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19.1: Enumeration - Mathematics


19.1: Enumeration - Mathematics

Classes

Define and Use Enumerations

Associate values with predefined names using constant properties or enumeration classes.

Define enumeration classes by creating an enumeration block in the classdef file.

Refer to enumeration members using the class name and the member name.

Use logical, set membership, and string comparison operations on enumerations.

Enumeration classes restrict certain aspects of their use and definition.

Use a handle enumeration to enumerate a set of objects whose state can change over time. Use a value enumeration to enumerate a set of abstract (and immutable) values.

The type of enumeration class determines the information MATLAB ® saves with the class.

Hiding enumeration members enables you to replace enumeration names without creating incompatibilities.

Specialized Enumeration Classes

Enumeration classes derived from built-in types inherit behaviors from these types.

Define properties in an enumeration class to associate specific data with enumeration members.

Use enumeration classes to restrict properties to a predefined set of values.


Gaussian Gamble (Concluded)

The Pre-Pravega event Gaussian Gamble is an online challenge series conducted by Enumeration. The series consists of the teams trying to solve exciting and challenging math problems which are released every fortnight. We will be sure to intrigue you with questions from a wide range of topics to catch your fancy. All you need is a teammate to bounce ideas off, an inquisitive approach to problem solving, and you will be all set to start!

A leaderboard will be maintained which will consist of the score obtained by the teams on each problem, so that you can compete with your friends across the country for the top spot!

Awards

The toppers of the leaderboard will be given lateral entry into the finals of the enumeration event to be conducted at IISc, Bangalore. The top 5 teams on the leaderboard will also receive a certificate.


Mathematics for System Safety Analysis (MATH 19-1)

Course Description

This course is focused on the mathematics used in system safety analysis. The purpose of this course is to provide the trainees with a working understanding of the mathematical theories underlying system safety analysis. From this course, the trainees will be able to properly interpret the results of a system safety analysis and use it in their intended applications. The course will begin with the fundamentals of probability theory and will cover the uses of that theory for solving various system safety problems. Statistical methods will also be covered in relations to establishing equipment failure frequencies. System safety examples will be used throughout the course. Each student should bring a calculator with statistical functions.

Objectives

To provide a level of understanding of the mathematical concepts used in conducting system safety analyses.

Who Should Attend

Individuals who intend to take the system safety course or would like to enhance their understanding of the fundamental mathematical theories used in system theory.


19.1: Enumeration - Mathematics

All of my videos were recorded in the Fall 2016 and Spring 2017 semesters at Tufts University. I did most of them myself, and unfortunately sometimes the microphone caused static noise. With Tufts Technology Services, I have edited most videos (multiple files pieced together, non-essential student questions removed, notes on screen regarding clarifications and corrections, background noise reduced, some contrast improvements, camera zoomed on screen). However, for some of these edited versions, my voice is sometimes a bit muffled because of the background noise elimination. If you find a particular clip to be unclear, please let me know. I can make original content available.

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    1. Useful definitions and properties (No video is planned for this material)
      • Class notes: PDF
      1. Main classification of numbers (just the definitions)
        • e.g., integer, natural, real, rational, irrational, even and odd , prime, etc.
          • Scheinerman 1(3)
      2. Common functions: pages 53-59 in CLRS.
          . (definition)
          • Scheinerman 2(9), Rosen p.151.
          . (definition)
          • Scheinerman 5(29), p.208, Rosen p.149
          . (definition and examples)
          • Scheinerman 7(37), MCS 9.6, Rosen 4.1.
          . (definition and arithmetic)
      3. Exponentiation
          . Pay attention to "Integer exponents", especially "Identities and properties" (3.1 to 3.4)
      4. Rosen, Appendix 2.
    2. Logarithm
        . Pay attention to "Definition" (1.2), "Examples" (1.3), and identities (2.1, 2.2).
    3. Rosen, Appendix 2.
    4. Fibonacci numbers
        . Besides the definition, check out sections 14-15 ("applications" and "in nature")
    5. A very nice, short TED talk.
    6. Fibonacci rabbits
    7. Series (and their sums)
      • Geometric series.
        • This is often crucial for the analysis of algorithms.
        • See p.1147 in CLRS (appendix A). In MCS see beginning of 14, and 14.1.4. Rosen, p.164.
        • xkcd 994 . xkcd 1153
        . We prove the equality in several ways, in section 3. Also see p.1146 in CLRS (Appendix A). : see p.1147 in CLRS.
          .
      • (definition and first example). See p.1148 in CLRS. MCS 14.4.2. Rosen, p.321.
      • Prerequisite knowledge: basic exponent rules (see first page of notes)
      • Class notes: Slideshow
      • Video:
          (propositions, conjectures, notation) [11.5 min] (theorems, if-then, IFF, straightforward proofs) [29 min]
        • MCS: 1.1 -- 1.7
        • Scheinerman: 1(1-6)
        • Rosen: First 10 pages.

        1. Proof by contrapositive, contradiction, and smallest counterexample
          • Prerequisite knowledge: section 2. A couple definitions about prime numbers are mentioned in one proof. See first page of the notes.
          • Class notes: Slideshow
          • Video:
              (describing proof by contrapositive) [16 min] (three proofs by contrapositive, three proofs by contradiction, including that &radic 2 is irrational) [25 min] (proof by contradiction: infinite number of primes. Smallest counterexample: sum of odd numbers) [17.5 min] (smallest counterexample: 2 n > n 2 , and Fibonacci numbers grow exponentially) [13.5 min]
          • To record: the proof about empty pentagons.
        2. Textbook chapters:
          • MCS: 1.5, 1.8, 2.2
          • Scheinerman: 4(20,21)
          • Rosen: chapter 1.7, p.83--88
        • Prerequisite knowledge: section 2. [factorial of zero and sum or zero objects appear in a proof see first page of notes]
        • Class notes: Slideshow
        • Video:
            [11min] (intro and a geometric series example) [32.5min] (six examples, including strong induction, and integers being products of primes) [20min] (three examples, including a lesson about failing and trying again) [21min] (two examples, focusing on recurrences)
          • MCS: 5
          • Scheinerman: 4(22)
          • Rosen: 5
          • CLRS: Appendix A.2
          • Prerequisite knowledge: number of subsets in a set, sum of powers. Section 3A. (only for some proofs see first page of notes).
          • Class notes: Slideshow
          • Video:
              [42.5min]
            • MCS: 15.8
            • Scheinerman: 5(25)
            • Rosen: chapter 6.2
            • Prerequisite knowledge: None
            • Class notes: PDF.
            • Video:
                [10min] [7min]
              • Rosen: p.96--99
              • Assuming what you want to prove.
                  . See circular reasoning as well.
              • This can be done by mistake: you might use a "well known" property B to prove property A, but the proof of B actually relies on knowing A is true.
              • Note that it is sometimes ok to assume something similar but weaker than what you're trying to prove, for instance with proof by induction.
                • Technically in a proof one can rely on extremely complicated results, as long as they are properly described and cited. In an introductory course (like one where you learn about proof techniques) you should avoid doing this, unless you're relying on something that has already been proved during the course, or is basic enough to be considered prerequisite material.
                • Example: assuming that a set of numbers are integers, when the statement to be proved made no such restriction.
                • In general, don't interpret problems in a way that is convenient to get a simple solution. For example, if I say "you have a deck of n cards", don't assume n=52 just because that's what a standard deck has. And of course, don't do this.
                • This is like "making assumptions" without knowing about it. It typically arises when one hasn't really understood what they're relying on, specifically when one hasn't understood what assumptions were made to obtain the result they're relying on. One such example is that students often state as a fact that hashing always takes constant time.
                • Also, "proof by picture". For instance, I ask you to prove a property about planar graphs. You draw a planar graph, assume it is general enough, show the property is true, and incorrectly conclude that the property holds in general.
                • Make sure that your claims hold in general, not for specific situations.
                • Not quite as bad, but still something to avoid: defining things and then not using them.
                • Often a proof will end up referring to something, like "therefore it is equal to 3", where "it" was actually last referred to about 10 lines and 5 sentences ago. This could be avoided by giving "it" a name. It doesn't help to be more specific about "it" by saying "that object that I was talking about ten lines above".
                • For example, you have a case analysis proof with several cases, but many of those cases could have been eliminated. A typical situation might be: "Case 1: If A is true and B is true, do C. Case 2: if A is true and B is not true, do C."
                • Basically, writing a whole lot without getting to the point, or purposely fitting in lengthy complicated math to hide the fact that a crucial step is missing somewhere.
                • Prerequisite knowledge: Just a couple definitions from section 2 (see first page of notes)
                • Class notes: Slideshow
                • Video:
                    To be recorded
                  • MCS: chapter 3
                  • Scheinerman: 1(7) and 2(11)
                  • Rosen: A lot of chapter 1 is related to this material. You could look at the beginning of chapter 12 as well.
                  • Prerequisite knowledge: none
                  • Class notes: Slideshow
                  • Video:
                      To be recorded
                    • MCS: 4.1 and 4.2
                    • Scheinerman: 2(8,10,12)
                    • Rosen: 2.1 and 2.2. Some of 2.5
                    • CLRS: Appendix B.1
                    • Prerequisite knowledge: definition of set.
                    • Class notes:
                    • Video:
                        To be recorded
                      • MCS: 4.3 and 4.4
                      • Scheinerman: Relations are covered in 3(14,15). Functions are covered in 5(24)
                      • Rosen: 2.3 for functions, 9.1 for relations.
                      • CLRS: Appendix B.2 (for Relations). Appendix B.3 (for Functions). See p.1152 for splitting summations.
                      • Prerequisite knowledge: it helps to be familiar with common functions (e.g., polynomial vs exponential vs logarithmic).
                      • Class notes:
                        • Slideshow
                        • A lesson about bounding functions: exaggerate and simplify
                          on Insertion sort and justification of big-O. (to be merged one day)
                        • MCS: 14.7
                        • Scheinerman: 5(29). Note that the definitions for big-O and big-Omega differ slightly from mine. I assume that functions are positive, which is quite standard for CS.
                        • Rosen: 3.2
                        • CLRS : 2, p.43-52. This textbook is consistent with how I define things.
                        • Prerequisite knowledge: sections 3B and 7. Geometric series.
                        • Class notes: full slideshow and condensed (the beginning of these notes are tied to Mergesort, you can ignore this and begin at "How to solve. ")
                        • Video (to be updated in the context of discrete math low priority)
                            describing mergesort and setting up its recurrence. This is just to give some context to the recurrence relation solved in the following links.
                        • My lecture on how to solve recurrences: part 1 and part 2. (45 min. total)
                          • CLRS: chapter 4, p.83-92
                          • Prerequisite knowledge:
                          • Class notes: Slideshow to be created
                          • Video:
                              To be recorded
                            • MCS: 16.4 and 22
                            • Scheinerman: 4(23)
                            • Rosen: 8
                            • Prerequisite knowledge:
                            • Class notes: Slideshow to be created
                            • Video:
                                To be recorded
                              • MCS: 14,15,16 (specifics TBD)
                              • Scheinerman: 3(16,17,18,19)
                              • Rosen: 6
                              • CLRS: Appendix C.1
                              • Prerequisite knowledge:
                              • Class notes: Slideshow to be created
                              • Video:
                                  To be recorded
                                • Scheinerman: 5(27)
                                • Rosen: 6
                                • CLRS: Appendix C.1

                                1. Intro (examples, representation, degree, regularity, isomorphism, subgraphs)
                                  • Prerequisite knowledge: very little: e.g., arithmetic series, notation for sets.
                                  • Class notes: Slideshow and condensed.
                                  • Video (based on other notes to be replaced by expanded version)
                                  • Textbook chapters:
                                    • MCS: 12.1 -- 12.4
                                    • Scheinerman: 9(47 and the first 2 pages of 48).
                                    • Rosen: 10.1 -- 10.3
                                    • CLRS: Appendix B.4, B.5. Also, chapter 22, p.589-592.
                                  • Links
                                    • Graph isomorphism
                                    • Graph enumeration
                                    • Miscellaneous

                                • Prerequisite knowledge: section 12A
                                • Class notes: Slideshow and condensed.
                                • Video:
                                    [4 min] [29 min]
                                  • Scheinerman: 9(48).
                                  • Rosen: p.404-405
                                  • Ramsey numbers (cliques vs independent sets)
                                    • R(x,y) represents the smallest integer N such that in any graph of size at least N there is either a clique of size x or an independent set of size y.
                                    • Here is a wiki about R(3,3). Here is a link showing R(4,3)=9, without resorting to a recurrence as in the previous link. (see 2 paragraphs before the 2nd figure). : requires knowledge of R(4,3).
                                    • R(5,5) is unknown! on Ramsey's theorem. Notice that there are also Ramsey numbers with more than 2 parameters.
                                    • Games
                                      • "Sim" (wiki) is a game where 2 players take turns coloring the edges of K6. One player uses red and the other uses blue. Whoever completes a triangle of their own color loses. An extended version of the game asks two players to color K18, while avoiding coloring a monochromatic K4. There is a 3-player version as well. Ramsey theory tells us that these games cannot end in a draw.
                                      • A variant of sim is Hajnal's triangle-free game, discussed here. Two players take turns adding edges, with the restriction that neither can complete a triangle. However, there are no colors in this game. A player wins if the other player cannot add an edge. The game can also be played using a score that is just the total number of edges added. The goal of one player is to maximize the score, and the goal of the 2nd player is to force a configuration where no edge can be added, as quickly as possible. Finally there is also a variant with an additional constraint, where the graph must remain connected at all times.
                                      • Prerequisite knowledge:
                                      • Class notes: Slideshow and condensed
                                      • Video:
                                          To be recorded
                                        • MCS: 12.8, 12.9
                                        • Scheinerman: 9(49,50).
                                        • Rosen: 10.4, 11.1 -- 11.4.
                                        • CLRS: parts of appendix B.4, B.5
                                        • Prerequisite knowledge:
                                        • Class notes: Slideshow and condensed
                                        • Video:
                                            [3.5 min] (until 20:22) [20.5 min]
                                          • MCS: 12.9.4 but I recommend a different source.
                                          • Rosen: 11.5
                                          • CLRS: chapter 23, p.624-629.
                                          • Prerequisite knowledge:
                                          • Class notes: Slideshow and condensed
                                          • Video:
                                              [7 min] [28 min]
                                            • MCS: 13.1 -- 13.5
                                            • Scheinerman: 9(53)
                                            • Rosen: 10.7
                                            • Euler's formula (for planar graphs)
                                              • The wiki on planar graphs contains a section on Euler's formula.
                                              • The formula V-E+F=2 is just a particular case of a far greater body of work. As mentioned in class, if you were to draw a connected graph on a sphere, you'd get the same relation The Euler characteristic would be 2. Similarly, count the vertices, edges and faces on any convex polyhedron. Other surfaces have other characteristic numbers.
                                              • The wiki on planar graphs starts out by mentioning the theorems of Kuratowski and Wagner, which essentially say that a graph G is non-planar if and only if the shape of a K5 or of a K3,3 is present in G. See here. Look at the two figures, showing two (non-planar) graphs not directly containing a K5 or a K3,3, but that implicitly contain those shapes.
                                                The section that follows in the wiki discusses other planarity criteria. Theorem 1 in particular is quite simple, and easy to prove.
                                              • (advanced) notes on Kuratowski's theorem. The advanced part is proving that any graph not containing K5 or K3,3 as a subgraph -- or as a minor -- must be planar. In other words, non-planarity implies finding such a subgraph/minor. In our class we only discuss the other direction: i.e., that containing one of those subgraphs implies non-planarity.
                                              • Prerequisite knowledge:
                                              • Class notes: vertex coloring
                                              • Video:
                                                  [29 min]
                                                  (you may ignore the two minutes between 14:00 and 16:00, they are more algorithmic)
                                                • MCS: 12.6 and 13.6
                                                • Scheinerman: 9(52)
                                                • Rosen: 10.8
                                                • wiki: 4 color theorem
                                                • wiki: Grotzsch's theorem
                                                • Hadwiger-Nelson problem: This is an awesome vetex coloring problem, with an infinite number of vertices: all points in the plane. It is explained nicely in this wiki.
                                                • Edge coloring
                                                  • The idea here is to color edges of a graph, so that no vertex is incident to 2 edges of the same color. Of course, you must minimize the number of colors used.
                                                  • wiki (see the applications section).
                                                  • Eulerian paths (covered in Scheinerman chapter 9(51), Rosen 10.5)
                                                  • Hamiltonian paths (covered in Rosen 10.5)
                                                  • Binary trees.

                                                  1. Intro
                                                    • Prerequisite knowledge:
                                                    • Class notes: Slideshow and condensed
                                                    • Video:
                                                        To be recorded
                                                  2. Textbook chapters:
                                                    • MCS: 17
                                                    • Scheinerman: 6(30,31).
                                                    • Rosen: 7.1, 7.2
                                                    • CLRS: Appendix C.2
                                                  3. Links
                                                      on poker probability.
                                                  4. The birthday problem
                                                      . Besides the sections on the basic problem, look at section 7.4 ("near matches"). This concerns the third bet mentioned in the notes.
                                                      Also note that section 7.6 deals with the average number of people you have to ask, to find a birthday match. It is a bit higher than 23. This is discussed in the section on IRVs (14-D).
                                                  5. Somewhat related: If we record birthdays of people queried iteratively (say, on the street we won't run out of people to ask), how many people do we expect to ask before recording all possible birthdays?
                                                    This is known as the Coupon Collector's Problem. Here, we have n "coupons". The answer is Theta(nlogn).
                                                  6. . Near the very bottom of the table, you'll find 10 80 , the estimated number of particles in the universe.
                                      • Prerequisite knowledge:
                                      • Class notes: Slideshow and condensed
                                      • Video:
                                          To be recorded
                                        • MCS: 18
                                        • Scheinerman: 6(32)
                                        • Rosen: 7.2, 7.3
                                        • CLRS: Appendix C.2
                                        • Monty Hall problem
                                            . some people can win this game 100% of the time.
                                        • BBC intro to the problem. Includes an introductory video. Read the last 3 short paragraphs on diagnostic tests. . If you find a neater one, let me know. , including video of a lecture, and an explanation of how the conditional probability formula can be misused. . A variation of the game, involving some prior knowledge.
                                        • A research paper on an extension of the game: many doors and many offers to switch.
                                          • Wiki intro and examples. "intuitive and short explanation". I would start reading at "Anatomy of a Test".
                                          • Prerequisite knowledge:
                                          • Class notes: Slideshow and condensed
                                          • Video:
                                              [see first 25 minutes] . [see first 24 minutes]
                                            • MCS: chapter 19.1, 19.2, 19.5 (and some other small parts of chapter 19)
                                            • Scheinerman: 6(33,34)
                                            • Rosen: 7.2, 7.4
                                            • CLRS: Appendix C.3
                                            • Prerequisite knowledge:
                                            • Class notes: Slideshow and condensed
                                            • Video:
                                                (start at 24:55) [16 min] [42.5min] (start at 23:45) [33 min]
                                            • 2016 lecture, part 2 [13.5 min]
                                              • MCS: chapter 19.1, 19.2, 19.5 (and some other small parts of chapter 19)
                                              • CLRS: chapter 5.1, 5.2
                                              1. The hat check problem.
                                                • FYI - Link 1 and link 2 calculate the probability of an outcome for this problem, without IRVs.
                                              2. The hiring problem.
                                                • Link. Note that the second problem in this link, involving birthdays, is mentioned below. (related to the hiring problem).
                                              3. Finding local maxima in permutations.
                                                • Jump to the 30:00 mark in this youtube video, which is part of Harvard's Stat 110 course. This part of the lecture lasts 9 minutes. After that is an explanation of the St. Petersburg paradox which is fun to think about. Here's the wiki on that.
                                              4. Counting inversions in permutations. (involves IRVs with 2 subscripts)
                                                • Look at problem 2 in this homework set from a course at WVU. This follows the hat check problem. Problem 3 is not related to IRVs, but is interesting.
                                              5. A birthday problem. (involves IRVs with 2 subscripts)
                                                • The 2nd example in this link is a variant of our good old birthday problem. I discuss this and one more variant here.
                                              6. A problem with balls and bins.
                                                • See the second example in this link. Evaluation of the expected value of each IRV is a bit more complicated in this example than in previous ones. Note that the first example in this link is equivalent to the hat check problem. It deals with fixed points in permutations. In a permutation of the integers 1. n, a fixed point occurs when integer k is placed at position k.
                                              • Monkeys and typewriters
                                                • wiki on infinite monkey theorem.
                                                • nice short explanation
                                                • A very different model: link 1link 2
                                                • Prerequisite knowledge:
                                                • Class notes: Slideshow to be created
                                                • Video:
                                                    To be recorded
                                                  • MCS: chapter 9
                                                  • Scheinerman: chapter 7 (TBD which parts will be used here)
                                                  • Rosen: 4
                                                  • CLRS: chapter 31

                                                  I have made an attempt to point to the right chapters or pages in some of the following books, for each of the sections above. Each book introduces discrete math in its own way, emphasizing different concepts. So what appears on page 1 in one book might not appear until chapters later in another, and coverage of some topic can range from nothing up to several chapters. This makes it a bit difficult to be accurate with referencing. Even if I point to some reference, it might not focus on the same things that I have.
                                                  Feel free to point out omissions or errors.


                                                  19.1: Enumeration - Mathematics

                                                  Http://www.trnicely.net Current e-mail address

                                                  Last revised 1000 GMT 18 January 2010.

                                                  Date of first release 23 August 1999.

                                                  The content of this document (other than the addendum, which was not part of the submission for publication) is essentially that of the original release, except that information rendered obsolete by subsequent events has been removed or modified, in both the main document and the addendum (this liberty is taken in view of the fact that the paper was never accepted for publication). There may also be differences in formatting, and in minor details and corrections, such as updated URLs.

                                                  Abstract

                                                  Mathematics Subject Classification 2000 (MSC2000)

                                                  Key Words and Phrases

                                                  1. Introduction

                                                  2. The Prime Quadruplets

                                                  In the same fashion, we define Brun's constant B_4 for the prime quadruplets as the limit of the sum of the reciprocals, As with B_2, we do not if the series is finite or infinite but as a consequence of Brun's proof of the convergence of the sum of the reciprocals of the twins, we do know that the series for B_4 is convergent for if the series is infinite, its terms are a proper subset of the series in (2.6). Since (2.6) is a positive series, its convergence is immune to deletion, rearrangement, or regrouping of terms thus the series (2.8) defining B_4 must also be convergent. In essence, the quadruplets are constructed from a proper subset of the twins.

                                                  Although (2.8) is convergent, the monotonically increasing partial sums approach the limit quite slowly. However, assuming the validity of the Hardy-Littlewood approximation (2.3), a more rapidly converging first order extrapolation may be derived as follows. Define S_4(x) as the partial sum of the reciprocals of the quadruplets, Then the remainder term in the series defining B_4 may be approximated by where we have employed the density (27/2)c_4*1/(ln(t))^4 of quadruplets implied by the Hardy-Littlewood approximation, approximated the sum of the reciprocals of a particular quadruplet by 4/t, and used the substitution u = ln(t). This produces a "first order" extrapolation of S_4(x) to B_4, which we denote by F_4(x), No effective second order extrapolation is known however, we will present evidence that the error term in (2.11) is O(1/(sqrt(x)*(ln(x))^2), so that implying that F_4(x) converges to B_4 at a rate O(sqrt(x)/ln(x)) faster than the partial sums S_4(x).

                                                  3. Computational Results

                                                  Table 1 contains a brief summary of the computational results, including the counts pi_4(x) of the prime quadruplets (q, q+2, q+6, q+8) the values of the discrepancy between pi_4(x) and the Hardy-Littlewood approximation, the partial sums S_4(x) of the reciprocals of the quadruplets and the first order extrapolations F_4(x) of S_4(x) to the limit, according to (2.11), members of a sequence believed to be converging to Brun's B_4 constant.

                                                  Rather than attempting to bound the error F_4(x) - B_4 directly, we consider the "scaled deviations" where the scaling factor sqrt(x)*(ln(x))^2 arises from the conjectured error term in (2.12). No rigorous derivation is known for this error term. One justification for it was given in the derivation of (3.3), based on the observed magnitude of delta_4(x) in the computed data. A second justification arises from further analysis of the data, which reveals that the resulting mean absolute value of P_4(x) (using our best estimate F_4(1.6e15) in place of B_4) remains of the same order of magnitude O(1) over most intervals of 10^12 from 0 through 1.6e15, as would be expected with a valid scaling factor (deviations at all values of x are thus given similar weight).

                                                  We now hypothesize that the deviations F_4(x) - B_4 (and consequently P_4(x) as well) change sign infinitely often more precisely, given any x_0, however large, there will exist integers x_1 > x_0 and x_2 > x_0 such that F_4(x_1) < B_4 and F_4(x_2) > B_4. We will refer to this as hypothesis [H4]. Note that although [H4] is neither necessary nor sufficient for the Hardy-Littlewood approximation (2.3), it will almost certainly fail (along with (4.1) in its entirety) if Hardy-Littlewood is false. In support of [H4], we note that if our best estimate F_4(1.6e15) is used for B_4, then F_4(x) - B_4 is observed to undergo 504 sign changes over the 160081 data points recorded in the present study, with 315 sign changes occurring beyond 10^15.

                                                  Given [H4], we then look for the maximum computable value of the function where x_1 and x_2 are integers such that x_2 > x_1 > 1. N_4 may be considered as a "normalized" measure of the amplitude of the oscillations in F_4(x), or alternately as a measure of the scaled deviation of a "predictor" value F_4(x_1) from a "terminal approximation" F_4(x_2). In contrast to P_4, N_4 has the virtue that it is independent of the uncertain value of B_4. If N_4 has a global maximum, or even an upper bound, then given [H4] this must also represent an upper bound on the absolute value of P_4, thus producing an unconditional error bound for any specific value |P_4(x_3)| will be exceeded by N_4(x_3, x_4), where x_4 is chosen so that x_4 > x_3 while F_4(x_3) - B_4 and F_4(x_4) - B_4 are of opposite sign ([H4] implies that there is an infinite sequence of such integers x_4 for any given integer x_3). In practice, we are unable to prove that N_4 has a global maximum, although (2.12) would imply that P_4 is bounded. Indeed, it is computationally impractical even to find the absolute maximum of N_4 over all integers in the range under investigation, 1 < x_1 < x_2 <= 1.6e15 this would involve the comparison of more than 1e30 data pairs (F_4(x_1), F_4(x_2)), a calculation of doubtful feasibility. What has been determined is that the absolute maximum of N_4 over all of the (more than 10^10 such) data pairs recorded in this study is However, additional computations reveal an even larger value of N_4 in the vicinity of this point, where the values in both (4.4) and (4.5) have been rounded up. Although the value 22.6687145 is still not established as the sought after upper bound on N_4 (and thus on |P_4|), the numerical evidence (zero exceptions among more than 1e10 data pairs extending over fifteen orders of magnitude) indicates that if any larger values of N_4 (and thus |P_4|) exist, they must be relatively rare. Our intuitive conclusion is that for the great majority (99 percent or more?) of integers x > 1, it will be true that |P_4(x)|


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                                                  An Introduction to Enumeration

                                                  Written for students taking a second or third year undergraduate course in mathematics or computer science, this book is the ideal companion to a course in enumeration. Enumeration is a branch of combinatorics where the fundamental subject matter is numerous methods of pattern formation and counting. An Introduction to Enumeration provides a comprehensive and practical introduction to this subject giving a clear account of fundamental results and a thorough grounding in the use of powerful techniques and tools.

                                                  Two major themes run in parallel through the book, generating functions and group theory. The former theme takes enumerative sequences and then uses analytic tools to discover how they are made up. Group theory provides a concise introduction to groups and illustrates how the theory can be used to count the number of symmetries a particular object has. These enrich and extend basic group ideas and techniques.

                                                  The authors present their material through examples that are carefully chosen to establish key results in a natural setting. The aim is to progressively build fundamental theorems and techniques. This development is interspersed with exercises that consolidate ideas and build confidence. Some exercises are linked to particular sections while others range across a complete chapter. Throughout, there is an attempt to present key enumerative ideas in a graphic way, using diagrams to make them immediately accessible. The development assumes some basic group theory, a familiarity with analytic functions and their power series expansion along with some basic linear algebra.

                                                  "I have adopted your text "An introduction to enumeration" for my

                                                  undergraduate combinatorics course. Overall, it is wonderful. I like how your book actually teaches something, beyond a loose tour of combinatorics that many other books offer. The ability to fluidly go back and forth between recurrences and generating functions, that sort of thing. It trades the broadest focus for specific techniques, in that way reminding me what's fun in say, an ODE course. My students actually learn something they couldn't do before, it makes them feel smart so they like the course " (Prof Dave Bayer, Barnard College, NY, USA)

                                                  “This work is a very basic, short introduction to combinatorial enumeration techniques. Camina (Univ. of East Anglia, UK) and Lewis (The Mathematical Association, UK) properly explain the fundamental concepts … . There are 20-25 exercises per chapter … . The numerical answers to the majority of these exercises are included at the end. … It may be useful for students who need very basic enumeration skills … . Summing Up … . Lower- and upper-division undergraduates.” (M. Bona, Choice, Vol. 49 (4), December, 2011)

                                                  “This book is written as an introduction to enumeration for second or third year undergraduate students in mathematics or computer science. Its theme is counting, using finite or infinite series … . The development is interspersed with exercises, linked to particular sections, or covering a complete chapter. Solutions are provided at the end of the book.” (Andreas N. Philippou, Zentralblatt MATH, Vol. 1230, 2012)

                                                  “This is an introductory text on counting and combinatorics that has good coverage … . It is aimed at a sophomore or higher level and has few prerequisites beyond power series. … There are numerous exercises, and all have brief solutions in the back. … It is unusual to see a book that combines such extensive coverage with so few prerequisites.” (Allen Stenger, The Mathematical Association of America, August, 2011)


                                                  (1) Visayas State University, Visayas State University
                                                  (2) Department of Statistics, Visayas State University
                                                  (3) Department of Mathematics and Physics, Visayas State University
                                                  (4) Department of Statistics, Visayas State University
                                                  (*) Corresponding Author

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