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 |
|---|---|---|---|---|
4,450,722 | <blockquote>
<p>Q: <span class="math-container">$f\in L^1(\mathbb{R}^n),\alpha>0$</span> and
<span class="math-container">$$\int_{\mathbb{R}^n} |f(x+h)-f(x)|\,dx\leq |h|^{1+\alpha},\ h\in\mathbb{R}^n.$$</span>
Show that <span class="math-container">$f=0$</span> a.e.</p>
</blockquote>
<p>My attempt:
Take <span class=... | Binxu Wang 王彬旭 | 504,153 | <p>This could be derived step by step, using chain rules</p>
<ol>
<li>Frobenious norm. Let's deal with the squared norm first! taking sqrt is easy.
<span class="math-container">$$
d(A,B)^2=\|A-B\|^2_F = tr((A-B)^T(A-B)) = \sum_{ij} (A_{ij}-B_{ij})^2
$$</span>
So if one of the matrices <span class="math-container">$A$</... |
3,912,109 | <p>Let <span class="math-container">$T$</span> be a complete first-order theory over a <em>countable</em> language <span class="math-container">$L$</span>, such that <span class="math-container">$T$</span> has an infinite model.</p>
<p>It is well-known that we can construct an elementary chain of size-<span class="math... | Claude Leibovici | 82,404 | <p>This is a transcendental equation; so, you will need numerical methods and then good starting guesses.</p>
<p>Consider that you look for the zero's of function <span class="math-container">$f(x)=3x^2+2\sin x-2$</span>.</p>
<p>For example, using Taylor series built around <span class="math-container">$x=0$</span>
<sp... |
3,289,226 | <p>I am unsure on how to prove this statement. I am relatively certain that it is true, any advice on how to start?</p>
<p>Edit: It has occurred to me that this will be significantly easier to show by restricting the simplex to be regular (and may be only true in this case).</p>
| sds | 37,092 | <p>If the "origin" is <em>arbitrary</em>, then the statement is clearly wrong: if it is sufficiently far enough from the simplex, the vectors from it to the vertices are almost collinear.</p>
<p>Even for a regular triangle and the origin <em>inside</em> it, but close to one of the vertices, the angle between the the v... |
1,056,194 | <p>With the regular trig functions, if I ever end up with something like $\operatorname{trig}_1(\operatorname{arctrig}_2(f(x))$, where $\text{trig}_1$ and $\text{trig}_2$ are two arbitrary trigonometric functions, I can draw a right triangle to find a formula for this that doesn't involve any trigonmetric functions.</p... | Kaynex | 296,320 | <p>This answer may be a little late, but I was wondering the same thing, and I think I may have come up with an answer.</p>
<p>Let's say we want to find <span class="math-container">$\sinh(\operatorname{artanh}(x))$</span>. Draw your triangle as per usual, putting x on the opposite, and 1 on the adjacent.</p>
<p>Howeve... |
2,235,645 | <p>I am struggling with solving sum of this alternate series:</p>
<p>$$
\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n(n+1)}\
$$</p>
<p>I know that:</p>
<p>$$
\log(1+x) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \cdot x^n\
$$</p>
<p>But It seems that I can't find a way to get to this form.
Thanks.</p>
| Olivier Oloa | 118,798 | <p><strong>Hint</strong>. Due to properties of power series, one is allowed to write
$$
\begin{align}
\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n(n+1)}&=\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\int_{0}^1t^{n}dt
\\\\&=\int_{0}^1\sum_{n=1}^\infty\frac{(-1)^{n+1}t^{n}}{n}\:dt
\\\\&=\int_0^1\ln(1+t)dt
\end{align}
$$ Can ... |
81,079 | <p>Some time ago I met a forester. He told that there are only larches and spruces in his forest.
He also said that there are exactly $10$ spruces at the distance of exactly 1 km from each larch. Next, he argued that there are more larches than spruces in his forest. Can he be right?</p>
| GEdgar | 442 | <p>Say there are exactly 10 spruces, all growing at the same point. And a million larches, all exactly 1 km from that point.</p>
|
81,079 | <p>Some time ago I met a forester. He told that there are only larches and spruces in his forest.
He also said that there are exactly $10$ spruces at the distance of exactly 1 km from each larch. Next, he argued that there are more larches than spruces in his forest. Can he be right?</p>
| ezekiel | 307,839 | <p>Yes, he could. 10 spruces are in the middle of a formation, each point one kilometer away from the spruces. Larches could be planted around them,so there could be 20 larches sharing 10 spruces. Also the spruces could be in one metre square. </p>
|
1,400,987 | <blockquote>
<p>Given $x\in \left(0; \frac\pi2\right)$. Prove that $$\sin x>\frac{2x}{\pi}$$</p>
</blockquote>
<p>This is my try:</p>
<p>Let $y=\sin x-\frac{2x}\pi\implies y ' = \cos x - \frac2\pi\implies y ''=-\sin x <0; \forall x\in \left(0; \frac\pi2\right)$
$\implies \left\{\begin{matrix} y '(x)> y '(\... | suncup224 | 61,149 | <p>Alternative solution: Let $y = \sin x$. Then $y''=-\sin x <0$ for $0<x<\pi /2$. So $y$ is strictly concave on that interval. Note that $y(0)=0$ and $y(\pi/2)=1$. Therefore, $y$ lies strictly above the line segment joining $(0,0)$ and $(\pi/2, 1)$. Therefore $y > 2x/ \pi$ for $0<x<\pi /2$. </p>
|
926,754 | <p>Find a polynomial with the maximum 1. degree which best approximates the $f(x)=e^x$ function in terms of minimax approximation in $[0,1]$.</p>
| Amzoti | 38,839 | <p>We are asked to find a polynomial of maximum degree $1$ which best approximates $f(x)=e^x$ function in terms of minimax approximation in the range $[0,1]$.</p>
<p>Consider the graph of $f(x) = e^x$ and a best possible linear approximation of $y = m x + b$. </p>
<p><img src="https://i.stack.imgur.com/e2O92.png" al... |
1,313,348 | <p>I am trying to figure out how $p$-adic numbers work and currently am having trouble wrapping my head around how they work, so I made a pun! HAH!</p>
<p>Jokes aside, I am working on this question</p>
<blockquote>
<p>Show that the sequence $(3,34,334,3334,.....)$ is equal to $2/3$ in $\hat{\mathbb{Z}}_5$</p>
</blo... | Dietrich Burde | 83,966 | <p>The sequence $(a_n)=(4,34,334,3334,\ldots)$ satisfies $3a_n-2\to 0$ for $n\to \infty$ in the $5$-adic topology. You can "see" this by considering
\begin{align*}
3\cdot 4 & = 12 \\
3\cdot 34 & = 102 \\
3\cdot 334 & = 1002 \\
3\cdot 3334 & = 10002
\end{align*}
etc. Hence for the metric $d$ we have $d(a... |
3,372,482 | <p>Looking at some of the more common pythagorean triples, I noticed a trend that there are usually two numbers which differ by only one or two, with more much smaller or larger number in comparison. For example:</p>
<p><span class="math-container">$$5,12,13$$</span>
<span class="math-container">$$8,15,17$$</span>
<sp... | José Carlos Santos | 446,262 | <p>This comes from the fact that, when <span class="math-container">$n$</span> is odd<span class="math-container">$$\left(n,\frac{n^2-1}2,\frac{n^2+1}2\right)$$</span>is a pythagorean triple and, when <span class="math-container">$n$</span> is even,<span class="math-container">$$\left(n,\left(\frac n2\right)^2-1,\left(... |
3,372,482 | <p>Looking at some of the more common pythagorean triples, I noticed a trend that there are usually two numbers which differ by only one or two, with more much smaller or larger number in comparison. For example:</p>
<p><span class="math-container">$$5,12,13$$</span>
<span class="math-container">$$8,15,17$$</span>
<sp... | hamam_Abdallah | 369,188 | <p>If you want for example that <span class="math-container">$$c=b+1$$</span></p>
<p>you choose <span class="math-container">$a$</span> such that</p>
<p><span class="math-container">$$a^2=2b+1=(2k+1)^2$$</span></p>
<p>or
<span class="math-container">$$b=2k^2+2k$$</span>
for <span class="math-container">$k=3$</span> ... |
2,156,709 | <ul>
<li>Prove-($\bigcup\limits_{i}A_i$) $\bigcap $($\bigcup\limits_{j}B_j$)=$\bigcup\limits_{i,j}(A_i \bigcap B_j)$</li>
<li>Prove-($\bigcap\limits_{i}A_i$) $\bigcup $($\bigcap\limits_{j}B_j$)=$\bigcap\limits_{i,j}(A_i \bigcup B_j)$</li>
<li>If {$I_j$} is a family of sets with domain J,say;$\Bbb K$=$\bigcup\limits_{... | David | 342,780 | <p>Call the salaries $X_1,X_2,X_3$. The first person generates a random number $R_1$ and tells the number $X_1+R_1$ to person two. Now, person two then does the same thing. Therefore, person three recieves the number $(X_1+R_1)+(X_2+R_2)$ Person three does the same thing and passes the result back to person one. Then t... |
2,716,545 | <p>Suppose that a point with co-ordinates $(X, Y)$ is chosen uniformly at random from the triangle $\{(x,y)\in\mathbb R^2 :0\leq x\leq1\,,0\leq y\leq\min(x,1−x)\}$. Find the density of the distribution of $X$.</p>
<p>I am struggling for when $x>1/2$.</p>
| zoli | 203,663 | <p>The right definition is: $Q << P$ means, that $P[A] = 0 \Rightarrow \color{red}Q[A] = 0$, for $A \in F$. Since $P[A]$ is never zero except if $A$ is the empty set, the implication holds. </p>
<p>As far as finiteness.</p>
<p>If $\Omega$ was not finite then we couldn't say so easily that $P[\{ω\}] >
0$ for... |
3,148,902 | <p>I'm trying to find an explicit formula for the following recurrence relation:</p>
<blockquote>
<p><span class="math-container">$a_0 = 1$</span>, <span class="math-container">$\forall n \ge 1: a_n = n^3a_{n-1} + (n!)^3$</span></p>
</blockquote>
<p>So far though, my attempts have been unsuccessful.</p>
<hr>
<p>... | Ben | 526,995 | <p>One simple way to find these sequences (so simple it is almost cheating!) is to generate the first few terms and search for the resulting sequence in the <a href="https://oeis.org/" rel="nofollow noreferrer">Online Encyclopedia of Integer Sequences (OEIS)</a>. In this case you have:</p>
<p><span class="math-contai... |
2,669,054 | <p>Question: What is the area of the region enclosed by the curves:</p>
<p>$$2y = 4\sqrt{x},\quad y = 3,\quad \text{and} \quad 2y + 2x = 6.
$$
I have tried calculate all the definite integrals but I am not sure which curve I am supposed to subtract and which one is supposed to come first. And also, I am a little confu... | Andrew Li | 344,419 | <p><strong>Hint:</strong> Split it into two integrals. One where $2y+2x=6$ is under $y=3$ and one where $2y=4\sqrt x$ is under $y=3$. Then find where $2y=4\sqrt x$ and $2y+2x=6$ intersect, and that's the bound between the two integrals.</p>
<p>If you don't want two integrals, you can also integrate with respect to $y$... |
1,850,977 | <p>I am creating "children" in script and would like to calculate the number of children before launching the script. I have "width" the number of child nodes and "depth" the depth of the nodes.</p>
<ul>
<li>5 wide and 1 deep = 5</li>
<li>5 wide and 2 deep = 30 = (5+25)</li>
<li>5 wide and 3 deep = 155 = (5+25+125)</l... | SchrodingersCat | 278,967 | <p><strong>HINT</strong>: $$\begin{align} & 3^{2n+5}+40n+13 \\ &=9\cdot 3^{2n+3}+9\cdot 40n -9\cdot 27 - 8\cdot 40n+256 \\ & =9(3^{2n+3}+40n -27) - 64\cdot 5n+ 64\cdot 4\end{align}$$</p>
|
452,570 | <p>I need help with this problem:</p>
<blockquote>
<p>Let $\{u_{n}\}$ and $\{v_{n}\}$, $(n \in \mathbb{N})$ be two different Hilbert bases in a Hilbert space $H$. Define a linear operator $T$ so that, for $x \in H$:
$$T(x) = \sum_{n=1}^{\infty} \lambda(n)u_{n}⟨v_{n} \mid x⟩$$ where $\lambda(n)$ is a fixed sequence... | Vishal Gupta | 60,810 | <p>Assuming that both these basis are orthonormal, we get</p>
<p>$$||T(x)||^{2} \leq \sum|<v_{n},x>|^{2} = ||x||^{2}$$</p>
<p>where the last inequality is due to Parseval's formula. Hence the operator is bounded.</p>
<p>For the adjoint, calculate $<Tx,y>$ and surmise what $T^{*}y$ should be to get the sa... |
353,576 | <p>I am looking at the proof of Bishop's Theorem on pages 122 and 123 of Rudin's Functional Analysis. The following quote is from the the last two sentences of the proof on pg. 123.</p>
<p>"Every continuous linear functional on $C(S)$ that annihilates $A$ also annihilates g. Hence $g\in A$ by the Hahn-Banach Separat... | Theo | 29,980 | <p>I assume that $A$ is a subspace in your context. If $g\notin \overline{A}$, then by HBST, there exist a bounded linear functional $F$ such that $F(g)$ and $F(\overline{A})$ are disjoint. Since $\overline{A}$ is a subspace, $F(\overline{A})$ is a subspace of the scalar field. As well, $F(\overline{A})$ is different ... |
842,911 | <p>What is the following product in terms of $n$?</p>
<p>$$\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8} \cdot \cdots \cdot \frac{2n-1}{2n}$$</p>
<p>Thank you.</p>
| Anurag A | 68,092 | <p>$$\frac{1}{2}\frac{3}{4}\frac{5}{6} \dotsb \frac{2n-1}{2n}=\frac{1\mathbf{2}3\mathbf{4}5 \dotsb (2n-1)\mathbf{2n}}{(2 \cdot 4 \cdot 6 \dotsb 2n)^2}=\frac{(2n)!}{2^{2n}(n!)^2}$$</p>
|
733,140 | <p>Suppose 26 athletes (Alice, Bob, Charlie, Derek.....Zak) run a race and each ordering of the athletes is equally likely. What's the probability that Alice finishes before Bob, and Bob finishes before Charlie. </p>
<p>For this I thought the answer was 1/4 because I thought that there are half of the possibilites whe... | user2566092 | 87,313 | <p>Hint: It's possible that $X$ only takes on values $0$ and $7$, with probabilities $P(X = 0) = 2/7$, $P(X = 7) = 5/7$. So does $X$ have to be geometric or binomial, and does $X$ have to equal $5$ with positive probability? If you can rule that out, then try to use inequalities on $E(X)$ in terms of $P(X \geq a)$ and ... |
627,620 | <p>I am wondering if you can compute the SVD/eigenvectors of a rank r matrix of size n*n in time O(n^2 r)? My understanding is that standard eigenvector computations involve bringing matrix into Hesseberg form which costs O(n^3) operations.</p>
<p>Thanks.</p>
| Martin Argerami | 22,857 | <p>Vectors, on their own, can be thought as starting anywhere in space. </p>
<p>But usually we want to do more than that. We want to algebraic operations with them: we want to think of $\mathbb R^n$ as a vector space, even a normed space, or even an inner product space. These notions are defined naturally when we thin... |
233,483 | <p>Now that <em>AlphaGo</em> <a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol" rel="noreferrer">has just beaten</a> Lee Sedol in Go and <em>Deep Blue</em> <a href="https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov" rel="noreferrer">has beaten</a> Garry Kasparov in chess in 1997, I wonder what... | joro | 12,481 | <p><strong>Added</strong> For me one advantage of humans is the experience in mathematics for thousands years, including teaching/collaborating.</p>
<p>On this scale, for me, computers are just "newborns".</p>
<p>Some of the older science fiction already came true.</p>
<p>Also, some predictions about computer... |
233,483 | <p>Now that <em>AlphaGo</em> <a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol" rel="noreferrer">has just beaten</a> Lee Sedol in Go and <em>Deep Blue</em> <a href="https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov" rel="noreferrer">has beaten</a> Garry Kasparov in chess in 1997, I wonder what... | Konstantinos Gaitanas | 38,851 | <p>I believe that our advantage is the following: </p>
<blockquote>
<p>The fact that we ask questions.</p>
</blockquote>
<p>Remember that mathematical induction was invented/discovered because we wanted to answer the following question:<br>
<em>how can we prove</em> $1+2+...+n=\frac{n(n+1)}{2}$ is valid for ever... |
233,483 | <p>Now that <em>AlphaGo</em> <a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol" rel="noreferrer">has just beaten</a> Lee Sedol in Go and <em>Deep Blue</em> <a href="https://en.wikipedia.org/wiki/Deep_Blue_versus_Garry_Kasparov" rel="noreferrer">has beaten</a> Garry Kasparov in chess in 1997, I wonder what... | Alex Gavrilov | 9,833 | <p>I believe this is a topic where it is better to steer clear of philosophy and be concrete. The natural way to consider this problem is to simply compare mathematics to those areas where computers have already proved to be superior to humans, such as board games. </p>
<p>It is difficult not to notice that mathemati... |
406,255 | <p>$ \int_ {-\pi}^{\pi}\ f(x) \ dx $ is equivalent to integrating $f(x)$ over $-\pi < x < \pi$ (strictly less than) and also $-\pi \le x \le \pi$ (less than or equal). So I just need to reassure myself that the bounds of integration don't change when considering either less than or less than or equal. A quick co... | amWhy | 9,003 | <p><strong>With respect to your edit</strong>, Yes, you can compute this particular integral (and evaluate "at" the bounds $I(x) \big|_b^a$, but keep in mind Jon Claus's answer: that what you are <strong><em>really</em></strong> doing is evaluating the limit of $I(x)$ as $x \to a^-, x\to b^+$, and that's crucial to rem... |
2,643,778 | <p>I can intuitively realize why this is true. All the points that are on the line $y=-x$ are in the set. How can I prove this with sequences?</p>
| Stefan4024 | 67,746 | <p>Let $a = (x,y)$ is an accumulation point of the set. Then there exists a sequence $(a_n) \in \mathbb{R}^{2}$ converging to it. Assume that $a = (x,y)$ isn't in the set. Then we have that $x+y = N < 0$. Now take a ball with radius $\frac{|N|}{2}$ around $a$. Obviously that the intersection with the set is empty, b... |
1,696,143 | <p>Prove that:</p>
<p>$$\int_0^B e^\left(x^2\right) \;dx \sim \frac{e^\left(B^2\right)}{2B}$$ where $$B \to \infty$$</p>
<p>We should prove the equivalence of the given integral to the expression in the right part. </p>
<p>It's obvious, that the integral in the given form cannot be expressed through known indefinite... | robjohn | 13,854 | <p>Using the same approach as in <a href="https://math.stackexchange.com/a/95454">this answer</a>, but leaving out the details, which are handled in a similar fashion, we get the asymptotic expansion
$$
\begin{align}
\int_0^xe^{t^2}\,\mathrm{d}t
&=e^{x^2}\int_0^xe^{t^2-x^2}\,\mathrm{d}t\\
&=e^{x^2}\int_0^xe^{-2... |
690,531 | <p>Is there some sort of intuition or a good ilustrative example for random variables being $\sigma$-algebra measurable? I understand the definition, but when looking at martingales, the meaning of random variables being measurable eludes me. So my question is mainly aimed on the case of martingales where sequence of r... | Bjørn Kjos-Hanssen | 61,578 | <p>The random variable $X$ is measurable with respect to $\sigma$-algebra $\mathfrak F$ if $X=\mathbb E(X\mid\mathfrak F)$.</p>
<p>One can understand this in a few steps:</p>
<ol>
<li>$\mathbb E(X\mid A)$, where $A$ is an event, is the expected value of $X$ given that $A$ occurs;</li>
<li>$\mathbb E(X\mid Y)$, where ... |
690,531 | <p>Is there some sort of intuition or a good ilustrative example for random variables being $\sigma$-algebra measurable? I understand the definition, but when looking at martingales, the meaning of random variables being measurable eludes me. So my question is mainly aimed on the case of martingales where sequence of r... | jpvigneaux | 126,596 | <p>Maybe this can help you to understand the concept of conditional expectation, behind your question.</p>
<p>Suppose you have a probability space $(\Omega, \mathcal P (\Omega), \mathbb{P})$, where $\mathcal P (\Omega)$ denotes the set of all possible subsets of $\Omega$ (evidently, a $\sigma$-algebra), and $\mathbb{... |
2,051,843 | <p>I am analyzing the below puzzle presented <a href="https://books.google.ie/books?id=8EDCAgAAQBAJ&pg=PA37&lpg=PA37&dq=in%20an%20election,%20two%20candidates,%20Albert%20and%20Benjamin&source=bl&ots=GGih2A4OTC&sig=had3_VHldWIeszXN1HtMpG31YnE&hl=en&sa=X&ved=0ahUKEwjghIXlk-jQAhVYF8AKH... | John | 7,163 | <p>The sequences show the order that the ballots are drawn, left being first and right being last.</p>
<p>For the bolded ones, there is no value $n$ with $1 \leq n \leq 5$ for which the first $n$ ballots have the same number of $A$ and $B$ ballots. Those bolded cases satisfy the condition.</p>
<p>Contrarily, looking... |
180,504 | <p>I want to plot a curve given by $$f(x,y)\equiv y-y^3+3yx^2=0,$$ that passes through $(0,0)$ and $f(x,y) \to + \infty$ as $|x|,|y| \to \infty$. I thought of using ContourPlot to do so but when I use ContourPlot[y-y^3+3yx^2==0,{x,x_min,x_max},{y,y_min,y_max}], there are three other curves in addition to the one I want... | kglr | 125 | <p>As an alternative to <code>HoldPattern[Evaluate[...]</code> you can use <a href="http://reference.wolfram.com/language/ref/PatternSequence.html" rel="noreferrer"><code>PatternSequence</code></a> which evaluates its argument:</p>
<pre><code>{Cases[{1, a -> 2 b}, PatternSequence[a -> 2 b]],
Cases[{1, a -> 1... |
180,504 | <p>I want to plot a curve given by $$f(x,y)\equiv y-y^3+3yx^2=0,$$ that passes through $(0,0)$ and $f(x,y) \to + \infty$ as $|x|,|y| \to \infty$. I thought of using ContourPlot to do so but when I use ContourPlot[y-y^3+3yx^2==0,{x,x_min,x_max},{y,y_min,y_max}], there are three other curves in addition to the one I want... | Henrik Schumacher | 38,178 | <p>Here is another way by using <code>Verbatim</code>. Maybe it feels a bit less hacky.</p>
<pre><code>{
Cases[{1, a -> 2 b}, Verbatim[a -> 2 b]],
Cases[{1, a -> 1/2 b}, Verbatim[a -> 1/2 b]],
Cases[{1, a -> π b}, Verbatim[a -> π b]],
Cases[{1, a -> c b}, Verbatim[a -> c b]]
}
</code></pre... |
207,871 | <p>I am writing a code to display longitudinal waves.
I started with the following, filling the slots using the Range function. </p>
<pre><code> l = {Blue, Disk[{3 Sin[(5) t + 0.1 #] + 1.4 #, 1}, 0.5]} & /@
Range[25];
Animate[Evaluate[Graphics[l, PlotRange -> {{-4, 44}, All}]], {t, 0,
20. \[Pi]}, Anim... | kglr | 125 | <h3><a href="https://reference.wolfram.com/language/ref/Array.html" rel="nofollow noreferrer"><code>Array</code></a>:</h3>
<pre><code>ClearAll[disks]
disks[t_, nc_, nr_] := Array[Disk[{3 Sin[5 t + 0.1 #] + 1.4 #, 1.5 #2}, 0.5] &, {nc, nr}];
Animate[Graphics[{Blue, disks[t, 25, 10]}, PlotRange -> {{-4, 44}, All}... |
3,043,144 | <p><strong>Problem</strong>:</p>
<p>Let <span class="math-container">$A$</span> be some linear operator such that:
<span class="math-container">$$ (A^{2006}-I)^{2006}-I=0. $$</span>
Determine all real numbers that can be eigenvalues of operator <span class="math-container">$A$</span>.</p>
<p><strong>Question</strong>: ... | Myunghyun Song | 609,441 | <p><span class="math-container">$\textbf{Hint:}$</span> Suppose <span class="math-container">$\lambda$</span> is an eigenvalue of <span class="math-container">$A$</span>. Is <span class="math-container">$p(\lambda)$</span> an eigenvalue of <span class="math-container">$p(A)$</span>? (<span class="math-container">$p(\cd... |
910,844 | <p>I need help evaluating this with contour integration
$$
\int_{0}^{1}{\ln\left(\,x\,\right)\over
\,\sqrt{\vphantom{\large A}\,1 - x^{2}\,}}\,{\rm d}x
$$
I am not sure as to how to work with the branch cuts of both
$\ln\left(\,x\,\right)$ and $\sqrt{1 - x^{2}}$</p>
<p>Second part is to evaluate
$$ \int_{0}^1 \frac{\s... | dustin | 78,317 | <p>For your second integral, we can use Ramanujan's method which I mentioned <a href="https://math.stackexchange.com/questions/1158046/integrating-int-0-pi-2-log2-sin2x-sin2x-rm-dx/1158340#1158340">here</a>. That is, let $x = e^{-t}$. Then we have
\begin{align}
\int_0^1\frac{\log^{1/2}(x)}{\sqrt{1-x^2}}dx &= \int_0... |
3,081,186 | <p>I am looking to figure out if the following integral converges or diverges.
<span class="math-container">$\int_{0}^\infty{\frac{dx}{\sqrt{x}*(1+x)}}$</span> = <span class="math-container">$\int_{0}^\infty{\frac{dx}{\mathrm{x}^{1/2}+\mathrm{x}^{3/2}}}$</span>.</p>
<p>I have set it less than or equal to <span class="... | jmerry | 619,637 | <p>The key to a split like this? Split the integral before you make a comparison, then compare to different things in the two pieces. If you were going to make the same comparison in both parts, there wouldn't be any reason to split it. (OK, that's a more advanced view - with an improper Riemann integral, we have to sp... |
1,338,050 | <p>I can visualize a first-order tensor as a segment (a vector), but I'm not sure how to visualize a second-order tensor. The book that I'm trying to study is "Vector and tensor analysis with applications" (A. Borisenko and I. Tarapov)</p>
| janmarqz | 74,166 | <p>For a vector space $V$ over a scalar field ${\Bbb F}$, you can see it as a bilinear transformation
$$B:V\times V\longrightarrow{\Bbb F}$$
which allow you to impose a metric in $V$.
If we consider that $V$ is spawned by a basis $\{b_i\}$,
then it is necessary that the matrix
$$[B]=[B(b_i,b_j)]$$
complies certain co... |
1,171,733 | <p>Let $\{x_n\}$ be a sequence in $\mathbb{R}$ where $\forall n\in\mathbb{N}:x_n\neq 0$ and it converges to some $x\neq 0$. If the sequence is NOT monotone is it ever true that $\frac{1}{x_n}\rightarrow\frac{1}{x}$? If so what other conditions (if any) are needed and how would you show it. </p>
<p>Thanks in advance a... | MPW | 113,214 | <p>It is <em>always</em> true, provided that $x\neq 0$ and each $x_n\neq 0$.</p>
<p>Generally, $f(x_n)\to f(x)$ if $f$ is continuous and $x$ and each $x_n$ are in the domain of $f$. This follows from continuity.</p>
|
3,088,503 | <p>For <span class="math-container">$r\in [0,1)$</span> define
<span class="math-container">$$
I(r) = \int_0^{2\pi}\frac{\cos(t) - r}{1 - 2r\cos t + r^2}\,dt.
$$</span>
Numerical experiments hint at <span class="math-container">$I(r) = 0$</span> for all <span class="math-container">$r\in [0,1)$</span> but I can't show ... | ComplexYetTrivial | 570,419 | <p>An alternative proof using complex methods:</p>
<p>For <span class="math-container">$0< r < 1$</span> let
<span class="math-container">$$ f_r \colon B_\frac{1}{r} (0) \to \mathbb{C} \, , \, f_r(z) = \frac{- \ln(1-rz)}{z} \, , $$</span>
where <span class="math-container">$f_r(0) = r $</span> . Then <span class... |
3,088,503 | <p>For <span class="math-container">$r\in [0,1)$</span> define
<span class="math-container">$$
I(r) = \int_0^{2\pi}\frac{\cos(t) - r}{1 - 2r\cos t + r^2}\,dt.
$$</span>
Numerical experiments hint at <span class="math-container">$I(r) = 0$</span> for all <span class="math-container">$r\in [0,1)$</span> but I can't show ... | Paramanand Singh | 72,031 | <p>Note that <span class="math-container">$$I(r) =\frac{-1}{2r}\int_{0}^{2\pi}\frac{1-2r\cos t+r^2+r^2-1}{1-2r\cos t+r^2}\,dt$$</span> which is same as <span class="math-container">$$-\frac{\pi} {r} +\frac{1-r^2}{r}\int_{0}^{\pi}\frac{dt}{1+r^2-2r\cos t} $$</span> The integral above is equal to <span class="math-contai... |
3,321,399 | <p><span class="math-container">$$
\lim\limits_{n \to \infty} \cos^2\left(\pi \sqrt[3]{n^3+n^2+2n}\right)
$$</span>
where <span class="math-container">$n \in \mathbb{N}$</span>.</p>
<p>In this question, what I thought was, since <span class="math-container">$n \to \infty$</span> and <span class="math-container">$\cos ... | eyeballfrog | 395,748 | <p>Use the binomial theorem on <span class="math-container">$(1+x)^{1/3}$</span> to get
<span class="math-container">$$
\sqrt[3]{n^3 + n^2 + 2n} = n\sqrt[3]{1 + n^{-1} + 2n^{-2}}= n\left[1 + \frac{n^{-1}+2n^{-2}}{3} - \frac{(n^{-1}+2n^{-2})^2}{9}+...\right] \\= n + \frac{1}{3} + \frac{5}{9n} + O(n^{-2})
$$</span>
So
<... |
124,694 | <p>I'm teaching an honors differential equations class and have been using linear algebra heavily. I thought it would be interesting to include a proof of the method of undetermined coefficients along the lines described below, which I figured is already known. Nonetheless, Google turns up no proof other than the obv... | Oswaldo R B de Oliveira | 33,413 | <p>Hi Ryan and everybody,</p>
<p>Besides the very beautiful proof by Tao, a very nice and easy linear algebra approach to the undetermined coefficients method can be found in C. C. Ross ``Why the method of undetermined coefficients works'', Am. Math. Monthly 98 (1991), pp. 747-749.</p>
<p>I also suggest R. C. Gupta `... |
4,086,718 | <p>Prove <span class="math-container">$\lim\limits_{(x,y) \to (0,0)} \frac{xy(y^2-x^2)}{x^2+y^2}=0$</span></p>
<p>My attempt: For all <span class="math-container">$\epsilon > 0$</span> there is a <span class="math-container">$\delta > 0$</span> such that <span class="math-container">$\left | \frac{xy(y^2-x^2)}{x^... | Koro | 266,435 | <p>Your last step is not correct as for example take <span class="math-container">$|2^2-5^2|\lt |5^2+2^2-5^2|$</span>, which is clearly not true.<br/></p>
<p>You may proceed like this also: <br/></p>
<p>For any <span class="math-container">$\epsilon\gt 0$</span>,<br/>
<span class="math-container">$\begin{align}|\frac{x... |
2,709,977 | <p>let $X$ be a compact Hausdorff space.consider commutative Banach algebra $C(X)$.</p>
<p>let for $x\in X$ is fixed. define$$M_x=\{f:f\in C(X)\ \text{and}\ f(x)=0\}$$.</p>
<p>then $M_x$ is proper two sided ideal. but is $M_x$ maximal ideal in $C(X)$</p>
<p>any hint</p>
| user284331 | 284,331 | <p>Let $A$ be the rotation to $\pi/4$ clockwise, then $(x,y)\rightarrow A(x,y)$ is homeomorphism from $S$ to the upper half plane $M:=\{(x,y): y>0\}$.</p>
<p>It suffices to prove that $M$ is path-connected. But this is clear because any point in the line segment of any two points of $M$ must have strictly positive ... |
164,114 | <p>Is a commutative ring a field? A set equipped with addition and multiplication which is abelian over those two operations and it holds distributivity of multiplication over addition?</p>
| Steven Gamer | 47,540 | <p>In a commutative ring not every nonzero element has a multiplicative inverse unlike the requirement in a field that every nonzero element has a multiplicative inverse.</p>
|
3,602,660 | <p>More clearly the question is:</p>
<p>Prove that R is a field by showing that <span class="math-container">$\mathbb{R}$</span> forms an isomorphisim to <span class="math-container">$\mathbb{C}$</span> as rings (explicitly find an isomorphism between them and prove it is an isomorphism).</p>
<p>Let <span class="math... | user764828 | 764,828 | <p>In the case of <span class="math-container">$x^3 = 0$</span>, there is only one root. If <span class="math-container">$x \neq 0$</span>, then we can divide both sides by <span class="math-container">$x$</span> to get <span class="math-container">$x^2 = 0/x = 0$</span>, and similarly, <span class="math-container">$x ... |
3,602,660 | <p>More clearly the question is:</p>
<p>Prove that R is a field by showing that <span class="math-container">$\mathbb{R}$</span> forms an isomorphisim to <span class="math-container">$\mathbb{C}$</span> as rings (explicitly find an isomorphism between them and prove it is an isomorphism).</p>
<p>Let <span class="math... | Matteo | 686,644 | <p>Solving this type of equation is very simple because you can always rewritw it as follows:
<span class="math-container">$$(x-0)^3=0$$</span>
From here, you can see that the only solution to the equation is:
<span class="math-container">$$x=0$$</span>
because being <span class="math-container">$x$</span> a general co... |
1,386,193 | <p>I am learning conditional expectation for fun. I cannot solve the following problem, I think it is just my vocabulary is limited. This problem came from a book on Stochastic Processes:</p>
<p>Let $\Omega = [0,1]\times [0,1]$ with the usual Lebesgue measure and $\sigma$-algebra of Borel sets. Let $X$ and $Y$ be rand... | Hasan Kadour | 304,309 | <p>we can write
$ E(X|Y)=\int_X xf(x|y)dx$</p>
<p>but $f(x|y)=\frac{f(x,y)}{f(y)}$</p>
<p>and
$ f(y)=\int_X f(x,y)dx=\int_{0}^{1} (x+y)dx=(x^2/2+xy |_0^1)=1/2+y$</p>
<p>Also</p>
<p>$$ E(X|Y)=\int_X xf(x|y)dx=\int_{0}^{1}x \frac{x+y}{1/2+y} dx$$</p>
<p>$$ =\int_{0}^{1} \frac{x^2+xy}{1/2+y} dx=\frac{1}{1/2+y} (x^3/... |
98,971 | <p>I'm working on an algorithm which should check if two polygons, described by their vertex coordinates, are: one inside the other, are intersecting or are separated</p>
<p>image below describe this three cases:
<a href="https://i.stack.imgur.com/hl6zL.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.co... | george2079 | 2,079 | <p>using this: <a href="https://mathematica.stackexchange.com/a/51425/2079">https://mathematica.stackexchange.com/a/51425/2079</a> </p>
<pre><code>segsegintersectionQ[lines_] := Module[{
md = Subtract @@ (Plus @@ # & /@ lines),
sub = Subtract @@ # & /@ lines, det}, det = -Det[sub];
TrueQ[And @@ (Abs[#... |
4,521,067 | <blockquote>
<p>If <span class="math-container">$$f(x)=\prod_{k=1}^\infty\left(1-\frac{x^3}{k^3\pi^3}\right)$$</span>
and if <span class="math-container">$$f(x)=\sum_{n=0}^\infty a_n x^n$$</span>
Then find <span class="math-container">$a_n$</span> or convert the above infinite product into an infinite sum.</p>
</block... | metamorphy | 543,769 | <p>An "algorithmic" way is via doing the <a href="https://en.wikipedia.org/wiki/Exponential_formula" rel="nofollow noreferrer">exponentiation</a> in <span class="math-container">$$f(x)=\exp\left[-\sum_{k=1}^\infty\sum_{n=1}^\infty\frac1n\left(\frac{x}{k\pi}\right)^{3n}\right]=\exp\left[-\sum_{n=1}^\infty\frac... |
2,146,192 | <p>My question is that I want to evaluate step-by-step the integral </p>
<blockquote>
<p>$$\int_0^1 \frac{\mathrm{d}x}{(a^2+x^2)\sqrt{1-x^2}}$$</p>
</blockquote>
<p>where $a$ is some constante in the real numbers and $x$ is also a real number. </p>
<p><strong>My attempt:</strong> If $x = \sin(u)$ then the integral... | Ahmed S. Attaalla | 229,023 | <p>With $x=\sin u$ we have,</p>
<p>$$\int_{0}^{\frac{\pi}{2}} \frac{1}{\sin^2(u)+a^2} du$$</p>
<p>Dividing top and bottom by $\cos^2 u$ gives,</p>
<p>$$\int_{0}^{\frac{\pi}{2}} \frac{\sec^2 u}{\tan^2 u+a^2 \sec^2 u} du$$</p>
<p>$$=\int_{0}^{\frac{\pi}{2}} \frac{\sec^2 u}{\tan^2 u+a^2(\tan^2 u+1)} du$$</p>
<p>$$=\i... |
2,146,192 | <p>My question is that I want to evaluate step-by-step the integral </p>
<blockquote>
<p>$$\int_0^1 \frac{\mathrm{d}x}{(a^2+x^2)\sqrt{1-x^2}}$$</p>
</blockquote>
<p>where $a$ is some constante in the real numbers and $x$ is also a real number. </p>
<p><strong>My attempt:</strong> If $x = \sin(u)$ then the integral... | David Quinn | 187,299 | <p>Hint...when you do the substitution $x=\sin u$ you get $$I=\int_0^{\frac{\pi}{2}}\frac{1}{a^2+\sin^2u}du$$</p>
<p>Now do the substitution $t=\tan u$ and the integral becomes$$I=\int_0^{\infty}\frac{1}{a^2+(a^2+1)t^2}dt$$</p>
<p>Can you finish this?</p>
|
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... | Noble Mushtak | 307,483 | <p>Let's say we have $(a, b) \in X$ and $(b, c) \in X$. Since $(a, b) \in X$, we have $a=1$. Since $a=1$, we have $(a, c) \in X$, proving transitivity.</p>
<p>The interesting thing here is that because we have $(b, c) \in X$, we have $b=1$. Therefore, transitivity only applies when $(a, b)=(1, 1)$.</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... | Calvin Khor | 80,734 | <p>I will use $\|f\|_X$ for function norms and $|f(x)|$ for absolute values.</p>
<p>You may have missed that the author states that $y$ was such that $|f(y)|= \min_x |f(x)| ≤ |f(z)|$ for every $z\in[0,1]$. Integrating in $z$,</p>
<p>$$ |f(y)| = \min_x |f(x)|\int_0^1 dz ≤ \int_0^1|f(z)| \, dz$$
You are correct for th... |
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... | Mostafa Ayaz | 518,023 | <p>We have <span class="math-container">$$f_n(x)=\int_0^x\{nu\}du=\begin{cases}
{nx^2\over 2}&,\quad 0\le x< {1\over n}\\
{1\over 2n}+{n\left(x-{1\over n}\right)^2\over 2}&,\quad {1\over n}\le x< {2\over n}\\
{2\over 2n}+{n\left(x-{2\over n}\right)^2\over 2}&,\quad {2\over n}\le x< {3\over n}\\
{3\... |
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... | kcrisman | 1,608 | <p>This is a contentious and highly individual thing, of course, so all answers in this should be taken with a grain of salt. But, if this is your <em>job</em> then:</p>
<p><strong><em>YES.</em></strong></p>
<p>Why? First, some practical reasons that apply to any job.</p>
<ul>
<li>You are not an island. Some topi... |
3,357,166 | <p>It is well-known that the image of a compact set is compact and the image of a connected set is connected under a continuous map. Here <span class="math-container">$A$</span> is perfect if <span class="math-container">$A$</span> it is equal to the derived set of itself.</p>
<p>I would like to ask if this property h... | Arthur | 15,500 | <p>No, it is not true. Consider the image of <span class="math-container">$\Bbb R$</span> under <span class="math-container">$\arctan$</span> (as a subset of <span class="math-container">$\Bbb R$</span>).</p>
<p>For compact sets, the boundedness property lets closed sets remain closed, so compactness is preserved. How... |
2,295,231 | <p>Let $A=(a_{ij}) $ be a $2\times 2$ real matrix such that $a_{11}^2+a_{22}^2+a_{21}^2+a_{12}^2 < 1/10 $. </p>
<p>Prove that $I+A$ is invertible .</p>
<p>My approach : Suppose $I+A$ were singular . Then it has a non-zero $2\times 1$ column vector such that $(I+A)v=0 $ i.e $Av=-v $. Hence $-1$ is an eigenvalue of ... | user49640 | 417,455 | <p>If $X = (x,y)^T$ is any column vector, you have $AX = (a_{11} x + a_{12} y, a_{21}x + a_{22}y)^T$. So
$$
\begin{align*}
|AX|^2 &= [(a_{11},a_{12}) \cdot (x,y)]^2 + [(a_{21},a_{22}) \cdot (x,y)]^2 \\ &\leq |(a_{11},a_{12})|^2|(x,y)|^2 + |(a_{21},a_{22})|^2|(x,y)|^2 \\
&=(a_{11}^2 + a_{12}^2 + a_{21}^2 + ... |
135,783 | <p>Assume that $ B \hookrightarrow A $ is an injective ring homomorphism. </p>
<blockquote>
<blockquote>
<p><b> Question: </b> If $\mathfrak{p}$ is an associated prime of $B$, is there an associated prime $ \mathfrak{q}$ of $A$ such that $ \mathfrak{q} \cap B = \mathfrak{p}$?</p>
</blockquote>
</blockquote>
<... | Steven Landsburg | 10,503 | <p>Put $B=k[x,y]/(xy,y^2)$ and $A=B[z]/(yz-x)$. Then $(y)$ is an associated (in fact minimal!) prime of $B$. But any prime of $A$ that contains $(y)$ must also contain $x$, and so cannot contract to just $(y)$.</p>
|
855,665 | <p>suppose that we have following binary digits</p>
<p>$00011001.110 $,we can do following thing</p>
<p>$00011001.110=1\cdot2^4+1\cdot2^3+1\cdot2^0+1\cdot2^{-1}+1\cdot2^{-2}=25.75$ </p>
<p>then what does means?</p>
<p>We then “float” the binary point:</p>
<p>$00011001.110 \Rightarrow 1.1001110 \cdot 2^4 $</p>
<... | acer | 12,448 | <p>That <a href="http://www.maplesoft.com/support/help/Maple/view.aspx?path=PolynomialTools/Shorten" rel="nofollow">PolynomialTools:-Shorten</a> Maple Library command is a renamed version of the <a href="http://www.math.fsu.edu/~hoeij/lib/NEW/shorten" rel="nofollow">shorten</a> command written by <a href="http://www.ma... |
2,702,833 | <p>Instead of using L'Hospital's rule can this be proved by using the definition of the limit and or the squeeze theorem.</p>
| user284331 | 284,331 | <p>Use Mean Value Theorem to deduce that $1+u+u^{2}/2\leq e^{u}\leq 1+u+u^{2}$ for $0\leq u\leq 1/2$, then for $n\geq 2$, we have $1/n\leq 2$ and hence $1+(1/n)+1/(2n^{2})\leq e^{1/n}\leq 1+(1/n)+1/n^{2}$, so $n(e^{1/n}-1)\rightarrow 1$.</p>
<p>The use of Mean Value Theorem here:</p>
<p>Let $\varphi(u)=1+u+u^{2}-e^{u... |
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>
| Marc | 132,141 | <p>No, say $a_1 = 1$, $b_1 = 0$, $a_2 = 10$, $b_2 = 1$, then
$$
a_1 - a_2 = 1-10 = -9
$$
and
$$
b_1 - b_2 = 0-1 = -1
$$</p>
|
66,178 | <p>Let $G$ be a finite group. Suppose $Z \le Z(G)$ and $G/Z \cong A\times B$, where $A$ is a $\pi$-group and $B$ is a $\pi'$-group for some set of primes $\pi$. Is it true that $G\cong C \times D$, where $C$ is a $\pi$-group and $D$ is a $\pi'$-group for the same set of primes $\pi$? </p>
| Jack Schmidt | 583 | <p>Hall's version of Sylow's theorem handles this.</p>
<p><em>Background</em>: A Hall <em>π</em>-subgroup is a <em>π</em>-subgroup whose index is a <em>π</em>′-number. If every composition factor of <em>G</em> is either a <em>π</em>-subgroup or a <em>π</em>′-subgroup (that is, <em>G</em> is <em>π</em>-separable), the... |
66,178 | <p>Let $G$ be a finite group. Suppose $Z \le Z(G)$ and $G/Z \cong A\times B$, where $A$ is a $\pi$-group and $B$ is a $\pi'$-group for some set of primes $\pi$. Is it true that $G\cong C \times D$, where $C$ is a $\pi$-group and $D$ is a $\pi'$-group for the same set of primes $\pi$? </p>
| the_fox | 11,450 | <p>To provide some context for this problem, I needed it as a lemma in order to do part b) of problem 5A.8 in Martin Isaac's book "Finite Group Theory". In page 151 of his book Isaacs gives the definition for the Schur multiplier and states (without proof of course) the existence and uniqueness of a largest second comp... |
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... | Shubham Johri | 551,962 | <p><span class="math-container">$\begin{bmatrix}\begin{array}{ccc|c}
1 & x & x^2 & x \\
0 & 1-x^2 & x-x^3 & 0\\
x^2 & x & 1 & x^3
\end{array}\end{bmatrix}\xrightarrow{R_3\to R_3-x^2R_1}\begin{bmatrix}\begin{array}{ccc|c}
1 & x & x^2 & x \\
0 & 1-x^2 & x-x^3 & ... |
322,365 | <p>Let $\phi:R[x]\rightarrow C$ defined for a fixed $z\in C$ as $\phi(p(x))=p(z)$ where $p\in R[x]$. Then this map is supposed to be surjective. Could someone explain me please why?</p>
| Andreas Caranti | 58,401 | <p>Just to provide a simple example, the $2 \times 2$ zero matrix is a root of all polynomials
$$
x^2 + a x,
$$
for all $a \in F$.</p>
<p>Its characteristic polynomial is $x^2$, of course.</p>
|
850,249 | <p>Express a complex number in modulus amplitude form</p>
<p>$\displaystyle 1+\sin \alpha +i\cos \alpha $</p>
<p>My Attempt: </p>
<p>$\displaystyle r\cos \theta= 1+\sin \alpha $</p>
<p>$\displaystyle r\sin \theta= \cos \alpha $</p>
<p>Squaring and adding.. $\displaystyle r^2= (1+\sin \alpha)^2+ \cos^2 \alpha$</p>
... | Pranav Arora | 117,767 | <p>It is better to solve it this way:</p>
<p>$$\begin{aligned}
1+\sin\alpha+i\cos \alpha &=1+\cos\left(\frac{\pi}{2}-\alpha\right)+i\sin\left(\frac{\pi}{2}-\alpha\right)\\
& \stackrel{*}{=}2\cos^2\left(\frac{\pi}{4}-\frac{\alpha}{2}\right)+i2\sin\left(\frac{\pi}{4}-\frac{\alpha}{2}\right)\cos\left(\frac{\pi}{4... |
2,552,118 | <p>I know this might be stupid for most of the people here. I am starting with set theory and I cannot prove this:A-(A∩B)=A-B</p>
<p>I start by assuming: 1) x∈A & ¬ (x∈A & x∈B)</p>
<pre><code> Therefore I understand I can infer:
2) x∈A from 1
3) ¬ (x∈A & x∈... | I was suspended for talking | 474,690 | <p>\begin{align*}
x \in A - (A\cap B)
&\iff x \in A \; \wedge \; x \notin A\cap B \\
&\iff x \in A \; \wedge \; (x\notin A \: \vee \: x \notin B) \\
&\iff x \in A \; \wedge \; x \notin B \\
&\iff x \in A -B.
\end{align*}</p>
|
2,354,218 | <p>On page 5 of Royden, Fitzpatrick's real analysis, the book defined cardinality of the set in the following way:</p>
<blockquote>
<p>Given a set $X$, the relation of equipotence (invertible map) is an equivalence relation on the collection of $2^X$ of all subsets of $X$. The equivalence class of a set with respect... | Diego Marcon | 435,506 | <p><em>Hint:</em> I guess one assumes $f$ is not identically zero to begin with. Consider a convergent sequence $z_n \to z_0$ (with distinct points). Then, if $c=0$, we conclude that either $f(z_n) = 0$ or $f(-z_n) = 0$ for infinitely many indices $n$. Why does that show that $f \equiv 0$?</p>
|
2,905,129 | <blockquote>
<p>If we have a matrix $A \in \mathbb{R}^{n \times n}$ and $A=T\Lambda T^{-1}$ for $T\in \mathbb{R}^{n \times n}$, how to prove that each column vector $t^i \in \mathbb{R}^n$ that $At^i = \lambda(t^i)$?</p>
</blockquote>
<p>I'm having trouble understanding how to prove that each element $t^i$ of $T$ is ... | Siong Thye Goh | 306,553 | <p>$$A=T\Lambda T^{-1}$$</p>
<p>Multiply $T$ on both sides from the right.</p>
<p>$$AT=T\Lambda$$</p>
<p>Look at the $i$ column or if it helps, multiply by $e_i$ where $e_i$ is the $i$-th standard unit vector, there you can observe that </p>
<p>$$At^i = \lambda_i t^i$$</p>
|
2,295,515 | <p>I was doing a problem and I ran into this triangle:</p>
<p><a href="https://i.stack.imgur.com/sFSxM.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sFSxM.png" alt="enter image description here"></a></p>
<p>The problem itself is not that important for this question, but rather the explanation of ... | JMP | 210,189 | <p>There are several proofs depending on how 'deep' you want to go.</p>
<p>One is that line $A$ perpendicular to a line $B$ which perpendicular to another line $C$ is parallel to $C$, and therefore preserves angles along yet another line $D$, which creates similar triangles.</p>
|
901,703 | <p>Let $G$ be a $p$-group and $H$ be a $p$-subgroup of $G$; $H$ is not normal in $G$. Then prove that the number of distinct conjugates of $H$ in $G$ divides $|G|$.
Notice that the conjugates may have nontrivial intersection.</p>
| mesel | 106,102 | <p>Let $\Omega$ be set of all conjugates of $H$ then $G$ act on $\Omega$ by conjugation. And since every element is a conjugate of $H$, this action is transitive.</p>
<hr>
<p><strong>By orbit stabilizer theorem</strong>; $$|\Omega|=|G:Stab(H)|$$</p>
<p>And $Stab(H)=\{g\in G| g^{-1}Hg=H\}=N_G(H)$ so we are done.</p>
... |
2,404,946 | <p>I am an architect, and just started using the parametric software "Grasshopper", trying to model and reshape surfaces based on geometric formulas.</p>
<p>That's where I came up with the terms "Mean Curvature" and "Gaussian Curvature",
and have no idea what they mean.</p>
<p>Can someone please tell me in plain Engl... | Christian Blatter | 1,303 | <p>Consider a smooth surface $S\subset{\mathbb R}^3$, say the surface of a car roof, or an ordinary egg. This surface is not uniformly curved, like a sphere is; instead we have a "curvature status" that changes from point to point on $S$. At each point $p\in S$ we have a tangential plane $T$. If we consider this plane ... |
18,231 | <p>I'm stuck on the following problem :</p>
<p>Does there exist a real borel measure $\mu$ on $[0,1]$ such that</p>
<p>$$\int_{0}^{1}x^n d\mu = e^{-n^2}$$
for all $n \geq 1$ ?</p>
<p>Does anyone have any hint?</p>
<p>Thank you</p>
| Arturo Magidin | 742 | <p>In addition to "general problem solving" (I second PEV's recommendation of Polya's book), much along the lines of this <a href="https://math.stackexchange.com/questions/4862/where-to-go-after-calculus/4865#4865">previous question about what to do after Calculus</a>, I would suggest to start with Number Theory and Hi... |
78,357 | <p>I would like to know whether a Heyting algebra gives rise to ring in a similar way that a Boolean algebra gives rise to a Boolean ring. In a Boolean algebra $(B,\lor,\land,\lnot,0,1)$ I can define multiplication and addition as follows:</p>
<p>$$a * b = a \land b$$
$$a + b = (a \land \lnot b) \lor (b \land \lnot a)... | Raeez | 39,254 | <p>Though not directly analogous to your example of a boolean ring, a <a href="http://link.springer.com/article/10.1007%2Fs00012-010-0092-1#page-1" rel="nofollow">2010 paper in <em>Algebra Universalis</em></a> approaches the problem of classifying linearly ordered Heyting Algebras via the concept of a Godel Ring. The i... |
2,118,974 | <p>Prove that:$$\frac {\sec (16A) - 1}{\sec (8A) - 1}=\frac {\tan (16A)}{\tan (4A)}$$.</p>
<p>My Attempt,</p>
<p>$$L.H.S= \frac {\sec (16A)-1}{\sec (8A)-1}$$
$$=\frac {\frac {1}{\cos (16A)} -1}{\frac {1}{\cos (8A)} -1}$$
$$=\frac {(1-\cos (16A)).(\cos (8A)}{(\cos (16A))(1-\cos (8A))}$$.</p>
<p>What should I do next?... | Kanwaljit Singh | 401,635 | <p>$\frac{\sec 16A -1}{\sec 8A -1}$</p>
<p>= $\frac{\frac{1}{\cos 16A}-1}{\frac{1}{\cos 8A}-1}$</p>
<p>= $\frac{\frac{1 - \cos 16A}{\cos 16A}}{\frac{1 - \cos 8A}{\cos 8A}}$</p>
<p>= $\frac{2 \sin^2 8A}{\cos 16A} × \frac{\cos 8A}{2 \sin^2 4A}$</p>
<p>= $\frac{2 \sin 8A \cos 8A}{\cos 16A} × \frac{\sin 8A}{2 \sin^2 4A... |
3,264,420 | <p>From Convex Optimization by Boyd and Vandenberghe:</p>
<blockquote>
<p>Let <span class="math-container">$T \in \Bbb R^{n \times n}$</span> be nonsingular. Let <span class="math-container">$f: \Bbb R^n
\rightarrow \Bbb R$</span> convex and twice continuously differentiable.</p>
<p>Define <span class="math-c... | peek-a-boo | 568,204 | <p>This is really the chain rule,which states <span class="math-container">$d(\varphi \circ \psi)_y = d \varphi_{\psi(y)} \circ d\psi_y$</span>. So in your case, we have
<span class="math-container">\begin{align}
\bar{f} = f \circ T.
\end{align}</span>
Hence,
<span class="math-container">\begin{align}
d\bar{f}_{y} = d... |
2,059,406 | <p>I know that any index 2 subgroup is normal.</p>
<p>But is it the biggest?</p>
| p Groups | 301,282 | <p>It would be appropriate to say "it is <em>maximal</em>" instead of <em>biggest</em>; this is then obvious. Then instead of $2$, a subgroup of index $p$, prime, is also maximal (=biggest in your sense), but not necessarily normal; Robert Israel's argument (2nd statement) explains why it should be maximal.</p>
<p>Bei... |
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... | Bernard | 202,857 | <p>First note it can't have a linear factor, as it would imply $X^4+a^2$ has a rational, hence real, root. Second it can be factored in $\mathbf R[X]$:
$$X^4+a^2=(X^2+a)^2-2aX^2=(X^2-\sqrt{2a}X+a)(X^2+\sqrt{2a}X+a),$$
so all we have to prove is $\sqrt{2a}$ is irrational, which is equiavalent to $2a$ being a square in ... |
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... | Yuriy S | 269,624 | <p>First, your homogeneous solution contains a mistake. It should be:</p>
<p>$$y_h = C_1e^{2t}\cos(t) + C_2e^{2t}\sin(t)$$</p>
<hr>
<p>Now, this equation allows for a simple particular solution, but a more general way to find a particular solution is making one of the constants depend on $t$ and substituting this in... |
533,720 | <p>Let $A$ be open in $\mathbb{R}^m$; let $g:A\rightarrow\mathbb{R}^n$. If $S\subseteq A$, we say that $S$ satisfies the <strong>Lipschitz condition</strong> on $S$ if the function $\lambda(x,y)=|g(x)-g(y)|/|x-y|$ is bounded for $x\neq y\in S$. We say that $g$ is <strong>locally Lipschitz</strong> if each point of $A$ ... | Rudolf L | 315,757 | <p>By Rademacher's theorem any Lipschitz continuous function is differentiable a.e. </p>
<p>The derivative
of a lipschitz function is a bounded measurable function, but this derivative need not be continuous.</p>
<p>Hence the locally lipschitz function need not be $C^1$. </p>
<p>For example $f(x) = \frac {x^i x^... |
583,544 | <p>Sometimes I see an exercise like this: </p>
<blockquote>
<p>Let $f:[a,b] \to \mathbb R$ be differentiable. (A few more givens here.) Show that $f'$ has such-and-such property.</p>
</blockquote>
<p>What is usually meant by that? Should $f'$ be defined on $[a,b]$ with one-sided derivatives in $a,b$? Or should one ... | 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... |
1,886,332 | <p>Let $f$ be a continuous function from the reals $\mathbb{R}$ onto $I=[0,1]$ with usual topology. Prove that if $C$ is a subset of $I$ and the preimage of $C$ is closed in $\mathbb{R}$ then $C$ is closed in $I$.</p>
<p>My attempt is to use normal space properties but it does not help. </p>
| Luiz Cordeiro | 58,818 | <p>Hint: There is a compact interval $J=[c,d]$ such that $f(J)=I$. If $(x_n)\subseteq C$ converges in $I$, take preimages and use compactness of $J$ to show that $\lim x_n\in C$.</p>
|
1,815,431 | <p>I'm currently trying to work out a function where</p>
<p>f(x) = 2x + 4</p>
<p>The question is to define f(x) <strong><em>recursively</em></strong></p>
<p>I'm unsure what this means, I have an idea of how to do it but I'm not sure if it's right. The way I thought of doing it was:</p>
<p>f(0) = 2*0 + 4 = 4<br>
f(1... | Ahmed S. Attaalla | 229,023 | <p>It means that given a term, can you define your function $f$ based on that term and some algebraic rule that will help you determine a proceding term.</p>
<p>Lines $mx+b$ can be defined recursively by,</p>
<p>$$f(x+1)=f(x)+m,x \geq 0$$</p>
<p>$$f(0)=b$$</p>
<p>Look at your sequence. Like you said it goes, </p>
... |
416,704 | <p><span class="math-container">$\newcommand{\R}{\mathbb{R}}$</span>I have a map <span class="math-container">$f\in C^0(X,\mathbb{R})$</span>, where <span class="math-container">$X$</span> is a compact and Hausdorff topological space, which is a manifold outside of a compact subset <span class="math-container">$K\subs... | Fernando Muro | 12,166 | <p>The <span class="math-container">$1$</span>-type of <span class="math-container">$M\mathbb{Z}/(2^r)$</span> does have an <span class="math-container">$E_\infty$</span>-ring structure for <span class="math-container">$r> 1$</span>. I'm going to show it by using the algebraic models for <span class="math-container"... |
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... | G Cab | 317,234 | <p>Besides the particular examples given in the precedent answer, for any function $y=f(x)$, the graph of its inverse function $f_{(-1)}(x)$ will be symmetric with respect to the diagonal line $y=x$.</p>
<p>If $f(x)$ does not cross the diagonal (as $e^x$), so won't the inverse. And any zero of $f(x)-x$ will be the sam... |
2,424,061 | <p>A question asks "Is $f(x)=x^2$ an onto function or not?" Here the domain and codomain are not mentioned specifically, so what codomain should I consider?</p>
| John Griffin | 466,397 | <p>It's likely the intended domain is all real numbers such that the expression makes sense, and the intended codomain is the set of real numbers. This function would be neither injective nor surjective under these assumptions.</p>
<p>However, the domain and codomain should always be specified. This is especially true... |
3,086,979 | <p>Attempt:
I've used that <span class="math-container">$\sin(x) = (2t)/(1+t^2)$</span> and <span class="math-container">$\cos(x) = (1-t^2)/(1+t^2)$</span>.
However I don't seem to get a perfect square, instead I get <span class="math-container">$$(2/((1+t^2)^2))(14t^2 +20t + 38)$$</span>.</p>
<p>I'm not sure if there... | J.Dane | 544,749 | <p><span class="math-container">$$\begin{align}
2(1+\cos(x))(5\sin(x)+12\cos(x)+13)&=2\left(1+\frac{1-t^2}{1+t^2}\right)\left(\frac{10t}{1+t^2}+\frac{12-12t^2}{1+t^2}+13\right)\\
&=\frac{4}{(1+t^2)^2}(t+5)^2
\end{align}$$</span></p>
|
423,172 | <p>Let us suppose you have a certain function $f(x)$ and you want to find out in which intervals this function is decreasing, constant or increasing. I know you need to follow these steps:</p>
<ol>
<li>Find out $f'(x)$.</li>
<li>Find out the values for which $f'(x)=0$. In other words, we need to find out the zeroes of... | DonAntonio | 31,254 | <p>In the example you give, we have</p>
<p>$$\frac{e^x}x\begin{cases}>0&\,\;\;x>0\\{}\\<0&,\;\;x<0\end{cases}$$</p>
<p>So the function of which the above is the derivative is monotone ascending if $\,x>0\;$ and monotone descending when $\,x<0\,$</p>
<p>The above is all you need in general, ... |
253,592 | <p>If $E\subset R^n$ be a Lebesgue measurable set and let $(f_k)$ be a sequence of non-negative Lebesgue measurable function on E such that $\lim f_k=f$ a.e.
I want to prove that if $\int_E fd\lambda<\infty$,and$$\lim_k \int_E f_kd\lambda=\int_E fd\lambda,$$then$$\lim_k \int_A f_kd\lambda=\int_A fd\lambda$$
for ever... | Davide Giraudo | 9,849 | <p>For the first question, use Fatou's lemma and the sequence $g_k:=f_k+f-|f-f_k|\geqslant 0$.</p>
<p>For the second part, take $f_n:=\chi_{(n,n+1)}+\chi_{(-1,0)}\frac 1x$ and $f:=\chi_{(-1,0)}\frac 1x$, $A:=[0,+\infty)$.</p>
|
253,592 | <p>If $E\subset R^n$ be a Lebesgue measurable set and let $(f_k)$ be a sequence of non-negative Lebesgue measurable function on E such that $\lim f_k=f$ a.e.
I want to prove that if $\int_E fd\lambda<\infty$,and$$\lim_k \int_E f_kd\lambda=\int_E fd\lambda,$$then$$\lim_k \int_A f_kd\lambda=\int_A fd\lambda$$
for ever... | Deven Ware | 14,334 | <p>Here is a counter example in the case when $\int f = \infty$:</p>
<p>Take $E = \mathbb{R}$ and set
$$f_k = 1_{(-\infty, 0)} + 1_{(k,k+1)} \rightarrow 1_{(-\infty, 0)}$$
And take $A = [0,\infty)$. </p>
|
477,695 | <p>I was asked to compute the 8-th roots of $i$. </p>
<p>Correct me if I am wrong, but am I looking for 8 different solutions to this? My thought was to first consider a root of the form $cos(x) + i sin(x)$. So, we are dealing with $cos(8x) + isin(8x)$. At this point I believe the $cos(8x)$ is $1$. Therefore we have $... | draks ... | 19,341 | <p>Use $i=e^{i\pi/2+i2\pi n}$ to get $8$ solutions
$$
(i)^{1/8}=e^{i(\pi/2+2\pi n)/8}=e^{i(\pi/16+n\pi/4 )},
$$
where $n$ runs from $0$ to $7$.</p>
|
1,703,100 | <p>A car starts from rest and accelerates in $a = \frac{2\cdot m}{3\cdot s^3}t$,</p>
<p>After $3$ seconds, The car will be $27$ metres from beginning.</p>
<p>Find distance as function of time.</p>
<p>I know i have to integral the acceleration,But i don't know how.</p>
<p>I found the Equation is $x_{t} = 24 + \frac{... | BruceET | 221,800 | <p>Comment: Simulation of a million plays of this game in R verifies the
answer by @Cm7F7Bb.</p>
<pre><code> B = 10^6
d1 = sample(1:6, B, rep=T)
d2 = sample(1:6, B, rep=T)
s = numeric(B) # vector of all 0's to start
s[d1 > 1] = d1[d1 > 1] # score if first die shows > 1
s[d1 == 1] = (d1+... |
1,949,133 | <p>$\newcommand{\Reals}{\mathbb{R}}\newcommand{\Vec}{\mathcal{T}}\newcommand{\Brak}[1]{\left\langle #1\right\rangle}$
Just a question about the following exercise.</p>
<p><strong>Exercise</strong></p>
<p>Let $C=\{(x,y,z)\in\mathbb{R}^3|x^2+y^2=1\}\subset\mathbb{R}^3$. The cylinder $C$ can be identified with $M=\mathb... | Andrew D. Hwang | 86,418 | <p>Secretly, the induced metric on the cylinder $C$ is the Euclidean (!) metric
$$
g_{C} = d\theta^{2} + d(\log \rho)^{2}
= d\theta^{2} + dz^{2}
$$
in the $(\theta, z)$-plane. Consequently, the geodesics are "straight lines parametrized at constant speed". (On the cylinder, these paths trace meridians, helices, or la... |
2,565,567 | <p>In a certain Algebra 2 class of 30 students, 22 of them play basketball and 18 of them play baseball. There are 3 students who play neither sport. What is the probability that a student chosen randomly from the class plays both basketball and baseball?</p>
<p>I know how to calculate the probability of students play... | RSA | 430,010 | <p>The events playing of basketball and baseball are not independent. Hence, you cannot use the formula that you suggested.</p>
<p>You need to use: $\Pr(Basketball \cap Baseball) = \Pr(Basketball) + \Pr(Baseball) - \Pr(Basketball \cup Baseball).$</p>
|
2,201,908 | <ol>
<li>$(P \vee Q)$ & $R$ </li>
<li>$(R$ & $P) \rightarrow S$</li>
<li>$(Q$ & $R) \rightarrow S$ </li>
</ol>
<p>Conclusion is $S$</p>
<p>without using conditional proof, please?</p>
<p>And perhaps if you have time, please show me the conditional proof method or at least which CP you would use.
Thank yo... | quasi | 400,434 | <p>\begin{align*}
\text{Given:}&&1.&\;\; (P \lor Q) \land R\\[2pt]
&&2.&\;\; (R \land P) \to S\\[2pt]
&&3.&\;\;(Q \land R) \to S\\[12pt]
\text{To prove:}&&S&\\[12pt]
\text{Proof:}&&4.&\;\; P \lor Q&&\text{[$1$, SIM]}\\[2pt]
&&5.&\;\; R&... |
2,881,871 | <p>Is it true that if $N/M$ is a normal subgroup of $G/M$ then $N$ is a normal subgroup of $G$ itself?</p>
<p>I would think that not necessarily because I would expect that under conjugation we could get a subgroup of $G$ isomorphic to $N$ but maybe not $N$ it self but a subgroup identical from $N$ except its elements... | Alan Wang | 165,867 | <p>I will show you another answer without using correspondence theoerem:</p>
<p>Let $g\in G$ and $n\in N$.<br>
Then $nM\in N/M$ and $gM\in G/M$.<br>
Since $N/M$ is normal in $G/M$, $$(gM)^{-1}(nM)(gM)\in N/M$$<br>
which means that $(g^{-1}ng)M\in N/M$<br>
Hence $g^{-1}ng\in N$, which means that $N$ is normal in $G$. ... |
2,079,246 | <p>Is (A vector +B vector) always perpendicular to (A vector - B vector)?</p>
| DonAntonio | 31,254 | <p>No, of course not. Why do you think it would? Try almost any two random vectors in the basic spaces $\;\Bbb R^n\;$ , for example, with the usual inner product:</p>
<p>$$a:=\binom10,\,\,b:=\binom11\implies a+b=\binom21\;,\;\;a-b=\binom0{-1}$$</p>
|
2,079,246 | <p>Is (A vector +B vector) always perpendicular to (A vector - B vector)?</p>
| guanglei | 152,216 | <p>No. </p>
<p>Let $a, b$ be two vectors in a vector space equipped with an inner product, the necessary condition would be: $\left\|a\right\| = \left\|b\right\|$. </p>
<p>The derivation is </p>
<p>\begin{align}
(a + b)(a - b)
= a*a - b*b
= \left\|a\right\| - \left\|b\right\|
= 0
\end{align}</p>
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.