What I learned today

  Turbulence as a Fracture of Navier–Stokes: Vorticity Stretching, Two Repair Channels, and 3D Failure Part I — The Apparent Closure of Navier–Stokes The compact PDE and the hidden dynamical system 1.1 Navier–Stokes as u_t + u·∇u = -∇p + νΔu 1.2 Incompressibility div u = 0 1.3 Local differential closure versus global regularity 1.4 Velocity as an infinite-dimensional state 1.5 Why smooth coefficients do not imply smooth evolution 1.6 The PDE as compressed representation 1.7 Turbulence as failure of representation-level smoothness The three competing operators 2.1 Nonlinear transport u·∇u 2.2 Pressure projection -∇p 2.3 Viscous diffusion νΔu 2.4 Transport creates geometry 2.5 Pressure enforces compatibility 2.6 Viscosity destroys fine scales 2.7 Why their balance is dynamically unstable Two supporting repair channels 3.1 Repair channel A: viscous diffusion 3.2 Repair channel B: incompressibility and pressure redistribution 3.3 Local damping versus nonlocal constraint repair 3.4 Why ...

  Why Smooth Flow Requires Complex Mathematics to Unravel Part I — The Compression Problem: A Short Equation, a Huge State Space The deceptive simplicity of a flow equation 1.1 Local differential rule versus global evolution 1.2 Why formula length says little about dynamical complexity 1.3 A velocity field as an infinite-dimensional state 1.4 Smoothness as a property of an entire orbit, not one instant 1.5 Initial data, forcing, geometry, and boundary conditions 1.6 Local solvability versus global continuation 1.7 Representation hides the evolving boundary of regularity From particles to fields 2.1 Lagrangian particle trajectories 2.2 Eulerian velocity fields 2.3 The flow map X(a,t) 2.4 Deformation gradient and volume distortion 2.5 Eulerian–Lagrangian conversion 2.6 Why the two descriptions expose different singular mechanisms 2.7 Particle smoothness versus field smoothness What “smooth flow” actually demands 3.1 Continuity 3.2 Differentiability 3.3 Higher derivatives 3.4 Sobolev ...

Convexity: An Analytic Viewpoint. Convex functions and sets — p. 1 1.1 Convex subsets of real vector spaces 1.2 Convex combinations and convex hulls 1.3 Convex and strictly convex functions 1.4 Epigraphs and geometric interpretation 1.5 One-dimensional convexity and secant slopes 1.6 Continuity and differentiability of convex functions 1.7 Supporting lines and tangent inequalities 1.8 Jensen’s inequality 1.9 Hölder and Minkowski inequalities from convexity 1.10 Gauges of convex sets 1.11 Legendre transforms 1.12 Hahn–Banach viewed through supporting tangents Simon explicitly identifies differentiability, Jensen, gauges, Legendre transforms, and the Hahn–Banach viewpoint as foundational themes   Orlicz spaces 2.1 Young functions 2.2 Convex growth functions beyond power laws 2.3 Modular functionals 2.4 Luxemburg-type norms 2.5 Orlicz classes and Orlicz spaces 2.6 Hölder-type inequalities in Orlicz spaces 2.7 Complementary Young functions 2.8 Duality phenomena 2.9 Comparison with...

  Distinguishability Transport and Reconstruction Theory — DTRT TOC Core spine: PAIR → CARRIER → TRANSFORMATION → DISTINCTION PROFILE → SURVIVAL LAW → LOSS PROFILE → RECOVERABILITY → ADEQUATE CARRIER . PART I — FOUNDATIONAL OBJECTS 1. Probability States 1.1 Probability measures as states 1.2 Sample spaces and measurable structure 1.3 Probability law versus density representation 1.4 Support and effective support 1.5 Joint and marginal states 1.6 Conditional states 1.7 Process-valued states 1.8 Model families as sets of probability states 1.9 State equivalence 1.10 State comparison 2. Ordered Probability Pairs 2.1 (P,Q) as the primitive comparison object 2.2 Directionality of comparison 2.3 Symmetric versus asymmetric comparison 2.4 Pair equivalence 2.5 Pair orbits under transformations 2.6 Pair invariants 2.7 Pairwise distinguishability 2.8 Family-relative distinguishability 2.9 Pairwise versus multi-state comparison 2.10 Comparison without scalarization 3. Carriers 3.1 Definition...

  Kullback–Leibler Divergence: Basic Geometry and Dynamics of Probability Distributions Under Inference, Transport, Coarse-Graining, and Model Change Prologue — From Relative Entropy to a Theory of Distinguishability 0.1 Probability distributions as states of uncertainty 0.2 Comparison before distance 0.3 Why KL divergence is directional 0.4 D(P‖Q) as expected log-likelihood ratio 0.5 Distinguishability rather than geometric distance 0.6 Probability law ≠ density ≠ parameterization ≠ statistical model 0.7 Transformation of distributions as the central dynamical operation 0.8 Four basic transformation regimes: inference, transport, coarse-graining, model change 0.9 Preservation, contraction, amplification, and destruction of distinguishability 0.10 Local geometry versus global divergence 0.11 Empirical fluctuations versus dynamical trajectories 0.12 Relative entropy as the canonical empirical-measure large-deviation cost 0.13 Beyond large deviations: path ensembles, thermodynamic s...