## The Vertices of a Graph and Its Dimension

We show that a graph’s dimension is smaller than or equal to the cardinality of its vertices. Author(S) Details Volker W. ThÃ¼rey Independent Researcher, Bremen, Germany. View Book:- https://stm.bookpi.org/RAMRCS-V4/article/view/4879

## Reliability Analysis of a Multi-State Delivery Network through the Symbolic Derivation of a Probability-Ready Expression

The dependability of a multi-state delivery network (MSDN) with different suppliers, transfer stations, and marketplaces (depicted as vertices) connected by branches of multi-state capabilities, delivering a specific commodity or service between their end vertices is the subject of this chapter. We use a symbolic logic interpretation of network success to meet market demand while staying […]

## Determining the Sufficiency in Optimal Control without the Strengthened Condition of Legendre

In this paper, we develop a sufficiency theorem for an unconstrained Lagrange fixed-endpoint problem that provides sufficient conditions for processes that do not satisfy the traditional nonsingularity assumption, i.e., the strengthened Legendre condition is not imposed by the new sufficiency theorem. Unlike numerous generalisations of conjugate points, solutions of specific matrix Riccati equations, invariant integrals, […]

## A Counterexample of the Statement P = NP: Scientific Explanation

By generating a very simple example of unsolvability in polynomial time in a well-known NP-complete problem, it is proven that the classes P and NP do not coincide in this work. A random process is built for a specific NP-complete issue that provides solutions with random numbers. The assumption that a concrete NP-complete problem belongs […]

## On the 4th Clay Millennium Problem for the Periodic Navier Stokes Equations

The Gagliardo-Nirenberg and Prekopa-Leindler inequalities are used to prove that the integrand of the integral form of the solution obtained can be set to zero everywhere in space and time, as well as results on the velocity-pressure distribution using Debreu’s, Brouwer’s, and Lusin’s theorems, and a final theorem proving no finite time blowup of the […]

## Channel Capacity Enhancement on Maximization of Modified Verma and Bose-Einstein Entropies by Gaussian Distribution

This chapter examined how to improve channel capacity by using a Gaussian distribution [5] and Kullback-[3] Leibler’s information measure of directed divergence to maximise modified Verma and Bose-Einstein entropies. Author(S) Details Rohit Kumar Verma Department of Mathematics, Bhilai Institute of Technology, Durg (C.G.), India. View Book:- https://stm.bookpi.org/RAMRCS-V4/article/view/4874

## Study on Dual Bi-Directional Heuristic Development Framework

A multi-start heuristic framework is developed in this study. Heuristic convergence to high-quality optimum solutions is accelerated using the heuristic development approach. The Dual Bi-Directional (DBD) heuristic is a multi-start heuristic that uses a pair search starting from both the beginning and ending nodes to find the global optimum solution (GOS) in four directions at […]

## A Phenomenological Relation for Decay Times: A Quantum Mechanical Approach

Georges Lemaitre was invited to a British Association symposium on the link between physicality and spirituality in London in 1931. In a report published in Nature titled ‘The Beginning of the World from the Point of View of Quantum Theory,’ he proposed that the Universe expanded from a starting point, which he dubbed the “Primeval […]

## Study on Fixed Effect versus Random Effects Modeling in a Panel Data Analysis; a Consideration of Economic and Political Indicators in Six African Countries

Individual heterogeneity in panel data analysis can be addressed to avoid bias in the final estimations. The data in a panel might be balanced or imbalanced, and the panel can be short or long. The fixed effects and random effects modelling methodologies were applied to an economic data set, “Africa,” in the R Amelia package, […]

## Study on Generalized Exponential Type Estimators for Ratio of Two Population Means Using Auxiliary Attribute in the Presence of Non-response

In this research, we offer generalised exponential type estimators for the ratio of two population means in the case of non-response utilising an auxiliary feature. Up to the first order of approximation, formulas for the mean square error of the suggested estimators are produced. There are also some specific examples of the proposed estimators. The […]