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 |
|---|---|---|---|---|
30,443 | <p>My aim is to plot a vector field of the following system with a few trajectories:</p>
<p>$$r'(t)=i-l.r(t)-\text{ux}. r(t). x(t)-\text{uy}. r(t). y(t) \\ x'(t)=\text{ex}. \text{ux}. r(t). x(t)-\text{mx}. x(t)\\y'(t)=\text{ey}.\text{uy}. r(t). y(t)-\text{my}. y(t)$$</p>
<p>$i,l,ux,uy,mx,my,ex,ey$ are parameters.</p... | Nasser | 70 | <p>You can't use <code>VectorPlot3D</code> if the independent variable is the same. <code>t</code> in your case. You'll get errors such as </p>
<pre><code>VectorPlot3D::glims: Range specifications {t,0,100} and {t,0,100} \
contain the same iteration variable.
</code></pre>
<p>So, without knowing what you are trying ... |
2,391,931 | <p>The rate of data transfer, $r$, over a particular network is directly proportional to the bandwidth, $b$, and inversely proportional to the square of the number of networked computers, $n$.</p>
<p><strong>Quantity A</strong> = The resulting rate of data transfer if the bandwidth is quadrupled and the number of net... | user345 | 395,145 | <p><strong>Quantity B is greater</strong> I would think about this in terms of direct and inverse relationships. If there is a direct relationship with bandwidth, then increasing bandwidth, will increase speed. And inverse relationship means that increasing the computers decreases the speed.</p>
<p>You are close to th... |
287,976 | <p>I want to prove that the sum of the fourth powers of the diagonals of a regular $n$-gon inscribed in the unit circle is equal to $6n$. I consider the distance from 1 to the other $n$th roots of unity given by $\omega^k$, $k=1,2,\dots, (n-1)$. So basically my working is
$$\sum_1^{n-1}|1-\omega^k|^4=\sum_1^{n-1}(|1-\o... | Gerry Myerson | 8,269 | <p>If $n$ is odd, then $\omega^2$ is also a primitive $n$th root of unity. If $n=2m$, then $\omega^2$ is a primitive $m$th root of unity. In either case, you can use formulas you already know. </p>
|
1,662,958 | <blockquote>
<p>Say Bob tosses his $n+1$ fair coins and Alice tosses her $n$ fair coins. Lets assume independent coin tosses. Now after all the $2n+1$ coin tosses one wants to know the probability that Bob has gotten more heads than Alice. </p>
</blockquote>
<p>The way I thought of it is this : if Bob gets $0$ heads... | lulu | 252,071 | <p>the answer is indeed $\frac 12$ .</p>
<p>As an alternative way to see that: let's pause just before Bob tosses his final (extra) toss. At this point, there are three possible states: either Bob is ahead, Alice is ahead, or they are tied. Let $p$ be the probability that Bob is ahead. By symmetry, $p$ is also th... |
1,662,958 | <blockquote>
<p>Say Bob tosses his $n+1$ fair coins and Alice tosses her $n$ fair coins. Lets assume independent coin tosses. Now after all the $2n+1$ coin tosses one wants to know the probability that Bob has gotten more heads than Alice. </p>
</blockquote>
<p>The way I thought of it is this : if Bob gets $0$ heads... | Nebojsa Gogic | 597,950 | <p>Bob wins if:
1) Bob and Alice have equal number of heads and Bob tosses his last coin and gets head
2) Bob is ahead after tossing <span class="math-container">$10$</span> coins</p>
<ul>
<li>Probability of the first event is <span class="math-container">$1/2 \cdot 1/2 = 1/4$</span></li>
<li>Probability of the second... |
1,753,901 | <p>The question says:
Let $U$ be a set $ \{ 1, 2, . . . , n \}$ for an arbitrary positive integer $n$. How many subsets are there? How many possible relations of the form $A \subseteq B$ are there? Can you make an informed guess as to how many of these relations are true?</p>
<p>Since the given set has $n$ elements, t... | almagest | 172,006 | <p>If the subset $A$ has $k$ elements, then there are $2^{n-k}$ subsets $B$ containing $A$ (because each of the remaining $n-k$ elements can be in or out). There are ${n\choose k}$ subsets with $k$ elements, so the total number of possibilities is $\sum_{k=0}^n{n\choose k}2^{n-k}=(1+2)^n=3^n$ by the binomial theorem.</... |
222,759 | <p>Here is the given series 3/(9n+1), decide whether it converges or diverges.
I used the ratio test only to end up with the ratio=1.
I know this is harmonic series but it is smaller than 1/n, therefore i cannot conclude it diverges.
Please help!!</p>
| david | 46,315 | <p>$$ \frac{3}{9n+1} \ge
\frac{1}{9n+1} \ge
\frac{1}{9n+9} \ge
\frac{1}{9(n+1)} \ge
\frac{1}{9} \frac{1}{n+1}
$$ </p>
<p>At which point you should be able to figure that out...</p>
|
222,759 | <p>Here is the given series 3/(9n+1), decide whether it converges or diverges.
I used the ratio test only to end up with the ratio=1.
I know this is harmonic series but it is smaller than 1/n, therefore i cannot conclude it diverges.
Please help!!</p>
| blindman | 35,965 | <p>We have
$$
\frac{3}{9n+1}\geq \frac{3}{9n+9}=\frac{1}{3(n+1)}.
$$
Since $\displaystyle\sum_{n=1}^{\infty}\frac{1}{3(n+1)}$ is disvergent, $\displaystyle\sum_{n=1}^{\infty}\frac{3}{9n+1}$.</p>
|
4,187,052 | <p>I have a problem that looks like a typical problem of maximizing functions in a compact interval. However, I am not being able to prove the bound I need.</p>
<blockquote>
<p>Let <span class="math-container">$n\geq 6$</span> be an integer number. Consider the function: <span class="math-container">$$f(t) = \frac{n^2}... | Andrew D. Hwang | 86,418 | <p>In case a less-clever estimate by separation of cases is useful:</p>
<p>Our goal is to show
<span class="math-container">$$
\frac{n^{2}}{2}\, t^{n-4}(1 - t^{2}) \left(t^{2} - \frac{n-3}{n}\right)
= \frac{n^{2}}{2}\, t^{n-4}(1 - t^{2}) \left(\frac{3}{n} - (1 - t^{2}\right)
\leq 1
$$</span>
for <span class="math-conta... |
1,882,401 | <p>Let $T$ be a linear operator on a finite dimensional vector space $V$ with dimension $n$. Then, is it necessary that $T$ has $n$ eigen values or could it be less than $n$?</p>
| Landon Carter | 136,523 | <p>Depends on your field. For an algebraically closed field, all eigenvalues can be found and the linear transformation will have $n$ eigenvalues (of course, including multiplicity). But you may not be able to "find" all eigenvalues, if your field is, say $\mathbb R$.</p>
|
312,249 | <p>I am attempting to solve an equation ${{n-2} \choose {2}} + {{n-3} \choose {2}} + {{n-4} \choose {2}} = 136$. With the formula for a combination being $\frac{n!}{r!(n - r)!}$, I simplified the given equation to:</p>
<p>$(n-2)! + (n-3)! + (n-4)! = 272n - 1088$ </p>
<p>However, I am not sure how I would solve for $\... | Community | -1 | <p>Note that
$$\dbinom{n-2}2 = \dfrac{(n-2)(n-3)}{2}$$
$$\dbinom{n-3}2 = \dfrac{(n-3)(n-4)}{2}$$
$$\dbinom{n-4}2 = \dfrac{(n-4)(n-5)}{2}$$
Hence,
$$\dbinom{n-2}2 + \dbinom{n-3}2 + \dbinom{n-4}2 = \dfrac{(n-2)(n-3)}{2} + \dfrac{(n-3)(n-4)}{2} + \dfrac{(n-4)(n-5)}{2}$$
Now solve the quadratic to get the answer.</p>
<p><... |
4,321,742 | <p>In Spivak's Calculus, Ch. 5 on Limits, there is the following theorem about the uniqueness of a limit of a function near a point:</p>
<blockquote>
<p>A function cannot approach two different limits near <span class="math-container">$a$</span>. In other
words, if <span class="math-container">$f$</span> approaches <sp... | Ricardo770 | 238,708 | <p>Recall the following two results:</p>
<p><span class="math-container">$$\int_0^\infty e^{-bt}\cos\left(xt \right)\,dt=\frac{b}{b^2+x^2}\tag{1}$$</span></p>
<p><span class="math-container">$$\int_0^\infty \frac{\cos\left(xt \right)}{\cosh\left(ax \right)}\,dx=\frac{\pi} {2a \cosh\left(\frac{\pi t}{2a} \right)} \tag... |
4,612,472 | <blockquote>
<p>Five friends <span class="math-container">$-\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}$</span> and <span class="math-container">$\mathrm{T}-$</span> go to a movie and sit next to each other in a row. <span class="math-container">$P$</span> can only sit in <span class="math-container">$k$</span> of t... | Alan Chung | 305,824 | <p>Try <span class="math-container">$$\cos\left( x + \pi/4 \right) = \frac{\sqrt{2}}{2} \cos(x) - \frac{\sqrt{2}}{2} \sin(x).$$</span></p>
<p>EDIT: my bad for the error</p>
|
3,409,012 | <p><a href="https://i.stack.imgur.com/CqTOf.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/CqTOf.png" alt="enter image description here"></a></p>
<p>For example, I'm asked to find the line integral of <span class="math-container">$C_1$</span> above for a vector field <span class="math-container">$\... | PrincessEev | 597,568 | <p>Your parameterization differs in that, for all intents and purposes, it's "backwards."</p>
<p>Plug in <span class="math-container">$t=0$</span>, the "start" point. Where your parameterization starts incorrectly at <span class="math-container">$2 \vec j$</span> in that parameterization, the suggested one correctly s... |
79,863 | <p>I've encountered this problem on my Non commutative algebra handouts wich says:</p>
<p>given $R,S$ rings and $f:R\to S\:$ a ring homomorphism, define a canonical functor $$F:\textbf{Mod-S}\to \textbf{Mod-R}.$$ Where $\textbf{Mod-S}$ is the category of right $S-$modules and similarly $\textbf{Mod-R}$ is the category... | Dan Petersen | 677 | <p>Actually there are canonical functors in both directions. To go from $R$-modules to $S$-modules, you need to use the tensor product (and this functor is not in general faithful), but the other direction (which is suggested in the homework) is easier. </p>
<p>One way to see what the construction is to note that an $... |
1,843,153 | <p>In the Plato's dialogue "Theaetetus", at a certain point, we have the following "problem"
\begin{align*}
5040 &= 7! \\
&= 1\times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \\
&= 2 \times 3 \times 2 \times 2 \times 5 \times 2 \times 3 \times 7 \\
&= 2^4 \times 3^2 \times 5 \times 7 \\
&= 2^... | Ashwin Ganesan | 157,927 | <p>There is a basic counting principle called the <em>product rule</em>, which says the following: in an experiment consisting of two steps, if the number of ways to do the first step is $n_1$, and the number of ways to do the second step is $n_2$ for each of the $n_1$ outcomes in the first step, then the total number ... |
3,282,719 | <p>In Javascript, the largest integer that can be represented exactly is <code>Number.MAX_SAFE_INTEGER</code>, with a value of <span class="math-container">$ 2^{53} - 1$</span>. What is the largest prime value that fits under this value threshold? I cannot find a suitable reference for this value on the internet.</p>
| Robert Soupe | 149,436 | <p>In <a href="https://www.wolframalpha.com/" rel="nofollow noreferrer">Wolfram Alpha</a>, you can type in <code>NextPrime[2^53 - 1, -1]</code> and it will give you the answer: 9007199254740881.</p>
<p>It's of course not a JavaScript runtime engine that comes up with this answer; the Wolfram Alpha server sends your qu... |
1,394,088 | <blockquote>
<p>What is the smallest possible natural number $$ for which the equation $x^{2}-nx+2014=0$ has integer roots?</p>
</blockquote>
<p>My idea was, If the roots are integers, then they are the divisors of $2014$, I don't know if it's true or not.</p>
| DeepSea | 101,504 | <p>We have: $\triangle = b^2-4ac = n^2 - 4(2014) = k^2 \to (n-k)(n+k) = 4\times 2014$. From this we can list all possible values of $n$, and select the smallest candidate.</p>
|
4,768 | <p>How we can show every permutation is either even or odd,but not both......I can't arrive at a proof for this ..... Can anybody give me the proof...</p>
<p>Thanks in advance...</p>
| Ted | 15,012 | <p>It is enough to show that the product of an odd number of transpositions cannot be the identity.</p>
<p>Every permutation of a finite set $S$ is a unique product of disjoint cycles in which every element of $S$ occurs exactly once (where we include fixed points as 1-cycles). Let $p$ be any permutation of $S$, let ... |
1,689,892 | <p>I understand how to check if something is a tensor (if it transforms like a tensor, it's a tensor).</p>
<p>I have this example of a rank 2 tensor: $$\tau_{ij}=r_if_j - r_jf_i$$
and I understand that it is a tensor because $$\sum_j\tau_{ij}e_j$$ is a vector (e is some vector). </p>
<p>What I'm trying to find is a m... | Mikhail Katz | 72,694 | <p>A typical example of an object that fails to behave like a tensor is the $\Gamma_{ij}^k$. If you prefer to have only two indices just contract one of them and use $\Gamma_{ij}^k v^i$ where $v$ is a vector.</p>
|
3,749,548 | <blockquote>
<p>Calculate:
<span class="math-container">$$\frac{d}{dx}(\cos(\sin(\cos(\sin(...(\cos(x)))))))$$</span></p>
</blockquote>
<p>This looks kind of daunting but I decided to see what happens to the derivative for a section of the function. If I consider:</p>
<p><span class="math-container">$$\frac{d}{dx}(\cos... | Chris Culter | 87,023 | <p>The equation <span class="math-container">$\cos(\sin(t)) = t$</span> has a single solution, <span class="math-container">$t=0.768\ldots$</span>. So if by <span class="math-container">$\cos(\sin(\cos(\sin(...(\cos(x))))))$</span> you mean this constant, then the derivative is the derivative of a constant, <span class... |
1,265,200 | <p>If a particle performs a random walk on the vertices of a cube, what is the mean number of steps before it returns to the starting vertex S? What is the mean number of visits to the opposite vertex T to S before its first return to S and what is the mean number of steps before its first visit to T?</p>
<p>Nobody ev... | Dan | 97,250 | <p>The vertices of a cube(oid) can be labelled with 3-digit binary numbers since there are $8$ corners and $2^3=8$. To keep things simple lets say that we are considering the unit cube and the binary number correspond directly to the coordinates, so that vertce $000$ is at $(0,0,0)$, vertex $010$ is at $(0,1,0)$ and so... |
666,821 | <p>Here is a standard identity:</p>
<p>$$\sum_{i=0}^{\infty}\frac{a^i}{i!}=e^a$$</p>
<p>Why does it hold true?</p>
| michael straws | 124,038 | <p>The Taylor series for $e^x$ is</p>
<p>$$
\sum_{k = 0}^{\infty} \frac{x^k}{k!}.
$$</p>
<p>Letting $x = a$, we obtain</p>
<p>$$
e^a = \sum_{k = 0}^{\infty} \frac{a^k}{k!}
$$</p>
<p>as desired.</p>
|
3,405,035 | <p>I have been reading Spivak's Introduction to Differential Geometry and I cannot get this. I have already seen other proofs of the following statement, and I get those, I just do not follow this one.</p>
<blockquote>
<p><span class="math-container">$d\theta$</span> is closed but it is not exact</p>
</blockquote>
<p>I... | Travis Willse | 155,629 | <p>This problem illustrates that <span class="math-container">$d\theta$</span> is a particularly misleading and hence poor notation for this <span class="math-container">$1$</span>-form: It isn't the exterior derivative of anything!</p>
<p>(More generally the common notation <span class="math-container">$dV$</span> fo... |
293,765 | <p>Find the local minima and maxima of function:</p>
<p>$$f(x,y) = x^2-2x+y^2$$</p>
<p>It's easy task with one-variable functions. What should I go about in this case? </p>
| Mikasa | 8,581 | <p>Hint: First of all find the critical points by doing : $$f_x=0,~f_y=0$$ Assume $(a,b)$ is such that oint. Now find the following terms: $$\Delta_1=f_{xx},~~\Delta_2=f_{xx}f_{yy}-f_{xy}$$ Now if $$\Delta_1|_{(a,b)}>0,~\Delta_2|_{(a,b)}>0$$ then $(a,b)$ will make $f$ minimum. If $$\Delta_1|_{(a,b)}<0,~\Delta... |
293,765 | <p>Find the local minima and maxima of function:</p>
<p>$$f(x,y) = x^2-2x+y^2$$</p>
<p>It's easy task with one-variable functions. What should I go about in this case? </p>
| Inquest | 35,001 | <p>\begin{align}
f(x,y)&=x^2 - 2x+y^2\\
\nabla f(x,y)&=\begin{bmatrix}
2x-2\\2y
\end{bmatrix}\\
\nabla^2f(x,y)&=\begin{bmatrix}
2&0\\0&2\end{bmatrix}\\
\text{Set : }\nabla f(x,y)&=0\\
\implies (x,y)&=(1,0)
\end{align}
This is <strong>Minima</strong>. (Hessian is Positive Definite)</p>
<p>Th... |
3,960,527 | <p>Let <span class="math-container">$X$</span> be a topological space and <span class="math-container">$A\subset X$</span>. Show that <span class="math-container">$X = int(A) \cup Fr(A)\cup int(X-A)$</span>, this being a union disjointed.</p>
<p>To show this equality I must show the inclusions:
<span class="math-contai... | Ottavio | 280,734 | <p>Hint: <span class="math-container">$\operatorname{Int}(A)\sqcup\operatorname{Fr(A)}=\operatorname{cl}(A)$</span>, which is (by definition? I guess it depends on your definition of closure and boundary of a subset) the smallest closed subset of <span class="math-container">$X$</span> containing <span class="math-cont... |
368,117 | <blockquote>
<p>Prove that sign$(\sigma \tau)$ = sign$(\sigma)$sign$(\tau)$ for any permutations $\sigma, \tau \in S_n$.</p>
</blockquote>
<p>I think the two thing's I'm trying to show are:</p>
<ul>
<li>If sign$(\sigma)$ = sign$(\tau) = \pm 1 \implies$ sign$(\sigma \tau)$ = $1$</li>
<li>Wlog, if sign$(\sigma) = 1$,... | shobon | 73,393 | <p>sign is defined as 1 or -1 depending on whether the number of transpositions you can write a permutation in is even or odd.</p>
<p>if $\sigma$ is written as $k$ transpositions and $\tau$ is written as $t$ then $\sigma \tau$ is $k+t$ transpositions.</p>
<p>This proves that $\text{sgn}(\sigma)\text{sgn}(\tau)=\text{... |
368,117 | <blockquote>
<p>Prove that sign$(\sigma \tau)$ = sign$(\sigma)$sign$(\tau)$ for any permutations $\sigma, \tau \in S_n$.</p>
</blockquote>
<p>I think the two thing's I'm trying to show are:</p>
<ul>
<li>If sign$(\sigma)$ = sign$(\tau) = \pm 1 \implies$ sign$(\sigma \tau)$ = $1$</li>
<li>Wlog, if sign$(\sigma) = 1$,... | Community | -1 | <p>Alternatively you can show that $\operatorname{sgn}$ factors through the group of $n \times n$ permutation matrices using the determinant map. Since $\det$ is a homomorphism we get the result.</p>
|
5,238 | <p>For my homework, I have been asked to rationalise and simplify this surd;</p>
<p><span class="math-container">$$\frac{11}{3\sqrt{3}+7}$$</span></p>
<p>Each time I do this I get the wrong answer. The method I am using is;</p>
<p><span class="math-container">$$ \frac{11}{3\sqrt3+7} \times \frac{3\sqrt3-7}{3\sqrt3-7... | Robin Chapman | 226 | <p>After multiplying the numerator and denominator by $3\sqrt3-7$
the new denominator is
$$(3\sqrt 3+7)(3\sqrt3-7)=(3\sqrt3)^2-7^2=27-49=-22$$
a nice integer to divide by.</p>
|
1,655,992 | <p>Give an example of a equicontinuous sequence of functions ($f_n$) over a non-compact set $S\subset\Bbb R^n$ converging pointwise to a function $f$ at each $x\in S$, but $f_n$ does not converge uniformly to $f$ over $S$.</p>
<p>I'm really stuck on this problem, and I thought about the cases of $f_n(x) = x^n$ with th... | Arthur | 15,500 | <p>Take the functions $f_n(x)=x/n$ over $\Bbb R$, converging non-uniformly to $f(x)=0$.</p>
|
2,714,756 | <p>Let $M$ be a smooth manifold and $J$ be an integrable almost complex structure on $M$. Let $f: M\to M$ be a diffeomorphism with $f_{*}:TM\to TM$ its tangent map. Then it is easy to see that $f_{*}Jf_{*}^{-1}$ is a new almost complex structure. </p>
<p>Question: Is $f_{*}Jf_{*}^{-1}$ an integrable almost complex str... | Wei Xia | 42,386 | <p>Note the following two interesting phenomena:</p>
<p>1.$f: (M,J)\to (M,f_*Jf_*^{-1})$ is a biholomorphic map but $f : (M, J) \to (M, J)$ may be not a biholomorphic map.</p>
<p>2.$Id: (M,J)\to (M,f_*Jf_*^{-1})$ may be not a biholomorphic map but $Id : (M, J) \to (M, J)$ is a biholomorphic map.</p>
<p>The point is ... |
1,868,755 | <p>I came across this in a set of notes. </p>
<blockquote>
<p>Let $K$ be a field of characteristic $p$ and let $\lambda\in K$. Then $$\lambda^{p-1}=1.$$ </p>
</blockquote>
<p>I've never seen this before. Is it correct?</p>
| Community | -1 | <p>The relevant fact here is that, for any field $F$ of characteristic $p$, there is a unique field homomorphism $\mathbf{F}_p \to F$, and that the nonzero elements of (the image of) $\mathbf{F}_p$ are precisely the $(p-1)$-th roots of unity in $F$.</p>
<p>$\mathbf{F}_p$ means the field of $p$ elements, which is isomo... |
221,279 | <blockquote>
<p>How does one show that $\chi_{[1, \infty)}1/x$ is not (Lebesgue) integrable?</p>
</blockquote>
<hr>
<p>What I could think of is as follows:</p>
<p>Letting $f(x)=1/x$ (defined for $x\geq 1$), define
$$
f_n(x)=f\chi_{[1, n)}(x).
$$ </p>
<p>Each $f_n$ is, therefore, Riemann integrable on $[1, n)$ wi... | kahen | 1,269 | <p>For additional insight, you should try to prove the following:</p>
<blockquote>
<p><strong>Proposition</strong>. If $(X,\mu)$ is an infinite measure space and $f \in \mathcal L_1(X,\mu)$ is positive, then $\displaystyle \int \frac1f\,d\mu = \infty$.</p>
</blockquote>
<p>Hint: $1 < f + \dfrac 1f$.</p>
|
221,279 | <blockquote>
<p>How does one show that $\chi_{[1, \infty)}1/x$ is not (Lebesgue) integrable?</p>
</blockquote>
<hr>
<p>What I could think of is as follows:</p>
<p>Letting $f(x)=1/x$ (defined for $x\geq 1$), define
$$
f_n(x)=f\chi_{[1, n)}(x).
$$ </p>
<p>Each $f_n$ is, therefore, Riemann integrable on $[1, n)$ wi... | AJY | 192,914 | <p>So we have that $\int f$ is the supremum of the integrals of simple functions of finite support no more than $f$. Let $\phi_{n} = \sum_{k = 1}^{n} \chi_{[k, k + 1)} / (k + 1)$. Then $\phi_{n}$ has measure $n$ and integral $\sum_{k = 1}^{n} 1/(k + 1)$. Thus $\int f \geq \sum_{k = 1}^{n} 1/(k + 1)$ for all $n$, so we ... |
300,181 | <p>If we have $n$ different numbers from the set $\mathbb N$ what is the maximum possible number of numbers that we can contruct from these numbers by performing $m$ successive operations, where operation is addition or multiplication? To be more precise about the problem I will clarify it further with some examples, ... | Ross Millikan | 1,827 | <p>A simple upper bound is that you choose a variable to start with ($n$ choices), then each operation gives $2n$ possibilities, as you can choose $n$ different variables and $2$ operations, so the total is $n(2n)^n$. The hard part is counting the number that must be equal due to commutivity. If you choose the $x_i$ ... |
4,645,763 | <p>Recently I played a little bit around with GeoGebra and I constructed the in- and circumcircle of a <span class="math-container">$\triangle ABC$</span> with <span class="math-container">$A=(0,0)$</span> and <span class="math-container">$B=(1,0)$</span> and I asked myself if it is possible to construct the area where... | Parcly Taxel | 357,390 | <p>I will first find a condition the side lengths <span class="math-container">$a,b,c$</span> must satisfy for the circumcentre to lie within the incircle. In trilinear coordinates <span class="math-container">$x:y:z$</span> the incircle has equation
<span class="math-container">$$ayz+xbz+xyc-(ax+by+cz)\left(\frac{x(b+... |
2,274,514 | <p>I want to know how to evaluate $\int \frac{\log x}{x^2}$.
Using by parts, and after moving terms, we get something like
$$2 \int\frac{\log x}{x^2} = \frac{(\log x)^2}{x} + \int \frac{(\log x)^2}{x^2}$$
Using by parts again gives
$$\int \frac{(\log x)^2}{x^2} = \frac{2}{3}(\log x)^3 - \int \frac{(\log x)^3}{x^2}$$
At... | Jan Eerland | 226,665 | <p>Well, using integration by parts:</p>
<p>$$\int\frac{\ln\left(x\right)}{x^2}\space\text{d}x=-\frac{\ln\left(x\right)}{x}+\int\frac{1}{x^2}\space\text{d}x\tag1$$</p>
|
4,470,269 | <p>While practicing from a book I found a product in the form <span class="math-container">$$(x^{a^1}+1)\cdot(x^{a^2}+1)\cdot(x^{a^3}+1)\cdot(x^{a^4}+1)$$</span> and was immediately curious if I could a formula to solve the product for <span class="math-container">$n$</span> terms, that is, a single formula for the pro... | Carl Schildkraut | 253,966 | <p>For the specific value of <span class="math-container">$a=1/2$</span> (as mentioned in the comments), one has
<span class="math-container">$$\prod_{j=1}^n \left(x^{2^{-j}}+1\right)=\frac{x-1}{x^{2^{-n}}-1},$$</span>
which can be easily proven by induction on <span class="math-container">$n$</span>: if this holds for... |
4,612,286 | <p>This is a bit of a soft question, but I am interested in a list of classes of structures (in the sense of model theory) which are "surprisingly" first-order axiomatizable classes. Meaning, the class of structures is defined in such a way that it is not at all obvious that it is in fact first-order axiomati... | tomasz | 30,222 | <p>How about the class of fields <span class="math-container">$K$</span> with a finitely generated (over <span class="math-container">$K$</span>) algebraic closure?</p>
<p>(It consists of the fields which are either real closed or algebraically closed.)</p>
<p>I think most equivalent definitions (<a href="https://en.wi... |
2,339,408 | <p>I see a question in Chinese senior high schools books:</p>
<blockquote>
<p>Throwing a fair coin until either there is one Head or four Tails.
Find the expectation of times of throwing.
(You start throwing a coin, if you see Head, then the game suddenly over; and if you see four Tail, the game is over too. Onl... | Matthew Leingang | 2,785 | <p>You can use as a sample space the set
$$
\Omega=\left\{H,TH,TTH,TTTH,TTTT\right\}
$$
The probabilities of these outcomes are $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, $\frac{1}{16}$, and $\frac{1}{16}$ respectively.</p>
<p>[You asked how to show this rigorously. Each throw is independent from every other th... |
378,028 | <p>Let us say that a polynomial with real coefficients is <strong>totally real</strong> if all its complex roots are real and distinct. Let <span class="math-container">$P \in \Bbb R [X]$</span> be totally real. Is it true that</p>
<p><span class="math-container">$$Q(X)=\int_0^XP(t)\,dt+aP(X)$$</span></p>
<p>is also to... | Max Kubierschky | 19,129 | <p>Nope. Lets look at the case a=0.</p>
<p>If you have a positive quadratic polynomial, at its second root the antiderivative has a minimum. It is easy to construct an example where this minimum is positive.</p>
<p>As an example take <span class="math-container">$P(x)=(x-3)^2-1=x^2-6x+8$</span>.
Its antiderivative is <... |
3,881,711 | <p>Sorry in advance, it is probably a stupid question.
I encountered it when I was thinking about the birthday problem. The probability of having at least one pair of the same birthday is
<span class="math-container">$$ 1- \frac{365\cdot364\cdot\ldots\cdot(365-n+1)}{365^n}$$</span> and it is above 0.5 for n>22.
Howe... | Kavi Rama Murthy | 142,385 | <p>The series is convergent. <span class="math-container">$e^{x}-1-2x$</span> is decreasing in <span class="math-container">$(0, \ln 2)$</span> since its derivative is negative here. Since this function vanishes at <span class="math-container">$0$</span> it follows that <span class="math-container">$e^{x}-1-2x <0$</... |
200,629 | <p>I am trying to find the characteristic function for Johnson's SU distribution by integrating the probability density function with <code>Exp[I*t*x]</code> but Mathematica is returning the input itself.</p>
<p>As the characteristic function always exists, I'm not able to understand why Mathematica is not finding the... | Slepecky Mamut | 48,033 | <p>The first assignment is incorrect, resp. I don't know what it should mean from a Wolfram Mathematica syntax perspective. It can mean </p>
<p>expression: a correct form is</p>
<pre><code> expr1 = PDF[JohnsonDistribution["SU", γ, δ, ξ, λ], x]
</code></pre>
<p>or function definition: a correct form is</p>
<pre><cod... |
2,117,784 | <p>A <strong>multilinear form</strong> is a mapping</p>
<p>\begin{align}
\Delta: V^n \rightarrow K
\end{align}</p>
<p>where $V$ is a finite-dimensional vector space over field $K$.</p>
<p>It must meet the following requirements:</p>
<ul>
<li>First:</li>
</ul>
<p>\begin{align}
&\Delta\left(a_1, \dots, a_{i-1}, ... | Ennar | 122,131 | <p><strong>Hint:</strong> Define maps $\varphi_{i_1,\ldots,i_n}(v_1,\ldots,v_n) = (e_{i_1}^*v_1)\cdot (e_{i_2}^*v_2)\cdots (e_{i_n}^*v_n)$ where $\{e_i\}$ is base for $V$ and $\{e_i^*\}$ its dual base. Show that these are multilinear, linearly independent and generate your space.</p>
|
1,739,867 | <p>I need some help to solve these integral:</p>
<p>$$\int \sin(x)[\sec(2x)]^{3/2}dx$$</p>
<p>Thank you.</p>
| zz20s | 213,842 | <p>$$\int \sin(x)[\sec(2x)]^{3/2}dx=\int\frac{\sin x}{(\cos^2x-\sin^2x)^{3/2}}dx=\int\frac{\sin x}{(2\cos^2x-1)^{3/2}}dx$$</p>
<p>Now, let $u=\cos x$ and $du=-\sin x dx$</p>
<p>$$-\int \frac{du}{(2u^2-1)^{3/2}}$$</p>
<p>Now, let $u=\frac{\sec t}{\sqrt{2}}$ and $du=dt\frac{\sec t \tan t}{\sqrt{2}}$</p>
<p>$$-\frac{1... |
1,739,867 | <p>I need some help to solve these integral:</p>
<p>$$\int \sin(x)[\sec(2x)]^{3/2}dx$$</p>
<p>Thank you.</p>
| MathematicsStudent1122 | 238,417 | <p>Firstly, note $$\sec 2x = \frac{1}{2\cos^2(x) - 1}$$ and hence </p>
<p>$$\int \sin(x)[\sec(2x)]^{3/2} \ dx= \int \sin(x) \left[{2\cos^2(x) - 1}\right]^{-3/2} \ dx $$</p>
<p>Taking $u = \cos x$ yields </p>
<p>$$-\int ({2u^2 - 1})^{-3/2} \ du$$</p>
<p>We can solve the above integral firstly for only nonnegative $u... |
982,386 | <p>I am quite a beginner in linear algebra and matrix calculus. I was wondering what is the derivative of the matrix inverse when the matrix is symmetric. More precisely, I'm looking for $\frac{\partial}{\partial \mathbf{X}} \mathbf{X}^{-1}$ when $\mathbf{X}$ is a symmetric matrix.</p>
<p>I am asking this because I ha... | Community | -1 | <p>Assume that $A,X\in Sym$ (the $n\times n$ real symmetric matrices) and $\det(X)>0$. The derivative of $f$ is $Df_X:H\in Sym \rightarrow -tr((X^{-1}AX^{-1}+X^{-1})H)$ (cf. the first part of the lynn's post). If $f$ reaches an extremum in $X$, then, for every $H\in Sym$, $tr((X^{-1}AX^{-1}+X^{-1})H)=0$. Chosing $H=... |
1,236,753 | <p>I am trying to compute $\int x\ln (x+1)\, dx$. I tried integrating by parts and ended up with:
$$\int x\ln(x+1)\,dx = \frac{1}{2}x^2\ln(x+1) - \frac{1}{2}\int\frac{x^2}{x+1}\,dx$$ but I'm stuck here.</p>
| Mark Viola | 218,419 | <p>Write $x^2=x^2-1+1$ so that</p>
<p>$$\frac{x^2}{x+1}=\frac{x^2-1}{x+1}+\frac{1}{x+1}=(x-1)+\frac{1}{x+1} $$</p>
|
1,236,753 | <p>I am trying to compute $\int x\ln (x+1)\, dx$. I tried integrating by parts and ended up with:
$$\int x\ln(x+1)\,dx = \frac{1}{2}x^2\ln(x+1) - \frac{1}{2}\int\frac{x^2}{x+1}\,dx$$ but I'm stuck here.</p>
| Paul | 17,980 | <p>Hints:View $x^2$ as $x^2-1+1$ in the second part.</p>
|
2,701,182 | <p>I must once again resort to the advice of this great community.</p>
<p>As I was reading about the pigeonhole principle something about its proof struck me as odd. Allow me to explain:</p>
<p>After reading the "The Foundations: Logic and Proofs" chapter in Rosen's "Discrete mathematics and its applications" book I ... | Kafein | 544,097 | <p>The advantages of the symbolic representation are that:</p>
<ul>
<li>You have a greater degree of confidence that what you're doing is actually formal logic and not wishy washy argumentation.</li>
<li>You are not forgetting something.</li>
<li>It's much more difficult to write ambiguously with mathematical symbols.... |
2,701,182 | <p>I must once again resort to the advice of this great community.</p>
<p>As I was reading about the pigeonhole principle something about its proof struck me as odd. Allow me to explain:</p>
<p>After reading the "The Foundations: Logic and Proofs" chapter in Rosen's "Discrete mathematics and its applications" book I ... | Chris Bouchard | 207,734 | <p>A lot of this also comes down to writing style. These sorts of proofs aren't just a computational exercise; they're meant to express an idea, to be read and understood, just like any prose writing in any philosophy book. Different writers will want to emphasize different parts, and the same writer may want to expres... |
2,701,182 | <p>I must once again resort to the advice of this great community.</p>
<p>As I was reading about the pigeonhole principle something about its proof struck me as odd. Allow me to explain:</p>
<p>After reading the "The Foundations: Logic and Proofs" chapter in Rosen's "Discrete mathematics and its applications" book I ... | alsowarwickstudent | 384,267 | <p>Just doing to add to something that's been said probably. </p>
<p>My lecturers have always said that the real aim of a proof is to convince the reader that they understand what's going on just as well as you. If you can do that using words alone, and symbols won't make it any easier, that's brilliant. If you feel y... |
2,285,021 | <p>I need to understand how to obtain $f/g$ and $f/h$ from the following equations:
$$
\frac{f^2\sin^2\alpha}{h^2}+\frac{f^2\cos^2\alpha}{g^2}=f^2/k^2
$$
$$
\frac{f^2\sin^2\beta}{h^2}+\frac{f^2\cos^2\beta}{g^2}=f^2/l^2
$$
Which must lead to the following expressions:
$$
(f/g)^2=\frac{(\frac{f\sin\alpha}{l}+\frac{f\sin\... | Narasimham | 95,860 | <p>Remove common $f^2$ as having no role in problem. </p>
<p>You are asking to just to solve two linear equations with two unknowns.</p>
<p>Let $$ k_1=1/h^2,\, k_2=1/g^2,\, 1/k^2 = x_1,\, 1/l^2 = x_2,\, $$</p>
<p>$$ k_1 \cos ^2 \alpha+ k_2 \sin^2 \alpha = x_1 \tag1$$
$$ k_1 \cos ^2 \beta+ k_2 \sin^2 \beta = x_2 \ta... |
541,940 | <p>I am having some issues with this problem in my Linear Algebra textbook. The goal is to either show that the given set, W, is a vector space, or to find a specific example to the contrary:</p>
<p>\begin{Bmatrix}
\begin{bmatrix}
a\\
b\\
c\\
d
\end{bmatrix} :
\begin{matrix}
3a + b = c\\
a + b + 2c = 2d
\end{matri... | sanjshakun | 73,938 | <p>Verify all conditions that define a vector space one by one. For example, you have to verify that if $u$ and $v$ are two vectors that satisfy the given equations and if $\alpha$ is a constant (an element of the underlying field) then $\alpha \times u$ is a solution and $u+v$ is a solution.</p>
|
541,940 | <p>I am having some issues with this problem in my Linear Algebra textbook. The goal is to either show that the given set, W, is a vector space, or to find a specific example to the contrary:</p>
<p>\begin{Bmatrix}
\begin{bmatrix}
a\\
b\\
c\\
d
\end{bmatrix} :
\begin{matrix}
3a + b = c\\
a + b + 2c = 2d
\end{matri... | Henry | 6,460 | <p>If you have a vector satisfying the two constraints then multiplying by $k$ you get $$3a+b=c \implies 3(ka)+(kb)=(kc)$$ $$ a + b + 2c = 2d \implies (ka) + (kb) + 2(kc) = 2(kd)$$ so you have closure under scalar multiplication. Similarly for addition $$3a_1+b_1=c_1 \text{ and }3a_2+b_2=c_2 \\ \implies 3(a_1+a_2)+(b_... |
541,940 | <p>I am having some issues with this problem in my Linear Algebra textbook. The goal is to either show that the given set, W, is a vector space, or to find a specific example to the contrary:</p>
<p>\begin{Bmatrix}
\begin{bmatrix}
a\\
b\\
c\\
d
\end{bmatrix} :
\begin{matrix}
3a + b = c\\
a + b + 2c = 2d
\end{matri... | VnK | 386,589 | <p>Algorithmic approach:
If the given set was to contain vectors u, v and constant c
Then go down the list below and make sure all constraints in your problem are met through trial. </p>
<p>Can u and v exist in this space? </p>
<ol>
<li>Addition:</li>
</ol>
<p>(a) u + v is a vector in V (closure under addition).</p>... |
3,205,693 | <p>Algebras are defined with respect to an underlying ring. Now it makes sense that the underlying ring is smaller. But I was wondering if it is possible to have an underlying ring which is larger. In this case the homomorphism from the ring <span class="math-container">$R$</span> to the algebra <span class="math-conta... | Dietrich Burde | 83,966 | <p>Substituting <span class="math-container">$y=7+x^2-x$</span> from the third equation, the first two equations are
<span class="math-container">\begin{align*}
3x^3 - x^2 + 22x - 6 & = 0\\
3x^3 - x^2 + 22x + 2 & = 0
\end{align*}</span>
Hence we obtain <span class="math-container">$8=0$</span>, which gives a co... |
63,974 | <p>Let $E$ be a spectrum. For any CW complex $X$, define $h_*=\pi_i(E\wedge X)$. Then we know that $h_*$ form a homology theory. In other words, there functors satisfy the homotopy invariance, maps a cofiber sequence of spaces to a long exact sequence of abelian groups, also satisfy the wedge axiom in the definition of... | Mark Grant | 8,103 | <p>The answer is yes, if you replace the wedge axiom with the stronger direct limit axiom
$h_{i}(X) = \mathrm{lim}\ h_{i}(X_{\alpha})$,
where $X$ is the direct limit of subcomplexes $X_{\alpha}$.</p>
<p>As well as Switzer, this is discussed in Chapter 4.F of Hatcher's "Algebraic Topology", Adams' little blue book "... |
1,046,229 | <p>For this problem in proving that the cardinality of <span class="math-container">$(0,1)$</span> is equal to that of the set of real numbers, would I just prove that <span class="math-container">$(0,1)$</span> is uncountable, and then use the theorem that the subset of an uncountable set is uncountable, by saying <sp... | Mark | 147,256 | <p>Just because two sets are both uncountable does not imply that they have the same cardinality. For example, $\mathbb{R}$ and $P(\mathbb{R})$ are both uncountable, but the cardinality of $P(\mathbb{R})$ is strictly greater than that of $\mathbb{R}$.</p>
<p>For your problem, I would try to construct a bijection from ... |
3,692,997 | <p>A three-digit integer is chosen at random. What is the probability that it is possible to add a digit to its right end such that the resulting four-digit number is a multiple of <span class="math-container">$45$</span>?</p>
<p>Edit: I got <span class="math-container">$\frac{1}{15}$</span> as my answer can anyone co... | SlipEternal | 156,808 | <p>How many 3-digit numbers are there? How many are of the form <span class="math-container">$9k$</span>? How many are of the form <span class="math-container">$9k+4$</span>? </p>
<p>For total 3-digit numbers, you have <span class="math-container">$999-99 = 900$</span>. </p>
<p>For 3-digit numbers of the form <span c... |
3,137,295 | <p>I am doing tasks from Concrete Mathematics by Knuth, Graham, Patashnik for trainning, but there are a lot of really tricky sums like that: <br><br>
Calculate sum <span class="math-container">$$S_n = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{2^k} + \frac{1}{2} \right\rfloor $$</span></p>
<h2>My idea</h2>
<p>I had th... | Community | -1 | <p>Let</p>
<p><span class="math-container">$$2^{b}\le n<2^{b+1}$$</span></p>
<p>For all <span class="math-container">$1\le k\le b$</span>,</p>
<p><span class="math-container">$$\left\lfloor\frac{n}{2^k}+\frac12\right\rfloor=\frac{2^b}{2^k}+\left\lfloor\frac{n-2^b}{2^k}+\frac12\right\rfloor.$$</span></p>
<p>For <... |
3,137,295 | <p>I am doing tasks from Concrete Mathematics by Knuth, Graham, Patashnik for trainning, but there are a lot of really tricky sums like that: <br><br>
Calculate sum <span class="math-container">$$S_n = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{2^k} + \frac{1}{2} \right\rfloor $$</span></p>
<h2>My idea</h2>
<p>I had th... | epi163sqrt | 132,007 | <p>Here we use a technique which is introduced in section 3.2 <em>Floor/Ceiling Applications</em> of OPs referred book <em><a href="https://www.csie.ntu.edu.tw/~r97002/temp/Concrete%20Mathematics%202e.pdf" rel="nofollow noreferrer">Concrete Mathematics</a></em>
by R. L. Graham, D. E. Knuth and O. Patashnik. We show ... |
278,669 | <p>$f:(0,\infty)\rightarrow\mathbb{R}$ is differentiable, $\lim_{x\rightarrow\infty}f(x)=1$, $\lim_{x\rightarrow\infty}f'(x)=c$ we need to show $c=0$</p>
<p>well, I tried like this $|f(x)-1|<\epsilon\forall x>M$, where $M$ is very large, $|f'(x)-c|<\epsilon\forall x>M$, what more I can say?thank you.</p... | Nameless | 28,087 | <p>Yet another proof:</p>
<p>Let $0<x$. By the MVT in $(x,x+1)$:
$$\exists \xi\in (x,x+1):f'(\xi)=\frac{f(x+1)-f(x)}{x+1-x}=f(x+1)-f(x)$$
Letting $x\to +\infty$, $\xi\to +\infty$ and so
$$\lim_{x\to +\infty}f'(x)=1-1=0$$</p>
|
4,152,323 | <p>Let
<span class="math-container">$$
a_n = \sum_{k=0}^n \binom{n + 1}{k}b_k.
$$</span>
I am trying to write <span class="math-container">$b_n$</span> in terms of <span class="math-container">$a_k$</span>.</p>
<p>Of course, if the binomial coefficient was <span class="math-container">$\binom{n}{k}$</span> instead of <... | Somos | 438,089 | <p>Given
<span class="math-container">$$ a_n = \sum_{k=0}^n \binom{n + 1}{k}b_k. \tag{1} $$</span>
Let the exponential generating functions (e.g.f.) for <span class="math-container">$\,a_n\,$</span> and <span class="math-container">$\,b_n\,$</span> be
<span class="math-container">$$ A(x) := \sum_{n=0}^\infty a_n x^n/n!... |
4,095,486 | <p>I am currently learning about Jacobians, and I need help on the following integral:</p>
<blockquote>
<p><span class="math-container">$$
\int_0^3 \int_{y^2}^9 y \cos(x^2) dx dy.
$$</span></p>
</blockquote>
| xupeng duan | 721,286 | <p><span class="math-container">$$ \int_0^3 \int_{y^2}^9 y \cos(x^2) dx dy=\int_{0}^{\sqrt{x}}\int_{0}^9y \cos(x^2)dxdy=\int_0^9\dfrac{1}{2}x\cos (x^2)dx=\dfrac{\sin 81}{4}. $$</span></p>
|
964,543 | <p>My question is what are the possible order of $A_6$? And how would I show I get $\frac{6!}{2}=360$. Any tips? I know that $A_6$ is the group of even permutations on six elements. I also know that $360=2^3*3^2*5^1$. How do I got about doing this?</p>
| Alastair Litterick | 6,667 | <p>Facts that you might already know, or might need to prove:</p>
<ul>
<li>The symmetric group $S_n$ has order $n!$</li>
<li>If I take an odd permutation $\sigma$ in $S_n$, then left-multiplication by $\sigma$ sends the set of even elements of $S_n$ to the set of odd elements, and vice-versa. So there are an equal num... |
2,943,037 | <p>Statement : Prove that <span class="math-container">$SL(n,\mathbb{Z})$</span> is generated by <span class="math-container">$(n^2-n)$</span> elements.</p>
<p>The determinant is a n linear function of the rows of the matrix. Given any matrix, if the determinant is nonzero, say <span class="math-container">$det(x_1, x... | Robert Lewis | 67,071 | <p>Here is a proof for real matrices <span class="math-container">$L$</span>, whether or not the <span class="math-container">$0$</span> eigevalue is simple:</p>
<p>Note that, for <em>any</em> real matrix <span class="math-container">$L$</span>, <span class="math-container">$L + L^T$</span> is symmetric, hence is diag... |
379,554 | <p>How can you fit a equilateral triangle on three arbitrary parallel lines with an edge and compass?</p>
<p><img src="https://i.stack.imgur.com/x8s9a.png" alt="enter image description here"></p>
| Ali N | 301,906 | <p>Here is a very simple construction:
Suppose the parallel lines are called a,b,c. A transversal crosses a at A, b at B and c at C. Let XYZ be an arbitrary equilateral triangle. Find point E on XY so that the ratio EX over EY is the same as BA over BC. From X and Y draw lines parallel to ZE. This is almost the answer.... |
3,667,131 | <p>I have this indefinite integral , with <span class="math-container">$a\in \Bbb R, \: a\neq 0$</span></p>
<p><span class="math-container">$$\int \frac{dx}{\sqrt{a^2+x^2}}, \tag 1$$</span></p>
<p>I solve the integral <span class="math-container">$(1)$</span> with <span class="math-container">$x=at$</span>, and using... | Mark Viola | 218,419 | <p>If your high school students have learned about the hyperbolic functions, then there is a natural substitution based on the identity <span class="math-container">$\cosh^2(x)-\sinh^2(x)=1$</span>. </p>
<p>Therefore, if we make the substitution <span class="math-container">$x=a\sinh(t)$</span> such that <span class... |
1,224,085 | <p>A series is an expression of the form
$$
\sum_{n=k}^{\infty} a_n
$$
where the $a_n$ are real numbers and they depend on $n$. If $a_n = b_n$ for all $n\geq k$, then I would assume that one would <em>say that</em> the two series
$$
\sum_{n=k}^{\infty} a_n\quad\text{and}\quad \sum_{n=k}^{\infty} b_n
$$
are the <em>same... | Stuart Jeckel | 65,348 | <p>Regarding your edited question:</p>
<p>Yes, it makes sense to talk about the set of all series, if the set is <a href="http://en.wikipedia.org/wiki/Well-defined" rel="nofollow">well-defined</a>. Your question implies that your interest is in whether the set is <a href="http://en.wikipedia.org/wiki/Well-order" rel=... |
3,412,063 | <p>I'm studying the Lorenz dynamical system, and I'm asking myself if the critical points are unstable critical points. </p>
<p>Considering the theory they are unstable - one eigenvalue <span class="math-container">$\in \mathbb{R}$</span> which is negative and 2 complex eigenvalues with a negative real part. But when ... | Iguana | 140,296 | <p>Consider the equation:
<span class="math-container">$$\lambda_0v+\lambda_1Tv+\cdots+\lambda_{n-1}T^{n-1}v = 0$$</span>
for <span class="math-container">$v$</span> such that <span class="math-container">$T^{n-1}(v) \neq 0$</span>. Apply <span class="math-container">$T^{n-1}$</span> on this equation. It will give you ... |
2,391,676 | <p>I have a difficulty when reading "Lectures in Abstract Algebra, II. Linear Algebra" by Nathan Jacobson (GTM #31). The section is excerpted at the end of the question. The sentence I have difficulty is underlined with red. My problem is: the definition of a bilinear form should be given based on a left and a right ve... | Ted | 15,012 | <p>It might be clearer to state the construction as follows.</p>
<p>We are given a right vector space $R'$ and left vector space $S$, over the field $F$.</p>
<p>Take <em>any</em> vector space $R$ over $F$ of the same dimension as $R'$. Take a non-degenerate bilinear form $g : R \times R' \to F$. Then we may define ... |
4,397,783 | <p>A car salesman can make a sale to 65% of his male customers but to only 45% of his female customers. All of his sales are independent. On Monday morning, the car salesman has two make and one female customer. Find the probability that he makes exactly two sales.</p>
<p>My Solution:
Lets Call the two male customers T... | true blue anil | 22,388 | <p>Since each person has two choices, buy / don't buy, there can only be a total of <span class="math-container">$2^3 = 8$</span> distinct events</p>
<p>Your case <span class="math-container">$3$</span> and <span class="math-container">$4$</span>, eg are exactly the same event, both males buy, and the female doesn't. W... |
4,397,783 | <p>A car salesman can make a sale to 65% of his male customers but to only 45% of his female customers. All of his sales are independent. On Monday morning, the car salesman has two make and one female customer. Find the probability that he makes exactly two sales.</p>
<p>My Solution:
Lets Call the two male customers T... | Adam Bailey | 22,062 | <p>There is no reason in this scenario why the order should matter. None of the probabilities depend in any way on the sequence of events.</p>
<p>Your cases 1-3 cover all the ways in which the salesman makes exactly two sales, so you simply need to sum them:</p>
<p><span class="math-container">$$P(\text{exactly 2 sales... |
691,497 | <p>Suppose that $M$ and $F$ are real matrices. Let $A$ be the block-matrix
$$
A=
\begin{pmatrix}
M & F \\
F & M
\end{pmatrix}
$$
If $\det(M)=0$ is $\det(A)\leq0$? If not, what conditions need to be satisfied?</p>
<p>Also, </p>
<p>Does A have a non-positive eigenvalue?</p>
| Vadim | 26,767 | <p>$$\det\begin{pmatrix}
M & F \\
F & M
\end{pmatrix}=\det(M-F)\det(M+F),$$</p>
<p>So, if, say $M=0$ and <strong>$F$ is <em>any</em> matrix of even order</strong>, $\det -F=\det F$, and</p>
<p>$$\det\begin{pmatrix}
0 & F \\
F & 0
\end{pmatrix}=(\det F)^2.$$</p>
<hr>
<p><strong>Edit.</strong>... |
2,754,985 | <p>I know that if $$F(x,y)=f(x)+g(x)$$, then $$\frac{\partial^2 F}{\partial x \partial y} = 0$$ but when is $\frac{\partial^2 F}{\partial x \partial y} \not= 0$</p>
<p>Is it when $$F(x,y) = f(x)g(x)\space \text{or} \space f(x)^{g(x)}\space \text{or} \space f\bigl(g(x)\bigr)$$
or what?</p>
| szw1710 | 130,298 | <p>We have here a simple partial differential equation. Since $$\frac{\partial^2 F}{\partial x\partial y}=0$$ then $$\frac{\partial F}{\partial x}=\varphi(x)$$ for some differentiable function $\varphi$ ($F$ is constant wrt. $y$). Hence $$F(x,y)=\Phi(x)+\Psi(y)$$ for some functions $\Phi,\Psi$ regular enough. So, $$\fr... |
2,754,985 | <p>I know that if $$F(x,y)=f(x)+g(x)$$, then $$\frac{\partial^2 F}{\partial x \partial y} = 0$$ but when is $\frac{\partial^2 F}{\partial x \partial y} \not= 0$</p>
<p>Is it when $$F(x,y) = f(x)g(x)\space \text{or} \space f(x)^{g(x)}\space \text{or} \space f\bigl(g(x)\bigr)$$
or what?</p>
| Community | -1 | <p>This holds for most functions of two variables (except in the special case that you give).</p>
<p>Try with $f(x,y)=xy$.</p>
|
317,904 | <p>Given, in 3D space: a point $P$ and a direction $v$, a point $Q$ and a direction $w$. So, two lines, $L_1 = P + tv$, $L_2 = Q + tw$.</p>
<p>I am looking for two points, one on each line, say P' and Q'. My requirement is that the distance from $P$ to $P'$ plus $Q$ to $Q'$ equals the distance from $P'$ to $Q'$:</p>
... | Community | -1 | <p>Since the exponential function is non-negative, we have for $x\geq 0$,
$$0\leq \int_0^x\int_0^y \exp(z)\,dz\,dy=\exp(x)-x-1. $$
Plugging into the inequality $1+x
\leq \exp(x)$ gives
$$1\leq \left(1+\frac{1}{n!}\right)^{2n}\leq \exp\left(2n/n!\right)=\exp\left(2/(n-1)!\right)\to 1$$</p>
|
3,502,232 | <p>I've tried to apply Bayes theorem to the following question, but I think I'm using it wrong.</p>
<p>2000 people take an exam and 1 person is cheating. A lie detector that is accurate 99% of the time is used to screen the candidates one by one. At some point during the screening the lie detector beeps to signal that... | ad2004 | 717,666 | <p>I think you need to modify your denominator. I believe the correct formulation is:</p>
<p><span class="math-container">$P\left(\text{cheated}|\text{detected}\right)=\frac{P\left(\text{detected}|\text{cheated}\right)P\left(\text{cheated}\right)}{P\left(\text{detected}|\text{cheated}\right)P\left(\text{cheated}\righ... |
92,654 | <p>Before my question proper, a little background: I'm wanting to optimise some computer rendering by eliminating the drawing of things that aren't visible given the current view.</p>
<p>Suppose we have a sphere. Consider a grid of points on its surface, lying on longitude and latitude lines, equally divided around th... | xpda | 14,448 | <p>Here is one way to do it, in general terms.</p>
<p>Think of the rectangular viewport as having four 3-dimensional arcs instead of four lines on a side, each arc is on the sphere's surface, and each arc is on the great circle between endpoints of the viewing rectangle, or viewport.</p>
<p>If the width of the viewpo... |
4,469,302 | <p>It is often written that all monads are functors, but it is quite hard to find an actual mathematical proof of it.</p>
<p>A functor is defined as a higher level type defining the <code>fmap</code> function:</p>
<pre><code>class Functor f where
fmap :: (a -> b) -> f a -> f b
</code></pre>
<p>It must als... | HallaSurvivor | 655,547 | <p>I'm not convinced that this is a math question... Mathematically (as Zhen Lin points out), it's <em>part of the definition</em> of a monad that it also be a functor.</p>
<p>The question is really about the haskell typeclasses "Monad" and "Functor", and while these are obviously made to imitate th... |
2,674,043 | <p>A linear model has been fitted under the usual assumptions, i.e. Y = Xβ + ε, with $ε ∼ N(0,σ^2I)$. How would the sketch of a residual plot look for residuals from an exponential distribution with expectation 0?</p>
| BruceET | 221,800 | <p>It is simpler to discuss residuals in a one-factor ANOVA, say with three levels $i = 1,2,3=g$ of the factor and $r=10$ replications per level. Suppose all $3n = 30$ observations are from $\mathsf{Exp}(\lambda = 1/5).$ The following
data have no main effect $(\mu_1 = \mu_2 = \mu_3 = 5)$ because this exponential
distr... |
184,210 | <p>Consider the set $S=\{1,2,\ldots, n\}$, and let $a<b<n$. What is the minimum number $f(a,b)$ such that there exist $f(a,b)$ subsets of $S$ of size $a$ for which any subset of $S$ of size $b$ contains at least one of the chosen subsets?</p>
<p>It is not hard to obtain a bound $f(a,b)\leq \dbinom{n}{a}-\dbinom{... | bof | 43,266 | <p>This is Turán's problem. I quote the opening paragraph of A. E. Brouwer and M. Voorhoeve, "Turán theory and the lotto problem", Mathematical Center Tracts 106 (1979), 99-105 = Chapter 7 of A. Schrijver, ed., <em>Packing and Covering in Combinatorics</em>, Mathematisch Centrum, Amsterdam, 1979, ISBN 90-6196-180-7.</p... |
20,807 | <p>我想要讓我的抽象代數班級的學生能夠透過網路向我詢問問題，
我可以建置一個私人的社群，
並且讓他們可以在這裡用中文問問題，
並且讓我用中文回答他們嗎？</p>
<hr>
<p>Google translate produces:</p>
<blockquote>
<p>I want to let my abstract algebra class of students through the Internet to be able to ask me questions, I can build a private community, and so that they can ask questions here ... | Martin Sleziak | 8,297 | <p>In my opinion the main issue here is not the language. The possibility of asking in other languages than English was discussed before and several older discussions are linked in the comments. I will not repeat here what was said there.</p>
<p>What I see as the main problem is that you want to create <em>private com... |
20,807 | <p>我想要讓我的抽象代數班級的學生能夠透過網路向我詢問問題，
我可以建置一個私人的社群，
並且讓他們可以在這裡用中文問問題，
並且讓我用中文回答他們嗎？</p>
<hr>
<p>Google translate produces:</p>
<blockquote>
<p>I want to let my abstract algebra class of students through the Internet to be able to ask me questions, I can build a private community, and so that they can ask questions here ... | Zach466920 | 219,489 | <p>(This post has been edited to sound less accusatory as several users pointed out)</p>
<p>I am American and my native language is English.</p>
<p>You can't make a private community within Math Stack Exchange, and I'm sensing that the community is against including other languages in a private forum setting. Using ... |
2,835,802 | <p>I'm struggling to come up with the method to find the number of ways to take $k$ objects from $n$ groups which at least one object from each group is taken and order matters.</p>
<p>More specifically, I'm trying to order $8$ digits from the digit pool of $5-10$ ($6$ digits) and each digit must appear at least once ... | Pepe | 571,449 | <p>You could try an induction on $n$. For $n=1$ it's easy. If you know the number (call it $\phi(r)$) for all $r<n$. Then, if I'm not mistaken,</p>
<p>$$\phi(n) = \psi(n) - \sum_{r=1}^{n-1} \binom{n}{r}\phi(r)$$</p>
<p>with $\psi(n)$ the number of ways to choose $k$ from $n$ groups without the imposed restriction.... |
3,826,237 | <p>The question asks to prove directly from the definition of a Cauchy sequence that <span class="math-container">$b_k$</span> is Cauchy, but I am hopelessly confused, these are evidently series approaching infinity</p>
| Jingeon An-Lacroix | 471,868 | <p>This may seem as cheating. But: <span class="math-container">$f(x)=1/(1-x)$</span> is continuous everywhere except on <span class="math-container">$x=1$</span>. Therefore since <span class="math-container">$a_k$</span> converges to <span class="math-container">$1/2$</span>, <span class="math-container">$$\lim_{k\rig... |
3,826,237 | <p>The question asks to prove directly from the definition of a Cauchy sequence that <span class="math-container">$b_k$</span> is Cauchy, but I am hopelessly confused, these are evidently series approaching infinity</p>
| Bernard | 202,857 | <p><strong>Hint</strong>:
To prove directly that <span class="math-container">$(b_k)$</span> is Cauchy, compute first
<span class="math-container">$$b_k-b_\ell=\frac1{1-a_k}-\frac 1{1-a_\ell}=\frac{a_k-a_\ell} {(1-a_k)(1-a_\ell)}$$</span>
and note that, since <span class="math-container">$(a_k)$</span> converges to <s... |
4,230,454 | <p>I have graphed two data sets, each with the same y-values but the two data sets have different x-values. I have graphed them and one had a positive correlation, while the other had no correlation.
in the x-axis I was given to label MPQ and in the y I was given to label T450. The question was whether increasing level... | Jacob A | 401,520 | <p>"The question was whether increasing levels of MPQ had also increased levels of T450."</p>
<p>Since the two datasets have two statistically differing outcomes, the only correct answer would involve separating the two cases:</p>
<p>For dataset A, since there is a positive correlation between the two variabl... |
1,808,222 | <p>While solving PhD entrance exams I have faced the following problem:</p>
<blockquote>
<p>Minimize the function $f(x)=- \sum_{i=1}^n \ln(\alpha_i +x_i)$ for fixed $\alpha_i >0$ under the conditions: $\sum_{i=1}^n x_i =1$ and $x_i \ge0$.</p>
</blockquote>
<p>I was trying to use <a href="https://en.wikipedia.org... | Rodrigo de Azevedo | 339,790 | <p>Let</p>
<p>$$f(x) := - \sum_{i=1}^n \ln(\alpha_i +x_i) = - \ln\left(\prod_{i=1}^n \alpha_i + x_i\right)$$</p>
<p>be the objective function. Using the AM-GM inequality,</p>
<p>$$\frac{1}{n}\sum_{i=1}^n \alpha_i + x_i = \frac{1}{n}\sum_{i=1}^n \alpha_i + \frac{1}{n}\underbrace{\sum_{i=1}^n x_i}_{=1} \geq \left(\pro... |
3,448,705 | <blockquote>
<p>Let <span class="math-container">$m,a_1,\dots,a_k\in\mathbb{N},$</span> show that:
<span class="math-container">$$m\mid \gcd(a_1,\dots,a_k)\text{ implies } \gcd(a_1,\dots,a_k)= m\gcd(\frac{a_1}{m}, \dots,\frac{a_k}{m}), \gcd\left(\frac{a_1}{\gcd(a_1,\dots,a_k)},\dots,\frac{a_k}{\gcd(a_1,\dots,a_k)}\... | Andronicus | 528,171 | <p>This implication does not make sense, because <span class="math-container">$m$</span> is never used. But the implication is always true. Suppose:</p>
<p><span class="math-container">$$\gcd\left(\frac{a_1}{\gcd(a_1,\dots,a_k)},\dots,\frac{a_k}{\gcd(a_1,\dots,a_k)}\right)=n$$</span></p>
<p>This would mean, that <spa... |
3,448,705 | <blockquote>
<p>Let <span class="math-container">$m,a_1,\dots,a_k\in\mathbb{N},$</span> show that:
<span class="math-container">$$m\mid \gcd(a_1,\dots,a_k)\text{ implies } \gcd(a_1,\dots,a_k)= m\gcd(\frac{a_1}{m}, \dots,\frac{a_k}{m}), \gcd\left(\frac{a_1}{\gcd(a_1,\dots,a_k)},\dots,\frac{a_k}{\gcd(a_1,\dots,a_k)}\... | user3257842 | 365,433 | <p>Using inequalities seems like overkill.</p>
<p>As a general property, we have and will use <span class="math-container">$$m\mid \gcd(a_1,\dots,a_k) \hspace{0.2cm}\Leftrightarrow \hspace{0.2cm} m \mid a_{p}, \forall p\in\{1..k\} \hspace{2cm} (1)$$</span></p>
<p>Assume <span class="math-container">$\gcd\left(\frac{a... |
1,920,125 | <p>I just came to a realization that my entire view of how functions work might not be completely sound. I'll explain:</p>
<p>Coming from more of a CS background, I see functions $f(x)$ as an example, as taking some input, "storing" it in $x$, substituting this new value wherever $x$ comes up, and produces one output.... | JJacquelin | 108,514 | <p>$g(a) = \frac{d}{dx}f(a)=0$ is wrong because it is the consequence of the misunderstanding of an ambiguous symbolism.</p>
<p>Come back to $\quad g(x) = \frac{d\,f(x)}{dx}$ . If fact, that is a contraction of :
$$g(x) = \left(\frac{d\,f(t)}{dt}\right)_{t=x} \quad \text{where }t\text{ is a dummy variable.}$$
This mea... |
473,446 | <p>Let $X$ be connected cubic $s$-regular graph then $|Aut(X)| = 2^{s-1}\cdot 3\cdot |V(X)|$. I want a reference for proof.</p>
| AG. | 80,733 | <p>By your assumption that the graph $X$ is cubic and $s$-regular, I assume you mean that $X$ is 3-regular and $s$-arc-regular; the latter means (by definition) that $Aut(X)$ acts regularly on the $s$-arcs of the graph. (A permutation group is regular if it is transitive and semiregular.) Thus, given any two $s$-ar... |
3,516,494 | <p><span class="math-container">$$2x^2 + 3x + 1$$</span></p>
<p>applying quadratic formula:</p>
<p><span class="math-container">$$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$</span></p>
<p><span class="math-container">$$a=2, b=3, c=1$$</span></p>
<p><span class="math-container">$$x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \c... | Matt Samuel | 187,867 | <p>Your error is in forgetting about the leading coefficient. If a <em>monic</em> polynomial <span class="math-container">$p(x) $</span> has roots <span class="math-container">$r_1,\ldots, r_n$</span>, with multiplicity, then
<span class="math-container">$$p(x) =(x-r_1) \cdots(x-r_n) $$</span>
If it is not monic but in... |
4,608,244 | <p>It is desirable that the volume of the three-dimensional object is located both inside the cylinder <span class="math-container">$x^2+y^2=1$</span> and inside the sphere <span class="math-container">$x^2+y^2+z^2=4$</span>.</p>
| Bemciu | 1,135,404 | <p><strong>Spherical coordinates are not necessary</strong>. Because of our first constraint, we will integrate over <span class="math-container">$x^{2}+y^{2} \leq 1$</span>, and because of the second constraint, we will integrate the function <span class="math-container">$z=2\sqrt{4-x^{2}-y^{2}}$</span> (<span class="... |
4,142,944 | <p><span class="math-container">$$ \int_0^t \frac{x \cos u - x^2}{1 - 2x \cos u + x^2} \, du\quad\text{with}\quad x \in ]-1,1[ $$</span></p>
<blockquote class="spoiler">
<p> <span class="math-container">$$ \int_0^t \frac{x \cos u - x^2}{1 - 2x \cos u + x^2} \, du = \arctan \left( \frac{x \sin t}{1- x \cos t} \right) $... | Community | -1 | <p>I'm not sure this helps but ...</p>
<p><span class="math-container">\begin{align}\mathcal{I}&=\displaystyle\int \frac{x\cos u-x^2}{(-2x)\cos u+(x^2+1)}\mathrm du\\&=\underbrace{\displaystyle\int \frac{x\cos u}{-2x\cos u+(x^2+1)}\mathrm du}_{\mathcal{I_1}}-x^2\overbrace{\displaystyle\int \frac{\mathrm du}{-2x... |
377,169 | <p>How does one calculate the value within range <span class="math-container">$-1.0$</span> to <span class="math-container">$1.0$</span> to be a number within the range of e.g. <span class="math-container">$0$</span> to <span class="math-container">$200$</span>, or <span class="math-container">$0$</span> to <span class... | Matt L. | 70,664 | <p>If you have numbers $x$ in the range $[a,b]$ and you want to transform them to numbers $y$ in the range $[c,d]$ you need to do this:</p>
<p>$$y=(x-a)\frac{d-c}{b-a}+c$$</p>
|
2,216,418 | <p>Liz and Sara start new jobs on the same day. Liz works three days in a row followed by $1$ rest day. Sara works $7$ days in a row followed by $3$ rest days. How many days between Day $1$ and Day $1000$ will they both have a rest day ? I know the answer is $100$, but how does this come about ?</p>
| Vidyanshu Mishra | 363,566 | <p>Well... I see this problem in the following way:</p>
<p><strong>Observation $1$</strong></p>
<p>Liz is working for three consecutive days and taking rest in fourth day, so the rest day for Liz occurs on multiples of all the multiples of $4$.</p>
<p><strong>Observation $2$</strong></p>
<p>Sara work for seven cons... |
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