A total $m$-labelling $\alpha$ on graph $\Gamma$ is called an edge irregular total $m$-labelling if for any two different edges of $\Gamma$, their weights respect to $\alpha$ are distinct, where the weight of any edge is defined as the sum of its label and the labels of its end vertices. We determine the minimum $m$ such that $\Gamma$ can be labelled by an edge irregular total $m$-labelling whenever $\Gamma$ is an odd staircase graph or an even staircase graph.

The Collatz conjecture is a well-known number theory puzzle that states that every positive integer would eventually converge to the trivial cycle of 1, 2, 1, 2,... when repeatedly exposed to a particular transformation. In this transformation, even numbers are divided in half, odd numbers are tripled, and one is added. In this study, we present a new method for creating a directed graph and using it to display and analyze Collatz sequences. Our technique creates what we call a Collatz directed graph by joining an endless number of simple directed graphs, each of which corresponds to a natural number. We show by careful mathematical analysis that all positive integers are included in this Collatz directed graph. Moreover, we give an evidence that verifies the Collatz conjecture by showing that the sole cycle in this graph is the trivial cycle of 1, 2, 1, 2,... We also prove that there is no sequence that diverges to infinity in this graph. Our results provide insights into the fundamental structure of Collatz sequences and further our knowledge of the Collatz conjecture .

In 1937,German mathematician L.Collatz proposed a conjeture:for any positive integer,if it is even,divide it by 2;if it is odd,multiply it by 3 and 1 to get an even number;keep going on and on, the final result is 1.This paper gives a sequence classification of odd numbers,and gives 1-1 mapping of odd numbers and odd number sequences,and uses these results to give a proof of this conjecture.

Build a special identical equation, use its calculation characters to prove and search for solution of any odd converging to 1 equation through (*3+1)/2^k operation, change the operation to (*3+2^m-1)/2^k, and get a solution for this equation, give a specific example to verify. Thus prove the Collatz Conjecture is true. Furthermore, analysis the sequences produced by iteration calculation during the procedure of searching for solution, build a weight function model, prove it decrease progressively to 0, build a complement weight function model, prove it increase to its convergence state. Build a (*3+2^m-1)/2^k odd tree, prove if odd in (*3+2^m-1)/2^k long huge odd sequence can not converge, the sequence must outstep the boundary of the tree after infinite steps of (*3+2^m-1)/2^k operation.

We report on recent experimental results on β decay into self-conjugate (N=Z) nuclei with mass number 58≤A≤70. Super-allowed β decays from the Jπ=0+ ground state of a Z=N+2 parent nucleus are to the isobaric analogue state, through so-called Fermi transitions, and to Jπ=1+ states, by way of Gamow–Teller (GT) transitions. The operator of the latter decay is a generator of Wigner’s SU(4) algebra and as a consequence GT transitions obey selection rules associated with this symmetry. Since SU(4) is progressively broken with increasing A, mainly as a consequence of the spin–orbit interaction, this symmetry is not relevant for the nuclei considered here. We argue, however, that the pseudo-spin–orbit splitting can be small in nuclei with 58≤A≤70, in which case nuclear states exhibit an approximate pseudo-SU(4) symmetry. To test this conjecture, GT decay strength is calculated with use of a schematic Hamiltonian with pseudo-SU(4) symmetry. Some generic features of the GT β decay due to pseudo-SU(4) symmetry are pointed out. The experimentally observed GT strength indicates a restoration of pseudo-SU(4) symmetry for A=70.

Deployment of Level 3 and Level 4 autonomous vehicles (AVs) in urban environments is significantly constrained by adverse weather conditions, limiting their operation to clear weather due to safety concerns. Ensuring that AVs remain within their designated Operational Design Domain (ODD) is a formidable challenge, making boundary monitoring strategies essential for safe navigation. This study explores the critical role of an ODD monitoring system (OMS) in addressing these challenges. It reviews various methodologies for designing an OMS and presents a comprehensive visualization framework incorporating trigger points for ODD exits. These trigger points serve as essential references for effective OMS design. The study also delves into a specific use case concerning ODD exits: the reduction in road friction due to adverse weather conditions. It emphasizes the importance of contactless computer vision-based methods for road condition estimation (RCE), particularly using vision sensors such as cameras. The study details a timeline of methods involving classical machine learning and deep learning feature extraction techniques, identifying contemporary challenges such as class imbalance, lack of comprehensive datasets, annotation methods, and the scarcity of generalization techniques. Furthermore, it provides a factual comparison of two state-of-the-art RCE datasets. In essence, the study aims to address and explore ODD exits due to weather-induced road conditions, decoding the practical solutions and directions for future research in the realm of AVs.

There is high prevalence of hypertension and is rapidly increasing around the world, despite the intervention programme implemented, this study aimed at estimating the prevalence rate, test of association between hypertension and risk factors and model hypertension rate. Data used was obtained from the health record of Federal Medical Centre, Keffi from January 2016 – January 2019. Ordinal logistic regression model was used; Model Fitting Information, Goodness-of-Fit, Pseudo R-Square and Test of Parallel Lines are fitted to the data sets to test the accuracy and correctness of the model. The results indicated that the overall prevalence of hypertension rate is high at 36.4%, among the adult population, body mass index and gender are statistically significant, and Age is not significant in the study. Individuals that are overweight are more likely to be hypertensive compare to other weights. At age 40 – 49 years which have the highest rate of 26.5% and the odd ratio is 0.75 compared to others. One year increase in age 30 – 39, the cumulative odd of being hypertensive is 0.91 while other independent variables are held constant. The odd ratio of female being hypertensive is 0.85, therefore the females are more likely to be hypertensive with 54.4% compared to the males at 45.6% . There is no presence of multicolinearity among the variables and Logit models were formulated to calculate probabilities of the various possible outcomes.

This paper definitively settles the longstanding conjecture regarding odd perfect numbers. A perfect number is one whose sum of divisors equals itself doubled. While Euclid's method for constructing even perfect numbers is well-known, the existence of odd ones has remained elusive. By employing elementary techniques and analyzing the properties of the divisor sum function, we conclusively prove that no odd perfect numbers exist.

The theory says that number interacts with spaces based on the meaning of mathematics, in this way it is possible to obtain numerical knowledge. The theory shows the possibility of using spaces related to each other to efficiently discover whether results and origins are even or odd, but always applying rules to make this work. the study uses equivalent representations of odd and even numbers. Thus, it is possible to discover the possibilities of origins and their results.

This work offers a new log generalized odd log-logistic exponential regression model for analyzing weekly dengue fever cases in 2022 with a location-systematic component. To achieve this, a data set of 49 observations of dengue fever cases in the Federal District of Brazil is employed. A review of the mathematical properties of the generalized odd log-logistic exponential distribution is provided, the maximum likelihood method is used to estimate the parameters, and, through Monte Carlo simulations, the accuracy of the estimators is investigated. The model's fit is assessed using global influence metrics and residual analysis. For the time scenario studied, the proposed regression identified factors that have an impact on dengue fever cases, which may contribute to improve disease control. Finally, several interpretations are addressed, and a discussion presents results that aid in better understanding the data set and future research on different data.

Pentadecanoic acid (C15:0) and heptadecanoic acid (C17:0) have been described as dietary biomarkers of dairy-fat consumption with varying degrees of reliability between studies. It remains unclear how the total amount of dietary fat; one of the main confounding factors in these biomarker investigations, affects C15:0 and C17:0 circulating compositions independently to their relative intake. Additionally, it is unknown how changes in the dietary total-fat affects other fatty acids in circulation. Through two dietary studies with different total-fat levels but maintaining individual fatty acid compositions we were able to see how the dietary total-fat affects the fatty acids in circulation. We saw that there was a significant, proportionate, and robust decrease in the endogenous C15:0 levels with an increase in dietary total-fat. However, there was an increase in the circulating C17:0 compositions as the total-fat increased. To conclude, the dietary total-fat content and fat-type have a very complex influence on the relative compositions of circulating fatty acids, which are independent to the actual dietary fatty acid composition. Knowing how to manipulate circulating C15:0 and C17:0 composition is far-reaching in nutritional/pathological research as they highlight a dietary route to attenuate the development of metabolic disease (both by reducing risk and improving prognosis).

The evolution of even and odd Schrödinger cat states of the inverted oscillator is obtained in the center-of-mass tomographic probability description of the two-mode oscillator.The notion of entangled probability distribution is reviewed. Evolution equations describing the time-dependence of probability distributions identified with quantum system states are discussed.The connection with the Schrödinger equation and the von Neumann equation is clarified.

Prime number-related issues can be viewed from drastically different perspectives by examining the close connections between prime numbers and composite numbers. We think that multiple perspectives are the pillars on the path to solutions so we have created this study. As a result of the study, we proposed two new formulas by presenting three theorems and one proof for each theorem, a total of three proofs. We proved that the formula p · n + p returns a composite number in the first of the theorems, which is the preliminary theorem. Our first theorem except the preliminary theorem is that the formula p · n + p returns all composite numbers, and we proved that too. Finally, we created Theorem II using Theorem I to use in our other work and proved that the formula 2 · n · p + p returns all odd composite numbers, which is Theorem II. Afterward, we presented the similarities of the 2 · n · p + p formula we put forth with another known formula.

The paper presents two novel mathematical approaches to prove Goldbach's conjecture. The conjecture states that every even number greater than or equal to 6 can be expressed as the sum of two odd prime numbers. We propose that the conjecture has only four genetic codes, and as long as one of them remains valid, the conjecture continues to hold. Through these new mathematical ideas, we aim to prove the validity of Goldbach's conjecture comprehensively.

Following technological advances, autonomous and remote-controlled ships are performing trials in some sea areas; a mandatory goal-based MASS code, expected to enter into force on 1 January 2032, is under discussion in the International Maritime Organization (IMO). However, there are no unified standards regarding verification methods to define operating limitations for cutting-edge technologies, which are expected to be elements of MASS, so it is difficult to assess the risk factors for the management and operation of MASS. This paper outlines the regulatory framework for MASS, followed by a discussion of its application, considering the differences between MASS and automated driving, and how to develop safety management and operation using standards for automated driving. A requirements management framework developed as a system and software engineering standards to build safety management and operation of MASS is proposed. It is also suggested that the requirements management framework could be used not only for MASS development but also for regulatory review from a requirements management view.

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.

Soft-Core architecture for Analogue to Digital Converter (ADC) sampling is useful for mixed signal applications. Soft-core architecture for cutting edge odd or even ADC sampling with interface to block RAM memory has not been found. Soft-core architecture as a concept has become popular due to the advantage of customization for different applications as compared to general-core architecture suited for single application. The latest generation of piece wise sampling is odd sampling and was introduced in the second decade of the 20th century. Odd and even order sampling techniques are analogue in nature driven by a tuned (tuned for odd or even) mixer. This paper proposes a third-generation piece wise sampling with soft-core architecture that enables an option to select both odd and even while interfacing to memory mapping. The proposed odd/even has superior SNR performance of 6 dB as compared to existing architecture such as Mod-∆ which recorded worst performance of 18 dB. Advances in soft-core technology have allowed a niche odd/even switching field to be identified and studied, the study has also been extended to include memory architecture.