Global optimal approximate solutions of best proximity points

In this paper, we introduce the concept of contractive pair maps and give some necessary and sufficient conditions for existence and uniqueness of best proximity points for such pairs. In our approach, some conditions have been weakened. An application has been presented to demonstrate the usability of our results. Also, we introduce the concept of cyclic ?-contraction and cyclic asymptotic ?-contraction and give some existence and convergence theorems on best proximity point for cyclic ?-contraction and cyclic asymptotic ?-contraction mappings. The presented results extend, generalize and improve some known results from best proximity point theory and fixed-point theory.

Global Climatologies of Fronts, Airmass Boundaries, and Airstream Boundaries: Why the Definition of “Front” Matters

Climatologies of fronts, airmass boundaries, and airstream boundaries can be calculated using automated approaches on gridded data. Such approaches may require choices to define a front, including a quantity (or quantities) to diagnose the front, a mathematical function(s) that operates upon the quantity to produce a diagnostic field, a level(s) at which the field is calculated, and a minimum threshold(s) in the magnitude of the field. To understand how resulting climatologies depend upon these choices using a consistent dataset, ERA-Interim reanalyses from 1979 to 2016 are used to construct global monthly climatologies for various definitions of fronts and airstream boundaries from potential temperature, equivalent potential temperature, water vapor mixing ratio, and wind, including gradients, thermal front parameter, frontogenesis, and asymptotic contraction rate at the surface and 850 hPa. Maps of automated fronts are similar to manual analyses when about 10% of the map is identified as a front. Definitions of fronts that use potential temperature or frontogenesis produce climatologies similar to those of manually analyzed fronts with maxima along the major storm tracks and their seasonal migrations. In contrast, definitions that use equivalent potential temperature or the thermal front parameter produce fewer fronts at higher latitudes and more fronts at lower latitudes, more akin to airmass boundaries than fronts. Although surface fronts defined by thermodynamic quantities are more infrequent over the oceans than at 850 hPa, they are more frequent when using metrics that include the wind field (e.g., frontogenesis, asymptotic contraction rate).

Three-Dimensional Storm Structure and Low-Level Boundaries at Different Stages of Cyclic Mesocyclone Evolution in a High-Precipitation Tornadic Supercell

Nearly continuous wind retrievals every three minutes for an unprecedented 90-minute period were constructed during multiple mesocyclone cycles in a tornadic high-precipitation supercell. Asymptotic contraction rate analysis revealed the relationship between the primary and secondary rear-flank gust fronts (RFGF and SRFGFs) and the rear-flank downdraft (RFD) and occlusion downdrafts. This is thought to be the first radar-based analysis where the relationship between the near-surface gust fronts and their parent downdrafts has been explored for sequential mesocyclones. Changes in the SRFGFs were associated with surges in the RFD. During part of the mesocyclone lifecycle, the SRFGF produced a band of low-level convergence and associated deep updraft along the southwestern side of the hook echo region that ingested the RFD outflow and limited both entrainment into the RFD and reinforcement of low-level convergence along the leading edge of the primary RFGF. The second mesocyclone intensified from stretching in an occlusion updraft rather than in the primary updraft. This low-level mesocyclone remained well separated from the updraft shear region vorticity that was associated with a more traditional midlevel mesocyclone. However, the third mesocyclone initiated in the vorticity-rich region of the primary updraft zone and was amplified by stretching in the primary updraft.

Fixed Point Theorems for Asymptotically Contractive Multimappings

We present fixed point theorems for a nonexpansive set-valued mapping from a closed convex subset of a reflexive Banach space into itself under some asymptotic contraction assumptions. Some existence results of coincidence points and eigenvalues for multimappings are given.

Fixed Points of Nonlinear and Asymptotic Contractions in the Modular Space

A fixed point theorem for nonlinear contraction in the modular space is proved. Moreover, a fixed point theorem for asymptotic contraction in this space is studied.

Contraction Rate and Its Relationship to Frontogenesis, the Lyapunov Exponent, Fluid Trapping, and Airstream Boundaries

Although a kinematic framework for diagnosing frontogenesis exists in the form of the Petterssen frontogenesis function and its vector generalization, a similar framework for diagnosing airstream boundaries (e.g., drylines, lee troughs) has not been constructed. This paper presents such a framework, beginning with a kinematic expression for the rate of change of the separation vector between two adjacent air parcels. The maximum growth rate of the separation vector is called the instantaneous dilatation rate and its orientation is called the axis of dilatation. Similarly, a maximum decay rate is called the instantaneous contraction rate and its orientation is called the axis of contraction. These expressions are related to the vector frontogenesis function, in that the growth rate of the separation vector corresponds with the scalar frontogenesis function, and the rotation rate of the separation vector corresponds with the rotational component of the vector frontogenesis function. Because vorticity can rotate air-parcel pairs out of regions of deformation, the instantaneous dilatation and contraction rates and axes may not be appropriate diagnostics of airstream boundaries for fluid flows in general. Rather, the growth rate and orientation of an airstream boundary may correspond better to the so-called asymptotic contraction rate and the asymptotic axis of dilatation, respectively. Expressions for the asymptotic dilatation and contraction rates, as well as their orientations, the asymptotic dilatation and contraction axes, are derived. The asymptotic dilatation rate is related to the Lyapunov exponent for the flow. In addition, a fluid-trapping diagnostic is derived to distinguish among adjacent parcels being pulled apart, being pushed together, or trapped in an eddy. Finally, these diagnostics are applied to simple, idealized, steady-state flows and a nonsteady idealized vortex in nondivergent, diffluent flow to show their utility for determining the character of air-parcel trajectories and airstream boundaries.

On the laminar flow in a free jet of liquid at high Reynolds numbers

This paper is concerned with the jet of liquid, open to the atmosphere, that emerges from a two-dimensional channel in which there is Poiseuille flow far upstream, the flow being driven by an applied pressure gradient. The problem is discussed with the aid of the method of matched asymptotic expansions; the small parameter involved is the inverse Reynolds number. A boundary layer forms adjacent to the free surface, and a classical boundary-layer analysis is applied to find the flow there (for moderate distances downstream); the influence of this boundary layer on the flow in the core of the jet is then investigated. Higher-order boundary-layer effects, such as indeterminacy and eigensolutions, are also discussed. The first few terms are found of an asymptotic expansion for the equation of the free surface, and considerations of momentum balance are applied to find the asymptotic contraction ratio of the jet.