Enhancing Critical Path Problem in Neutrosophic Environment Using Python

In the real world,one of the most common problems in project management is the unpredictability of resources and timelines.An efficient way to resolve uncertainty problems and overcome such obstacles is through an extended fuzzy approach,often known as neutrosophic logic.Our rigorous proposed model has led to the creation of an advanced technique for computing the triangular single-valued neutrosophic number.This innovative approach evaluates the inherent uncertainty in project durations of the planning phase,which enhances the potential significance of the decision-making process in the project.Our proposed method,for the first time in the neutrosophic set literature,not only solves existing problems but also introduces a new set of problems not yet explored in previous research.A comparative study using Python programming was conducted to examine the effectiveness of responsive and adaptive planning,as well as their differences from other existing models such as the classical critical path problem and the fuzzy critical path problem.The study highlights the use of neutrosophic logic in handling complex projects by illustrating an innovative dynamic programming framework that is robust and flexible,according to the derived results,and sets the stage for future discussions on its scalability and application across different industries.

Existence of solutions for fourth-order boundary value problem with parameter

This paper is a sequel to a previous paper （Yang, Y. and Zhang, J. H. Existence of solutions for some fourth-order boundary value problems with parameters. Nonlinear Anal. 69（2）, 1364-1375 （2008）） in which the nontrivial solutions to the fourthorder boundary value problems were studied. In the current work with the same conditions near infinity but different near zero, the positive, negative, and sign-changing solutions are obtained by the critical point theory, retracting property, and invariant sets.

A new perturbation to a critical elliptic problem with a variable exponent Dedicated to Professor Daomin Cao on the Occasion of His Sixtieth Birthday

In this paper,we study the following critical elliptic problem with a variable exponent:{-Δu=u^(p+ϵa(x))inΩ,u>0 inΩ,u=0 on∂Ω,where a(x)2∈C^(2)(Ω),p=N+2/N-2,∈>0,andΩis a smooth bounded domain in R^(N)(N>4).We show that for∈small enough,there exists a family of bubble solutions concentrating at the negative stable critical point of the function a(x).This is a new perturbation to the critical elliptic equation in contrast to the usual subcritical or supercritical perturbation,and gives the first existence result for the critical elliptic problem with a variable exponent.

Exploratory Analysis on Dictated Handwriting Samples

Dictated handwriting samples are widely used in practice due to their simplicity,convenience,and practicality.However,dictation is typically listed as one of the many collection methods in textbooks and monographs,and there is usually no separate section focusing on dictated handwriting samples.Therefore,further study of dictated handwriting samples will have important practical significance.Consideration of the definition,existing problems,collection techniques,and critical aspects of dictated handwriting samples willsupport investigators and document examiners in their professional abilities and contribute to the theoretical system of document examination.In this article,an exploratory analysis will be conducted and ideas about dictated handwriting samples will be shared,including the definition of dictated samples,their relationship to experimental samples,practical problems,feasible collection methods,and some critical points that require special attention.Dictation is widely used but problematic because of a lack of quantity and low levels of comparability.Those difficulties are mainly caused by a lack of theoreticalstudy and understanding of the requirements and collection techniques of dictated handwriting samples among first‑line investigators.Dictated samples should be collected based on subjective and objective conditions of formation with aims to improve comparability and five similarities.Further studies are needed to improve the theoretical system and practical use of dictated samples so that they can contribute to successfully reaching conclusions in investigations.

Multiple Solutions for a Quasilinear Second Order Differential Equation Depending on a Parameter

The purpose of this paper is to use a very recent three critical points theorem due to Bonanno and Marano to establish the existence of at least three solutions for the quasilinear second order differential equation on a compact interval[a,b] R{-u＇＇=（λf（x,u）＋g（u））h（u＇）,in（a,b）,u（a）=u（b）=0under ppropriate hypotheses.We exhibit the existence of at least three（weak）solutions and,and the results are illustrated by examples.