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| viXra.org > Functions and Analysis > 2606 |
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[8] viXra:2606.0110 [pdf] submitted on 2026-06-29 20:37:01
Authors: Yuhan Wei
Comments: 35 Pages.
We prove the existence of finite-dimensional inertial manifolds forthe dyadic model of turbulence for all dissipation exponents α≥1/3.For α = 1/3 and α > 1/3 the proof is unified by working in the Hα-norm and employing a generalized cone method. The dimension scales as N∼1 2αlog λ log ν−1, matching the optimal upper bounds for shell models. The construction relies on a low-mode cut-off, a forward cascade estimate that exploits the monotone structure of the dyadic model, and a modified strong squeezing property of Koksch (2000). The resulting inertial manifold is Lipschitz and C1+ϵ-smooth, and satisfies the exponential tracking property. This provides a rigorousfinite-dimensional reduction for the entire supercritical range α≥1/3. We also answer an open question by Cheskidov (2008) regarding theexistence of strong compact global attractors for α<1/2.
Category: Functions and Analysis
[7] viXra:2606.0105 [pdf] submitted on 2026-06-29 20:17:00
Authors: Lucas Brandt
Comments: 35 Pages. (Note by viXra Admin: Please submit article written with AI assistance to ai.viXra.org)
The collapse of Silicon Valley Bank in March 2023 unfolded in forty-eight hours, following months of visible but apparently tolerated balance-sheet deterioration. This pattern, prolonged stability followed by sudden and discontinuous collapse, is difficult to reconcile with the canonical Diamond-Dybvig model of bank runs, which treats a bank failure as an instantaneous jump between two static states. This paper proposes a dynamical systems extension of the Diamond-Dybvig framework. We model the joint evolution of depositor beliefs and bank fragility as a coupled pair of nonlinear differential equations, map the full geometry of the system, and establish two threshold results: a saddle-node bifurcation at a critical interest rate r*, at which the system changes from having one possible resting state to having two, and a Hopf bifurcation at r** > r*, at which the calm resting state loses stability and a self-sustaining oscillation appears. That oscillation formalizes the idea of a bank in chronic stress: repeatedly approaching, but not yet crossing, the edge of collapse. Therefore, a bank failure is reinterpreted not as a choice between equilibria but as the crossing of a tipping line in the space of possible states. We apply the framework to Silicon Valley Bank, arguing that the Federal Reserve’s 2022 to 2023 hiking cycle pushed the bank past the saddle-node threshold, that the ensuing months of intermittent stress correspond to the oscillatory regime, and that the capital-raise announcement of March 8th, 2023 was the discrete shock to beliefs that pushed the system across the tipping line into irreversible collapse. Policy implications for deposit insurance design, capital requirements, and depositor-network regulation follow directly from the geometry.
Category: Functions and Analysis
[6] viXra:2606.0075 [pdf] submitted on 2026-06-20 21:33:52
Authors: Julinho Jorge Luís
Comments: 14 Pages.
The Bohr-Mollerup Theorem establishes the uniqueness of the Gamma function on the right half-plane, but makes no assertion regarding extensions to the left half-plane that preserve the integral form. This work proposes a Dual Architecture consisting of two integral operators with complementary domains: the Classical Gamma and a Symmetric Factorial, connected by an operator derived from the Hankel contour integral. When applied to the Riemann zeta function, this architecture replaces the classical functional equation—which exhibits indeterminate forms at integer points—with a formulation that is directly evaluable and preserves all values of the Dirichlet series. Analysis of the connection operator reveals that its real part vanishes exclusively on the critical line within the critical strip. Assuming a non-trivial zero off this line leads, via a closed cycle of the dual functional equation, to a contradiction involving the modulus of the Gamma function, as established independently through the Weierstrass representation and the maximum modulus principle. The result forces all non-trivial zeros of the Riemann zeta function to lie on the critical line.Keywords: Gamma function, Riemann zeta function, critical line, Hankel contour, Weierstrass representation.
Category: Functions and Analysis
[5] viXra:2606.0074 [pdf] replaced on 2026-06-24 05:20:36
Authors: Deepak Ponvel Chermakani
Comments: 2 Pages. Clarified on the new LRC variant called equi-split-LRC
Consider an instance of the shifted Lonely Runner Conjecture (shifted-LRC) where n runners (except the stationary runner 0) have integer speeds and start from real values in [0,1[ at time t=0. We show that one can derive an alternative vector of starting points that can be made to be arbitrarily close to the initial vector of starting points. The alternative starting point of each runner i is a rational in [0,1[ and is expressible as (qi / P) where P is a large prime and qi is an integer in [0, P-1]. We then introduce a new LRC variant called the equi-split-LRC. The LRC instance allows a minimal loneliness gap of f for runner 0 from the remaining n-1 runners, if and only if, the equi-split-LRC instance with the same vector of speeds and alternative vector of starting points simultaneously allows a minimal loneliness gap of f/P for the arc-center of each of P sectors into which the circle is divided from the remaining n-1 runners. Here, f is a desired fraction in ]0,1[. This finding is important in the light of recent counter-examples to the shifted-LRC.
Category: Functions and Analysis
[4] viXra:2606.0053 [pdf] submitted on 2026-06-14 21:00:26
Authors: Vladyslav Vasilache
Comments: 2 Pages. (Note by viXra Admin: Please cite and list scientific reference and submit article written with AI assistance to ai.viXra.org)
This paper introduces a new, highly accurate approximation for the function $e^{-x^2}$. By differentiating a known error function approximation and optimizing its parameters, we drastically reduce the maximum absolute error from $1.88%$ to less than $0.09%$ without using any exponential terms.
Category: Functions and Analysis
[3] viXra:2606.0035 [pdf] submitted on 2026-06-10 10:49:40
Authors: Masatoshi Ohrui
Comments: 23 Pages.
This is an application of functional analysis to the existence and smoothness of the Navier—Stokes equations using elementary weak solutions in Sobolev spaces.We solve the problem in mathematics. The problems are not in physics, so we do not use any physics or assumptions-falsified mathematics, such as other papers. We use mathematics only. We can solve the problem by using an exactly and completely FALSIFIED resolution, where large initial values destroy the earth, because uniqueness does NOT hold, or SMALL initial values love your cup of coffee.There are no long or complicated calculations; semi-groups, a priori estimates, and boundary conditions are not used at all. We apply the local solvability of linear partial differential operators with constant coeficients.
Category: Functions and Analysis
[2] viXra:2606.0034 [pdf] submitted on 2026-06-10 10:55:32
Authors: Masatoshi Ohrui
Comments: 2 Pages.
We can prove Hartog’s phenomenon by solving the ∂-bar equation for compactly supported forms. To solve the equation, we construct the solution using convolution.
Category: Functions and Analysis
[1] viXra:2606.0017 [pdf] submitted on 2026-06-06 18:49:02
Authors: Richard J. Mathar
Comments: 16 Pages.
Hastings and later Cody tabulated minimax polynomial approximations for the Complete Elliptic Integral of the First Kind. The simplicity of this representation by polynomials and polynomials times a logarithm allows to integrate their terms analytically. We demonstrate how integrals of the Complete Elliptic Integral times a power of its argument achieve double precision accuracy for powers from 0 to 2 based on Cody's polynomials up to 9th order.
Category: Functions and Analysis
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