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Classically, B\'ezout's theorem says that an intersection of hypersurfaces in a projective space is rationally equivalent to a number of copies of a smaller projective space, the number depending on the degrees of the hypersurfaces. We give a generalization of that result to the context of $C_2$-equivariant hypersurfaces in $C_2$-equivariant linear projective space, expressing the intersection as a linear combination of equivariant Schubert varieties., Comment: 45 pages

Nanoscale precipitates in the microstructure of nickel-based superalloys hinder dislocation motion, which results in an extraordinary strengthening effect at elevated temperatures. We used molecular dynamics (MD) with classical effective potential to observe the movement of an $\frac{a}{2}\langle110\rangle\{111\}$ edge dislocation under shear in pure Ni, which represents the Ni solid solution matrix, and extracted the locations of the dislocations. We show how a Differential Evolution Monte Carlo (DE-MC) analysis is an effective way to find the parameters of an equation of motion for the dislocation lines with quantified uncertainties. The parameters of interest were the effective mass, drag coefficient, and force experienced by the dislocation. The marginal parameter and joint posterior distributions were estimated from the accepted samples produced by the DE-MC algorithm. The equation of motion and parameter distributions were used to predict the dislocation positions and velocities at the simulation timesteps, and the mean fit was found to match the MD trajectories with a root mean square error (RMSE) of \SI{0.2}{\nano\metre}. We also discuss how the selected model can be extended to account for the presence of multiple dislocations as well as dislocation-precipitate interactions. This work serves as the first step towards building a predictive surrogate model that describes the deformation behaviour of Ni-based superalloys., Comment: 19 pages, 7 figures

This article presents a weakly supervised machine learning method, which we call DAS-N2N, for suppressing strong random noise in distributed acoustic sensing (DAS) recordings. DAS-N2N requires no manually produced labels (i.e., pre-determined examples of clean event signals or sections of noise) for training and aims to map random noise processes to a chosen summary statistic, such as the distribution mean, median or mode, whilst retaining the true underlying signal. This is achieved by splicing (joining together) two fibres hosted within a single optical cable, recording two noisy copies of the same underlying signal corrupted by different independent realizations of random observational noise. A deep learning model can then be trained using only these two noisy copies of the data to produce a near fully-denoised copy. Once the model is trained, only noisy data from a single fibre is required. Using a dataset from a DAS array deployed on the surface of the Rutford Ice Stream in Antarctica, we demonstrate that DAS-N2N greatly suppresses incoherent noise and enhances the signal-to-noise ratios (SNR) of natural microseismic icequake events. We further show that this approach is inherently more efficient and effective than standard stop/pass band and white noise (e.g., Wiener) filtering routines, as well as a comparable self-supervised learning method based on masking individual DAS channels. Our preferred model for this task is lightweight, processing 30 seconds of data recorded at a sampling frequency of 1000 Hz over 985 channels (approx. 1 km of fiber) in $<$1 s. Due to the high noise levels in DAS recordings, efficient data-driven denoising methods, such as DAS-N2N, will prove essential to time-critical DAS earthquake detection, particularly in the case of microseismic monitoring., Comment: Submitted for publication to Geophysical Journal International. For the purpose of open access, the author(s) has applied a Creative Commons Attribution (CC BY) licence to the Author Accepted Manuscript version arising from this submission

The paper computes the Witt-sheaf cohomology rings of partial flag varieties in type A in terms of the Pontryagin classes of the subquotient bundles. The proof is based on a Leray-Hirsch-type theorem for Witt-sheaf cohomology for the maximal rank cases, and a detailed study of cohomology ring presentations and annihilators of characteristic classes for the general case. The computations have consequences for the topology of real flag manifolds: we show that all torsion in the integral cohomology is 2-torsion, which was not known in full generality previously. This allows for example to compute the Poincar\'e polynomials of complete flag varieties for cohomology with twisted integer coefficients. The computations also allow to describe the Chow-Witt rings of flag varieties, and we sketch an enumerative application to counting flags satisfying multiple incidence conditions to given hypersurfaces., Comment: 58 pages

Coarse-graining or model reduction is a term describing a range of approaches used to extend the time-scale of molecular simulations by reducing the number of degrees of freedom. In the context of molecular simulation, standard coarse-graining approaches approximate the potential of mean force and use this to drive an effective Markovian model. To gain insight into this process, the simple case of a quadratic energy is studied in an overdamped setting. A hierarchy of reduced models is derived and analysed, and the merits of these different coarse-graining approaches are discussed. In particular, while standard recipes for model reduction accurately capture static equilibrium statistics, it is shown that dynamical statistics such as the mean-squared displacement display systematic error, even when a system exhibits large time-scale separation. In the linear setting studied, it is demonstrated both analytically and numerically that such models can be augmented in a simple way to better capture dynamical statistics., Comment: 33 pages, 8 figures. Version accepted for publication in SIAM Multiscale Modeling and Simulation

B\'ezout's theorem, nonequivariantly, can be interpreted as a calculation of the Euler class of a sum of line bundles over complex projective space, expressing it in terms of the rank of the bundle and its degree. We give here a generalization to the $C_2$-equivariant context, using the calculation of the cohomology of a $C_2$-complex projective space from an earlier paper. We use ordinary $C_2$-cohomology with Burnside ring coefficients and an extended grading necessary to define the Euler class, which we express in terms of the equivariant rank of the bundle and the degrees of the bundle and its fixed subbundles. We do similar calculations using constant $\mathbb{Z}$ coefficients and Borel cohomology and compare the results., Comment: 20 pages

This work introduces a model for large-strain, geometrically nonlinear elasto-plastic dynamics in single crystals. The key feature of our model is that the plastic dynamics are entirely driven by the movement of dislocations, that is, $1$-dimensional topological defects in the crystal lattice. It is well known that glide motion of dislocations is the dominant microscopic mechanism for plastic deformation in many crystalline materials, most notably in metals. We propose a novel geometric language, built on the concepts of space-time "slip trajectories" and the "crystal scaffold" to describe the movement of (discrete) dislocations and to couple this movement to plastic flow. The energetics and dissipation relationships in our model are derived from first principles drawing on the theories of crystal modeling, elasticity, and thermodynamics. The resulting force balances involve a new configurational stress tensor describing the forces acting against slip. In order to place our model into context, we further show that it recovers several laws that were known in special cases before, most notably the equation for the Peach-Koehler force (linearized configurational force) and the fact that the combination of all dislocations yields the curl of the plastic distortion field. Finally, we also include a brief discussion on how a number of other effects, such as hardening, softening, dislocation climb, and coarse-graining, could be incorporated into our model., Comment: 44 pages, 5 figures

Lattice defects in crystalline materials create long-range elastic fields which can be modelled on the atomistic scale using an infinite system of discrete nonlinear force balance equations. Starting with these equations, this work rigorously derives a novel far-field expansion of these fields: The expansion is computable and is expressed as a sum of continuum correctors and discrete multipole terms which decay with increasing algebraic rate as the order of the expansion increases. Truncating the expansion leaves a remainder describing the defect core structure, which is localised in the sense that it decays with an algebraic rate corresponding to the order at which the truncation occurred.

We show that Witt groups of spinor varieties (aka.\ maximal isotropic Grassmannians) can be presented by combinatorial objects called even shifted young diagram. Our method relies on the Blow-up setup of Balmer-Calm\`es, and we investigate the connecting homomorphism of the localization sequence via the projective bundle formula of Walter-Nenashev, the projection formula of Calm\`es-Hornbostel and the excess intersection formula of Fasel., Comment: An appendix by Heng Xie; Final version. Generalized to the case of general bundles with complete flags

Recent advances in deep reinforcement learning (RL) have led to considerable progress in many 2-player zero-sum games, such as Go, Poker and Starcraft. The purely adversarial nature of such games allows for conceptually simple and principled application of RL methods. However real-world settings are many-agent, and agent interactions are complex mixtures of common-interest and competitive aspects. We consider Diplomacy, a 7-player board game designed to accentuate dilemmas resulting from many-agent interactions. It also features a large combinatorial action space and simultaneous moves, which are challenging for RL algorithms. We propose a simple yet effective approximate best response operator, designed to handle large combinatorial action spaces and simultaneous moves. We also introduce a family of policy iteration methods that approximate fictitious play. With these methods, we successfully apply RL to Diplomacy: we show that our agents convincingly outperform the previous state-of-the-art, and game theoretic equilibrium analysis shows that the new process yields consistent improvements.

This paper focuses on the connections between four stochastic and deterministic models for the motion of straight screw dislocations. Starting from a description of screw dislocation motion as interacting random walks on a lattice, we prove explicit estimates of the distance between solutions of this model, an SDE system for the dislocation positions, and two deterministic mean-field models describing the dislocation density. The proof of these estimates uses a collection of various techniques in analysis and probability theory, including a novel approach to establish propagation-of-chaos on a spatially discrete model. The estimates are non-asymptotic and explicit in terms of four parameters: the lattice spacing, the number of dislocations, the dislocation core size, and the temperature. This work is a first step in exploring this parameter space with the ultimate aim to connect and quantify the relationships between the many different dislocation models present in the literature., Comment: Version 2, 63 pages

In this paper, we construct stable Bott--Samelson classes in the projective limit of the algebraic cobordism rings of full flag varieties, upon an initial choice of a reduced word in a given dimension. Each stable Bott--Samelson class is represented by a bounded formal power series modulo symmetric functions in positive degree. We make some explicit computations for those power series in the case of infinitesimal cohomology. We also obtain a formula of the restriction of Bott--Samelson classes to smaller flag varieties., Comment: 23 pages. Comments are welcome

We consider crack propagation in a crystalline material in terms of bifurcation analysis. We provide evidence that the stress intensity factor is a natural bifurcation parameter, and that the resulting bifurcation diagram is a periodic "snaking curve". We then prove qualitative properties of the equilibria and convergence rates of finite-cell approximations to the "exact" bifurcation diagram.

In this paper we express the class of the structure sheaves of the closures of Deligne--Lusztig varieties as explicit double Grothendieck polynomials in the first Chern classes of appropriate line bundles on the ambient flag variety. This is achieved by viewing such closures as degeneracy loci of morphisms of vector bundles., Comment: 8 pages, minor changes

We compute the equivariant cohomology of complex projective spaces associated to finite-dimensional representations of $C_2$, using ordinary cohomology graded on representations of the fundamental groupoid, with coefficients in the Burnside ring Mackey functor. This extension of the $RO(C_2)$-graded theory allows for the definition of Euler classes, which are used as generators of the cohomology of the projective spaces. As an application, we give an equivariant version of Bezout's theorem., Comment: 57 pages, this is a major update. The proof of the main result has been simplified. As an application an equivariant version of Bezout's theorem was added

The Mori-Zwanzig formalism is applied to derive an equation for the evolution of linear observables of the overdamped Langevin equation. To illustrate the resulting equation and its use in deriving approximate models, a particular benchmark example is studied both numerically and via a formal asymptotic expansion. The example considered demonstrates the important of memory effects in determining the correct temporal behaviour of such systems., Comment: 19 pages, 5 figures

We develop a model for an anti-plane crack defect posed on a square lattice under an interatomic pair-potential with nearest-neighbour interactions. In particular, we establish existence, local uniqueness and stability of solutions for small loading parameters and further prove qualitatively sharp far-field decay estimates. The latter requires establishing decay estimates for the corresponding lattice Green's function, which are of independent interest.

We study the double Grothendieck polynomials of Kirillov--Naruse for the symplectic and odd orthogonal Grassmannians. These functions are explicitly written as sums of Pfaffian and are identified with the stable limits of the fundamental classes of Schubert varieties in the torus equivariant connective K-theory of these isotropic Grassmannians. We also provide a combinatorial description of the ring formally spanned by double Grothendieck polynomials., Comment: This article was originally a part of our previous paper [arXiv:1504.02828]. It was separated from its published version and became an independent paper

We present a mathematical framework within which Discrete Dislocation Dynamics in three dimensions is well-posed. By considering smooth distributions of slip, we derive a regularised energy for curved dislocations, and rigorously derive the Peach-Koehler force on the dislocation network via an inner variation. We propose a dissipative evolution law which is cast as a generalised gradient flow, and using a discrete-in-time approximation scheme, existence and regularity results are obtained for the evolution, up until the first time at which an infinite density of dislocation lines forms., Comment: 30 pages, 3 figures

Significant efforts have been devoted in the last decade towards improving the predictivity of coarse-grained models in molecular dynamics simulations and providing a rigorous justification of their use, through a combination of theoretical studies and data-driven approaches. One of the most promising research effort is the (re-)discovery of the Mori-Zwanzig projection as a generic, yet systematic, theoretical tool for deriving coarse-grained models. Despite its clean mathematical formulation and generality, there are still many open questions about its applicability and assumptions. In this work, we propose a detailed derivation of a hybrid multi-scale system, generalising and further investigating the approach developed in [Espa\~{n}ol, P., EPL, 88, 40008 (2009)]. Issues such as the general co-existence of atoms (fully-resolved degrees of freedom) and beads (larger coarse-grained units), the role of the fine-to-coarse mapping chosen, and the approximation of effective potentials are discussed. The concept of an approximate projection is introduced along with a discussion of its use as measure of the error committed with the approximation of the true interactions among the beads. The theoretical discussion is supported by numerical simulations of a monodimensional non-linear periodic benchmark system with an open-source parallel Julia code, easily extensible to arbitrary potential models and fine-to-coarse mapping functions. The results presented highlight the importance of introducing, in the macroscopic model, a non-constant dissipative term, given by the Mori-Zwanzig approach, to correctly reproduce the reference fine-grained results without requiring \emph{ad-hoc} calibration of interaction potentials and thermostats., Comment: 29 pages, 9 figures

Motivated by rate-independent stress-strain hysteresis observed in filled rubber, this article considers a scalar viscoelastic model in which the constitutive law is random and varies on a lengthscale which is small relative to the overall size of the solid. Using stochastic two-scale convergence as introduced by Bourgeat, Mikelic and Wright, we obtain the homogenized limit of the evolution, and demonstrate that under certain hypotheses, the homogenized model exhibits hysteretic behaviour which persists under asymptotically slow loading. These results are illustrated by means of numerical simulations in a particular one-dimensional instance of the model., Comment: 54 pages, 4 figures

In this paper we provide generalisations of Pfaffian formulas for the degeneracy loci classes in the algebraic cobordism of symplectic/odd orthogonal Grassmann bundles., Comment: 19 pages

In this paper, the dynamics of a system of a finite number of screw dislocations is studied. Under the assumption of antiplane linear elasticity, the two-dimensional dynamics is determined by the renormalised energy. The interaction of one dislocation with the boundary and of two dislocations of opposite Burgers moduli are analysed in detail and estimates on the collision times are obtained. Some exactly solvable cases and numerical simulations show agreement with the estimates obtained., Comment: 25 pages, 4 figures

In this paper, we prove determinant formulas for the $K$-theory classes of the structure sheaves of degeneracy loci classes associated to vexillary permutations in type $A$. As a consequence we obtain determinant formulas for Lascoux-Sch\"utzenberger's double Grothendieck polynomials associated to vexillary permutations. Furthermore, we generalize the determinant formula to algebraic cobordism., Comment: 12 pages

This paper studies the boundary behaviour at mechanical equilibrium at the ends of a finite interval of a class of systems of interacting particles with monotone decreasing repulsive force. Our setting covers pile-ups of dislocations, dislocation dipoles and dislocation walls. The main challenge is to control the nonlocal nature of the pairwise particle interactions. Using matched asymptotic expansions for the particle positions and rigorous development of an appropriate energy via Gamma-convergence, we obtain the equilibrium equation solved by the boundary layer correction, associate an energy with an appropriate scaling to this correction, and provide decay rates into the bulk., Comment: 41 pages, 13 figures

In this paper, we prove a generalization of Kempf-Laksov formula for the degeneracy loci classes in even infinitesimal cohomology theories of the Grassmannian bundle and the Lagrangian Grassmannian bundle., Comment: 22 pages. arXiv admin note: text overlap with arXiv:1602.05704

In this paper, we study Segre classes in algebraic cobordism. We also prove a generalization of Kempf-Laksov formula for the degeneracy loci classes in the algebraic cobordism of the Grassmannian bundle., Comment: 14 pages. The exposition and some arguments have been modified