G. William Moore's Commentary on R. A. Struble's Infinite Products. 11/20/2005.

PICTURES, PATHOLOGY, AND MATHEMATICS:
G. WILLIAM MOORE'S COMMENTARY ON
RAIMOND A. STRUBLE'S
INFINITE PRODUCTS.
DRAFT COPY ONLY.
11/20/2005.
http://www.infiniteproduct.info/struppma.htm
© 2005, G. William Moore, MD, PhD.
All Rights Reserved.

Send comments and correspondence to: George.Moore4@va.gov
See also: http://www.infiniteproduct.info/strupict.htm ............. http://www.infiniteproduct.info/struifpr.htm ............. http://www.infiniteproduct.info/struitgr.htm ............. http://www.infiniteproduct.info/struppma.htm ............. http://www.infiniteproduct.info/infnpapl.htm

TABLE OF CONTENTS.

Table of Contents.
Abstract.
Preface. An Appreciation to Prof. Struble.
Chapter 1: Introduction.
Chapter 2: Pathology and Mathematics.
Chapter 3: Pathology Pictures.
Chapter 4: Pathology and Medical Reasoning.
Chapter 5: Mathematics Pictures.
Chapter 6: Arithmetic.
       Section 6A: Definition of Arithmetic.
       Section 6B: Pictures of Arithmetic.
       Section 6C: Arithmetic in Pathology: Computer Security.
Chapter 7: Geometry.
       Section 7A: Definition of Geometry.
       Section 7B: Pictures of Geometry.
       Section 7C: Geometry in Pathology: Topography of Pathology.
Chapter 8: Algebra.
       Section 8A: Definition of Algebra.
       Section 8B: Pictures of Algebra.
       Section 8C: Algebra in Pathology: Probability and Statistics.
Chapter 9: Analysis.
       Section 9A: Definition of Analysis.
       Section 9B: Pictures of Analysis.
       Section 9C: Analysis in Pathology: Image Analysis. Fourier Series. Contraction Maps.
Chapter 10: Logic.
Chapter 11: Conclusions.
Chapter 12: Outlines of Human Pathology.
Chapter 13: Proofs.
Chapter 14: Examples.
Chapter 15:
Chapter 16:
Chapter 17:
Chapter 18:
Chapter 19:
Chapter 20:
Chapter 21:
Chapter 22: Infinite Papillomas.
Chapter 23: Glossary.
Chapter 24: History of Biomathematics.
Chapter 25: Sample Calculations.
Chapter 26: References.

ABSTRACT.

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The purpose of this on-line manuscript is to present a brief survey of mathematics and human pathology, with an emphasis on their areas of commonality. This purpose is best achieved with numerous diagrams, or cartoons, which highlight the important didactic points. The five major branches of mathematics are: arithmetic; geometry; algebra; analysis, including calculus; and logic. Each branch has a corresponding application in human pathology: computer security; topography; quality monitors; image analysis; and medical reasoning, respectively.

Mathematics Branch.	Main Idea.	Pathology Application.
1. Geometry.	Spatial Relations.	Topography.
2. Algebra.	Solve for X.	Probability, Statistics, Syntax.
3. Analysis.	Area Approximation.	Image Analysis.
4. Logic.	Syntax.	Medical Reasoning.
5. Arithmetic.	Number Theory.	Computer Security.

PREFACE. AN APPRECIATION TO
PROF. STRUBLE.

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In 1967, when I was a first-year graduate student in Biomathematics at North Carolina State University at Raleigh, I attended Prof. Struble's beginning course in Mathematical Analysis (i.e., the theoretical underpinnings of calculus), with some trepidation. I had majored in biology in undergraduate school, and I was the lone student of biomathematics in a sea of engineering, physics, and mathematics students. I had taken two years of calculus and differential equations, and I had memorized enough of the basic definitions and formulas of calculus to pass the examinations, but I always felt ill-at-ease with the underlying ideas of calculus.

In the first week of the course, Prof. Struble told us not to take notes, but to listen carefully. We would not be tested on the material from the first week. Then, entirely from memory and without notes, Prof. Struble built up all the basic ideas of mathematics from high school algebra and geometry through undergraduate calculus. It was a magical week for me. For the first time ever, I began to feel at home with the episilons and deltas that had had befuddled me four years earlier. The lessons that Prof. Nicholas D. Kazarinoff and Prof. Allen L. Shields had painfully drilled into me four years earlier at the University of Michigan unfolded before me. Perhaps it was Prof. Struble's illuminating didactic approach; perhaps I had matured mathematically in those four years; perhaps the ideas that were swimming in my brain finally fell together; or perhaps it was some combination of the above.

This manuscript is an attempt to capture some of the magic of that week, and to show how these ideas impact upon my chosen field, human pathology informatics.

PICTURES, especially cartoons (i.e., hand-drawings designed to highlight/emphasize a particular point; not mere photographs) have an important but undeservedly disreputable role in both mathematics and pathology pedagogy. Nobody could imagine teaching plane geometry without diagrams, and Euclid's (230 - 275 BC) original Greek textbook, Elements from the third century BC, is richly illustrated. Proof of the Pythagorean theorem, perhaps the greatest achievement of ancient mathematics, begins with a cartoon (see below). Unfortunately, some mathematicians make almost a fetish about not showing pictures in their proofs. The non-pictorial approach is sufficient for building a rigorous proof; but it often fails as a teaching tool.

Likewise, professional anatomic pathology texts are typically replete with photographs. Several well-published pathologists have told me how difficult it is to prepare just the right photograph, that illustrates a point but isn't full of distracting features. Well-drawn cartoons do not have this disadvantage. And, historically, Rudolf Virchow's [1821-1902] original textbook of cellular pathology (Cellulärpathologie, 1858) is replete with instructive line drawings.

I am not suggesting that we dispense with formulas in mathematics or photographs in anatomic pathology. It's too easy to fool oneself with a distorted diagram masquerading as a proof. And part of a diagnostic pathologist's job is to recognize images, for which there is no substitute for looking at many glass slides and photographs. Still, diagrams and cartoons have their place, as I shall attempt to show in this manuscript.

CHAPTER 1. INTRODUCTION.

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INTRODUCTION.

Mathematicians are the provers of science. Pathologists are the provers of medicine. A few, well-chosen pictures are the key to understanding many of the important concepts of both fields.

HUMAN PATHOLOGY studies the etiology (cause), pathogenesis (stepwise progression), and diagnosis (recognition) of human disease. MATHEMATICS is the study of proof, and the application of proven computing methods to practical problems.

ANATOMIC PATHOLOGY is the study of disease-processes, as they are reflected in normal and altered anatomy. Traditionally, anatomic pathology consists of autopsy pathology (study of deceased patients); surgical pathology (study of surgical specimens obtained at operation); and cytopathology (study of cells in fluids, usually for the detection of cancer).

The first recorded human dissection was performed by Susutra, in India in 5000 BC (Geller, 2003). The three ancient civilizations (Mediterranean, Indian, Chinese) all recognized that anatomy relates to illness in the body, and that human anatomy is similar to the anatomy of other animals. Some early human dissections were performed to answer forensic questions, such as suspected poisonings. In ancient Greece, the predominant/reigning/current theory of human disease was that it resulted from imbalances in the four humors (blood, bile, urine, and phlegm). (Zadeh, 2000; Hippocrates).

Altered anatomy as a cause or manifestation of disease was recognized in the mid-18th century, by Giovanni Battista Morgagni [1682-1771], in his famous work, De Sedibus et Causis Morborum per anatomen indigatis [1761] (Latin: On the sites and causes of diseases, as indicated by anatomy.)

Nineteenth century Austrian pathologist Carl Freiherr von Rokitansky [1804-1878], who personally performed 30,000 autopsies, and examined an additional 60,000 autopsies, catalogued/described the anatomic changes in all major human organ-systems. German pathologist Rudolf Virchow [1821-1902] used microscopic findings to study changes in tissues and cells. (Virchow, 1858).

Surgical Pathology, which examines the microscopic changes in specimens taken at surgery, began in the late nineteenth century, under the leadership of Sir James Paget, .............

Cytopathology began in the mid-twentieth century, under the leadership of Greek-born American physician, George N. Papanicolaou. Prof. Papanicolaou recognized that cancer cells shed more readily from tissue-surfaces than normally-growing cells (see also: Watson JD, Molecular Biology of the Gene); and therefore that exfoliative cytology (i.e., study of cells shed from tissue-surfaces) would tend to concentrate cancer and pre-cancer cells for microscopic examination. Cancer of the uterine cervix, a major cause of death among young women early in the twentieth century, was nearly eliminated by Prof. Papanicolaou's tireless efforts in the area of cervical cytology, the "Pap smear". It can be fairly said that Prof. Papanicolaou's methods saved more lives than any other advance in anatomic pathology in the twentieth century.

MATHEMATICS was invented in ancient Mediterranean cultures during the Bronze Age [3000 B.C.], as methods for accounting (trade and taxation) and spatial relations (surveying, navigation, architecture).

MATHEMATICS can be divided into five major branches/disciplines/pillars: ARITHMETIC, including number theory; GEOMETRY, including topology; ALGEBRA, including matrix and abstract algebra; ANALYSIS, including limits and calculus; and LOGIC, including set theory.

Arithmetic and geometry were known to ancient civilizations in the Bronze Age, who used them for monetary transactions, taxation, land surveying, architecture, and navigation. Algebra was invented by Arabic, Persian, and Indian scholars in the late first millennium. Analysis came of age in 17th century Europe, although rudiments of analysis were known to Archimedes (287-212 BC).

The concept of ZERO was unknown, or more correctly not tolerated, in ancient Mediterranean and Chinese cultures (Seife, 2000). The idea traveled to ancient India (Brahmagupta (598-670)) by way of the Babylonian/Persian Empires, whre it had received a better/warmer reception/response. Once you have zero, it's a short step/distance to negative numbers and simple algebra, which was invented by Arab/Persian scholars, especially Al-Khawárizmi (?780-?845). The words ALGEBRA and ALGORITHM are derived from Arabic words. European continued to be hobbled by its antiquated/obsolete system of Roman numerals, until Fibonacci (?1170-?1240) (an Italian trained as a teenager by Arab mathematicians in North Africa) brought Arabic numerals to Europe in the early 13th century.

The publication of Lord John Napier's Table of Logarithms (Maor, 1998) in the 16th century instantly lighted the computational burden for mathematicians, by converting multiplication problems into addition problems; and long division problems into subtraction problems.

The need for better navigation for exploration and trade with the new world led to Newton's (1642-1727) Principia Mathematica Naturalis Philosophiae, and the co-invention with Gottfried Leibniz (1646-1716) of CALCULUS, the fourth discipline of mathematics, based upon earlier work by Bonaventura Cavalieri (1598-1647), Pierre de Fermat (1601-1665), and Isaac Barrow (1630-1677). There are three essential ideas of calculus: LIMIT, DERIVATIVE, and INTEGRAL. Ancient Greek mathematicians danced around the idea of limit, but were hobbled by their unwillingness to accept ZERO and its intimidating companion, INFINITY. Early 17th century European mathematicians worked with FLUXIONS (derivatives or differentials) and QUADRATURES (integrals), but it fell upon Newton's (1642-1727) and Leibniz (1646-1716) to prove that derivatives and integrals are the opposite of one another, the FUNDAMENTAL THEOREM OF DIFFERENTIAL AND INTEGRAL CALCULUS. All the rest of calculus is just applications of algebra and trigonometry, and advanced topics (topology, measure theory, real and complex variables, etc.).

LOGIC, including set theory, is the most fundamental branch of mathematics, but historically logic was the last to develop as a mature discipline in mathematics. While the basic ideas of logic were known to Aristotle (384-322 BC) and was explored by Leibniz (1646-1716), formal logic did not become a computational method in mathematics until the nineteenth century, in the hands of George Boole (1815-1864), Georg Ferdinand Ludwig Philipp Cantor (1845-1918), Jan Lukasiewicz (1878-1956), Kurt Gödel (1906-1978), and Lotfi A. Zadeh (1912-). Mathematicians are trained to think logically, so for a long time, formal logic seemed like a made-up discipline, designed to belabor the obvious. Even today, some mathematicians in other fields take the attitude: it's there; use it; don't obsess over it. Why make a formal apparatus for the obvious? Among some mathematicians who do not work in the area of logic, there is an attitude that: logic is there; use it when you need it; don't obsess over it. Eventually, Cantor's (1845-1918) proof that there are distinct sizes of infinity; Gödel's (1906-1978) proof that certain true statements in mathematics are unprovable; and the demonstration that there are consistent, János Bolyai (1802-1860) and Nikolai Ivanovich Lobachevsky's (1792-1856) non-Euclidean geometries, finally settled the question of whether logic is "obvious".

Similarly, medical reasoning has a stepchild status, in my opinion undeserved, in pathology informatics. Perhaps one of my readers can change all this....

In ancient Mediterranean (Greco-Roman) culture, logic was formalized by Aristotle (384-322 BC), who introduced two quintessentially, but sometimes troublesome, Western philosophical views, which have either driven us forward or befuddled us, depending upon one's perspective. They are: the Law of Excluded Middle (LXM) and the Law of Contradiction (LC), also known as Ex Falso Quod Libet (XFQL, Latin: from contradiction, anything that you please).

According to LXM, any statement, x, is either true (+x) or (-x), but not both and not neither. This means that if one conclusively excludes the possibility of -x (i.e., if -x is false), the one is left with +x (i.e., +x is true), as the only other possibility.

This binary view of the universe is the driving force behind many proofs in mathematics, and by extension, much of the progress that mathematics has brought to the modern world, starting with Euclid (230-275 BC) and Archimedes (287-212 BC). In a typical proof, Euclid begins with the negation, -x, of what he wishes to prove, +x, often accompanied by a rickety/bogus diagram or cartoon; he shows that the negation/cartoon, -x, is absurd (Reductio ad absurdum; Latin: reduction to absurdity); and what's left is the desired statement, +x (QED = Quod est demonstrandum; Latin: that which must be demonstrated). This general property of mathematical systems is termed COMPLETENESS. That is, after one has settled the status of +x and -x, then there is nothing else possible regarding x. We shall revisit this issue/idea anon/later.

According to LC, if any statemnt, x, is both true and false, then every possible statement, y, is true. The general property of mathematical systems, where some, but not all, statements are true, is termed, CONSISTENCY. Consistency is a draconian/terrible/very-tight straightjacket: it meansthat one little contradictory statement, no matter how apparently insignificant, can collapse an entire mathematical system. The late Stanislaw Ulam (1909-1981), a Manhattan Project mathematician, likened the situation to cheating at poker [Ulam, Adventures...]. For the nineteenth century American Old West, perhaps the analogy is apt: when the cheating is uncovered, the injured cowboy guns down the cheater.

The world part/features of the consistency dilemma is that it is not ultimately known whether any of the major branches of mathematics, namely, arithmetic, geometry, algebra, analysis, and logic itself, are themselves consistent! The only real proof of consistency for all of mathematics is that, in 2500 years of scrutiny by many of the greatest intellects on our planet, no inconsistency has yet been found. There have been some close calls! Some otherwise near-disasters in mathematics have been averted by Brahmagupta (invention of zero); Georg Ferdinand Ludwig Philipp Cantor (1845-1918) (transfinite sets); Henri Lebesgue (1875-1941) (rescue of Riemann integration); Jan Lukasiewicz (1878-1956) (modal logic); Gödel (1906-1978) (unprovability); Paul Joseph Cohen (1934-, Fields Medal, 1966) (independence proofs); and Lotfi A. Zadeh (1912-) (fuzzy set theory).

In my opinion, with Zadeh's introduction of fuzzy set theory, formal logic has now become mature enough to tackle the complex arena/issues of medical reasoning.

In European culture, Aristotle's philosophy outlived its useful intellectual life, largely due to the genius of St Thomas Aquinas (1225-1274), and the temporal power and intransigency of the medieval Roman Catholic Church. St Thomas Aquinas created a model of Christian theology based upon Aristotle's philosophy. Lesser intellects and power-hungry clergy expanded the purview of Aquinas's masterpiece into a global imposition of all of Aristotle's views to the exclusion of all others, enforced by the Inquisition, with dissent punishable by death (Giordano Bruno, 1548-1600), death by burning at the stake), by imprisonment (Galileo Galilei, 1564-1642), or by other persecution (Johannes Kepler, 1571-1630). Many of Aristotle's concepts, such as abhorrence of zero and infinity, flat earth, geocentrism, and the four elements (earth, air, fire, water), were all reasonable ideas in the fourth century BC, but were resuscitated long beyond their natural death, by the Roman Catholic theocracy.

Leibniz (1646-1716) tinkered with logic in his Ratio universalis (Latin: universal reasoning), but was befuddled by using exclusive-or, rather than inclusive-or, in his logic calculations. Logic reawakened in the hands of George Boole (1815-1864), Felix Hausdorff (1868-1942), Georg Ferdinand Ludwig Philipp Cantor (1845-1918) , Bertrand Russell (1872-1970), Jan Lukasiewicz (1878-1956), and Kurt Gödel (1906-1978) , built logic and set theory into a modern mathematical discipline.

Two features of mathematical logic have special relevance in medical reasoning: fuzzy logic and the distributive property of implication/hierarchy.

George Boole (1815-1864) advanced Aristotelian logic from a scholarly plaything ("All Greeks are mortal; Socrates is a Greek; ...) into a reliable calculation method, the basis for modern computers, and particularly for Boolean search algorithms. In Boolean algebra, logical-or ∨ behaves approximately like addition; logical-and ∧ behaves approximately like multiplication; and logical-not (-) behaves approximately like negation. (However, there are no "negative numbers" in a logic expression.) Boole overcame the obstacles of earlier systems by using inclusive-or, rather than exclusive-or, that had been used by Leibniz. Then a few techniques allow one to manipulate complex expressions in a systematic fashion:

--x = +x
(+x ⇒ +y) = (-x ∨ +y)
(+x ∧ +y) = - (-x ∨ -y)
(+x ∨ +y) = - (-x ∧ -y)

Boole's work was followed by a school of logic in Warsaw, Poland, headed by Jan Lukasiewicz (1878-1956), inventor of "Polish logic" or "Polish notation", that requires no parentheses, and is more convenient than conventional notation for some proofs and computer calculations. In the period between World Wars I and II, work in this area was conducted by the Vienna Circle of Exact Logic, in Vienna, Austria, that included Rudolf Carnap (1891-1970), Willard Van Orman Quine (1908-2000), and Kurt Gödel (1906-1978). Hitler's invasion of Poland on September 1, 1939, ended these academic pursuits, and dispersed their participants, some to Nazi death camps.

Two stunning proofs from this era elevated logic from the realm of the everyday: Cantor's demonstration that there are different levels of infinity; and Gödel's proof that there are some true statements that are formally unprovable("formal unentscheidbar"). Cantor's proof allowed Lebesgue to plug off one of the leaks in Riemann integration, so that even badly-behaved/wierd functions, including some Fourier series expansions, could be integrated. Gödel's proof settled a long-standing controversy regarding the completeness of formal logic, somewhat surprisingly: not all true statements in logic are provable. Furthermore, Gödel introduced a novel calculation method, that was later exploited in John von Neumann's (1903-1957) design of the first digital computer, ENIAC, at Princeton University in 1948. That is, the idea of using numbers to represent mathematical terms, such as NOT, AND, OR, IMPLIES, THERE-EXISTS, FOR-EVERY, etc. It is not coincidental that von Neumann was sitting in the audience when Gödel first presented his celebrated proof in public.

For this purpose, Gödel invented his own, new language, or GÖDELIZATION, in which one could translate any sentence in mathematics as a GÖDEL NUMBER. Gödel assigned a number to each mathematical term, as for example:

1 .
2 not
3 and
4 or
5 implies
6 there-exists
7 for-every

In a more ordinary example, consider the following, five-word dictionary:

1 .
2 be
3 not
4 or
5 to

where the punctuation-mark . counts as a word. Now construct the famous sentence from Shakespeare's (1564-1616): Hamlet, in which the position of each word in the sentence is assigned to the next consecutive prime number:

to be or not to be . 2 3 5 7 11 13 17

Now, for each prime number, assign a power corresponding to the dictionary-numbering:

to be or not to be . 2⁵ 3² 5⁴ 7³ 11⁵ 13² 17¹

Then 2⁵ × 3² × 5⁴ × 7³ × 11⁵ × 13² × 17¹ = 32 × 9 × 625 × 343 × 161051 × 169 × 17 = 28,567,068,000,000,000 is a Gödel number for this set over this dictionary, and the assignment of Gödel numbers to all sentences in the set is a Gödelization. As long as one agrees upon the number-assignments in the dictionary, then each Gödel number has an unambiguous meaning, and can be decoded by factoring it into its constituent prime numbers.

A number this large, where roundoff is not allowed, doesn't seem very practical to carry around for ordinary calculations, even on modern computers. In fact, the very difficulty in computing such numbers (specifically, factorization back down to the component prime numbers) is used in modern computer security methods, as described herein.

Gödel's achievement was that he constructed a Gödelization for all of formal logic (recently published in the Principia Mathematica by Alfred North Whitehead (1861-1947) and Bertrand Russell (1872-1970)), and showed that the system must contain a statement that means I am not provable.

As a teaser, consider the following short dictionary:

1 ,
2 .
3 adenocarcinoma
4 of
5 prostate
6 prostatic

Then:

adenocarcinoma of prostate . 2 3 5 7

has Gödel number: 2³ × 3⁴ × 5⁵ × 7² = 99,225,000;

and:

adenocarcinoma , prostate . 2 3 5 7

has Gödel number: 2³ × 3¹ × 5⁵ × 7² = 3,675,000;

and:

prostate adenocarcinoma . 2 3 5

has Gödel number: 2⁵ × 3³ × 5² = 21,600;

and:

prostatic adenocarcinoma . 2 3 5

has Gödel number: 2⁶ × 3³ × 5² = 43,200, which all mean the same thing, but have different Gödel numbers. How do we deal with this?

Cantor (1845-1918) demonstrated that there is more than one size of infinity in our number system. Size, or CARDINALITY, is easy to understand for ordinary, finite numbers, such as 1, 2, 3, .... But how large is infinity? Are all infinities the same size? If not, then when can you say that one infinity is the same size as another? Suppose you have a stadium that seats, say, 48,876 fans (the published seating of Oriole Park at Camden Yards, Baltimore, MD). Then one may conclude that, if every place is filled for a sell-out crowd, there are exactly 48,876 guests present. That is, there is a ONE-TO-ONE CORRESPONDENCE between the number of places and the number of fans, if the stadium is filled.

Now let us construct HILBERT'S HOTEL (David Hilbert (1862-1943)), with an infinite number of rooms, numbered 1, 2, 3, ...:

590.

We say that Hilbert's hotel has a COUNTABLY INFINITE NUMBER OF ROOMS, denoted

₀ (aleph-naught; aleph is the first letter of the Hebrew alphabet; Cantor's father was Jewish).
The hotel is asked to provide rooms for a countably infinite number of guests:

591.

Do the guests exactly fit into the rooms, with none left over? It would seem so:

592.

For rooms numbered 1, 2, 3, ...:

593.

obviously, if the hotel has guests named Guest₁, Guest₂, Guest₃, ..., then one can assign exactly one guest to each room, from which we conclude that Guest₁, Guest₂, Guest₃, ..., is a countably infinite number of guests,

₀.

594.

But what if only even-numbered guests, Guest₂, Guest₄, Guest₆, ..., check into Hilbert's hotel? Will there be rooms left over? The answer is NO! The hotel registration desk assigns Guest₂ to Room 1, Guest₄ to Room 2, Guest₆ to Room 3, .... Again, the guests eactly fill the rooms! That is, the cardinality of the set of even-numbered guests equals the cardinality of the set of all positive-numbered guests.

595.

Remarkably, the set of all rational numbers, i.e., all numbers of the form p/q, where p and q are integers, is also [א]₀. That is, it is possible to assign every rational number to exactly one room in Hilbert's hotel. However, the set of all real numbers, that includes numbers such as √2, π, e, that do not have the form p/q, form a larger infinity, for which no room-assignment at Hilbert's hotel is possible. This larger infinity is called

₀. It can be demonstrated that there is a countable number of

's, namely,

₁,

₂,

₃, .... (Incidentally, Hippasus of Metapontum was EXECUTED by Pythagoras (560-480) in 600 BC, for publicizing the fact that √2 is not a rational number.

Cantor's work allowed Lebesgue to describe and plug up a significant leak in Riemann's theory of integration. (See Asimov's short story: Not Final!)

Integral calculus determines the area or volume under a particular function. The traditional method for determining the integral of a function is Riemann Integration, R, in which one sets up a collection of vertical bars underneath the function (so-called Lower Darboux sum (Gaston Darboux, 1842-1917):

29.
A corresponding upper Darboux sum of vertical bars is formed above the function. One makes the bars thinner and thinner, until the two Darboux sums meet. Then one adds up the areas of the blocks. Since the formula for calculating the area of a block has been known since Euclidean (325-265BC) geometry, namely, area, A = bh, where b is the base and h is the height of the block, this process turns a potentially difficult mathematics problem into a collection of simple problems.

Riemann integration, R, works well if the functions are smooth and simple ("well-behaved"), but breaks down for extremely irregular curves, as for example:

44.

Nineteenth century French mathematician, Henri Lebesgue (1875-1941) proposed LEBESGUE INTEGRATION, L, a system for adding up horizontal slabs, to deal with the exceptional cases, particularly problems involving statistical and probability distributions, and image analysis involving the Fourier series. Whenever Riemann integration is possible, Lebesgue integration yields the same result as Riemann integration, i.e., L = R.

The main issue is as follows. In Riemann integration, one places one set of vertical bars UNDERNEATH the function, and another set of vertical bars ON TOP OF the function. The area of the lower vertical bars is called the LOWER DARBOUX SUM (illustrated above), and the area of the upper vertical bars is called the UPPER DARBOUX SUM (Gaston Darboux, 1842-1917) also Thomas Joannes Stieltjes, 1856-1895) . As the vertical bars become THINNER AND THINNER, the two Darboux sums tend toward a common limit, namely, the Riemann integral, R. These two Darboux sums converge toward one another if the function is well-behaved. However, there are some badly-behaved functions (for example, the Fourier series approximating the Heaviside function, which is relevant to pathology image analysis), for which the Darboux sums do not converge to a common limit. To deal with this anomaly, there is an enormous mathematical apparatus of transfinite set theory, topology, measure theory, normed linear spaces, open/closed sets, and the Heine-Borel theorem (Eduard Heine, 1821-1881; Émile Borel, 1871-1956).

The HEAVISIDE FUNCTION (Oliver Heaviside, 1850-1925) is a function, h(x), whose value is zero for x < 0 and one for x > 0. The first derivative (actually, all derivatives) of h(x) is zero everywhere, except at x = 0, where no derivative exists. You can create a GENERALIZED FUNCTION, the so-called DIRAC DELTA FUNCTION (Paul Adrien Maurice Dirac, 1902-1984, Nobel Prize for Physics, 1933), which behaves everywhere like the first derivative of h(x).

Since the Riemann integral exists for every step-function, then the Riemann integral exists for h(x) over every finite interval. For example:

∫_-π^+π h(x) dx (Riemann notation) = ∫_[-π,+π] (Lebesgue notation) = π

The FOURIER SERIES (Jean Baptiste Joseph Fourier, 1768-1830) is a sum of sines and cosines, of the form:

F(x) = ∑_n=0^n=∞ a_n sin nx + b_n cos nx.

where F(x) can be used to fit any function ARBITRARILY CLOSELY, over a finite interval. The customary domain interval for discussing a Fourier series is [-π,+π], i.e., the interval from -π to +π, although the Fourier series domain may be scaled upward or downward, to any desired interval. The customary range of the Fourier series is [-1,+1], but again, this range may be scaled. The issue of what is meant by "arbitrarily closely" is the subject of MEASURE THEORY.

At least one pundit, at URL:

http://mathworld.wolfram.com/RiemannIntegral.html

states baldly that Lebesgue integration is purely a mathematical fairy-tale; that there is no function in nature whose area cannot be determined by Riemann integration. The pundit is right, but he misses an important point. Yes, the Heaviside function is obviously R-integrable over any interval, certainly without resorting to Fourier-series; whereas Cantor's interval-function arose exclusively in Cantor's addled brain, and certainly exists nowhere else in nature. On the other hand, Fourier-series is an incredibly powerful tool in applied mathematics, that curve-fits everything from planetary orbits to pathology images. Why abandon Fourier series? Why not have a general theory of integration that includes Fourier series in its toolbox, especially since L=R whenever R exists, anyway? Why not spend a few more weeks in your mathematics course, learning about Lebesgue integration?

And remember, infinity itself is an abstract concept, with no existence in nature outside the mathematical mind.

Cantor proposed the following, extremely pathologic function, f(x), over the real line from 0 to 1, denoted [0,1]. Let the value of f(x) be 1 for all RATIONAL NUMBERS in this interval (i.e., numbers that can be expressed as the ratio of two integers, p/q, where p, q are integers.

But what if the upper/lower Darboux sums never "get close"? Then Riemann integration could have a lot of different values. That's not good. You want a definition for Riemann integration where everyone gets the same, unique answer. The problem comes up in the Fourier series approximation of Heaviside functions.

ANALYSIS, including calculus, is the mathematical theory of approximation, including LIMITS, DERIVATIVES, and INTEGRALS. As with all other branches of mathematics besides algebra, analysis known in rudimentary form to the ancient Greeks. A LIMIT (Latin: limes=boundary) is the ultimate upper boundary (or ultimate lower boundary) of an infinite sequence of numbers, if such a boundary exists. For example, the ultimate upper (and lower) boundary of the infinite sequence, 1, 1, 1, 1, ..., is 1. Furthermore, ....................

GEOMETRY, including topology ....................

ALGEBRA, including group theory ....................

ARITHMETIC, including number theory ....................

ANALYSIS, including calculus ....................

All the five major branches of mathematics have significant applications in pathology, as follows:

1. Arithmetic: computer security, patient privacy.
2. Geometry: anatomic relations, specimen orientation.
3. Algebra: probability, statistics.
4. Analysis: Fourier series, image analysis.
5. Logic: computer programming, quality assurance.

In mathematics, the most important proof-picture of all time is the proof of the Pythagorean Theorem:

441.

Expanding, (a+b)² = a² + 2ab + b² - 2ab = c², so that a² + b² = c², Q.E.D.

The pyramids of Giza in Egypt were designed with an intuitive understanding of the Pythagorean Theorem, one millennium before a proof was known and published, by Euclid.

In pathology, the most important picture is Homo supinus (supine human). This is more-or-less equivalent to Leonardo Da Vinci's much published VITRUVIAN MAN:

600.

601.

604.
A pathology pundit once said to me that all diagnostic pathology amounts to memorizing one hundred pictures, and being able to recall them at will, in various combinations. This is somewhat of an exaggeration..........

Mathematics has five branches: arithmetic; geometry; algebra; analysis (=limits and calculus); and logic. ..................

ARITHMETIC: 3-4-5 rectangle.
Sieve of Eratosthenes:

448.

GEOMETRY: Proof of the Pythagorean Theorem:

441.
Trigonometry:

453.

ALGEBRA: Picture of a semigroup.

ANALYSIS: Zeno's Paradox:

438.

Archimedes (287-212 BC) n-gon:

443.

444.

445.

LOGIC: Truth table. Borkowski's binary logic operator. Venn diagram.

HUMAN PATHOLOGY examines/studies the etiology (cause), pathogenesis (stepwise progression), and diagnosis (recognition) of human disease. Human pathology is different from other areas of biopathology, in that humans can describe their experiences and symptoms; and there are collectively more data on human disease than on any other species; but/however the study of humans is subject to significant ethical constraints.

There are two cognitive/intellectual branches/components of human pathology: image recognition and medical reasoning. Both major branches of human pathology have corresponding mathematical models: mathematical analysis (=limits and calculus) for image recognition; and set-theory/logic/foundations for medical reasoning.

..................... Human pathology involves two two major processes: NORMAL ANATOMY and PATHOLOGIC PROCESSES. That is, pathologic processes occur in the context of normal anatomy, and can only be understood fully in the anatomic context. Therefore, any reasonably complete model of human pathology must explicitly/implicitly contain a model of human anatomy.

Normal anatomy.

1. Cardiovascular.
Heart.
Right Atrium.
Tricuspid valve.
Right Ventricle.
Pulmonic valve.
Left Atrium.
Mitral valve.
Left Ventricle.
Aorta.
Pulmonary Artery.
Middle-sized arteries.
Arterioles.
Capillaries.
Venules.
Middle-sized veins.
2. Respiratory.
Mouth.
Nasopharynx.
Larynx.
Trachea.
Bronchus.
Bronchiole.
Alveolus.
3. Gastrointestinal.
Tubular gastrointestinal tract.
Mouth.
Oropharynx.
Esophagus.
Cervical esophagus.
Mid-esophagus.
Lower esophagus.
Gastroesophageal junction.

Stomach.
Cardia.
Fundus.
Body.
Antrum.
Pylorus.

Small intestine.
Proximal duodenum.
Mid-duodenum.
Distal duodenum.
Jejunum.
Proximal ileum.
Mid-ileum.
Terminal ileum.

Large intestine.
Ileocecal valve.
Cecum.
Appendix.
Ascending Colon.
Hepatic Flexure.
Transverse Colon.
Splenic flexure.
Descending colon.
Sigmoid colon.
Rectum.
Anus.

Hepatobiliary system.
Liver.
Biliary tree.
Gallbladder.
Pancreas.

4. Genitourinary.

Common.
Kidney.
Ureter.
Urinary Bladder.

Female.
Uterine corpus:endometrium; myometrium; mixed mesodermal; gestational-trophoblastic.
Fallopian tube.
Ovary.
Uterine cervix.
Vagina.
Vulva.

Male.
Testis.
Prostate.
Penis.

5. Endocrine.
Thyroid.
Parathyroid.
Adrenal cortex.
Adrenal Medulla.
Anterior pituitary.
Posterior pituitary.
Pineal body.
6. Integumentary.
Inflammatory dermatoses.
Keratinocyte proliferation.
Appendageal proliferation.
Fibrous proliferation.
Melanocytic lesions.
7. Musculoskeletal.

8. Hemolymphatic.
Blood.
Lymph.
Lymph nodes.
Spleen.
9. Central Nervous System.
Glioma.
Glioblastoma multiforme.
Astrocytoma.
Oligodendroglioma.
Ependymoma.
Choroid plexus papilloma.
Mixed glioma.
Glioneuronal tumor.
Medulloblastoma.
Meningioma.
Ancillary neuronal tumor: neurilemoma; neurofibroma; craniopharyngioma; hemangioblastoma; pituitary adenoma; hematopoietic tumors.....
Pineal gland tumor: germ cell tumor; pinealoma.
CNS Metastasis.

Diagnostic human pathology has two major intellectual components: image recognition and medical reasoning. Neither field is mature enough as an applied mathematical discipline for use in diagnostic pathology, but medical reasoning is advanced enough for some quality assurance processes in highly-computerized medical centers, such as the Baltimore VA Maryland Health Care System.

MATHEMATICS is the study of proof and application of computing methods to practical problems. Traditionally, mathematics has been applied to problems of land management, accounting, architecture, and physics. The ancient Egyptian pharoahs hired court mathematicians to survey riparian farmlands after the annual flooding of the Nile River, so that the Pharoah's government could collect its taxes. The Pyramids of Giza are masterpieces of practical mathematics.

Mathematics is increasingly applied to medicine in general and to pathology in particular, as will be shown herein.

HUMAN PATHOLOGY is the study of the etiology (root-cause), pathogenesis (stepwise progression) and diagnosis (recognition) of human disease. Medicine has its origins in ancient Greece (Hippocrates) and China (), but pathology did not become a defined specialty until 17th century English coroners (Latin: corona=crown) began to perform autopsies for cause-of-death investigations, again, so that the English King could collect his inheritance taxes. Chinese medicine has its origins in military medicine, as we see for the Chinese ideogram for "physician", which depicts a combat-zone doctor pulling an arrow out of a soldier. ......... Pathology as a biological science was developed in the 18th-19th centuries: Morgagni (De Causis and Sedibus Morborum: On the causes and locations of diseases); Virchow [1821-1902] and Rokitansky [1804-1878].

Today, practicing pathologists recognize the altered appearances of diseased tissue, and place these findings in the larger context of medical reasoning. Pathologists are increasingly called upon to perform institutional quality assurance functions, such as detection of unusual events; turnaround time and other statistical studies; and preparation of performance reports to outside regulatory agencies.

Any MATHEMATICAL SYSTEM is a collection of UNDEFINED CONCEPTS, or PRIMITIVES; UNPROVEN RULES, or AXIOMS; concepts (DEFINITIONS) defined in terms of primitives; and proven statements, or THEOREMS, proved from the axioms. Classical Euclidean geometry is a model still valid today in broad terms. Points, lines are primitives; intersection, parallel, perpendicular are concepts defined in terms of the primitives; axioms..... theorems.......

All mathematics may be classified broadly as: arithmetic, geometry, algebra, analysis (i.e., calculus); and logic. Analysis and logic, in particular, have important applications in pathology.

PATHOLOGY examines morbid (i.e., diseased; Latin: morbus=disease) anatomy. All anatomy may be broadly classified as normal, degenerative, or proliferative. Pathologists use tissue-appearances and other, contextual information to arrive at a diagnosis, and possibly prognosis, of disease. In broad strokes, the pathologist combines image data, logic, and established experience, to formulate a diagnosis.

WHY SHOULD PATHOLOGISTS
CARE ABOUT MATHEMATICS?.

Why should pathologists care about mathematics?

1. Medicine (Hippocrates) and Mathematics (Pythagoras of Samos (560 - 480 BC), Euclid (230-275 BC), Archimedes (287-212 BC).. Sun-Tse) are common legacies of ancient Mediterranean (Greco-Roman) and Far Eastern (Chinese) cultures.

2. Both fields have common bonds of picturial demonstrations: geometry; gross anatomy and microanatomy.

3. Mathematicians are the provers of physics. Pathologists are the provers of medicine.

4. Applications in: quality assurance (statistics); medical logic (logic, artificial intelligence, EMRs); image analysis (Fourier analysis).

WHY SHOULD MATHEMATICIANS
CARE ABOUT PATHOLOGY?

Why should mathematicians care about pathology?

1. For all of the above reasons....

2. In addition, pathology and quality assurance processes in medicine are destined to add many new interesting areas into applied mathematics.

Why pathologists should be leaders in medical quality assurance?

1. Pathologists are natural record-keepers in medicine. As a consultant specialty that often doesn't see the patient, pathologists are particularly

2. In many patients, and in essentially all cancer patients, the pathology report is the most important record in the medical chart.

3. Pathologists handle more patient-records than any other medical specialty, and pathologists are trained in epidemiologic and statistical methods. Essentially every medical chart contains a pathology report (typically, serum chemistries); and one-third of medical charts contain an anatomic pathology report.

WHAT IS PATHOLOGY?

1. Pathology is the study of the etiology, pathogenesis, and diagnosis of disease.

2. Anatomy (gross anatomy, microanatomy:
2.1. Cardiovascular.
2.2. Respiratory.
2.3. Gastrointestinal.
2.4. Genitourinary.
2.5. Endocrine.
2.6. Integumentary.
2.7. Musculoskeletal.
2.8. Hemolymphatic.
2.9. Central Nervous System.
3. Pathophysiology. All diseases may be classified as one-or-more of the following pathophysiologic processes:
3.1. Metabolic.
3.2. Infectious.
3.3. Inflammatory/immunologic.
3.4. Ischemic/vascular.
3.5. Congenital/neonatal.
3.6. Neoplastic.
3.7. Trauma.
3.8. Systemic.

4. Medical logic.
4.1. Artificial intelligence.
4.2. Computer surveillance.
4.3. Spreadsheets.

5. Image analysis.

5.1. Fourier analysis.
5.2. Epicycles/Ptolemaic cosmology.

6. Privacy and computer security.
6.1. Public/private keys.
6.2. Social aspects of computer security: picking a password; keeping it secret.

WHAT IS MATHEMATICS?

1. Pure mathematics is the study of proof.

2. Applied mathematics is the use of these proofs in practical/real-world problems.

CORE CONCEPTS OF MATHEMATICS:

1. Solving problems. Mathematics is the study of reducing a hard problem to a collection of easier problems. Historically, mathematics has addressed problems of land surveying and accounting; but it now touches nearly every aspect of modern 21st century life.

2. Well-formed statements. All mathematics consists of well-formed (i.e., syntactically correct) statements, that may be either true, false, or indeterminate. These well-formed statements may be represented as statements in a natural language (such as English or French); statements in an abstract language, such as equations in algebra; or in pictures. While a picture, in itself, is not a proof, a picture may illuminate the otherwise inscrutable reasoning processes in a proof.

3. Consistency. No well-formed statement can be both true and false. It has been said that ..... a mathematician would sooner not believe a direct observation than abandon the notion of consistency.

4. Substitution of Equals. If x = y, then you may substitute x into any mathematical expression where y appears.

5. Major branches of mathematics. The major branches of mathematics are: arithmetic; geometry; algebra; analysis (limits and calculus); and logic. All these branches of mathematics, except for algebra, were known to ancient Mediterranean, Chinese, and Indian cultures.

6. All mathematics is built up from two, fundamental ideas: numbers and operations, corresponding loosely to nouns and verbs in linguistics.

7. The oldest system of numbers is the counting numbers or natural numbers, namely, 1, 2, 3, .... (starting at 1), and known to all three ancient cultures: Mediterranean, Chinese, and Indian. ZERO was developed by the eighth-century Indian mathematician, Brahmagupta, although glimmerings of zero are present as the place-holder on an abacus in ancient Babylonian/Persian mathematics.

8. The oldest mathematical operations is addition over natural numbers. This operation has properties of closure, associativity, and commutativity (q.v.), which were later abstracted in many modern concepts of mathematics. However, making sense out of subtraction, which is the inverse of addition, awaited the invention of zero and algebra in the Indo-Mediterranean cultures.

9. Numbers have two general meanings: ORDER (ordinal numbers): (first, second, third, .... and SIZE (cardinal numbers): (one, two, three, ....).

10. INTEGERS include the natural numbers, zero, and the negative natural numbers: ... -3, -2, -1, 0, 1, 2, 3, ... With this enlarged idea of numbers, one can build a system of addition and subtraction over integers that includes IDENTITY (ZERO) and INVERSE-ADDITION (SUBTRACTION)

11. IDENTITY (zero for addition). The identity can be added to any number, and the result is always the same, e.g., 5 + 0 = 0 + 5 = 5.

12. INVERSE (minus is the inverse for addition) is the property that if you add a number to its inverse, then the result is the identity, e.g., -5 is the inverse of 5, so that -5 + 5 = 0.

13. CLOSURE is the property that, if you add two numbers, the result it still a number. Thus, if 3,4 are numbers, 3+4 = 7 is a number. This is a fairly obvious property for addition over integers, but you have to be careful about more abstract operations.

14. ASSOCIATIVITY is the property that (a o b) o c) = (a o (b o c)).

15. COMMUTATIVITY is the property that (a o b) = (b o a).

16. GROUP is a mathematical system with properties 1,2,3,4:

1. Closure.
2. Associativity.
3. Identity.
4. Inverse.
5. Commutativity.

A SEMIGROUP has properties 1,2 only.

17. RING

18. FIELD

19. FOURIER SERIES is a finite (or possibly infinite) sum of sines and cosines (q.v.) of the form:

F(x) = a₀ + a₁ sin(x) + b₁ cos(x) + a₂ sin(2x) + b₁ cos(2x) + a₃ sin(3x) + b₃ cos(3x) + ....

that fits a periodic function to arbitrary accuracy. Actually, ANY SMOOTH FUNCTION over a finite interval, such as [0, 2π] can be fit to arbitrary accuracy with a Fourier series that contains enough terms, simply by declaring that the function has a single period.

Ancient Mediterranean mathematicians used Fourier series to calculate the pathways of stars and planets, to aid in navigation at sea. The major cycle, a₁ sin(x) + b₁ cos(x), is the primary cycle, and subsequent cycles, a_n sin(nx) + b_n cos(nx), form epicycles to a desired level of approximation.

A PERIODIC FUNCTION, f(x), is any function for which f(x+p) = f(x), at all x, where p is called the PERIOD.

21. π Most important number in mathematics. Ratio of circumference divided by diameter of a circle. Value = 3.141592........

22. HYPERBOLA is the function: f(x) = 1/x, defined over the entire real line, except for zero.

23. e is the second-most important number in mathematics, after π, named in honor of Leonhard Euler, eighteenth century Swiss mathematician. Defined either based upon area under a hyperbola or as an infinite product. Value = 2.71828..........
In the hyperbola definition, e is the number such that the area under the hyperbola from x=1 to x=e equals 1.
In the infinite-product definition, e is the limit as n approaches infinity of (1 + 1/n)ⁿ. The infinite product definition is particularly valuable in understanding installment loans, which take the form of (1 + r/n)ⁿ for n payments; and in deriving the theory of infinite papillomas.

24. TRIGONOMETRY is the study of triangle measurement (Greek = trigon = triangle; = metron = measurement). The TRIGONOMETRIC FUNCTIONS are based upon a right triangle with angle x at the origin. The three sides of this triangle are labeled A = adjacent side, O = opposite side, and H = hypoteneuse.

There are six trigonometric functions, defined as follows:

sin x = O/H
cos x = O/H
tan x = O/H
cot x = A/O
sec x = H/A
csc x = H/O

Note that sin x is the reciprocal of csc x; cos x is the reciprocal of sec x; etc. (Reciprocal is NOT the same as inverse!)

Traditionally, angles are measured in degrees, where a full circuit of angles is 360˚. Then, for example:

sin 0˚ =
sin 30˚ =
sin 45˚ =
sin 60˚ =
sin 90˚ =
sin 120˚ =
....................

For many mathematics dicussions, it is convenient to measure angles in radians where a full circuit of angles (i.e., circumference for circle of radius 1) is 2π = 6.283184....

MATHEMATICAL ANALYSIS Study of infinity and limits (infinitesimals).

EXPONENTIAL

LOGARITHM The inverse of EXPONENTIAL

INSTALLMENT LOAN

COMPOUND INTEREST.

INVERSE. A very broad concept in mathematics, the idea underlying all SOLUTIONS of a mathematical problem, at all levels of mathematical generality. Inverse underlies the practical basis for the bread-and-butter of applied mathematicians, i.e., getting the answer. For example, the inverse of addition is subtraction. If the addition, x + 6 = 10, then what is x? The solution is: x = 10 - 6 = 10. Following the same idea, the inverse of multiplication is division. If the multiplication, x × 6 = 24, then what is x? The inverse of multiplication is division, and the solution is x = 24 " 6 = 4.

The concept of inverse has broad applications in mathematics. For example, the inverse of a derivative in calculus is the integral (Fundamental Theorem of Calculus, invented independently by Newton's (1642-1727) and Leibniz (1646-1716)).

SOLUTION. Latin: solvere=to break apart.

ADDITION

SUBTRACTION

MULTIPLICATION

DIVISION

CONVERGENCE TEST

HEAVISIDE FUNCTION

CONVOLUTION RING

CONVOLUTION SPACE

CONVOLUTION PRODUCT

CONVOLUTION INFINITE PRODUCT

CONTINUOUS FUNCTION

DISTRIBUTION

TEMPERED DISTRIBUTION

INTEGRABLE DISTRIBUTION

MIKUSIŃSKI OPERATOR RING

MIKUSIŃSKI OPERATOR

FOURIER TRANSFORM

INVERSE FOURIER TRANSFORM

OPERATOR

IDENTITY OPERATOR

OPERATOR CALCULUS

OPERATOR QUOTIENT

OPERATOR PRODUCT

DIFFERENTIAL OPERATOR

LINEAR FUNCTION

POLYNOMIAL

FUBINI THEOREM The assertion that a multiple integral can be re-expressed as a single integral.

DOUBLE INTEGRAL

RECTANGULAR FILLING

ISOSCELES TRIANGLE

SLIM TRIANGLE

FAT TRIANGLE

DECREASING PRODUCT

INCREASING PRODUCT

GENERALIZE, PARTICULARIZE Words of advice given to me by a college mathematics professor, for solving a mathematics problem.

GROUP A mathematical group consists of a set of elements, an identity element, an operation and its inverse operation. Axioms:

1: Closure.
2: Associative.
3: Identity element.
4: Inverse.

An Abelian or commutative group has an additional axiom:

5: Commutativity.

Example: Set of integers, under addition, inverse=subtraction.
Example: Set of integers, under multiplication, inverse=division.

A SEMIGROUP has only axioms 1 and 2 (closure and associative).

A RING.........

A FIELD.........

RING. An algebraic ring consists of a set, an identity element, two operations and the inverse operation of the first operation. Example: integers, addn, sbtr, mult.

FIELD. An algebraic field consists of a set of elements; two operations; an identity element for each operation; and inverse operations for both primary operations. Example: integers, addn, sbtr, mult, divn.

Mathematical Function. One of the most fundamental ideas of mathematics.

MATHEMATICAL FUNCTION, f. In simplest terms, a mathematical function, f, is a curve on the xy-plane:
427.

The abscissa is the horizontal-axis or x-axis. The ordinate is the vertical-axis or y-axis.
In mathematical terms, a mathematical function, f, is a collection of ordered-pairs, (x,y) ∈ f, also denoted f(x)=y, where x is the argument and y is the value. For each argument, x there exists at most one value, y=f(x). (However, one value of y may have many x's.) Familiar functions include f(x)=x (identity function); f(x)=ax+b (linear function); f(x)=ax²+bx+c (quadratic function); f(x)=e^x (exponential function), etc.
The DOMAIN OF f is the set of values taken by the x's, i.e., is the set of x for which y=f(x) exists. The RANGE OF R is the set of values taken by the y's, i.e., the set of y for which there exists some x such that y=f(x) exists.
A mathematical relation, R, is a collection of ordered-pairs, (x,y) ∈ R, in which one value of x may have many y's; and conversely, one value of y may have many x's.

MATHEMATICAL RELATION, R. A mathematical relation, R, is a collection of ordered-pairs, (x,y) ∈ R, also denoted xRy, where x is the argument and y is the value. A relation in which, for each argument x, there exists at most one value y is a function. The DOMAIN OF R is the set of values taken by the x's, i.e., the set of x for which there exists some y such that xRy. The RANGE OF R is the set of values taken by the y's, i.e. the set of y for which there exists some x such that xRy.

BRANCHES OF MATHEMATICS:

1. Arithmetic.
1. Addition, Multiplication.
The most fundamental operation of arithmetic is ADDITION. Ancient arithmetic (Greek, Chinese) was greatly advanced by ZERO (Brahmagupta, 598-670) and by ARABIC NUMERALS (Al-Khawárizmi, ?780-?845). Addition is a FUNCTION, denoted +, of two (numeric) ARGUMENTS or ADDENDS, and one (numeric) VALUE, or SUM. For example, 2+3=5, or, in functional notation, +(2,3)=5.

Addition over integers, rational numbers, or reals, has five properties:
1. Closure.
2. Associativity.
3. Identity.
4. Inverse.
5. Commutativity.
Semigroup: property 1 only. Group: properties 1,2,3,4. The group is a very general concept in mathematics, and may be applied to logic, matrix algebra, differential equations, and modal arithmetic (in computer security).

2. Inverse: Subtraction, Division.

3. Number Theory.
Basic properties of numbers, typically whole numbers. Developed by the Ancient Chinese. Core concepts include PRIME NUMBERS and MODAL ARITHMETIC.

4. Computer security: public/private keys. Based upon the (unproven, but very credible) assertion that it is much more computer-intensive solve for the prime-factors of a large composite number, than to multiply those two prime-factors. ..............
2. Geometry. Intuitively, known to ancient Egyptians, at least two millennia B.C.E., for design and construction of the pyramids of Giza, an impressive accomplishment for any era, and even more impressive before labor-saving machines. Pythagoras (560 BC-480 BC) Theorem is the centerpiece of ancient geometry. Plane (classical) geometry was formalized and summarized by Euclid (230-275).
Euclid's contribution was to reduce all geometric ideas to five axioms, and deduce all the rest from logical, step-by-step arguments.
In classical geometry, there are three primitive concepts: point, line, and between. Geometric objects are invariant under TRANSLATION (movement along a straight line) and ROTATION.

Geometry. The rigid geometric figure.
439.

Geometry. Translation. (Motion along a straight line.)
448.

Geometry. Rotation.
446.

TOPOLOGY is a generalization of geometric ideas, starting with Leonhard Euler (1707-1783), not constrained by rigidity of geometric figures. In topology (a non-classical geometry), geometric objects are invariant under INSIDE and OUTSIDE. (Inside/outside is a surprisingly subtle concept.) Thus, a teacup is the topologic equivalent to a doughnut. Teacup morphs to a doughnut:

3. Algebra. Developed by Uzbek/Arab/Persian mathematician, Muhammed Al-Khawárizmi, ?780-?845).
Group Theory. Generalization of algebraic ideas,..........

4. Analysis. Calculus. Limits, Derivatives, Integration.

Weierstrass (1815-1897) revolutionized mathematical analysis in the late 19th century, ...........

5. LOGIC. Generalization of Aristotle's (384-322 BC) "Law of the Excluded Middle" (every statement is either true or false, not both, not neither). Historically, logic was the last formalized branch of mathematics, but philosophically, logic is the most fundamental branch of mathematics. It is fair to say that the ancient mathematicians and philosophers understood/employed logic implicitly but not formally.
Gottfried Leibniz (1646-1716), the contemporary of Newton (1642-1727) and co-inventor of calculus, sketched out the basic ideas of formal logic (so-called Ratio universalis, Latin: universal reasoning), but Leibniz got stuck because he used EXCLUSIVE-OR for his calculations, whereas INCLUSIVE-OR is much easier and more productive/rewarding to work with. George Boole (1815-1864) developed the essential ideas of formal logic, which are paralleled in SET THEORY.

PICTURES OF MATHEMATICS:

Arithmetic. The straight line.
437.
Arithmetic. Multiplication.
439.

Geometry. The rigid geometric figure.
439.

Geometry. Translation. (Motion along a straight line.)
448.

Geometry. Rotation.
446.

Limit. Zeno's paradox.
438.

Limit. Archimedes (287-212 BC). . circle, as limit of n-gons:
443. 444. 445.

HISTORY OF MATHEMATICS:

1. Arithmetic. Tax-accounting for the Egyptian Pharoah's taxes.
Comment: It has been said that two-thirds of the parables in the New Testament of the Holy Bible are, broadly speaking, about money, borrowing, lending, taxing, etc.

2. Geometry. Resurveying land after the annual flood of the Nile River.
Comment: The two ancient branches of mathematics (arithmetic, geometry) were developed in the Mediterranean civilization so that the Egyptian Pharoah could collect his taxes.

3. Algebra. Group Theory.

4.

5.

6.

7.

CHAPTER 1. INTRODUCTION.

Next Chapter.
Previous Chapter.
Return to Table of Contents.

The finite product, P_n, for a sequence of numbers, r₁, r₂, r₃, ..., r_n, where 0 < r_k < 1 for all k = 1, 2, 3, ..., n, is defined as:

(1.1) P_n = (1-r₁) × (1-r₂) × (1-r₃) × ... × (1-r_n), for n = 1, 2, 3, ....

Since each factor, (1-r_k), is positive but less than 1, the sequence of numbers, P_n, is decreasing with increasing n; and the P_n possess a limit, P, as n ⇒ ∞, lying between 0 and 1, or possibly at 0 itself. For now, let's not worry about the fact that we're working with decreasing products. At some point, we will effectively dispense with this constraint.

The limit, P, is called the infinite product, and is written:

(1.2) P = (1-r₁) × (1-r₂) × (1-r₃) × ...
or, simply:
P = (1-r₁)(1-r₂)(1-r₃) ...

A central question to ask is: Does P=0 or is P>0? A related question to ask is: What is the limit, S, of the increasing sequence of partial sums:

(1.3) S_n = r₁ + r₂ + r₃ + ... + r_n.

of the same numbers as n⇒∞? Does S=∞ or is S<∞? S is called the infinite series, and is written:

(1.4) S = r₁ + r₂ + r₃ + ....

The amazing fact is that, as Prof. Struble proved over two decades ago, P=0 if and only if S diverges!

This commentary has two purposes: first, to show that there is a rich history of mathematical understanding and development in the field of infinite products; and second, to suggest that there is a model for malignant growth in surface tumors.

SAMPLE CALCULATIONS.

You can get an idea about how infinite products behave from the following:

Enter your own numbers and click on SUBMIT. The internet computer program is limited to positive values of r_k between r₁ and r₂₀. Other values will cause an error-message.

BARN PAINTING PROBLEM.

JOHN WALLIS. Sixteenth century English mathematician. The name WALLIS means Welshman in Latin. Wallis introduced the symbol for infinity, ∞.

A Wallis product ...........

SIEVE OF ERATOSTHENES (Eρατοσθενης): A method for determining prime numbers, known to the ancient Greeks.

448.
A PRIME NUMBER is a whole number that is divisible with no remainder only by one and by itself. The early prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ..... A COMPOSITE NUMBER is whole number that can be factored into two or more smaller numbers, eventually into primes. The early composite numbers are: 4=2², 6=2×3, 8=2³, 9=3², 10=2×5, 12=2²×3, 14=2×7, 15=3×5, ....

In the Sieve of Eratosthenes, one examines whole numbers one-by-one to see whether it has a non-remainder factor If the given number has a non-remainder factor, then it is composite, and the given number falls through the sieve. If the given number has no non-remainder factor, then it is prime, and the given number remains in the sieve.

There are a few shortcuts: don't examine even numbers (2 is the only even prime); and don't examine factors greater than the square root of the given number (why is that?).

In this cartoon, drawn by G. Vincent Moore, the water flowing into the sieve contains rocks (primes) and fluid (composites). The primes are caught in the sieve, and the composites flow through.

BILLBOARD .............

TERMINAL VOLUME .............

CATHEDRAL (Latin: cathedra=seat. In Christianity, the seat of the Bishop. .............

OVERPAINT THE BARN

CONDITIONAL CONVERGENCE.

ABSOLUTE CONVERGENCE.

TENT FUNCTION.

NATURAL NUMBERS.

INTEGERS.

RATIONAL NUMBERS.

REAL NUMBERS.

ALGEBRAIC NUMBERS.

TRANSCENDENTAL NUMBERS. The most important transcendental numbers are π = 3.141592... and e = 2.7182818....

IMAGINARY NUMBERS.

PRIME NUMBERS.

MATHEMATICAL EQUATION.

DERIVATIVE/DIFFERENTIAL. Basic definition. Average rate-of-change. Instantaneous rate-of-change. Tangent (geometry). Examples of Derivatives: Powers of x. Fractions with variable-denominator. Square roots.

LIMIT. Basic definition. Weierstrass interpretation. Complications of limits.

DIVISION BY ZERO. 0/0.

CONTINUITY.

NEOCONTINUITY.

CHAIN RULE.

MEAN VALUE THEOREM.

INTEGRATION/QUADRATURES.

BASIC DEFINITION. The area under a curve, such as:

427.
Here, the area is colored red:

433.

Notation: Leibniz [1646-1716], Newton [1642-1727] .

INDEFINITE INTEGRAL.

DEFINITE INTEGRAL.

RIEMANN INTEGRAL.

In Riemann integration, the area under the curve, f(x), is obtained as a progressive approximation, with VERTICAL BARS placed underneath the curve.

427.

The vertical bars get progressively thinner:

428.

429.

430.

In the limit, the sum of the areas of the vertical bars equals the area underneath the curve.

In many curves of mathematical interest, particularly those involving image analysis, The model of progressively thinner vertical bars fails, and we must resort to more sophisticated means to find the area. Examples are: Lebesgue integration, Mikusiński integration, and Struble-Mikusiński integration.

LEBESGUE INTEGRAL. In Lebesgue integration, the area under the curve, f(x), is obtained as a progressive approximation, with horizontal slabs placed underneath the curve.

431.

MIKUSIŃSKI INTEGRAL. In Mikusiński integration, the area under the curve, f(x), is obtained as a progressive approximation, with BRICKS placed underneath the curve.

432.

STRUBLE-MIKUSIŃSKI INTEGRAL. In Struble-Mikusiński integration, the area under the curve, f(x), is obtained as a progressive approximation, with ARBITRARY AREAS placed underneath the curve.

432.
The only constraint on Struble-Mikusiński integration is that the remaining-areas removed by the arbitrary areas at each step should form a DIVERGENT INFINITE SERIES. If these remaining-areas form a CONVERGENT INFINITE SERIES, then integration fails.

FUNDAMENTAL THEOREM OF CALCULUS. Statement. Proof. Examples of antiderivatives. Antiderivatives and Chain Rule.

TRIGONOMETRIC FUNCTIONS. Basic definitions.

Identities.

Derivatives.

Antiderivatives.

453.

454.

DERIVATIVES OF TRIGONOMETRIC FUNCTIONS.

Trig Function.	Derivative.
sin	cos
cos	sin
tan	...

EXPONENTIALS AND LOGARITHMS. Basic definitions. Identities. Derivatives. Antiderivatives.

DECREASING INFINITE PRODUCTS.

FILLING LEMMA.

MIXED INFINITE PRODUCTS.

QUOTIENT INFINITE PRODUCTS.

MIKUSINSKI THEOREM.

MATHEMATICS is the study of proof. The chief tool of mathematics is CONSISTENCY. That is, a statement is true if its exact negation cannot be true. A philosopher has stated that if a mathematician observed an event that appeared inconsistent, then he/she would sooner disavow the observation than accept the violation of consistency [1].

Mathematics has five major branches [2]: Arithmetic; Geometry; Algebra; Analysis; and Logic.

ARITHMETIC involves the basic properties of the number system, including: natural numbers, integers, rational numbers, real numbers, and imaginary numbers; and the basic operations of addition, subtraction, multiplication, and division.

Included under arithmetic are ...............

ANALYTIC GEOMETRY was invented by the French mathematician and philosopher, René Descartes [1596-1650]. Pointwise correspondence to Euclidean geometry [3]. In a flash, Descartes converted all the theorems of Euclidean geometry into theorems of algebra, and vice-versa. In mathematical analysis, almost all proofs involve algebraic-style statements, but much of our intuition about what these proofs mean, and why they are true, is based upon visualizations involving Euclidean geometry.

XY-PLANE:

421.

423.

VECTOR ON XY-PLANE:

424.

425.

TRIANGLE ON XY-PLANE:

426.

CURVE/FUNCTION ON XY-PLANE:

427.

The major role of LOGIC in mathematical analysis is the formation of exact negations. In proof by contradiction, known to Euclid, the exact negation of a statement-to-be-proved is set up as a straw man. Then the negation is shown to be false (so-called: Reductio ad absurdum Latin: Reduction to absurdity), from which the statement-to-be-proved is thus proved true. For example: this statement (Weierstrass [1815-1897] definition) of continuity at X): for every ε > 0, there exists a δ > 0 such that for every x where |x - X| < δ, then |f(x) - f(X)| < ε, has the following exact negation: there exists an ε > 0 such that for every δ > 0 there exists an x such that |x - X| < δ and |f(x) - f(X)| > ε.

Almost everywhere (a.e.):

RECTANGLE FILLING:

ISOSCELES TRIANGLE:

FAT TRIANGLE:

MAX TRIANGLE:

SLIM TRIANGLE:

CHAPTER 10. SAMPLE CALCULATIONS.

Next Chapter.
Previous Chapter.
Return to Table of Contents.

The present report addresses mathematical issues regarding infinite products and infinite series. However, it is sometimes helpful to get a flavor of the infinite process by trying out a few finite examples. For the sequence of r_n, the finite product, P_n, is defined as: P_n = (1 - r₁)(1 - r₂)(1 - r₃) ... (1 - r₁); and the finite sum S_n, is defined as: S_n = r₁ + r₂ + r₃ + ... + r_n). Then the infinite product, P, is defined as S = lim_{n ⇒ ∞} S_n); and the infinite sum, S, is defined as S = lim_{n ⇒ ∞} S_n).

Here are three examples of the P-calculation for r₁, r₂, r₃, r₄, r₅, r₆, r₇. To calculate the value for P, the product, click on SUBMIT. Three examples are presented: a convergent (geometric) series; a divergent (constant) series; and a divergent (harmonic) series. You may wish to alter the numbers, and examine the effects of the series on the product calculations:

CONVERGENT SERIES (GEOMETRIC).

DIVERGENT SERIES (CONSTANT=1/10).

DIVERGENT SERIES (HARMONIC).
NOW TRY YOUR OWN, by changing the values of r_n in the boxes, and clicking on SUBMIT.

Note N1. Sketch of proof that: 1-x < e^-x holds for all x.
For x=0: 1-x = 1-0 = 1 = e⁰ = e^-x.
For x gt; 0, it suffices to show that: de^-x/dx > 0. But de^-x/dx = e^-x > 0. Q.E.D.

Note N2. Proof that: (1-R)^x/R < (1-x)..
First, the two boundary cases:

At x=0, 1 = (1-R)^0/R < (1-0) = 1.
At x=R, 1-R = (1-R)^R/R < (1-R) = 1-R.

Now, for 0 < x < R, let: y = R-x > 0; and z = 1-R > 0. Since x, y, z, > 0, clearly: z ^x < (y + z)^x+y. Take both sides to power 1/(x+y):

z^x/(x+y) < (y + z)

Substituting:

(1-R)^x/R < (1-x).

Q.E.D.

Note N3. Proof that: a approaches e (a⇒e): Let R = 1/(n+1). Then 1/(1-R) = 1/(1 - (1/(n+1))) = (n+1) / (n+1-1) = n+1 / n = 1 + 1/n.

Scalar. One-dimensional variable, say x.

Vector. A vector is ann-dimensional variable, say (x₁, x₂, x₃, ..., x_n).

Time. In physics and biomedicine, time is a scalar quantity that moves ever-forward. In mathematics, the process may be reversed, at least in broad theoretical strokes. Thus, mathematics allows one to second-guess what one has already done; or partially control what one is about / intends to do (if one believes in human free-will). .............

Measure Theory.

Group Theory.

MATHEMATICAL GROUP..

FACTORIAL, n!. For any whole number, n, the factorial of n, or n factorial, denoted n!, equals 1 × 2 × 3 × ... × n!. By convention, 0! = 1.

Mathematical operator. Generalization of a mathematical function, in which ......

Continuity. In simplest terms, a continuous function (curve) is one that can be drawn without lifting up one's pencil. In intuitive mathematical terms, function f(x) is continuous at X if, when you get x close enough to X, then f(x) gets arbitrarily close to f(X). In formal mathematical terms, function f(x) is continuous at point X if and only if for every ε > 0, no matter how small, there exists a small enough δ> 0, such that for every h < δ, |f(X±h) - f(X)| < ε

Cartesian plane. Analytic geometry: one-to-one correspondence between points in the Euclidean plane and (x,y) pairs in algebra. Descartes didn't just come up with some new applications of algebra in the xy-plane. In an instant, Descartes co-opted the entire body of geometry, theorems known heuristically to the ancient Egyptians and proved by Euclid (230-275) and made these into theorems of algebra as well. Thus, the repertoire of theorems of seventeenth century mathematics immediately doubled in size.
Named after: René Descartes [1596-1650], French mathematician and philosopher.

INFINITE PRODUCT. Product of an infinite sequence.

INFINITE SERIES. Sum of an infinite sequence.

Fanciful Applications. My favorite fanciful application of infinite products is an infinite papilloma. In tumor biology, a papilloma is an abnormal upward growth of surface tissue, as might occur on the skin or other tissue-surface (lining of the mouth, windpipe, foodpipe, or colon):

Squamous papilloma:
225.
Some of these growths are benign (i.e., do not grow without limit); other growths are malignant (i.e., grow without limit, until they are removed or kill the patient).

In general, upward-growth from a tissue-surface is called exophytic growth; the entire structure is called a papilloma; and a single, tall cone within the papilloma is called a papilla. Downward-growth from a tissue-surface is called endophytic growth; and the entire structure is called an acanthoma.

Exophytic growth:         Endophytic growth:
49.         50.

The potential volume, possibly infinite, into which the papilloma grows, is the billboard, possibly infinite. The papilloma may either fill or not-fill the billboard. (See: Section 4A. Rectangle-filling).

Hypothesis: a non-filling papilloma is benign; a filling papilloma is malignant.

Result: a non-filling-papilloma (benign) results from a convergent series of papillae; a filling-papilloma (malignant) results from a divergent series of papillae.

Geometric (benign) papilloma:         Harmonic (malignant) papilloma:
322.         321.

It is a commonplace observation in human pathology that a geometric papilloma resembles a pedunculated papilloma, which has a relatively better prognosis; and that a harmonic papilloma resembles a sessile papilloma, which has a relatively poorer prognosis.

IS VOLUME-FILLING A PLAUSIBLE BIOLOGICAL MECHANISM FOR CANCER? Yes, volume-filling is how wound-healing works, a normal physiological process. Cancer is just a permanent, genetically-programmed form of volume-filling. Virchow (1858) first proposed this Stimulus Theory (Reiztheorie) of cancer.

Elementary arithmetic. The theory of addition, multiplication, subtraction, and division.

Al-Khawárizmi. The inventor of many methods in solving linear and quadratic equations was ninth century Arab-Persian mathematician, Abu Abdullah Muhammad bin Musa Al-Khawarizmi [?780 - ?845]:

أبو عبد الله محمد بن موسى الخوارزمي

[Farsi: `man from Khawarizm', now in Uzbeckistan]. for whom the term algorithm is named. Al-Khawarizmi's team of mathematicians developed Arabic numerals and the mathematical discipline of algebra [Arabic: al-jabbar = the way], for whom the term algorithm is named. Al-Khawarizmi's team of mathematicians developed Arabic numerals (invented by Indian mathematician, Brahmagupta, see [7.54]) and the mathematical discipline of algebra [Arabic: al-jabar = the way]. Many authors transliterate the Arabic script for Al-Khawarizmi's name as Al-Khwarizmi, or even Al-Kwarizmi, but I am told that this is incorrect (Al-Ubaydli, 2005).

Infinity, ∞.

Algebraic ring.

Convergence.

Convolution. [Latin: convolvere = to turn together.]

Compact support.

Exponential.

Logarithm.

Natural logarithm.

Calculus. [Latin: stone]. In general, a method of calculation, which in ancient times was performed with an abacus, consisting of moveable stones. Nowadays, refers to .............................

Convergence Test.

e. Defined as the limit, as n approaches infinity, of (1+n^-1)ⁿ, i.e., e = Lim_n⇒∞(1+n^-1)ⁿ, valued at 2.718281828459045.....
e is the upper limit of the integral, ₁∫^e x^-1dx = 1.
Also, e has the property thatde^x/dx = e^x.
Named after eighteenth-century Swiss mathematician, Leonhard Euler[17??-17??].
See Compound interest.

Show that...:

RIEMANN ZETA-FUNCTION, ζ(s): Defined as: ζ(s) = ∑_n=1^∞ n^-s.
Note that when s=1, then ζ(1) is exactly the harmonic series.

Integration ....:

Differentiation ....:

Georg Friedrich Bernhard Riemann [1826-1866] ....:

Henri Lebesgue [1875-1941]. ....:

Salomon Bochner [1881-1982]. ....:

Fundamental theorem of differential and integral calculus: The assertion that, for well-behaved functions (i.e., continuous, differentiable), the operations of integration and differentiation are inverse operations of one another. Analogous to the properties of arithmetic, where addition/subtraction are inverse operations of one another; and multiplication/division are inverse operations of one another.
In mathematical terms, if F(x) = _a∫^x f(t) dt is differentiable, then d F(x) / dx = f(x).
The theorem was proved independently in the late seventeenth century by British mathematician Sir Isaac Newton (1642-1727) and by German mathematician Gottfried Leibniz (1646-1716). The academic priority battle that ensued, and the accompanying large professorial and nationalistic egos, retarded the development of European mathematics for much of the following century.

r₁, r₂, r₃, ... r_n, ...: The fraction of the unpainted portion of the barn that is newly painted at step n.

t₁, t₂, t₃, ....: Tent-functions. ...........................

S₁, S₂, S₃, ....: Terms of the infinite series.....

P₁, P₂, P₃, ... P_n,...: Terms of the infinite product, P_n, where:

P_n = (1-r₁)(1-r₂)(1-r₃) ... (1-r_n), for n = 1, 2, 3, ....

(See: (1.1)).

P₊: The product of terms of the the infinite product, i.e., P = P_∞, that take the form, (1 + r_k) ...............

P_- ....: The product of terms of the the infinite product, P that take the form, (1 - r_k).

S₊ ....: The sum of positive terms of the the infinite series, S, that take the form, +r_k.

S_-....: The sum of negative terms of the the infinite series, S, that take the form, - r_k

Mathematical induction. ............................

Heaviside step function. ............................

Distribution. ............................

Operator. ............................

Natural numbers. [Latin: natura = nature.] The counting numbers, 1, 2, 3, 4, 5, 6, ..., known to ancient cultures. [560 B.C. - 480 B.C.]

Ordinal numbers. [Latin: ordo = order.] Numbers in order: 1st, 2nd, 3rd, 4th, 5th, 6th, .... There is nothing between a pair of consecutive ordinal numbers, e.g., there is no one-and-a-halfth.

Cardinal numbers. [Latin: cardo = hinge or linchpin. Signifying great importance, since a door without its hinge will not function as a door. A cardinal in the Roman Catholic Church is a priest of great importance.] Numbers of magnitude: 1, 2, 3, 4, 5, 6, .... Between any two cardinal numbers, there is always another cardinal number.

Prime numbers. [Latin: primus = first.] A prime number is a whole number (i.e., natural number) divisible without remainder only by itself and 1. The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, .... [Notice: 2 is a prime number; but 1 is not a prime number.]

Limit, L. [Latin: limes = boundary.] Traditionally, that number, L, on the real line, to which a sequence, x₁, x₂, x₃, ... x_n, ..., approaches, as n ⇒ ∞. For a finite limit, x_n ⇒ L, the x_n get "ever so close to" L. For an infinite limit, the x_n ⇒ ∞, i.e., the x_n "gallop off to" ∞.
Weierstrass [1815-1897] interpretation. For a finite limit, x_n ⇒ L, for every ε>0, there exists an N>0 such that for every n>N, |x_n-L|<ε.
Weierstrass interpretation. For an infinite limit, x_n ⇒ ∞ and for every n>N, x_n > ∞.

Calculus. [Latin: stone, used in an abacus, for calculation.] Any method of computation. Calculus has come to mean differential and integral calculus.

INFINITY, ∞. The entity greater than any real number. That is, for every N, N<∞.

Integral, ∫. [Latin: integer = whole, unbroken.] In simplest terms, the integral of a curve/function is the area under that curve. In mathematical terms:

_A∫^B f(x) dx

denotes the area under f(x) between x=A and x=B:

23.
Traditionally, this area-value is obtained by dividing the area into very thin, vertical bars, and summing up the bars Riemann integration.
Alternatively, one may useLebesgue integration (thin horizontal slabs); Mikusinski integration (tiny bricks); ?painting integration (tiny, arbitrary blobs). The calculation of area under a function, f(x), between the range where x=a and x=b, denoted _a∫^b f(x)dx.

23. Integration is performed by making smaller and smaller areas under the function, that get closer and closer to the actual curvature of the function.

Riemann integral, ∫. An integral where the limiting areas are vertical columns. In Riemann integration, the area under the curve, f(x), is obtained as a progressive approximation, with VERTICAL BARS placed underneath the curve.

427.

The vertical bars get progressively thinner:

428.

429.

430.

In the limit, the sum of the areas of the vertical bars equals the area underneath the curve.

LEBESGUE INTEGRAL, ∫. An integral where the limiting areas are horizontal slabs.

431.

Every Lebesgue integrable function is Riemann integrable, but not necessarily vice-versa.

BOCHNER INTEGRAL, ∫. An integral where the limiting areas are bricks.

432.

MIKUSIŃSKI INTEGRAL, ∫. An integral where the limiting areas are bricks.

432.

PAINTING INTEGRAL, ∫. An integral where the limiting areas are triangles or arbitrary

SUMMATION, ∑. For a sequence of values, x₁, x₂, x₃, ..., x_n, the notation ∑_i=1ⁿ x_i denotes x₁+x₂+x₃+...+x_n, that is, the sum of x_i from i=1 to i=n. For example, the notation ∑_i=1⁵ 1/n_i denotes 1 + 1/2 + 1/3 + 1/4 + 1/5 = 1 + 0.5 + 0.33333 + 0.25 + 0.2 = 2.28333.... These are the first five terms of the harmonic series.

PRODUCT, Π. For a sequence of values, x₁, x₂, x₃, ..., x_n, the notation Π_i=1ⁿ x_i denotes x₁×x₂×x₃× ...×x_n, that is, the product of x_i from i=1 to i=n. For example, the notation Π_i=1⁵ 1/n_i denotes 1 × 1/2 × 1/3 × 1/4 × 1/5 = 1 × 0.5 × 0.33333 × 0.25 × 0.2 = 0.0083332. These are the first five terms of the harmonic product.

Weierstrass [1815-1897] PARADIGM. Statements about differential and integral calculus of the form: "for every ε>0, however small, there exists a δ>0, however small, or an N>0, however large, such that...". This paradigm replaces the otherwise sloppy concepts of "getting ever so close to a limit" and "galloping off to infinity". See Courant, Robbins, Stewart. What is Mathematics?

DERIVATIVE/DIFFERENTIAL. The derivative/differential of a function at point x, denoted df(x)/dx, is the tangent of the function at point x. Definition:

Lim_{Δx → 0} (f(x+Δx) - f(x))/Δx.

THEOREMS FOR DERIVATIVES. The derivative of a constant is zero:

Theorem: dc/dx = 0.
Proof. (c - c)/Δx = 0.

The derivative of x itself is one:

Theorem: dx/dx = 1.
Proof. ((x+Δx - x)/Δx = Δx/Δx = 1.

The derivative of x² is 2x. At this point, we will do something slightly slippery. Since Δx is small, it follows that terms (Δx)² , (Δx)³,..., are very small. In an expression that contains a Δx-term, the terms (Δx)², (Δx)³,..., may be discarded.
Caution: Indiscriminate use of this trick leads to some neat conjectures, but also to some serious mistakes. Newton [1642-1727] got the wrong answer for d(f(x)+g(x))/dx, and many eighteenth century mathematicians stumbled about using this technique. Weierstrass [1815-1897] cleaned everything up in the mid-nineteenth century.

Theorem: dx²/dx = 2x.
Proof. ((x+Δx)² - x²)/Δx = (x² + 2xΔx + Δx² - x²)/Δx = 2xΔx / Δx = 2x, since Δx² is a very small term.

The derivative of x³ is 3x².

Theorem: dx³/dx = 3x²
Proof. ((x+Δx)³ - x³)/Δx ... as above.

The derivative of any power of x, namely xⁿ, is nx^(n-1).

Theorem: dxⁿ/dx = nx^(n-1).
Proof. By induction.

The derivative of the sum of two functions, f and g, is the sum of their derivatives:

Theorem: d(f(x)+g(x))/dx = df(x)/dx + dg(x)/dx.
Proof. ((f(x+Δx) + (g(x+Δx)) - (f(x)+g(x)))/Δx = ((f(x+Δx) - f(x))/Δx + (g(x+Δx)) - g(x)))/Δx.

The derivative of the products of two functions, f and g, is fdg/dx + gdf/dx.

Theorem: d(f(x)×g(x))/dx = f(x)(dg(x)/dx) + g(x)(df(x)/dx).
Proof. (Slippery proof, but the result is correct). d(f(x)×g(x)) = f(x+Δx) g(x+Δx) = f(x)g(x) = + g(x) Δf(x) + f(x) Δg(x) Δf(x) Δg(x) - f(x)g(x) = = f(x)dg(x) + g(x)df(x) since Δf(x) Δg(x) is a very small term.

ANTIDERIVATIVE. Reverse of derivative. That is, if ........... In well-behaved functions, the anti-derivative equals the integral. (This is the Fundamental Theorem of Calculus.)

HARMONIC SERIES. ∑_n=1^∞ n^-1 = (1/2) + (1/3) + (1/4) + (1/5) + = ∞. Yes, amazingly, the harmonic series does not converge! A nice, easy proof of this assertion is included in: Derbyshire J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Plume Books. 2004. ISBN: 0452285259.

Proof that the Harmonic Series is Infinite:

Group the terms of the series as follows:

½ = (1/2)
½ > (1/3) + (1/4)
½ > (1/5) + (1/6) + (1/7) + (1/8)
½ > (1/9) + (1/10) + (1/11) + (1/12) + (1/13) + (1/14) + (1/15) + (1/16)
....

Obviously, the harmonic series adds up to even greater than: ½ + ½ + ½ + ½ + ... = ∞.

GEOMETRIC SERIES. ∑_n=1^∞ 2^-n = (1/2) + (1/4) + (1/8) + (1/16) + .... = 1. That is, the geometric series converges to one.

Lim _{n → ∞} ∑_n=1^∞ 2^-n = 1

CONVERGENCE. A convergent sequence is an infinite sequence with a finite limit, L.

DIVERGENCE. A divergent sequence is an infinite sequence that does not converge to a finite limit, L. The sequence may either gallop off to infinity, such as 1, 2, 3, 4, 5, 6, 7, 8,..., or else vacillate indefinitely, so that it never converges, as for example, +1, -1, +1, -1, +1, -1, +1, -1,....

RIEMANN HYPOTHESIS. Named after its author, Bernhard Riemann, the Riemann Hypothesis is the assertion that all the non-trivial roots of the Riemann zeta function have real part ½. (See: [Derbyshire, 2004]).

e Named after Leonhard Euler, e is the upper limit of the integral, ₁∫^e x^-1dx = 1, valued at 2.718281828459045..... Alternatively defined as the limit, as n approaches infinity, of (1+n^-1)ⁿ, i.e., e = Lim_n⇒∞(1+n^-1)ⁿ.
The derivative of e^x is e^x.
See Compound interest.

COMPOUND INTEREST. If you place $1 in a savings-account at 100% annual interest, and the bank pays the interest exactly once at the end of the year, then you receive $1(1+100%) = $2 at year's end. If the bank pays once at six months and once-again at year's end, then you receive $1(1+100%/2)(1+100%/2) = $2.25 at year's end. If the bank pays once each at four months, eight months, and at year's end, then you receive $1(1+100%/3)(1+100%/3)(1+100%/3) = $2.37037037037037037... at year's end. If the bank pays once each at three months, six months, nine months, and at year's end, then you receive $1(1+100%/4)(1+100%/4)(1+100%/4)(1+100%/4) = $2.44140625 at year's end. If the bank compounds interest instantaneously, then you receive lim_{n => ∞} $1(1+1/n)ⁿ = 2.718281828459045.... = e at year's end.
If the interest-rate is r, then the year-end sum, compounded instantaneously, is (1+r/n)ⁿ = e^-r. If the interest-rate at the nth computing-cycle is r_n, then the year-end payment is .......................... See the connection with infinite products?

π. Pi (pronounced "pie"), π, is the ratio of the circumference of a circle to twice its radius; or, the value of the circumference of the unit circle, 3.1415927..... See Beckmann P. A History of Pi. 1967.

Infinite sequence. An infinite sequence of values, x₁, x₂, x₃, .... Both the geometric sequence, (1/2), (1/4), (1/8), (1/16), ... 2^-n,..., and the harmonic sequence, (1/2), (1/3), (1/4), (1/5), (1/6), ... 1^-n,..., are infinite sequences that converge to zero.

Infinite series. An infinite sum of values, x₁ + x₂ + x₃ + ... = ∑_i=1^∞ x_i. The geometric series, (1/2), (1/4), (1/8), (1/16), ... 2^-n,..., is an infinite series that converges to 1. The harmonic series, (1/2), (1/3), (1/4), (1/5), (1/6), ... n^-1,..., is an infinite series that does not converge.

Infinite product. An infinite product of values, x₁ × x₂ × x₃ × ... = Π_i=1^∞ x_i. The geometric product, (1/2), (1/4), (1/8), (1/16), ... 2^-n,..., and the harmonic product, (1/2), (1/3), (1/4), (1/5), (1/6), ... n^-1,..., are infinite products that converge to 0.

Aristotle [384 B.C. - 322 B.C.]. Greek philosopher, who compiled an encyclopedia of all scientific and other human knowledge available at that time. Aristotle's Rule: for every positive y such that x > y, there exists an n > 0 such that y > x. Loosely speaking, Aristotle was the first biostatistician, since his works contain discussions of biology and chance phenomena.

1. Archimedes (Αρχιμεδης) (287? BCE-212 BCE). Ancient Greek mathematician, one of the three greatest mathematicians of all time (Archimedes, Newton [1642-1727] , Gauss). Archimedes' The Sand Reckoner is the first serious effort to deal with large-number problems, in this case, the number of grains of sand on the entire Earth. The elements of calculus are present in this document. Believe it or not, Archimedes was pretty close to the right number of grains! Statistics is the mathematical study of repeated sampling from a larger, possibly infinite, population. Archimedes laid the foundation for such large-number studies.

Euclid (Ευκλιδης) (230-275 BC). Ancient Greek mathematician, who summarized the rules of geometry, or Elements, which had been known as an empirical science by the ancient Egyptians for at least a millennium previously. Euclid's main contribution is that he collected the known truths of geometry and derived all geometric theorems from two undefined concepts (point, line) and five postulates, using the idea of proof by deduction. The Pythagorean theorem, namely that in a right triangle with legs a, b, and hypoteneuse c:

a² + b² = c²

is the basis for the Theory of Least Squares (see below), one of the bedrock methods of statistics.

PROOF OF THE PYTHAGOREAN THEOREM (PICTURE):

440.
The area of the entire figure is W+X+Y+Z, which is equivalent to (a+b)². The areas of the sub-rectangles are: W = b×b; X = a×b; Y = a×b; Z = a×a. Therefore, we conclude that

(a+b)² = W+X+Y+Z = b×b + a×b + a×b + a×a = a² + 2ab + b²

This is the usual binomial expansion:

(*) (a+b)² = a² + 2ab + b².

441.
By a similar argument, the area of the above figure: R+S+T+U+V, is equivalent to:

(**) 2×a×b + c²,

since the area of each right-triangle is (1/2)×a×b. Subtracting 2×a×b from expressions (*) and (**) yields the Pythagorean theorem, namely that in a right triangle with legs a, b and hypoteneuse c:

a² + b² = c²

This is essentially the same picture that appeared in Euclid's elements, handed down from over two millennia ago.

Leonardo Bigollo Pisano Fibonacci (1170?-1240?). Leonard of Pisa, pre-renaissance Italian mathematician. Fibonacci's Liber Abaci (1202) (Latin: Book of the Abacus) introduces Arabic numerals to European mathematics, namely: ٠١٢٣٤٥٦٧٨٩ Yes, the real Arab numerals are: 0=٠, 1=١, 2=٢, 3=٣, 4=٤, 5=٥, 6=٦, 7=٧, 8=٨, 9=٩. Do you see the resemblance? These numerals are used today in the Middle-East, and are used to enumerate verses in the Holy Quran (written in 7th century, Classical Arabic). Europe had been in an intellectual dark ages between the sacking of Rome in 476 C.E. until the Renaissance (13th-15th centuries).

Fibonacci imported Arabic numerals to 13th century Italy. Arabic numerals, including zero, are far superior to Roman numerals (or the even more primitive numerals of Ancient Greek, Hebrew, and Chinese cultures) for both accounting and mathematics. Fibonacci was the son of a merchant from Pisa, Italy, living in Algeria, where he was educated by Arab teachers, who were familiar with the Arab numeral system, and the great Arab masters of mathematics, such as Al-Khwárizmí and Abu-Kamil.

The Fibonacci sequence is: 1, 1, 2, 3, 5, 8, 13, 21, 34, ..., where each number in the sequence is the sum of the two previous numbers. The ratio of a given Fibonacci number to its immediate predecessor approaches the so-called Golden Ratio or Divine Proportion, (1+√5)/2 = 1.6181...., for large Fibonacci numbers (in the language of calculus, Fibonacci numbers approaching infinity). The Golden Ratio is observed in many areas of nature and esthetics. For example, the petals of a rose and a sunflower are arranged in this ratio; allegedly the ratio of body-height to the height of the umbilicus (belly-button) of a beautiful woman satisfy this ratio; and the dimensions of nearly all national flags (including the U.S. flag, of course) are in the Golden Ratio.

Occam. William of Ockham (1300?-1349). English philosopher. Occam's Razor: Entia praeter necessitatem non sunt multiplicanda. Latin: Entities should not be multiplied beyond necessity. Paraphrased: The simplest explanation is best.

John Napier, Laird of Merchiston (1550-1617). Sixteenth century Scottish nobleman, gentleman of leisure, and the inventor of logarithms, which transformed multiplication problems into addition problems, and division problems into subtraction problems. This advance in calculation methods led the way to increased accuracy of navigation and astronomy. Essentially there would never have been a Galileo [1564-1642], a Kepler [1571-1630], or a Newton [1642-1727], without Napier's incredibly powerful advance in calculation, the 16th century equivalent of the digital computer.
Napier's discovery is a simple extension of the idea that:

bⁱ×b^j = b^i+j

which had been known since Euclid (230-275 BC) and Archimedes(287-212 BC). For example:

8×16 = 2³×2⁴ = 2³⁺⁴ = 2⁷ = 128.

That is, if you want to multiply the two numbers, x and y, where x = bⁱ and y = b^j, all you have to do is look up the value of i for which x = bⁱ and the value of j for which y = b^j. Then add i+j (a lot easier than multiplication), and look up the value z = b^i+j. Napier spent twenty years developing a table of LOGARITHMS, where the value of i for which x = bⁱ is called the logarithm of x to the base b, or i = log_bx. A reversal of this same process converts long division into subtraction, which is worlds of difficulty easier.
Napier used 10,000,000 as the base for his logarithm table, and a contemporary mathematician, Wolfgang Burgi, converted Napier's tables into base-10, which is to be found in most high-school algebra texts. The so-called BASE OF NAPIERIAN LOGARITHMS, e, is the value 2.71......., has a deep significance in calculus, is named in honor of Napier, but was not known during Napier's lifetime.

Nicolaus Copernicus [1473-1543]. (Latin for Mikolaj Kopernik). Polish physicist who proposed the hypothesis that the sun, not the earth, lay at the center of the universe.

Galileo Galilei [1564-1642]. Italian physicist and astronomer, who discovered four of the moons of Jupiter, and provided a mathematical basis for Polish physicist Copernicus' [1473-1543] hypothesis that the sun, not the earth, lay at the center of the universe. For making the latter assertion, Galileo was placed under house arrest by the Papal Court in 1630 (a better fate than that of Italian physicist Giordano Bruno, who was burnt at the stake in 1630 for a similar crime). In the late 17th century, Catholic university physics departments began to teach Galileo's doctrines. In 1993, more than three-and-a-half centuries after Galileo's arrest, Pope John Paul II exonerated him [Beckmann, 1960]. Galileo died the same year that Newton [1642-1727] was born.

René Descartes (1596-1650). Inventor of Analytic Geometry, an algebraic mirror of Euclidean geometry. In an instant, all the true statements of geometry were true for algebra, and all the true statements of algebra were true for geometry, thus doubling the body of provable statements (theorems) in mathematics. This exact relationship between the statements of algebra and the statements of geometry is called a ONE-TO-ONE MAPPING. Dr. Lawrence A. Brown has suggested a Biblical prophesy foreshadowing Descartes's Theory is Matt 16:18 (known to all educated Roman Catholics): "Thou art Peter, and upon this rock shall I build my church, and I shall give thee the Keys to the Kingdom of Heaven, and whatsoever is bound on earth will be bound in heaven; and whatsoever is loosed on earth will be loosed in heaven." St. Peter, the first Roman Catholic pope, serves as a one-to-one mapping between heaven and earth.

Sir Isaac Newton [1642-1727]. Seventeenth century British physicist and mathematician, one of the three greatest mathematicians of all time ( Archimedes(287-212 BC). Newton (1642-1727), Gauss). Inventor of Classical Physics, in his classic book: Principia Mathematica Philosophiae Naturalis (Latin: Mathematical Principles of Natural Philosophy). Co-inventor, with Blaise Pascal, of the Binomial Formula, which is used to build the Method of Least Squares. Co-inventor, with Gottfried Leibniz (1646-1716), of differential and integral calculus, which are used in proving the Methods of Least Squares.

Abraham DeMoivre (1667-1754). Inventor of the Normal Distribution.

Blaise Pascal (1623-1662). Seventeenth century Swiss mathematician and philosopher.
Pascal's Wager. Pascal's Wager is a bet with the Lord God that He exists, using risk-benefit analysis. That is, the risk of unbelief (infinite) is so great, compared to the cost of belief (i.e., regular religious observance), that one is better off being religiously observant. This analysis is considered to be one of the first formal, mathematical treatments of risk-benefit analysis.
Pascal's Triangle. The expansion of the binomial, (a + b)², for arbitrary powers (i.e., polynomial). Polynomials are the bread-and-butter of calculus problems, as well as numerical analysis, used by modern computers to solve practical calculus problems.

PASCAL'S TRIANGLE.
Pascal's Triangle is a method for determining the coefficients of powers, n, of the binomial expansion, (a+b)ⁿ. Pascal's triangle is as follows:

(a+b)¹	.	.	.	.	.	.	.	.	.	1	.	1	.	.	.	.	.	.	.	.	.
(a+b)²	.	.	.	.	.	.	.	.	1	.	2	.	1	.	.	.	.	.	.	.	.
(a+b)³	.	.	.	.	.	.	.	1	.	3	.	3	.	1	.	.	.	.	.	.	.
(a+b)⁴	.	.	.	.	.	.	1	.	4	.	6	.	4	.	1	.	.	.	.	.	.
(a+b)⁵	.	.	.	.	.	1	.	5	.	10	.	10	.	5	.	1	.	.	.	.	.
(a+b)⁶	.	.	.	.	1	.	6	.	15	.	20	.	15	.	6	.	1	.	.	.	.
(a+b)⁷	.	.	.	1	.	7	.	21	.	35	.	35	.	21	.	7	.	1	.	.	.
(a+b)⁸	.	.	1	.	8	.	28	.	56	.	70	.	56	.	28	.	8	.	1	.	.
(a+b)⁹	.	1	.	9	.	36	.	84	.	126	.	126	.	84	.	36/td>	.	9	.	1	.
(a+b)¹⁰	1	.	10	.	45	.	120	.	210	.	252	.	210	.	120	.	45	.	10	.	1

Every coefficient-number is the sum of its left-parent and its right-parent.

Coulter.

Beckmann.

Paul Ehrlich. (1854-....)

Louis Pasteur.

W. Edwards Deming. (1900-1993).

Rosalind Yalow. [1921-]. Nobel Prize, Physiology or Medicine, 1977, for her work in radioimmune assay.

Shapiro (flow).

Wallace H. Clark.

Walter Lever.

Arthur T. Hertig.

Mallory.

Wilhelm His.

Franklin P. Mall

Zeno of Elea. (Ζενο) [490 B.C. - 425 B.C.] . Ancient Greek philosopher and mathematician, who ...........

Aristotle (384-322 BC)

Diophantus. Ancient Greek mathematician, famed for his work with integer arithmetic.

Pythagoras (Ρυθαγορας) (560 - 480 BC). The Pythagorean theorem, namely that in a right triangle with legs a and b, and hypoteneuse c:

a² + b² = c²

is the basis for the Theory of Least Squares, one of the bedrock methods of statistics.

Sun-Tse. First Century Chinese mathematician. Inventor of the Chinese Remainder Theorem. (See [Schneier, 1996]).

Brahmagupta (598-670). Indian Mathematician. Inventor of Zero. Brahmagupta wrote important works on mathematics and astronomy, especially Brahmasphutasiddhanta (The Opening of the Universe) in 628. Brahmagupta became the head of the astronomical observatory at Ujjain which was the foremost mathematical center of ancient India at this time. Outstanding mathematicians built up a strong school of mathematical astronomy. Topics covered are: mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; risings and settings; the moon's crescent; the moon's shadow; conjunctions of the planets with each other; and conjunctions of the planets with the fixed stars. Brahmagupta's understanding of the number systems went far beyond that of others of the period. In the Brahmasphutasiddhanta (628), he defined zero as the result of subtracting a number from itself. (See [Seife, 2000].)

Abu Abdullah Muhammad bin Musa al-Khawárizmi (أبو عبد الله محمد بن موسى الخوارزمي) Seventh Century Arab Mathematician. Inventor of the Algorithm. Note the similarity of name. (See [Seife, 2000].)

Leonardo Bigollo Pisano Fibonacci (1170?-1240?). Leonard of Pisa, pre-renaissance Italian mathematician. Fibonacci's Liber Abaci (1202) [Latin: Book of the Abacus] introduces Arabic numerals to European mathematics, namely: ٠١٢٣٤٥٦٧٨٩ Yes, the real Arab numerals are: 0=٠, 1=١, 2=٢, 3=٣, 4=٤, 5=٥, 6=٦, 7=٧, 8=٨, 9=٩. Do you see the resemblance? These numerals are used today in the Middle-East, and are used to enumerate verses in the Holy Quran (written in 7th century, Classical Arabic). Europe had been in an intellectual dark ages between the sacking of Rome in 476 A.D. until the Renaissance (13th-15th centuries).

Fibonacci imported Arabic numerals to 13th century Italy. Arabic numerals, including zero, are far superior to Roman numerals (or the even more primitive numerals of Ancient Greek, Hebrew, and Chinese cultures) for both accounting and mathematics. Fibonacci was the son of a merchant from Pisa, Italy, living in Algeria, where he was educated by Arab teachers, who were familiar with the Arab numeral system, and the great Arab masters of mathematics, such as Al-Khwárizmí and Abu-Kamil.

The Fibonacci sequence is:

1, 1, 2, 3, 5, 8, 13, 21, 34, ...

where each number in the sequence is the sum of the two previous numbers. The ratio of a given Fibonacci number to its immediate predecessor approaches the so-called Golden Ratio or Divine Proportion, (1+√5)/2 = 1.6181..., for large Fibonacci numbers (in the language of calculus, Fibonacci numbers approaching infinity). The Golden Ratio is observed in many areas of nature and esthetics. For example, the petals of a rose and a sunflower are arranged in this ratio; allegedly the ratio of body-height to the height of the umbilicus (belly-button) of a beautiful woman satisfy this ratio; and the dimensions of nearly all national flags (including the U.S. flag, of course) are in the Golden Ratio. (See [Livio, 2003].)

John Napier, Laird of Merchiston (1550-1617). Sixteenth century Scottish nobleman, gentleman of leisure, and the inventor of logarithms, which transformed multiplication problems into addition problems, and division problems into subtraction problems. This advance in calculation methods led the way to increased accuracy of navigation and astronomy. Essentially there would never have been a Galileo [1564-1642], a Kepler [1571-1630], or a Newton [1642-1727], without Napier's incredibly powerful advance in calculation, the 16th century equivalent of the digital computer.
Napier's discovery is a simple extension of the idea that:

bⁱ×b^j = b^i+j

which had been known since Euclid (230-275) and Archimedes(287-212 BC). For example:

8×16 = 2³×2⁴ = 2³⁺⁴ = 2⁷ = 128.

That is, if you want to multiply the two numbers, x and y, where x = bⁱ and y = b^j, all you have to do is look up the value of i for which x = bⁱ and the value of j for which y = b^j. Then add i+j (a lot easier than multiplication), and look up the value z = b^i+j. Napier spent twenty years developing a table of LOGARITHMS, where the value of i for which x = bⁱ is called the logarithm of x to the base b, or i = log_bx. A reversal of this same process converts long division into subtraction, which is worlds of difficulty easier.
Napier used 10,000,000 as the base for his logarithm table, and a contemporary mathematician, Wolfgang Burgi, converted Napier's tables into base-10, which is to be found in most high-school algebra texts. The so-called BASE OF NAPIERIAN LOGARITHMS, e, is the value 2.718281828..., and has a deep significance in calculus, is named in honor of Napier, but was not known during Napier's lifetime.

Tycho Brahe [1546-1601]. Danish-born astronomer who had collected extensive astronomical data on the orbit of Mars.

Giordano Bruno (1548-1600) Sixteenth century philosopher and free-thinker who ran afoul with the Roman Catholic Church's Inquisition, and was burnt at the stake in February, 1600. Considered a martyr to the Enlightenment.

Johannes Kepler. (1571-1630). German astronomer.. Discoverer of the three laws of planetary motion. Kepler gave the first mathematical treatment of close packing of equal spheres; gave the first proof of how logarithms worked; devised a method of finding the volumes of solids of revolution; and calculated the most exact astronomical tables then known (Rudolphine Tables, Ulm, Germany, 1627). Kepler entered Prague on January 1, 1600, to begin a fruitful collaboration with Tycho Brahe, a Danish-born astronomer who had collected extensive astronomical data on the orbit of Mars.

William Shakespeare (1564-1616). Sixteenth century English poet and playwright, during the high-water-mark of English power in Europe. Many of the commonplace quotations used in English are either from Shakespeare or from the King James Version of the Holy Bible.

Galileo Galilei [1564-1642]. Seventeenth century Italian physicist and astronomer, and beneficiary of the patronage of the Medici family, who discovered four of the moons of Jupiter, and provided a mathematical basis for Polish physicist Copernicus' hypothesis that the Sun, not the Earth, lay at the center of the universe. For making the latter assertion, Galileo was placed under house arrest by the Papal Court in 1630 (a better fate than that of Italian physicist Giordano Bruno, who was burnt at the stake in 1630 for a similar crime). In the late 17th century, Catholic university physics departments began to teach Galileo's doctrines. In 1993, more than three-and-a-half centuries after Galileo's arrest, Pope John Paul II exonerated him [Beckmann, 1960].

René Descartes (1596-1650). Inventor of Analytic Geometry, an algebraic mirror of Euclidean geometry. In an instant, all the true statements of geometry were true for algebra, and all the true statements of algebra were true for geometry, thus doubling the body of provable statements (theorems) in mathematics. This exact relationship between the statements of algebra and the statements of geometry is called a ONE-TO-ONE MAPPING. Dr. Lawrence A. Brown has suggested a Biblical prophesy foreshadowing Descartes's Theory is Matt 16:18 (known to all educated Roman Catholics): "Thou art Peter, and upon this rock shall I build my church, and I shall give thee the Keys to the Kingdom of Heaven, and whatsoever is bound on earth will be bound in heaven; and whatsoever is loosed on earth will be loosed in heaven." St. Peter, the first Roman Catholic pope, serves as a one-to-one mapping between heaven and earth.

Bonaventura Cavalieri (1598-1647). Italian mathematician who contributed to the understanding of calculus.

Pierre de Fermat (1601-1665). Seventeenth century French civil servant and amateur mathematician. In Fermat's literary estate, there is an assertion written in one of the books in Fermat's library (the works of Diophantus, the ancient Greek mathematician, famed for his work with integer arithmetic), that for integers x, y, z, and k, the equation:

x^k = y^k + z^k

is only true for k<2, so-called Fermat's Last Theorem. When k=2, there are numerous examples of this equation, the most being the famous 3-4-5 triangle, for which 3² = 4² + 5², where 3 and 4 are legs of a right triangle with hypoteneuse equal to 5. Fermat wrote in the margin that he knew a proof, but didn't have room to write it into the book margins. No proof and no counterexample were found for the next 350 years.

Isaac Barrow (1630-1677). English mathematician who contributed to the understanding of calculus.

Sir Isaac Newton [1642-1727]. Seventeenth century British physicist and mathematician, one of the three greatest mathematicians of all time (Archimedes (287-212 BC), Newton, Gauss). Inventor of Classical Physics, in his classic work: Principia Mathematica Philosophiae Naturalis [Latin: Mathematical Principles of Natural Philosophy]. Co-inventor, with Blaise Pascal, of the Binomial Formula, which is used to build the Method of Least Squares. Co-inventor, with Gottfried Leibniz (1646-1716), of differential and integral calculus, which are used in proving the Methods of Least Squares. 1642 was a big year in physics and mathematics: the death of Galileo [1564-1642] and the birth of Newton [1642-1727].

14. Gottfried Leibniz [1646-1716] . Seventeenth century German philosopher and mathematician. Co-inventor, with Sir Isaac Newton [1642-1727], of differential and integral calculus.

Giovanni Battista Morgagni [1682-1771]. Italian physician. De Sedibus et Causis Morborum. (Latin: On the seats and causes of diseases).

Daniel Bernoulli [1700-1782]. Swiss mathematician. Bernoulli Trial.

Leonhard Euler (1707-1783). Swiss mathematician. Inventor of topology (the famous Königsberg bridge problem). It is said that every calculus textbook is either Euler, a copy of Euler, or a copy of a copy of Euler [Agnew, 1962]. Worked extensively with Pascal's triangle and the binomial distribution. The Swiss government office has honored Prof. Euler on their ten-franc note.

John Graunt [1620-1674]. Seventeenth century British gentleman, who first organized death statistics, London Bills of Mortality. and used them for descriptive and predictive purposes. Graunt documented the plague epidemic in London, and was able to conclude that the outbreak originated from unsanitary conditions of the poorer districts of London, leading to social reforms in public health.

Rev. Thomas Bayes (1702-1761). British Anglican priest who developed the theory of conditional probability.

Karl F. Gauss [1777-1855]. Nineteenth century German mathematician, one of the three greatest mathematicians of all time (Archimedes(287-212 BC), Newton (1642-1727), Gauss). Gauss invented the method of least squares, which forms the backbone of modern statistical theory. (Actually, Gauss always claimed that he copied the method from a mathematical colleague [Bühler, 1980].) Gauss also created a mathematical framework for arithmetic and classical physics. Gauss attempted to verify from actual physical measurements whether Euclid's controversial theorem, that the angles of a triangle always add up to 180^o, was actually true.

Gauss was equally fluent in German, his mother-tongue, and in Latin, the scientific language of Europe in the eighteenth century. His prolific diaries, the two languages are continuously intermingled. Gauss wrote his most important papers in Latin, and his lesser works in German.

Nikolai Ivanovich Lobachevsky (1792-1856)

János Bolyai (1802-1860)

Carl Freiherr von Rokitansky [1804-1878]. Austrian pathologist.

Benjamin Disraeli, Earl of Beaconsfield (1804-1881). Conservative British Prime Minister during the Victorian Era. "There are lies, damn lies, and statistics." It is not an accident that statistics developed in Great Britain, and that the world's best statisticians still live and work there. Great Britain is an island nation, and has always made its national livelihood from maritime trade. Ships at sea, like dice at a gaming table, are subject to chance occurrences. In his career, Disraeli must have seen more than his share of deceptive statistics.

Sir James Paget [1814-1899]. British surgeon and pathologist.

George Boole (1815-1864)

Weierstrass (1815-1897).

Eduard Heine (1821-1881)

Rudolph Ludwig Karl Virchow [1821-1902]. German pathologist.

Gregor Mendel [1822-1884]. Czech-German geneticist, who first discerned the principles of inheritance, from experiments on pea-plants: Law of Recessive Inheritance. Law of Segregation. Sir Ronald A. Fisher [1890-1962] later demonstrated statistically that Mendel had probably fudged his data a little bit. Copies of the paper (English and the original German) are available on the Internet.

Sir Francis Galton [1822-1911]. Nineteenth century British statistician and biologist, who studied the statistical behavior of inherited traits in human populations. Galton's name is (falsely) associated with racist doctrines common in 19th century Britain, regarding the supposed genetic superiority of the British people.

Samuel Clemens (Mark Twain) [1835-1910]. American humorist. Very few professions were spared from Mark Twain's acerbic wit, including statisticians. In his book, Life on the Mississippi, Twain speculates on the fact that the Mississippi River is continually becoming shorter (because stagnant ox-bow lakes in the Mississippi delta become truncated). According to statisticians (using linear regression methods, see below), Twain says, during the Roman Empire, New Orleans must have been as distant from St. Louis as it was from the moon; whereas in the 21st century, we can expect that New Orleans will become a suburb of St. Louis.
The Mississippi River begins in St. Louis at the confluence of the .... rivers, and ends south of New Orleans, where the Mississippi River empties into the Gulf of Mexico. The flow of the Mississippi River slows down near its termination, so that the river becomes very tortuous (twisted) near its mouth, the Mississippi Delta. The twist in the river forms a so-called oxbow lake. Everytime this happens, the length of the entire Mississippi river shortens by a few miles, so that during Mark Twain's lifetime, The Mississippi River lost several hundred miles of its entire length. Using linear regression analysis, you would predict that the Mississippi River was tens of thousands of miles long during the Roman Empire, but would become only several miles south of New Orleans in the next millennium. Of course, statisticians..........

Gaston Darboux (1842-1917)

Elie Metchnikoff [1845-1916]. Russian biologist and immunologist.

Georg Ferdinand Ludwig Philipp Cantor (1845-1918).

Oliver Heaviside (1850-1925). The HEAVISIDE FUNCTION is a function, h(x), whose value is zero for x < 0 and one for x > 0. The first derivative (actually, all derivatives) of h(x) is zero everywhere, except at x = 0, where no derivative exists.

Thomas Joannes Stieltjes (1856-1895)

Karl Pearson (1857-1936). British statistician, who introduced the correlation coefficient, or Pearson's r. Father of Egon S. Pearson (1895-1980), another twentieth century statistical giant.

David Hilbert (1862-1943)

James Ewing [1866-1943]. American pathologist.

Karl Landsteiner (1868-1943). Winner of the 1930 Nobel Prize in Physiology or Medicine, for his discovery of blood groups.

Émile Borel (1871-1956)

Bertrand Russell (1872-1970),

Henri Lebesgue (1875-1941)

William S. Gossett (Student) [1876-1937]. An employee of the Guinness Brewery in Dublin, Ireland, who wrote the ground-breaking papers in the British journal, Nature, about the Student t test. Gossett was a student of Karl Pearson (1857-1936), but because Gossett did his work as an employee for a commercial firm, he concealed his identity. His papers were signed, simply, Student. The Guinness Book of World Records was written by the Guinness Brewery as an aid to settle arguments in British bars where Guinness products were served.

Jan Lukasiewicz (1878-1956)

Albert Einstein (1879-1955). Swiss-American Physicist, and winner of the 1921 Nobel Prize in Physics. Einstein's theory of relativity revolutionized our concepts of space and time in physics, but Einstein was always suspicious of non-deterministic, i.e., probabilistic, methods to describe physical observations. His famous statement, in a letter to Max Born: Der Herrgott würfelt nicht (German: The Lord God does not play dice.) I have heard four interpretations of this famous saying. First, that Einstein deeply believed that every move made by the Lord God was planned and pre-determined. Not a swallow falls from the sky without knowledge of the Lord God. Second, that Einstein did not understand probability theory very well, and was trying to rescue physics from the evils of quantum mechanics, a probabilistic-statistical theory of physics invented by Erwin Schrödinger (1887-1961) in the 1930s. (Einstein was an extremely smart man; this is an unlikely interpretation.) Third, that Einstein lined up with the 19th-20th century anti-gambling activists in Europe and North America, who believe that games of chance are morally wrong. Fourth, that Einstein, not a man with a small ego, was giving marching orders to the Lord God.

John Maynard Keynes (1883-1946). "In the long run, we're all dead." British economist, who developed concepts of national fiscal and monetary policy. Many economic theories distinguish between short-run and long-run processes, without really specifying how long is long-run. This quote is Keynes's ridicule of this particular paradox of academic economics.

George N. Papanicolaou (1883-1962). Inventor of the gynecologic cytology screening test for cervical cancer. Saved more lives than any other discovery in anatomic pathology in the twentieth century. Erwin Schrödinger. (1887-1961) German physicist. Inventor of Quantum mechanics. Co-winner of 193x Nobel Prize in physics, with Paul Dirac.

Sir Ronald Aylmer Fisher [1890-1962]. Greatest British statistician of the twentieth century. Sir Ronald corrected a small error in a formula for variance that had originally been promulgated by Karl Pearson [1857-1936]. Fisher proved that the correct formula for the sample variance is: s² = (∑ⁿ_i=1 (x_i) - x)²)/(n-1), not s² = (∑ⁿ_i=1 (x_i) - x)²)/n, as Pearson had thought.
Sir Ronald was the scientist who demonstrated statistically that Mendel had probably fudged his data.
The F-test for the analysis of variance is named in honor of Fisher.
However, Sir Ronald sold out to the tobacco industry. When the reports first emerged that tobacco use was bad for your health, Fisher defended the tobacco industry by asserting that the cause-effect relationship was not conclusively demonstrated. Sir Ronald developed the concept of CONFOUNDING, in which he argued that tobacco users might have some other mysterious quality that caused them to develop tobacco-related illnesses, apart from the tobacco use. Sir Ronald's prominence in the field of statistics helped the tobacco industry hide from its responsibilities for a number of years. Sir Ronald's assertion was eventually rebuffed by the fact that tobacco users who quit experienced subsequent decrease in tobacco-related illnesses.

Jerzy Neyman (1894-1981). Polish-American statistician, who developed a method for assessing the robustness of a statistical formula, the Neyman-Pearson Theorem. Neyman was a playful genius, who called the null hypothesis, "the devil". I once attended a lecture that he gave at North Carolina State University Department of Statistics, where I received my PhD. Egon S. Pearson (1895-1980). Twentieth century statistical giant. Co-author of the Neyman-Pearson condition in statistics.

Joseph Berkson [1899-1982]. British statistician. Berkson's Paradox. In short, if you don't die of one thing, you'll die of another. For example, if you collect a sample of autopsied patients who died of cancer, you will discover that a lower proportion of these patients have significant atherosclerosis (hardening of the arteries, which can lead to heart disease, stroke, kidney failure, etc.) than the proportion of atherosclerotic patients in the general population. Since cancer and atherosclerosis can both lead to death, the lower prevalence of atherosclerosis among cancer patients is explained by the fact that cancer patients die of their cancer before they acquire significant atherosclerosis. Before Berkson made this observation, there was a lot of nonsense in the medical literature that cancer somehow protected you against atherosclerosis, and therefore it was somehow good for your heart if you got cancer.

Paul Adrien Maurice Dirac, 1902-1984, Nobel Prize for Physics, 1933. The so-called DIRAC DELTA FUNCTION is a generalized function that behaves everywhere like the first derivative of h(x), the Heaviside function.

Aleksander Nikolaevich Kolmogorov (1903-1987). Russian statistician and mathematician, who introduced many non-parametric methods in statistics, including the Kolmogorov-Smirnov test.

John von Neumann (1903-1957) Hungarian-born American Mathematician who designed the first digital computer, ENIAC.

Kurt Gödel (1906-1978)

Lotfi A. Zadeh (1912-).

Stanislaw Ulam (1909-1981).

Jan Mikusiński [1913-1987]. Polish mathematician .............

Paul Joseph Cohen (1934-, Fields Medal, 1966)

Sir Andrew Wiles, mathematician from Cambridge and Princeton, and his student, Dr. Richard Taylor, finally proved Fermat's Last Theorem this theorem in the early 1990s, three-and-a-half centuries, after it had tormented the minds of virtually every great mathematician living during those years. The Wiles/Taylor proof was not simple, and we may never know whether Fermat had a correct proof of the assertion. Leonhard Euler (1707-1783) is said to have paid to have Fermat's literary effects searched after his death, in a vain effort to find Fermat's proof.

ARITHMETIC. Mathematical study of the fundamental numbering system, including natural numbers (1, 2, 3, ...), integers (..., -3, -2, -1, 0, 1, 2, 3, ...), rational numbers, real numbers, and complex numbers. Number theory is devoted largely to the study of prime numbers (q.v.).

GEOMETRY. Mathematical study of spatial relations.

ALGEBRA. Mathematical study of the operations of addition and multiplication, and related operations.

ANALYSIS. Mathematical study of limits and infinitesimals.

FOUNDATIONS OF MATHEMATICS. Mathematical study of sets and logic.

NUMBERS/NUMERALS. A NUMBER is an abstract concept, that includes all instances of anything that can be counted. That is, the number one, or one-ness, includes every occurrence of things that are one; number two, or two-ness, includes every occurrence of things that are two or paired; number three, or three-ness, includes every occurrence of things that are three, etc. r₁ + r₂ + r₃ +... diverges.

A NUMERAL is the physical representation of a number, which is different in different cultures. All modern cultures use so-called ARABIC NUMERALS, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,..., although these numerals actually appear as 1=١, 2=٢, 3=٣, 4=٤, 5=٥, 6=٦, 7=٧, 8=٨, 9=٩, in Arabic script. Ancient Chinese/Far-Eastern numerals are: 1=一, 2=二, 3=三, 4=..., 5=五, 10=十. The ancient Phoenicians, Hebrews, and Greeks reused their alphabets as numerals. Thus, the ancient Greek numerals, seen in the works of Euclid and Archimedes (287-212 BC), are: 1=α, 2=β, 3=γ, 4=δ, 5=ε, 6=ζ, 7=η, 8=θ, 9=ι, 10=κ. The ancient Roman numerals, of course, are: 1=Ⅰ, 2=Ⅱ, 3=Ⅲ, 4=Ⅳ, 5=Ⅴ, 6=Ⅵ, 7=Ⅶ, 8=Ⅷ, 9=Ⅸ, 10=Ⅹ. The Hindu/Sanskrit numerals are: 0=०, 1=१, 2=२, 3=३, 4=४, 5=५, 6=६, 7=७, 8=८, 9=९. The Gujarati numerals are: 0=૦, 1=૧, 2=૨, 3=૩, 4=૪, 5=૫, 6=૬, 7=૭, 8=૮, 9=૯. However it is represented, whether numeral 3, ٣, 三, ३, ૩, or Ⅲ, all represent the same number, namely three. The classic philosophical work, Principia Mathematica, by Alfred North Whitehead and Bertrand Russell, devotes 345 pages to the number three.

If you think of mathematical numbers as being merely an academic abstraction, consider this: EVERYTHING in the universe that occurs as one (e.g., one partridge, one pear tree) belongs to the number one; everything that occurs as two (e.g., two turtle doves, two eyes, two hands) belongs to the number two, etc. With this philosophical definition, mathematicians have seized ownership of everything in the universe that can be counted.

Concepts such as number and numeral, which are so obvious to those of us living in the twenty-first century, were great mysteries to the ancient peoples. The ancient Greeks had no zero, and had a great deal of difficulty with one. After that, they were OK up to about a million.... No ancient culture (Mediterranean, Indian, Chinese) had a concept or name for zero, and the Greeks abhorred zero. (See [Seife, 2000].) Zero was introduced in eighth century India by Brahmagupta.

NATURAL NUMBERS. The counting numbers, 1, 2, 3, 4, 5, 6, ..., known to ancient cultures.

ORDINAL NUMBERS. [Latin: ordo = order.] Numbers in order: 1st, 2nd, 3rd, 4th, 5th, 6th, .... There is nothing between a pair of consecutive ordinal numbers, e.g., there is no one-and-a-halfth.

CARDINAL NUMBER. [Latin: cardo = hinge or linchpin. Signifying great importance, since a door without its hinge will not function as a door. A cardinal in the Roman Catholic Church is a priest of great importance.] Numbers of magnitude: 1, 2, 3, 4, 5, 6, .... Between any two cardinal numbers, there is always another cardinal number.

PRIME NUMBERS. A PRIME NUMBER is a whole number (i.e., natural number) divisible without remainder only by itself and 1. The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, ...,

INTEGERS. The natural numbers, the negative natural numbers, and zero: ..., -3, -2, -1, 0, 1, 2, 3, ....

RATIONAL NUMBERS. The values of the ratios of two integers, i.e., integers divided by integers.

REAL NUMBERS. Cuts formed from the ordering of rational numbers.

COMPLEX NUMBERS. Symbolized as (x+iy), where x and y are real, and are formally added and multiplied, with i²=-1.

TRIANGLE. A three-sided, planar figure, consisting of three lines (EDGES), connected at three points (VERTICES). Each vertex lies opposite of an edge; and each edge lies opposite of an vertex. In this report, one edge of the triangle is placed horizontally, called the BASE, b. The vertex opposite to the base is the APEX. A line that is dropped from the apex to the line through the base is the HEIGHT, h. The rules of plane geometry were known on an intuitive basis by the ancient Egyptians (second millennium B.C.), and were systematized and summarized by Euclid (230-275 B.C.)

ISOSCELES TRIANGLE. A triangle with two equal edges. By convention, the unequal edge is placed at the base. Proof that the base angles of an isosceles triangle are equal is called PONS ASINORUM [Latin: bridge of asses], and was one of the first successes attributed to computer programs for discovering mathematical proofs.

FRACTAL. Fractals are endlessly repeated geometric figures, as for example, Sierpinski's triangle. (See [Lauwerier, 1995].) The barn-painting problem is an example of a fractal.

SIERPINSKI'S TRIANGLE. "...Sierpinski's Triangle (or gasket), after the Polish mathematician, Waclaw Sierpinski, who described some of its interesting properties in 1916. Among these is its fractal or self-similar character. The large blue triangle consists of three smaller blue triangles, each of which itself consists of three smaller blue triangles, each of which ..., a process of subdivision which could, with adequate screen resolution, be seen to continue indefinitely. Fractals and self-similarity are of considerable interest in their own right, but our interest here is in how to construct Sierpinski's triangle. One way to do so is to inscribe a second triangle inside the original one, by joining the midpoints of the three sides, and then repeat the process for the resulting three outer triangles, for the three outer triangles that result from that, and so forth...." (See [Sierpinski, 1916].)

FAT ISOSCELES TRIANGLE. An isosceles triangle in which the base, b, is much greater than the height, h.

SLIM ISOSCELES TRIANGLE. An isosceles triangle in which the height, h, is much greater than the base, b.

MAXIMUM ISOSCELES TRIANGLE. An isosceles triangle in which the height, h, and the base, b are balanced, to give maximum area, when placed within an inverted triangle.

LIMIT, L. Traditionally, that number, L, on the real line, to which a sequence, ℓ₁, ℓ₂, ℓ₃, ... ℓ_n, ..., approaches, as n → ∞. For a finite limit, ℓ_n → L, the x_n get "ever so close to" L. For an infinite limit, the ℓ_n → ∞, i.e., the ℓ_n "gallop off to" ∞.
Weierstrass interpretation. For a finite limit, ℓ_n → L, for every ε>0, there exists an N>0 such that for every n>N, |ℓ_n-L|<ε.
Weierstrass interpretation. For an infinite limit, ℓ_n → ∞ and for every n>N, ℓ_n > ∞.

CALCULUS. [Latin: stone, used in an abacus, for calculation.] Any method of computation. Calculus has come to mean differential and integral calculus.

INFINITY, ∞. The entity greater than any real number. That is, for every N, N<∞.

INTEGRAL, ∫. The calculation of area under a function, f(x), between the range where x=a and x=b, denoted _a∫^b f(x)dx.

71. Integration is performed by making smaller and smaller areas under the function, that get closer and closer to the actual curvature of the function.

RIEMANN INTEGRAL, ∫. An integral where the limiting areas are vertical columns.

71.
WEIERSTRASS [1815-1897] INTERPRETATION.

LEBESGUE INTEGRAL, ∫. An integral where the limiting areas are horizontal slabs.

BOCHNER INTEGRAL, ∫. An integral where the limiting areas are bricks.

PAINTING INTEGRAL, ∫. An integral where the limiting areas are triangles or arbitrary

SUMMATION, ∑. For a sequence of values, x₁, x₂, x₃, ..., x_n, the notation ∑_i=1ⁿ x_i denotes x₁+x₂+x₃+...+x_n, that is, the sum of x_i from i=1 to i=n. For example, the notation ∑_i=1⁵ 1/n_i denotes 1 + 1/2 + 1/3 + 1/4 + 1/5 = 1 + 0.5 + 0.33333 + 0.25 + 0.2 = 2.28333.... These are the first five terms of the harmonic series.

PRODUCT, Π. For a sequence of values, x₁, x₂, x₃, ..., x_n, the notation Π_i=1ⁿ x_i denotes x₁×x₂×x₃× ...×x_n, that is, the product of x_i from i=1 to i=n. For example, the notation Π_i=1⁵ 1/n_i denotes 1 × 1/2 × 1/3 × 1/4 × 1/5 = 1 × 0.5 × 0.33333 × 0.25 × 0.2 = 0.0083332. These are the first five terms of the harmonic product.

WEIERSTRASS [1815-1897] PARADIGM. Statements about differential and integral calculus of the form: "for every ε>0, however small, there exists a δ>0, however small, or an N>0, however large, such that...". This paradigm replaces the otherwise sloppy concepts of "getting ever so close to a limit" and "galloping off to infinity". See Courant, Robbins, Stewart. What is Mathematics?

DERIVATIVE/DIFFERENTIAL. The derivative of a function at point x, denoted df(x)/dx, is the tangent of the function at point x.

HARMONIC SERIES. ∑_n=1^∞ n^-1 = (1/2) + (1/3) + (1/4) + (1/5) + = ∞. Yes, amazingly, the harmonic series does not converge! A nice, easy proof of this assertion is included in: Derbyshire J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Plume Books. 2004. ISBN: 0452285259.

GEOMETRIC SERIES. ∑_n=1^∞ 2^-n = (1/2) + (1/4) + (1/8) + (1/16) + .... = 1. That is, the geometric series converges to one.

CONVERGENCE. A convergent sequence is an infinite sequence with a finite limit, L.

DIVERGENCE. A divergent sequence is an infinite sequence that does not converge to a finite limit, L. The sequence may either gallop off to infinity, such as 1, 2, 3, 4, 5, 6, 7, 8,..., or else vacillate indefinitely so that it never converges, such as +1, -1, +1, -1, +1, -1, +1, -1,....

RIEMANN ZETA FUNCTION, ζ(s) = ∑_n=1^∞ n^-s.
Note that when s=1, then ζ(1) is exactly the harmonic series.

RIEMANN HYPOTHESIS. All the non-trivial roots of the Riemann zeta function have real part ½. (See [Derbyshire, 2004]).

CRYPTOGRAPHY. The study of encryption of documents. (See [Schneier, 1996]).

CRYPTANALYSIS. The study of decryption of documents.

ZEROS/ROOTS. A zero/root, r, of a function, f(x), is a value of x where the function has value zero.

FRACTION, F. The final fraction of the barn painted in the barn-painting program.

PRODUCT, P. The infinite product giving the final unpainted portion of the barn.

e Named after Leonhard Euler, e is the upper limit of the integral, ₁∫^e x^-1dx = 1, 2.718281828459045..... Alternatively defined as the limit, as n approaches infinity, of (1+n^-1)ⁿ, i.e., e = Lim_n⇒∞(1+n^-1)ⁿ.
See Compound interest.

COMPOUND INTEREST. Compound interest.....

π. Pi (pronounced "pie"), π, is the ratio of the circumference of a circle to its radius, or, the value of the circumference of the unit circle, 3.1415927..... See Beckmann P. A History of Pi. 1967.

INFINITE SEQUENCE. An infinite sequence of values, x₁, x₂, x₃, .... Both the geometric sequence, (1/2), (1/4), (1/8), (1/16), ... 2^-n,..., and the harmonic sequence, (1/2), (1/3), (1/4), (1/5), (1/6), ... 1^-n,..., are infinite sequences that converge to zero.

INFINITE SERIES. An infinite sum of values, x₁ + x₂ + x₃ + ... = ∑_i=1^∞ x_i. The geometric series, (1/2), (1/4), (1/8), (1/16), ... 2^-n,..., is an infinite series that converges to 1. The harmonic series, (1/2), (1/3), (1/4), (1/5), (1/6), ... n^-1,..., is an infinite series that does not converge.

INFINITE PRODUCT. An infinite product of values, x₁ × x₂ × x₃ × ... = Π_i=1^∞ x_i. The geometric product, (1/2), (1/4), (1/8), (1/16), ... 2^-n,..., and the harmonic product, (1/2), (1/3), (1/4), (1/5), (1/6), ... n^-1,..., are infinite products that converge to 0.

MATHEMATICAL RELATION, R. A mathematical relation, R, is a collection of ordered-pairs, (x,y) ∈ R, also denoted xRy, where x is the argument and y is the value. A relation in which, for each argument x, there exists at most one value y is a function. The DOMAIN OF R is the set of values taken by the x's, i.e., the set of x for which there exists some y such that xRy. The RANGE OF R is the set of values taken by the y's, i.e. the set of y for which there exists some x such that xRy.

MATHEMATICAL FUNCTION, f. A mathematical function, f, is a collection of ordered-pairs, (x,y) ∈ f, also denoted f(x)=y, where x is the argument and y is the value, and for each argument, x there exists at most one value, y. The DOMAIN OF f is the set of values taken by the x's, i.e., is the set of x for which y=f(x) exists. The RANGE OF R is the set of values taken by the y's, i.e., the set of y for which there exists some x such that y=f(x) exists.

r₁, r₂, r₃, ..., r_n, .... The fraction of the unpainted portion of the barn that is newly painted at step n.

X-AXIS, ABSCISSA, HORIZONTAL AXIS. The horizontal axis on the xy-plane.

Y-AXIS, ORDINATE, VERTICAL AXIS. The vertical axis on the xy-plane.

FORMAL LOGIC. Developed by 19th century British mathematician, George Boole (1815-1864). Logic-operations bear a superficial resemblance to addition in arithmetic, except instead of numbers, for arguments and values, there are true-false statements. Instead of addition, there are LOGIC-OPERATORS. The commonly-used logic-operators are: NOT, AND, INCLUSIVE-OR, EXCLUSIVE-OR, IMPLIES. The operator NOT is a unary operator, because it takes one argument, for example, NOT X, denoted ~X, where X is a true-false statement, and ~X is true if and only if X is false. The other operators are binary operators, because they take two arguments, for example, X AND Y, denoted X&Y (true if X and Y are both true); X INCLUSIVE-OR Y, denoted X ior Y (true if either X and Y or both are true); X EXCLUSIVE-OR Y, denoted X xor Y (true if either X and Y but not both are true); X IMPLIES Y, denoted X ⇒ Y (true if either X is false or Y is true).

Summary of eight commutative, binary operators are given in the table:

.	TT	TF	FT	FF
VERUM	TRUE	TRUE	TRUE	TRUE
IOR	TRUE	TRUE	TRUE	FALSE
NXOR	TRUE	FALSE	FALSE	TRUE
AND	TRUE	FALSE	FALSE	FALSE
NAND	FALSE	TRUE	TRUE	TRUE
XOR	FALSE	TRUE	TRUE	FALSE
NIOR	FALSE	FALSE	FALSE	TRUE
FALSUM	FALSE	FALSE	FALSE	FALSE

The complete set of eight commutative, binary operators are given in the tables:

Logical-and: X AND Y is true if and only if X and Y are both true:

Operation	Arg #1	Arg #2	Value
AND	TRUE	TRUE	TRUE
AND	TRUE	FALSE	FALSE
AND	FALSE	FALSE	FALSE
AND	FALSE	FALSE	FALSE

Logical-not-and: X NAND Y is true if and only if X and Y are both false:

Operation	Arg #1	Arg #2	Value
NAND	TRUE	TRUE	FALSE
NAND	TRUE	FALSE	TRUE
NAND	FALSE	FALSE	TRUE
NAND	FALSE	FALSE	TRUE

Inclusive-or: X IOR Y is true if and only if either X or Y or both are true:

Operation	Arg #1	Arg #2	Value
IOR	TRUE	TRUE	TRUE
IOR	TRUE	FALSE	TRUE
IOR	FALSE	TRUE	TRUE
IOR	FALSE	FALSE	FALSE

Not-or: X NIOR Y is true if and only if either X or Y or both are false:

Operation	Arg #1	Arg #2	Value
NIOR	TRUE	TRUE	FALSE
NIOR	TRUE	FALSE	FALSE
NIOR	FALSE	TRUE	FALSE
NIOR	FALSE	FALSE	TRUE

Exclusive-or: X XOR Y is true if and only if either X or Y but not both are true:

Operation	Arg #1	Arg #2	Value
XOR	TRUE	TRUE	FALSE
XOR	TRUE	FALSE	TRUE
XOR	FALSE	TRUE	TRUE
XOR	FALSE	FALSE	FALSE

Verum: X VERUM Y is always true:

Operation	Arg #1	Arg #2	Value
VERUM	TRUE	TRUE	TRUE
VERUM	TRUE	FALSE	TRUE
VERUM	FALSE	TRUE	TRUE
VERUM	FALSE	FALSE	TRUE

Falsum: X FALSUM Y is always false:

Operation	Arg #1	Arg #2	Value

FALSUM	TRUE	TRUE	FALSE
FALSUM	TRUE	FALSE	FALSE
FALSUM	FALSE	TRUE	FALSE
FALSUM	FALSE	FALSE	FALSE

Non-commutative:

Operation	Arg #1	Arg #2	Value
IMPLIES	TRUE	TRUE	TRUE
IMPLIES	TRUE	FALSE	FALSE
IMPLIES	FALSE	TRUE	TRUE
IMPLIES	FALSE	FALSE	TRUE

The most interesting of these operators is NAND, also known as Sheffer-δ (pronounced Sheffer-dee). All other classical-logic-operators may be built up from δ, as follows:

NOT-X	XδX
X & Y	(XδY)δ(XδY)
X IOR Y	*{[(XδX)δ(YδY)]δ[(XδX)δ(YδY)]} δ{(same)}*

POLISH LOGIC.

Invented by Jan Łukasiewicz [1878-1956] in the early twentieth century. This is parenthesis-free notation for logic, which makes it easier to set up and understand certain proofs. Instead of X IMPLIES Y, one writes, IMPLIES X Y, The trick applies equally well to arithmetic: instead of X PLUS Y, one writes, PLUS X Y. So-called Polish-notation was used for complicated calculations in late twentieth-century hand-held calculators.

NANDSET ADDITION.

SPREADSHEET LOGIC.

The intellectual content of pathology, beyond image-recognition, may be described in hierarchical tables. This development in pathology doctrine/pedagogy is apparent in academic presentations using media such as PowerPoint; and in textbooks whose content is largely contained in numerous diagrams and charts (Bostwick, Sinard, Barnhill, Kao). In a simple example, all anatomy is either gross anatomy or microanatomy; and all anatomy may be crudely classified as either normal, degenerative, or proliferative, as follows:

 Anatomy.
    Gross anatomy.
    Microanatomy.
 Anatomy.
    Normal anatomy.
    Degenerative.
       Ischemic.
       Inflammatory.
       Infectious.
       Systemic.
       Metabolic.
    Proliferative.
       Hyperplasia.
          Reactive hyperplasia.
          Idiopathic hyperplasia.
       Benign proliferation.
       Dysplasia.
          Low-grade.
          High-grade.
       Low malignant potential.
       Malignant proliferation.
          Low-grade.
          High-grade.

This information may be organized into a rectangular table, such as Microsoft® Excel:

.	.	.	.	.	.
İ	.	.	.	.	.
.	Anatomy.	.	.	.	.
.	.	Gross Anatomy.	.	.	.
.	.	Microanatomy.	.	.	.
.	Anatomy.	.	.	.	.
.	.	Normal anatomy.	.	.	.
.	.	Degenerative.	.	.	.
.	.	.	Ischemic.	.	.
.	.	.	Inflammatory.	.	.
.	.	.	Infectious.	.	.
.	.	.	Systemic.	.	.
.	.	.	Metabolic.	.	.
.	.	Proliferative.	.	.	.
.	.	.	Hyperplasia..	.	.
.	.	.	.	Reactive hyperplasia..	.
.	.	.	.	Idiopathic hyperplasia.	.
.	.	.	Benign proliferation.	.	.
.	.	.	Dysplasia.	.	.
.	.	.	.	Low-grade dysplasia.	.
.	.	.	.	High-grade dysplasia.	.
.	.	.	Low malignant potential.	.	.
.	.	.	Malignant proliferation.	.	.
.	.	.	.	Low-grade malignant proliferation.	.
.	.	.	.	High-grade malignant proliferation.	.

With an additional mathematical twist, this table can be viewed as an expression in formal logic. We place the origin, Į ("importance", because every patient is important), in the upper-left corner. Cell İ is the ultimate parent, and its first daughter, D₁, is placed one-down-one-right. Multiple daughters, D₁, D₂, ..., D_n, are placed in the same column below the first daughter. In general, if a parent, P, has daughters, D₁, D₂, ..., D_n, then the corresponding logic expression is: P implies D₁ ior D₂ ior ... ior D_n.

According to the HIERARCHY REVERSAL THEOREM:

.	.	.	.
İ	.	.	.
.	İ	.	.
.		A	.
.		B	.
.	İ	.	.
.		C	.
.		D	.

is equivalent to:

.	.	.	.
İ	.	.	.
.	A	.	.
.		C	.
.		D	.
.	B	.	.
.		C	.
.		D	.

This is a powerful result, that allows one to compact hierarchies under certain circumstances.

HEINE-BOREL THEOREM. The missed points suggest an amusing fly-swatting game. Imagine that there are many, many flies, all attracted to the centers of our filling circles. A single fly lands on the center of the unit circle prior to our placing it there. The unit circle becomes our fly-swatter, but the fly escapes just before we are able to swat it. The fly lands on one of the second-stage centers, and is joined by 2 other flies at the other 2nd-stage centers. These 3 flies then escape just before we are able to swat them with the 3rd-stage fly-swatter circles. They land on the 4th-stage centers, and are joined by 6 other flies. The game continues by the addition of more and more escaping flies. Those flies with homing instincts, who never leave a spike once entered, chart direct paths to missed points. Other flies chart indirect paths, or else continually flit about without landing anywhere.

In a similar vein, did you know that there is an article, I believe published in the American Rifleman (one of my late father's favorite magazines; he pointed out the article to me about 35 years ago when I was a biomath grad student), entitled something like "The Mathematical Theory of Wild Animal Capture", basically a take-off on the Heine-Borel Theorem, that every open cover of a closed, bounded set has a a finite subcover. I believe that Dr. Bob Ramsay had shown me a similar tongue-in-cheek article in an otherwise serious mathematics journal somewhat earlier. You have the wild animals in a closed, bounded hunting area, and you remove an infinite number of open sets until you are able to limit the wild animals within one of the finite number of sets that remain. My recollection is that the American Rifleman write-up was serious, a demonstration that animal capture (and, by extension, gun ownership) was a respectable discipline in mathematics. I have tried to locate the article, but so far no luck.

Heine-Borel is one of (classically) three manifestations of what is called a compact space: (1) In R_n, a compact space is a closed, bounded set. More generally, the other two are: (2) Heine-Borel (finite subcovering of every open covering); and (3) the "sequential version", due to Weierstrass, I think, namely, every (bounded) sequence has a convergent subsequence. I am using a stronger fact that a sequence of (bounded) telescoping (U_n ≽ U_n+1 ≽ ...) sequence of sets (they happen to be open, if you like) shrinking down (in size), must converge onto a point. This is saying, in effect, that the plane has NO HOLES, i.e., the set cannot shrink down to nothing! Of course, by just following the centers of the circles in a telescoping sequence of shrinking sets, is something like "sequential" compactness business. But we need to know more than the fact that a bounded sequence of points has a convergent subsequence. We have here, many convergent subsequences of many different points.

In the fly arena, my high-school geometry teacher, the late Mrs. Ruth H. Murphy, presented a sort of geodesic fly-swatting problem, where a fly could only escape death if it traveled the shortest possible distance between where it was now (and where the fly-swatter was aimed), and an escape-hatch on the other side of a rectangular room. The fly could only crawl along the surface of the room. I struggled with the problem a few hours that evening after school, and finally constructed a paper model of the rectangular room. By trial-and-error, I cut the edges of the room in various ways, and finally found the hypoteneuse of a triangle with the desired minimum distance. When you put the room back together, the fly's path was very angular and unintuitive. Barb, Geoff, Vince, and I visited Mrs. Murphy about twenty years later (she was in the same Detroit nursing as my great-aunt Carol), and Mrs. Murphy said that I was one of only a handful of students who had solved the problem in her 30+ years of teaching high-school geometry.

CHAPTER 24. APPENDIX H: SKIN CANCER MODEL.
by G. William Moore, MD, PhD.

Previous Page.
Return to Table of Contents.
The SKIN may be regarded, simply and abstractly, as a 2-layered chocolate layer-cake.

224. The cake-parts are: the EPIDERMIS (on the top); and the DERMIS (on the bottom). [Greek: επι = epi = on-top-of; δερμις = dermis = skin]. (See picture.)

48. The cake sits on top of underlying soft tissues (muscle, fibrous tissue, blood vessels, nerves, etc.). The middle filling is the boundary between the epidermis and dermis, the DERMAL-EPIDERMAL JUNCTION. The top frosting is the air. The epidermis, in turn, is comprised of four layers (from top to bottom): cornified layer; granular layer; prickle layer; and basal layer. All cell-division among epidermal skin-cells takes place in the basal layer. The dermis is comprised of three layers (from top to bottom): papillary dermis; reticular dermis; and deep dermis.

.	.	.
SKIN	.	.
.	EPIDERMIS.	.
.	.	Cornified Layer.
.	.	Granular Layer.
.	.	Prickle Layer.
.	.	Basal Layer.
.	DERMAL-EPIDERMAL JUNCTION.	.
.	DERMIS.	.
.	.	Papillary Dermis.
.	.	Reticular Dermis.
.	.	Deep Dermis.

Under ordinary circumstances, the dermis replaces itself by cell division very slowly; whereas the epidermis replaces itself in about one month, faster if the skin has been injured or irritated. The skin is the largest organ of the body, by weight or by volume. Skin, along with blood-forming-cells and the surface of the gastrointestinal tract, is the most rapidly growing tissue in the body. Because of its location, skin is the organ most vulnerable to external toxins/insults. For all these reasons, skin cancer is the most common class of cancers, affecting about 1% of the U. S. population each year. (Most cases arise on sun-exposed skin, and are obvious and curable.)

SKIN CANCERS AND PRECANCERS usually arise from the rapidly-growing cells in the basal layer of the epidermis. In skin-proliferative disorders, including cancers and precancers, the epidermal basal cells reproduce faster (or die more slowly, see Moore and Berman, 1991; Berman and Moore, 1992) than is necessary to replace skin cells lost by ordinary wear-and-tear. A simple model for skin cancer is just that: the "thermostat" for cell-death is altered. Evidence for this cell-death-model is the presence of "programmed-cell-death" in the form of apoptotic cells, or suicide cells, during ordinary cell proliferation. This programmed-cell-death process is altered or inhibited by genes that are present in some cancers (altered bcl-2 genes). The growth-rate of all known cancers can be explained/accounted-for by postulating a cell-death rate between 43% (very rapidly-growing tumors, such as Burkitt's lymphoma) and 49% (very indolent tumors, such as chronic lymphocytic leukemia). Slightly less than 50% death-rate after each cell-division would maintain the status-quo.

Since the epidermal basal-cells grow more rapidly than cells of the underlying dermis, what happens to the relative excess of basal-cells? Answer: they heap up as folded sheets of cells, either upward or downward. Skin proliferation and cancer growth may be either EXOPHYTIC (implantation outward); or ENDOPHYTIC (implantation inward) [Greek: φυτο = phyto = implant; εξο = exo = outward; ενδο = endo = inward]. (See picture.)
EXOPHYTIC:

49.
ENDOPHYTIC:

50.

.	.
SKIN CANCER GROWTH.	.
.	EXOPHYTIC.
.	ENDOPHYTIC.

If the basal-cells grow upward (exophytic growth), then they may cause a cosmetically unsightly bump on the skin, but no significant harm comes to the patient. If the basal-cells grow downward (endophytic growth), then the unchecked, rapidly-growing cells eventually invade deep tissues, and significantly harm or kill the patient, unless they are removed surgically. Interestingly, both the exophytic growths observed in humans (verruca vulgaris, squamous papilloma, verrucous carcinoma) and the endophytic growths (squamous cell carcinoma, basal cell carcinoma, malignant melanoma) resemble the spire-paintings described in Appendix B. Overpainting the Barn.
SQUAMOUS PAPILLOMA:

225.
OVERPAINTING THE BARN:

170.

Since the spire-painting-model permits generalized shapes (globs, triangles, bricks, profiles, ...), there must be some r₁, r₂, r₃, ... that correspond to the proliferative behavior of actual exophytic and endophytic growths, including all skin cancers and precancers.

Now let's take a leap: suppose that the genetic process that makes the skin grow (i.e., programmed-cell-death and all the rest) corresponds to the painting process described herein. If the series is convergent, then the barn/skin-cross-section never gets painted/filled-with-cells. If the series is divergent and the tower is finite, then the barn/skin-cross-section eventually gets painted/filled-with-cells, but not infinitely overpainted (benign tumor). If the series is divergent and the tower is infinite, then the barn/skin-cross-section eventually gets infinitely overpainted (malignant tumor). The malignant tumor corresponds to an infinite Tower of Babel (barn that reaches the sky, and beyond). In real life, of course, the malignant tumor, unchecked, eventually invades deep tissues, and kills the patient. In this model, unchecked malignancy is the barn-equivalent of the Lord God confusing construction-workers of the Tower of Babel with a multitude of tongues (Genesis 11:1-9). The special appeal of the painting model is twofold: the tower looks like an exo/endophytic cancer; and the values of r₁, r₂, r₃,... are arbitrary and variable, just as the papillae appear in an epidermal cancer.

GEOMETRIC PAPILLOMA:

322.
A GEOMETRIC PAPILLOMA has r₁=1, r₂=1/2, r₃=1/4, r₄=1/8, .... A geometric papilloma resembles a pedunculated polyp, with a relatively good prognosis.

HARMONIC PAPILLOMA:

321.
A HARMONIC PAPILLOMA has r₁=1, r₂=1/2, r₃=1/3, r₄=1/4, .... A harmonic papilloma resembles a sessile polyp, with a relatively poorer prognosis.

General Remarks. Over 90% of all malignancies (cancers) in the USA occur on flat tissue surfaces, called epithelium. Cancers of blood (lymphoma, leukemia), bone, soft tissues, and brain, are relatively infrequent by comparison. The largest epithelial surface over the human body is the epidermis of the skin. However, there are many other epithelial surfaces that undergo malignant degeneration: the respiratory tract (nose, throat, larynx, bronchi, and lungs); the gastrointestinal tract (mouth, esophagus, stomach, small and large bowel); the duct and gland surfaces of the urinary bladder, breast, liver, pancreas, female genitalia (vagina, uterus, Fallopian tube, ovary) and male genitalia (penis, prostate, testis). All these epithelial surfaces where cancer arises show either exophytic or, more commonly, endophytic growth of malignant cells, and downward growth of malignant epithelium eventually threatens the life of the patient. The current conventional wisdom is that an epithelial tumor becomes potentially lethal when it breaches its lower boundary, the basement membrane, and enters deeper tissues. Pathologists who diagnose tumors invest a great deal of effort and expense (additional tissue sections; special tissue stains) to determine whether a tumor on a particular patient has broken through this basement membrane boundary. What if there is another process that predicts invasive growth and malignancy? What if exo/endophytic growth is a painting problem? What if you could use the shape of the tumor at a given moment to predict whether the tumor-tower will be finite (benign) or infinite (malignant)? Hmmmm.

Dr. Moore received his Ph.D. in biomathematics in 1971, from North Carolina State University, Raleigh, NC; and his M.D. in 1976, from Wayne State University, Detroit, MI. He is a staff pathologist at the Baltimore Veterans Affairs Maryland Health Care System, associate professor of pathology at the University of Maryland School of Medicine, and visiting assistant professor of pathology at The Johns Hopkins Medical Institutions.

CHAPTER 20. AFTERWARD.

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This informal manuscript has taken you through a tour of human pathology and mathematics over the past two millennia, in 100 pages.

It has been my general observation that physicians hate mathematics, and took as little of it as they could get by with in their premedical training. Alarmingly, many physicians have a very casual knowledge of statistics, despite the cardinal importance of this subject in evaluating new clinical regimens.

Similarly, young mathematicians may regard biological subjects as "a lot of memorization", with no connection to deep principles that characterize physics and mathematics.

CHAPTER 21. APPENDIX A. SUMMARY OF DIAGNOSTIC HUMAN ANATOMIC PATHOLOGY.

Next Chapter.
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Normal anatomy.

1. Cardiovascular.
Heart.
Right Atrium.
Tricuspid valve.
Right Ventricle.
Pulmonic valve.
Left Atrium.
Mitral valve.
Left Ventricle.
Aorta.
Pulmonary Artery.
Middle-sized arteries.
Arterioles.
Capillaries.
Venules.
Middle-sized veins.
2. Respiratory.
Mouth.
Nasopharynx.
Larynx.
Trachea.
Bronchus.
Bronchiole.
Alveolus.
3. Gastrointestinal.
Tubular gastrointestinal tract.
Mouth.
Oropharynx.
Esophagus.
Cervical esophagus.
Mid-esophagus.
Lower esophagus.
Gastroesophageal junction.

Stomach.
Cardia.
Fundus.
Body.
Antrum.
Pylorus.

Small intestine.
Proximal duodenum.
Mid-duodenum.
Distal duodenum.
Jejunum.
Proximal ileum.
Mid-ileum.
Terminal ileum.

Large intestine.
Ileocecal valve.
Cecum.
Appendix.
Ascending Colon.
Hepatic Flexure.
Transverse Colon.
Splenic flexure.
Descending colon.
Sigmoid colon.
Rectum.
Anus.

Hepatobiliary system.
Liver.
Biliary tree.
Gallbladder.
Pancreas.

4. Genitourinary.

Common.
Kidney.
Ureter.
Urinary Bladder.

Female.
Uterine corpus:endometrium; myometrium; mixed mesodermal; gestational-trophoblastic.
Fallopian tube.
Ovary.
Uterine cervix.
Vagina.
Vulva.

Male.
Testis.
Prostate.
Penis.

5. Endocrine.
Thyroid.
Parathyroid.
Adrenal cortex.
Adrenal Medulla.
Anterior pituitary.
Posterior pituitary.
Pineal body.
6. Integumentary.
Inflammatory dermatoses.
Keratinocyte proliferation.
Appendageal proliferation.
Fibrous proliferation.
Melanocytic lesions.
7. Musculoskeletal.

8. Hemolymphatic.
Blood.
Lymph.
Lymph nodes.
Spleen.
9. Central Nervous System.
Glioma.
Glioblastoma multiforme.
Astrocytoma.
Oligodendroglioma.
Ependymoma.
Choroid plexus papilloma.
Mixed glioma.
Glioneuronal tumor.
Medulloblastoma.
Meningioma.
Ancillary neuronal tumor: neurilemoma; neurofibroma; craniopharyngioma; hemangioblastoma; pituitary adenoma; hematopoietic tumors.....
Pineal gland tumor: germ cell tumor; pinealoma.
CNS Metastasis.

CHAPTER 22. REFERENCES.

Next Chapter.
Previous Chapter.
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1. Human Pathology.
http://www.pathology.wisc.edu/pathcourses/path703/about/intro.html
http://www.pathology.wisc.edu/pathcourses/path703/

2. Mathematics.
http://mathworld.wolfram.com/

3. Kumar et al. Pathologic Basis of Disease.

4. Derbyshire J. Prime Obsession.

Kazarinoff ND.
Analytic Inequalities.
New York: Dover Publications. 2003;:.
ISBN: 0486432440, 96 pages.

Shields AL.
A note on invariant subspaces.
Michigan Math J. 1970;17(3):231-233.

Geller SA.
A Short History of the Autopsy. Chapter 2.
In: Collins KA, Hutchins G, eds. Autopsy Performance & Reporting. Second Edition.
Northfield, IL: College of American Pathologists. 2003;2:13-16.
ISBN 0-930304-78-0, 397 pages.

Collins KA, Hutchins G, eds.
Autopsy Performance & Reporting. Second Edition.
Northfield, IL: College of American Pathologists. 2003;2:.
ISBN 0-930304-78-0, 397 pages.

Moore GW.
Autopsy Reporting. Chapter 28.
In: Collins KA, Hutchins G, eds. Autopsy Performance & Reporting. Second Edition.
Northfield, IL: College of American Pathologists. 2003;2:265-274.
ISBN 0-930304-78-0, 397 pages.

Sadegh-zadeh K. Goodbye to the Aristotelian Weltanschauung.

Morgagni GB.

De Sedibus et Causis Morborum per anatomen indigatis. (Latin: On the sites and causes of diseases, as indicated by anatomy.)
1761;:.

Agnew RP.
Calculus. Analytic Geometry and Calculus, with Vectors.
New York: McGraw-Hill Book Company, Inc. 1962;:.
U.S. Library of Congress No: 61-18624, ISBN not stated, 738 pages.

Mayr E.
What evolution is.
New York: Basic Books. A member of the Perseus Books Group. 2001;:.
ISBN 0-465-04426-3, 318 pages.

Shakespeare W.
Hamlet.
Act 3, Scene 1.
"To be or not to be, that is the question...."

Al-Ubaydli M.
Free Software for Busy People.
Idiopathic Publishing; 2005;:.
ISBN 0-9544157-3-6, pages.
For further information:
http://www.freedomsoftware.info
To download a free copy in Adobe pdf format:
http://www.freedomsoftware.info/book.pdf
This is a fabulous book, easy to read, and well worth the read. Helps the average person understand the Free Software movement.

Brahmagupta. Brahmasphutasiddhanta.
[Sanskrit: The Opening of the Universe.]
628;:.

Brahmagupta. Khandakhadyaka.
665;:.
Second work on mathematics and astronomy.

Fibonacci.
Liber Abaci. (Latin: Book of the Abacus.)
1202;:.

Livio M.
The Golden Ratio.
2003;:.

Ulam S.
Adventures of a Mathematician.

Last updated: 11/20/2005, by G. William Moore, MD, PhD.

PICTURES, PATHOLOGY, AND MATHEMATICS: G. WILLIAM MOORE'S COMMENTARY ON RAIMOND A. STRUBLE'S INFINITE PRODUCTS. DRAFT COPY ONLY. 11/20/2005. http://www.infiniteproduct.info/struppma.htm © 2005, G. William Moore, MD, PhD. All Rights Reserved.

TABLE OF CONTENTS.

ABSTRACT.

PREFACE. AN APPRECIATION TO PROF. STRUBLE.

CHAPTER 1. INTRODUCTION.

CHAPTER 1. INTRODUCTION.

CHAPTER 10. SAMPLE CALCULATIONS.

CHAPTER 24. APPENDIX H: SKIN CANCER MODEL. by G. William Moore, MD, PhD.

CHAPTER 20. AFTERWARD.

CHAPTER 21. APPENDIX A. SUMMARY OF DIAGNOSTIC HUMAN ANATOMIC PATHOLOGY.

CHAPTER 22. REFERENCES.

PICTURES, PATHOLOGY, AND MATHEMATICS:
G. WILLIAM MOORE'S COMMENTARY ON
RAIMOND A. STRUBLE'S
INFINITE PRODUCTS.
DRAFT COPY ONLY.
11/20/2005.
http://www.infiniteproduct.info/struppma.htm
© 2005, G. William Moore, MD, PhD.
All Rights Reserved.

PREFACE. AN APPRECIATION TO
PROF. STRUBLE.

CHAPTER 24. APPENDIX H: SKIN CANCER MODEL.
by G. William Moore, MD, PhD.