The pervasive use of screens, averaging nearly 7 hours per day globally across mobile phones, computers, notebooks and TVs, has sparked a growing desire to minimize reflections from ambient lighting and enhance readability in harsh lighting conditions, without the need to increase screen brightness. This demand highlights a significant need for advanced anti-glare (AG) technologies, to increase comfort and eventually reduce energy consumption of the devices. Currently used production technologies are limited in their texture designs, which can lead to suboptimal performance of the anti-glare texture. To overcome this design limitation and improve performance of the anti-glare feature, this work reports a new, cost-effective high-volume production method that enables the much needed design freedom on a large area. This is achieved by combining mastering via large-area Laser Beam Lithography (LBL) and replication by Nanoimprint Lithography (NIL) processes. The environmental impact of the production method, such as material consumption, are considered and the full cycle from design to final imprint is discussed.

For a left module RM over non-commutative ring R, the notion for the class of nilpotent elements (nilR(M)) was first introduced and studied by SSevviiri and Groenewald . Moreover, Armendariz and semicommutative modules are generalizations of reduced modules and nilR(M)=0 in the case of reduced modules. Thus, the nilpotent class plays a vital role in these modules. Motivated by this, we present the concept of nil-Armendariz modules as a generalization of reduced modules and a refinement of Armendariz modules, focusing on the class of nilpotent elements. Further, we demonstrate that the quotient module M/N is nil-Armendariz if and only if N is within the nilpotent class of RM. Additionally, we establish that the matrix module Mn(M) is nil-Armendariz over Mn(R) and explore conditions under which nilpotent classes form submodules. Finally, we prove that nil-Armendariz modules remain closed under localization.

Occasionally, officials of the world’s regulatory agencies embark upon attempts at bringing previously unregulated physical systems under regulation by them. Each such attempt raises an epistemological issue. At issue is whether enough information about the outcomes of the events of the future, given the outcomes of the events of the present, will be in the hands of a would-be regulator for this regulator to regulate effectually. If present, this information is provided by runs of a model of the physical system that is slated for regulation. Ideally, this model makes an argument that draws its conclusion from the evidence presented to it. If so, this argument is of the form of a predictive inference. However, the process by which an argument draws its conclusion from the evidence may go awry. This happens, for example, when the axiom of probability theory called unit measure is falsified by a conclusion that is drawn from the evidence by this argument.A method is derived from first principles for determination of whether unit measure is satisfied or falsified by an argument made by a model, given that this argument may attach unusual meanings to statistical terms. This method is used in a study of whether unit measure is satisfied or falsified by the arguments that are made by a pair of models. Both models are in active use by regulatory agencies around the world. Under neither argument do runs of the model provide an official of a regulatory agency with the information gain aka mutual information that he or she would need to regulate effectually. Under both arguments, attachment of unusual meanings to statistical terms creates the illusion that such an official can and does regulate effectually.

In this note we consolidate and give a brief description of several recently published results in ring theory having disproof. These results have been published during 2016 to 2021 in the so called non-predatory reputed mathematical journals indexed in the well known database like Scopus. We have considered results on rings in which each element is a sum of two idempotents appeared in Canad. Math. Bull. (2016), weakly tripotent rings appeared in Bull. Korean Math. Soc. (2018) and Rendiconti Sem. Mat. Univ. Pol. Torino (2021), involution t-clean rings appeared in Eur. J. Pure Appl. Math (2022) and strongly 2-nil clean rings with units of order two appeared in Eur. J. Pure Appl. Math (2023).

The primary purpose of this research article is to study the famous K$\ddot{o}$the's Conjecture for the special cases - when a ring $\mathfrak{R}$ is commutative, and when $\mathfrak{R}$ is an Algebra over a field. Furthermore, some open problems involving tensor product of Algebras over a field has also been discussed extensively along with their proofs. Subsequently, an equivelent statement of K$\ddot{o}$the's problem in terms of algebra of matrices has been established.