The Okubo-Weiss criterion in hydrodynamic flows:Geometric aspects and further extension

The Okubo [5]-Weiss [6] criterion has been extensively used as a diagnostic tool to divide a two-dimensional (2D) hydrodynamical flow field into hyperbolic and elliptic regions and to serve as a useful qualitative guide to the complex quantitative criteria. The Okubo-Weiss criterion is frequently validated on empirical grounds by the results ensuing its application. So, we will explore topological implications into the Okubo-Weiss criterion and show the Okubo-Weiss parameter is, to within a positive multiplicative factor, the negative of the Gaussian curvature of the underlying vorticity manifold. The Okubo-Weiss criterion is reformulated in polar coordinates, and is validated via several examples including the Lamb- Oseen vortex, and the Burgers vortex. These developments are then extended to 2D quasi- geostrophic (QG) flows. The Okubo-Weiss parameter is shown to remain robust under the -plane approximation to the Coriolis parameter. The Okubo-Weiss criterion is shown to be able to separate the 2D flow-field into coherent elliptic structures and hyperbolic flow configurations very well via numerical simulations of quasi-stationary vortices in QG flows. An Okubo-Weiss type criterion is formulated for 3D axisymmetric flows, and is validated via application to the round Landau-Squire Laminar jet flow.

SYSTEMATIZATION OF IDENTIFICATION CRITERIA AND POSITIONING PARAMETERS OF TERRITORIAL COMMUNITIES: INNOVATIVE ASPECT

The article deepens theoretical and methodical provisions on the attractive image formation for the territorial communities in terms of systematization of their identification criteria and positioning parameters, strengthened in this context local innovative aspect of the implementation of sustainable development concept. The theoretical and methodical provisions regarding the marketing tools of economic entities at the local level have been analysed, which made it possible to research the territorial community as a complex and innovatively active system characterized by emergence and synergistic metasystem connections between its elements that require ordering. This ordering became possible due to the methodological research platform, which based on the positions of the systemic and integrated approach, D. Forrester counterintuitive method of system dynamics, A. Marshall partial equilibrium approach and the principle of stable disequilibrium. The authors proved that the process of determining and ordering identification criteria and positioning parameters depends on the existing institutional frames for the functioning and development of territorial communities, which have a narrowing effect on the search opportunities and alternatives in choice. The discreteness of the coverage of the unique characteristics, which made it possible to distinguish the community among many others based on the established scientific hypothesis, justified the choice of 15 identification criteria and 13 positioning parameters that form the idea of a territorial unit. The provisions of the article systematized of certain criteria and parameters by constructing appropriate classification schemes based on a descending hierarchical chain of classification taxa “class – type – criterion / parameter – result (value) of criterion / parameter”. This made it possible to distinguish between the classes of identification criteria into key and auxiliary ones, and the classes of positioning parameters into single (quantitative) and complex (qualitative) ones. The article proves importance of the implementation of identification criteria and positioning parameters in the practice of the territorial communities strategizing to improve their innovative activity and determine key development positions in a strategic perspective.

On Uniform Nonintegrability and Weak Uniform Nonintegrability of a Sequence of Random Variables with Respect to a Nonnegative Array

Chandra, Hu and Rosalsky [1] introduced the notion of a sequence of random variables being uniformly nonintegrable and they established a de La Vallée Poussin type criterion for this notion. Inspired by the Chandra, Hu and Rosalsky [1] article, Hu and Peng [2] introduced the weaker notion of a sequence of random variables being weakly uniformly nonintegrable and they also established a de La Vallée Poussin type criterion for this notion using a modification of the Chandra, Hu and Rosalsky [1] argument. In this correspondence, we introduce the more general notion of uniform nonintegrability and weak uniform nonintegrability with respect to an array of nonnegative real numbers together with a de La Vallée Poussin type criterion for this notion. This criterion immediately yields as particular cases the criteria of Chandra, Hu and Rosalsky [1] and Hu and Peng [2] , and it has a substantially simpler and more straightforward proof.

Use of the Modified Riccati Technique for Neutral Half-Linear Differential Equations

We study the second-order neutral half-linear differential equation and formulate new oscillation criteria for this equation, which are obtained through the use of the modified Riccati technique. In the first statement, the oscillation of the equation is ensured by the divergence of a certain integral. The second one provides the condition of the oscillation in the case where the relevant integral converges, and it can be seen as a Hille–Nehari-type criterion. The use of the results is shown in several examples, in which the Euler-type equation and its perturbations are considered.

Oscillation Criteria of Higher-order Neutral Differential Equations with Several Deviating Arguments

This work is concerned with the oscillatory behavior of solutions of even-order neutral differential equations. By using the technique of Riccati transformation and comparison principles with the second-order differential equations, we obtain a new Philos-type criterion. Our results extend and improve some known results in the literature. An example is given to illustrate our main results.