1
vote
2answers
276 views

How can I find an upper bound for the radius of an arc, given arc length and chord length?

This is similar to this question, except I am finding the radius now using the Bisection method and then Newton's method for finding a zero. This is a computer science for a Numerical Methods course. ...
3
votes
2answers
134 views

Different ways to simplify the structure of a graph

Let $G = (V,E)$ be an undirected finite simple graph with no loops. Definition Let's call a subset $U\subset V$ autonomous if every vertex of $V\ \setminus U$ is either adiacent to every vertex of ...
5
votes
2answers
380 views

Correlation Coefficient between these two random variables

Suppose that $X$ is real-valued normal random variable with mean $\mu$ and variance $\sigma^2$. What is the correlation coefficient between $X$ and $X^2$?
0
votes
1answer
74 views

Cylindrical Shells problem (can't find region)

"Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis." I have two equations : x= 2y^(2)-y^(3) and x= 0. ...
4
votes
1answer
154 views

Are there disjoint closed sets which contain sequences that get arbitrarily close to each other?

I had a question of curiosity. Take the interval $(0,1)$ with the usual metric in $\mathbb{R}$. Is it possible to find closed sets $X$ and $Y$ with $X\cap Y=\varnothing$ such that there is a sequence ...
3
votes
1answer
181 views

Limit involving the exponential of an expression plus a small-o

I'm trying to formalize the following limit: $$\lim_{x \to \infty} \left[1+\frac{a}{x}+o\left(\frac{1}{x}\right) \right]^x= e^a$$ however I can't see how to rigorously show this identity. Any ...
2
votes
3answers
334 views

Finding an $f(x)$ that satisfies $f(f(x)) = 4 - 3x$

I need to find $f(f(x)) = 4 - 3x$ In other examples, such as $f(2)$, I can see that the result equates to $-2$ or $f(x^2)$ becomes $-3x^2 + 4$. Do I really just substitute $f(x)$ for $x$ and ...
1
vote
1answer
59 views

Let $f(t) = \frac{t - x}{t + y}$

I have the problem. Let $$f(t) = \frac{t - x}{t + y}.$$ Show that $$f(x + y) + f(x - y) = \frac{-2y^2}{x^2 + 2xy}.$$ I know that this is just some substitution followed by simplification, ...
6
votes
5answers
422 views

Logical NOT of an implication

I was looking through my notes but I was unable to find the answer to this, which I need to start am assignment question. What would the following be, in terms on moving the negation inside the ...
4
votes
1answer
165 views

Dimension of compositum of field extensions

Suppose we have fields $L$, $M$ and $N$ all infinite algebraic Galois extensions of a field $k$ such that $L \cap M$, $L \cap N$ and $N \cap M$ are finite dimensional extensions of $k$. Then is $L ...
2
votes
1answer
114 views

Minimum spanning tree assumption

If I have a weighted graph $G$, with minimum spanning tree $M$, is the following true: If a new edge $e$ is added to $G$, in order to check whether $M$ is still the MST of $G$, it suffices to check ...
2
votes
1answer
66 views

Scale-invariant geometries where $C/d \neq \pi$

Is there a geometry where everywhere, or locally: $$ \frac{C}{d} = \mathrm{constant} \neq \pi$$ $C, d$ being the circumference and diameter of a circle?
3
votes
1answer
236 views

Proof of identity involving binomial coefficients

I'll be happy if you could help me prove this argument with algebraic tools: $${N\choose 0}a^N+{N\choose 1}a^{N-2}+{N\choose 2}a^{N-4}+{N\choose 3}a^{N-6}+\dots = \frac{a^2+1}{a}$$ Thanks, Don
4
votes
1answer
109 views

Measurability of a mapping to a product space and of its component mappings

In Section 13 of Probability and Measure by Billingsley, it has been shown that for a measurable space $(F, \mathcal{F})$, $g:F\rightarrow \mathbb{R}^m$ and $g_i: F\rightarrow \mathbb{R}$ with $g(x) = ...
0
votes
0answers
71 views

Question on p-divisible groups

Given a $p$-divisible group $(G_n)_{n \in \mathbf{N}}$, $G_n$ a $\mathbf{Z}/p^n$-module, do we have $G_{n+1}/p^nG_{n+1} \to G_n$ is an isomorphism? My idea: One has $0 \to G_1 \to G_{n+1} \to G_n \to ...
2
votes
4answers
343 views

Evidence of Absence = Absence of Evidence?

Any clever-cloggs out there who can explain the formula below in more simple English please? - Do you agree with the formula?
1
vote
2answers
95 views

Question about topology on $K^\times$ in local CFT

I'm trying to parse a page in Milne's CFT notes. The local reciprocity law gives us isomorphisms $$\phi_{L/K}:K^\times/Nm(L^\times)\to \textrm{Gal}(L/K)$$ for all abelian extensions $L$ of a ...
2
votes
2answers
220 views

The sign of a given permutation

If I have a permutation $\sigma$ on the set $A$ written by disjoint cycles. There are $n$ disjoint cycles can I then write the sign of the permutation as: $\operatorname{sign}(\sigma) = ...
6
votes
3answers
260 views

When does $a + b$ divide $a^p + b^p$?

I came across a problem in Niven's number theory text (problem 51 on page 20) that asks the following: Show that if $(a, b) = 1$ and $p$ is an odd prime, then $$\left(a + b, \frac{a^p + b^p}{a + ...
9
votes
3answers
321 views

Why does $a^n - b^n$ never divide $a^n + b^n$?

I'm working through the problems in Niven's number theory book, and problem 46 in section 1.2 (page 19) has me stumped. Prove that there are no positive integers $a, b, n > 1$ such that $(a^n - ...
2
votes
1answer
97 views

Opposite and connected quivers problem

Here are two problems from Elements of the Representation Theory of Associative Algebras by D. Simson, et. al (Page $65$). $1$. Let $Q=(Q_{0},Q_{1},s,t)$ be a quiver. Prove $(KQ)^{op} \cong KQ^{op}$ ...
1
vote
1answer
416 views

Extending the cube root function to $\mathbb{C}$

On $\mathbb{R}$, the cube-root function (call it $f(x)$) is a well-defined single-valued function which is $C^\infty$ except at the origin. ($f(27)=3$, $f(-27)=-3$, etc.) The complex cube root (call ...
2
votes
1answer
170 views

A number system

Can we have a number system $S$ of cardinality continuum such that for every $x \in S$, there is a unique $y \in S$, such that for all $z>x$ in S, $x<y\le z$ holds?
5
votes
1answer
514 views

Prove there is no element of order 6 in a simple group of order 168

Let $G$ be a simple group of order 168. Let $n_p$ be the number of Sylow $p$ subgroups in $G$. I have already shown: $n_7 = 8$, $n_3 = 28$, $n_2 \in \left\{7, 21 \right\}$ Need to show: $n_2 = 21$ ...
0
votes
1answer
84 views

Set for which two characteristic functions are equal

I am stuck on this problem involving characteristic functions. Say you have two characteristic functions, $\phi_1$ and $\phi_2$, and you are looking at the set $A =\{t : \phi_1(t) = \phi_2(t)\}$. How ...
13
votes
1answer
1k views

Is chess Turing-complete?

Is there a set of rules that translates any program into a configuration of finite pieces on an infinite board, such that if black and white plays only legal moves, the game ends in finite time iff ...
4
votes
3answers
351 views

Power function inequality

Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ . I recently discovered this result. I am sure it is known, but it is new to me. It is ...
3
votes
0answers
52 views

Proving $\mathscr L(C_G(H)) \subseteq \mathfrak c_{\mathfrak g}(\mathfrak h) = \{ \mathrm x \in \mathfrak g \mid [\mathrm x, \mathfrak h] = 0\}$

Let $H$ be a closed subgroup of the algebraic group $G$, $C = C_G(H)$. Prove that $\mathfrak{c} = \mathscr{L}(C_G(H)) \subseteq \mathfrak{c}_{\mathfrak{g}}(\mathfrak{h}) = \{ \mathrm{x} \in ...
0
votes
0answers
280 views

Map from $SL(2,\mathbb C)/SU(2)$ to $SL(2,\mathbb C)$

I am trying to write a map $s$, such that $s: SL(2,\mathbb{C})/SU(2) \to SL(2,\mathbb C)$. Using Iwasawa decomposition, I see that any element $g$ in $SL(2,\mathbb{C})$ can be written as $g = su$, ...
6
votes
3answers
300 views

Can we say (for sure) that "the function is increasing” to mean that the first derivative is positive?

Can we say (for sure) that "the function is increasing” to mean that the first derivative is positive? Whenever $f'$ (the first derivative) is positive the function is increasing,but does that ...
0
votes
1answer
41 views

How to calculate the point where a straight will pass?

I have a straight that pass through the points (a,b) and (c,d). If I want to extend this length in an arbitrary value "V", how can I calculate the additional points where the straight will pass ...
0
votes
0answers
38 views

calculating the arc of a wall anchored elastic object

how can i calculate the arc that an object has if it is anchored to a wall and hangs into the room. To describe it with an example. i have a wooden stick (from thin/leightweight wood) that is anchored ...
3
votes
4answers
306 views

Find the limit as $x$ tends to $-\infty$

Find the limit as $x$ tends to $-\infty$ of $$ f(x)=\frac{\sqrt{x^2+1}}{x+1} $$ I did $$ f(x) = \frac{\sqrt{1+1/x^2}}{1+1/x}\to \frac{\sqrt{1+0}}{1-0} =1 $$ (as $x$ tends to -infinity, $1/x^2$ tends ...
16
votes
2answers
1k views

necessary and sufficient condition for trivial kernel of a matrix over a commutative ring

In answering Do these matrix rings have non-zero elements that are neither units nor zero divisors? I was surprised how hard it was to find anything on the Web about the generalization of the ...
2
votes
2answers
110 views

Find the limit of a function

$$\lim_{x\to 3}\frac{\sqrt{3x} - 3}{\sqrt{2x-4} - \sqrt{2}}.$$ Letting $$F(x) = \frac{\sqrt{3x} - 3}{\sqrt{2x-4}-\sqrt{2}},$$ we have $$F(x) = \frac{\sqrt{3}(\sqrt{x} - ...
-1
votes
1answer
114 views

Best method for this example to get from transfer function to state space

I have this system here: In this example the state space representation $ \frac{dx}{dt} = Ax + bu $ and the corresponding transition matrix $\Phi(t)$ is asked for. So to get the state space, I ...
37
votes
2answers
5k views

Mathematical explanation behind a picture posted (lifted from facebook)

In this image given below, there is an actor's (famous south Indian actor Rajinikanth) image which can be seen only if you shake your head ! I had lifted this from Facebook. I am just curious to ...
3
votes
1answer
186 views

Basic property of homotopy

Suppose $ f : X \to Y$ and $g,h : Y \to Z$ are continuous, with $g \simeq h$. Prove that $ gf \simeq hf $. My attempt: Suppose $ L(x,t) $ gives a homotopy from $g$ to $h$, i.e. $ L(x,t) : Y \times ...
8
votes
3answers
531 views

Uncountability of countable ordinals

According to Wikipedia, there are uncountably many countable ordinals. What is the easiest way to see this? If I construct ordinals in the standard way, $$1,\ 2,\ \ldots,\ \omega,\ \omega +1,\ \omega ...
0
votes
1answer
401 views

How to solve such an equation ? (Line-Plane Intersection)

I don't know how to solve such an equation: $$ t = - \frac{ ...
3
votes
2answers
897 views

How to find whether it is possible for each vertex of a graph to have a different degree?

I want to prove whether it is possible for a graph to have different degrees for each vertex. I think that it can be possible with an example, but I can't prove it with mathematics.
2
votes
1answer
123 views

Find all integers $n\in\mathbb{N}^*$ such that any integer $k$ satisfying $1\le k\le \sqrt{n}$ divides $n$

I've found following: 1, 2, 3, 4, 6, 8, 12, 24 and suspect that no integer larger than 24 satisfies the requirements. How do I prove that or can you find a counterexample?
4
votes
1answer
265 views

Need help with Applications of Differentiation Problem

Question My Working Following the hint + some help I got from tutorial below is what I got ... but I believe I did something wrong ... its not the answer below the question yet $0.955\text{ ...
13
votes
2answers
440 views

Motivation behind the definition of Prime Ideal

Can someone explain what's the motivation behind the definition of a prime ideal? Or why is it exactly called a prime ideal? Has it anything to do it prime numbers?
1
vote
1answer
103 views

Polynomial equations in 2 variables with symmetry

Suppose $P(x,y)$ is a polynomial with real coefficients. Is it true that any solution $(x_0,y_0)$ of the system $P(x,y)=P(y,x)=0$ has the property that $y_0 = \overline{x_0}$ (i.e. they are ...
2
votes
1answer
3k views

Using Fourier series to calculate an infinite sum

Given the Fourier series of the $2\pi$-periodic function defined for $$-\pi\leqslant x \leqslant \pi$$ by $$f(x) = |x|$$ is $$ \frac{\pi}{2} -\frac{4}{\pi} \sum_{k\geq 1, k\ odd}^{\infty} ...
0
votes
0answers
59 views

What is this special type of factor called?

I'm wondering if there's a special term for the following: The (special factor) of a number $x$ is a pair of numbers that multiply to give $x$ but has the smallest difference compared to other ...
16
votes
5answers
1k views

How do I prove that $\cos{\frac{2\pi}{7}}\notin\mathbb{Q}$?

How do I prove that $\cos{\frac{2\pi}{7}}\notin\mathbb{Q}$? Should I use some geometrical approach or apagoge?
1
vote
1answer
259 views

Solving a system of equations in Maple

I have a set of equations which I would like to symbolically solve for a defined set of variables using the solve command in Maple. However, being not familiar with the program, it does not give me ...
1
vote
2answers
188 views

Show that rays of the form $(-\infty, a)$ and $(b, \infty)$ ; $a,b \in \mathbb R$, are a sub-basis for the standard topology on $\mathbb R$?

Show that rays of the form $(-\infty, a)$ and $(b, \infty)$ ; $a,b \in \mathbb R$, are a sub-basis for the topology generated by open intervals of $\mathbb R$ on $\mathbb R$? I'd just like to know if ...

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