arXiv subjects

John Stogin

Publications and source records attributed to John Stogin.

3 recordsLinked to original sources

arXiv2026

Nonlinear Wave Dynamics in Black Hole Spacetimes

This manuscript is a lightly reformatted version of my 2017 PhD thesis. I am posting it on arXiv at the request of my advisor, Sergiu Klainerman, who noted that it has been useful to some students. The content largely reflects the thesis in its original form. This thesis details a method for proving global boundedness and decay results for nonlinear wave equations on black hole spacetimes. The method is applied to five example problems of increasing difficulty. The first problem, which addresses the semilinear wave equation on Minkowski space, is quite simple and should be accessible to a reader who is still new to the field of partial differential equations. The final problem, which was posed by Ionescu and Klainerman in [IK14], constitutes a step toward proving stability for slowly rotating Kerr black holes. The remaining intermediate problems are: a semilinear wave equation on the Schwarzschild spacetime, a semilinear wave equation on any subextremal Kerr spacetime with the additional assumption of axisymmetry, and a restriction of the final problem to the Schwarzschild case. The method used in this thesis is based on a few particular developments that may be useful for other related problems. These include: a new method for constructing Morawetz-type estimates that is fairly robust (insofar as it may be successfully applied to all five problems), a strategy based on a decay hierarchy for energy estimates on uniformly spacelike hypersurfaces using, in particular, a notion of weak decay, and a technique for handling certain terms with factors that are singular on an axis of symmetry.

math.AP↗

arXiv2022

A provably stable neural network Turing Machine

We introduce a neural stack architecture, including a differentiable parametrized stack operator that approximates stack push and pop operations for suitable choices of parameters that explicitly represents a stack. We prove the stability of this stack architecture: after arbitrarily many stack operations, the state of the neural stack still closely resembles the state of the discrete stack. Using the neural stack with a recurrent neural network, we introduce a neural network Pushdown Automaton (nnPDA) and prove that nnPDA with finite/bounded neurons and time can simulate any PDA. Furthermore, we extend our construction and propose new architecture neural state Turing Machine (nnTM). We prove that differentiable nnTM with bounded neurons can simulate Turing Machine (TM) in real-time. Just like the neural stack, these architectures are also stable. Finally, we extend our construction to show that differentiable nnTM is equivalent to Universal Turing Machine (UTM) and can simulate any TM with only \textbf{seven finite/bounded precision} neurons. This work provides a new theoretical bound for the computational capability of bounded precision RNNs augmented with memory.

cs.LG↗

arXiv2016

Global Stability of the Nontrivial Solutions to the Wave Map Problem from Kerr $|a|\ll M$ to the Hyperbolic Plane under Axisymmetric Perturbations Preserving Angular Momentum

This document proves global boundedness and decay for axisymmetric perturbations of a known solution to the wave map problem from a slowly rotating $|a|\ll M$ Kerr spacetime to the hyperbolic plane. This problem is motivated by the general axisymmetric stability of Kerr conjecture and was first posed by Ionescu and Klainerman in [IK14]. Two particular developments in this paper, the treatment of terms near the axis of symmetry and the use of a decay hierarchy for energy estimates on uniformly spacelike hypersurfaces, can be used for a variety of similar problems.

math.AP↗