Functions and Analysis

2609 Submissions

[2] ai.viXra.org:2609.0069 [pdf] submitted on 2026-09-27 17:45:05

A Spatial-Intersection Criterion for Global Smoothness of Three-Dimensional Euler Flow

Authors: Thomas Birnie
Comments: 54 pages; English; CC BY 4.0. Prepared with assistance from OpenAI ChatGPT and Codex. Revised edition of DOI 10.6084/m9.figshare.33479899.v1; current DOI 10.6084/m9.figshare.33479899.v2.

We prove a global smoothness criterion for three-dimensional incompressible Euler flow on R^3. For a smooth finite-energy solution, whole-interval L^2_t H^K_x membership of the velocity at every finite spatial order generates every temporal velocity and canonical pressure derivative. The proof combines the full differentiated Euler recurrence with an energy interpolation inequality; each output requires only finitely many spatial orders. Adjacent temporal rows give strong Sobolev traces and continuity of every mixed derivative on the closed finite slab. One local continuation through a proposed finite maximal time then proves global smoothness under the stated membership hypothesis. Neither a sum over derivative orders nor an infinite sum of local lifespans is required. We also construct the full intersection on one common local interval from arbitrary smooth data. Extending that construction through an entire candidate finite maximal lifespan remains unresolved.Further results examine the fluid's material and pressure evolution. From energy alone we construct a unique weak terminal velocity, a terminal transport isometry with an exact orthogonal energy defect, and a measure-preserving particle limit whose secants recover terminal momentum. Circulation reaches a terminal current; at strict times it combines with the deformed material metric to reconstruct velocity. The first temporal response has a sharp energy-level weak topology, but an explicit Fourier-packet construction makes the second response unbounded at a fixed output frequency. Separate pressure arguments give continuation under absolute and directional inverse-square bounds with coefficient below 2, and under a small signed weighted matrix primitive along vortical trajectories. We compare these deductions with existing pressure criteria and retain the spherical contact term in the coupled Hessian evolution. We describe normalized clocks, material anisotropy, cross-Leray exchange, and circulating surface stacks recruited by localized receivers. Fixed-mass recurrence at escaping frequencies forces infinite accumulated activity, and fixed-mass shrinking spatial cores are excluded under a Type-I gradient bound. Fragmentation and exact countertests identify the limits of that hypothesis and of proposed scalar budgets. These constructions and obstructions retain their stated hypotheses; they do not remove the spatial membership condition from the global theorem.
Category: Functions and Analysis

[1] ai.viXra.org:2609.0054 [pdf] submitted on 2026-09-23 11:45:16

On the Conceptual Limitations of the Theoretical Substantiation of the Derivatives of Inverse Trigonometric and Hyperbolic Functions

Authors: Vitaliy Boltian
Comments: 3 Pages.

This paper addresses the conceptual limitations in classical mathematical analysis regarding the differentiation of inverse trigonometric and hyperbolic functions. The traditional approach, which relies heavily on the inverse function theorem and complex virtual transformations, is reconsidered. The author introduces a novel theoretical framework based on a system of real companion functions and defines the concept of a "transition coefficient". This approach establishes direct, invariant relations between trigonometric, hyperbolic, and algebraic function systems within the real domain. The practical application of the proposed apparatus is demonstrated through specific integration examples, showcasing a significant simplification of mathematical operations and analytical structures.
Category: Functions and Analysis