1,350 research outputs found

    Probabilistic Programming Interfaces for Random Graphs::Markov Categories, Graphons, and Nominal Sets

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    We study semantic models of probabilistic programming languages over graphs, and establish a connection to graphons from graph theory and combinatorics. We show that every well-behaved equational theory for our graph probabilistic programming language corresponds to a graphon, and conversely, every graphon arises in this way.We provide three constructions for showing that every graphon arises from an equational theory. The first is an abstract construction, using Markov categories and monoidal indeterminates. The second and third are more concrete. The second is in terms of traditional measure theoretic probability, which covers 'black-and-white' graphons. The third is in terms of probability monads on the nominal sets of Gabbay and Pitts. Specifically, we use a variation of nominal sets induced by the theory of graphs, which covers Erdős-Rényi graphons. In this way, we build new models of graph probabilistic programming from graphons

    Optimal Transport with Defective Cost Functions with Applications to the Lens Refractor Problem

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    We define and discuss the properties of a class of cost functions on the sphere which we term defective cost functions. We discuss how to extend these definitions and some properties to cost functions defined on Euclidean space and on surfaces embedded in Euclidean space. Some important properties of defective cost functions are that they result in Optimal Transport mappings which map to points along geodesics, have a nonzero mixed Hessian term, among other important properties. We also compute the cost-sectional curvature for a broad class of cost functions using the notation built from defining defective cost functions and apply the formulas to a few known examples of cost functions. Finally, we discuss how we can construct a regularity theory for defective cost functions by satisfying the Ma-Trudinger-Wang (MTW) conditions on an appropriately defined domain. As we develop the regularity theory of defective cost functions, we discuss how the results apply to a particular instance of the far-field lens refractor problem and to cost functions that already fit into the preexisting regularity theory, but now by employing simple formulas derived in this paper.Comment: 32 pages, 9 figure

    Convergence of Dynamics on Inductive Systems of Banach Spaces

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    Many features of physical systems, both qualitative and quantitative, become sharply defined or tractable only in some limiting situation. Examples are phase transitions in the thermodynamic limit, the emergence of classical mechanics from quantum theory at large action, and continuum quantum field theory arising from renormalization group fixed points. It would seem that few methods can be useful in such diverse applications. However, we here present a flexible modeling tool for the limit of theories: soft inductive limits constituting a generalization of inductive limits of Banach spaces. In this context, general criteria for the convergence of dynamics will be formulated, and these criteria will be shown to apply in the situations mentioned and more.Comment: Comments welcom

    A fixed-point formula for Dirac operators on Lie groupoids

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    We study equivariant families of Dirac operators on the source fibers of a Lie groupoid with a closed space of units and equipped with an action of an auxiliary compact Lie group. We use the Getzler rescaling method to derive a fixed-point formula for the pairing of a trace with the K-theory class of such a family. For the pair groupoid of a closed manifold, our formula reduces to the standard fixed-point formula for the equivariant index of a Dirac operator. Further examples involve foliations and manifolds equipped with a normal crossing divisor.Comment: 50 page

    Uniform syndeticity in multiple recurrence

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    The main theorem of this paper establishes a uniform syndeticity result concerning the multiple recurrence of measure-preserving actions on probability spaces. For any integers d,l≥1d,l\geq 1 as well as any ε>0\varepsilon > 0, we demonstrate the existence of δ\delta and η\eta (dependent only on dd, ll, and ε\varepsilon) such that the following holds: Consider a solvable group Γ\Gamma of derived length ll, a probability space (X,μ)(X, \mu), and dd pairwise commuting measure-preserving Γ\Gamma-actions T1,…,TdT_1, \ldots, T_d on (X,μ)(X, \mu). Let EE be a measurable set in XX with μ(E)≥ε\mu(E) \geq \varepsilon. Then, we have: \begin{equation*} \underline{\mathtt{BD}}_\Gamma\left(\left\{\gamma\in\Gamma\colon \mu(T_1^{\gamma^{-1}}(E)\cap T_2^{\gamma^{-1}} \circ T^{\gamma^{-1}}_1(E)\cap \cdots \cap T^{\gamma^{-1}}_d\circ T^{\gamma^{-1}}_{d-1}\circ \ldots \circ T^{\gamma^{-1}}_1(E))\geq \delta \right\}\right)\geq \eta \end{equation*} Here, BD‾Γ(A)\underline{\mathtt{BD}}_{\Gamma}(A) represents the lower Banach density of a set A⊂ΓA\subset \Gamma. This result significantly generalizes and refines two earlier uniformity results by Furstenberg and Katznelson.Comment: [v3]: Strengthened the main uniformity result and improved presentatio

    T-Dualities and Courant Algebroid Relations

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    We develop a new approach to T-duality based on Courant algebroid relations which subsumes the usual T-duality as well as its various generalisations. Starting from a relational description for the reduction of exact Courant algebroids over foliated manifolds, we introduce a weakened notion of generalised isometries that captures the generalised geometry counterpart of Riemannian submersions when applied to transverse generalised metrics. This is used to construct T-dual backgrounds as generalised metrics on reduced Courant algebroids which are related by a generalised isometry. We prove an existence and uniqueness result for generalised isometric exact Courant algebroids coming from reductions. We demonstrate that our construction reproduces standard T-duality relations based on correspondence spaces. We also describe how it applies to generalised T-duality transformations of almost para-Hermitian manifolds.Comment: 68 page

    Geodesic bi-angles and Fourier coefficients of restrictions of eigenfunctions

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    This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace eigenfunctions ϕj\phi_j of a compact Riemannian manifold to a submanifold H⊂MH \subset M. We fix a number c∈(0,1)c \in (0,1) and study the asymptotics of the thin sums, Nϵ,Hc(λ):=∑j,λj≤λ∑k:∣μk−cλj∣<ϵ∣∫Hϕjψk‾dVH∣2 N^{c} _{\epsilon, H }(\lambda): = \sum_{j, \lambda_j \leq \lambda} \sum_{k: |\mu_k - c \lambda_j | < \epsilon} \left| \int_{H} \phi_j \overline{\psi_k}dV_H \right|^2 where {λj}\{\lambda_j\} are the eigenvalues of −ΔM,\sqrt{-\Delta}_M, and {(μk,ψk)}\{(\mu_k, \psi_k)\} are the eigenvalues, resp. eigenfunctions, of −ΔH\sqrt{-\Delta}_H. The inner sums represent the `jumps' of Nϵ,Hc(λ) N^{c} _{\epsilon, H }(\lambda) and reflect the geometry of geodesic c-bi-angles with one leg on HH and a second leg on MM with the same endpoints and compatible initial tangent vectors ξ∈SHcM,πHξ∈B∗H\xi \in S^c_H M, \pi_H \xi \in B^* H, where πHξ\pi_H \xi is the orthogonal projection of ξ\xi to HH. A c-bi-angle occurs when ∣πHξ∣∣ξ∣=c\frac{|\pi_H \xi|}{|\xi|} = c. Smoothed sums in μk\mu_k are also studied, and give sharp estimates on the jumps. The jumps themselves may jump as ϵ\epsilon varies, at certain values of ϵ\epsilon related to periodicities in the c-bi-angle geometry. Subspheres of spheres and certain subtori of tori illustrate these jumps. The results refine those of the previous article (arXiv:2011.11571) where the inner sums run over k:∣μkλj−c∣≤ϵk: | \frac{\mu_k}{\lambda_j} - c| \leq \epsilon and where geodesic bi-angles do not play a role.Comment: 51 pages. Referee's comments incorporated. To appear in Pure and Applied Analysi

    Rectilinear approximation and volume estimates for hereditary bodies via [0,1]-decorated containers

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    We use the hypergraph container theory of Balogh--Morris--Samotij and Saxton--Thomason to obtain general rectilinear approximations and volume estimates for sequences of bodies closed under certain families of projections. We give a number of applications of our results, including a multicolour generalisation of a theorem of Hatami, Janson and Szegedy on the entropy of graph limits. Finally, we raise a number of questions on geometric and analytic approaches to containers.Comment: 25 pages, author accepted manuscript, to appear in Journal of Graph Theor

    Smooth rigidity for very non-algebraic Anosov diffeomorphisms of codimension one

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    In this paper we introduce a new methodology for smooth rigidity of Anosov diffeomorphisms based on "matching functions." The main observation is that under certain bunching assumptions on the diffeomorphism the periodic cycle functionals can provide such matching functions. For example we consider a sufficiently small C^1 neighborhood of a linear hyperbolic automorphism of the 3-dimensional torus which has a pair of complex conjugate eigenvalues. Then we show that two very non-algebraic (an open and dense condition) Anosov diffeomorphisms from this neighborhood are smoothly conjugate if and only they have matching Jacobian periodic data. We also obtain a similar result for certain higher dimensional codimension one Anosov diffeomorphisms.Comment: Version3: exposition improvement and minor errors corrected thanks to the referee report. Version 2: included more results in dimension 3 and in higher dimensions. Version 1 could still be a valuable resource for those who would like to read the basic 3-dimensional cas
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