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 |
|---|---|---|---|---|
886,935 | <p>Let $A_1,A_2,A_3,\dots$ be a sequence of sequences where each
$$A_i = a_{i,1},a_{i,2},a_{i,3},\dots$$</p>
<p>Each sequence $A_i$ converges and in particular as $t \rightarrow \infty$, $a_{i,t} \rightarrow L_i$ for every $i$. We also have that in the limit as $i \rightarrow \infty$ the limits of these sequences con... | Chen Jiang | 167,817 | <p>Usually this is not true. You must take care when consider the limit of a limit thing.</p>
<p>Here is a counterexample.</p>
<p>We define $a_{i,t}$ inductively. Let $a_{1,1}=1$ and $a_{1,t}=2$ if $t>1$. Assume that we defined $a_{j,t}$ for all $j<i$. Then we define
$$
a_{i,t}=
\begin{cases}
1 &t<i\\
\... |
1,761,527 | <p>Let $f,g,h:X\to\mathbb{R}$ such that $f(x)\leq g(x)\leq h(x)$ for all $x\in X$. If $f$ and $h$ are differentiable in $a$ and $h(a)=f(a)$. Then $g$ is differentiable and $g'(a)=f'(a).$</p>
<p>How can I can prove that $g$ is differentiable in $a$.? Thanks</p>
| Tryss | 216,059 | <p>Without loss of generality, let $a=0$ and $f(0) = g(0) = h(0) = 0$</p>
<p>Then</p>
<p>$$\frac{f(\epsilon)}{\epsilon} \leq \frac{g(\epsilon)}{\epsilon} \leq \frac{h(\epsilon)}{\epsilon} $$</p>
<p>Now, take the limit when $\epsilon \to 0$, and as $f$ and $h$ are differentiable at $0$, you have that $\lim_{\epsilon ... |
2,650,634 | <p>EDIT: I know how to integrate the last part. I'm just try to find mistake in converting Sum to integral</p>
<p>Question: </p>
<blockquote>
<p>$$a_n=\left(\left(1+\left(\frac1n\right)^2\right)\left(1+\left(\frac2n\right)^2\right)\cdots\left(1+\left(\frac{n}n\right)^2\right)\right)^n$$ find<br>
$$\lim_{n\to\inf... | Wouter | 89,671 | <p>You're right, your book is wrong.</p>
<p><a href="https://i.stack.imgur.com/R4hyF.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/R4hyF.png" alt="enter image description here"></a></p>
|
2,725,839 | <p>The question is below.<a href="https://i.stack.imgur.com/k3UMf.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/k3UMf.png" alt="enter image description here"></a></p>
<p>I was able to solve part (a) because the $x$-coordinate would just be the circumference of the circle, which is $2\pi$. Therefor... | Narasimham | 95,860 | <p>The parametric equation of point $P$ of cycloid on rolling wheel circle radius $a=1$</p>
<p>$$ x= a (\theta- \sin \theta),\quad y= a(1-\cos \theta) $$</p>
<p>With a bit of calculus horizontal slope is </p>
<p>$$ \frac{dy/d \theta}{dx/d \theta}= \tan \phi_h = \cot \phi =\frac {\sin \theta }{1- \cos \theta}= \cot \... |
1,396,449 | <p>How does the Newton Interpolation work? The definition can be found here: <a href="http://www.nptel.ac.in/courses/122104018/node109.html" rel="nofollow">http://www.nptel.ac.in/courses/122104018/node109.html</a></p>
<p>Not how it's defined since that's mathematically clear, but I'm trying to grasp the general intuit... | Olivier Oloa | 118,798 | <p><strong>Hint.</strong> Here is an approach using the <a href="http://dlmf.nist.gov/5.12">Euler beta function</a>. </p>
<p>By the change of variable, $u=e^{-x}$, you get
$$
\begin{align}
\int_0^{\infty} \frac{\log^2(1 - e^{-x})x^5}{e^x - 1} \, dx&=-\int_0^1\frac{\log^2(1-u)\log^5u}{1-u}\:du\\\\
&=-\left.\par... |
1,396,449 | <p>How does the Newton Interpolation work? The definition can be found here: <a href="http://www.nptel.ac.in/courses/122104018/node109.html" rel="nofollow">http://www.nptel.ac.in/courses/122104018/node109.html</a></p>
<p>Not how it's defined since that's mathematically clear, but I'm trying to grasp the general intuit... | Leucippus | 148,155 | <p>Given
$$I = \displaystyle \int\limits_0^{\infty} \frac{\log^2(1 - e^{-x})x^5}{e^x - 1} \, dx $$
then let $t = e^{-x}$ to obtain
\begin{align}
I &= - \int_{1}^{0} ( - \ln t)^{5} \, \ln^{2}(1-t) \, \frac{dt}{1-t} \\
&= - \int_{0}^{1} \frac{\ln^{5}t \, \ln^{2}(1-t)}{1-t} \, dt \\
&= \int_{0}^{1} \ln^{5}t \,... |
794,842 | <p>Let the statement $?PQR$ be determined by the following truth-table.</p>
<pre><code>P Q R ?PQR
T T T T
T T F F
T F T F
T F F T
F T T T
F T F T
F F T F
F F F T
</code></pre>
<ol>
<li>After ‘Answer:’ below, give a logically equivalent sentence of ?PQR in FOL. Bu... | Peter Smith | 35,151 | <p>@AsafKaragila, being a nice guy, has given you a hint, and patiently explained more in his comments. Being not such a nice guy, can I point out that your question rather suggests that you haven't been to the library and done your basic groundwork? More or less any elementary logic text will have a section, under the... |
131,842 | <p>Let $X,Y$ be normed linear spaces. Let $T: X\to Y$ be linear. If $X$ is finite dimensional, show
that $T$ is continuous. If $Y$ is finite dimensional, show that $T$ is continuous if and only if $\ker T$ is closed. </p>
<p>I am able to show that $X$, finite dimensional $\implies$ $T$ is bounded, hence continuous.... | azarel | 20,998 | <p>If $\ker(T)$ is closed then $X/\ker(T)$ is a normed vector space. Observe that the map $\overline{T}:X/\ker(T)\to Y$ given by $\overline{T}(x+\ker(T))=T(x)$ is a well-defined linear map by the first part $\overline T$ is continuous since $X/\ker(T)$ is a finite dimensional vector space (as it is isomorphic to a subs... |
241,210 | <p>I am confused with the concept of topology base. Which are the properties a base has to have?</p>
<p>Having the next two examples for $X=\{a,b,c\}$:</p>
<p>1) $(X,\mathcal{T})$ is a topological space where $\mathcal{T}=\{\emptyset,X,\{a\},\{b\},\{a,b\}\}$. Which is the general procedure to follow in order to get a... | murad.ozkoc | 301,250 | <p>Let $(X,\tau)$ be a topological space and $\mathcal{B}\subseteq 2^X$.
$$\mathcal{B} \text{ is a base for topology } \tau \text { on } X$$
$$:\Leftrightarrow$$
$$1) \mbox{ } \mathcal{B}\subseteq \tau$$
$$2) \mbox{ } (\forall A\in\tau)(\exists\mathcal{A}\subseteq\mathcal{B})(A=\cup\mathcal{A})$$</p>
|
307,458 | <p>Let <span class="math-container">$\mathbf{x}_1, \dots, \mathbf{x}_n \in \mathbb{R}^d$</span> be <span class="math-container">$n$</span> given vectors. Define the function</p>
<p><span class="math-container">$$ \mathcal{K}(\mathbf{x},\mathbf{y}) := \alpha\exp\left(-\frac{\|\mathbf{x}-\mathbf{y}\|^2}{2\sigma^2}\right)... | Arash | 124,260 | <p>You are looking for a stochastic optimizer <a href="https://en.wikipedia.org/wiki/Stochastic_optimization" rel="nofollow noreferrer">[1]</a>.</p>
<p>This optimization is stochastic. Thus, it requires you optimize the expectation $E[.]$ of a particular value.</p>
<p>In an engineering problem, they often have some c... |
804,871 | <p>Prove that if x and y are odd natural numbers, then $x^2+y^2$ is never a perfect square.</p>
<p>Let $x=2m+1$ and $y=2l+1$ where m,l are integers.</p>
<p>$x^2+y^2=(2m+1)^2+(2l+1)^2=4(m^2+m+l^2+l)+2$</p>
<p>Where do I go from here?</p>
| Achari S Ganesha | 86,508 | <p>When a perfect square number is divided by 4 it doesn't leave a remainder of 2- GANESHA A S</p>
|
1,811,081 | <blockquote>
<p>Let $1,x_{1},x_{2},x_{3},\ldots,x_{n-1}$ be the $\bf{n^{th}}$ roots of unity. Find: $$\frac{1}{1-x_{1}}+\frac{1}{1-x_{2}}+......+\frac{1}{1-x_{n-1}}$$</p>
</blockquote>
<p>$\bf{My\; Try::}$ Given $x=(1)^{\frac{1}{n}}\Rightarrow x^n=1\Rightarrow x^n-1=0$</p>
<p>Now Put $\displaystyle y = \frac{1}{1-x... | Nate | 91,364 | <p>Let $f(x) = \frac{(x^n-1)}{(x-1)} = x^{n-1}+x^{n-2}+...+x+1$</p>
<p>Then $\frac{f'(x)}{f(x)} = \frac{1}{x-x_1} + \frac{1}{x-x_2}+ ... + \frac{1}{x-x_{n-1}}$</p>
<p>So what you want is just $\frac{f'(1)}{f(1)}$, which is easy to compute as $\frac{n-1}{2}$.</p>
|
3,424,720 | <p>I want to calculate the above limit. Using sage math, I already know that the solution is going to be <span class="math-container">$-\sin(\alpha)$</span>, however, I fail to see how to get to this conclusion.</p>
<h2>My ideas</h2>
<p>I've tried transforming the term in such a way that the limit is easier to find:
... | user | 505,767 | <p>You are right and from here </p>
<p><span class="math-container">$$ x\frac{\cos(\alpha)(\cos(x)-1)}{x^2} - \sin(\alpha) \to0\cdot \left(-\frac{\cos \alpha}2\right) -\sin(\alpha)=-\sin \alpha$$</span></p>
<p>indeed recall </p>
<p><span class="math-container">$$\frac{\cos(x)-1}{x^2} \to -\frac12$$</span></p>
<p>As... |
1,569,476 | <p>We throw fair dice until $6$ will appear. Let $X$ denote total number of throws and $Y$ - number of $5$ we received.</p>
<ol>
<li>Find distribution $(X,Y)$</li>
<li>Are variables $X$ and $Y$ independent?</li>
</ol>
<p>I have to say that I have utterly no idea how to proceed with this question, detailed explanation... | Alain Remillard | 278,299 | <p>$X$ follow a distribution called geometric
$$P(X= x) = \left(\frac56\right)^{x-1} \times \frac16$$
Having something other than 6 for the first $x - 1$ throw, then a 6.</p>
<p>Of course, $Y$ and $X$ are dependant. If you get a 6 on the first throw ($X = 1$), the $Y = 0$, you can't have any five.</p>
<p>Knowing the... |
3,797,724 | <blockquote>
<p>Find all real continuous functions that verifies :
<span class="math-container">$$f(x+1)=f(x)+f\left(\frac{1}{x}\right) \ \ \ \ \ \ (x\neq 0) $$</span></p>
</blockquote>
<p>I found this result <span class="math-container">$\forall x\neq 1 \ \ f(x)=f\left(\frac{x}{x-1} \right)$</span> and I tried ... | md2perpe | 168,433 | <h3>Some results</h3>
<p>Assume that <span class="math-container">$f$</span> is defined and continuous on <em>all</em> of <span class="math-container">$\mathbb{R}$</span> even if the equation <span class="math-container">$f(x+1) = f(x) + f(\frac{1}{x})$</span> is not defined for <span class="math-container">$x=0.$</spa... |
621,109 | <p>I need to find all the numbers that are coprime to a given $N$ and less than $N$.
Note that $N$ can be as large as $10^9.$ For example, numbers coprime to $5$ are $1,2,3,4$.</p>
<p>I want an efficient algorithm to do it. Can anyone help? </p>
| Robert Israel | 8,508 | <p>Once you find the prime factors $p_1, \ldots, p_k$ of $N$, you could use a sieve: start with $1 \ldots N-1$, delete all multiples of $p_1$, then all multiples of $p_2$, etc. What's left is coprime to $N$.</p>
|
96,657 | <p>I'm trying to understand an alternative proof of the idea that if $E$ is a dense subset of a metric space $X$, and $f\colon E\to\mathbb{R}$ is uniformly continuous, then $f$ has a uniform continuous extension to $X$.</p>
<p>I think I know how to do this using Cauchy sequences, but there is this suggested alternativ... | yunone | 1,583 | <p>I had a lot of help on this question in chat from users Srivatsan and t.b. the other day. I tried my best to write up what was said as an answer here.</p>
<hr>
<p>Notice that the sets $\overline{f(V_n(p))}\supseteq\overline{f(V_{n+1}(p))}\supseteq\cdots$ form a nested sequence of closed sets. Moreover, let $\epsil... |
2,289,777 | <p>I have a function $f(x)=(8x^2+7)^3(x^3-7)^4$</p>
<p>I have differentiated it using the chain rule and arrived at:</p>
<p>$3(8x^2+7)^2 \cdot 16x \cdot 4(x^3-7)^3 \cdot 3x^2$ And apparently this is wrong?</p>
<p>What am I missing here? </p>
| Mike Miller | 67,263 | <p>If $p(x)=(8x^2+7)^3(x^3-7)^4$ then by the product rule $(fg)' = f'g + fg'$ we have $f(x) = (8x^2+7)^3$ and $g(x) = (x^3-7)^4$ so by the product and chain rule</p>
<p>$$p'(x) = 48x(8x^2+7)^2(x^3-7)^4 + 12x^2(8x^2+7)^3(x^3-7)^3\\=12x(8x^2+7)^2(x^3-7)^3(12x^3+7x-28).$$</p>
|
3,927,845 | <p>n is a natural number. Prove <span class="math-container">$6^n \geq n3^n$</span> holds for every natural number.</p>
<p><span class="math-container">$n = 1:$</span></p>
<p><span class="math-container">$$6 \geq 3 $$</span></p>
<p><span class="math-container">$n \rightarrow n + 1:$</span></p>
<p><span class="math-cont... | hamam_Abdallah | 369,188 | <p>Let us prove that <span class="math-container">$$(\forall n\ge 1)\;\; P_n : 2^n\ge n$$</span></p>
<p><strong>first step</strong>
<span class="math-container">$$2^1=2\ge 1 \implies P_1$$</span></p>
<p><strong>second step</strong></p>
<p>Let <span class="math-container">$ n\ge 1 $</span> such that <span class="math-co... |
59,965 | <p>If I have a function $f(x,y)$, is it always true that I can write it as a product of single-variable functions, $u(x)v(y)$?</p>
<p>Thanks.</p>
| André Nicolas | 6,312 | <p>Unfortunately, the answer is no. For example, let $f(x,y)=x-y$.</p>
<p>Note that $f(1,1)=0$. If $f(x,y)=u(x)v(y)$ for all $x$, $y$, then $0=f(1,1)= u(1)v(1)$. But if $u(1)v(1)=0$, then $u(1)=0$ or $v(1)=0$. </p>
<p>Suppose for example that $u(1)=0$. Then $u(1)v(y)=0$ for all $y$. If $f(x,y)=u(x)v(y)$ for all $x... |
2,496,114 | <p>Show that $ a \equiv 1 \pmod{2^3 } \Rightarrow a^{2^{3-2}} \equiv 1 \pmod{2^3} $</p>
<p>Show that$ a \equiv 1 \pmod{2^4 } \Rightarrow a^{2^{4-2}} \equiv 1 \pmod{2^4} $</p>
<p><strong>Answer:</strong></p>
<p>$ a \equiv 1 \pmod{2^3} \\ \Rightarrow a^2 \equiv 1 \pmod{2^3} \\ \Rightarrow a^{2^{3-2}}=a^{2^1} \equiv... | Guy Fsone | 385,707 | <p>See that, $\displaystyle\frac{\sin x}{x} =f(x) = \frac{1}{2}\int_{-1}^{1} e^{-itx} dt$ Then
$$|f^{(n)}(x)| =\left|\frac{1}{2}\int_{-1}^{1} (-it)^ne^{-itx} dt\right| \le\frac{1}{2}\int_{-1}^{1} |t|^n dt=\int_0^1t^n\,dt=\frac1{n+1}.$$</p>
|
134,523 | <p>Is there any recursively axiomized system with infinitely many proofs for some propositions or a proposition? So we will have at least one proposition which is deduced from the recursively axiomatic system in infinite ways.Could any one give an example or proof?</p>
<p><strong>EDIT</strong>:We define a proof or a... | Andreas Blass | 6,794 | <p>In any of the usual axiomatic systems (ZF set theory, Peano arithmetic, etc.), any proposition that has a proof at all will have infinitely many. The reason is that you can add irrelevant padding or stupid detours to any argument. To make a real question along these lines, you'd have to specify some notion of "gen... |
134,523 | <p>Is there any recursively axiomized system with infinitely many proofs for some propositions or a proposition? So we will have at least one proposition which is deduced from the recursively axiomatic system in infinite ways.Could any one give an example or proof?</p>
<p><strong>EDIT</strong>:We define a proof or a... | none | 35,233 | <p>The question of whether two proofs are essentially the same is sometimes called "proof identity". It is an active topic in proof theory and there is an old MO thread with some references:</p>
<p><a href="https://mathoverflow.net/questions/3776/when-are-two-proofs-of-the-same-theorem-really-different-proofs">When a... |
908,083 | <p>I'd like to know what methods can I apply to simplify the fraction $\frac{4x + 2}{12 x ^2}$ </p>
<p>Is it valid to divide above and below by 2? (I didn't know it but Geogebra's Simplify aparantly does this)</p>
<p>Thanks in advance</p>
| 2'5 9'2 | 11,123 | <p>If you are looking for a formula that can be evaluated with a minimal number of bit operations for any given $x$, it might be best to multiply by $\frac{\frac14}{\frac14}$ and get $$\frac{x+\frac12}{3x^2}$$ which in binary would be $$\frac{x+0.1}{11\cdot x^{10}}$$</p>
|
4,013,796 | <p>If we make a regular polygon with n vertices (n edges) and triangulate on the inside with n-3 edges, then triangulate on the outside with (n-3) edges (or draw dotted lines inside again), a Maximal Planar Graph is formed. Edges shouldn't be repeated and there's no loops or directions.</p>
<p>How many distinct graphs... | Will Orrick | 3,736 | <p>This is too long to be a comment, but only provides a modest reduction in the <span class="math-container">$C_{n-2}^2$</span> upper bound.</p>
<p>For <span class="math-container">$n\ge3$</span> let <span class="math-container">$a_n$</span> be the number of unordered pairs of triangulations of the <span class="math-c... |
1,982,216 | <p>Consider the operator $B: L^1\left(\mathbb{R^+} \right)\to L^1\left(\mathbb{R^+} \right)$ defined for each $f\in L^1\left(\mathbb{R^+} \right)$ by
$$(Bf)(t)=\int_0^\infty\alpha (t,s)f(s)ds, \ \ \ \text{for} \ \ t\geq 0$$
where $\alpha:\mathbb{R^+} \times \mathbb{R^+}\to\mathbb{R}$ is a real function satisfying
$$\le... | Davide Giraudo | 9,849 | <p>Following the references in <a href="https://math.stackexchange.com/questions/35115/classifying-the-compact-subsets-of-lp">this thread</a> about the characterization of compact subsets of $L^p\left(\mathbb R^n\right) $, $1\leqslant p\lt \infty$, the condition
$$\lim_{\delta\to 0}\sup_{\substack{ t,t'\geqslant 0\\ |t... |
536,073 | <p>I came across several questions like this in the problem section of a book on coding theory & cryptography and I have no idea how to tackle them. There must be a certain trick that allows for efficiently solving such problems by hand.</p>
| Ekaveera Gouribhatla | 31,458 | <p>$$2^9\equiv-1(mod)19$$ $\implies$
$$2^{162}\equiv1(mod)19$$ $\implies$
$$2^{170}\equiv256(mod)19=9(mod)19$$ Also
$$3^3\equiv 2^3(mod)19$$ So
$$3^{63}\equiv2^{63}(mod)19=-1(mod)19$$ So
$$2^{170}+3^{63}\equiv 8(mod)19$$</p>
|
2,832,614 | <blockquote>
<p>Prove by induction that
$$\lim_{x \to a} \frac{x^n-a^n}{x-a}=na^{n-1}.$$</p>
</blockquote>
<p>I did a strange proof using two initial results: We know that result is true for $n=1$ and $n=2$. Assuming the result is true for $n=k-1$ and $n=k$, I can prove the result for $n=k+1$. For this I used my a... | Doug M | 317,162 | <p>My algebra would be:</p>
<p>Suppose $\lim_\limits{x\to a} \frac {x^n-a^n}{x-a} = na^{n-1}$</p>
<p>Show that $\lim_\limits{x\to a} \frac {x^{n+1}-a^{n+1}}{x-a} = na^{n}$</p>
<p>$\frac {x\cdot x^{n} - a\cdot a^{n}}{x-a}$</p>
<p>Add and subtract the same term such that your expression will factor into something tha... |
3,436,891 | <p>I have the following stupid question in my mind while i am studying for exams.
Does <span class="math-container">$X<\infty \ a.s$</span>, implies that <span class="math-container">$\mathbb E(X)<\infty$</span>? </p>
<p>Further on this, is the converse of the above statement true? Do give me a bit summary on th... | Simon Segert | 497,898 | <p>An even simpler counterexample is to take <span class="math-container">$P(X=n)=C/n^2$</span> where n is a positive integer and <span class="math-container">$C$</span> is a normalizing constant.</p>
<p>However, the converse is true. Usually, the expectation is only defined when <span class="math-container">$X$</span... |
115,269 | <p>I'm studying the remainder class $\mathbb{Z}_n$, I've grabbed something, but something else is unfocused. Let
$$20x \equiv 4\pmod{34}$$
then GCD(20,34)=2 so I rewrite as:
$$10x \equiv 2\pmod{17}$$
and successively:
$$10x \equiv 1\pmod{17}$$
Now I know $\gcd(10, 17)=1$</p>
<blockquote>
<p>Question 1: Why? Is this... | Manolito Pérez | 13,293 | <p>More generally, you can set $f(k) = \sum_{i=1}^k a_i$, and then you get the recurrence equation $f(k) = f(k-1) + a_k$, with $f(1)$ known. Now solve the recurrence equation, and you have the formula you're looking for. </p>
|
115,269 | <p>I'm studying the remainder class $\mathbb{Z}_n$, I've grabbed something, but something else is unfocused. Let
$$20x \equiv 4\pmod{34}$$
then GCD(20,34)=2 so I rewrite as:
$$10x \equiv 2\pmod{17}$$
and successively:
$$10x \equiv 1\pmod{17}$$
Now I know $\gcd(10, 17)=1$</p>
<blockquote>
<p>Question 1: Why? Is this... | Brian M. Scott | 12,042 | <p>There are indeed methods of dealing with many more complicated summations; the excellent book <em>Concrete Mathematics</em>, by Graham, Knuth, & Patashnik, is full of such techniques. One of them is <em>finite calculus</em>; <a href="http://www.cs.purdue.edu/homes/dgleich/publications/finite-calculus.pdf">this P... |
115,269 | <p>I'm studying the remainder class $\mathbb{Z}_n$, I've grabbed something, but something else is unfocused. Let
$$20x \equiv 4\pmod{34}$$
then GCD(20,34)=2 so I rewrite as:
$$10x \equiv 2\pmod{17}$$
and successively:
$$10x \equiv 1\pmod{17}$$
Now I know $\gcd(10, 17)=1$</p>
<blockquote>
<p>Question 1: Why? Is this... | Norbert | 19,538 | <p>My approach will look like overkill for your particular problem, but you asked for the general method of summation such series.</p>
<p>If you can find explicit formula for series of the form $S_r(k)=\sum\limits_{i=0}^k i^r$, $r\in\mathbb{Z}_+$ then you can find formula for the series of the form $\sum\limits_{i=1}^... |
88,159 | <p>Define $\omega=e^{i \pi /4}$. Is there an elegant way of showing that $20^{1/4} \omega^3$ is not inside $\mathbb{Q}(20^{1/4} \omega)$?</p>
<p>The way i am doing it is by observing that $20^{1/4} \omega$ is algebraic over $\mathbb{Q}$ with minimal polynomial $x^4+20$ and i assume that $20^{1/4} \omega^3$ is in the s... | Georges Elencwajg | 3,217 | <p>Let $K= \mathbb{Q}(20^{1/4}\omega) $, <em>a field of dimension $4$ over $\mathbb Q$</em> .<br>
Clearly $\omega^2\in K \iff K \;$is the splitting field of $f(X)=X^4+20$ over $\mathbb Q$ .<br>
It suffices to show that this is not the case.<br>
The resolvent of $f(X)$ is $r(Y)=Y^3-80Y $ . This implies that t... |
64,977 | <p>Suppose I had a complete bipartite graph with edges each given some numerical "cost" value. Is there a way to select a subset of those edges such that each vertex on each side of the graph is mapped to each vertex on the other (one to one) and the total "costs" is maximized (or minimized)?</p>
<p>Has anyone ever f... | Or Zuk | 1,778 | <p>You can find a set $\Omega$ with size $O(d)$. Assume for simplicity that $d=2^k-1$ for an integer $k$. Let $y_1,..,y_k$ be binary variables and define
$x_j = \sum_i y_i \alpha_i(j)$ where $\alpha_i(j)$ is simply the $i$-th digit of $j$ when represented in binary. (For example, if $n=7$, you'll get $y_1, y_2, y_3, y... |
2,995,495 | <p>I'm trying to prove that, for every <span class="math-container">$x \geq 1$</span>:</p>
<p><span class="math-container">$$\left|\arctan (x)-\frac{π}{4}-\frac{(x-1)}{2}\right| \leq \frac{(x-1)^2}{2}.$$</span> </p>
<p>I could do it graphically on <span class="math-container">$\Bbb R$</span>, but how to make a formal... | Barry Cipra | 86,747 | <p>It is convenient to let <span class="math-container">$x=u+1$</span> and rewrite the inequality to be proved as a pair of inequalities:</p>
<p><span class="math-container">$${u\over2}-{u^2\over2}\le\arctan(u+1)-\arctan1\le{u\over2}+{u^2\over2}$$</span></p>
<p>for <span class="math-container">$u\ge0$</span>. Now</p>... |
480,195 | <p>Three friends brought 3 pens together each 10 dollars. Next day they got 5 dollars cash back so they shared each 1 dollar and donated 2 dollars. Now the pen cost for each guy will be 9 dollars (\$10 -\$1).</p>
<p>But if you add all 9+9+9 = 27 dollars and donated amount is 2 dollars so total 29 dollars. </p>
<p>Whe... | stig | 92,473 | <p>10 x 3 = 30.
30 - 5 = 25.
25 + 3 = 28.
28 + 2 = 30.
so, no missing $.</p>
|
526,837 | <p>Let $(\Omega, {\cal B}, P )$ be a probability space, $( \mathbb{R}, {\cal R} )$ the usual
measurable space of reals and its Borel $\sigma$- algebra, and $X : \Omega \rightarrow \mathbb{R}$ a random variable.</p>
<p>The meaning of $ P( X = a) $ is intuitive when $X$ is a discrete random variable, because it's the d... | lab bhattacharjee | 33,337 | <p>As $$\frac{-4x}{1+2x}=\frac{-2(1+2x)+2}{1+2x}=-2+\frac2{2x+1},$$</p>
<p>$$\int\frac{-4x}{1+2x}dx=-2\int dx+\int\frac2{1+2x}dx=-2x+\ln|1+2x|+C$$ </p>
<p>putting $1+2x=u$ in the second integral
and where $C$ is an arbitrary constant of indefinite integral</p>
<p>In your case $C=-1$</p>
<p>In fact, if $$f'(x)=g'(x)... |
2,032,711 | <p>In a triangle $ABC$, if $\sin A+\sin B+\sin C\leq1$,then prove that $$\min(A+B,B+C,C+A)<\pi/6$$
where $A,B,C$ are angles of the triangle in radians.</p>
<p>if we assume $A>B>C$,then $\sum \sin A\leq 3 \sin A$,and $ A\geq \frac{A+B+C}{3}=\pi/3$.also $\sum \sin A\geq 3\sin C$ and $ C\leq \frac{A+B+C}{3}=\pi... | Djura Marinkov | 361,183 | <p>A must be larger than $\frac {5\pi} 6 $ cos if it's not, then $\sin A\ge\frac 1 2$ so $\sin B+\sin C\le\frac 1 2$ but $B+C\ge\frac {\pi} 6 $ which is not possible since $\sin B+\sin C>\sin(B+C)\ge\frac 1 2$</p>
|
878,115 | <p>Question1:
I found 30 boxes. In 10 boxes i found 15 balls. In 20 boxes i found 0 balls.
Afer i collected all 15 balls i put them randomly inside the boxes.</p>
<p>How much is the chance that all balls are in only 10 boxes or less?</p>
<p>Question2:
I found 30 boxes. In 10 boxes i found 15 balls. In 20 boxes i foun... | Mr.Spot | 155,516 | <p><strong>Solution of Question 1:</strong></p>
<p>This is an occupancy problem with $n=30$ boxes and $k=15$ balls.</p>
<p>Let's first consider the expected number of empty boxes. That is much easier to obtain. The exact answer is $30(1-1/30)^{15}=18.04.$ This is approximately $30/\sqrt e.$ See the answer by Mr.Spot ... |
3,371,302 | <p>trying to find all algebraic expressions for <span class="math-container">${i}^{1/4}$</span>.</p>
<p>Using. Le Moivre formula , I managed to get this : </p>
<blockquote>
<p><span class="math-container">${i}^{1/4}=\cos(\frac{\pi}{8})+i \sin(\frac{\pi}{8})=\sqrt{\frac{1+\frac{1}{\sqrt{2}}}{2}} + i \sqrt{\frac{1-... | J. W. Tanner | 615,567 | <p><strong>Hint:</strong>. If <span class="math-container">$\exp(i\theta)^4=\exp(i4\theta)=i$</span>, then <span class="math-container">$4\theta+2\pi n=\dfrac {\pi}2$</span>, where <span class="math-container">$n\in\Bbb Z$</span>. If you solve for <span class="math-container">$\theta, $</span> you should get the othe... |
3,371,302 | <p>trying to find all algebraic expressions for <span class="math-container">${i}^{1/4}$</span>.</p>
<p>Using. Le Moivre formula , I managed to get this : </p>
<blockquote>
<p><span class="math-container">${i}^{1/4}=\cos(\frac{\pi}{8})+i \sin(\frac{\pi}{8})=\sqrt{\frac{1+\frac{1}{\sqrt{2}}}{2}} + i \sqrt{\frac{1-... | Rhys Hughes | 487,658 | <p>If <span class="math-container">$(a+bi)^2=c+di$</span>, then:</p>
<p><span class="math-container">$$a=\pm\sqrt{\frac{c\pm |c+di|}{2}},b=\pm\sqrt{\frac{-c\pm |c+di|}{2}}$$</span></p>
<p>First we get the result: <span class="math-container">$$\sqrt{\lambda i}=\pm\sqrt{\frac\lambda 2}(1+i)$$</span></p>
<p>and then <... |
1,650,204 | <p>I was given this problem and I can't seem to think of a solution.</p>
<p>Here is a possibly helpful graphic:</p>
<p><a href="https://i.stack.imgur.com/VKZkv.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/VKZkv.png" alt="Here is a possibly helpful link:" /></a></p>
<blockquote>
<p>Given two parall... | P Vanchinathan | 28,915 | <p>One talks of eigenvalues etc for linear transformations that are functions from a vector space to itself. $T(u+v)= T(u)+T(v),\ T(av)= aT(v)$.</p>
<p>Having studied physics you must have a good understanding of the 2 and 3-dimensional (euclidean) spaces. I'll stick to them. Any three dimensional rotation around or... |
3,381,939 | <p>Be T tree order n given:</p>
<p><span class="math-container">$(a)$</span> <span class="math-container">$ 95 < n < 100$</span></p>
<p><span class="math-container">$(b)$</span> Just have vertices with degree 1,3,5</p>
<p><span class="math-container">$(c)$</span> T has twice the vertices of degree 3 that degre... | J.G. | 56,861 | <p>If such a formula existed, one with a minimal number of implications would contain at least one <span class="math-container">$\to$</span> (because you've already checked the case with none), and we could write some such contradiction as <span class="math-container">$a\to b$</span>. (If <span class="math-container">$... |
2,437,983 | <p>What is the chance that at least two people were born on the same day of the week if there are 3 people in the room?</p>
<p>I know how to get the answer which is 19/49 when considering all 3 people <strong>not being born on the same day</strong>. However, when I try to calculate the answer directly I seem to get it... | mechanodroid | 144,766 | <p>Let $\mathcal{I}$ be the collection of all open intervals, and $\mathcal{A}$ be the collection of all half open-intervals $[a,b)$. We shall prove $\mathcal{B}_\mathbb{R} = \sigma(\mathcal{I})$ and $\sigma(\mathcal{I}) = \sigma(\mathcal{A})$.</p>
<p>Open intervals are open sets so $\mathcal{I} \subseteq \mathcal{B}_... |
1,058,831 | <p>Let $A$ and $B$ be two Hermitian matrices. I wanted to know if there is any relation between the maximum eigenvalue of $AB$ and that of $A$ and $B$. Is the following relation true?
$\lambda_{\text{max}}\left(AB\right)\le \lambda_{\text{max}}\left(A\right)\lambda_{\text{max}}\left(B\right)$</p>
<p>If it is true, the... | copper.hat | 27,978 | <p>Let $A=B=\operatorname{diag}(-2,1)$, then $AB = \operatorname{diag}(4,1)$, but
$\lambda_\max (AB) = 4$, $\lambda_\max (A) = \lambda_\max (B)= 1$.</p>
<p>If the matrices are positive semi-definite, then $\|A\|=\lambda_\max(A)$ (since
$A$ is unitarily diagonalisable) and the
spectral norm is submultiplicatve, hence t... |
1,058,831 | <p>Let $A$ and $B$ be two Hermitian matrices. I wanted to know if there is any relation between the maximum eigenvalue of $AB$ and that of $A$ and $B$. Is the following relation true?
$\lambda_{\text{max}}\left(AB\right)\le \lambda_{\text{max}}\left(A\right)\lambda_{\text{max}}\left(B\right)$</p>
<p>If it is true, the... | Algebraic Pavel | 90,996 | <p>If $A$ and $B$ are $\color{blue}{\rm HPD}$, then
$$
\begin{split}
\color{red}{\lambda_{\max}(AB)}
&=
\lambda_{\max}(B^{1/2}AB^{1/2})
\\&=
\max_{x\neq 0}\frac{x^*B^{1/2}AB^{1/2}x}{x^*x}
\\&=
\max_{x\neq 0}\frac{x^*Ax}{x^*B^{-1}x}
\\&=
\max_{x\neq 0}\left(\frac{x^*Ax}{x^*x}\right)\left(\frac{x^*x}{x^*B... |
2,849,017 | <p>\begin{align}
dA & = 2RR\,dv = 2R^2\,dv \\[8pt]
A & = \int_0^\pi 2R^2\,dv \\[8pt]
\text{arclength} & = R\,dv
\end{align}</p>
<p><a href="https://i.stack.imgur.com/YBIX5.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YBIX5.png" alt="enter image description here"></a></p>
<p>Area of a... | Michael Hardy | 11,667 | <p>The length of the red line parallel to the diameter is $2R\cos v.$</p>
<p>Its distance from that diameter is $R\sin v.$ Therefore its infinitesimal change is $R\cos v \,dv.$</p>
<p>And $v$ needs to go from $-\pi/2$ to $\pi/2,$ for reasons that should be apparent from the picture.</p>
<p>So the area is $\displayst... |
686,981 | <p>So I have to solve the equation $$y^2=4\tag{1.9.88 unit 3*}$$</p>
<p>I did this: $$y^2=4 \text{ means } \sqrt{y^2}=\sqrt{4}=>y=2$$</p>
<p>But I have a problem, $y$ can be either negative or positive so I need to do: $$\sqrt{y^2}=|y|=2=>y=2- or- y=-2$$</p>
<p>Is it right?</p>
| Hakim | 85,969 | <p>Never forget this rule: <em>For all $x\in\mathbb{R}$, the following holds:</em> $$\sqrt{x^2}=|x|=\begin{cases}x & x\gt 0\\ -x & x\leqslant0\end{cases}$$
Applying that to the equation $y^2=4:$ $$\begin{align}
\sqrt{y^2}=\sqrt{4}&\iff |y|=2\\ &\iff y=2\,\,\mathrm{or}\,\,y=-2
\end{align}$$
So you're rig... |
995,159 | <p>I have a matrix $(a_{j,k})_{j,k\in\mathbb{N}}$ given by:</p>
<p>$ a_{j,k} = \dfrac{1 -e^{-jk}}{jk + 1}$</p>
<p>and I need to show that this induces a bounded operator on $\ell^2$. I'm pretty sure Schur's test is inconclusive. So my guess is to use the Hilber-Schmidt test, which states that if,</p>
<p>$\sum\limits... | Winther | 147,873 | <p>We have</p>
<p>$$\left|\frac{1-e^{-jk}}{jk+1}\right| < \frac{1}{jk}$$</p>
<p>so</p>
<p>$$\sum_{j,k=1}^\infty \left|\frac{1-e^{-jk}}{jk+1}\right|^2 < \sum_{j,k=1}^\infty \frac{1}{j^2k^2} = \left(\sum_{j=1}^\infty \frac{1}{j^2}\right)^2 < \infty$$</p>
<p>since $\sum_{j=1}^\infty \frac{1}{j^2} = \frac{\pi... |
3,101,286 | <p>I would like to get this text translated from Dutch to English:</p>
<p><a href="https://i.stack.imgur.com/0IWlQ.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/0IWlQ.png" alt="enter image description here"></a></p>
<p>I tried using Google translator but the result is confusing me:</p>
<p><a hre... | drhab | 75,923 | <p>(a) Let <span class="math-container">$(y_n)_n$</span> be a bounded sequence in <span class="math-container">$\mathbb C$</span>. Show that for every sequence <span class="math-container">$(x_n)_n$</span> in <span class="math-container">$\mathbb C$</span> for which the series <span class="math-container">$\sum_nx_n$</... |
3,583,475 | <p>Write as a single fraction:</p>
<p><span class="math-container">$(4x+2y)/(3x) - (5x+9y)/(6x) + 4$</span></p>
<p>Simplify your answer as much as possible.</p>
<p>The answer that I got from when I did the math was: (27x-5y)/(6x). But I have asked some of my friends who some got a different answer from mine. Plea... | trula | 697,983 | <p>your answer is right, don't believe the others.</p>
|
2,777,982 | <p>I was asked to describe the surface described by</p>
<p>$${\bf r}^\top {\bf A} {\bf r} + {\bf b}^\top {\bf r} = 1,$$</p>
<p>where $3 \times 3$ positive definite matrix ${\bf A}$ and vector $\bf b$ are given.</p>
<p>My intuition tells me that it is a rotated ellipsoid with a centre that is off the origin. However,... | mvw | 86,776 | <blockquote>
<p>However, I am told to show this via the substitution
${\bf r} = {\bf x} + {\bf a}$, with $\bf a$ being a constant vector, and dictate the
conditions on this vector to obtain a new quadric surface
${\bf x}^\top{\bf A}{\bf x} = C$.</p>
</blockquote>
<p>Following your advice:
\begin{align}
1
&am... |
1,723,942 | <p>The four color theorem declares that any map in the plane (and, more generally, spheres and so on) can be colored with four colors so that no two adjacent regions have the same colors. </p>
<p>However, it's not clear what constitutes a map, or a region in a map. Is this actually a theorem in graph theory, something... | Robert Israel | 8,508 | <p>The theorem is a statement about planar graphs. In the application to geography, you have a finite collection of "countries" (disjoint connected open subsets of the plane with, let's say, piecewise-smooth boundaries). Two countries are considered to be adjacent if the intersection of their boundaries has nonzero l... |
1,944,628 | <p>Let $(X,{\mathcal T}_X)$ and $(Y,{\mathcal T}_Y)$ be topologiclal spaces, and let $f,g:(X,{\mathcal T}_X)\to(Y,{\mathcal T}_Y)$ be continuous maps. </p>
<p>Define the equality set as $$E(f,g) = \{x\in X \ | \ f(x) = g(x) \}$$</p>
<p>I have worked out that if $(Y,{\mathcal T}_Y)$ is Hausdorff, then $E(f,g)$ is $... | Hermès | 127,149 | <p>Define the <strong>Sierpinski space S</strong> to be the set $X = \{0,1\}$, together with the topology $\mathscr{T}_X = \{\emptyset, \{1\}, \{0,1\}\}$. There clearly $S$ is not Hausdorff, as we cannot isolate $0$ and $1$ using two disjoint open sets.
Let $\mathbb{R^*_+}$ be equipped with its natural topology. Defin... |
669,582 | <p>Let $X,τ$ be a topological space. Prove that a subset $A$ of $X$ is dense if and only if every open subset of $ X$ contain some point of $A$</p>
<p>this is what I got</p>
<p>Let $X,τ$ be a topological space</p>
<p>Part 1: Assume that a subset $A$ of $X$ is dense, show that every open subset of $X$ contain some po... | aphorisme | 127,196 | <p>Since every (non-empty) open subset of $X$ contains a point from $A$ it has to hold that </p>
<p>(1.) $\forall U \in \mathcal{O}(X)\setminus\{\emptyset\}: U \cap A \neq \emptyset$. </p>
<p>Because of $A \subseteq \mathcal{Cl}A$ it holds also that </p>
<p>(2.) $\complement \mathcal{Cl}A \cap A = \emptyset$</p>
<p... |
2,130,911 | <p>I'm unsure how to compute the following : 3^1000 (mod13)</p>
<p>I tried working through an example below,</p>
<p>ie) Compute $3^{100,000} \bmod 7$
$$
3^{100,000}=3^{(16,666⋅6+4)}=(3^6)^{16,666}*3^4=1^{16,666}*9^2=2^2=4 \pmod 7\\
$$</p>
<p>but I don't understand why they divide 100,000 by 6 to get 16,666. Where di... | Mark | 310,244 | <p><a href="https://en.wikipedia.org/wiki/Fermat%27s_little_theorem" rel="nofollow noreferrer">Fermat's Little Theorem</a> says that:
$$a^p \equiv a\pmod{p}$$
Or, that:
$$a^{p-1}\equiv 1\pmod{p}$$
You're looking at this mod $7$, so $3^{7-1} = 3^6\equiv 1\pmod{p}$.
So, we're trying to split $100,000$ into $6k+r$ where $... |
1,447,852 | <p>Compute this sum:</p>
<p><span class="math-container">$$\sum_{k=0}^{n} k \binom{n}{k}.$$</span></p>
<p>I tried but I got stuck.</p>
| Brian Cheung | 248,555 | <p>$$\sum_{k=0}^nk\binom{n}{k}=\sum_{k=1}^nk\binom{n}{k}=\sum_{k=1}^nn\binom{n-1}{k-1}=n\sum_{k=0}^{n-1}\binom{n-1}{k}=n2^{n-1}$$<br>
Identities used:<br>
1)$$\binom{n}{k}=\dfrac{n!}{k!(n-k)!}=\dfrac nk\dfrac{(n-1)!}{(k-1)!(n-k)!}=\dfrac nk\binom{n-1}{k-1}$$<br>
2)$$\sum_{k=0}^n\binom nk=2^n$$
which can be proved by ex... |
1,082,390 | <p>$$\lim_{x \to \infty} \left(\sqrt{4x^2+5x} - \sqrt{4x^2+x}\ \right)$$</p>
<p>I have a lot of approaches, but it seems that I get stuck in all of those unfortunately. So for example I have tried to multiply both numerator and denominator by the conjugate $\left(\sqrt{4x^2+5x} + \sqrt{4x^2+x}\right)$, then I get $\di... | Bumblebee | 156,886 | <p>As $x\to\infty$ $$\displaystyle \frac{4x}{\sqrt{4x^2+5x} + \sqrt{4x^2+x}}=\displaystyle \frac{4}{\sqrt{4+\dfrac{5}{x}} + \sqrt{4+\dfrac{1}{x}}}\to1$$</p>
|
2,426,897 | <p>Let $\mathbb{H}$ be the ring of real quaternions and $Z(\mathbb{H})$ be its center. Of course $Z(\mathbb{H})=\mathbb{R}$. </p>
<p>Suppose $a+bi+cj+dk$, $x+yi+zj+wk \in \mathbb{H}$ such that $(a+bi+cj+dk)(x+yi+zj+wk) \in Z(\mathbb{H})$. </p>
<p>Does it imply $(x+yi+zj+wk)(a+bi+cj+dk) \in Z(\mathbb{H})$?</p>
| Dr. Sonnhard Graubner | 175,066 | <p>it is $$44<\sqrt{2017}<45$$ since $$44^2=1936$$ and $$45^2=2025$$</p>
|
2,426,897 | <p>Let $\mathbb{H}$ be the ring of real quaternions and $Z(\mathbb{H})$ be its center. Of course $Z(\mathbb{H})=\mathbb{R}$. </p>
<p>Suppose $a+bi+cj+dk$, $x+yi+zj+wk \in \mathbb{H}$ such that $(a+bi+cj+dk)(x+yi+zj+wk) \in Z(\mathbb{H})$. </p>
<p>Does it imply $(x+yi+zj+wk)(a+bi+cj+dk) \in Z(\mathbb{H})$?</p>
| Will Jagy | 10,400 | <p>As to the next question, Pell -1 and Pell +1..Hmm, seems to be showing the exponents incorrectly.</p>
<p>$$ {106515299132603184503844444}^2 - 2017 \cdot 2371696115380807559791481^2 = -1 $$</p>
<p>$$ 22691017898615873418283839489716246568157231499338273^2 - 2017 \cdot 5052438423628393473350846837568851798192790... |
2,426,897 | <p>Let $\mathbb{H}$ be the ring of real quaternions and $Z(\mathbb{H})$ be its center. Of course $Z(\mathbb{H})=\mathbb{R}$. </p>
<p>Suppose $a+bi+cj+dk$, $x+yi+zj+wk \in \mathbb{H}$ such that $(a+bi+cj+dk)(x+yi+zj+wk) \in Z(\mathbb{H})$. </p>
<p>Does it imply $(x+yi+zj+wk)(a+bi+cj+dk) \in Z(\mathbb{H})$?</p>
| kingW3 | 130,953 | <p>Knowing the answer lies between $40$ and $50$ the rational thing to do would be to try the number in the middle of $40$ and $50$ i.e $45$. </p>
<p>If you don't feel comfortable with multiplying $45\cdot 45$ you can use the
formula for $(40+5)^2=40^2+2\cdot 40\cdot 5+5^2=1600+400+25=2025$</p>
<p>Now you can use th... |
2,426,897 | <p>Let $\mathbb{H}$ be the ring of real quaternions and $Z(\mathbb{H})$ be its center. Of course $Z(\mathbb{H})=\mathbb{R}$. </p>
<p>Suppose $a+bi+cj+dk$, $x+yi+zj+wk \in \mathbb{H}$ such that $(a+bi+cj+dk)(x+yi+zj+wk) \in Z(\mathbb{H})$. </p>
<p>Does it imply $(x+yi+zj+wk)(a+bi+cj+dk) \in Z(\mathbb{H})$?</p>
| mathreadler | 213,607 | <p>You can use the Newton square root method <a href="https://en.wikipedia.org/wiki/Methods_of_computing_square_roots" rel="nofollow noreferrer">wikipedia</a> for integers:</p>
<p>$$x_{n+1} = \frac 1 2 \left(x_n+ \frac S {x_n}\right)$$</p>
<p>Let us start with a crappy guess </p>
<ol>
<li>$x_1 = 500$:</li>
<li>$x_2 ... |
2,426,897 | <p>Let $\mathbb{H}$ be the ring of real quaternions and $Z(\mathbb{H})$ be its center. Of course $Z(\mathbb{H})=\mathbb{R}$. </p>
<p>Suppose $a+bi+cj+dk$, $x+yi+zj+wk \in \mathbb{H}$ such that $(a+bi+cj+dk)(x+yi+zj+wk) \in Z(\mathbb{H})$. </p>
<p>Does it imply $(x+yi+zj+wk)(a+bi+cj+dk) \in Z(\mathbb{H})$?</p>
| Simon Fraser | 717,270 | <p><span class="math-container">$\sqrt{1600} = 40$</span> and <span class="math-container">$\sqrt{2500} = 50$</span>. <span class="math-container">$(40+4)^2 = 40^2 + 8\cdot 40 + 16 = 1936$</span> and <span class="math-container">$(40+5)^2 = 40^2 + 10\cdot 40 +25 = 2025$</span>. Hence the desired answer is <span class="... |
317,160 | <p>If $ f(x) = \frac{1}{(1+x)\sqrt x} $ how to find all $ p > 0 $ such that
$$ \int^{\infty}_0 |f(x)|^p dx < \infty $$
The integral is with respect to lebesgue measure. Any solution or hints would be helpful. The answer is the integral converges iff $ p\in (\frac{2}{3}, 2) $.</p>
| Mhenni Benghorbal | 35,472 | <p>Using the change of variables $ z=\frac{1}{1+x} $ and the <a href="http://en.wikipedia.org/wiki/Beta_function" rel="nofollow">beta function</a></p>
<p>$$ \beta(x,y) = \int_{0}^{1}t^{x-1}(1-t)^{y-1}dt,\quad Re(x),\,Re(y)>0, $$</p>
<p>we have</p>
<p>$$ \int_{0}^{\infty} \frac {1}{(1+x)^p x^{\frac{p}{2}}} dx = \i... |
3,393,466 | <p>I am in final year of my undergraduate in mathematics from a prestigious institute for mathematics. However a thing that I have noticed is that I seem to be slower than my classmates in reading mathematics. As in, how muchever I try, I seem to finish my works at the last moment and I rarely find any time for extra r... | Toney Leung | 645,097 | <p>I'm similar to you.While reading mathematics,I always think it over deeply until I thoroughly understand the full meaning.Knowing the connections between different math concepts clearly will be good for your further study in mathematics.Because undergraduate mathematics is just the beginning of real mathematics，and ... |
1,905,863 | <p>I'm on the section of my book about separable equations, and it asks me to solve this:</p>
<p>$$\frac{dy}{dx} = \frac{ay+b}{cy+d}$$</p>
<p>So I must separate it into something like: $f(y)\frac{dy}{dx} + g(x) = constant$</p>
<p>*note that there are no $g(x)$</p>
<p>but I don't think it's possible. Is there someth... | Claude Leibovici | 82,404 | <p>For your specific example, extracting the horizontal asymptote you can also write $$\frac{cy+d}{ay+b}=\frac c a+\frac{a d-b c}{a (a y+b)}$$ Then $$I=\int\frac{cy+d}{ay+b}\,dy=\frac c a \int dy+\frac{a d-b c} a\int \frac {dy} { (a y+b)}=\frac c a \int dy+\frac{a d-b c} {a^2}\int \frac {a\,dy} { (a y+b)}$$ I am sure t... |
522,289 | <p>It is an exercise on the lecture that i am unable to prove.</p>
<p>Given that $gcd(a,b)=1$, prove that $gcd(a+b,a^2-ab+b^2)=1$ or $3$, also when will it equal $1$?</p>
| lab bhattacharjee | 33,337 | <p>HINT:</p>
<p>Let prime $p$ divides $a+b, a^2-ab+b^2$</p>
<p>$\implies p$ divides $\{(a+b)^2-(a^2-ab+b^2)\}=3ab$</p>
<p>If $p|a,$ as $p|(a+b),p$ must divide $(a+b)-a=b\implies p|(a,b)$ </p>
<p>But as $(a,b)=1,p$ can not divide $a$</p>
<p>Similarly, $p$ can not divide $b$</p>
<p>$\implies p|3$</p>
<p>$\implies... |
580,974 | <blockquote>
<p>Show that the following equalities hold true for every $n$ from $\mathbb{N}$ and
every $x$ from $\mathbb{R}$</p>
<p>$$\sin^{(n)}(x)=\sin(x+n\fracπ2)$$</p>
</blockquote>
<p>How do I solve this?</p>
| Tim Ratigan | 79,602 | <p>$$\begin{align}
\sin\left(x+\frac{(2n+1)\pi}{2}\right)&=\sin x\cos\left(\frac{(2n+1)\pi}{2}\right)+\cos x\sin\left(\frac{(2n+1)\pi}{2}\right) \\
&=\sin x\cos(n\pi+\pi/2)+\cos x\sin(n\pi+\pi/2) \\
&=-\sin x\sin(n\pi)+\cos x \cos(n\pi) \\
\end{align}$$</p>
<p>You can clearly see that this is not $\sin^n(x... |
580,974 | <blockquote>
<p>Show that the following equalities hold true for every $n$ from $\mathbb{N}$ and
every $x$ from $\mathbb{R}$</p>
<p>$$\sin^{(n)}(x)=\sin(x+n\fracπ2)$$</p>
</blockquote>
<p>How do I solve this?</p>
| hhsaffar | 104,929 | <p><strong>Hint:</strong></p>
<p>Show $sin^{(1)}(x)=sin(x+\frac\pi2)$</p>
<p>Then suppose for $n$ we have $sin^{(n)}(x)=sin(x+n\fracπ2)$, show that $sin^{(n+1)}(x)=sin(x+(n+1)\fracπ2)$. Do this by differentiating $sin^{(n)}(x)$ once. </p>
|
201,381 | <p>I have basic training in Fourier and Harmonic analysis. And wanting to enter and work in area of number theory(and which is of some interest for current researcher) which is close to analysis. </p>
<blockquote>
<p>Can you suggest some fundamental papers(or books); so after reading these I can have, hopefully(p... | arsmath | 3,711 | <p>An older book that I really liked was Ayoub's <i>An Introduction to the Analytic Theory of Numbers</i>. It describes several famous classical theorems proved by analytical methods. It requires very little background beyond analysis. You can read the introduction <a href="http://www.ams.org/books/surv/010/surv010-... |
201,381 | <p>I have basic training in Fourier and Harmonic analysis. And wanting to enter and work in area of number theory(and which is of some interest for current researcher) which is close to analysis. </p>
<blockquote>
<p>Can you suggest some fundamental papers(or books); so after reading these I can have, hopefully(p... | Bobby Grizzard | 37,644 | <p>An area where analysis (especially the Fourier kind) is used quite heavily is in the study of so-called "Beurling-Selberg maximal functions," which have applications to several areas in number theory, such as counting lattice points and studying the behavior of zeta- and $L$-functions. The basic idea is that there ... |
2,062,398 | <p>May you tell me if my translation to symbolic logic is correct? </p>
<p>Thank you so much! Here is the problem:</p>
<p>To check that a given integer $n > 1$ is a prime, prove that it is enough to show that $n$ is not divisible by any prime $p$ with $p \le \sqrt{n}$.</p>
<p>$$\forall p \in P ~\forall n \in N ~(... | Community | -1 | <p><strong>Hint.</strong> Try to prove by contradiction. i.e., assume that $\sqrt{2} + \sqrt[3]{5}$ is of the form $p/q$ and do some algebraic manipulations to obtain a contradiction.</p>
|
4,294,577 | <p>If I have a function with all positive integer for the coefficients, is there a way to have a lower bound? Zero isn't an option, because I've done the rational root theorem and found all possible roots. If you need, I can provide the function and its list of possible roots below:</p>
<p><span class="math-container">... | Torsten Schoeneberg | 96,384 | <p>This is part of proposition 4 in Bourbaki's books on <em>Lie Groups and Algebras</em>, ch. VIII §5 no. 2. Since Bourbaki's setting is slightly more general (they are looking at a semisimple <span class="math-container">$\mathfrak g$</span> with a splitting Cartan subalgebra <span class="math-container">$\mathfrak h$... |
2,596,098 | <p>For a square matrix $A$ and identity matrix $I$, how does one prove that $$\frac{d}{dt}\det(tI-A)=\sum_{i=1}^n\det(tI-A_i)$$ Where $A_i$ is the matrix $A$ with the $i^{th}$ row and $i^{th}$ column vectors removed?</p>
| A.Γ. | 253,273 | <p>One way to prove this claim is to take the matrix $tI-A$ and replace the first $t$ on the main diagonal with $t_1$, the second one with $t_2$ etc. Let the resulting determinant be $p(t_1,t_2,\ldots,t_n)$. Then
$$
\det(tI-A)=p(t,t,\ldots,t),
$$
and by the chain rule
$$
\frac{d}{dt}\det(tI-A)=\frac{\partial}{\partial ... |
2,431,548 | <p>Okay, so, my teacher gave us this worksheet of "harder/unusual probability questions", and Q.5 is real tough. I'm studying at GCSE level, so it'd be appreciated if all you stellar mathematicians explained it in a way that a 15 year old would understand. Thanks!</p>
<p>So, John has an empty box.
He puts some red cou... | MEK MEK g | 622,454 | <p>John has a empty box. He puts some red counters and some blue counters. The ratio of the number of red counters to blue counters is 1:4. Linda takes at random 2 counters. The probability that she took 2 red counters is 6/155. How many red counters did John put in the box?</p>
<p>Let there be <span class="math-conta... |
853,774 | <blockquote>
<p>If $(G,*)$ is a group and $(a * b)^2 = a^2 * b^2$ then $(G, *)$ is abelian for all $a,b \in G$.</p>
</blockquote>
<p>I know that I have to show $G$ is commutative, ie $a * b = b * a$</p>
<p>I have done this by first using $a^{-1}$ on the left, then $b^{-1}$ on the right, and I end up with and expres... | Ivo Terek | 118,056 | <p>In one side, $(a \ast b)^2 = a^2 \ast b^2$ (hypothesis). On the other side, $(a \ast b)^2 = a \ast b \ast a \ast b $. So: $$a^2 \ast b^2 = a \ast b \ast a \ast b \\ a \ast b = b \ast a $$
In groups there is only <strong>one</strong> operation, which we often <em>think</em> as multiplication. It could be well additio... |
3,407,368 | <p>Please help me to think through this.</p>
<p>Take Riemann, for example. Finding a non-trivial zero with a real part not equal to <span class="math-container">$\frac{1}{2}$</span> (i.e., a counterexample) would disprove the conjecture, and also so it to be decidable.</p>
<p>How about demonstrating that Riemann is u... | Shiranai | 580,826 | <p>It is important to differentiate two aspects of mathematics, the deductive system (which is about what can be proved or not) and the model (about what is true or what is false). They are related, as everything that can be proved is true, but the converse does not hold: being true does not imply it can be proved.</p>... |
440,583 | <p><strong>Question.</strong> Is there an entire function <span class="math-container">$F$</span> satisfying first two or all three of the following assertions:</p>
<ul>
<li><span class="math-container">$F(z)\neq 0$</span> for all <span class="math-container">$z\in \mathbb{C}$</span>;</li>
<li><span class="math-contain... | Pavel Gubkin | 498,423 | <p>Here is a slightly different construction, from <a href="https://math.stackexchange.com/a/4642796/1048496">this answer</a> by @reuns , the function built this way has a <span class="math-container">$\exp(\exp(|z|))$</span> growth compared to triple exponent in the <a href="https://mathoverflow.net/a/440610/498423">... |
1,648,587 | <blockquote>
<p><strong>Problem.</strong> Consider two arcs <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> embedded in <span class="math-container">$D^2\times I$</span> as shown in the figure. The loop <span class="math-container">$\gamma$</span> is obviously nullhomotopic ... | Ian Mateus | 17,751 | <p>If you picture the solid cylinder as a solid ball, you will be able to deform $X$ into a solid ball without two parallel chords, which deformation retracts to the figure eight.</p>
|
2,472,746 | <p>I have that $f_n$ is measurable on a finite measure space.</p>
<p>Define $F_k=\{\omega:|f_n|>k \}$</p>
<p>$F_k$ are measurable and have the property $F_1 \supseteq F_2\supseteq\cdots$</p>
<p>Can I then claim that $m\left(\bigcap_{n=1}^\infty F_n\right) = 0$?</p>
| Tsemo Aristide | 280,301 | <p>If $f$ is real valued, $\cap F_k$ is empty. so its measure is zero.</p>
|
1,437,287 | <p>On <a href="https://en.wikipedia.org/wiki/Geometric_series#Geometric_power_series" rel="nofollow">Wikipedia</a> it is stated that by differentiating the following formula holds:</p>
<p>$$ \sum_n n q^n = {1\over (1-q)^2}$$</p>
<p>Does this not require a proof? It seems to me because the series is infinite it is not... | ncmathsadist | 4,154 | <p>Power series are termwise differentiable in their open interval of convergence. This is a basic property of power series.</p>
|
3,344,728 | <p>Let <span class="math-container">$S$</span> be the set of all real numbers except <span class="math-container">$-1$</span>. Define <span class="math-container">$*$</span> on <span class="math-container">$S$</span> by
<span class="math-container">$$a*b=a+b+ab.$$</span></p>
<p>Goal: Show that <span class="math-contai... | Community | -1 | <p>Hint: Show that <span class="math-container">$a*b=(a+1)(b+1)-1$</span></p>
|
4,467,841 | <p>For a complex number <span class="math-container">$z=a+bi$</span> and a positive real value <span class="math-container">$R$</span>, we have <span class="math-container">$e^{Rbi}=\cos(Rb)+i\sin(Rb)$</span>. I am struggling to understand this since no matter how large <span class="math-container">$b$</span> or <span ... | Nuke_Gunray | 1,060,681 | <p>Starting with the basics, <span class="math-container">$t\mapsto e^{it} = \cos t +i\sin t$</span> just describes the complex unit circle, which for <span class="math-container">$t\in\mathbb{R}$</span> is run through counter-clockwise with constant speed <span class="math-container">$1$</span>. This just means that o... |
4,304 | <p>I am trying to understand <a href="http://en.wikipedia.org/wiki/All-pairs_testing" rel="nofollow noreferrer"><strong>pairwise testing</strong></a>.</p>
<p>How many combinations of tests would be there for example, if</p>
<blockquote>
<p><code>a</code> can take values from 1 to m</p>
<p><code>b</code> can take values... | castal | 1,123 | <p>After reading <a href="http://www.satisfice.com/tools.shtml" rel="nofollow">this page</a>, it seems that pairwise testing requires a set of test cases in which every pair of values from any two of the n categories occurs at least once among the test case n-tuples. In the present case, the problem is to find a minima... |
2,735,984 | <p>I tried to solve this recurrence by taking out $n+1$ as a common in the RHS, but still have $n \cdot a_n$ and $a_n$</p>
| Tal-Botvinnik | 331,471 | <p><strong>HINT</strong></p>
<p>Assuming you meant $\frac{n+1}{n}$ (since you stated that you factored $n+1$ from the rhs), you can divide by $n+1$ and define
$$b_n=\frac{a_n}{n+1}.$$</p>
<p>Now, what is the equation satisfied by $b_n$? Do you know how to solve such an equation?</p>
|
4,030,359 | <p>Consider the abstract von Neumann algebra
<span class="math-container">$$M:= \ell^\infty-\bigoplus_{i \in I} B(H_i)$$</span>
which consists of elements <span class="math-container">$(x_i)_i$</span> with <span class="math-container">$\sup_i \|x_i\| < \infty$</span> and <span class="math-container">$x_i \in B(H_i)$... | daw | 136,544 | <p>Does this functions fit to your conditions?
<span class="math-container">$$
f(x) = \sqrt{|x|}
$$</span></p>
|
4,030,359 | <p>Consider the abstract von Neumann algebra
<span class="math-container">$$M:= \ell^\infty-\bigoplus_{i \in I} B(H_i)$$</span>
which consists of elements <span class="math-container">$(x_i)_i$</span> with <span class="math-container">$\sup_i \|x_i\| < \infty$</span> and <span class="math-container">$x_i \in B(H_i)$... | Adam Rubinson | 29,156 | <p>An example similar to daw's is:</p>
<p><span class="math-container">$$f(x)=\begin{cases}\arcsin(-2x-1) + \frac{\pi}{2} & -1\leq x< 0\\ \arcsin(2x-1) + \frac{\pi}{2} & 0\leq x\leq 1 \end{cases}$$</span></p>
|
1,569,543 | <blockquote>
<p>Prove or disprove: $(\ln n)^2 \in O(\ln(n^2)).$ </p>
</blockquote>
<p>I think I would start with expanding the left side. How would I go about this?</p>
| Casteels | 92,730 | <p>No. The path on $4$ vertices is known to have (Laplacian) eigenvalues $2-2\cos\left(\frac{k\pi}{4}\right)$ for $k\in\{0,1,2,3\}$. </p>
<p>The second smallest eigenvalue is $2-\sqrt{2}\neq 1$. In fact none of the eigenvalues are $1$.</p>
|
206,825 | <p>Let's say i have</p>
<p>N1 = -584</p>
<p>N2 = 110</p>
<p>Z = 0.64 </p>
<p>How do i calculate from Z which value is it in range of N1..N2? Z is range from 0 to 1.</p>
| André Nicolas | 6,312 | <p>We want to map numbers in the interval $[0,1]$ to numbers in your interval $[-584,110]$ by a function $fx)$ of the shape $f(x)=px+q$.</p>
<p>We want $f(0)=-584$, and $f(1)=110$, so $q=-584$ and $p=694$.</p>
<p>So our function is $694x-584$. Plug in $x=0.64$.</p>
<p><strong>Remark:</strong> Exactly the same idea ... |
2,808,159 | <p><strong>The question is:</strong> </p>
<blockquote>
<p>A half cylinder with the square part on the $xy$-plane, and the length $h$ parallel to the $x$-axis. The position of the center of the square part on the $xy$-plane is $(x,y)=(0,0)$.
<img src="https://i.stack.imgur.com/fB5le.jpg" alt="Image description"> ... | Arthur Sauer | 563,844 | <p>The lower and upper boundaries for $y$ in $S_1$ should be $-r$ and $r$ instead of $0$ and $r$. I was thinking in polar coordinates instead of in Cartesian coordinates... </p>
<p>I'll edit the original post accordingly</p>
|
109,754 | <p>Please help me get started on this problem:</p>
<blockquote>
<p>Let <span class="math-container">$V = R^3$</span>, and define <span class="math-container">$f_1, f_2, f_3 ∈ V^*$</span> as follows:<br />
<span class="math-container">$f_1(x,y,z) = x - 2y$</span><br />
<span class="math-container">$f_2(x,y,z) = x + y + ... | KnifeWrench | 61,233 | <p>Proving $\lbrace f_1,f_2,f_3\rbrace$ is a basis for $V^*$ can be done by row reducing the coefficients of $\lbrace f_1,f_2,f_3 \rbrace$ and showing that it has a rank of $3$.</p>
<p>The dual basis of $ \lbrace f_1,f_2,f_3 \rbrace $ is found by calculating the inverse of coefficients of $f_i$ which is:</p>
<p>$x_{1... |
3,281,828 | <p>I am new to permutation and combination and am looking for guidance in the following example:</p>
<p>We have 3 people - A, B, C</p>
<p>How many ways are there to arrange them into Rank 1,2,3</p>
<p>Looking at the example, it is clear that No repetitions are allowed and that ordering is not important (in the sense... | Vedvart1 | 354,933 | <p>A <em>permutation</em> asks how many ways there are to <em>permute</em>, or order, some number of elements from a larger set of elements. For instance, suppose you're organizing a photo with family members. Only <span class="math-container">$5$</span> of them will fit in the photo, but you have <span class="math-con... |
4,454,630 | <p>Is it true that for integers <span class="math-container">$i+j+k= 3m = n$</span> where <span class="math-container">$i , j, k , m , n\ge 0$</span> the inequality holds ?
<span class="math-container">$$
\binom{n}{i\;j\;k} \le \binom{n}{m\;m\;m}
$$</span>
I tried to show
<span class="math-container">$$
\frac{n!}{m!m!... | user26857 | 121,097 | <p>It is well known that <span class="math-container">$$\bigcap_{i=1}^\infty I^n=\{x\in R\mid\exists a\in I\text{ such that }x=ax\} $$</span>
(See <a href="https://math.stackexchange.com/a/18850/121097">here</a>.)</p>
<p>Let <span class="math-container">$\mathrm{Min}(R)=\{P_1,\dots,P_m\}$</span>. If <span class="math-c... |
1,242,075 | <p>I have some function $g$. I know that $g \in C^1[a,b]$, so $g'(x)$ exists. I want to know, if $g: [a,b] \to [a,b]$ is onto. How can I find out if this is true or not?</p>
<p>P.S. I am not saying all $g$ have the said property, I want to have some kind of test to distinguish functions with this property from functio... | Blind | 207,277 | <p>We provide a class of functions $g\in C^1[a,b]$ such that $g(x)\in [a,b]$ for all $x\in[a,b]$.</p>
<p>(1) $g(a), g(b)\in[a,b]$;</p>
<p>(2) $|g^\prime(x)|\leq \alpha \; \forall x\in (a,b)$, where
$$
\alpha\leq\min\left\{\frac{b-g(a)}{b-a};\frac{g(b)-a}{b-a}\right\}.
$$</p>
<p>Indeed, if $x\in [a, b]$ then there ex... |
3,236,205 | <p>I'm studying for a qualifying exam in algebra and I've come across the following problem:</p>
<blockquote>
<p>Let <span class="math-container">$G$</span> be a finite group with a subgroup <span class="math-container">$N$</span>. Let <span class="math-container">$Aut(G)$</span> be the group of automorphisms of <sp... | Santana Afton | 274,352 | <p>Hint:</p>
<p>First, note that <span class="math-container">$|N|$</span> is coprime to <span class="math-container">$|\!\operatorname{Aut}(G)|$</span> if and only if the order of every <span class="math-container">$n\in N$</span> is coprime to <span class="math-container">$|\!\operatorname{Aut}(G)|$</span>. </p>
<p... |
186,726 | <p>Just a soft-question that has been bugging me for a long time:</p>
<p>How does one deal with mental fatigue when studying math?</p>
<p>I am interested in Mathematics, but when studying say Galois Theory and Analysis intensely after around one and a half hours, my brain starts to get foggy and my mental condition d... | unclejamil | 11,743 | <p>Often just a change in the way you are thinking about the subject is enough to wake you up and continue learning so let me pass on my own trick to keeping fresh when I get burned out reading through a book. Take a break for 10 minutes and then switch over to learning how to use the software associated with what you... |
4,576,565 | <p>How would you decide whether a tridiagonal matrix with all ones in the diagonals has only a trivial solution (as matrix b is zero in the equation Ax=b)?</p>
<p>Edit: So, a general solution to an n by n matrix of the following appearance:</p>
<p><span class="math-container">$\begin{bmatrix}
1& 1& 0& 0... | P.S. Dester | 386,763 | <p>The reason <span class="math-container">$\alpha$</span> can't be greater or equal than <span class="math-container">$2$</span> is analogous to the reason <span class="math-container">$\beta$</span> can't be greater or equal than <span class="math-container">$1$</span> in this other one-dimensional integral:
<span cl... |
2,223,577 | <p>$\mathbb{Q}[e^{\frac{2\pi i}{5}}]$ is an extension of $\mathbb{Q}$ of degree 4, since $x^4+x^3+x^2+x+1$ is the irreducible polynomial of $\theta=e^{\frac{2\pi i}{5}}$ over $\mathbb{Q}$.</p>
<p>I'm asked if there is a quadratic extension $K$ of $\mathbb{Q}$ inside $\mathbb{Q}[e^{\frac{2\pi i}{5}}]$.
I suspect that ... | Paramanand Singh | 72,031 | <p>We can get an answer in this simple case by solving the equation $$x^{4}+x^{3}+x^{2}+x+1=0$$ Dividing by $x^{2}$ and putting $y=x+1/x$ we get $y^{2}+y-1=0$ so that $$y=\frac{-1\pm\sqrt{5}}{2}$$ and $$x=\frac{y\pm\sqrt{y^{2}-4}} {2}$$ and then clearly we can see that there is a tower of field extensions $$\mathbb{Q} ... |
59,888 | <p>Very similar to </p>
<p><a href="https://mathematica.stackexchange.com/questions/11638/parameters-in-plot-titles">Parameters in plot titles</a></p>
<p>in which I want to call a parameter from an array using <code>PlotLabel</code> in my plot using <code>Manipulate</code>. I've tried all of the suggestions in the ab... | ubpdqn | 1,997 | <p>Just another way using v10 <code>StringTemplate</code></p>
<pre><code>Manipulate[
Plot[Sin[f[[g]] x], {x, -2, 2},
PlotLabel -> TemplateApply[s, g]], {g, Range[6]},
Initialization :> (f = Range[6];
s = StringTemplate["f=<*f[[`1`]]*>"])]
</code></pre>
<p><img src="https://i.stack.imgur.com/llgX... |
2,319,126 | <p>I have to find dimension of $V+W,$ where$ V$ is a vector subspace given by solutions of the linear system:</p>
<p>$$x+2y+z=0$$
$$3y+z+3t=0$$</p>
<p>and $W$ is the subspace generated from vectors
$(4,0,1,3)^T,(1,0,-1,0)^T$.</p>
<p>I don't know how to combine the two subspaces and calculate the dimension.</p>
| Siong Thye Goh | 306,553 | <p>Hint:</p>
<p>The dimension of the basis of the solution space is $2$. </p>
<p>Also, none of the vectors in the basis of $W$ is a solution to the linear system.</p>
<p>Can you compute the dimension of $V+W$ now?</p>
|
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