Implicit representation conjecture for semi-algebraic graphs

The implicit representation conjecture concerns hereditary families of graphs. Given a graph in such a family, we want to assign some string of bits to each vertex in such a way that we can recover the information about whether 2 vertices are connected or not using only the 2 strings of bits associated with those two vertices. We then want to minimise the length of this string. The conjecture states that if the family is hereditary and small enough (it only has $2^{O(n\ln(n))}$ graphs of size $n$), then $O(\ln(n))$ bits per vertex should be sufficient. The trivial bounds on this problem are that: (1) some families require at least $\ln_2(n)$ bits per vertex ; (2) $(n-1)/2+\ln_2(n)$ bits per vertex are sufficient for all families. In this paper, we will be talking about a special case of the implicit representation conjecture, where the family is semi-algebraic (which roughly means that the vertices are points in some euclidean space, and the edges are defined geometrically, or according to some polynomials). We will first prove that the `obvious' way of storing the information, where we store an approximation of the coordinates of each vertex, doesn't work. Then we will come up with a way of storing the information that requires $O(n^{1-\epsilon})$ bits per vertex, where $\epsilon$ is some small constant depending only on the family. This is a slight improvement over the trivial bound, but is still a long way from proving the conjecture.Comment: 17 page

Applications of entropy to extremal problems

The Sidorenko conjecture gives a lower bound on the number of homomorphisms from a bipartite graph to another graph. Szegedy [28] used entropy methods to prove the conjecture in some cases. We will refine these methods to also give lower bounds for the number of injective homomorphisms from a bipartite graph to another bipartite graph, and a lower bound for the number of homomorphisms from a k-partite hypergraph to another k-partite hypergraph, as well as a few other similar problems. Next is a generalisation of the Kruskal Katona Theorem [19, 17]. We are given integers k 4 we will make a lot of progress towards finding a solution. The next chapter is to do with Turán-type problems. Given a family of k-hypergraphs F, ex(n;F) is the maximum number of edges an F-free n-vertex k-hypergraph can have. We prove that for a rational r, there exists some finite family F of k-hypergraphs for which ex(n;F) = Ɵ(nk-r) if and only if 0 < r < k - 1 or r = k. The final chapter will deal with the implicit representation conjecture, in the special case of semi-algebraic graphs. Given a graph in such a family, we want to assign a name to each vertex in such a way that we can recover each edge based only on the names of the two incident vertices. We will first prove that one `obvious' way of storing the information doesn't work. Then we will come up with a way of storing the information that requires O(n1-E) bits per vertex, where E is some small constant depending only on the family

Compton scattering in strong magnetic fields: Spin-dependent influences at the cyclotron resonance

The quantum electrodynamical (QED) process of Compton scattering in strong magnetic fields is commonly invoked in atmospheric and inner magnetospheric models of x-ray and soft gamma-ray emission in high-field pulsars and magnetars. A major influence of the field is to introduce resonances at the cyclotron frequency and its harmonics, where the incoming photon accesses thresholds for the creation of virtual electrons or positrons in intermediate states with excited Landau levels. At these resonances, the effective cross section typically exceeds the classical Thomson value by over 2 orders of magnitude. Near and above the quantum critical magnetic field of 44.13 TeraGauss, relativistic corrections must be incorporated when computing this cross section. This paper presents formalism for the QED magnetic Compton differential cross section valid for both subcritical and supercritical fields, yet restricted to scattered photons that are below pair creation threshold. Calculations are developed for the particular case of photons initially propagating along the field, mathematically simple specializations that are germane to interactions involving relativistic electrons frequently found in neutron star magnetospheres. This exposition of relativistic, quantum, magnetic Compton cross sections treats electron spin dependence fully, since this is a critical feature for describing the finite decay lifetimes of the intermediate states. The formalism employs both the Johnson and Lippmann (JL) wave functions and the Sokolov and Ternov (ST) electron eigenfunctions of the magnetic Dirac equation. The ST states are formally correct for self-consistently treating spin-dependent effects that are so important in the resonances. Relatively compact analytic forms for the cross sections are presented that will prove useful for astrophysical modelers.Comment: 45 pages, 10 figures, accepted for publication in Phys. Rev.

Poster 259: A Quantitative Study of Morphologic Changes in the Lumbar Spine After Medial Branch Radiofrequency Neurotomy

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/146961/1/pmr2s116b.pd

A Services\u27 Frameworks And Support Services For Environmental Information Communities

For environmental datasets to be used effectively via the Internet, they must present standardized data and metadata services and link the two. The Open Geospatial Consortium\u27s (OGC) web services (WFS, WMS, CSW etc.), have seen widespread use over many years however few organizations have deployed information architectures based solely on OGC standards for all their datasets. Collections of organizations within a thematically-based community certainly cannot realistically be expected to do so. To enable service use flexibility we present a services framework - a Data Brokering Layer (DBL). A DBL presents access to data and metadata services for datasets, and links between them, in a standardized manner based on Linked Data and Semantic Web principles. By specifying regular access methods to any data or metadata service relevant for a dataset, community organizers allow a wide range of services for use within their community. Additionally, a community service profile testing service – a Conformance Service – may be run that reveals the day-to-day status of all of a community’s services to be known allowing both better end-user experiences and also that data providers’ data is acceptable to a community and continues to remains available for use. We present DBL and Conformance Service designs as well as a whole-of-community architecture that facilitates the use of the two. We describe implementations of them within two Australian environmental information communities: eReefs and Bioregional Assessments and plans for wider deployment

Energy-aware wireless networked control using radio-mode management in the case of a finite horizon

Energy efficiency is one of the main issues in wireless Networked Control Systems. The control community has already shown large interest in the topics of intermittent estimation and control, allowing to turn off the radio of the nodes which is the main energy consumer. While the existing literature only addresses policies using two radio-modes (ON/OFF), this paper considers intermediate radio-modes (\eg IDLE), which consume less energy than the ON mode and offer better reactivity than the OFF mode, but introduce transition costs. The objective of the paper is to discuss the relevance and benefit to use low consuming radio-modes and to propose a mode switching policy to perform a trade-off between energy savings and performance of the control application in the case where we have a finite horizon. We propose two possible algorithms which solve this using dynamic programming, and then test them with an example of an application.La consommation d'énergie est l'un des principaux problèmes dans les Systèmes de Contrôle en Réseaux. La communauté de contrôle a déjà montré beaucoup d'intérêt pour les estimations et contrôles intermittents, en permettant d'éteindre la radio qui est le composant qui consomme le plus. La littérature existante n'adresse que les systèmes avec deux modes (ON,OFF).Ce papier considère des modes intermédiaires (\eg IDLE), qui consomment moins d'énergie que ON et offrent une meilleure réactivité que OFF, mais qui introduit des coups de transition. L'objectif de ce papier est d'étudier les effets et les avantages d'utiliser des modes intermédiaires et de proposer une méthode pour décider quand changer de mode pour avoir un On propose deux algorithmes qui résolvent ce problème et ensuite on les teste avec un exemple d'une application

Optimized unconventional superconductivity in a molecular Jahn-Teller metal

Understanding the relationship between the superconducting, the neighboring insulating, and the normal metallic state above Tc is a major challenge for all unconventional superconductors. The molecular A3C60 fulleride superconductors have a parent antiferromagnetic insulator in common with the atom-based cuprates, but here, the C603– electronic structure controls the geometry and spin state of the structural building unit via the on-molecule Jahn-Teller effect. We identify the Jahn-Teller metal as a fluctuating microscopically heterogeneous coexistence of both localized Jahn-Teller–active and itinerant electrons that connects the insulating and superconducting states of fullerides. The balance between these molecular and extended lattice features of the electrons at the Fermi level gives a dome-shaped variation of Tc with interfulleride separation, demonstrating molecular electronic structure control of superconductivity