Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Three couples sit at random in a line of six seats, probability that no couple sits together? If three married couples (so 6 people) sit in a row of six seats at random, what is the probability that no couples sit together?
Another way to think about it (couples are AB, CD, and EF)
| There are $6!$ possible seating arrangements. From these, we must exclude those in which one or more couples sit in adjacent seats.
There are three ways to select a couple who sit in adjacent seats. That gives us five objects to arrange, the couple and the other four people. The objects can be arranged in $5!$ ways.... | {
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Let $A$ be a $2\times 2$ matrix with eigenvalues $1,2$. What is $\det(A^3-3A^2+A+5I)$ Let $A$ be a $2\times 2$ matrix with eigenvalues $1,2$. What is $\det(A^3-3A^2+A+5I)$
I know that $\det(A)=$ product of eingenvalues = $2$.
I also know that since $A$ has 2 distinct eiginvalues, it is diagonalizable.
$A=UDU^T$, where ... | By Cayley–Hamilton, $A^2-3A+2I=0$ and so $$A^3-3A^2+A+5I=A(A^2-3A+2I)-A+5I=-A+5I$$ The eigenvalues of $-A+5I$ are $-1+5=4$ and $-2+5=3$ and so $$\det(A^3-3A^2+A+5I) = \det(-A+5I) = 4 \cdot 3 = 12$$
Or you could argue directly that the eigenvalues of $P(A)$ are $P(1)$ and $P(2)$ and so $\det P(A) = P(1)P(2)$.
| {
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l'Hôpital vs Other Methods Consider the first example using repeated l'Hôpital:
$$\lim_{x \rightarrow 0} \frac{x^4}{x^4+x^2} = \lim_{x \rightarrow 0} \frac{\frac{d}{dx}(x^4)}{\frac{d}{dx}(x^4+x^2)} = \lim_{x \rightarrow 0} \frac{4x^3}{4x^3+2x} = ... = \lim_{x \rightarrow 0}\frac{\frac{d}{dx}(24x)}{\frac{d}{dx}(24x)} =... | After doing derivative one more time you get $12x^2 +2 $ which is not $0$ when $x$ goes to $0$.
| {
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Difference between torsion and Lie bracket After viewing a lecture on torsion, the lecturer said that the torsion is the failure of curves to close.
Since this is almost also what I have read about the Lie bracket, I want to know their difference and also understand it geometrically.
The Lie bracket is included in the ... | In simple words(not formal): The torsion describes how the tangent space twisted when it is parallel transported along a geodesic. The Lie bracket of two vectors measures, as you said, the failure to close the flow lines of these vectors.
The main difference is that torsion uses parallel transport whereas Lie bracket u... | {
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Maximum value of $\gcd(u^2-v^2, 2uv, u^2+v^2)$ What is the maximum of $\gcd(u^2-v^2, 2uv, u^2+v^2)$, where $u$ and $v$ are integers and $u\neq v\neq 0$ ? I have tried to find it but I haven't got anywhere. I got this question when I was watching this video, where he says, at 9:36, that we need not scale down by less th... | Let $d=\gcd(u,v)$, $u=dx$, $v=dy$. Then $$\gcd(u^2-v^2,2uv,u^2+v^2)=d^2\gcd(x^2-y^2,2xy,x^2+y^2).$$
Now if $p\mid x$ then $p\nmid x^2+y^2$ and if $p\mid y$ then $p\nmid x^2+y^2$. Hence $\gcd(x^2-y^2,2xy,x^2+y^2)$ can at most be $2$, and this happens iff $x,y$ are both odd.
| {
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Number of Integer Solutions for $x_1 + x_2 + x_3 + x_4 = 15$ where $-5 \le x_i \le 10$ I am trying to find the number of Integer Solutions for $x_1 + x_2 + x_3 + x_4 = 15$ where $-5 \le x_{i_{\in [4]}} \le 10$
I know if $x_i$s are all non-negative integers, it is a number partition of 15 however, this case a bit tri... | Let $y_i = x_i + 5$ for each $i$. Then you're trying to find the number of integer solutions to $y_1 + y_2 + y_3 + y_4 = 35$ with $0 \le y_i \le 15$.
| {
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Find real and complex $A_{m \times n}$ such that $\operatorname{Ran} A = \operatorname{Ker} A^T$ where $\operatorname{Ran} A$ is column space of $A$. $\newcommand{\Ran}{\operatorname{Ran}} \newcommand{\Ker}{\operatorname{Ker}}\newcommand{\b}{\mathbf}$
If $\b y \in \Ran A$ and $\b y \in \Ker A^T$, then $\b y = A \b x$ a... | Here is an argument for a real Euclidean space. Note that if $A \neq 0$ then $\langle A v, A v \rangle > 0$ for some $v$. Then since $\langle A^{\mathrm{t}} A v, v \rangle = \langle A v, A v \rangle > 0$ it follows that $A^{\mathrm{t}} A v \neq 0$. So $A v$ is in the range of $A$ but not in the kernel of $A^{\mathrm{t}... | {
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Does $f_n$ uniformly converge to $f$? $f_n :[0, 1]\rightarrow\mathbb{R} \qquad x \mapsto x^n - x^{n+1}$
The sequence converges pointwise to the zero function. It converges uniformly if
$$\sup_{x \in [0, 1]} \; \big| \, f_{n}(x) - f(x) \, \big|$$
tends to zero. But I am not sure if it does or how to prove.
| Notice that $f_{n+1} = x f_n$ as $x\in[0,1]$ you have that $f_n$ is monotone. Use Dini's theorem Link to complete the proof.
| {
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Determining Whether a function is linear I'm positive this function is linear but am having trouble showing it:
Determine whether $T:V \to W$ defines a linear transformation:
$V=R^3$, $W=R$
$T((a_1, a_2, a_3)) = 3a_1 +2a_2 + a_3 $
I know I have to show that $T(u+v) = T(u) + T(v)$ and $T(cu) = cT(u)$ but am unsure how ... | These are the steps you should take to prove $T(u+v)=T(u)+T(v)$.
*
*Set $u=(u_1,u_2,u_3)$ and $v=(v_1,v_2,v_3)$.
*Write down what $T(u)$ is.
*Write down what $T(v)$ is.
*Write down what $T(u)+T(v)$ is.
*Write down what $u+v$ is.
*Write down what $T(u+v)$ is.
*Compare results from (3) and (5).
Which of these ... | {
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Finding the Lagrange Dual I'm working on the following (convex) optimization problem:
Let $Q$ be an $n \times n$ positive semidefinite matrix, $A$ an $m\times n$ matrix and $b\in \mathbb{R}^m$. Determine the Lagrange dual of
\begin{align}
\min\{x^TQx \ | \ Ax\leq b , \ x\in\mathbb{R}^n \}
\end{align}
My problem is esp... | The primal model is
\begin{array}{ll}
\text{minimize} & x^T Q x \\
\text{subject to} & A x \leq b
\end{array}
The Lagrangian is
$$L(x,u) = x^TQx - u^T(b-Ax)$$
where the Lagrange multiplier $u$ is nonnegative. The dual function is
$$g(u) = \inf_x x^TQx - u^T(b-Ax)$$
The optimality conditions for the infimum are indeed... | {
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An example of a unipotent matrix which is NOT upper triangular Define::= $I_n$ -- the $n \times n$ identity matrix.
Let $A$ be an $n \times n$ real matrix.
Define::= Nilpotent matrix -- an $n \times n$ real matrix $X$ such that $X^n = $ the zero matrix for some $n$ in the positive integers.
Define::= unipotent matrix... | $$A=\left[ \begin{array}{ccc}1&1&0\\ 0&1&0\\ 0&1&1\end{array}\right]$$is unipotent because $(A-I)^2=0$ and $(A-I)$ is nilpotent where $I$ is the identity (or one can also argue that the characteristic polynomial of $A$ is $(\lambda-1)^3$) whereas A is neither upper nor lower triangular.
| {
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Prove 11 does not divide $3^{3k-1}+5*3^k$ for any odd k. First I did an induction proof that it does work for even k. Then I started the proof as so.
Suppose there exists a k of the form 2n+1, s.t 11 divides $3^{3k-1}+5*3^k$.
After some algebra I can arrive at this point $5*2^{6n-1}+3(2^{6n-1}+5*3^{2n})$
Since I proved... | $3^{3k-1}+5\times 3^k=3^{k-1}(3^{2k}+15)=3^{k-1}(x^2+15)$ with $x=3^k$
Also $x^2+15\equiv x^2+4\pmod{11}$
$\begin{array}{l}
k=0: & 3^0\equiv 1\pmod{11} & x^2+4\equiv 5\pmod{11}\\
k=1: & 3^1\equiv 3\pmod{11} & x^2+4\equiv 13\equiv 2\pmod{11}\\
k=2: & 3^2\equiv 9\pmod{11} & x^2+4\equiv 85\equiv 8\pmod{11}\\
k=3: & 3^3\eq... | {
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Stirling numbers of first kind proof For the Stirling numbers of the first kind, show that
\begin{align*}
(x)^{(n)} =\sum_{k=0}^n s'(n,k)x^k.
\end{align*}
For this proof we can proceed with induction, by proving the base cases first $n=0$ and $n=1$:
\begin{eqnarray*}
(x)^{(0)}=s'(0,0)x^0 = 1 \\
(x)^{(1)}=s'(1,1)x^... | The Stirling numbers of the first kind $s(n,k)$ satisfy the recurrence:
$$ s(n,k) = s(n-1,k-1) + (n-1) s(n-1,k) $$
The rising factorials are defined by:
$$(x)^n = x(x+1)(x+2) \cdots (x+n-1) $$
We wish to show that:
$$ (x)^n = \sum_k s(n,k) x^k $$
The induction hypothesis is that:
$$ (x)^{n-1} = \sum_k s(n-1,k) x^k $$
U... | {
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Reference request: Toroidal graph I have asked a similar question here but not sure if it has reached the right community.
I need reference to learn about graphs that have genus 1 i.e. toroidal graphs. Specifically, i am trying to find answers to the questions below.
*
*Since toroidal graphs can be recognized in po... | For (2), take a look at this paper:
Myrvold, Wendy; Woodcock, Jennifer, A large set of torus obstructions and how they were discovered, Electron. J. Comb. 25, No. 1, Research Paper P1.16, 17 p. (2018). ZBL1380.05134.
Abstract
We outline the progress made so far on the search for the complete set of torus obstructions ... | {
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Elementary asymptotics of $\sum_{k=0}^\infty \sqrt k \frac{x^k}{k!}$ as $x\to \infty$
Consider the power series $\sum_{k=0}^\infty \sqrt k \frac{x^k}{k!}$. It is easily seen that its radius of convergence is $\infty$. I'm looking for an elementary proof of the asymptotic expansion $$\sum_{k=0}^\infty \sqrt k \frac{x^k... | This answer deals with different approaches to the asymptotics of the series in question. Here I will try to give a probabilistic argument.
This series has the following probabilistic interpretation: If $N_t$ is a Poisson distribution of rate $t > 0$ then we have
$$ \mathbb{E}[\sqrt{N_t}] = \sum_{k=0}^{\infty} \sqrt{k... | {
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Compact Hausdorff Topologies on a set are all incomparable. Is the same true for the $\sigma$-compact Hausdorff case? It is well known that compact Hausdorff topologies on a set are all incomparable; i.e. if $X$ is a set and $\tau_1$ and $\tau_2$ are compact Hausdorff topologies on $X$ then the identity map $(X,\tau_1)... | You can use $\mathbb{Z}$ with the discrete and the finite complement topologies.
They are both $\sigma$-compact and Haussdorff, but not the same.
| {
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Pythagorean Triple where $a=b$? I cannot find even a single webpage mentioning this topic. I'm a programmer and I'm looking for a 45-45-90 triangle where all of the sides are whole numbers. In the video I am watching, they say to use $ a = 10 $, $ a = 10 $, $ c = 14 $ because $ 10 \sqrt{2} $ is close enough to $ 14 $.... | Yes, it is just the fact that $\sqrt{2}$ is irrational. Suppose there would be such a right-triangle. Then $n=1$ would be a congruent number, i.e., the area of an right-triangle with rational sides. By Fermat, for exponent $4$, it isn't.
| {
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Proving $\lim_{x \to \infty}\frac{\ln x}{x^r}=0$ and $\lim_{x \to 0^+}x^r\ln x=0$ for $r>0$ This is the question I'm trying to answer
This is how I went about proving i) and ii):
i) $$\lim_{x \to \infty}\frac{\ln x}{x^r}=\lim_{x \to \infty}(\frac{1}{x^{r-1}}\cdot\frac{\ln x}{x})=\lim_{x \to \infty}(\frac{1}{x^{-1}}\c... | Using $\infty\cdot 0$ in limit formulae is not valid. Rather, perhaps, you need to show that, for $r>0$, there exists some constant $C_{r}>0$ such that $\log x\leq C_{r}x^{r/2}$ for $x\geq1$. Then $\left|\dfrac{\log x}{x^{r}}\right|\leq C_{r}\dfrac{1}{x^{r/2}}$, taking $x\rightarrow\infty$ finishes the job. For the sec... | {
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There is no straightline in $C\setminus\{3i\}$ which is mappeed onto a straight line in C by f(True/false)
Let $\displaystyle f \colon \Bbb C\setminus \{3i\} \to \Bbb C$ be defined by $$f(z)=\frac{z-i}{iz+3}.$$ which of the following statement is true ?
1) f map circles in $C\setminus\{3i\}$ onto circles in C
2)There... | Hint:
$$
\frac{z-i}{iz+3}=-i+\frac{2}{z-3i}
$$
If $z-3i$ is bounded away from $0$, $f$ is bounded.
| {
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the set of compact operators on $H$ is nonunital How to prove $K(H)$ is a nonunital ,where $K(H)$ is the set of compact operators on $H$,$H$ is a infinite dimensional Hilbert space? Can anyone give me some hints?Thanks
| For a projection $P\in B(H)$ to be compact, it has to be finite-rank. This is because its rank is $PH$, and the unit ball of a subspace will be compact if and only if said subspace is finite-dimensional.
Since $I$ is also a projection, it can only be compact when its image is finite-dimensional; that is, when $H$ is f... | {
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Covariance and contravariance Given a vector space $V$, a vector $v \in V$ can be written in components with respect to different bases, say $X$ and $Y$. Now when i make a transformation from $X$ to $Y$, the components of the vector are transforming contravariantly.
Now the dual space$V^*$ of $V$ is also a vector space... | The condition "$V$ left or right vector space over a field $F$" is purely algebraic. It tells us nothing on the type of transformations of tensors on it. That is actually due to further definitions, i.e. those of vectors/vector fields and covectors/covector fields on $V$. In other words, being a vector space doesn't im... | {
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Solution of $\cos(2x) - A\sin(2x) = 0?$ The physics exercises of today's lecture made me face following equation:
$$\cos(2x) - A\sin(2x) = 0$$
I was not able to solve for $x$? Do you know how to proceed?
(Note: A is $\approx -1/7$)
| You can rearrange to get
$$\tan(2x)=\frac{1}{A}\,.$$
The general solution is
$$x=\frac{1}{2}\arctan\left(\frac{1}{A}\right)+\frac{n\pi}{2},$$
where $n$ is an integer.
| {
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help me verify my proof ($\lfloor x-1 \rfloor = \lfloor x \rfloor - 1$) prove the following statement:
$\forall x \in \mathbb{R}, \lfloor x-1 \rfloor = \lfloor x \rfloor - 1$
suppose $x \in \mathbb{Z}$, then $\lfloor x-1 \rfloor = x-1 $ and $ \lfloor x \rfloor -1 = x-1 $ since the floor of any integer is itself.
suppos... | Good start.
You have shown that for $x \in \mathbb{Z}$, $\lfloor x-1 \rfloor = x-1 $ and $ \lfloor x \rfloor -1 = x-1 $.
For $x \in \mathbb{R} $, let $x=y+\delta$, where $y \in \mathbb{Z}$ and $0 \le \delta <1$.
Then $\lfloor x-1 \rfloor = \lfloor y+ \delta -1 \rfloor = \lfloor y-1 + \delta \rfloor = y-1$.
and $\lfloor... | {
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Tricky induction proof: pile of stones split into n groups So I understand setting up this proof and the basis step just fine, however, it is the induction step where I am completely lost. I went and asked the math tutors at my school, the the tutor that spoke with me even had a tough time with this problem... it is su... | I am sure your instructor will want an algebraic proof, so don't turn this in as your proof, but here's a 'Proof by Picture' for the inductive step:
Explanation:
The claim is that $n$ stones you will eventually end up with $\frac{n(n-1)}{2}$ stones, which is the sum of all numbers $1$ through $n-1$ ... which is the nu... | {
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How to find $x$ given $\log_{9}\left(\frac{1}{\sqrt3}\right) =x$ without a calculator? I was asked to find $x$ when: $$\log_{9}\left(\frac{1}{\sqrt3}\right) =x$$
Step two may resemble:
$${3}^{2x}=\frac{1}{\sqrt3}$$
I was not allowed a calculator and was told that it was possible. I put it into my calculator and found... | $\log_{9}\left(\frac{1}{\sqrt3}\right) = $
$\log_{9}(3^{-\frac 12})=$
$\log_{9}((\sqrt{9})^{-\frac 12})=$
$\log_{9}((9^{\frac 12})^{-\frac 12}) = $
$\log_{9}(9^{\frac 12*(-\frac 12}) =$
$\log_9(9^{-\frac 14}) =$
$-\frac 14$
..... or .....
$\log_9(\frac 1{\sqrt 3}) =x$ So
$9^x = \frac 1{\sqrt 3}= \frac 1{\sqrt{\sqrt 9}... | {
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joint pdf of two random variables A pair of random variable $(X, Y)$ is uniformly distributed in the quadrilateral region with $(0,0),(a,0),(a,b),(2a,b)$, where $a,b$ are positive real numbers.
What is the joint pdf $f(X,Y.)$
Find the marginal probability density functions $f_X (x)$ and $f_Y (y)$.
Find $E(X)$and $E(Y)... | Note that \begin{align}f_X(x) &= \int_{0}^b f_{X,Y}(x,y) \,dy \\
&= \begin{cases} \int_0^{\frac{b}{a}x} f_{X,Y}(x,y) \, dy & , 0 \leq x \leq a \\
\int_{\frac{b}{a}x-b}^b f_{X,Y}(x,y) \, dy &, a < x \leq 2a\end{cases}\end{align}
Recheck your value value for $f_Y$ as well. You should get a simpler expression. Notice th... | {
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How prove this $\sum_{k=2}^{n}\frac{1}{3^k-1}<\frac{1}{5}$ show that $$\sum_{k=2}^{n}\frac{1}{3^k-1}<\dfrac{1}{5}\tag1$$
I try to use this well known: if $a>b>0,c>0$,then we have
$$\dfrac{b}{a}<\dfrac{b+c}{a+c}$$
$$\dfrac{1}{3^k-1}<\dfrac{1+1}{3^k-1+1}=\dfrac{2}{3^k}$$
so we have
$$\sum_{k=2}^{n}\dfrac{1}{3^... | Since $3^k > 6$ for $k \geq 2$, $$\frac{1}{3^k-1}<\frac{1}{3^k-\frac{3^k}{6}}=\frac{6}{5}\cdot\frac{1}{3^k}$$ so $$\sum_{k=2}^{n}\dfrac{1}{3^k-1}<\frac{6}{5}\sum_{k=2}^{n}\dfrac{1}{3^k}<\frac{6}{5}\sum_{k=2}^\infty\dfrac{1}{3^k}=\dfrac{1}{5}$$ as required.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2502560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
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Show that $T$ is an isomorphism if and only if $T$ is invertible
Let $T : V → W$ be a linear transformation of vector spaces. We say that
$T$ is invertible if and only if there exists a map $S : W → V$ such that
$S \circ T = 1_W$ and $T \circ S = 1_V$ . Show that $T$ is an isomorphism if and only
if $T$ is inver... | Let $V$ and $W$ be two vector spaces and let $T$ be a Linear Transformation. T is said to be an isomorphism if T is also a Bijection. Bijection implies that there exists another Linear Transformation $S:W\to V$ such that S and T are each other's inverse.
So essentially the statements that T is an Isomorphism and T is I... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why are corner points of feasible region candidates in solving linear programming problem? In a linear programming problem, when the goal is to optimize a linear combination of variables with some constraints, it is said that the corners of feasible solution (the Polyhedron determined by constraints) are candidates for... | Here are a few pictures that might help. If we are working with a system that has three constraints such that our feasible space is inside this triangle.
Then, if we look at any point on the interior of our space
We see that we can improve (get a more extreme value of) our function by moving closer to one of the bou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2502781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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show that $\sum_{k=1}^{n}(1-a_{k})<\frac{2}{3}$ Let $a_{1}=\dfrac{1}{2}$, and such $a_{n+1}=a_{n}-a_{n}\ln{a_{n}}$,show that
$$\sum_{k=1}^{n}(1-a_{k})<\dfrac{2}{3}$$
My attemp: let $1-a_{n}=b_{n}$,then we have
$$b_{n+1}=b_{n}+(1-b_{n})\ln{(1-b_{n})}<b^2_{n}<\cdots<(b_{1})^{2^{n}}=\dfrac{1}{2^{2^n}}$$
where use $\ln{(1+... | We can prove that $0< a_{n}< 1$ inductively by making use of the graph$:\quad y= x\left ( 1- \ln x \right ).$ We let
$$b_{n}:= 1- a_{n}, \left \{ b_{n} \right \}_{n= 1}^{\infty}\Leftrightarrow b_{1}= \frac{1}{2}, b_{n+ 1}= b_{n}+ \left ( 1- b_{n} \right )\ln\left ( 1- b_{n} \right )$$
Well$,\quad a_{n+ 1}- a_{n}= -a_{n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Prove that: $1-\frac 12+\frac 13-\frac 14+...+\frac {1}{199}- \frac{1}{200}=\frac{1}{101}+\frac{1}{102}+...+\frac{1}{200}$ Prove that:
$$1-\frac 12+\frac 13-\frac 14+...+\frac {1}{199}- \frac{1}{200}=\frac{1}{101}+\frac{1}{102}+...+\frac{1}{200}$$
I know only this method:
$\frac {1}{1×2}+\frac {1}{2×3}+\frac {1}{3×4... | My method :
$$\left\{ 1-\frac {1}{2}-\frac {1}{4}-...-\frac {1}{128} \right\}+\left\{ \frac {1}{3}-\frac {1}{6}- \frac{1}{12}-...- \frac{1}{192}\right\}+\left\{\frac {1}{5}-\frac{1}{10}-\frac{1}{20}-...- \frac{1}{160}\right\}+...+\left\{ \frac{1}{99}-\frac{1}{198}\right\}+\left\{ \frac{1}{101}+\frac{1}{103}+\frac{1}{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2502982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
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Linear Algebra Challenge Question involving Vector Spaces. Problem:
Let $V$ be the space of continuously differentiable maps $f:\mathbb{R}\to\mathbb{R}$ and let $W$ be the subspace of those maps $f$ for which $f(0)=f^\prime(0)= 0.$ Let $Z$ be the subspace of $V$ consisting of maps $x\to ax+b$, with $a,b \in \mathbb{R}$... | Hint Suppose there is a decomposition $V = W \oplus Z$, so that for any $v(x) \in C^1(\Bbb R)$ we have
$$v(x) = w(x) + z(x)$$
for $w(x) \in W, z(x) \in Z$.
Now, we want to express $w(x)$ and $z(x)$ in terms of $v(x)$, and the only information available is the definitions of $W, Z$. Since $W$ is characterized by the eva... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Airy's equation 2nd order ODE Suppose we have the equation
$$ \frac{ d^2 y}{d z^2}= z\, y $$
for function $y(z).$
We would like to find the 6 first coefficients of the equation given the boundary conditions
$\dfrac{dy}{dz}=1$ at $ z=0,y=2,\, y(z)=a_0+a_1z+a_2z^2+a_3z^3+...+a_5z^5.$
I've arrived at the recurrence re... | You have a third order recurrence relation for the coefficients. Since the recursion is third order, it needs three initial conditions. $a_0$ and $a_1$ come directly from $y(0),y'(0)$. (If you had different boundary conditions, the problem would turn out differently.)
To find $a_2$, notice that putting the expansion in... | {
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Understanding Laurent and Taylor series I know that we should use Laurent series to expand a function around a singularity and Taylor series otherwise. But there is a few aspects that I don´t understand.
Imagine $\sin\left(\frac{1}{z}\right)$.
I know $\sin\left(\frac{1}{z}\right)$ has an essential singularity at $z=0$... | $ \sin z = \displaystyle \sum_{n=0}^{\infty} \frac{ (-1)^n z^{2n+1}}{(2n+1)!}$ if $|z|< \infty$
Then if $|z| < \infty \rightarrow \displaystyle 0<{|\frac{1}{z}|} < \infty$
$ \sin{\frac{1}{z}} = \displaystyle \sum_{n=0}^{\infty} \frac{ (-1)^n}{z^{2n+1}(2n+1)!}$
And $0$ is a essential singularity because we have infin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Hermitian form induces an isomorphism between $V$ and $V^*$? Let $H(-,-):V \times V \rightarrow \mathbb{C}$ be an Hermitian form, linear in the first factor. My question is, how can one show that this induces an isomorphism from $V$ to it's dual space $V^*$? Here's what I've tried so far:
We must check that $H(-,v):V \... | There's nothing wrong in your derivation. Actually, what the Hermitian form $H$ induces is a conjugate linear bijection, not a vector space isomorphism. However, in some sense we can say that $V$ and $V^*$ are "the same". In an infinite-dimensional Hilbert space $H$, we have analogous one-to-one correspondence between ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proofs on Convex Sets.
Prove that the set $C:=\{ x : x+S_2 \subset S_1 \}$, with $S_1,S_2\subset \Bbb R^n$ is convex if $S_1$ is convex.
I understant that a vectorial space is a convex set. So $S_1$ and $S_2$ are both convex sets. But I do not understand how to continue with the idea.
| Suppose $x, y \in C$ and $0 \le t \le 1$. You want to show $tx + (1-t)y \in C$, i.e. $tx + (1-t)y + S_2 \subset S_1$.
If $s_2 \in S_2$, $tx + (1-t) y + s_2 = t (x + s_2) + (1-t) (y + s_2)$ with
$x + s_2 \in S_1$ and $y + s_2 \in S_1$. Since $S_1$ is convex, you are done.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Difficulty with a "trivial" probability question: drawing balls from an urn --- is event probability affected by order in which balls are drawn? Say you have an urn with $2$ red balls, $2$ white balls, and $2$ black balls.
Draw $3$ balls from this urn.
What is the probability that all balls have a different colour?
I ... | There is yet another way to get the probability:
label the two balls of each color differently (as in your method 2)
so that each of the six balls is uniquely identified and
you can distinguish exactly which three of them were selected,
but do not consider the order in which the three balls were drawn.
There are $\bin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How can I prove $\begin{equation*} \lim_{n \rightarrow \infty} \frac{n}{\sqrt{n^2 + n+1}}=1 \end{equation*}$ How can I prove(using sequence convergence definition):
$$\begin{equation*} \lim_{n \rightarrow \infty} \frac{n}{\sqrt{n^2 + n+1}}=1 \end{equation*}$$
I need to cancel the n in the numerator, any hint will be a... | Note that$$\lim\limits_{x\to\pm\infty}\left(\frac 1{x}\right)^n=0$$With simple limits to infinity type problems, we can divide both the numerator and denominator by the greatest power and take the limit as each term tends towards infinity$$\begin{align*}\lim\limits_{x\to\infty}\frac x{\sqrt{1+x+x^2}} & =\lim\limits_{x\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove a property of the weight decomposition of representation of $\mathfrak{sl}(2,\mathbb{C})$ I try to prove the following property of complete decomposition of the representation of $\mathfrak{sl}(2,\mathbb{C})$:
Show if $V$ is a finite-dimensional representation of $\mathfrak{sl}(2,\mathbb{C})$, then
$$V\cong \... | I assume that $V_k$ are the standard irrep of weight $k$. Then, consider multiplication by $x : V[k] \to V[k+2]$. Clearly, $x$ is surjective and moreover, $x(v) = 0$ iff $v$ is a heighest weight vector of weight $k$. So $\dim \ker(x) = \dim V[k] - \dim V[k+2]$ is indeed $n_k$.
Here is an intuitive explanation : there a... | {
"language": "en",
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Every day a student solves one, two or three problems. Find the number of distinct ways
Every day a student solves one, two or three problems. Find the number of distinct ways
a) he solves problems in 30 days, ... | Hint. As regards b) note that if for $d_i$ days he solves $i$ problems for $i=1,2,3$ then $d_1+2d_2+3d_3=50$ and this can be done in
$$\frac{(d_1+d_2+d_3)!}{d_1!\cdot d_2!\cdot d_3!}.$$
For c) we have another constraint: $d_1+d_2+d_3=30$.
P.S. Generating function approach. The answers for b) and c) are given respective... | {
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Irreducible cubic curve in normal form I have been given a cubic curve $F : X^3 - XZ^2 + YZ^2 = 0$ in the projective plane. I have that a singular point of the curve is $[0,1,0]$. I am asked to find a change of variable which puts $F$ into 'normal form', i.e $Y^2Z - G(X,Z)$ where $G(X,Z) = X^3 + bX^2Z + cXZ^2 + dZ^3$ f... | There is an "algorithm" for transforming cubic curves to the (long) Weierstrass form, and from that to the short Weierstrass form, which is usually introduced when studying elliptic curves. You can find it in its various forms here or here. But that involves a lot of tedious work.
Here is how I'd solve it manually: Us... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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pigeonhole principle, for a chessboard problem. assume we have a chess-board 8 by 8 squares.
lets define the "prince" play tool:
the "prince" can move from any given square to any other square the Queen can move to or to any other square the Horse can move to.
means that if from some square on the chess board the set o... | HINT
Note that a prince ends up attacking all of the 24 squares around it with itself being in the middle of a $5 \times 5$ square. So think of it how you can place such $5 \times 5$ squares on a board, noting of course that a prince placed along the side of the board threatens less than $24$ actual squares of the boar... | {
"language": "en",
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Help solving variable separable ODE: $y' = \frac{1}{2} a y^2 + b y - 1$ with $y(0)=0$ I am studying for an exam about ODEs and I am struggling with one of the past exam questions. The past exam shows one exercise which asks us to solve:
$$y' = \frac{1}{2} a y^2 + b y - 1$$with $y(0)=0$
The solution is given as
$$y(x) =... | You complete the square $\frac12ay^2+by-1=\frac12a(y+\frac ba)^2-1-\frac{b^2}{2a}$ and use this to inspire the change of coordinates $u=ay+b$ leading to
$$
\int \frac{dy}{\frac12ay^2+by-1}=\int\frac{2\,du}{u^2-2a-b^2}
$$
and for that your integral tables should give a form using the inverse hyperbolic tangent. Or you p... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does a partition of unity has the finite intersection property of compact sets? I have a question regarding partitions of unity. I use the same definition and notation as 2. here (Wikipedia).
Let $K \subseteq X$ be a compact set. Why do we have that $K \cap \text{supp}\rho_j \neq \varnothing$ for only finitely man... | For each $x\in K$, there exists an open neighbourhood $U_x$ of $x$ where all but finitely many of the functions are $=0$. By compactness, $K$ is covered by finitely many of the $U_x$. All but the finitely many functions that are non-zero in at least one of these finitely many $U_x$ are $=0$ on all of $K$. This does not... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Clean way to expand product $\prod_{k=1}^n (1 + x_k)$ Is there a clean way to write the expansion of:
$$\prod_{k=1}^n (1 + x_k) = (1 + x_1)(1 + x_2)\dots(1 + x_n)$$
The expansion may be written:
$$1+\sum_{1\le i \le n} x_i +\sum_{i \le n}\sum_{j\lt i} x_j x_i + \sum_{i \le n}\sum_{j\lt i}\sum_{k\lt j} x_kx_jx_i+\cdots... | As explained in Richard Stanley book we have:
$$\prod_{k=1}^n (1 + x_k) = (1 + x_1)(1 + x_2)\dots(1 + x_n)
=\sum_{A\subseteq [n]}\prod_{i\in A}x_i$$
where $\displaystyle{\prod_{i\in\varnothing}x_{i}:=1}$ as Ethan remarks.
More generally, you can consider even the multiset case:
$$\prod_{k=1}^n (1 + x_k+x_k^{2}+\cdots) ... | {
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Let $\gamma=[-1, 1+i]+\beta+[-1+i,1]$, where $\beta(t)=i+e^{it}$ for $0\leq t\leq 3\pi.$ Compute: $\int_{\gamma}(z^2+1)^{-1}dz$ Let $\gamma=[-1, 1+i]+\beta+[-1+i,1]$, where $\beta(t)=i+e^{it}$ for $0\leq t\leq 3\pi.$ Compute: $\int_{\gamma}(z^2+1)^{-1}dz$
I have come to the following but I do not know what else to do:
... | Let's make this problem easier by deforming your contour $\gamma$ into $\gamma^*$ where $\gamma^*=[-1,0]+\beta^*+[0,1]$ and $\beta^*=i+e^{it}$ for $-\frac{\pi}{2}\leq t \leq \frac{7\pi}{2}$ (draw a picture!). We can do this because our integrand is analytic everywhere besides $z=\pm i$. Now we have
\begin{align}
\in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the largest interval for solution If $y(x)$ is the solution of differential equation$\frac {dy}{dx}=2(1+y)(\sqrt y)$ satisfying $y(0)=0,y(\frac {\pi}{2})=1 $ then the largest interval (to the right of origin ) on which the solution exists is
(a)$[0,\frac {3\pi}{4}]$
(b)$[0,\pi)$
(c)$[0,2\pi)$
(d) )$[0,\frac {2\pi... | The curve $$y=\tan^2 x\qquad(x\geq0)\tag{1}$$
and all its horizontal translates are solution curves of this ODE. To this family of curves its envelope $y(x)\equiv0$ has to be added, since it is a solution as well. But this is not all: Since the ODE does not satisfy a crucial assumption of the existence and uniqueness t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2505434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Asymptotic solutions to ODE Wondering where I can find resources to do more questions of the following type and also if you guys can help me answer this problem.
Consider the differential equation:
$$u'' + \left( 1-\frac{\gamma}{x^2} \right) u = 0$$
for $x > 4$. Obtain the first two terms of the asymptotic solution fo... | An expansion of $u $ derived from the perturbation theory would be $u=u_0 +\gamma u_1 + \dots$ The $0$th-order and $1$st-order equations (in terms of powers of $\gamma$) are respectively
\begin{aligned}
u''_0 + u_0 &= 0 \, ,\\
u''_1 + u_1 &= \frac{u_0}{x^2} \, .
\end{aligned}
The $0$th-order solution is $$u_0(x) = a_0\... | {
"language": "en",
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Can we prove that $1/\sqrt2\in$ Cantor set on $[0,1]$? Today, I was trying to prove Cantor set is uncountable and I completed it just a while ago.
So, I know that the end-points of each $A_n$ are elements of $C$ and those end-points are rational numbers. But since $C$ is uncountable, $C$ must contain uncountable number... | Any number in $[0, 1]$ which has (at least one) base $3$ expansion without a $1$ will be in the Cantor set. More precisely,
$$\mathcal{C} = \left\{ \sum_{n=1}^{\infty} \frac{c_n}{3^n} : (\forall n)( c_n \in \{ 0, 2 \} ) \right\}.$$
It is known that a number as above is irrational if and only if the expansion is non-rec... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proving that $s$ is a reflexive closure of $r$ I have this problem that I cannot figure out. Could you please help me out with this?
Let $r$ be a relation on the set $X$ and let $$R:=\{t:t \text{ is a reflexive relation on } X \text{ with } r\subseteq t \}.$$ For $x,y \in X$ we have $x \, s \, y$ iff for each $t\in R$... | Yeah, this has some problems.
Since $r\subseteq t$ for all $t\in R$, it follows $r \subseteq s$ and also $s \subseteq t$.
How does it follow that $r \subseteq s$? That's not clear.
Fortunately, to show that $s$ is reflexive we don't need it ... yes, we need that $s$ is reflexive, but given its definition that's the e... | {
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How many Functions/ Bijections X → X do exist, if the set X has 4 elements? So I belive the amount of functinos is $4^4$. But I can't seem to find the right explaination to prove it.
My idea was, that
$a\mapsto a,b,c,d$
$b \mapsto a,b,c,d$
...
...
are the possible ways the elements are related.
... and if you change/... | Suppose $X = \{a,b,c,d\}$. Consider a bijection $f\colon X \rightarrow X$. You are looking for all the possible combinations of $f(a),f(b),f(c),f(d)$. Start with $f(a)$. You have $4$ possibilities for it, i.e. $a,b,c$ or $d$. Now $f(a)$ is fixed, so $f(b)$ will have to be different from $f(a)$ in order for $f$ to be in... | {
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Book recommendation about mathematics I'm looking for a book about mathematics, but not about calculus, algebra or any field of math. I want to read a book about the use of mathematics in the world, for example, Why is math so useful. Or maybe something about the point of view of a mathematician in the world or even a ... | A good idea-since you are after all a Maths student-would be "Mathematics Made Difficult" by Carl E. Linderholm.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2506120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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$\varepsilon$-$\delta$ definition of a limit [Calculus] $$\lim_{x\to 2} x^2 - 8x + 8= -4 $$
Now to express $ f(x) - l $ it as $x - x_0$ , we write $ x = (x - x_0) + x_0$.
So \begin{align}
f(x) - l &= x^2 - 8x + 8 + 4\\
&= (x - 2 + 2)^2 - 8(x - 2 + 2)+ 12\\
&= (x - 2)^2 +4(x-2) + 4 - 8(x - 2) - 16+ 12 \\
&= (x - 2)^2 - ... | No, it is not correct. If you choose $\delta\leqslant-\frac\varepsilon3$, then $\delta<0$. And it is part of the $\varepsilon-\delta$ definition of limite that $\delta>0$.
Note that the inequality $\bigl|f(x)-l\bigr|\leqslant|x-2|^2-4|x-2|$ is false, since it leads to situations in which $\bigl|f(x)-l\bigr|<0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2506205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show $\int_0^{2\pi}\cos(n\phi')\cos^l(\phi-\phi')\mathrm{d}\phi=\frac{2\pi}{2^l}\cos(l\phi)\delta_{l,n}$ I have to show $$\int_0^{2\pi}\cos(n\phi')\cos^l(\phi-\phi')\mathrm{d}\phi=\frac{2\pi}{2^l}\cos(l\phi)\delta_{l,n}$$
where $l,n$ are positive integers such that $l\leq n$
I'm supposed to use the fact
$$ \int_0^{2\p... | Use $\cos x=(e^{ix}+e^{-ix})/2$ to rewrite
$$
\int_0^{2\pi}\cos(n\phi')\cos^l(\phi-\phi')\mathrm{d}\phi'=\frac{1}{2^{l+1}}\int_0^{2\pi}(e^{in\phi'}+e^{-in\phi'})(e^{i(\phi-\phi')}+e^{-i(\phi-\phi')})^l d\phi'\ .
$$
Then use the binomial theorem to rewrite
$$
\frac{1}{2^{l+1}}\sum_{k=0}^l {l\choose k}\int_0^{2\pi}(e^{in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2506310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The limit of $f_n = \frac {e^x \sin(x) \sin(2x) \cdots \sin(nx)}{\sqrt n}$
I want to calculate a limit of $f_n = \frac {e^x \sin(x) \sin(2x) \cdots
\sin(nx)}{\sqrt n}$ if $n$ goes to an infinity.
I was wondering if I can use this: $ \lim_{n\to \infty} \frac {\sin(nx)}{n} = 0$, because I have doubts.
I know that $$ ... | The pointwise convergence towards zero is trivial, we may actually prove we have uniform convergence over any compact subset of $\mathbb{R}$.
Let
$$ g_n(x) = \sin(x)\sin(2x)\cdots \sin(nx). $$
The supremum of $g_n$ over $\mathbb{R}$ is attained at a point of the interval $\left(0,\frac{\pi}{n}\right)$. Due to the appro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2506439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Compute $m$ so that the polynomial $R(x)$ has a root at $x=-3$ Apologies upfront, this is quite an elementary question, but I'm stuck here.
I'm asked to figure out the value of $m$ so that this polynomial: $$R(x)=x^2-mx+3$$ has a root at $x=-3$.
Following the definition of the root of a polynomial, $x=-3$ is a root if ... | It looks like you incorrectly performed your original calculation using the definition of a root of a polynomial. The calculation should look like this:
$$(-3)^2 - m(-3) + 3=0$$
$$9 + 3m + 3 = 0$$
$$12 + 3m = 0$$
$$12 = -3m$$
$$m=-4$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2506667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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verfiy my proof (Prove that for all sets A, B, C, $(A \cup B)-C = (A-C) \cup (B-C)$) Prove that for all sets A, B, C, $(A \cup B)-C = (A-C) \cup (B-C)$
suppose $x \in A,B,C$
if $x\in A$ or $B$ or both , then by definition of union, $x\in A\cup B$.
Since $x\in (A \cup B) $ and $x\in C$, by definition of subtraction, $x\... | after rethinking my proof i came up with:
suppose $x \in (A \cup B) -C$
if $x \in (A \cup B)-c $ then $x \notin C$ this $x \in A$ and/or $x\in B$
Since $x \notin C$ and $x\in (A \cup B)-C$ , $x \in (A-C)$ and/or $x \in (B-C)$
thus if $x \in (A-C)$ and/or $x \in (B-C)$ then $x \in (A-C) \cup (B-C)$
by the definition o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2506790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Divergence of reciprocal of primes, Euler On Wikipedia at link currently is:
\begin{align}
\ln \left( \sum_{n=1}^\infty \frac{1}{n}\right) & {} = \ln\left( \prod_p \frac{1}{1-p^{-1}}\right)
= -\sum_p \ln \left( 1-\frac{1}{p}\right) \\
& {} = \sum_p \left( \frac{1}{p} + \frac{1}{2p^2} + \frac{1}{3p^3} + \cdots \righ... | Hint :
$$\sum_p \frac {1}{p^s} < \sum_{n=1}^{\infty} \frac 1{n^s} = \zeta(s) $$
$\zeta(s)$ is Riemann Zeta function, and $\zeta(s)$ converges for each $s \in \Bbb R, s>1$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Question on the proof that a closed subset of a compact set is compact I was reading the proof here on the claim that a closed subset of a compact set is compact, which reads:
Say F ⊂ K ⊂ X where F is closed and K is compact. Let $\{V_α\}$ be an open
cover of F. Then $F^c$ is a trivial open cover of $F^c$. Consequen... | You don't have to know that $V_{alpha}$ doesn't cover $K$, if it happens to cover $K$ then still adding $F^C$ wouldn't alter anything - the resulting cover would still cover $K$.
Now of course in the subcover of $K$ including $F^C$ we must know that the subcover excluding $F^C$ would cover $F^C$, but this is rather obv... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What are vertex fields, gradient and divergence on graphs?
I have a few questions these two slides on the topic of calculus on graphs:
*
*What are the vertex fields defined here? My understanding is that it is a set of functions that takes in a vertex and gives a real number output. And because each vertex may nee... | *
*A vertex field is a square-summable function from the set of vertices into $\mathbb{R}$. (If there are only finitely many vertices, saying "square-summable" is unnecessary.) Imagine a graph, say the triangle $K_3$. Put a number next to each vertex, say $3, 6, -2$. You have a vertex field.
*The concept of inner pr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Simple group problem. I am reading a book of algebra and I have a difficulty in understanding one thing.
In the book it is written there is no simple group of order 528 and the explanation is as follows.
Let $H$ be a Sylow 11-subgroup. Then $n_{11}=12$, which I understood why, and $|N(H)|=44$ and $G$ is isomorphic to a... | For the subgroup of $A_{12}$ part, note that $G$ acts transitively by conjugation on the $12$ Sylow $11$-subgroups. Consider the Kernel of this action - it is a normal subgroup and the action is non-trivial, so given $G$ is simple, the Kernel must be the trivial subgroup, and the image is isomorphic to $G$. The image i... | {
"language": "en",
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Is the derived category of a full subcategory full? Certainly not, but I cannot find a good counterexample. I tried to do something like $\mathbf Z\text{-Mod}\subseteq \mathbf Z[x]\text{-Mod}$, but without success. Does anyone have a short counterexample?
| Thanks to the comments by @MarianoSuárez-Álvarez, I can formulate the following answers:
Consider a field $k$ and the ring $k[x]$. Of course, $\hom_{D(k)}(k, k[i])=\begin{cases}k & \text{if i=0,}\\0& \text{o/w}\end{cases}$ in the derived category $D(k)$. Now in the category $k[x]\text{-Mod}$, there is a nontrivial exte... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Trying to solve differential equation $y'=\frac{3x-y+4}{x+y}$ ÊDIT: Found the second mistake (failed by calculating $u_{1,2}$)!
I might have made a mistake, however, I am not able to detect it. Here we go:
$$y'=\frac{3x-y+4}{x+y}, \quad y(1)=1$$
1.) set $x = X+a$ and $y=Y+b$, so $$\frac{dY}{dX} = \frac{3X-Y+(3a-b+4)}... | Your mistake started from 6 (I'm writing $c=C^2$ here)
$$ u^2 + 2u = 3 - \frac{1}{cX^2} $$
$$ (u+1)^2 = 4 - \frac{1}{cX^2} $$
$$ u= -1 \pm \sqrt{4-\frac{1}{cX^2}} $$
or
$$ \frac{y-1}{x+1} =-1 \pm \sqrt{4-\frac{1}{c(x+1)^2}} $$
Using the condition $x = 1, y = 1$ we get
$$ -1 \pm \sqrt{4-\frac{1}{4c}} = 0 $$
Only the pl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2507701",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove this sequent by natural deduction? How do I prove
$$\forall x\forall y\forall z(S(x, y)\land S(y, z) \Rightarrow S(x, z)), \forall x\neg S(x, x) \vdash \forall x\forall y(S(x, y) \Rightarrow \neg S(y, x)).$$
by natural deduction?
1 $\quad \forall x\forall y\forall z(S(x, y)\land S(y, z) \Rightarrow S(x, z)... | Hint: Assume $S(x,y)$ and $S(y,x)$. Then $S(x,x)$ by the first premise, contradicting the second premise.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2507808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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How to take a partial derivative of $\|y - Xw\|^2$ with respect to w? So I've tried to solve this problem where we are asked to solve the partial derivative of function $\sum(y - Xw)^2$ or $\|y-Xw\|^2$ and then minimize it. I've never done any linear algebra aside some really basic stuff and I can't seem to find any i... | An easy way to do it is write it like scalar product and then use the properties of the scalar product. I assume you are working with real matrix and vectors (it really seems that you are dealing with Ordinary Least Squares).
\begin{align}
\|y-Xw\|^2 &=\langle y-Xw,y-Xw \rangle \\
&=y^Ty-2w^TX^Ty+w^TX^TXw.
\end{align}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2507923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Showing that ${1 + i,−1 + i}$ is a basis for the vector space $\mathbb C$ over $\mathbb R$ I feel my justification for this is weak, and I'm seeking for improvement.
$$Span((1+i), (-1+i)) = a(1+i) + b(-1+i)$$
I have two conclusions from this:
1.
$$a(1+i) + b(-1+i) = a + ai -b + bi$$
$$a(1+i) + b(-1+i) = (a - b) + i (a ... | By definition, the numbers $1$ and $i$ are a basis the vector space $\mathbb C$ over $\mathbb R$, so this is a two-dimensional $\mathbb R \text{-}$vector space.
In general, a nonzero vector $\vec{w}$ is linearly independent from a nonzero $\vec{v}$ iff for any $\lambda$, $\; \vec{w} \ne \lambda \vec{v}$. In particular... | {
"language": "en",
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"question_score": "2",
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Calculating variance for a sequence of i.i.d. variables which seems non-partitionable
Let $(Z_n : 1 \leq n < \infty )$ be a sequence of independent and identically distributed (i.i.d.) random variables with $$\mathbb{P}(Z_n=0)=q=1-p,\quad \mathbb{P}(Z_n=1)=p.$$ Let $A_i$ be the event $$\{Z_i=0\}\cap\{Z_{i-1}=1\}.$$ I... | The following is an incorrect treatment (incorrectly assuming independence)... just leaving in place for reference.
Let $J$ be the number of zeros in the sequence, not counting the first element. $J$ is binomial with $n-1$ trials and probability q.
For a sample with $j$ zeros beyond the first element, the number that p... | {
"language": "en",
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Final step in proof of the Hellinger-Toeplitz Theorem Consider the following statement:
(Hellinger-Toeplitz Theorem) Let $H$ be a Hilbert space and $A : H \to H$ be linear and symmetric,
i.e. $$\langle x,Ay \rangle = \langle Ax,y \rangle$$ holds for all
$x,y \in H$. Then $A$ is bounded.
I prove this using the Ban... | Because $\varphi_{x/ \|x\|}(A(x)) \leq c \|A(x)\|$ where $c = \sup_{ T \in \mathcal{F}} \|T\|$ we have
$$
\|A(x)\|^2 \leq c \|x \| \|A(x)\|.
$$
Then dividing by $\|A(x)\|$ gives you that $A$ is bounded.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2508235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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How do I use Leibniz formula to solve this difficult equation? Suppose there exists a $y$ such that $$y \equiv \frac{d^n}{dx^n}e^{-x^2/2}$$.
Prove that $$\frac{d^2y}{dx^2} + x\frac{dy}{dx} + (n+1)y = 0$$
I'm not sure where to start as Leibniz formula require at least 2 functions to begin with. There are no clear two f... | Using the product rule iteratively, we have
\begin{eqnarray*}
x \frac{d^{n+1}}{dx^{n+1}} e^{-x^2/2} &=&\frac{d}{dx} \left(x \frac{d^n}{dx^n} e^{-x^2/2} \right) - \frac{d^{n}}{dx^{n}} e^{-x^2/2} \\
&=&\frac{d^2}{dx^2} \left(x \frac{d^{n-1}}{dx^{n-1}} e^{-x^2/2} \right) - 2\frac{d^{n}}{dx^{n}} e^{-x^2/2} \\
& \vdots & \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Let $\sum_{n=-\infty}^{\infty}a_{n}(z+1)^n$ be the laurent's series expansion of $f(z)=\sin(\frac{z}{z+1})$. Then $a_{-2}$ is
Let $\sum_{n=-\infty}^{\infty}a_{n}(z+1)^n$ be the laurent's series
expansion of $f(z)=\sin(\frac{z}{z+1})$. Then $a_{-2}$ is
(a)$1$
(b)$0$
(c)$\cos(1)$
(d)$-\frac{1}{2}\sin(1)$
Reference: C... | HINT:
$$\sin\left( \frac {z}{z+1}\right)=\sin(1)\cos\left( \frac1{z+1}\right)-\cos(1)\sin\left( \frac1{z+1}\right)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2508490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Olympiad inequality
Let $a,b,c,d \in [0,1]$.
Prove
$$ \frac {a}{1+b}+\frac{b}{1+c}+\frac{c}{1+d}+\frac{d}{1+a}+abcd\le3$$
Hi. Im a school student so ive been unable to look for ways to apply this problem. I attempted to use AM-GN on 1+a etc but the bound $2\sqrt{a}$ was too large and was easily larger than 3. Th... | We need to prove that
$$\sum_{cyc}\left(\frac{a}{1+b}-a\right)+abcd\leq3-a-b-c-d$$ or
$$3-a-b-c-d+\sum_{cyc}\frac{ab}{1+b}\geq abcd.$$
Now, by C-S and the given condition $$\sum_{cyc}\frac{ab}{1+b}=\sum_{cyc}\frac{abcd}{cd+bcd}\geq\frac{16abcd}{\sum\limits_{cyc}(cd+bcd)}\geq\frac{16abcd}{8}=2abcd.$$
Thus, it's enough t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A nonzero unital division ring $D$ A nonzero unital ring $D$ in which every nonzero element is
invertible is called a division ring.
I think when the ring is divided, the following two are equivalent?Is it right?
1: For all $a, b \in D$ with $a \neq 0$, the equations $ax = b$ and $ya = b $ have
unique solutions i... | As I showed you here, (2) implies there are only trivial right ideals.
Then here I showed you that as long as multiplication isn't zero ($D^2\neq \{0\}$) that having only trivial right ideals implies $D$ has an identity.
Then obviously there are no proper left ideals either, and $ya=b$ has a solution when $a$ is nonzer... | {
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"timestamp": "2023-03-29T00:00:00",
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Counterexample to Bolzano-Weierstrass in infinite dimension From Bolzano-Weirerstrass we can demonstrate that in a normed vector space $E$ of finite dimension, every bounded sequence admits a limit point.
What are some counterexamples in infinite dimension? Does there exist a counterexample in every infinite dimensiona... | Yes, that's the obvious counter example. You don't need to require finite support, when using supremum norm you only need the sequences to be bounded. Let the $j$th sequence be $k\to \delta_{jk}$ where $\delta_{jk}$ is the Kronecker delta. Every element in the sequence has norm $1$ and the distance between any two ele... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Show that $n(n^2-1)$ is divisible by 24, if $n$ is an odd integer greater than $2$. How can I show that $n(n^2-1)$ is divisible by 24, if $n$ is an odd integer greater than $2$?
I am thinking that since odd numbers have the form of $2n-1$ in which if it is to be more than $2$, it will be $2n-1+1 = 2n+1$. So would it be... | Easier with congruences:
*
*$n^3-n\equiv 0\mod 3$ for all $n$ (that's Little Fermat's theorem),
*If $n$ is odd, $n\equiv \pm 1,\pm3\mod 8$, so $\;n^2\equiv 1\mod8$,
*last, use the Chinese remainder theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2508927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
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Distribution function related with the limit from left show that the distribution fun $F$ of random variable $f$ takes only values $0$ and $1$ iff there exists areal number $c$ such that $P(f=c)=1$ ??
I begin solve the ex and i prove $F$ takes two value $0$ and $1$ (one side )
another side I had problem
suppose $F$ ... | Hint:
Define $c:=\inf\{x\in \mathbb R\mid F(x)=1\}$
Then, since $F$ is continuous from the right, we have $F(c)=1$.
What can be said about $F(x)$ if $x<c$?
What does that say about $P(f=c)$?
| {
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"url": "https://math.stackexchange.com/questions/2509057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Piecewise step function for changing dollar amounts? Visiting the Wikipedia article for continuous functions I found this:
As an example, consider the function h(t), which describes the height
of a growing flower at time t. This function is continuous. By
contrast, if M(t) denotes the amount of money in a bank acc... | The jumps come in increments of time, not in increments of dollars. You can use the floor or ceiling to describe the function. Say I open a bank account for $100$ that pays $5$ simple interest every year. Let $t$ be the time in years. The account value is then $100+5\lfloor t \rfloor$. You can also use the floor a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to express the size of a dimension of a matrix Let's say I have a matrix $A\in \mathbb{C}^{4\times6}$.
How can I express mathematically that I want to input $A$ and output $4$ as an answer, for example?
In code we might do size(A), A.shape, A.RowCount(), depending on the computer language we are using. But what abo... | You would say that's the number of rows of the matrix $A$.
EDIT: If roundabout ways are okay, one route is to identify a $m\times n$ matrix $M$ as some linear transformation $T:\mathbb{R}^n\longrightarrow\mathbb{R}^m$.
Then we can say $\dim(\text{Codomain}(M))=m$ and $\dim(\text{Domain}(M))=n$.
If you think thre's no a... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Intermediate value theorem for $\sin x.$ How is intermediate value theorem valid for $\sin x$ in $[0,\pi]$?
It has max value $1$ in the interval $[0,\pi]$ which doesn't lie between values given by $\sin0$ and $\sin\pi$.
| In general, any time you have a theorem, "If ____, then ____," it really means, "If ____, then ____ and maybe some other stuff not mentioned here also happens."
Because when you're working in just about any branch of mathematics,
no matter how thoroughly you describe the implications of any mathematical fact there is a... | {
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Weird Combinatorial Identitiy A friend of mine came across this rather odd combinatorial identity. We've spent a while but haven't been able to prove it. Any ideas?
The following holds exactly for even integers $n$, and is approximately true for odd integers $n$: $$n = \dfrac{n+1}{n^n - 1} \sum_{k=1}^{n/2} \dbinom{n-k}... | So we want to prove that for even $n$
$$
\frac{n^n-1}{n+1}=\sum_{l\ge0}\binom{n-1-l}l n^l(n-1)^{n-1-2l}.
$$
The RHS is the coefficient of $t^{n-1}$ in $\frac1{1-(n-1)t-nt^2}$. But
$$
\frac1{1-(n-1)t-nt^2}=\frac1{(1+t)(1-nt)}=(1-t+t^2-\ldots)(1-nt+n^2t^2-\ldots),
$$
so this coefficient is a sum of a geometric progressio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2509693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Finding $cdf$ of the sample minimum, $X_{(1)}$
Consider iid random variables $X_1$ and $X_2$, having $pdf$ $$f_X(x) =
4(1−2x)I_{(0,1/2)}(x)$$ Give the $cdf$ of the sample minimum,
$X_{(1)}$.
$$\begin{align*}
F_{X(1)}(x)
&= P(X_{(1)} \leq x) \\\\
&= 1 - P(min{\{X_1, X_2}\} \gt x) \\\\
&= 1 - P(X_1 \gt x, X_2 \gt x... | seems good to me:
Just a baseline check: $F(1/2) = 1$, and as long as you understand that this is valid for $x \in (0, 1/2)$ we are ok. What would the values of $F$ be for negative $x$ and $x \ge 1/2$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2509739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that $f\in C([-1,1],\mathbb{R})$ and that $\int_{-1}^{1}f(x)\,dx=0$. Consider $X=C([-1,1],\mathbb{R})=\{f:[-1,1]\rightarrow \mathbb{R}\,:\,f \text{ is continuous}\}$ with the supremum metric defined by $$d(f,g)=\text{sup}_{x\in[-1,1]}|f(x)-g(x)|$$ for all $f,g \in X$.
Show that if $(f_n)_{n=1}^{\infty}$ is a sequ... | If $(f_n)_{n \in \mathbb{N}}$ converges in $(X,d)$ to some function $f$ that means:
$$\sup_{x \in [-1,1]} |f_n(x)-f(x)| \stackrel{n \rightarrow +\infty}\rightarrow 0$$
which implies $|f_n(x)-f(x)| \rightarrow 0$ for any $x \in [-1,1]$. This is called uniform convergence (because you are not fixing $x$ in your domain, a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2509861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Projection onto the second-order cone I'm having difficulties in proving that the projection of $$(s,y)\in R \times R^{n}$$ onto the second-order cone $$Q^{n+1} = \{(t,x) \in R \times R^n : \|x\|_2 \leq t \}$$ is $$ \frac{s+\|y\|_2}{2\|y\|_2} (\|y\|_2,y)$$ when $\|y\|_2 > s $ and $ \|y\|_2 > -s $.
I tried to first sho... | Let $(t^*,x^*)$ be the projection.
The optimality condition implies that for any $(t,x) \in Q^{n+1}$ we must have
$$\langle(t-t^*, x-x^*), (s-t^*, y-x^*)\rangle \le 0.$$
Applying this inequality to $(t,x)=(0,0)$ and $(t,x) = 2(t^*,x^*)$ shows that in particular we have
$$\langle(t^*, x^*), (s-t^*, y-x^*)\rangle = 0\tag... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2509986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Determine galois group $Gal \Big(\frac{\mathbb{Q}(\sqrt[3]{3},\sqrt{-3})}{\mathbb{Q}}\Big)$ I've had some hard time determining Galois group $Gal \Big(\frac{\mathbb{Q}(\sqrt[3]{3},\sqrt{-3})}{\mathbb{Q}}\Big)$ because I didn't know exactly how to compute the order of the elements. See here for the computation of the o... | Wouldn't it be simpler to remark that by definition, $\omega$ is a primitive cubic root of $1$, so that your field $K$ is just the splitting field of the polynomial $X^3 - 3$ ? As such, $K/\mathbf Q$ is normal, with Galois group $G$ isomorphic to the permutation group of the roots, so $G\cong S_3 \cong D_6$, generated ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2510098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Unique Cyclic Subgroups If we let $$S = \{ e^{q\pi i} : q\in Q \} $$
Prove that for each $ n \ge 1$ there is a unique cyclic subgroup of order $n$ in $S$ and the union of these cyclic subgroups is $S$. Any help on this?
| The elements of $S$ are complex numbers. The group operation is complex multiplication and the identity element is $1 = e^{2\pi i}$. So $x \in S$ lies in a copy of $\mathbb Z/ n \mathbb Z$ if and only if it is an $n$th root of unity. We have explicit formulas for the roots of unity. This will prove existence. From the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2510208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Every order interval in $l^1$ is norm compact Let $l^1$ denote the space of sequences $(x_n)\subset \mathbb{R}$ with $\Vert (x_n)\Vert_1:=\sum_{n\geq 1} |x_n|<\infty$. We say that $(x_n^1)\leq (x_n^2)$ whenever $x_n^1\leq x_n^2$ for every $n\in\mathbb{N}$. It is well-known that $(l^1,\Vert\cdot\Vert_1)$ is a Banach sp... | We may consider the set $S=[0, (x_n)]$ only without loss of generality.
Let $(y_n^k) \in S$. We want to show that it has a convergent subsequence in $S$.
A short but not elementary proof: The set $S$ is compact in the product topology. Thus, there is a point wise convergent subsequence $(y_n^{k_m})$ with a limit $(y_n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2510328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
If $AT=TA$ with $A\geq0$. Why $A^{1/2}T=TA^{1/2}$? Let $\mathcal{H}$ be a complex Hilbert space. Let $T\in \mathcal{B}(\mathcal{H})$ and let $A\in \mathcal{B}(\mathcal{H})^+$ (i.e. $A^*=A$ and $\langle Ax\;| \;x\rangle \geq0,\;\forall x\in \mathcal{H}$).
Assume that $AT=TA$. Why $A^{1/2}T=TA^{1/2}$?
Thank you
| You can check that $p(A)T=Tp(A)$ for any polynomial $p$.
For any continuous function $f$ on $\sigma(A)$, there is a sequence $\{p_n\}$ of polynomials such that $p_n$ is convergent to $f$ uniformly. Hence $p_n(A)$ is convergent to $f(A)$ and thus $f(A)T=Tf(A)$.
Specially, $f(x)=\sqrt{x}, x\in \sigma(A)\subset [0,\inft... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2510496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to calculate $a,b,c,d$ given the eigenvectors
Given the matrix
$$A=\begin{bmatrix}
a & b & 2 \\
c & d & 6 \\
3 & 4 & -3
\end{bmatrix}$$
with eigenvectors
$$v_1=\begin{bmatrix}
5 \\
1 \\
3
\end{bmatrix}
\quad\text{and}\quad v_2=\begin{bmatrix... | $A*V_1 = \lambda*V_1$
$A*V_1$ is
\begin{bmatrix}
5a+b+6 \\
5c+d+18 \\
10 \\
\end{bmatrix}
$\lambda*V_1$ is
\begin{bmatrix}
5*\lambda \\
1*\lambda \\
3*\lambda \\
\end{bmatrix}
On equating, we see that $\lambda = \frac{10}{3}$
Similarly solve for other eige... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2510634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Does there exist a non-measurable function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f ^{−1} (y) $is measurable for any $y \in \mathbb{R}$? Does there exist a non-measurable function
$f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f ^{−1} (y) $is measurable for any $y \in \mathbb{R}$?
| Let $A \subseteq \mathbb{R}$ be a non-measurable set such that $|A| = |\mathbb{R} \setminus A| = \mathfrak{c}$.
Let $f : A \to \{x \in \mathbb{R} \mid x < 0\}$ and $g : \mathbb{R} \setminus A \to \{x \in \mathbb{R} \mid x \geq 0\}$ be bijections. Then $(f \cup g) : \mathbb{R} \to \mathbb{R}$ is a non-measurable biject... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2510877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove: $\int_a^b f(x) g(x) dx = f(a) \int_a^c g(x) dx + f(b) \int_c^b g(x) dx$
Let $f$ and $g$ be real-valued continuous functions on a closed and bounded interval $[a,b]$. Given that $f$ is
increasing and $g$ is positive on $[a,b]$. Prove the following: $$
\int_a^b f(x) g(x) dx = f(a) \int_a^c g(x) dx + f(b) \int... | Write $H(x)=f(a)\int_a^xg(x)dx+f(b)\int_x^bg(x)dx-\int_a^bf(x)g(x)dx$. $H(a)\geq 0$ and $H(b)\leq 0$.
$H(a)=\int_a^b(f(b)-f(x))g(x)\geq 0$ since $f(b)\geq f(x)$ and $g(x)\geq 0$ $H(b)=\int_a^b(f(a)-f(x))g(x)\leq 0$, so there exists $c$ such that $H(c)=0$ i.e
$f(a)\int_a^cg(x)d(x)+f(b)\int_c^bg(x)dx=\int_a^bf(x)g(x)dx$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2511007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$\sum_{n=1}^{\infty}\frac{1}{n^2} <\frac{33}{20}$ using elementary inequalities There are many ingenious ways for proving $$\zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6} \approx 1.6449$$
Using the inequality $$\frac{1}{n^2} < \frac{1}{n^{2}-\frac{1}{4}} =\frac{1}{n-\frac{1}{2}}-\frac{1}{n+\frac{1}{2}}$$ we ... | You only need a few more terms:
$$\zeta(2)<1+\frac14+\frac19+\sum_{n=4}^\infty\frac1{n^2-1/4}
=1+\frac14+\frac19+\frac27$$
which is already less than $33/20$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2511115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
For a fixed element of a ring R, show that the set is a subring of R. The question follows. For a fixed element of a ring R, show that the set $S=\{x\in \mathbb{R}|ax=0\}$ is a subring of R.
To show that S is a subring of R, it must meet the following conditions:
$(i) S \neq \emptyset$
$(ii) x \in S$ and $y \in S$ impl... | You have the right ideas, you just need to write them down better. For instance, you want to prove that $S$ is a subring, so you cannot start with “since $S$ is a subring of $R$”. Also the argument in (ii) is backwards.
Avoid those pesky arrows! ;-) It seems that you start from $a(xy)=0$ (which instead is what you need... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2511263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that $d_1\sim d_2$ and whether $(X,d)$ is complete Let $X = (-\pi/2,\pi/2), d_1(x,y) = | \tan x - \tan y|,\ d_2(x,y) = | x - y|$. Show that $d_1\sim d_2$, the space $(X,d_1)$ is complete, whereas the space $(X,d_2)$ is not. Conclude that completeness is not topological invariant.
For the first metric, I know that ... | To show $d_1 \sim d_2$ it is enough to show that for an arbitrary open ball $B_2(x_0, r)$ with respect to $d_2$, there exists $r' > 0$ such that $B_1(x_0, r') \subseteq B_2(x_0, r)$, and vice versa.
Let $x, y \in X$. Using the mean value theorem, we have:
$$\left|\tan x - \tan y\right| = \frac1{\cos^2\theta}|x - y|$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2511362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to expand $(x^{n-1}+\cdots+x+1)^2$ (nicely) sorry if this is a basic question but I am trying to show the following expansion holds over $\mathbb{Z}$:
$(x^{n-1}+\cdots+x+1)^2=x^{2n-2}+2x^{2n-3}+\cdots+(n-1)x^n+nx^{n-1}+(n-1)x^{n-2}+\cdots+2x+1$.
Now I can show this in by sheer brute force, but it wasn't nice and ce... | HINT.-Why not using $x^{n-1}+\cdots+x+1=\dfrac{x^n-1}{x-1}$ and dividing
$x^{2n}-2x^n+1$ by $x^2-2x+1$?
You will easily find successively the coefficients $$1,2,3,4,\cdots,(n-2),(n-1) ,n,(n-1),(n-2)\cdots,4,3,2,1$$ with a symmetry like the binomial coefficients.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2511494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 4
} |
if $\int_Q f$ exists then $\int_{y \in B} f(x, y)$ exists for each $x \in A − D$, where $D$ is of measure zero. Let $f : Q = A \times B \to \Bbb R$ be bounded, where $A, B$ are respectively rectangles in $\Bbb R^l$ and $\Bbb R^k$. Show that if $\int_Q f$ exists then
$\int_{y \in B} f(x, y)$ exists for each $x \in A − D... | Hints: (if you are working with Riemann integration)
(1) For $f$ bounded and Riemann integrable on $Q = A \times B$ show that
$$\int_Q f = \int_{x \in A} \underline{\int}_{y \in B}f(x,y) = \int_{x \in A}\overline{\int}_{y \in B}f(x,y)$$
where $\underline{\int}$ and $\overline{\int}$ denote lower and upper Darboux inte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2511584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Probability that sum is odd but not divisible by $3$ Out of $20$ consecutive natural numbers two are chosen randomly.Find Probability that sum is odd but not divisible by $3$.
We have denominator as $\binom{20}{2}$.
Now any number will be of the form $3k$, $3k+1$ or $3k-1$.
Since sum of $3p+1$ and $3q-1$ is divisible ... | Because with $20$ numbers you do not have the same number of numbers of the form $3k$, $3k+1$, and $3k+2$, I would recommend considering three separate cases, namely where the series starts with a number of the form $3k$, where it starts with $3k+1$, and where it starts with $3k+2$. For each, figure out how many pairs ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2511696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
show this inequality $\left(\sum_{i=1}^{n}x_{i}+n\right)^n\ge \left(\prod_{i=1}^{n}x_{i}\right)\left(\sum_{i=1}^{n}\frac{1}{x_{i}}+n\right)^n$ Let $x_{i}\ge 1$,show that
$$\left(\sum_{i=1}^{n}x_{i}+n\right)^n\ge \left(\prod_{i=1}^{n}x_{i}\right)\left(\sum_{i=1}^{n}\dfrac{1}{x_{i}}+n\right)^n$$
or
$$\left(\dfrac{\sum_{... | we have to prove :
$$\left(\dfrac{\sum_{i=1}^{n}x_{i}+n}{\sum_{i=1}^{n}\dfrac{1}{x_{i}}+n}\right)^n\ge \prod_{i=1}^{n}x_{i}$$
Let $x_i\geq 1$ be real numbers so we have :
$$\frac{1}{\sum_{i=1}^{n}S_i}\left(\sum_{i=1}^{n}S_i[(-x_i+\frac{(\sum_{i=1}^{n}x_i)+n}{(\sum_{i=1}^{n}\frac{1}{x_i})+n})(n)x_i^{n-1}+x_i^n]\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2511793",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Show that $|\sin x−\cos x|≤ 2$ for all $x.$ Please help me out in answering this:
Show that $|\sin x−\cos x|≤ 2$ for all $x.$
I don't know where to start and I think we have to use the mean value theorem to show this.
| The mean value theorem also works, since
$$
\sin(x+\pi/2)-\sin x=-\frac{\pi}{2}\cos\xi,
$$
where $\xi\in(x,x+\pi/2)$. It follows that $|\cos x-\sin x|\le \pi/2$. [This is, of course, not as sharp as achille hui's approach.]
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2511917",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 8,
"answer_id": 6
} |
Prove that $\langle H, K\rangle = HK$ This is a question from an exam I sat about two hours ago, and I still couldn't figure it out on the bus home so I've decided to ask for a clue here.
Let $G$ be a group, and $H, K \leq G$. Let $HK := \{hk: h\in H, k\in K\}$, and $KH := \{kh: h\in H, k\in K\}$. Suppose $HK = KH$.
... | Let me first introduce a standard notation in computer science. Given a subset $S$ of $G$, let $S^0 = \{1\}$ and $S^{n+1} = S^nS$ for all $n \geqslant 0$. Finally, let $S^* = \cup_{n \geqslant 0} S^n$.
Hint. First prove that $(HK)^*$ is equal to $\langle H, K\rangle$. Next, using the relations $HK = KH$, $HH = H$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2512042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Integral of a power combined with a Gaussian hypergeometric function I think the following is true for $k \ge 3$,
$$
\int_0^{\infty } (w+1)^{\frac{2}{k}-2} \, _2F_1\left(\frac{2}{k},\frac{1}{k};1+\frac{1}{k};-w\right) \, dw
=
\frac{\pi \cot \left(\frac{\pi }{k}\right)}{k-2}
.
$$
I have checked Table of Integrals, Seri... | Using the integral representation 15.6.6
$$
\, _2F_1\left(\frac{2}{k},\frac{1}{k};1+\frac{1}{k};-w\right)=\frac{\Gamma\left(1+1/k\right)}{2\pi i\Gamma\left(1/k\right)\Gamma\left(2/k\right)}\int_{-i\infty}^{i\infty}\frac{\Gamma\left(1/k+t\right)\Gamma\left(2/k+t\right
)\Gamma\left(-t\right)}{\Gamma\left(1+1/k+t\right)}w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2512169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Can we split a function while doing differentiation? I have a silly question here. Assume a function $$f(x)= \frac{x}{(x-1)^2} \cdot (x+2)$$
Can I write
$$\frac{d}{dx} f(x) = \frac{d}{dx} x \cdot \frac{d}{dx} (x-1)^{-2} \cdot \frac{d}{dx} (x+2)^{-1}$$
If not, then why?
| Yes, you can "split" a function; in fact, that's the standard way for computing derivatives algebraically.
However, you have to know what to do with the pieces.
Some examples (the "addition rule", "product rule", and "chain rule") are
$$ \frac{d}{dx} \big( f(x) + g(x) \big) = f'(x) + g'(x) $$
$$ \frac{d}{dx} \big( f(x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2512258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
} |
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