The discrete fragmentation equations : semigroups, compactness and asynchronous exponential growth

In this paper we present a class of fragmentation semigroups which are compact in a scale of spaces defined in terms of finite higher moments. We use this compactness result to analyse the long time behaviour of such semigroups and, in particular, to prove that they have the asynchronous growth property. We note that, despite compactness, this growth property is not automatic as the fragmentation semigroups are not irreducible

Key strategies for managing acid sulphate soil (ASS) problems on the southeastern coast of New South Wales, Australia

The acidification of Australian coastal waterways as a result of the oxidation of acid sulphate soil (ASS) containing appreciable quantities of sulphidic material (e.g. pyrite) has well recognised environmental, economic and social effects including the loss of fish, biodiversity and agricultural productivity as well as the corrosion of concrete and steel infrastructure by acidic drainage. Largescale artificial drainage and one-way floodgates in low-lying coastal floodplains has lowered the groundwater table, thus enhancing pyritic oxidation and increasing the distribution, magnitude and frequency of acid generation and release of toxic metals such as aluminium (Al3+) and iron (total Fe) from ASS. Engineering strategies implemented on the Shoalhaven Floodplain, southeast New South Wales, Australia have been designed to remediate ASS. These include: (1) fixed-level v-notch weirs, which raise the groundwater table above the pyritic layer and reduce the rate of discharge of acidic products from the groundwater into the drains; (2) modified two-way floodgates, which allow for tidal buffering of acidic drainage; (3) a subsurface alkaline horizontal impermeable lime-fly ash barrier, which prevents pyrite oxidation and neutralises acidic groundwater and (4) an alkaline permeable reactive barrier (PRB) using recycled materials, which significantly increases groundwater pH and reduces Al and Fe concentrations within and down-gradient of the PRB. A critical review of each of these strategies will outline their role in remediating ASS and their respective benefits and limitations

Coagulation and fragmentation processes with evolving size and shape profiles : a semigroup approach

We investigate a class of bivariate coagulation-fragmentation equations. These equations describe the evolution of a system of particles that are characterised not only by a discrete size variable but also by a shape variable which can be either discrete or continuous. Existence and uniqueness of strong solutions to the associated abstract Cauchy problems are established by using the theory of substochastic semigroups of operators

Integral representation of the linear Boltzmann operator for granular gas dynamics with applications

We investigate the properties of the collision operator associated to the linear Boltzmann equation for dissipative hard-spheres arising in granular gas dynamics. We establish that, as in the case of non-dissipative interactions, the gain collision operator is an integral operator whose kernel is made explicit. One deduces from this result a complete picture of the spectrum of the collision operator in an Hilbert space setting, generalizing results from T. Carleman to granular gases. In the same way, we obtain from this integral representation of the gain operator that the semigroup in L^1(\R \times \R,\d \x \otimes \d\v) associated to the linear Boltzmann equation for dissipative hard spheres is honest generalizing known results from the first author.Comment: 19 pages, to appear in Journal of Statistical Physic

Recycling of end-of-life tyres in seismic isolation foundation systems

Over 6.3 million waste tyres are produced annually in New Zealand (Tyrewise, 2021), leading to socioeconomic and environmental concerns. The 2010-11 Canterbury Earthquake Sequence inflicted extensive damage to ~6,000 residential buildings, highlighting the need to improve the seismic resilience of the residential housing sector. A cost-effective and sustainable eco-rubber geotechnical seismic isolation (ERGSI) foundation system for new low-rise buildings was developed by the authors. The ERGSI system integrates a horizontal geotechnical seismic isolation (GSI) layer i.e., a deformable seismic energy dissipative filter made of granulated tyre rubber (GTR) and gravel (G) – and a flexible rubberised concrete raft footing. Geotechnical experimental and numerical investigations demonstrated the effectiveness of the ERGSI system in reducing the seismic demand at the foundation level (i.e., reduced peak ground acceleration) (Hernandez et al., 2019; Tasalloti et al., 2021). However, it is essential to ensure that the ERGSI system has minimal leaching attributes and does not result in long-term negative impacts on the environment

Quantum Kinetic Evolution of Marginal Observables

We develop a rigorous formalism for the description of the evolution of observables of quantum systems of particles in the mean-field scaling limit. The corresponding asymptotics of a solution of the initial-value problem of the dual quantum BBGKY hierarchy is constructed. Moreover, links of the evolution of marginal observables and the evolution of quantum states described in terms of a one-particle marginal density operator are established. Such approach gives the alternative description of the kinetic evolution of quantum many-particle systems to generally accepted approach on basis of kinetic equations.Comment: 18 page

Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers

The rotated multipliers method is performed in the case of the boundary stabilization by means of a(linear or non-linear) Neumann feedback. this method leads to new geometrical cases concerning the "active" part of the boundary where the feedback is apllied. Due to mixed boundary conditions, these cases generate singularities. Under a simple geometrical conditon concerning the orientation of boundary, we obtain a stabilization result in both cases.Comment: 17 pages, 9 figure

Distributionally chaotic families of operators on Fréchet spaces

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Communications on Pure and Applied Analysis (CPAA) following peer review. The definitive publisher-authenticated version Conejero, J. A., Kostić, M., Miana, P. J., & Murillo-Arcila, M. (2016). Distributionally chaotic families of operators on Fréchet spaces.Communications on Pure and Applied Analysis, 2016, vol. 15, no 5, p. 1915-1939, is available online at: http://dx.doi.org/10.3934/cpaa.2016022The existence of distributional chaos and distributional irregular vectors has been recently considered in the study of linear dynamics of operators and C-0-semigroups. In this paper we extend some previous results on both notions to sequences of operators, C-0-semigroups, C-regularized semigroups, and alpha-timesintegrated semigroups on Frechet spaces. We also add a study of rescaled distributionally chaotic C-0-semigroups. Some examples are provided to illustrate all these results.The first and fourth authors are supported in part by MEC Project MTM2010-14909, MTM2013-47093-P, and Programa de Investigacion y Desarrollo de la UPV, Ref. SP20120700. The second author is partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia. The third author has been partially supported by Project MTM2013-42105-P, DGI-FEDER, of the MCYTS; Project E-64, D.G. Aragon, and Project UZCUD2014-CIE-09, Universidad de Zaragoza. The fourth author is supported by a grant of the FPU Program of Ministry of education of Spain.Conejero, JA.; Kostic, M.; Miana Sanz, PJ.; Murillo Arcila, M. (2016). Distributionally chaotic families of operators on Fréchet spaces. Communications on Pure and Applied Analysis. 15(5):1915-1939. https://doi.org/10.3934/cpaa.2016022S1915193915