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H: Probability, making a selection of 5 people from 10, with two married couples with restrictions
10 people. must make a committee of 5 people
So the restrictions are
1) Mr and Mrs Q can't be separated
2) Mr and Mrs P can't be in the same committee.
So how many possible committees can you form.
4 CASES:
CASE 1: N... |
H: Need help applying the root test for: $\sum\limits_{n=1}^{\infty}\left(\frac{2}{e^{-8n}-1}\right)^n$
I'm not sure if I am doing something wrong, or not... I've got an answer but it doesn't look right to me.
Given the following series, determine if it is convergent or divergent using the root or ratio test. If the ... |
H: Prove that one of the following sets is a subspace and the other isn't?
OK, here goes another.
Prove that $ W_1 = ${$(a_1, a_2, \ldots, a_n) \in F^n : a_1 + a_2 + \cdots + a_n = 0$} is a subspace of $F^n$ but $ W_2 = ${$(a_1, a_2, \ldots, a_n) \in F^n : a_1 + a_2 + \cdots + a_n = 1$} is not.
OK. Any subspace has ... |
H: Enveloping Algebra $U(L \oplus L')$
I'm having trouble understanding part of a proof of the following statement
Let $L,L'$ be Lie algebras and $L \oplus L'$ their direct sum. Then $$ U(L \oplus L') \cong U(L) \otimes U(L')$$
Let $i_L : L \to U(L)$ denote the natural inclusion into the enveloping algebra, and sim... |
H: Limit points and interior points in relative metric
Let $M$ be a metric space and let $X$ be a subset of $M$ with the relative metric. If $Y$ is a subset of $X$, let $\overline{Y}^{(X)}$ denote the closure of $Y$ in the metric space $X$. Prove that $\overline{Y}^{(X)}=\overline{Y}\cap X$. State and prove a corresp... |
H: The simplest nontrivial (unstable) integral cohomology operation
By an integral cohomology operation I mean a natural transformation $H^i(X, \mathbb{Z}) \times H^j(X, \mathbb{Z}) \times ... \to H^k(X, \mathbb{Z})$, where we restrict $X$ to some nice category of topological spaces such that integral cohomology $H^n(... |
H: About rationalizing expressions
For example, rationalizing expressions like
$$\frac{1}{\pm \sqrt{a} \pm \sqrt{b}}$$
Is straightforward. Moreover cases like
$$\frac{1}{\pm \sqrt{a} \pm \sqrt{b} \pm \sqrt{c}}$$
and
$$\frac{1}{\pm \sqrt{a} \pm \sqrt{b} \pm \sqrt{c} \pm \sqrt{d}}$$
Are still easy to rationalize. But m... |
H: Is there an axiom of ZFC expressing that GCH fails as badly as possible?
The GCH axiom basically says that for all infinite cardinal numbers $\kappa$, the number of cardinals lying strictly between $\kappa$ and $2^\kappa$ is as small as possible. Namely, there are none.
Is there an axiom which claims the opposite, ... |
H: Quadratic form $\mathbb{R}^n$ homogeneous polynomial degree $2$
Could you help me with the following problem?
My definition of a quadratic form is: it is a mapping $h: \ V \rightarrow \mathbb{R}$ such that there exists a bilinear form $\varphi: \ V \times V \rightarrow \mathbb{R}$ such that $h(v)=\varphi(v,v)$.
Cou... |
H: Find minimum value of a
Find the minimum value of $a$ if there's a differentiable function in $\mathbb{R}$ for which :
$$e^{f'(x)}= a {\frac{|(f(x))|}{|(1+f(x)^2)|}}$$ for every $x$
pretty much stuck. I think the minimum value should be $1$ but not sure.
AI: Let $f(x)$ be any solution of the DE for any $a$ over all... |
H: What increases faster: $(\log n)^2$ or $n^{1/3} + \log n$
What increases faster: $(\log n)^2$ or $n^{1/3} + \log n$ and why? And also, what increases faster $\log n$ or $n^{x}$, where $x$ is a random positive constant number?
AI: Recall that:
$$
f(n)=O(g(n)) \iff \text{For some } C\ge0,\quad\lim_{n\to\infty} \dfra... |
H: Proving replacement theorem?
I want to see if I am understanding the proof of the replacement theorem correctly.
Let $V$ be a vector space that is spanned by a set $G$ containing $n$ vectors.
Let $L \subseteq V$ be a linearly independent subset containing $m$ vectors.
Then $m\leq n$ and there exists a subset H o... |
H: If $p(2x+1)=p(x^2)$ for all $x\in\mathbb{R}$, then $p\equiv\text{const.}$
Let $p\in \Bbb{R}[x]$ (polynomial) with $\deg(p)=n$. Suppose that $p(2x+1)=p(x^2)$ for all $x\in\mathbb{R}$. Prove that $p\equiv\text{const.}$
AI: Hint: If $p(2x + 1) = p\left(x^2\right)$ for all $x\in\Bbb{R}$, $\deg(p(2x + 1)) = \deg\left(p... |
H: Which integral theorem to use to evaluate this triple integral?
Take the normal pointing outwards from the surface. Use an appropriate integral theorem
$$\iint_S \textbf{F}\cdot d\textbf{S} \space \space where \space \space \textbf{F} (x,y,z)=(x^3,3yz^2,3y^2z+10) $$ and $S$ is the surface $z=-\sqrt{4-x^2-y^2}$
My ... |
H: I think that the limit of $\frac{x-1}{xy-2x-y+2}$ as $(x,y) \rightarrow (1,-2)$ exists, and equals $-1/4$. How to confirm this?
This question is on a practice exam; unfortunately, I have no idea how to solve it!
Let $$f(x,y)=\frac{x-1}{xy-2x-y+2}.$$
Does the limit of $f(x,y)$ as $(x,y) \rightarrow (1,-2)$ exist? I ... |
H: Convert two points to line eq (Ax + By +C = 0)
Say one has two points in the x,y plane. How would one convert those two points to a line? Of course I know you could use the slope-point formula & derive the line as following:
$$y - y_0 = \frac{y_1-y_0}{x_1-x_0}(x-x_0)$$
However this manner obviously doesn't hold whe... |
H: Solving Pell's equation(or any other diophantine equation) through modular arithmetic.
Let us take a solution of Pell's equation ($x^2 - my^2 = 1$) and take any prime $p$. Then we have found a solution of the Pell's equation mod $p$.
Now, conversely, for any prime $p$, we can find a solution of Pell's equation. My ... |
H: Convergence in $\mathbb{Z}_p$
Here is my question:
Let $\alpha_0, \dots, \alpha_{p-1} \in \mathbb{Z}_p$ be such that $\alpha_i \equiv i \pmod{p}$ for all $i = 0,\dots, p-1$. Show that, for any $x\in \mathbb{Z}_{p}$, you can find an infinite sequence $(a_{n})_{n\geq 0}$, where each $a_n \in \lbrace\alpha_0,\alpha_1,... |
H: multiplication in GF(256) (AES algorithm)
I'm trying to understand the AES algorithm in order to implement this (on my own) in Java code.
In the algorithm all byte values will be presented as the concatenation of its individual bit values (0 or 1) between braces with the most significant bit first.
So bytes are int... |
H: How to diagonalize $f(x,y,z)=xy+yz+xz$
Could you tell me how to diagonalize $f(x,y,z)=xy+yz+xz$.
I know I can rewrite it as $(x+ \frac{1}{2}y + \frac{1}{2}z)^2 - x^2 - \frac{1}{4}(y-z)^2$
What do I do next?
Could you help me?
AI: I have explained the general procedure in my comments in your previous thread. So, I w... |
H: Defining infinitesimals
Can such definition of infinitesimals hold?
$$\mathrm{d} x :=a:(a>0 \;\And\; \forall b \in \mathbb{R}^+\backslash \{ a \}\;(a<b))$$
And, if the above definiton works, then obviously
$$\mathrm{d} f(x) := f(x+{\mathrm{d} x})-f(x)$$
This definiton is basically a try not to delve into the realms... |
H: Finite generation of Hom between cyclic and artinian module
Let $R$ be a Noetherian ring with unit, and $I$ be a nonzero ideal of $R$. Let $M$ be an artinian $R$ module. Is $\operatorname{Hom}(R/I, M)$ finitely generated?
Thanks.
AI: There are many counterexamples. If $IM=0$, then $\hom(R/I,M)=M$. Now take any arti... |
H: Create Fourier-Series of f(x) = x if 0 < x < Pi and 0 if Pi < x < 2*Pi
I tried the following to create the Fourier-series of the function:
$$ f(x) = \begin{cases} x & 0<x<\pi \\ 0 & \pi < x < 2 \pi \end{cases}$$
This is what I tried:
$$a_0 = \frac{1}{\pi}\int\limits_0^\pi x dx = \frac{1}{\pi}\cdot\left(\frac{x^2... |
H: Does $X_n \stackrel{Prob}{\longrightarrow} X$, $X \in L^2$ imply $X_n \stackrel{L^2}{\longrightarrow} X$?
Let $X_n$, $n\in \mathbb{N}$ be a sequence of random variables which converges in probability to $X$, i.e. $X_n \stackrel{Prob}{\longrightarrow} X$. Furthermore it is known that $X \in L^2$. Does this imply $X_... |
H: Hartshorne 8.9.1 $\mathcal O_{\Delta X}$-module structure on $\mathcal I$ := the kernel of the diagonal morphism
Hartshorne asserts in 8.9.1 that $\mathcal I$, the kernel of the diagonal morphism $X \to X \times_Y X$, has a natural $\mathcal O_{\Delta X}$-module structure.
My problem is that $\mathcal I$ is an idea... |
H: Solve : $x^2-92 y^2=1$
As some of you might know,this is Brahmagupta's equation . How to find solution for this ?
I mean integral solution? How to solve it using programming ?
I tried something like $x^2=1+92y^2$
$x=\sqrt{1+92y^2}$
Use brute force approach to check for every y ? Is there any better answer ?
AI: A... |
H: Solve $-1+B^{\prime}(r)r+B(r)=\frac{Q^2}{4 \pi r^2}$ analytically
I need to solve $$-1+B^{\prime}(r)r+B(r)=\frac{Q^2}{4 \pi r^2}$$ , $Q=const$. The boundary condition is $B(r)\to 1$ as $r \to \infty$. I am faced with this equation while solving for the spherically symmetric metric with a charge $Q$. Though, I can f... |
H: How is the real number line a second axiom space?
Out of all the stupid questions I may have asked, this surely may be the stupidest. Any help would be appreciated.
My book says "The real number line is a second axiom space"
How??
Sure one might say $(-n,n), n\in\mathbb{Z}$ could be described as the sets in the bas... |
H: Show the eigenspaces of $T$ are all $1$-dimensional
Let $V$ be a finite-dimensional complex vector space and $T:V\to V$ a linear transformation. Suppose there exists $v\in V$ such that $\{v,Tv,T^2v,\ldots,T^{n-1}v\}$ is a basis for $V$. Show that the eigenspaces of $T$ are all $1$-dimensional.
I want to use the min... |
H: how to work out $14^{293}-12^{26}\pmod{13}$
How can I work this out without a calculator?
$$14^{293}-12^{26} \pmod{13}$$
I just couldn't figure out a way to do this.
AI: $14\equiv 1\mod 13$ and $12\equiv -1\mod 13$
$14^{293}\equiv 1\mod 13$ and $12^{26}\equiv (-1)^{26}=1\mod 13$
$(14^{293}-12^{26})\equiv 0\mod 9$ |
H: 17! mod 13, How do I do this without a calculator
So I know $$17! = 17 \times16\times15...\times1$$
So I was thinking maybe go $$17mod(13)\equiv4 \space \space and \space 16mod(13)\equiv3 ...$$
add all that together but that is too much work so I went
17-13 = 4
$$4! mod 13 \equiv 4 mod(13) \space 3 mod(13)... etc$... |
H: Homeomorphism example
I'm reading some notes about topology and homeomorphisms and there is an example of a homeomorphism from the unit ball in $\mathbb R^n$ to $\mathbb R^n$. The map $x \mapsto {x \over 1 - |x|^2}$. I assume absolute value means Euclidean norm here. What I don't understand is the square in the def... |
H: Height of a prime ideal and number of generators of its localization
This question is very related to this one: generators of a prime ideal in a noetherian ring.
Let $\mathfrak{p}$ be a prime ideal in a Noetherian ring and let $k$ be its height. Further suppose that $f_{1},\dots, f_{k} \in \mathfrak{p}$ generate th... |
H: The probability that an event with exponential distribution will happen before an event with a Poisson distribution
I have two variables depicting arrival. One (lets call it $A$) has a Poisson distribution, so the probability of $n$ elements arriving in time period $\tau$ is: $P_n(\tau)=\frac{\left(\lambda t\right)... |
H: congruence proof: Prove that there is no integer such that $x≡2 \pmod 6$ and $x≡3 \pmod 9$ are both true.
Prove that there is no Integer such that $x≡2 \pmod 6$ and $x≡3 \pmod 9$ are both true.
How should I approach this question?
I attempted using contra-positive proof, so $x=6p+2$ and $x=9q+3$ where $p,q$ are i... |
H: Programmatically solving a system of nonlinear equations over GF(2)
I have the following relatively large system of nonlinear equations over $GF(2)$:
$
0 = w_7x_7 + w_7x_5 + w_7x_4 + w_7x_0 + w_6x_6 + w_6x_5 + w_6x_1 + w_5x_7 + w_5x_6 + w_5x_2 + w_4x_7 + w_4x_3 + w_3x_4 + w_2x_5 + w_1x_6 + w_0x_7 \\
0 = w_7x_6 + w_... |
H: Why is $\sum_{g \in G} \rho(g) =0$ for any nontrivial irreducible representation
Let $F$ be an arbitrary field, and $(\rho, V)$ be an irreducible representation of $G$. Then $$\sum_{g \in G} \rho(g) = \begin{cases}
0 & \text{ if } \rho \neq 1_G, \\
|G|1_V & \text{ if } \rho = 1_G.
\end{cases}$$
The case when $\rh... |
H: Inverse of a polynomial function
I want to find the inverse of $f(x)=\frac{3}{4}x^2-\frac{1}{4}x^3 $ when $0<x<2$.
According to wolfram the answer is inverse
I would like to know how can I find wolfram's inverse.
AI: The correct formula is $$(f^{-1})'(y) = \dfrac{1}{f'(f^{-1}(y))}$$ |
H: Prove $A\in \mathbb R^{n\times n}$ is antisymmetric iff...
Prove that $A\in \mathbb R^{n\times n}$ is antisymmetric iff $ \forall v\in\mathbb R^n:\langle v,Av\rangle=0 $
$\langle \cdot,\cdot\rangle$ is just the dot product.
I'm a little stumped by this problem. I'm fairly sure that the $\Leftarrow$ part of the proo... |
H: In how many ways can you rearrange CANADA?
I'm trying to solve the following question which is in the permutations unit:
In how many ways can all the letters of the word CANADA be arranged if the consonants must always be in the order in which they occur in the word itself?
I have no idea where to start, any hint... |
H: Symmetric and exterior powers of a projective (flat) module are projective (flat)
Assume that $R$ is a commutative ring with unity and $P$ a projective (flat) $R$-module. Why $\mathrm{Sym}^n(P)$ and $\Lambda^n(P)$ are projective (flat) for every $n$?
AI: Here's one approach using the fact that projective modules ar... |
H: Calculating interior angles of quadrilateral
stupid question... but:
I've a polygon which has the points $(a_x,a_y),(b_x,b_y),(c_x,c_y), (d_x,d_y)$
How can I calculate each interior angle of this quadrilateral? I know that in sum, it has to be $360^\circ$.
Thanks
AI: If you know about vectors, the dot product can h... |
H: Limit problem in the process of integration?
I am trying to solve this integral
$$\int_{0}^{+\infty }\frac{x\ln(x)}{(1+x^2)^2}$$
where
$$F(x)=\int_{}^{}\frac{x\ln(x)}{(1+x^2)^2}=\frac{1}{4}\left( \frac{-2\ln(x)}{1+x^2}+\ln\frac{x^2}{1+x^2} \right).$$
I think the best way to solve our integral $\int_{0}^{+\infty }\... |
H: How many divsors of $4725$ are there?
I need to solve the following problem:
How many divsors of $4725$ are there?
I found the number of divsors between $0-9$ that can divide $4725$ which are: $3,5,7,9$ but how do I find the others? Also, what is a good way to approach such problems?
Thanks!
AI: Here's a proof I... |
H: Polytopes characterization in $\mathbb R^n$
Given $P = \{x \in\mathbb R^n \mid a_1x_1 + \ldots + a_nx_n = \text{constant}\}$, $(a_1, \ldots , a_n) \ne 0$.
Can $P$ be a polytope? I think that with $N = 1$, $P$ is a point. Can a point in $\mathbb R^1$ be a polytope?
Thank you all!
AI: This is essentially an issue o... |
H: Area between $y=e^{-x}$ and $y=e^{-x}\sin x$
Let $x_n$ denote the $x$-coordinate of the $n$th point of contact between the curves $y=e^{-x}$ and $y=e^{-x}\sin x$, with $0<x_1<x_2<\cdots$, and let $A_n$ denote the area of the region enclosed by the two curves between $x_n$ and $x_{n+1}$. Show that $$\sum_{n=1}^\inf... |
H: Find maximum and minimum of $f(x, y) = xy$ on $D = \left\{ (x,y) \in \mathbb{R}^2: x^2+2y^2 \leq 1 \right\}$
I'm kinda stuck on this one :
Find the minimum and maximum of the given function $f$ on $D$, where $$f(x, y) = xy$$ and $$D = \left\{(x,y) \in \mathbb{R}^2 : x^2+2y^2 \leq 1 \right\}$$
I don't know what t... |
H: Solving Systems of Equations Question
Saw this question and have been unsure of how to solve it properly. Any help would be appreciated!
A pilot of a downed airplane fires the emergency flare into the sky. The path of the flare is modeled by the equation $y =-0.096(x-25)^2+60$, where $y$ is the height of the flare ... |
H: Conversion Calculations
I'm trying to create an app for a workshop that I'll be running.
It's going to be a variation of the temperature converter app that tends to be the "Introduction to programming" default.
Anyway, I was wondering about conversions.
If you take, temperature, time, pressure, distance... and the ... |
H: Find the limit $\displaystyle\lim_{n\rightarrow\infty}{(1+1/n)^{n^2}e^{-n}}$?
Find the limit $\displaystyle\lim_{n\rightarrow\infty}{(1+1/n)^{n^2}e^{-n}}$?
I found the limit as $e^{-1/2}$ using l'Hospital rule. I guess I made a mistake. Because the limit seems to be 1. Also, can we find the limit without L'Hospita... |
H: trigonometric inequality - how to prove it?
Let $ 0 < x < \frac {\pi}{2}$
How to prove it?
$$2 \sin x \le x- \frac {\pi}{3} + \sqrt {3} $$
AI: The following is a standard mechanical approach. (But mechanical is often not best.) Let
$$f(x)=x-\frac{\pi}{3}+\sqrt{3} -2\sin x.$$
Then $f'(x)=1-2\cos x$.
The derivative i... |
H: Using FFT in matlab
I am not completely sure if this is where a MatLab question belongs, so if not, please direct me where I should ask.
But onto my question. I am working on trying to deconvolution a signal with noise. So I have $h(x)=f(x) \ast g(x) +n(x) $. I want to find what the function g is. n(x) is whit... |
H: If totally disconnectedness does not imply the discrete topology, then what is wrong with my argument?
Assume that $X$ is a totally disconnected space. Then every two-point set is disconnected, which implies that every singleton is open in the topology of $X$ (because the one-point subsets of two-point sets form a ... |
H: Totally geodesic immersions
Let $ x: M \rightarrow \overline{M} $ be a totally geodesic immersion, where $ M $ is a $ k- $ dimensional Riemannian manifold and $ \overline{M} $ is a $ n- $ dimensional Riemannian manifold. Is it true that $ x $ is an embedding?
Thanks
AI: Not as it stands. Consider the usual irratio... |
H: Draw two or three balls from an urn with ten balls
My urn contains two black balls and eight white balls. What is the probability that I get the two black balls
a) after two draws
b) after three draws?
My approach is to draw a decision tree.
draw 1: W B
draw 2: W B W B
dr... |
H: Proportion and inverse proportion
It is given that $x$ is directly proportional to $y^2$ and $y$ is inversely proportional to $z$. If $x=20$ and $y=2$ when $z=5$
(A) the value of $y$ when $z=20$
(B) the value of $x$ when $y=3$
(C) an equation relating $x$ and $z$
(D) the value of $z$ when $x = \frac 5 4$, given tha... |
H: What is limit superior and limit inferior?
I've looked at the Wikipedia article, but it seems like gibberish. The only thing I was able to pick out of it was the concept of infimum (greatest lower bound) and supremum (least upper bound), as I had learned them previously in an intro discrete math course.
The limit ... |
H: What defines the dimension of a representation?
For example, if I have trivial representation of $S_3$, why does it have dimension 1? Why can't I take a vector space of dimension 2 and map all the vectors identically so I would have a representation of dimension 2?
Thanks in advance!
AI: A trivial representation do... |
H: Definition of compact set/subset
I found one exercise in my book
Let $X$ be a compact subset of a metric space $M$. Prove that $X$ is closed.
In the definitions, the book only mentions compact space and never compact set.
An open cover of a metric space $M$ is a collection $U$ of open subsets of $M$ such that $... |
H: Finding the definite integral $\int_0^1 \log x\,\mathrm dx$
$$\int_{0}^1 \log x \,\mathrm dx$$
How to solve this? I am having problems with the limit $0$ to $1$. Because $\log 0$ is undefined.
AI: Yet another approach:
\begin{align}
\int_0^{1}\ln x\,dx=\left[\frac{\partial}{\partial s}\left(\int_0^1x^s dx\right)\ri... |
H: Derived functors - how is natural transformation between $L_0T$ and $T$ constructed?
For simplicity's sake, consider the categories $R\text{-Mod}, S\text{-Mod}$ of left $R$-modules and left $S$-modules, respectively, and let $\mathcal{F}$ be some precovering class in $R\text{-Mod}$. Then given a functor $T:R\text{-... |
H: Splitting fields of symmetric groups
Is it true that $k$ is a splitting field of $S_n$ if and only if the characteristic $p$ of $k$ is zero or larger than $n$? The fact that the character table (over $\mathbb C$) has only integer entries smaller or equal to $n$, seems to imply this, or am I mistaken?
If the stateme... |
H: Ordinary Differential Equation
Consider a system of differential equation
y'(t) = By(t) where B is a 2X2 matrix defined as: B = [α , -β ; β , α]
where α,β ∈ ℝ and (α,β) ≠ (0,0).
A) If the roots of B are purely imaginary verify that the the solution of this system y=( y1 , y2 ) is of the form y1(t) = c1 cos(c2 + βt... |
H: How to find the coordinates of the point on a sphere closest to another point?
Take the sphere $x^2 + y^2 + z^2 = 4$ and find the point on it that is closest to the point $(3,1,-1)$ without using calculus.
AI: The point on a surface closest to a point not on the surface (called an "external point") lies on a normal... |
H: Congruence relation possible typo?
Is the following a typo? If $a \equiv b \pmod{m}$, then for some scalar $c>0$, $ac \equiv bc \pmod{mc}$
Or should it be $\pmod{m}$?
AI: No typo in the congruence equations or implication:
Theorem: $$\text{If}\; a \equiv b \pmod{m},\;\text{ then for any scalar }\;c \neq 0,\; ac \e... |
H: Is a bounded sequence Cauchy if the element come closer?
MISSED THE CONDITION ON THE SUP....
I try to prove the existence of a limit in a Banach space. I have a sequence $\{x_n\}$ and I have managed to prove that $\limsup_{n\to\infty}\|x_n\|= C<\infty$ and $\|x_n-x_{n-1}\|>\|x_{n+1}-x_n\|$, $\forall n\in\mathbb{N}$... |
H: Simpsons rule & Lagrange?
What is the relation between Lagrange interpolation and Simpson's rule to integrate some function with some points $x_0,f(x_0)$; ... $x_n, f(x_n)$ ?
AI: There are various approaches to deriving Simpson's Rule. A common one uses a special case of Lagrange interpolation. Recall that we used ... |
H: How to show convergence or divergence of a series when the ratio test is inconclusive?
Use the ratio or the root test to show convergence or divergence of the following series. If inconclusive, use another test:
$$\sum_{n=1}^{\infty}\frac{n!}{n^{n}}$$
So my first instinct was to try the ratio test due to the exis... |
H: How can evaluate $\lim_{x\to0}\frac{3x^2}{\tan(x)\sin(x)}$?
I know this
$$\lim_{x\to0}\frac{3x^2}{\tan(x)\sin(x)}$$
But I have no idea how make a result different of:
$$\lim_{x\to0}\frac{3x}{\tan(x)}$$
I would like understand this calculation without using derivation or L'hôpital's rule. Thank you.
AI: Since $\tan ... |
H: Algebra and skills needed for Hatcher
A couple of months ago I asked a professor by e-mail to mentor me on topology during the summer. He advised me to study general topology (Hausdorff spaces, connectedness, compactness) and algebraic topology (from Hatcher) before we meet, as time permits.
Yesterday I finished ch... |
H: Parent function of $\sqrt{x^2 - 4}$?
Does this particular function($\sqrt{x^2 - 4}$) have a parent, such that it can be represented as a translation, compression, rotation, stretching, etc, of the parent graph?
AI: $y=\sqrt{x^2-4}$ is the top half of the two branches of the hyperbola $x^2-y^2 = 4$, with standard fo... |
H: Clarification on quotient groups
I've only recently started looking at quotient groups, so I don't know if this question will make sense...
In this wiki article, $G/H$ is defined as the set of left cosets of $H$ in $G$, without any reference to whether or not $H$ is normal. In the quotient group article, however, t... |
H: Path of particle under gravity
If a particle is subjected to gravity then
$$\frac{∂^2 u}{∂\theta^2} +u = \frac{GM}{h^2} $$
where
$$ u = \frac{1}{r}$$
and
$$h = r^2\dot{\theta}.$$
If you solve this you get
$$u = A\sin\theta+B\cos\theta + \frac{GM}{h^2}.$$
But the general solution for... |
H: Archimedean Proof?
I've been struggling with a concept concerning the Archimedean property proof. That is showing my contradiction that For all $x$ in the reals, there exists $n$ in the naturals such that $n>x$.
Okay so we assume that the naturals is bounded above and show a contradiction.
If the naturals is b... |
H: Converting Ranges
I am writing a program for the arduino that takes a number as input and displays colors based on that input. At any given time, I know the value of the variables min and max where min is the minimum value of the input range and max is the maximum value of the input range. The problem is, in orde... |
H: limit of expression with first and second derivatives
I got the expression: $$\frac{X'(t) X''[t]+Z'(t) Z''(t)}{\sqrt{X'(t)^2+Z'(t)^2}}$$
How do I find the limit when I get to a point with $X'(t)=Z'(t)=0$, it's like everytime it happens, the limit is $\pm\sqrt{X^{\prime\prime }(t)^2+Z^{\prime\prime }(t)^2}$ is this ... |
H: Finding distance from point to line
Knowing the position of 3 points($A, B, C$) , how can I get the distance from $A$ to the line $\overline {BC}$ if I know the angle?
AI: If you have the positions as vectors, compute $\frac{|(A-B)\times (C-B)|}{|C-B|}$. |
H: Characterization of $T+T^*\geq 0$, for $T$ a bounded operator on Hilbert space
(This is Exercise 3.2.1 in Pedersen's book Analysis Now.) Let $T$ be a bounded operator on a complex Hilbert space $H$. I want to prove that $T+T^*\geq 0$ if and only if $T+I$ is invertible in $\mathcal{B}(H)$ with $\|(T-I)(T+I)^{-1}\|\l... |
H: Consequence of injectivity of projections from covering spaces
We have the theorem which says that the induced homomorphism $p_* : \pi_1(\tilde X,\tilde x_0)\rightarrow \pi_1(X,x_0)$ is injective (hence a monomorphism). Here $\tilde X$ is a covering space of $X$.
I am just trying to digest this fact (the injectivit... |
H: Inequality holding for complex numbers in the unit disk
In Nehari's book Conformal Mapping he gives it as an exercise to prove that for $a,b\in \mathbb{C}$, $|a|, |b| <1$ we have $$\frac{|a|-|b|}{1-|ab|} \leq \left|\frac{a-b}{1-\overline{a}b}\right| \leq \frac{|a|+|b|}{1+|ab|}.$$I've been trying to prove this ineq... |
H: Lines and planes at space
First of all, sorry for my poor English.
Can someone please help me?
How can I find the parametric equation of the line that have $ A=(1, -2, -1) $ and passes through the skew lines
$r:$
$ x= z -1$
$ y= 2z - 3 $
$s:$
$x= z-2$
$y= -z + 1$
Thanks!
AI: The parametric equa... |
H: What is combinatorics? How is it related to Ramsey theory? What is the background needed to study it?
What is combinatorics? How is it related to Ramsey theory? What is the background needed to study it?
When I was reading about Ramsey theory in some reviews on some books, many people mentioned this branch of math... |
H: Relation between differentiable,continuous and integrable functions.
I have been doing lots of calculus these days and i want to confirm with you guys my understanding of an important concept of calculus.
Basically, in the initial phase,students assume that integration and differentiation are always associated to ... |
H: A finite-type quiver has no self-loops
I am reading through Etingof et al's notes on representation theory, and they assert in Exercise 5.4(c) on page 80 that a finite-type quiver has no self-loops. I think the way to show this is to consider the simplest case: a quiver with one vertex and one edge - the self-loop... |
H: Help with implicit differentiation: $e^{9x}= \sin(x+9y)$
Find $\;\dfrac{dy}{dx}\;$ given $\;e^{9x}= \sin(x+9y)$
the answer is $\;\displaystyle\frac{e^{9x}}{\cos(x+9y)}- \frac{1}{9}$.
Can you show the process of how this is worked?
thanks.
AI: Differentiate both sides of your equation using chain rule
$$
\left... |
H: Derivation of Pythagorean Triple General Solution Starting Point:
I was reading on proof wiki about the derivation of the general solution to the pythagorean triple diophantine equation:
$$
x^2 + y^2 = z^2,
$$
where $x,y,z > 0$ are integers.
I came across the following general solution to the primitive function:
\... |
H: prime notation clarification
When I first learned calculus, I was taught that $'$ for derivatives was only a valid notation when used with function notation: $f'(x)$ or $g'(x)$, or when used with the coordinate variable $y$, as in $y'$.
But I have seen on a number of occasions, both here and in the classroom, where... |
H: Pointwise convergence and uniform convergence
Suppose that $f$ has a uniformly continuous derivative. We define $\ f_n: \Bbb R\to\Bbb R $ by
$$\ f_n(x) = n \left( f \left(x + \frac{1}{n}\right) - f(x)\right) $$
Find a pointwise convergence $\ f_n$. Prove that the sequence $\ f_n$ converges uniformly to its limit.... |
H: Systems of Quadratic Equations Question
looking for help on this question.
Solve the following systems of equations algebraically using the quadratic formula.
$$\begin{align} y& =-x^2+2x+9\\ y& =-5x^2+10x+12\end{align}$$
Any help would be appreciated!
AI: Hint: put both equations equal to one another. You'll have a... |
H: Prove the Equality of Two Integrals
This is what I've done so far:
$V_1 = \pi\int_0^af(x)^2dx = -\pi\int_0^by^2 (1/f'(x))\ dy = -\pi\int_0^by^2 (1/f'(g(y)))\ dy = -\pi\int_0^by^2 g'(y)\ dy$
Integrating by parts:
$u=y^2,\ du=2y\ dy, \ v=g(y), \ dv = g'(y)\ dy$
$y^2g(y)|_0^b -2\pi\int_0^by\ g(y)\ dy = -ab^2 -2\pi\in... |
H: Radius of convergence of $\sum_{n = 0}^{\infty} (a_1^n + \dots + a_k^n)z^n$, where $|a_1| = |a_2| = \dots = |a_k| = 1$
Here's the problem: Find the radius of convergence of $f(z) = \sum_{n = 0}^{\infty} (a_1^n + \dots + a_k^n)z^n$, where $|a_1| = |a_2| = \dots = |a_k| = 1$, and $a_i \in \mathbb{C}$.
Since the serie... |
H: Lines and Planes
can someone please help me? How can I find the equation of the plane which contains the line
$r:$ $ x= 2 + 2\lambda$
$ y= 3 - \lambda$
$ z= -3\lambda$
($\lambda$ is a Real number)
and makes an angle of $\pi$/4 with the line
$ s:$ $x= 1 - 2t $
$y= 2 - t$
$z=3 - t$
(t is a Real number)
Th... |
H: Finding the derivative of a relational problem
I am self studying some calculus and I have gotten really stuck! I thought I had the right idea but I keep getting the answer totally wrong. I am sure I am missing something important. Here is the problem:
For the equation $6x^{\frac{1}{2}}+12y^{-\frac{1}{2}} = 3xy$, f... |
H: What is some application of variance in actuarial science/insurance risk?
What is some application of variance in actuarial science/insurance risk?
I learn that a lot of applied math book of actuarial science have variance of probability distribution frequently.
Don't the books already have its probability for di... |
H: $M$ compact iff every open additive cover of $M$ contains $M$
Call an open over $\mathscr{U}$ of a metric space $M$ an additive cover if whenever $U,V\in\mathscr{U}$, we have $U\cup V\in \mathscr{U}$. Prove that $M$ is compact if and only if every additive open cover of $M$ contains $M$.
To begin with, I'm quite ... |
H: Help with differentiation of natural logarithm
Find $\;\dfrac{dy}{dx}\;$ given $y=\frac{\ln(8x)}{8x}$.
The answer is $\;\dfrac{1-\ln(8x)}{8x^2}\;$.
Can you show the process of how this is worked?
Thanks.
AI: Here we can use the quotient rule and the chain rule:
Quotient rule: $\quad$ If $y = \dfrac{f(x)}{g(x)... |
H: What could be a homeomorphism from the circle to a triangle?
I'm looking for a bijection from a circle to a triangle that is continuous with a continuous inverse. What could be one?
AI: I will assume that by circle you mean the curve, not the disk, and also that by triangle you mean the "curve" consisting of three ... |
H: What is wrong with my thinking, simple groups order $168$
How many elements of order $7$ are there in a simple group of order $168$?
I will work on this more but I have seen some solutions out there. My only question is regarding what is wrong what my thinking here:
An element of order $7$ produces a cyclic group... |
H: Normal Vector to a Sphere
I'm having kind of a problem on calculating the normal vector to a sphere using a parameterization. Consider a unit-radius sphere centered at the origin.
One can parameterize it using the following:
$$P(\phi, \theta)=(\sin(\phi)\cos(\theta),\,\sin(\phi)\sin(\theta),\,\cos(\phi)) $$
My Vect... |
H: Chain of compact subsets and their intersection
Let $\{X_n\}$ be a sequence of compact subsets of a metric space $M$ with $X_1\supset X_2\supset X_3\supset\dotsm$. Prove that if $U$ is an open set containing $\bigcap X_n$, then there exists $X_n\subset U$.
So suppose for contradiction that there doesn't exist $X_... |
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