Wikipedia:Reference desk/Archives/Mathematics/2006 November 14

= November 14 =

Why generalised functions are called distributions
I would like to know whether there is any (historical) reason for using the term distributions for generalised functions. How do generalised functions generalize probability distributions, as claimed in distributions. Mairan 04:22, 14 November 2006 (UTC)
 * I don't think it answers your search for historical reasons, except that it gives you another link to look for your answer in, but I changed the redirect on generalised function so that it is now the same as generalized function. Perhaps this and distribution deserve to be merged, but generalised function and generalized function shouldn't go to different places. —David Eppstein 04:38, 14 November 2006 (UTC)

Perhaps, it is because it generalises measure in some sense and probability distributions are measures. Mairan 07:07, 14 November 2006 (UTC)

How to calculate Volume.
I want the formula or calculation to convert fuel level in a cylinderical tank lying horizontally having dimension of 1825mm Diameter and 3960mm long.We have a dip stick which gives the level in millimeter but we want that with the calculation or formula we can get that millimeter reading in liters. An answer to this Question will be highly appreciated.


 * The answer is not very simple. I hope you have some familiarity with algebra and trigonometry. I use radians below for angles, where the size of a right angle is π/2, instead of 90°. Also, for keeping the formulas general and readable, let R = 0.9125m be the radius (half the diameter) and L = 3.960m the length. If we take a cross section of the tank, we get a circle of radius R in which the fuel occupies a circular segment (see the yellow part of the picture there; you have to turn it upside down in your mind). If A is the area of that circular segment, then we have that the volume V = A × L. So the issue is how to calculate A. Using the same notation as in the article Circular segment, the angle θ is determined by d/R = cos(θ/2), where d = R − h. Therefore:
 * θ = 2 arccos((R − h) / R)
 * A = 1/2R2(θ − sin θ)
 * V = 1000 A L l/m3
 * where the last factor 1000 comes from going from m3 (cubic metres) to l (litres).
 * Two examples. First, for h = 100mm:
 * h = 0.10000
 * (R − h) / R = (0.9125 − 0.1) / 0.9125 = 0.89041
 * θ = 2 arccos 0.89041 = 0.94510
 * sin 0.94510 = 0.81055
 * A = 0.5 × 0.91252 × (0.94510 − 0.81055) = 0.05601
 * V = 1000 × 0.05601 × 3.960 = 222
 * The second example is with h = 1100m, so h > R:
 * h = 1.10000
 * (R − h) / R = (0.9125 − 1.1) / 0.9125 = −0.20548
 * θ = 2 arccos(−0.20548) = 3.55550
 * sin 3.55550 = −0.40219
 * A = 0.5 × 0.91252 × (3.55550 − (−0.40219)) = 1.64770
 * V = 1000 × 1.64770 × 3.960 = 6525
 * --Lambiam Talk 10:11, 14 November 2006 (UTC)

help i want to pass in maths exams of mvita school based kenya
Help wed i will be doing MATHS AND ENGLISH i want to pass in them so please i want some formulas for maths
 * The plural of "formula" is "formulae". That should help with the English exam. Tompw 11:54, 14 November 2006 (UTC)
 * Please see pedantic. Both "formulas" and "formulae' are acceptable plurals of "formula". --Lambiam Talk  13:41, 14 November 2006 (UTC)


 * It is very difficult to give you advice, because you have not indicated at what level the exams are. There are dozens of formulas at each level, altogether hundreds. And if your exams are tomorrow, you have little time left to study. You need not only to know the formulas, but also how to use them.
 * For an elementary level, look at Fraction, Elementary algebra, Linear equation, and Quadratic equation.
 * Slightly more advanced: Logarithm, Logarithmic identities, Trigonometry, List of trigonometric identities.
 * For texts to study, visit the Mathematics bookshelf. --Lambiam Talk  14:01, 14 November 2006 (UTC)


 * I would suggest talking to your teachers for advice for exams. Formulas is not correct; it is not acceptable in english, it may be where you are, but that makes it in no way less wrong.Englishnerd 18:49, 14 November 2006 (UTC)


 * Formulas is not acceptable in Latin. In English, for words with a Latin root, it is usually correct to use either the S pluralization or the original Latin plural. Check a dictionary, and you'll find both listed. - Rainwarrior 18:54, 14 November 2006 (UTC)
 * It actually is acceptable in Latin in the sentence Nonnullas formulas desidero. There used to be a time when scholars, when employing a Latin or Greek noun in a text in the vernacular, would put it in the appropriate grammatical case of its declensional paradigm. --Lambiam Talk  07:47, 15 November 2006 (UTC)
 * A dictionary? Whose authors probably speak some vulgar language like *gasp* English? I've always said that if you want to know how to speak English, there's no better source than a bunch of Romans who died two millennia ago. Millenniums. Sigh. Tesseran 05:33, 16 November 2006 (UTC)
 * BTW, formulas gets 36 million hits on Google, formulae 11 million. I'm not sure if Google's plural+singular function might be checking in and I normally detest Google for this kind of thing but it does agree with my expectation. Nil Einne 17:22, 20 November 2006 (UTC)

Solution of Travelling Salesman Problem using Hopfield nets
Has anyone solved the Travelling Salesman problem using Hopfield nets.If so then would you be kind enough to answer the following questions. 1.What is the probability of arriving at the shortest distance path for 5,8,10,12,15,18,20 cities? 2.Is it possible to attain 90% chance of finding the shortest path for 10 cities ? 3.How does the probability of finding the shortest path vary with the number of cities ? 4.Are there algorithms and heuristics to drastically increase the chance of finding the correct path ? 5.Is there any mathematical proof or theorem to suggest that the path lines which form the shortest path do not intersect inside the polygon formed but only at the vertices or outside the polygon.

It was difficult to find resources on these questions. If you would be kind enough to answer any one of them please e-mail me the answers at [e-mail address blanked].


 * Did you follow the link given at Hopfield net? Apparently someone has. The questions concerning the probabilities can only be answered in regard to a specific collection of data sets, or a specified process for generating input data sets. --Lambiam Talk  18:45, 14 November 2006 (UTC)
 * Concerning question 4: Yes, there is an algorithm that gives you a very good chance indeed of finding the true shortest path. It is knowns as brute-force search. Unfortunately, it takes very long. No algorithms or heuristics are known to drastically increase that chance while taking polynomial time. As to question 5, if this is about the Euclidean TSP where the cities are points in the plane and the distances are the Euclidean distances, then it is not hard to show that if two segments intersect, say A1—A2 intersect B1—B2, where the direction of travel is the same for both, you get a shorter tour by "rewiring" this as A1—B2 and B1—A2 (thereby reversing the direction for part of the tour). Basically, in a convex quadrilateral, the sum of the length of the diagonals exceeds the sum of the lengths of two opposite sides.  --Lambiam Talk  19:29, 14 November 2006 (UTC)

pi
Does pi ever end


 * Nope. Splintercellguy 23:55, 14 November 2006 (UTC)


 * That was the short answer. Here is a bit longer answer. All numbers with finite decimal representations are rational numbers, which means they can be written as a fraction p / q for integers p and q. (And in the special case of a finite decimal representation q is or divides a power of 10.) The number pi, on the other hand, is known to be irrational. This was already proved in 1761 by Lambiam Lambert. See further Pi and Pi. --Lambiam Talk  07:31, 15 November 2006 (UTC)


 * Just to clarify on Lambiam's response, some numbers without finite decimal representations can still be rationals. For example, 1/3 = 0.333..., etc. In fact, all recurring decimals are rational and all rationals have decimal expansions that either terminate or are recurring. The numbers which do not terminate and are not recurring are, as mentioned, the irrational numbers. Popular examples of irrational numbers are pi, e, and the square root of 2. Maelin (Talk | Contribs) 08:48, 15 November 2006 (UTC)


 * Just to clarify on Maelin's response, the same applies not only to decimal representation, but to every positional numeral system:
 * for every natural radix p&gt;1, each rational number has a recurring representation in the system with that radix p (this includes finite representations, which are recurring, too, with repeating zeros, like 1.17 = 1.1700000...),
 * and each irrational number has infinite, nonrecurring representation in every positional numeral system.
 * --CiaPan 16:31, 15 November 2006 (UTC)