Stabbing at ‘Shakespeare Cipher Systems’ and
other Parasites
W. Ron Hess (BeornsHall@earthlink.net),
assisted
by Dr. Winston Chow (WChow@rappahannock.edu)
Written May 2009
Introduction:
This article examines a range of the "dead-end" distractions that once infested an earlier Anti-Stratfordian
movement, particularly the claims that "Shakespeare Cipher Systems" reside inside the Bard's works and other places. It distinguishes between "ciphers" and "codes," showing that simple types of codes
were actually used by Elizabethans, including by Oxford and his circle. It debunks two recent Oxfordian cipher systems, and
throws doubt on a third. But, most of all, it introduces Oxfordians to sound cipher principles, so that we can be inoculated
against, and more suspicious of, future claims of discovered ciphers.
A century and a quarter ago, Anti-Stratfordians were mostly Baconians. But by 1880 Baconianism had already begun to
deteriorate. Spiritualists were starting to write that Sir Francis Bacon could be channeled; and Rosicrucian and Freemason
secret societies were pointing to their variety of fabricated evidence, claiming that Bacon was a Grand or Ascendant Master
in their orders -- heir to secret knowledge of antiquity.
The disgraceful end was hastened 1884-88 when an American retired
politician, Ignatius Donnelly, revealed a system that he claimed was a variation of the "bilateral cipher" Bacon had described
(Friedmans 27-33). Serious cryptologists criticized Donnelly's system as not what Bacon described (33-35), saying that although
Bacon's simple system was valid (247, 287), Donnelly's was not valid (35-36), and that only Donnelly himself could make sense
of his system (36-37).
Still, Donnelly made a fortune from his The Great Cryptogram best seller,
so other Baconians spawned many systems which they fervently claimed Bacon had put into Shakespeare's works. A few amateur
cryptologists had even earned a patina of semi-veracity. This continued until in 1957 a range of Baconian systems were debunked
by the Friedmans' book.
Perhaps the worst damage from Donnelly and his ilk came from their contrived reasoning about why Bacon needed to be
so cryptic. Baconian systems yielded results in which Bacon allegedly confessed incest in his family (157, 163). And Bacon
supposedly claimed to be Prince Tudor, the eldest of several brothers who were fathered by the Earl of Leicester upon Queen
Elizabeth (63, 131, 178, 194, 202), and then left in the nests of prominent Elizabethans (157, 166, 194).
Parasites
infect Oxfordianism: Eaten from within by parasitical
pet theories, fanatical Baconianism became hostile to serious research, favoring the weird -- such as summoning Bacon's ghost
(9, 14, 282). The Baconian cause became odder with each passing decade, losing members to emerging theories – such as
to Oxfordianism.
Nevertheless, the parasites were adept at jumping from host to host. Almost as soon as Oxfordianism was founded, ex-Baconians
tried to transfer fanaticisms mentioned above -- including Spiritualists holding séances in which they channeled Oxford and
others of his time to learn that Oxford was the father of Prince Tudor (as with Percy Allen in Talks with Elizabethans).
In addition to Allen, other Oxfordians adopting pseudo-Baconian pet theories included Dorothy Ogburn and B.M. Ward,
although they produced creditable works as well. Ogburn's This Star is still a
font of information, but it eagerly tipped its hat to an Oxfordian version of Prince Tudor. Ward wrote the original biography
of Oxford, and yet in 1926 he had decided to emulate Baconian cipher systems. He contrived an alleged string-cipher in the
"L'Escu d'Amour" poem of 1573 A Hundreth Sundrie Flowres, claiming Oxford's name
was spelled out. The Friedmans debunked the cipher by showing "Lewis Carroll" was an alternative solution (132-36). Still,
I argue that the Friedmans erred somewhat by not citing more and better objections. Such as:
·
the cipher's selective decision-making about which lines to choose from,
·
variability about choices for how many letters to choose per line, and
·
variability about lines-up using the same choices as lines-down.
Unfortunately, despite all these objections, Ward's
invalid string-cipher has remained a staple Oxfordian myth which we really should discard.
The Friedmans’
rules for valid ciphers: The Friedmans were two
of the best-respected cryptologists from World War II and the Cold War. Yet, even after their book refuted Ward's and other
ciphers, intelligent but misguided enthusiasts have persisted in proposing
new ones.
Enthusiasts have often given lip-service to the Friedmans, but have never really adhered to principles the book espoused.
A valid cipher system must meet requirements in various categories, such as these (from the index, 292):
1. results should be within an allowable percentage of error,
2. a correct solution mustn't be a matter of a decoder's opinion,
3. the certainty of a solution should not be in doubt,
4. the law of probability should not be extreme in a solution,
5. length of results need to be long enough to constitute a deliberate
vs. random result (pg. 23 suggests "about 25 letters"),
6. the result must be unambiguous and unique,
7. there must be a credible relationship between plain and cryptic versions
whereby the decoder is tipped off that a cipher resides within,
8. two or more valid solutions to a cipher constitute an absurdity (as
with their dismissing of anagrams on pg. 113), and
9. the Friedmans specially denounced selective decision-making (pp. 18-20),
or what many call cherry-picking.
Since we amateurs aren't familiar with valid cipher principles, enthusiasts may convince us with
convoluted excuses for why Elizabethan decoders would logically make one choice versus another in order to proceed to directed
conclusions from their own pet theories. Although their systems inevitably have intrinsic flaws, enthusiasts can be adept
at hiding their flaws beneath jargon, overwhelming detail, and/or deflection. Worst of all, enthusiasts' dead-ends may distract
us from more promising research.
Systems based
on Equidistant Letter Sequencing (ELS): Over
the past dozen years, Oxfordian systems have been proposed by Dr. John Rollett (who is now an outspoken critic of our Oxfordian
theory, so presumably he no longer believes in his system) and Cornwall's David Roper. Both systems use a methodology of putting
texts into matrices, shuffling each matrix in ways that make allegedly plausible codes pop out, and rationalizing that the
results were deliberately encoded there -- essentially by resorting to claims that the results were otherwise statistically
improbable. But, in the hands of clumsy amateurs or malevolent deceivers, statistics can be abused (my
friend Dr. Winston Chow has prepared a side bar at the end below for discussing this point).
As one proverb says, "Statistics never lie, but Statisticians do!" Or the saying attributed to Benjamin Disraeli, Mark Twain,
and others, "There are three kinds of lies: lies, damned lies, and statistics." Our problem is
telling the distortions from those that may be sound.
Rollett, who began writing on the topic circa 1997, apparently never admitted that his system was quite similar to
a 1994 so-called discovery by Jewish occultists which was dubbed Equidistant
Letter Sequencing (ELS). In contrast, Roper openly used ELS as his methodology. Yet, ELS has been dismissed by cryptologists
as deeply flawed when used for systems which occultists dub "the Bible Code" or "Torah Code." Occultists claim that God wrote
the Old Testament and the Cabala book
of demons, inserting prophesies for ELS users to detect. Yet, their prophesies are generally discovered only after the prophesied
events. A serious evaluation of these systems is at the Religious Tolerance website.
When cryptology journal editors refuse to consider
reviewing Shakespearean ciphers, particularly when ELS is used, they are not just displaying prejudice against Oxfordians.
The editors rightly believe that such systems were already debunked by either the Friedmans or more recent cryptologists.
So, the only way enthusiasts can get their systems adequately peer reviewed is to first cultivate the interest and intercession
of genuinely well-respected cryptologists. To appeal to amateurs like us, instead of to experts, is the same flaw as that
used by advocates of "Cold Fusion," where they were already claiming to the press that they had solved the world's energy
crisis before any peer reviews had been done. Un-validated self-promotion is frowned on. Especially when claims are made --
as Roper does -- that his work is mathematically proven, or otherwise substantiated, without having adequately obtained proper
peer review.
Improbable Claims of Esoteric Knowledge from Antiquity: A flaw in many of the
irregular approaches above has been claims of present legitimacy from past secret knowledge stretching to sages from the renaissance,
middle ages, and antiquity. The implication is that those who don't recognize their wisdom do so out ignorance of that esoteric
knowledge. We who are properly skeptical may be accused of "closed mindedness" or "bigotry." Just as there's no solid evidence
the Freemasons existed prior to the 1720s, or the Rosicrucians prior to 1610 (see the Catholic Encyclopedia websites), there's
no good evidence that anybody used ELS prior to modern times. Except where amateurs and occultists warp the facts to
purportedly find ELS almost everywhere they look.
No one ever mentioned or claimed the existence
of ELS, or anything like it, before Jewish occultists did so in modern times. When later enthusiasts tried to use ELS for
their own contrived systems, they encountered the problem that they had to accept the “Bible Code" as valid, or
risk undermining their own pet theories. So, they searched for ELS in Elizabethan and earlier texts, and not surprisingly,
claimed to find it. That's because ELS is an invalid methodology which can be misused to find instances almost anywhere one
looks. Authentic cryptologists derisively find ELS in Melville's Moby Dick, Tolstoy's
War and Peace, and on cereal boxes (as described in the AWitness website).
From these contrivances, enthusiasts claim that
ELS was well known and often used in earlier times, because they allegedly found it there. For example, they claim that Sir
Francis Bacon used ELS, but they don't show that Bacon ever wrote about ELS, even though he wrote about another cipher. They
circularly reason that because ELS was allegedly used in antiquity, this substantiates their methodology. They want us to
believe ELS was used in the 1609 Sonnets dedication or 1623 Shaxpere effigy text,
similar to "Bible Code" claims that God used ELS in the Cabala.
Simple, easily detected, "Codes" were actually used: So, beware of parasites, occultists,
and ELS-based systems. Yet, Oxford's circle apparently loved simple codes, and to play upon his name, Edward de Vere
(where, for a multi-lingual audience, Vere = Spring, green, truth, youth, etc.; and with Oxford's "E" in front, it could be
invoked in English as "ever" or "never"). Here are a few examples of what Oxfordians
like me believe were used:
· Alan Tarica and I wrote in
the 2006 Summer & Fall Shakespeare Oxford Newsletter that a young Oxford apparently
encoded his own name, reading "iptsubion Edward," scribbled into a margin of Halles
Chronicles.
· I've repeatedly pointed out
in e-mails that Oxford was the probable author of 1584 Pandora (where
in Middle English "Soowthern" or "Soothern" = of Truth = de Vere = Oxford's name).
·
We know of "Ever or
Never" from the end of 1594 Willobie His Avisa.
·
We know of "A Never
Writer to an Ever Reader" from the preamble to 1609 Troilus
& Cressida.
·
We know "that everie
word doth almost tell my name" from Sonnet # 76.
·
In Vol. II, pg. 426, of my Dark Side trilogy, I showed that the title of Oxford's ex-servant Anthony Munday's 1600 Bel-vedére (= beautiful to view), which claimed to have poetry by both Oxford and Sh., was a perfect anagram
for "Bel-DeVere" (= fair or warlike Oxford).
·
We know that while he was Oxford's servant, Munday
richly used simple acrostic codes in his 1579 Mirrour of Mutabilitie, lavishly
dedicated to Oxford.
·
My website article about two Latin poems dedicated
to Oxford by Munday and Robert Greene shows a Latin acrostic I discovered in one of them, and double entendre on VERE,
the Beloved Youth (or Cupid), and other themes.
·
My website article about Tempest opposed Stratfordian claims that "Caliban" is an imperfect anagram of "cannibal";
I showed it is in fact a perfect anagram for "in Cabal,"
an alternative name of the Cabala occult book of demons.
· Prof. Alan Nelson identified an
acrostic poem dedicated to Oxford's daughter in 1606 Ourania, composed by Oxford's
one-time servant Nathaniel Baxter.
·
And we Oxfordians typically accept all of the poetry
signed "EOX" as written by Oxford, even though Prof. Steven May rejects some of the best of those.
The
fact is, even Stratfordians occasionally look into Shakespeare's works for simple, silly code systems, such as anagrams. For example, as reported on "Hardy's List," Dr. Roy Winnick has proposed anagrams
in some of the Sonnets (see the oxford journals website). He cites sonnet # 17's "should liue twise in it,
and in my rime" as containing all letters to spell "Wriothesley-twise." He cites
# 58's "Be where you list" as an anagram for "Be U Wriothesley." And of course we know that Henry Wriothesley, 3rd Earl of
Southampton, was the dedicatee of Shakespeare's 1593 V&A and 1594 RofL poems. Which opens the questions of why Southampton couldn't be openly acknowledged in the 1609 Sonnets, and if a poetical mind really would be concerned with injecting codes rather than beauty?
Although the Friedmans admitted
some valid use of simple systems, like short anagrams (93-4, 111), they also decried these in general as abused by amateurs
(212-13) through selective decision-making to
derive a message from large swathes of text. So, I doubt that Winnick's weak anagrams would meet their approval, if only for
the silly results and loose context (e.g., in sonnet #17, a selection of 16 letters out of 30 hardly constituted an intentional
solid system).
Certainly, we Oxfordians generally believe that
Oxford loved to pun by using simple codes and pseudonyms, for reasons which often we can only guess at. I go a bit farther and feel it's possible that as a studious youth, Oxford may have drilled himself in
certain tricks or habits related to his name and to fanciful identities of the "pulp fiction" of his time (such as the internationally
celebrated knights-errant "Astolfo" or "Palladine" [= "of the Spear-shaker"]),
both of them invincible English paladins. I believe that he appears to have used them later as an adult, and so did his circle.
So, am I a codes enthusiast myself to some extent?
Distinction between "ciphers" and "codes": Actually, the Friedmans (15) made
a careful distinction between flexible "codes" and far more exacting "ciphers":
"...we shall not discuss codes in this book, but since the word 'code' is often used by laymen... in contexts where
they clearly mean 'cipher,' it is worth drawing the technical distinction between the two... In code systems, the units or
symbols to be translated can be of different lengths: a letter, a syllable, a word, a sentence, or just a string of letter
or numbers is agreed to stand for a particular word or a whole phrase in the message... In contrast, the units in cipher systems
are of uniform length and bear a uniform relationship to the units of the plain text.... we shall be discussing the conditions
which any cipher has to satisfy if it is to count as a valid cryptogram.... based on common sense..."
So, what's to keep enthusiasts from arguing that their systems aren't ciphers, and should be accepted
as more flexible codes instead? My interpretation is that, for complex systems to be taken seriously, they must be subjected
to the strict regime laid out for ciphers. But simpler, less pretentious systems, like acrostics, short perfect anagrams,
and word play, were actually used, without doubt, and can be accepted to some extent without rigorous validation, within bounds
of "common sense."
For example, Roper begins defense of his system
by pointing to a code system written about in the mid-1500s by Girolamo Cardano, dubbed "Cardano Grilles" (CG). Because Oxford
wrote a preface to a 1573 Cardanus Comforte translation, Roper implies that Oxford
used CG, even though if we look carefully, Roper's ELS-based system actually does not use
CG at all (he apparently mentions CG only as a deflection from his own system's flaws, or to prop
up his system by showing that real codes did exist in the 16th century). In my opinion, Roper's system would have been
stronger if he had demonstrated its reliance on CG, rather than on invalid ELS, because CG as Cardano described it was definitely
used. And CG constituted a simple code system. Basically, two conspirators meet and agree on a popular book they will both
have, and they each carry with them cards with holes in them. They then pass messages to each other in which they secretly
tell very simple things, such as which page of the common book and which card to use. Then the card is placed over the page,
and the holes yield a text with a secret message. But CG is a "private code," completely inadequate for Roper's complex system,
where every visitor to the Shaxper effigy must be expected to divine that a code was used, that it was a "Cardano Grille,"
magically conjure up the exact card to be used, and then apply it to the text on the effigy. That absurd complexity certainly
violates the Friedmans' "common sense."
To me, the distinction between ciphers and codes
means that we should be able to recognize simple, common sense codes, while rejecting purported complex codes or un-validated
ciphers. Thus, although I believe there aren't yet any valid Shakespeare ciphers, what were just listed above for Oxford and
his circle constitute simple codes. And those can occur often enough in relevant texts for a pattern worth study, possibly
for identifying anonymous or pseudonymous works as authored by Oxford (as I would do for 1584 Pandora and 1594 Willobie).
"Diagonal Vere Acrostics": For the past 7 years, I've been intrigued by an unpublished system
proposed by Prof. Albert Burgstahler (awburg@ku.edu), called "Diagonal Vere Acrostics" (or DVAs, with "Alignments" often substituted
for the last word). Initially, it appears to be a simple code system, finding VERE, EVER, VEER, etc. diagonally and equi-spaced
across texts of printed poetry, both upwards and downwards. Given my belief that in his boyhood Oxford might have developed
such simple codes, and given that these DVAs constitute a simple acrostics innovation (which Burgstahler credits to the
late Oxfordian Ralph Tweedale), I was inclined to study further, even though I saw problems. For one thing, there was
an obvious need to involve printers, compositors, etc. into the scheme (for which I nominated Munday, since he was apprenticed
as a printer in 1576 to c. 1584).
But, worse problems arose when it turned out
DVAs, which Burgstahler admits often occur randomly, are not an end in themselves. Rather, they are a means for deriving "block-letters"
inside the poems, spelling out "E-O-X," "O-X," and a few other figures. This second phase adds layers of subjectivity, complexity,
and even improbability. I believe that its complexity requires Burgstahler to subject it to cipher validation. And that's
an uphill battle that I fear his system can't win.
Let's close on a more positive note for Burgstahler.
If his DVAs cipher can be validated by a peer reviewed cryptology journal, we may then be able to use it to identify anonymous
or pseudonymous works that were by Oxford. His hope is that if his system is validated, it will tip the authorship question
toward Oxford. For, while interpretations of history and literature are often debated, the Friedmans (286-88) noted that a
properly validated cipher system may be undeniable in terms of the information it conveys.
But, until any cipher system is validated, it's
more likely that it will be a hopeless dead-end, distracting quick minds from more valuable pursuits. Simply speaking, complex
ciphers are not likely worth the trouble -- for their discoverers, or for us!
Bibliography:
Allen, Percy & Dowden, Hester,Talks with Elizabethans revealing the mystery of William Shakespeare, London, 1947,
Rider.
Friedman, William and Elizabeth, The Shakespearean Ciphers Examined,
1957, Cambridge U. Press; at www.questia.com/PM.qst?a=o&d=11853542 it is free online for a trial use.
Hess, W. Ron and Tarica, Alan,
"Did Shakespeare read from the 17th Oxford's Personal Library," Pt. 1 Shakespeare Oxford Newsletter, 42:2, Sum
2006, 1+; Pt. 2 42:3, Fall, 25-28. Also see review "Annotated Chronicle?," Sh. Matters 6:2, 6 &32 and "Letter to Editor" SOS News, Sum 2007, 43:2.
Hess, W. Ron, The Dark Side
of Shakespeare, Vol. II of III: An Elizabethan Courtier, Diplomat, Spymaster, & Epic Hero, Lincoln, NE, 2003,
Writers Club Press, ISBN # 0-595-29390-5.
Hess, W. Ron, "Tempest's Red Herrings," 2003, at website
http://home.earthlink.net/~beornshall/index.html/id13.html 10/16/09.
Hess, W. Ron, "Translations and examination of two mysterious Latin poems dedicated
to the 17th Earl of Oxford," 2009, at website
http://home.earthlink.net/~beornshall/index.html/id25.html 10/24/09.
Ogburn Sr., Dorothy & Charlton,
This Star of England: William Shakespeare, Man of the Renaissance, NY, 1952, Coward-McCann.
Roper, David, Proving Shakespeare, self-published, ISBN 978-0557012619,
see www.dlroper.shakespearians.com.
Ward, B.M., The Seventeenth Earl of Oxford. . . From Contemporary Documents, London, 1928, Murray.
Winnick, Roy, "'Loe, here in one
line is his name twice writ': Anagrams, Shakespeare's Sonnets, and the Identity of the Fair Friend," at website
http://litimag.oxfordjournals.org (click on "Advance Access") 10/16/09.
The Bible Code in War & Peace
and Moby Dick,
www.awitness.org/essays/bibcode.html 10/1/09.
New Advent Catholic Encyclopedia/
Freemasonry,
www.newadvent.org/cathen/09771a.htm 10/1/09.
New Advent Catholic Encyclopedia/
Rosicrucians,
www.newadvent.org/cathen/13193b.htm 10/1/09.
Religious
Tolerence/ Bible Code,
www.religioustolerance.org/biblcode.htm
10/1/09.
________________________________________________
Potential Misuse of Mathematical,
Statistical, and Probability theory
Winston C. Whow, PhD
I do not believe that statistics
lies. Statistics is a branch of mathematics science whose foundation is probability theory. Statistics is science because
it uses empirically observed data; it is mathematics because analysis performed on the data is solidly mathematical; and it
is an art because there are tools and disciplines that a practitioner can master. So, as with any branch of mathematics, statistics
is based on definitions, assumptions, and laws known as theorems that can be deductively proved and cannot lie.
However, true
to its art side, statistics can be misused by practitioners, as among definitions, assumptions, and theorems, one can
make incorrect assumptions or misinterpret mathematical principles. One concern
is that data upon which statistics is to be performed must be collected properly, or else the proverbial "garbage-in, garbage-out"
will occur, which is a form of bad mathematical assumption, as data is “assumed” to be good. Also, improper analysis
of data is a form of misinterpreting mathematical ideas, as a mathematical principle is broken. Even in these cases, it is
not often due to intentional lies. Two examples from Mr. Hess' commentaries and attached paper illustrate my idea. The
first is due to misinterpreting a mathematical principle and the second is due to incorrect assumption(s).
Mr. Hess originally
wanted to use his own made-up example of how statistics can be misused. He said,
"There are 7 billion people on planet Earth, and only a one in 7 billion chance that Hess wrote his paper. The chances
are astronomically slim that Hess was the author. Ergo, it is statistically proven that Hess was not the author."
Advised by friends that his example was weak, he consulted me to see if better examples could be suggested. To say
that someone did not write an article because there are 7 billion people in the world, and the chance of a particular
person writing the article is almost nil, is a misinterpretation of a mathematical idea called conditional
probability. Unless some deception is afoot (e.g., Ron Hess is a pseudonym used by Prof. Stephen Hawking, an incorrect
mathematical assumption), it is already given that the author wrote the article. So one can only make inferences of uncertainty based
on the hypothesis that the author already wrote it. The probability that the author already wrote the article given (conditioned
on) that he wrote it is 100-percent. Thus, weak as this 7 billion example may be, it successfully illustrates a misuse of
statistics and probability theory.
Another illustration
of misuse, in principle similar to the above paragraph, is to say a coin will be heads if flipped twice because there is a
50% chance of getting heads each time. The fact is that there is a 25% chance of two heads in two flips, 25% of two tails,
25% of a heads and a tails, and 25% of a tails and a heads. Thus, there is a 75% chance of at least one heads, but also a
25% chance that no heads will occur at all. This is probability at its simplest.
Another mistake
is committed by some proponents of secret coding in literature, or ciphers, where a conclusion is claimed to be proven
by statistical analysis of the written lines or text. Thus, the statistical input data are perceived to have peculiar
patterns in written lines of the work of the literature in question. Yet, interpreting certain words or phrases
as an idiosyncrasy can be very subjective, and the analyst may perceive patterns that are in fact non-existent. Thus, the
data is perhaps unintentionally contrived by the wishful thinking of the analyst, and selective decision-making (or cherry
picking) occurs in order to arrive at a directed conclusion. Thus, the
"mathematical assumption that the data is objective" can be a wrong assumption, and results of the study are flawed. This
example illustrates the misuse of statistics by incorrect assumptions.
Another illustration
of the same use of non-objective data, and thus misuse of statistics by bad assumptions, is where a conclusion is made that
all mountains are over 10,000 feet above sea level. Data: Pike's Peak, Mount Rainer, Mount McKinley, and Mount Everest. Measurement:
all over 10,000 feet. Analysis and Conclusion: all mountains are over 10,000 feet. This is wrong due to a bad assumption
that good data has been harvested. In fact, anyone can easily see that the data in this case was cherry-picked, eliminating examples contradicting the conclusion, and
arriving at a bogus conclusion. So, the science and art of data selection has prejudiced the statistical outcome.
This short
commentary focused on simple, readily understood examples. Still, a bungling amateur or skillful expert can purposefully or
accidentally produce such errors, and only an expert can uncover them, sometimes
with difficulty. This often happens when selected data, abstruse formulae, expert tools, or copious tables and graphs have
been employed. In scientific publications for peer-reviewed journals, such data, formulae, tools, tables, and graphs are expected.
But in publications for scientific amateurs, for non-scientists, or even scientists in very unrelated fields, this can lead
readers down the wrong path!
Winston C. Chow received a B.S.
and M.S. in mathematics from West Virginia Wesleyan College and the University of Cincinnati, respectively. He received a
Ph.D. in information technology, specializing in statistical sciences with emphasis on stochastic processes from George Mason
University, located in Virginia. His interests are in the statistical analysis and applications of random processes. He has
published articles on subjects related to Brownian Motion. He is a member of Sigma Xi, The Scientific Research Society.