c(M)$ of a manifold $M$ is usually defined as the maximum number of disjoint two-sided codimension-1 submanifolds whose complement is connected. By collapsing such a configura... |
268,525 | <p>I have heard stated the following</p>
<p><strong>Theorem.</strong> If $\Sigma$ is a (orientable) surface, then $\mathrm b_1(\Sigma)$ counts the maximum number of "circular cuts" (embedded circles $C_1,\ldots,C_m$) that you can make on $\Sigma$ without disconnecting it (i.e. with $\Sigma\smallsetminus (C_1\cup\ldots... | Tyler Lawson | 360 | <p>Here is one way that your observation about the first Betti number generalizes to higher-dimensional manifolds.</p>
<p>Suppose that $M$ a compact, orientable smooth manifold and $W$ is a regularly embedded, closed, orientable submanifold of codimension 1. Then if the number of path components of $W$ is greater than... |
1,280,140 | <p>If $V$ is a $k$-vector space and $T : V \to k$ is a homomorphism (linear map) that is not the zero map, is it true that some element of $V$ is mapped to $1$ of $k$? Of course $k$ is a 1-dimensional $k$-vector space generated by $1$ but how can I see that there exists a $v \in V$ such that $T(v) = 1$?</p>
| J126 | 2,838 | <p>The image of any linear map has to be a subspace. How many subspaces does $k$ have?</p>
|
748,588 | <p>Imagine you have two finite alternating series.</p>
<p>$$S_a=a_1-a_2+a_3-a_4+\cdots+a_n$$</p>
<p>$$S_b=b_1-b_2+b_3-b_4+\cdots+ b_n$$</p>
<p><strong>Question:</strong>
If $|a_i|>|b_i|$ is $S_a>S_b$?</p>
| user2566092 | 87,313 | <p>No, suppose all $a_i$ are equal and set to a large positive number $N$, and $n$ is even. Then $S_a = 0$. Certainly you can define an alternating sequence of numbers with absolute value less than $N$ such that the sum $S_b$ is positive.</p>
|
3,033,628 | <p>I have Ax=b,
<span class="math-container">$$
\begin{bmatrix}
1 & x & x^2 \\
x & 1 & x \\
x^2 & x & 1
\end{bmatrix}
\begin{bmatrix}
a \\ b \\ c
\end{bmatrix} =
\begin{bmatrix}
x \\ x^2 \\ x^3
\end{bmatrix}
$$</span>
and I transform it into an augmented matrix:</p>
<p><span class="math-contain... | José Carlos Santos | 446,262 | <p>We only can divide a row by <span class="math-container">$x$</span> if <span class="math-container">$x\neq0$</span>. Therefore, the case in which <span class="math-container">$x=0$</span> should be treated as a special case. More generaly, you can only divide a row by <span class="math-container">$f(x)$</span> if <s... |
4,087,664 | <p>Solve the inequality <span class="math-container">$\frac{7 - |2x+1|}{2\cos(x)-7} < 0 $</span>.<br />
I have no clue how to solve this inequality. Please give a hint.</p>
| Aryaman Maithani | 427,810 | <p>My other answer finishes it in the way you were trying and also answers your generalisation.</p>
<p>However, here's another way of doing that for this simple case:<br />
Note that we have the tower of subfields <span class="math-container">$$\Bbb F_2 \subset \Bbb F_{2^2} \subset \Bbb F_{2^4}$$</span>
and there are n... |
1,554,108 | <p>Suppose $f$ is a nonnegative measurable function on a measure space $(X,M,\mu)$ satisfying $\int f d\mu < \infty$. Show that for every $\epsilon > 0 $ there exists a $\delta > 0 $ such that if $\mu(E) < \delta$ then $$\int_{E}f d\mu < \epsilon$$</p>
<p>Attempted proof: Let $E_n = \{f \geq 1/n\}$, the... | Matt Dickau | 273,941 | <p>After the first rotation you have a vector at a 45 degree angle from the z-axis; rotating that vector around the z-axis with the second rotation will trace out a cone. The z-component of the vector does not change with the second rotation which is why your method doesn't work.</p>
<p>To get the desired result, your... |
2,377,728 | <p>Suppose $F$ is a family of sets. Then the intersection of $F$ is the set $\cap F$ defined as follows:</p>
<p>$$ \cap F = \{x \mid \forall A. A \in F \rightarrow x \in A)\}$$</p>
<p>I am having trouble understanding this. This is how I am currently reading it:</p>
<p>The intersection of $F$ (which is a family of s... | Kiryl Pesotski | 417,344 | <p>You have
$$a_{n+1}=c_{n}a_{n}$$
Devide by $\prod_{k=0}^{n}c_{k}$ to give
$$\frac{a_{n+1}}{\prod_{k=0}^{n}c_{k}}=\frac{a_{n}}{\prod_{k=0}^{n-1}c_{k}}$$
Let
$$\frac{a_{n}}{\prod_{k=0}^{n-1}c_{k}}=A_{n}$$
Then
$$A_{n+1}=A_{n}$$
Hence $A_{n}=A$, where $A$ is constant
$$A=\frac{a_{n}}{\prod_{k=0}^{n-1}c_{k}}$$
So
$$a_{n}... |
2,552,118 | <p>I know this might be stupid for most of the people here. I am starting with set theory and I cannot prove this:A-(A∩B)=A-B</p>
<p>I start by assuming: 1) x∈A & ¬ (x∈A & x∈B)</p>
<pre><code> Therefore I understand I can infer:
2) x∈A from 1
3) ¬ (x∈A & x∈... | Mariah | 319,890 | <p>You'd do this by set inclusion.
If we can show that $A - (A \cap B) \subset A-B$, and that $A-B \subset A - (A \cap B)$, we'd conclude equality.</p>
<p>So let $x \in A-B$. Then $x \in A, x \notin B$, so $x \notin A \cap B$, and so $x \in A - (A \cap B)$.</p>
<p>If $x \in A - (A \cap B)$, $x\in A$ but not in $ A \c... |
2,295,515 | <p>I was doing a problem and I ran into this triangle:</p>
<p><a href="https://i.stack.imgur.com/sFSxM.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sFSxM.png" alt="enter image description here"></a></p>
<p>The problem itself is not that important for this question, but rather the explanation of ... | fleablood | 280,126 | <p><a href="https://i.stack.imgur.com/LhIvj.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/LhIvj.jpg" alt="enter image description here"></a> </p>
<p>Notice the red triangle and the blue triangle are similar as the two sides of the blue triangle (6.4 and 6.0) are half of the sides of the red triang... |
3,561,155 | <p>For real numbers <span class="math-container">$a$</span> and <span class="math-container">$b$</span> define</p>
<p><span class="math-container">$$f(x) = \frac1{ax+b}$$</span></p>
<p>For which <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are there three distinct real numbers <... | Batominovski | 72,152 | <p>Here is a linear algebra approach. It is long because I am trying to explain what is going on. However, the approach of pairing a map of the form <span class="math-container">$f(x)=\frac{ax+b}{cx+d}$</span> with the matrix <span class="math-container">$A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$</span> is very... |
440,806 | <p>I've been trying to solve this problem and I always get 1, but the answer is $1 - 2x$.</p>
<p>If $x<\frac12$ then what is $\left|x-\sqrt{(x-1)^2}\right|$ ?</p>
| André Nicolas | 6,312 | <p>Note that if $c\ge 0$, the square root of $c$ is defined as the <strong>non-negative</strong> number whose square is $c$.</p>
<p>Thus $\sqrt{a^2}=|a|$. You can check this with for example $a=-3$. We have $\sqrt{(-3)^2}=\sqrt{9}=3$. So $\sqrt{(-3)^2}\ne -3$. </p>
<p>In particular, if $x\lt 1$, then $\sqrt{(x-1)^2}=... |
440,806 | <p>I've been trying to solve this problem and I always get 1, but the answer is $1 - 2x$.</p>
<p>If $x<\frac12$ then what is $\left|x-\sqrt{(x-1)^2}\right|$ ?</p>
| egreg | 62,967 | <p>The fundamental thing you have to use is
$$
\sqrt{a^2}=|a|
$$
So your expression is
$$
\bigl|x-|x-1|\bigr|=\bigl|x-(1-x)\bigr|=\bigl|2x-1\bigr|=1-2x
$$
Indeed $x<\frac{1}{2}$ entails $x<1$, so $|x-1|=1-x$. Also $2x-1<0$, so $|2x-1|=1-2x$.</p>
|
2,404,946 | <p>I am an architect, and just started using the parametric software "Grasshopper", trying to model and reshape surfaces based on geometric formulas.</p>
<p>That's where I came up with the terms "Mean Curvature" and "Gaussian Curvature",
and have no idea what they mean.</p>
<p>Can someone please tell me in plain Engl... | Kajelad | 354,840 | <p>I'll assume you are considering only 2-dimensional surfaces, since higher dimensional cases shouldn't come up very often in architecture. First, however, we can examine the one dimensional case.</p>
<p>For a smooth curve, at a point $P$ we can draw a circle that best "fits" the local curvature of the curve. We call... |
1,682,306 | <p>I am looking to prove, by induction, the following equality:$${n \choose k} = {n \choose k-1}\frac{n-k+1}{k}$$ From Pascal's identity, I know we have that $${n+1 \choose k} = {n \choose k} + {n \choose k-1}$$ The problem gives the following hint: get ${n \choose k-1}$ and ${n+1 \choose k-1}$ in terms of ${n \choose ... | Eleven-Eleven | 61,030 | <p>$$\binom{n}{k}=\frac{n!}{(n-k)!k!}=\frac{(n-k+1)n!}{(n-k+1)!k!}=\frac{n!(n-k+1)}{(n-k+1)!(k-1)!k}=\frac{n!}{(n-k+1)!(k-1)!}\frac{(n-k+1)}{k}=\binom{n}{k-1}\frac{n-k+1}{k}$$</p>
|
982,582 | <p>Assume that $W$ and $V$ are two subspace of $X$. Is their union a subspace of $X$ too? I think it is not true unless under certain conditions but I do not know what conditions...</p>
| Eclipse Sun | 119,490 | <p>A necessary and sufficient condition is that<br>
$$W\subseteq V \text{ or } V\subseteq W.$$The sufficiency is clear.<br>
If $W\bigcup V$ is a subspace, suppose that there exist an $x\in W-V$, and a $y\in V-W$. Then we have $x+y\notin W\bigcup V$, contradiction! Hence, either $x$ or $y$ does not exist, which leads to... |
2,828,070 | <p>If $$\log_{3x+5} (ax^2+8x+3)>2$$ find the interval in which x lies.</p>
<p>Answer is $\left(-\dfrac43 , -\dfrac{23}{22}\right)$. Now hint me how to solve the problem please.</p>
| Alejandro Sánchez Yalí | 217,786 | <p>You can suposse that we have the space $\Omega = [-1, 1]\times [-1, 1]$ and the events $A = [0, 1]\times [0, 1]$ and $B = [-1, 0]\times[-1, 0]$. It's clear that $A\cap B = \{(0, 0)\}$. If you consider the geometric probability, then $P(A)=P(B)=1/4$ and $P(A\cap B) = 0$ and you can see that $A\cap B\neq \emptyset$. G... |
29,435 | <p>Suppose $f$ and $g$ are two embeddings of $S^1$ in $R^3$ or in $S^3$. How do I show whether they form a link or not?</p>
| Carl Brannen | 8,536 | <p><a href="http://en.wikipedia.org/wiki/Linking_number" rel="nofollow">Gauss's linking integral</a> will give the linking number:
$$N = \frac{1}{4\pi}\int\int \frac{\vec{f}(\sigma)-\vec{g}(\tau)}{|\vec{f}(\sigma)-\vec{g}(\tau)|^3}\cdot \left(\frac{d\vec{f}}{d\sigma}\times \frac{d\vec{g}}{d\tau}\right)\;d\sigma\;d\tau$... |
2,321,285 | <p>For my abstract algebra course I have to decide whether
$$(*)\qquad X^4 + a^2, \; a \in \mathbb{Z}\;\text{odd}$$ is irreducible over $\mathbb{Q}[X]$ and $\mathbb{Z}[X]$. </p>
<p>Since the degree of the polynomial is 4 and thus even I should be able to split it into two polynomials of degree two $if$ the coefficie... | Paolo Leonetti | 45,736 | <p>Since $d=-b$ you have $e+c=b^2$ and $ec=a^2$, with $a$ odd (hence $b$ even). Then $e,c$ are roots of $x^2-b^2x+a^2=0$, i.e.,
$$
(2x-b^2)^2=b^4-4a^2.
$$
But $b^4-4a^2$ cannot be a square (of a even integer $2c$). Indeed, in the opposite case, setting $b=2B$ you would have
$$
(2B)^4-4a^2=(2c)^2 \implies a^2+c^2=4B^4 \... |
2,634,752 | <p>So the question is to solve this differential equation:
$y'' - 4y' + 5y = 1 + t$.</p>
<p>I've got the general solution to $\ C_1e^{2t}\cos(2t) + C_2e^{2t}\sin(2t) $</p>
<p>However, I don't really know how to proceed with the particular solution.</p>
<p>I know that t is a polynomial of the first degree and should... | PM. | 416,252 | <p>The method of undetermined coefficients has been called 'educated guesswork' for finding particular solutions. One attempts trial solutions based on the form of the right-hand-side (RHS), bearing in mind the form of the complementary function.
In this case, the RHS is a polynomial of degree 1, and as it does not app... |
1,936,963 | <p>Let all other solutions diverge from $y = \frac{3}{4}$. write a differential equation $(\frac{dy}{dt} = ay +b)$ whose solutions have the required behavior as t goes to infinity. Explain how you came up with the differential equation).</p>
<p>I think the answer is $y' = 4y - 3$ but I am not sure. Can someone help me... | Stefan4024 | 67,746 | <p>As the sequence is positive we have that:</p>
<p>$$x_{n+1} \ge \sqrt{2} \iff \frac{x_n^2 + 2}{2x_n} \ge \sqrt{2} \iff (x_n - \sqrt{2})^2 \ge 0$$</p>
<p>Using this we have that</p>
<p>$$x_{n+1} - x_{n} = \frac{2-x_n^2}{2x_n} \le 0$$</p>
<p>Therefore the sequence is bounded from below and it's decreasing, hence it... |
38,721 | <p>I encountered a really strange problem with <code>BarLegend</code>.
Say you have a user-defined color function:</p>
<pre><code>cf[z_]:=RGBColor[RandomReal[z, 3]]
</code></pre>
<p>When you want to use it in <code>BarLegend</code> as below:</p>
<pre><code>BarLegend[{cf, {1, 10}}]
</code></pre>
<p>Mathematica will ... | Meir Bar-Tal | 46,717 | <p>With this response I tried:</p>
<pre><code>maxmin = {10, 20};
BarLegend[{Function[x, Hue[0.8 Rescale[x, maxmin]]], maxmin}]
</code></pre>
<p>which worked nicely to truncate the Hue function to remove the red at the top</p>
|
2,676,479 | <p>Say you have $f(x) $ and $g(x)$ and $g(x) = f^{-1}(x) $.</p>
<p>I observed that these two curves need not intersect, for example with $f(x) = e^x$ and $g(x) = \ln x $ never intersecting each other.</p>
<p>I also observed that a function can either have one, two, or three intersections with its inverse, but I was u... | Community | -1 | <p>Consider $f(x) = x$.</p>
<p>Note that $g(x) = f^{-1}(x) = x = f(x)$.</p>
<p>So, there are infinitely many intersections!</p>
<blockquote>
<p>Good solution. I am aware of the infinite intersections solution, but does anyone have any functions which have 4 or more intersections with
their inverse (but not an i... |
1,401,972 | <p>I have the sequence of functions: $s_n(x)= \displaystyle\sum_{k=0}^n \frac{k^2x}{1+k^4x^2}$ where $x \in [0,1]$</p>
<p>Consider$ [\delta, 1]$ for any $\delta \in (0,1]$</p>
<p>Now $0 \leq \frac{k^2x}{1+k^4x^2}\leq \frac{1}{k^2x} \leq \frac{1}{k^2\delta} $
So for any $x \in [\delta,1]$ The series converges uniform... | Joel | 85,072 | <p>Hint: Consider the sequence $\{ 1/n^2 \}_{n=1}^\infty$. Can we get a lower bound bigger than $0$ on $s(1/n^2)$? If so, then $\lim_{x\to 0} s(x) \neq 0 = s(0)$.</p>
<p>The point to demonstrating that this is not continuous on $[0,1]$ is to show that $s_n(x)$ does not converge uniformly to $s(x)$, since otherwise $s(... |
23,704 | <p>I hit a snag whilst revising some log rules, could anyone confirm my suspicion:</p>
<p>$$\log _b \left( x \right) = \log _b \left( y \right) \rightarrow x = y ?$$</p>
| yoyo | 6,925 | <p>$\log(x)$ is increasing, for instance its derivative $1/x$ is positive.</p>
|
4,412,423 | <p>Hello I was trying to prove</p>
<p><span class="math-container">$$\sum_{0 \leq k \leq n}^{} \binom{n-k}{k} = \operatorname{Fib}_{n+1}$$</span>
which the approach I made was</p>
<ol>
<li>Prove that it holds for <span class="math-container">$n=0$</span>, <span class="math-container">$n=1$</span>.</li>
<li>Then use the... | Egg Man | 1,006,586 | <p>The proof of this went like :</p>
<p><span class="math-container">$$(\nabla _a \nabla _b -\nabla_b \nabla )(fw^c)=f((\nabla _a \nabla _b-\nabla _b \nabla _a) w^c))$$</span></p>
<p><span class="math-container">$f$</span> is a scalar field. Take a special case of this identity. Take <span class="math-container">$f$</s... |
51,301 | <p>With <code>n = 17</code> I would like to get </p>
<blockquote>
<p>{4, 1}</p>
</blockquote>
<p>and with <code>n = 999</code></p>
<blockquote>
<p>{31, 6, 1, 1}</p>
</blockquote>
<p>so that, for example,</p>
<pre><code>Total[#^2 & /@ {31, 6, 1, 1}]
</code></pre>
<p>gives</p>
<blockquote>
<p>999</p>
</b... | kglr | 125 | <pre><code>f = Module[{n = 1, res}, While[(res = Reduce`SumOfSquaresReps[n, #]) == {}, n++]; res]&;
f[17]
(* {{1, 4}} *)
f[999][[1]]
(* {1, 1, 6, 31} *)
f[999]
(* {{1, 1, 6, 31}, {1, 6, 11, 29}, {1, 7, 7, 30}, {1, 7, 18, 25},
{1, 10,13, 27}, {1, 14, 19, 21}, {1, 15, 17, 22}, {2, 3, 5, 31},
{2, 3, 19, 25... |
2,349,503 | <p>Let $f(x)=\frac{1}{1-x^2}$. </p>
<p>With a $u=x^2$ substitution, $f(x)$ becomes $g(u)=\frac{1}{1-u}$, which being the limit of a geometric series has the form $$g(u)=\sum_{k=0}^\infty u^k$$</p>
<p>Plugging the substitution back in yields a power series representation for $f(x)$</p>
<p>$$\frac{1}{1-x^2}=\sum_{k=0}... | hamam_Abdallah | 369,188 | <p>Starting with</p>
<p>$$f (x)=\frac {1}{1-x^2}=\sum_{i=0}^{+\infty}a_{2i}x^{2i} $$</p>
<p>we have</p>
<p>$$1=(1-x^2)f (x)=$$
$$a_0+\sum_{i=1}^{+\infty}(a_{2i}-a_{2i-2})x^{2i}$$</p>
<p>thus
$$a_{2i}=a_{2i-2}=...=a_0=f (0)=1$$</p>
|
4,451,700 | <blockquote>
<p>Calculate the line integral
<span class="math-container">$$
\int_{L} \frac{-y \,d x+x \,d y}{x^{2}+y^{2}}
$$</span>
where <span class="math-container">$L$</span> is the line segment from <span class="math-container">$(1,0)$</span> to <span class="math-container">$(0,1)$</span> parametrized by
<span clas... | Travis Willse | 155,629 | <p>The "change in argument" is, informally, just the total net angle that the path winds around the origin anticlockwise (a negative change in argument indicates a net clockwise winding). We can see this from a computation similar to the one that gave you a potential of a restriction of <span class="math-cont... |
1,631 | <p>What's a good way to format a citation to a MathOverflow post in a paper?
Here is a first draft (with hyperlink, and assuming it's reference number 16, say), but maybe people can think of improvements:</p>
<blockquote>
<p>[16] Pietro Majer, answer of April 8, 2014, 17:26pm HST to <em>MathOverflow</em> question <... | Gerhard Paseman | 3,206 | <p>I am using a nonstandard citation format, primarily for simplicity. An example is:</p>
<p>MathOverflow Questions (Number refers to URL, so 37679
expands to "http:// mathoverflow.net/questions/37679") [Note: the current display scripts
replace the URL with a visible title and hidden link: I've added a space to brea... |
173,107 | <p>are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$\Big(\int_a^b{|f(x)|^p dx}\Big)^\frac{1}{p} = f(p)$$ $$\forall p\in(a,b)$$<br>
This question came up when thinking about $L_p$ norms as functions of $p$ and thus as a mappi... | Narutaka OZAWA | 7,591 | <p>The answer is NO and here's a counterexample.</p>
<p>Let $\prod_{n\in{\mathbb N}}{\mathbb M}_n$ be the $\ell_\infty$-direct sum of the full matrix algebras ${\mathbb M}_n$, and let $X\subset\prod_{n\in{\mathbb N}}{\mathbb M}_n$ be the weak$^*$-closed subspace defined by $$X=\{ (x_n)_{n=1}^\infty : \Theta_{m,n}(x_m)... |
3,789,885 | <p>I need help proving the following trig inequality:</p>
<blockquote>
<p><span class="math-container">$$\cos a-\cos b-\cos c\geq -\frac{3}{2}$$</span> where <span class="math-container">$a+b+c=2\pi$</span> and <span class="math-container">$a,b,c>0$</span>.</p>
</blockquote>
<p>I've found that equality occurs when <... | DeepSea | 101,504 | <p>Put <span class="math-container">$t = \cos\left(\frac{b+c}{2}\right) \implies LHS = 2t^2 -2t\cos\left(\frac{b-c}{2}\right) + \dfrac{1}{2} \ge 0$</span>. But this is clear viewed as a quadratic in <span class="math-container">$t$</span> and equality occurs at <span class="math-container">$\cos\left(\frac{b-c}{2}\righ... |
44,740 | <p>This question is closely related to MO <a href="https://mathoverflow.net/questions/4347/ffxexpx-1-and-other-functions-just-in-the-middle-between-linear-and-expo">f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential.</a> Consider $e^{e^x-1}$, this is the generating function of the ... | Gottfried Helms | 7,710 | <p>1) Concerning Bell-numbers and generalizations: you might be interested in the treatize</p>
<p><a href="http://go.helms-net.de/math/binomial_new/04_5_SummingBellStirling.pdf" rel="nofollow">http://go.helms-net.de/math/binomial_new/04_5_SummingBellStirling.pdf</a></p>
<p>where I deal with continuous interpolations ... |
44,740 | <p>This question is closely related to MO <a href="https://mathoverflow.net/questions/4347/ffxexpx-1-and-other-functions-just-in-the-middle-between-linear-and-expo">f(f(x))=exp(x)-1 and other functions “just in the middle” between linear and exponential.</a> Consider $e^{e^x-1}$, this is the generating function of the ... | Anixx | 10,059 | <p>Let $\sigma(x)=\exp(x)-1$ We know that $e^{\sigma(x)-1}$ is a generating function for Bell numbers</p>
<p>$$\exp(\sigma^{[p]}(t))=\sum_{n=0}^{\infty}B_n^p\frac{t^n}{n!}$$</p>
<p>where $B_n^p$ are the Bell's numbers of p-th order.</p>
<p>So to find $\sigma^{[1/2]}(t)$ we have to generalize Bell's numbers to fracti... |
2,885,412 | <ol>
<li>In a math textbook I found a statement that every empty set is equal.</li>
<li>There are such sets that it is impossible for them not to be empty: set of all natural numbers between 10 and 11.</li>
<li>There are such sets that it is possible for them not to be empty: all the people from the Earth, who are on... | Aniruddha Deshmukh | 438,367 | <p>Every empty set is same in the sense that if you take two empty sets, say $\emptyset_1$ and $\emptyset_2$, then they are contained in one another. You can in fact give a logical argument for this.</p>
<p>If you take any element $x \in \emptyset_1$ (which is none) it is also contained in $\emptyset_2$ and vice - ver... |
2,885,412 | <ol>
<li>In a math textbook I found a statement that every empty set is equal.</li>
<li>There are such sets that it is impossible for them not to be empty: set of all natural numbers between 10 and 11.</li>
<li>There are such sets that it is possible for them not to be empty: all the people from the Earth, who are on... | saulspatz | 235,128 | <p>Two sets are equal if and only if they contain the same elements. Obviously, there cannot be two different empty sets under this definition. The mistake in your description of three is that you say, that the set of all Earthling who are on Mars now is empty, but in the future there may be Earthlings on Mars. In t... |
1,293,971 | <p>Let $R = \mathbb{C}[x]$, $M = \mathbb{C}^2$. Does there exist a module structure over $M$ as an $R$-module or not?</p>
| P Vanchinathan | 28,915 | <p>To make an abelian group $M$ into a module over $R$ we need a ring homomorphism
from $R$ into the (non-commutative) ring of group endomorphisms of $R$. To make $\mathbf{C^2}$, a module we can construct a homomorphism that goes into the subring of vector space endomorphisms.</p>
<p>Here we can easily specify a ring ... |
1,293,971 | <p>Let $R = \mathbb{C}[x]$, $M = \mathbb{C}^2$. Does there exist a module structure over $M$ as an $R$-module or not?</p>
| Martin Brandenburg | 1,650 | <p>If $R$ is a commutative ring and $M$ is some $R$-module, then $R[x]$-module structures on $M$ extending the given $R$-module structure correspond 1:1 to $R$-linear endomorphisms of $M$.</p>
<p>So, any $\mathbb{C}$-linear map $\mathbb{C}^2 \to \mathbb{C}^2$ (and surely, there are plenty of them!) corresponds to a $\... |
2,344,414 | <p>Can anyone tell me how I should calculate this problem?</p>
<p>You are on vacation in a foreign country, and you need help. There are 50 people nearby, and you know <strong>25% of the population of this country speaks English</strong>. </p>
<p>If you were to call for help: </p>
<p>(a) <strong>What is the probabil... | Michael | 459,626 | <p>The probability of all $50$ not understanding is $(1-.25)^{50}$.</p>
|
1,552,992 | <p>Let $G$ be abelian. Let $n \in \mathbb{Z}^+$ such that the order of $G$ and $n$ are relatively prime. Show that the function $\varphi : G \rightarrow G$ defined by $\varphi (a) = a^n $ is an automorphism for $a \in G$.</p>
<p>I'm stuck in proving 1-1-ness of the function. Let $a,b$ in $G$ such that $\varphi (x) = \... | Nicky Hekster | 9,605 | <p>Fun fact to know that it works also <em>the other way around</em>: let $G$ be abelian, if $\varphi(a)=a^n$ is an automorphism of $G$, then gcd$(n,|G|)=1$. <p>Proof: assume that gcd$(n,|G|) \neq 1$ and let $p$ be a prime dividing both $n$ and $|G|$. By Cauchy's Theorem, there is an $a \neq 1$, such that $a^p = 1$. Bu... |
318,740 | <p>How to find the projection(the closest point) of point $a$ to half-space $(p,x)\le\alpha$, where $p,x\in\mathbb R^n$,$\alpha \in \mathbb R$? $(p,x)$ is dot product.</p>
| Royi | 33 | <p>The problem is given by:</p>
<p><span class="math-container">$$\begin{aligned}
\arg \min_{x} \quad & \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} \\
\text{subject to} \quad & {a}^{T} x \leq b
\end{aligned}$$</span></p>
<p>Solving the KKT conditionn will yield:</p>
<p><span class="math-container">$$ x = \b... |
580,592 | <p>The RSA values that state require factoring of two primes, does this mean that "the RSA number" can only have two prime factors but may have other factors that are not prime as well?</p>
<p>Is the fastest known algorithm to determine if a int1024 value is Prime slow or quick?</p>
| Clayton | 43,239 | <p><strong>Hint:</strong> What does it mean to be in the annihilator? Write this down and use the fact that $f$ is injective.</p>
|
580,592 | <p>The RSA values that state require factoring of two primes, does this mean that "the RSA number" can only have two prime factors but may have other factors that are not prime as well?</p>
<p>Is the fastest known algorithm to determine if a int1024 value is Prime slow or quick?</p>
| DonAntonio | 31,254 | <p>You have that</p>
<p>$$M\cong f(M)\le N$$</p>
<p>and thus your work basically reduces (though it is <em>not</em> the same exactly!) to show that</p>
<p>$$M\le N\implies \text{ann}\,(N)\le\text{ann}\,(M)$$</p>
<p>which seems to be fairly simple.</p>
|
4,149,316 | <p>Is <span class="math-container">$f(x,y) = x^2 - 2e^y$</span> concave or convex?</p>
<p>My thought process is as such:</p>
<p><span class="math-container">$f'(x,y)=2x - 2e^y$</span>;</p>
<p><span class="math-container">$f''(x,y)=2 - 2e^y$</span></p>
<p>Thus when <span class="math-container">$y<0$</span> (<span cla... | occassional user | 362,290 | <p>Indeed, this is a good example of a function with a <a href="https://en.wikipedia.org/wiki/Saddle_point" rel="nofollow noreferrer">saddle point</a> at 0. Mind you, the expression <span class="math-container">$f^\prime$</span> is meaningless as the function depends on two variables, hence we need to take partial deri... |
4,149,316 | <p>Is <span class="math-container">$f(x,y) = x^2 - 2e^y$</span> concave or convex?</p>
<p>My thought process is as such:</p>
<p><span class="math-container">$f'(x,y)=2x - 2e^y$</span>;</p>
<p><span class="math-container">$f''(x,y)=2 - 2e^y$</span></p>
<p>Thus when <span class="math-container">$y<0$</span> (<span cla... | Calvin Khor | 80,734 | <p>There is a way to proceed without derivatives. Note that if a function is concave, then the function restricted to any line is also concave. Similarly with convex.</p>
<p>The contrapositive of the above is that if there is a line such that the restriction of <span class="math-container">$f$</span> to it is not conca... |
2,387,402 | <p>Is it possible in n dimensions that 3 vectors have a 45-degree angle with each other?</p>
<pre><code><(x,y)= <(x,z) = <(z,y) = 45
</code></pre>
<p>is there also any condition for it? </p>
| Andrei | 331,661 | <p>It is possible. I will show how to create such a vector for $n=3$. Start with vector $\vec{x}=(1,0,0)$. A vector at $45$ degrees is $\vec{y}=\frac{1}{\sqrt 2}(1,1,0)$. The reason I choose the $\frac{1}{\sqrt 2}$ factor is to have $|\vec x|=|\vec y|=1$. Now assume that $\vec z$ has a component out of the plane of $\v... |
113,266 | <p>One of the problems of my Statistics assignment requires me to calculate the expected value of a minimum. Here's the situation:</p>
<p>The following density function is given: $f(x) = \frac{\theta}{x^2}$ where $x\ge\theta$ and $\theta>0$</p>
<p>I have to calculate: E[min{$X_i$}]</p>
<p>My initial guess was tha... | savick01 | 18,493 | <p>Let's compute the PDF of the minimum:
$$\phi(t) = P(\min \leq t) = 1 - P(\min \gt t) = 1 - P(\forall_i X_i \gt t) = 1 - \prod_i P(X_i > t) = \\ = 1 - P(X_1 \gt t)^n.$$</p>
<p>So the density $g$ of the minimum is $\frac{d\phi}{dt}$:
$$g(t) = \phi'(t) = - n \cdot P(X_1 \gt t)^{n-1} \cdot P(X_1 \gt t)' = - n \cdot ... |
2,130,527 | <p>I am trying to plot the function $(-2(\alpha-1)/\alpha)^{(-2(\alpha+2)/\alpha)}$ in maple,
but I got the following warning</p>
<pre><code>plot( (-2*(alpha-1)/alpha)^(-2*(alpha+2)/alpha) ,alpha=5..10);
</code></pre>
<blockquote>
<p>Warning, unable to evaluate the function to numeric values in the
region; comple... | s.harp | 152,424 | <p>You have:
$$a_n=\sum_{k=1}^n(a_k-a_{k-1}) +a_0\qquad |a_k-a_{k-1}|≤c|a_{k-1}-a_{k-2}|≤...≤c^{k-1}|a_1-a_0|$$</p>
<p>And so
$$\sum_{k=1}^n|a_k-a_{k-1}|≤\sum_{k=1}^n c^{k-1} |a_1-a_0|$$
Since $0<c<1$ the sum $\sum_k a_k-a_{k-1}$ converges absolutely, being majorised by the absolutely convergent sum $\sum_{k=1}^... |
1,796,056 | <p>I'm reading <a href="https://proofwiki.org/wiki/Non-Trivial_Discrete_Space_is_not_Connected" rel="nofollow noreferrer">this proof</a> that says that a non-trivial discrete space is not connected. I understood that the proof works because it separated the discrete set into a singleton <span class="math-container">${x... | Community | -1 | <p>I will only try to answer your question in update because I think that you have already gotten the answer of your other questions. If I am wrong, let me know.</p>
<p>This claim follows immediately if you assume the following as a definition of Connected Space.</p>
<blockquote>
<p><strong>Definition.</strong> A t... |
3,738,646 | <p>I think that a constant sequence is a Cauchy sequence because of the definition of Cauchy sequence:</p>
<p>The sequence <span class="math-container">$\{q_n\}$</span> of rationals is a Cauchy sequence if, for every <span class="math-container">$\epsilon>0$</span>, where <span class="math-container">$\epsilon$</spa... | ilovebulbasaur | 586,948 | <p>I believe this is Rudin Exercise 3.3? To show that <span class="math-container">$\{s_n\}$</span> is monotonic, you can use induction. Indeed, for <span class="math-container">$n=1$</span> we have <span class="math-container">$s_1=\sqrt{2}$</span> and <span class="math-container">$s_2=\sqrt{2+\sqrt{\sqrt{2}}}$</span>... |
5,197 | <p>Is it possible to split a list into two lists at a specific position? The main list is for example: </p>
<pre><code>data={{xa,ya},{xb,yb},{xc,yc},...,{xz,yz}}.
</code></pre>
<p>I want to split this list into two new lists: </p>
<pre><code>data1={{xa,ya},{xb,yb},...,{xi,yi}}
</code></pre>
<p>and </p>
<pre><cod... | ybeltukov | 4,678 | <p>Starting from <strong>10.2</strong> you can use <code>TakeDrop</code></p>
<pre><code>data = {{x1, y1}, {x2, y2}, {x3, y3}, {x4, y4}, {xz, yz}};
{data1, data2} = TakeDrop[data, 3]
(* {{{x1, y1}, {x2, y2}, {x3, y3}}, {{x4, y4}, {xz, yz}}} *)
</code></pre>
|
751,293 | <p>I'm nearing the end of the semester of an introductory-level complex variables class. (Very introductory -- it's the version of the class that's only required for engineering and physics majors, as it doesn't require two semesters of undergrad analysis that are prerequisite to the complex variables class for math ma... | Qiaochu Yuan | 232 | <p>A harmonic function is a function whose value at a point is always equal to the average of its values on a sphere centered at that point (<a href="http://en.wikipedia.org/wiki/Harmonic_function#The_mean_value_property" rel="noreferrer">reference</a>). This is why they show up as steady-state solutions to the heat eq... |
225,367 | <p>Prove that a group of order 36 must have a normal subgroup of order 3 or 9.</p>
<p>Let n2 be the number of 2-Sylow subgroups of G (with |G|=36). Then
n must be 1 or 3.
Let n3 be the number of 3-Sylow subgroups of G. then n3=1 or n3=4
if n3=1 we have 1 3-sylow group of order 9. and it is also a normal group(from sy... | SBR KUMAR | 475,341 | <p>The normal subgroup of order 3 in a group of order 36 is guaranteed. By Sylow theorem the subgroup of order 9 is possible. Also index of the subgroup is 4. And 36 does not divide 4!=24. Hence there is a non trivial normal subgroup contained in a subgroup of order 9.By lagrange theorem those normal subgroup order is ... |
2,152,736 | <p>With respect to a matrix $A\in\mathbb{R}^{n\times n}$, is there any if and only if conditions that make sure the eigen values of $A$ remain real and distinct? $A$ here is not symmetric but is stable and also there exists a $P$ such that $PAP^{-1}$ is symmetric.
For having real eigen values, $A$ needs to symmetric or... | Kiryl Pesotski | 417,344 | <p>For the eigenvalues to be real matrix has to be Hermitian $v^{\dagger}Aw=w^{\dagger}A^{\dagger}v$
For any $v, w\in\mathbb{C}$. And here $F^{\dagger}=\bar{F}^{T}$. Don't know what to say about them being distinct.</p>
|
1,884,791 | <p>I'm trying to prove the convergence of </p>
<p>$$\int^{\infty}_{0} \frac{1}{e^x+1}dx$$</p>
<p>Any suggestion or hint for how to deal with it?</p>
<p>Thanks in advance!</p>
| Extremal | 188,110 | <p>Use the substitute $e^x+1=t$. Then your integral becomes $\displaystyle \int_2^{\infty}\frac{1}{t(t-1)}dt.$</p>
<p>For $t>2$, $t^2>2t$. Hence, $\displaystyle \frac{1}{t(t-1)}=\frac{1}{t^2-t}\leq \frac{1}{t^2-t^2/2}=\frac{2}{t^2}.$</p>
<p>$\displaystyle \int_2^{\infty}\frac{1}{t^2}dt $ converges since $2>1... |
2,483,254 | <p>I have to compute the limit</p>
<p>$$
\lim\limits_{N \to +\infty} \sqrt{N+1} \log \left(1+\frac{x}{N+1}\right)
$$
where $x \ge 0$ is fixed. I tried to see the previous as</p>
<p>$$
\log \lim\limits_{N \to +\infty} \left(1+\frac{x}{N+1}\right)^{\sqrt{N+1}}
$$
and to change variable, but it doesn't work. Intuitively... | marty cohen | 13,079 | <p>The "+1" doesn't matter.</p>
<p>For large $N$ and fixed $x$</p>
<p>$\sqrt{N} \log (1+\frac{x}{N})
=\sqrt{N} (\frac{x}{N}+O((\frac{x}{N})^2))
=\frac{x}{\sqrt{N}}+O(\frac{1}{N^{3/2}})
\to 0
$.</p>
<p>This holds with
$N^{c}$
(instead of $\sqrt{N}$)
for all
$0 < c < 1$.</p>
<p>If
$c=1$,
we have
$N \log (1+\fra... |
2,998,342 | <p>Find <span class="math-container">$y$</span> such that </p>
<p><span class="math-container">$$\lfloor y \rfloor + \lfloor 3y \rfloor = 5$$</span></p>
<p>First, I use the properties that <span class="math-container">$$n \leq y <n+1$$</span>
And suppose </p>
<p><span class="math-container">$$\lfloor y\rfloor = 5... | Servaes | 30,382 | <p>These are both special cases of <a href="https://en.wikipedia.org/wiki/Catalan%27s_conjecture" rel="nofollow noreferrer">Mihăilescu's theorem</a>, which says that the only solution to
<span class="math-container">$$x^a-y^b=1,$$</span>
in integers <span class="math-container">$a,b>1$</span> and <span class="math-c... |
2,121,733 | <p>$F(x)=\begin{cases} x-1 & \text{if } x \text{ is even}\\ 2x &\text{if } x \text{ is odd}\end{cases}$</p>
<p>I need to exhibit the left inverse of this function. I know it exists because the function is one-to-one. Now, it's easy for me to switch the $x$'s and $y$'s to get:
$$g(x)=\begin{cases} \frac{x}{2} \... | Olba12 | 229,160 | <p>Notice at first that if $x$ is even, then $x-1$ is odd. And if $x$ is odd, then $2x$ is even. Thus you can some what think of the function as $f(\text{odd})=\text{even}$ and $f(\text{even})=\text{odd}$. Now when applying the inverse, then we get $f^{-1}f(\text{odd}) = \text{odd} = f^{-1}(\text{even})$ and $f^{-1}f(\... |
3,151,329 | <p>How to find range of this function, it looks easy but somehow I am not able to do that . <br> What is more important to me is , why is wolframalpha unable to find the range , is it not a simple enough function ?
<img src="https://i.stack.imgur.com/pwGkm.png" alt="enter image description here"></p>
| sgrmshrsm7 | 417,228 | <p>Domain of <span class="math-container">$f$</span> is <span class="math-container">$[-1,1]$</span> since <span class="math-container">$1-x²≥0$</span> because it is under square root.
The function is continuous so find critical points using <span class="math-container">$\frac{df}{dx} = 0$</span> and then find the valu... |
4,198,816 | <p>I just got introduced to Bayes theorem, and tried solving the following question:</p>
<blockquote>
<p>Two balls are drawn in succession from a jar containing 3 red balls and 4 white balls. What is the probability that the first ball was white given the second ball was red.</p>
</blockquote>
<p>I used the formula giv... | diracdeltafunk | 19,006 | <p>This is true. It suffices to show that every finite irreducible space has a generic point, since <span class="math-container">$T_0$</span> implies that generic points are unique. So, let <span class="math-container">$X$</span> be a finite irreducible space. Then <span class="math-container">$X$</span> is the union o... |
129,655 | <p>Here is a nice <em>sufficient (but not necessary)</em> condition on whether a quintic is solvable in radicals or not. Given,</p>
<p>$x^5+10cx^3+10dx^2+5ex+f = 0\tag{1}$ </p>
<p>If there is an ordering of its roots such that,</p>
<p>$x_1 x_2 + x_2 x_3 + x_3 x_4 + x_4 x_5 + x_5 x_1 - (x_1 x_3 + x_3 x_5 + x_5 x_2 +... | Riemann | 27,899 | <p>The parametric representation of line from $(1,0)$ to $(0,1)$ is $y=-x+1, 0\leq x \leq1$, so $$\int_{\gamma}y^2dx+x^2dy=\int_1^0[(-x+1)^2-x^2]dx=0.$$</p>
|
4,022,066 | <p>I don't quite understand how <span class="math-container">$\dfrac1 x$</span> is unbounded while <span class="math-container">$\dfrac{x}{1+x^2}$</span> is not.</p>
| user0102 | 322,814 | <p>Here it is an alternative approach for the sake of curiosity.</p>
<p><span class="math-container">\begin{align*}
|x^{2} + 1| = x^{2} + 1 \geq 2\sqrt{x^{2}} = 2|x| \Longleftrightarrow \left|\frac{x}{x^{2}+1}\right| \leq \frac{1}{2} \Longleftrightarrow -\frac{1}{2} \leq \frac{x}{x^{2}+1}\leq\frac{1}{2}
\end{align*}</s... |
299,156 | <p>We say that paths $\alpha, \beta: I \to X$ with common initial point $\alpha(0)=\beta(0)$ and common terminal point $\alpha(1)=\beta(1)$ are homotopic provided that there is a continuous function $H:I \times I \to X$ such that $H(t,0)=\alpha(t)$ and $H(t,1)=\beta(t)$ for $t \in I$ and $H(0,s)=\alpha(0)=\beta(0)$ and... | Avi Steiner | 13,487 | <p>It is in fact guaranteed by the continuity of $H$. To see this, let's fix an $s\in I$, and define a map $i_s\colon I\to I$ by $i_s(t)=(t,s)$. This is clearly continuous. Then the "in between" path $h_s:=H(-,s)$ is just the composition of the continuous functions $i_s$ and $H$. Hence, $h_s$ is itself continuous.</p>
|
4,516,512 | <p>The finite subgroups of <span class="math-container">$\mathrm{GL}_2(\mathbb{C})$</span> are pretty much classified in several expository papers in the web as well as posts on this site. However, I would like to know that how does one show that the Heisenberg group <span class="math-container">$H_3$</span> cannot be ... | David A. Craven | 804,921 | <p>Here is a completely different argument, because we seem to be collecting proofs here.</p>
<p>Let <span class="math-container">$G$</span> be the Heisenberg group, and suppose that <span class="math-container">$G$</span> has a faithful <span class="math-container">$2$</span>-dimensional representation <span class="ma... |
2,678,895 | <p>I was wondering how to map the set $\mathbb{Z}^+$ to the sequence $1, 2, 3, 1, 2, 3, \ldots$. I thought it would be easy, but I was only able to obtain an answer through trial and error.</p>
<p>For a function $f \colon \mathbb{Z}^+ \rightarrow \mathbb{Z}$, we have that</p>
<p>$f(x) = x \bmod 3$ gives the numbers $... | dxiv | 291,201 | <p>You figured out already that $\,x \bmod n\,$ gives the periodic sequence $\,(1, 2, \ldots, n-1, \color{red}{0}, 1, \ldots)\,$ then the rest is just "shifting" it to fit the desired start/end values. To match $\,(1, 2, \ldots, \color{red}{n}, 1, \ldots)\,$ for example, you need to shift $\,x \bmod n\,$ up by $\color{... |
2,678,895 | <p>I was wondering how to map the set $\mathbb{Z}^+$ to the sequence $1, 2, 3, 1, 2, 3, \ldots$. I thought it would be easy, but I was only able to obtain an answer through trial and error.</p>
<p>For a function $f \colon \mathbb{Z}^+ \rightarrow \mathbb{Z}$, we have that</p>
<p>$f(x) = x \bmod 3$ gives the numbers $... | epi163sqrt | 132,007 | <p>Let's assume we want to find a formula for the sequence
\begin{align*}
\color{blue}{3,1,\frac{1}{3},\frac{1}{3},1,3,3,1,\frac{1}{3},\frac{1}{3},1,3,3,1,\frac{1}{3},\frac{1}{3},1,3,\ldots}
\end{align*}
repeating in cycles of $6$ elements.</p>
<blockquote>
<p>The $r$-th roots of unity $\omega_j=\exp\left(\frac{2\pi... |
2,678,895 | <p>I was wondering how to map the set $\mathbb{Z}^+$ to the sequence $1, 2, 3, 1, 2, 3, \ldots$. I thought it would be easy, but I was only able to obtain an answer through trial and error.</p>
<p>For a function $f \colon \mathbb{Z}^+ \rightarrow \mathbb{Z}$, we have that</p>
<p>$f(x) = x \bmod 3$ gives the numbers $... | John | 538,403 | <p>f(x) = ((x−1) mod n) + 1</p>
<p>Step 1: You want a function with a period of n, i.e. it repeats every n numbers. Then necessarily you are modding something by n. g(x) = h(x) mod n
Step 2: You want the function to reset after every nth number. But the nature of modular division is that it resets on every nth num... |
2,590,359 | <p>I have to find a distance between a plane represented with a triangle and a point in $3D$ space. I searched other questions but the plane is usually represented by a plane equation or a normal, so I am having trouble finding this distance when the plane is represented by three points.</p>
| BallBoy | 512,865 | <p>You've implied that it would help to have a normal. Given three points forming a triangle, we can find two nonparallel vectors in the plane. Then their cross product would be a normal to the plane because it would be orthogonal to both vectors.</p>
|
208,419 | <blockquote>
<p>Every group of order $231$ is the direct product of a group of order $11$ and a group of order $21$.</p>
</blockquote>
<p>By Sylow's theorem, we know there are one Sylow-7 subgroup, one Sylow-11 subgroup (these two are normal, for sure), and some Sylow-2 subgroups.</p>
<p>What does the "direct produ... | MJD | 25,554 | <p>It means there is a group $\langle G, \star\rangle $ of order 11, and a group $\langle H, \ast\rangle$ of order 21, so that your group of order 231 has elements of the form $(g, h)$ where $g\in G$ and $h\in H$, and a group operation that operates componentwise, so that $(g,h)\cdot(g', h') = (g\star g', h\ast h')$, a... |
65,183 | <p>Let $p$ be an odd prime. I am interested in how many quadratic residues $a$ sre there such that $a+1$ is also a quadratic residue modulo $p$. I am sure that this number is
$$
\frac{p-6+\text{mod}(p,4)}{4},
$$
but I have neither proof nor reference. It is a particular case of the question in the title: if $a$ and $b... | Felipe Voloch | 2,290 | <p>It is easy to write this number (of $a$ such that $a,a+1$ are squares) in terms of the number of solutions of $x^2-y^2=1$. This is a conic which has $p+1$ projective points over the field of $p$ elements (since it is isomorphic to $\mathbb{P}^1$). It has two points at infinity, two points with $y=0$ and two or zero ... |
65,183 | <p>Let $p$ be an odd prime. I am interested in how many quadratic residues $a$ sre there such that $a+1$ is also a quadratic residue modulo $p$. I am sure that this number is
$$
\frac{p-6+\text{mod}(p,4)}{4},
$$
but I have neither proof nor reference. It is a particular case of the question in the title: if $a$ and $b... | Sonia Balagopalan | 262 | <p>Here's a copy-paste of something I wrote up a while ago:</p>
<p>Lemma: Let $q$ be odd, and let $Q$ be the set of quadratic residues (including $0$) in $\mathbb F_q$. Then the number of elements $s_q(c)$ in $\{x^2+c|x \in \mathbb{F}_q\} \cap Q$ is given by
\begin{array}{|c|c|c|}
\hline
& c \in Q & c \notin ... |
3,358,850 | <p>I am pretty sure the general idea of my proof is correct, but I am not sure if it is coherent. For example, is something like <span class="math-container">$i \in I$</span> , which I mean by for any number <span class="math-container">$i$</span> in the index (<span class="math-container">$I$</span>), common enough no... | BaronVT | 39,526 | <p>No, this doesn't work.</p>
<p>Things to address:</p>
<p>1) What is <span class="math-container">$I$</span>?</p>
<p>2) 'Clearly <span class="math-container">$v$</span> is not a linear combination...' This is not clear. In fact it's the opposite: you have written down <span class="math-container">$v$</span> as a li... |
3,358,850 | <p>I am pretty sure the general idea of my proof is correct, but I am not sure if it is coherent. For example, is something like <span class="math-container">$i \in I$</span> , which I mean by for any number <span class="math-container">$i$</span> in the index (<span class="math-container">$I$</span>), common enough no... | Rodolfo Ferreira de Oliveira | 705,451 | <p>Well, I think <span class="math-container">$\mathbb{F}$</span> is a field and you're considering:
<span class="math-container">$$
\mathbb{F}^\infty = \{(x_1,x_2,...): x_i \in \mathbb{F}\}
$$</span>
Suppose that <span class="math-container">$\dim(\mathbb{F^\infty})<+\infty$</span>. Then, exists <span class="math-c... |
2,485,325 | <p>Let $a$ and $b$ be two positive integers such that gcd$(a,b)=1$. Prove that there exists $(x,y)\in\mathbb{Z}\times \mathbb{Z}$ satisfying </p>
<p>(1) $ax-by=1$</p>
<p>(2) $0<x<b$ and $0<y<a$.</p>
<p>I know (1) is by the fact that gcd$(a,b)=1$ but I don't know how to prove (2). Any hint? </p>
| Angina Seng | 436,618 | <p>The condition (2) cannot be satisfied if $a=1$ or $b=1$, so assume
$a>1$ and $b>1$.</p>
<p>Choose $x_0$, $y_0\in\Bbb Z$ with $ax_0-by_0=1$. Then $x_0\not\equiv0\pmod{b}$ and so there is $t\in\Bbb Z$ with $x_1=x_0+tb$ satisfying
$0<x_1<b$. Then $ax_1-by_1=1$ with $y_1=y_0-ta$. Then
$$y_1=\frac{ax_0-1}{b}... |
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