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H: Derivations on semisimple Lie algebra
First recall some definitions :
Let $B$ be a Killing form on Lie algebra $\mathfrak{g}$ over ${\bf R}$ such that
$B(X,Y)\doteq Tr(ad_Xad_Y)$.
$\mathfrak{g}$ is semisimple if $B$ is non-degenerate.
Define $\partial \mathfrak{g} \doteq \{ D | D[X,Y]=[DX,Y] + [X,DY] \}$ Cle... |
H: Proving that Euclidean space having the infinity metric is a complete metric space (stuck)
I am trying to prove that the space ${\mathbb{R}}^k$ with the $\infty$-metric is a complete metric space.
I know that I need to show that every Cauchy sequence in the metric space ${\mathbb{R}}^k$ with the $\infty$-metric con... |
H: Proof of Bienayme Inequality
I have a bit of trouble about the proof of Bienayme Inequality.
Bienayme Inequality is as follows:
If X has mean $\mu$ and variance $\sigma^2$, then
$$\mathbb{P}\left(\frac{|X-\mu|}{\sigma}\ge k\right)\le\frac{1}{k^2}.$$
Bienayme's Proof:
Let $B = \{|X-\mu|\ge k\sigma\}$ and $\mathbb{1... |
H: Show relation for integrals
Let $f \in C^{1}([a,b];\mathbb{R})$ and $|f'(x)-f'(y)| \le L |x-y|$
then we have $|\int_a^b f(x) dx -f(\frac{a+b}{2})(b-a)| \le L\frac{(b-a)^3}{4}$.
I have troubles to show this inequality. the problem is that i need to have a difference of derivatives of the function in order to use th... |
H: How to graph the equation: $y=\frac {x-2}{x+1}$?
the title says it all.
I'm pretty sure this is a hyperbola, but is there an alternative way of doing this besides a table of values?
"Graph the equation $y=\frac {x-2}{x+1}$"
I know that $x$ cannot equal $-1$ but I'm not sure how to carry on from there.
Any help woul... |
H: the domain for $\dfrac{1}{x}\leq\dfrac{1}{2}$
What is the domain for $$\dfrac{1}{x}\leq\dfrac{1}{2}$$
according to the rules of taking the reciprocals, $A\leq B \Leftrightarrow \dfrac{1}{A}\geq \dfrac{1}{B}$, then the domain should be simply $$x\geq2$$
however negative numbers less than $-2$ also satisfy the origin... |
H: Compute value of $\pi$ up to 8 digits
I am quite lost on how approximate the value of $\pi$ up to 8 digits with a confidence of 99% using Monte Carlo. I think this requires a large number of trials but how can I know how many trials?
I know that a 99% confidence interval is 3 standard deviations away from the mean ... |
H: The harmonic conjugate of $\Im e^{z^2}$?
It is obvious that $e^{z^2}$ is analytic, right? So the harmonic conjugate of $\Im e^{z^2}$ is $\Re e^{z^2}$, isnt' it?
However, the solutions manual I'm consulting gives the answer as $\Im (-ie^{z^2})$, which is not the same function, and I don't understand.
AI: For real-va... |
H: Find $p$ if $(x + 3)$ is a factor of $x^3 - x^2 + px + 15$.
I'm just making sure I answered this correctly.
If $(x+3)$ is a factor, then $P(-3)$ would equal $0$, correct?
AI: Yes, that's correct. If $(x+3)$ is a factor, then $P(-3) = 0$ by the Factor Theorem. So
\[P(-3) = (-3)^3-(-3)^2-3p+15 = -27-9-3p+15 = -3p-21 ... |
H: Evaluate $\int \dfrac{1}{\sqrt{1-x}}\,dx$
Find $$\int \dfrac{1}{\sqrt{1-x}}\,dx$$
I did this and got $\dfrac23(1-x)^{\frac32} + c$
But a online calculator is telling me it should be $-2(1-x)^{\frac12}$
What one is on the money and if not me why?
AI: $$
\int \frac{1}{\sqrt{1-x}}~dx=\int(1-x)^{-1/2}~dx
$$
Let $u=1-x$... |
H: Find the minimum values of $a,b,c,d,e,f$ that satisfy following equations
${ a }^{ 2 }+{ b }^{ 2 }={ c }^{ 2 }\\ { a }^{ 2 }+{ \left( b+c \right) }^{ 2 }={ d }^{ 2 }\\ { a }^{ 2 }+{ \left( b+c+d \right) }^{ 2 }={ e }^{ 2 }\\ { a }^{ 2 }+{ \left( b+c+d+e \right) }^{ 2 }={ f }^{ 2 }$
where $a,b,c,d,e,f$ positive i... |
H: What is the professional term for the combination of the selection in n out of the total m elements?
I know the number of combinations is called ${}_nC_r$, but what about all the exact outcomes?
For example: I have $3$ elements $a,b,c$ and for the parameter $2$, I will have outcomes
$$ab,\quad ac,\quad ba,\quad bc... |
H: How to Find the Center of a Parallelogram
I want to find the center of a parallelogram in order to use it in my java program. I have four coordinates of the parallelogram and I want to find the center coordinate of the parallelogram. It seems I need to find the intersection point of the diagonal lines that I couldn... |
H: If a number is a square modulo $n$, then it is also a square modulo any of $n$'s factors
Say we have $a \equiv x^2 \bmod n$. How would we prove that this implies:
$$\forall \text{ prime }p_i\text{ such that }\,p_i\mid n,\;\exists y\,\text{ such that }\, a \equiv y^2 \bmod p_i$$
AI: If $n$ is a natural number and $a... |
H: Show $\|f\|_p\leq \lim\inf\|f_n\|$
$\Omega$ is a bounded domain of $\mathbb R^n$. If $\{f_n\}\subset L^p(\Omega)$ and $f_n\rightarrow f\in L^p(\Omega)$ weakly, then
$$\|f\|_p\leq \lim_{n\rightarrow\infty}\inf\|f_n\|$$
AI: You are asking to prove why $$\left(\int |f|^p\right)^{1/p} \leq \lim_{n\rightarrow \infty} \... |
H: Find the antiderivative of $\sqrt{3x-1} dx$
Find the antiderivative of $\sqrt{3x-1} dx$.
I got $\frac{2}{3}(3x-1)^{3/2}+c$ but my book is saying $\frac{2}{9}(3x-1)^{3/2}+c$
Can some one please tell me where the $2/9$ comes from?
AI: $$
\int \sqrt{3x-1}~dx=\int(3x-1)^{1/2}~dx
$$
Let $u=3x-1$, $du=3~dx$, so
$$
\in... |
H: How to find inverse of the function $f(x)=\sin(x)\ln(x)$
My friend asked me to solve it, but I can't.
If $f(x)=\sin(x)\ln(x)$, what is $f^{-1}(x)$?
I have no idea how to find the solution. I try to find
$$\frac{dx}{dy}=\frac{1}{\frac{\sin(x)}{x}+\ln(x)\cos(x)}$$
and try to solve it for $x$ by some replacing and ot... |
H: 3rd grade exercise: "make your own turning pattern"
My 8 year old has been given a worksheet of numeric sequences, e.g. "what are the next numbers in the sequence 11, 12, 14, 15, 17, ..." and "make your own number pattern" and "make your own colour repeating pattern". I've had no problem helping her with these.
The... |
H: How to show that $f : {}^\ast \Bbb R \to {}^\ast \Bbb R$ is bounded, if it obtains a limited value everywhere?
Let $f : {}^\ast \Bbb R \to {}^\ast \Bbb R$. For all $x \in {}^\ast \Bbb R $, there exists $y \in \Bbb R $ such that $f(x) \leq y$. How to show that there exists $r \in \Bbb R$, such that for all $z \i... |
H: Logarithm question involving different base
Calculate the values of $z$ for which $\log_3 z = 4\log_z3$.
AI: Hint
$$\log_3 z = 4\log_z3 \Rightarrow \dfrac{\log z}{\log 3}=4\dfrac{\log 3}{\log z}$$
Answer (don't look until you try the hint)
$$\color{lightgrey}{\dfrac{\log z}{\log 3}=4\dfrac{\log 3}{\log z}\Rightarro... |
H: Is my solution correct? Generating functions question: How many non-negative solutions does the equation $x_1+x_2+x_3+x_4+x_5+x_6=12$ have?
so we began studying this subject, and I tried solving this question: How many non-negative and whole ($\in \Bbb Z$) solutions does the equation $x_1+x_2+x_3+x_4+x_5+x_6=12$ h... |
H: Uniformly regular measure "Babiker"
A regular Borel (Radon) probability measures $\mu$ on compact Hausdorff space $X$ is called uniformly regular if:
There is a countable family $\mathcal{A}$ of compact $G_\delta$-subsets of $X$ such that for every open set $U\subseteq X$ and every $\epsilon >0$, there is $A\in\ma... |
H: Uniform distribution on the n-sphere.
I have the next RV:
$$\underline{W}=\frac{\underline{X}}{\frac{||\underline{X}||}{\sqrt{n}}}$$
where $$X_i \tilde \ N(0,1)$$
It's a random vector, and I want to show that it has a uniform distribution on the n-sphere with radius $$\sqrt{n}$$.
I understand that it has this radi... |
H: The negative square root of $-1$ as the value of $i$
I have a small point to be clarified.We all know $ i^2 = -1 $ when we define complex numbers, and we usually take the positive square root of $-1$ as the value of "$i$" , i.e, $i = (-1)^{1/2} $.
I guess it's just a convention that has been accepted in maths and... |
H: having trouble intuiting analyticity
My textbook seems to suggest that the analytic functions are precisely the functions that can be written in terms of $z$ alone (no $x$ or $y$ or conjugate-$z$).
Am I inferring correctly?
Does this mean that $\sin (z+x)$ is not analytic? [where $z=x+iy$]
AI: More precisely, an ... |
H: Multiplication properties in rings of matrices
Let $R$ be an arbitrary ring and let $M_n(R)~~(n>1)~$ be the ring of all $n ~ х ~ n$ matrices with elements from $R$ with usual matrix addition and multiplication.
1) Is it right that there are zero divisors in $M_n(R)$ iff $R$ is non-trivial?
2) Which necessary and su... |
H: Geometric question?
First of all, is it Geometric?
Image of the drafted:
I need help solving this question, and I am completely lost on how can I solve this.
Could anyone explain the way of solving this geometric question?
Here is a drafting of the two lines I and II.
With 3 formulas (a), (b), (c).
(a) $y = 2x + ... |
H: How do I show that the degree $n$ Taylor polynomials of $f$ about two points are equal?
Question
Suppose that $f(x)$ is a polynomial of degree $d$, and that $n \ge d$. Let $x_0 \neq x_1$. Show that the degree $n$ Taylor polynomials of $f$ about $x_0$ and $x_1$ are equal.
Attempt
Let the polynomial be $f(x) = \sum_{... |
H: How to show this matrix is invertible?
Let $f:H \times H \to \mathbb{R}$ be a mapping with $H$ a Hilbert space.
Let $A$ be a matrix with entries $a_{ij}=f(b_i, b_j)$ with
$$a_{ii}=f(b_i, b_i) \geq C\lVert b_i\rVert_{H}^2.$$
Suppose $b_i \neq b_j$ and $(b_i, b_j)_H = 0$ for $i \neq j$ .
How do I show that $A$ is in... |
H: Why is Lie derivative smooth?
Let $G$ be a linear Lie group, say $\mathrm{SL}(2,\mathbb{R})$. Suppose $X\in\mathfrak{g}$ and $f:G\to\mathbb{R}$ is smooth. The Lie derivative of $f$ with respect to $X$ is the function $\mathcal{L}_X f:G\to\mathbb{R}$ defined as $$\mathcal{L}_X f(y):=\left.\frac{\mathrm{d}}{\mathrm{d... |
H: Eigenvectors and differential equations
I was able to find part (a), and I got 4 and -1 for the eigenvalues and from these values I got eigenvectors of [1,1] and [-3,2], but I don't know what to do for part (b) and (c)
AI: Let ${\bf y}=(x,y)$. Then the coupled system can be written as $${\bf y}'=A{\bf y}$$ The gen... |
H: Conditional joint probability and independence
Let's have a joint probability of three events, $\mathbf{P}(X,A,B)$. If $\mathbf{P}(X|A) = \mathbf{P}(X)$, can we show that $\mathbf{P}(X|A,B) = \mathbf{P}(X|B)$? If so, how?
AI: Toss a fair coin twice, and let $A$ be the event that the first toss is a head and $B$ the... |
H: What is the idea behind sheaves of rings on distinguished open sets
In the book on algebraic geometry of Mumford (which can be found here), he said :
We want to enlarge the ring $R$ into a whole sheaves of rings on $SpecR$, written $\mathcal{O}_{SpecR}=\mathcal{O}_{X}$.
So he need to define $\mathcal{O}_{X}(U)$ f... |
H: Given $x,y,z\geq0$ and $x^2+y^2+z^2+x+2y+3z=13/4$. Find the minimum of $x+y+z$.
Given $x,y,z\geq0$ and $x^2+y^2+z^2+x+2y+3z=13/4$. Find the minimum of $x+y+z$.
I tried many method, such as AM-GM, but all of them failed.
Thank you.
AI: use this
$$(x+y+z)^2+3(x+y+z)\ge x^2+y^2+z^2+x+2y+3z$$
and let $x+y+z=t$,
then... |
H: Series of $\int_0^z \zeta^{-1} \sin \zeta d \zeta$
This is a homeworkquestion so I would appreciate some good hints. I have $f(z) = \int_0^z \zeta^{-1} \sin \zeta d \zeta$. Can this be written as a power-series in $\mathbb C$ around $z = 0$?
AI: Hint: What is the power series for $\frac{\sin z}{z}$ around $z=0$? |
H: Rings with zero divisors
Is there a ring $~R~$ with non-trivial multiplication (i.e. $~\exists a,b\in R ~~~ ab\neq 0$) such that each non-zero element of $~R~$ is a zero-divisor?
AI: The simplest example might be $R=2\mathbb Z/8\mathbb Z$. Then $2\cdot 2\neq 0$ but $a\cdot 4=0$ for all $a\in R$. |
H: Basic concepts in finite fields
I need some help with clearing up some some basic concepts in finite fields.
I understand that $\mathbb{F}_p = GF(p)$ where $p$ is a prime is a finite field, which is isomorphic to $\mathbb{Z}/p\mathbb{Z}$. However, I get quite confused with $GF(p^n)$.
Since $GF(p)$ is a finite field... |
H: The algebraic possibilities of the (topological) procedure of the compactification of a space
If $X$ is locally compact $K$-vector space, then $X\cup \{\infty\}$ is via the Alexandroff-compactification a compact space.
But this purely topological procedure tells me nothing about the algebraic relationship of $\inft... |
H: Simplify square of sinc functions
I need to simplify if possible the following:
$$\left(i^n\cdot \operatorname{sinc}\big(\pi(x-\tfrac{n}{2})\big)+(-i)^n\cdot \operatorname{sinc}\big(\pi(x+\tfrac{n}{2})\big)\right)^2$$
with $n \in \mathbf{N}$ and $\operatorname{sinc}(x)=\sin(x)/x$.
Thanks
AI: Note that $\;\pi\left(x... |
H: Finding the range of a vector valued function
For a single valued function, I can infer if the function is monotone from its derivative.
For a vector valued function, is it possible to infer monotonicity from the directional derivative?
For example, define
$$
D=[1,2]\times[1,2],
$$
and
$$
f(x,y)=\left( \frac{2}{1... |
H: Integrate: $\int_0^{\pi} \log ( 1 - 2 r \cos \theta + r^2)d\theta$
If $r \in \Bbb R$ how to integrate $\displaystyle \int_0^{\pi} \log ( 1 - 2 r \cos \theta + r^2)d\theta$?
I need some hints. Special case, if $r = 1$ then I know the above integral is zero.
Here is my working
\begin{align*}
\int_0^{\pi}\log (1 - 2... |
H: Diffeomorphic surfaces and Jacobian
Suppose $S$ and $T$ are bounded (open) surfaces in $\mathbb{R}^n.$ Let them have boundary $\partial S$ and $\partial T$.
Suppose $F:S \to T$ is a $C^k$ diffeomorphism.
Under what conditions on $F$ and $S$ and $T$ and their boundaries do we get that the determinant of the Jacobia... |
H: Prime with digits reversed is prime?
Well, just another idea came up into my mind and i have no idea how to solve it :D
Is there infinitely many prime numbers, which are not repunits and their inverse is also prime? (For example, inverse of 31 is 13 which is also prime. i didn't have any other name to describe the ... |
H: Find the value of $\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx$
I'm trying to figure out how to evaluate the following:
$$
J=\int_{0}^{\infty}\frac{x^3}{e^x-1}\ln(e^x - 1)\,dx
$$
I'm tried considering $I(s) = \int_{0}^{\infty}\frac{x^3}{(e^x-1)^s}\,dx\implies J=-I'(1)$, but I couldn't figure out what $I(s)$ ... |
H: Apparently simple integral
I am having trouble solving this apparently simple integral:
$\int\frac{x}{3+\sqrt{x}}dx$
Hints would be preferable than complete answer...
Thanks!
AI: Substitute $x = y^2$ and use long division to simplify the integrand.
Thus, we have $dx = 2 y dy$. Substituting in the original integrand... |
H: The value of $\lim_{x\to +\infty} \dfrac{e^{x^2}}{10^{|x|}}$
I want to find the value of
$$\lim_{x\to +\infty} \dfrac{e^{x^2}}{10^{|x|}}.$$
Since $x \rightarrow + \infty$, I only consider the value of the function for $x \ge 0$, i.e.
$$\lim_{x\to +\infty} \dfrac{e^{x^2}}{10^{x}}.$$
Noticing that the numerator and d... |
H: Group extension of $\mathbb Z_4$ by $\mathbb Z_2$
Let $f : G →\mathbb Z_2$ be an extension of $\mathbb Z_4$ by $\mathbb Z_2$. Suppose that the induced action $α_f :\mathbb Z_2 →\mathbb Z^{\times}_4$ carries the generator of $\mathbb Z_2$ to $−1$. Then $G$ is isomorphic to either $D_8$ or $Q_8$.
Proof
Let $\mathbb ... |
H: $ e^{At}$ for $A = B^{-1} \lvert \cdots \rvert B $
For a homework problem, I have to compute $ e^{At}$ for
$$ A = B^{-1} \begin{pmatrix}
-1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 3 \end{pmatrix} B$$
I know how to compute the result for $2 \times 2$ matrices where I can calculate the eigenvalues, but this is $3 \times 3$,... |
H: Binomial sum of derivatives
I would like to know the result of the following sum:
$$\sum_{p=0}^m \binom{m}{p}(-1)^{p-1}\frac{\partial^{p-1}}{\partial x^{p-1}}f(x)\cdot(-1)^{m-p-1}\frac{\partial^{m-p-1}}{\partial x^{m-p-1}}g(x)$$
with $$\frac{\partial^{-1}}{\partial x^{-1}}=\int dx$$
Thanks
AI: You can pull out the ... |
H: space of riemann integrable functions not complete
Define norm as $\int |f|$ (Riemann integral) on $\mathcal R^1[0,1]$, the space of riemann integrable functions on $[0,1]$ with identification $f=g$ iff $\int |f-g|=0$.
Let $\{ r_1,r_2,\cdots \}$ be the rationals in $[0,1]$, and let $f_n=1_{\{r_1,\cdots,r_n\}}$. Th... |
H: Permutation combination problem
This is how Edward’s Lotteries work. First, 9 different numbers are selected. Tickets with exactly 6
of the 9 numbers randomly selected are printed such that no two tickets have the same set of numbers.
Finally, the winning ticket is the one containing the 6 numbers drawn from the 9 ... |
H: Pigeon holes principle
Let $P$ be a group that it's elements are 257 sentences in which only atomic sentences from $A,B,C$ exist (i.e. $A \iff B,\space\space A \wedge B \wedge C, \space\space...$) Show that there exists two different $p_1, p_2 \in P$ so that the sentence $p_1 \iff p_2$ is a tautology.
Pigeons are t... |
H: Getting rid of a floor function in the next expression:$\left\lfloor\frac{(x-2)^2}{4}\right\rfloor $, It is known x is odd.
I was wondering if there's a way in which you can get rid of a floor function in the next expression:$$\left\lfloor\frac{(x-2)^2}{4}\right\rfloor $$ It is known x is odd.
AI: Since $x$ is odd,... |
H: Study of Set theory: Book recommendations?
Can you suggest a good book for set theory?
I have just started reading about Group theory and want to learn set theory on my own.
Thanks in advance
AI: A couple of ‘entry level’ treatments that can be confidently recommended.
Herbert B. Enderton, Elements of Set Theory (... |
H: How can I solve this Laws of Sines problem?
This is a homework question that was set by my teacher, but it's to see the topic our class should go over in revision, etc.
I have calculated $AB$ to be 5.26m for part (a). I simply used the law of cosines and plugged in the numbers.
part (b) is the question I have bee... |
H: proving $n!>2^n\;\;\forall \;n≥4\;$ by mathematical induction
My teacher proved the following $n!>2^n\;\;\;\forall \;n≥4\;$ in the following way
Basis step: $\;\;4!=24>16$ ok
Induction hypothesis: $\;\;k!>2^k$
Induction step: $\qquad\qquad(k+1)!=k!(k+1)>(k+1)2^k>2^k\cdot 2=2^{k+1}$
I wonder how did he assume tha... |
H: Two questions on finite abelian groups
Which of the following are true?
1.Every group of order $6$ abelian.
2.Two abelian groups of the same order are isomorphic
AI: none of them is true.
for 1. consider $S_3$.
for 2. consider $\mathbb{Z}_4$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ |
H: Finite group is generated by a set of representatives of conjugacy classes.
Could you tell me how to prove that a finite group is generated by a set of representatives of conjugacy classes?
I've read this https://mathoverflow.net/questions/26979/generating-a-finite-group-from-elements-in-each-conjugacy-class but I ... |
H: What kind of functions can be moment-generating functions for a random variable?
Given an infinitely differentiable function $ g: \mathbb{R} \rightarrow \mathbb{R}$, can we always find a distribution function $f_X$ of some random variable $X$ so that
$g(t) = \int_{-\infty}^\infty e^{tx}f_X(x) dx$?
If my question i... |
H: Image of a union of collection of sets as union of the images
I am having problems with establishing the following basic result. Actually, I found a previous post that is close in nature (it is about inverse image), but I was interested in this specific one, with the following notation, because it is the one I foun... |
H: Easy way to check for a valid solution in this triple equality?
Let's say I have the following equalities
$a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 = b_1x_1 + b_2x_2 + b_3x_3 + b_4x_4 = c_1x_1 + c_2x_2 + c_3x_3 + c_4x_4$
Where the $a$'s, $b$'s, and $c$'s are known, non-negative integers.
Is there an efficient way to check... |
H: Find the first 5 terms of the expansion in a power series
Find the first 5 terms of the expansion in a power series
$$y′=xe^{x}+2y^{2}$$
I've got a riccati equation $$ x e^{x}+2y^{2}, y(0)=0$$
After solving: $$y=e^{x}(x-1)+\frac{2}{3}y^{3} - 1$$
And I don't know how to go forward. Please help me.
AI: You assume th... |
H: Dimension of the set of self-adjoint operators
I'm trying to figure out what the dimension of the set of self-adjoint operators on V would be, or in more concrete terms:
Let $dim V =n$. Let $S(V)$ denote the set of self-adjoint linear operators on V. What is its dimension?
The only thing I know that somewhat resemb... |
H: Confusing math problem
How would I solve this question? I came across it and is really confused.
The payment of Jon was bigger by $960$ than the payment of David.
After the payment of David got increased by $10\%$, Jon and David got the same payment amount.
$\text{A}$. We will mark with $X$ the payment of David in ... |
H: Results following from Analyticity on a domain
This is part of an old Oxford exam paper (1997 2602 Q2) I'm working on for revision.
Suppose we have a function $f$ which is holomorphic on the disc radius $R$ about $0$. We want to show that there is a sequence $\{p_n\}$ of polynomials such that $\{p_n\} \rightarrow f... |
H: Application of the Identity Theorem to $|x|^3$ for $-1
Oxford Exam $2602$ $1997$ $Q3$
We want to show that there is no function $f$ which is holomorphic in $D(0;1)$ and such that $f(x)=|x|^3$ for $-1<x<1$.
Here are my thoughts thus far:
Suppose there is. Then $g(x)=f(x)-|x|^3=0$ for $-1<x<1$. Then by the Identity ... |
H: General Linear Groups with Homomorphisms
Let $G=\mathrm{GL}_n(\mathbb R)$ and $H=\mathbb R^*$. Let $\phi : G=\mathrm{GL}_n(\mathbb R) \rightarrow \mathbb R^*$ be the map defined by $\phi(A)=\det(A)$. Show that $\phi$ is a group homomorphism.
If $\phi : G \rightarrow H$ is a group homomorphism, then the set $\{g \i... |
H: Rotation of a point in 3d space
I'm trying to rotate a point around a single axis of a 3D system.
Given $P=\begin{pmatrix}
101 \\
102 \\
103
\end{pmatrix}
$,
And the rotation matrix formula for rotation around the X axis only, I get:
$Rx(\psi=90^\circ)=
\begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & -1 \\
0 & 1 & 0
\end{pmat... |
H: Why the root of this tree has to be "1"?
Arrange $2^{n-1}-1$ zeroes and $2^{n-1}$ ones in a balanced full binary tree of depth $n$. If we want the number of edges that connect the same (and respectively different) digits are the same, then one claims that the root of this tree has to be a one. Why is that?
For exam... |
H: For x < 5 what is the greatest value of x
It can't be $5$. And it can't be $4.\overline{9}$ because that equals $5$. It looks like there is no solution... but surely there must be?
AI: There isn't one. Suppose there were; let's call it $y$, where $y<5$.
Let $\epsilon = 5 -y$, the difference between $y$ and 5. $... |
H: Proof by induction that $\left(\sum_{k=1}^nk\right)^2\ge\sum_{k=1}^nk^2$
I am a bit new to logical induction, so I apologize if this question is a bit basic.
I tried proving this by induction:
$$\left(\sum_{k=1}^nk\right)^2\ge\sum_{k=1}^nk^2$$
Starting with the base case $n=1$:
$$1^2\ge1^2$$
Then to prove that $P(n... |
H: Trigonometric substitution integral
Trying to work around this with trig substitution, but end up with messy powers on sines and cosines... It should be simple use of trigonometric properties, but I seem to be tripping somewhere.
$$\int x^5\sqrt{x^2+4}dx $$
Thanks.
AI: You don't even need a trigonometric substituti... |
H: First derivative of $\sqrt[\large 5]{\frac{t^3 + 1}{t + 1}}$
I have yet another derivative I need help with. I have to differentiate :
$$\sqrt[\uproot{3}{\Large 5}]{\frac{t^3 + 1}{t + 1}}$$
with respect to $t$.
I had two thoughts about this, use the chain rule then the quotient rule and multiply out, but then I am... |
H: Find integer solutions of $x^2 -px +q=0$, where $p$ and $q$ are prime
Quick number theory question that I have just come across, was wondering if anyone could shed some light on it.
So $p$ and $q$ are given to be prime numbers, and we are told that the equation $x^2 -px +q=0$ has two integer solutions. How can mi... |
H: Inequality proof 2
How to prove the inequality : for real numbers $\alpha_1, \ldots \alpha_n, \beta_1, \ldots \beta_n$:
$$\sqrt{(\alpha_1 + \beta_1)^2+\cdots+(\alpha_n + \beta_n)^2} \leq \sqrt{\alpha_1^2 + \cdots + \alpha_n^2}+\sqrt{\beta_1^2+\cdots+\beta_n^2}.$$
Thanks!
AI: Hint: Square both sides
$(\alpha_1 + \be... |
H: Confused on definition of strong induction
I found the following statement in Munkres' Topology:
Theorem 4.2 (Strong induction principle). Let $A$ be a set of positive integers. Suppose that for each positive integer $n$, the statement $S_n \subset A$ [here $S_n = \{1, 2, \dots, n\}$] implies the statement $n \in ... |
H: Prove or disprove: $\sum\limits_{i=1}^n i^2 = O(n^2) $
Prove or disprove:
$$\sum_{i=1}^n i^2 = O(n^2) $$
If we want to prove this, find some summation that we know the $ O(n)$ runtime for, and is $ O(n^2) $ or smaller.
Otherwise, we could disprove this by finding some summation that is less than this one, but has ... |
H: Convergence of these series
$$\sum_{n=1}^\infty \frac{2nx^{n}}{(n+1)^{2}3^{n}} \tag{1}$$
$$\sum_{n=1}^\infty x^{n}\tan\frac{x}{4^{n}}\tag{2}$$
Is there any good article that describes an equivalents like if $$ \lim_{n\to\infty} \sin\frac{2\pi}{3^{n}} \sim \frac{2\pi}{3^{n}}\tag{3}$$
(am I right about $(3)$?)
AI: Th... |
H: What is the name of a game that cannot be won until it is over?
Consider the following game:
The game is to keep a friend's secret. If you ever tell the secret, you lose. As long as you don't you are winning. Clearly, it's a game that takes a lifetime to win.
Another example would be honor. Let's say, for our purpo... |
H: Extension of valuation to the algebraic extension of a number field.
I am trying to get the idea how we can extend the $p$-adic valuation on $\mathbb Q$ to an algebraic extension. In particular, how to extend the $p$-adic valuation for $p = 5$ from $\mathbb Q$ to $\mathbb Q(5^{1/3})$. Thank you.
AI: The analogue o... |
H: Some questions about variations of fixed point method
I'm doing some excercises in Fixed Point Iteration methods with Matlab. I have to find roots for $f(x)=e^x -x -1.9\cos x$ by using $x_{n+1}=g(x_n)$. I know how to choose $g(x)$ such that I can find both roots. The following part of the exercise is as follows:
In... |
H: 1st derivative of $\frac{2x}{\sqrt{x^2 + 1}}$
Another simple question that I can't work out today, yet I would work it out two weeks ago!
I need to find the 1st derivative of $$\frac{2x}{\sqrt{x^2 + 1}}$$.
So I use the Quotient rule and I get: $$\frac{(x^2 + 1)^.5 (2) - (2x)(0.5x^2 + 0.5)^-5}{x^2+1}$$
Am I heading ... |
H: Probability of two random n-digit numbers dividing each other
Let $n$ be a positive integer. Suppose $a$ and $b$ are randomly (and independently) chosen two $n$-digit positive integers which consist of digits 1, 2, 3, ..., 9. (So in particular neither $a$ nor $b$ contains digit 0; I am adding this condition so that... |
H: Radioactive isotopes differential equationa
I am having a hard time finding the correct differential equation to my problem. The problem is :
There's 2 isotopes: A and B. A is is transforming into B to a rate proportional to its quantity and B is decreasing to a rate proportional to its quantity.
I need to find an ... |
H: How to show in a clean way that $z^4 + (x^2 + y^2 - 1)(2x^2 + 3y^2-1) = 0$ is a torus?
How to show in a clean way that the zero-locus of $$z^4 + (x^2 + y^2 - 1)(2x^2 + 3y^2-1) = 0$$ is a torus?
AI: This took me longer than it should have, given the simplicity of the answer.
First, analyze the equation for when it h... |
H: Riemann integral show $f(x)=g(x)$ for at least 1 $x$ in [a,b]
Let $f$ and $g$ be continuous functions on $[a,b]$ such that $\int_a^b f = \int_a^b g$. Show that there exists $x\in [a,b]$ such that $f(x) = g(x) $.
I want to assume not and then show that the integrals cannot be equal. but perhaps an argument on an upp... |
H: Confusion about Lemma 13.2 in Munkres' topology (property which implies that a collection is a basis for a topology)
Lemma 13.2 and its proof confuse me.
$X$ is a topological space and $C$ is a collection of open sets of $X$ satisfying a property. A specific topology is not mentioned in the lemma. In the proof, h... |
H: Laurent expansion problem
Expand the function $$f(z)=\frac{z^2 -2z +5}{(z-2)(z^2+1)} $$ on the ring $$ 1 < |z| < 2 $$
I used partial fractions to get the following $$f(z)=\frac{1}{(z-2)} +\frac{-2}{(z^2+1)} $$
then
$$ \frac{1}{z-2} = \frac{-1}{2(1-z/2)} = \frac{-1}{2} \left[1+z/2 + (z/2)^2 + (z/2)^3 +\cdots\righ... |
H: Closed form of an integral
Is there a closed form of $$\int\limits_0^1\frac{\arctan(\sqrt{x^2+2})}{(1+x^2)\cdot(\sqrt{x^2+2})}dx \quad \text{?}$$
I just know that $\int\limits_0^1\frac{\arctan(\sqrt{x^2+2})}{(1+x^2)\cdot(\sqrt{x^2+2})}dx = 0.514042...$
AI: This is called the Ahmed integral.
$$ \int_{0}^{1} \frac{\t... |
H: Diagonalizable matrices in $M_{2\times 2}(\mathbb{F}_2)$
List all diagonalizable $2\times 2$ matrices over the a field $F$ consisting of two elements $0$ and $1$.
I want to try and do this using C++, but perhaps this isn't the place to ask. I have an idea as to how I'd do it.
AI: If $M$ is diagonalizable, we have $... |
H: How to find the unknown values in this Numerical Integration type?
Given the following type of numerical integration:
$$I(f)=\int_0^1 f(x) \, dx \approx \frac 12 f(x_{0}) +c_1 f(x_1) $$
a) Find the values of: the coefficient $c_1$ and points $x_0$ and $x_1$ so that the above formula numerical integration to be ... |
H: Ring Inside an Algebraic Field Extension
Let $E|F$ be an algebraic field extension and a ring $K$ such that $F\subseteq K\subseteq E$. It is true that $K$ is a field?
AI: Yes. Suppose $0\ne k\in K$. Since $k\in E$ and $E/F$ is algebraic, we have some minimal polynomial $x^n+a_{n-1}x^{n-1}+\cdots+a_0$ with coefficie... |
H: Question regarding $\lim_{x \to 0} \left(\exp(\sin (x)) + \exp \left(\frac{1}{\sin (x)}\right)\right)$
I wanted to find out whether the following limit exists, and find the value if it does.
$$\lim_{x \to 0} \left(\exp(\sin (x)) + \exp \left(\frac{1}{\sin (x)}\right)\right).$$
Attempt
After many attempt to prove t... |
H: Why is $2\pi i \neq 0?$
We know that $e^{\pi i} = -1$ because of de Moivre's formula. ($e^{\pi i} = \cos \pi + i\sin \pi = -1).$
Suppose we square both sides and get $e^{2\pi i} = 1$(which you also get from de Moivre's formula), then shouldn't $2\pi i=0$? What am I missing here?
AI: You have shown that $e^{2\pi i} ... |
H: Help with modular arithmetic
If$r_1,r_2,r_3,r_4,\ldots,r_{ϕ(a)}$ are the distinct positive integers less than $a$ and coprime to $a$, is there some way to easily calculate, $$\prod_{k=1}^{\phi(a)}ord_{a}(r_k)$$
AI: The claim is true, with the stronger condition that there is some $i$ with $e_i=1$ and all other exp... |
H: Weierstrass $M$-test problem, $f_n(x)=(nx^2)/(n^3+x^3)$
Use the Weierstrass M-test to show $$f(x)=\sum_{n=1}^\infty \frac{nx^2}{n^3+x^3}$$ converges uniformly on any finite interval $[-R,R]$.
This was an exam question I had. My attempt was to find an upper bound for $\frac{nx^2}{n^3+x^3}$ by taking the derivative... |
H: $\int^{\pi/2}_{0}\log|\sin x| \,dx = \int^{\pi/2}_{0}\log|\cos x| \,dx $
Prove that :
$$\int^{\pi/2}_0 \log|\sin x| \,dx = \int^{\pi/2}_0 \log|\cos x| \,dx $$
I tried to cut the integral into a sum of parts and changing variable but it didn't work out right, i dont know how to solve this kind of problems in any ... |
H: limsup and liminf and the product of sequences
I'm trying to show that if $ \limsup s_n = +\infty$ and $\liminf t_n > 0$, then $\lim\sup s_n t_n = +\infty$.
Could someone check my proof/give feedback?
Since $\lim\inf t_n > 0$, we know that there is a natural $N_1$ such that $m = \inf \{t_n \ | \ n > N_1 \} > 0$. Al... |
H: Proof on showing $\frac{(b-a)}{2}(f(a) + f(b)) \leq \int_a^b f \leq (b-a) f(\frac{a+b}{2})$ for class $C^2$ function $f$
The task is as follows:
Given:
(a) function $f \in C^2$
(b) $f \geq 0$ and (c) $f'' \leq 0$ on $[a,b]$
Goal:
Show
$$\frac{(b-a)}{2}(f(a) + f(b)) \leq \int_a^b f \leq (b-a) f(\frac{a+b}{2})$$
... |
H: Can an algebraic structure have indistinguishable elements?
Sometimes, a topological space has indistinguishable points - we call those spaces non-$T_0$. But given such a space, we can always identify indistinguishable points, thereby yielding a $T_0$ space. (Technically, we've taken the Kolomogorov quotient).
Does... |
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