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This paper explores the implications of blocking duality---pioneered by Fulkerson et al.---in the context of $p$-modulus on networks. Fulkerson's blocking duality is an analogue on networks to the method of conjugate families of curves in the plane. The technique presented here leads to a general framework for studying families of objects on networks; each such family has a corresponding dual family whose $p$-modulus is essentially the reciprocal of the original family's. As an application, we give a modulus-based proof for the fact that effective resistance is a metric on graphs. This proof immediately generalizes to yield a family of graph metrics, depending on the parameter $p$, that continuously interpolates among the shortest-path metric, the effective resistance metric, and the mincut ultrametric. In a second application, we establish a connection between Fulkerson's blocking duality and the probabilistic interpretation of modulus. This connection, in turn, provides a straightforward proof of several monotonicity properties of modulus that generalize known monotonicity properties of effective resistance. Finally, we use this framework to expand on a result of Lov\'asz in the context of randomly weighted graphs., Comment: Added a new proof of the fact that effective resistance is a metric on graphs, as an application of the theory developed in this paper. As a result, we changed the title, rewrote the abstract and introduction, and added a co-author

This volume contains the proceedings of TERMGRAPH 2016, the Ninth International Workshop on Computing with Terms and Graphs which was held on April 8, 2016 in Eindhoven, The Netherlands, as a satellite event of the European Joint Conferences on Theory and Practice of Software (ETAPS 2016).

We study the structure of loops in networks using the notion of modulus of loop families. We introduce a new measure of network clustering by quantifying the richness of families of (simple) loops. Modulus tries to minimize the expected overlap among loops by spreading the expected link-usage optimally. We propose weighting networks using these expected link-usages to improve classical community detection algorithms. We show that the proposed method enhances the performance of certain algorithms, such as spectral partitioning and modularity maximization heuristics, on standard benchmarks., Comment: 8 pages, 20 figures

Using the worldline formalism of the Dirac field with a non-Abelian gauge symmetry we show how to describe the matter field transforming in an arbitrary representation of the gauge group. Colour degrees of freedom are carried on the worldline by auxiliary fields, responsible for providing path ordering and the Wilson-loop coupling. The Hilbert space of these fields is reducible but we make use of recent work in order to project onto a single, arbitrary, irreducible representation. By functionally quantising the resulting theory we show that this procedure correctly generates the Wilson-loop interaction between the gauge field and the matter field taken to transform in a chosen representation. This work has direct application to physical observables such as scattering amplitudes in the presence of such a matter multiplet and lifts the restriction on the type of matter that has previously featured in worldline calculations., Comment: 47 Pages, 2 Appendices. Update to version accepted for publication

The notion of $p$-modulus of a family of objects on a graph is a measure of the richness of such families. We develop the notion of minimal subfamilies using the method of Lagrangian duality for $p$-modulus. We show that minimal subfamilies have at most $|E|$ elements and that these elements carry a weight related to their "importance" in relation to the corresponding $p$-modulus problem. When $p=2$, this measure of importance is in fact a probability measure and modulus can be thought as trying to minimize the expected overlap in the family., Comment: Corrected several typos

We consider the first quantised approach to quantum field theory coupled to a non-Abelian gauge field. Representing the colour degrees of freedom with a single family of auxiliary variables the matter field transforms in a reducible representation of the gauge group which - by adding a suitable Chern-Simons term to the particle action - can be projected onto a chosen fully (anti-)symmetric representation. By considering F families of auxiliary variables, we describe how to extend the model to arbitrary tensor products of F reducible representations, which realises a U(F) "flavour" symmetry on the worldline particle model. Gauging this symmetry allows the introduction of constraints on the Hilbert space of the colour fields which can be used to project onto an arbitrary irreducible representation, specified by a certain Young Tableau. In particular the occupation numbers of the wavefunction - i.e. the lengths of the columns (rows) of the Young Tableau - are fixed through the introduction of Chern-Simons terms. We verify this projection by calculating the number of colour degrees of freedom associated to the matter field. We suggest that, using the worldline approach to quantum field theory, this mechanism will allow the calculation of one-loop scattering amplitudes with the virtual particle in an arbitrary representation of the gauge group., Comment: 1+32 pages

This e-book is a celebration of and reflection on 10 years of the The Centre for Languages, Linguistics and Area Studies (LLAS) elearning symposium, an event which is run by the Centre for Languages, Linguistics and Area Studies based at the University of Southampton, UK. (LLAS) was founded in 2000 as one of 24 Higher Education Academy subject centres. Over time it has built a reputation for providing support to the wider languages, linguistics and area studies communities through its regular conferences, workshops and innovative projects which provide professional development and networking opportunities to teachers in the higher education and school sectors. More recently it has also become a successful enterprise unit within Modern Languages at the University of Southampton, which works in partnership with universities around the world to deliver bespoke professional development courses or develop collaborative projects. This ebook is structured around four sections. Each section begins with an invited contribution and contains articles related to the broad thematic area introduced in the invited piece. This structure deliberately reflects the alternation of keynote plenary sessions and talks which has been the distinct format of the elearning symposium in the past ten years. The book is aimed at practitioners and as such recounts real, practical experiences of innovation in using technology in language teaching or professional practice. The authors' work is based in research, but they all present lived examples of excellent practice in action. The range of projects and case studies presented in this e-volume is a testament to the irrepressible creativity of language teachers in their adoption of technology to achieve their pedagogic aims. [For the complete book, "10 Years of the LLAS eLearning Symposium: Case Studies in Good Practice," see ED577046.]

These are notes of lectures on spinning particles and the worldline formalism originally given by Olindo Corradini and Christian Schubert at the School on Spinning Particles in Quantum Field Theory: Worldline Formalism, Higher Spins, and Conformal Geometry, held at Morelia, Mexico, from November 19 through November 23, 2012. The lectures were addressed to graduate level students with a background in relativistic quantum mechanics and at least a rudimentary knowledge of field theory. They have since been updated to include a further set of lecture notes on tree level processes from a worldline perspective based on a mini-course by James P. Edwards at the Instituto de Fisica y Matematicas in Morelia, Mexico given to graduates and visiting professors during July 2017 and in various later classes, complemented by a series of three lectures titled New techniques for amplitude calculation in QED given by Naser Ahmadiniaz at the Center for Relativistic Laser Science (CoReLS), Institute for Basic Science(IBS), November 2015, Gwangju, South Korea and as an invited lecturer at the Helmholtz International Summer School (HISS) - Dubna International Advanced School of Theoretical Physics (DIAS-TH): 'Quantum Field Theory at the Limits: from Strong Fields to Heavy Quarks', July-August 2019, Dubna, Russia. These additional notes complete the picture of first quantised techniques and bring the worldline description up to date. The extended Abstract can be found in the manuscript., Comment: 1+ 169 pages, 8 captioned figures. New parts on the worldline approach to dressed propagators added, authorship changed, conventions updated in Chapters 1 and 2, typos corrected, references added

In this work, we use molecular dynamics simulations to study the thermodynamics, structure and dynamics of the Li$_2$CO$_3$-K$_2$CO$_3$ (62:38 mol%) eutectic mixture. We present a new classical non-polarizable force field for this molten salt mixture, optimized using experimental and first principles molecular dynamics simulations as reference data. This simple force field allows efficient molecular simulations of phenomena at long timescales. We use this optimized force field to describe the behavior of the eutectic mixture in the 900-1100 K temperature range, at pressures between 0 and 5 GPa. After studying the equation of state in these thermodynamic conditions, we present molecular insight into the structure and dynamics of the melt. In particular, we present an analysis of the temperature and pressure dependence of the eutectic mixture's self diffusion coefficients, viscosity and ionic conductivity.

This study develops the epidemic hitting time (EHT) metric on graphs measuring the expected time an epidemic starting at node $a$ in a fully susceptible network takes to propagate and reach node $b$. An associated EHT centrality measure is then compared to degree, betweenness, spectral, and effective resistance centrality measures through exhaustive numerical simulations on several real-world network data-sets. We find two surprising observations: first, EHT centrality is highly correlated with effective resistance centrality; second, the EHT centrality measure is much more delocalized compared to degree and spectral centrality, highlighting the role of peripheral nodes in epidemic spreading on graphs., Comment: 6 pages, 1 figure, 3 tables, In Proceedings of 2015 Asilomar Conference on Signals, Systems, and Computers

Algebraic connectivity, the second eigenvalue of the Laplacian matrix, is a measure of node and link connectivity on networks. When studying interconnected networks it is useful to consider a multiplex model, where the component networks operate together with inter-layer links among them. In order to have a well-connected multilayer structure, it is necessary to optimally design these inter-layer links considering realistic constraints. In this work, we solve the problem of finding an optimal weight distribution for one-to-one inter-layer links under budget constraint. We show that for the special multiplex configurations with identical layers, the uniform weight distribution is always optimal. On the other hand, when the two layers are arbitrary, increasing the budget reveals the existence of two different regimes. Up to a certain threshold budget, the second eigenvalue of the supra-Laplacian is simple, the optimal weight distribution is uniform, and the Fiedler vector is constant on each layer. Increasing the budget past the threshold, the optimal weight distribution can be non-uniform. The interesting consequence of this result is that there is no need to solve the optimization problem when the available budget is less than the threshold, which can be easily found analytically.

We study the propagator of a colored scalar particle in the background of a non-abelian gauge field using the worldline formalism. It is obtained by considering the open worldline of a scalar particle with extra degrees of freedom needed to take into account the color charge of the particle, which we choose to be in the fundamental representation of the gauge group. Specializing the external gauge field to be given by a sum of plane waves, i.e. a sum of external gluons, we produce a master formula for the scalar propagator with an arbitrary number of gluons directly attached to the scalar line, akin to similar formulas derived in the literature for the case of the scalar particle performing a loop. Our worldline description produces at the same time the situation in which the particle has a color charge given by an arbitrarily chosen symmetric or antisymmetric tensor product of the fundamental., Comment: 21 pages, 1 figure; title modified, discussion improved, references added, main results unchanged. Matches version published in PRD

We study pure noncommutative U(1) gauge theory representing its one-loop effective action in terms of a phase space worldline path integral. We write the quadratic action using the background field method to keep explicit gauge invariance, and then employ the worldline formalism to write the one-loop effective action, singling out UV-divergent parts and finite (planar and non-planar) parts, and study renormalization properties of the theory. This amounts to employ worldline Feynman rules for the phase space path integral, that nicely incorporate the Fadeev-Popov ghost contribution and efficiently separate planar and non-planar contributions. We also show that the effective action calculation is independent of the choice of the worldline Green's function, that corresponds to a particular way of factoring out a particle zero-mode. This allows to employ homogeneous string-inspired Feynman rules that greatly simplify the computation., Comment: 27+1 pages. Minor corrections; matches published version

It is postulated that quantum gravity is a sum over causal structures coupled to matter via scale evolution. Quantized causal structures can be described by studying simple matrix models where matrices are replaced by an algebra of quantum mechanical observables. In particular, previous studies constructed quantum gravity models by quantizing the moduli of Laplace, weight and defining-function operators on Fefferman-Graham ambient spaces. The algebra of these operators underlies conformal geometries. We extend those results to include fermions by taking an osp(1|2) "Dirac square root" of these algebras. The theory is a simple, Grassmann, two-matrix model. Its quantum action is a Chern-Simons theory whose differential is a first-quantized, quantum mechanical BRST operator. The theory is a basic ingredient for building fundamental theories of physical observables., Comment: 4 pages, LaTeX

Relativistic particle actions are a useful tool to describe quantum field theory effective actions using a string-inspired first-quantized approach. Here we describe how to employ suitable particle actions in the computation of the scalar contribution to the one-loop gluon effective action. We use the well-known method of introducing auxiliary variables that create the color degrees of freedom. In a path integral they implement automatically the path ordering needed to ensure gauge invariance. It is known that the color degrees of freedom introduced this way form a reducible representation of the gauge group. We describe a method of projecting onto the fundamental representation (or any other chosen irrep, if desired) of the gauge group. Previously, we have discussed the case of anticommuting auxiliary variables. Choosing them to be in the fundamental representation allows to obtain, without any extra effort, also the situation in which the color is given by any antisymmetric tensor product of the fundamental. Here, we describe the novel case of bosonic auxiliary variables. They can be used equivalently for creating the color charges in the fundamental representation. In addition one gets, as a byproduct, the cases where the particle can have the color sitting in any symmetric tensor product of the fundamental. This is obtained by tuning to a different value a Chern Simons coupling, present in the model, which controls how the projection is achieved., Comment: 11 pages, to appear in the Proceedings of the Victor Villanueva Memorial Workshop

Relativistic particles with higher spin can be described in first quantization using actions with local supersymmetry on the worldline. First, we present a brief review of these actions and their use in first quantization. In a Dirac quantization scheme the field equations emerge as Dirac constraints on the Hilbert space, and we outline how they lead to the description of higher spin fields in terms of the more standard Fronsdal-Labastida equations. Then, we describe how these actions can be extended so that the propagating particle is allowed to take different values of the spin, i.e. carry a reducible representation of the Poincar\'e group. This way one may identify a four dimensional model that carries the same degrees of freedom of the minimal Vasiliev's interacting higher spin field theory. Extensions to massive particles and to propagation on (A)dS spaces are also briefly commented upon., Comment: 10 pages, updated references. To appear in Proceedings of Victor Villanueva Memorial Workshop

This paper presents new results for the modulus of families of walks on a graph---a discrete analog of the modulus of curve families due to Beurling and Ahlfors. Particular attention is paid to the dependence of the modulus on its parameters. Modulus is shown to generalize (and interpolate among) three important quantities in graph theory: shortest path, effective resistance, and max-flow or min-cut., Comment: Updated with referee's comments. To appear in ECGD

The several algebraic approaches to graph transformation proposed in the literature all ensure that if an item is preserved by a rule, so are its connections with the context graph where it is embedded. But there are applications in which it is desirable, for example when cloning an item, to specify different embeddings for the original and for the copy. Therefore we propose a conservative extension of these approaches where a rule can specify how the embedding of a preserved item should be changed, typically by removing certain connections.

We study the combined effects of lattice deformation, e-e interaction and spin-orbit coupling in a two-dimensional (2D) honeycomb lattice. We adopt different kinds of hopping modulation--generalized dimerization and a Kekule distortion--and calculate topological invariants for the non-interacting system and for the interacting system. We identify the parameter range (Hubbard U, hopping modulation, spin-orbit coupling) where the 2D system behaves as a trivial insulator or Quantum Spin Hall Insulator., Comment: 8 pages, 4 figures: discussion improved, typos corrected, references updated. Matches version published in PRB

We propose a model of quantum gravity in arbitrary dimensions defined in terms of the BV quantization of a supersymmetric, infinite dimensional matrix model. This gives an (AKSZ-type) Chern-Simons theory with gauge algebra the space of observables of a quantum mechanical Hilbert space H. The model is motivated by previous attempts to formulate gravity in terms of non-commutative, phase space, field theories as well as the Fefferman-Graham curved analog of Dirac spaces for conformally invariant wave equations. The field equations are flat connection conditions amounting to zero curvature and parallel conditions on operators acting on H. This matrix-type model may give a better defined setting for a quantum gravity path integral. We demonstrate that its underlying physics is a summation over Hamiltonians labeled by a conformal class of metrics and thus a sum over causal structures. This gives in turn a model summing over fluctuating metrics plus a tower of additional modes-we speculate that these could yield improved UV behavior., Comment: 22 pages, LaTeX, 3 figures, references added, version to appear in PRD

We study actions for massive bosonic particles of higher spins by dimensionally reducing an action for massless particles. For the latter we take a model with a SO(N) extended local supersymmetry on the worldline, that is known to describe massless (conformal) particles of higher spins in flat spacetimes of even dimensions. Dimensional reduction produces an action for massive spinning particles in odd dimensions. The field equations that emerge in a quantization a la Dirac are shown to be equivalent to the Fierz-Pauli ones. The massless limit generates a multiplet of massless states with higher spins, whose first quantized field equations have a geometric form with fields belonging to various types of Young tableaux. These geometric equations can be partially integrated to show their equivalence with the standard Fronsdal-Labastida equations. We covariantize our model to check whether an extension to curved spacetimes can be achieved. Restricting to (A)dS spaces, we find that the worldline gauge algebra becomes nonlinear, but remains first class. This guarantees consistency on such backgrounds. A light cone analysis confirms the presence of the expected propagating degrees of freedom. A covariant analysis is worked out explicitly for the massive case, which is seen to give rise to the Fierz-Pauli equations extended to (A)dS spaces. It is worth noting that in D=3 the massless limit of our model when N goes to infinity has the same field content of the Vasiliev's theory that accommodates each spin exactly once., Comment: 31 pages

In this work, we explore a mechanism for abelian gauge field localization on thick branes based on a five-dimensional Stueckelberg-like action. A normalizable zero mode is found through the identification of a suitable coupling function between the brane and the gauge field. The same mechanism is studied for the localization of the abelian Kalb--Ramond field., Comment: 9 pages, revised final version, accepted for publication in EPJC

Using Cluster Perturbation Theory we calculate Green's functions, quasi-particle energies and topological invariants for interacting electrons on a 2-D honeycomb lattice, with intrinsic spin-orbit coupling and on-site e-e interaction. This allows to define the parameter range (Hubbard U vs spin-orbit coupling) where the 2D system behaves as a trivial insulator or Quantum Spin Hall insulator. This behavior is confirmed by the existence of gapless quasi-particle states in honeycomb ribbons. We have discussed the importance of the cluster symmetry and the effects of the lack of full translation symmetry typical of CPT and of most Quantum Cluster approaches. Comments on the limits of applicability of the method are also provided., Comment: 7 pages, 7 figures: discussion improved, one figure added, references updated. Matches version published in New J. Phys

A highly fluorinated anatase lattice has been recently reported, providing a new class of materials whose general chemical formula is $\rm Ti_{1- \mathit x}\square_{\mathit x}X_{4\mathit x}O_{2- 4\mathit x}$ (X$^-$ = F$^-$ or OH$^-$). To characterise the complex structural features of the material and the different F environments, we here apply a computational screening procedure. After deriving a polarisable force--field from DFT simulations, we screen in a step-wise fashion a large number of possible configurations differing in the positioning of the titanium vacancies ($\square$) and of the fluorine atoms. At each step only 10% of the configurations are retained. At the end of the screening procedure, a configuration is selected and simulated using DFT-based molecular dynamics. This allows us to analyse the atomic structure of the material, which is strongly disordered, leading to a strong decrease (by 0.8~eV) of the band gap compared to conventional anatase., Comment: 13 pages, 5 figures, 5 supplementary figures, 2 supplementary tables. Updated version. Accepted for publication in Scientific Reports

Molten fluorides are known to show favorable thermophysical properties which make them good candidate coolants for nuclear fission reactors. Here we investigate the special case of mixtures of lithium fluoride and thorium fluoride, which act both as coolant and fuel in the molten salt fast reactor concept. By using ab initio parameterized polarizable force fields, we show that it is possible to calculate the whole set of properties (density, thermal expansion, heat capacity, viscosity and thermal conductivity) which are necessary for assessing the heat transfer performance of the melt over the whole range of compositions and temperatures. We then deduce from our calculations several figures of merit which are important in helping the optimization of the design of molten salt fast reactors., Comment: 6 pages, 4 figures

Classical molecular dynamics simulations are performed on LiF in the framework of the polarizable ion model. The overlap-repulsion and polarization terms of the interaction potential are derived on a purely non empirical, first-principles basis. For the dispersion, three cases are considered: a first one in which the dispersion parameters are set to zero and two others in which they are included, with different parameterizations. Various thermodynamic, structural and dynamic properties are calculated for the solid and liquid phases. The melting temperature is also obtained by direct coexistence simulations of the liquid and solid phases. Dispersion interactions appear to have an important effect on the density of both phases and on the melting point, although the liquid properties are not affected when simulations are performed in the NVT ensemble at the experimental density., Comment: 8 pages, 5 figures

We introduce the notion of modulus of families of walks on graphs. We show how Beurling's famous criterion for extremality, that was formulated in the continuous case, can be interpreted on graphs as an instance of the Karush-Kuhn-Tucker conditions. We then develop an algorithm to numerically compute modulus using Beurling's criterion as our guide., Comment: Fixed a minor mistake and updated a reference