Newest Questions
1,698,195 questions
1
vote
0
answers
45
views
geometric problem for spiral similarity
We consider a quadrilateral $ABCD$ inscribed in a circle $\omega$. Let $P$ be a point inside $\omega$ and the following equalities are satisfied
$$\angle PAD = \angle PCB,\ \angle ADP = \angle CBP.$$ ...
- 5,246
1
vote
0
answers
92
views
Is it plausible to express $1\cdot 5\cdot 9\cdots(4n-3)$ and $7\cdot 11\cdot 15\cdots(4n+3)$ into a single factorial?
This came up when I tried to solve an ODE: $4xy''+y'-y=0$ using Frobenius method. I'm stuck at rewriting this into a compact form:
$$z(x,0)=a_0\left[1+\frac{x}{1\cdot1}+\frac{x^2}{1\cdot2\cdot1\cdot5}+...
- 2,501
0
votes
1
answer
59
views
will EVT work in non-Hausdorff spaces where compact sets are not necessarily closed?
Suppose $X$ is a compact topological space and $f : X \rightarrow \mathbb{R}$ a function.
Suppose $X$ is not in an Hausdorff space i.e. that the Heine-Borel theorem does not work and where
...
2
votes
1
answer
50
views
Simplify expression for lemniscate tangent function
$\def\sl{\operatorname{sl}}\def\cl{\operatorname{cl}}\def\tl{\operatorname{tl}}\def\cscl{\operatorname{cscl}}\def\secl{\operatorname{secl}}\def\cotl{\operatorname{cotl}}\def\d{\,\mathrm{d}}$
The ...
- 63
1
vote
1
answer
57
views
Does there exist a prime decomposition for every possible combination of ramification and inertia degree
I just finished the treatment of quadratic fields and cyclotomic fields in Marcus' Number Fields and I decided to approach biquadratic fields as a fun exercise. From the Ram-Rel identity we know that ...
- 193
0
votes
0
answers
45
views
Equisatisfiable and its conditions
I need to understand the condition of Proposition 3.3.8 from the book: Logic in Computer Science by Hantao Zhang, Jian Zhang in page 97, that: $x$ is avariable not appearing in $S$. Then $S\approx S^{...
- 27
0
votes
1
answer
46
views
Mac Lane's characterization of bifunctors in terms of one-variable functors
I am trying to understand Mac Lane's characterization of bifunctors in terms of one-variable functors (proposition 1 in §1.3. of Categories for the Working Mathemathician). The theorem is as follows:
...
- 2,584
1
vote
1
answer
95
views
Find an interval for which $f_n(x) = \frac{x}{1+nx}$ does not converge uniformly
I am studying for my Real Analysis course and one of my practice problems asks us to "prove the sequence of functions $f_n(x) = \frac{x}{1+nx} \to f$ uniformly on certain intervals."
I've ...
1
vote
0
answers
22
views
When do coequalizers of algebraic stacks exist?
This question might need some work to actually get a "good" answer.
Here's the background motivation: the $2$-category of algebraic stacks has fibre products and products and so has ...
- 427
2
votes
2
answers
197
views
Maximum Travelling Salesman Problem on the integer line
I am trying to solve the following problem:
Let's say a "frog" is jumping on the numberline starting at $0$, jumps randomly on every integer from $1,\dots,n$ and then comes back to 0. What ...
- 225
0
votes
0
answers
43
views
Digital Roots in Pascal's Triangle
I know that in Pascal's Triangle there are wonderful patterns (Sierpinski's Triangle one example) that result from highlighting the multiples of a certain prime number, for example only highlighting ...
- 1
-1
votes
0
answers
129
views
Solving the equation $(2^{x}-1)^2 = \log_{2}\!\big((1+\sqrt{x})^2\big)$ [closed]
the problem
$\text{Solve the equation} \qquad (2^{x}-1)^2 = \log_{2}\!\big((1+\sqrt{x})^2\big).$
My idea
Define
$$
f(x) = (2^{x}-1)^2 - \log_{2}\!\big((1+\sqrt{x})^2\big), \qquad x \ge 0.
$$
The ...
- 539
1
vote
0
answers
35
views
Calculation of lower branch of Lambert W-function based on root of principal branch
Let's say I have a known root of $$x e^x = a$$ for $a \in (-\frac{1}{e}, 0)$ on a principal branch of Lambert W-function. In other words, I have a $W_0(a)$ on my hands due to the structure of the ...
0
votes
1
answer
82
views
Maximisation of functions of the form $f(x) = \sqrt{1 - x^2} + (ax+b)x$
I am studying a function arising in the analysis of robust aggregation rules in distributed learning, but the question is purely analytical. The function I am facing depends on parameters $a, b > 0$...
- 936
1
vote
4
answers
83
views
Is there an official name for a modulo-like function with result in the range $-m/2$ to $m/2$? [duplicate]
(NB: edited after the criticism of the first answer)
The standard result will be
$$
n \ \text{modulo} \ m
= n - m \left\lfloor\frac{n}{m}\right\rfloor\
\ \in \ \{0,\, \dots,\, m\!-\!1\}
$$
and ...
- 648