In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.

In this paper, we introduce a comprehensive solution aimed at enhancing the understanding of three-dimensional atmospheric pollutant dispersion. This innovative solution develops a generalized model that extends previous research and is applicable to all parameterization schemes of the equation, including wind speed profiles and turbulent diffusion coefficients, while incorporating the dry deposition criterion. Our methodology involves subdividing the atmospheric boundary layer into distinct sublayers, which facilitates a detailed examination of pollutant dispersion dynamics. Extensive validation with data from the Hanford experiment has demonstrated the accuracy of this solution in simulating pollutant concentrations. The results exhibit a strong correlation between projected and observed concentrations, underscoring the statistical reliability of our approach. This validation situates the statistical indices of our solution within an acceptable range, confirming its accuracy in predicting atmospheric pollutant dispersion. These findings thus establish our solution as a valid and effective method for studying complex environmental phenomena.

1,3,5-Tris-(α-naphthyl)benzene is an organic non-electrolyte with notable stability of an amorphous phase. Its glassy and supercooled liquid states were previously studied by spectroscopic and calorimetric methods. However, the thermodynamic parameters of phase transitions available from the literature are inconsistent. In this work, the heat capacities of crystalline and liquid phases, the temperature dependence of the saturated vapor pressures, fusion and vaporization enthalpies were determined using differential and fast scanning calorimetry and verified using the estimates based on solution calorimetry. The structural features of 1,3,5-tris-(α-naphthyl)benzene are discussed based on the computations performed and the data on the molecular refractivity. The consistency between the values obtained by independent techniques was demonstrated. The deviations in the previously published data were attributed to possible polymorphism.

There are laws and rights aimed at people with disabilities, especially those with Mild Intellectual Disability (LDD), who actively participate in society but face barriers in learning and participating in various activities. In Colombia, where 1.3 million people with disabilities are registered, 16.8% correspond to the category of DIL, some of whom are part of the economically active population. Despite this, significant employment gaps persist.In order to address this gap, this research focuses on the analysis of the specific problems related to the labor inclusion of people with IDL in logistics environments, particularly in cold rooms. The aim is to prevent occupational hazards through the application of Industry 4.0 technologies.Through the collaboration of the Analytics and Design Research Seedbed, with entrepreneurs and experts in the field, significant challenges in labor inclusion in low-temperature environments were identified. The agile Scrum methodology was used to develop a prototype of an application that addresses these issues. This prototype meets the functional and non-functional requirements necessary to ensure the protection and prevention of risks for workers with DIL who perform functions in refrigerated rooms.This prototype manages to provide an inclusive alternative for people with disability Intellectual Light, challenging business cultural paradigms, by integrating mobile devices with Industry 4.0 technologies, encouraging greater interaction between machines and human beings, thus prioritizing the health and safety of employees in their daily activities.

Fabrics comprised of porous fibers could provide effective passive protection against chemical and biological (CB) threats whilst maintaining high air permeability (breathability). Here, we fabricate hierarchically porous fibers consisting of regenerated silk fibroin (RSF) and activated-carbon (AC) prepared through two fiber spinning techniques in combination with ice-templating – namely cryogenic solution blow spinning (Cryo-SBS) and cryogenic wet-spinning (Cryo-WS). The Cryo-WS RSF fibers had exceptionally small macropores (as low as 0.1 µm) and high specific surface areas (SSAs) of up to 79 m2 g-1. The incorporation of AC could further increase the SSA to 210 m2 g-1 (25 wt. % loading) whilst also increasing adsorption capacity for volatile organic compounds (VOCs).

Childhood diarrhea continues to be a major cause of under-five (U-5) mortality globally and in India. In this study, 1571 U-5 children residing in nine rural villages and four urban slums in Ujjain, India were included with the objective to use community participation and drug utilization research to improve diarrheal case management. The mean age was 2.08 years, with 297 (19%), children living in high diarrheal index households. Most mothers (70%) considered stale food, teething (62%), and hot weather (55%) as causes of diarrhea. Water, sanitation, and hygiene (WASH)-related characteristics revealed that most (93%) households had toilets, but only 23% of the children used them. The study identified ineffective household water treatment by filtration through cloth by most (93%) households and dumping of household waste on the streets (89%). The results revealed low community awareness of correct causes of diarrhea (poor hand hygiene, 21%; littering around the household, 15%) and of correct diarrhea treatment (oral rehydration solution (ORS) and zinc use, 29% and 11%, respectively) and a high antibiotic prescription rate by healthcare providers (83%). Based on the results of the present study, context-specific house-to-house interventions will be designed and implemented.

Four methods in two different families have been constructed to derive the exact solutions to Benjamin-Bona-Mahony equation in two space dimensions. Simply defined hyperbolic tangent, hyperbolic secant and hyperbolic cosecant ansatzes and the expansion method based on the Sine-Gordon equation in two dimensions are directly substituted into the governing ODE reduced from the two dimensional BBM equation. Classical algebraic method is used to find the relations among the target parameters representing the nonzero coefficients in the predicted solutions and the wave transform parameters. Some complex and real solutions have been constructed in explicit forms.

The purpose of the research is to investigate the unique solvability of a nonlocal boundary value problem for a loaded equation of parabolic-hyperbolic type in a special domain. Methods: Using representations of a general regular solution, the existence and uniqueness of the problem posed is proved. Results: A new method for proving the unique solvability of nonlocal boundary value problems for one loaded equation of mixed type in a special domain has been formed and developed. Conclusions: The obtained results make it possible to solve some problems of genetics, immunology, transonic gas dynamics and thermal physics, which are associated with displacement.

Given several nonnegative matrices with a single pattern of allocation among their zero/nonzero elements, the average matrix should have the same pattern, too. This is the first tenet of the pattern-multiplicative average (PMA) concept, while the second one suggests the multiplicative (or geometric) nature of averaging. The original concept of PMA was motivated by the practice of matrix population models as a tool to assess the population viability from long-term monitoring data. The task has reduced to searching for an approximate solution to an overdetermined system of polynomial equations for unknown elements of the average matrix (G), hence to a nonlinear constrained minimization problem for the matrix norm. Former practical solutions faced certain technical problems, which required sophisticated algorithms but returned acceptable estimates. Fresh idea to use the eigenvalue approximation ensues also from the basic equation of averaging, which determines the exact value of λ1(G), the dominant eigenvalue of matrix G, too, and the corre-sponding eigenvector. These are bound by the well-known linear equations, which reduce the task to a standard problem of linear programing (LP). The LP approach is realized for 13 fixed-pattern matrices gained in a case study of Androsace albana, an alpine short-lived perennial, monitored on permanent plots during 14 years. A standard software routine reveals the unique exact solution, rather than an approximate one, to the PMA problem, which turns the LP approach into ‘’the best of versatile optimization tools”. The exact solution turns out to be peculiar in reaching zero bounds for certain nonnegative entries of G, which deserves a modified problem formulation separating the lower bounds from zero.

Today, more disciplines are intercepting each other, giving rise to “cross-disciplinary” research. Technological advancements in material science and device structure and production have paved the way towards development of new classes of multi-purpose sensory devices. Organic phototransistors (OPTs) are photo-activated sensors based on organic field-effect transistors that convert incident light signals into electrical signals. The organic semiconductor (OSC) layer and three-electrode structure of an OPT offer great advantages for light detection compared to conventional photodetectors and photodiodes, due to their signal amplification and noise reduction characteristics. Solution processing of the active layer enables mass production of OPT devices at significantly reduced cost. The chemical structure of OSCs can be modified accordingly to fulfil detection at various wavelengths for different purposes. Organic phototransistors have attracted substantial interest in a variety of fields, namely biomedical, medical diagnostics, healthcare, energy, security, and environmental monitoring. Lightweight and mechanically flexible and wearable OPTs are suitable alternatives not only at clinical levels but also for point-of-care and home-assisted usage. In this review, we aim to explain different types, working mechanism and figures of merit of organic phototransistors and highlight the recent advances from the literature on development and implementation of OPTs for a broad range of research and real-life applications.

The paper describes an application of machine learning approach for parameter optimization. The method is demonstrated for the elasto-viscoplastic model with both isotropic and kinematic hardening. It is shown that the proposed method based on long-short term memory networks enabled to obtain reasonable agreement of stress-strain curves for cyclic deformation in low-cycle fatigue regime. The main advantage of the proposed approach over traditional optimization schemes lies in the possibility to get parameters for a new material without the necessity to conduct any further optimizations. As the power and robustness of the developed method was demonstrated on the very challenging problem (cyclic deformation, crystal plasticity, self-consistent model, isotropic and kinematic hardening), it is directly applicable to other experiments and models.

Thin twin-roll cast strips from a model Al-Cu-Mg-Li-Zr alloy with a small addition of Sc were prepared. A combination of a fast solidification rate and a favorable effect of Sc microalloying impacts the size and distribution of primary phase particles and the dimension of eutectic cells. Small interdendritic spacing in the range of 10-15 um enables substitution of the initial long-term homogenization annealing by a two-step annealing at 300 and 450 °C and a short solution treatment at 530 °C before quenching and the final age-hardening procedure. A dense dispersion of small Al3(Sc,Zr) precipitates intensively inhibiting recrystallization forms during this treatment. Further refining and fragmentation of the structure were performed using constrained groove pressing before the solution treatment. Small pre-straining of quenched specimens and subsequent age-hardening simulating T8 temper were used to promote the formation of a homogeneous distribution of strengthening particles and to suppress preferential heterogeneous precipitation on subgrain and grain boundaries. Microstructural features and microhardness evolution during annealing were critically examined and confronted with microstructural characteristics observed on reference mold cast alloy or alloys without Sc alloying. The proposed treatment represents a model near net shape procedure for manufacturing thicker sheets for the aerospace industry.

Intestinal failure (IF) is characterized by a critical reduction of functional gut mass below the minimum needed for optimal growth in children. It requires parenteral nutrition (PN) and home-PN (HPN) which is challenging in terms of meeting nutritional needs according to age, growth velocity, clinical situation, and rapid changes in fluid and electrolyte requirements. Due to these complex requirements, age-adapted multi-chamber bags (MCBs) are important additions to the nutrition armamentarium. The launch of composite fish oil (FO)-containing intravenous lipid emulsions (ILEs) heralded the development of MCBs containing these ILEs in combination with a crystalline amino acids solution (CAAS) adapted for pediatric use. The safety and efficacy of lipid and amino acid components in this context have been widely documented in numerous published studies. This manuscript includes a review of the articles published in PubMed, Embase, and Google Scholar until June 2022, exploring studies in the age groups from term infants to children and adolescents. Preterm infants with their highly specific demands are not included. It aims to offer an overview of the clinical experience regarding the use of a composite FO-based ILE and a specific CAAS developed

The three dimensional conformable time fractional Kadomtsev-Petviashvili and the conformable time fractional modified Kawahara equations are solved by implementing the Kudryashov's procedure. The corresponding wave transformation reduces both equations to some ODEs. Balancing the nonlinear and the highest order derivative terms gives the structure of the solutions in the finite series form. The useful symbolic tools are used to solve the resultant algebraic systems. The solutions are expressed in explicit forms.

Hydroponic farming systems play an increasingly important role in the sustainable production of nutrient-rich foods. The contamination of surfaces in hydroponic fresh produce production poses risks to the food safety of crops, potentially endangering public health and causing economic losses in the industry. While sanitizers are widely used in commercial hydroponic farms, their effectiveness against human pathogens on surfaces, and their impact on plant health and quality are not known. In this study, we evaluated the efficacy of chemical sanitizers in eliminating Salmonella Typhimurium from inanimate surfaces in commercial hydroponic Nutrient Film Technique (NFT) systems. Further, we assessed the impact of sanitizers on the yield, quality, and nutritional value of lettuce and basil. Sanitizers (Virkon, LanXess; Sanitate 12.0, Bi-oSafe Systems; Kleen Grow, Pace Chemical Ltd.; Green Shield, United Labs Inc.; Zerotol, BioSafe Systems; Bleach, Pure Bright), were tested against Salmonella Typhimurium inoculated on NFT surfaces (nutrient reservoirs, growing channels, top covers, drain lines). The effective treatments were then tested for their impact on lettuce and basil in a split-plot experiment conducted in commercial NFT units. Crop yield, color, and nutrient content were measured throughout the crop life cycle. While all quaternary ammonium compounds (QAC), SaniDate 12.0 (200ppm), Zorotol (5%), and Virkon (1%) eliminated Salmonella Typhimurium from commercial NFT surfaces, chlorine-based sanitizer treatments were statistically similar to water treatments on most surfaces. All chemical sanitizers impacted the yield, color, and nutritional value of lettuce and basil. SaniDate 12.0 (200ppm) was the least detrimental to crops and was identified as a potential candidate for further validation in commercial hydroponic settings. The findings of this study will be translated into recommendations for the industry and will contribute to the development of future food safety guidelines and policies.

BKW is a highly efficient Storage Function-type rainfall-runoff model that utilizes the steady-state storage (S) and outflow (Q) of a catchment at different rainfall intensities as the S-Q relation of the catchment. For the inundation nowcasting of urban areas under high-intensity, short-duration rainfall, a Storage Function-type model is a judicious choice. However, S-Q hysteresis exists in the catchment S-Q relation, and BKW is unable to exhibit hysteresis due to assuming that the S-Q relation of a catchment is a nonlinear relation. The S-Q hysteresis can influence the outflow estimation of BKW, reducing the model's accuracy. Through the evaluation of analytical solutions, we found that the change in rainfall intensity dominates the S-Q relation of a catchment. The S-Q relation during a rainfall event is a dynamic process rather than a monotonic nonlinear relation. By applying the dynamic S-Q relation in BKW, we improved BKW to become a highly efficient and accurate rainfall-runoff model for small urban catchments.

Evaporating liquid sessile drop deposited on horizontal surface is an important object of applications (printing technologies, electronics, sensorics, medical diagnostics, hydrophobic coatings, etc.) and theoretical investigations (microfluidics, self-assembly of nanoparticle, crystallization of the solute, etc.). The arsenal of formulas for calculating the slow evaporation of an axisymmetric drop of capillary dimensions deposited on a flat solid surface is reviewed. Such characteristics as vapor density, evaporation flux density, total evaporation rate are considered. Exact solutions obtained in the framework of the Maxwellian model, in which the evaporation process of the drop is limited by vapor diffusion from the drop surface to the surrounding air, are presented. The summary covers both well-known results obtained during the last decades and new results published by us in the last few years, but practically unknown to the wide scientific community. The newest formulas, not yet published in refereed publications, concerning exact solutions for a number of specific contact angles are also presented. In addition, new approximate solutions are presented for the first time (total evaporation rate and mass loss per unit surface area per unit time in the whole range of contact angles ), which can be used in modeling without requiring significant computational resources.

How to disentangle the possible genuine quenching of gA caused by scale anomaly of QCD parameterized by the scale-symmetry-breaking quenching factor qssb from nuclear correlation effects is described. This is done by matching the Fermi-liquid xed (FLFP) point theory to the "Extreme Single Particle (shell) Model" (acronym ESPM) in superallowed Gamow-Teller transitions in heavy doubly-magic shell nuclei. The recently experimentally observed indication for (1-qssb) ≠0 - that one might identify as "fundamental quenching (FQ)" - in certain experiments seems to be alarmingly signi cant. I present arguments how symmetries hidden in the matter-free vacuum can emerge and suppress such FQ in strong nuclear correlations. How to con rm or refute this observation is discussed in terms of the superallowed Gamow-Teller transition in the doubly-magic nucleus 100Sn and in the spectral shape in the multifold forbidden β decay of 115In.

In this paper, we study the deflection angle for wormhole-like static aether solution by using Gibbons and Werner technique in non-plasma, plasma and dark matter mediums. For this purpose, we use optical spacetime geometry to calculate the Gaussian optical curvature, then implement the Gauss-Bonnet theorem in weak field limits. Moreover, we compute the deflection angle by using a technique known as Keeton and Petters technique. Furthermore, we analyze the graphical behaviour of the bending angle ψ with respect to the impact parameter b, mass m as integration constant and parameter q in non-plasma and plasma mediums. We examine that deflection angle is exponentially increasing as direct with charge. Also, we observe that for small values of b, ψ increases and for large values of b the angle deceases. We also considered an analysis to the shadow cast of the wormhole relative to an observer at various locations. Comparing it the the Schwarzschild shadow, shadow cast is possible for wormhole as r<2m. At r>2m, the Schwarzschild is larger. As r → ∞, we have seen that the behavior of the shadow, as well as the weak deflection angle, approaches that of the Schwarzschild black hole. Overall, the effect of plasma tends to decrease the value of the observables due to the wormhole geometry.

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to get two new representations of the fundamental solution in form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed form formulas for particular cases of the fundamental solution are derived. In particular, we solve an open problem of representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.

Molecular dynamics simulations were performed to examine the impact of temperature (ranging from 250K to 350K) on the structural behavior of cellulose chains when dissolved in an aqueous cuprammonium hydroxide Cuam solvent system. By monitoring the root mean square deviation (RMSD) and radius of gyration (Rg) values at each temperature, we found that the geometric deformation of the cellulose chains changed significantly at T=300K. In contrast, Rg and RMSD values remained stable at temperatures in the range of 250K, 270K and the range of 330K, 350K. Calculating the interaction energy between the solvent and cellulose chains revealed that temperature significantly influences the cellulose dissolution process in the Cuam solvent system and that T=300K is an efficient temperature for its dissolution. The number of intra- and interchain hydrogen bonds was also calculated as a function of temperature, and the analysis confirmed that there is no breakage of these bonds at temperatures below 300 K (i,e: 250K, 270K) and above 310 K (I,e, : 330K and 350K) either between two native chains or between a native chain and another derivative.

A paper impregnated with silver nanoparticles (AgNPs) has been prepared. For the preparation of the substrates, aqueous suspensions of pulp fines, a side product from the paper production, have been mixed with Ag nanoparticles (AgNP) suspensions. The nanoparticle synthesis thereof was carried out via laser ablation of pure Ag in water. After the sheet formation process, the leaching of the AgNPs was determined to be low while the sheets exhibited antimicrobial activity towards E. Coli.

This multi-center retrospective study evaluated the diagnostic performance of a deep learning (DL)-based application for detecting, classifying, and highlighting suspected aortic dissections (ADs) on chest and thoraco-abdominal CT angiography (CTA) scans. CTA scans from over 200 U.S. and European cities acquired on 52 scanner models from 6 manufacturers were retrospectively collected and processed by CINA-CHEST (AD) (Avicenna.AI, La Ciotat, France) device. The di-agnostic performance of the device was compared with the ground truth established by the majority agreement of three US board-certified radiologists. Furthermore, the DL-algorithm's time-to-notification was evaluated to demonstrate clinical effectiveness. The study included 1,303 CTAs (mean age 58.8 ± 16.4 years old, 46.7% male, 10.5% positive). The device demonstrated a sensitivity of 94.2% [95% CI: 88.8% – 97.5%] and a specificity of 97.3% [95% CI: 96.2% - 98.1%]. The application classified positive cases by AD types with an accuracy of 99.5% [95% CI: 98.9% – 99.8%] for type A and 97.5 [95% CI: 96.4% – 98.3%] for type B. The application did not miss any type A cases. The device flagged 32 cases incorrectly, primarily due to acquisition artefacts and aortic pathologies mimicking AD. The mean time to process and notify of potential AD cases was 27.9 ± 8.7 seconds. This deep learning-based application demonstrated strong performance in detecting and classifying aortic dissection cases, potentially enabling faster triage of these urgent cases in clinical settings.

In real-life scenario, there are many mathematical tools to handle the incomplete and imprecise data. One of them is the fuzzy approach. This article aims to contribute to the literature of fuzzy multi-objective dynamic programming issues involving the fuzzy objective functions. The piecewise quadratic fuzzy numbers characterize these fuzzy parameters. Some basic notions in the problem under the α-pareto optimal solution concept is redefined and analyzed to study the stability of the problem. Furthermore, a technique, named as enhancing decomposition approach, is presented for achieving a subset for the parametric space that contains the sameα-pareto optimal solution. For a better understanding and comprehension of the suggested concept, a numerical example is provided.

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.

(1) Teeth, represent in humans the most resilient tissues. However, exposure to concentrated acids might lead to their obliteration, thus making human identification difficult. Teeth often contain dental restorations from materials that are even more resilient to acid impact. This paper introduces novel method of 3D reconstruction of dental patterns as a crucial step for digital identification with dental records.; (2) With combination of modern methods of Micro-Computer Tomography, Cone Beam Computer Tomography, Attenuated Total Reflection in conjunction with Fourier-Transform Infrared Spectroscopy and Artificial Intelligence Convolutional Neural Network algorithms, the paper presents the way of 3D dental pattern reconstruction and human remains identification. Research studies morphology of teeth, bone, and dental materials (Amalgam, Composite, Glass-ionomer cement) under different periods of exposure to 75% sulfuric acid; (3) Results reveal significant volume loss in bone, enamel, dentine, and as well glass-ionomer cement. Results also reveal significant resistance of composite and amalgam dental materials to sulfuric acid impact, thus serving as strong parts in the dental pattern mosaic. Paper also introduces probably first successful artificial intelligence application in automated forensic CBCT segmentation.; (4) Interdisciplinary cooperation utilizing mentioned technologies can solve problem of human remains identification with 3D reconstruction of dental patterns and their 2D projections over existing ante-mortem records.

If we assume that: a. The four fundamental forces of nature are independent waves without rest masses, and their speeds are constant in a vacuum, just like light. b. Light or electromagnetic waves and gravity are comparable in structures. Weak and strong interactions are similar in structures. c. Light and weak interaction have the same speed c_L with spin number +1 or –1. d. Gravity and strong interaction have the same speed c_G without spin. e. The primary particles, namely electrons, electron neutrinos, and dark neutrinos in this paper, are made by the above four waves. We can find and describe some fundamental characteristics of the primary particles (e.g., their sizes, energies, and interactions) and introduce new attractive results from them (e.g., the source of the Pauli exclusion principle, the solution to the Einstein-Podolsky-Rosen paradox, and c_G slightly faster than c_L).

Let $f\left(1\right)=1$ , and let $f(n+1)=2^{2^{f\left(n\right)}}$ for every positive integer n. We consider the following hypothesis: if a system S ⊆ {x_i · x_j = x_k : i, j, k ∈ {1, . . . , n}} ∪ {x_i + 1 = x_k : i, k ∈{1, . . . , n}} has only finitely many solutions in non-negative integers x₁, . . . , x_n, then each such solution (x₁, . . . , x_n) satisfies x₁, . . . , x_n ≤ f (2n). We prove: (1) the hypothesisimplies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that there exists an algorithm for listing the Diophantine equations with infinitely many solutions in non-negative integers; (3) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (4) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (5) the hypothesis implies that if a set $\mathfrak{M}$ ⊆ has a finite-fold Diophantine representation, then $\mathfrak{M}$ is computable.

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 notion of Critical Point Theory has extensive application in the filed of Partial Differential Equations, one of which is studying the existence and commenting on the uniqueness and multiplicity (in case when the solution is not unique) of weak solutions to Elliptic PDEs under certain boundary conditions.\par This article provides a brief survey about specific concepts like : differentiation on Banach Spaces, maxima and minima and its applications to PDEs. Furthermore, we've discussed the notion of defining a weak topology on Banach Spaces, before introducing the Variational Principle and its applications.\par The epilogue of the article primarily deals with results which studies the existence of weak solutions to Dirichlet Boundary Value Problems under specific conditions. The most notable aspect of this article is the proof of \textit{Rabinowitz's Saddle Point Theorem} using the method of \textit{Brouwer Degree}. Readers who are highly motivated in pursuing research in any of the topics relevent to the contents of this paper will surely find the \texttt{References} section to be extremely resourceful.

Abstract:Shown is the derivation of Lorentz-Einstein k-factor in SRT as an amplitude-term of oscillation-differential equations of second order.This case is shown for classical Lorentz-factor as solution of an equation for undamped oscillation as well as the developed theorem as a second solution for advanced SRT of fourth order with an equation for damped oscillation-states.This advanced term allows a calculation for any velocities by real rest mass