arXiv subjects

Netzer Moriya

Publications and source records attributed to Netzer Moriya.

17 recordsLinked to original sources

arXiv2026

A Gaussian-Perron Prime-Side Defect and Local Profiles Near Critical-Line Zeros of the Riemann Zeta Function

We introduce a Gaussian--Perron prime-force defect that compares a smoothed prime-side logarithmic force with the logarithmic derivative of the Riemann zeta function. The construction turns the explicit formula into a local diagnostic for zero geometry. Its kernel produces an error-function prime weight and an anisotropic zero-side damping law, with an explicit boundary separating amplified and suppressed nonlocal zero contributions. We prove an exact zero-side formula, derive a universal selected-zero profile on the logarithmic scale, and formulate a finite-window damping certificate for non-selected residues. Under explicit damping, pole, and contour-regularity hypotheses, these ingredients localize the full defect near a selected zero. Assuming the Riemann Hypothesis and the stated pole-damping condition, the full defect near each fixed simple critical-line zero has the selected-zero profile up to an exponentially small nonlocal remainder. The framework provides a local diagnostic for zero geometry associated with the Riemann zeta function.

math.NT↗

arXiv2026

Photonic Chirality for Braiding and Readout of Non-Abelian Anyons

We propose a cavity-based scheme that uses photonic chirality to control braiding and read out non-Abelian anyons in a fractional quantum Hall platform. Counter-propagating cavity modes interfere with a classical reference tone to create a rotating pinning landscape whose direction is set by photon circulation, so that opposite photonic branches drive opposite anyon loops. This realizes a branch-conditioned braid operation and maps the resulting braid response onto cavity intermode coherence. We derive the rotating pinning term and the readout relation at the effective-theory level, identify an operating window set by subgap driving, adiabatic transport, localization, and cavity coherence, and provide phenomenological diagnostics of transport locking. In the minimal four-anyon Ising realization, the leading signal reduces to a calibrated phase; more generally, the same readout structure becomes state dependent when the relative braid operator is non-scalar. The scheme provides a cavity route to braid-sensitive readout of non-Abelian anyons without relying on fragile electronic interference fringes.

cond-mat.mes-hall↗

arXiv2025

Normalized Ensemble-Averaged OAM Spectrum in Disordered Statistical Q-Plates

This work presents an exact, analytical derivation of the ensemble-averaged orbital angular momentum (OAM) power spectrum for a circularly polarized Gaussian beam traversing a statistical Q-plate with Gaussian spatial disorder. Utilizing the Gaussian moment theorem, a new closed-form expression for the averaged mutual coherence is obtained. This coherence function is then rigorously projected onto OAM modes, yielding an exactly normalized series representation whose absolute convergence is formally proven. The analysis meticulously resolves limiting disorder regimes: for coarse disorder, an ideal OAM spectrum, sharply peaked at its nominal OAM mode, is demonstrated. Conversely, for fine disorder, a specific fraction of total power exponentially decays with disorder variance into the nominal OAM mode, with remaining power distributed among nearby OAM modes, exhibiting an effective width inversely proportional to the dimensionless correlation length. Crucially, a new universal scaling framework, defined by a master control parameter and a universal coordinate, is introduced. This framework, rigorously derived from the exact solution, unifies the spectrum's description across all relevant disorder regimes and enables robust data collapse. Monte Carlo simulations, implemented with the same normalization and OAM projection, substantiate these claims by corroborating the predicted scaling and universal behavior, offering a foundational, device-level perspective on OAM universality in complex media.

physics.optics↗

arXiv2025

Quantized Orbital Angular Momentum from Discrete Chaotic Phase Surfaces

We present a new theory for orbital angular momentum (OAM) generation by chaotic phase surfaces with discrete integer bias distributions. We derive fundamental selection rules that determine which OAM modes can be coherently generated. Our analysis shows that ensemble-averaged OAM exists only when the bias parameter takes integer values that match the discrete OAM eigenspace, creating "allowed" and "forbidden" OAM levels. We derive analytical expressions for the OAM power spectrum and demonstrate universal caling behavior within the allowed manifold. These theoretical predictions are validated by comprehensive Monte Carlo simulations, which confirm the selection rules with a forbidden-level suppression factor exceeding 10^4 and demonstrate the universal scaling with exceptional accuracy.

physics.optics↗

arXiv2025

A Hybrid Zernike-Lyapunov Framework for Aberration-Based Statistical Wavefront Reconstruction of Chaotic Optical Surfaces

We present a comprehensive theoretical framework that unifies chaotic wavefront dynamics with classical aberration theory through a Statistical Wavefront Reconstruction Framework (SWRF) formalism. By establishing rigorous connections between ray trajectory deflections and wave-optical phase perturbations through the eikonal equation, we decompose chaotic wavefront perturbations into modified Zernike-Lyapunov hybrid expansions, establishing mathematical equivalences between Lyapunov exponents, fractal dimensions, and traditional aberration coefficients. This chaotic aberration theory enables systematic incorporation of non-integrable wavefront dynamics into deterministic design frameworks, providing a rigorous foundation for controlled chaos in optical systems. We derive analytical relationships connecting surface chaos parameters to optical performance metrics, demonstrate the framework's validity through phase space analysis, and establish convergence criteria for the chaotic expansion. The theory reveals how chaotic surface geometries can be intentionally designed to achieve specific optical functionalities, including beam homogenization, speckle reduction, and novel wavefront shaping capabilities, while maintaining mathematical rigor comparable to classical aberration analysis.

physics.optics↗

arXiv2025

Polynomial Eigenfunctions and Matrix Lyapunov Equations from Energy Balance Integrals

We establish a unified theoretical framework that connects classical orthogonal polynomial systems to matrix Lyapunov equations through the fundamental physics of energy dissipation in stochastic dynamical systems. Starting from the energy balance principle in infinite-dimensional Hilbert spaces, we derive a master integral representation that naturally encompasses both spectral geometry and covariance dynamics. The theory reveals that established orthogonal polynomials (Zernike, Hermite, spherical harmonics) and matrix Lyapunov equations are dual manifestations of the same underlying energy dissipation structure. We provide rigorous mathematical foundations showing how finite-dimensional projections of infinite-dimensional energy integrals reproduce classical matrix equations, with specific structure determined by the symmetries of noise processes. The framework demonstrates that adding uniform dissipation to classical differential operators preserves their polynomial eigenfunction structure while ensuring the energy balance conditions required for physical consistency.

math.OC↗

arXiv2025

Surface-Encoded Partial Coherence Transformation: Modeling Source Coherence Effects in Wave Optics

We present a new mathematical framework for incorporating partial coherence effects into wave optics simulations through a comprehensive surface-to-detector approach. Unlike traditional ensemble averaging methods, our dual-component framework models partial coherence through: (1) a surface-encoded transformation implemented via a linear integral operator with a spatially-dependent kernel that modifies coherence properties at the reflection interface, followed by (2) a propagation component that evolves these coherence properties to the detection plane. This approach differs fundamentally from conventional models by explicitly separating surface interactions from propagation effects, while maintaining a unified mathematical structure. We derive the mathematical foundation based on the coherence function formalism, establish the connection to the Van Cittert-Zernike theorem, and prove the equivalence of our framework to conventional partial coherence theory. The method reduces the dimensional complexity of coherence calculations and offers potential computational advantages, particularly for systems involving multiple surfaces and propagation steps. Applications include optical testing and astronomical instrumentation. We provide rigorous mathematical proofs, demonstrate the convergence properties, and analyze the relative importance of surface and propagation effects across different optical scenarios.

physics.optics↗

arXiv2025

Bridging Statistical Scattering and Aberration Theory: Ray Deflection Function -- I: Theoretical Framework

This paper introduces a new conceptual framework that recasts surface roughness effects as a "ray deflection function" (RDF) which can be statistically represented through a modified Zernike-Fourier hybrid approach that directly connects the PSD with statistical aberration coefficients through spectral overlap integration. By establishing a direct mathematical relationship between the power spectral density (PSD) of surface imperfections and the statistical distribution of aberration coefficients, we develop a formalism that bridges known probabilistic scattering theory with deterministic aberration analysis. This transformation allows surface roughness to be seamlessly integrated with other optical aberrations by expressing its effects through equivalent modifications to the ideal mirror shape. This framework provides computational advantages for ray-tracing simulations while maintaining statistical fidelity to established scattering models, particularly for predicting the three-dimensional structure of imperfect focal bodies in optical systems.

physics.optics↗

arXiv2025

Bridging Statistical Scattering and Aberration Theory: Ray Deflection Function -- II: Numerical Validation

This paper presents a comprehensive experimental validation of a recently developed Ray Deflection Function (RDF) approach, which offers a new framework for modeling surface roughness effects in optical systems. Through detailed geometrical ray tracing simulations, we demonstrate that the RDF methodology successfully bridges two traditionally separate domains: statistical scattering models and deterministic aberration analysis. We implement and compare the two approaches for modeling a parabolic mirror with surface imperfections with three cases: (1) an ideal parabolic mirror baseline, (2) the conventional Harvey-Shack (HS) statistical scattering theory applied to ray perturbations, and (3) the newly proposed aberration term method based on the RDF theory. Our results confirm the statistical equivalence between the HS approach and the RDF-based aberration term method, with both producing close near-focal-plane distributions and focal volume characteristics. By establishing this equivalence, we validate that surface roughness effects can be accurately represented as deterministic aberration terms while maintaining fidelity to established statistical scattering models.

physics.optics↗

arXiv2025

Flexible Foil Mesh Generation for Spatial Focal-Body Modeling of a Spherical Mirror

We present a novel application of the Flexible Foil Mesh Generation (FFMG) method to model the $3D$ Focal Body generated by a spherical mirror collecting light from an infinitely distant source on its optical axis. The study addresses the challenge of accurately representing highly concave structures formed by the focusing effect. Through theoretical analysis and numerical simulations, we demonstrate the effectiveness of the FFMG method in capturing the intricate geometry of the Focal Body, with implications for computational geometry, $3D$ reconstruction, and optical system modeling.

math.OC↗

arXiv2025

Physically-Based Mesh Generation for Confined 3D Point Clouds Using Flexible Foil Models

We propose a method for constructing high-quality, closed-surface meshes from confined 3D point clouds via a physically-based simulation of flexible foils under spatial constraints. The approach integrates dynamic elasticity, pressure-driven deformation, and adaptive snapping to fixed vertices, providing a robust framework for realistic and physically accurate mesh creation. Applications in computer graphics and computational geometry are discussed.

cs.GR↗

arXiv2024

An $O(N)$ Algorithm for Solving the Smallest Enclosing Sphere Problem in the Presence of Degeneracies

Efficient algorithms for solving the Smallest Enclosing Sphere (SES) problem, such as Welzl's algorithm, often fail to handle degenerate subsets of points in 3D space. Degeneracies and ill-posed configurations present significant challenges, leading to failures in convergence, inaccuracies or increased computational cost in such cases. Existing improvements to these algorithms, while addressing some of these issues, are either computationally expensive or only partially effective. In this paper, we propose a hybrid algorithm designed to mitigate degeneracy while maintaining an overall computational complexity of $O(N)$. By combining robust preprocessing steps with efficient core computations, our approach avoids the pitfalls of degeneracy without sacrificing scalability. The proposed method is validated through theoretical analysis and experimental results, demonstrating its efficacy in addressing degenerate configurations and achieving high efficiency in practice.

cs.CG↗

arXiv2024

Ill-Posed Configurations in Random and Experimental Data Points Collection

Ill-posed configurations, such as collinear or coplanar point arrangements, are a persistent challenge in computational geometry, complicating tasks as in triangulation and convex hull construction. This paper discusses the probability of such configurations arising in two scenarios: (1) data sampled randomly from a uniform distribution, and (2) data collected from physical systems, such as reflective surfaces or structured environments. We present a probabilistic framework, analyze the geometric and sampling constraints, and provide some mathematical insights into how data acquisition processes influence the likelihood of degeneracies. Notably, our findings reveal that degeneracies occur more frequently in physical systems than in purely random simulations due to systematic biases introduced by instrumental setups and environmental structures, emphasizing the risks of drawing conclusions solely based on assumptions derived from random data.

math.OC↗

arXiv2024

Form Convex Hull to Concavity: Surface Contraction Around a Point Set

This paper investigates the transformation of a convex hull, derived from a d-dimensional point cloud, into a concave surface. Our primary focus is on the development of a methodology that ensures all points in the point cloud are encapsulated within a closed, non-intersecting concave surface. The study begins with the initial convex hull and employs an iterative process of facet replacement and expansion to evolve the surface into Scc, which accurately conforms to the complex geometry of the point cloud.

math.OC↗

arXiv2024

The Largest Empty Sphere Problem in 3D Hollowed Point Clouds

We introduce a new approach for the adaptation of the Maximal Internal Envelope method, extended to address the Largest Empty Sphere problem within unstructured 3D point clouds. We explore the identification of the Largest Empty Sphere by computing Convex Hull vertices and employing a Voidness Score based on Minimal Distance Scoring for optimal segment selection. The integration of Delaunay triangulation and Voronoi diagrams facilitates the initial identification of potential Largest Empty Sphere candidates. Our analysis reveals the method's efficacy and efficiency, often locating the Largest Empty Sphere in initial computational stages, suggesting a lower complexity than initially projected.

math.OC↗

arXiv2023

Void Shape Identification in a 2D Point Distribution

We introduce a new approach for identifying and characterizing voids within two-dimensional (2D) point distributions through the integration of Delaunay triangulation and Voronoi diagrams, combined with a Minimal Distance Scoring algorithm. Our methodology initiates with the computational determination of the Convex Hull vertices within the point cloud, followed by a systematic selection of optimal line segments, strategically chosen for their likelihood of intersecting internal void regions. We then utilize Delaunay triangulation in conjunction with Voronoi diagrams to ascertain the initial points for the construction of the maximal internal curve envelope by adopting a pseudo-recursive approach for higher-order void identification. In each iteration, the existing collection of maximal internal curve envelope points serves as a basis for identifying additional candidate points. This iterative process is inherently self-converging, ensuring progressive refinement of the void's shape with each successive computation cycle. The mathematical robustness of this method allows for an efficient convergence to a stable solution, reflecting both the geometric intricacies and the topological characteristics of the voids within the point cloud. Our findings introduce a method that aims to balance geometric accuracy with computational practicality. The approach is designed to improve the understanding of void shapes within point clouds and suggests a potential framework for exploring more complex, multi-dimensional data analysis.

cs.CG↗

arXiv2023

Smallest Enclosing Sphere in 3D -- Particle Swarm Optimization Approach

We have employed Particle Swarm Optimization to address a stochastic variant of the Smallest Enclosing Sphere estimation problem. An efficient algorithm has been developed to ascertain the optimal center and radius of a sphere encompassing a cloud of points within a three-dimensional space. Our findings are benchmarked against simulated scenarios of the classical problem. Additionally, we elucidate several benefits of our proposed algorithm over Welzl's Algorithm.

math.OC↗