Related papers: Entropy versus influence for complex functions of …
Complex numbers play a pivotal role in both mathematics and physics, particularly in quantum mechanics, and are extensively utilized to depict the behavior of microscopic particles. Recognizing the significance of complex numbers, a…
We survey a few concentration inequalities for submodular and fractionally subadditive functions of independent random variables, implied by the entropy method for self-bounding functions. The power of these concentration bounds is that…
We study the problem of estimating a monotone function $f:\{0,1\}^d\to[0,1]$ from noisy observations at uniformly random vertices of the Boolean hypercube. As a measure of complexity for the target~$f$, we use the total $L^1$-influence…
The (e,n)-complexity functions describe total instability of trajectories in dynamical systems. They reflect an ability of trajectories going through a Borel set to diverge on the distance $\epsilon$ during the time interval n. Behavior of…
The subject is the overview of the use of quasi-entropy in finite dimensional spaces. Matrix monotone functions and relative modular operators are used. The origin is the relative entropy and the f-divergence, monotone metrics, covariance…
Let B be the open unit ball in C^2 and let a, b, c be three points in C^2 which do not lie in a complex line, such that the complex line through a and b meets B and such that <a|b> is different from 1 if one of the points a, b is in B and…
Let $f: \mathbb{N}^2 \mapsto \mathbb{C}$ be an arithmetic function of two variables. We study the existence of the limit: \[\displaystyle \lim_{x \to \infty} \frac{1}{x^2 (\log x)^{k-1}} \sum_{n_1 , n_2 \le x} f (n_1, n_2) \] where $k$ is a…
We study the regularity of exceptional actions of groups by $C^{1,\alpha}$ diffeomorphisms on the circle, i.e. ones which admit exceptional minimal sets, and whose elements have first derivatives that are continuous with concave modulus of…
Given a continuous real-valued function on [0, 1], and a closed subset E \subset [0, 1] we denote by f E the restriction of f to E, that is, the function defined only on E that takes the same values as f at every point of E >. The…
We consider a family of probability distributions depending on a real parameter and including the binomial, Poisson and negative binomial distributions. The corresponding index of coincidence satisfies a Heun differential equation and is a…
By considering a least squares approximation of a given square integrable function f:[0,1]^n --> R by a shifted L-statistic function (a shifted linear combination of order statistics), we define an index which measures the global influence…
We evoke situations where large fluctuations in the entropy are induced, our main example being a spacetime containing a potential black hole whose formation depends on the outcome of a quantum mechanical event. We argue that the…
We review recent progress in understanding certain aspects of the thermodynamics of black holes and other horizons. Our discussion centers on various ``entropy bounds'' which have been proposed in the literature and on the current…
We prove a general lower bound of quantum decision tree complexity in terms of some entropy notion. We regard the computation as a communication process in which the oracle and the computer exchange several rounds of messages, each round…
The effect of a spherical monodromy defect on the entanglement entropy and central charge $C_T$ of a free conformal scalar field propagating on an odd-dimensional sphere is investigated. As on even spheres the central charge becomes…
It is a classical theorem that if a function is integrable along the boundary of the unit circle, then the function is the nontangential limit of a holomorphic function on the open disc if and only if its Fourier coefficients for…
We compute the polynomial entropy of $C(f)$ where $f$ is any circle homeomorphism.
Quantitative predictions are presented of a depletion-induced torque and force acting on a single colloidal hard rod immersed in a solvent of hard spheres close to a planar hard wall. This torque and force, which are entirely of entropic…
It is known that a subharmonic function of finite order $\rho$ can be approximated by the logarithm of the modulus of an entire function at the point $z$ outside an exceptional set up to $C\log|z|$. In this article we prove that if such an…
It is well known that the entropy $H(X)$ of a finite random variable is always greater or equal to the entropy $H(f(X))$ of a function $f$ of $X$, with equality if and only if $f$ is one-to-one. In this paper, we give tights bounds on…