Eleftherios Gkioulekas

The University of Texas Rio Grande Valley · School of Mathematical and Statistical Sciences

Using the fusion-rules hypothesis for three-dimensional and two-dimensional Navier-Stokes turbulence, we generalize a previous nonperturbative locality proof to multiple applications of the nonlinear interactions operator on generalized structure functions of velocity differences. We call this generalization of nonperturbative locality to multiple...

When confronted with a public health emergency, significant innovative treatment protocols can sometimes be discovered by medical doctors at the front lines based on repurposed medications. We propose a statistical framework for analyzing the case series of patients treated with such new protocols, that enables a comparison with our prior knowledge...

A cohort of 30,423 Covid-19 patients treated between March 2020 and December 2021 at the IHU-Méditerranée Infection in Marseille (France) was retrospectively analyzed in terms of treatment attempted and disease worsening factors to quantify efficacy with respect to the composite endpoint of transfer to intensive care unit or death, within a couple...

A cohort of 30,423 Covid-19 patients treated between March 2020 and December 2021 at the IHU-Méditerranée Infection in Marseille (France) was retrospectively analyzed in terms of treatment attempted and disease worsening factors to quantify vaccination efficacy with respect to the composite endpoint of transfer to intensive care unit or death, with...

We review the available evidence supporting the use of hydroxychloroquine-based multidrug protocols in the treatment of COVID-19, in response to a recently published editorial by TMJ.

When confronted with a public health emergency, significant innovative treatment protocols can sometimes be discovered by medical doctors at the front lines based on repurposed medications. We propose a statistical framework for analyzing the case series of patients treated with such new protocols, that enables a comparison with our prior knowledge...

When confronted with a public health emergency where the pre-existing standard of care is inadequate, significant innovative treatment protocols can sometimes be discovered by medical doctors at the front lines based on repurposed medications. We propose a statistical framework for analyzing the case series of patients treated with such new protoco...

In the two-layer quasi-geostrophic model, the friction between the flow at the lower layer and the surface boundary layer, placed beneath the lower layer, is modeled by the Ekman term, which is a linear dissipation term with respect to the horizontal velocity at the lower layer. The Ekman term appears in the governing equations asymmetrically; it i...

We review the history and previous literature on radical equations and present the rigorous solution theory for radical equations of depth 2, continuing a previous study of radical equations of depth 1. Radical equations of depth 2 are equations where the unknown variable appears under at least one square root and where two steps are needed to elim...

In the two-layer quasi-geostrophic model, the friction between the flow at the bottom layer and the surface layer, placed beneath the bottom layer, is modeled by the Ekman term, which is a linear dissipation term with respect to the horizontal velocity at the bottom layer. The Ekman term appears in the governing equations asymmetrically, it is plac...

The standard technique for solving equations with radicals is to square both sides of the equation as many times as necessary to eliminate all radicals. Because the procedure violates logical equivalence, it results in extraneous solutions that do not satisfy the original equation, making it necessary to check all solutions against the original equ...

We present the basic theory of denesting nested square roots, from an elementary point of view, suitable for lower level coursework. Necessary and sufficient conditions are given for direct denesting, where the nested expression is rewritten as a sum of square roots of rational numbers, and for indirect denesting, where the nested expression is rew...

We investigate an inequality constraining the energy and potential enstrophy flux spectra in two-layer and multi-layer quasi-geostrophic models. Its physical significance is that it can diagnose whether any given multi-layer model that allows co-existing downscale cascades of energy and potential enstrophy can allow the downscale energy flux to bec...

We give a detailed derivation of a generalization of the second derivative test of single-variable calculus which can classify critical points as local minima or local maxima (or neither), whenever the traditional second derivative test fails, by considering the values of higher-order derivatives evaluated at the critical points. The enhanced test...

Many limits, typically taught as examples of applying the ‘squeeze’ theorem, can be evaluated more easily using the proposed zero-bounded limit theorem. The theorem applies to functions defined as a product of a factor going to zero and a factor that remains bounded in some neighborhood of the limit. This technique is immensely useful for both sing...

A detailed development of the theory of convex functions, not often found in complete form in most textbooks, is given. We adopt the strict secant line definition as the definitive definition of convexity. We then show that for differentiable functions, this definition becomes logically equivalent with the first derivative monotonicity definition a...

In the Nastrom-Gage spectrum of atmospheric turbulence we observe a $k^{-3}$ energy spectrum that transitions into a $k^{-5/3}$ spectrum, with increasing wavenumber $k$. The transition occurs near a transition wavenumber $k_t$, located near the Rossby deformation wavenumber $k_R$. The Tung-Orlando theory interprets this spectrum as a double downsca...

In previous papers I have argued that the fusion rules hypothesis, which was originally introduced by L'vov and Procaccia in the context of the problem of three-dimensional turbulence, can be used to gain a deeper insight in understanding the enstrophy cascade and inverse energy cascade of two-dimensional turbulence. In the present paper, we show t...

We investigate and clarify the notion of locality as it pertains to the cascades of two-dimensional turbulence. The mathematical framework underlying our analysis is the infinite system of balance equations that govern the generalized unfused structure functions, first introduced by L'vov and Procaccia. As a point of departure we use a revised vers...

We elaborate further on a hypothesis by Winterberg that turbulent fluctuations of the zero point field may lead to a breakdown of the superluminal quantum correlations over very large distances. A phenomenological model that was proposed by Winterberg to estimate the transition scale of the conjectured breakdown, does not lead to a distance that is...

A general proof that more energy flows upscale than downscale in two-dimensional turbulence and barotropic quasi-geostrophic (QG) turbulence is given. A proof is also given that in surface QG turbulence, the reverse is true. Though some of these results are known in restricted cases, the proofs given here are pedagogically simpler, require fewer as...

In this paper, we revisit the claim that the Eulerian and quasi-Lagrangian same time correlation tensors are equal. This statement allows us to transform the results of an MSR quasi-Lagrangian statistical theory of hydrodynamic turbulence back to the Eulerian representation. We define a hierarchy of homogeneity symmetries between incremental homoge...

Two-dimensional turbulence appears to be a more formidable problem than three-dimensional turbulence despite the numerical advantage of working with one less dimension. In the present paper we review recent numerical investigations of the phenomenology of two-dimensional turbulence as well as recent theoretical breakthroughs by various leading rese...

This thesis is concerned with two foundational challenges in our understanding of turbulence: the elimination of sweeping interactions from theories of three-dimensional turbulence, and reconciling 2D models with atmospheric turbulence. The problem of eliminating the sweeping interactions has been a thorn on all efforts aimed at developing analytic...

In systems governing two-dimensional turbulence, surface quasi-geostrophic turbulence, (more generally $\alpha$-turbulence), two-layer quasi-geostrophic turbulence, etc., there often exist two conservative quadratic quantities, one ``energy''-like and one ``enstrophy''-like. In a finite inertial range there are in general two spectral fluxes, one a...

This paper is concerned with three interrelated issues on our pro-posal of double cascades intended to serve as a more realistic theory of two-dimensional turbulence. We begin by examining the approach to the KLB limit. We present improved proofs of the result by Fjortoft. We also ex-plain why in that limit the subleading downscale energy cascade a...

The Kraichnan-Leith-Batchelor scenario of a dual cascade, con-sisting of an upscale pure energy cascade and a downscale pure enstrophy cascade, is an idealization valid only in an infinite domain in the limit of in-finite Reynolds number. In realistic situations there are double cascades of energy and enstrophy located both upscale and downscale of...

As a first step to predicting the 3D conformation of a protein given the amino acid sequence, we have addressed the problem of classifying protein sequences to structural families. To test whether a protein sequence belongs to a certain group, we use decision algorithms that decide between the group and a group of randomly generated sequences. Prot...

Thesis (Ph. D.)--University of Washington, 2006. Vita. Includes bibliographical references (p. 116-132).