All Questions

3answers
2k views

Find the 2nd order Taylor Polynomial of y(x) about x = 0, given:

I am given the equation: $$y^3 + y^2 + y - 3 = x$$ with $y(0)=1$. I am wondering if what I have done is valid, given that this is a homework question for an Integral Calculus class, but I seem to ...
1answer
29 views

Euler's derivation of e?

Does anyone know where I can read Euler's original derivation of the infinite series used to define $e$? I mean the series as defined in the wikipedia page about $e$.
0answers
8 views

Why does the follsolution of the ODE $x'=(t.cos(t)+sin(t),t^2.cos(t)+2t.sin(t))$, $x(0)=(0,0)$ doesn't contradict Picard?

I have found the following solution: $\varphi(t)=(t.sin(t),t^2.sin(t))$, but $\varphi(0)=\varphi(2\pi)$, while $\varphi'(0)$ and $\varphi'(2\pi)$ are linearly independent. My professor said that this ...
2answers
16 views

what is the chromatic index of this graph

I am trying to figure out the chromatic index of this graph. I thought that it is 4, however in the solutions that I have it says that the chromatic index is only 3. Which is the correct answer?
1answer
18 views

Quotient of $\mathbb{P}^n\times\cdots\times \mathbb{P}^n$ by $S_d$: is it projective?

Let us have $\mathbb{P}^n\times\cdots\times \mathbb{P}^n$ ($d$ copies of $\mathbb{P}^n$), and we have the symmetric group $S_d$ act by permuting the factors. Is the quotient projective? I have the ...
1answer
12 views

2answers
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2answers
16 views

Finding all solutions to 2SAT

Let's say there are four variables A,B,C,D with values False, True, True, False. How do you find all satisfying truth? I know there's an approach of drawing a directed graph. How do you do that?
1answer
28 views

Prove that the open interval $(0,1)\subset \mathbb{R}$ is uncountable if and only if $\mathbb{R}$ is uncountable.

$I = (0,1) = {x \in \mathbb{R} : 0 < x < 1}$ $\mathbb{N} \nsim I \iff \mathbb{N} \nsim \mathbb{R}$ Forward $\mathbb{N} \nsim I \Rightarrow \mathbb{N} \nsim \mathbb{R}$ Union of countable ...
4answers
35 views

How many group homomorphisms they are?

How many group homomorphisms there are: $$\phi:\mathbb{Z}_{16}\to \mathbb{Z}_{20}$$ So from exercises that I saw I want to say only the trivial, and that makes them 2, but is the id here a ...
0answers
9 views

Equicontinuity of a set of functions

Let $E$ be a compact space and $X,Y \subset E$ compact subsets of $E$. I wonder if the family of continous functions $C(X,Y) := \{f:X\rightarrow Y \mid f\ \text{is continuous} \}$ is ...
2answers
15 views

Disjoint union topology vs Product topology

I am trying to understand the differens between the two more precisely. I am aware that the product topology is the product in category while the disjoint union topology is the coproduct. ...
2answers
14 views

When solving a simultaneous equation like this:

When solving a simultaneous equation like this: $2y - x = 4$ $2x² + 3y² = x + 4y = 17$ How do you express this second equation? I know how to solve simultaneous equations. I'm not just sure of ...
1answer
35 views

Mistake in Conway's Analysis book

I'm reviewing Conway's complex analysis book on page 5 and I think he made a mistake: Following my calculations the cube of the second solution is $-2\sqrt 2$ instead of $1$. Thanks
0answers
10 views

Help with these calculations in Conway's complex analysis book

I'm studying Conway's complex analysis book and on page 9 he gets this relation between the point $(x_1,x_2,x_3)$ in the Riemann sphere and $z$ in the extended plane $\mathbb C_{\infty}$: ...
1answer
7 views

Polynomial rings which are Dedekind domains.

Let $K$ be a field. Is $K[X,Y]$ a Dedekind domain?
1answer
22 views

Simple Proof of FT of Algebra

Which proof of the Fundamental Theorem of Algebra requires minimum mathematical maturity and has the best chances to be understood by an amateur with knowledge of complex numbers and polynomials?
0answers
8 views

How to find reflexive, symmetric and transitive closure of a relation R?

I have to solve this question. Any hints or what closure actually means?
0answers
2 views

Invariant subspaces of L \sb{\infty} G

I am reading a book named "Lectures on Amenability". The author Dr. Volkar Runde defines the following : Let $G$ be a locally compact group, and let $E$ be a subspace of $L^{\infty}( G)$ containing ...
0answers
9 views

Find a line with maximum points from N points

You are given N points and you want to draw a line such that maximum points lie on the line. What is the efficient way to find the number of maximum points and find those points?
1answer
24 views

Vector calculus - Material derivative in spherical coordinates…

So this one might be a little simple for some of you but I was hoping I could get all the nuts and bolts needed to show this for myself. I have the following relationship, which makes use of the the ...
1answer
32 views

If a subspace of $L^p\cap L^q$ is closed with respect to both norms, the norms are equivalent on this subspace

Let $\mu$ be a positive measure, $1< p,q<\infty$ and let $X$ be a linear subspace of $L^p(\mu)\cap L^q(\mu).$ Suppose $X$ is closed in $L^p(\mu)$ and also $X$ is closed in $L^q(\mu)$. Prove ...
0answers
7 views

Iterating adjunctions in two different ways

Let $F$ and $G$ be two endofunctors of a category $\mathcal C$ such that $(F,G, \eta, \epsilon)$ is an adjunction, i.e. I denote the unit by $\eta$ and the counit by $\epsilon$. Then we may form two ...
0answers
12 views

Iterating the suspension-loop adjunction in two different ways

Let $X$ be a sufficiently nice topological space (i.e. an object of a category of spaces where the reduced suspension-loops, $(\Sigma, \Omega)$, holds.) There are two directed systems of spaces one ...
1answer
8 views

Grade of a maximal ideal in a polynomial ring

Let $M$ be maximal ideal in $R[x]$, the ring of polynomials over a commutative ring $R$, and let $P=M∩R$. If $a_1,...,a_n$ is a maximal R-sequence in $P$, it is clearly an R-sequence in $PR[x]$. Let ...
0answers
15 views

Differentiability question wouldn't f(0,0) be undefined because

Hi i was given a set of solutions i couldn't and got stuck at this part. Please My question is not the given question but rather the given solution. I know that differentiability is to show that ...
0answers
6 views

Show that the interest rate sequence is defined

I have a problem where I have to show that the interest rate sequence defined by $x_n := (1+\frac{1}{n})^n$ is Cauchy. I think I can apply the binomial theorem to show that but I have no idea how ...
0answers
11 views

Number of circular seating arrangements if each person can't sit next to two other people

There are 30 people sitting around a round table, and those 30 people all arrived to the event in 10 groups of 3. How many different ways can those people sit around the table if no-one is allowed to ...
0answers
13 views

How can I make these two functions coincide?

This function is the arc of an ellipse, but is generated by a semicircle plus a trig function (arcsin or arccos or arctan) $$(\sqrt{(x(10-x))} + 5\pi -10\arctan(\sqrt{(10-x)/x)})/2$$ or ...
0answers
18 views

Excursion of random walk conditioning on return

Consider a simple random walk in one dimension starting from the origin. Let $\epsilon>0$. How to prove that, conditioning on the event that the random walk is at the origin at time $n$, the ...
3answers
164 views
+500

Systematic solution to my soccer ball puzzle

I once received a puzzle that can be described as follows: There are $12$ black pentagonal and $20$ white hexagonal pieces. The goal is to form a soccer ball from these (aka. truncated icosahedron). ...
2answers
60 views

For which orthogonal matrices does the matrix exponential converge?

Part (a) For which 2×2 orthogonal matrices A does $\large e^A=I+\frac{A^1}{1!}+\frac{A^2}{2!}+…$ converge? Part(b) For what A does the series converge to an orthogonal matrix? My work: Let A ...
1answer
27 views

Find all homomorphisms from $A_4, D_{2n}$ to $\mathbb C^\times$

I have a question that how to find all homomorphisms from dihedral group $D_{2n}=\langle s,r : s^2=r^n=1 , srs=r^{-1} \rangle$ to the multiplicative group $\mathbb C^\times$? alternating group ...
0answers
22 views

0answers
4 views

i need help in searching for peaks and subtracting them iteratively

assuming a matrix (5,5) with a peak value at the center, another matrix say (20,20) with randomly distributed peaks of different magnitudes. how can i iteratively subtract the peaks in the bigger ...
1answer
58 views

Partial Integral of an ellipse

this is my first question on stack exchange so please bear with me. I am trying to generate a synthetic image of an ellipse in Matlab where each pixel is shaded according to how much of that pixel ...
0answers
14 views

Relations in $U_q(sl(3))$ by induction

I am looking to the quantum group $U_q(sl(3))$ with generators $E_1,E_2,F_1,F_2,K_1$ and $K_2$. I want to find out what the elements to to the elements of the form $F_2^mF_1^nv$ where v is a vector ...
1answer
131 views

Theories with countably many countable models

Having another question in mind (which is not yet fully worked out, but will come soon) I'd like to gather some examples of (interesting) theories with countably many countable models ...
1answer
9 views

How to mathematically phrase a logical “or” over multiple options.

I apologize if this is off topic but I think it fits. I am trying to succinctly describe a logical check for an algorithm which is a logical or. That is I want to say that we remove any cells $k$ ...
0answers
56 views
+50

Minkowski-like inequality for the trace of outer products of random vectors

I am wondering if the following inequality is correct and can be shown? Let $A$ and $B$ be random vectors of dimension $n$. Then for $p \ge 1$ \begin{align} E^{\frac{1}{2p}} \left[ \left| Tr ...
0answers
8 views

Value at Risk, Confidence level, infimum

Given an uncertain future loss L which is modelled as a random variable with cdf Fl , the value at risk (VaR) at confidence level α is defined as $VaR_\alpha (L) = \inf\{l \in R | Fl(l) \ge \alpha\}$. ...

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