Remarks on the Cauchy functional equation and variations of it

This paper examines various aspects related to the Cauchy functional equation $f(x+y)=f(x)+f(y)$, a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as a one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation.Comment: To appear in Aequationes Mathematicae (important remark: the acknowledgments section in the official paper exists, but it appears before the appendix and not before the references as in the arXiv version); correction of a minor inaccuracy in Lemma 3.2 and the initial value proof of Theorem 2.1; a few small improvements in various sections; added thank

The impact of diabetes on the pathogenesis of sepsis

Diabetes is associated with an increased susceptibility to infection and sepsis. Conflicting data exist on whether the mortality of patients with sepsis is influenced by the presence of diabetes, fuelling the ongoing debate on the benefit of tight glucose regulation in patients with sepsis. The main reason for which diabetes predisposes to infection appears to be abnormalities of the host response, particularly in neutrophil chemotaxis, adhesion and intracellular killing, defects that have been attributed to the effect of hyperglycaemia. There is also evidence for defects in humoral immunity, and this may play a larger role than previously recognised. We review the literature on the immune response in diabetes and its potential contribution to the pathogenesis of sepsis. In addition, the effect of diabetes treatment on the immune response is discussed, with specific reference to insulin, metformin, sulphonylureas and thiazolidinediones

Cole-Cole Plots for Linear and Nonlinear Dielectric Relaxation in Solutions of Rigid Highly Dipolar Symmetric-Top Molecules in Spherical Solvents

The graphical analysis of the influence of the rotational diffusion tensor anisotropy and the orientation of the permanent dipole moment on the linear and nonlinear dielectric relaxation is shown. The solution of Smoluchowski-Debye rotational diffusion equation for rigid, and noninteracting polar, symmetric-top molecules, in the "weak molecular reorientation approximation", was used. In order to highlight the influence of the symmetric shape of molecule, in comparison with classical, spherical-top Smoluchowski rotational diffusion, we present sets of Argand-type plots and three-dimensional Cole-Cole diagrams for linear and nonlinear electric susceptibilities. The results indicate that, in describing the nonlinear dielectric relaxation, the simplest spherical-top rotational diffusion model may be a sufficient approximation in some special cases only

Kinetic Theory of the Langevin Saturation in Dilute Solution of Dipolar Symmetric-Top Molecules in Spherical Solvents

Assuming the low molecular reorientation approximation, the formulae for the third-order electric polarization induced in liquids composed of rigid noninteracting dipolar, symmetric-top molecules in spherical solvents were derived. Our medium is acted on by a strong external dc bias electric field superimposed on a weak ac electric field, and the classical Smoluchowski-Debye equation for rotational diffusion of the symmetric-top molecules is applied. In order to highlight the influence of the anisotropy of rotational diffusion tensor components and the orientation of permanent dipole moment of the molecule on the complex linear and nonlinear electric susceptibilities, we present three-dimensional plots of the linear and nonlinear dispersion and absorption spectra, for different values of the frequency of ac electric field

Ensemble Averages Calculated for Two-Dimensional Smoluchowski-Debye Rotational Diffusion in DC Electric Field

Ensemble averages ⟨cos nϑ(t)⟩ (n = 1, 2, 3, 4) are calculated for the two-dimensional Smoluchowski-Debye rotational diffusion model in the presence of a DC reorienting electric field. The time dependence of ⟨cos nϑ(t)⟩ is plotted for the special cases of (i) dipolar nonpolarizable molecules (p ≠ 0), and (ii) nondipolar anisotropically polarizable molecules (p = 0) and rectangular reorienting pulses