Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms

The local minimum degree of a graph is the minimum degree that can be reached by means of local complementation. For any n, there exist graphs of order n which have a local minimum degree at least 0.189n, or at least 0.110n when restricted to bipartite graphs. Regarding the upper bound, we show that for any graph of order n, its local minimum degree is at most 3n/8+o(n) and n/4+o(n) for bipartite graphs, improving the known n/2 upper bound. We also prove that the local minimum degree is smaller than half of the vertex cover number (up to a logarithmic term). The local minimum degree problem is NP-Complete and hard to approximate. We show that this problem, even when restricted to bipartite graphs, is in W[2] and FPT-equivalent to the EvenSet problem, which W[1]-hardness is a long standing open question. Finally, we show that the local minimum degree is computed by a O*(1.938^n)-algorithm, and a O*(1.466^n)-algorithm for the bipartite graphs

The Parameterized Complexity of Domination-type Problems and Application to Linear Codes

We study the parameterized complexity of domination-type problems. (sigma,rho)-domination is a general and unifying framework introduced by Telle: a set D of vertices of a graph G is (sigma,rho)-dominating if for any v in D, |N(v)\cap D| in sigma and for any $v\notin D, |N(v)\cap D| in rho. We mainly show that for any sigma and rho the problem of (sigma,rho)-domination is W[2] when parameterized by the size of the dominating set. This general statement is optimal in the sense that several particular instances of (sigma,rho)-domination are W[2]-complete (e.g. Dominating Set). We also prove that (sigma,rho)-domination is W[2] for the dual parameterization, i.e. when parameterized by the size of the dominated set. We extend this result to a class of domination-type problems which do not fall into the (sigma,rho)-domination framework, including Connected Dominating Set. We also consider problems of coding theory which are related to domination-type problems with parity constraints. In particular, we prove that the problem of the minimal distance of a linear code over Fq is W[2] for both standard and dual parameterizations, and W[1]-hard for the dual parameterization. To prove W[2]-membership of the domination-type problems we extend the Turing-way to parameterized complexity by introducing a new kind of non deterministic Turing machine with the ability to perform `blind' transitions, i.e. transitions which do not depend on the content of the tapes. We prove that the corresponding problem Short Blind Multi-Tape Non-Deterministic Turing Machine is W[2]-complete. We believe that this new machine can be used to prove W[2]-membership of other problems, not necessarily related to dominationComment: 19 pages, 2 figure

Global linear stability analysis of kinetic Trapped Ion Mode (TIM) turbulence in tokamak plasma using spectral method

Trapped ion modes (TIM) which belong to the family of ion temperature gradient (ITG) modes, is one of the important ingredients in heat turbulent transport at the ion scale in tokamak plasmas. It is essential to properly estimate their linear growth rate to understand their influence on ion-scale turbulent transport. A global linear analysis of a reduced gyro-bounce kinetic model for trapped particle modes is performed, and a spectral method is proposed to solve the dispersion relation. Importantly, the radial profile of the particle drift velocity is taken into account in the linear analysis by considering the exact magnetic flux {\psi} dependency of the equilibrium Hamiltonian H_{eq}({\psi}) in the quasi-neutrality equation and equilibrium gyro-bounce averaged distribution function F_{eq} . Using this spectral method, linear growth-rates of TIM instability in presence of different temperature profiles and precession frequencies of trapped ions, with an approximated constant Hamiltonian and the exact {\psi} dependent equilibrium Hamiltonian, are investigated. The growth-rate depends on the logarithmic gradient of temperature \kappa_{T} , density \kappa_{n} and equilibrium Hamiltonian \kappa_{\Lambda} . With the exact {\psi} dependent Hamiltonian, the growth rates and potential profiles are modified significantly, compared to the cases with approximated constant Hamiltonian. All the results from the global linear analysis agree with a semi-Lagrangian based linear Vlasov solver with a good accuracy. This spectral method is very fast and requires very less computation resources compared to a linear version of Vlasov-solver based on a semi-Lagrangian scheme

Periodic harmonic functions on lattices and points count in positive characteristic

This survey addresses pluri-periodic harmonic functions on lattices with values in a positive characteristic field. We mention, as a motivation, the game "Lights Out" following the work of Sutner, Goldwasser-Klostermeyer-Ware, Barua-Ramakrishnan-Sarkar, Hunzikel-Machiavello-Park e.a.; see also 2 previous author's preprints for a more detailed account. Our approach explores harmonic analysis and algebraic geometry over a positive characteristic field. The Fourier transform allows us to interpret pluri-periods of harmonic functions on lattices as torsion multi-orders of points on the corresponding affine algebraic variety.Comment: These are notes on 13p. based on a talk presented during the meeting "Analysis on Graphs and Fractals", the Cardiff University, 29 May-2 June 2007 (a sattelite meeting of the programme "Analysis on Graphs and its Applications" at the Isaac Newton Institute from 8 January to 29 June 2007

Fermi-Edge Singularities in AlxGa1-xAs Quantum Wells : Extrinsic Versus Many-Body Scattering Processes

A Fano resonance mechanism is evidenced to control the formation of optical Fermi-edge singularities in multi-subband systems such as remotely doped AlxGa1-xAs heterostructures. Using Fano parameters, we probe the physical nature of the interaction between Fermi-sea electrons and empty conduction subbands. We show that processes of extrinsic origin like alloy-disorder prevail easily at 2D over multiple diffusions from charged valence holes expected by many-body scenarios.Comment: 4 pages, 3 figures, accepted for publication in Physical Review Letter

Spin-transfer in an open ferromagnetic layer: from negative damping to effective temperature

Spin-transfer is a typical spintronics effect that allows a ferromagnetic layer to be switched by spin-injection. Most of the experimental results about spin transfer are described on the basis of the Landau-Lifshitz-Gilbert equation of the magnetization, in which additional current-dependent damping factors are added, and can be positive or negative. The origin of the damping can be investigated further by performing stochastic experiments, like one shot relaxation experiments under spin-injection in the activation regime of the magnetization. In this regime, the N\'eel-Brown activation law is observed which leads to the introduction of a current-dependent effective temperature. In order to justify the introduction of these counterintuitive parameters (effective temperature and negative damping), a detailed thermokinetic analysis of the different sub-systems involved is performed. We propose a thermokinetic description of the different forms of energy exchanged between the electric and the ferromagnetic sub-systems at a Normal/Ferromagnetic junction. The corresponding Fokker Planck equations, including relaxations, are derived. The damping coefficients are studied in terms of Onsager-Casimir transport coefficients, with the help of the reciprocity relations. The effective temperature is deduced in the activation regime.Comment: 65 pages, 10 figure

Exercise Dose Equalization in High-Intensity Interval Training: A Scoping Review

Based on comparisons to moderate continuous exercise (MICT), high-intensity interval training (HIIT) is becoming a worldwide trend in physical exercise. This raises methodological questions related to equalization of exercise dose when comparing protocols. The present scoping review aims to identify in the literature the evidence for protocol equalization and the soundness of methods used for it. PubMed and Scopus databases were searched for original investigations comparing the effects of HIIT to MICT. A total of 2041 articles were identified, and 169 were included. Of these, 98 articles equalized protocols by utilizing energy-based methods or exercise volume (58 and 31 articles, respectively). No clear consensus for protocol equalization appears to have evolved over recent years. Prominent equalization methods consider the exercise dose (i.e., energy expenditure/production or total volume) in absolute values without considering the nonlinear nature of its relationship with duration. Exercises resulting from these methods induced maximal exertion in HIIT but low exertion in MICT. A key question is, therefore, whether exercise doses are best considered in absolute terms or relative to individual exercise maximums. If protocol equalization is accepted as an essential methodological prerequisite, it is hypothesized that comparison of program effects would be more accurate if exercise was quantified relative to intensity-related maximums

Spin relaxation in low-dimensional systems

We review some of the newest findings on the spin dynamics of carriers and excitons in GaAs/GaAlAs quantum wells. In intrinsic wells, where the optical properties are dominated by excitonic effects, we show that exciton-exciton interaction produces a breaking of the spin degeneracy in two-dimensional semiconductors. In doped wells, the two spin components of an optically created two-dimensional electron gas are well described by Fermi-Dirac distributions with a common temperature but different chemical potentials. The rate of the spin depolarization of the electron gas is found to be independent of the mean electron kinetic energy but accelerated by thermal spreading of the carriers.Comment: 1 PDF file, 13 eps figures, Proceedings of the 1998 International Workshop on Nanophysics and Electronics (NPE-98)- Lecce (Italy

Negatively Charged Excitons and Photoluminescence in Asymmetric Quantum Well

We study photoluminescence (PL) of charged excitons ($X^-$) in narrow asymmetric quantum wells in high magnetic fields B. The binding of all $X^-$ states strongly depends on the separation $\delta$ of electron and hole layers. The most sensitive is the ``bright'' singlet, whose binding energy decreases quickly with increasing $\delta$ even at relatively small B. As a result, the value of B at which the singlet--triplet crossing occurs in the $X^-$ spectrum also depends on $\delta$ and decreases from 35 T in a symmetric 10 nm GaAs well to 16 T for $\delta=0.5$ nm. Since the critical values of $\delta$ at which different $X^-$ states unbind are surprisingly small compared to the well width, the observation of strongly bound $X^-$ states in an experimental PL spectrum implies virtually no layer displacement in the sample. This casts doubt on the interpretation of PL spectra of heterojunctions in terms of $X^-$ recombination

Photoluminescence investigations of 2D hole Landau levels in p-type single Al_{x}Ga_{1-x}As/GaAs heterostructures

We study the energy structure of two-dimensional holes in p-type single Al_{1-x}Ga_{x}As/GaAs heterojunctions under a perpendicular magnetic field. Photoluminescence measurments with low densities of excitation power reveal rich spectra containing both free and bound-carrier transitions. The experimental results are compared with energies of valence-subband Landau levels calculated using a new numerical procedure and a good agreement is achieved. Additional lines observed in the energy range of free-carrier recombinations are attributed to excitonic transitions. We also consider the role of many-body effects in photoluminescence spectra.Comment: 13 pages, 10 figures, accepted to Physical Review