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Patrick Erik Bradley

Publications and source records attributed to Patrick Erik Bradley.

At least 37 records · Page 2Linked to original sources
arXiv2024

Heat Equations and Hearing the Genus on p-adic Mumford Curves via Automorphic Forms

A self-adjoint operator is constructed on the $L_2$-functions on the $K$-rational points $X(K)$ of a Mumford curve $X$ defined over a non-archimedean local field $K$. It generates a Feller semi-group, and the corresponding heat equation describes a Markov process on $X(K)$. Its spectrum is non-positive, contains zero and has finitely many limit points which are the only non-eigenvalues, and correspond to the zeros of a given regular differential 1-form on $X(K)$. This allows to recover the genus of X from the spectrum. The hyperelliptic case allows in principle an explicit genus extraction.

math.NT↗
arXiv2023

Hearing Shapes via p-Adic Laplacians

For a finite graph, a spectral curve is constructed as the zero set of a two-variate polynomial with integer coefficients coming from p-adic diffusion on the graph. It is shown that certain spectral curves can distinguish non-isomorphic pairs of isospectral graphs, and can even reconstruct the graph. This allows the graph reconstruction from the spectrum of the associated p-adic Laplacian operator. As an application to p-adic geometry, it is shown that the reduction graph of a Mumford curve and the product reduction graph of a p-adic analytic torus can be recovered from the spectrum of such operators.

math.SP↗
arXiv2022

Heat equations and wavelets on Mumford curves

A general class of heat operators over non-archimedean local fields acting on $L_2$-function spaces on affinoid domains in the local field are developed. $L_2$-spaces and integral operators invariant under the action of a non-archimedean Schottky group are constructed in order to have nice function spaces and operators on Mumford curves. General wavelets are constructed which, together with functions coming from certain graph Laplacian eigenvectors, diagonalise these operators. The corresponding Cauchy problems for the heat equations with these operators are solved in the affirmative, and properties of wavelet eigenvalues and the spectral gaps of the operators on Mumford curves are studied.

math.AG↗
arXiv2019

Generalised diffusion on moduli spaces of $p$-adic Mumford curves

A construction of a pseudo-differential operator on non-archimedean local fields invariant under a finite group action is given together with the solution of the corresponding Cauchy problem. This construction is applied to parts of the Gerritzen-Herrlich Teichmüller space in order to obtain a self-adjoint operator whose spectrum can decide about certain properties of the reduction graph of the corresponding Mumford curves.

math.AG↗
arXiv2019

A scaled space-filling curve index applied to tropical rain forest tree distributions

In order to be able to process the increasing amount of spatial data, efficient methods for their handling need to be developed. One major challenge for big spatial data is access. This can be achieved through space-filling curves, as they have the property that nearby points on the curve are also nearby in space. They are able to handle higher dimensional data, too. Higher dimensional data is widely used e.g. in CityGML and is becoming more and more important. In a laboratory experiment on a tropical rain forest tree data set of 2.5 million points taken from an 18-dimensional space, it is demonstrated that the recently constructed scaled Gray-Hilbert curve index performs better than its standard static version, saving a significant amount of space for a projection of the data set onto 8 attributes. The implementation is based on a binary tree in a data-driven process, in a similar way as e.g. the R-tree. Its scalability allows the handling of different kinds of data distributions which are reflected in the tree structure of the index. The relative efficiency of the scaled Gray-Hilbert curve in comparison with the best static version is seen to depend on the distribution of the point cloud. A local sparsity measure derived from properties of the corresponding trees can distinguish point clouds with different tail distributions. The different resulting binary trees are visualised to illustrate the influences of the different tail distributions they have been built on.

cs.DS↗
arXiv2019

p-Adic scaled space filling curve indices for high dimensional data

Space filling curves are widely used in Computer Science. In particular Hilbert curves and their generalisations to higher dimension are used as an indexing method because of their nice locality properties. This article generalises this concept to the systematic construction of p-adic versions of Hilbert curves based on affine transformations of the p-adic Gray code, and develops an efficient scaled indexing method for data taken from high-dimensional spaces based on these new curves, which with increasing dimension is shown to be less space consuming than the optimal standard static Hilbert curve index. A measure is derived which allows to assess the local sparsity of a data set, and is tested on some data.

cs.DS↗
arXiv2018

Homology of Finite Topological Spaces

A new method is given for computing generators of the homology groups with integer coefficients for any finite $T_0$-space. An important role in this method is played by irreducible cycles which are defined here and give rise to continuous injective maps between spaces called immersions. The so-called orb complex which is much smaller than the order complex induces a surjective map of its homology to the simplicial homology of the space. An analysis of the kernel of this map allows to define an effective algorithm for computing the homology groups, whose time complexity is polynomial in the size of the space.

math.AT↗
arXiv2018

Homology under monotone maps between finite topological spaces

It is shown that a surjective monotone map $X\to Y$ between finite $T_0$-spaces induces a surjective map on homology. As such a map turns out to be a sequence of edge contractions in the Hasse diagram of $X$, followed by a homeomorphism, this leads to an explicit relation between the Betti numbers of $X$ to those of $Y$ and the cokernels of the edge contraction maps on the order complexes.

math.AT↗
arXiv2013

Dimension of Alexandrov Topologies

We prove that the Krull dimension of an Alexandrov space of finite height can be characterised with the specialisation preorder of its associated $T_0$-space.

math.GN↗
arXiv2013

Integrating Space, Time, Version and Scale Using Alexandrov Topologies

This article introduces a novel approach to spatial database design. Instead of extending the canonical Solid-Face-Edge-Vertex schema by, say, "hypersolids" these classes are generalised to a common type SpatialEntity, and the individual BoundedBy relations between two consecutive classes are generalised to one BoundedBy relation on SpatialEntity instances. Then the pair (SpatialEntity, BoundedBy) is a so-called incidence graph. The novelty about this approach uses the observation that an incidence graph represents a topological space of SpatialEntity instances because the BoundedBy-relation defines a so-called Alexandrov topology for them turning them into a topological space. So spatial data becomes part of mathematical topology and topology can be immediately applied to spatial data. For example, continuous functions between two instances of spatial data allow the consistent modelling of generalisation. Further, it is also possible to establish a formal topological definition of spatial data dimension, and every topological data model of arbitrary dimension gets a simple uniform data model. This model covers space-time, and the version history of a spatial model can be represented by an Alexandrov topology, too. By integrating space, time, version, and scale into one single schema, topological queries across those aspects are enabled through topological constructions. In fact, the topological constructions cover a relationally complete query language for spaces and can be redefined to operate accordingly on their graph representations. With these observations a relational database schema for a spatial data model of dimension 6 and more is developed. The schema seamlessly integrates 4D space-time, levels of detail and version history, and it can be easily expanded to also contain non-spatial information or be linked to other data sources.

cs.DB↗
arXiv2009

A $p$-adic RanSaC algorithm for stereo vision using Hensel lifting

A $p$-adic variation of the Ran(dom) Sa(mple) C(onsensus) method for solving the relative pose problem in stereo vision is developped. From two 2-adically encoded images a random sample of five pairs of corresponding points is taken, and the equations for the essential matrix are solved by lifting solutions modulo 2 to the 2-adic integers. A recently devised $p$-adic hierarchical classification algorithm imitating the known LBG quantisation method classifies the solutions for all the samples after having determined the number of clusters using the known intra-inter validity of clusterings. In the successful case, a cluster ranking will determine the cluster containing a 2-adic approximation to the "true" solution of the problem.

cs.CV↗
arXiv2009

A dyadic solution of relative pose problems

A hierarchical interval subdivision is shown to lead to a $p$-adic encoding of image data. This allows in the case of the relative pose problem in computer vision and photogrammetry to derive equations having 2-adic numbers as coefficients, and to use Hensel's lifting method to their solution. This method is applied to the linear and non-linear equations coming from eight, seven or five point correspondences. An inherent property of the method is its robustness.

cs.CV↗
arXiv2009

On $p$-adic Classification

A $p$-adic modification of the split-LBG classification method is presented in which first clusterings and then cluster centers are computed which locally minimise an energy function. The outcome for a fixed dataset is independent of the prime number $p$ with finitely many exceptions. The methods are applied to the construction of $p$-adic classifiers in the context of learning.

cs.LG↗
arXiv2008

Counting plane Mumford curves

A $p$-adic version of Gromov-Witten invariants for counting plane curves of genus $g$ and degree $d$ through a given number of points is discussed. The multiloop version of $p$-adic string theory considered by Chekhov and others motivates us to ask how many of these curves are Mumford curves, i.e.\ uniformisable by a domain at the boundary of the Bruhat-Tits tree for $\PGL_2(\mathbb{Q}_p)$. Generally, the number of Mumford curves depends on the position of the given points in $\mathbb{P}^2$. With the help of tropical geometry we find configurations of points through which all curves of given degree and genus are Mumford curves. The article is preceded by an introduction to some concepts of $p$-adic geometry and their relation to string theory.

math-ph↗
arXiv2008

Mumford dendrograms and discrete p-adic symmetries

In this article, we present an effective encoding of dendrograms by embedding them into the Bruhat-Tits trees associated to $p$-adic number fields. As an application, we show how strings over a finite alphabet can be encoded in cyclotomic extensions of $\mathbb{Q}_p$ and discuss $p$-adic DNA encoding. The application leads to fast $p$-adic agglomerative hierarchic algorithms similar to the ones recently used e.g. by A. Khrennikov and others. From the viewpoint of $p$-adic geometry, to encode a dendrogram $X$ in a $p$-adic field $K$ means to fix a set $S$ of $K$-rational punctures on the $p$-adic projective line $\mathbb{P}^1$. To $\mathbb{P}^1\setminus S$ is associated in a natural way a subtree inside the Bruhat-Tits tree which recovers $X$, a method first used by F. Kato in 1999 in the classification of discrete subgroups of $\textrm{PGL}_2(K)$. Next, we show how the $p$-adic moduli space $\mathfrak{M}_{0,n}$ of $\mathbb{P}^1$ with $n$ punctures can be applied to the study of time series of dendrograms and those symmetries arising from hyperbolic actions on $\mathbb{P}^1$. In this way, we can associate to certain classes of dynamical systems a Mumford curve, i.e. a $p$-adic algebraic curve with totally degenerate reduction modulo $p$. Finally, we indicate some of our results in the study of general discrete actions on $\mathbb{P}^1$, and their relation to $p$-adic Hurwitz spaces.

cs.DM↗
arXiv2008

$p$-adic Hurwitz numbers

We introduce stable tropical curves and use these to count covers of the $p$-adic projective line of fixed degree and ramification types by Mumford curves in terms of tropical Hurwitz numbers. Our counts depend on the branch loci of the covers.

math.AG↗
arXiv2007

Families of dendrograms

A conceptual framework for cluster analysis from the viewpoint of p-adic geometry is introduced by describing the space of all dendrograms for n datapoints and relating it to the moduli space of p-adic Riemannian spheres with punctures using a method recently applied by Murtagh (2004b). This method embeds a dendrogram as a subtree into the Bruhat-Tits tree associated to the p-adic numbers, and goes back to Cornelissen et al. (2001) in p-adic geometry. After explaining the definitions, the concept of classifiers is discussed in the context of moduli spaces, and upper bounds for the number of hidden vertices in dendrograms are given.

stat.ML↗