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 |
|---|---|---|---|---|
633,223 | <p><img src="https://i.stack.imgur.com/xVS2C.png" alt="enter image description here"></p>
<p>This one has a great degree of self-evidence. Paradoxically, I find it difficult to deduce it from primitive propositions. The book only hinted ❋4.21 and ❋4.22.</p>
| Albert Steppi | 7,323 | <p>$\textbf{Edit:}$ My original proof had a gap in the derivation of what I know call statement $B$. I've added the relevant propositions that would allow one to fill the gap.</p>
<p>I own the second edition, but I don't think that should matter too much. I'm not going to give a full Russell-Whitehead style proof, but... |
149,830 | <p>As we know, the eigen vectors and eigen values of a real symmetric matrix always is are real numbers. I am trying to use the <em>Mathematica</em> to verify this theory. Suppose I have a matrix <code>A</code></p>
<pre><code>A={{6585, 7579, 6717}, {7579, 11002, 12324}, {6717, 12324, 17030}}
Eigenvalues[A]
</code></pr... | LCarvalho | 37,895 | <p>It's just an idea because it lacks information:</p>
<pre><code>Points = {{1, 2}, {2, 4}, {3, 7}, {4, 12}};
MapThread[f, Points];
f[x_, y_] := {x, y*200}
ListLogPlot[f @@@ Points]
</code></pre>
|
2,756,744 | <p>It is easy to derive from $AB=-BA$ that $\mathrm{tr}(AB)=0$ since $\mathrm{tr}(AB)=\mathrm{tr}(-BA)=-\mathrm{tr}(BA)=-\mathrm{tr}(AB)$. However, I cannot get that $\mathrm{tr}(A)=\mathrm{tr}(B)=0$ without the fact that $A$ and $B$ are invertible. </p>
<p>My Professor suggested I use the Cayley-Hamilton Theorem. How... | egreg | 62,967 | <p>The Cayley-Hamilton theorem says that
$$
A^2-\operatorname{tr}(A)A+\det(A)I=0
$$
If we multiply by $B$ on the right:
$$
A^2B-\operatorname{tr}(A)AB+\det(A)B=0
$$
If we multiply by $B$ on the left:
$$
BA^2-\operatorname{tr}(A)BA+\det(A)B=0
$$
Subtracting the two relations:
$$
A^2B-BA^2-2\operatorname{tr}(A)AB=0
$$
On... |
2,756,744 | <p>It is easy to derive from $AB=-BA$ that $\mathrm{tr}(AB)=0$ since $\mathrm{tr}(AB)=\mathrm{tr}(-BA)=-\mathrm{tr}(BA)=-\mathrm{tr}(AB)$. However, I cannot get that $\mathrm{tr}(A)=\mathrm{tr}(B)=0$ without the fact that $A$ and $B$ are invertible. </p>
<p>My Professor suggested I use the Cayley-Hamilton Theorem. How... | user1551 | 1,551 | <p>Here is an alternative proof without using Cayley-Hamilton theorem. We assume that the underlying field has characteristic $\ne2$, otherwise $A=B=\operatorname{diag}(1,0)$ would give a counterexample. As you said, $\operatorname{tr}(AB)=0$. It remains to show that $\operatorname{tr}(A)=\operatorname{tr}(B)=0$.</p>
... |
452,803 | <p>Test the convergence of improper integrals :</p>
<p>$$\int_1^2{\sqrt x\over \log x}dx$$</p>
<p>I basically have no idea how to approach a problem in which log appears. Need some hint on solving this type of problems.</p>
| Ron Gordon | 53,268 | <p>Consider the following:</p>
<p>$$\lim_{\epsilon \to 0} \int_{\epsilon}^1 dx \frac{\sqrt{1+x}}{\log{(1+x)}}$$</p>
<p>Now, for the bottom limit of the integral, note that</p>
<p>$$\log{(1+\epsilon)} \sim \epsilon$$</p>
<p>so that, near this limit, the integrand behaves as $1/\epsilon$ as $\epsilon \to 0$, or as $1... |
452,803 | <p>Test the convergence of improper integrals :</p>
<p>$$\int_1^2{\sqrt x\over \log x}dx$$</p>
<p>I basically have no idea how to approach a problem in which log appears. Need some hint on solving this type of problems.</p>
| Kunnysan | 84,764 | <p>Let, $I(\delta)=\displaystyle\int_{1+ \delta}^2{\sqrt x\over \log x}dx$</p>
<p>$$I(\delta) \ge \displaystyle\int_{1+ \delta}^2{1\over \log x} \geq \displaystyle\int_{1+ \delta}^2{1\over x-1}=-\log\delta$$</p>
<p>As, $\log x \le x-1 $ for $x\ge 1$.</p>
<p>Letting $\delta \to 0$ you can have that your integral dive... |
3,418,526 | <p>The problem is as follows:</p>
<blockquote>
<p>The figure from below shows the squared speed against distance
attained of a car. It is known that for <span class="math-container">$t=0$</span> the car is at <span class="math-container">$x=0$</span>.
Find the time which will take the car to reach <span class="m... | AgentS | 168,854 | <p>Given graph has the equation:
<span class="math-container">$$v^2 = 1+x$$</span></p>
<p>Implicitly differentiate both sides with respect to <span class="math-container">$x$</span> :</p>
<p><span class="math-container">$$2v\dfrac{dv}{dx} =1$$</span></p>
<p>Multiply left side by <span class="math-container">$1=\colo... |
941,182 | <p>If I know <span class="math-container">$\text{Im}(T)$</span> and <span class="math-container">$\text{Ker }(T)$</span>, is <span class="math-container">$\text{Im}(T)+\text{Ker }(T)$</span> the union of the two vector space?</p>
<p>If not, how do I find the addition of the two vector space. It is best if examples can ... | karakusc | 176,950 | <p>No, <span class="math-container">${\rm Im}(T)+{\rm Ker}(T)$</span> is not the same as <span class="math-container">${\rm Im}(T)\cup {\rm Ker}(T)$</span>. The former is defined as</p>
<p><span class="math-container">$$
{\rm Im}(T)+{\rm Ker}(T) = \{x+y: x \in {\rm Im}(T), y \in {\rm Ker}(T) \}
$$</span></p>
<p>To visu... |
941,182 | <p>If I know <span class="math-container">$\text{Im}(T)$</span> and <span class="math-container">$\text{Ker }(T)$</span>, is <span class="math-container">$\text{Im}(T)+\text{Ker }(T)$</span> the union of the two vector space?</p>
<p>If not, how do I find the addition of the two vector space. It is best if examples can ... | Kamster | 159,813 | <p>For an example, let <span class="math-container">$T:\mathbb{R}^2\rightarrow\mathbb{R}^2$</span> be defined such that
<span class="math-container">$$T(x,y)=T(x,0)$$</span>
so <span class="math-container">$T$</span> is essentially the projection of a vector in <span class="math-container">$\mathbb{R}^2$</span> on the ... |
2,919,683 | <p>I am using Monte Carlo method to evaluate the integral above:
$$\int_0^\infty \frac{x^4sin(x)}{e^{x/5}} \ dx $$
I transformed variables using $u=\frac{1}{1+x}$ so I have the following finite integral:
$$\int_0^1 \frac{(1-u)^4 sen\frac{1-u}u}{u^6e^{\frac{1-u}{5u}}} \ du $$
I wrote the following code on R:</p>
<p>set... | Matematleta | 138,929 | <p>Use the following easily proved facts: a set $V\subseteq X$ is saturated with respect to $f$ if and only if there is a $U\subseteq Y$ such that $f^{-1}(U)=V,$ and then $V=f^{-1}(f(V))$</p>
<p>The claim is that if $f:X\to Y$ is continuous and surjective, then $f$ is a quotient map if and only if it takes saturated o... |
335,651 | <p>I'm having trouble proving $$\left(\frac{\sin(\frac{n\theta}{2})}{\sin(\frac{\theta}{2})}\right)^2=\left|\sum_{k=1}^{|n|}e^{ik\theta}\right|^2$$ where $n\in\mathbb{Z}$ and $\theta\in\mathbb{R}$. Can anyone suggest a hint?</p>
| Ángel Mario Gallegos | 67,622 | <p>In order to proof
$$\sum_{k=1}^{n}{\sin \left(\varphi + k\alpha \right)}=\frac{\sin\left(\frac{\left(n+1\right)\alpha}{2}\right)\cdot\sin\left(\varphi+\frac{n\alpha}{2}\right)}{\sin \frac{\alpha}{2}}$$
observe
$$\sin \frac{\alpha}{2}\cdot \sin \left(\varphi + k\alpha\right)=\frac{1}{2}\left[\cos \left(\varphi + k\a... |
1,531,755 | <p>Let $a\in [0,1)$. I want to show that $$\lim_{n\to \infty}{na^n}=0$$</p>
<p>My try : $$na^n={n\over e^{-(\log{a})n}}$$ and the limit is $${+\infty\over +\infty}$$
Hence by l'Hopital's rule we have that
$$\lim_{n\to \infty}{1\over -(\log{a})e^{-(\log{a})n}}={1\over -\infty}=0$$</p>
<p>Is there any other way to com... | draks ... | 19,341 | <p><strong>HINT</strong> Expand $e^{-(\log a)n}$ and divide by $n$...</p>
|
4,004,827 | <p>I need to calculate:
<span class="math-container">$$\displaystyle \lim_{x \to 0^+} \frac{3x + \sqrt{x}}{\sqrt{1- e^{-2x}}}$$</span></p>
<p>I looks like I need to use common limit:
<span class="math-container">$$\displaystyle \lim_{x \to 0} \frac{e^x-1}{x} = 1$$</span></p>
<p>So I take following steps:</p>
<p><span c... | Restless | 880,216 | <p><span class="math-container">$$\lim_{x \to 0^+} \frac{3x + \sqrt{x}}{\sqrt{1- e^{-2x}}} =\lim_{x \to 0^+}\frac{\sqrt{x}}{\sqrt{1- e^{-2x}}} \cdot(3\sqrt{x} + 1) = \lim_{x \to 0^+} \sqrt{\frac{x}{1- e^{-2x}}} \cdot\lim_{x \to 0^+} (3\sqrt{x} + 1)$$</span></p>
<p><span class="math-container">$$\lim_{x \to 0^+} \sqrt{\... |
200,093 | <p>I have a BLDC electric motor, I'm currently trying to control via a <code>PIDTune</code>. This is mostly an attempt to reduce (remove) a small run away drift that ends up showing up in the motor signal <code>u[t]</code>.</p>
<p>I've modelled this via:</p>
<pre><code>ssm = StateSpaceModel[\[ScriptCapitalJ] \[Phi]''... | MK. | 61,732 | <p>Not an answer, but too long for a comment.</p>
<p>Your definition of <code>ssm</code> seems to not comply with the syntax listed in the documentation. It can be changed, for example, to</p>
<pre><code>ssm = StateSpaceModel[
\[ScriptCapitalJ] \[Phi]''[t] + \[ScriptCapitalR] \[Phi]'[
t] == \[ScriptCapitalT] u[t],
{{... |
789,407 | <p>If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. </p>
<p><strong>Edit:</strong> I forgot to mention in the question that $a$, $b$, and $c$ are natural numbers. Sorry for the inconvenience.<br>
<strong>Edit 2:</strong> As Hagen von Eitzen said abo... | vadim123 | 73,324 | <p>If you multiply the equation by $k$, you get $$(ka)x^2-(bk)x+(ck)=0$$
This new equation has the same roots as the original, hence in $(0,1)$, but has the product of <em>its</em> coefficients $k^3abc$. By letting $k\to\pm \infty$ (depending on whether $abc>0$ or $abc<0$), you can make this product as small as ... |
3,272,738 | <p>I've been trying to make sense of these two integrals, somehow the result seems intuitive, yet very hard to compute. We define</p>
<p><span class="math-container">$$
f(x)=\frac{1}{4\pi}\delta(|x|-R)$$</span>
and then note that
<span class="math-container">$$
-\frac{1}{2}\int\int\frac{f(x)f(y)}{|x-y|}=-\frac{1}{2R}$... | StubbornAtom | 321,264 | <p>By the <a href="https://en.wikipedia.org/wiki/Law_of_total_expectation" rel="nofollow noreferrer">law of total expectation</a>,</p>
<p><span class="math-container">$$E[XY]=E\left[E\,[XY\mid X]\right]=E\left[XE\,[Y\mid X]\right]$$</span></p>
<p>, where you are given that <span class="math-container">$E\,[Y\mid X]=1... |
3,386,371 | <p>Find the explicit form of
<span class="math-container">$$
\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n(n+2)}x^{n-1}.
$$</span></p>
<p>Let <span class="math-container">$S(x)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n(n+2)}x^{n-1}$</span>. It has radius of convergence <span class="math-container">$1$</span>.</p>
<p>Let <span ... | Ross Millikan | 1,827 | <p>Stirling is a reasonable approach here. We have
<span class="math-container">$$\frac{100!}{(50!)^2 2^{100}}\approx \frac {100^{100}e^{50}e^{50}\sqrt{2\pi 100}}{50^{50}50^{50}e^{100}2^{100}(2\pi 50)}=\frac 1{\sqrt{50\pi}}\approx \frac 1{7\cdot 1.8}=\frac 1{12.6}$$</span>
Where I took <span class="math-container">$\... |
3,386,371 | <p>Find the explicit form of
<span class="math-container">$$
\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n(n+2)}x^{n-1}.
$$</span></p>
<p>Let <span class="math-container">$S(x)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n(n+2)}x^{n-1}$</span>. It has radius of convergence <span class="math-container">$1$</span>.</p>
<p>Let <span ... | Brian Tung | 224,454 | <p>Completely aside from the mental math estimation, you may see that the standard deviation of the number of heads is <span class="math-container">$\sqrt{100(0.5)(0.5)} = 5$</span>, which means that the probability of being between <span class="math-container">$45$</span> and <span class="math-container">$55$</span> i... |
3,832,684 | <p>Does the following inequality hold?
<span class="math-container">$$\sqrt {x-z} \geq \sqrt x -\sqrt{z} \ , $$</span>
for all <span class="math-container">$x \geq z \geq 0$</span>.</p>
<p>My justification
<span class="math-container">\begin{equation}
z \leq x \Rightarrow \\ \sqrt z \leq \sqrt {x} \Rightarrow \\ 2\sqr... | user21820 | 21,820 | <p>While the other existing answers give simple algebraic reasons for this fact, it is actually far more useful in general to see this fact as a special case of the <strong>smoothing</strong> technique. In particular, for any concave function <span class="math-container">$f$</span> on domain <span class="math-container... |
4,033,831 | <p>Example:</p>
<p><img src="https://i.stack.imgur.com/nPzJb.png" alt="Notation" /></p>
<p>From this we can tell no negative real number can be the image of any element of the domain. Thus not surjective because the range is not equal to the codomain, which means the function associates a any real number to a positive ... | Theo Bendit | 248,286 | <p>It depends on the context, but you're right that writing <span class="math-container">$f : \Bbb{R} \to \Bbb{R}^+$</span> contains superior information about the rule <span class="math-container">$x \mapsto |x|$</span>. There are a few reasons why you might choose to write a more general codomain than what is called ... |
2,120,539 | <p>Find the points of local maximum and minimun of the function:
$$f(x)=\sin^{-1}(2x\sqrt{1-x^2})~~~~;~~x\in (-1,1)$$
I know
$$f'(x)=-\frac{2}{\sqrt{1-x^2}}$$</p>
<p>How to find the local maximum and minimum? I have drawn the fig and seen the points of local maximum and minimum. But how to find then analytically?
<a h... | egreg | 62,967 | <p>First of all, a bit of theory.</p>
<p>Suppose you have a function $F$ defined over the interval $(a,b)$ and increasing thereon. Suppose also that $G$ takes on values in $(a,b)$. Then, in order to find the points where $F(G(x))$ has a local maximum or minimum, it's sufficient to find the points where $G$ has a local... |
1,392,205 | <p>The equation of line $A$ is $3x + 6y - 1 = 0$. Give the equation of a line that passes through the point $(5,1)$ that is</p>
<ol>
<li><p>Perpendicular to line $A$.</p></li>
<li><p>Parallel to line $A$.</p></li>
</ol>
<p>Attempting to find the parallel,</p>
<p>I tried $$y = -\frac{1}{2}x + \frac{1}{6}$$</p>
<p>$$... | Michael Hardy | 11,667 | <p>The parameters $p$ if that is construed in the usual way, would remain the same, and $r$ would be $36$.</p>
<p><b>But</b> that's not the best way to proceed. Find the expected value and variance of your geometric distribution. Multiply them by $36$. Those would be the expected value and variance of the distributi... |
1,817,681 | <p>Let $n=|\mathbb{Z}[i]/(1+i)|$</p>
<p>Need to show it is isomorphic to a field of order n once I find n.</p>
<p>I know $(1+i)$ is maximal in $\mathbb{Z}[i]$.</p>
<p>Kind of confused on quotients like this. Saw the other solution: 2/(1+i)=(1-i) etc. No idea what it means.</p>
<p>I know I might need to use isomorph... | Crostul | 160,300 | <p>Since you know that $2=(1-i)(1+i)$, you have that in the quotient
$$ \overline{a+bi} = \overline{a} + \overline{i}\overline{b} = \overline{a}-\overline{b} = \overline{a-b} = (a-b) \mod{2}$$
So that the elements of $\Bbb{Z}[i]/(i+1)$ are $\overline{0},\overline{1}$.</p>
|
1,817,681 | <p>Let $n=|\mathbb{Z}[i]/(1+i)|$</p>
<p>Need to show it is isomorphic to a field of order n once I find n.</p>
<p>I know $(1+i)$ is maximal in $\mathbb{Z}[i]$.</p>
<p>Kind of confused on quotients like this. Saw the other solution: 2/(1+i)=(1-i) etc. No idea what it means.</p>
<p>I know I might need to use isomorph... | Brent Kerby | 218,224 | <p>Hint: $\mathbb Z[i]\cong \mathbb Z[x]/(x^2+1)$. The image of the ideal $(1+i)$ under this isomorphism is $(1+x,x^2+1)/(x^2+1)$, hence $$\mathbb Z[i]/(1+i) \cong [\mathbb Z[x]/(x^2+1)]/[(1+x,x^2+1)/(x^2+1)] \cong \mathbb Z[x]/(1+x,x^2+1)$$</p>
|
634,132 | <p>Let $G$ be a cyclic group with $N$ elements. Then it follows that</p>
<p>$$N=\sum_{d|N} \sum_{g\in G,\text{ord}(g)=d} 1.$$</p>
<p>I simply can not understand this equality. I know that for every divisor $d|N$ there is a unique subgroup in $G$ of order $d$ with $\phi(d)$ elements. But how come that when you add all... | arkadeep | 120,499 | <p>See it is a bi variate function that you have given here.
Just think of a 3-D(2 dimentional system of co-ordinate).
Now if a function is a bivariate one then the functional value will have the value on axis which one is mutually perpendicular to the other two axis.
Now you have the equation as f(a,b)=f(a,c).Now we h... |
3,041,632 | <p><span class="math-container">$X_n=4X_{n-1}+5$</span></p>
<p>How come the solution of this recurrence is this? </p>
<p><span class="math-container">$X_n=\frac834^n+\frac53$</span></p>
<p>I also have that <span class="math-container">$X_0=1$</span>.</p>
<p>I am using telescoping method and I am trying to solve it ... | Matthias | 626,460 | <p>If you calculate the first few terms explicitly, you will find that the <span class="math-container">$n$</span>th term is the sum of an exponential and a geometric series. For example,</p>
<p><span class="math-container">$$
X_3 = 4^3 + 5(4^2+4+1).
$$</span>
So in general,
<span class="math-container">$$
X_n = 4^n +... |
315,551 | <p>So I'm going over my practice midterms (which all seem to have solutions like this one), </p>
<p><img src="https://i.stack.imgur.com/fC8Gu.png" alt="Image"></p>
<p>Can anyone help clarify this for me? I understand that you multiply by the reciprocal to get to line two. But after that I'm completely lost, I don't u... | Cameron Buie | 28,900 | <p>This is one of those "verification left to the reader" moments.</p>
<p>If it helps, we can use the intermediate step that $$x^2-1-[(x+h)^2+1]=x^2-(x+h)^2,$$ so the conclusion follows from the fact that $a^2-b^2=(a-b)(a+b).$</p>
<p>Too, they didn't explain how they got from the third line to the fourth, but since $... |
1,063,352 | <p>$A$ and $B$ are sets and $\mathcal{F}$ is a family of sets. I'm trying to prove that</p>
<p>$\bigcap_{A \in \mathcal{F}}(B \cup A) \subseteq B \cup (\cap \mathcal{F})$</p>
<p>I start with "Let $x$ be arbitrary and let $x \in \bigcap_{A \in \mathcal{F}}(B \cup A)$, which means that $\forall C \in \mathcal{F}(x \in... | Jack D'Aurizio | 44,121 | <p>Since:
$$\frac{5^n}{25^n+1}=\frac{1}{5^n}-\frac{1}{125^n}+\frac{1}{3125^n}-\ldots $$
we have:
$$\sum_{n=0}^{+\infty}\frac{5^n}{25^n+1}=\frac{1}{2}+\left(\frac{1}{4}-\frac{1}{124}+\frac{1}{3124}+\ldots\right)=\frac{1}{2}+\sum_{k=0}^{+\infty}\frac{(-1)^k}{5^{2k+1}-1}.$$</p>
<hr>
<p>Despite being easy to compute thro... |
3,911,548 | <p><strong>If a,b,c,d are real numbers and <span class="math-container">$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=17$</span> and <span class="math-container">$\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}=20$</span>, then find the sum of all possible valuse of <span class="math-container">$\frac{a}{b}$</span>+... | Ben Grossmann | 81,360 | <p><strong>Hint:</strong> Note that
<span class="math-container">$$
\frac ac + \frac ca + \frac bd + \frac db = 20 \implies\\
\frac ab \cdot \frac bc + \frac cd \cdot \frac da + \frac bc \cdot \frac cd + \frac da \cdot \frac ab = 20.
$$</span>
With that in mind, compare the expanded sums
<span class="math-container">$$... |
3,282,895 | <p>How do I calculate <span class="math-container">$$\int_{0}^{2\pi} (2+4\cos(t))/(5+4\sin(t)) dt$$</span></p>
<p>I've recently started calculating integral via the residue theorem. Somehow I'm stuck with this certain integral. I've substituted t with e^it and received two polynoms but somehow I only get funny solutio... | Maurizio Moreschi | 578,146 | <p>As mentioned in the comment, the following gives you an alternative way to compute the integral without using complex analysis.</p>
<p>I sketch my idea and let you finish the computation yourself. Please tell me in you need more details.</p>
<p>First, using that <span class="math-container">$\cos(t+\pi)=-\cos(t)$<... |
83,246 | <p>Let H be a separable and infinite-dimensional Hilbert space and let B be the closed ball
of H having unit radius, whose center is at the origin h of H. Suppose one would like to
know how much of B can be "filled up" by any of its compact subsets-since B itself
(although closed and bounded) is not compact. Let E be t... | Alessandro Sisto | 9,342 | <p>Instead of using compact subsets one can just use finite subsets. Now, consider any finite subset of $B$: it is contained in a finite dimensional subspace $V$. There is a unit vector perpendicular to $V$, and such vector has distance at least $1$ from the given finite subset.</p>
|
3,528,237 | <p>I am just being introduced to quantifiers in logic and my lecturer was going through the following two statements. The question is to determine which, if any, is/are true.</p>
<ol>
<li><span class="math-container">$(\forall x \in \mathbb{R})(\exists y \in \mathbb{R})[x + y = 0]$</span></li>
<li><span class="math-co... | Mostafa Ayaz | 518,023 | <p>The second one is false because <strong>there is no real number <span class="math-container">$x$</span> such that its addition with all real numbers results in <span class="math-container">$0$</span>.</strong> You can take <span class="math-container">$y=x^2+1$</span> for each <span class="math-container">$x$</span>... |
2,791,087 | <p>I have the following density function:
$$f_{x, y}(x, y) = \begin{cases}2 & 0\leq x\leq y \leq 1\\ 0 & \text{otherwise}\end{cases}$$</p>
<p>We know that $\operatorname{cov}(X,Y) = E[(Y - EY)(X - EX)]$, therefore we need to calculate E[X] and E[Y]. </p>
<p>$$f_x(x)=\int_x^1 2\,\mathrm dy = \big[2y\big]_x^1 =... | caverac | 384,830 | <p>In general</p>
<p>$$
\mathbb{E}[g(X,Y)] = \iint_{\mathbb{R}^2}{\rm d}x{\rm d}y~ g(x,y)\color{blue}{f_{X,Y}(x,y)} = \color{blue}{2}\int_0^1 {\rm d}x\int_x^1{\rm d}y~ g(x,y)
$$</p>
<p>so that</p>
<p>\begin{eqnarray}
\mathbb{E}[X] = 2\int_0^1 {\rm d}x\int_x^1{\rm d}y~ x = 2\int_0^1{\rm d}x~x(1-x) = \frac{1}{3}
\e... |
250,119 | <p>I'd like to show that if a set $X$ is Dedekind finite then is is finite if we assume $(AC)_{\aleph_0}$. As set $X$ is called Dedekind finite if the following equivalent conditions are satisfied: (a) there is no injection $\omega \hookrightarrow X$ (b) every injection $X \to X$ is also a surjection.</p>
<p>Countable... | Zhen Lin | 5,191 | <p>You will have to restrict the language, obviously, because $P$ could be the proposition "the base field is $\mathbb{C}$". As André Nicolas has already mentioned, if $P$ is a proposition in the <em>first-order</em> language of fields then anything that holds for $\mathbb{C}$ holds for any algebraically closed field o... |
890,313 | <p>Say the probability of an event occurring is 1/1000, and there are 1000 trials.</p>
<p>What's the expected number of events that occur? </p>
<p>I got to an answer in a quick script by doing the above 100,000 times and averaging the results. I got 0.99895, which seems like it makes sense. How would I use math to ge... | beep-boop | 127,192 | <p>In general, the expectected value (i.e. the expected number of events), </p>
<h2>$$\boxed{E=np},$$</h2>
<p>where $n$ is the number of trials and $p$ is the probability of success during each trial.</p>
|
5,586 | <p>I'm in my last year of highschool. And I'm aiming for a perfect grade in maths. The problem is that this year is the hardest year of maths I have ever faced in my entire life. Especially derivation and limits as its the first time I am studying it. Here are the lessons that are required to study for the first semest... | alternative | 3,250 | <p>Solve problems.</p>
<p>That's literally the only way to get better in math. Do more problems. Don't do the same problem with different numbers, do harder and harder problems until you are mentally exhausted, take a break, and delve in again.</p>
|
5,586 | <p>I'm in my last year of highschool. And I'm aiming for a perfect grade in maths. The problem is that this year is the hardest year of maths I have ever faced in my entire life. Especially derivation and limits as its the first time I am studying it. Here are the lessons that are required to study for the first semest... | Axiomaric | 7,893 | <p>I feel that the excellent answers given already may not do enough to discourage the idea that you <em>need</em> to be perfect. While I do understand your aim towards a perfect grade, you need to embrace the possibility that this may not happen and to not let that deter you from continuing your math education with th... |
649,239 | <p>By <a href="http://en.wikipedia.org/wiki/Post%27s_theorem" rel="nofollow">Post's Theorem</a> we know that a set $A\subseteq\mathbf{N}$ is recursively enumerable iff it is definable by a $\Sigma_1$-formula, i.e. there exists a $\Sigma_1$-formula $\varphi(x)$ with $x$ free such that for every number $n$:
\[
n\in A\lon... | Andreas Blass | 48,510 | <p>The answer to this question is that it depends on how we are "given" the r.e. set <span class="math-container">$A$</span>. In most situations, the answer is yes. For example, if we are given a Turing machine (or a C++ program or anything like that) to list all the elements of <span class="math-container">$A$</span>,... |
530,484 | <p>Let $f:\mathbb{R}\rightarrow\mathbb{R}^2$ be a $C^1$ function. Prove that the image of $f$ contains no open set of $\mathbb{R}^2$.</p>
<p>So say $f(x)=(g(x),h(x))$. Since $f$ is $C^1$, we have that $g'(x),h'(x)$ both exist and are continuous functions in $x$. To show that $f$ contains no open set of $\mathbb{R}^2$,... | Luiz Cordeiro | 58,818 | <p>Let $F:\mathbb{R}^2\rightarrow\mathbb{R}^2$ be given by $F(x,y)=f(x)$. Then every point of $\mathbb{R}^2$ is a critical point of $F$ (that is $\det J(F)(x)=0$ for every $x\in\mathbb{R}^2$, where $J(F)(x)$ denotes the Jacobian of $F$ at $x$). By Sard's theorem, $f(\mathbb{R})=F(\mathbb{R}^2)$ has zero measure. Since ... |
54,311 | <p>I found <a href="http://www.rle.mit.edu/dspg/documents/HilbertComplete.pdf" rel="nofollow">this paper</a> on Hilbert Transform, which is a very nice read. I've studied signal processing, but from a more practical than mathematical perspective. Can someone explain to me how we arrive at equation (2) in this paper?</p... | Michael Hardy | 11,667 | <p>We have
$$
\oint X(v) H\left(\frac z v\right) v^{-1} \, dv.
$$
Let $u = \dfrac z v$, so that $du = \dfrac{-z}{v^2}\,dv$. Then $v$ becomes $\dfrac z u$ and $v^{-1}\,dv$ becomes $\dfrac{-du}{u}$. But as $v$ goes around the unit circle in the counterclockwise direction, $u$ goes around in the clockwise direction. ... |
105,868 | <p>Let $f(x)$ be a continuous probability distribution in the plane. It is obvious that if $X$ and $X'$ are two independent random samples from $f$, then $\mathbf{E}(\|X - X'\|) \leq 2 \mathbf{E}(\|X\|)$ by the triangle inequality. Can this upper bound be made tighter if we assume that $f$ is rotationally symmetric a... | Robert Israel | 13,650 | <p>The conditional expectation
$$\eqalign{E[ \|X - X'\| | \|X\| = r, \|X'\| = s] &= \frac{1}{2\pi} \int_0^{2\pi} \sqrt{r^2 + s^2 - 2 r s \cos \theta}\ d\theta \cr &= \frac{2(r+s)}{\pi} EllipticE(2 \sqrt{rs}/(r+s))\cr}$$
where EllipticE is Maple's version of the complete elliptic integral of the second kind.
No... |
213,665 | <p><strong>I've tried 3 methods but all failed to do that.</strong></p>
<p>1st Method</p>
<pre><code>Apply[Flatten, {1, {2, {3, 4}, 5}, 6}, {2}]
</code></pre>
<p>2nd Method</p>
<pre><code>Map[Flatten, {1, {2, {3, 4}, 5}, 6}, {2}]
</code></pre>
<p>3rd Method</p>
<pre><code>Flatten[{1, {2, {3, 4}, 5}, 6}, {2}]
</co... | Sjoerd Smit | 43,522 | <p>We've got a few answers already, but here's my 2 cents:</p>
<pre><code>Replace[l_List :> Flatten[l]] /@ {1, {2, {3, 4}, 5}, 6}
</code></pre>
<blockquote>
<p>{1, {2, 3, 4, 5}, 6}</p>
</blockquote>
|
213,665 | <p><strong>I've tried 3 methods but all failed to do that.</strong></p>
<p>1st Method</p>
<pre><code>Apply[Flatten, {1, {2, {3, 4}, 5}, 6}, {2}]
</code></pre>
<p>2nd Method</p>
<pre><code>Map[Flatten, {1, {2, {3, 4}, 5}, 6}, {2}]
</code></pre>
<p>3rd Method</p>
<pre><code>Flatten[{1, {2, {3, 4}, 5}, 6}, {2}]
</co... | Mr.Wizard | 121 | <p>A couple more:</p>
<pre><code>list = {1, {2, {3, 4}, 5}, 6};
Apply[## &, list, {2}]
Flatten[{##}] & @@@ list
</code></pre>
<p>Also if you are looking for specific level control consider <em>levelspec</em> in <a href="https://reference.wolfram.com/language/ref/Replace.html" rel="nofollow noreferrer"><code... |
2,538,297 | <p>This is my first question here so I hope I'm doing it right :) sorry otherwise!</p>
<p>As in the title, I was wondering if and when it is OK to calculate a limit i three dimensions through a substitution that "brings it down to two dimensions". Let me explain what I mean in a clearer way through an example. I was c... | Dr. Sonnhard Graubner | 175,066 | <p>use that $$x^2+y^2\geq 2|xy|$$ then $$\frac{\ln(1+\sin^2(xy))}{x^2+y^2}\le \frac{\ln(1+\sin^2(xy))}{2|xy|}$$ substituting $$xy=t$$ then we have $$\frac{\ln(1+\sin^2(t))}{2t}$$ with L'Hospital we can prove that $$\lim_{t\to 0}\frac{\ln(1+\sin^2(t))}{2t}=\lim_{t\to0}\frac{\sin(2t)}{2(1+\sin^2(t))}=0$$</p>
|
4,610,394 | <p>Clearly, none of the roots are in <span class="math-container">$\mathbb{Q}$</span> so <span class="math-container">$f(x) = x^4 + 1$</span> does not have any linear factors. Thus, the only thing left to check is to show that <span class="math-container">$f(x)$</span> cannot reduce to two quadratic factors.</p>
<p>My ... | user2661923 | 464,411 | <p>Alternative approach:</p>
<blockquote>
<p>My proposed solution was to state that <span class="math-container">$f(x) = x^4 + 1 = (x^2 + i)(x^2 - i)$</span> but <span class="math-container">$\pm i \not\in \mathbb{Q}$</span> so <span class="math-container">$f(x)$</span> is irreducible.</p>
</blockquote>
<p>This isn't q... |
4,610,394 | <p>Clearly, none of the roots are in <span class="math-container">$\mathbb{Q}$</span> so <span class="math-container">$f(x) = x^4 + 1$</span> does not have any linear factors. Thus, the only thing left to check is to show that <span class="math-container">$f(x)$</span> cannot reduce to two quadratic factors.</p>
<p>My ... | reuns | 276,986 | <p>A general method: assume that <span class="math-container">$f\in \Bbb{Q}[x]$</span> monic of degree <span class="math-container">$1$</span> or <span class="math-container">$2$</span> divides <span class="math-container">$x^4+1$</span>. The roots of <span class="math-container">$x^4+1$</span> have absolute value <spa... |
3,884,659 | <p>Let <span class="math-container">$H$</span> be a graph, and let <span class="math-container">$n > |V(H)|$</span> be an integer.Suppose there is a graph on <span class="math-container">$n$</span> vertices and <span class="math-container">$t$</span> edges containing no copy of <span class="math-container">$H$</span... | saulspatz | 235,128 | <p>About <span class="math-container">$3$</span> years ago, I downloaded a solution manual from <a href="https://radimentary.files.wordpress.com/2017/03/solutions_compilation.pdf" rel="nofollow noreferrer">https://radimentary.files.wordpress.com/2017/03/solutions_compilation.pdf</a>, but this link no longer works. The... |
3,884,659 | <p>Let <span class="math-container">$H$</span> be a graph, and let <span class="math-container">$n > |V(H)|$</span> be an integer.Suppose there is a graph on <span class="math-container">$n$</span> vertices and <span class="math-container">$t$</span> edges containing no copy of <span class="math-container">$H$</span... | Vezen BU | 823,641 | <p>Below is my solution when I met the problem in my homework before.</p>
<p>When <span class="math-container">$|V(H)| \leq 1$</span>, it's not so meaningful, so we only need to care about the cases when <span class="math-container">$|V(H)| \geq 2$</span>, and then <span class="math-container">$n \geq |V(H)| + 1 \geq 3... |
2,637 | <p>Trying to round-trip expressions through JSON, I'm getting unexpected errors for held expressions, and would be grateful for advice or clues. Consider, first, something that works well</p>
<pre><code>Export[Environment["USERPROFILE"] <> "\\AppData\\Local\\test.json", {1, 2, 3},"JSON"]
</code></pre>
<p>and re... | Leonid Shifrin | 81 | <p>The short answer is that, as @FJRA noted in the comment, only certain types are supported. Which types? Enter the long answer.</p>
<h3>Why the converter behaves as it does</h3>
<p>Long answer: JSON supports only certain types, and their nested combinations, as defined e.g. <a href="http://www.json.org/" rel="nofollo... |
230,154 | <p><strong>Question.</strong> Is it true that to check that a model category is right proper, it suffices to check the property for weak equivalences with fibrant codomain ? (if the domain is also fibrant, the pullback is always a weak equivalence). Or is there a close statement that I can't remember (browsing nLab did... | Karol Szumiło | 12,547 | <p>To complete the argument you need to apply K. Brown's Lemma. Call your model category $\mathcal{M}$, then the map $Z \to Y$ induces a pullback functor $\mathcal{M} \downarrow Y \to \mathcal{M} \downarrow Z$ and the lemma implies that it preserves weak equivalences between fibrations over $Y$. If you define $V \to Y$... |
1,260,260 | <blockquote>
<p>Find, with proof, the smallest value of $N$ such that $$x^N \ge \ln x$$ for all $0 < x < \infty$. </p>
</blockquote>
<p>I thought of adding the natural logarithm to both sides and taking derivative. This gave me $N \ge \frac 1{\ln x}$. However, is there a better way to this?</p>
<p>Please note... | abel | 9,252 | <p>first you can show that the line $ y = kx $ touches at $x = e, y = 1, k = \frac 1 e.$ that is $$kx > \ln(x) \text{ for } k > \frac1e \text{ and } k x = \ln x \text{ for } x = e, k = \frac 1e.\tag 1$$</p>
<p>we have $$x^N > \ln x \implies N\ln x > \ln(ln(x) \implies N > \frac 1e. $$</p>
<p>$$\text{ ... |
2,767,070 | <p>The intuition for $E[g(Y)|Y=y]$ would be that $g(Y)$ would play the role of a constant once $Y$ is fixed to a certain $y$ value. But how to show this more formally ? I can't seem to expand the equation below.</p>
<p>$E[g(Y)|Y=y]=\sum_{y} g(y)P[g(y)=y'|Y=y]$</p>
| heropup | 118,193 | <p>Your error arises from using the same variable name for the index of summation as the given condition. More precisely, for a discrete random variable $Y$ with support $S$, $$\operatorname{E}[g(Y) \mid Y = y] = \sum_{a \in S} g(a) \Pr[Y = a \mid Y = y].$$ Since $\Pr[Y = a \mid Y = y] = \mathbb 1(a = y)$, it follow... |
3,032,258 | <p>Assume 5 out of 100 units are defective. We pick 3 out of the 100 units at random. </p>
<p>What is the probability that exactly one unit is defective?</p>
<hr>
<p>My answer would be </p>
<p><span class="math-container">$P(\text{Defect}=1) = P(\text{Defect})\times P(\text{Not defect})\times P(\text{Not defect}) =... | trancelocation | 467,003 | <p>Here is a suggestion how to proceed as ordering does not play a role</p>
<ul>
<li>Choose one defective item: <span class="math-container">$\binom{5}{1}$</span></li>
<li>Choose two non-defective ones: <span class="math-container">$\binom{95}{2}$</span></li>
<li>Chose any three: <span class="math-container">$\binom{1... |
163,672 | <p>Is there a characterization of boolean functions $f:\{-1,1\}^n \longrightarrow \{-1,1\}$,
so that $\mathbf{Inf_i}[f]=\frac{1} {2}$, for all $1\leq i\leq n$? Is it known how many such functions there are? </p>
| Brendan McKay | 9,025 | <p>I didn't really understand the original question, but from the comments of others you are looking for correlation-immune boolean functions of order 1. These go under many other names as well, including binary orthogonal arrays of strength 1 and balanced hypercube colourings. The last image is the simplest to under... |
3,253,891 | <p>I'm a complete n00b at math, but I'm wondering how one would go about determining the value of <code>n</code> in the following comparison.</p>
<p><code>n * 1.5 + 12.5 = 12.5 / 2 + n</code></p>
<p>I'm new to the math StackExchange, so I'm also not sure how to properly format this question. Feel free to edit.</p>
<p><... | BigSlurm | 833,788 | <p>Elliptical PDEs can be defined by functions that do not have any characteristic lines/surfaces. That is, there are no functions <span class="math-container">$F(x)$</span> or <span class="math-container">$G(y)$</span> such that <span class="math-container">$u''(x,y)=F(x)$</span> or <span class="math-container">$u''(x... |
127,322 | <p>Being a new member, I am not yet sure whether my question will be taken as a research level question (and thus, appropriate for MO). However, I have seen similar questions on MO, couple of which led me asking mine, and I seem to not be able to find many resources except discussion on FOM and MO. So, any references t... | Emil Jeřábek | 12,705 | <p>$\def\zfc{\mathrm{ZFC}}\def\pr{\operatorname{Prov}\nolimits}$The statement</p>
<blockquote>
<p>$\zfc\vdash\pr_\zfc(\ulcorner\varphi\urcorner)$ implies $\zfc\vdash\varphi$ for every sentence $\varphi$ in the language of $\zfc$</p>
</blockquote>
<p>is equivalent to the statement that $\zfc$ is either inconsistent ... |
787,894 | <p>Find the values of $x,y$ for which $x^2 + y^2$ takes the minimum value where $(x+5)^2 +(y-12)^2 =14$.</p>
<p>Tried Cauchy-Schwarz and AM - GM , unable to do.</p>
| lab bhattacharjee | 33,337 | <p>Any point satisfying $\displaystyle(x+5)^2 +(y-12)^2 =14$ can be expressed as $\sqrt{14}\cos\phi-5,\sqrt{14}\sin\phi+12$ </p>
<p>$\displaystyle x^2 + y^2=14(\cos^2\phi+\sin^2\phi)+2\sqrt{14}(12\sin\phi-5\cos\phi)+5^2+12^2$
$\displaystyle=14+12^2+5^2+2\sqrt{14}(12\sin\phi-5\cos\phi)$ </p>
<p>This will attain minimu... |
1,614,989 | <p>A portion of a $30$m long tree is broken by
tornado and the top struck up the ground
making an angle $30^{\circ}$ with ground
level. The height of the point where the tree
is broken is equal to:</p>
<p>$a.)\ \dfrac{30}{\sqrt{3}}m$ $~~~~~~~~~~$ $\color{green}{b.)\ 10m} \\$
$~~~~~~~~~~$ $c.)\ 30\sqrt{3}m$ ... | Eric Haney | 203,977 | <p><a href="https://i.stack.imgur.com/9xrRe.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/9xrRe.png" alt="enter image description here"></a></p>
<p>The tree AB has been split at the point D. The top of the tree has fallen to point K making a 30 degree angle with the ground. Since the tree's height... |
3,506,316 | <p>I am trying to evaluate this limit:</p>
<p><span class="math-container">$$\lim_{x\to0^{+}}(x-\sin x)^{\frac{1}{\log x}}$$</span></p>
<p>It's a <span class="math-container">$0^0$</span> intedeterminate form, and I am unsure how to deal with it. I have a feeling that if I could turn it to a form where L'Hopital's ru... | Paramanand Singh | 72,031 | <p>The best option is to take logs. If <span class="math-container">$L$</span> is the desired limit then we have <span class="math-container">$$\log L=\lim_{x\to 0^{+}}\frac{\log(x-\sin x)} {\log x} $$</span> The expression under limit above can be rewritten as <span class="math-container">$$3+\dfrac{\log\dfrac{x-\sin ... |
4,246,726 | <p>For the system of linear equations <span class="math-container">$Ax = b$</span> with <span class="math-container">$b =\begin{bmatrix}
4\\
6\\
10\\
14
\end{bmatrix}\\
$</span>. The set of solutions is given by- <span class="math-container">$\left... | greg | 357,854 | <p><span class="math-container">$
\def\a{\alpha}\def\b{\beta}
\def\o{{\tt1}}\def\p{\partial}
\def\E{{\cal E}}\def\F{{\cal F}}\def\G{{\cal G}}
\def\L{\left}\def\R{\right}\def\LR#1{\L(#1\R)}
\def\vec#1{\operatorname{vec}\LR{#1}}
\def\trace#1{\operatorname{Tr}\LR{#1}}
\def\grad#1#2{\frac{\p #1}{\p #2}}
\def\c#1{\color{red... |
738,083 | <blockquote>
<p>Show that if two random variables X and Y are equal almost surely, then they
have the same distribution. Show that the reverse direction is not correct.</p>
</blockquote>
<p>If $2$ r.v are equal a.s. can we write $\mathbb P((X\in B)\triangle (Y\in B))=0$ (How to write this better ?)</p>
<p>then </... | user521337 | 521,337 | <p>If <span class="math-container">$X$</span> is a random variable following uniform <span class="math-container">$\mathcal U(-1,1)$</span> distribution, then <span class="math-container">$X$</span> and <span class="math-container">$-X$</span> are identically distributed, but obviously <span class="math-container">$X$<... |
990,930 | <p>Graphing this function is difficult as many overlaps exist and finding a viewing window is hard.</p>
<p>What's a good algebraic method to solve this problem? </p>
| Community | -1 | <p><strong>Hint</strong></p>
<p>Use the relation $\cos^2x+\sin^2 x=1$ to find a quadratic equation with unkown $\cos x$. Solve it and find the value of $x$ in the desired interval.</p>
|
29,766 | <p>I'm looking for a news site for Mathematics which particularly covers recently solved mathematical problems together with the unsolved ones. Is there a good site MO users can suggest me or is my only bet just to google for them?</p>
| Helge | 3,983 | <p>Another suggestion <a href="http://pjm.math.berkeley.edu/scripts/coming.php?jpath=annals" rel="nofollow"> Annals: to appear</a>. Also other top journals. If a big problem gets solved, its solution probably gets submitted to a journal of this type, so its to appear lists are what you are looking for. Of course, you o... |
174,165 | <p>I have Maths test tomorrow and was just doing my revision when I came across these two questions. Would anyone please give me a nudge in the right direction?</p>
<p>$1)$ If $x$ is real and $$y=\frac{x^2+4x-17}{2(x-3)},$$ show that $|y-5|\geq2$ </p>
<p>$2)$ If $a>0$, $b>0$, prove that $$\left(a+\frac1b\right)... | hxthanh | 58,554 | <p>$\displaystyle \cos\frac{n\pi}{2}=\{1,0,-1,0\}=2\left\lfloor\frac{n}{4}\right\rfloor+2\left\lfloor\frac{n+1}{4}\right\rfloor+1-n$</p>
<p>$\displaystyle \sin\frac{n\pi}{2}=\{0,1,0,-1\}=n-2\left\lfloor\frac{n+1}{4}\right\rfloor-2\left\lfloor\frac{n+2}{4}\right\rfloor$</p>
|
1,149,561 | <p>I've tried using mods but nothing is working on this one: solve in positive integers $x,y$ the diophantine equation $7^x=3^y-2$.</p>
| HSN | 58,629 | <p>The Euler characteristic is multiplicative in the sense that if $M$ and $N$ are two spaces $\chi(M\times N) = \chi(M)\cdot\chi(N)$. Thus, if a space $M$ has Euler characteristic $0$ or $1$, this means that for any other space $N$ either $\chi(M\times N) = 0\cdot\chi(N)=0 = \chi(M)$, or $\chi(M\times N) = 1\cdot\chi(... |
3,568,050 | <blockquote>
<p>Let <span class="math-container">$R$</span> be an equivalence relation in the set <span class="math-container">$A$</span> and <span class="math-container">$a,b \in A$</span>. Show that <span class="math-container">$R(a)=R(b)$</span> <strong>iff</strong> <span class="math-container">$aRb$</span>.</p>
<... | mathcounterexamples.net | 187,663 | <p>I would do something like that.</p>
<p>Suppose first that <span class="math-container">$R(a)=R(b)$</span>.
As <span class="math-container">$R$</span> is reflexive, we have <span class="math-container">$aRa$</span> which implies <span class="math-container">$a\in R(a)$</span>. By hypothesis, we also have <span class... |
4,002,458 | <p>I'm a geometry student. Recently we were doing all kinds of crazy circle stuff, and it occurred to me that I don't know why <span class="math-container">$\pi r^2$</span> is the area of a circle. I mean, how do I <em>really</em> know that's true, aside from just taking my teachers + books at their word?</p>
<p>So I t... | J.G. | 56,861 | <p>In the limit as your mini squares shrink to no size, the area goes from being a sum of their areas to an integral. Consider the circle <span class="math-container">$x^2+y^2=r^2$</span>. In the positive quadrant, one quarter of its area is<span class="math-container">$$\int_0^r\sqrt{r^2-x^2}dx=\int_0^{\pi/2}r^2\cos^2... |
3,578,191 | <p>Without tables or a calculator, find the value of <span class="math-container">$\displaystyle\frac{(\sqrt5 +2)^6 - (\sqrt5 - 2)^6}{8\sqrt5}$</span>.</p>
<p>I do not understand how the positive/negative signs are obtained as shown in the book; is there a formula for expanding these kind of things (what kind of expre... | trancelocation | 467,003 | <p>It might be interesting that you may avoid tedious calculations with roots if you use recurrence relations:</p>
<ul>
<li>Set <span class="math-container">$t_1 =2+\sqrt 5$</span> and <span class="math-container">$t_2 = 2-\sqrt 5$</span>.</li>
</ul>
<p>So, the searched for value is
<span class="math-container">$$\f... |
4,506,151 | <p>Determine all functions <span class="math-container">$f:\mathbb{R} \to \mathbb{R}$</span> such that <span class="math-container">$$f(x f(x+y))+f(f(y) f(x+y))=(x+y)^{2}, \forall x,y \in \mathbb{R} \tag1)$$</span></p>
<p>My approach:
Let <span class="math-container">$x=0$</span>, we get
<span class="math-container">$$... | Bruno B | 1,104,384 | <p>By taking <span class="math-container">$f(y)^2 = y^2$</span> in <span class="math-container">$(3)$</span>, we get:
<span class="math-container">$$\forall y \in \mathbb{R},\quad f(y^2) = y^2$$</span>
Thus: <span class="math-container">$$\forall x \in \mathbb{R_+},\quad f(x) = x$$</span>
Now, let <span class="math-con... |
2,943,790 | <p>A function is said to be <em>continuous at zero</em> iff:</p>
<p><span class="math-container">$\lim_{x \rightarrow 0}{f(x)} = f(0)$</span></p>
<p>Could this be the same as saying:</p>
<ul>
<li>Let <span class="math-container">$\Delta$</span> = <em>the smallest open set containing zero</em></li>
<li><span class="m... | Mohammad Riazi-Kermani | 514,496 | <p>There is no smallest open set containing zero. </p>
<p>Any open set containing zero by definition contains an open interval containing zero and that interval contains a smaller open interval containing zero.</p>
|
2,943,790 | <p>A function is said to be <em>continuous at zero</em> iff:</p>
<p><span class="math-container">$\lim_{x \rightarrow 0}{f(x)} = f(0)$</span></p>
<p>Could this be the same as saying:</p>
<ul>
<li>Let <span class="math-container">$\Delta$</span> = <em>the smallest open set containing zero</em></li>
<li><span class="m... | José Carlos Santos | 446,262 | <p>There is no such thing as “the smallest open set containing <span class="math-container">$0$</span>”. That's so because if <span class="math-container">$A$</span> is an open subset of <span class="math-container">$\mathbb R$</span> (I'm assuming that you're working in <span class="math-container">$\mathb... |
1,715,265 | <p>I've tried a method similar to showing that $\mathbb{Q}(\sqrt2, \sqrt3)$ is a primitive field extension, but the cube root of 2 just makes it a nightmare.</p>
<p>Thanks in advance </p>
| Golan Levy | 664,446 | <p>I think there is a simpler solution.</p>
<p>Let <span class="math-container">$a = \sqrt{2} \sqrt[3]{2}$</span>.</p>
<p><span class="math-container">$a$</span> is clearly in the field extension and both generators can be generated by it:</p>
<p><span class="math-container">$\sqrt{2} = (\frac{a}{2})^{-3}$</span></p... |
3,360,396 | <p>In trying to answer <a href="https://math.stackexchange.com/q/1854193/104041">this question</a> on MSE, I got stuck. This taunts me because I think I should be able to do it.</p>
<h2>The Question:</h2>
<blockquote>
<p>Let <span class="math-container">$\phi : G\twoheadrightarrow H$</span> be an epimorphism of gro... | Robert Shore | 640,080 | <p>Choose <span class="math-container">$x \in \phi^{-1}(W).$</span> Then <span class="math-container">$\phi(x) \in W$</span>, so <span class="math-container">$\exists h \in H \text{ such that } h\phi(x)h^{-1} \in K.$</span> Because <span class="math-container">$\phi$</span> is an epimorphism, <span class="math-contai... |
1,893,540 | <p>I've been asked to prove the following,
if $x - ε ≤ y$ for all $ε>0$ then $x ≤ y$.
I tried proof by contrapositive, but I keep having trouble choosing the right $ε$. Can you guys help me out? </p>
| fleablood | 280,126 | <p>For shits and giggles and the learning experience it brings, here's a proof intended to reinforce your concepts of sup/inf.</p>
<p>$x - \epsilon \le y \forall \epsilon >0$</p>
<p>$x - y \le \epsilon \forall \epsilon > 0$</p>
<p>So $x-y$ is a lower bound for $\{\epsilon > 0\} = (0,\infty) $.</p>
<p>So $x... |
2,913,974 | <p>In an additive category, we say that an object $A$ is compact if the functor $\text{Hom}(A, -)$ respects coproducts. That is, if the canonical morphism
$$
\coprod_{i} \text{Hom} \left( A, X_{i} \right) \longrightarrow \text{Hom} \left( A, \coprod_{i} X_{i} \right)
$$
is a bijection. Suppose $A \oplus B$ is compact. ... | Community | -1 | <p>More generally, a retract of a compact object is compact. Recall that expressing $U$ as a retraction of $V$ is to give morphisms</p>
<p>$$ U \xrightarrow{i} V \xrightarrow{p} U $$</p>
<p>whose composite is the identity on $U$. This is also called a split idempotent, since $ip$ is an idempotent map $V \to V$.</p>
... |
2,435 | <p>I'm not sure we already have something similar, but I'm working on more code inspections for the IntelliJ plugin and it's always a good idea to ask the community. Since it doesn't really fit on main, I'm posting it here on Meta.</p>
<p>Linting is an excellent way to point the developer to probable errors that he mi... | Roman | 26,598 | <blockquote>
<h1>Status Completed</h1>
</blockquote>
<p>I agree with @CE that precedence issues would be great to point out, and maybe suggest to the programmer to use more parentheses. The relative precedences of <code>@</code>, <code>@@</code>, <code>@@@</code>, <code>/@</code>, <code>/.</code>, <code>//.</code>, ... |
2,435 | <p>I'm not sure we already have something similar, but I'm working on more code inspections for the IntelliJ plugin and it's always a good idea to ask the community. Since it doesn't really fit on main, I'm posting it here on Meta.</p>
<p>Linting is an excellent way to point the developer to probable errors that he mi... | user42582 | 42,582 | <p>I'm not sure if this qualifies as a separate claim related to operator precedence. I include it as an answer simply to facilitate the discussion format, even though I believe it should be a comment, instead.</p>
<p>I consistently have had a hard time with <code>expr//f/*g</code> when <code>g</code> is an anonymous ... |
2,300,613 | <p>I tried to calculate few derivatives, but I cant get $f^{(n)}(z)$ from them. Any other way? </p>
<p>$$f(z)=\frac{e^z}{1-z}\text{ at }z_0=0$$</p>
| Simply Beautiful Art | 272,831 | <p>Hint:</p>
<p>$$\frac1{1-z}=\sum_{n=0}^\infty z^n$$</p>
<p>$$e^z=\sum_{n=0}^\infty\frac{z^n}{n!}$$</p>
<p>Now apply <a href="https://en.wikipedia.org/wiki/Cauchy_product#Cauchy_product_of_two_power_series" rel="nofollow noreferrer">Cauchy products</a> to see that</p>
<p>$$\frac{e^z}{1-z}=\sum_{n=0}^\infty z^n\sum... |
957,400 | <p>S: Every employee who is honest and persistent is successful or bored.</p>
<p>Would this statement be the negations, converse, or contrapositive of S?</p>
<p>-> All employees who are dishonest or not persistent must be unsuccessful and not bored.</p>
| amWhy | 9,003 | <p>The altered statement is the converse of the contrapositive of $S$.</p>
<p>Contraposive of $S$: "All employees who are unsuccessful and not bored are dishonest or not persistent."</p>
<p>Converse of the contrapositive of $S$: "All employees who are dishonest or not persistent are unsuccessful and not bored."</p>
|
598,962 | <p>I have to determine the following:</p>
<p>$$\lim_{x \rightarrow 0}\frac{9}{x}\left(\frac{3}{(x+3)^3}-\frac{1}{9}\right)$$</p>
<p>I've got so far:</p>
<p>$$\lim_{x \rightarrow 0}\frac{9}{x}\left(\frac{3}{(x+3)^3}-\frac{1}{9}\right)= \lim_{x \rightarrow 0}\left(\frac{27}{x(x+3)^3}-\frac{1}{x}\right)=\lim_{x \righta... | DeepSea | 101,504 | <p>Let $L$ = the limit in question, then use definition of derivative for $f(x) = \frac{3}{(x + 3)^3}$, then $L = 9f'(0) = 9\left(\frac{-9}{(0 + 3)^4}\right) = -1$.</p>
|
90,070 | <h2>Question:</h2>
<p>Let <span class="math-container">$A\in\mathbb{R}^{n \times n}$</span> be an orthogonal matrix and let <span class="math-container">$\varepsilon>0$</span>. Then does there exist a rational orthogonal matrix <span class="math-container">$B\in\mathbb{R}^{n \times n}$</span> such that <span class="... | Igor Rivin | 11,142 | <p>Yes. It is a theorem of Cayley that the mapping $S \rightarrow (S-I)^{-1}(S+1)$ gives a correspondence between the set of $n\times n$ skew-symmetric matrices over $\mathbb{Q}$ and the set of $n\times n$ orthogonal matrices which do not have one as an eigenvalue. Since the mapping is nice, and rational skew-symmetric... |
4,515,488 | <p>I am making a computer program to play cards, for this algorithm to work I need to deal cards out randomly.
However, I know that some people cannot have some cards due to the rules of the card game.</p>
<p>To elaborate on this, imagine we have 3 players: <em>a</em>, <em>b</em> and <em>c</em>. Also, there are 4 cards... | jorisperrenet | 1,049,661 | <p>So, I've been busy with implementing the algorithm of @kodlu.
Here is my Python code for anyone walking into a similar problem.</p>
<pre><code>import random
random.seed(0)
### Which card each player can receive
poss = [{1, 2, 4}, {1, 2, 3, 4}, {3, 4}]
N = [1, 2, 1]
num_players = len(poss) # in my case: 3 players
... |
3,395,098 | <p>I am trying to work out for what <span class="math-container">$\lambda_1, \lambda_2 > 0$</span> is it true that <span class="math-container">$f(y) = \lambda_1 e^{y-\lambda_1 e^y} + \lambda_2 e^{y-\lambda_2 e^y}$</span> is unimodal?</p>
<p>Experimentally it seems it is unimodal when <span class="math-container">$... | River Li | 584,414 | <p><strong>The proof is not hard but long</strong>. </p>
<p>Let <span class="math-container">$w = \lambda_1 \mathrm{e}^y$</span> and <span class="math-container">$c = \frac{\lambda_2}{\lambda_1}$</span>.
Let <span class="math-container">$$g(w) = w\mathrm{e}^{-w} + cw \mathrm{e}^{-cw}.$$</span></p>
<p>We first give th... |
1,081,021 | <p>In what follows I'm only considering positive real valued functions.</p>
<p>Everywhere I look about the definition of the Lebesgue integral it is required to consider a measurable function. Why do we not define the integral for non-measurable functions? From what I see we require measurablility of the simple functi... | Darrin | 117,053 | <p>In addition to the previous advice, note that the function you gave does not "approximate" $f$. An approximating sequence $\{s_n\}$ for $f$ (which $f$ would have iff $f$ were measurable provided the measure space for the domain of $f$ is complete) would need to be within a distance of $\epsilon$ from $f$ for any giv... |
434,290 | <p>According to the <a href="http://arxiv.org/abs/0910.5922" rel="nofollow">equation 4</a>,
$$\phi(0,t)= \frac{A_0}{(1+\frac{2t^2}{R^4})^{3/4}}\cos \left(\sqrt2 t+ \frac{3}{2}\tan^{-1}\left[\frac{\sqrt2 t}{R^2}\right]\right)\tag{1}$$
what conditions makes, $$\cos \left(\sqrt2 t+ \frac{3}{2}\tan^{-1}\left[\frac{\sqrt2 ... | xpaul | 66,420 | <p>We can use the following way to solve. It is very simple. In fact
\begin{eqnarray}
I&=&\int_0^\infty \frac{\ln x}{1+x^4}dx=\int_0^1 \frac{\ln x}{1+x^4}dx+\int_1^\infty \frac{\ln x}{1+x^4}dx\\
&=&\int_0^1 \frac{\ln x}{1+x^4}dx-\int_0^1 \frac{x^2\ln x}{1+x^4}dx\tag{1}\\
&=&\int_0^1\sum_{n=0}^\i... |
680,205 | <p>Milnor lemma 2 pg 34
"Any orientation preserving diffeomorphism f on $R^m$ is smoothly homotopic to the identity"</p>
<p>So he proves that $f\simeq df_0$ ,which he says is clearly homotopic to the identity.
Can you explain me why?</p>
<p>Here I found two explanations I don't understand:
1) $Gl^{+}(m,\mathbb{R})$ i... | Wisław | 78,540 | <p>This is not really an answer to your specific question, but another way of proving the lemma.</p>
<p>Since $f$ is an orientation preserving diffeomorphism on $\mathbb R^m$ it must have degree 1. By the Hopf degree theorem (p. 51 in Milnor), it is smoothly homotopic to the identity.</p>
|
945,104 | <p>7 people are attending a concert.</p>
<p>(a) In how many different ways can they be seated in a row?</p>
<p>(b) Two attendees are Alice and Bob. What is the probability that Alice sits next to Bob?</p>
<p>(c) Bob decides to make Alice a rainbow necklace with 7 beads, each painted a different
colour on one side (r... | André Nicolas | 6,312 | <p>For Question (b), we want to count the number of seatings in which Alice and Bob are neighbours. We are assuming the seating is random. That may not be reasonable, if we consider Bob's actions described in (c).</p>
<p>The leftmost of the two seats occupied by our two heroes can be chosen in $6$ ways. For each such ... |
213,198 | <p>Given the expression:</p>
<pre><code>Simplify[Reduce[Exists[{x},a x^2+b x+ c==0],x,Reals]]
</code></pre>
<p>The answer comes out as:</p>
<pre><code>(b==0 && ((c>0 && a<0)||(a>0 && c<0)))||(b!=0 && 4 a c<=b^2)||c==0
</code></pre>
<p>However, surely this is just the s... | Carl Woll | 45,431 | <p>The first predicate is false for a->0, b->0, c->1:</p>
<pre><code>(b==0 && ((c>0 && a<0)||(a>0 && c<0)))||(b!=0 && 4 a c<=b^2)||c==0 /. {a->0, b->0, c->1}
</code></pre>
<blockquote>
<p>False</p>
</blockquote>
<p>The second predicate is true for this case:<... |
213,198 | <p>Given the expression:</p>
<pre><code>Simplify[Reduce[Exists[{x},a x^2+b x+ c==0],x,Reals]]
</code></pre>
<p>The answer comes out as:</p>
<pre><code>(b==0 && ((c>0 && a<0)||(a>0 && c<0)))||(b!=0 && 4 a c<=b^2)||c==0
</code></pre>
<p>However, surely this is just the s... | user13892 | 13,892 | <p><code>Exists</code> and <code>ForAll</code> are qualifier statements and you can attempt to <code>Resolve</code> them to remove the qualifiers as follows:</p>
<pre><code>qualifierStatement=Exists[x,a x^2+b x+c==0]
</code></pre>
<p>Now to resolve it under the real domain as follows:</p>
<pre><code>resolvedStatemen... |
815,195 | <p>I am working on an old qualifying exam problem and I can't seem to really get anywhere. I would love some help. Thank you.</p>
<p>Let $f$ be a polynomial such that
$|f(z)| ≤ 1 − |z|^2 + |z|^{1000}$
for all $z ∈ C.$ Prove that $|f(0)| ≤ 0.2.$</p>
| Umberto P. | 67,536 | <p>The minimum value of $\phi(t) = 1 - t^2 + t^{1000}$ on the set $[0,\infty)$ is found easily enough: since $\phi'(t) = -2t + 1000 t^{999} = -2t(1-500t^{998})$ the minimum occurs at $t_0 = \sqrt[998]{1/500}$. Thus if $|z| = t_0$, we have
$$|f(z)| \le 1 - t_0^2 + t_0^{1000}$$
which is (computation omitted thanks to Wol... |
1,447,547 | <p>$$x^3>x$$</p>
<p>Steps I took:</p>
<p>$$x^{ 3 }-x>0$$</p>
<p>$$x(x^{ 2 }-1)>0$$</p>
<p>$$x(x-1)(x+1)>0$$</p>
<p>Now I see that all three linear factors must equal a positive value when multiplied. </p>
<p>I took each linear factor and set each to either greater than or less than $0$ since the solut... | Curious | 141,191 | <p>By solving corresponding equality, we get 3 values -1,0,1. This will give us 4 intervals $(-\infty,-1), (-1,0), (0,1), (1, \infty)$. By checking on this intervals, the inequality is satisfied for intervals $(1,\infty)$ and $(-1,0)$</p>
|
2,904,912 | <p>$$24a(n)=26a(n-1)-9a(n-2)+a(n-3)$$
$$a(0)=46, a(1)=8, a(2)=1$$
$$\sum\limits_{k=3}^{\infty}a(k)=2^{-55}$$
How can I prove it?</p>
| lhf | 589 | <p>Let $S=\sum\limits_{k=3}^{\infty}a(k)$. Then
$$
24S = 26(S+a(2))-9(S+a(1)+a(2))+(S+a(0)+a(1)+a(2))
$$
This gives $S=0$.</p>
|
1,178,080 | <p>How to calculate the number of solutions of the equation $x_1 + x_2 + x_3 = 9$ when $x_1$, $x_2$ and $x_3$ are integers which can only range from <code>1</code> to <code>6</code>.</p>
| AvZ | 171,387 | <p>We can find the number of solutions using binomial theorem.<br>
The coefficient of $x^9$ in the following will be the required answer.
$$(x+x^2+\cdots+x^6)^3$$
This above, is a Geometric Progression. Therefore,
$$=\left (\frac{x-x^7}{1-x}\right )^3$$
$$=(x-x^7)^3(1-x)^{-3}$$
Now apply binomial theorem to get the coe... |
1,178,080 | <p>How to calculate the number of solutions of the equation $x_1 + x_2 + x_3 = 9$ when $x_1$, $x_2$ and $x_3$ are integers which can only range from <code>1</code> to <code>6</code>.</p>
| Lorence | 220,813 | <p>There are six situations.
126,135,144,225,234 and 333,it's easy to know there is no other situation.
So,we can add each of them together using permutation theorem.
$$P_3^3+P_3^3+\frac{P_3^3}{P_2^2}+\frac{P_3^3}{P_2^2}+P_3^3+\frac{P_3^3}{P_3^3}=25$$</p>
|
563,499 | <p>What's the summation of the following expression;</p>
<p>$$\sum_{k=1}^{n+3}\left(\frac{1}{2}\right)^{k}\left(\frac{1}{4}\right)^{n-k}$$
The solution is said to $$2\left(\frac{1}{4} \right)^{n}\left(2^{n+3}-1\right)$$</p>
<p>But I'm getting $$\left(\frac{1}{4} \right)^{n}\left(2^{n+3}-1\right).$$
How is this possi... | Steve Kass | 60,500 | <p>Let $n$=0. The sum is then $\sum_{k=1}^{3}\left(\frac{1}{2}\right)^{k}\left(\frac{1}{4}\right)^{-k}= \frac{1}{2}\cdot4+\frac{1}{4}\cdot16+\frac{1}{8}\cdot64=2+4+8=14$. This equals $2\left(\frac{1}{4} \right)^{0}\left(2^{0+3}-1\right)=2\cdot 7$, and it does not equal $\left(\frac{1}{4} \right)^{n}\left(2^{n+3}-1\righ... |
563,499 | <p>What's the summation of the following expression;</p>
<p>$$\sum_{k=1}^{n+3}\left(\frac{1}{2}\right)^{k}\left(\frac{1}{4}\right)^{n-k}$$
The solution is said to $$2\left(\frac{1}{4} \right)^{n}\left(2^{n+3}-1\right)$$</p>
<p>But I'm getting $$\left(\frac{1}{4} \right)^{n}\left(2^{n+3}-1\right).$$
How is this possi... | Steve Kass | 60,500 | <p>The formula $$\sum_{k=1}^{n}ar^{k}=a\left(\frac{1-r^{n}}{1-r}\right)$$ is incorrect. The correct formula is $$\sum_{{\Large k=}{\Huge0}}^{\Huge n-1}ar^{k}=a\left(\frac{1-r^{n}}{1-r}\right)$$ or $$\sum_{k=1}^{n}ar^{k}=a{\Large{r}}\left(\frac{1-r^{n}}{1-r}\right)$$. With $a=1$ and $r=2$, you should have $\sum_{k=1}^{n... |
3,156,643 | <blockquote>
<p>Prove that <span class="math-container">$\sin(x) < x$</span> when <span class="math-container">$0<x<2\pi.$</span></p>
</blockquote>
<p>I have been struggling on this problem for quite some time and I do not understand some parts of the problem. I am supposed to use rolles theorem and Mean v... | little o | 543,867 | <p>Let <span class="math-container">$f(x) = \sin x, x \in \Bbb R.$</span> Let <span class="math-container">$x \in (0,1)$</span> then using Lagrange's MVT on the interval <span class="math-container">$[0,x]$</span> we get <span class="math-container">$$\frac {f(x)-f(0)} {x-0} = f'(c)$$</span> where <span class="math-co... |
555,239 | <p>Since the polynomial has three irrational roots, I don't know how to solve the equation with familiar ways to solve the similar question. Could anyone answer the question?</p>
| Robert Israel | 8,508 | <p>These are the equations of (fairly small) ellipses in the $x-y$ plane. Plot and count.</p>
|
1,878,573 | <p><a href="https://i.stack.imgur.com/3iZQ8.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/3iZQ8.png" alt="enter image description here"></a></p>
<p>I cannot get the $f'(0)$ by using L'Hôpital's rule, because it appears recurrence item. Can you help me?</p>
| Barry Cipra | 86,747 | <p>By the definition of the derivative, a function $f$ is differentiable at $0$ if and only if the limit</p>
<p>$$\lim_{x\to0}{f(x)-f(0)\over x}$$</p>
<p>exists. In this case, $f(0)$ is <em>defined</em> to be $0$, so the question of differentiability boils down to examining</p>
<p>$$\lim_{x\to0}{(e^{x^2}-e^{-x^2})\... |
3,861,324 | <p>Given a nonzero column vector <span class="math-container">$A$</span>=<span class="math-container">$[a_1 a_2.......a_n]^T$</span>. Find the non zero eigen values and eigen vectors for <span class="math-container">$A$$A^T$</span>.</p>
<p>I have no idea.what theorem should I apply or what I have to do to solve this. ... | Brian Fitzpatrick | 56,960 | <p>Suppose <span class="math-container">$\boldsymbol{v},\boldsymbol{w}\in\Bbb R^n$</span> are represented as column vectors. Recall that the <em>dot product</em> of <span class="math-container">$\boldsymbol{v}$</span> and <span class="math-container">$\boldsymbol{w}$</span> may be written as <span class="math-containe... |
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