One of the ramifications of the COVID-19 pandemic is that it has lent urgency to ongoing discussions on mental well-being. While standard techniques are available to diagnose mental health disorders such as depression, anxiety, and stress, ambiguity persists regarding the emotional aspect of well-being. Emotional well-being (EWB) is a recently developed concept that seeks to understand the contribution of emotions to one’s well-being. Interactive approaches for such investigations are recommended to understand people’s contextual experiences. In this study, we use a qualitative approach to examine the psychological impact of the built environment on its residents to formulate the desiderata for an ideal place based on their perceptions. We use personal construct theory and the associated qualitative repertory grid technique. We recruited 15 students from the United Arab Emirates University and obtained information on their perceptions of three built environments they experienced. The findings revealed information on the students’ emotional constructs that were associated with a set of design-related factors, and the way the ideal hostel should be characterized in response to these associations. These findings enrich our knowledge of EWB within built environments and can inform their future design by considering the emotional aspect of the well-being of residents.

Let $R$ be a commutative ring with identity. A proper ideal $I$ of a ring $R$ is called a square-difference factor absorbing primary ideal of $R$ if for $a,b\in R$, whenever $a^{2}-b^{2}\in I$, then $a+b\in\sqrt{I}$ or $a-b\in I$. Several characterizations and properties of this class of ideals are presented.Various examples are provided to illustrate the obtained results and demonstrate the applicability of our findings. Furthermore, the properties of this class of ideals are investigated in extensions of rings.

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. The purpose of this paper is to introduce the concept of weakly S-primary ideals as a new generalization of weakly primary ideals. An ideal I of R disjoint with S is called a weakly S-primary ideal if there exists s∈S such that whenever 0≠ab∈I for a,b∈R, then sa∈√I or sb∈I. The relationships among S-prime, S-primary, weakly S-primary and S-n-ideals are investigated. For an element r in any general ZPI-ring, the (weakly) S_{r}-primary ideals are charctarized where S={1,r,r²,⋯}. Several properties, characterizations and examples concerning weakly S-primary ideals are presented. The stability of this new concept with respect to various ring-theoretic constructions such as the trivial ring extension and the amalgamation of rings along an ideal are studied. Furthermore, weakly S-decomposable ideals and S-weakly Laskerian rings which are generalizations of S-decomposable ideals and S-Laskerian rings are introduced.

In this paper, $(\alpha,\beta)$-fuzzy sublattices and $(\alpha,\beta)$-fuzzy ideals of a lattice, where $\alpha\in\{ \in_{\gamma}, q_{\delta}, \in_{\gamma}\vee q_{\delta}\}$, $\beta\in\{ \in_{\gamma}, q_{\delta}, \in_{\gamma}\vee q_{\delta},\in_{\gamma}\wedge q_{\delta}\}$, are introduced and studied. In the special case, $(\in_{\gamma}, \in_{\gamma}\vee q_{\delta})$-fuzzy sublattices and $(\in_{\gamma}, \in_{\gamma}\vee q_{\delta})$-fuzzy ideals of a lattice are presented and their characterizations by three level sublattices and ideals are given.

Let R be a commutative ring with identity and m, n be positive integers. In this paper, we introduce the class of (m,n)-prime ideals which lies properly between the classes of prime and (m,n)-closed ideals. A proper ideal I of R is called (m,n)-prime if for a,b∈R, a^{m}b∈I implies either aⁿ∈I or b∈I. Several characterizations of this new class with many examples are given. Analougus to primary decomposition, we define the (m,n)-decomposition of ideals and show that every ideal in an n-Noetherian ring has an (m,n)-decomposition. Furthermore, the (m,n)-prime avoidance theorem is proved.

Let $a, b$ and $n > 1$ be three positive integers such that $a$ and $\sum_{j=0}^{n-1} b^j$ are relatively prime. In this paper, we prove that the toric ideal $I$ associated to the submonoid of $\mathbb{N}$ generated by $\{\sum_{j=0}^{n-1} b^j\} \cup \{\sum_{j=0}^{n-1} b^j + a\, \sum_{j=0}^{i-2} b^j \mid i = 2, \ldots, n\}$ is determinantal. Moreover, we prove that for $n > 3$, the ideal $I$ has a unique minimal system of generators if and only if $a < b-1$.

In this paper we present two ways to solve the Jacobian conjecture. The first way is an equivalent statement to the Jacobian conjecture using idempotent ideals. The second way is to use an equivalent thesis to the thesis of the Jacobian conjecture. In both cases we will show that the Jacobian conjecture is true.

In this paper we define families of open sets as sums of idempotent ideals and sums of square-free ideals in a ring. We investigate some topological properties of such topologies. We define the square-free spectrum of a ring as the set of all square-free ideals and examine what topological properties it has for a given monoid.

The aim of this paper is to investigate the amenability modulo an ideal of Banach algebras with emphasis on applications to homological algebras. In doing so, we show that amenability modulo an ideal of A^** implies amenability modulo an ideal of A. Finally, for a large class of semigroups, we prove that l¹(S)^** is amenable modulo Iσ^** if and only if an appropriate group homomorphic image of S is finite where Iσ is the closed ideal induced by the least group congruence σ.

Building on the Sustainable Happiness Model, this study examined how congruency between ideal self-image and brand image influence a sense of happiness. The findings show that when ideal self-image and ideal social self-image are congruent with brand image a sense of happiness can be enhanced through brand identification and positive emotions. This study contributes to literature as it reveals the mechanism of how congruency between ideal self-image and brand image positively affect happiness.

In this paper the one-to-one correspondences and equivalences between ideals, primals, filters and grills are introduced. It is shown the local functions and the topological spaces induced by them are the same. From this point of view, the topological properties with respect to one topology can be derived from topological properties valid in the corresponding topology.

Let $n \geq 2$ be a fixed integer and $\mathcal{A}$ be a $C^*$-algebra. A permuting $ n $-linear map $ \mathcal{G} : \mathcal{A} ^{n} \rightarrow \mathcal{A} $ is known to be symmetric generalized $n$-derivation if there exists a symmetric $n$-derivation $ \mathfrak{D} : \mathcal{A} ^{n} \rightarrow \mathcal{A} $ such that $ \mathcal{G} \left(x_{1}, x_{2}, \ldots, x_{i} x_{i}^{\prime}, \ldots, x_{n}\right)= \mathcal{G} \left(x_{1}, x_{2}, \ldots, x_{i}, \ldots, x_{n}\right) x_{i}^{\prime}+x_{i} \mathfrak{D} (x_{1}, x_{2},\ldots, x_{i}^{\prime}, \ldots, x_{n})$ holds for all $x_{i}, x_{i}^{\prime} \in \mathcal{A} $. In this paper, we investigate the structure of $C^*$-algebras involving generalized linear $n$-derivations. Moreover, we describe the forms of traces of linear $n$-derivations satisfying certain functional identity.

This paper focuses on examining a new type of n-additive maps called the symmetric reverse n-derivations. As implied by its name, it combines the ideas of n-additive maps and reverse derivations, with a 1-reverse derivation being the ordinary reverse derivation. We explore several findings that expand our knowledge of these maps, particularly their presence in semiprime rings and the way rings respond to specific functional identities involving elements of ideals. Also, we provide examples to help clarify the concept of symmetric reverse n-derivations. This study aims to deepen the understanding of these symmetric maps and their properties within mathematical structures.

In this paper, we generalize the concept of I-statistical convergence, which is a recently introduced summability method, with an infinite matrix of complex numbers and a modulus function. Lacunary sequences will also be included in our definitions. The name of our new method will be A^{I}-lacunary statistical convergence with respect to a sequence of modulus functions. We also define strongly A^{I}-lacunary convergence and we give some inclusion relations between these methods.

This paper focuses on examining a new type of $n$-additive maps called the symmetric reverse $n$-derivations. As implied by its name, it combines the ideas of $n$-additive maps and reverse derivations, with a $1$-reverse derivation being the ordinary reverse derivation. We explore several findings that expand our knowledge of these maps, particularly their presence in semiprime rings and the way rings respond to specific functional identities involving elements of ideals. Also, we provide examples to help clarify the concept of symmetric reverse $n$-derivations. This study aims to deepen the understanding of these symmetric maps and their properties within mathematical structures.

We study the generalized χ and η cross helicities for non-ideal non-barotropic magnetohydrodynamics (MHD). χ and η, the additional label translation symmetry group, have been used to generalize the cross helicity in ideal flows. Both new helicities are additional topological invariants of ideal MHD. To study there behaviour in non-ideal MHD, we derive the time derivative of both helicities using the non-ideal MHD equations taking viscosity, finite resistivity and heat conduction into account. Physical variables are divided into ideal and non-ideal quantities separately during the mathematical analysis for simplification. The analytical results indicate that χ and η cross helicities are not strict constant of motion in the non-ideal MHD and show the rate of dissipation which is compared to the dissipation of other topological constants of motion.

Crop domestication and breeding considerably increased productivity over centuries, but simultaneously involved unconscious selection against ‘selfish plant behavior’. Paradoxically, modern-day crop breeding largely enhances individual plant-fitness. As agriculture relies on community performance, embracing an “Agroecological Genetics and Genomics” viewpoint might maximize communal yield by matching crop genotypes to target environments.

In our paper, we develop the hypothesis of a general call for high mobility and discuss the consequences of it. First, we examine the contemporary social representations of mobility, looking for a definition of what is seen as being properly mobile. We will then show how intertwined social representations of space and time result in the prevalent vision of an inevitable and constant mobility. Next, we will thus discuss the importance of seeing mobility as much more than its material facet. Our third step will be to propose a formalisation of the contemporary requisite for mobility. Through four imperatives (activity, activation, participation, adaptation), the mobilitarian ideal requires each person and organisation to be constantly active, mobile, flexible, networking, etc. We argue that, today, we are all meant to be highly mobile. We will illustrate this point with the example of the parliamentary documents of the Belgian Prison Act (2005) in which prison is open and porous, good inmates are described as dynamic individuals on the move and the legitimate penitentiary system is a paradoxical mobilisation system. We will conclude by discussing the need to reshape our vision of the prison, considering its apparently paradoxical relation with mobility.

Perfect cardiovascular health (CVH) has been defined by the American Heart Association as the lack of cardiovascular disease and the presence of seven key factors and health behaviors. In this study, we aimed to estimate the prevalence of ideal and poor CVH among the Polish adult population, taking the chosen socio-demographic factors into consideration. This is a cross-sectional study conducted among 3,901 adults without cardiovascular diseases, aged between 35 and 64 years. Participants completed a questionnaire, and they had anthropometric and physiological measurements taken. Blood samples were analyzed for fasting glucose and cholesterol levels. Ideal CVH was found in 5.4% of the participants, with the advantage of being toward city dwellers over those living in the rural areas (6.3% vs. 5.0%) p = 0.02. In case of the residents of rural areas, their likelihood of having an ideal body mass index (BMI) was found to be 22% lower (odds ratio (OR) = 0.78; 95% CI: 0.66–0.92), their likelihood of having an ideal diet was found to be 27% lower (OR = 0.71; 95% CI: 0.54–0.94), their likelihood of having perfect blood pressure was found to be 29% lower (OR = 0.71; 95% CI: 0.56–0.89), and their likelihood of having the perfect glucose levels were found to be 28% lower (OR = 0.72; 95% CI: 0.63–0.84), than the residents of urban areas. The prevalence of ideal cardiovascular behaviors and factors is lower in the rural community compared with people living in the city. Results indicate that more effort should be dedicated toward the country’s health policy, specifically concerning primary prevention. Preventive actions in the field of cardiovascular disease should be addressed to the residents of rural areas to a larger extent.

This paper introduces a class of symmetric ciphering systems with a finite secret key, which provides ideal secrecy, autonomy in key generation and distribution, and robustness against the probabilistic structure of messages (Ideally Secret Autonomous Robust – ISAR system). The ISAR system is based on wiretap polar codes constructed over an artificial wiretap channel with a maximum secrecy capacity of 0.5. The system autonomously maintains a minimum level of key equivocation by continuously refreshing secret keys without needing additional key generation and distribution infrastructure. Thus, this class of systems generalizes Shannon's ideal and strongly ideal ciphering systems. Additionally, it can transform any stream ciphering system with a finite secret key of known length into an ISAR system without the necessity of knowing and/or changing its algorithm. Therefore, this class of system strongly supports privacy, a critical requirement for contemporary security systems. The ISAR system's reliance on wiretap polar coding for strong secrecy ensures resistance to passive attacks with known plaintext messages. Furthermore, resistance to passive attacks on generated keys follows directly from ideal secrecy and autonomy. The results presented offer an efficient methodology for synthesizing this class of systems with predetermined security margins and a complexity of the order of nlogn, where n is the block length of the applied polar code.

In this paper for every number field K generated by a root α of a trinomial x⁷ + ax + b ∈ Z[x] and for every prime integer p, we calculate ν_p(i(K)), the highest power of p dividing the index i(K) of the field K. In particular, we calculate the index i(K). As application, when the index of K is not trivial, then K is not monogenic.

Throughout this paper, we will establish a comprehensive theoretical foundation and rigorously develop the methodology to investigate the structure of quotient ring S/P under the influence of symmetric generalized n-derivations, where S represents an arbitrary ring, and P signifies a prime ideal of S satisfying certain algebraic identities acting on prime ideal P without relying on the assumption of primeness or semi-primeness of the ring.

A well-known result of Posner’s second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation is central, then the ring is commutative. In the present paper, we extend this bluestocking theorem to an arbitrary ring with involution involving prime ideals. Further, apart from proving several other interesting and exciting results, we establish *-version of Vukman’s theorem [48, Theorem 1]. Precisely, we describe the structure of quotient ring A/L, where A is an arbitrary ring and L is a prime ideal of A. Further, by taking advantage of the *-version of Vukman’s theorem, we show that if a 2-trosion free semiprime ring A with involution admits a nonzero *-centralizing derivation, then A contains a nonzero central ideal. This result is in a spirit of the classical result due to Bell and Martindale [19, Theorem 3]. As the applications, we extends and unify several classical theorems proved in [6],[25,[42], and [48] . Finally, we conclude our paper with a direction for further research.

The continuous casting tundish is non-isothermal due to heat losses and temperature variation from the inlet stream generating relevant convection forces. This condition is commonly avoided through qualitative fluid dynamic analysis only. This work searches to establish the conditions for which non-isothermal simulations are mandatory or for which isothermal simulations are enough to accurately describe the fluid dynamics inside the tundish by quantifying the buoyant and inertial forces. The mathematical model, simulated by CFD software, considers the Navier-Stokes equations, the realizable k- model for solving the turbulence, and the Lagrangian discrete face to track the inclusion trajectories. The results show that temperature does not significantly impact the volume fraction percentages or the mean residence time results, nevertheless, bigger velocity magnitudes under non-isothermal conditions than in isothermal conditions and noticeable changes in the fluid dynamics between isothermal and non-isothermal cases in all the zones where buoyancy force dominates over inertial forces were observed. Because of the results, it is concluded that isothermal simulations can accurately describe the flow behavior in tundishes when the flow control devices control the fluid dynamic, but the simulation without control devices or with a weak fluid dynamic dependence on the control devices requires non-isothermal simulations.

Understanding the stability limit of crystalline materials under variable tensile stress conditions is of capital interest for their technological applications. In this study, we present results from first-principles density functional theory calculations that quantitatively account for the response of selected covalent and layered materials to general stress conditions. In particular, we have evaluated the ideal strength along the main crystallographic directions of 3C and 2H polytypes of SiC, hexagonal ABA stacking of graphite and 2H-MoS2. Transverse superimposed stress on the tensile stress was taken into account in order to evaluate how the critical strength is affected by these multi-load conditions. In general, increasing transverse stress from negative to positive values leads to the expected decreasing of the critical strength. Few exceptions found in the compressive stress region correlate with the trends in the density of bonds along the directions with the unexpected behavior. In addition, we propose a modified spinodal equation of state able to accurately describe the calculated stress-strain curves. This analytical function is of general use and can also be applied to experimental data anticipating critical strengths and strains values and providing informattion on the energy stored in tensile stress processes.

For the problem of poor accuracy of the existing multi-rate simulation methods, this paper proposes a multi rate real-time simulation method based on the Norton equivalent, compared with multi-rate simulation method based on the ideal source equivalent. After the Norton equivalence of the fast subsystem and the slow subsystem, they are obtained simultaneously at the junction nodes. In order to reduce the amount of simulation calculation, the Norton equivalent circuit is obtained by incremental calculation. The data interface between the fast subsystem and the slow subsystem is realized by extrapolation method. For ensuring the real-time performance of the simulation, the method that the slow subsystem calculates ahead of the fast subsystem is given for the slow subsystem with a large amount of calculation. Finally, the AC/DC hybrid power system was simulated on the real-time simulation platform (FRTDS), and the simulation results were compared with the single-rate simulation, which verified the correctness and accuracy of the method.

I present in this paper Diatomic, an open-source Excel application that calculates molar ther-modynamic properties for diatomic ideal gases. This application is very easy to use and requires only a limited number of molecular constants, which are freely available online. Despite its sim-plicity, Diatomic provides methodologies and results that are usually unavailable in general quantum chemistry packages. This application uses the general formalism of statistical mechanics, enabling two models to describe the rotational structure and two models to describe the vibra-tional structure. In this work, Diatomic was used to calculate standard molar thermodynamic properties for a set of fifteen diatomic ideal gases. Special emphasis was placed on the analysis of four properties (standard molar enthalpy of formation, molar heat capacity at constant pressure, average molar thermal enthalpy and standard molar entropy), which were compared with ex-perimental values. The results obtained lead to the following main conclusions: (i) the theoretical model used to describe rotational structure has a small effect on the accuracy of the results, (ii) at moderate and high temperatures, the Morse model gives better results than the harmonic model, and (iii) at the lowest temperatures both models give similar results.

This Paper intends to introduce the notion of prime ideals of Intuitionistic fuzzy normed Rings and to establish basic properties related to it. It investigates these notions and shown anew Result using intuinistic fuzzy points and non membership function incorparating with t-norm and s-norm to show aome results of fuzzy prime ideal.

We calculate the hydrodynamic forces exerted on an oscillating circular cylinder when it moves translationally perpendicular to its axis in the infinitely deep water covered by compressed ice. The cylinder can oscillate both horizontally and vertically. In the linear approximation, we find a solution for the steady wave motion generated by the cylinder within the hydrodynamic set of equations for the incompressible ideal fluid. We show that depending on the rate of ice compression, the normal and anomalous dispersion can occur in the system. In the latter case, the group velocity can be opposite to the phase velocity in a certain range of wavenumbers. We investigate the dependences of the hydrodynamic loads (added mass, damping coefficients, wave resistance, and lift force) exerting on the cylinder on the translational velocity and frequency of oscillation. It is shown that there is a possibility of the appearance of negative values for the damping coefficients at the relatively big cylinder velocity; then the wave resistance decreases with increasing of cylinder velocity. The theoretical results are underpinned by the numerical calculations for the real parameters of ice and cylinder motion.

We present detailed models at the hydrodynamics and decoherence phenomena observed in ideal quantum gases. The model is expressed in terms of a quantum field theory, where the action involves both hydrodynamic and decoherence terms. The density-current relation is derived, and the current Green function is introduced. The effective action for two fields is presented, and the physical field is defined in terms of the bare action. Identities such as CTP symmetry, transposition, and the Ward identity are discussed. The Green function and its inverse for different components are also explored. These notes provide a comprehensive understanding of the theoretical framework for studying the hydrodynamics and decoherence of ideal quantum gases.

We propose a correspondence between partition functions of ideal gases consisting of both bosons and fermions and algebraic bases of supersymmetric polynomials on the Banach space of absolutely summable two-sides sequences ℓ1(Z0). Such an approach allows us to interpret some combinatorial identities for supersymmetric polynomials from a physical point of view. We consider a relation of equivalence on ℓ1(Z0) induced by the supersymmetric polynomials, and semiring algebraic structures on the quotient set with respect to this relation. The quotient set is a natural model for the set of energy levels of a quantum system. We introduce two different topological semiring structures on this set and discuss their possible physical interpretations.

The objective of the paper is to introduce new similarity measure for single valued neutrosophic sets based on logarithm function. We define logarithm similarity measure and their weighted similarity measure for single valued neutrosophic sets. Then we define hybrid logarithm similarity measure and weighted hybrid logarithm similarity measure for single valued neutrosophic sets. We prove the basic properties of the proposed measures. We then define an entropy function using logarithm function to determine unknown attribute weights. We develop a novel multi attribute group decision making strategy for single valued neutrosophic sets based on the weighted hybrid logarithm similarity measure. We address an illustrative example to demonstrate the effectiveness and aptness of the proposed strategies. We conduct a sensitivity analysis of the developed strategy. We also make a comparison between the obtained results from proposed strategies and different existing strategies in the literature.

This article is the first part of a scientific project under the general title "Geometrized Vacuum Physics". This study is based on two main postulates: 1) independence of the propagation velocity of electromagnetic waves in vacuum from their frequency; 2) the constancy of the averaged zero vacuum balance, associated with the assertion that only mutually opposite formations are born from vacuum, so that on average they completely compensate for each other's manifestations. In this part of the "Geometrized Vacuum Physics" the foundations of the Algebra of Stignatures are laid, which is the mathematical and logical foundation of the entire project. In the next articles of this project it will be shown that the Algebra of Stignatures can be used for the development of "zero" (vacuum) technologies, algebraic genetics, vacuum cosmology, the vacuum standard model of "elementary particles", vacuum gravity and levitation, vacuum energy, ethics and aesthetics and many other branches of knowledge. At the end of this article, one of the many possible applications of the Algebra of Stignatures is given, in particular, the basics of the stignature-spectral analysis are outlined, with the help of which the possibilities of communication channels can be significantly expanded.

Considering increasingly ambitious pledges by countries, pressure from society, investors, and clients further up the supply chain, the question for companies is not so much whether to take decarbonisation action, but what action and by when. However, determining an ideal mix of measures to apply ‘decarbonisation efficiency’ requires more than knowledge of technically feasible measures and how to combine them to achieve the most economic outcome: In this paper, the author describes seven aspects which heavily influence the composition of an ‘ideal mix’ that executive leadership needs to take a (strategic) position on. These aspects consider underlying motivations and span across (socio-)economic, technical, regulatory, strategic, corporate culture and environmental factors and further underline the necessity of clarity of definitions. How these decisions influence the determination of the decarbonisation-efficient ideal mix of measures, both in principle but also in terms of specific impact is further explored by providing examples. What choices are taken by German manufacturers in several of the aspects is disclosed by insights from about 850 responses to the ‘Energy Efficiency Index of German Industry’. Knowledge of the status quo, and clarity in definitions, objectives, time frames and scope are key.

Here we investigate how the local properties of particles in a thermal bath may influence the thermodynamics of the bath, and consequently alter the statistical mechanics of subsystems that comprise the bath. We are guided by the theory of small-system thermodynamics, which is based on two primary postulates: that small systems can be treated self-consistently by coupling them to an ensemble of similarly small systems, and that a large ensemble of small systems forms its own thermodynamic bath. We adapt this “nanothermodynamics” to investigate how a large system may subdivide into an ensemble of smaller subsystems, causing internal heterogeneity across multiple size scales. For the semi-classical ideal gas, maximum entropy favors subdividing a large system of “atoms” into an ensemble of “regions” of variable size. The mechanism of region formation could come from quantum exchange symmetry that makes atoms in each region indistinguishable, while decoherence between regions allows atoms in separate regions to be distinguishable by their distinct locations. Combining regions reduces the total entropy, as expected when distinguishable particles become indistinguishable, and as required by a theorem in quantum mechanics for sub-additive entropy. Combining large volumes of small regions gives the usual entropy of mixing for a semi-classical ideal gas, resolving Gibbs paradox without invoking quantum symmetry for atoms that may be meters apart. Other models presented here are based on Ising-like spins, which are solved analytically in one dimension. Focusing on the bonds between the Ising-like spins we find similarity in the equilibrium properties of a two-state model in the nanocanonical ensemble and a three-state model in the canonical ensemble. Thus, emergent phenomena may alter the thermal behavior of microscopic models, and the correct ensemble is necessary for fully-accurate predictions. Another result using Ising-like spins involves simulations that include a nonlinear correction to Boltzmann’s factor, which mimics the statistics of indistinguishable states by imitating the dynamics of spin exchange on intermediate lengths. These simulations exhibit 1/f-like noise at low frequencies (f), and white noise at higher f, similar to equilibrium thermal fluctuations found in many materials.

The diverse socio-economic and environmental impacts that the set-up of a new photovoltaic installation has must be weighed carefully in order to reach the best possible solution. Among the different photovoltaic systems there are several classification criteria, depending on the technology, application and the size of the modules that define them. The size (installed nominal capacity) stands out as an impartial and critical measure in the decision making process. In this article we use a multi-criteria decision making method to analyse the responses of five experts to a detailed questionnaire in which several different criteria are correlated with various photovoltaic installation sizes. The limitation associated to the low number of experts is addressed with a robustness and sensitivity analysis. With this study we seek first, to apply and demonstrate the feasibility of a methodology which combines technical information with multi-criteria decision making methods, and second, to obtain a clear result oriented that increases the benefits of a forthcoming photovoltaic installation of modules in distributed generation adding up to 1GW total peak power in standard conditions. We observe a consistent result in which smaller photovoltaic modules are the ideal solution that maximises the socio-economic benefits of any installation. If a decision has to be taken about the type of photovoltaic plant to be installed, the conclusion is clear: given a certain size, small, easily scalable installations are the best solution for stake-holders, for the inhabitants, and for the environment.

Safety limits application has always been a traditional approach to ensure the safe operation of electro-mechanical systems within many industries including automated vehicles, robotics, aerospace, traditional automotive, railways, manufacturing, etc. In all of these applications, control and safety limits are usually hard-coded into the production firmware and are fixed throughout the entire product life cycle. Currently, due to the evolving needs of automated systems like automated vehicles and robots, this traditional approach does not address all the use cases and scenarios to ensure safe operation. Especially for data-driven machine learning applications that constantly evolve and learn over time, it is important to be able to adjust the safety limits application strategy to be more flexible and adaptable based on different Operational Design Domains (ODDs) and scenarios. Our ITSC conference paper ~\cite{4} introduced the concept of a new safety limits application strategy called the Dynamic Control Limits Application (DCLA) strategy that supports the flexible application of diverse limits profiles based on the parameters involved within the dynamic scenario at different layers of the Autonomy software stack.This paper extends the DCLA strategy to derive the complete methodology for safety limits application based on ODD elements, scenario identification and scenario classification using Decision Making Engines. It leverages the layered architecture introduced in the ITSC conference paper to implement the Decision Making (DM) algorithms. Another important extension is the use of cloud infrastructure that is based on the Vehicle-to-Infrastructure (V2I) technology to store the scenarios and the limits mapping to serve as a ground truth and/or a backup mechanism in case of errors or failures associated with the main Decision Making (DM) Engine. There is also a focus on providing a more comprehensive list of scenarios and a custom built experimental dataset to cover the maximum ODD elements, and multiple tables of safety limits to be chosen from, which eventually helps in creating different safety limits application profiles. These distinct safety limits application profiles are based on the scenario parameters that are perceived by the system upon which the Decision Making algorithms are applied or trained. This systematic approach can be used within the industry for any of the future automated vehicles and systems until Level 5 Autonomy.

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.

Quantum mechanics could be studied through polynomial algebra, as has been demonstrated by a work (“An approach to first-principles electronic structure calculation by symbolic-numeric computation” by A. Kikuchi). We carry forward the algebraic method of quantum mechanics through algebraic number theory; the basic equations are represented by the multivariate polynomial ideals; the symbolic computations process the ideal and disentangle the eigenstates as the algebraic variety; upon which one canbuild the Galois extension of the number field, in analogy with the univariate polynomial case, to investigate the hierarchy of solutions; the Galois extension is accompanied with the group operations, which permute the eigenstates from one to another, and furnish the quantum system with a non-apparent symmetry. Besides, this sort of algebraic quantum mechanics is an embodiment of the class field theory; some of the important consequences of the latter emerge in quantum mechanics. We shall demonstrate these points through simple models; we will see the use of computational algebra facilitates such sort of analysis, which might often be complicated if we try to solve them manually.

A new framework in quantum chemistry has been proposed recently ("An approach to first principles electronic structure calculation by symbolic-numeric computation'' by A. Kikuchi). It is based on the modern technique of computational algebraic geometry, viz. the symbolic computation of polynomial systems. Although this framework belongs to molecular orbital theory, it fully adopts the symbolic method. The analytic integrals in the secular equations are approximated by the polynomials. The indeterminate variables of polynomials represent the wave-functions and other parameters for the optimization, such as atomic positions and contraction coefficients of atomic orbitals. Then the symbolic computation digests and decomposes the polynomials into a tame form of the set of equations, to which numerical computations are easily applied. The key technique is Grröbner basis theory, by which one can investigate the electronic structure by unraveling the entangled relations of the involved variables. In this article, at first, we demonstrate the featured result of this new theory. Next, we expound the mathematical basics concerning computational algebraic geometry, which are necessitated in our study. We will see how highly abstract ideas of polynomial algebra would be applied to the solution of the definite problems in quantum mechanics. We solve simple problems in "quantum chemistry in algebraic variety'' by means of algebraic approach. Finally, we review several topics related to polynomial computation, whereby we shall have an outlook for the future direction of the research.