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Results tagged with gm.general-mathematics
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user 36721
Questions about mathematics which don't fall into the other arXiv categories. If you have a general question about mathematics but it is not research level, it's off-topic but it might be welcomed on Mathematics Stack Exchange.
2
votes
Accepted
What's the lower bound of the correlation coefficient?
$\newcommand\P{\operatorname P}\newcommand\E{\operatorname E}\newcommand\Var{\operatorname{Var}}\newcommand\Cov{\operatorname{Cov}}$As you noted, necessarily $\rho\ge0$, so that $0$ is a lower bound o …
- 128k
3
votes
Accepted
Any theorem shows that flowmap $\phi_{\sum_{i=1}^n a_i f_i(x)}^\tau$ can be approximated by ...
$\renewcommand{\a}{\mathbf a}$The conclusion is indeed very intuitive. However, the proof of it is rather tedious, along the lines of the proof of the Picard–Lindelöf theorem.
Accordingly, assume the …
- 128k
1
vote
Accepted
What's the lower bound for this quantity?
The exact lower bound on $k$ is $0$.
Indeed, let $a:=\alpha$. By Jensen's inequality, $\ln Ee^{ax}\ge0$ for all real $a$. So, $k\ge0$ when $k$ is defined.
On the other hand, let $y$ be any random vari …
- 128k
3
votes
Accepted
Is the right-hand term of the autonomous dynamic system equivalent to the original system af...
In the autonomous case the answer is yes.
Indeed, suppose that
$$\text{$z'(t)=f(z(t))$ for $t\in[0,\tau']=[0,a\tau]$ and $z(0)=x_0$.}$$
For $t\in[0,\tau]$, let $X(t):=z(at)$. Then
$$\text{$X'(t)=az'(a …
- 128k
2
votes
Accepted
Does this inequality hold for the cumulant generating function?
This is not true in general. Indeed, let $X$ be a zero-mean random variable (r.v.) such that $Ee^{tX}<\infty$ for $t\in[0,\tau)$ but $Ee^{\tau X}=\infty$. Then for all $t\in(0,\tau)$ the left-hand sid …
- 128k
2
votes
Accepted
Does this KL divergence inequality hold?
The answer is no. E.g., if $p_1=1/2$, $p_2=1/2$, $q_1=1/100$, $q_2=99/100$, and $\beta=1/10$, then the ratio of the left-hand side of the conjectured inequality to its right-hand is $0.00877\ldots<1$. …
- 128k
-7
votes
Examples of bad notation and its consequences
$\newcommand\E{\mathsf E}\newcommand\P{\mathsf P}\newcommand\Eb{\mathbb E}\newcommand\Pb{\mathbb P}\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}$ChatGPT gives a few examples of bad notation used m …
- 128k
7
votes
Progress in robustifying mathematics - i.e. making mathematical theorems robust to small cha...
There is a direction of research in statistics called robust statistics.
There also is a direction of research in probability, initiated by Zolotarev, concerned with stability problems in probability …
- 128k
15
votes
Accepted
Is it ever unnecessary to mathematically formalize a concept?
Your question is
Are there ever cases where formally defining physical phenomena in mathematical language is unnecessary?
It is never possible to define physical phenomena directly in mathematical l …
- 128k
2
votes
How to prove this high-degree inequality
This is to complement the nice answer by Fedor Petrov by a calculus proof of
the inequality
$$ (2 x^{10} + x^{-20})^2\ge3 (2 x^{13} + x^{-26}),$$
for real $x>0$.
Rewrite this inequality as
$$f(x):=4 …
- 128k
9
votes
Accepted
How to prove this high-degree inequality
These inequalities are algebraic and thus can be proved purely algorithmically.
Mathematica takes a minute or two for this proof of your first inequality:
Here is a "more human" proof:
Substituting …
- 128k
2
votes
Existence of functions satisfying a homogeneity condition
$\newcommand\R{\mathbf R}$Here we shall interpret the condition that $g$ be nontrivial as the condition that $g$ be non-constant.
We have
\begin{equation}
g(af(a)s)=f(a)g(s) \quad\text{for all real $a …
- 128k
1
vote
Accepted
Closed-form for recursive "geometric-like" recursion
It seems extremely unlikely that a simple "non-recursive" expression for $x_n$ is possible. However, let us obtain an exact upper bound on the $x_n$'s.
Let $a:=\alpha\in(0,1]$ and $b:=k|C|^a\in(0,\inf …
- 128k