A numerical study of entropy and residual entropy estimators based on smooth density estimators for non-negative random variables

In this paper, we are interested in estimating the entropy of a non-negative random variable. Since the underlying probability density function is unknown, we propose the use of the Poisson smoothed histogram density estimator to estimate the entropy. To study the per- formance of our estimator, we run simulations on a wide range of densities and compare our entropy estimators with the existing estimators based on different approaches such as spacing estimators. Furthermore, we extend our study to residual entropy estimators which is the entropy of a random variable given that it has been survived up to time $t$.

A Note on Distributions in the Second Chaos

In this article we study basic properties of random variables X, and their associated distributions, in the second chaos, meaning that X has a representation X = ∑ k ≥ 1 λ k ( ξ k 2 − 1 ) , where ξ k ∼ N ( 0 , 1 ) are independent. We compute the Lévy-Khintchine representations which we then use to study the smoothness of each density function. In particular, we prove the existence of a smooth density with asymptotically vanishing derivatives whenever λ k ≠ 0 infinitely often. Our work generalises some known results presented in the literature.

ASYMPTOTIC EXPANSION OF THE DENSITY FOR HYPOELLIPTIC ROUGH DIFFERENTIAL EQUATION

We study a rough differential equation driven by fractional Brownian motion with Hurst parameter $H$ $(1/4<H\leqslant 1/2)$ . Under Hörmander’s condition on the coefficient vector fields, the solution has a smooth density for each fixed time. Using Watanabe’s distributional Malliavin calculus, we obtain a short time full asymptotic expansion of the density under quite natural assumptions. Our main result can be regarded as a “fractional version” of Ben Arous’ famous work on the off-diagonal asymptotics.

The chemistry in clumpy AGB outflows

The chemistry within the outflow of an AGB star is determined by its elemental C/O abundance ratio. Thanks to the advent of high angular resolution observations, it is clear that most outflows do not have a smooth density distribution, but are inhomogeneous or “clumpy”. We have developed a chemical model that takes into account the effect of a clumpy outflow on its gas-phase chemistry by using a theoretical porosity formalism. The clumpiness of the model increases the inner wind abundances of all so-called unexpected species, i.e. species that are not predicted to be present assuming an initial thermodynamic equilibrium chemistry. By applying the model to the distribution of cyanopolyynes and hydrocarbon radicals within the outflow of IRC+10216, we find that the chemistry traces the underlying density distribution.

Absolute continuity of the law for solutions of stochastic differential equations with boundary noise

We study the existence and regularity of the density for the solution [Formula: see text] (with fixed [Formula: see text] and [Formula: see text]) of the heat equation in a bounded domain [Formula: see text] driven by a stochastic inhomogeneous Neumann boundary condition with stochastic term. The stochastic perturbation is given by a fractional Brownian motion process. Under suitable regularity assumptions on the coefficients, by means of tools from the Malliavin calculus, we prove that the law of the solution has a smooth density with respect to the Lebesgue measure in [Formula: see text].

Existence and regularity of the density for solutions of stochastic differential equations with boundary noise

We study the existence and regularity of densities for the solution of a nonlinear heat diffusion with stochastic perturbation of Brownian and fractional Brownian motion type: we use the Malliavin calculus in order to prove that, if the nonlinear term is suitably regular, then the law of the solution has a smooth density with respect to the Lebesgue measure.

Convolution in perfect Lie groups

LetGbe a connected perfect real Lie group. We show that there exists α < dimGandp∈$\mathbb{N}$* such that if μ is a compactly supported α-Frostman Borel measure onG, then thepth convolution power μ*pis absolutely continuous with respect to the Haar measure onG, with arbitrarily smooth density. As an application, we obtain that ifA⊂Gis a Borel set with Hausdorff dimension at least α, then thep-fold product setApcontains a non-empty open set.

Research of Topology Optimization of Nodal Density Based on Bilinear Interpolation

With the purpose to overcome the numerical instabilities and to generate more distinct structural layouts in the topology optimization, by using bilinear interpolation function, a topology optimization model of density interpolation based on nodal density is established, smooth density field is constructed. This method can ensure that the density field in the fixed design domain owns C0 continuity, and checkerboard patterns are naturally avoided in the nature of mathematics. After adding the sensitivity filtering, the optimal structures are smoother and have lesser details, which is helpful for manufacturing. Two numerical examples show that not only checkerboard pattern can be solved by the proposed method, but also the middle density nodal can be suppressed effectively.