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Universal Successor Features ApproximatorsDec 18 2018The ability of a reinforcement learning (RL) agent to learn about many reward functions at the same time has many potential benefits, such as the decomposition of complex tasks into simpler ones, the exchange of information between tasks, and the reuse ... More

Transfer in Deep Reinforcement Learning Using Successor Features and Generalised Policy ImprovementJan 30 2019The ability to transfer skills across tasks has the potential to scale up reinforcement learning (RL) agents to environments currently out of reach. Recently, a framework based on two ideas, successor features (SFs) and generalised policy improvement ... More

Unit Tests for Stochastic OptimizationDec 20 2013Feb 25 2014Optimization by stochastic gradient descent is an important component of many large-scale machine learning algorithms. A wide variety of such optimization algorithms have been devised; however, it is unclear whether these algorithms are robust and widely ... More

Successor Features for Transfer in Reinforcement LearningJun 16 2016Transfer in reinforcement learning refers to the notion that generalization should occur not only within a task but also across tasks. Our focus is on transfer where the reward functions vary across tasks while the environment's dynamics remain the same. ... More

Prioritized Experience ReplayNov 18 2015Feb 25 2016Experience replay lets online reinforcement learning agents remember and reuse experiences from the past. In prior work, experience transitions were uniformly sampled from a replay memory. However, this approach simply replays transitions at the same ... More

Residual Policy LearningDec 15 2018Jan 03 2019We present Residual Policy Learning (RPL): a simple method for improving nondifferentiable policies using model-free deep reinforcement learning. RPL thrives in complex robotic manipulation tasks where good but imperfect controllers are available. In ... More

Dynamics of Twisted Alexander InvariantsJan 14 2008Apr 30 2009The Pontryagin dual of the twisted Alexander module for a d-component link and GL(N,Z) representation is an algebraic dynamical system with an elementary description in terms of colorings of a diagram. In the case of a knot, its associated topological ... More

Twisted Alexander polynomials detect the unknotApr 04 2006May 21 2009The group of a nontrivial knot admits a finite permutation representation such that the corresponding twisted Alexander polynomial is not a unit.

Graph complexity and Mahler measureJan 21 2017The (torsion) complexity of a finite edge-weighted graph is defined to be the order of the torsion subgroup of the abelian group presented by its Laplacian matrix. When G is d-periodic (i.e., G has a free action of the rank-d free abelian group by graph ... More

Few-Shot Bayesian Imitation Learning with Logic over ProgramsApr 12 2019We describe an expressive class of policies that can be efficiently learned from a few demonstrations. Policies are represented as logical combinations of programs drawn from a small domain-specific language (DSL). We define a prior over policies with ... More

Successor Features for Transfer in Reinforcement LearningJun 16 2016Apr 12 2018Transfer in reinforcement learning refers to the notion that generalization should occur not only within a task but also across tasks. We propose a transfer framework for the scenario where the reward function changes between tasks but the environment's ... More

Learning Continuous Control Policies by Stochastic Value GradientsOct 30 2015We present a unified framework for learning continuous control policies using backpropagation. It supports stochastic control by treating stochasticity in the Bellman equation as a deterministic function of exogenous noise. The product is a spectrum of ... More

FeUdal Networks for Hierarchical Reinforcement LearningMar 03 2017Mar 06 2017We introduce FeUdal Networks (FuNs): a novel architecture for hierarchical reinforcement learning. Our approach is inspired by the feudal reinforcement learning proposal of Dayan and Hinton, and gains power and efficacy by decoupling end-to-end learning ... More

Reinforcement Learning with Unsupervised Auxiliary TasksNov 16 2016Deep reinforcement learning agents have achieved state-of-the-art results by directly maximising cumulative reward. However, environments contain a much wider variety of possible training signals. In this paper, we introduce an agent that also maximises ... More

A Bayesian Approach to Constraint Based Causal InferenceOct 16 2012We target the problem of accuracy and robustness in causal inference from finite data sets. Some state-of-the-art algorithms produce clear output complete with solid theoretical guarantees but are susceptible to propagating erroneous decisions, while ... More

A Logical Characterization of Constraint-Based Causal DiscoveryFeb 14 2012We present a novel approach to constraint-based causal discovery, that takes the form of straightforward logical inference, applied to a list of simple, logical statements about causal relations that are derived directly from observed (in)dependencies. ... More

Continuous control with deep reinforcement learningSep 09 2015Feb 29 2016We adapt the ideas underlying the success of Deep Q-Learning to the continuous action domain. We present an actor-critic, model-free algorithm based on the deterministic policy gradient that can operate over continuous action spaces. Using the same learning ... More

Twisted magnetic structures emerging from buoyancy instabilitiesOct 30 2012We here report calculations of magnetic buoyancy instabilities of a sheared magnetic layer where two separate zones are unstable. The idea is to study the possible generation of large-scale helical structures which could then rise through a stellar convection ... More

Three dimensions of knot coloringJan 23 2013May 12 2016This survey article discusses three aspects of knot colorings. Fox colorings are assignments of labels to arcs, Dehn colorings are assignments of labels to regions, and Alexander-Briggs colorings assign labels to vertices. The labels are found among the ... More

On reciprocality of twisted Alexander invariantsMay 15 2009Given a knot and an SL(n,C) representation of its group that is conjugate to its dual, the representation that replaces each matrix with its inverse-transpose, the associated twisted Reidemeister torsion is reciprocal. An example is given of a knot group ... More

Euclidean Mahler measure and twisted linksDec 28 2004Mar 10 2009If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Similarly, ... More

The Euler characteristic of a category as the sum of a divergent seriesJul 05 2007The Euler characteristic of a cell complex is often thought of as the alternating sum of the number of cells of each dimension. When the complex is infinite, the sum diverges. Nevertheless, it can sometimes be evaluated; in particular, this is possible ... More

Perfect numbers and groupsApr 01 2001A number is perfect if it is the sum of its proper divisors; here we call a finite group `perfect' if its order is the sum of the orders of its proper normal subgroups. (This conflicts with standard terminology but confusion should not arise.) The notion ... More

Up-to-Homotopy MonoidsDec 10 1999Feb 22 2000Informally, a homotopy monoid is a monoid-like structure in which properties such as associativity only hold `up to homotopy' in some consistent way. This short paper comprises a rigorous definition of homotopy monoid and a brief analysis of some examples. ... More

A general theory of self-similarityOct 21 2010Nov 09 2010A little-known and highly economical characterization of the real interval [0, 1], essentially due to Freyd, states that the interval is homeomorphic to two copies of itself glued end to end, and, in a precise sense, is universal as such. Other familiar ... More

Higher Operads, Higher CategoriesMay 02 2003Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. It draws its inspiration from areas as diverse as topology, quantum algebra, mathematical physics, logic, and theoretical ... More

Structures in higher-dimensional category theorySep 04 2001This paper, written in 1998, aims to clarify various higher categorical structures, mostly through the theory of generalized operads and multicategories. Chapters I and II, which cover this theory and its application to give a definition of weak n-category, ... More

The triangle-free processJun 26 2008Consider the following stochastic graph process. We begin with the empty graph on n vertices and add edges one at a time, where each edge is chosen uniformly at random from the collection of potential edges that do not form triangles when added to the ... More

TASI Lectures on Holographic Space-Time, SUSY and Gravitational Effective Field TheoryJul 22 2010Sep 23 2010I argue that the conventional field theoretic notion of vacuum state is not valid in quantum gravity. The arguments use gravitational effective field theory, as well as results from string theory, particularly the AdS/CFT correspondence. Different solutions ... More

M Theory and CosmologyNov 10 1999This is a series of lectures on M Theory for cosmologists. After summarizing some of the main properties of M Theory and its dualities I show how it can be used to address various fundamental and phenomenological issues in cosmology.

Quantum Hair on D-branes and Black Hole Information in String TheoryJun 05 1996Jun 07 1996We introduce a notion of quantum hair which completely characterizes the state of a D-brane in perturbative string theory. The hair manifests itself as a phase (more generally a unitary matrix in subspaces of degenerate string eigenstates) in the scattering ... More

Majority Rule at Low Temperatures on the Square and Triangular LatticesMay 16 1996We consider the majority rule renormalization group transformation applied to nearest neighbor Ising models. For the square lattice with 2 by 2 blocks we prove that if the temperature is sufficiently low, then the transformation is not defined. We use ... More

Conformal Invariance and Stochastic Loewner Evolution Predictions for the 2D Self-Avoiding Walk - Monte Carlo TestsJul 25 2002Feb 05 2003Simulations of the self-avoiding walk (SAW) are performed in a half-plane and a cut-plane (the complex plane with the positive real axis removed) using the pivot algorithm. We test the conjecture of Lawler, Schramm and Werner that the scaling limit of ... More

On the Bogolyubov-Ruzsa lemmaOct 30 2010Dec 03 2012Our main result is that if A is a finite subset of an abelian group with |A+A| < K|A|, then 2A-2A contains an O(log^{O(1)} K)-dimensional coset progression M of size at least exp(-O(log^{O(1)} K))|A|.

Roth's theorem in Z_4^nJul 31 2008Apr 01 2010We show that if A is a subset of Z_4^n containing no three-term arithmetic progression in which all the elements are distinct then |A|=o(4^n/n).

A Freiman-type theorem for locally compact abelian groupsOct 12 2007Apr 01 2010We prove a Freiman-type theorem for locally compact abelian groups. If A is a subset of a locally compact abelian group with Haar measure m and m(nA) < n^dm(A) for all n>d log d then we describe A in a way which is tight up to logarithmic factors in d. ... More

A note on Freiman's theorem in vector spacesMay 18 2006Apr 01 2010We show that if A is a subset of F_2^n and |A+A| < K|A| then A is contained in a subspace of size at most 2^{O(K^{3/2}log K)}|A|. This improves on the previous best of 2^{O(K^2)}.

The Littlewood-Gowers problemMay 18 2006Apr 01 2010We show that if A is a subset of Z/pZ (p a prime) of density bounded away from 0 and 1 then the A(Z/pZ)-norm (that is the l^1-norm of the Fourier transform) of the characterstic function of A is bounded below by an absolute constant times (log p)^{1/2 ... More

Approximate (Abelian) groupsDec 03 2012ECM survey article discussing the structure of subsets of Abelian groups which behave `a bit like' cosets (of subgroups).

A quantitative version of the non-abelian idempotent theoremDec 02 2009Dec 10 2010Suppose that G is a finite group and A is a subset of G such that 1_A has algebra norm at most M. Then 1_A is a plus/minus sum of at most L cosets of subgroups of G, and L can be taken to be triply tower in O(M). This is a quantitative version of the ... More

Chowla's cosine problemJul 31 2008Oct 14 2010Suppose that G is a discrete abelian group and A is a finite symmetric subset of G. We show two main results. i) Either there is a set H of O(log^c|A|) subgroups of G with |A \triangle \bigcup H| = o(|A|), or there is a character X on G such that -wh{1_A}(X) ... More

Affine Grassmannians in A^1-algebraic topologyJan 25 2018Let k be a field. Denote by Spc(k)_* the unstable, pointed motivic homotopy category and by Omega_Gm: Spc(k)_* \to Spc(k)_* the Gm-loops functor. For a k-group G, denote by Gr_G the affine Grassmannian of G. If k is infinite and G is split reductive, ... More

Riemann-Hilbert problems from Donaldson-Thomas theoryNov 11 2016We study a class of Riemann-Hilbert problems arising naturally in Donaldson-Thomas theory. In certain special cases we show that these problems have unique solutions which can be written explicitly as products of gamma functions. We briefly explain connections ... More

Note on a Paper by Ooguri and VafaNov 28 2016Nov 30 2016In a recent paper, Ooguri and Vafa [arXiv:1610.04564] argued that a mild extension of the Weak Gravity conjecture\cite{weakgrav} led to the conclusion that the only models of quantum gravity in AdS space with "radius large compared to the string scale" ... More

Kähler structures on spaces of framed curvesJan 11 2017We consider the space $\mathcal{M}$ of Euclidean similarity classes of framed loops in $\mathbb{R}^3$. Framed loop space is shown to be an infinite-dimensional K\"{a}hler manifold by identifying it with a complex Grassmannian. We show that the space of ... More

Water alignment, dipolar interactions, and multiple proton occupancy during water-wire proton transportOct 29 2003A discrete multistate kinetic model for water-wire proton transport is constructed and analyzed using Monte-Carlo simulations. The model allows for each water molecule to be in one of three states: oxygen lone pairs pointing leftward, pointing rightward, ... More

Enhancement of charged macromolecule capture by nanopores in a salt gradientMay 08 2009Nanopores spanning synthetic membranes have been used as key components in proof-of-principle nanofluidic applications, particularly those involving manipulation of biomolecules or sequencing of DNA. The only practical way of manipulating charged macromolecules ... More

Kelvin-Helmholtz instabilities across periodic platesSep 14 2006We consider the linear stability of two inviscid fluids, in the presence of gravity, sheared past each other and separated by an flexible plate. Conditions for exponential growth of velocity perturbations are found as functions of the flexural rigidity ... More

Kinetics of nanopore transportJan 31 1998A nonlinear kinetic exclusion model is used to study osmosis and pressure driven flows through nearly single file pores such as antibiotic channels, aquaporins, zeolites and nanotubules. Two possible maxima in the steady state flux as a function of pore-solvent ... More

Filtered ends of infinite covers and groupsDec 04 2005Nov 17 2006Let f:A-->B be a covering map. We say A has e filtered ends with respect to f (or B) if for some filtration {K_n} of B by compact subsets, A - f^{-1}(K_n) "eventually" has e components. The main theorem states that if Y is a (suitable) free H-space, if ... More

Bootstrapping partition regularity of linear systemsApr 16 2019Suppose that $A$ is a $k \times d$ matrix of integers and write $\mathfrak{R}_A:\mathbb{N} \rightarrow \mathbb{N}\cup \{ \infty\}$ for the function taking $r$ to the largest $N$ such that there is an $r$-colouring $\mathcal{C}$ of $[N]$ with $\bigcup_{C ... More

The distribution of the number of parts of $m$-ary partitions modulo $m$Feb 29 2016We investigate the number of parts modulo $m$ of $m$-ary partitions of a positive integer $n$. We prove that the number of parts is equidistributed modulo $m$ on a special subset of $m$-ary partitions. As consequences, we explain when the number of parts ... More

Givental's Lagrangian Cone and S^1-Equivariant Gromov-Witten TheoryJul 31 2006Oct 25 2007In the approach to Gromov-Witten theory developed by Givental, genus-zero Gromov-Witten invariants of a manifold X are encoded by a Lagrangian cone in a certain infinite-dimensional symplectic vector space. We give a construction of this cone, in the ... More

Stability conditions and Kleinian singularitiesAug 15 2005Sep 23 2005We describe the spaces of stability conditions on certain triangulated categories associated to Dynkin diagrams. These categories can be defined either algebraically via module categories of preprojective algebras, or geometrically via coherent sheaves ... More

Supports of irreducible characters of p-groupsJul 22 2013If chi is an irreducible character of a finite group G then the support of chi is the subset of G on which chi does not vanish. In this note, we study the supports of characters of certain classes of p-groups (a p-group is a finite group of prime power ... More

Conditions for (No) Eternal InflationMay 13 2019May 25 2019We construct analytic and numerical solutions of the Fokker-Planck equation that arises in the context of stochastic inflation. We use these solutions to derive necessary conditions for eternal inflation on the higher derivatives of the scalar field potential ... More

Orders of elements and zeros and heights of characters in a finite groupApr 14 2006Let \chi be an irreducible character of the finite group G. If g is an element of G and \chi(g) is not zero, then we conjecture that the order of g divides |G|/\chi(1). The conjecture is a generalization of the classical fact that irreducible p-projective ... More

Bell's Experiment in Quantum Mechanics and Classical PhysicsAug 21 2013Both the quantum mechanical and classical Bells experiment are within the focus of this paper. The fact that one measures different probabilities in both experiments is traced back to the superposition of two orthogonal but nonentangled substates in the ... More

Pseudo-Orbit Tracing and Algebraic actions of countable amenable groupsJan 05 2017Feb 06 2017Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite ... More

Coset decision trees and the Fourier algebraMay 06 2018We show that if G is a finite group and f is a {0,1}-valued function on G with Fourier algebra norm at most M then f may be computed by a coset decision tree (that is a decision tree in which at each vertex we query membership of a given coset) having ... More

Analytic solution and stationary phase approximation for the Bayesian lasso and elastic netSep 25 2017Nov 28 2018The lasso and elastic net linear regression models impose a double-exponential prior distribution on the model parameters to achieve regression shrinkage and variable selection, allowing the inference of robust models from large data sets. However, there ... More

The Hammersley-Welsh bound for self-avoiding walk revisitedAug 30 2017Nov 22 2017The Hammersley-Welsh bound (1962) states that the number $c_n$ of length $n$ self-avoiding walks on $\mathbb{Z}^d$ satisfies \[ c_n \leq \exp \left[ O(n^{1/2}) \right] \mu_c^n, \] where $\mu_c=\mu_c(d)$ is the connective constant of $\mathbb{Z}^d$. While ... More

The Erdos-Moser sum-free set problemApr 10 2018May 20 2018We show that if A is a finite set of integers then it has a subset S of size \log^{1+c} |A| (c>0 absolute) such that s+s' is never in A when s and s' are distinct elements of S.

A Datamining Approach to the Short Title Catalogue Flanders: the Case of Early Modern Quiring PracticesJun 28 2017Nov 19 2018This paper contains a data mining approach to the Short Title Catalogue Flanders (http://www.stcv.be/), which aims to record all books printed in Flanders up to 1801 (24.850 editions, per 31/08/2018). More specifically, it aims to analyse the Early Modern ... More

Indistinguishability of collections of trees in the uniform spanning forestOct 15 2018We prove the following indistinguishability theorem for $k$-tuples of trees in the uniform spanning forest of $\mathbb{Z}^d$: Suppose that $\mathscr{A}$ is a property of a $k$-tuple of components that is stable under finite modifications of the forest. ... More

A formula for the Jacobian of a genus one curve of arbitrary degreeOct 14 2015We extend the formulae of classical invariant theory for the Jacobian of a genus one curve of degree $n \le 4$ to curves of arbitrary degree. To do this, we associate to each genus one normal curve of degree $n$, an $n \times n$ alternating matrix of ... More

The Riemann-Hilbert approach to obtain critical asymptotics for Hamiltonian perturbations of hyperbolic and elliptic systemsNov 15 2011The present paper gives an overview of the recent developments in the description of critical behavior for Hamiltonian perturbations of hyperbolic and elliptic systems of partial differential equations. It was conjectured that this behavior can be described ... More

Generalized solutions of nonlinear differential equations A nonstandard jets approachMay 02 2012Using the rudiments of pde jets theory in a nonstandard setting, we first deepen and extend previous nonstandard existence results for generalized solutions of linear differential equations and second extend the previous results for linear differential ... More

On some algebras associated to genus one curvesJul 26 2017Haile, Han and Kuo have studied certain non-commutative algebras associated to a binary quartic or ternary cubic form. We extend their construction to pairs of quadratic forms in four variables, and conjecture a further generalisation to genus one curves ... More

The Euler characteristic of a categoryOct 08 2006The Euler characteristic of a finite category is defined and shown to be compatible with Euler characteristics of other types of object, including orbifolds. A formula for the cardinality of the colimit of a diagram of sets is proved, generalizing the ... More

A general theory of self-similarity II: recognitionNov 15 2004This paper concerns the self-similarity of topological spaces, in the sense defined in math.DS/0411344. I show how to recognize self-similar spaces, or more precisely, universal solutions of self-similarity systems. Examples include the standard simplices ... More

Topology and Higher-Dimensional Category Theory: the Rough IdeaJun 27 2001Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. Although it can be treated purely as an algebraic subject, it is inherently topological in nature: the higher-dimensional ... More

General self-similarity: an overviewNov 15 2004Informal seminar notes explaining the ideas in math.DS/0411344 and math.DS/0411345.

Scattering diagrams, Hall algebras and stability conditionsMar 01 2016To any quiver with relations we associate a consistent scattering diagram taking values in the motivic Hall algebra of its category of representations. We show that the chamber structure of this scattering diagram coincides with the natural chamber structure ... More

Stability conditions on K3 surfacesJul 11 2003Nov 16 2006This paper contains a description of one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic K3 surface.

Stability conditions on triangulated categoriesDec 17 2002Feb 08 2006This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from M.R. Douglas's notion of $\Pi$-stability. From a mathematical point of ... More

The Globular Cluster Luminosity Function: New Progress in Understanding an Old Distance IndicatorApr 16 2003I review the Globular Cluster Luminosity Function (GCLF) with emphasis on recent observational data and theoretical progress. As is well known, the turn-over magnitude (TOM) is a good distance indicator for early-type galaxies within the limits set by ... More

Phase separation in the neutral Falicov-Kimball modelMay 30 1997The Falicov-Kimball model consists of spinless electrons and classical particles (ions) on a lattice. The electrons hop between nearest neighbor sites while the ions do not. We consider the model with equal numbers of ions and electrons and with a large ... More

Motivic and Real Etale Stable Homotopy TheoryAug 31 2016Let X be a Noetherian scheme of finite dimension and denote by rho the (additive inverse of the) morphism in SH(X) from S to Gm corresponding to the unit -1. Here SH(X) denotes the motivic stable homotopy category. We show that the category obtained by ... More

Numerical Evaluation of Two-Loop Integrals in FDRDec 15 2015Feb 05 2016We present a method to numerically evaluate infrared-finite one- and two-loop integrals within the Four-Dimensional Regularization/Renormalization approach, in which a small mass serves as regulator. Typical integrals exhibit a logarithmic dependence ... More

A new strain energy function for modelling ligaments and tendons whose fascicles have a helical arrangement of fibrilsAug 17 2015A new strain energy function for the hyperelastic modelling of ligaments and tendons whose fascicles have a helical arrangement of fibrils is derived. The stress-strain response of a single fascicle whose fibrils exhibit varying levels of crimp throughout ... More

Wired Cycle-Breaking Dynamics for Uniform Spanning ForestsApr 15 2015We prove that every component of the wired uniform spanning forest (WUSF) is one-ended almost surely in every transient reversible random graph, removing the bounded degree hypothesis required by earlier results. We deduce that every component of the ... More

Constraints on Axion Inflation from the Weak Gravity ConjectureMar 03 2015Sep 08 2015We derive constraints facing models of axion inflation based on decay constant alignment from a string-theoretic and quantum gravitational perspective. In particular, we investigate the prospects for alignment and `anti-alignment' of $C_4$ axion decay ... More

A Note on the Kullback-Leibler Divergence for the von Mises-Fisher distributionFeb 25 2015We present a derivation of the Kullback Leibler (KL)-Divergence (also known as Relative Entropy) for the von Mises Fisher (VMF) Distribution in $d$-dimensions.

Anonymized e-mail interviews with R package maintainers active on CRAN and GitHubJun 17 2016This technical report accompanies a research article that empirically studies the problems related to inter-repository package dependencies in the R ecosystem of statistical computing, with a focus on R packages hosted on CRAN and GitHub. The current ... More

Divisorial Extractions from Singular Curves in Smooth 3-Folds, IMar 29 2014Consider a singular curve $\Gamma$ contained in a smooth 3-fold $X$. Assuming the general elephant conjecture, the general hypersurface section $\Gamma\subset S\subset X$ is Du Val. Under that assumption, this paper describes the construction of a divisorial ... More

Hall algebras and Donaldson-Thomas invariantsNov 11 2016This is a survey article on Hall algebras and their applications to the study of motivic invariants of moduli spaces of coherent sheaves on Calabi-Yau threefolds. It is a write-up of my talks at the 2015 Salt Lake City AMS Summer Research Institute and ... More

Codensity and the Giry monadOct 16 2014The Giry monad on the category of measurable spaces sends a space to a space of all probability measures on it. There is also a finitely additive Giry monad in which probability measures are replaced by finitely additive probability measures. We give ... More

On certain other sets of integersJul 30 2010Jan 31 2011We show that if A is a subset of {1,...,N} containing no non-trivial three-term arithmetic progressions then |A|=O(N/ log^{3/4-o(1)} N).

On a non-abelian Balog-Szemeredi-type lemmaDec 02 2009Oct 14 2010We show that if G is a group and A is a finite subset of G with |A^2| < K|A|, then for all k there is a symmetric neighbourhood of the identity S with S^k a subset of A^2A^{-2} and |S| > exp(-K^{O(k)})|A|.

On a theorem of ShkredovJul 31 2008Jan 26 2011We show that if A is a finite subset of an abelian group with additive energy at least c|A|^3 then there is a subset L of A with |L|=O(c^{-1}\log |A|) such that |A \cap Span(L)| >> c^{1/3}|A|.

Bounds in the idempotent theoremOct 22 2016We show that if G is a finite Abelian group and f is an integer-valued map on G with algebra norm at most M then there is some L < \exp(M^{4+o(1)}), cosets of (possibly different) subgroups W_1,...,W_L, and s_1,...,s_L \in {-1,1} such that f=\sum_i{s_i1_{W_i}}. ... More

Learning to Learn Neural NetworksOct 19 2016Meta-learning consists in learning learning algorithms. We use a Long Short Term Memory (LSTM) based network to learn to compute on-line updates of the parameters of another neural network. These parameters are stored in the cell state of the LSTM. Our ... More

Radiative Balance between Photon Emission from Surface and in VolumeDec 02 2016An expression is derived which links the probability of photon reabsorption (recycling) in a material with other optical parameters. We discuss the application of this relation in several typical situations which are relevant in the operation of devices ... More

Implicit higher derivatives, and a formula of Comtet and FioletMay 17 2008Let F(x,y) be a function of two variables, and suppose y = f(x) satisfies F(x,y)=0 in some range. Then dy/dx = -Fx/Fy, where Fx and Fy denote the partial derivatives of F with respect to x and y. It is natural to seek a general expression for the higher ... More

Hyperbolic localization of intersection cohomologyFeb 24 2002Jul 29 2003For a normal variety X defined over an algebraically closed field with an action of the multiplicative group G_m, we consider the ``hyperbolic localization'' functor from D^b(X) to D^b(X^T), which localizes using closed supports in the directions flowing ... More

Expansions for Droplet States in the Ferromagnetic XXZ Heisenberg ChainOct 27 2003We consider the highly anisotropic ferromagnetic spin 1/2 Heisenberg chain with periodic boundary conditions. In each sector of constant total z component of the spin, we develop convergent expansions for the lowest band of eigenvalues and eigenfunctions. ... More

Power residues of Fourier coefficients of modular formsSep 30 2003Oct 09 2003Let r : G_Q -> GL_n Q_l be a motivic l-adic Galois representation. For fixed m > 1 we initiate an investigation of the density of the set of primes p such that the trace of the image of an arithmetic Frobenius at p under r is an m^th power residue modulo ... More

Conditions for (No) Eternal InflationMay 13 2019We construct analytic and numerical solutions of the Fokker-Planck equation that arises in the context of stochastic inflation. We use these solutions to derive necessary conditions for eternal inflation on the higher derivatives of the scalar field potential ... More

Integral Canonical Models for Automorphic Vector Bundles of Abelian TypeMay 09 2016Aug 03 2017We define and construct integral canonical models for automorphic vector bundles over Shimura varieties of abelian type. More precisely, we first build on Kisin's work to construct integral canonical models over rings of integers of number fields with ... More