All Questions

3
votes
1answer
75 views

Understanding torsion from a presentation

Let $F_2 = \langle a,b \rangle$ be the free group on two generators, and for each word $w \in F_2$, let $G(w) = \langle a, b \ | \ w \rangle$. Is the following statement true? $G(w)$ is torsion free ...
0
votes
1answer
39 views

Which matrix operation should I use.

The title is quite vague, but I don't see how to phrase it. I'm new to MatLab and have very little experience with matrix calculation. Suppose a matrix "a" : ...
1
vote
1answer
108 views

Plotting a complex argument arc

I am having trouble sketching a complex argument arc $$ \text{Sketch the following on an arcand diagram:}\\ \arg\left(\frac{w+1}{w}\right)=\frac{\pi}{6}$$ I've tried to devise a method on my own ...
6
votes
2answers
89 views

limit of derivatives of a function

I wanna show that for $f:(0,\infty)\rightarrow\mathbb R, x\mapsto\exp(-\frac1{x^2})$ the sum of the derivates of $f$, so $\sum\limits_{n=0}^\infty f^{(n)}(x)$, converges to $0$, so ...
0
votes
1answer
68 views

Euclidean Geometry in Classical Thought - Used for Realization or Representation?

I posted this in the Physics.SE Forum but I figured I'd ask this here as well since it's relevant to the forum subject :] Taken from John J. Roche's "The Mathematics of Measurement: A Critical ...
2
votes
1answer
3k views

Probability a coin comes up heads more often than tails

I am told that a fair coin is flipped $2n$ times and I have to find the probability that it comes up heads more often that it comes up tails. Please, how do I find the required probability?
1
vote
0answers
213 views

Proof of smoothness of solution to a parabolic non-linear PDE (edited with images)

Edit: the images are from the paper by Sigurd Angenent called Parabolic Equations for Curves on Surfaces. You shouldn't need any more information to answer the question (I think..) They define the ...
5
votes
2answers
358 views

Primitive element of $\mathbb{Q}(\sqrt{2}+i,\sqrt{3}-i)/\mathbb{Q}$

Is there a clever way to determine a primitive element of the finite extension $$F=\mathbb{Q}(\sqrt{2}+i,\sqrt{3}-i)/\mathbb{Q} \text{ ?}$$ On simpler examples, I've been able to find one by ...
2
votes
1answer
307 views

Prove that a conic section is symmetrical with respect to its principal axis.

A Calculus book that I'm self-studying is asking me to prove the following theorem about conic sections: A conic section is symmetrical with respect to its principal axis. Here is my attempt at ...
2
votes
2answers
414 views

Why is the matrix representing a non-degenerate sesquilinear form invertible?

Let's consider a finite-dimensional vector space $E$ on the field $\mathbb{K}$ (where $\mathbb{K}=\mathbb{C} \ \text{or}\ \mathbb{R}$) and a sesquilinear (or bilinear if $\mathbb{K}=\mathbb{R}$) form ...
1
vote
1answer
292 views

Tensor product of sets

The cartesian product of two sets $A$ and $B$ can be seen as a tensor product. Are there examples for the tensor product of two sets $A$ and $B$ other than the usual cartesian product ? The context ...
5
votes
1answer
922 views

Countably Compact vs Compact vs Finite Intersection Property

There is this exercise: Show that countable compactness is equivalent to the following condition. If ${C_n}$ is a countable collection of closed sets in S satisfying the finite intersection ...
3
votes
2answers
458 views

How to find extrema on a triangle

Let $T\subset\mathbb{R}^2$ be the (closed) triangle bounded by the lines $x+y=4$, $x\ge-1$ and $y\ge-1$. I want to find and classify all the extrema of the function $f(x,y)=-x^2y(x+y-2)$ on the ...
1
vote
1answer
40 views

Is there any way to check if one graph is the result of identification and/or splitting on another graph?

Does some algorithm exist that can be used to check if graph $A$ and graph $B$ are related only by combining or separating vertices? Also, would this be possible if vertices had values (a vertex's ...
3
votes
1answer
667 views

Division into strictly isosceles acute triangles

What is the smallest number of strictly isosceles acute triangles that an equilateral triangle can be divided into? The following construction is by WR Somsky, with 13 triangles. Is this minimal?
8
votes
1answer
562 views

Group presentation for semidirect products

If $G$ and $H$ are groups with presentations $G=\langle X|R \rangle$ and $H=\langle Y| S \rangle$, then of course $G \times H$ has presentation $\langle X,Y | xy=yx \ \forall x \in X \ \text{and} \ y ...
2
votes
0answers
91 views

Why is there no alpha-approximation algorithms for k-center problem where $\alpha<2$?

On page 39 of Design of Approximation Algorithms, the author argues this case with the dominating set problem. I can't understand it. A dominating set problem is a special case of the $k$-center ...
1
vote
3answers
195 views

Diagonal Lemma justification

Given the diagonal lema stated as above: Diagonal Lema. Let $\mathfrak{T}$ be a theory wich is capable of representing the primitive recursive functions, and a codification schema for formulas in ...
2
votes
1answer
263 views

Changing order of summation

I would like to rewrite the sum $$\sum_{i=1}^K \sum_{l=-\infty}^\infty \sum_{j=-\infty}^\infty f(i+lK;j-l)$$ In the form $$ \dots\sum_{s=-\infty}^\infty \sum_{w=-\infty}^\infty f(s,w)$$ where ...
4
votes
1answer
107 views

Width and height of partial ordered sets

The width $w$ of a partial ordered set(poset) is defined as the cardinality of the maximum antichain. By Dilworth Theorem, we know it is equivalent to the minimum number of chains in any partition. ...
7
votes
2answers
100 views

Can anyone give any insight on this group given these generators and relations?

$G = \langle x,y | x^3 = 1, y^3 = 1, (xy)^3 = 1, (xy^2)^n = 1 \rangle$ I am studying this group and I can't seem to get anywhere with it. I've tried making a Cayley Table but it's getting pretty ...
22
votes
1answer
605 views

Connectedness of the spectrum of a tensor product.

Let $A$, $B$ be finite free $\mathbb{Z}$-algebras such that $\operatorname{Spec}(A)$ and $\operatorname{Spec}(B)$ are both connected. Is $\operatorname{Spec}(A\otimes_{\mathbb{Z}} B)$ connected?
1
vote
0answers
126 views

Pyramid and perpendicular planes

I need help with the following problem. Given a pyramid ABCDM. The opposite planes (ABM) and (DCM) are perpendicular to the base ABCD. The base ABCD is trapezoid (AD || BC) and AD=3 cm., BC=5 cm. If ...
3
votes
2answers
77 views

$PGL_2(q)$ acts on $\Omega$ $3-$transitively?

Anyone who studies Permutation Groups will be encountering the following definition: A group $G$ acting on a set $\Omega$ is said to be “Sharply m-Transitive” iff $$\forall (a_1,a_2…,a_m) , ...
1
vote
2answers
173 views

A sufficient condition for order isomorphism of posets?

Let $\mathfrak{A}$ be a poset. For $a, b \in \mathfrak{A}$ we will denote $a \not\asymp b$ if only if there are a non-least element $c$ such that $c \leqslant a \wedge c \leqslant b$. Let ...
3
votes
1answer
226 views

Boundedness of multiplication operator on $L^p$ spaces.

I am asking myself when a $L^p\to L^q$ multiplication operator is continuous. The following should be true: Let $a:[0,1]\to\mathbb{C}$ be a measurable function. Let $T_a: L^p([0,1])\to L^q([0,1])$, ...
25
votes
2answers
1k views

Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$

Let $f(x) \in \mathbb{Z}[x]$. If we reduce the coefficents of $f(x)$ modulo $p$, where $p$ is prime, we get a polynomial $f^*(x) \in \mathbb{F}_p[x]$. Then if $f^*(x)$ is irreducible and has the same ...
2
votes
3answers
132 views

Division by $2p+1$

Can $\left\lfloor{\dfrac{x}{2p+1}} \right\rfloor$ be expressed in terms of $\left\lfloor{\dfrac{x}{p}} \right\rfloor$ for prime $p$? How to divide by $2p+1$ by only using division by $p$? EDIT: The ...
4
votes
0answers
136 views

Set theoretic arguments to prove the existence of a certain null set

Let me recall the well-known Carleson's theorem. Theorem (Carleson). Let $f$ be any periodic $L^2[0, 2\pi]$ function. Let $\hat{f}(n)$ be its Fourier coefficients. Then we have $$\lim_{N \to ...
0
votes
1answer
37 views

index $ n(F;D)$ is odd integer

Let $ F: U \subset \mathbb R ^2 \to \mathbb R^2$ be a map with $ F(x,y) = (f(x,y) , g(x,y))$ ,satisfies $F(-q)=-F(q) \quad \forall q \in D \subset U$ where $D$ is a closed disk with center the origin ...
1
vote
1answer
137 views

Inequalities for a Sum over Elementary Symmetric Polynomials

Define $f(x_1, \dots, x_n) = \sum_{l = 2}(l - 1) \sigma_{l}(x_1, \dots, x_n)$, where $\sigma_{l}$ is the $l^{\text{th}}$-elementary symmetric polynomial and $(x_1, \dots, x_n)$ is non-negative. Beyond ...
0
votes
1answer
280 views

How do I handle image gradient calculation at the edge of images?

The image gradient is the rate of change over any given pixel of an image, either in the horizontal or vertical direction. An image can be thought of as a large matrix of values [0, 255]. A common ...
4
votes
3answers
163 views

Lower bound for $\|A-B\|$ when $\operatorname{rank}(A)\neq \operatorname{rank}(B)$, both $A$ and $B$ are idempotent

Let's first focus on $k$-by-$k$ matrices. We know that rank is a continuous function for idempotent matrices, so when we have, say, $\operatorname{rank}(A)>\operatorname{rank}(B)+1$, the two ...
3
votes
1answer
216 views

System of equations of 3rd degree

I need help with the following system of equations: $ 2y^3 +2x^2+3x+3=0 $ $ 2z^3 + 2y^2 + 3y + 3= 0 $ $2x^3 + 2z^2 + 3z + 3 = 0$
2
votes
4answers
16k views

what is the use of derivatives

Can any one explain me what is the use of derivatives in real life. When and where we use derivative, i know it can be used to find rate of change but why?. My logic was in real life most of the ...
5
votes
2answers
241 views

Example of profinite groups

Could someone help me with an simple example of a profinite group that is not the p-adics integers or a finite group? It's my first course on groups and the examples that I've found of profinite ...
3
votes
1answer
356 views

Second Bianchi identity

This is q. 7 of ch. 4 from Do Carmo's book on Riemannian Geometry . Prove that: $$ \nabla R(X,Y,Z,W,T) + \nabla R(X,Y,W,T,Z) + \nabla R(X,Y,T,Z,W)=0.$$ Let $\{e_i\}$ a geodesic frame on $p$ , it is ...
0
votes
1answer
222 views

Conditional Combinations

Asking this question on SO, I have been advised to post it here. I will be using Javascript to implement : Please consider a row of size 12. On that row, I want to place some items that have 3 ...
5
votes
3answers
218 views

Has anyone ever tried to develop a theory based on a negation of a commonly believed conjecture?

I know that plenty of theorems have been published assuming the Riemann hypothesis to be true. I understand that the main goal of such research is to have a theory ready when someone finally proves ...
-1
votes
3answers
4k views

Condition for commuting matrices

Let $A,B$ be $n \times n$ matrices over the complex numbers. If $B=p(A)$ where $p(x) \in \mathbb{C}[x]$ then certainly $A,B$ commute. Under which conditions the converse is true? Thanks :-)
2
votes
0answers
173 views

Sum and product of an ultrafilter

I know the following simple fact is true, but I can't find a good proof: Over the naturals, the only ultrafilter $\mathcal U$ such that $\mathcal U \oplus \mathcal U = \mathcal U \odot \mathcal U$ is ...
3
votes
2answers
381 views

Monte Carlo algorithm that determines if a permutation of the integers 1 through $n$ has already been sorted.

This question is from "Discrete Mathematics and Its Applications", from Kenneth Rosen, 6th Edition. Devise a Monte Carlo algorithm that determines whether a permutation of the integers 1 through ...
6
votes
1answer
143 views

An “independence” condition on two algebraic elements over $K$.

Let $K$ be a field and let $a,b\in \overline K$ be algebraic elements. I've stumbled upon a certain condition on $a,b$, which I feel could be considered an "independence" condition. I would like to ...
1
vote
1answer
101 views

analysing time series

I'm analysing time series and have a question related to the dependency between the elements. Lets assume I have a time series and want to extrapolate future values. For this purpose I want to know if ...
3
votes
2answers
527 views

Module isomorphic to second dual

Is there a simple condition on a module $M$ over a ring $R$ which will ensure that $M$ is isomorphic to its double dual, $M^{**} = \operatorname{Hom} (\operatorname{Hom}(M,R),R)$? What about a ...
2
votes
1answer
245 views

How many maximal consistent sets are there on a $\mathscr{FOL}$

Let $\mathfrak L$ be a $\mathscr{FOL}$ with completeness and soundness. My question is how many maximal consistent sets on it? I know that every maximal consistent set can be dealt as an ...
11
votes
3answers
658 views

In axiomatization of propositional logic, why can uniform substitution be applied only to axioms?

I'm reading an introductory book about mathematical logic for Computation (just for reference, the book is "Lógica para Computação", by Corrêa da Silva, Finger & Melo), and would like to ask a ...
2
votes
1answer
122 views

Interval scheduling by minimum spanning tree

This is a homework and I'd like your feedback on whether I'm on the right track. Thank you. Problem: There's a project to build a railroad to connect $n$ cities. The railroad that connects any two ...
3
votes
3answers
2k views

What is the importance of determinants in linear algebra?

In some literature on linear algebra determinants play a critical role and are emphasized in the earlier chapters. (See books by Anton & Rorres, and Lay). However in other literature it is totally ...
3
votes
2answers
165 views

Dedekind complete ⇒ Sequentially complete

Let F be an ordered field with least upper bound property. 1.Let $\alpha: \mathbb{N} \to F$ be a Cauchy sequence. Since F is an ordered field, $x$ is bounded both above and below. 2.By assumption and ...

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