Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems

In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study

Nomenclature of the hydrotalcite supergroup: Natural layered double hydroxides

Layered double hydroxide (LDH) compounds are characterized by structures in which layers with a brucite-like structure carry a net positive charge, usually due to the partial substitution of trivalent octahedrally coordinated cations for divalent cations, giving a general layer formula [( M 2+ 1-x M 3+ x )(OH)2] x +. This positive charge is balanced by anions which are intercalated between the layers. Intercalated molecular water typically provides hydrogen bonding between the brucite layers. In addition to synthetic compounds, some of which have significant industrial applications, more than 40 mineral species conform to this description. Hydrotalcite, Mg6Al2(OH) 16[CO3]•4H2O, as the longest-known example, is the archetype of this supergroup of minerals. We review the history, chemistry, crystal structure, polytypic variation and status of all hydrotalcite-supergroup species reported to date. The dominant divalent cations, M 2+, that have been reported in hydrotalcite supergroup minerals are Mg, Ca, Mn, Fe, Ni, Cu and Zn; the dominant trivalent cations, M 3+, are Al, Mn, Fe, Co and Ni. The most common intercalated anions are (CO3)2-, (SO4)2- and Cl -; and OH-, S2- and [Sb(OH)6] - have also been reported. Some species contain intercalated cationic or neutral complexes such as [Na(H2O)6]+ or [MgSO4]0. We define eight groups within the supergroup on the basis of a combination of criteria. These are (1) the hydrotalcite group, with M 2+:M 3+ = 3:1 (layer spacing ∼7.8 Å); (2) the quintinite group, with M 2+:M 3+ = 2:1 (layer spacing ∼7.8 Å); (3) the fougèrite group, with M 2+ = Fe2+, M 3+ = Fe3+ in a range of ratios, and with O2- replacing OH- in the brucite module to maintain charge balance (layer spacing ∼7.8 Å); (4) the woodwardite group, with variable M 2+:M 3+ and interlayer [SO4] 2-, leading to an expanded layer spacing of ∼8.9 Å; (5) the cualstibite group, with interlayer [Sb(OH)6]- and a layer spacing of ∼9.7 Å; (6) the glaucocerinite group, with interlayer [SO4]2- as in the woodwardite group, and with additional interlayer H2O molecules that further expand the layer spacing to ∼11 Å; (7) the wermlandite group, with a layer spacing of ∼11 Å, in which cationic complexes occur with anions between the brucite-like layers; and (8) the hydrocalumite group, with M 2+ = Ca2+ and M 3+ = Al, which contains brucite-like layers in which the Ca:Al ratio is 2:1 and the large cation, Ca2+, is coordinated to a seventh ligand of 'interlayer' water. The principal mineral status changes are as follows. (1) The names manasseite, sjögrenite and barbertonite are discredited; these minerals are the 2H polytypes of hydrotalcite, pyroaurite and stichtite, respectively. Cyanophyllite is discredited as it is the 1M polytype of cualstibite. (2) The mineral formerly described as fougèrite has been found to be an intimate intergrowth of two phases with distinct Fe 2+:Fe3+ ratios. The phase with Fe2+:Fe 3+ = 2:1 retains the name fougèrite; that with Fe 2+:Fe3+ = 1:2 is defined as the new species trébeurdenite. (3) The new minerals omsite (IMA2012-025), Ni 2Fe3+(OH)6[Sb(OH)6], and mössbauerite (IMA2012-049), Fe3+ 6O 4(OH)8[CO3]•3H2O, which are both in the hydrotalcite supergroup are included in the discussion. (4) Jamborite, carrboydite, zincaluminite, motukoreaite, natroglaucocerinite, brugnatellite and muskoxite are identified as questionable species which need further investigation in order to verify their structure and composition. (5) The ranges of compositions currently ascribed to motukoreaite and muskoxite may each represent more than one species. The same applies to the approved species hydrowoodwardite and hydrocalumite. (6) Several unnamed minerals have been reported which are likely to represent additional species within the supergroup. This report has been approved by the Commission on New Minerals, Nomenclature and Classification (CNMNC) of the International Mineralogical Association, voting proposal 12-B. We also propose a compact notation for identifying synthetic LDH phases, for use by chemists as a preferred alternative to the current widespread misuse of mineral names. © 2012 Mineralogical Society.Fil: Mills, S.J.. Museum Victoria; AustraliaFil: Christy, A.G.. Australian National University. Centre for Advanced Microscopy; AustraliaFil: Génin, J. M. R.. CNRS-Université de Lorraine; FranciaFil: Kameda, T.. Tohoku University. Graduate School of Environmental Studies; JapónFil: Colombo, Fernando. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigaciones en Ciencias de la Tierra. Universidad Nacional de Córdoba. Facultad de Ciencias Exactas Físicas y Naturales. Centro de Investigaciones en Ciencias de la Tierra; Argentin

An Analytical Framework to Describe the Interactions Between Individuals and a Continuum

We consider a discrete set of individual agents interacting with a continuum. Examples might be a predator facing a huge group of preys, or a few shepherd dogs driving a herd of sheeps. Analytically, these situations can be described through a system of ordinary differential equations coupled with a scalar conservation law in several space dimensions. This paper provides a complete well posedness theory for the resulting Cauchy problem. A few applications are considered in detail and numerical integrations are provided

Electrochromic orbit control for smart-dust devices

Recent advances in MEMS (micro electromechanical systems) technology are leading to spacecraft which are the shape and size of computer chips, so-called SpaceChips, or ‘smart dust devices’. These devices can offer highly distributed sensing when used in future swarm applications. However, they currently lack a feasible strategy for active orbit control. This paper proposes an orbit control methodology for future SpaceChip devices which is based on exploiting the effects of solar radiation pressure using electrochromic coatings. The concept presented makes use of the high area-to-mass ratio of these devices, and consequently the large force exerted upon them by solar radiation pressure, to control their orbit evolution by altering their surface optical properties. The orbital evolution of Space Chips due to solar radiation pressure can be represented by a Hamiltonian system, allowing an analytic development of the control methodology. The motion in the orbital element phase space resembles that of a linear oscillator, which is used to formulate a switching control law. Additional perturbations and the effect of eclipses are accounted for by modifying the linearized equations of the secular change in orbital elements around an equilibrium point in the phase space of the problem. Finally, the effectiveness of the method is demonstrated in a test case scenario

Schur functions and their realizations in the slice hyperholomorphic setting

we start the study of Schur analysis in the quaternionic setting using the theory of slice hyperholomorphic functions. The novelty of our approach is that slice hyperholomorphic functions allows to write realizations in terms of a suitable resolvent, the so called S-resolvent operator and to extend several results that hold in the complex case to the quaternionic case. We discuss reproducing kernels, positive definite functions in this setting and we show how they can be obtained in our setting using the extension operator and the slice regular product. We define Schur multipliers, and find their co-isometric realization in terms of the associated de Branges-Rovnyak space

Impact hazard protection efficiency by a small kinetic impactor

In this paper the ability of a small kinetic impactor spacecraft to mitigate an Earth-threatening asteroid is assessed by means of a novel measure of efficiency. This measure estimates the probability of a space system to deflect a single randomly-generated Earth-impacting object to a safe distance from the Earth. This represents a measure of efficiency that is not biased by the orbital parameters of a test-case object. A vast number of virtual Earth-impacting scenarios are investigated by homogenously distributing in orbital space a grid of 17,518 Earth impacting trajectories. The relative frequency of each trajectory is estimated by means Opik’s theory and Bottke’s near Earth objects model. A design of the entire mitigation mission is performed and the largest deflected asteroid computed for each impacting trajectory. The minimum detectable asteroid can also be estimated by an asteroid survey model. The results show that current technology would likely suffice against discovered airburst and local damage threats, whereas larger space systems would be necessary to reliably tackle impact hazard from larger threats. For example, it is shown that only 1,000 kg kinetic impactor would suffice to mitigate the impact threat of 27.1% of objects posing similar threat than that posed by Apophis