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2021 ◽  
Author(s):  
Jackeline del Carmen Huaccha Neyra ◽  
Aurelio Ribeiro Leite de Oliveira

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1931
Author(s):  
Eva Trojovská ◽  
Kandasamy Venkatachalam

Let (Fn)n≥0 be the sequence of Fibonacci numbers. The order of appearance of an integer n≥1 is defined as z(n):=min{k≥1:n∣Fk}. Let Z′ be the set of all limit points of {z(n)/n:n≥1}. By some theoretical results on the growth of the sequence (z(n)/n)n≥1, we gain a better understanding of the topological structure of the derived set Z′. For instance, {0,1,32,2}⊆Z′⊆[0,2] and Z′ does not have any interior points. A recent result of Trojovská implies the existence of a positive real number t<2 such that Z′∩(t,2) is the empty set. In this paper, we improve this result by proving that (127,2) is the largest subinterval of [0,2] which does not intersect Z′. In addition, we show a connection between the sequence (xn)n, for which z(xn)/xn tends to r>0 (as n→∞), and the number of preimages of r under the map m↦z(m)/m.


2021 ◽  
Vol 23 (07) ◽  
pp. 1158-1164
Author(s):  

In Numerical Analysis and various uses, including operation testing and processing, Newton’s method may be a fundamental technique. We research the history of the methodology, its core theories, the outcomes of integration, changes, they’re worldwide actions. We consider process implementations for various groups of optimization issues, like unrestrained optimization, problems limited by equality, convex programming, and methods for interior points. Some extensions are quickly addressed (non-smooth concerns, continuous analogue, Smale’s effect, etc.), whereas some others are presented in additional depth (e.g., variations of the worldwide convergence method). The numerical analysis highlights the quicker convergence of Newton’s approach obtained with this update. This updated sort of Newton-Raphson is comparatively straightforward and reliable; it’d be more probable to converge into an answer than either the upper order strategies (4th and 6th degree) or the tactic of Newton itself. Our dissertation could be about the Convergence of the Newton-Raphson Method which is a way to quickly find an honest approximation for the basis of a real-valued function g(m) = 0. The derivation of the Newton Raphson formula, examples, uses, advantages, and downwards of the Newton Raphson Method has also been discussed during this dissertation.


Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1657
Author(s):  
Jochen Merker ◽  
Benjamin Kunsch ◽  
Gregor Schuldt

A nonlinear compartment model generates a semi-process on a simplex and may have an arbitrarily complex dynamical behaviour in the interior of the simplex. Nonetheless, in applications nonlinear compartment models often have a unique asymptotically stable equilibrium attracting all interior points. Further, the convergence to this equilibrium is often wave-like and related to slow dynamics near a second hyperbolic equilibrium on the boundary. We discuss a generic two-parameter bifurcation of this equilibrium at a corner of the simplex, which leads to such dynamics, and explain the wave-like convergence as an artifact of a non-smooth nearby system in C0-topology, where the second equilibrium on the boundary attracts an open interior set of the simplex. As such nearby idealized systems have two disjoint basins of attraction, they are able to show rate-induced tipping in the non-autonomous case of time-dependent parameters, and induce phenomena in the original systems like, e.g., avoiding a wave by quickly varying parameters. Thus, this article reports a quite unexpected path, how rate-induced tipping can occur in nonlinear compartment models.


2021 ◽  
Vol 53 (2) ◽  
pp. 335-369
Author(s):  
Christian Meier ◽  
Lingfei Li ◽  
Gongqiu Zhang

AbstractWe develop a continuous-time Markov chain (CTMC) approximation of one-dimensional diffusions with sticky boundary or interior points. Approximate solutions to the action of the Feynman–Kac operator associated with a sticky diffusion and first passage probabilities are obtained using matrix exponentials. We show how to compute matrix exponentials efficiently and prove that a carefully designed scheme achieves second-order convergence. We also propose a scheme based on CTMC approximation for the simulation of sticky diffusions, for which the Euler scheme may completely fail. The efficiency of our method and its advantages over alternative approaches are illustrated in the context of bond pricing in a sticky short-rate model for a low-interest environment and option pricing under a geometric Brownian motion price model with a sticky interior point.


Author(s):  
Tim A. Hartmann ◽  
Stefan Lendl ◽  
Gerhard J. Woeginger

AbstractWe study a continuous facility location problem on undirected graphs where all edges have unit length and where the facilities may be positioned on the vertices as well as on interior points of the edges. The goal is to cover the entire graph with a minimum number of facilities with covering range $$\delta >0$$ δ > 0 . In other words, we want to position as few facilities as possible subject to the condition that every point on every edge is at distance at most $$\delta $$ δ from one of these facilities. We investigate this covering problem in terms of the rational parameter $$\delta $$ δ . We prove that the problem is polynomially solvable whenever $$\delta $$ δ is a unit fraction, and that the problem is NP-hard for all non unit fractions $$\delta $$ δ . We also analyze the parametrized complexity with the solution size as parameter: The resulting problem is fixed parameter tractable for $$\delta <3/2$$ δ < 3 / 2 , and it is W[2]-hard for $$\delta \ge 3/2$$ δ ≥ 3 / 2 .


Author(s):  
Jinho Song ◽  
Junhee Lee ◽  
Kwanghee Ko ◽  
Won-Don Kim ◽  
Tae-Won Kang ◽  
...  

Abstract In this paper, a method for classifying 3D unorganized points into interior and boundary points using a deep neural network is proposed. The classification of 3D unorganized points into boundary and interior points is an important problem in the nonuniform rational B-spline (NURBS) surface reconstruction process. A part of an existing neural network PointNet, which processes 3D point segmentation, is used as the base network model. An index value corresponding to each point is proposed for use as an additional property to improve the classification performance of the network. The classified points are then provided as inputs to the NURBS surface reconstruction process, and it has been demonstrated that the reconstruction is performed efficiently. Experiments using diverse examples indicate that the proposed method achieves better performance than other existing methods.


Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2045
Author(s):  
Zaheer-ud-Din ◽  
Muhammad Ahsan ◽  
Masood Ahmad ◽  
Wajid Khan ◽  
Emad E. Mahmoud ◽  
...  

In this work, meshless methods based on a radial basis function (RBF) are applied for the solution of two-dimensional steady-state heat conduction problems with nonlocal multi-point boundary conditions (NMBC). These meshless procedures are based on the multiquadric (MQ) RBF and its modified version (i.e., integrated MQ RBF). The meshless method is extended to the NMBC and compared with the standard collocation method (i.e., Kansa’s method). In extended methods, the interior and the boundary solutions are approximated with a sum of RBF series, while in Kansa’s method, a single series of RBF is considered. Three different sorts of solution domains are considered in which Dirichlet or Neumann boundary conditions are specified on some part of the boundary while the unknown function values of the remaining portion of the boundary are related to a discrete set of interior points. The influences of NMBC on the accuracy and condition number of the system matrix associated with the proposed methods are investigated. The sensitivity of the shape parameter is also analyzed in the proposed methods. The performance of the proposed approaches in terms of accuracy and efficiency is confirmed on the benchmark problems.


Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1813 ◽  
Author(s):  
Cheng-Yu Ku ◽  
Chih-Yu Liu ◽  
Jing-En Xiao ◽  
Shih-Meng Hsu

In this article, we present multiquadric radial basis functions (RBFs), including multiquadric (MQ) and inverse multiquadric (IMQ) functions, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme. Different from the conventional collocation method that assigns the RBF at each center point coinciding with an interior point, we separated the center points from the interior points, in which the center points were regarded as the fictitious sources collocated outside the domain. The interior, boundary, and source points were therefore collocated within, on, and outside the domain, respectively. Since the radial distance between the interior point and the source point was always greater than zero, the MQ and IMQ RBFs and their derivatives in the governing equation were smooth and globally infinitely differentiable. Accordingly, the shape parameter was no longer required in the MQ and IMQ RBFs. Numerical examples with the domain in symmetry and asymmetry are presented to verify the accuracy and robustness of the proposed method. The results demonstrated that the proposed method using MQ RBFs without the shape parameter acquires more accurate results than the conventional RBF collocation method with the optimum shape parameter. Additionally, it was found that the locations of the fictitious sources were not sensitive to the accuracy.


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