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A systematic construction approach for all 4×4 involutory MDS matrices
No full textMaximum distance separable (MDS) matrices play a crucial role not only in coding theory but also in the design of block ciphers and hash functions. Of particular interest are involutory MDS matrices, which facilitate the use of a single circuit for both encryption and decryption in hardware implementations. In this article, we present several characterizations of involutory MDS matrices of even order. Additionally, we introduce a new matrix form for obtaining all involutory MDS matrices of even order and compare it with other matrix forms available in the literature. We then propose a technique to systematically construct all 4×4 involutory MDS matrices over a finite field F2m. This method significantly reduces the search space by focusing on involutory MDS class representative matrices, leading to the generation of all such matrices within a substantially smaller set compared to considering all 4×4 involutory matrices. Specifically, our approach involves searching for these representative matrices within a set of cardinality (2m-1)5. Through this method, we provide an explicit enumeration of the total number of 4×4 involutory MDS matrices over F2m for m=3,4,…,8
Algebraic Aspects and Functoriality of the Set of Affiliated Operators
No full textIn this article, we aim to provide a satisfactory algebraic description of the set of affiliated operators for von Neumann algebras. Let be a von Neumann algebra acting on a Hilbert space, and let denote the set of unbounded operators of the form for with, where denotes the Kaufman inverse. We show that is closed under sum, product, Kaufman-inverse, and adjoint, and has the structure of a (right) near-semiring. Moreover, the above quotient representation of an operator in is essentially unique. Thus, we may view as the multiplicative monoid of unbounded operators on generated by and. We further show that our definition of affiliation, as reflected in, subsumes the traditional one. Let be a unital normal ∗-homomorphism between represented von Neumann algebras and. Using the quotient representation, we obtain a canonical extension of to a mapping which is a near-semiring homomorphism that respects Kaufman-inverse and adjoint; in addition, respects Murray-von Neumann affiliation of operators and also respects strong sum and strong product. Thus, is intrinsically associated with and transforms functorially as we change representations of. Furthermore, preserves operator properties such as being symmetric, or positive, or accretive, or sectorial, or self-adjoint, or normal, and also preserves the Friedrichs and Krein-von Neumann extensions of densely defined closed positive operators. As a proof of concept, we transfer some well-known results about closed unbounded operators to the setting of closed affiliated operators for properly infinite von Neumann algebras, via abstract nonsense
Application of artificial intelligence tools in wastewater and waste gas treatment systems: Recent advances and prospects
No full textThe non-linear complex relationships among the process variables in wastewater and waste gas treatment systems possess a significant challenge for real-time systems modelling. Data driven artificial intelligence (AI) tools are increasingly being adopted to predict the process performance, cost-effective process monitoring, and the control of different waste treatment systems, including those involving resource recovery. This review presents an in-depth analysis of the applications of emerging AI tools in physico-chemical and biological processes for the treatment of air pollutants, water and wastewater, and resource recovery processes. Additionally, the successful implementation of AI-controlled wastewater and waste gas treatment systems, along with real-time monitoring at the industrial scale are discussed
Bacteria as ecosystem engineers: Unraveling clues through a novel functional response and tritrophic model
No full textBacteria play a crucial role in regulating the nutrient cycle of ecosystems, making their abundance essential for the sustainability of these environments. Maintaining a thriving bacterial population is achieved through complex nutrient–bacteria–protozoa interactive dynamics, which can be likened to a tritrophic food chain model. The interaction between bacteria and protozoa is governed by a critical factor known as the bacteria–protozoa functional response, which serves as the foundation for developing this tritrophic model. However, existing functional responses have shown limitations in accurately describing the intricacies of the bacteria–protozoa interaction. One significant drawback is the neglect of bacterial behavioral traits. To address this issue, we consider the concept of cooperation as a group defense mechanism employed by bacteria facilitated through a quorum-sensing communication process. By incorporating the cooperation trait into the functional response, our model offers a more comprehensive understanding of the complex tritrophic food chain dynamics. We evaluate the stability of different equilibrium points, along with Hopf-bifurcation around the coexistence equilibrium point. We find that a balance between strong group defense and moderate cooperation is essential for bacteria sustainability and overall system stability. Our results also elaborately address the effects of the increasing group defense through the bistable equilibrium followed by a branch point and saddle–node bifurcation. Through comprehensive analyses and simulations, we examine the paradox of enrichment in nutrition flow at the community level and explore how nutrient washout controls system stability. This innovation not only enhances our comprehension of ecosystem sustainability but also opens up new avenues for studying the intricate relationships that govern the overall balance of nature
Change in Religious Composition across Districts in India from 2001 to 2011 A Descriptive Analysis of the Religion Census
No full textCharacterization of genuine ramification using formal orbifolds
No full textWe give a characterization of genuinely ramified maps of formal orbifolds in the Tannakian framework. In particular we show that a morphism is genuinely ramified if and only if the pullback of every stable bundle remains stable in the orbifold category. We also give some other characterizations of genuine ramification. This generalizes the results of [BKP1] and [BP1]. In fact, it is a positive characteristic analogue of results in [BKP2]
Coalgebraic fuzzy geometric logic
No full textA generalized form of modal logic can be created within the context of coalgebraic logic. Coalgebraic geometric logic was recently developed by adding modalities to the language of propositional geometric logic using the coalgebra approach. However, as far as we are aware, no studies have been done specifically on fuzzy geometric modal logic. This study is the first step towards developing fuzzy geometric modal logic using coalgebra theory. This new logic might potentially be used to model and reason about transition systems that involve uncertainty in behaviour. We propose a theoretical framework based on coalgebra theory to add modalities into the language of fuzzy geometric logic. Coalgebras for an endofunctor on a category of fuzzy topological spaces and fuzzy continuous maps serve as the foundation for models of this logic. Our key finding is the existence of a final model in the category of models for endofunctors defined on sober fuzzy topological spaces. Furthermore, we present a comparative analysis of the notions of behavioural equivalency, bisimulation, and modal equivalency on the resulting class of models
Community electrification and women\u27s autonomy
No full textThis study examines the effects of community-level electrification on women\u27s social autonomy in India using panel household survey, administrative and satellite data spanning over two decades. Using flexible difference-in-difference estimators, we find that higher community-level electricity hours improve women\u27s autonomy captured via reduced incidence of sexual violence against women, improved mobility, fertility choices, and access to health care. Results are robust to use of night-time luminosity as an alternative indicator for community electrification, most recent data on electrification and alternative longitudinal estimation techniques. A heterogeneity analysis shows that the effects are stronger in rural areas, where women\u27s autonomy is more restricted. We identify four main channels of the impact: paid employment, education, exposure to mass media and safety
Commuting tuple of multiplication operators homogeneous under the unitary group
No full textLet (Formula presented.) be the group of (Formula presented.) unitary matrices. We find conditions to ensure that a (Formula presented.) -homogeneous (Formula presented.) -tuple (Formula presented.) is unitarily equivalent to multiplication by the coordinate functions on some reproducing kernel Hilbert space (Formula presented.), (Formula presented.). We describe this class of (Formula presented.) -homogeneous operators, equivalently, nonnegative kernels (Formula presented.) quasi-invariant under the action of (Formula presented.). We classify quasi-invariant kernels (Formula presented.) transforming under (Formula presented.) with two specific choice of multipliers. A crucial ingredient of the proof is that the group (Formula presented.) has exactly two inequivalent irreducible unitary representations of dimension (Formula presented.) and none in dimensions (Formula presented.), (Formula presented.). We obtain explicit criterion for boundedness, reducibility, and mutual unitary equivalence among these operators
Compact embeddings, eigenvalue problems, and subelliptic Brezis–Nirenberg equations involving singularity on stratified Lie groups
No full textThe purpose of this paper is twofold: first we study an eigenvalue problem for the fractional p-sub-Laplacian over the fractional Folland–Stein–Sobolev spaces on stratified Lie groups. We apply variational methods to investigate the eigenvalue problems. We conclude the positivity of the first eigenfunction via the strong minimum principle for the fractional p-sub-Laplacian. Moreover, we deduce that the first eigenvalue is simple and isolated. Secondly, utilising established properties, we prove the existence of at least two weak solutions via the Nehari manifold technique to a class of subelliptic singular problems associated with the fractional p-sub-Laplacian on stratified Lie groups. We also investigate the boundedness of positive weak solutions to the considered problem via the Moser iteration technique. The results obtained here are also new even for the case p=2 with G being the Heisenberg group