On generalized γµ-closed sets and related continuity

In this paper our main interest is to introduce a new type of generalized open sets defined in terms of an operation on a generalized topological space. We have studied some properties of this newly defined sets. As an application, we have introduced some weak separation axioms and discussed some of their properties. Finally, we have studied some preservation theorems in terms of some irresolute functions.

Applications of operations on generalized topological spaces

In this paper γµ -open sets and γµ -closed sets in a GTS (X, µ) have been studied, where γµ is an operation from µ to P(X). In general, collection of γµ -open sets is smaller than the collection of µ-open sets. The condition under which both are same are also established here. Some properties of such sets have been discussed. Some closure like operators are also defined and their properties are discussed. The relation between similar types of closure operators on the GTS (X, µ) has been established. The condition under which the newly defined closure like operator is a Kuratowski closure operator is given. We have also defined a generalized type of closed sets termed as γµ -generalized closed set with the help of this newly defined closure operator and discussed some basic properties of such sets. As an application, we have introduced some weak separation axioms and discussed some of their properties. Finally, we have shown some preservation theorems of such generalized concepts.

The scenery flow of self-similar measures with weak separation condition

We show that self-similar measures on $\mathbb R^d$ satisfying the weak separation condition are uniformly scaling. Our approach combines elementary ergodic theory with geometric analysis of the structure given by the weak separation condition.

The adaptation front equation explains diversification hotspots and living-fossilization

Taxonomic turnovers are common in the evolutionary histories of biological communities. Such turnovers are often associated with the emergence and diversification of groups that have achieved fundamental innovations beneficial in various ecological niches. In the present study, we show that such innovation-driven turnovers could be analyzed using an equation that describes the dynamics of zero-fitness isoclines in a two-dimensional trait space comprising a "fundamental trait" (describing fundamental innovation) and a "niche trait" (describing niche position) or with its higher-dimensional extensions. Our equation allows analytical prediction of evolutionary source-sink dynamics along the niche axis for an arbitrary unimodal (or multimodal with weak separation) carrying capacity distribution. The prediction was confirmed by numerical simulation under different assumptions for resource competition, reproduction, and mutation. In the simulated evolution, biodiversity sources are the central niches having higher carrying capacities than the outer niches, allowing species there the faster evolutionary advancement in fundamental traits and their repeated diversification into outer niches, which outcompete the indigenous less advanced species. The outcompeted species go extinct or evolve directionally toward the far outer niches of the far slower advancement because of the far lower carrying capacities. In consequence of this globally acting process over niches, species occupying peripheral (i.e., the outermost) niches can have significantly primitive fundamental traits and deep divergence times from their closest relatives, and thus, they correspond to living fossils. The extension of this analysis for multiple geographic regions showed that living fossils are also expected in geographically peripheral regions for the focal species group.

Prosecutorial Independence in Lesotho: A Critique of the Model

The Constitution of Lesotho is substantially cast on the Westminster prototype. As such, its institutions, by and large, reflect the structure of similar institutions at Westminster. The institution of the Director of Public Prosecutions (DPP) is no different: it has been designed to mirror its namesake under the British constitutional design. The underlying feature of classic British-based constitutions is the weak separation of powers and the predominance and condescending nature of the executive branch of government, as incarnated by the office of the Prime Minister. As such, most institutions within the design are beholden to the executive in general and to the Prime Minister in particular. The institution of the DPP is integral in the administration of criminal justice. Hence, its independence and its accountability in the discharge of this important constitutional mandate are of paramount importance. This notwithstanding, the Constitution of Lesotho is generally weak on safeguarding the independence of the office of the DPP and ensuring its corresponding duty of accountability. The purpose of this article is to critique the constitutional design in relation to the office of the DPP and to expose the deficiency of the constitutional clause establishing the Lesotho DPP office. The article contends that while the Constitution, under section 141, provides for some small measure of independence of this office, the broader schematisation of the Constitution is feeble on the independence and accountability of the office. The article analyses the constitutional design of the Lesotho DPP office in comparison with international developments.

A note on one-loop cluster adjacency in $$ \mathcal{N} $$ = 4 SYM

We study cluster adjacency conjectures for amplitudes in maximally supersymmetric Yang-Mills theory. We show that the n-point one-loop NMHV ratio function satisfies Steinmann cluster adjacency. We also show that the one-loop BDS-like normalized NMHV amplitude satisfies cluster adjacency between Yangian invariants and final symbol entries up to 9-points. We present conjectures for cluster adjacency properties of Plücker coordinates, quadratic cluster variables, and NMHV Yangian invariants that generalize the notion of weak separation.

Image space analysis for uncertain multiobjective optimization problems: Robust optimality conditions

<p style='text-indent:20px;'>We introduce the <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{C} $\end{document}</tex-math></inline-formula>-robust efficient solution and optimistic <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{C} $\end{document}</tex-math></inline-formula>-robust efficient solution of uncertain multiobjective optimization problems (UMOP). By using image space analysis, robust optimality conditions as well as saddle point sufficient optimality conditions for uncertain multiobjective optimization problems are established based on real-valued linear (regular) weak separation function and real-valued (vector-valued) nonlinear (regular) weak separation functions. We also introduce two inclusion problems by using the image sets of robust counterpart of (UMOP) and establish the relations between the solution of the inclusion problems and the <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{C} $\end{document}</tex-math></inline-formula>-robust efficient solution (respectively, optimistic <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{C} $\end{document}</tex-math></inline-formula>-robust efficient solution) of (UMOP).</p>

Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.