1,350 research outputs found
Probabilistic Programming Interfaces for Random Graphs::Markov Categories, Graphons, and Nominal Sets
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
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The geometric evolution of aortic dissections: Predicting surgical success using fluctuations in integrated Gaussian curvature
Clinical imaging modalities are a mainstay of modern disease management, but the full utilization of imaging-based data remains elusive. Aortic disease is defined by anatomic scalars quantifying aortic size, even though aortic disease progression initiates complex shape changes. We present an imaging-based geometric descriptor, inspired by fundamental ideas from topology and soft-matter physics that captures dynamic shape evolution. The aorta is reduced to a two-dimensional mathematical surface in space whose geometry is fully characterized by the local principal curvatures. Disease causes deviation from the smooth bent cylindrical shape of normal aortas, leading to a family of highly heterogeneous surfaces of varying shapes and sizes. To deconvolute changes in shape from size, the shape is characterized using integrated Gaussian curvature or total curvature. The fluctuation in total curvature (δK) across aortic surfaces captures heterogeneous morphologic evolution by characterizing local shape changes. We discover that aortic morphology evolves with a power-law defined behavior with rapidly increasing δK forming the hallmark of aortic disease. Divergent δK is seen for highly diseased aortas indicative of impending topologic catastrophe or aortic rupture. We also show that aortic size (surface area or enclosed aortic volume) scales as a generalized cylinder for all shapes. Classification accuracy for predicting aortic disease state (normal, diseased with successful surgery, and diseased with failed surgical outcomes) is 92.8±1.7%. The analysis of δK can be applied on any three-dimensional geometric structure and thus may be extended to other clinical problems of characterizing disease through captured anatomic changes
Optimal Transport with Defective Cost Functions with Applications to the Lens Refractor Problem
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
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
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
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 as well as any , we
demonstrate the existence of and (dependent only on , ,
and ) such that the following holds:
Consider a solvable group of derived length , a probability space
, and pairwise commuting measure-preserving -actions
on . Let be a measurable set in with
. 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,
represents the lower Banach density of a
set . 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
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
This article concerns joint asymptotics of Fourier coefficients of
restrictions of Laplace eigenfunctions of a compact Riemannian
manifold to a submanifold . We fix a number and
study the asymptotics of the thin sums, where
are the eigenvalues of and are the
eigenvalues, resp. eigenfunctions, of . The inner sums
represent the `jumps' of and reflect the
geometry of geodesic c-bi-angles with one leg on and a second leg on
with the same endpoints and compatible initial tangent vectors , where is the orthogonal projection of
to . A c-bi-angle occurs when .
Smoothed sums in are also studied, and give sharp estimates on the
jumps. The jumps themselves may jump as varies, at certain values of
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
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
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
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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