Search

It was proved in Forti and Schwaiger (C R Math Acad Sci Soc R Can 11(6):215–220, 1989), Schwaiger (Aequ Math 35:120–121, 1988) and with different methods in Schwaiger (Developments in functional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Bȩdlewo, Poland, May 17–23, 2015, Springer, Cham, pp 275–295, 2017) that under the assumption that every function defined on suitable abelian semigroups with values in a normed space such that the norm of its Cauchy difference is bounded by a constant (function) is close to some additive function, i.e., the norm of the difference between the given function and that additive function is also bounded by a constant, the normed space must necessarily be complete. By Schwaiger (Ann Math Sil 34:151–163, 2020) this is also true in the non-archimedean case. Here we discuss the situation when the bound is a suitable non-constant function. [ABSTRACT FROM AUTHOR]

In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields Ξ g , a and bi-Laplacian fields Ξ b , a . They can be characterized as follows: for f = δ the solution u of ∇ · a ∇ u = f , a is a uniformly elliptic random environment, is the covariance of Ξ g , a . When f is the white noise, the field Ξ b , a can be viewed as the distributional solution of the same elliptic equation. Our results characterize the scaling limit of such fields on both, a sufficiently regular domain D ⊂ R d , or on the discrete torus. Based on stochastic homogenization techniques applied to the eigenfunction basis of the Laplace operator Δ , we will show that such families of fields converge to an appropriate multiple of the GFF resp. bi-Laplacian. The limiting fields are determined by their respective homogenized operator a ¯ Δ , with constant a ¯ depending on the law of the environment a . The proofs are based on the results found in Armstrong et al. (in: Grundlehren der mathematischen Wissenschaften, Springer International Publishing, Cham, 2019) and Gloria et al. (ESAIM Math Model Numer Anal 48(2):325-346, 2014). [ABSTRACT FROM AUTHOR]

At various times in his life, Louis de Broglie firmly believed in the coexistence of both particle and an associated wave referring to "the theory of the double solution", and an equation which he called "the guidance formula". In an attempt to account for both particle and wave, the author has proposed a Lorentz invariant alternative to Newton's second law which is developed in Hill (Zeitschrift fur angewandte Mathematik und Physik 69:133–145, 2018; Zeitschrift fur angewandte Mathematik und Physik 70:5–14, 2019; Zeitschrift fur angewandte Mathematik und Physik 70:131–153, 2019; Math Mech Solids 26:263–284, 2020; Math Mech Solids 25: 1763–1777, 2020; Zeitschrift fur angewandte Mathematik und Physik 72:1–14; Mathematics of particle–wave mechanical systems, Springer, Cham, 2022). Here, we summarise some of the major outcomes of this approach, including simple solutions of the proposed model exhibiting both sub-luminal and superluminal behaviour dependent upon the region of space-time, and two symmetrical sets of rate-differential relations for the two Lorentz invariants for a single spatial dimension x. One set follows the particle, while the other follows the wave, revealing a complete symmetry between the one-dimensional spatial physical force f and the force g in the "direction of time". The existence of these symmetrical equations reinforces the parity and interchangeability of particles and waves. [ABSTRACT FROM AUTHOR]