INFINITE PAPILLOMA:
MODEL FOR UNBOUNDED TUMOR GROWTH.
DRAFT COPY ONLY.
11/20/2005.
G. William Moore, MD, PhD.[1,2,3]
Raimond A. Struble, PhD.[4]
Lawrence A. Brown, MD.[1,2]
Grace F. Kao, MD.[1,5]
Grover M. Hutchins, MD.[3]
http://www.medparse.com/infnpapl.htm
http://www.medparse.com/infnpapl.ppt

From the Pathology and Laboratory Medicine Service, Veterans Affairs Maryland Health Care System, Baltimore, Maryland [1]; Department of Pathology, University of Maryland Medical System, Baltimore, Maryland [2]; Department of Pathology, The Johns Hopkins Medical Institutions, Baltimore, Maryland [3]; Department of Mathematics, North Carolina State University, Raleigh, NC [4]; and Department of Dermatology, George Washington University School of Medicine, Washington, DC [5].

Send comments and correspondence to: George.Moore4@med.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

Context: Surface tumors in skin and mucus membrane are the most common human tumors, and typically arise as an upward, exophytic growth, or papilloma; or as a downward, endophytic growth, or acanthoma. In benign growth, such as wound-healing, proliferation stops after the injured surface tissue has been replaced. In malignant growth, proliferation continues indefinitely.

Technology: This report proposes a mathematical model, using infinite products and Lebesgue integration, in which papillae/acanthi seek to fill a potential volume, above/below the normal tissue surface.

Design: For a potential tumor volume normalized to 1, one considers the n-product, Pn=(1-r1)×(1-r2)× ... ×(1-rn), where each ri represents the fraction of remaining volume removed by the ith papilla/acanthus. The infinite product, P, is the limit of the n-products as n approaches infinity.

Results: Mathematically, an infinite number of papillae fill the potential volume if and only if the infinite sum, S=r1+r2+... is DIVERGENT, an apparently paradoxical mathematical result. It is proposed that a convergent series corresponds to benign proliferation; whereas a divergent series corresponds to malignancy. Thus, a malignancy keeps trying to fill the potential volume; whereas benign proliferation is satisfied and ceases after the volume is partially filled.

Conclusion: The theory is completely general, because the exact functional form or values of the ri are not specified by the model. Mathematical models can be used to propose alternatives to conventional wisdom in pathology, and explore general properties.

1. Surface tumors are the most common human tumors, including epithelial, mesothelial, endothelial tumors, in skin and mucus membrane, and accounting for over twenty million new cases per year worldwide.

2. Surface tumors typically arise as an upward (exophytic) growth, or papilloma; or as a downward (endophytic) growth, or acanthoma:
574.                                        575.
Exophytic growth/papilloma.                                                   Endophytic growth/acanthoma.

In benign growth, such as wound-healing, proliferation stops after the injured surface tissue has been replaced. In malignant growth, proliferation continues indefinitely.

3. We propose a mathematical model in which papillae (or acanthi) seek to fill a potential volume, or BILLBOARD, above (or below) the normal tissue surface. The actual height (or depth) of the tumor is the TOWER. We propose that there is TERMINAL VOLUME, wherein the tumor-cells know that they no longer wish to continue dividing. This is not a completely far-fetched idea, since epithelial cells involved in wound-healing seem to know when to stop; but malignant epithelial cells do not. Growth ceases either when the terminal volume is filled, or in the case of a malignancy, when the tumor invades a vital structure in the patient.

4. Let the terminal tumor volume be normalized to 1, without loss of generality. At the nth step in papilloma-growth, let the nth papilla/acanthus occupy rn of the volume remaining from the prior step, where P0=1. Then the volume of tumor remaining-to-be-filled after step n is the n-product, Pn, where Pn = (1-r1) × (1-r2)× ... × (1-rn)

In this drawing, the first (red) papilla consumes half, r1 = 1/2, of the terminal volume, starting at 1, i.e., P1 = (1-r1) = 1/2 is the volume as yet unfilled by first papilla/red-triangle:

563.


The second (green), third (blue), and fourth (yellow) papillae each consume 1/2 of the remaining terminal volume, i.e., r2 = 1/2, r3 = 1/2, r4 = 1/2, so that:
P2 = (1-r1) × (1-r2) = (1 - 1/2) × (1 - 1/2) = 1/4 = 0.25;
P3 = (1-r1) × (1-r2) × (1-r3) = (1 - 1/2) × (1 - 1/2) × (1 - 1/2) = 1/8 = 0.125;
P4 = (1-r1) × (1-r2) × (1-r3) × (1-r3) = (1 - 1/2) × (1 - 1/2) × (1 - 1/2) × (1 - 1/2) = 1/16 = 0.0625,...:

564. 565. 566.


It is apparent that this somewhat abstract-appearing papilloma, with a corresponding infinite series of
r1 + r2 + r3 + r4 + ... = ∞
will completely fill the terminal volume, i.e., P = P∞ = 1/2 × (1-1/2)∞ = 0:

What happens when the fraction, i.e., rn, is smaller than 1/2, say 1/4, but is still a constant, r>0, i.e., the terminal volume fills steadily but more slowly. Does the terminal volume still fill entirely? This time, r1 = 1/4, r2 = 1/4, r3 = 1/4, r4 = 1/4, so that: Pn = 1/4 × (1-r)n-1, and again, P = P∞ = 1/4 × (1-1/4)∞ = 0:

570. 571. 572. 573.


Clearly, the remaining volume, Pn, keeps on getting smaller, but does it ever reach zero? The answer for this example is YES!

What if the fraction, rn, is not constant, but rather, gets a little bit smaller at each step? This time, r1 = 1/2, r2 = 1/4, r3 = 1/8, r4 = 1/16, ..., the GEOMETRIC SERIES.

563. 567. 568. 569.


Clearly, the remaining volume, Pn, keeps on getting smaller, but does it ever reach zero? The answer for this example is NO!

5. At this point, things get tricky. What if the fraction, rn, is not constant, but rather, gets a little bit smaller at each step? The answer is: the terminal volume fills if and only if the corresponding INFINITE SERIES, S = r1 + r2 + r3 + ... DIVERGES.

For example, the HARMONIC SERIES, namely 1/2 + 1/3 + 1/4 + ..., DIVERGES, i.e., 1/2 + 1/3 + 1/4 + ... = ∞. That is, if you add up a large enough sequence of the fractions, 1/2, 1/3, 1/4, ..., you can always get a bigger sum than any a-priori number that you choose. The harmonic series creeps along slowly but inexorably:
S1 = 0.5.
S2 = 0.83333....
S3 = 1.08333....
S4 = 1.28333....
S5 = 1.5
S50 = 4.99205....
S500 = 6.79282....
S5,000 = 9.09451....
S50,000 = 11.39700....
S500,000 = 13.69958....
That is, the remaining volume, P, for this series becomes zero.

On the other hand, the GEOMETRIC SERIES, namely 1/2 + 1/4 + 1/8 + ..., CONVERGES, i.e., 1/2 + 1/4 + 1/8 + ... = 1. For this case, the remaining volume, P, for this series is: 0.288788112299715..... For negative powers of 3, i.e., 1/3 + 1/9 + 1/27 + ..., P = 0.560126094.... For negative powers of 4, i.e., 1/4 + 1/16 + 1/64 + ..., P = 0.688537557.... For a general evaluation process, see Struble [1].

6. The mathematical question is: do an infinite number of papillae fill the billboard? The reason why mathematicians care about this question is that finding the sum of the papillae in the terminal volume, which might be a relatively easy problem in addition, is equivalent to finding the terminal volume, which may be a much harder problem, and which is the job of integral calculus. If the volume is unknown but one can calculate the sum of the papillae, then one can calculate the terminal volume, BUT ONLY IF THE TERMINAL VOLUME IS FILLED.

Historically, the ancient Greeks (Archimedes (287BC-212BC)) first estimated the value of π (i.e., the ratio of circumference-to-diameter of a circle, 3.141592...) from the sum of ever-more-sided n-polygons, which are nothing but collections of triangles (with properties and area-formulas well-known to ancient Greek mathematicians).

7. For rn, defined as the infinite product equals zero if and only if the infinite sum, ∑ ri, diverges. Equivalently, the infinite product is greater than zero if and only if the infinite sum, ∑ ri, converges (Struble Theorem, [1]).

8. We propose that a CONVERGENT SERIES corresponds to BENIGN PROLIFERATION; whereas a DIVERGENT SERIES corresponds to MALIGNANCY. That is, a malignancy keeps trying to fill the potential volume, representing unbounded growth, constrained only if the tumor is treated or kills the patient. On the other hand, benign proliferation is SATISFIED, and ceases after the volume is partially filled.

9. A geometric series, such as r1=1/2, r2=1/4, r3=1/8, r4=1/16, ..., is convergent, and corresponds to a benign growth, where the cells are satisfied and do not attempt to fill the potential volume above the surface.

The harmonic series, namely r1=1/2, r2=1/3, r3=1/4, r4=1/5, ..., is divergent, and corresponds to a malignant growth, where the cells are "grabby" and attempt to fill the potential volume above the surface.

10. Morphologically, a geometric papilloma resembles a pedunculated polyp, with less malignant potential; whereas a harmonic papilloma resembles a sessile polyp, with greater malignant potential:

557.                                        556.
Geometric/pedunculated papilloma.                                                   Harmonic/sessile papilloma.

Unbounded growth results in an "infinite papilloma/acanthoma":


170.


11. The theory is completely general, because the exact form or values of the rn are not overspecified by the model. One is not obligated to assume that there is an underlying Gaussian, binomial, Poisson, or other statistical process. The only assumption is that the infinite series, ∑ rn, is either convergent or divergent.

12. Potential pitfalls. There is no such thing as infinity in biology, but infinity is a necessary part of the present mathematical argument. In particular, there is no way to determine the tail of an infinite sequence, based upon the first few terms in the sequence, but all you can really measure on an actual specimen are the leading terms of a sequence.

Infinity, however, is a commonly-used tool in the theory of physics and chemistry, so why not in biology? Actually, Malthusian growth is an example of infinity in biology, that is a very useful theoretical tool. Malignancy, if you will, is Malthusian growth at the cellular level. All one has to do is determine whether the first few papillae seem to fit into a geometric (convergent) or harmonic (divergent) series.

The fact that a population of persons can exhibit exponential growth was appreciated by Leonhard Euler(1748), who ridiculed the objections of those persons who "denied that the whole earth could be filled in a short time with inhabitants descended from a single man": "Quam ob causam maxime ridiculae sunt eorum incredulorum hominum objectiones, qui negant tam brevi temporis spatio ab uno homine universam terram incolis impleri potuisse." Furthermore, Thomas Malthus (1817) noted that despite a substantial probability of extinction for any particular ancestor in that population, the entire population can exhibit exponential growth.

13. Why are there no data in this manuscript? Actually, there is the general observation that sessile polyps have a worse behavior than pedunculated polyps. It seems like a waste of energy to collect data that would verify this commonplace finding. However, the primary purpose of this manuscript is to explore the general properties of a sequential, space-filling process, and relate these properties to properties of individual papillae. Suppose that cells in a papilla are genetically programmed to reproduce into a potential space, in a manner analogous to programmed cell death (apoptosis), which is already known to have implications for malignancy [8,9,10,11,12,13].

14. Mathematical theories can be used to propose alternatives to conventional wisdom in pathology, and explore their general properties. In the conventional model of cancer, invasion is heralded by the surface-tumor breaking through the basement membrane. In this model, tumor proliferation is a property of cells attempting to fill a potential volume.

15. Possible implications for diagnosis. Is there a geometric property of malignant tumors (other than inexorable growth) that could be used in diagnosis?

16. Possible implications for therapy. Struble's papers (1, 2) demonstrate that a SPACE-FILLING infinite product can be converted into a NON-SPACE-FILLING infinite product, merely by expanding the potential volume. Could this be a useful trick for cancer therapy? That is, is there a means by which one could fool the cancer-cells into believing that the potential volume is larger than it is, and thus convert a malignant tumor into a benign tumor?
REFERENCES.

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Can one do serious mathematics using pictures and calculus?
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9. Miettinen M.
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Excellent presentation of diagnostic soft tissue pathology for the practicing pathologist.

10. Miettinen M.
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In: Miettinen M, Diagnostic Soft Tissue Pathology. New York: Churchill Livingstone. 2003;:. ISBN 0-443-006611-6, 593 pages.
pp. 73-74:
"The bcl2 gene product is a 25-kd protein in the mitochondrial, microsomal, and some inner membranes. It has an apoptosis preventing function and has complex interactions with other apoptosis-modulating proteins (Hockenberry, 1995). This gene for bcl2 was originally known from follicular lymphoma, where it is overexpressed as a result as a result of the t(14;18) translocation, which causes juxtaposition of the bcl2 gene with the promoter of the immunoglobulin heavy chain gene (Tsujimoto, 1986). Bcl2 is constitutively expressed in many long-lived cell types, such as neurons (Lebrun, 1993).

"Of soft tissue tumors, bcl2 has been widely expressed in the tested tumors. Strongly positive are Kaposi sarcoma, GISTs, solitary fibrous tumor, synovial sarcoma (especially spindle cell components), whereas nodular fasciitis and desmoid and GI leiomyomas are negative. These findings may be of some differential diagnostic value (Suster, 1998, Miettinen, 1998).

"Although there are indications for the use of bcl2 as a prognostic/biologic potential marker for breast and some other carcinomas, no such applications have been validated for soft tissue tumors."

11. Hockenberry DM.
bcl-2, a novel regulator of cell death.
Bioessays. 1995;17:631-638.
(cited in: Miettinen M, Diagnostic Soft Tissue Pathology.)

12. Tsujimoto Y, Croce CM.
Analysis of the structure, transcripts, and protein products of bcl-2, the gene involved in human follicular lymphoma.
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13. LeBrun DP, Warnke RA, Cleary ML.
Expression of bcl-2 in fetal tissues suggests a role in morphogenesis.
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14. Suster S, Fisher C, Moran CA.
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(cited in: Miettinen M, Diagnostic Soft Tissue Pathology.)

15. Miettinen M, Sarlomo-Rikala M, Kovatich AJ.
Cell-type and tumor-type related bcl-2 reactivity in mesenchymal cells and soft tissue tumors.
Virchows Arch. 1998;433-255-260.
(cited in: Miettinen M, Diagnostic Soft Tissue Pathology.)

16. U. S. National Cancer Institute. U. S. National Institutes of Health.
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Cell growth simulations predicting polyclonal origins for 'monoclonal' tumors.
Cancer Lett. 1991 Nov;60(2):113-119.
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PubMed Entry
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Public-domain source code:
http://www.netautopsy.org/monoclon.htm#table1

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Spontaneous regression of residual tumour burden: prediction by Monte Carlo simulation.
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PMID: 1445794.
PubMed Entry
Full Text of Article:
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Berman JJ, Moore GW.
The role of cell death in the growth of preneoplastic lesions: a Monte Carlo simulation model.

Cell Prolif. 1992 Nov;25(6):549-557.
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PubMed Entry
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http://www.netautopsy.org/celdeath.htm

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e: The Story of a Number.
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NOTES.

1. Most common Human Tumors....
Skin tumors are the most common tumors, probably because of three factors:
1.1. Skin is the largest organ, by volume or by weight, of the body.
1.2. Skin is most exposed to environmental insults.
1.3. Skin growths are most easily recognized, and come early to medical attention.
There are one million new diagnoses of skin cancer annually in the USA, more than all other body-sites combined. The big four, besides skin, are: colon, breast, prostate, lung, all on the order of 100,000 new cases annually in the USA, and all predominantly epithelial (surface) tumors. Non-surface tumors (hematolymphatic, bone, and soft-tissue) are all comparatively uncommon. Extrapolating, based upon the world having 20x the population of the USA, then there are at least 20 million new surface-cancer cases annually in the world.

2. Exophytic/endophytic.... (Greek: εξο = exo = outward; ενδο = endo = inward; φυτων = phyton = implant).

3. TERMINAL VOLUME. In the model, papillae seek to fill a potential volume, or BILLBOARD; the height above the of the tumor is the TOWER; and there is TERMINAL VOLUME, wherein the tumor-cells know that they no longer continue to divide. This is not a completely far-fetched idea, since epithelial cells involved in wound-healing seem to know when to stop; but malignant epithelial cells do not. For example, cells control their growth by PROGRAMMED CELL DEATH or APOPTOSIS, involving gene-products such as bcl2, which is widely expressed in malignant tumors (Miettinen, 2003). That is, unsuppressed bcl2 expression results in unbounded growth of tumors. Moore and Berman (Moore and Berman, 1991; Berman and Moore, 1992) proposed a simple random-number-generator model for cell-division, in which one could slightly alter a single parameter, namely the CELL DEATH RATE for cells after a single cell-division, and account for the growth of all human tumors, ranging from 0.43 for very rapidly-growing tumors, such as Burkitt's lymphoma, to 0.49 for relatively indolent tumors. (Normal tissue would have a cell death rate of 0.5.)

4. DEFINITION OF INFINITE PRODUCT. We consider a sequence of numbers satisfying 0 < rk < 1, for all k = 1, 2, 3, ..., and the corresponding n-products:
             Pn = (1-r1)(1-r2)(1-r3) ... (1-rn),        for n = 1, 2, 3, ....
In the model Pn represents the proportion of terminal volume remaining after the nth papilla has entered the terminal volume. Then the sequence of numbers, Pn, is decreasing with increasing n; and the Pn possess a limit, P, as n → ∞, lying between 0 and 1, or possibly at 0 itself. P is called the infinite product, and is written:
             P = P∞ (1-r1)(1-r2)(1-r3) ...
The obvious 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:
             Sn = r1 + r2 + r3 + ... + rn.
of the same numbers, as n⇒∞? Does S=∞ or is S<∞? S is called the corresponding infinite series, and is written:
             S = r1 + r2 + r3 + ....
It is shown that P is sandwiched ("STRUBLE SANDWICH") in between two decaying exponential functions of S:
             a-S < P < e-S


PROOF OF STRUBLE'S FILLING THEOREM.

The central theorem of this report is the assertion that the limit of the infinite product, P = P∞, corresponding to unfilled terminal volume, is zero (i.e., the terminal volume is filled) if and only if the infinite series, S = S∞, is infinite (malignant papillae in the present model). This theorem was proven by Struble (1984), and is sketched herewith. The proof has two parts: (i) IF and (ii) ONLY IF. That is, (i) P = 0 if S = ∞, and (ii) P = 0 only if S = ∞, (i.e., P > 0 if S < ∞).

Proof. (i) P = 0 if S = ∞.

Part (i) is easy, and depends upon a simple property of the base of the natural logarithm system, e = 2.718281828...., defined as the limit of (1 + (1/n))n as n → ∞, as shown in the following graph:
301.

That is, for x=0, (1-x) = e-x; and for x > 0, (1-x) < e-x. You can satisfy yourself that this graph is accurate by noting that (-1/(1-x)) < -1 for 0 < x < 1. Therefore, the antiderivative (i.e., integral for a well-behaved function) is (log(1-x)) < -x, and its corresponding exponential is (1-x) < e-x. Q.E.D.

Then P1 = (1 - r1) < e-r1, P2 = (1 - r1) (1 - r2) < e-r1 e-r2 = e-r1-r2, ..., and P = P∞ = (1 - r1)(1 - r2) ... < e-r1 e-r2... = e-r1-r2... = U, which serves as the upper-bound for P. In particular, for ∑rn = ∞, 0 < P = P∞ = < e-∞ = 0, i.e., P = 0. But even if ∑rn < ∞, it is still true that P < U.

Proof (ii). P > 0 if S < ∞.

To form a lower-bound, L, for P, let R = max1<k<∞ rk<1, and note that:
             (1 - R)x/R < 1-x
holds for all x, 0 < x < R, as shown in the graph:
302.



You can satisfy yourself that this graph is accurate by observing that
       (1) for x=0, (1-R)0 = (1-0) = 1;
       (2) for x=R, (1-R)1 = (1-R); and
       (3) for x=0,
Exponentiating both sides by x yields:
             (1 - R)x/R < 1-x
Q.E.D.

Therefore:
             (1-R)(1/R)rk < 1 - rk
holds for all k, so that:
             (1-R)(1/R)(r1 + r2 + r3 + ... rn) < (1-r1)(1-r2)(1-r3) ... (1-rn) = Pn
holds for all n, and:
(1-R)(1/R)S < P.
It is useful to recast this inequality in the form:
             L = a-S < P < e-S = U,
where a = 1/(1-R)(1/R) > e. Therefore, P is sandwiched between two decaying exponential functions of S:
             L = a-S < P < e-S = U,
It can be shown that the sandwiching is tight if R is small []. In any event, P > 0 if S < ∞.

The easiest way to understand e is as instantaneously compounded interest on a bank account. For example, if you deposit $1 in a bank that offers 100% interest compounded annually, then at the end of one year, the account will have $(1+1)=$2 on deposit. If the bank compounds semi-annually, then at year-end, the account will have $(1+1/2)(1+1/2) = $2.25 on deposit. If the bank compounds quarterly, then at year-end the account will have $(1+1/4)(1+1/4)(1+1/4)(1+1/4) = $2.4414062 on deposit. If the bank compounds monthly, then at year-end the account will have $(1+1/12)12 = $2.613033 on deposit. If the bank compounds n-annually, then at year-end the account will have $(1+1/n)n on deposit, depending upon the value of n. If the bank compounds instantaneously, then at year-end the account will have limn → ∞ $(1+1/n)n = 2.718281828... on deposit, where the constant e = 2.718281828... is named in honor of Leonhard Euler (1707-1783)

5. REMOVAL OF THE BILLBOARD BY PAPILLAE.
The billboard is a square with base 1 unit and height 1 unit, for a total area of 1 × 1 square-units.
In the first step, 1 square-unit remains behind from the zeroth step (i.e., the unpainted billboard), and half the billboard area is painted, yielding a remaining area of 1/2 square-units.
In the second step, 1/2 square-unit remains behind from the first step, and 1/16 of the total billboard area (i.e., 1/16 ÷ 1/2 = 1/8 of the remaining billboard area is painted, yielding a remaining unpainted area of 1/2 - 1/16 = 7/16 = q2 square-units.

6. WHY DO MATHEMATICIANS CARE ABOUT INFINITE PRODUCTS? One of the bedrock disciplines of mathematics, invented by Sir Isaac Newton (1642-1727) and Gottfried Wilhelm Leibniz (1646-1716) in the late 17th century, is differential and integral calculus. Differential calculus examines the slope of a particular curve, or MATHEMATICAL FUNCTION at any given point. 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:

29.

Then one adds up the areas of the blocks. Since the formula for calculating the area of a block has been known since Euclidean (325BC-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. 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 (from Georg Cantor, 1845-1918), topology, measure theory, normed linear spaces, open/closed sets, and the Heine-Borel theorem.

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=0n=∞ an sin nx + bn 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.

John Wallis (1616-1703) ......... invented the "lazy eight" notation for infinity, namely, ∞. He also invented the WALLIS PRODUCT.

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. (Incidentally, Hippasus of Metapontum was executed in 600 B.C., by Pythagoras, for publicizing the fact that 2 is NOT rational.) and let the value of f(x) be 0 for all the remaining numbers on the [0,1] interval). (RATIONAL NUMBERS)

In the 20th century, Polish mathematician, Ian Mikusiński (1913-1987), American mathematician, Raimond A. Struble proposed models, respectively, for adding up bricks and arbitrary areas.

So, demonstrating that the infinite product equals zero for a particular terminal volume means that the papilla summation equals the area/volume under the curve. This is a fantastic, neat method for determining the integral under some very complex curves.

7. THIS IS REALLY KIND OF AN AMAZING MATHEMATICAL RESULT.
That is, if the sum of papillae is DIVERGENT, i.e., adds up to infinity, then the infinite product becomes zero; if the sum of papillae is CONVERGENT, i.e., does not add up to infinity, then the infinite product never reaches zero.

8. CONVERGENT=BENIGN; DIVERGENT=MALIGNANT.
We are assuming that a malignant cancer-cell population has a genetic mandate to proliferate, unconstrained by the usual limiting processes, such as p53; or equivalently, a genetic mandate to not-to-die (apoptosis), unconstrained by the usual limiting processes, such as bcl-2.

9. As shown in Struble (2005a), the geometric series is approximated by
             P = (1/2)(3/4)(7/8) ... (2n-1)/(2n) e(2n-1)/(2n)
             = 0.2887881


Proof that the geometric series converges. The geometric series is defined as: 1/2 + 1/4 + 1/8 + .... First, observe that:
             1/(1-x) = 1 + x + x2 + x3 + ...
This can be verified by multiplying both sides of the equation by (1-x). Therefore, 2 = 1/(1 - 1/2) = 1 + 1/2 + 1/4 + 1/8 + .... Subtracting 1 from both sides of the equation yields: 1 = 1/2 + 1/4 + 1/8 + .... Q.E.D.

Proof that the harmonic series diverges. The harmonic series is defined as: 1/2 + 1/3 + 1/4 + .... Organize members of the series as follows:
1/2 < 1/2,
1/3 + 1/4 < 1/2,
1/5 + 1/6 + 1/7 + 1/8 < 1/2,
1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16 < 1/2,
Since 1/2 + 1/2 + 1/2 + 1/2 + ... = ∞, the harmonic series is divergent. Q.E.D.

10. Pedunculated versus sessile polyp.
A pedunculated polyp has a broad top and a thin stalk, like a mushroom. Note that papillae-heights calculated from the geometric series, 1/2, 1/4, 1/8, 1/16,..., have the appearance of a pedunculated polyp.

A sessile polyp has a broad top that merges with the base, without an intervening stalk. Note that papillae-heights calculated from the harmonic series, 1/2, 1/3, 1/4, 1/5,..., have the appearance of a sessile polyp.

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17. Perl source code for calculation of P. 

#/usr/local/bin/perl
 $iii=0; $hhh=0; $dnm=1;
 $head=1; $tail=2.718281828;
 while($iii<25){$iii++;
 $tail=sqrt($tail);
 $dnm=$dnm*2; $num=$dnm-1;
 $head=$head*($num/$dnm);
 $hhh = $head*$tail;
 print "\n $iii h = $hhh";};
exit;

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