Jonathan R. Landers

Stochastic Redistribution vs. LP Surrogates for Circuit Balancing

A work-related exploration of optimal energy grid allocation framed through the lens of stochastic optimization. This investigation models service-point imbalances driven by lognormal load deviations and shows that, under exogenous dynamics, a seemingly discrete redistribution problem collapses to an exact convex program. The result bridges practical grid engineering with elegant stochastic theory, revealing when linear relaxations remain integral, and when real-world coupling demands MILP or MIQP extensions.

Forrelation, Bent Functions, and a Spectral Criterion for Hardness

This note unifies the random-oracle and bent-function regimes of the Forrelation problem into a continuous spectral framework. By introducing a quantitative criterion based on Fourier flatness and resilience, it defines a smooth spectrum of classical hardness and quantum advantage. The discussion bridges Aaronson–Ambainis and Girish–Servedio, extending Forrelation theory through a geometric and information-theoretic lens.

Decomposing Time Series into Marginal and Dependence Components

This note presents a compact factorization of a univariate time series into a marginal law of values and a dependence process of ranks, formalizing the informational split between “what occurs” and “how it is arranged in time.” Using Sklar’s theorem, it connects copula decomposition to information-theoretic and PAC-learnability frameworks, showing how modular learning of marginals and dependence composes into learning the full process.

Opposite-Skewness Symmetry & TV Bounds

A short, self-contained note showing how two scaled Beta distributions can share the same support, mean, and median while exhibiting opposite skewness via centered reflection. It gives a clean total-variation expression in terms of the density’s overlap with its mirror and a small-imbalance approximation that links TV directly to parameter differences. In ticket-pricing applications, this symmetry reflects opposite-skew pricing distributions for the same event snapshot, highlighting why full-shape geometry (not just mean/variance) matters for model robustness and divergence-based evaluation.

For related work and broader context, see my arXiv paper .

Neural Net Dropout Viewed as Probability-Mass Dilution

This short note extends the implicit regularization mechanism proved in my Random Forests work to neural networks. By viewing dropout as random thinning of active units, it derives a concise odds-compression inequality showing how dropout reduces dominance of high-scoring units. The note directly connects to Section 5 of my arXiv paper .

On the Geometry of Fractal Boundaries in ReLU Networks

This paper develops a constructive and theoretical account of fractal geometry in neural network decision boundaries, introducing explicit ReLU modules that realize tent-map and Cantor-style refinements with provable dimensions. It proves that exact self-similarity arises only under measure-zero weight settings, while empirical probes reveal finite-range fractal mimicry and propose boundary fractal dimension as a diagnostic of overfitting.

Quantum-Accelerated Stabilization for Markov Chains

This note introduces a band–window stabilization criterion for Markov-chain averaging and demonstrates a quadratic quantum speedup via amplitude estimation. It contrasts classical and quantum complexities, presenting both immediate and anytime stabilization theorems with clear operational interpretations.

Limits of Hawking-Induced Magnetism

This series of short notes explores the boundary between quantum evaporation effects and classical plasma dynamics around Kerr black holes. Each note tackles a different facet of why Hawking-induced magnetic fields remain negligible and structurally undetectable compared to accretion-disk–driven fields.

Magnetic Fields from Hawking Radiation vs. Accretion Disk Dynamics – Scaling arguments show Hawking-radiated charges cannot compete with disk dynamo fields across astrophysical regimes.
Unphysical Magnetic Field Parity – Proves impossibility in two directions: capped growth forbids parity in forward models, and inverse decompositions cannot recover Hawking modes.
Non-Identifiability of Subcomponents – A general theorem on information loss in aggregates, applied to Hawking fields, showing zero channel capacity for detection.

Misspecification, Quantile Mobility, and Arc Length

This note unifies three threads: distributional misspecification bounds for a lognormal truth fit by a moment-matched normal; quantile mobility difficulty under additive vs. multiplicative geometries; and a geometric relation between quantile shifts and the arc length of the PDF, clarifying why mobility differs across shapes.

Magnetic Reactor Clock

A minimalist physics clock driven by an electron orbiting a loop. The simulated current and Biot–Savart field evolve with real time, with a live readout and cumulative |B|. Blends clean WebGL shaders with an elegant EM formulation for a memorable, functional demo.

Mark Twain fMRI

A playful reading visualization: as a Twain passage scrolls and words highlight, simulated brain regions (frontal, visual, auditory, motor, emotional, spatial) light up in a 3D point cloud. Technically polished and human-centered, showcasing lateral thinking and real-time rendering.

Research Notes