The effect of new building regulations on indoor radon in radonprone municipalities

Radon is an important contributor to public radiation dose and it is important to monitor levels in homes and introduce measures to reduce radon concentration levels, both overall and where levels are especially high. In Norway, new building regulations were introduced in 2010, which required balanced ventilation and preventive measures to reduce indoor radon levels, including a radon barrier toward the ground and pressure reducing features beneath the building that prevent soil gas from entering (radon sump). Investigations of randomly selected homes all across Norway have shown that houses built under these new regulations have significantly lower radon levels. However, a few municipalities in Norway are especially radon-prone and have houses with particularly high levels. It is crucial to verify the effect of the new regulations in these municipalities, which we have done in this study. Here, we show that both preventive radon measures and balanced ventilation and the building regulations of 2010 have significant effects on reducing the radon levels in the houses of the public. Noticeably for management, houses with a well-ventilated crawl space, which have been exempt from the required preventive measures, still in some cases have levels above action and maximum recommended levels

Relaxation of functionals with linear growth: Interactions of emerging measures and free discontinuities

For an integral functional defined on functions ( u , v ) ∈ W 1 , 1 × L 1 {(u,v)\in W^{1,1}\times L^{1}} featuring a prototypical strong interaction term between u and v, we calculate its relaxation in the space of functions with bounded variations and Radon measures. Interplay between measures and discontinuities brings various additional difficulties, and concentration effects in recovery sequences play a major role for the relaxed functional even if the limit measures are absolutely continuous with respect to the Lebesgue one.

Structured coagulation-fragmentation equation in the space of radon measures: Unifying discrete and continuous models

We present a structured coagulation-fragmentation model which describes the population dynamics of oceanic phytoplankton. This model is formulated on the space of Radon measures equipped with the bounded Lipschitz norm and unifies the study of the discrete and continuous coagulation-fragmentation models. We prove that the model is well-posed and show it can reduce down to the classic discrete and continuous coagulation-fragmentation models. To understand the interplay between the physical processes of coagulation and fragmentation and the biological processes of growth, reproduction, and death, we establish a regularity result for the solutions and use it to show that stationary solutions are absolutely continuous under some conditions on model parameters. We develop a semi-discrete approximation scheme which conserves mass and prove its convergence to the unique weak solution. We then use the scheme to perform numerical simulations for the model.

Integration Theory

The Lebesgue measure on ?n (presented in section 8.4) is a pivotal component of this chapter. The approach in the chapter is to extend the positive linear functional provided by the Riemann integral on the space of continuous, compactly supported functions on ?n (presented in section 8.1). An excursion on Radon measures is included at the end of section 8.4. The rest of the sections are largely independent of sections 8.1 and 8.4 and constitute a deep introduction to general measure and integration theories. Topics include measurable spaces and measurable functions, Carathéodory’s theorem, abstract integration and convergence theorems, complex measures and the Radon-Nikodym theorem, Lp spaces, product measures and Fubini’s theorem, and a good collection of approximation theorems. The closing section of the book provides a glimpse of Fourier analysis and gives a nice conclusion to the discussion of Fourier series and orthogonal polynomials started in section 4.10.

Horospherically invariant measures and finitely generated Kleinian groups

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \Gamma &lt; {\rm{PSL}}_2( \mathbb{C}) $\end{document}</tex-math></inline-formula> be a Zariski dense finitely generated Kleinian group. We show all Radon measures on <inline-formula><tex-math id="M2">\begin{document}$ {\rm{PSL}}_2( \mathbb{C}) / \Gamma $\end{document}</tex-math></inline-formula> which are ergodic and invariant under the action of the horospherical subgroup are either supported on a single closed horospherical orbit or quasi-invariant with respect to the geodesic frame flow and its centralizer. We do this by applying a result of Landesberg and Lindenstrauss [<xref ref-type="bibr" rid="b18">18</xref>] together with fundamental results in the theory of 3-manifolds, most notably the Tameness Theorem by Agol [<xref ref-type="bibr" rid="b2">2</xref>] and Calegari-Gabai [<xref ref-type="bibr" rid="b10">10</xref>].</p>

Distributions of anisotropic order and applications to H-distributions

We define distributions of anisotropic order on manifolds, and establish their immediate properties. The central result is the Schwartz kernel theorem for such distributions, allowing the representation of continuous operators from [Formula: see text] to [Formula: see text] by kernels, which we prove to be distributions of order [Formula: see text] in [Formula: see text], but higher, although still finite, order in [Formula: see text]. Our main motivation for introducing these distributions is to obtain the new result that H-distributions (Antonić and Mitrović), a recently introduced generalization of H-measures are, in fact, distributions of order 0 (i.e. Radon measures) in [Formula: see text], and of finite order in [Formula: see text]. This allows us to obtain some more precise results on H-distributions, hopefully allowing for further applications to partial differential equations.