Two-part and k-Sperner families: New proofs using permutations

This is a paper about the beauty of the permutation method. New and shorter proofs are given for the theorem [P. L. Erdős and G. O. H. Katona, J. Combin. Theory. Ser. A,4

Sperner type theorems with excluded subposets

Let F be a family of subsets of an n-element set. Sperner's theorem says that if there is no inclusion among the members of F then the largest family under this condition is the one containing all ⌊ frac(n, 2) ⌋-element subsets. The present paper surveys certain generalizations of this theorem. The maximum size of F is to be found under the condition that a certain configuration is excluded. The configuration here is always described by inclusions. More formally, let P be a poset. The maximum size of a family F which does not contain P as a (not-necessarily induced) subposet is denoted by La (n, P). The paper is based on a lecture of the author at the Jubilee Conference on Discrete Mathematics [Banasthali University, January 11-13, 2009], but it was somewhat updated in December 2010. © 2011 Elsevier B.V. All rights reserved

Dispersive stabilization of the inverse cascade for the Kolmogorov flow

It is shown by perturbation techniques and numerical simulations that the inverse cascade of kink-antikink annihilations, characteristic of the Kolmogorov flow in the slightly supercritical Reynolds number regime, is halted by the dispersive action of Rossby waves in the beta-plane approximation. For beta tending to zero, the largest excited scale is proportional to the logarithm of one over beta and differs strongly from what is predicted by standard dimensional phenomenology which ignores depletion of nonlinearity.Comment: 4 pages, LATEX, 3 figures. v3: revised version with minor correction

On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations

We consider an active scalar equation that is motivated by a model for magneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast, the critically diffusive equation is well-posed. In this case we give an example of a steady state that is nonlinearly unstable, and hence produces a dynamo effect in the sense of an exponentially growing magnetic field.Comment: We have modified the definition of Lipschitz well-posedness, in order to allow for a possible loss in regularity of the solution ma

Stability of Rossby waves in the beta-plane approximation

Floquet theory is used to describe the unstable spectrum at large scales of the beta-plane equation linearized about Rossby waves. Base flows consisting of one to three Rossby wave are considered analytically using continued fractions and the method of multiple scales, while base flow with more than three Rossby waves are studied numerically. It is demonstrated that the mechanism for instability changes from inflectional to triad resonance at an O(1) transition Rhines number Rh, independent of the Reynolds number. For a single Rossby wave base flow, the critical Reynolds number Re^c for instability is found in various limits. In the limits Rh --> infinity and k --> 0, the classical value Re^c = sqrt(2) is recovered. For Rh --> 0 and all orientations of the Rossby wave except zonal and meridional, the base flow is unstable for all Reynolds numbers; a zonal Rossby wave is stable, while a meridional Rossby wave has critical Reynolds number Re^c = sqrt(2). For more isotropic base flows consisting of many Rossby waves (up to forty), the most unstable mode is purely zonal for 2 <= Rh < infinity and is nearly zonal for Rh = 1/2, where the transition Rhines number is again O(1), independent of the Reynolds number and consistent with a change in the mechanism for instability from inflectional to triad resonance.Comment: 56 pages, 31 figures, submitted to Physica

Considering Fluctuation Energy as a Measure of Gyrokinetic Turbulence

In gyrokinetic theory there are two quadratic measures of fluctuation energy, left invariant under nonlinear interactions, that constrain the turbulence. The recent work of Plunk and Tatsuno [Phys. Rev. Lett. 106, 165003 (2011)] reported on the novel consequences that this constraint has on the direction and locality of spectral energy transfer. This paper builds on that work. We provide detailed analysis in support of the results of Plunk and Tatsuno but also significantly broaden the scope and use additional methods to address the problem of energy transfer. The perspective taken here is that the fluctuation energies are not merely formal invariants of an idealized model (two-dimensional gyrokinetics) but are general measures of gyrokinetic turbulence, i.e. quantities that can be used to predict the behavior of the turbulence. Though many open questions remain, this paper collects evidence in favor of this perspective by demonstrating in several contexts that constrained spectral energy transfer governs the dynamics.Comment: Final version as published. Some cosmetic changes and update of reference

Sharp Lower Bounds for the Dimension of the Global Attractor of the Sabra Shell Model of Turbulence

In this work we derive a lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing and for sufficiently small viscosity term $\nu$, the Sabra shell model has a global attractor of large Hausdorff and fractal dimensions proportional to $\log_\lambda \nu^{-1}$ for all values of the governing parameter $\epsilon$, except for $\epsilon=1$. The obtained lower bounds are sharp, matching the upper bounds for the dimension of the global attractor obtained in our previous work. Moreover, we show different scenarios of the transition to chaos for different parameters regime and for specific forcing. In the ``three-dimensional'' regime of parameters this scenario changes when the parameter $\epsilon$ becomes sufficiently close to 0 or to 1. We also show that in the ``two-dimensional'' regime of parameters for a certain non-zero forcing term the long-time dynamics of the model becomes trivial for any value of the viscosity

Power and Resourse Efficient Envoronmentally Safe Technology for Processing Dumps of Technogenic Waste From Ore-Dressing and Processing Enterprises

This research proposes a systematic approach for the analysis of volumes, physicochemical, granulometric, lithologic and thermal characteristics of waste from ore-dressing and processing enterprises stored in the dumps (tailing dumps) of ore-dressing and processing plants to assess the economic potential of its use in the system of complex power and resource efficient environmentally safe processing including palletizing machines, conveyor indurating machines and ore–thermal furnaces. The obtained results allow the authors to formulate the basic engineering, technological, economic and environmental requirements for complex chemical and power engineering systems of processing technogenic waste from ore-dressing and processing plants, these results make it also possible to define the degree of variability for the characteristics of the waste lots from various dumps. The paper describes the developed intensional and mathematical formulations for the multiscale problem of optimizing chemical and power engineering processes of technogenic raw materials processing in a complex chemical and power engineering system as a problem for discrete dynamic programming. The distinctive feature of this problem is to take into account the spatio-temporal multistage processing in a moving multilayer mass of pelletized raw material, the intensity of the process of internal moisture transfer and the variables for the control flow of the heat carrier gas. It allows increasing power efficiency by intensifying heat and mass transfer processes of multilayer drying, calcination and sintering. The criterion of the efficiency is the minimum cost of electric and thermal energy spent on processing. The obtained results were used to calculate power efficient environmentally safe processing of technogenic waste from ore-dressing and processing enterprises dumps. It was defined that heat and mass transfer processes are intensified, power consumption is reduced and the quality of the finished product is increased in the conditions of optimal power and resource efficient operation for the processing system. Keywords: tecnhogenic waste, waste processing, ore-dressing and processing plant, power and resource efficiency, optimization, system analysis, environmentally safet

MRI channel flows in vertically-stratified models of accretion disks

Simulations of the magnetorotational instability (MRI) in 'unstratified' shearing boxes exhibit powerful coherent flows, whereby the fluid vertically splits into countermoving planar jets or `channels'. Channel flows correspond to certain axisymmetric linear MRI modes, and their preponderance follows from the remarkable fact that they are approximate nonlinear solutions of the MHD equations in the limit of weak magnetic fields. We show in this paper, analytically and with one-dimensional numerical simulations, that this property is also shared by certain axisymmetric MRI modes in vertically-stratified shearing boxes. These channel flows rapidly capture significant amounts of magnetic and kinetic energy, and thus are vulnerable to secondary shear instabilities. We examine these parasites in the vertically stratified context, and estimate the maximum amplitudes that channels attain before they are destroyed. These estimates suggest that a dominant channel flow will usually drive the disk's magnetic field to thermal strengths. The prominence of these flows and their destruction place enormous demands on simulations, but channels in their initial stages also offer a useful check on numerical codes. These benchmarks are especially valuable given the increasing interest in the saturation of the stratified MRI. Lastly we speculate on the potential connection between 'run-away' channel flows and outburst behaviour in protostellar and dwarf nova disks.Comment: 17 pages, 12 figures. MNRAS, accepted

Hydrodynamic fluctuations in the Kolmogorov flow: Linear regime

The Landau-Lifshitz fluctuating hydrodynamics is used to study the statistical properties of the linearized Kolmogorov flow. The relative simplicity of this flow allows a detailed analysis of the fluctuation spectrum from near equilibrium regime up to the vicinity of the first convective instability threshold. It is shown that in the long time limit the flow behaves as an incompressible fluid, regardless of the value of the Reynolds number. This is not the case for the short time behavior where the incompressibility assumption leads in general to a wrong form of the static correlation functions, except near the instability threshold. The theoretical predictions are confirmed by numerical simulations of the full nonlinear fluctuating hydrodynamic equations.Comment: 20 pages, 4 figure