scholarly journals On Extraneous Values in Simultaneous Equations

1914 ◽  
Vol 16 ◽  
pp. 183-184
Author(s):  
G. D. C. Stokes
Keyword(s):  
Simultaneous Equations ◽  
Geometrical Meaning ◽  
The Subject ◽  
Straight Lines

Mr Ridley's note in the January number giving an algebraic explanation of extraneous values suggests an elementary treatment of the subject on the geometrical side. It must be assumed that the pupil is able to trace from their equations straight lines, circles, and parabolas of the type y = ax2 + bx + c. He should be made to draw the curves in cases like those given here, and to state the geometrical meaning of every distinct step in the algebra.

VII. On the theory of local probability, applied to straight lines drawn at random in a plane; the methods used being also extended to the proof of certain new theorems in the integral calculus

1868 ◽  
Vol 158 ◽  
pp. 181-199 ◽  
Keyword(s):  
Integral Calculus ◽  
Past Generation ◽  
Differences Of Opinion ◽  
Local Probability ◽  
Modern Analysis ◽  
The Subject ◽  
Latent Error ◽  
Straight Lines ◽  
Distinguishing Features ◽  
Inherent Ambiguity

1. The new Theory of Local or Geometrical Probability, so far as it is known, seems to present, in a remarkable degree, the same distinguishing features which characterize those portions of the general Theory of Probability which we owe to the great philosophers of the past generation. The rigorous precision, as well as the extreme beauty of the methods and results, the extent of the demands made on our mathematical resources, even by cases apparently of the simplest kind, the subtlety and delicacy of the reasoning, which seem peculiar to that wonderful application of modern analysis— ce calcul delicat , as it has been aptly described by Laplace—reappear, under new forms, in this, its latest development. The first trace which we can discover of the Theory of Local Probability seems to be the celebrated problem of Buffon, the great naturalist— a given rod being placed at random on a space ruled with equidistant parallel lines, to find the chance of its crossing one of the lines. Although the subject was noticed so early, and though Buffon’s and one or two similar questions have been considered by Laplace, no real attention seems to have been bestowed upon it till within the last few years, when this new field of research has been entered upon by several English mathematicians, among whom the names of Sylvester and Woolhouse are particularly distinguished. It is true that in a few cases differences of opinion have arisen as to the principles, and discordant results have been arrived at, as in the now celebrated three-point problem, by Mr. Woolhouse, and the four-point problem of Professor Sylvester; but all feel that this arises, not from any inherent ambiguity in the subject matter, but from the weakness of the instrument employed; our undisciplined conceptions of a novel subject requiring to be repeatedly and patiently reviewed, tested, and corrected by the light of experience and comparison, before they are purged from all latent error. The object of the present paper is, principally, the application of the Theory of Probability to straight lines drawn at random in a plane; a branch of the subject which has not yet been investigated. It will be necessary to begin by some remarks on the general principles of Local Probability. Some portion of what follows I have already given elsewhere.

On the convergence of iterative processes for the solution of simultaneous equations in several variables

1960 ◽  
Vol 56 (3) ◽  
pp. 286-289 ◽  
Author(s):  
W. F. Bodmer
Keyword(s):  
Detailed Analysis ◽  
Iterative Process ◽  
Order Of Convergence ◽  
Simultaneous Equations ◽  
Iterative Processes ◽  
Several Variables ◽  
The Subject ◽  
Newton Raphson

The notion of the order of convergence of iterative processes may well have been known to Gauss or even Newton, but to the author's knowledge it was first considered in detail in 1870 by Schröder(5). The subject was later investigated independently by Hartree(4) and Bodewig(1). Hartree alone mentions, briefly, the extension from one to several variables, but gives no detailed analysis. Domb and Fisher(2) consider a particular type of iterative process in several variables, which they call ‘degenerate’, in which all the variables tend to the same value. We shall follow essentially Hartree's treatment, in considering the solution of r simultaneous equations in r variables and applications to a generalization of the Newton-Raphson process.

scholarly journals Religiosity of pilgrims in Serbia: Case study of three sanctuaries

2012 ◽  
Vol 23 (1) ◽  
pp. 53-68 ◽  
Author(s):  
Dragana Radisavljevic-Ciparizovic
Keyword(s):  
International Communication ◽  
Family Influence ◽  
Religious Expression ◽  
Self Assessment ◽  
Religious Tourism ◽  
Religious Formation ◽  
The Subject ◽  
Straight Lines ◽  
Depth Interviews

Pilgrimage is an ancient form of religious expression, inherent in almost every confession. Modern pilgrimage differs from past pilgrim travels in various attributes. Pilgrimages contribute to the tourism development because they affect interreligious and international communication. In the first place, significance and topicality of the subject is explained, and basic concepts are defined (pilgrimage, religious tourism, contemporary pilgrim). After that, a theoretical and methodological framework has been provided. Research relays to religious and ethical ?mixed? pilgrimages and also includes Orthodox chancels: St. Petka?s chapel in Kalemegdan and Madonna of Djunis monastery and catholic chancel Madonna of Tekije near Petrovaradin. These places are visited regardless of the visitor's faith. In depth interviews were conducted in 2007 in Belgrade, and the sample consisted of 25 Orthodox and 25 Catholic interviewees. We were monitoring religiosity of pilgrims through time by using three categories: upbringing, conversion, and self-assessment of religiosity. Regarding the category of upbringing, three groups were identified: those with a traditional religious formation, an irreligious formation, and those from devotional families. According to the self-assessement of religiosity, the following typology was formed: assured and practical believer, missionary, and traditionalist. We noticed three straight lines in believers? religious life: progression, stagnation and regression. The results of this exploration confirm that family influence on examinees? religiosity is strong, but the key category for progression in their religious lives is conversion understood as a dynamic category.

scholarly journals II. Additional Observations on the Runic Obelisk at Ruthwell; the Poem of the Dream of the Holy Rood; and a Runic Copper Dish found at Chertsey. By John M. Kemble, Esq. in a Letter to Sir Henry Ellis, K.H., F.R.S. Secretary

Archaeologia ◽  
1844 ◽  
Vol 30 ◽  
pp. 31-46 ◽  
Author(s):  
John M. Kemble
Keyword(s):  
Great Degree ◽  
Peculiar Form ◽  
The Subject ◽  
Almost All ◽  
Straight Lines ◽  
Personal Feeling

It gives me very sincere pleasure to be able to offer you undeniable confirmation of the justice of my views respecting the Runic Obelisk at Ruthwell. I would not, indeed, have you suppose that I ever entertained the slightest doubt of having seized the general sense of the inscription; but, taking into consideration the fragmentary state of the legend itself, as well as the abrasion of many characters, which justified, indeed rendered necessary, a somewhat bold method of proceeding, I could not venture to hope that I had entirely escaped errors, which are in some degree inseparable from conjectural criticism. From the peculiar form of the Runic characters, which consist principally of straight lines and angles, they are especially liable to confusion when the slightest portion is abraded by age: and this is the case with the Ruthwell inscription in a very great degree. The task of restoring readings so injured by lapse of time, though not a hopeless, might fairly be considered a difficult one; and in this conviction, I was prepared to admit the probability that better versions than my own might at some time be substantiated. Circumstances have, however, now placed within my reach a complete, though modern, copy of the whole inscription, parts of which it cost me so much serious labour to decypher: and it is highly satisfactory to discover that in almost all the details of interpretation, I was proceeding upon sound and safe grounds. This naturally affords a gratification of a far higher character than the mere selfish pleasure derived from any personal feeling: and it is in the hope that some of your readers may be induced to bestow attention upon a system whose results are so strongly confirmed by further experience, that I again bring the subject before your notice.

The Variation with Temperature of the Dynamic Properties of Rubber and Synthetic Rubberlike Materials

1945 ◽  
Vol 18 (2) ◽  
pp. 306-317
Author(s):  
W. P. Fletcher ◽  
J. R. Schofield
Keyword(s):  
Natural Rubber ◽  
Freezing Point ◽  
Dynamic Properties ◽  
Dynamic Compression ◽  
Smooth Curves ◽  
The Subject ◽  
Region 10 ◽  
Increasing Temperature ◽  
Straight Lines ◽  
Rubberlike Materials

Abstract (1) Over the range from 5° to 40° C, the temperature coefficients of the dynamic compression moduli of all the rubber and rubberlike materials studied are negative and fall numerically with increasing temperature. (2) The highest numerical value of this coefficient for natural rubber is −2.7×10−3 per ° C. Neoprene-Gn has a coefficient 3 to 4 times this value and Buna-S about 5 times. Hycar OR-15 shows the highest coefficient of −1.3×10−1 per ° C from 10 to 20° C, the value changing sharply at 20.2° C to −0.14×10−1, which is maintained up to 40° C. (3) Results for Neoprene-E were not reproducible, owing to a type of slow freezing effect. (4) In all cases but Thiokol-RD resilience tended to increase with increasing temperature throughout the range. (5) Resilience-temperature curves for natural rubber, Neoprene-GN, Neoprene-YD, and Buna-S take the form of straight lines intersecting at 20° to 22.5° C. Neoprene-Z shows a similar effect with intersection at 30.5° C. and Hycar OR-15 similarly at 31.5° C. (6) Copolymers of butadiene and acrylonitrile show increasing modulustemperature coefficient, and in the region 10° to 20° C decreasing resilience with increasing acrylonitrile content. The resilience-temperature diagrams for these polymers, except Hycar OR-15, are smooth curves which appear to reach steady values towards the upper end of the temperature range. (7) Thiokol-RD appears to have a freezing-point, under the dynamic conditions employed, somewhere in the region of 15–20° C; there is evidence that Hycar OR-15 shows a similar effect between 0° and 10° C. (8) The relationships enumerated above refer to basic compounds of the various materials. How far the temperature effects may be reduced or modified by suitable compounding is the subject of continuing investigation.

scholarly journals The skin friction of flat plates to Oseen's approximation

1933 ◽  
Vol 140 (842) ◽  
pp. 543-561 ◽  
Keyword(s):  
Boundary Layer ◽  
Integral Equation ◽  
Recent Observation ◽  
Reynolds Numbers ◽  
Simultaneous Equations ◽  
Viscous Effects ◽  
Flat Plates ◽  
Thin Boundary Layer ◽  
The Subject ◽  
Opposite Extreme

Recent observation that flow past tangential flat plates may remain steady up to Reynolds’ numbers so great as 5 x 10 5 has renewed interest in the problem of calculating the motion. For large motions, such as are characterized by Prandtl’s thin boundary layer of viscous effects, there has long existed the well-known theory of Blasius which recent experiments by Hansen tend to confirm. Approaching the problem from the opposite extreme, Bairstow and Misses Cave and Lang have obtained a solution according to Oseen’s approximation to the equations of viscous flow. Their result is given in the form of an integral equation for the distribution of doublets along the plate which will exactly satisfy Oseen’s suggestion and the boundary conditions for an infinite fluid. But the solution of the equation has depended so far upon constructing a group of simultaneous equations with numerical coefficients determined by graphical means. The process is cumbersome and only two evaluations have been attempted, viz., at Reynolds’ numbers 4 and 4 x 10 4 . Exact treatment of Bairstow and Misses Cave and Lang’s integral equation presents difficulties, but it is possible to find an analytical solution of the equation whose errors throughout the experimental range are probably less than those involved in graphical manipulation. This enquiry is the subject of Section I of the present paper. Section II gives the streamlines and other details of the flow.

scholarly journals II. On the theory of probability, applied to random straight lines

1868 ◽  
Vol 16 ◽  
pp. 266-269
Keyword(s):  
Local Probability ◽  
The Subject ◽  
Straight Lines

This paper relates to the Theory of Local Probability—that is, the application of Probability to geometrical magnitude. This inquiry seems to have been originated by the great naturalist Buffon, in a celebrated problem proposed and solved by him. Though the subject has been more than once touched upon by Laplace, yet the remarkable depth and beauty of this new Calculus seem to have been little suspected till within the last few years, when the attention of several English mathematicians has been directed to it, and results of a most singular character have been obtained. The problems on Local Probability which have been hitherto treated relate almost exclusively to points taken at random. The object of the present paper is to show how the Theory of Probability is to be applied to straight lines whose position is unknown, or, in other words, which are taken at random.

scholarly journals THE COMMON ERRORS IN THE LEARNING OF THE SIMULTANEOUS EQUATIONS

2020 ◽  
Vol 9 (2) ◽  
pp. 263
Author(s):  
Pg. Mohammad Adib Ridaddudin Pg. Johari ◽  
Masitah Shahrill
Keyword(s):  
Linear Equations ◽  
Simultaneous Equations ◽  
Government Schools ◽  
Brunei Darussalam ◽  
Non Linear ◽  
The Subject ◽  
The Common ◽  
Non Linear Equations ◽  
Mathematical Error

The purpose of this study is to understand the causes of common errors and misconceptions in the learning attainment of simultaneous equations, specifically on linear and non-linear equations with two unknowns. The participants consisted of 30 Year 9 students in one of the elite government schools in Brunei Darussalam. Further analyses of their work led to the categorisation of four factors derived from the recurring patterns and occurrences. These four factors are complicating the subject, wrong substitution of the subject, mathematical error and irrational error in solving the question. These factors usually cause participants to make errors or simply misconceptions that usually led them to errors in solving simultaneous equations.

The Resistance of Rubber to Dynamic Forces Part I

1940 ◽  
Vol 13 (1) ◽  
pp. 81-91 ◽  
Author(s):  
R. Ariano
Keyword(s):  
Elastic Recovery ◽  
Strain Curve ◽  
Stress Strain Curve ◽  
Stress Strain ◽  
Joule Effect ◽  
Dynamic Forces ◽  
Widespread Belief ◽  
The Subject ◽  
Service Conditions ◽  
Straight Lines

Abstract The subject of the present paper, which is of great interest on account of the numerous service conditions under which rubber is subjected to dynamic forces, has received little attention, perhaps because of the complexity of the phenomena and the consequent difficulty of coming to any definite and significant conclusions from experimental data. It is a widespread belief, for instance, that in static tension plastic flow takes place and that this is responsible for the Joule effect and that it modifies the shape of the stress-strain curve. By working at high velocities of extension, Williams proved that at room temperature and also at 60° C the stress-strain curves are straight lines and that complete elastic recovery takes place. The importance of verifying such a conclusion as this is obvious. Since, in fact, the elongations for a given load found by Williams were in every case greater when the stress was static, one is led to the conclusion that the deformation brought about by a given load is the sum of two components; one a perfectly elastic component, which obeys Hooke's law and which therefore is applicable to the established science of construction; a second component, which, in contrast to the first, is plastic in character and consequently depends on the duration of application of the load and on the loads previously applied. In brief, the law of deformation should be capable of reduction to the laws of two types of systems, viz., an elastic system and a plastic system. Unfortunately however this assumption could not be confirmed.

scholarly journals Intelligent Auxiliary Artificial Wood Plank Pattern Design Based on the Subject Search Algorithm of Multimedia Resources

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Lihua Zhao
Keyword(s):  
Time Use ◽  
Calculation Method ◽  
Search Algorithm ◽  
Pattern Design ◽  
Wood Processing ◽  
Multimedia Resources ◽  
The Subject ◽  
Straight Lines ◽  
Self Learning ◽  
Multimedia Resource

Regarding the restriction of the wood processing enterprises in the market, intelligent artificial wood materials are mainly based on the demand for pattern quality levels, and the calculation method of multimedia resource theme search is used to achieve the pattern design of intelligent auxiliary artificial wood materials. First, analyze the pattern characteristics of intelligent auxiliary artificial wood materials. After analyzing the characteristics, use the multimedia resource subject search calculation method to carry out the binarization design. At the same time, use the self-learning method to optimize the convergence efficiency and reduce the design time. Finally, pass the softmax designer extracts design schemes for patterns and straight lines.

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