Abstract: We validly ignore even prime number 2. Based on all arbitrarily large number of even Prime gaps 2, 4, 6, 8, 10...; the complete set and its derived subsets of Odd Primes fully comply with the Prime number theorem for Arithmetic Progressions, and our derived Generic Squeeze theorem and Theorem of Divergent-to-Convergent series conversion for Prime numbers. With these conditions being satisfied by all Odd Primes, we argue Polignac's and Twin prime conjectures are proven to be true when they are usefully treated as Incompletely Predictable Problems. In so doing [and with Riemann hypothesis being a special case], this action also support the generalized Riemann hypothesis formulated for Dirichlet L-function. By broadly applying Hodge conjecture, Grothendieck period conjecture and Pi-Circle conjecture to Dirichlet eta function (proxy function for Riemann zeta function), Riemann hypothesis is separately proven to be true when it is usefully treated as Incompletely Predictable Problem. Crucial connections exist between these Incompletely Predictable Problems and Quantum field theory whereby eligible (sub-)functions and (sub-)algorithms are treated as infinite series. PubDate: Fri, 03 May 2024 10:47:35 +000

Abstract: The multiquadric radial basis function method has been widely used to solve partial differential equations-based problems regarding its flexibility and meshfree characteristics. The accuracy and stability of this method are derived and based on the use of a free-shape parameter that sensibly controls the comportment of the technique. Significant improvements have already been reported and show that variable shape parameters conduct the method to handle problems with striking results compared to global-based techniques. Nevertheless, choosing a suitable set of shape parameters is still an open topic because of the complexity of the method when the number of collocation points increases. The current work proposes a variant particle swarm optimization based on local displacement with attractors to determine the multi-quadratic function's ``best'' optimal variable shape parameter in solving boundary value problems. Based on an initially random set of variable shape parameters, the proposed algorithm first performs and evaluates the errors between the expected exact solution and the approximate solution thoroughly. In the first stage, the particle swarm algorithm search for an optimal set of shape parameters that minimize the error and the conditioning number of the radial basis system matrix. In the second stage, the obtained optimal set of shape parameters is applied to solve the considered problem. In this way, when the number of collocation points increases, the first stage based on particle swarm optimization stabilizes the strategy. It proposes an ``acceptable'' set of shape parameters for the given problem. The proposed method is applied to a set of well-known boundary value problems in one and two-dimensional spaces and compared to other techniques published in the literature. The results show that the proposed method achieves more accurate solutions than recently proposed in the literature. PubDate: Fri, 03 May 2024 08:05:46 +000

Abstract: This article presents a mathematical model for tensor complete contraction utilizing permutation group theory. The model helps in identifying the index class that dictates the complete contraction of even-order tensors. Additionally, the concept of a complete contraction closed loop is introduced along with a method for calculating the number of completely contracted images of any even-order tensor under any λ type permutation. The model is applied to investigate the algebraic structure of global conformal invariants on three-dimensional hypersurfaces. Moreover, it is shown that the generalized Willmore functional is the sole global conformal invariant in three-dimensional hypersurfaces when the difference is a constant multiple. PubDate: Mon, 29 Apr 2024 18:32:28 +000

Abstract: The main purpose of this project is to study the Rusty the Robot problem where a robot travels equidistantly from the starting point back to the starting point. We mainly use mathematical models such as trigonometric functions, recursive sequences, and mathematical induction to explore, deduce and demonstrate findings, while using mathematical computer software for calculation and verification. In the beginning, Geogebra was used to explore the original question and allow us to obtain some properties and results. Then, the topic was extended and the relationship between the number of steps and angles was explored, as well as the recursive relationship between the landing points during the travel process. Next, we altered the starting positions of Rusty to see how this affects our findings. Finally, the two intersecting straight lines are extended into three straight lines intersecting at the same point. Based on the angles between them, we discuss the number of returning paths from the starting point to the starting point in an equidistant traveling method. Based on the mathematical model constructed, this study constructs a return path diagram for the robot to return to the starting point in an equidistant manner according to different starting point positions and obtains the recursive relationship of the relative positions on this return path. During the research process of this study, some interesting mathematical theories were obtained. It is expected that these results can be applied to related fields of AI robot movement patterns in the future. PubDate: Mon, 29 Apr 2024 18:28:49 +000

Abstract: In this paper, the averaging principle for BSDEs and one barrier RBSDEs, with non Lipschitz coefficients, is investigated. An averaged BSDEs for the original BSDEs is proposed, as well as the one barrier RBSDEs, and their solutions are quantitatively compared. Under some appropriate assumptions, the solutions to original systems can be approximated by the solutions to averaged stochastic systems in the sense of mean square. PubDate: Mon, 29 Apr 2024 18:13:07 +000