Newest Questions
1,698,195 questions
-6
votes
0
answers
39
views
The Imitation Number Formula in the Collatz Conjecture is it novel [closed]
I have been analysing the Collatz Conjecture and have identified an infinite family of numbers, which I call 'Imitation Numbers' (N), that share an identical initial trajectory structure with a ...
2
votes
0
answers
38
views
Area estimate of the intersection of ball and sphere
The following estimate arises in the proof of Tomas-Stein restriction theorem.
$$
\sigma_{\mathbb{S}^{d-1}} (B(x,r)\cap \mathbb{S}^{d-1}) \leq C r^{d-1}
$$
The estimate is very intuitive, and I have a ...
- 666
0
votes
1
answer
31
views
Base of the line integration [closed]
Does line integral integrate over the projection of a 3d curve onto the x-y plane or over the 3d curve itself as the base of the integration?
Thanks
0
votes
0
answers
33
views
Construction of Base Change/Comparison Map $g^*f_*F \to f'_* g'^*F$
Let us consider a cartesian diagram of schemes
$$
\require{AMScd}
\begin{CD}
X'=X \times_S S' @>{g'} >> X \\
@VVf'V @VVfV \\
Y' @>{g}>> Y
\end{CD}
$$
and let $F$ a sheaf on $X$.
...
- 10.1k
1
vote
0
answers
58
views
Quotient of $\mathrm{GL}_2(\mathbb{C})$ by a finite group
Let us consider the algebraic group $G=\mathrm{GL}_2(\mathbb{C})$ and consider the $S_2$-action given by conjugation with $P_0=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, that is, the $S_2$-...
- 181
0
votes
0
answers
92
views
Getting different answers for integration problem: $\int_0^2 x d( \{x\} )$
My teacher used integration by parts to solve the problem like so:
$$\int_0^2 xd(\{x\})
=[x\{x\}]_0^2-\int_0^2 \{x\}dx\\
=0-\int_0^1 xdx-\int_1^2 (x-1)dx$$
which comes out to -1. But when I was ...
3
votes
3
answers
143
views
Complex logarithm base 1
Is a logarithm with base 1 defined in the field of complex numbers? I have not found any information about this. In real numbers, this is uncertain because $ \ln(1) = 0 $ and
$ \log_a(b)= \frac {\ln(...
- 97
2
votes
0
answers
17
views
Solution to a Constrained Optimal Control Problem: Is $v(t) = S/T$ Optimal
The problem is stated as:
$\min_{v}\int^T_0f(v(t),t)dt$
Subject to the following constraints:
$s'(t)=v(t)$,
$s(0)=0$,
$s(T)=S$,
$v_{\min}\le v(t),S/T\le v_{\max}$
where:
$T$ and $S$ are given ...
- 23
1
vote
0
answers
106
views
How to integrate $\int \frac{4x^5 + 3x^2 - 1}{(2x^6 + x^3 - x + 7)^4}\,\mathrm{d}x$ analytically?
I'm trying to solve the integral
$$\int \frac{4x^5 + 3x^2 - 1}{(2x^6 + x^3 - x + 7)^4}\,\mathrm{d}x$$
I do know that a similar integral
$$\int \frac{12x^5 + 3x^2 - 1}{(2x^6 + x^3 - x + 7)^4}\,\mathrm{...
- 13
0
votes
0
answers
33
views
Find the locus that divides the line joining a point and any arbitrary point on the circumference of a circle in a fixed ratio
Given a point and a circle, find the locus of points that divide the line joining the given point and an arbitrary point on the circumference of the circle in a fixed ratio.
(If A is a point and C(O, ...
0
votes
0
answers
24
views
A surjective homomorphism of $\mathbb R$-groups that is not surjective on $\mathbb R$-points.
$\newcommand{\R}{{\mathbb R}}
\newcommand{\C}{{\mathbb C}}
$Consider a non-connected reductive group $G$ over the field $\R$ of real numbers.
Write $S=G^0$ for the identity component of $G$, and ...
- 1,408
0
votes
0
answers
22
views
Lyapunov Approximation of Separation of Trajectories
I am self-studying dynamical systems, and I came across a property that I am unsure if I correctly identified how it is found. That is,
$$
\phi_t(x+\epsilon) - \phi_t(x) \approx \epsilon e^{t\lambda}
$...
- 1
0
votes
1
answer
50
views
Fundamental group of planar topological graphs
I am trying to do the following exercise on homotopy theory:
“Prove that every finite, connected topological graph $\Gamma\subset \mathbb{R}^2$ is homotopically equivalent to the wedge sum (pointed ...
- 871
3
votes
1
answer
85
views
Compute $\int\left(5\cos^3(t)-6\cos^4(t)+5\cos^5(t)-12\cos^6(t)\right)\,\mathrm{d}t$ using some results in Linear Algebra
This problem appears in the book:
Linear Algebra and its applications - David C. Lay - Fourth Edition
It appears in: Chapter 4 (Vector Spaces), Section 4.7 (Change of Basis), Exercise 18
$(4.7), \...
- 5,844
1
vote
0
answers
67
views
Solving second order differential equation graphically
I am stuck on the following problem, hoping that someone will be able to help me.
I have a following second order differential equation:
$$y''=-\sqrt{y}+0.5y'$$
with the following initial conditions: $...
- 71