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Mathematica By Example (rev. Ed.)

A locus is the set of all points (usually forming a curve or surface) satisfying some condition. For example, the locus of points in the plane equidistant from a given point is a circle, and the set of points in three-space equidistant from a given point is a sphere.

Mathematica by Example (rev. ed.)

for are distinct (Guy 1994). An example is 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290, 361, ... (OEIS A005282). Halberstam and Roth (1983) contains an accessible account of most known results up to around 1965. Recent advances have been made by Cilleruelo, Jia, Kolountzakis, Lindstrom, and Ruzsa.

A second aspect of the new method concerns the use of mathematicaltheory not to derive testable conclusions from hypotheses, as Galileoand Huygens had done, but to cover a full range of alternativetheoretical possibilities, enabling the empirical world then to selectamong them. This new approach is spelled out most forcefully at theend of Book 1, Section 11:

A third aspect of the new method, which proved most controversial atthe time, was the willingness to hold questions about the mechanismthrough which forces effect their changes in motion in abeyance, evenwhen the mathematical theory of the species and proportions of theforces seemed to leave no alternative but action at a distance. Thisaspect remained somewhat tacit in the first edition, but then, inresponse to criticisms it received, was made polemically explicit inthe General Scholium added at the end of the second edition:

During most of the eighteenth century the primary challenge thePrincipia presented to philosophers revolved around what tomake of a mathematical theory of forces in the absence of a mechanism,other than action at a distance, through which these forces work. Bythe last decades of the century, however, little room remained forquestioning whether gravity does act according to the laws that Newtonhad set forth and does suffice for all the motions of the heavenlybodies and of our sea. No one could deny that a science had emergedthat, at least in certain respects, so far exceeded anything that hadever gone before that it stood alone as the ultimate exemplar ofscience generally. The challenge to philosophers then became one ofspelling out first the precise nature and limits of the knowledgeattained in this science and then how, methodologically, thisextraordinary advance had been achieved, with a view to enabling otherareas of inquiry to follow suit.

The contention that the empirical reasoning in the Principiadoes not presuppose an unbridled form of absolute time and space shouldnot be taken as suggesting that Newton's theory is free of fundamentalassumptions about time and space that have subsequently proved to beproblematic. For example, in the case of space, Newtonpresupposes that the geometric structure governing which lines areparallel and what the distances are between two points isthree-dimensional and Euclidean. In the case of time Newtonpresupposes that, with suitable corrections for such factors as thespeed of light, questions about whether two celestial events happenedat the same time can in principle always have a definite answer.And the appeal to forces to distinguish real from apparent non-inertialmotions presupposes that free-fall under gravity can always, at leastin principle, be distinguished from inertial motion.[20]

A fundamental contrast between Newton's mathematical theory of motionunder centripetal forces and the mathematical theories of motiondeveloped by Galileo and Huygens is that Newton's is generic. Galileoand Huygens examined one kind of force, uniform gravity, with a goalof deriving testable consequences. Newton's theory covers not onlyforces that vary as 1/r2, for which thePrincipia is famous, but also forces that vary as r,as 1/r3, and even as any arbitrary function ofr. At the end of Section 11 he gives a reason, quotedearlier:

Newton illustrates the value of Proposition 6 with a series ofexamples, the two most important of which involve motion in anellipse. If the force center is at a focus S of the ellipse, thenthe limit of (QR/QT2) is everywhere equal to half theconstant latus rectum of the ellipse, and hence the force varies as1/SP2, or 1/r2. But if the forcecenter is at the center C of the ellipse, the force turns out to varyas PC, that is, linearly with r. This contrast raises aninteresting question. What conclusion can be drawn in the case ofmotion in an ellipse for which the foci are very near the center, andthe center of force is not known to be exactly at the focus?Newton clearly noticed this question and supplied the means foranswering it in the Scholium that ends Section 2.

The group of propositions following the deduction of universal gravitygives indications of the evidential strategy that lies behind the leapto taking this law to be exact. Immediately upon concluding firstthat the planets would sweep out equal areas in equal times in exactellipses and then that the orbits would be exactly stationary were itnot for the gravitational interactions among the planets, Newton callsattention to the easiest to observe deviations from this idealization,the then still mysterious vagaries in the motions of Jupiter and Saturnwhich Newton attributes to their gravitational interaction.Because, according to the theory, the idealization would hold exactlyin the specified circumstances, these and all other deviations mustresult from further forces not taken into account in the idealizedcase. Identifying these forces and showing that, according to thetheory, they do produce the deviations is a way for ongoing research tomarshal continuing evidence to bear on the theory of gravity. Toput the point differently, the initial idealizations that Newtonidentifies can serve as the starting point for a process of successiveapproximations that should yield increasingly close agreement with thecomplex true motions. These idealizations are especially wellsuited for this purpose precisely because, according to the theory,they would hold exactly were no other forces at work, and hence everydeviation from them should be physically telling, and not just, forexample, an accidental feature of a curve-fit. Pursuit of such aresearch program of successive approximations promises to yield eitherfurther evidence for the theory of gravity when the program issuccessful or the exceptions Newton speaks of in Rule 4 that requirethe theory to be revised.

Newton's account of the tides in Propositions 24, 36, and 37 wasmuch heralded not only at the time, but still today. He isnevertheless receiving more credit for this than he is due. Hedid identify solar and lunar gravity as the forces driving the tides,but this is all he did. He ignored the rotation of the Earth, andworse he considered only the radial component of the solar and lunargravitational forces in these three propositions. In fact, theradial component of these forces has a very small effect compared withthe transradial component, that is, the component perpendicular to theradial component. All of this became clear in the 1770s whenLaplace developed the mathematical theory of tidal motion from which allsubsequent work has proceeded.

In two passages that remained word for word the same in all threeeditions Newton announced that the Principia was meant toillustrate a new approach to empirical inquiry. Neither theremark about deriving forces from phenomena of motion and then motionsfrom these forces in the Preface to the first edition nor the remarkabout comparing a generic mathematical theory of centripetal forceswith the phenomena in order to find out which conditions of forceactually hold at the end of Book 1, Section 11, however, shed muchlight on just what this new approach is supposed to be. Otherthan these two passages, the only notable remark about methodology isthe famous passage, quoted earlier, from the General Scholium added inthe second edition as a final, parting statement:

The difference between the two notions can be clarified by a simple example. The set with the trivial topology is pathwise-connected, but not arcwise-connected since the function defined by for all , and , is a path from to , but there exists no homeomorphism from to , since even injectivity is impossible.

We present the tensor computer algebra package xPert for fast construction and manipulation of the equations of metric perturbation theory, around arbitrary backgrounds. It is based on the combination of explicit combinatorial formulas for the nth order perturbation of curvature tensors and their gauge changes, and the use of highly efficient techniques of index canonicalization, provided by the underlying tensor system xAct, for Mathematica. We give examples of use and show the efficiency of the system with timings plots: it is possible to handle orders n = 4 or n = 5 within seconds, or reach n = 10 with timings below 1 h.

Babylonian mathematics is a range of numeric and more advanced mathematical practices in the ancient Near East, written in cuneiform script. Study has historically focused on the Old Babylonian period in the early second millennium BC due to the wealth of data available. There has been debate over the earliest appearance of Babylonian mathematics, with historians suggesting a range of dates between the 5th and 3rd millennia BC.[6] Babylonian mathematics was primarily written on clay tablets in cuneiform script in the Akkadian or Sumerian languages.

Most clay tablets that describe Babylonian mathematics belong to the Old Babylonian, which is why the mathematics of Mesopotamia is commonly known as Babylonian mathematics. Some clay tablets contain mathematical lists and tables, others contain problems and worked solutions.

The Babylonians used pre-calculated tables to assist with arithmetic. For example, two tablets found at Senkerah on the Euphrates in 1854, dating from 2000 BC, give lists of the squares of numbers up to 59 and the cubes of numbers up to 32. The Babylonians used the lists of squares together with the formulae: 041b061a72

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