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We show how to reduce the computational time of the practical implementation of the Raviart-Thomas mixed method for second-order elliptic problems. The implementation takes advantage of a recent result which states that certain local subspaces of the vector unknown can be eliminated from the equations by transforming them into stabilization functions; see the paper published online in JJIAM on August 10, 2023. We describe in detail the new implementation (in MATLAB and a laptop with Intel(R) Core (TM) i7-8700 processor which has six cores and hyperthreading) and present numerical results showing 10 to 20% reduction in the computational time for the Raviart-Thomas method of index $k$, with $k$ ranging from 1 to 20, applied to a model problem.

Praeger-Xu graphs are connected, symmetric, 4-regular graphs that are unusual both in that their automorphism groups are large, and in that vertex stabilizer subgroups are also large. Determining number and distinguishing number are parameters that measure the symmetry of a graph by investigating additional conditions that can be imposed on a graph to eliminate its nontrivial automorphisms. In this paper, we compute the values of these parameters for Praeger-Xu graphs. Most Praeger-Xu graphs are 2-distinguishable; for these graphs we also provide the cost of 2-distinguishing., Comment: 23 pages, 4 figures

The derived graph of a voltage graph consisting of a single vertex and two loops of different voltages is a circulant graph with two generators. We characterize the automorphism groups of connected, two-generator circulant graphs, and give their determining and distinguishing number, and when relevant, their cost of 2-distinguishing. We do the same for the subdivisions of connected, two-generator circulant graphs obtained by replacing one loop in the voltage graph with a directed cycle.

A graph automorphism is a bijective mapping of the vertices that preserves adjacent vertices. A vertex determining set of a graph is a set of vertices such that the only automorphism that fixes those vertices is the identity. The size of a smallest such set is called the determining number, denoted Det$(G)$. The determining number is a parameter of the graph capturing its level of symmetry. We introduce the related concept of an edge determining set and determining index, Det$'(G)$. We prove that Det$'(G) \le \text{Det}(G) \le 2\text{Det}'(G)$ when Det$(G) \neq 1$ and show both bounds are sharp for infinite families of graphs. Further, we investigate properties of these new concepts, as well as provide the determining index for several families of graphs., Comment: 24 pages, 8 figures

In this paper we study the symmetry parameters determining number, distinguishing number, and cost of 2-distinguishing, for some variations on hypercubes, namely Hamming graphs, powers of hypercubes, folded hypercubes, enhanced hypercubes, augmented hypercubes and locally twisted hypercubes.

The Mycielskian construction, denoted $\mu(G)$, takes a finite simple graph $G$ to a larger graph with of the same clique number but larger chromatic number. The generalized Mycielskian construction, denoted $\mu_t(G)$, takes $G$ to a larger graph with the same chromatic number but with larger odd girth. In this chapter we look at symmetry parameters of $\mu(G)$ and $\mu_t(G)$ in terms of the same parameters of $G$. These symmetry parameters include determining number, distinguishing number, and cost of distinguishing., Comment: arXiv admin note: text overlap with arXiv:2007.15284

In J. Sci. Comput., 81: 2188-2212, 2019, we considered a superconvergent hybridizable discontinuous Galerkin (HDG) method, defined on simplicial meshes, for scalar reaction diffusion equations and showed how to define an interpolatory version which maintained its convergence properties. The interpolatory approach uses a locally postprocessed approximate solution to evaluate the nonlinear term, and assembles all HDG matrices once before the time intergration leading to a reduction in computational cost. The resulting method displays a superconvergent rate for the solution for polynomial degree $k\ge 1$. In this work, we take advantage of the link found between the HDG and the hybrid high-order (HHO) methods, in ESAIM Math. Model. Numer. Anal., 50: 635-650, 2016 and extend this idea to the new, HHO-inspired HDG methods, defined on meshes made of general polyhedral elements, uncovered therein}. We prove that the resulting interpolatory HDG methods converge at the same rate as for the linear elliptic problems. Hence, we obtain superconvergent methods for $k\ge 0$ by some methods. We present numerical results to illustrate the convergence theory.

A set $S$ of vertices is a determining set for a graph $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The size of a smallest determining set for $G$ is called its determining number, $Det(G)$. A graph $G$ is said to be $d$-distinguishable if there is a coloring of the vertices with $d$ colors so that only the trivial automorphism preserves the color classes. The smallest such $d$ is the distinguishing number, $Dist(G)$. If $Dist(G) = 2$, the cost of 2-distinguishing, $\rho(G)$, is the size of a smallest color class over all 2-distinguishing colorings of $G$. The Mycielskian, $\mu(G)$, of a graph $G$ is constructed by adding a shadow master vertex $w$, and for each vertex $v_i$ of $G$ adding a shadow vertex $u_i$ with edges so that the neighborhood of $u_i$ in $\mu(G)$ is the same as the neighborhood of $v_i$ in $G$ with the addition of $w$. That is, $N(u_i)=N_G(v_i)\cup\{w\}$. The generalized Mycielskian $\mu^{(t)}(G)$ of a graph $G$ is a Mycielskian graph with $t$ layers of shadow vertices, each with edges to layers above and below, and $w$ only adjacent to the top layer of shadow vertices. A graph is twin-free if it has no pair of vertices with the same set of neighbors. This paper examines the determining number and, when relevant, the cost of 2-distinguishing for Mycielskians and generalized Mycielskians of simple graphs with no isolated vertices. In particular, if $G \neq K_2$ is twin-free with no isolated vertices, then $Det(\mu^{(t)}(G)) = Det(G)$. Further, if $Det(G) = k \geq 2$ and $t \ge k-1$, then $Dist(\mu^{(t)}(G))=2$, and $Det(\mu^{(t)}(G)) = \rho(\mu^{(t)}(G))= k$. For $G$ with twins, we develop a framework using quotient graphs with respect to equivalence classes of twin vertices to give bounds on the determining number of Mycielskians. Moreover, we identify classes of graphs with twins for which $Det(\mu^{(t)}(G)) = (t{+}1) Det(G)$.

A graph $G$ is $d$-distinguishable if there is a coloring of the vertices with $d$ colors so that only the trivial automorphism preserves the color classes. The smallest such $d$ is the distinguishing number, $\operatorname{Dist}(G)$. The Mycielskian $\mu(G)$ of a graph $G$ is constructed by adding a shadow vertex $u_i$ for each vertex $v_i$ of $G$ and one additional vertex $w$ and adding edges so that $N(u_i)~=~N_G(v_i)~\cup~\{w\}$. The generalized Mycielskian $\mu_t(G)$ is a Mycielskian graph with $t$ layers of shadow vertices, each with edges to layers above and below. This paper examines the distinguishing number of the traditional and generalized Mycielskian graphs. Notably, if $G~\neq ~K_1,~K_2$ and the number of isolated vertices in $\mu_t(G)$ is at most $\operatorname{Dist}(G)$, then $\operatorname{Dist}(\mu_t(G)) \le \operatorname{Dist}(G)$. This result proves and exceeds a conjecture of Alikhani and Soltani.

A graph $G$ is said to be $d$-distinguishable if there is a labeling of the vertices with $d$ labels so that only the trivial automorphism preserves the labels. The smallest such $d$ is the distinguishing number, Dist($G$). A subset of vertices $S$ is a determining set for $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The size of a smallest determining set for $G$ is called the determining number, Det($G$). The orthogonality graph $\Omega_{2k}$ has vertices which are bitstrings of length $2k$ with an edge between two vertices if they differ in precisely $k$ bits. This paper shows that Det($\Omega_{2k}$) $= 2^{2k-1}$ and that if $\binom{m}{2} \geq 2k$ then $2<$ Dist($\Omega_{2k}$) $\leq m$., Comment: 17 pages, 5 figures

Neural network training entails heavy computation with obvious bottlenecks. The Compute Unified Device Architecture (CUDA) programming model allows us to accelerate computation by passing the processing workload from the CPU to the graphics processing unit (GPU). In this paper, we leveraged the power of Nvidia GPUs to parallelize all of the computation involved in training, to accelerate a backpropagation feed-forward neural network with one hidden layer using CUDA and C++. This optimized neural network was tasked with predicting the level of glycated hemoglobin (HbA1c) from non-invasive markers. The rate of increase in the prevalence of Diabetes Mellitus has resulted in an urgent need for early detection and accurate diagnosis. However, due to the invasiveness and limitations of conventional tests, alternate means are being considered. Limited Joint Mobility (LJM) has been reported as an indicator for poor glycemic control. LJM of the fingers is quantified and its link to HbA1c is investigated along with other potential non-invasive markers of HbA1c. We collected readings of 33 potential markers from 120 participants at a clinic in south Trinidad. Our neural network achieved 95.65% accuracy on the training and 86.67% accuracy on the testing set for male participants and 97.73% and 66.67% accuracy on the training and testing sets for female participants. Using 960 CUDA cores from a Nvidia GeForce GTX 660, our parallelized neural network was trained 50 times faster on both subsets, than its corresponding CPU implementation on an Intel Core (TM) i7-3630QM 2.40 GHz CPU., Comment: 5 pages, 6 figures, conference paper: International Conference on Information System and Data Mining, published at https://dl.acm.org/citation.cfm?id=3325940

In our earlier work [8], we approximated solutions of a general class of scalar parabolic semilinear PDEs by an interpolatory hybridizable discontinuous Galerkin (Interpolatory HDG) method. This method reduces the computational cost compared to standard HDG since the HDG matrices are assembled once before the time integration. Interpolatory HDG also achieves optimal convergence rates; however, we did not observe superconvergence after an element-by-element postprocessing. In this work, we revisit the Interpolatory HDG method for reaction diffusion problems, and use the postprocessed approximate solution to evaluate the nonlinear term. We prove this simple change restores the superconvergence and keeps the computational advantages of the Interpolatory HDG method. We present numerical results to illustrate the convergence theory and the performance of the method.

We establish that the Weak Galerkin methods are rewritings of the hybridizable discontinuous Galerkin methods.

The production of a Higgs boson in association with at least two jets receives contributions both from the fusion of weak vector bosons (VBF) and from QCD processes, especially gluon fusion (GF). The former process is important for measuring the coupling of the Higgs boson to weak bosons, whereas the latter process plays an important role in determining any $CP$-admixtures in the Higgs sector. In this paper we go beyond the current state-of-the-art for fixed order calculations of the GF process (i.e. one loop $H+2j$ including full quark mass effects) by including the all-order effects in leading $\log(\hat s/p_t^2)$, together with full quark mass and loop-propagator kinematic effects. We calculate the mass-dependent components and implement the resummation within the framework of High Energy Jets. The high energy effects suppress the prediction compared to fixed order at large $\Delta y_{12}$ and $m_{jj}$ (and therefore within the usual VBF cuts of widely separated jets), just as found in the limit of $m_t\to \infty$. The mass dependence is more significant than at fixed order, because the systematic inclusion of the leading logarithms in $\hat s/p_t^2$ results in a hardening of the transverse momentum of the Higgs boson, which in turn probes in more detail the loop-structure of the coupling. In particular, the full mass dependence reduces the cross section within VBF cuts by 11\% compared to a calculation based just on the infinite top mass limit, but the impact of the bottom quark remains small. This all implies that the gluon-fusion contribution within VBF-cuts is less severe than current estimates suggest., Comment: v2: New appendix (D) shows the earlier results with an alternative central scale choice, matches published version

We propose the interpolatory hybridizable discontinuous Galerkin (Interpolatory HDG) method for a class of scalar parabolic semilinear PDEs. The Interpolatory HDG method uses an interpolation procedure to efficiently and accurately approximate the nonlinear term. This procedure avoids the numerical quadrature typically required for the assembly of the global matrix at each iteration in each time step, which is a computationally costly component of the standard HDG method for nonlinear PDEs. Furthermore, the Interpolatory HDG interpolation procedure yields simple explicit expressions for the nonlinear term and Jacobian matrix, which leads to a simple unified implementation for a variety of nonlinear PDEs. For a globally-Lipschitz nonlinearity, we prove that the Interpolatory HDG method does not result in a reduction of the order of convergence. We display 2D and 3D numerical experiments to demonstrate the performance of the method.

We find new discrete $H^1$- and Poincar\'e-Friedrichs inequalities by studying the invertibility of the DG approximation of the flux for local spaces admitting M-decompositions. We then show how to use these inequalities to define and analyze new, superconvergent HDG and mixed methods for which the stabilization function is defined in such a way that the approximations satisfy new $H^1$-stability results with which their error analysis is greatly simplified. We apply this approach to define a wide class of energy-bounded, superconvergent HDG and mixed methods for the incompressible Navier-Stokes equations defined on unstructured meshes using, in 2D, general polygonal elements and, in 3D, general, flat-faced tetrahedral, prismatic, pyramidal and hexahedral elements., Comment: 22 pages

The Large Hadron Collider (LHC) has provided, and will continue to provide, data for collisions at the highest energies ever seen in a particle accelerator. A strong knowledge of the properties of amplitudes for Quantum Chromodynamics in the High Energy Limit is therefore important to interpret this data. We study this limit in the context of the High Energy Jets (HEJ) formalism. This formalism resums terms in the perturbative expansion of the cross-section that behave like $\alpha_s^n \log \left(\frac{s}{-t}\right)^{n-1}$, which are enhanced in this limit. Understanding this region is particularly important in certain key analyses at the LHC: for example, Higgs-boson-plus-dijet analyses where cuts are applied to pick out events with a large $m_{jj}$ and in many searches for new physics.

We propose a new tool, which we call $M$-decompositions, for devising superconvergent hybridizable discontinuous Galerkin (HDG) methods and hybridized-mixed methods for linear elasticity with strongly symmetric approximate stresses on unstructured polygonal/polyhedral meshes. We show that for an HDG method, when its local approximation space admits an $M$-decomposition, optimal convergence of the approximate stress and superconvergence of an element-by-element postprocessing of the displacement field are obtained. The resulting methods are locking-free. Moreover, we explicitly construct approximation spaces that admit $M$-decompositions on general polygonal elements. We display numerical results on triangular meshes validating our theoretical findings., Comment: 45 pages, 2 figures

We present a systematic construction of finite element exact sequences with a commuting diagram for the de Rham complex in one-, two- and three-space dimensions. We apply the construction in two-space dimensions to rediscover two families of exact sequences for triangles and three for squares, and to uncover one new family of exact sequence for squares and two new families of exact sequences for general polygonal elements. We apply the construction in three-space dimensions to rediscover two families of exact sequences for tetrahedra, three for cubes, and one for prisms; and to uncover four new families of exact sequences for pyramids, three for prisms, and one for cubes., Comment: 37pages

We investigate the dynamics of BPS vortices in the presence of magnetic impurities taking the form of axially-symmetric localised lumps and delta-functions. We present numerical results for vortices on flat space, as well as exact results for vortices on hyperbolic space in the presence of delta-function impurities. In fact, delta-function impurities of appropriate strength can be captured within the moduli space approximation by keeping one or more of the vortices fixed. We also show that previous work on vortices on the 2-sphere extends naturally to the inclusion of delta-function impurities., Comment: 17 pages, 8 figures

Stable ultrashort light pulses and frequency combs generated by mode-locked lasers have many important applications including high-resolution spectroscopy, fast chemical detection and identification, studies of ultrafast processes, and laser metrology. While compact mode-locked lasers emitting in the visible and near infrared range have revolutionized photonic technologies, the systems operating in the mid-infrared range where most gases have their strong absorption lines, are bulky and expensive and rely on nonlinear frequency down-conversion. Quantum cascade lasers are the most powerful and versatile compact light sources in the mid-infrared range, yet achieving their mode locked operation remains a challenge despite dedicated effort. Here we report the first demonstration of active mode locking of an external-cavity quantum cascade laser. The laser operates in the mode-locked regime at room temperature and over the full dynamic range of injection currents of a standard commercial laser chip.

A graph G is a homomorphic preimage of another graph H, or equivalently G is H-colorable, if there exists a graph homomorphism from G to H. A classic problem is to characterize the family of homomorphic preimages of a given graph H. A geometric graph is a simple graph G together with a straight line drawing of G in the plane with the vertices in general position. A geometric homomorphism (resp. isomorphism) is a graph homomorphism (resp. isomorphism) that preserves edge crossings (resp. and non-crossings). The homomorphism posetof a graph G is the set of isomorphism classes of geometric realizations of G partially ordered by the existence of injective geometric homomorphisms. A geometric graph G is H-colorable if there is a geometric homomorphism from G to some element of the homomorphism poset of H. We provide necessary and sufficient conditions for a geometric graph to be C_n-colorable for n less than 6., Comment: 11 pages, 9 figures

The geometric phase analysis (GPA) algorithm is known as a robust and straightforward technique that can be used to measure lattice strains in high resolution transmission electron microscope (TEM) images. It is also attractive for analysis of aberration-corrected scanning TEM (ac-STEM) images that resolve every atom column, since it uses Fourier transforms and does not require real-space peak detection and assignment to appropriate sublattices. Here it is demonstrated that in ac-STEM images of compound materials (i.e. with more than one atom per unit cell) an additional phase is present in the Fourier transform. If the structure changes from one area to another in the image (e.g. across an interface), the change in this additional phase will appear as a strain in conventional GPA, even if there is no lattice strain. Strategies to avoid this pitfall are outlined., Comment: 9 pages, 7 figures, Preprint before review, submitted to Ultramicroscopy 7 April 2015

We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order $0<\alpha<1$. For each time $t \in [0,T]$, the HDG approximations are taken to be piecewise polynomials of degree $k\ge0$ on the spatial domain~$\Omega$, the approximations to the exact solution $u$ in the $L_\infty\bigr(0,T;L_2(\Omega)\bigr)$-norm and to $\nabla u$ in the $L_\infty\bigr(0,T;{\bf L}_2(\Omega)\bigr)$-norm are proven to converge with the rate $h^{k+1}$ provided that $u$ is sufficiently regular, where $h$ is the maximum diameter of the elements of the mesh. Moreover, for $k\ge1$, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for $u$ converging with a rate $h^{k+2}$ (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed.

We study the use of the hybridizable discontinuous Galerkin (HDG) method for numerically solving fractional diffusion equations of order $-\alpha$ with $-1<\alpha<0$. For exact time-marching, we derive optimal algebraic error estimates {assuming} that the exact solution is sufficiently regular. Thus, if for each time $t \in [0,T]$ the approximations are taken to be piecewise polynomials of degree $k\ge0$ on the spatial domain~$\Omega$, the approximations to $u$ in the $L_\infty\bigr(0,T;L_2(\Omega)\bigr)$-norm and to $\nabla u$ in the $L_\infty\bigr(0,T;{\bf L}_2(\Omega)\bigr)$-norm are proven to converge with the rate $h^{k+1}$, where $h$ is the maximum diameter of the elements of the mesh. Moreover, for $k\ge1$ and quasi-uniform meshes, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for $u$ converging with a rate of $\sqrt{\log(T h^{-2/(\alpha+1)})}\, \,h^{k+2}$.

Hyperbolic monopole solutions can be obtained from circle-invariant ADHM data if the curvature of hyperbolic space is suitably tuned. Here we give explicit ADHM data corresponding to axial hyperbolic monopoles in a simple, tractable form, as well as expressions for the axial monopole fields. The data is deformed into new 1-parameter families preserving dihedral and twisted-line symmetries. In many cases explicit expressions are presented for their spectral curves and rational maps of both Donaldson and Jarvis type., Comment: 20 pages, 1 figure

A large class of explicit hyperbolic monopole solutions can be obtained from JNR instanton data, if the curvature of hyperbolic space is suitably tuned. Here we provide explicit formulae for both the monopole spectral curve and its rational map in terms of JNR data. Examples with platonic symmetry are presented, together with some one-parameter families with cyclic and dihedral symmetries. These families include hyperbolic analogues of geodesics that describe symmetric monopole scatterings in Euclidean space and we illustrate the results with energy density isosurfaces. There is a metric on the moduli space of hyperbolic monopoles, defined using the abelian connection on the boundary of hyperbolic space, and we provide a simple integral formula for this metric on the space of JNR data., Comment: 30 pages, 2 figures

Company mergers and acquisitions are often perceived to act as catalysts for corporate growth in free markets systems: it is conventional wisdom that those activities lead to better and more efficient markets. However, the broad adoption of this perception into corporate strategy is prone to result in a less diverse and more unstable environment, dominated by either very large or very small niche entities. We show here that ancestry, i.e. the cumulative history of mergers, is the key characteristic that encapsulates the diverse range of drivers behind mergers and acquisitions, across a range of industries and geographies. A long-term growth analysis reveals that entities which have been party to fewer mergers tend to grow faster than more highly acquisitive businesses., Comment: 13 pages and 6 figures

Thin magnetic multilayered films containing FePt have attracted a lot of attention recently due to their possible usage in ultra-high density magnetic storage. Although structure defects play a dramatic role in the magnetization process and influence magnetic properties in general this dependence haven't been studied thoroughly. The main aim of this work was to perform theoretical investigation of the magnetic properties of FePt and Fe/FePt thin films with high coercivity with respect to the structure defects such as anisotropy constant, magnetization saturation, exchange constant fluctuations and easy axis deviation. For selected defect patterns the coercive field dependence on layer thicknesses was analysed. Numerical study of the bilayer with hard magnetic layer having the planar anisotropy was carried on using micromagnetic calculations. Values of layers thickness have been found optimal for perspective applications, the dependence of the hysteresis loop shape upon the magnetization process has been shown and analysed., Comment: 9 pages, 4 figures

A geometric graph \G is a simple graph drawn in the plane, on points in general position, with straight-line edges. We call \G a geometric realization of the underlying abstract graph G. A geometric homomorphism from \G to \H is a vertex map that preserves adjacencies and crossings (but not necessarily non-adjacencies or non-crossings). Geometric homomorphisms can be used to define a partial order on the set of isomorphism classes of geometric realizations of an abstract graph G. In this paper, the homomorphism poset of K_{3,3} is determined., Comment: 8 pages, 4 figures

We review the stochastic Gross-Pitaevskii approach for non-equilibrium finite temperature Bose gases, focussing on the formulation of Stoof; this method provides a unified description of condensed and thermal atoms, and can thus describe the physics of the critical fluctuation regime. We discuss simplifications of the full theory, which facilitate straightforward numerical implementation, and how the results of such stochastic simulations can be interpreted, including the procedure for extracting phase-coherent (`condensate') and density-coherent (`quasi-condensate') fractions. The power of this methodology is demonstrated by successful ab initio modelling of several recent atom chip experiments, with the important information contained in each individual realisation highlighted by analysing dark soliton decay within a phase-fluctuating condensate., Comment: Unedited version of chapter to appear in Quantum Gases: Finite Temperature and Non-Equilibrium Dynamics (Vol. 1 Cold Atoms Series). N.P. Proukakis, S.A. Gardiner, M.J. Davis and M.H. Szymanska, eds. Imperial College Press, London (in press). See http://www.icpress.co.uk/physics/p817.html

The stochastic Gross-Pitaevskii equation and modified Popov theory are shown to provide an ab initio description of finite temperature, weakly-interacting two-dimensional Bose gas experiments. Using modified Popov theory, a systematic approach is developed in which the momentum cut-off inherent to classical field methods is removed as a free parameter. This is shown to yield excellent agreement with the recent experiment of Hung et al. [Nature, 470, 236 (2011)], verifying that the stochastic Gross-Pitaevskii equation captures the observed universality and scale-invariance., Comment: 5 pages, 4 figures

We perform an ab initio analysis of the temperature dependence of the phase coherence length of finite temperature, quasi-one-dimensional Bose gases measured in the experiments of Richard et al. (Phys. Rev. Lett. 91, 010405 (2003)) and Hugbart et al. (Eur. Phys. J. D 35, 155-163 (2005)), finding very good agreement across the entire observed temperature range ($0.8

The fluctuating and dissipative dynamics of matter-wave dark solitons within harmonically trapped, partially condensed Bose gases is studied both numerically and analytically. A study of the stochastic Gross-Pitaevskii equation, which correctly accounts for density and phase fluctuations at finite temperatures, reveals dark soliton decay times to be lognormally distributed at each temperature, thereby characterizing the previously predicted long lived soliton trajectories within each ensemble of numerical realizations (S.P. Cockburn {\it et al.}, Phys. Rev. Lett. 104, 174101 (2010)). Expectation values for the average soliton lifetimes extracted from these distributions are found to agree well with both numerical and analytic predictions based upon the dissipative Gross-Pitaevskii model (with the same {\it ab initio} damping). Probing the regime for which $0.8 k_{B}T < \mu < 1.6 k_{B}T$, we find average soliton lifetimes to scale with temperature as $\tau\sim T^{-4}$, in agreement with predictions previously made for the low-temperature regime $k_{B}T\ll\mu$. The model is also shown to capture the experimentally-relevant decrease in the visibility of an oscillating soliton due to the presence of background fluctuations., Comment: 17 pages, 14 figures

A geometric graph G(bar) is a simple graph drawn in the plane, on points in general position, with straight-line edges. We call G(bar) a geometric realization of the underlying abstract graph G. A geometric homomorphism is a vertex map that preserves adjacencies and crossings (but not necessarily non-adjacencies or non-crossings). This work uses geometric homomorphisms to introduce a partial order on the set of isomorphism classes of geometric realizations of an abstract graph G. We say G(bar) precedes G(hat) if G(bar) and G(hat) are geometric realizations of G and there is a vertex-injective geometric homomorphism from G(bar) to G(hat). This paper develops tools to determine when two geometric realizations are comparable. Further, for 3 \leq n \leq 6, this paper provides the isomorphism classes of geometric realizations of P_n, C_n and K_n, as well as the Hasse diagrams of the geometric homomorphism posets of these graphs. The paper also provides the following results for general n: the poset of P_n and C_n has a unique minimal element and a unique maximal element; if k \leq n then the poset of P_k (resp., the poset of C_k) is a subposet of the poset for P_n (resp., C_n); and the poset for K_n contains a chain of length n-2., Comment: 42 pages, 25 figures; co-author added in replaced version

The stochastic Gross-Pitaevskii equation is shown to be an excellent model for quasi-one-dimensional Bose gas experiments, accurately reproducing the in situ density profiles recently obtained in the experiments of Trebbia et al. [Phys. Rev. Lett. 97, 250403 (2006)] and van Amerongen et al. [Phys. Rev. Lett. 100, 090402 (2008)], and the density fluctuation data reported by Armijo et al. [Phys. Rev. Lett. 105, 230402 (2010)]. To facilitate such agreement, we propose and implement a quasi-one-dimensional stochastic equation for the low-energy, axial modes, while atoms in excited transverse modes are treated as independent ideal Bose gases., Comment: 10 pages, 5 figures; updated figures with experimental data

We analyze the equilibrium properties of a weakly interacting, trapped quasi-one-dimensional Bose gas at finite temperatures and compare different theoretical approaches. We focus in particular on two stochastic theories: a number-conserving Bogoliubov (ncB) approach and a stochastic Gross-Pitaevskii equation (sGPe) that have been extensively used in numerical simulations. Equilibrium properties like density profiles, correlation functions, and the condensate statistics are compared to predictions based upon a number of alternative theories. We find that due to thermal phase fluctuations, and the corresponding condensate depletion, the ncB approach loses its validity at relatively low temperatures. This can be attributed to the change in the Bogoliubov spectrum, as the condensate gets thermally depleted, and to large fluctuations beyond perturbation theory. Although the two stochastic theories are built on different thermodynamic ensembles (ncB: canonical, sGPe: grand-canonical), they yield the correct condensate statistics in a large BEC (strong enough particle interactions). For smaller systems, the sGPe results are prone to anomalously large number fluctuations, well-known for the grand-canonical, ideal Bose gas. Based on the comparison of the above theories to the modified Popov approach, we propose a simple procedure for approximately extracting the Penrose-Onsager condensate from first- and second-order correlation functions that is computationally convenient. This also clarifies the link between condensate and quasi-condensate in the Popov theory of low-dimensional systems., Comment: Detailed comparison of equilibrium properties, correlation functions and condensate statistics between stochastic Gross-Pitaevskii, stochastically-sampled number-conserving Bogoliubov and modified Popov theory; Relation between quasi-condensate and Penrose-Onsager condensate (35 pages, 22 figures)

A geometric graph is a simple graph G together with a straight line drawing of G in the plane with the vertices in general position. Two geometric realizations of a simple graph are geo-isomorphic if there is a vertex bijection between them that preserves vertex adjacencies and non-adjacencies, as well as edge crossings and non-crossings. A natural extension of graph homomorphisms, geo-homomorphisms, can be used to define a partial order on the set of geo-isomorphism classes of realizations of a given simple graph. In this paper, the homomorphism poset of the complete bipartite graph K_{2,n} is determined by establishing a correspondence between realizations of K_{2,n} and permutations of S_n, in which crossing edges correspond to inversions. Through this correspondence, geo-isomorphism defines an equivalence relation on S_n, which we call geo-equivalence. The number of geo-isomorphism classes is provided for all n <= 9. The modular decomposition tree of permutation graphs is used to prove some results on the size of geo-equivalence classes. A complete list of geo-equivalence classes and a Hasse diagrams of the poset structure are given for n <= 5., Comment: 31 pages, 16 figures; added connections to permutation graphs; added a new section 4; new co-author

The dynamics of dark matter-wave solitons in elongated atomic condensates are discussed at finite temperatures. Simulations with the stochastic Gross-Pitaevskii equation reveal a noticeable, experimentally observable spread in individual soliton trajectories, attributed to inherent fluctuations in both phase and density of the underlying medium. Averaging over a number of such trajectories (as done in experiments) washes out such background fluctuations, revealing a well-defined temperature-dependent temporal growth in the oscillation amplitude. The average soliton dynamics is well captured by the simpler dissipative Gross-Pitaevskii equation, both numerically and via an analytically-derived equation for the soliton center based on perturbation theory for dark solitons., Comment: 4 pages, 3 figures. Added several references

The stochastic Gross-Pitaevskii equation represents a versatile approach for studying the dynamics of trapped degenerate ultracold Bose gases in the presence of large phase and density fluctuations. Following a brief review of the original formulation of Stoof, which highlights the benefits of this approach and its relation to alternative theories, we present selected applications for the dynamics of effectively one-dimensional systems, and briefly discuss the generalization to two-dimensional systems, highlighting certain potential pitfalls in their numerical implementations., Comment: 14 pages, 9 figures. Corrected typos

Preparing courses for flexible delivery and distance education is normally a time-consuming and expensive process. This paper describes the design and evaluation of a system that automatically captures and indexes audio and video streams of traditional university lectures without demanding any changes in the style or tools used by teachers. Using a "Wizard-of-Oz" technique to simulate the automatic indexing, a 4-month trial of the system in a large (746 students) first year Computer Studies course was run. The results reveal some surprising social implications of making flexible delivery available to students at a residential university. Early in the trial, many students expressed an intention to use the system, but few used it. Late in the course, many students stated that they urgently needed the system for revision, but even fewer used it. At the same time, lecture attendance appeared to be lower than normal. It is hypothesized that the availability of a flexible alternative to lectures removed the necessity of attending lectures, and that students deceived themselves about their intentions to catch up using the digital medium. (Contains 18 references.) (Author/AEF)

Polaron dephasing processes are investigated in InAs/GaAs dots using far-infrared transient four wave mixing (FWM) spectroscopy. We observe an oscillatory behaviour in the FWM signal shortly (< 5 ps) after resonant excitation of the lowest energy conduction band transition due to coherent acoustic phonon generation. The subsequent single exponential decay yields long intraband dephasing times of 90 ps. We find excellent agreement between our measured and calculated FWM dynamics, and show that both real and virtual acoustic phonon processes are necessary to explain the temperature dependence of the polarization decay., Comment: 10 pages, 4 figures, submitted to Phys Rev Lett

We show that mid infrared transmission spectroscopy of a quantum cascade laser provides clear cut information on changes in charge location at different bias. Theoretical simulations of the evolution of the gain/absorption spectrum for the $\lambda \sim$ 7.4 $\mu$m InGaAs/AlInAs/InP quantum cascade laser have been compared with the experimental findings. Transfer of electrons between the ground states in the active region and the states in the injector goes in hand with a decrease of discrete intersubband absorption peaks and an increase of broad high-energy absorption towards the continuum delocalised states above the barriers., Comment: 6 pages, 5 figures

This article looks at the recent contributions made by feminists who advocate a distinctive 'ethic of care' to replace the conventional 'ethic of rights'. The article explores ways in which the ethic of care could be utilized and applied to the children's rights context. After looking at the important feminist criticisms of conventional rights-based approaches, it is argued that there needs to be some caution applied to the feminist ethic of care, if it is to be successfully applied to the context of children. These cautions are that it is important to recognize the contested nature of care and not to valorize the perspectives of carers over those being cared for. Second, the feminist ethic of care might lead to a 'needs-based' discourse, an approach that is unsatisfactory in its implications for children's rights. Finally, conceptions of justice and equality must not be dropped from political arguments. Rather, their limitations must be acknowledged and then used strategically and partially. However, despite these cautions, the feminist ethic of care remains a constructive approach to the children's rights context as it emphasizes responsibilities and relationships, the concrete contexts of caring interdependencies, and allows children to be active social players with a voice rather than passive recipients of care and rights. It is hoped that this article might serve as both a corrective and conceptual enrichment of the feminist ethic of care.