Questions tagged [ds.dynamical-systems]
Dynamics of flows and maps (continuous and discrete time), including infinite-dimensional dynamics, Hamiltonian dynamics, ergodic theory.
2,482 questions
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“Cohomological equation” in dynamical systems
Let $$\dot{x}=Ax+v_r(x)+v_{r+1}(x)+ \dots$$ with $x \in \mathbb{C}^n$ and $v_r: \mathbb{C}^n \to \mathbb{C}^n$ a homogenous, polynomial function of order $r.$
Then, being able to find a suitable $h$ ...
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97
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Bounds and repulsion domains for the Dirichlet eta function $\eta(\sigma+it)$, for fixed $\sigma$
Let $\eta(\sigma+it)$ be the Dirichlet eta function, with $t>0$ (the variable) and $\sigma$ be fixed, with $\frac{1}{2}\leq \sigma <2$. I define the hole $\Omega_T
=\Omega_T(\sigma)$ as the ...
- 3,098
11
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1
answer
521
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Cohomology for extension problems in symbolic/topological dynamics?
Context: I know essentially nothing about cohomology of any kind, but I have a problem involving classifying obstructions to extensions of certain maps or covers, and I have heard that cohomology is ...
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2
votes
1
answer
166
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In general is $\frac{d\,\mu_1}{d\,\mu_2}\circ T = \frac{d\,T\mu_1}{d\,T\mu_2}$?
Given an ergodic and non-singular dynamic system (definition provided here) $(X, \mathcal{B}, \mu_1, T)$ where $(X, \mathcal{B}, \mu_1)$ is a measure space and $T$ is a fixed transformation, we then ...
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1
answer
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How to find an optimal sequence of merging operations?
Given a set of items, each characterized by a quality $q_i\in(0,1)$. We can merge two items of quality $q_i$ and $q_j$ to a single item $k$ of quality $q_k=f(q_i,q_j)$, where $f$ is increasing in $q_i$...
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2
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0
answers
141
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Choosing the derivative of a flow
I am looking for something like the Franks' Lemma for flows. The celebrated Franks' Lemma states that: Let $f:M \rightarrow M$ be a $C^1$ diffeomorphism and $S=\{p_1,...,p_k\}$ be a finite set of ...
- 121
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Unique solution to nonlinear optimization through gradient descent
I am trying to estimate the path of a random walk described by the following SSM
$$
\begin{align}
x_{t+1} &= x_{t} + q_{t+1} \newline
y_{t+1} &= h(x_{t+1}) + r_{t+1}
\end{align}
$...
- 11
2
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0
answers
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Lyapunov function utility in stochastic optimal control
The article Optimal strategy of vaccination and treatment in an SIRS model with Markovian switching by (X.Mu, Q.Zhang) studies necessary and sufficient conditions on near-optimal controls.
In both ...
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Stochastic dynamics: how do the random matrix $J_{ij}$ and coupling strengh $g$ affect the variance of the local field $h_i$?
Context: Q3 in How to understand the largest Lyapunov exponent?
We know $g$ is proportional to (square root of) the variance of $J$'s every entry ($J_{ij}\sim \mathcal{N}(0,g^2/N)$).
Why is it also ...
- 139
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0
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Complex Hölder space
I already posted this question on math.stackexchange, but got no response and was suggested to post it here.
I came across a space in an ergodic theory paper, which I am calling here a (complex) ...
- 136
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1
answer
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Surjectivity for distal continuous functions on a compact metric space
Where can I find a proof that a distal continuous function of a compact metric space is surjective?
PS:
The person asking the question Is there an elementary proof that distal maps are invertible? ...
- 11
6
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Construction of minimal zero entropy measure-theoretically strong mixing subshift?
Does anyone know of a construction of a subshift (over $\mathbb{Z}$) which is
(1) minimal
(2) zero (topological) entropy
(3) measure-theoretically strong mixing (for some measure)?
I am in particular ...
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3
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1
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131
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Questions about ratio set in a dynamical system
Given a dynamical system $(X, \Omega, \mu)$ ($\Omega$ the $\sigma$-algebra and $\mu$ a measure), we assume there is a group $G$ acting on the system in the sense that, for each group element $g$, $g$ ...
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1
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1
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Computing kneading sequences for renormalizations of Lorenz maps
I am stuck trying to understand certain claims made in this paper, and for completeness I will reproduce some definitions from it.
A Lorenz map $f$ on $I = [0,1]$ is a monotone increasing function ...
- 13
3
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2
answers
415
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Asymptotic behavior of system of differential equations
Let $f:[0,1]^2 \rightarrow \mathbb{R}^2,$ where $f_1(x,y) = g(y)-x$ and $f_2(x,y) = g(x)-y.$ Here $g(\cdot)$ is a strictly decreasing polynomial function such that $g(0)=1$ and $g(1)=0.$ I am ...
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votes
2
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The mower's challenge
Weeds have taken over the paths (two squares). If mowed, they don't grow back, but unmowed weeds spread at speed $1$ along the road. What's the minimum speed of the mower to get rid of all the weeds? ...
- 2,619
6
votes
1
answer
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Does length spectrum determine a hyperbolic 3-manifold? What if we also know holonomies?
Question 1: Suppose that two hyperbolic 3-manifolds $M_1$ and $M_2$ with finitely generated fundamental group satisfy the property that for every closed geodesic in $M_1$, there is a closed geodesic ...
- 327
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1
answer
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Boundedness of orbits and limit sets
Let $T: {\bf R}^n \rightarrow {\bf R}^n$ be an homeomorphism and $x$ a point in ${\bf R}^n$.
The positive orbit of $x$ is the set $\{T^n(x) \mid n \in {\bf N}\}$ and its
$\omega$-limit set is the set ...
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83
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Distortion estimates to control Hausdorff measure of a curve
I am studying the paper Blumenthal - Statistical properties for compositions of standard maps with increasing coefficent.
I have a problem to understand how the distortion estimates are used. The ...
20
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2
answers
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Find $Y\in\operatorname{GL}_n(\mathbb{Z})$ such that all eigenvalues of $YX$ are nonnegative
I saw this problem some years ago and I would greatly appreciate any reference or solution.
Let $X \in \operatorname{M}_n ( \mathbb{R} )$. Prove that there is $Y \in \operatorname{M}_n ( \mathbb{Z} )$...
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5
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2
answers
256
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Can a holomorphic vector field have an attractor homoclinic loop?
It is well known that a holomorphic vector field $z'=f(z), z\in \mathbb{C}$ does not have any limit cycle.See the last paragraph of this post
Orbits space of real-analytic planar foliations
One can ...
- 356
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0
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43
views
Persistence of planar trajectory converging to a node / focus
I consider a planar system $\dot u =F(u,p)$ where p is a scalar parameter. Suppose that the flow $\phi^t(u_0; 0)$ from $u_0$ converges to a stable node / focus $x^{eq}_0$ for the parameter value $p=0$....
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149
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Polynomial entropy of topological dynamical systems
For a continuous mapping $\varphi: X \to X$ on a compact Hausdorff space $X$ we define the entropy as follows:
Given open covering $\mathcal{U}$, $\mathcal{V}$ of $X$, we call $\mathcal{V}$ a ...
- 467
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0
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Relation between the distance projective maps and their angles
Let $f:N \to \mathbb{R}^2$
be a differentiable map of smooth manifolds. Let $\mathbb{R}^2$ be decomposed as a direct sum of line bundles, i.e. $\mathbb{R}^2=E(x) \oplus F(x)$, where $F(x)$ and $E(x)$ ...
- 1,043
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Clarification on the proof of Lyapunov-Razumikhin asymptotic stability theorem for delayed differential equations
this is my first question here, hope I am in the right place :)
Recently I have been looking at the proof of theorem 4.2 on Razumikhin stability for RFDEs in the book by Jack Hale and Lunel Verduyn: ...
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9
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1
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380
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Equivalent definitions of topological weak mixing
A dynamical system $f:X\to X$ is said to be topologically transitive if for any two nonempty open sets $U,V$ there exists $n \in \mathbb{Z}$ such that $f^{\circ n}(U) \cap V \neq \emptyset$. The ...
- 91
3
votes
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answer
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Extension of Khintchine's recurrence in a simple case
Suppose an ergodic system $(X,\mathcal{B},\mu,T)$ has a Kronecker factor that is isomorphic to an ergodic rotation, say on the Torus.
How can one prove that the large intersection property holds for $...
- 195
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Exponential map of cotangent bundle and Morse theory on based loop space
Consider $M$ to be a compact manifold and $q_0,q_1\in M$. Let $L_t$ be a lagrangian and $\mathcal{E}_L$ the lagrangian action functional on the based loop space $\Omega(M,q_0,q_1)$ defined as $\...
- 791
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1
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Chaotic complex dynamics and Newton's method
I'm trying to understand Harold E. Benzinger, Scott A. Burns, and Julian I. Palmore's work on one-parameter families of Julia sets arising from Newton's method in the complex domain. They show the ...
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3
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Does mixing automatically imply this seemingly stronger "uniform modulo re-ordering" version of mixing?
THE QUESTION
Let $(X,\mathcal{X})$ be a standard Borel space, $T \colon X \to X$ a measurable map, and $\mu$ a $T$-mixing probability measure.
Is it necessarily the case that for all $A \in \mathcal{...
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Show that two matrices are strongly shift equivalent
The following question is from Introduction to dynamical systems, written by Michael Brin and Garrett Stuclk.
Given two non-negative integer square matrices $A, B$, we say $A, B$ are elementarily ...
- 307
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0
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84
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The closure of the subgroup generated by a vector field may not be compact
Suppose $X$ is a vector field on a manifold $M$, consider the one parameter group:
$$L=\left\{\phi^t_X: t\in\mathbb{R}\right\}$$
where $\phi^t_X$ is the flow of the vector field $X$, which sends $p\in ...
- 101
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votes
2
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435
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The (last step of the) proof that the set of badly approximable matrices has measure zero
An $m \times n$ matrix $A$ is called badly approximable if there exists $c > 0$ such that for all integer vectors $p \in \mathbb Z^m$ and $q \in \mathbb Z^n-\{0\}$ we have
$$ \|A q + p \| \ge c \| ...
- 1,565
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0
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views
How are synchrony and stability conceptually related in complex systems?
Consider two models. Firstly, a set of $n$ variables which satisfy a set of differential equations
$$
\frac{d \mathbf{x}}{d t} = \mathbf{A x}
$$
where $\mathbf{x}$ is an $n \times 1$ column vector, ...
- 640
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2
answers
294
views
$\lim\inf_{t\to \infty} f(x(t))=0\Longrightarrow \lim\inf_{t\to\infty} \|x(t)\|=0$
I am a PhD student and during my research I was presented to the claim that
For a positive definite function $f:\mathbb{R}\to \mathbb{R}$ continuous in $0$, with $0$ a stable point at $t=0$ for $x$, ...
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0
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56
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Minimising risk in dynamical systems
I have been reading the paper of Goerner and Ulancowicz - "Quantifying economic sustainability" in which it is suggested that there is a tradeoff between sustainability and efficiency. ...
user478214
0
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2
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205
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Special topological equivalence
Are there known examples of a smooth ($C^{\infty}$) or analytic autonomous ordinary differential equation $\dot{x}=f(x,y)$, $\dot{y}=g(x,y)$ in the upper half-plane $\{y\geqslant 0\}$ such that the ...
- 163
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1
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Multidimensional intersection property
Consider the multidimensional annulus $\{(p,\theta)\} = \mathbb R^n\times\mathbb T^n$ endowed by the $1$-form $\omega=p\,d\theta$. A diffeomorphism $A$ of this annulus onlo itself is said to be exact ...
- 163
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168
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Mañé's example of an attractor with no natural measure
I'm reading Milnor's notes on dynamical systems and in Lecture 3 he gives an example of an attractor with no natural measure, which he attributes to Mañé. I can find no other reference in which this ...
- 63
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1
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196
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Stable periodic orbits for three equal masses
For three equal masses in any number of dimensions (this might not be important,
but 2D or 3D or 4D is fine) under just classical gravity (i.e., inverse-square force law),
what stable periodic orbits ...
- 1,547
4
votes
1
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349
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Poincaré–Bendixson Theorem on a compact, connected, orientable, two-dimensional manifold
I'm currently reading the article "A Generalization of a Poincaré–Bendixson Theorem to Closed Two-Dimensional Manifolds" by Arthur Shwartz. The paper first establishes a result for minimal ...
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2
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1
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113
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A lemma in approximating sequences
Consider the circle $\mathbb{T}^1= \frac{\mathbb{R}}{\mathbb{Z}}$. We represent it as a union of disjoint subsegments $M_j=[t_j,t_{j+1})$, $j = 0, \cdots, n$, $t_0=t_n$ and define the map $S$ by the ...
- 145
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0
answers
74
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Equicontinuity, Beltrami coefficients and Sequence of top/bottom semi-annuli
In the Beltrami equation literature, one approach to showing equicontinuity for pairs $(f_{n},\mu_{n})$ (where $\partial_{\bar{z}}f_{n}=\mu_{n}(z)\partial_{z}f_{n}$) is via the relations of moduli and ...
- 5,474
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votes
0
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167
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Why were these constants picked in this Lyapunov function and how did the author arrive at the final form of the Lyapunov function?
Consider the following paper:
"A note on global stability for a tuberculosis model" by Gao and Huang: https://doi.org/10.1016/j.aml.2017.05.004
The methodology is understood in this paper ...
- 185
2
votes
0
answers
192
views
Almost periodicity and approximation in tracial von Neumann algebra
Let $N$ be a von Neumann algebra with a faithful normal tracial state $\tau$. For a countable group $G$, let $\sigma: G\rightarrow \text{Aut}(N)$ be a $G$- action on $N$ which preserves the tracial ...
- 73
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0
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80
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Cyclicity of composition operators
Let $E=C([0,1]^m,\mathbb{R}^n)$ where $K=[0,1]^m$ where $E$ has the compact convergence topology. Recall that for a function $f:[0,1]^m\rightarrow [0,1]^m$ the associated composition operator $C_f$ ...
- 5,405
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3
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247
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Equivalence problem of classifying heat equations
I have tried to search for references online but I am unable to do so.
I am looking for references that uses Cartan's method of moving frames to classify heat equations.
Also are there references that ...
- 135
6
votes
1
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571
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Direct proof that the set of badly approximable numbers have full Hausdorff dimension without using Schmidt games
A badly approximable number is an $x$ for which there is a positive constant $c$ such that for all rational $p/q$ we have
$$\left|{ x - \frac{p}{q} }\right| > \frac{c}{q^2} \ . $$
The set of badly ...
- 1,565
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votes
0
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258
views
Minimal actions commuting with amenable actions of $\mathbb{F}_2$
For a countable discrete group $G$ acting by homeomorphisms on a compact metrizable space $X$, we say that $G\curvearrowright X$ is (topologically) amenable if there exists a sequence of continuous ...
- 211
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votes
6
answers
1k
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A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?
Let $f(x)$ be sufficiently regular (e.g. a smooth function or a formal power series in characteristic 0 etc.). In my research the following recursion made a surprising entrance
$$
f_1(x) = f(x),\ f_{n+...
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