1
vote
2answers
24 views

Triangles - sin, cos etc.

I know this is a quite simple question for most of you out there. However it has been a little troubling for me, and would like to get a little help if possible. I have a triangle $ABC$ where I know ...
0
votes
3answers
31 views

integrate $\int \frac{x\cos x}{\sin^2x}dx$

$$\int \frac{x\cos x}{\sin^2x}dx$$ $$\int \frac{x\cos x}{\sin^2x}dx=\int \frac{x\cos x}{1-\cos^2x}dx=\int \frac{x\cos x}{(1-\cos x)(1+\cos x)}dx$$ How can I find the two fractions? if there are ...
0
votes
0answers
2 views

Paradox on a linear system with semi-orthogonal matrix

Let $XA^{T} = B$ be a linear system on $X$ with $A \in \mathbb{R}^{m \times n}$ a semi-orthogonal matrix. Hence, $A^{T}A = I$, but $AA^{T} \neq I$ if $m > n$. Assume the case $m > n$. By ...
3
votes
2answers
51 views

Solve $e^{\sin(z)}=1$ in the complex plane

I am trying to solve $e^{\sin(z)}=1$ in the complex plane. I know that this means that $\sin(z)=2k\pi i$ for some integer $k$. This is equivalent to saying that $$\frac{e^{iz}- e^{-iz}}{2i}=2k \pi ...
1
vote
2answers
34 views

How do I get $\|x\|\le C\|y\|$ in this case?

I feel that the title is a bit uninformative, please feel free to edit it. This is a problem related to the Open Mapping Theorem. Let $T:X\to Y$ be a bounded linear operator from a Banach space X to ...
1
vote
2answers
60 views

Errata in Prof. Rotman AIHA book about projectives in the chain complex category

EDIT After thinking carefully with the help of the clear answer of ZhenLin, I think I will reformulate my question the following way. The text of my original question is kept below. When I look at ...
0
votes
0answers
31 views

What does power of a point represents?

I am really unable to understand what does power of a point shows. We know if there is a point P outside or inside the circle then the power of a point equation says that ...
3
votes
2answers
265 views

How to generalize the Thue-Morse sequence to more than two symbols?

The Thue-Morse sequence is defined as a binary sequence and can be generated like 0, 01, 01 10, 01 10 10 01, 01 10 10 01 10 01 01 10, ... . So the second half of the series is always the binary ...
2
votes
3answers
43 views

What is the sum of all the natural numbers between $500$ and $1000$.

What is the sum of all the natural numbers between $500$ and $1000$ (extremes included) that are multiples of $2$ but not of $7$?
0
votes
1answer
7 views

Proximal operator to Huber function

I want to solve the following problem: $$ \arg\min_x |x|_\mu + \frac{1}{2\sigma} |x-x^k|^2 $$ , where $$|x|_\mu = \begin{cases} \frac{|x|^2}{2\mu}, & |x|<\mu \\ |x|-\frac \mu 2 & |x|\geq ...
0
votes
0answers
6 views

How could someone conclude $\check{H}^i (M, \mathbb{R}) = 0$ for arbitrary $M$?

sorry if this is a very stupid question and I'm missing something very trivial, though I could not solve it after thinking for a while. In page 18-19 of ...
2
votes
1answer
40 views

Is there a closed form expression for the following sum?

Is there a closed form expression for the following sum? $$\sum_{0\le i_1<i_2<\cdots<i_k\le n}r^{i_1+i_2+\cdots+i_k}$$ I can understand that there are $\binom{n}{k}$ such terms and the ...
3
votes
1answer
28 views

Is $\pi^k$ any closer to $[\pi^k]$ than expected?

Particular questions such as Why is $\pi$ so close to $3$? or Why is $\pi^2$ so close to $10$? may be regarded as the first two cases of the question sequence Why is $\pi^k$ so close to $[\pi^k]$? ...
1
vote
0answers
30 views

difference between slopes of lines represented by an equation

The question is find the difference between slopes of lines give represented by equation of pair of lines which is $$x^2(\tan^2(\theta)+\cos^2(\theta))-2xy\tan(\theta)+y^2(\sin^2(\theta))=0$$ i have ...
0
votes
0answers
8 views

UFDs with exactly two irreducibles

Let $p$ be a prime number in $\mathbb{Z}$. Let $R = R_p = \{x \in \mathbb{Q}\ |\ ord_p(x)\geq0\}$, which is a subring of $\mathbb{Q}$. Can you generalise this to construct UFDs with exactly ...
0
votes
0answers
22 views

Several options using Black-Scholes equation(s)

Could someone provide me some information about the modelling of several options at the same time by using Black-Scholes (probably coupled) equations? Any reference to papers and/or books shall be ...
0
votes
0answers
5 views

Convergent Bounded Linear Maps

I'm not sure how to show that the composition of two convergent bounded linear maps converges to the composition of their limits. I've shown that the composition of bounded linear maps is a bounded ...
1
vote
1answer
89 views

A thief and a policeman [closed]

A policeman desperately tries to catch a thief that is $a$ meters away. The thief has the constant velocity $v$, and the policeman has the constant velocity $k\cdot v$, with $k > 1$. The policeman ...
0
votes
0answers
16 views

Group of finite order where every element has infinite order

An often given example of a group of finite order where every element has infinite order is the group $\dfrac{\mathbb{Q, +}}{\mathbb{Z, +}}$. But I don't see why every element necessarily has finite ...
0
votes
0answers
18 views

Eigen vectors and determinant of a block matrix

I have two questions regarding matrix $A$. The matrix $A$ can be partitioned into four tridiagonal matrices $A_1$, $A_2$, $A_3$ and $A_4$. $$A=\begin{pmatrix} A_1&A_2\\A_3&A_4 ...
0
votes
0answers
5 views

“linear independence” of characters? Artin's theorem.

Let $G$ be a monoid and $K$ be a field. We define a character of $G$ in $K$ as a monoid homomorphism $f\colon G\to K^{\times}$, where $K^{\times}$ denotes the multiplicative group of $K$. One fact ...
1
vote
0answers
12 views

Which are the most used and correct nomenclature for gradient, divergence, curl and Laplace operator in differents contexts?

I used to write these operator in this way: $\vec{\nabla}$ for divergence and gradient and for Laplace operator $\vec{\nabla}^2$. But I have noticed that in some books and website divergence is ...
3
votes
4answers
2k views

Book with lots of geometry theorems

I want to study geometry and was looking for some book that has lots of theorems and covers almost all Euclidean geometry that is needed for High School and Maths Olympiads. Thanks.
0
votes
0answers
11 views

Use Gröbner bases to count the $3$-edge colorings of planar cubic graphs…

I found a nice introduction on how to Use Gröbner bases to construct the colorings of a finite graph. Now my graphs $G=(V,E)$ are the line graphs planar cubic graphs, so they are $4$-regular. The ...
1
vote
1answer
23 views

Kernel of a map $\phi: \mathbb{C^*}^3 \to \mathbb{C^*}^2$

I cannot understand which is the kernel of the following map $\phi: \mathbb{C^*}^3 \to \mathbb{C^*}^2$ with $$ (t_1,t_2,t_3) \mapsto \left(\frac{t_2}{t_1}, \frac{t_3}{t_1}\right) $$ In other words I ...
5
votes
3answers
272 views

An algorithm for making conditionally convergent series take arbitrary values?

This thread reminded me of an old unsettled question I have. Given an arbitrary conditionally convergent series $\beta=\sum\limits_{k=1}^\infty a_k$ and a target value $\alpha$, is there an algorithm ...
1
vote
0answers
13 views

Limit theorems in measure theory

From probability theory/measure theory we know set of theorems such as Monotone convergence, dominated convergence or conditions like uniform integrability which deals with the general question of ...
0
votes
1answer
25 views

analytic result of an integral

I am working on a physics project and I encounter an integral that I need to get analytic results about. Otherwise I will have to numerically compute the second integral, which significantly increases ...
1
vote
0answers
7 views

Number of points satisfying a quadratic equation over $GF(q)$

I am stuck on the following problem. Let $GF(q)$ be the finite field of order $q$, where $q$ is an odd prime power. How can I show by elementary methods that the number of points $(x,y,z)$ satisfying ...
0
votes
1answer
14 views

Connectedness proof of a metric space.

Assume that there is a $p$ in a metric space $(\chi, d)$ such that the function $f(q) = d(p, q)$, $q \in \chi$ omits the value $c > 0$, but takes values greater than $c$. Show that $(\chi, d)$ is ...
2
votes
2answers
31 views

Positive definite matrix meaning in human language? “Definite”?

I have to consult Wikipedia every time to re-learn what is positive (semi) definite. So that I am sure I will be able to decompose it further in some ways. Wikipedia Now I am trying to truly ...
2
votes
1answer
75 views

Find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $f (x+xy+f(y) )= (f(x)+ \frac 12 )\ (f(y)+ \frac 12 \ ).$

Find all functions $f:\mathbb R \rightarrow \mathbb R$ such that $$f\left (x+xy+f(y) \right )=\left (f(x)+ \frac 12 \right )\left (f(y)+ \frac 12 \right ).$$ for every $x,y \in \mathbb R$. My work ...
0
votes
0answers
16 views

Represetation of a smooth function in the neighborhood of its zero-set

Consider $k$ smooth functions $g_i(x)$, $x\in \mathbb{R}^n$, $k<n$. The set $G$ is defined as $G=\{x\in \mathbb{R}^n|g_i(x)=0, i=1,...,k\}$. We also assume that the Jacobi matrix $\frac{Dg}{Dx}$ is ...
3
votes
1answer
612 views

2-increasing functions

I'm trying to learn about copulas. Two definitions I've come across are the H-volume of a rectangle, which is defined as $V_H=H(x_2,y_2)-H(x_2,y_1)-H(x_1,y_2)+H(x_1,y_1)$. The function H, whose ...
1
vote
0answers
17 views

Criterion for isomorphism of two groups given by generators and relations

When are two presentations of groups are isomorphic? In this post it is said: [...] find a set of generators of the first group that satisfies the relations of the second group [...] But I doubt ...
2
votes
1answer
183 views

If a group mod its commutator is cyclic, then the group is abelian

If $G/G'$ is cyclic then G is abelian. Let G be a group and suppose $G/G'$ is cyclic.Then for all g in G we have $ g ∈ gG' = x^k G'$ for some integer k. In particular, $g=x^kz$ for some integer k and ...
0
votes
1answer
11 views

free semi-group

I'm trying to prove the following : Let S be the free semi-group on the alphabet $ A$ and let T be an arbitrary semi-group. Assume that $ g : A \rightarrow\ T $ is any mapping. Prove that there is ...
0
votes
1answer
12 views

topological invariance of being contained in a set of given dimension

Suppose $U$ is contained in $E^n$ ($n$-dimensional Euclidean space) and is homeomorphic to a set $V$ in $E^m$, where $m>n$. Is there a topological manifold in $E^m$ of dimension $n$ that contains ...
1
vote
3answers
43 views

Find the general solution of the ODE $xy′′ − y′ + 4x^3y = 0$

Can someone help me figure out this ODE, its driving me crazy. I dont need a full solution beacuse that would take hours but maybe just the final answer? Find the general solution of the ODE $xy′′ − ...
2
votes
1answer
25 views

Birth-death Process/Extinction

Random processes in Continuous time. Given that $\beta = \frac{4}{5}*\mu$ I have calculated that the birth rate $= 0.4$ and the death rate $= 0.5$. If the initial population $X(0)=6$, how many events ...
1
vote
2answers
26 views

Existence of a sequence related to the convergence of a series

Trying to prove an exercise, I arrived at the following question: Let $\{a_j\}\subseteq\mathbb{R}^+$ be a monotone increasing sequence with limit $+\infty$. Suppose that there is a $D>0$ such that ...
-1
votes
0answers
27 views

Series converges point-wise

$$f_{n}=\sum_{n=1}^{\infty }\frac{x^{4}}{(1+x^{4})^{n}}$$ Show that it converges point-wise on $\mathbb{R}$, but not uniformly on $\mathbb{R}$. My attempt: I think, we should use Weierstrass's M ...
2
votes
0answers
24 views

A subset of easily solved quartic polynomials

I've found (maybe, maybe not, but it's not on this Wikipedia or this Wikipedia) that there is a subset of easily solved quartic polynomials of the form ...
6
votes
1answer
48 views

Is every topological group the topological fundamental group of an space?

The fundamental group $\pi_{1}(X)$ of a path connected topological space $X$ is the image of $Hom(S^{1},X)$. So the fundamental group can be topologized with quotient topology where ...
0
votes
1answer
5 views

prove that a process with independent increments and a constant mean is a martingle

how to prove that a process with independent increments and a constant mean is a martingle? in a solution to this problem i found this : $X_t - X_s$ is independent from $F_s$ hence : ...
16
votes
7answers
674 views

A logarithmic integral $\int^1_0 \frac{\log\left(\frac{1+x}{1-x}\right)}{x\sqrt{1-x^2}}\,dx$

How to prove the following $$\int^1_0 \frac{\log\left(\frac{1+x}{1-x}\right)}{x\sqrt{1-x^2}}\,dx=\frac{\pi^2}{2}$$ I thought of separating the two integrals and use the beta or hypergeometric ...
1
vote
2answers
39 views

subobject classifier for partial orders

Does the category of partial orders have a subobject classifier? (Edit: No, see Eric's answer.) If not, what is a category which is "close" to the category of partial orders (e.g. it should consists ...
0
votes
0answers
17 views

Subgroup of square generators. [duplicate]

Let $N$ be a subgroup of $G$ with $N$ being generated by $\{x^2|x \in G\}$. Prove that $N$ is a normal subgroup of $G$. And that $[G, G] \subset N$. My idea was to look at $G/N$ and take $f:G ...
0
votes
3answers
21 views

Riemann Lebesgue Lemma application?

Riemann Lebesgue Lemma. shows that if $f \in L^1 ( \bf R)$ then the Fourier transform of $f$ goes to $0$. Does this also implies that $f(x) \to 0$ as $\vert x \vert \to \infty$
2
votes
0answers
29 views

Roots of trigonometric function.

I have a function: $ f(x) = (ax+b) \cdot \sin(x) + (cx+d) \cdot \cos(x) + e$ for which I want to determine the roots. I know that for $ax \cdot \sin(x) + cx \cdot \cos(x)$ the roots are ...

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