Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Example of hypergraph Definition: A hypergraph $\Gamma=(V,\mathcal{E})$ is a set of vertices $V$ and a collection $\mathcal{E}$ of subsets of $V$ such that for every $E\in \mathcal{E}$, we have $|E|\geq 2$. The members of $\mathcal{E}$ are called hypergraphs.
Example: Let $V=\{1,2,3,4\}$ and consider the collection $\m... | The two main philosophies of hypergraph drawing are:
*
*The Set-Circling School (shown on the left). Here, you just take every hyperedge and circle all the vertices involved with a continuous region.
*
*Pros: Each edge is definitely unambiguously drawn.
*Cons: Way more going on in the diagram, especially when lo... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Simultaneous Equations with a Modulus I must confess I'm finding it pretty hard to get my head round these equations, much less solve them. So any help would be enormously appreciated. I'd appreciate if you'd show the logic of how to solve these, and keep it simple without explicitly mentioning the Extended Euclidean w... | So you will have $10$ answer pairs for the first equation.
Each of these is obtained by getting the inverse of some chosen value of $z_1$ and then solving for
$(8z_2 + 2) \equiv 3z_1^{-1} \bmod 11$ which calculates out to $z_2\equiv 7(3z_1^{-1}-2) \bmod 11$ since $8^{-1}\equiv 7 \bmod 11$.
You know in advance that $... | {
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"url": "https://math.stackexchange.com/questions/2668187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Vector-valued forms inside the first jet bundle On page 433 of "Self-duality in four-dimensional Riemannian geometry" by Atiyah, Hitchin and Singer, it is written that $p^*(E \otimes \Lambda^1) \subset p^*J_1(E)$, where $\Lambda^1 \to X$ is the bundle of $1$-forms, $p : E^* \to X$ is a vector bundle and $J_1(E) \to X$ ... | An element of $J_1(E)$ can be thought of simply as the value and first derivative of a section of $E$ at a single point. When $E$ is a trivial bundle $M \times V$ so that sections of $E$ are just vector-valued functions $M \to V,$ this is made very explicit by the canonical isomorphism
\begin{eqnarray}J_1(E) &=& E \op... | {
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"timestamp": "2023-03-29T00:00:00",
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Example of non-isomorphic sheaves, with isomorphic stalks at every point? Basically what the title asks. I'd like to see an example of two sheaves on a topological space which have the same (isomorphic) stalks at every point, but are not isomorphic as sheaves.
| Take as the topological space $S^1$ the circle. Take as stalks, $\mathbb{Z}_3$. There is an automorphism switching the two non identity elements. One sheaf is the trivial sheaf, the other is twisted like a Möbius strip.
| {
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"source": "stackexchange",
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Converting from Polar form to Cartesian Form I am just starting with complex numbers and vectors. The question is:
Convert the following to Cartesian form.
a) $8 \,\text{cis} \frac \pi4$
The formula given is:
$$z = x +yi = r\space(\cos\theta + i\sin\theta)$$
With $r=8$ and $\theta = \frac\pi4$, I did:
$$z=8\left(\c... | Knowing the $\pi\over4$ family:
$$\cos \dfrac{\pi}{4} = \sin \dfrac{\pi}{4} = \dfrac{1}{\sqrt 2}$$
Then applying this result to $8\,\text{cis} \frac{\pi}{4}$:
$$z=8\left(\dfrac{1}{\sqrt 2} + i\dfrac{1}{\sqrt 2}\right) = 8\cdot \dfrac{1}{\sqrt 2}(1+i)$$
And then multiply:
$$8\cdot \dfrac{1}{\sqrt 2} = \dfrac{8\sqrt 2}{2... | {
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What is the Laplace Inverse Transform of $\ln(s)/(s(s+a))$? I've come across a term $\frac{\ln(s)}{s(s+a)}$ in a transformed differential equation.
I am trying to take the inverse transform of it, but I have no idea how to approach it.
If anyone knows how to solve this, that would be greatly appreciated.
| Let $$F(s)=\ln s\\G(s)=\dfrac{1}{s(s+a)}$$therefore we're gonna find the Inverse Laplace Transform (ILT) of $F(s)G(s)$, also if we denote the ILTs of $F(s)$ and $G(s)$ with $f(t)$ and $g(t)$ respectively the $ILT$ of $F(s)G(s)$ is $f(t)*g(t)$ where $*$ denotes the convolution operator. Also we have$$f(t)=\dfrac{u(t)}{t... | {
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Why is it that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$ is not less than $\int_1^\infty \frac{dx}{x^2} = 1$? So according to Euler's proof of the Basel problem,
$$\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6},$$
But only for $n \in \mathbb{Z}$.
But if $n$ was a positive real and $n \geqslant 1$, then woul... | In the following picture, the pink area is the left-hand part of the integral $\displaystyle \int_{x=1}^\infty \frac1x \, dx$ while the green and pink areas together are left-hand part of the sum $\displaystyle \sum_{n=1}^\infty \frac1{n^2}$
The green area is the difference, which is clearly positive and is in fact $\d... | {
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Finding a set of vectors that is not a vector space I am asked to find a set of vectors in $\mathbb{R}^2$ such that if $y$ is in the set, then $b\cdot y$ is in the set for every real number $b$. However, I am told that this set cannot be a vector space.
The set would not be a vector space if the zero vector weren't inc... | Let $S$ be the union of the first and third quadrants (along with the $x$- and $y$-axes, let's say). Scaling by a real number $r$ will either keep each vector in its same quadrant (if $r>0$), or send the first and third quadrants to each other (if $r<0$), or send all vectors to the origin (if $r=0$). So $S$ is closed... | {
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>Finding range of $f(x)=\frac{\sin^2 x+4\sin x+5}{2\sin^2 x+8\sin x+8}$
Finding range of $$f(x)=\frac{\sin^2 x+4\sin x+5}{2\sin^2 x+8\sin x+8}$$
Try: put $\sin x=t$ and $-1\leq t\leq 1$
So $$y=\frac{t^2+4t+5}{2t^2+8t+8}$$
$$2yt^2+8yt+8y=t^2+4t+5$$
$$(2y-1)t^2+4(2y-1)t+(8y-5)=0$$
For real roots $D\geq 0$
So $$16(2y-1)... | Hint: Use
$$\sin^2 x +4\sin x+5 = (\sin x +2)^2 +1$$
and
$$2\sin^2 x +8 \sin x +8 = 2\left(\sin x + 2 \right)^2.$$
Also, break the fraction into two pieces
$$\dfrac{(\sin x +2)^2 +1}{2\left(\sin x + 2 \right)^2}=\dfrac{1}{2}+\dfrac{1}{2}\dfrac{1}{\left(\sin x + 2 \right)^2}$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is $F(c\vec{v})$ = $cF(\vec{v})$ condition of linear transformation the same thing as $F(\vec{a})+F(\vec{b})=F(\vec{a}+\vec{b})$? Is $F(c\vec{v})$ = $cF(\vec{v})$ condition of linear transformation the same thing as $F(\vec{a})+F(\vec{b})=F(\vec{a}+\vec{b})$?
I am a physics undergraduate student and studying linear alg... | You have proved that if $F(a) + F(b) = F(a+b)$ then given any $n \in \mathbb{N}$ and any real $c$ and any vector $v$ $F(cv) = nF(\frac{cv}{n})$. Following the same argument you can prove that $F(nv) = nF(v)$ with $n \in \mathbb{N}$. Using this two proofs you managed to do it with rational numbers. As you say, you miss ... | {
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Alternative way to calculate the sequence Given $a>0$ that satisfies $4a^2+\sqrt 2 a-\sqrt 2=0$.
Calculate $S=\frac{a+1}{\sqrt{a^4+a+1}-a^2}$.
Attempt:
There is only one number $a>0$ that satisfies $4a^2+\sqrt 2 a-\sqrt 2=0$, that is
$a=\frac{-\sqrt{2}+\sqrt{\Delta }}{2\times 4}=\frac{-\sqrt{2}+\sqrt{2+16\sqrt{2}}}{8}$... | First, we have
$$\begin{align}S&=\frac{a+1}{\sqrt{a^4+a+1}-a^2}\\\\&=\frac{a+1}{\sqrt{a^4+a+1}-a^2}\cdot\frac{\sqrt{a^4+a+1}+a^2}{\sqrt{a^4+a+1}+a^2}\\\\&=\frac{(a+1)(\sqrt{a^4+a+1}+a^2)}{a+1}\\\\&=\sqrt{a^4+a+1}+a^2\tag1\end{align}$$
We have
$$4a^2+\sqrt 2 a-\sqrt 2=0\implies (4a^2)^2=(\sqrt 2-\sqrt 2\ a)^2\implies a^... | {
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Prove log(log(n)) is Big-O (log(n)) I need to show that:
$g(n) = \log(\log(n))= O(\log(n))$
This is what I have so far:
Choose $k = 1$
Suppose $n > 1$ then:
$\log(n) < n$
$\log(\log(n)) < \log(n)$
But I can't figure out what my C should be if this is the correct answer?
| If $0<f(n)<g(n)$ then $c=1$ works to show $f(n)=O(g(n))$.
| {
"language": "en",
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Coset proof: $aH=bH$ if and only if $H=a^{-1}bH$ In Gallian's Contemporary Abstract Algebra, 9TH edition, on page 140, he is trying to prove $aH=bH$ if and only if $a^{-1}b\in H$.
And he says to observe that:
$aH=bH$ if and only if $H=a^{-1}bH$.
How is this true?
"$\rightarrow$"
Assume $aH=bH$, so let $t\in aH$. Then $... | Maybe this way of seeing it is clearer ?
$\begin{eqnarray*}
aH = bH &\iff& \{ ah_1, \dots, ah_n \} = \{bh_1, \dots, bh_n\} \\
& \iff & \{a^{-1}ah_1, \dots, a^{-1}ah_n \} = \{a^{-1}bh_1, \dots, a^{-1}bh_n \} \\
& \iff & H = a^{-1}bH
\end{eqnarray*}$
| {
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Composition of two pivot-angle rotations in 2D Suppose I have a set of points $P$ in the 2D plane.
Let $(\theta_1, p_1)$ and $(\theta_2, p_2)$ be two axis-pivot rotation, where for each $i \in \{1, 2\}$, $\theta_i$ is a rotation angle and $p_i$ the pivot point of this rotation.
I would like to rotate each point of $P$... | A rotation $R(\theta,P)$ can be decomposed as the product of two reflections $P_r$ and $P_s$, about any two lines $r$ and $s$ intersecting at $P$ and forming an angle $\theta/2$ between them: $R(\theta,P)=P_s\circ P_r$.
If you have two rotations $R(\theta_1,P_1)$, $R(\theta_2,P_2)$, and $r$ is line $P_1P_2$, you can t... | {
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Find limit of $\lim\limits_{x \to\infty}{\left({{(x!)^2}\over{(2x)!}}\right)}$ I'm practising solving some limits and, currently, I'm trying to solve $\lim\limits_{x\to\infty}{\left({{(x!)^2}\over{(2x)!}}\right)}$.
What I have done:
*
*I have attempted to simplify the fraction until I've reached an easier one to sol... | Stirling formula may be difficult to remember, but the simpler one below is extremely useful and allows you to solve most asymptotic results with factorials:
Factorial Inequality problem $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$
You get $\quad\dfrac{n^{2n}}{9^n}< (n!)^2 < \dfrac{n^{2n}}{4^n}$
And also $... | {
"language": "en",
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How many functions under these conditions? So in my new discrete mathematics class, I had the following exercise.
Let $\Bbb{S} = \{1,2,3,4,5,6\}$. How many functions $f : S → S$ can we find such that
a)$ \;\; \forall y \in \Bbb{S} $ there is $ \;\;\ x \in \Bbb{S} $ such that $f(x)=y$
b)$ \;\; \forall y \in $ {2,4,6... | For the second question, note that if there are two inputs mapping to each of 2, 4 and 6, then there are no inputs that map to 1, 3 or 5. So in other words, how many ways can you allocate two inputs to 2, two inputs to 4 and two inputs to 6? (Alternatively, how can you pick two elements of the set to map to 2, then pic... | {
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Determinant of matrix whose $(i,j)$ entry is $\min(i,j)$ Let $n \in\mathbb{N}$ and define $A \in M_{n}(\mathbb{R})$ by $A(i,j)= \min(i,j)$ for $i,j \in \{1, 2 ,3, 4,\cdots, n\}$.Compute $\det(A)$.
My try is with a example:
Given a $n\times n$ matrix whose $(i, j)$-th entry is the lower of $i,j$, eg.
$$\begin{pmatrix}1 ... | By looking at your example, it seems that by using Laplace expansion in the last column, you would get twice your induction assumption. Let me ellaborate: your determinant would be
$$-1\left|\begin{array}{ccc} 1 & 2 & 2 \\
1 & 2 & 3 \\
1 & 2 & 3\end{array}\right|+2\left|\begin{array}{ccc} 1 & 1 & 1 \\
1 & 2 & 3 \\
1 & ... | {
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Example of infinite dimensional linear spaces where the space is equal to its dual. My understanding is that in finite dimensions, every linear space $V$ is isomorphic to its dual $V^\ast$.
In infinite dimensions, we have that any Hilbert space $\mathcal{H}$ is isomorphic (specifically, anti-isomorphic) to its dual $\m... | There's some confusion here:
*
*In the context of Hilbert spaces, $\mathcal{H}^*$ is not the full (algebraic) dual of $\mathcal H$. It's the topological dual, that is, the space of all continuous linear forms.
*It is not true that every infinite-dimensional Hilbert space is isomorphic to the space $\ell^2$ of squar... | {
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Calculating $\lim_{x\to\infty} \frac{x^{5}}{2^{\sqrt{x}}}$ I'm interested in calculating
$$\lim_{x\to\infty}\frac{x^{5}}{2^{\sqrt{x}}}$$
I'm thinking of using L'Hopital's Rule here, which gives:
$$\lim_{x\to\infty}\frac{5x^{4}\sqrt{x}}{\ln(2)\cdot
2^{\sqrt{x}-1}}$$
But it doesn't look so great, is this the wrong approa... | This is essentially boils down to making the right substitutions. First we try $x \rightarrow x^2$ to get rid of the square root. Now we have $\frac{x^{10}}{2^x}.$ At this point, you can finish the proof by using the fact that exponentials always outpace polynomials.
Your job is rigorize my 2 statements to an acceptabl... | {
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Evaluating sum of binomial coefficients This sum popped out of one of my calculations, I know what it should evaluate to, but I have no idea how to prove it.
$$\sum_{i=0}^{r}{n \choose 2i } - {n\choose 2i -1}$$
I know that $2i-1$ is negative for $i=0$, but for the purpose of this sum, we will say that ${n \choose x}=... | Consider the coefficient of $x^{m}$ in the expansion of
$$(1-x)^n\frac{1}{1-x}=\sum_{r=0}^{n}(-1)^r{n \choose r}x^r\sum_{r=0}^{∞}x^r$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Roots of $x^3+5x-18=0$ using Cardano's method Given that $x^3+5x-18=0$. We have to solve it using Cardano's method.
Using trial $x=2$ will be a root. Dividing the equation by $x-2$ we shall get the other quadratic equation and solving that one, we shall obtain all the roots.
But when I am trying to solve the equation... | By your work we obtain:
$$x=\sqrt[3]{9+\sqrt{81+\frac{125}{27}}}+\sqrt[3]{9-\sqrt{81+\frac{125}{27}}}=$$
$$=
\sqrt[3]{9+\frac{34}{3}\sqrt{\frac{2}{3}}}+\sqrt[3]{9+\frac{34}{3}\sqrt{\frac{2}{3}}}=\frac{1}{3}\left(\sqrt[3]{243+102\sqrt6}+\sqrt[3]{243-102\sqrt6}\right)=$$
$$=\frac{1}{3}\left(\sqrt[3]{(3+2\sqrt6)^3}+\sqrt[... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Showing $x^4-x^3+x^2-x+1>\frac{1}{2}$ for all $x \in \mathbb R$
Show that
$$x^4-x^3+x^2-x+1>\frac{1}{2}. \quad \forall x \in \mathbb{R}$$
Let $x \in \mathbb{R}$,
\begin{align*}
&\mathrel{\phantom{=}}x^4-x^3+x^2-x+1-\frac{1}{2}=x^4-x^3+x^2-x+\dfrac{1}{2}\\
&=x^2(x^2-x)+(x^2-x)+\dfrac{1}{2}=(x^2-x)(x^2+1)+\dfrac{1}{2... | We have to prove $$x^4-x^3+x^2-x+0.5>0\forall x\in\mathbb{R}$$
So let $$f(x)=\frac{1}{2}\bigg[2x^4-2x^3+2x^2-2x+1\bigg]$$
$$f(x)=\frac{1}{2}\bigg[x^4+(x^2-x)^2+(x-1)^2\bigg]>0\forall x\in\mathbb{R}$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Does $A\in B$ imply $A\subset B$? Question: Does $A\in B$ imply $A\subset B$ and does $A\in B$ and $B\in C$ imply $A\in C$?
I've been trying to find examples to get some intuition for this and I've come up with the following:
Example 1: Suppose that $A = \{1\}$, $B = \{\{1\},2\}$. I'd say that $A$ is an element of $B$ ... | No. Taking $A=\{\emptyset\}$ and $B=\{A\} = \{\{\emptyset\}\}$, you have $A\in B$, but not $A\subseteq B$.
Furthermore, in Example $1$, $A=\{1\}$ is not a subset of the set $B=\{\{1\},2\}$, because it is not true that every element of $A$ is also an element of $B$. Specifically, $1$ is an element of $A$, but $1$ is no... | {
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Generating a finite group with a subset of more than $\frac{1}{p}$ elements Sorry if this is a duplicate, but I couldn't find any related question. I'm working on the following exercise
Let $G$ be a finite group with more than $1$ element, en $S\subset G$ a subset such that $\#S>\frac{1}{p}\#G$, with $p$ the smallest... | Put $H=\langle S \rangle$ then $H$ is a subgroup and $S \subseteq H$. If $H \subsetneq G$, then the index $|G:H| \gt 1$. But $|G:H| \leq |G|/\#S \lt p$. Since $p$ is the smallest prime dividing $|G|$ and $|G:H|$ divides $|G|$, this is a contradiction.
| {
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Evaluate the limit $\lim_{n\to \infty}\sqrt{n^2 +2} - \sqrt{n^2 +1}$ I know that
$$\lim_{n\to \infty}(\sqrt{n^2+2} - \sqrt{n^2+1})=0.$$
But how can I prove this?
I only know that $(n^2+2)^{0.5} - \sqrt{n^2}$ is smaller than $\sqrt{n^2+2} - \sqrt{n^2}$ = $\sqrt{n^2+2} - n$.
Edit:
Thank Y'all for the nice and fast... | Note that
$$( \sqrt{n^2+2}-\sqrt{n^2+1})(\sqrt{n^2+2}+\sqrt{n^2+1})=1$$
Thus
$$\sqrt{n^2+2}+\sqrt{n^2+1} \to \infty \implies \sqrt{n^2+2}-\sqrt{n^2+1}\to 0$$
| {
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Applications of perturbation expansion of eigenvectors Partially citing this thread, consider $A_{\varepsilon}=A+\varepsilon V$, a perturbation of $A$, where $\varepsilon$ is a small number (All matrices are symmetric).
A known result derived in some books that I've read is the following perturbation expansion for the... | The analytic dependence of an eigenvalue from a perturbation of the linear operator is a question that has many applications in the study of dynamical systems, and it extension to linear operators in function (Hilbert) spaces is very important in quantum mechanics applications. A classical book on this subject is here.... | {
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} |
Putnam 1957, 17th Problem I have spent the better part of today trying to prove the following question from Putnam 1957
If $\cos B\neq \cos A$, and $k>1$ is any natural number, prove that $$\left|\frac{\cos(kB)\cos(A)-\cos(kA)\cos(B)}{\cos(B)-\cos(A)}\right|<k^2-1$$
I tried using the mean value theorem for the functi... | The given inequality is equivalent to
$$|\cos(kB)\cos A-\cos(kA)\cos B|<(k^2-1)|\cos B-\cos A|$$
Using the formula $2\cos x\cos y=\cos(x-y)+\cos(x+y)$ we can rewrite the last inequality as
$$|\cos(kB-A)+\cos(kB+A)-\cos(kA-B)-\cos(kA+B)|<2(k^2-1)|\cos B-\cos A|$$
which is equivalent to
$$|[\cos(kB-A)-\cos(kA-B)]+[\cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2670994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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All roots $\lambda$ of $\det(A-\lambda B)=0$ are $\ge1$ when $B$ is p.d and $A-B$ is n.n.d. I am trying to prove the following statement:
Let $A,B\in M(n,\mathbb{R})$.
If $B$ is positive definite and $(A-B)$ is non-negative definite, then $\det(A-\lambda B)=0$ has all its roots $\lambda\geqslant1$ and conversely, if a... | You can finish your own proof with the expression $QD$ using this answer. Since $Q$ is p.d., $QD$ has the same eigenvalues as $Q^{1/2}DQ^{1/2}$. Since $x^TQ^{1/2}DQ^{1/2} x = (Q^{1/2}x)^TD(Q^{1/2}x) \geq 0$ (because $D$ is n.n.d.), we get that $QD$ is indeed n.n.d.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2671111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
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Proving a polynomial has x amount of zeros I am new to this thread so sorry if I violate any rules or whatever, but anyway in Calculus right now we are doing stuff related to Fermat's Theorem, Rolle's Theorem, and Intermediate value theorem. I am very confused by Rolles and Fermats and totally don't understand them I t... | i) $f(x) = x^3-6x^2+12x-8$.
Suppose that $f$ has $3$ zeroes in $a<b<c$.
How $f(a)=f(b)=f(c)=0%$ then, by Rolle's Theorem, there is a zero of $f'(x)$ in interval $(a, b)$ and another zero in $(b, c)$. In other words, $f'(x)$ has two zeroes. But $f'(x) = 3x^2-12x+12 = 3(x-2)^2$ has only one zero. Contradiction!
Therefore... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2671251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Stochastic Process: Is this a mistake or am I misunderstanding? (general two-state markov chain and finding $P(T\ge n)$ Consider the general two-state chain where p and q are not both 0. Let $T$ be the first reutrn time to state 1, for the chain started in 1.
(a) Show that $P(T\ge n) = p(1-q)^{n-2}$, for $n\ge 2$.
$$
... | The quantity you have written is the probability that $T=n$ exactly. For the event $\{T\geq n\}$, all that is required is that at the first $n-1$ steps you are not at $1$, which happens with probability $p(1-q)^{n-2}$ (jump to $2$ first, then stay there the next $n-2$ steps).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2671447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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eigenvalue test for linear time-varying systems Consider the linear time-varying system given by
$\dot x(t)=A(t)x(t)$
Denote by $\lambda_\max(t)$ the maximum eigenvalue of $A(t)+A^T(t)$. Suppose that there exist constants $\alpha\gt0$ and $\gamma$ such that $\lambda_\max(t)$ satisfies
$\int_\tau^t\lambda_\max(s)ds\le-\... | Following the hint we examine the derivative of the square of the Euclidean norm. I use $\langle \cdot, \cdot \rangle$ to denote the standard inner product on $\mathbb{R}^n$. We then have for all $t$
$$\frac{d}{dt}\|x(t)\|^2 = \frac{d}{dt} \langle x(t), x(t) \rangle
= \langle A(t)x(t), x(t) \rangle + \langle x(t), A(t)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2671562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why does expanding $\lim_{p \rightarrow 0}\frac{1-p-(1-p)^3}{1-(1-p)^3}$ give a different limit from just substituting? $$\lim_{p \rightarrow 0}\frac{1-p-(1-p)^3}{1-(1-p)^3}$$
$$= \frac{1-0-(1-0)^3}{1-(1-0)^3}$$
$$=\frac{0}{0}$$
$$\lim_{p \rightarrow 0}\frac{1-p-(1-p)^3}{1-(1-p)^3}$$
$$=\lim_{p \rightarrow 0}\frac{p^2... | A little bit of context:
Let $f, g$ be continuos in $D$, $a \in D$, and $g(a)\not =0.$
Then $\lim_{x \rightarrow a} \dfrac{f(x)}{g(x)} = \dfrac{f(a)}{g(a)}$.
Your functions $f$, in the numerator, and g, in the denominator, are continuos but : $g(a)=0$!!.
Hence the above is not applicable.
Another example:
$f(x)=x$, $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2671760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the probability that the first 2 balls are the same color while the last 2 balls are different colors? A box contains 3 Blue balls, 4 Green balls and 5 Red balls. 4 balls were picked at random without replacement. What is the probability that the first 2 balls are the same color while the last 2 balls are diffe... | I tried $2$ different methods and got $\frac{67}{330}$ both times. The denominators of the probabilities at each draw are always $12*11*10*9=11880$ so we only have to keep track of the numerators. We also only have to draw one order of the last $2$ balls, then multiply by $2$.
$$\begin{array}{c|c}\text{Config}&\text{Wa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2671839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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If $\alpha$ is the angle between two curves, find $\cos (\alpha)$ in terms of parametric representation of the curves. If $\alpha$ is the angle between two curves, find $\cos (\alpha)$ in terms of parametric representation of the curves. Answer in book:
Is this answer correct? I keep getting a minus sign between the... | Assume that
$$\gamma:\quad u\mapsto{\bf z}(u)=\bigl(x(u),y(u)\bigr)$$
is the parametric representation of a curve $\gamma\subset{\mathbb R}^2$. Then for any parameter value the vector ${\bf z}'(u)=\bigl(x'(u),y'(u)\bigr)$, if $\ne{\bf 0}$, is a tangent vector to $\gamma$ at the point ${\bf z}(u)$. In the problem at ha... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Representations of superalgebras If I have a superalgebra $A$ and consider the category of finite-dimensional $A$-modules.
Is this category the same as the category of finite-dimensional $A$-modules which are super-vector spaces?
I.e. is every representation of $A$ a super-vector space?
| Let $A=k[x]$ graded so that a monomial $x^i$ is even or odd according to the partity of $i$. Let $V=k^2$ and view $V$ as an $A$-module in such a way that multiplying by $x$ has matrix $\begin{pmatrix}1&1\\0&1\end{pmatrix}$. It is easy to see that $V$ does not have a nontrivial direct sum decomposition preserved by $x^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2672065",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Polynomially sized Folner sets Let $\Gamma$ be a finitely-generated group with a fixed finite generating set $S$.
Then, $\Gamma$ is amenable if and only if it there is a sequence $(F_n)_{n=1}^{\infty}$ of finite subsets of $\Gamma$, whose union is $\Gamma$, such that the boundary of $F_n$ with respect to $S$ is of size... | No: the Følner function grows at least as fast as the word growth: this is due to Varopoulos. Hence a group with polynomially bounded Følner function has polynomially bounded growth (this is purely analytic, not related to Gromov's theorem).
See Drutu's slides:
http://people.maths.ox.ac.uk/drutu/tcc2/TCC5-slides.pdf
PS... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Rubik's cube function I'm thinking in a function and if it's possible to solve that.
I have been playing with the cube using the following move: $R U L' U'.$
I notice that the cube solves itself with a certain number of moves: 28 moves to $2\times2\times2$ and 112 to $3\times3\times3.$ (If the cube already is solved).
... | Have you tried it on a $4\times4\times4$ cube? If not there are online rubiks cubes you can play around with.
But to answer your question my guess is that for any cube size $3$ and up you'll solve it again in $112$ moves. That is because R U L' U' doesn't touch any of the "inner" edges so that means larger cubes will... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2672360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Use Green's Theorem to Find the Area Use Green's Theorem to find the area enclosed by:
$$y=9-x^{2},y=8x, y=\frac{2}{5}x$$
(The area in Quadrant 1)
In class we only did examples of this type of problem that were very simple (eg. area under $\ x^{2}\ $ from 0 to 2), which made setting up the equation for area using Gr... | This really isn’t any different from the simpler examples that you’ve seen, or, for that matter, from the way you’d approach this in first-year Calculus: find the intersections of the curves and integrate piecewise. Remember that even in the example you cite of finding the area under $x^2$, the region is bounded by thr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2672517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does omitting the multiplication operator have an effect on order of operations When you write a mathematical expression like this:
$4:2(1+1)$,
does the fact that the multiplication operator is not explicitly written has any bearing on the precedence? What is the order of operations in this case?
Is it:
$4:4=1$ (order:... | The right way to do it, I think, would be as follows:
$$ 4 \div 2 (1 + 1) = 4 \div 2 \times (2) = 4 \div 2 \times 2 = (4 \div 2) \times 2 = 2 \times 2 = 4. $$
However, one can also proceed as follows:
$$ 4 \div 2(1+1) = (4 \div 2) \times (1+1) = 2 \times (1+1) = 2 \times 2 = 4. $$
Note that parentheses carry the topm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2672699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is the derivative directional? I was met with a surprising face when I assumed that a derivative is a directional change, i.e. that $$\frac{df(x)}{dx}$$
describes the change in $f(x)$ following an positive change in $x$. Moreover, the negative derivative describes the change in $f(x)$ following a negative change in $x$... | You have to be careful with statements like $-\frac{df(x)}{dx}$ being the change in $f(x)$ following a negative change in $x$
Take for example $f(x)=x^2$ at $x=3$ so $f(x)=9$. Then $\frac{df(x)}{dx}=2x$ which is $6$ when $x=3$.
A small positive change in $x$ changes $f(x)$ by about the change multiplied by $6$: so f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2672761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Expected Value of a Ratio of Log-Normal Variables Suppose $a$ and $b$ are constants (with $a\neq b$) and $X$ is a log-normal random variable, i.e. $\ln(X) \sim \mathcal N(\mu,\sigma^2)$. Suppose that
$$ \mathbb E \left[ \frac{X-a}{X-b} \right]=0.$$
Is it possible to solve for $\mu$ as an explicit function of $a$, $b$ a... | I think that you'd need to restrict the potential values of $b$ such that $b \le 0$ as otherwise the expectation doesn't exist.
And I'm not convinced that there is a closed-form solution other than for $b=0$. Here is some Mathematica code to find $\mu$ in terms of $a$ and $\sigma^2$ when $b=0$:
b = 0
expectation = Int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2672878",
"timestamp": "2023-03-29T00:00:00",
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Give an example of a topology that is not compact, not connected, and not Hausdorff here are my questions:
1) Give an example of a topology that is not compact, not connected, and not Hausdorff.
2) Give an example that is connected, but not compact and not Hausdorff.
This is the hint we have:
*i) Let $(X,\mathcal{T})$... | *
*N with the base { {1,2}, {n} : n in N, n > 2 }.
*N with the base { {1,n} : n in N }.
Details left to the diligent reader.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2672997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many different three-digit numbers contain both the digit $2$ and the digit $6$?
How many different three-digit numbers contain both the digit $2$ and the digit $6$?
There are six possible permutations: $$
\overline{26x},\ \overline{62x},\ \overline{2x6},\ \overline{6x2},\ \overline{x26},\ \overline{x62}.
$$
In ... | You should also take into account the repetition of a number. The following numbers are repeated:
$226({x26\text{ and }2x6})$
$262(26x\text{ and }x62)$
$266(26x\text{ and }2x6)$
$622(62x\text{ and }6x2)$
$626(62x\text{ and }x26)$
$662(6x2\text{ and }x62)$
So after subtracting the number of repeated numbers (i.e. $6$) f... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A problem of combinatorics (reduced from a problem of computing probability generating function) This is a problem involving computation of probability generating function (PGF).
I have reduced the problem into a problem of combinatorics, as given below.
Let $f(n,k)$ be defined as
$$f(n+1,k):=f(n,k)+f(n,k-1)+\cdots+f(... | The problem can be rewritten as
$$
\left\{ \matrix{
f(n,k) = \sum\limits_{0\, \le \,j\, \le \,n - 1} {f(n - 1,k - j)} \hfill \cr
f(n,k) = 0\quad \left| {\,n,k < 0} \right. \hfill \cr
f(0,k) = f(1,k) = \left[ {0 = k} \right] \hfill \cr} \right.
$$
and more compactly as
$$
f(n,k) = \sum\limits_{0\, \le \,j\, \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2673252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Fundamental Theorem of Calculus For Piecewise Continuous Functions Recall that the first fundamental theorem of calculus says that if $f$ is continuous real values function defined on $[a,b]$. Then the function defined by $F(x)= \int_ a^x f(t) dt$ is differentiable and $F’(x)= f(x) $
Does the same result hold if $f$ ... | The answer is no.
Consider the function $f(x)$ defined on $[-1,1]$ as $ f(x)=1$ for $-1\le x\le 0$ and $f(x)=-1$ for $0<x\le 1$
Then $F(x)= \int _{-1}^x f(t)dt =x+1$ if $x \le 0$ and $F(x)= \int _{-1}^x f(t)dt =1-x$ if $x > 0$
Note that F(x) is a tent map which is not differentiable at $x=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2673386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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How to get $\lim_{x\to 0}\frac{x}{\sin x} =1$ from $\lim_{x\to 0}\frac{\sin x}{x} =1$? We know the identity that
$$\lim_{x\to 0}\frac{\sin x}{x} =1$$
However in many solved examples that I was going through , I came across the identity
$$\lim_{x\to 0}\frac{x}{\sin x} =1$$
Although it was never formally mentioned any... | Simply note that
$$\frac{x}{\sin x}=\frac1{\frac{\sin x}{x}}\to \frac11=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2673493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving finite limit without L'Hôpital I have come across a problem which requires solving the following limit without L'Hôpital rule:
$$\lim_{x\to\infty} x^2\cdot(e^\frac{1}{x-1}-e^\frac{1}{x})$$
It is obvious from the graphic plot (or using L'Hôpital rule) that the limit is 1. I have tried a few algebraic manipulatio... | By the mean value theorem,
$$e^{\frac{1}{x-1}}-e^{\frac{1}{x}} = e^{c_x}\left(\frac{1}{x-1} -\frac{1}{x}\right
)=e^{c_x}\left(\frac{1}{(x-1)x}\right
)$$
for some $c_x$ between $1/x$ and $1/(x-1).$ Thus as $x\to \infty, c_x\to 0,$ hence $e^{c_x}\to 1.$ Multiplying by $x^2$ then gives a limit of $1.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2673590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
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Colimit of collection of finite sets Situation: Let $\mathbb{T}$ be an endofunctor on $\mathsf{Set}$, the category of sets an suppose that $\mathbb{T}$ restricts to an endofunctor on $\mathsf{FinSet}$, the category of finite sets. That is $\mathbb{T}Y$ is finite whenever $Y$ is finite. Moreover assume $\mathbb{T}$ is m... | No, of course not, at least not without more assumptions on $\mathbb{T}$.
For instance take $\mathbb{T}$ to be the covariant powerset functor. Then clearly it satisfies your conditions, but for infinite $X$, the power set of $X$ is much larger than the union of the power sets of $Y$ for $Y\subset X$ finite.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Perform the modified Euler's Method given a point and a stepsize Consider the following ODE
$$\dfrac{dy}{dx} = -\dfrac{2x}{y}~~~~\text{where}~~~~y(0)=1$$
Given that $y(0.7)=0.141421$ to $6$ digit precision, use the modified Euler's method to estimate $y(0.8)$ using $h=0.1$ and work to $5$ digit precision.
How do I do t... | Given the ODE $y'=f(x,y)$ then Euler's method says $y(x_{n+1})=y(x_n)+h\cdot f(x_n,y(x_n))$ where in your case $f(x,y) = -\frac{2x}{y}$, $x_{n+1}=0.8$, $x_n=0.7$ and $h=x_{n+1}−x_n=0.1$. The modified Euler's method says
$$y(x_{n+1}) = y(x_n) + \frac{h}{2}\cdot[f(x_n,y(x_n)) + f(x_{n+1},\tilde{y}(x_{n+1})) ]$$
where $ỹ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2673795",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $|\det(A_n)|=n^{n/2}$ For k $\ge2$ we recursively define $A_{2^k}$ as $\begin{bmatrix} A_{2^{k-1}} & A_{2^{k-1}} \\ A_{2^{k-1}} & -A_{2^{k-1}} \end{bmatrix}$ and $A_1=[1]$
The problem is to show that $|\det(A_n)|=n^{n/2}$
My attempt: we do an induction on $k$
$|\det(A_2)|=2=2^{2/2}$. Induction hypothesis: $|\... | Your proof looks good.
Alternatively, for the induction-step, notice that
$$
\begin{vmatrix}
A_n & A_n \\
A_n & -A_n
\end{vmatrix}
=
\begin{vmatrix}
A_n & A_n \\
0 & -2A_n
\end{vmatrix},
$$
since the matrices
$$
\begin{bmatrix}
A_n & A_n \\
A_n & -A_n
\end{bmatrix}
\text{and}
\begin{bmatrix}
A_n & A_n \\
0 & -2A_n
\end... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2673937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How would the sketch of a residual plot look for residuals from an exponential distribution with expectation 0? 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 ... | Here is a model with $\beta_0 = -8$ and $\beta_1$. As said @BruceET, in order to maintain expectation of $0$, I have to set $\lambda = 1/8$ in the exponential noise. In this case you can view $\epsilon_i \sim \mathcal{E}xp(\lambda) -\beta_0$, thus $\mathbb{E}{\epsilon_i}=0$. What you see on both plots is a random sampl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2674043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Ways to people can get exactly 4 beers
Let $b ≥ 1$ and $c ≥ 1$ be integers. Elisa’s neighborhood pub serves $b$ different types of
beer and $c$ different types of cider. Elisa invites $6$ friends to this pub and orders $7$ drinks,
one drink (beer or cider) for each friend, and one cider for herself. Different peop... | $\binom{6}{4}$ is the number of ways to choose the 4 of Elisa's friends who will get a beer.
$b^4$ is the number of ways those 4 friends can choose their beers: Each of those friends can choose one of $b$ different beers, so there are $b^4$ different assignments of friends to beers.
$c^3$ is the number of ways Elisa ... | {
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"timestamp": "2023-03-29T00:00:00",
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Proof verification: For isomorphism $\phi: G\to H$, $G$ is abelian iff $H$ is abelian. May someone please verify my following proof?
For isomorphism $\phi: G\to H$ of the two groups $G$ and $H$, prove that $G$ is abelian iff $H$ is abelian.
Proof: (assume $G$ is abelian) Let $h_{1},h_{2}\in H$. Since $\phi$ is biject... | It is correct but, in the first paragraph, you shouldn't have written “for all $h_1,h_2\in H$”. The elements $h_1$ and $h_2$ were given at the beginning of the proof.
And it is a waste of time assirting that $\phi^{-1}$ is onto. Every inverse has that property.
| {
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How would one solve a linear equation in two integer variables? For example, how would I find integers $a$ and $b$ that satisfy the following equation?
$$5a - 12b = 13$$
I always resorted to trial and error when doing something like this and more often than not I would finally reach my answer. But for this one I just k... | The Euclidean algorithm is best for large numbers, but for a small example like this, trial and error, combined with a little modular arithmetic works just fine, and is less work. Reducing $5a−12b=13$ modulo $12$, we have $a\equiv 1 \pmod{12},$ and $a\equiv 5 \pmod{12},$ by trial. Then it's clear that $a=5,$ $b=1$ is... | {
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Query about streamlines of the velocity function $\textbf{q} = k^2 \frac{x \textbf{j} - y \textbf{i}}{x^2 + y^2} , k =$ constant. If we consider the velocity potential $\textbf{q} = k^2 \frac{x \textbf{j} - y \textbf{i}}{x^2 + y^2} , k =$ constant.
The streamlines are given by the circles $x^2 + y^2 =$ constant, the ci... | This is the velocity field induced by a line vortex and there is a singularity at the origin in the plane. Because of this singularity, the potential is, in fact, not defined at the line $\{(0,0,z): -\infty < z < \infty\}$ and the equipotential surfaces converge here.
The flow is irrotational everwhere except at $(x,y... | {
"language": "en",
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Herbrand theorem works only with clauses I read that Herbrand's theorem is valid only for a set of clauses. I already know how to convert every formula in a clause.
But, what I haven't understood is why Herbrand's theorem doesn't work with a formula that's not a clause.
Can you show me ?
| See: Mordechai Ben-Ari, Mathematical Logic for Computer Science, Springer (3rd ed 2012), page 180:
Herbrand’s Theorem: A set of clauses $S$ is unsatisfiable if and only if a finite set of ground instances of clauses of $S$ is unsatisfiable.
From this we have that: a set of clauses $S$ is unsatisfiable iff $S$ has no ... | {
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if $i$ is a number then what is its numerical value? $ i $ is the unit imaginary part of complex number , but there is a question which it is mixed me probably i missed the definition of a number , wolfram alpha $ i $ is assumed to be a number , and others assumed it to be variable because it satisfies $ \sqrt{i^2}$ ... | Asking what's the value of $i$ is like asking what's the value of $2$. And, just like $i^2$ has two square roots, $i$ and $-i$, $2^2$ has two square roots, $2$, and $-2$.
And yes, it is a number, not a variable.
| {
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Can the product of three complex numbers ever be real? Say I have three numbers, $a,b,c\in\mathbb C$. I know that if $a$ were complex, for $abc$ to be real, $bc=\overline a$. Is it possible for $b,c$ to both be complex, or is it only possible for one to be, the other being a scalar?
| Another approach: suppose $a, b$ are complex and not real and $ab$ isn't real. Then let $c=\overline{ab}$.
Note that in a precise sense this is universal: if $abc$ is real (and each is nonzero), then $c$ is a real multiple of $\overline{ab}$.
| {
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How do I determine whether the orientation of a basis is positive or negative using the cross product I know that if I have an orthonormal base in $\mathbb{R}^3$, namely $e_1$, $e_2$ & $e_3$, then it is positively oriented if
$$e_1 \times e_2 = e_3$$
$$e_2 \times e_3 = e_1$$
$$e_3 \times e_1 = e_2$$
It would make sense... | A basis $u$, $v$, $w$ of $\mathbb{R}^3$ is positively oriented if $(u\times v)\cdot w > 0$, and negatively oriented if $(u \times v) \cdot w < 0$. In the case of orthonormal vectors, that is probably equivalent to what you have.
| {
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Equation and polynomial a) For $a,\,b,\,c\in \mathbb{R}$ , let $f(x)=x^3+ax^2+bx+c$ and $M=\max\{1,|a|+|b|+|c|\}$. Show that $f(x)>0$ for $x>M$ and $f(x)<0$ for $x<-M$
b) Consider the following polynomial with integer coefficients $a_1,...,a_n$: $P(x)=x^n+a_1 x^{n-1}+...+a_n$. Show that every rational root of $P$ is an... | Not a rigorous proof:
$x^3 > ax^2+ bx + c$ for some $x > r$
Consider the scenario where $r$ will be greatest. Then
$f(x) = x^3 - |a|x^2 - |b|x - |c|$
For $x = 1$, we get $f(1) = 1 - |a| - |b| - |c|$.
So if $1 \gt |a| + |b| + |c|$, then for all $x \ge 1$, $f(x) > 0$
and if $1 \lt |a| + |b| + |c|$, then for all $x \ge ... | {
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Prove that $\lim_{y \to x-}f(y) = a$ if for every sequence $(x_n)_n$ in $\mathbb{R}$ that increases towards x, the sequence $f(x_n)$ converges to $a$
Let $f: \mathbb{R} \to \mathbb{R}$ be a function.
Prove that: $$\forall (x_n)_n \subseteq \mathbb{R}: (x_n \to x \land
\forall n: x_n \leq x_{n+1} \implies f(x_n) \to ... | Attempt to fix it , correct me if wrong:
You have:
1)$0<x-y_n \lt 1/n$ and
2)$|f(y_n)-a| \ge \epsilon$.
1) Implies $\lim_{ n \rightarrow \infty} y_n =a.$
You need to choose an increasing subsequence such that
$y_{n_k} \le y_{n_{k+1}} $.
Inductively:
0) Start: $x-y_0\lt 1/n_0.$
1) Choose $n_1$ such that :
$1/n_1 \lt ... | {
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How to derive the formula for the expected value of a function of a continuous random variable If $X$ is a r.v. with density $f$ and $Y = g(X)$ then $$E[Y] = \int_{\Bbb R}g(x)f(x)$$
My text offers no demonstration of this. I am familiar with the Lebesgue integral in case the proof relies on measure-theoretic notions. A... | Here's a derivation for the Riemann Integral:
Let $F_X$/$F_Y$ and $f(x)$/$h(x)$ denote the CDF and pdf of $X$ and $Y = g(X)$ respectively. Recall that, by definition, the pdf is the derivative of the CDF. Thus we find the density $h(x)$ by finding $F_Y$ and using the fact that $h = F'_Y$. We thus have:
$F_Y(y) = Pr[Y \... | {
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If $P$ is a polynomial with $|P(1)| = \max\limits_{|z| =1} |P(z)|$, then its root on the unit circle is separated away from 0 Let $P(z)$ be a nonzero polynomial of degree $n$ such that $$|P(1)| = \max\limits_{|z| =1} |P(z)|.$$
Furthermore let $z_0 = e^{i\varphi_0}$, $\varphi_0 \in [-\pi,\pi]$ be a root of $P$ on the un... | EDIT. Mulpiplying on a constant, we may assume wlog that $P(1)=1$.
Consider the function $$f(t)=\frac{1}{2}\left( P(e^{it})+\overline{ P(e^{it})}\right)=\frac{1}{2}\left( P(e^{it})+P^*(e^{-it})\right),$$
where $P^*$ is the polynomial with complex conjugate coefficients.
This $f$ is an entire function of exponential typ... | {
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If $f(x) = { x^2 \sin \frac 1 x \text{ for } 0 < x \leq 1}$ and $f(0)=0$, prove $f$ is rectifiable. If $f(x) = { x^2 \sin \frac 1 x \text{ for } 0 < x \leq 1}$ and $f(0)=0$, prove $f$ is rectifiable.
I tried calculating the length, but couldn't do it. The actual integral should be $\int _0^1\sqrt{1+\left(2x\sin \left(... | You don't have to calculate it, you just have to show it's finite. Hint:
$$(2x\sin(1/x)-\cos(1/x))^2\le 4$$
| {
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if $d$ divides $n$, why is $d^{n-1} \not\equiv 1 \pmod n$? For the Fermat test it is stated that $a^{n-1} \equiv 1 \pmod n$ implies that $\gcd(a, n) = 1$ even when $n$ is not prime (the case for prime $n$ is obvious). I want to know why is this true.
If I can prove the above statement then it will prove this statemen... | If $\gcd(a,n)\ne 1$, then there is a prime $p$ with $p|a$ and $p|n$. $a^{n-1}\equiv 1\mod n$ means $n|a^{n-1}-1$ and because of $p|n$ this implies $p|a^{n-1}-1$.
But we also have $p|a^{n-1}$ because of $p|a$. This is a contradiction because a prime can never divide two consecutive integers simultaneously.
| {
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Can any plane in $\mathbb{R}^3$ be described using a two vectors that only move on two axes? Let's say I have the plane $2x-y-z=11$, and when I want the parametric form I get $u_1=(1,2,0)$ & $u_2=(0,-2,1)$. Aren't all vectors spanning the same plane parallel to one of these vectors? I can't make sense of this because t... | Considering the example you have provided, assuming the standard basis $\left\{\mathbb{\hat{i}}, \mathbb{\hat{j}}, \mathbb{\hat{k}}\right\}$, the two vectors $u_1$ and $u_2$ have between them non-zero contributions in all components necessary to span the space.
In this manner, the plane cannot be contained solely with... | {
"language": "en",
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Find radius of circle which is tangential to parabola.
Given $r < 57$, let $C$ denote a circle centered at ($57$, $r$) with
radius $r$.
$C$ is tangent to a parabola, $y=x^2+r$, from the outside in the first
quadrant.
Find the value of $r$.
I am not sure how to approach the above problem. I started with implicit ... | Let point of intersect be $(h,k)$. We get two equations, one from equating slopes and other from equating $y$ coordinate. $(h,k)$ satisfies both equations.
$$\frac{114-2h}{2k-2r} = 2h \tag{1}$$
$$(h-57)^2 + (k-r)^2 = r^2 \tag{2}$$
From first equation we can write:
$$114-2h = 4h^3$$
Since this is a cubic representing $x... | {
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Diffusion equation if f and φ are periodic the solution is also periodic If I have the diffusion equation
$$ \frac{\partial u}{\partial t}-D \frac{\partial^{2}u}{\partial x^{2}}=f(x,t) $$ with the initial condition $$u(x, 0) = φ(x).$$
How would I prove that if $f$ and $φ$ are p-periodic in $x$, that is for some $p > 0$... | Let $v(x, t) = u(x+p,t)-u(x,t)$, thus $v(x, 0)=0$ and
$$
\frac{\partial v} {\partial t} -D \frac{\partial^2 v} {\partial x^2}=0
$$
from which we conclude that $v(x, t) =0$ and thus $u(x+p,t)=u(x,t)$.
Edit: (more detailed)
1) Using periodicity of initial condition $\phi$ to find initial condition for $v$
$$
v(x, 0)= u(... | {
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Solve $n * \lceil{\frac{R}{T}}\rceil - \lceil{\frac{R*n}{T} - \frac{x*n}{T}}\rceil = \frac{x * n}{C}$ for x and n, $n \in Z^+$ and $x,R,T,C \in R^+$ I want to find the minimum value of n which satisfy given equation.
(Also, not stated in title but given is that $C < T$)
So far, I have been able to find the following pr... | Trivial but perhaps not efficient: Iterate over $n=1,2,\dots$ until you get a solution. For each $n$, substitute it (so that the only variable is $x$) and try to solve equation for $x$. Note that the left side is decreasing in $x$ and right side is increasing in $x$. Something "primitive" like bisection search should w... | {
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Solving area of a triangle where medians are perpendicular. Medians $\overline{AD}$ and $\overline{BE}$ of a $\triangle ABC$ are perpendicular. If $AD= 15$ and $BE = 20$, then what is the area of $\triangle ABC$?
Note: A lot of my work can have inaccuracies and is based off a diagram. It is very helpful if you draw a d... | Let $AD$ and $BD$ meet at $G$ = gravity center. Remember that $AG:GD =2:1$ so $AG=10$.
Area of the triangle $ABD$ which is half of the area of whole triangle $ABC$ is $${EB \cdot AG \over 2} = {20 \cdot 10\over 2} =100$$ so the whole triangle has area $200$.
| {
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Closed and bounded but not compact subset of $\ell^1$ I need to prove that
$$A=\{a\in \ell^1:\sum_{i=1}^{\infty}|a_n| \le 1\}$$
is closed, bounded and not a compact subset in $\ell^1$. Boundedness is trivial, but I get stuck in the other two. Proving subsets of $l^1$ are not closed seems easy, because one sequence whos... | PART 1. In any normed linear space, if $\lim_{n\to \infty}\|v_n-v\|=0 $ then $\lim_{n\to \infty}\|v_n\|=\|v\|.$ PROOF:
(i). $\|v_n\|=\|(v_n-v)+v\|\leq \|v_n-v\|+\|v\|. \;$.... So $\|v_n\|-\|v\|\leq \|v_n-v\|.$
(ii). $\|v\|=\|(v-v_n)+v_n\|\leq \|v-v_n\|+\|v_n\|.\; $.... So $\|v\|-\|v_n\|\leq \|v-v_n\|.$
(iii). From ... | {
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The coprimality of 2 integers that are divisors of 2 larger coprimes? Given an example with $(a,b)=1$ where $a=ux$ and $b=vy$ (with all variables being integers), obviously $(ux,vy) = 1$ directly, but does $(u,v) = 1$ as well? I am pretty sure it should but I am unsure if this is actually a true statement and how a rig... | If $u$ and $v$ share a prime factor then so do $ux$ and $vy$. So if the latter two are coprime, so are the former.
| {
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Adding Absolute value to a complex number: $ z+| z|=2+8i$ I would like to know my error in this problem.
Find the complex number such that:
$$ z+|z|=2+8i$$
So far, I have:
$$
\begin{split}
a+bi+\sqrt{a^2+b^2} &= 2 + 8i\\
a^2-b^2+a^2+b^2&=4-64\\
2a^2 -b^2 + b^2&=-60\\
a^2&=-30
\end{split}
$$
But I should end up with
$$a... | $a+bi+\sqrt{a^2+b^2} = 2 + 8i$ so
$a + \sqrt{a^2 + b^2} = 2$ and $b = 8$.
So $a + \sqrt{a^2 + 64} = 2$
So $\sqrt{a^2 + 64} = 2- a$
$a^2 + 64 = 4 -4a + a^2$
$4a = -60$
$a = -15$.
$z = -15 + 8i$.
....
To do what you were attempting
You have to realize that the $Re(z) = a + \sqrt{a^2 + b^2}$ and $Im(z) = b$. I think some... | {
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Permutation function on $\mathbb{Z}_p$ I'm curious about whether the function $\varphi:\mathbb{Z}_p\to\mathbb{Z}_p$ such that $\varphi(x)=x^3$ defines a permutation iff $p=5\pmod{6}$ (for $p\geq5$ of course). I have some evidence about it but it seems harder than I expected, I hope I'm not missing something important. ... | Since $x^3=0$ if and only if $x=0$, the question boils down to determining for which $p$ the map $x\mapsto x^3$ is an automorphism of $(\mathbb{Z}/p\mathbb{Z})^{\times}$.
And because $(\mathbb{Z}/p\mathbb{Z})^{\times}$ is a cyclic group of order $p-1$, $x\mapsto x^3$ is an automorphism if and only if $p-1$ and $3$ are... | {
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Solving the differential equation $(xy^{4}+y)dx - xdy = 0$ The differential equation is :
$(xy^{4}+y)dx - xdy = 0$
I am trying to simplify the equation to the form $\dfrac{fdg-gdf}{f^{2}}$ so that I can reduce it to $d(\dfrac{g}{f})$ but I am unable to do it.
Any ideas are appreciated.
| If you instead let $y(x)=x u(x)$ your differential equation takes the form
$$
\bigl(x(xu)^4+xu)\bigr)\,dx-x(x\,du+u\,dx)=0
$$
that is
$$
x^5u^4\,dx=x^2\,du.
$$
I'm sure you can solve this differential equation.
| {
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Reversed binary representation of primes I discovered a very interesting fact that if a prime is represented in a binary form , and if that binary is rewritten in the reverse order and converted back to base 10 decimal and if the resulting number is not divisible by 5 or 7 then it is a prime or a square number or a pro... | More counterexamples: $139$ is 10001011, reversed it's 11010001 which is $209 = 11 \times 19$, not divisible by $5$ or by $7$, not a square. $191$ is 10111111, reversed that's 11111101 which is $253 = 2^8 - 3 = 11 \times 23$.
You've been fooled by what humans call Richard K. Guy's law of small numbers. Mwahahahaha!
| {
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Determine if improper integral is convergent or divergent Determine if $$\int_1 ^\infty \frac {dx}{x^2+x} $$ is divergent or convergent. If convergent: determine its value.
Tip: When $ x\ge1 $ is $ \frac 1 {x^2} \ge \frac 1 {x^2+x} = \frac 1 {x} - \frac {1} {x+1} $
Don't really know where to start here. Finding conver... | HINT
Note that for $x\to \infty$
$$\frac {1}{x^2+x}\sim \frac{1}{x^2}$$
then use limit comparison test with $\int_1 ^\infty \frac {1}{x^2}dx$.
| {
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Product of two Abelian group My Question :
$G=AB$, where $A,\, B$ are abelian groups. Are $A$ and $B$ normal?
The problem is a very basic, and very much possibly duplicate of an existing question in this community.
But my question actually might have gone a different way......
I post this today. When I check it, I f... | Not necessarily. For example, the Klein bottle group (that is, the fundamental group of a Klein bottle) is given by $G=\langle a, b; a^{-1}ba=b^{-1}\rangle$. Here, $A=\langle a\rangle$ and $B=\langle b\rangle$. Clearly, $G=AB$. However, $A$ is not normal in $G$.
This group can be viewed as $\mathbb{Z}\rtimes\mathbb{Z}$... | {
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Devise a Newton iteration formula for computing $\sqrt[3]{R}$ where $R>0$. Perform a graphical analysis of your function $f(x)$ to determine Devise a Newton iteration formula for computing $\sqrt[3]{R}$ where $R>0$. Perform a graphical analysis of your function $f(x)$ to determine the starting values for which the iter... | You can prove by induction that the Newton iteration sequence is decreasing and always greater than $\sqrt[3]{R}$ if you start at $x_1 > \sqrt[3]{R}$:
*
*If $x_n > \sqrt[3]{R}$ then $x_{n+1}-x_n=-(\frac{x_n^3-R}{3x_n^2}) < 0$, hence $x_{n+1} < x_n$.
*As $f$ is convex for $x > \sqrt[3]{R}$, you get that $x_{n+1} > \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2678015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Moving limit into sup norm I need help understanding where I'm going wrong with this line of thought:
Assume $f_n$ converges pointwise to $f$, so $\lim \limits_{n \rightarrow \infty}f_n(x) = f(x) \forall x \in X$, then since the suprumum norm is a norm therefore continuous we can move a limit inside, like this:
$\lim \... | The norm is only necessarily continuous with respect to the topology that it induces. You need to be careful since you have two different topologies present here - the topology of pointwise convergence and the topology induced by $\| \cdot \|_\infty$ (which is the topology of uniform convergence).
As a result, in orde... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2678107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
RREF Practice Check I am going over the following exercise
Find a condition on $a,b,c$ so that $(a,b,c)$ in $\mathbb{R}^{3}$
belongs to the space spanned by $u = (2,1,0)$, $v=(1,-1,2)$, and $w =
(0,3,-4)$.
I write out the span of $u,v,w$ and set it equal to $a,b,c$. Then I reduce that linear system to
The last ro... | You row reduce the matrix
\begin{align}
\begin{bmatrix}
2 & 1 & 0 & a \\
1 & -1 & 3 & b \\
0 & 2 & -4 & c
\end{bmatrix}
&\to
\begin{bmatrix}
1 & 1/2 & 0 & a/2 \\
0 & -3/2 & 3 & b-a/2 \\
0 & 2 & -4 & c
\end{bmatrix}
\\&\to
\begin{bmatrix}
1 & 1/2 & 0 & a/2 \\
0 & 1 & -2 & -\frac{2}{3}(b-a/2) \\
0 & 2 & -4 & c
\end{bmatr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2678283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How do you generate a random number in real life from $1$ to $k$. $k\leq 4$ I've been in some multiple choice exams (4 choices, no penalty for incorrect answers) where I have $2$ minutes on the clock, and $10$ questions to go. According to probability, if I randomly chose one of the $4$ answers in each question, on exp... | I would agree with @Remy. I believe that choosing the 3rd option as the correct one for all remaining questions is a better strategy. The reason I feel is that the answers seldom have a "random" pattern!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2678554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Is there a formal name for this type of knot? My (undergrad) research group and I were working with this one specific class of knots, and we don't know quite how to search up their qualities to find out if they have a name.
Basically, they are knots with only one twist in their center, with some odd number x of crossin... | This is the (9,2) torus knot, which you can draw this way:
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2678678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Does the following limit exists? $$
\lim_{a\to 0}\left\{a\int_{-R}^{R}\int_{-R}^{R}
\frac{\mathrm{d}x\,\mathrm{d}y}
{\,\sqrt{\,\left[\left(x + a\right)^{2} + y^{2}\right]
\left[\left(x - a\right)^{2} + y^{2}\right]\,}\,}\right\}
\quad\mbox{where}\ R\ \mbox{is a}\ positive\ \mbox{number.}
$$
The integral exists, since ... | We may assume $a>0$ without loss of generality.
We may decompose $D=[-R,R]^2$ as the union of $B^-,B^+$ and $C=D\setminus\left(B^+ \cup B^-\right)$, where
$$B^-=\left\{(x,y)\in D : \left\|(x,y)-(-a,0)\right\|\leq\frac{a}{2}\right\} $$
$$B^+=\left\{(x,y)\in D : \left\|(x,y)-(a,0)\right\|\leq\frac{a}{2}\right\}. $$
If s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2678757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
How to find no. of polynomials over a finite field satisfying given condition. If i am given with a polynomial $$g(x) = \sum_{i=0}^4 a_ix^i$$ where $a_i \in \mathbb{F}_{2^k}$, finite field with $2^k$ elements. How to find out the no of such polynomials $g(x)$ such that either $g(w) = 0$ or $g(w^{-1}) = 0$, where $w$ is... | You are asking for the number of polynomials $g(x)$ of degree $\le4$ such that $g(x)$ is divisible by the minimal polynomial of either $w$ or $w^{-1}$ over $K=\Bbb{F}_{2^k}$. The answer depends on the degree of those minimal polynomials (they actually coincide for three quarters of choices of $k$), so we split the trea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2678831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Function which creates the sequence 1, 2, 3, 1, 2, 3, ... I was wondering how to map the set $\mathbb{Z}^+$ to the sequence $1, 2, 3, 1, 2, 3, \ldots$. I thought it would be easy, but I was only able to obtain an answer through trial and error.
For a function $f \colon \mathbb{Z}^+ \rightarrow \mathbb{Z}$, we have that... | I'm surprised no one has mentioned composition. You've identified that you need to shift the range
$$f_1(x) = x - 1$$
then take the value modulo $n$
$$f_2(x) = x \mod n$$
and finally shift the result again
$$f_3(x) = x + 1$$
The function you want is just the composition $f = f_3 \circ f_2\circ f_1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2678895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
"answer_count": 13,
"answer_id": 0
} |
to find probability of the given problem or
I solved this question like this but still doubt is there in my mind in the part where London is mentioned.
let us take first equally likely probability that letter could be from London or Washington
now, let $p(E)$ represent the probability that on is the only word legible ... | This problem is massively underspecified - you need to make lots of assumptions in order to get an answer, and different assumptions produce different answers.
The first question is the prior probability. In the absence of any other information, $P(L)=P(W)=1/2$ might be a reasonable prior (although the population of Lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2678977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Laplace transform of $\ddot{x} +4x = f(t)$ I am stuck in an excercise that on first sight didn't look that strange:
Given the initial problem:
$$\ddot{x} +4x = f(t), x(t=0)=3, \dot{x}(t=0)=-1$$
So I started:
$$s^2(X(s) -sx(0)-\dot{x}(0) +4X(s) = \mathcal{L}(f(t))$$
Now substitute the given values:
$$s^2X(s) -3s-(-1) +4... | Hint
You were almost done put all the s terme at the right side then use the convolution formula
$$X(s)(s^2+4) -3s+1 = \mathcal{L}(f(t))$$
$$X(s)(s^2+4) =3s-1+ \mathcal{L}(f(t))$$
For convenience I substitute $h(s)=\mathcal{L}(f(t))$
$$X(s) =\frac {3s-1}{s^2+4}+ h(s) * \, \frac 1 {s^2+4}$$
$$X(s) =3\frac {s}{s^2+4}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2679100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Coefficients of characteristic polynomial of a $3\times 3$ matrix Let $A$ be a $3\times 3$ matrix over reals. Then its characteristic polynomial $\det(xI-A)$ is of the form $x^3+a_2x^2+a_1x+a_0$. It is well known that
$$-a_2=\mbox{trace}(A) \mbox{ and } -a_0=\det(A).$$
Note that these constants are expressed as functi... | For invertible $A$, we have Adj$(A)=det(A)\cdot A^{-1}$ so the trace of the Adj$(A)$ is $det(A)\cdot (\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3})=x_1\cdot x_2+x_1\cdot x_3+x_2\cdot x_3=a_1 $ where $x_i$ are the eigenvalues. Since it holds for invertible $A$, it holds for all matrices.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2679233",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Set theory bracket notation, what is excluded $X=\{\emptyset,\{\emptyset\},\{\{\emptyset\}\}\}$ and $Y=X\setminus\{\{\emptyset\}\}$ If $X=\{\emptyset,\{\emptyset\},\{\{\emptyset\}\}\}$ and $Y=X\setminus\{\{\emptyset\}\}$
then what element is excluded from $X$? Is it $\{\{\emptyset\}\}$, or $\{\emptyset\}$?
In a simila... | $$X=\{\emptyset,\{\emptyset\},\{\{\emptyset\}\}\}$$
and $$ Y=X\setminus\{\{\emptyset\}\}= \{\emptyset,\{\{\emptyset\}\}\} $$ because $\{\emptyset\}$ is removed from your $X$.
For your next question regarding $$ Z=\{a, b, c\}$$ $Z\setminus a$ does not make sense unless $a$ is a set.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2679393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proof of the derivative of the compostion of functions. This question comes from the proof of the derivative of a composite function which I lifted straight from Proof Wiki
https://proofwiki.org/wiki/Derivative_of_Composite_Function:
Let $f, g, h$ be continuous real functions such that: $\forall x \in \mathbb R: h \le... | If $g'(x)$ $\neq$ $0$
Since $g'(x) = \frac{dy}{dx}$, and we have also assumed $dx$ is small but non-zero. So, the product of two non-zero values yield non-zero result. implies that $dy$ $\neq$ $0$
Since $dy$ $\neq$ $0$, by our definition of $dy$, we know this is equivalent to saying ${g(x+δx)−g(x)}$$\neq$ $0$. Then,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2679502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proving if $X_n$ are uniformly integrable and $X_n \Rightarrow X$, then $EX_n \to EX$ Below is the proof from Billingsley's Convergence of Probability Measures. However, in the proof, I don't understand the final step.
That is, we want to show that$$
\int_0^\alpha P[t<X_n<\alpha] \,\mathrm{d}t \to \int_0^\alpha P[t<X<... | $\def\dto{\xrightarrow{\mathrm{d}}}\def\d{\mathrm{d}}$Suppose $D = \{x > 0 \mid F_X \text{ is not continuous at } x\}$, then $D$ is a countable set, and for any fixed $α > 0$ and $t \in [0, α]$,$$
P(X = t) > 0 \Longrightarrow t \in [0, α] \cap D.
$$
If $P(X = α) = 0$, then for any $t \in [0, α] \setminus D$, there is $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2679616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Numerical evaluation of the period of a limit cycle How can I calculate all the periods of the limit cycle of the Ueda-Duffing equation with forcing:
$\ddot{x} + k \dot{x} + x^3 = B \cos(t) $
for each set of parameters $(k, B)$ ?
Edit:
The equation exhibits sub-harmonic resonance for some sets of parameters (and chaoti... | I have done some exploratory python script that uses at its core a boundary value solver to find the closed loops from a systematic sweep of the relevant part of the phase space:
k=0.08; B=0.2
def odesys(u,t): return [u[1], B*np.cos(t)-k*u[1]-u[0]**3 ]
def bc(ya, yb): return yb-ya
def norm(a): return max(abs(a))
row... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2679760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Volume of region inside sphere and cone
Let $R$ consist of the points lying inside of the sphere $$ x^2 + y^2 + z^2 = 3^2 $$ and inside the cone $$ z = \cot(\alpha)\sqrt{x^2 +y^2} $$ where $\alpha$ is $\arccos(\frac15)$ Find the volume of $R$.
So I used cylindrical coordinates and set the two equations for $z$ equal ... | HINT
*
*The intersection between the cone and the sphere is for $z=\cot \alpha \sqrt{9-z^2}\ge0\\\implies z=\frac{3\cot \alpha}{\sqrt{1+\cot^2 \alpha}}$
*For symmetry we can set up the integral in two parts for $z\ge0$, notably
$$V=\int_0^{2\pi} \int_0^{\frac{3\cot \alpha}{\sqrt{1+\cot^2 \alpha}}}\int_0^{r_1(z)}r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2679888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
I know $\mathbb{R}$ is the real number line. What really is $\mathbb{R}^n$? I know $\mathbb{R}$ is the real number line. What really is $\mathbb{R}^n$?
EDIT (based on comments below): What actually is the result of a cartesian product? That is something I failed to grasp too. What do you get when you multiply the num... | It is Euclidean space, which can be thought of as ordered $n$-tuples of real numbers.
For example, $\mathbb{R}^3$ is the set of all ordered triples $(a,b,c)$, where $a,b,c\in\mathbb{R}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2680085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
covariance of two points of an empirical cdf Probably it is a simple question but I was not able to find the answer somewhere.
Assume that $\hat{F}_{X}(\cdot)$ is the empirical cdf estimator that refers to the continuous random variable $X$. We know that the variance of this estimate at a point, say $x_{1}$, is $Var(\h... | Perhaps it's a bit late, but here's help at a solution.
Recall that $\mathbb{E}[\hat{F}_n(x)] = F(x)$. If we expand out the expression for covariance, we are trying to compute
\begin{align*}
\mathbb{E}\left[ \left(\hat{F}_n(x_1) - F(x_1)\right)\left(\hat{F}_n(x_2) - F(x_2)\right)\right] = \mathbb{E}\left[ \hat{F}_n(x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2680244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Quadratic equations with complex coefficients I am stuck on the final bit of this question. There is an answer for this on stack exchange but is different from two other answers on others sites which are also different to each other.
Solve the equation $z^2=-\sqrt3 + i$
So far i've done:
$$|z| = 2$$
$$\theta = \dfrac{... | Nearly or I should say almost perfect.
$$z^2=2e^{i(\frac {5\pi}{6}+2\pi k)}$$
Hence $$z=\sqrt 2e^{i(\frac {5\pi}{12}+\pi k)}$$
For $k=0,1$
We consider only 2 values because since we are dealing with a quadratic we must get only two roots.
In general if we find solution to some $z^n=re^{i\theta+2\pi k}$
Then $k=0,1,2,.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2680324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Show that system is linearly dependent We have that the system of vectors $$x, y, z$$
is linearly independent. Show that the system $$x-y, y-z, z-x $$ is linearly-dependent.
Here is my try. As the first system is independent, it means that
$$a_1x+b_1y+c_1z=0 => a_1=b_1=c_1=0$$
In the second system, writing the combina... | The mistake lies in the passage in which you jump from$$a-b=b-c=c-a=0\tag1$$to $a=b=c=0$. Note that if, say, $a=b=c=1$, then $(1)$ still holds.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2680493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
What is the order of operations for $p \implies q \implies r$ I've been studying mathematical logic recently and we have briefly covered the order of operations for operators like AND/OR/IMPLIES, etc.
However, we have a challenge question regarding how the following statement should be interpreted in terms of order of ... | The answer for the purposes of your course (or if you were to see it in a paper specifically on logic) may be different, but in general mathematical usage this is shorthand for "$p\Rightarrow q$ and $q\Rightarrow r$", similar to constructs like $a\leq b\leq c$.
(If something other than this is meant, I think parenthes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2680615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
On the proof that, for every $L^2$-bounded continuous martingale $M$ starting at $0$, the martingale $M^2-[M]$ is uniformly integrable I am trying to prove that, for every $L^2$-bounded continuous martingale $M$ starting at $0$, the process $M^2-[M]$ is a uniformly integrable martingale. In the proof I am currently re... | For every $Y$ in $L^1$, the set of random variables $S_Y=\{X\in L^1\mid |X|\leqslant Y\}$ is uniformly integrable.
Apply this fact to $Y=\sup\limits_{t\geqslant0}\,(M_t^2-[M]_t)$ and to the set $\{M_t^2-[M]_t\mid t\geqslant0\}\subseteq S_Y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2680708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
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