Accumulation and diffusion of charges in kinetics of polymer electric breakdown

In this paper, the distribution in electric durability [Formula: see text] of polymers in constant electric field at low temperatures has been measured. The results of continuous experiments and experiments of field discontinuous effect on polymer samples being remained not disruptive after exposure for time equal to mean value [Formula: see text] have been compared. During the interval, we have varied the time interval, temperature [Formula: see text] and electric intensity of opposite sign [Formula: see text] according to which various degree of regeneration of polymer electric strength properties has been observed. By the degree of regeneration, relaxation time [Formula: see text] of the accumulated charges in polymers causing breakdown has been found. It is established that the process of charge diffusion, which accumulation leads to breakdown has a thermofluctuation behavior and the activation energy of given process depends on the counter field intensity magnitude.

A mean value formula for the variational p-Laplacian

We prove a new asymptotic mean value formula for the p-Laplace operator, $$\begin{aligned} \Delta _pu=\text{ div }(|\nabla u|^{p-2}\nabla u), \quad 1<p<\infty \end{aligned}$$ Δ p u = div ( | ∇ u | p - 2 ∇ u ) , 1 < p < ∞ valid in the viscosity sense. In the plane, and for a certain range of p, the mean value formula holds in the pointwise sense. We also study the existence, uniqueness and convergence of the related dynamic programming principle.

The Harmonicity of Slice Regular Functions

In this article, we investigate harmonicity, Laplacians, mean value theorems, and related topics in the context of quaternionic analysis. We observe that a Mean Value Formula for slice regular functions holds true and it is a consequence of the well-known Representation Formula for slice regular functions over $${\mathbb {H}}$$ H . Motivated by this observation, we have constructed three order-two differential operators in the kernel of which slice regular functions are, answering positively to the question: is a slice regular function over $${\mathbb {H}}$$ H (analogous to an holomorphic function over $${\mathbb {C}}$$ C ) ”harmonic” in some sense, i.e., is it in the kernel of some order-two differential operator over $${\mathbb {H}}$$ H ? Finally, some applications are deduced such as a Poisson Formula for slice regular functions over $${\mathbb {H}}$$ H and a Jensen’s Formula for semi-regular ones.

Almansi Theorem and Mean Value Formula for Quaternionic Slice-regular Functions

We prove an Almansi Theorem for quaternionic polynomials and extend it to quaternionic slice-regular functions. We associate to every such function f, a pair $$h_1$$ h 1 , $$h_2$$ h 2 of zonal harmonic functions such that $$f=h_1-\bar{x} h_2$$ f = h 1 - x ¯ h 2 . We apply this result to get mean value formulas and Poisson formulas for slice-regular quaternionic functions.

Some Remarks on Harmonic Functions in Minkowski Spaces

We prove that in Minkowski spaces, a harmonic function does not necessarily satisfy the mean value formula. Conversely, we also show that a function satisfying the mean value formula is not necessarily a harmonic function. Finally, we conclude that in a Minkowski space, if all harmonic functions have the mean value property or any function satisfying the mean value formula must be a harmonic function, then the Minkowski space is Euclidean.

A Mean Value Formula and a Liouville Theorem for the Complex Monge–Ampère Equation

In this paper, we prove a mean value formula for bounded subharmonic Hermitian matrix valued function on a complete Riemannian manifold with nonnegative Ricci curvature. As its application, we obtain a Liouville type theorem for the complex Monge–Ampère equation on product manifolds.