# Questions tagged [entropy]

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### Are these topological sequence entropy definition equivalent?

I am working on Möbius disjointness for models of topological dynamic systems. In that purpose, I try to understand the notion of topological entropy. We know, for a t.d.s $(X,T)$ that it is defined ...
1 vote
82 views

### Entropy condition for quasi-linear evolution equations

Let's consider the problem: $$\partial_t u + \partial_x(f(u)) = 0, (x,t)\in \mathbb R \times \mathbb R^+.\\ u|_{t=0}=u_0.$$ I have seen three formulations for the entropy condition of this equation. ...
1 vote
57 views

### Discrete Orlicz space estimate

We consider the discrete LlogL space of sequences $x=(x_i)$ such that $$\Vert x\Vert_{LL}:=\sum_i \vert x_i \log(x_i)\vert <\infty.$$ Let $x=(x_i)$ and $y=(y_i)$ two sequences in the above LlogL ...
39 views

### Maximum entropy distribution in the hyperbolic plane with given "mean" and "variance"

On the Euclidean sphere, by defining suitable analogs of the mean and variance in terms of the first circular moment, it can be shown that the von Mises distribution is the maximum entropy ...
222 views

### Is total variation distance of normalized sum of random variables to Gaussian monotonic decreasing?

Let $X_1, X_2, \ldots$ be independent and identically distributed random variables with mean $0$ and variance $1$ and let $S_n = (X_1 + \cdots + X_n)/\sqrt{n}$ to be their normalized sum. Define $D_n$ ...
113 views

### Existence of sequence of distributions

This question concerns distributions $\mu$ over the naturals $\mathbb{N}=\{1,2,\ldots\}$. For $q\ge1$, let us define the $q$th moment of entropy: $$H_q(\mu)=\sum_{i=1}^\infty \mu(i)|\log\mu(i)|^q,$$ ...
131 views

### Non-minimal system in which every point is a full entropy point

Is there a discrete topological dynamical system $(X,f)$, where $X$ is a compact metric space (with distance $d$), which is transitive but not minimal, such that $h(f)>0$ and every point is a full ...
1 vote
33 views

### Marginal distribution of $I$-projection

I am reading this paper by Csiszar. Given a probability measure $R$ and a convex subset $\mathcal{E}$ of probability distributions, it defines ‘I-projection of R on $\mathcal{E}$’ (provided there ...
355 views

### Relative Entropy and p-norm

I asked this question on StackExchange but could not get any answer, therefore, I am posting it here. I am currently reading the book "A Dynamical Approach to Random Matrix Theory". The ...
244 views

### Integrability of $\int \log(f(x)) f(x) dx$ for a probability density function $f$

I am looking for weak conditions when a probability density function $f$ on $\mathbb{R}^d$ has a finite integral $$\int_{\mathbb{R}^d} \log(f(x)) f(x) dx.$$ Any references would be appreciated.
24 views

### Proof of property for Fiedland entropy

I am working with Friedland entropy and there is a proof I cannot figure out how to do. Friedland entropy is defined for $\mathbb{Z}^k$ continuos actions $\mathcal{T}$ on a topological metric space $X$...
386 views

### Is there a quantum analog of Kolmogorov Complexity?

Kolmogorov Complexity (interpreted in terms of shortest program computing a string) and Shannon Entropy are quite similar. Since there is a quantum entropy is it reasonable to ask if there is quantum ...
360 views

### The entropy of a partition of unity

A partition of unity can be thought of as a blurred out set partition: identify the fibers of the set partition with their indicator functions; the sum of these indicators is $1$. I would like to ...
127 views

### Entropy of a refinement of a partition

We consider a probability space $(X, B, \mu)$. Let $\alpha$ and $\beta$ be countable partitions of X. We suppose $\beta$ is a refinement of $\alpha$, ie that every set in $\alpha$ is a union of sets ...