Convexity
View article: Wern–Widgren Angle Trisection Method: From Geometric Insight to a Modern Analytic Approximation Framework (Version 1.1)
Wern–Widgren Angle Trisection Method: From Geometric Insight to a Modern Analytic Approximation Framework (Version 1.1) Open
This preprint presents an analytic study and refinement of a geometric angle trisection method introduced by Carl R. Wern (1925–). While Wantzel's theorem (1837) rules out exact trisection with straightedge and compass, Wern's construction…
View article: Existence Results for Nonconvex Nonautonomous Differential Inclusions in Hilbert Spaces
Existence Results for Nonconvex Nonautonomous Differential Inclusions in Hilbert Spaces Open
We establish a solvability criterion for nonautonomous time-evolution inclusions governed by the right-hand side without the convexity assumption. In this study, we examine the problem x˙(t)∈F(t,x(t))a.e.onI0:=[0,T0], for some T0>0, where …
View article: Wern–Widgren Angle Trisection Method: From Geometric Insight to a Modern Analytic Approximation Framework (Version 1.1)
Wern–Widgren Angle Trisection Method: From Geometric Insight to a Modern Analytic Approximation Framework (Version 1.1) Open
This preprint presents an analytic study and refinement of a geometric angle trisection method introduced by Carl R. Wern (1925–). While Wantzel's theorem (1837) rules out exact trisection with straightedge and compass, Wern's construction…
View article: Paper 32A: Timing Continua Meet Symmetry Selection From Weihs Peak–Shoulder–Cutoff to a T4–Math Selector on S2
Paper 32A: Timing Continua Meet Symmetry Selection From Weihs Peak–Shoulder–Cutoff to a T4–Math Selector on S2 Open
This paper connects finite timing structure in the Weihs Bell-test data to a symmetry-driven calibration map on the measurement sphere. Using only the official Alice/Bob archives, we rebuild the timing analysis with a minimal, audit-ready …
View article: Nash Equilibrium of Bi-objective Optimal Control of Fractional Space-Time Parabolic PDE
Nash Equilibrium of Bi-objective Optimal Control of Fractional Space-Time Parabolic PDE Open
This work investigates the existence and uniqueness of the Nash equilibrium (solutions to competitive problems in which individual controls aim at separate desired states) for a bi-objective optimal control problem governed by a fractional…
View article: Journal of Operator Theory
Journal of Operator Theory Open
We initiate the study of matrix convexity for operator spaces. We define the\nnotion of compact rectangular matrix convex set, and prove the natural analogs\nof the Krein-Milman and the bipolar theorems in this context. We deduce a\ncanoni…
View article: Paper 32A: Timing Continua Meet Symmetry Selection From Weihs Peak–Shoulder–Cutoff to a T4–Math Selector on S2
Paper 32A: Timing Continua Meet Symmetry Selection From Weihs Peak–Shoulder–Cutoff to a T4–Math Selector on S2 Open
This paper connects finite timing structure in the Weihs Bell-test data to a symmetry-driven calibration map on the measurement sphere. Using only the official Alice/Bob archives, we rebuild the timing analysis with a minimal, audit-ready …
View article: Nash Equilibrium of Bi-objective Optimal Control of Fractional Space-Time Parabolic PDE
Nash Equilibrium of Bi-objective Optimal Control of Fractional Space-Time Parabolic PDE Open
This work investigates the existence and uniqueness of the Nash equilibrium (solutions to competitive problems in which individual controls aim at separate desired states) for a bi-objective optimal control problem governed by a fractional…
View article: On Conditional Independence Graph Learning From Multi-Attribute Gaussian Dependent Time Series
On Conditional Independence Graph Learning From Multi-Attribute Gaussian Dependent Time Series Open
Estimation of the conditional independence graph (CIG) of high-dimensional multivariate Gaussian time series from multi-attribute data is considered. Existing methods for graph estimation for such data are based on single-attribute models …
View article: Comparing BFGS and OGR for Second-Order Optimization
Comparing BFGS and OGR for Second-Order Optimization Open
Estimating the Hessian matrix, especially for neural network training, is a challenging problem due to high dimensionality and cost. In this work, we compare the classical Sherman-Morrison update used in the popular BFGS method (Broy-den-F…
View article: Multi-dimensional EoS Merger via a Statistical Mixture
Multi-dimensional EoS Merger via a Statistical Mixture Open
We present results from a new general method to merge multidimensional equations of state (EoSs) by combining them in a two-fluid equilibrium statistical mixture in the grand canonical ensemble. The merged grand potential density $\omega$ …
View article: Comparing BFGS and OGR for Second-Order Optimization
Comparing BFGS and OGR for Second-Order Optimization Open
Estimating the Hessian matrix, especially for neural network training, is a challenging problem due to high dimensionality and cost. In this work, we compare the classical Sherman-Morrison update used in the popular BFGS method (Broy-den-F…
View article: Multi-dimensional EoS Merger via a Statistical Mixture
Multi-dimensional EoS Merger via a Statistical Mixture Open
We present results from a new general method to merge multidimensional equations of state (EoSs) by combining them in a two-fluid equilibrium statistical mixture in the grand canonical ensemble. The merged grand potential density $\omega$ …
View article: Synthetic Curvature from Optimal Transport on Metric Measure Spaces
Synthetic Curvature from Optimal Transport on Metric Measure Spaces Open
This paper explores the construction and implications of synthetic notions of Ricci curvature on general metric measure spaces through the lens of optimal transport theory. Traditional differential geometric definitions of curvature rely o…
View article: Synthetic Curvature from Optimal Transport on Metric Measure Spaces
Synthetic Curvature from Optimal Transport on Metric Measure Spaces Open
This paper explores the construction and implications of synthetic notions of Ricci curvature on general metric measure spaces through the lens of optimal transport theory. Traditional differential geometric definitions of curvature rely o…
View article: A Low-rank Augmented Lagrangian Method for Polyhedral-SDP and Moment-SOS Relaxations of Polynomial Optimization
A Low-rank Augmented Lagrangian Method for Polyhedral-SDP and Moment-SOS Relaxations of Polynomial Optimization Open
Polynomial optimization problems (POPs) can be reformulated as geometric convex conic programs, as shown by Kim, Kojima, and Toh (SIOPT 30:1251-1273, 2020), though such formulations remain NP-hard. In this work, we prove that several well-…
View article: THE MICROECONOMIC EQUILIBRIUM OPTIMIZER (MEO)
THE MICROECONOMIC EQUILIBRIUM OPTIMIZER (MEO) Open
This document presents the Microeconomic Equilibrium Optimizer (MEO), an interpretable, microeconomically grounded framework for multi-channel resource allocation. Instead of maximizing an explicit global objective function, MEO drives sys…
View article: Generative Geometry as Canonical Stratified Second-Variation Structure
Generative Geometry as Canonical Stratified Second-Variation Structure Open
This paper formulates a geometric theory of generative behavior in deep learning, rooted in the second-variation structure of a functional on a configuration manifold. It decomposes variational flows into reversible (antisymmetric) and dis…
View article: A Low-rank Augmented Lagrangian Method for Polyhedral-SDP and Moment-SOS Relaxations of Polynomial Optimization
A Low-rank Augmented Lagrangian Method for Polyhedral-SDP and Moment-SOS Relaxations of Polynomial Optimization Open
Polynomial optimization problems (POPs) can be reformulated as geometric convex conic programs, as shown by Kim, Kojima, and Toh (SIOPT 30:1251-1273, 2020), though such formulations remain NP-hard. In this work, we prove that several well-…
View article: THE MICROECONOMIC EQUILIBRIUM OPTIMIZER (MEO)
THE MICROECONOMIC EQUILIBRIUM OPTIMIZER (MEO) Open
This document presents the Microeconomic Equilibrium Optimizer (MEO), an interpretable, microeconomically grounded framework for multi-channel resource allocation. Instead of maximizing an explicit global objective function, MEO drives sys…
View article: Generative Geometry as Canonical Stratified Second-Variation Structure
Generative Geometry as Canonical Stratified Second-Variation Structure Open
This paper formulates a geometric theory of generative behavior in deep learning, rooted in the second-variation structure of a functional on a configuration manifold. It decomposes variational flows into reversible (antisymmetric) and dis…
View article: Optimal Transport and the Geometry of Radon Measures in $L^p$ Spaces
Optimal Transport and the Geometry of Radon Measures in $L^p$ Spaces Open
This paper explores the intricate relationship between optimal transport theory and the geometric properties of Radon measures when embedded in $L^p$ spaces. We investigate how the optimal transport cost, specifically the $p$-Wasserstein d…
View article: Optimal Transport and the Geometry of Radon Measures in $L^p$ Spaces
Optimal Transport and the Geometry of Radon Measures in $L^p$ Spaces Open
This paper explores the intricate relationship between optimal transport theory and the geometric properties of Radon measures when embedded in $L^p$ spaces. We investigate how the optimal transport cost, specifically the $p$-Wasserstein d…
View article: An Integrated Control-Equilibrium Framework for Socio-Ecological Systems
An Integrated Control-Equilibrium Framework for Socio-Ecological Systems Open
We develop a computable and mathematically rigorous framework for socio-ecological systems (SES) that combines dynamic policy design with decentralized equilibrium behavior. First, we represent the economic, social, and environmental pilla…
View article: Optimal Transport on Metric Measure Spaces with Non-Smooth Ricci Bounds
Optimal Transport on Metric Measure Spaces with Non-Smooth Ricci Bounds Open
This paper explores the intricate relationship between optimal transport theory and geometric analysis on metric measure spaces equipped with non-smooth lower Ricci curvature bounds. In classical Riemannian geometry, Ricci curvature plays …
View article: Optimal Transport on Metric Measure Spaces with Non-Smooth Ricci Bounds
Optimal Transport on Metric Measure Spaces with Non-Smooth Ricci Bounds Open
This paper explores the intricate relationship between optimal transport theory and geometric analysis on metric measure spaces equipped with non-smooth lower Ricci curvature bounds. In classical Riemannian geometry, Ricci curvature plays …
View article: A Discrete Phase–Metric Coupling Model on Graphs: Small–Coupling Regime and a Fully Worked Two–Node Prototype
A Discrete Phase–Metric Coupling Model on Graphs: Small–Coupling Regime and a Fully Worked Two–Node Prototype Open
We study a finite–dimensional gradient system on a finite graph, where a discrete “metric” g on edges and a scalar phase field \psi on vertices evolve together under the gradient flow of a coupled energy. The free energy splits as F = F_0 …
View article: A Discrete Phase–Metric Coupling Model on Graphs: Small–Coupling Regime and a Fully Worked Two–Node Prototype
A Discrete Phase–Metric Coupling Model on Graphs: Small–Coupling Regime and a Fully Worked Two–Node Prototype Open
We study a finite–dimensional gradient system on a finite graph, where a discrete “metric” g on edges and a scalar phase field \psi on vertices evolve together under the gradient flow of a coupled energy. The free energy splits as F = F_0 …
View article: Geometric Flow Approaches to Curvature-Dimension Conditions in Metric Measure Spaces
Geometric Flow Approaches to Curvature-Dimension Conditions in Metric Measure Spaces Open
The study of curvature-dimension conditions has revolutionized the understanding of geometry in non-smooth settings, providing powerful tools to extend classical Riemannian geometric concepts to spaces with singularities. This paper explor…
View article: Geometric Flow Approaches to Curvature-Dimension Conditions in Metric Measure Spaces
Geometric Flow Approaches to Curvature-Dimension Conditions in Metric Measure Spaces Open
The study of curvature-dimension conditions has revolutionized the understanding of geometry in non-smooth settings, providing powerful tools to extend classical Riemannian geometric concepts to spaces with singularities. This paper explor…