False Position Research Articles

An improved class of regula falsi methods of third order for solving nonlinear equations in R

A class of quadratically convergent regula falsi iterative methods for solving nonlinear equations f(x)=0 is proposed in Chen and Li (Appl. Numer. Math. 57:80–88, 2007). It is also shown there that both the sequences of diameters and iterative points sequence converge to zero simultaneously. The aim of this paper is to accelerate further the convergence of these methods from quadratic to cubic. This is done by replacing the parameter p in the iteration of Chen and Li (Appl. Numer. Math. 57:80–88, 2007) by a function p(x) defined suitably. A convergence theorem for establishing the cubic convergence of both the sequence of iterates and the sequence of diameters to the root is given. The numerical examples are worked out to demonstrate that our modified methods are more effective and comparable to those given in Chen and Li (Appl. Numer. Math. 57:80–88, 2007) as well as Newton’s method, Steffensen’s method and regula falsi method.

Navigation of artificial disc replacement: evaluation in a cadaver study

Introduction: Previous studies have shown that total disc replacement (TDR) resulted in significantly better restoration of disc-space height and significantly less subsidence than anterior interbody fusion with BAK cages. Clinical outcomes and flexion/extension range of motion correlated with the accuracy of surgical placement of the CHARITÉ™ artificial disc. False positioning of the artificial disc leads to spondylarthrosis and disc degeneration of the adjacent segment, and exclusive use of a C-arm could cause such false positioning (due to the parallax effect). The objective of this study was to test and evaluate the accuracy of navigated artificial disc replacement as performed by a spine surgeon without a prior learning curve. In each case, the placement position achieved by the surgeon was compared with the preoperatively planned position for that specimen.Materials and Methods: Lumbar intervertebral disc prostheses (CHARITÉ™, DePuy Spine) were placed using an image guidance technique (BrainLAB VectorVision system) in ten human cadaveric spine specimens. A total of 15 such disc replacements were performed using navigation. Post-instrumentation accuracy was assessed by a computer on the basis of CT scans.Results: The placement of the disc was assessed as ideal (<3 mm from the planned position), suboptimal (3–5 mm from the planned position) or poor (>5 mm from the planned position). Only three disc prostheses were placed suboptimally, and none was poorly placed. Placement in the coronal plane was significantly better than in the other planes.Discussion: Navigation is a useful instrument in the hands of the spine surgeon, enabling an ideal placement of the disc prosthesis. Navigation offers greater accuracy and less inter-procedural variation than standard fluoroscopy (due to the parallax effect). As accurate (ideal or suboptimal) placement correlates with good clinical outcome, further clinical studies on the navigation of TDR are essential. In this present study, the disc replacement was performed by a surgeon without experience in total disc replacement, indicating that prior completion of a learning curve was not essential.

Computer Model For Sieves' Vibrations Analysis, Using an Algorithm Based on the False-Position Method

The analysis of the sieves vibrations in the case of screening civil engineering construction bulk materials is usual made by using some differential equations depending on parameters related to different material and sieves characteristics. One of those parameters is the throwing coefficient -c- that is the ratio between the force capable to throw up the particle from the sieve surface, and the gravity of this particle. The throwing coefficient is one of the most important characteristics of a sieve dynamic behavior and its values are often used to establish a particular case to the sieve oscillations. In order to find the position of a particle that jump on the screen surface, a system of 6 differential equations with 6 unknown integrating constants can be established. All the involved equations are in transcendent form and it is necessary to solve the system by computer algorithms. First of all, the 6 unknown integrating constants are replaced with related linear relations depending on the throwing coefficient. Secondly, an original computer algorithm based on the so-called false-position method is proposed. In order to validate it, the new system of 6 differential equations depending on the throwing coefficient is solved for some particular cases of the particle jumps. Finally, the solutions are compared for the same conditions of the initial used system. The conclusion is that in the case of the construction bulk materials, the two systems give almost similar solutions. In this case, the new system depending on the throwing coefficient is much easier to work with that the initial system. Another advantage is that in the very first steps one can choose the throwing coefficient and establish the best vibrating regime.

Various Circular Arc Radii of the Distal Volar Radius and the Implications on Volar Plate Osteosynthesis

The purpose of this anatomical study was to explore the different circular arc radii of the distal volar radius and provide more detailed anatomic information that will further the understanding of volar plate osteosynthesis. The profiles of the volar distal radii of 100 cadaver specimens were measured with a common profile gauge. Profiles were copied onto paper and then matched to a best-fit circular arc template to determine the radius of curvature on the radial and ulnar sides of the distal volar radius. The mean circular arc radius of the distal volar surface was 2.6 cm (+/-1 cm, 1-6 cm) on the radial side and 2.3 cm (+/-1 cm, 1-6 cm) on the ulnar side. A significant difference (P<.01) was noted in the radii of curvature of the distal radius in 55% of the study population. In 37% of these cases, the circular arc radius flattens toward the ulnar side. In 63%, it flattens toward the radial side. This characteristic may lead to a false rotation position of the distal fracture fragment following volar plate osteosynthesis. In addition, suboptimal or incorrect plate position may result due to the discrepancy between the plate radius of curvature and the radius of curvature of either the radial or ulnar volar radius.

Using the Ancient Method of False Position to Find Solutions

During my twenty-four years as a middle school and high school teacher, I observed that students were often fascinated by vignettes from the history of mathematics. When the vignette had an ancient Egyptian setting, that background added a certain mystique.

Using, Seeing, Feeling, and Doing Absolute Value for Deeper Understanding

Several activities that are based on the ancient method of false position, also called false assumption, are presented in this article as a way to motivate students to find the solution of literal equations in beginning algebra. With a historical connection, the activities described engage students in solving linear equations in a novel way. Extension exercise included.

The difference [formula omitted] in a problem of rations from the Rhind mathematical papyrus

In an ancient Egyptian problem of bread distribution from the Rhind mathematical papyrus (dated between 1794 and 1550 B.C.), a procedure of “false position” is used in the calculation of a series of five rations. The algorithm is only partially illustrated in the problem text, and last century's prevailing interpretations suggested a determination of the series by trial and error. The missing part of the computational procedure is reconstructed in this article as an application of the algorithm, exemplified in the preceding section of the papyrus, to calculate an unknown quantity by means of the method of “false position.”

Decentralized position verification in geographic ad hoc routing

AbstractInter‐vehicle communication is regarded as one of the major applications of mobile ad hoc networks (MANETs). Compared to MANETs or wireless sensor networks (WSNs), these so‐called vehicular ad hoc networks (VANETs) have unique requirements on network protocols. The requirements result mainly from node mobility and the demands of position‐dependent applications. On the routing layer, those requirements are well met by geographic routing protocols. Functional research on geographic routing has already reached a considerable level, whereas security aspects have only been recently taken into account. Position information dissemination has been identified as being crucial for geographic routing since forged position information has severe impact regarding both performance and security.In this work, we first summarize the problems that arise from falsified position data. We then propose a framework that contains different detection mechanisms in order to mitigate or lessen these problems. Our developed mechanisms are capable of recognizing nodes cheating about their position in beacons (periodic position dissemination in most single‐path geographic routing protocols, e.g., GPSR). Unlike other proposals described in the literature, our detection system does not rely on additional hardware or special nodes, which would contradict the ad hoc approach. Instead, we use a number of different independent sensors to quickly give an estimation of the trustworthiness of other nodes' position claims. The different sensors run either autonomously on every single node, or they require cooperation between neighboring nodes.The simulation evaluation proves that the combination of autonomous and cooperative position verification mechanisms successfully discloses most nodes disseminating false position information, and thereby widely prevents attacks using position cheating. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Gridline methods for detection of object displacements and rigid body movements

Abstract If measurements are performed without artificial targets, deformation analysis cannot refer to the points on the objects defined at an initial or prior measurement epoch. This paper describes how an arrangement of virtual lines, so-called grid lines (GL), offers a solution. A total station, measuring distances without a reflector, or a theodolite measurement system can be used to make the measurements. The GLs should intersect the object to be measured, as far as possible, at right angles and they can be arranged in such a way that each of them runs through one of the object points chosen for the initial measurements. Subsequently deformations of the object can be described along the GLs. The general idea behind this method is to track the virtual lines with the optical axis of a theodolite, until it aims at the points where the GLs intersect the surface. Tracking starts where the GLs intersected the surface of the object during the last measurement epoch. However, if the surface moved this point will be no longer an intersection point and the optical axis hits the surface at another point. Consequently there will be a separation of these two points, and if the theodolite starts tracking the GL, a distance function is obtained. This can be used to determine the new intersection point of the GL if that line is tracked until the value of the distance function is zero. The following sections describe how this process can be iteratively controlled by algorithms for an one-dimensional search space. The algorithms are based on the binary-search-, so-called parabola-, and regula-falsi method. Simulations and practical tests are used to evaluate the different search algorithms. The accuracy of the intersection points and the number of necessary iteration steps, describing the speed of the algorithms, are used as quality assessment parameters. Both quality characteristics depend on the truncation criteria: a distance and a length of an interval.

A robust localization algorithm in wireless sensor networks

Most of the state-of-the-art localization algorithms in wireless sensor networks (WSNs) are vulnerable to various kinds of location attacks, whereas secure localization schemes proposed so far are too complex to apply to power constrainedWSNs. This paper provides a distributed robust localization algorithm called Bilateration that employs a unified way to deal with all kinds of location attacks as well as other kinds of information distortion caused by node malfunction or abnormal environmental noise. Bilateration directly calculates two candidate positions for every two heard anchors, and then uses the average of a maximum set of close-by candidate positions as the location estimation. The basic idea behind Bilateration is that candidate positions calculated from reasonable (i.e., error bounded) anchor positions and distance measurements tend to be close to each other, whereas candidate positions calculated from false anchor positions or distance measurements are highly unlikely to be close to each other if false information are not collaborated. By using ilateration instead of classical multilateration to compute location estimation, Bilateration requires much lower computational complexity, yet still retains the same localization accuracy. This paper also evaluates and compares Bilateration with three multilateration-based localization algorithms, and the simulation results show that Bilateration achieves the best comprehensive performance and is more suitable to real wireless sensor networks.

A new modification of false position method based on homotopy analysis method

A new modification of false position method for solving nonlinear equations is presented by applying homotopy analysis method (HAM). Some numerical illustrations are given to show the efficiency of algorithm.

Numerical mathematics with GeoGebra in high school

We have prepared a suite of motivational examples which illustrate numerical methods for equation solving. Fixed point iteration, Newton's method, secant method and regula falsi method are implemented as GeoGebra tools. Our experience in teaching of numerical mathematics in "Jovan Jovanovic Zmaj" high school in Novi Sad is presented. We have tested pupil proficiency in numerical equation solving with and without use of a computer and the results are presented.

Direct methods and the solution of organic structures from powder data

The electron density map produced after the application of direct methods to powder diffraction data of organic compounds is usually very approximated: some atoms are missed, other atoms are in false positions, some atoms are imperfectly located and the connectivity is quite low. A new procedure able to recover the complete structure model is described. In this procedure, a better interpretation of the map is combined with geometrical techniques for generating new atomic positions. The application of the new procedure may lead to the recovery of the complete crystal structure.

False position in Leonardo of Pisa's Liber Abbaci

We examine the rhetorical methods of Leonardo of Pisa in his exposition of single false position in Liber Abbaci. For example, Leonardo makes extensive use of formulaic phrases in his solutions. Some of these formulas also seem to indicate whether a particular solution needs further justification. Although he prefers proofs in terms of the pseudo-Euclidean canon of al-Khwārizmī, sometimes such proof eludes Leonardo and he resorts instead to justification by experiment. We also look at the extent to which using symbolic representations might distort our view of Leonardo's thinking.

On third-order convergent regula falsi method

An improved regula falsi iterative formula for solving nonlinear equations is proposed in this paper. This method is effective and promising with its third-order convergence of both the sequences of diameters and iterative points sequence.

Mathematics in Context: A Case in Early Nineteenth-Century Korea

ArgumentThis paper aims to show how a nineteenth-century Korean scholar's mathematical study reflects the Korean intellectual environment of his time by focusing on the rule of false double position and the method of root extraction. There were two major trends in Korean mathematics of the early nineteenth century: the first was “Tongsan,” literally “Eastern Mathematics,” which largely depended on Chinese mathematics of the Song and Yuan period adopting counting rod calculation; the second trend was Western mathematics, which was transmitted by the Jesuits and their Chinese collaborators from the late sixteenth century. There was also an intellectual transition in late eighteenth-century Korea when mathematics, which had been of only minor interest for Confucian scholars, became an important part of Confucian pursuits. We can gain an insight into the history of mathematics in Korea by examining and understanding Hong Kil-chu's (1786–1841) mathematical studies and the context of the academic world of his time.

FoxN3 is required for craniofacial and eye development of Xenopus laevis

A functional knockdown of FoxN3, a member of subclass N of fork head/winged helix transcription factors in Xenopus laevis, leads to an abnormal formation of the jaw cartilage, absence or malformation of distinct cranial nerves, and reduced size of the eye. While the eye phenotype is due to an increased rate of apoptosis, the cellular basis of the jaw phenotype is more complex. The upper and lower jaw cartilages are derivatives of a subset of cranial neural crest cells, which migrate into the first pharyngeal arch. Histological analysis of FoxN3-depleted embryos reveals severe deformation and false positioning of infrarostral, Meckel's, and palatoquadrate cartilages, structural elements derived from the first pharyngeal arch, and of the ceratohyale, which derives from the second pharyngeal arch. The derivatives of the third and fourth pharyngeal arches are less affected. FoxN3 is not required for early neural crest migration. Defects in jaw formation rather arise by failure of differentiation than by positional effects of crest migration. By GST-pulldown analysis, we have identified two different members of histone deacetylase complexes (HDAC), xSin3 and xRPD3, as putative interaction partners of FoxN3, suggesting that FoxN3 regulates craniofacial and eye development by recruiting HDAC.

Simulation of dynamics of a boiler steam superheater with an attemperator

This paper presents a numerical method to determine the proper rate of spray water mass flow to be supplied to an injection attemperator. The basic aim of this method is to keep the steam temperature of the superheater outlet at a constant (assumed) level, when the parameters of steam at the inlet to the attemperator and combustion gases at the inlet to the superheater vary. The presented method is suggested for digital control of superheated steam temperature. A crucial condition for correct control is setting up a precise numerical model of the superheater, which would provide fast and accurate determination of the cooling water mass flow supplied to the injection attemperator. For that purpose, a model of a superheater with distributed parameters was developed, which comprises solving equations describing the laws of mass, momentum, and energy conservation. The proposed method to determine the mass flowrate of spray water is based on computed steam temperatures at the inlet and outlet of the analysed superheater stage. The so formulated problem, being an inverse one, was solved using the Regula Falsi method. A computational test verifying the proposed solution is presented. A comparison of numerical calculation results and measurements, taken on the platen superheater of a steam boiler type OP-210 with 210 × 103 kg/h capacity, is also presented. In both cases, a fully satisfactory agreement of the computed steam temperature at the superheater outlet with the assumed value was achieved. The suggested solution to the inverse problem, combined with the proposed model of superheater dynamics, enables one to determine the proper mass flow of spray water in on-line mode and improve the control quality.

일정체적 고정-회전 기둥의 동적안정 해석

This paper deals with the dynamic stability analysis of clamped-hinged columns with constant volume. Numerical methods are developed for solving natural frequencies and buckling loads of such columns, subjected to an axial compressive load. The parabolic taper with the regular polygon cross-section is considered, whose material volume and column length are always held constant. Differential equations governing both free vibrations and buckled shapes of such columns are derived. The Runge-Kutta method is used to integrate the differential equations, and the Regula-Falsi method is used to determine natural frequencies and buckling loads, respectively. The numerical methods developed herein for computing natural frequencies and buckling loads are found to be efficient and robust. From the numerical results, dynamic stability regions, dynamic optimal shapes and configurations of strongest columns are reported in figures and tables.

POSITION VERIFICATION APPROACHES FOR VEHICULAR AD HOC NETWORKS

Intervehicle communication is regarded as one of the major applications of mobile ad hoc networks (MANETs). Compared to MANETs, these so-called vehicular ad hoc networks (VANETs) have special requirements in terms of node mobility and position-dependent applications, which are well met by geographic routing protocols. Functional research on geographic routing has already reached a considerable level, whereas security aspects have been vastly neglected so far. Since position dissemination is crucial for geographic routing, forged position information has severe impact regarding both performance and security. In this work, we first analyze the problems that may arise from falsified position data. Then, in order to lessen these problems, we propose detection mechanisms that are capable of recognizing nodes cheating about their location in position beacons. In contrast to other position verification approaches, our solution does not rely on special hardware or dedicated infrastructure. Evaluation based on simulations shows that our position verification system successfully discloses nodes disseminating false positions and thereby widely prevents attacks using position cheating