arXiv subjects

Patrick Erik Bradley

Publications and source records attributed to Patrick Erik Bradley.

At least 19 recordsLinked to original sources

arXiv2026

Searching Schemes With $p$-Adic Neumann Boundary Value Problems

Firstly, for branched covering maps $f\colon Y\to X$ between smooth, separated schemes of finite type over the integral ring $O_K$ of a non-archimedean local field $K$, the ramification divisor is found to coincide with the divisor of the Radon-Nikodym derivative of the Radon measure associated with an algebraic differential form on $Y$ against the pullback measure of one on $X$, both taking Borel sets of the space $Y(O_K)$ $O_K$-rational points of $Y$ as input values. Secondly, a series of $p$-adic Neumann Boundary Value Problems, depending on algebraic and pluricanonical differential forms with poles on spaces $X(O_K)$ coming from schemes is formulated and solved, extending previous work of the author. Thirdly, these are then used to solve reconstruction problems on schemes: the divisor of a pluricanonical form with poles with at worst log-terminal singularities, as well as Weierstrass points of projective algebraic curves can be reconstructed via repeatedly finding weak solutions of $p$-adic Neumann Boundary Value Problems.

math.AG↗

arXiv2026

$p$-Adic Diffusion and Random Walks on $[0, 1]$

Integral operators on the real unit interval are constructed as transported from ones on the $p$-adic unit disc via the Monna map. This gives rise to strong Markov processes on $[0, 1]$, whose paths are right-continuous and have no discontinuities other than jumps. The spectra of the corresponding diffusion operators, whose kernel functions depend on a $p$-adic distance, are calculated. Further, the solution to the Cauchy problem for their heat equations are approximated via continuous- time random walks on finite sets coming from a hierarchical partition of the unit interval induced by the $p$-adic distance. The transport of $p$-adic diffusion to the real domain via the Monna map gives rise to a simple visualisation method. Illustrations of concrete examples are undertaken in the end.

math.PR↗

arXiv2026

Hearing Tamagawa Factors modulo $q - 1$

The Tamagawa factor w.r.t.\ the prime $p$ of an abelian variety $A$ over a non- archimedean local field $K$ is found to be congruent modulo $q - 1$ to the wavelet eigenvalues of a $p$-adic Laplacian integral operator on the $O_K$-rational points of its Néron model, where $q = p^f$ is the cardinality of the residue field of $K$. The method is to express the volume of the $K$-rational points of $A$ w.r.t.\ the canonical measure in terms of the local $L$-factor given by the Frobenius action on $\ell$-adic cohomolgy, and the Tamagawa factor; and then observe that this coincides with the Serre invariant of that compact $p$-adic analytic manifold modulo $q - 1$. A previous result by Á.M.\ Ledezma and the author on hearing Serre invariants then yields the asserted congruence.

math.NT↗

arXiv2026

Neumann Boundary Value Problems on $p$-Adic Analytic Manifolds

Inspired by graph theory, Neumann Boundary Value Problems on $p$-adic analytic manifold are formulated. Existence and uniqueness of weak solutions is proven in the ultracontractive case. For this task, a $p$-adic Gauss-Green identity and an extended eigenvalue formula for wavelets with small supports on the manifold are proven. As an application, the closeness to a zero of an analytic function on the manifold is shown to be detectable via a Neumann Boundary Value Problem.

math.AP↗

arXiv2026

Boundary Value Problems on $p$-Adic Analytic Manifolds

An account is given on newest developments on $p$-adic boundary value problems on $p$-adic analytic manifolds and their relationship with diffusion. In particular, novel coordinate Laplacians on $p$-adic analytic $n$-manifolds constructed with the help of frame bundles, are introduced in this context. These are used to construct elliptic operators. Related Dirichlet problems are formulated and solved, generalising results on compact subdomains of $p$-adic $n$-space. In the end, an outlook towards number-theoretic applications as well as extensions of this theory to ultrametric analytic manifolds is given. This is a substantial upgrade of the presentation given at Branko's 80-th Birthday Conference in Belgrade, May 2025.

math.NT↗

arXiv2025

Diffusion operators on $p$-adic analytic manifolds

Kernel functions for Laplacian integral operators are constructed on $p$-adic analytic manifolds using charts and transition maps from an atlas with connected nerve complex. In the compact case, an operator of Vladimirov-Taibleson type parametrised by a real parameter $s$ is defined. Its kernel function uses a geodetic-like distance function on the nerve complex of its atlas. The $L^2$-spectrum of this operator is established, and it is shown that it gives rise to a Feller semigroup. In this way, the Cauchy problem for the corresponding heat equation is solved in the positive by a transition function of a Markov process. The existence of a heat kernel function and a Green function in the case $s > 1$ is proven. As an application, it is shown how to express the number of points on the reduction curve defined over the residue field of an elliptic curve with good reduction in terms of the eigenvalues of a Vladimirov-Taibleson-like operator. This provides for an alternative way of counting points on elliptic curves defined over finite fields.

math.AP↗

arXiv2025

Hearing the Serre invariant of a compact $p$-adic analytic manifold

Using a previous novel way of defining kernel functions for Laplacian integral operators on a compact $p$-adic analytic manifold $X$, one such operator $Δ_0^s$ with $s\in\mathds{R}$ is applied to hearing the Serre invariant $i(X)$ by showing that a wavelet eigenvalue is always congruent to $i(X)$ modulo $q-1$, where $q$ is the cardinality of the residue field $k$ attached to a $p$-adic number field $K$. It is shown how the number of $k$-rational points of the special fibre of the Néron model of an elliptic curve defined over $K$ relates to the wavelet spectrum of $Δ^s_0$, and this then leads to the realisation that the Serre invariant $i(E(X))$ in the case of an elliptic curve $E$ with split multiplicative reduction vanishes modulo $q-1$.

math.NT↗

arXiv2025

Vladimirov-Pearson Operators on $ζ$-regular Ultrametric Cantor Sets

A new operator for certain types of ultrametric Cantor sets is constructed using the measure coming from the spectral triple associated with the Cantor set, as well as its zeta function. Under certain mild conditions on that measure, it is shown that it is an integral operator similar to the Vladimirov-Taibleson operator on the p-adic integers. Its spectral properties are studied, and the Markov property and kernel representation of the heat kernel generated by this so-called \emph{Vladimirov-Pearson} operator is shown, viewed as acting on a certain Sobolev space. A large class of these operators have a heat kernel and a Green function explicitly given by the ultrametric wavelets on the Cantor set, which are eigenfunctions of the operator.

math.AP↗

arXiv2025

Boundary Value Problems for p-Adic Elliptic Parisi-Zúñiga Diffusion

Elliptic integral-differential operators resembling the classical elliptic partial differential equations are defined over a compact d-dimensional p-adic domain together with associated Sobolev spaces relying on coordinate Vladimirov-type Laplacians dating back to an idea of Wilson Zúñiga-Galindo in his previous work. The associated Poisson equations under boundary conditions are solved and their $L_2$-spectra are determined. Under certain finiteness conditions, a Markov semigroup acting on the Sobolev spaces which are also Hilbert spaces can be associated with such an operator and the boundary condition. It is shown that this also has an explicitly given heat kernel as an $L_2$-function, which allows a Green function to be derived from it.

math.AP↗

arXiv2025

Predicting Air Temperature from Volumetric Urban Morphology with Machine Learning

In this study, we firstly introduce a method that converts CityGML data into voxels which works efficiently and fast in high resolution for large scale datasets such as cities but by sacrificing some building details to overcome the limitations of previous voxelization methodologies that have been computationally intensive and inefficient at transforming large-scale urban areas into voxel representations for high resolution. Those voxelized 3D city data from multiple cities and corresponding air temperature data are used to develop a machine learning model. Before the model training, Gaussian blurring is implemented on input data to consider spatial relationships, as a result the correlation rate between air temperature and volumetric building morphology is also increased after the Gaussian blurring. After the model training, the prediction results are not just evaluated with Mean Square Error (MSE) but some image similarity metrics such as Structural Similarity Index Measure (SSIM) and Learned Perceptual Image Patch Similarity (LPIPS) that are able to detect and consider spatial relations during the evaluation process. This trained model is capable of predicting the spatial distribution of air temperature by using building volume information of corresponding pixel as input. By doing so, this research aims to assist urban planners in incorporating environmental parameters into their planning strategies, thereby facilitating more sustainable and inhabitable urban environments.

cs.LG↗

arXiv2024

Local ultrametric approximation of graph distance based Laplacian diffusion

The error estimation for eigenvalues and eigenvectors of a small positive symmetric perturbation on the spectrum of a graph Laplacian is related to Gauß hypergeometric functions. Based on this, a heuristic polynomial-time algorithm for finding an optimal locally ultrametric approximation of a graph-distance power Laplacian matrix via the Vietoris-Rips graph based on the graph distance function is proposed. In the end, the error in the solution to the graph Laplacian heat equation given by extension to a locally p-adic equation is estimated.

math.AP↗

arXiv2024

Schottky invariant diffusion on the transcendent p-adic upper half plane

The transcendent part of the Drinfeld p-adic upper half plane is shown to be a Polish space. Using Radon measures associated with regular differential 1-forms invariant under Schottky groups allows to construct self-adjoint diffusion operators as Laplacian integral operators with kernel functions determined by the p-adic absolute value on the complex p-adic numbers. Their spectra are explicitly calculated and the corresponding Cauchy problems for their associated heat equations are found to be uniquely solvable and to determine Markov processes having paths which are cadlag. The heat kernels are shown to have explicitly given distribution functions, as well as boundary value problems associated with the heat equations under Dirchlet and von Neumann conditions are solved.

math.NT↗

arXiv2024

Schottky-Invariant $p$-Adic Diffusion Operators

A parametrised diffusion operator on the regular domain $Ω$ of a $p$-adic Schottky group is constructed. It is defined as an integral operator on the complex-valued functions on $Ω$ which are invariant under the Schottky group $Γ$, where integration is against the measure defined by an invariant regular differential 1-form $ω$. It is proven that the space of Schottky invariant $L^2$-functions on $Ω$ outside the zeros of $ω$ has an orthonormal basis consiting of $Γ$-invariant extensions of Kozyrev wavelets which are eigenfunctions of the operator. The eigenvalues are calculated, and it is shown that the heat equation for this operator provides a unique solution for its Cauchy problem with Schottky-invariant continuous initial conditions supportes outside the zero set of $ω$, and gives rise to a strong Markov process on the corresponding orbit space for the Schottky group whose paths are càdlàg.

math.AG↗

arXiv2024

Approximating Diffusion on Finite Multi-Topology Systems Using Ultrametrics

Motivated by multi-topology building and city model data, first a lossless representation of multiple $T_0$-topologies on a given finite set by a vertex-edge-weighted graph is given, and the subdominant ultrametric of the associated weighted graph distance matrix is proposed as an index structure for these data. This is applied in a heuristic parallel topological sort algorithm for edge-weighted directed acyclic graphs. Such structured data are of interest in simulation of processes like heat flows on building or city models on distributed processors. With this in view, the bulk of this article calculates the spectra of certain unbounded self-adjoint $p$-adic Laplacian operators on the $L^2$-spaces of a compact open subdomain of the $p$-adic number field associated with a finite graph $G$ with respect to the restricted Haar measure. as well as to a Radon measure coming from an ultrametric on the vertices of $G$ with the help of $p$-adic polynomial interpolation. In the end, error bounds are given for the solutions of the corresponding heat equations by finite approximations of such operators.

cs.DM↗

arXiv2024

Theta-Induced Diffusion on Tate Elliptic Curves over Non-Archimedean Local Fields

A diffusion operator on the $K$-rational points of a Tate elliptic curve $E_q$ is constructed, where $K$ is a non-archimedean local field, as well as an operator on the Berkovich-analytification $E_q^{an}$ of $E_q$. These are integral operators for measures coming from a regular $1$-form, and kernel functions constructed via theta functions. The second operator can be described via certain non-archimedan curvature forms on $E_q^{an}$. The spectra of these self-adjoint bounded operators on the Hilbert spaces of $L^2$-functions are identical and found to consist of finitely many eigenvalues. A study of the corresponding heat equations yields a positive answer to the Cauchy problem, and induced Markov processes on the curve. Finally, some geometric information about the $K$-rational points of $E_q$ is retrieved from the spectrum.

math.NT↗

arXiv2024

On the Local Ultrametricity of Finite Metric Data

New local ultrametricity measures for finite metric data are proposed through the viewpoint that their Vietoris-Rips corners are samples from p-adic Mumford curves endowed with a Radon measure coming from a regular differential 1-form. This is experimentally applied to the iris dataset.

cs.IR↗

arXiv2024

A non-autonomous p-Adic diffusion equation on time changing graphs

Motivated by the recently proven presence of ultrametricity in physical models (certain spin glasses) and the very recent study of Turing patterns on locally ultrametric state spaces, first non-autonomous diffusion operators on such spaces, where finitely many compact p-adic spaces are joined by a graph structure, are studied, including their Dirichlet and van Neumann eigenvalues. Secondly, the Cauchy problem for the heat equation associated with these operators is solved, its solution approximated by Trotter products, and thirdly, the corresponding Feller property as well as the Markov property (a Hunt process) is established.

math.AP↗

arXiv2024

Topological Applications of p-Adic Divergence and Gradient Operators

$p$-Adic divergence and gradient operators are constructed giving rise to $p$-adic vertex Laplacian operators used by Zúñiga in order to study Turing patterns on graphs, as well as their edge Laplacian counterparts. It is shown that the Euler characteristic of a finite graph can be expressed via traces of certain heat kernels associated with these new operators. This result is applied to the extraction of topological information from Mumford curves via heat kernels.

math.AP↗