This is the latest of my web pages. It was started on the 4

my e-mail
contact is: kurzweg@ufl.edu and home-page at:
https://mae.ufl.edu/~uhk/HOMEPAGE.html

The recent BP oil spill in the Gulf of
Mexico and the unabated leakage of oil and methane (about 60,000
barrels a day) from the broken well for a period of now more
than 70 days has got me thinking about the origin of this oil
located at least 13,000 ft below the bottom of the Gulf. The
standard view held by most oil geologists is that it has a
biogenic origin and was produced by decaying vegetation under
high temperature and pressure. If this is so then the world is
indeed running out of oil and we may already have reached a peak
in oil recovery.
However, there is a small minority (Russian School especially
Mendeleev of periodic table fame and Kurdryavtsev,
plus some individuals in the United States including Gold and
Kenney) who cite evidence for an abiogenic origin involving the chemical
conversion of methane arising
from deep within the earth mantle being converted to larger
chain hydrocarbons by the high pressures and temperatures
existing at greater depth. If the latter group is correct, then
it would appear there is an unlimited supply of hydrocarbon oils
and gases available, provided one drills deep enough into
pockets capable of holding such products, especially methane.
After some thinking on the subject, I find myself siding with
the minority in believing that most hydrocarbons are indeed of abionic origin. Go HERE to
see my points supporting an abiogenic
origin.

**July 8, 2010-How much oil can be
expected to leak from a blowout involving an oil well of
cross section A and depth L?**

**July
23,
2010-Why is human vision confined the very narrow wavelength
electromagnetic range of 4000A<λ<7000A and its hearing
confined to a frequency range of 20Hz<f<20,000Hz?**

Of the five human senses of vision,
hearing, taste, smell, and touch it is the first two which are
perhaps the most important for survival. Our homo sapiens ancestors and those species
preceding clearly required acute seeing and hearing capabilities
to function in the environment in which they existed. The
question that arises is how were these
senses developed in the process of evolution. Go HERE for our
thoughts on the matter.

**July
29,
2010-What is the rate of energy consumption in the **United States
and the World and what are the energy sources available now
and in the future?

The world consumption of energy is
estimated to be about 474 exajoules=474x10^{18}J
of which 88% comes from fossil fuels.

This implies an energy consumption rate of-

[474x10^18J]/[365x24x3600]s=1,50x10^13W=15TeraWatts

The sources for this energy are oil, coal, gas, hydro, nuclear, and renewables in descending order. Go HERE for further discussions.

**July 30, 2010-Was Malthus right?**

A little over two-hundred years ago the Anglican
clergyman Thomas Malthus (1766-1834) published his famous book *ï¿½An Essay on the Principles of Populationï¿½. *In
it he pointed out that (1)human population will increase with an
increase of subsistence resources,(2) that such an increase will
inevitable produce a population over shoot, and (3) that this
will lead to a decrease in population
levels until it falls below the subsistence level. After that
the process repeats itself. Go HERE for details
of why I think he was right.

### August 25, 2010- What are the energy
densities of various energy conversion processes and how do
their magnitudes suggest a direction the world will have to take
to meet its ever increasing energy demands?

Most of us back in our elementary physics courses
in high school and undergraduate college learned about the
various different forms of energy ranging from kinetic energy E_{k}=(1/2)mv^{2}, through chemical energy E_{c},
to nuclear where E_{n}=mc^{2}. We want here to
look at the concept of energy content per volume and use it to
point out some reasonable directions the world might take to
meet its future energy needs. Go HERE for the
discussion.

**September 23,
2010- How can one determine in a non-destructive manner if a
gold bar is made of pure 24K gold?**

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### September 25,
2010-What is the origin and purpose of the ancient Trident
Geoglyph found along the Bay
of Paracas in Peru

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Several decades ago while travelling in southern
Peru I came across the well known rock carving in the shape of a
600 ft long trident geoglyph located on a sloping hill off of
the Bay of Paracas. We discuss HERE our
thoughts on what its purpose might be.

###

### October 12, 2010 ï-What is
the Goldilocks zone for planets orbiting about a central star?

In the last decade or so astronomers have found
the presence of several hundred planets orbiting about stars
using optical occultation techniques. Most of these planets lie
in regions about the central star incapable of supporting life,
however, some probably do. The presence of such planets lends
strong support for the nebular hypothesis of planetary formation
first proposed by Kant and Laplace. In this theory a distributed
disc of matter collapses by gravitational collapse leaving a
central star plus a rotating disc of matter which eventually
condenses to planets and moons all lying in essentially the same
plane(the ecliptic). By the conservation of angular momentum
most of the resultant planets will be spinning with a rotation
axis nearly perpendicular to the ecliptic. This rotation allows
a nearly uniform distribution in temperature at a given latitude
of the planet and should make the probability of
extraterrestrial life quite likely provided the planet is not
too close or too far away form the central star. The
zone where life is possible has been termed in the literature as
the Goldilocks Zone. The term is taken
from the Grimmï¿½s brothers fairy tale ï¿½Goldilocks and the
Three Bearsï¿½.Go HERE to see our
discussions on these Goldilocks Zones about stars in our galaxy.

### October 16, 2010-What is the
approximate speed versus height of a free-fall parachutist
jumping from a high altitude balloon?

About 50 years
ago air force colonel Joseph Kittinger (alumni of UF) carried
out a free fall parachute jump from a balloon at an altitude of
102,800 ft. He reached a maximum speed of about 600mph(268m/s)
before deploying his chute at 18,000ft and landing safely. This
record jump is to be finally challenged in the next few months
by Felix Baumgartner who, with sponsorship of Red Bull, will
attempt to jump from a height of H=120,000ft(36.6km) and hopes
to reach the speed of sound before chute deployment. We
want here to examine the speed as a function of height a
parachutist will experience during a jump from heights above
100,000ft. Go HERE
for the details.

November 1, 2010- How did early stone carvers construct massive cyclopean walls using boulders of irregular shape and how where they able to produce extremely tight fits between neigboring stones without the use of mortar?

One finds throughout the world in Greece, Ireland, Stonehenge
and Avebury, Incan Peru, and Easter Islands stone structures
built with huge non-rectangular cross-section boulders
with very little modification of the original stone surfaces. Of
particular interest are the cyclopean structures found near
Cuzco, Peru. There huge irregular shaped boulders were cut in
such a manner that they fit together like pieces of a jigsaw
puzzle. We discuss HERE how this
was most likely accomplished.

December 17, 2010-What is the origin of the Decimal Numbering System?

With only a few exceptions, the base ten number system is the dominant one used throughout the world for commerce and measurements. Why is that? Clearly it stems from the fact that early man used his ten fingers to count objects and that such finger counting was soon replaced by symbols such as sticks and by marks recorded on surfaces. The Roman number system of R={I, II, III, IV, V, VI, VII, VIII, IX, X}clearly hints at such stick figures as does the earlier Chinese method of representing numbers by bamboo sticks. We want HERE to discuss the probable origin of the Decimal Numbering System and indicate how the basic mathematical operations of addition, subtraction, multiplication, and division are accomplished. A few other number systems are also discussed.

December 22, 2010-Why are the most prosperous and technologically advanced and creative countries around the world located within two relatively narrow temperature bands ranging from yearly lows no lower than -10degC and highs no higher than +30degC ?

If one looks at the most technologically advanced countries around the world it is clear that these lie within two temperature bands which I call the creativity(or JUST RIGHT) bands. These zones have produced and are producing the majority of the literature, art, science, and technology and have been the home of such creative individuals as Plato, Confucius, Michelangelo, daVinci, Shakespeare, Newton, Rembrandt, Bach, Tolstoy, Beethoven,Voltaire, Edison, Picasso, and Einstein.The northern hemisphere band includes western Europe, the western part of Russia , Iran, northern India, China, Korea, Japan, the southern part of Canada, and the United States. The southern hemisphere band includes the southern and eastern parts of Australia, New-Zealand, Chile, the northern parts of Argentina, the southern part of Brazil, and South Africa. We discuss HERE why it is that within these bands most of the advancements in the arts, literature, and science have been and are being made and why most regions outside of these bands remain underdeveloped exibiting very little in the form of advanced human intellectual activity or creativity.

**2011**

January 22, 2011- What is the big deal with Binary and how does one manipulate things mathematically in this Number System?

Although most of us were brought up in school counting things in a base ten (decimal) number system it is clear that other number systems and especially the Binary System are becoming more and more important. Most computer manipulations, data storage, and the transmission of writing and music are now pretty much handled in binary because of the conveneince of a base requiring just two digits instead of the ten associated with decimal. We dicuss HERE some of the properties of the Binary Number System, look at several problems which are well handled in binary, and show how one can mechanically perform basic mathematical operations using a binary abacus.

March 1, 2011-What is relative time and how is it measured?

Time in an absolute sense does not exist as made clear by numerous philosophers(Kant, Heidegger) and by modern technical analysis including relativity. It can however be measured in a relative sense by comparing the interval beteen two events to that of a known time interval such as the period of one earth orbit about the sun. We discuss HERE how relative time measurements were first developed using astronomical observations and later employing atomic clocks. We use a modified Deborah number to measure relative time. An example of a digital combination of date and time is also presented.

March 27,2011- What are the Economics of Solar Energy Conversion using Photovoltaic Methods?

The recent events inJapan
will probably set-back the use of nuclear energy for electricity
generation by a decade or more. In the meanwhile there appears
to be sufficient fossil fuel supplies(especially coal and
natural gas)available to muddle through into the near future
despite of the clear rise in costs of these available energy
sources and their possible contribution to global warming. If the nuclear option cannot be
revived because of political and economic reasons and also the
recovery of fossil fuels becomes economically prohibitive, then
one needs to go to renewable energy sources such as wind,
biofuels, and solar. Of these renewable energy sources, solar
energy via photovoltaic energy conversion appears to be most
promising. This energy source is plentiful throughout much of
the world but has the disadvantage that one is dealing with a
very diffuse energy source requiring multiple square kilometers
just to supply a large city with its energy needs. We
want HERE to re-look
at the economics of solar energy using photovoltaic conversion
in order to see if it would make sense as a viable and economic
energy source.

May 5, 2011- What is the BMI and how is it Measured?

The ever increasing problem of obesity in this country has increased interest in having a quick way to estimate the fat content of the human body. Clearly too much fat is unhealthy and indicates a calorie food intake above that required for the individuals energy output. An excellent approximate measure of fat content is the Body Mass Index(BMI) first proposed by the Belgium polymath A. Quetelet in which one looks at the ratio of a persons weight W ( kg)divided by the square of the heigh H( meters). Population statistics show that the normal range lies between 20 and 25, with values above 25 considered overweight up to 30 and obese above 30. Today nearly 70% of the United States population has a BMI above 25. Go HERE to see our discussion on this index and how one can construct a simple circular sliderule to quickly determine ones BMI.

MAY 17, 2011- What is Symmetry and where is it Encountered?

Most recognize symmetry in an object , be it in the human face, in architecture, or mathematics , but often find it difficult to pinpoint exactly what makes the object so. We discuss HERE some of the characteristics of line, plane, and rotational symmetry by analyzing several different examples. Bilateral symmetry is demonstrated by looking at mirror images of the face of Charlize Theron and of the Taj Mahal. Next hexagons, circles, ellipsoids, cubes , the Laplace equation , and the five petal Rhodonea are examined for their symmetry.

July 1, 2011-Do Stock Indexes World-Wide Correlate?

We have been aware for a long time that stock averages throughout the world have a tendency to correlate . This has been especially true in recent years probably do to the increasing speeds of electronic communications and the use of computer arbitrage. We present HERE a graph of the excellent correlation observed between the S&P500, the German DAX Index, the DOW World Index, the Hang Sang Index, and the ILF ETF over a ten year period. Also a discussion of why such correlations should exist is presented.

July 12, 2011-What is RSA Cryptography and its Connection to Large Prime Numbers?

One of the best and most used cryptography techniques is the RSA approach which uses both public and private keys. It involves a public key which relies on the use of two large prime numbers p and q. The product N=pq is essentially impossible to factor with even the fastest electronic computers in any finite amount of time. Go HERE to see our explanation of the RSA method in simple terms, view an unbreakable public key which I constructed in just a few minutes, and find an example of sending a short encoded message from party B to party A.

July 26, 2011-How does an Energy Balance relate to Weight Control?

It is well known that a large fraction of the American public are overweight according to BMI measurements. This fact has become more prevalent in recent years and has been attributed to overeating accompanied by insufficient energy expenditure due to sedentary working conditions. Mathematically one can say that DW/dt=I-E=B, where W is a person's weight, I is the food intake, E the energy output and B the energy balance factor. Go HERE for a discussion of this problem and how it can most effectively be solved. Decrease in calorie intake and increase in physical activity are the well known factors making for weight loss ( B<0). Weight gain occurs when B>0.

August 1, 2011-What is meant by Exponential Growth and Decay and what are the Properties of an Exponential Function?

Although one often hears and reads about exponential growth in the literature, it is not always clear that the authors know precisely what they are talking about. In discussing true exponential behaviour, it is first necessary to understand what is meant by an exponential function, how it arises and what its properties are. We discuss HERE some of these. Both population growth and radioactive decay are described and we show how one can quickly estimate the value of an exponential function via a quotient of two polynomials.

August 4, 2011-Could the US experience Hyperinflation?

Actions by the Federal Reserve in recent years has essentially tripled the monetary base to about 2.5trillion dollars by simply printing money without any hard assets backing up these dollars. This is a recipe for hyperinflation as witnessed in Weimar during the 1920s and more recently in Zimbabwe. The reason we have not yet seen any major spike in inflation is due to the fact that banks are reluctant to loan out this new money as they have a risk free way to make money directly of the government via government bonds. Once house prices reverse their downtrend and the velocity of this extra money increases, large increases in inflation can be expected. Go HERE for our discussion of some of the histrorical aspects of hyperinflation and the possibility that it could happen in this country.

August 27, 2011-What are Self-Similar Patterns and how are they Generated?

Recent increased interest in fractals and how they may be used to explain many natural phenomena has got me to thinking more about how intricate 2D patterns can be generated from some very simple laws of replication. In particular, we look HERE at self-similar patterns which are generated from elementary geometrical figures such as squares, triangles and hexagons. The problem is treated as one of biological generation.

September 1, 2011-Can one create an easy to understand 3D Fractal based on a simple Reproduction Law?

Still thinking about fractals from our August 27th note, we recently posed ourselves the question is there an elementary three dimensional fractal whose basic initiator structure is simple and one where this structure is replicated forever through an infinite number of generations? We answer this question in the affirmative using a simple cube to generate smaller and smaller self-similar cubes. Go HERE for the details. We have not seen this type of 3d fractal before although some other more complicated 3D fractals have received attention earlier. It is hoped that this type of discussion might lead to a future increased emphasis on 3D fractal structures with the anticipation that they might prove helpful in describing the geometry of certain viruses. We also show how a simply cubic fractal model can be constructed from wood with the right tools and a lot of patience.

October 3, 2011-What is a Pelekinon and how is it used to measure Date and Time?

Before the advent of mechanical and electronic clocks, people depended on sundials to determine the time of day and used elaborate structures (such as Stonehenge) to measure the time of year when the solstices occur. The ancient Greeks used one of the simplest sundials called the Pelekinon. It was capable of telling both the hour of the day and the day of the year. The Pelekinon consists esentially of a vertical pole(the gnomon) and a flat plane on which they empirically observed and marked down the locus of the shadow produced by the pole throughout the year. We want here to discusss the mathematics behind the Pelekinon and show how one can use spherical trigonometry involving the astronomical triangle to accurately predict the movement of the shadow. Go HERE for the discussion.

October 23, 2011-What is the impact speed of a bullet fired straight up upon its return to earth?

Recent vidios showing Libyan freedom fighters firing their guns randomly into the air during victory celebrations, has brought the question to my attention of whether or not such acts pose dangers to individuals nearby. We analyse the problem of the returning bullets by first calculating what height H a bullet , fired from a standard Kalashnikov AK-47, will reach and then determine the impact speed V_{i} with which it impacts the ground on its
return. Go HERE
for the details. We will show that the returning bullets do
not pose a lethal threat. In addition, we put to rest
an urban legend that dropping a penny off of the Empire
State Building can kill someone walking on the sidewalk below.

December 12, 2011-What is Geometric Art what are the Basic Elements used in its Creation?

It is possible to create appealing abstract art work based on the superposition of simple geometrical shapes. I term this Geometric Art and give examples arising from religion, philosopy, and other movements. In addition I show how this type of art , involving straight lines, circles, squares and other simple 2D geometric figures, can readily be created by computer graphics or via handicraft approaches involving, for example, wood work. Go HERE for the details.

**2012**** **

February 1, 2012-How can an Abacus be used to Increase one's Ability to quickly Add and Multiply Numbers?

The standard method of adding, subtrtacting, multiplying and dividing numbers by the standard techniques taught in schools is very often not the quickest way to obtain an answer. Rather using an Abacus which requires a decimal concept of numbers can produce much faster results. We discuss HERE how this is accomplished by looking at how elementary mathematical operations are handled with an Abacus. Several different shortcuts involving multiplication based on recognizable powers of intergers and expresing numbers as differences of simpler numbers are discussed.

February 3, 2012-What are Logarithms and how may they be Used to Multiply and Divide Numbers?

Prior to the advent of hand calculators and electronic computers, the way to handle multiplication and divisions arising in complex mathematical expressions was to first convert the numbers contained therein to logarithms , perform the logarithmic operations, and then invert to get the final answer. We discuss HERE what is meant by a logarithm, how one manipulates them, and how slide rules can be used to quickly calculate approximate answers .

February 26, 2012-How does one express Large and Small Numbers?

Although most readers will be familiar with the exponetial notation for number such as 1000 being equvalent to 10^{3}
and 0.000001 equal to 10^{-6}, they are less likely to
be familiar with the designations involving Greek and Latin
prefixes for both very large and very small numbers. Such
designations are becoming more and more commenly used and one
needs to become familiar with them not just for understanding
the scientific literature. Go HERE for a
discussion on this topic. As examples we show that a light
year equals 9.46 Petameters, the worlds energy consumtion in
2008 was 474 Exajoules, and the Big Bang occured 13 Gigayears
ago. A circular slide rule capable of performing
multiplications and divisions over a wide 48 order of
magnitude range is also described.

March 20, 2012-How does Facial Recognition by Eletronic Computer work and what steps might be taken to improve the Procedure?

One of the more important and growing tasks in digital recognition is how to quickly identify a face from a collection of millions of other faces electronially and do so with one hundred percent accuracy. So far such approaches have been only partially successful and will fail when simple changes in hairstyle, aging, or facial orientation are introduced. This seems strange in view of the fact our brain allows facial identification in split seconds by an as yet incompletely understood process. We suggest here that improved computer recognition in shorter times might be accomplished by comparing a given face with the norm of a human face and then identifying things by only the differences. Go HERE for our thoughts on such a process.

May 17, 2012-How does the Concept Life and Evolution not violate the Law of Entropy?

The concept of life and its accompanyting evolution seems to be in contradiction to the Law of Entropy which requires that all systems move in time from an ordered state (low Entropy) to one of disorder (high Entropy). After first discussing the very slow rate of change produced by evolution over thousands of generations , we show how life and accompanying evolution is indeed possible without a violation of the Entropy Law by considering the entire environment . Go HERE for a discussion of the details.

May 27, 2012-What is the Difference between a Mean and a Median?

When reading articles in the popular press concerning population trends, wealth distribution , and weather records one quite often encounters the terms mean and median without the articles explaining the difference . This can lead to confusion on part of the readers and in addition is sometimes used to hide certain facts such as the true unemployment rate and the great wealth disparity existing between diffenert segments of the US population. We discuss HERE these two concepts in greater detail and show how highly skewed input data can lead to large differences between the median and mean of a data set. The AGM method for integral evaluation and the Pareto Curve used by economists is also discussed.

June 17, 2012-What are Pareto Curves and how are they be used to discuss Wealth Distribution?

Over a hundred years ago the Italian engineer and economist Vilfredo Pareto noticed that some 20% of any population typically controls about 80% of the wealth. His statistical analysis led to the power law curve y=x^{n},
where y is the fraction of the wealth owned by fraction x of
the population. The constant n is a measure of the inequality
of the wealth distribution with n=1 corresponding to an
egalitarian society where wealth is evenly distributed to
n>>1 for increasing income inequality. According to
Pareto, the n=8 curve appeares closest to equilibrium
conditions. At the present time the value for US wealth
distribution lies between n=16 and n=32 and hence is out of
wack with equilibrium condition and calls for adjustments. Go HERE for
further discussions.

July 4, 2012-What is the Relation between Dip-Stick Readings and the Volume of Gasoline left in a Storage Tank?

Several years ago one of our undergraduate students who was working part-time at a local gas station, asked me to explain to him the relation between dip-stick readings and the volume of gasoline remaing in an underground storage tank. In looking at the problem it became clear that one should expect a non-linear relation between fluid level and fluid volume. The relation depends very much on the type of tank cross-section one is dealing with. Go HERE to see some detailed calculations for both cylindrical cross-section and spherical cross-section tanks.

July 17, 2012-How were the Pyramids at Giza constructed using only 2500 BC Technology?

One of the most impressive sights in the world are the three pyramids on the Giza Plateau just west of Cairo, Egypt. People have speculated for years how a civilization having not yet invented the wheel nor familiar with iron tools and other simple machines such as pulleys could possibly have been able to construct such massive monuments. We show** ****HERE** how
this was indeed possible and how primitive technologies
including the use of ropes, inclined planes, sleds, and oil
lubrication were used in conjunction with a masive labour
force to construct these pyramids. We also show how a limited
knowledge of elementary astronomy allowed the builders to allign
these pyramids precisely with a north-south axis.

**August 14, 2012-What are the characteristics of an ICBM's
Trajectory in its Flight from Launch to Impact ?**

We examine the laws of conservation of energy and of momentum to discuss the properties of an ICBM trajectory. Using a minimum of mathematics , we obtain explicit formulas for the height H reached by the missile and the time to impact . An important parameter entering the analysis is the non-dimensional parameter alpha=2gR/V_{o}^{2}, where R is the earth
radius, V_{o }the launch speed and g the acceleration
of gravity. Go **HERE** for details of the
discussion including the effect of launch angle beta.

**August 19, 2012-What shaped Tiles can be used to cover a
Flat Surface without leaving Gaps?**

An interesting problem in 2D geometry is what type of tiles can be used to cover a flat surface (such as a floor) without leaving empty spaces between the tiles. Square and rectangular tiles are obvious examples which can do this. But as we will discuss**HERE** there are an infinity
number of other configurations which can produce a contiguous
array. We show how one can use a square and rhomboidal base
pattern to generate some intricate tile patterns which meet this
no gap condition. Also regular polygons are used to generate two
tile configurations in the shape of kites and arrows.

**September 26, 2012-What is the Sagitta and how
can it be used to calculate certain Dimensions associated
with Circles and Spheres?**

When dealing with geometrical figures such as circles and spheres one is often interested in the maximum distance from the figure edge to a chord running beteween two points A and B also located on the edge. This quantity is known as the Sagitta (from the Latin word sagittarius for arrow) and has important applications in architecture and line of sight radar among other areas. Go**HER****E** for details of
our discussion an this topic. Among other subjects, we briefly
look at the maximum depth a straight line tunnel of length 2L
dug between two points on earth will have.

**September 27, 2012-What are the Characteristics of a
Straight-Line Tunnel dug through the Earth between Two Points
on its Surface?**

Most of you who have taken an introductory physics
course during your life know that if a mass is dropped
into an imaginary shaft drilled through the earth center that
the mass will undergo simple harmonmic motion converting
potential energy to kinetic energy and visa versa and will make
the round trip in about 85 minutes. The same continues to hold
true for a mass moving in an off-center shaft . Such a shaft may
be looked at as a tunnel connecting two points A and B on the
earth's surface. We discuss **HERE **some of the
properties of such a tunnel and look in detail at the case of a
204 mile long tunnel between Washinghton DC and New York
City and passing some 1.3 miles under
Philadeplphia.Theoretically a mass moving through such a tunnel
would be cost free assuming all friction can be eliminated.

November 26, 2012-How does one distinguish between a Composite and a Prime Number when the Number under consideration becomes large?

**November 28, 2012-How much will
the Sea Level rise if all the ice in the Polar Caps were to
melt?**

Although the validity of the theory of global warming is still in some dispute, there is little question that ocean levels have varied from much below the present level during the ice ages and above the present during the times of the dinosauers. We look**HERE** at what the
maximum sea level rise would be if all ice in the Arctic and
Antarctic regions were to melt. The answer turns out to be
99 meters. This would put most coastal regions and cities under
water. However, for this to happen it would take thousands of
years giving man sufficient time to evacuate and rebuild.
We also briefly discuss beneficial effects of global warming
including increased agricultural production and recovery of
fresh water from icebergs.

**December 14,
2012-What is a Fractal Square and how is it Constructed?**

We examine a new
type of fractal consisting of the superposition of
progressively smaller squares whose locations are dictated
by very simple rules. Termed a Fractal Square it is an
extention of the well-known Koch Snowflake but unlike it has
finite area and either infinite or finite perimeter
depending on the magnitude of the fraction factor f between
neighboring generations. Some interesting graphics are
obtained including an intriguing figure which I name the
Black Snowflake. In addition, we show how such fractal
squares can be constructed by simple genetic algorithms. Go **HERE**
for the details.

December 28, 2012- What are the four basic Temperature Scales and how does one convert between them?

**February
25, 2013-What is Mental Arithmetic and how does it work?**

One often encounters situations where the ability to make quick mental math calculations without the aid of pocket calculators or pen and paper come in handy. For example , calculating the amount of an 18% tip on a dinner at a restaurant or the miles per gallon one's car has achieved during a recent trip. We discuss HERE how such simple mathematical calculations involving just addition, subtraction, multiplication, and division are performed mentally. Also we show how quick approximations to certain mathematical questions can be obtained without outside help or before someone pulls out an iphone to calculate or google things.

**April
3, 2013-What are the Negative Effects
of Quantitative Easing to Infinty? **

Those believing in Keynesian Economivcs claim that when a country is in recession or depression the way to get out of it is to devalue the currency by running the printing presses. Ben Bernanke and the Federal Reserve have been doing this since 2008 at an ever increasing rate with little to show for it except a dilution of the currencty by over 300%, the stoking of a stock market mania , and an artificial increase in bank earnings. Their policies have not helped unemployment and have had the direct effect of depriving retirees of a reasonable return on their life savings by running an essentially zero interest rate policy. It is time for the Fed to change course and to do so before international currency wars erupt and the dollar looses the little remaining value it has left. The best solution would be to immediately clean house at the Fedreal Reserve and in particular remove Chairman Bernanke before further damage is done. Go HERE for the details of the discussion.

**April 8, 2013-How do the Seasons
come about and what is the procedure
for locating the North and South
Celestial Poles?**

If you ask individuals how is the earth's tilt axis is related to seasonal changes and how one locates the north and south poles on the Celestial Sphere, most will not be able to answer. We give HERE a brief answer to these questions including how to determine the local latitude by measuing the altitude of the celestial poles above the horizon. A time lapse photo showing the circular paths stars take in the night sky about the south celestial pole is also presented.

**April 14, 2013-What are the Platonic Solids and how may
they be constructed by Electronic Computer using a Guiding
Sphere?**

** **

There are five basic Platonic Convex Polyhedra known as the Tetrahedron, the Hexahedron, the Octahedron, the Dodecahedron, and the Icosahedron. They were discovered by the ancient Greeks and form the starting point for most most discussions on 3D geometry. We show how these solids can be constructed via computer by using a guiding sphere on which all vertices of these polyhedra are located. Go HERE for the details. We also show how one can use cardboard cut-outs and wood polygons to construct these polyhedra .

**June 6, 2013-What is the Volume and Surface Area of
Standard Pyramids?**

** **

We use simple geometry and elementary calculus to determine the volume and surface areas of pyramids consisting of a regular polygon base and a vertex placed at height H above the centroid of the base. The results show that all of these pyramids have a volume equal to the product of the base area times the height divided by three. The properties of the square base Great Pyramid of Cheops at Giza are also discussed including the volume of such pyramids during construction. Go HERE for the details.

**June 19, 2013-Does the recently revealed Program of
Massive Surveilance by NSA constitute a violation of the 4th
Amenme**nt?

It has now been several weeks since the Guardian Newspaper's revalation of massive warantless spying by the National Security Agency(NSA) on all Americans . A heavy defense of these actions has been mounted by the mass media and the Executive, Legislative and Judicial Branches of the US Government. The essence of their arguments being that such surveilance is neseccary to protect against acts of terrorism. We are expected to believe the recent testimony by General Keith Alexander that this program has indeed protected us from over fifty attacks although he could not tell us what they were because its 'secret'. There seems to be only a minority including myself who recognize that there has been a violation of the 4th Amenment to the US Constitution and the__ right to privacy__. Go HERE to see our discussion on
this matter as I wrote about it several weeks ago.

**June 24, 2013-Does it make Economic Sense
to build new
Ocean-Connecting
Canal across
Nicaragua ?**

The Nicaraguan government has recently agreed to the building of a canal across southern Nicaragua by a Hong Kong based company. We discuss HERE the ecomomics of such a proposal and compare it with the cost of modernizing the Panama Canal. The difficulties with the construction of a trans-Nicaraguan canal, which would be about twice the length of the Panama Canal, would be the need to cut through higher terrain than in Panama and also be subject to potential earthquake damage, not to mention border disputes with Coasta Rica. We conclude that a more economic approach for all involved would be to not undertake the construction of such a canal but rather increase the size of existing locks of the Panama Canal in order to accomidate larger ships. The financial support for such an undertaking could come from a world wide consortium of companies, not unlike the AirBus consortium, if Panama can be made to agree.

July 20, 2013-At what height H above the equator must a satellite be placed in order to be Geosynchronous?

A geosynchronous satellite is one whose angular velocity matches that of the earth's rotation rate of

ω=2π /(365.25x24x3600) r/s. Using Newton's second law one finds that such a match occurs when the satellite height H above the equator is H=R{-1+[g/ωR]^{1/3}},
where g is the
acceleration
of gravity and
R the earth
radius. The
height turns
out to be
about 22
thousand
miles. Go HERE
for the
details of the
discussion,
including
finding the
orbital period
for satellites
in a circular
orbit at any
height and the
expected delay
in two-way
conversations
between any
two points on
earth using
geosynchronous
satellites.

**July 24, 2013-What are the Conic Sections and how
are they derived mathematically?**

**August 21, 2013-What is Number Theory and what are some of
its most important Results?**

** **

###

###

Number Theory is that branch of mathematics which deals with the relation between the positive integers 1,2,3,4,5.. . It has become of more practical importance in recent years with the advent of high speed electronic computers and the use of digital encryption. Basically the integers break up into two groups- the prime numbers such as 3,5,7,11,13.. and the composite numbers 4,6,8,9,10,.. We state some of the known properties for these groups and introduce some new concepts such as the number fraction f(n) and the plotting of Q primes along a hexagonal spiral. Go HERE for the details.

###

###

**August
25, 2013-What is the Volume of an Irregular Hexahedron such as
those seen in cut Cantaloupe Pi**

###

Have you ever noticed that the cut melon pieces
one has for breakfast are often in the form of irregular
six-sided polyhedra ? It is natural to ask what is their
volume. Clearly the problem is a lot more involved than just
calculating the volume of a rectangular solid and will require
some vector analysis to obtain the answer. We carry out such an
analysis HERE. It is shown that the
irregular hexahedron can be broken up into six irregular
pyramids having quadrangle bases and a common vertex placed at a
point within the hexahedron. The volume of each pyramid is found
to be Vol=(1/6){|V_{2}ï¿½(V_{1}xV_{3})|+|V_{4}ï¿½(V_{1}xV_{3})|},
where V_{1},V_{2},V_{3}, and V_{4}
are vectors defining the side lengths of a pyramid.

###

**September 14, 2013-What
is the Struve Geodetic Arc?**

Prior to the advent of
earth satellites , laser reflectors and radar, the standard way
to determine the precise distance between two points on
earth was by means of triangulation based on a known base
line and the ability of theodolites to accurately measure
the angles from the ends of the base line to a third
point. We discuss HERE
how such measurements are carried out and pay
special attention to one of the longest triangle chains ever
divised and now referred to as the Struve Geodetic Arc. A
simulation of the triagulation procedure using just three
oblique triangles is discussed in detail. Also we indicate how
the elevations of mountains such as Mt. Everest or Mt.McKinley
were originally obtained.

**September 16, 2013-What
is the Area of An Irregular S-Sided Polygon?**

We examine the area of
irregular polygons with S sides by evaluationg the areas of
T=S-2 sub-triangles constructed by L=S-3 straight lines. Several
different examples are considered and a brief discussion of how
land areas are determined knowing only the length between
neighboring vertices and the angles the connecting lines between
vertices make relative to a north-south line. It is also shown
how these area calculations can be greatly simpilfied by making
use of symmetry. Go HERE for
the details.

October 3, 2013-What is the Future of the US National Debt?

Perhaps the most critical economic problem facing this country at the present time is a run-away national debt which has increased by a factor of over fifty in the last half century and at the present time stands at 17.065 trillion dollars and accelerating. We make a linear extrapolation of this debt into the near future and show that in about seven years the US Treasury will be paying well over a trillion dollars a year just to service this debt. Clearly this is unsustainable and will lead to a default of the dollar. Printing of money by Ben Bernanke will work only as long as the lenders don't recognize the futility of his actions and recognize that their dollar holdings are being depreciated on a daily basis. A calculation based on the assumption that the US monetary system will collapse when the debt reaches 130% of GDP, suggests this point will be reached shortly before 2020. Go**HERE** for the details of the discussion.

**October
28, 2013-How can one use Generalizations to establish
Mathematical Principles?**

In the history of mathematics and especially in number theory investigators have established many mathematical principles based on the generalization of observations based on special cases. In many instances such generalizartion have proven to be valid for all integers and have led to the establishment of certain universal principles while in other cases the generalizations have failed. We want here to discuss some of the more important results obtained by generalization in number theory and then add a few thoughts on our own generalizations involving prime numbers. Go**HERE** for the discussions.

**November 13, 2013-What is the Future of Fission
Generated Nuclear Power after Chernobyl and Fukushima?**

The recent nuclear reactor accident at Fukushima and
earlier ones at Three-Mile Island and Chernobyl have called into
question the future use of nuclear fission as a viable energy
source. Since many of the pro and con arguments appearing in the
literature and on the internet are based on an incomplete
understanding of the facts, we want **HERE** to help clarify things
by discussing the basic process involved in nuclear power
generation, the radioactive species produced in the process,
their half-lives, detection methods , and the basic units of
radiation measurement. After presenting these in some detail, we
conclude with a short summary of our own views on the future of
nuclear power and its necessary co-existence with fossil fuel
energy generation.

November 28, 2013 (Thanksgiving)-How does one use Vector Operations to determine Angles between Straight Lines in Space?

It is known that any straight line connecting points [x_{1},y_{1},z_{1}]
and [x_{2},y_{2},z_{2}] in space can be
represented by the vector V=i(x_{2}-x_{1})+j(y_{2}-y_{1})+k(z_{2}-z_{1}).
Furthermore the scalar product of any two vectors normalized to
unit length equals the cosine of the angle between them. We use
this fact to treat a variety of problems requiring a precise
determination of the angle between lines. Among the problems
discussed are those of Eratosthenes for determing the
circumference of the earth, finding the shortest distance
between a parabola and a straight line, and locating the center
of mass of a regular tetrahedron. Go **HERE
**for the details.ast

December 6, 2013-How does one determine the Volume of an N sided Polyhedron?

A polyhedron is any 3D solid which has four or
more flat faces. The cube, icosaheron and dodecahedron are
examples of regular polyhedra. We show **HERE** how to calculate the
volume of any polyghedron with a number of sub-tetrahedra whose
exact volume are given by one sixths of the scalar triple
product of the three vectors representing its three lateral
edges. Several different examples are examined including the
volume of a triangle base truncated pyramid.

**2014****
**

anuary 1, 2014-What is the Relationship** between Integer
Spirals and Spider Webs?**

In earlier articless we have studied the properties of
certain spirals and radial lines intersecting them. Such curves
can be used to distinguish composite from prime numbers. It is
our purpose here to show the similarities of these curved
structures to spider webs by approximating an actual spider web
by an Integer Spiral pattern. Go **HERE** for the details. Also
a brief discussion is given of how such a mathematical
approximation may lead to a new type of electrostatic dust
cleaner.

**January 23, 2014-What is the Mathematics behind the
popular Numbers Game Sudoku?**

As many of you are aware, Sudoku is a very popular and addictive number puzzle in which one is asked to find all integers in a square array given a certain number of starting values. It is governed by several simple rules and is related to Euler's Latin Square.We show that the solution procedure is simply an extensive manipulation of elements in an n x n square matrix and involves finding the compliment to known numbers found in the row, column and sub-matrix which contains the element a_{i,j}. After the possible solutions for an
element have been found, one eliminates most of these by simple
rules to yield just one final answer. The sum of the n
elements in any row, column or submatrix will always be equals
to S(n)=n(n+1)/2. Go **HERE** for the details of
our discussion.

February 1, 2014-How did the Ancient Egyptians measure Slopes during Pyramid Construction?

It is well known that about 4000 years ago the
ancient Egyptians started building tombs for their kings in the
form of huge square base pyramids with precise equal slopes for
the slanting four sides. How was this accomplished with the very
primative tools at their disposal ? The answer for slope
determination was use of the Egyptian Level and Square which was
essentially an A-frame structure where the two sides intersect
at a right angle and a blumb-bob is suspended from this
intersection. We describe how this instrument was used to
measure slopes(and hence angles) of surfaces relative to the
horizontal. We also show how their use of length measurements
based on cubits, palms and digits allowed for the definition of
a slope in terms of the Seked. Go **HERE** for details of the
discussion and also to view a modern version of the Egyptian
Level-Square which we just constructed in our workshop.

February 20, 2014-What are Magic Squares and how are they Constructed?

A magic square is an array of integers having n
rows and n columns. The square obeys the law that the integer
elements in each row, column, and diagonal add up to S=n(n^{2}+1)/2
and that no integer appears more than once in the n x n array.
We discuss how these squares are constructed using both a Sudoku
approach and one involving adding together smaller magic squares
to form larger ones. We discuss in detail n=3, 4 and 6 magic
squares. Go **HERE** for details of the
discussions.

March 1, 2014-What is the relation between Grade and Angle of an Inclined Surface?

The standard way of measuring the incline of a
surface is to define its grade which represents the ratio G of
the vertical height 'b' of the incline to the
corresponding horizontal distance 'a'. It is essentially a
tangent measure which converts to the incline angle in radians
by the simple formula
α=arctan(G).
We graph this function and its approximation plus present a
table relating the grade to the incline angle. Also the
grades of the PA turnpike, railway beds, and pyramid
surfaces are discussed. Go **HERE**** **for the
details.

March 9, 2014-What is a Temperature Well and how can its properties be used to build a better Heat Exchanger?

We examine the heat flow into a 1D Temperature
Well and** **show** **that the early phases produce extremely
large heat transfers. This information is used to suggest a
way to quickly remove heat from a solid in a way analogoues
to the quenching of a hot iron slab by dunking it into a
batch of cold water. A design involving periodic injections
of fluid into the well is then used to suggest a possible
design for an improved a fluid-fluid or fluid solid heat
exchanger. Go **HERE** for the details.

**May 7, 2014-Is there an alternate way to generate and
easily store large Prime Numbers?**

We discuss a new way to generate and then store in compact
form large prime numbers. The technique depends on using the
first m digits of combinations of several different irrational
numbers including Pi ,sqrt(2), ln(2) and exp(1). Calling the
combination of a group of these mathatical constants taken out
to m places A and then eliminating the decimal point by
multiplying by 10^{m}, produces a large number N
whose digits are essentially random.The N mod(6) value is noted
and a small constant 'a' is added to N to make it confirnm to
the fact that all primes above three must have the form 6n+1 or
6n-1. Next carrying out a search of the sequence N+a+6k over a
small range for integer k will produce primes for certain
values of k. We show how to generate a number of large primes
and demonstrate how they may can be stored and also how they are
connected with public key cryptograhpy. Go HERE for the details.

**May 25, 2014-What are the
Characteristics of Financial Bubbles?**

We look at the characteristics of finacial
bubbles starting with the Dutch Tulip Mania in Holland during
the 1630s, through the South-Sea Bubble of 1720, to the 1929
Market Crash and the latest Bernanke-Yellen Bubble. Typically
one finds that these bubbles are characterized by a steep
acceleration in price followed a rapid decline which
catches most investors and speculators by surprise and can
lead to financial ruin. Such finacial bubbles can occur
for metals, stocks, other commodities, art, and home prices.
Typically they last just a few years, but new and different
bubbles will always re-emerge. Gold at $1800 per oz in 2011
and the dot.com peak of technollogy stocks in 2000 are clear
examples of earlier financial bubbles. Go **HERE** for details of our
discussion.

May 27, 2014-What is the Area of any N sided Irregular Polygon?

A polygon is a closed 2D figure consisting of
N vertexes connected to each other by straight line
boundaries. If the lines all have equal lengths then one has
a regular polygon such as a standard hexagon or octagon.
Irregular polygons have border lines of different lengths.
It is shown that any polygon can have its area expressed as
the sum of the areas of sub-triangles which in turn can be
evaluated quickly using a vector product of two of its
sides. We discuss in some detail the area of a hexagon, a
pentagram, and a four sided irregular quadrangle. Go **HERE** for the details

**J****UNE 1, 2014-What is
Thermal Pumping?**

About 30 years ago I found a new way to
transfer heat from a hot to a cold reservoir by oscillating
a fluid in a bundle of open ended capillaries connecting the
reservoirs and have termed the process Thermal Pumping. It
works by the coupling a large periodic transverse conduction
heat flow with a time-averaged axial convection process
confined mainly to the Stokes boundary layers formed along
the conduit walls. Go **HERE** to understand the
details of this heat transfer process.

JUNE 11, 2014-What are Sequences and how are they Generated?

A sequence is an array of numbers whose
elements f[n] are generated by formulas of the form
f[n]=F(n), where F is a specified function of the integer n.
Simple examples are the sequence {1, -1, 1, -1, 1, -1, 1,...}
generated by F=(-1)^{n} and the sequence {1, 4, 9, 16,
25,...} generated by F=n^{2}. We look **HERE** at more complicated
sequences including ones where the elements are complex numbers.
One of the more interseting new sequences examined has
F(n)=sum((n+k)!/(n-k)!, k=0..n-1).

**June 22, 2014-How does one generate Sudoku Squares?**

Sudoku is a math game in which one is given a few numbers in a square array and asked to find the remaining ones in order to generate a complete Sudoku Square. These squares have the property that all rows and columns in the n x n array contain each of the n numbers only once. The n sub-matrices of the square also contain each of the n symbols only once. It is our purpose here to look at the reverse (and much easier) problem of first generating some generic Sudoku Squares, examine their properties, and then use them to construct Sudoku Puzzles. Go**HERE
**for the details of the discussion.

July 6, 2014-How does one determine Latitude and Date using only Astronomical Observations?

Suppose one was stranded on a desert island without the
benefit of a GPS receiver, a nautical Almanac, and any form of
electronic receiver or transmitter. How would such an individual
(say Robinson Crusoe) find his latitude, the current date, and
hour of the day?** **The answer is via
astronomical observations. We discuss **HERE** how this would be
accomplished using only the most primitive observation methods
consisting of a pair of vertical posts oriented in a north-south
direction as defined by Polaris.

**July 9, 2014-How does one use the Big Dipper in Ursa
Major to determine Time?**

We show how a pointer defined by a line going
through the North Star Polaris and the two front stars of the
Big Dipper yield an accurate 24 hour clock. The pointer moves
counter-clockwise at the rate of one degree evry four minutes(15
degrees per hour). This astronomical clock is very precise in
measuring time increments since one does not need an accurate
local time to carry out such measurements.Go **HERE** for the details of the
discussion.

**July 16, 2014-What are Sun-Shadow Trajectories and how
are they calculated?**

The shawdow projected on a horizontal
surface by a vertical pole depends on the local latitude(LAT),
hour angle(HA), and the sun's declination(DEC). We examine the
time-dependent trajectory of the length of this shadow at local
noon(HA=0) throughout the year. Shadow trajectories for
Gainesville, FL, Washington, DC and Moscow, Russia are presented
in detail. The larger the latitude of the observation point the
longer the shadow line becomes at Winter Solstice. It is also
shown how the shadow length between two different places in the
Norethern Hemisphere along the same longitude can be used to
calculate the distance between them. Go **HERE** for the
details.

**August 2, 2014-What are Reuleaux Trian****gles
and how may they be used to make Square Holes? **

A Reuleaux Triangle is constructed from an equilateral triangle of side-length s by drawing circular arcs of this radius from each of the three vertexes. When such a triangle has its centroid follow a closed path about a fixed rotation axis, the triangle can sweep out a near perfect square. This allows one to use such triangles to drill holes of nearly square cross-section. We show**HERE** how such drills
function by looking at both the mathematics behind Reuleaux
Triangles and by demonstrating their rotational behavior
via a wooden model.

August 12, 2014-How does one determine the Curvature of a Function f(x) and its corresponding Evolute?

In introductory calculus one
learns about the curvature of 2D functions y=f(x). The
treatment is many times incomplete and students often have
trouble understanding the 3/2 power term arising in the
discussions. We help clarify this problem here by deriving
in detail the radius of curvature
**and the evolute of
any continuous function f(x). Among other curves
discussed ** **in detail are the parabola and
also the Witch of Agnesi. Go**** ****HERE**** ****for
details of the **** ****discussion.**

**August 30, 2014-What i s the Gaussian and how is it
Derived?**

We look at a modified Pascal triangle to derive the Normal
Gaussian Distribution for a continuous function. After taking
the appropriate limit, on finds the function F(z)={1/sqrt(Pi)}
exp(-z^{2}). Its properties are discussed, some
integrals involving F(z) are presented, and finally its relation
to the probability density distribution P(w) are looked at.
Values under the P(w) curve at one, two , and three standard
deviations are also discussed. Go **HERE** for the details.

**September 1, 2014-What is an Asymmetric Pascal Triangle
and how can it be Used to Predict Gambling Odds?**

We examine a modification of the Pascal Triangle where each
element is defined by the binomial expression
C[n,m,a,b]=n!/{m!(n-m)!}{a^{n}b^{(n-m)}.
Asymmetry is induced into the triangle whenever 'a' does not
equal 'b'. Gambling involving Coin Flipping and Roulette are
discussed in some detail including the probability of winning or
loosing and the payouts for wins. Go**
HERE** for details of the discussion.

**September 4, 2014-What was Stonehenge's Purpose?**

The Neolithic monument Stonehenge near Salisbury England was built about 4500 years ago. It's reason for being have fluctuated over the years from that of a primitive astronomical observatory to a place of worship and burials. We look here at the basic equations for determining the azimuth at sunrise and sunset during the solstices, to show that the orientation of the monument strongly supports its role as an instrument for measuring the time of the Summer and Winter Solstice, but nothing more elaborate than that. We find. among other things, that the hour angle (HA) along sun's path during the winter solstice is very close to one radian (3hrs 49min) after local noon. Go**HERE
**for the discussion.

**HERE** for** **the details.

December 18, 2014-What are the Dimensions of the Great Pyramid of Cheops?

We re-look at the dimensions of the Great Pyramid
at Giza and show that there is no need for hidden mathematical
constants to account for its shape. All that is required to
build these is the capability to accurately measure
length(cubit) and angle(seked) already known to the pyramid
builders in 2500 BC. Furthermore we show how knowledge of
a 3-4-5 right triangle is sufficient to build a pyramid very
close in shape to that of the Great Pyramid. Go** HERE**
for details of the discussion.

**2015**

January 28, 2015- How much does a Football's Pressure drop with Decrease in Air Temperature?

A major controversy among football fans at the moment
concerns what has been termed Deflategate. It involves the
question of whether the underinflated footballs used by
the New England Patriots in their recent trouncing of the
Indianapolis Colts was due to temperature changes or deliberate
deflation of several of the footballs by the Patriot
staff. We look here at the problem from a technical viewpoint to
see if the measured 2lb/in2 deflation in some of the gameballs
could have been caused by temperature changes alone as claimed
by Patriots' head coach Bill Belichick. Go **HERE** for the discussions
concerning this problem and some suggestions for future ball
handling protocol.

January 31, 2015-How does one Construct 2D Figures using only Straight Edge and Compass?

It is known that one can construct an infinite number of
two-dimensional figures using just a straight edge and compass.
Classic examples of such figures include n sided regular
polygons and squares and their vaiations including the Rhombus.
We want here to show how such constructions are carried out.
Among other figures we show how a pentagon can be generated and
how a combination of straight lines and circles can lead
to some interesting computer art. Go **HERE** for the details.

February 7, 2015-What is the Gaussian, its Properties and better known Applications?

We examine the Gaussian y(x)=a exp-{(x-b)^{2}/c}
including the special case of the Probability Density Function.
The Gaussian is shown to have a bell shape with its
maximum at x=b and a standard deviation of sigma=sqrt(c/2). It
is shown how one measures IQ in terms of the number of standard
deviations from the peak of the Gaussian. Several identities
involving y(x) are derived. Go **HERE**
for the details.

February 9, 2015-What is an Integer Spiral and what are some of its more important Properties?

We examine the complex function F[z,n,m]=n exp(i n Pi/m) for
integer n and m. It produces a series of points in the complex
plane which represent all positive integers n=1,2,3,..
when m is set to an integer. Among other results, we detail the
historical development of integer spirals starting with (1+i)^{n}.
In particular, we concentrate on F[z,n,4] and F[z,n,3] .
For the first time it is shown that F[z,n,3] represents all
positive integers with all primes above n=3 falling along just
two straight lines instead of being scrambled over the entire z
plane as they are in a standard Ulam spiral. Go** HERE**
for the details.

February 23, 2015-What are some of the Areas which can be created by the Intersection of Circles and Straight Lines?

We look at some of the areas which can be created by
combining circles and straight lines. In particular, we
emphasize different types of crescents. Also Heron's Formula for
triangles is derived and used to find the area of an equilateral
triangle incribed in a circle. In addition it is shown how to
find the central triangular scalloped triangle formed by three
equal radius circles just touching each other. Go **HERE** for the details.

**April 15, 2015-How can one Construct an Infinite Number of 2D
Curves using only the Length and Orientation of its Straight
Line Edge Components** **?**

**April 20, 2015-What are the basic Mathematical Constants and
how are they Derived?**

We look at the basic six mathematical constants, show how
they are derived, and how they can be evaluated to any desired
degree of accuracy using some new approaches. We give their
values to 100 digit accuracy using various approaches including
iteration, continued fractions, and integrals involving Legendre
polynomials. Go **HERE** for details of the
discussion.

**May 8, 2015-What are some of the Shortcuts one can Employ
to speed up Mental Arithmetic Calculations?**

We look at the four basic mathematical manipulations of addition, subtraction, multiplication, and subtraction and show how these operations can be performed quickly and accurately in one's mind without use of pencil and paper or electronic calculator. Also we indicate how to take powers and roots of integers plus calculate percentages of numbers. Go**HERE** for details of the
discussions.

May 13, 2015-How does one find the Focal Points of Parabolic and Elliptic Reflectors?

We use the vector form of the reflection law to
find the focal point of a parabolic reflector and the two focal
points of an ellipsoid. Also we discuss how such focusing
capabilities are used in reflector telescopes such as the Hubble
and in kidney stone pulverizers. Our own work on centrifugally
spun large diameter parabolic mirrors is also discussed. Go **HERE** for the details.

**May 17, 2015-What are some additional properties of 2D curves
generated by Genetic like Algorithms?**

We construct some two dimensional curves formed by the concatenation of straight line segments connected to each other by specified angles. Figures like a Swiss cross, a multiple pulse function, a magnified and upside down pentagon, and a hexagonal spriral are generated. Closed curves are formed when the sum of the connection angles after m elements add up to 2 pi. Go**HERE** for the details.

**May 20, 2015-How can a Train's Speed and other factors cause
Derailment in a Curve?**

The recent train derailment near Philadelphia which caused
eight deaths and injuries to many others was a clear case of a
train's excessive speed entering a sharp curve. We discuss **HERE** what the forces are
involved in such a derailment and what speeds can be considered
safe for a given radius curve. The use of track embankment
at a curve is also discussed. The main non-dimensional
quantities encountered in the problem are V^{2}/gR
and w/2H. Here V is the train's speed, g the acceleration of
gravity, R the curve radius, w the axle width, and H the
distance from the track to the center of mass.

August 1, 2015-How can one use a Square to generate intricate Fractal Patterns?

We show how one can take a simple square to construct
different fractal patterns. Among the more interesting are the
Black Snowflake and the Dancing Clown. The fractals are
generated by a generation growth process . We show how the nth
generation of squares must be smaller than a certain amount to
prevent overlap in the (n+1) generation. Go to **MORE-FRACTAL-SQUARES.pdf**
for further details.

August 21, 2015-How does one generate complicated 2D curves via a concatination of straight line elements?

We re-examine the problem of generating two dimensional
curves by genetic codes which define only the straight line
increments and their orientaion. In addition to regular
polygons, we generate a rectangular pulse train, a staircase
function and several versions of a Swiss cross. Also more
complicated figures such as a square spiral and a crosslet are
discussed. Go **HERE** for the details.

October 1, 2015-How does one determine the Area of an N sided Polygon?

We look some more at determining the area of both regular
and irregular N sided polygons. It is shown that partitioning
such polygons into T=N-2 oblique triangles and then finding the
areas of such triangles, allows one to uniquely determine the
area of any polygon. Several different examples are considered
including an irregular quadrangle and also N sided regular
polygons. Go **HERE** for the details.

October 15, 2015-What is a Cruise Missile, its development History, and future Directions?

A cruise missile is any unmanned aerodynamic vehicle capable
of delivering a warhead over long distances flying at low
altitude but at high speed. We briefly discuss its history
starting with the V-1 and take the discussion through the latest
and most sophisticated supersonic BrahMos misile. Also,
the US Tomahawk sea-launched missile is discussed and its needed
improvements pointed out. Go **HERE** for the details.

**December 1, 2015- How can one transmit a large Semi-Prime
N and its prime components p and q in
secret?**

We define any semi-prime N=pq via two new parameters
M=(p+q)/(2sqrt(N)) and K=(q-p)/(2N) making it possible to
transmit N, p, and q in secret. When M and K are used as a
public key in cryptography only a receiver familiar with these
parameters will be able to recover the prime p involved
and use it to send encrypted messages back to the sender without
a third party being able to figure out the message being sent.
Go **HERE** for details of the
discussion.

December 12, 2015-How can one find Super-Composites?

It is well known that all positive integers are either
primes or composites. Those numbers N which can be divided
only by themselves and one are primes such as 5, 11, and 41
while numbers having three or more divisors are composites such
as 6,12,60. We can distinguish any number by looking at its
unique number fraction f(N)={sigma(N)-N-1}/N, where sigma(N) is
the divisor function of number theory. Primes have f(N)=0 while
composites will typically have values of f(N) near 0.5. A small
fraction of the composites will yield values of f(N) in excess
of 1,2. We call these numbers super-composites. It is our
purpose here to determine some aditional properties of f(N) and
in in particular find some formulas capable of generating
super-composites.Go **HERE** for the details.

December 20, 2015-What is the Black Snowflake and Cellular Automata generated by it?

A cellular automaton is a structure defined by n generations
beginning with a single closed area as the first
generation followed by later generations obeying a certain
specified growth program. A good example of such a structure is
the Black Snowflake which uses a square of side-length s as its
starting point with later generations consisting of ever smaller
squares with side-length sf , where 0<f<1. An infinite
number of different intricate patterns can result. We develop
several additional cellular automata using the Black Snowflake
as the template. Go **HERE** for the** **discussion.

**
2016**
**
**

January 15, 2016-What are the Seven Rules for Successful Stock Market Investing?

We discuss the basic investment rules I follow for successful investment in equity markets. These rules have evolved over a fifty year period ever since my first stock purchases as a teenager in the mid fiftees. The basic idea is to be long stocks only in bull markets and short stocks only during bear markets. The type of market one is in is determined by looking at the longer term price trend of the market averages when compared to a running lag curve. Investment risks are lowered by not going on margin and dealing with ETFs as opposed to individual stocks. Go**HERE** to learn more about
the details of my investment observations.

**February 1, 1016-What are the Properties of Oblique
Triangles?**

We derive the well known laws for triangles including a
determination of their areas by both geometric and vector means.
Also the formulas for the radii and centers of circles

circumscribing and inscribing a triangle are derived. Next the possibility of tiling with oblique triagular tiles is demonstrated . Also a tile flooring involving the simultaneous use of equilateral and isosceles triangles are shown. Finally we discuss how oblique triangles are important in the functioning of trusses and tripods. Go**HERE** for the details.

**April 17, 2016-How do Probabilities enter Games of Chance?**

We examine the probabilities associated with games of
chance. Starting with the most elementary of such games, namely,
coin flipping, it is shown that to get n heads or tails in a row
has the very low probability of (1/2)^{n}. Although the
probability of any number t through 6 coming up is always p=1/6
when one casts a single die, conditions change dramatically when
dealing with two or more dice being thrown simultaneously. For a
two dice game such as craps the probability of making snake
eyes(1+1) or boxcars(6+6) is only p=1/36. We also look at the
probability of a favorable outcome in both American and European
Roulette .To only play red-black or even-odd is probably an
optimum strategy considering the house advantage. A brief
discussion of short term stock movements indicate that this
is essentually a random process and has no predictive
value.. Go** HERE
**for further details.

May 31,2016-How does one determine the Properties of an Equilateral Triangle using only Intersecting Interior Lines?

We show how, using simple geometry, algebra, and some
analytical methods, to solve a mathematical puzzle recently
posted in the Wall Street Journal. The problem involves finding
the value of the constant d knowing that the line lengths of
lines emanating from the three vertexes of an equilateral
triangle have lengths d, 2d, and 3d/2 when all meet at a fixed
interior point P(x,y). Simple geometry is used to solve part of
the problem while more advanced methods based on Heron's
triangle and analytic forms for circles are also employed. Go **HERE** for details of the
discussion.

June 6, 2016-How do the Pascal Triangle and the Gaussian relate to Games off Chance?

We discuss further how games of chance can be described as
face value S versus possible combinations T and how such
relationships lead to the probability of an event occurring.
Multiple coin flipping, marble withdrawals from a jar, and
rolling of dice are all analyzed in detail.It is shown how
multiple coin tosses lead naturally to the Pascal Triangle and
that the combinations involving multiple die games produce a
Gaussian distribution. The probabilities involved with Russian
Roulette are also discussed. Go **HERE** for the details.

**June 21, 2016-How can one quickly construct segments
of Irrational Numbers to find their neighboring Primes?**

We show how one can take the products of certain irrational
numbers a_{k} taken to their p_{k}th power to
construct an irrational number L of infinite length. By
choosing a segment of length m from this number, one obtains a
large finite length number M which can be conveniently
represented by a short number code. With appropriate
manipulations one can then find the neighboring primes to M.
Calling these p_{1},p_{2},..., one can generate
semi-primes such as N=(p_{a })(p_{b}). Such
numbers can be conveniently used as public keys in RSA
cryptography. Go **HERE** for the details.

**July 9, 2016-What is the Orbit Period of two Gravitational
Masses rotating about their common Mass Center?**

We examine the period of a binary mass system rotating about
its center of mass. Special cases considered are binary stars,
the Earth moving about the Sun, and the behavior of earth
satellites. By balancing the gravitational attractive force with
the centrifugal forces about the barypoint one is able to relate
orbit period to distance between the two masses and the total
value of the masses. The discussions are confined to only
circular orbits which are simpler to discuss than the more
general case of elliptic trajectories. Go **HERE** for the details.

**July 28, 2016-What are Power Towers and the Tetration of
Complex Numbers?**

We consider the complex numbers N=a+ib and ask if
Z=z[infinity] exists when undergoing the iteration

z[n+1}=(a+ib)^{z[n]} , subject to z[0]=1. Such
iterations are equivalent to power towers such as
(1/e)^(1/e)^(1/e)^ etc. For such iterates to converge at
z[infinity] it is necessary that z[n+1]=z[n] for large n.
Once this is achieved one has the explicit form for Z
expressed in terms of the Lambert Function result
Z=W(ln(1/(a+ib)))/ln(1/(a+ib)). The value of Z for sqrt(2) is 2
, for 1/e is W[1]=0.56714329, and for (1+i) is 0.6410+i0.5236.
Go **HERE** for the details.

**August 16, 2016-What is the increment ratio b/a for a
Chord of a Circle inscribing an Equilateral Triangle?**

In this weekend's Wall Street Journal they posed in their
Puzzle Page the problem of finding how the chord to a
circle breaks up into two small increments a+a and one larger
one b when an equilateral triangle is inscribed in the circle..
The question asked was to find the ratio of b/a. The ratio turns
out to equal the Golden Ratio phi=[1+sqrt(5]/2. Go **HERE** for the details of the
solution. We also extend the discussion to ratios obtained for
other inscribed regular polygons including for a standard
square.

**August 25, 2016-What are the Properties of a Cam?**

We examine the properties of cams as they arise in
connection with the opening and closing of intake and
exhaust valves in internal combustion engines. Starting with an
examination of the simplest cam, consisting of a circular disc
rotated at constant angular velocity about an off-center
rotation axis, we proceed on to elliptical type cams capable of
creating conditions where a valve driven by a follower is
capable of remaining shut during a good part of the rotation
cycle. Also we show how the shape of cams with desired on and
off capabilities can be constructed graphically starting with a
chosen elliptical shape as a first approximation. Go **HERE** for the details.

**September 4, 2016-What is the Volume of a glass
Aquarium having twelve identical Rhombic Faces ?**

We examine the latest math problem posted in this week's
Wall Street Journal Puzzle Page. The problem is to find the
fluid volume inside an aquarium constructed from twelve
identical rhombic faces. The edges of the individual faces each
have length 5sqrt(3) and the length ratio of the two diagonals
of a face go as sqrt(2). The problem is readily solved by just a
little geometry and some spatial visualization. Also we work out
a related problem of the filling of a spherical tank, This
latter analysis is related to the low tech dip-stick
measurements used by garage owners to measure the remaining
gasoline in an underground storage tank. Go **HERE** for the details.

**September 14, 2016- What is the Length of the shortest side
of a Right Triangle given the Hypotenuse and Median Lengths ?**

We examine the latest math problem posed in this week's Wall
Street Journal Puzzle Page. The problem is to find the length of
the shortest leg 'a' of a right triangle given a hypotenuse
length of L=10 and a median length of sqrt(ab), where b is the
second leg of the triangle. We use two approaches to solve the
problem. The first is an intuitive approach using symmetry and
simple geometry. The second makes full use of the basic
trigonometric formulas for two oblique sub-triangles. Both yield
the same answer but the first can be considered the more
elegant. Go **HERE **for the details.

**October 1, 2016- How many Diagonals fit into a regular
N-sided Polygon? **We look at a generalization of this
week's Wall Street Puzzle Page question dealing with the number
of unique diagonals D one can draw inside a regular n-sided
polygon.The answer turns out to be D=(n/2)[n-3]. A decagon with
ten sides will have 35 unique diagonals. The lowest number
of diagonals occur for the square where D=2. Go **HERE** for the details of the
discussion.

**October 12, 2016-How many unique Diagonals and Sub-Areas can
one create inside an N sided Polygon?**

** **This is an extention of an article we wrote on this
web page a few days ago. We are interested in finding the number
of unique diagonals D one can draw inside an N sided polygon and
then use this information to determine the number of sub-areas A
created. As expected the number of diagonals and number of
sub-areas increases rapidly with increasing N, It is shown that
N is a quadratic function of diagonal number D given by
D=N(N-3)/2. Although no direct functional relation between N and
A was found, we show how to find A by a counting process
involving the use of colorization. Also, some interesting
designs are created using the sub-areas A. Go **HERE
**for the details.

**October 27, 2016-What is the shortest Path between N Vertexes
of a regular Polygon and its Central Point?**

We examine the shortest path connecting multiple points in
the x-y plane. The particular problem considered deals with
finding the optimum trajectory between n vertexes of a regular
polygon and its center using an approach where the area in the
polygon is broken up into smaller sub-triangles into some of
which we position Steiner Forks which are known to minimize the
path between the three vertexes of any triangle. Squares,
hexagons, and dodecagons are considered. Go **HERE** for the details.

**November 26, 2016-What are the Angles and Areas created by
the Intersection of a Regular Polygon with multiple Circles?**

We solve the latest puzzle posed in the mathematical recreation page of the Nov.19, 2016 issue of the Wall Street Journal and then consider several variations thereof including the spacing left by a bundle of seven circles tightly bound by a regular hexagon. Among other problems considered are Heron's Formula for any oblique triangle whose sides are tangent to a single radius R circle placed within the triangle. Also we show how the Golden Ratio follows naturally from the radius ratio of two different circles intersecting a pentagon. Go**HERE** for the details.

**December 5, 2016-What are the Characteristics of Two-Stroke
Internal Combustion Engines?**

We discuss the properties of two-cycle ( also known
two-stroke ) engines. These devices consist of a single cylinder
inside of which the upward movement of a piston causes a
fuel-air mixture to be compressed. At the maximum compression
point a spark plug ignites the mixture and a downward power
stoke ensues. At the end of the power stroke the burnt gases are
expelled by one port while simultaneously a new atomized fuel
mixture is injected through a second port. These piston
movements are repeated and produce, with aid of a flywheel, a
continuous rotation movement of a connected crank shaft. Heat
removal is achieved via air-cooled fins. All higher powered
four-cycle engines require more intricate liquid cooling.
Two-cycle engines are used mainly for lower power needs such as
for motor-cycles, outboard motors, lawn mowers and chain
saws. Go **HERE** for further details
of the discussion.

**December 19, 2016-How does one Express Numbers in Different
Base Systems?**

We show how one can express a number N in different base b
systems.The main emphasis is on decimal(b=10), binary(b=2) and
hexadecimal(b=16)forms. After showing how one adds, subtracts,
multiplies and divides in such systems, we give the formula for
quickly converting a number between any two different bases.
Finally we also look at the historical used bases of b=20
of the Mayans and the base b=60 of the Sumarians. Go **HERE** for the details of
our discussion.

**December 29, 2016-What is the Relation between a Hexagonal
Spiral and the Prime-Numbers?**

We show that all primes five or greater have the form 6n$\pm $1. This fact makes it possible to locate all but the primes 2 and 3 along two radial lines 6n+1 and 6n+5 which cross the vertexes of a hexagonal spiral. This geometrical picture of all positive integers allows one to quickly distinguish between prime and composite numbers. The combination of hexagonal spiral and radial line is reminiscent of a spider web and is the reason we often refer to this pattern as such. Any odd number of the form 6n+3 and all even numbers must always be composite numbers no matter how large N becomes. Go**HERE** for details of the
discussion.

**
** **2017**

January 20, 2017-What are the Properties of Squares?

We examine the properties of squares, give their
mathematical representation in polar coordinates, their use in
deriving the Pythagorean Theorem, use in construction of more
intricate patterns, and the construction of spirals. In addition
we consider the stability of towers built with cubes and
determine the area ratios when cutting any square with a
straight line. Finally we show a couple paintings by famed
modern artist Jopseph Albers on homage to the square. Go **HERE** for the details.

February 6, 2017-What are the Laws of Geometrical Optics?

We use the Fermat Principle to derive the basic laws of
reflection and refraction for a ray of light striking the
interface between different index of refraction media. After
deriving the reflection law we demonstrate how it may be used to
explain the properties of corner reflectors. Next the
Snell's(alias Reflection) Law is derived and used to show how ,
via ray tracing, one is able to calculate ray deflection as it
propagates through a glass wedge. Finally we conclude by
deriving the first Fresnel formula for the reflection
coefficient at an interface. Go **HERE** for the discussions.

**March 18, 2017-What is the Relation between Archimedes and
Van Ceulen and the Constant Pi?**

Undoubtebly, the most famous mathematical constant is the
number Pi =3.145159265.. It measures the length ratio
between a circle's circumference and its diameter. Archimedes
was the first some 2200 years ago to introduce a rigorous
mathematical method to calculate its value By inscribing
and circumscribing a 96 sided polygon by circles, he was able to
show that Pi has a value somewhere between 3+10/71 and
3+1/7. Although the value of 3+1/7 is good to only two decimal
places it was taught to students in middle school for the next
2000 years. Many mathematicians, before the invention of
calculus, attempted to increase the digit accuracy by using
larger n sided polygons. The most heroic of these attempts was
by the Dutch mathematician Ludolph Van Ceulen who in the 16
hundreds was able to find Pi accurate to 35 places using the
Archimedes Method. The calculations took him essentially his
entire life and led to Pi being named the Ludolph number in
Germany until the end of the 19th century. Today of course this
method for finding Pi has become obsolete and replaced by artan
formulas, AGM methods, and simple iteration. We show here a new
method of Pi calculation which is related to the
Archimedes method but differs from the former by using an
integral involving even Legendre polynomials instead of
polygons. The technique converges much faster than the
Archimedes approach and does so with a minimum of mathematical
effort. Go **HERE** for the details.

**April 11, 2017-What are Difference Equations?**

A difference equation is an expression where the value
F[n+1] at n+1 is directly related to the function F[n] and
satisfies an initial condition F[0]=const. Equations of this
type arise in a variety of different areas including
economics, radioactive decay, compound interest, and
iteration formulas for various different mathematical constants
such as
$\pi $ the
golden ratio$\phi $.
and the exponential e. We discuss here various specific
solutions of such equations. In the process we show , among
other things, how they may be used to sum the pth powers
of the first n integers.Also we discuss special forms for which
F[n1] yields integer values. Go **HERE
**for the details.

**April 17, 2017-How does one factor Large Semi-Primes? **

We show how to factor large semi-primes N into its prime
components p and q using an approach involving the right
choice of k in the integer evaluation of two specific radicals R
and S. The new method used generates starting values k1[alpha)
depending on the fraction alpha=p/sqrt(N)=sqrt(N)/q. For
N mod(6)=1 primes a lower bound on the value of k1[alpha]m
is sqrt(N)/18. Go** ****HERE**** **fore the
details of the discussions.

**April 27, 2017-How can alpha(N) and beta(N) functions be used
to locate Primes, Twin Primes, and Rich Numbers? **

We introduce two new functions alpha(N)={sigma(N)-N-1}/N and beta(N)={tau(N)-2}/N^{0.25} , where sigma(N) and
tau(N) are standard point functions found in number theory.
These new functions have the interesting property that they have
zero value whenever N is a prime number. The functions exhibit
local maxima for N referred to rich numbers. Special rich
numbers N are shown to produce twin (or double) primes at N±1.
Go** ****HERE** for details of the
discussions.

**April 30, 2017-What is the Solution to the April 29th math
Puzzle in the Wall Street Journal? **

The mathematics problem posed in the April 29th issue
of the WSJ is to find the length of the fourth segment
of the hypotenuse of a right triangle . The triangle
has a superimposed circle centered along the diagonal
and just tangent to the two opposite sides of the right
triangle. We solve the problem in generic form. Go
**HERE** for the
details. The fourth segment is found to have length 135 when
then circle radius is 120 and the smallest segment along the
diagonal is 16.

**MAY 2, 2017-What is the Shape of a Hanging Rope under its
own Weight and when Weights are Hung from it? **

We examine two classical statics problems, namely, the
shape a rope assumes when suspended between two fixed end
points and the shape when the rope becomes weightless but
weights are suspended at regular intervals along the
rope. The first problem leads to the catenary governed by
the differential equation y"=sqrt[1+(y')^{2}] . The
second problem leads to straight line segments between
suspended weights. Examples of catenaries include the St.
Louis Arch and the attached weight solution includes
the mechanic of tightrope walking. Go **HERE** for the details.

**MAY 6, 2017-What are the Properties of Cannonball
Stacks? **

We develop the formulas for stacked cannonballs
giving not only the number of balls per layer but also the
number for the entire stack. Close-packed configurations for
both triangular and square bases are considered. In
the analysis one comes up with triangular numbers (1, 3, 6,
10,....) and square numbers (1, 4, 9, 16,...) . The angles
associated with square base stacks are shown to be
remarkably similar to those existing for the Great Pyramid
in Giza, Egypt. Go **HERE **for the details.

**MAY 29, 2017-What are my latest thoughts on factoring
large Semi-Primes ? **** **

Over the last few years I have been very interested
in finding a way to quickly factor large semi-primes . If
this could be accomplished in a simple manner then
conventional public key cryptography would become obsolete
since an adversary could then break any code involving
public keys. Although elaborate procedures for factoring
large semi-primes have been proposed in the literature these
methods still require an inordinate amount of computer time
using even the fastest supercomputers. We summarize **HERE** a simple way to
quickly factor larger semi-primes N=pq using only the
simplest of mathematical techniques. After deriving the
formulas we use for achieving the factorization we also show
how they may be applied to quickly solve four specific
semi-primes.

**JUNE 2, 2017-What is the shortest Path connecting a
Source with n Sinks? **

We examine the problem of finding the shortest distance
which connects a point source with n sinks using only
straight lines? This problem was first looked at by Jacob
Steiner (1796-1863). He showed that connecting three points
leads to a characteristic y shaped path with two of
the points located at the upper end of the two pronged y and
the remaining point at the bottom of the y. For obvious
reasons we term this structure a Steiner Fork. It is shown
in this article how one can combine several Steiner
Forks to create optimum paths between one source and n
sinks. Go **HERE **for the details.
Note that this is not the same as the equally complicated
traveling salesman problem but the two do share some
similarities.

**June 10, 2017-How can one use a Dip-Stick to measure the
Volume of Beer in a Barrel? **

We determine the volume of beer in a standard wooden
barrel and then show how a dip-stick inserted at the bung
hole can be used to find this volume. This is a classic
problem dating back to J. Kepler in the sixteen hundreds .
It is shown via simple calculus that dip-stick measurements
are quite precise if the sticks are appropriately graduated.
The relation between fluid level and fluid volume is shown
to be non-linear. Go **HERE** for the details.

**J****une 14, 2017-How does one determine longer term
Price Trends for Commodities? **

One of the more intriguing problems in economics is how to distinguish longer term price trends compared to short term price fluctuations. One way to accomplish this is to first look at a longer term price history and use this to filter out the higher frequency components. Once this has been done one can then draw a type of long term moving average referred by us as a lag curve, When the lag curve lies below the smoothed price one has a buy signal and when the lag curve lies above the smoothed price one has a sell condition. As we will show**HERE** this action will
lead more often then not to profitable transactions both on
the long and short sides of a market. We demonstrate the
existence of long term price trends by examples from the
housing market, behavior of GE stock, and the price of
West Texas Crude Oil.

**June 27, 2017-What are the Properties of the Exponential
Function Exp(x)? **

One of the best known functions in mathematics is f(x)=e^{x}=exp(x).
We show in this article how it arises naturally when taking
the derivative of the function f(x)=a^{x }. In
addition, the properties of exp(x) and the related
hyperbolic functions are determined. Also we introduce a new
technique based on an integral containing even Legendre
polynomials times cosh(x/2) to find the approximate value of
e=exp(1) to sixty place accuracy. Go **HERE
**for the details.

**June 29, 2017-What is the Ulam Spiral and how can it be
simplified by a Morphing Procedure?**

The Ulam Spiral is a spiral array of integers which
exhibit a distinct and non-periodic distribution of primes.
We show how this rather intricate distribution of primes
within the spiral essentially offers no new information on
primes other than they are odd numbers for p=3 or greater.
This fact is established by a simple transformation where
the spacing between integer n and n+1 equals exactly n. The
resultant morphed pattern places all primes(with the
exception of two), along two left leaning diagonals. An even
better distribution of primes occurs using our own
hexagonal integer spiral. Go **HERE** for the details.

**July 18, 2017-What is a quick way to determine the
numerical value of the basic six mathematical**

constants to 100 place accuracy?

We show how one can quickly evaluate the irrational
numbers sqrt(2), phi, exp(1), Pi,
$\gamma $, and
ln(2) by a new iteration approach and via integrals
involving the Legendre polynomials. Unlike series
expansions, the methods used converge much faster. For
example the root of two follows from the iteration
S[n+1]=(A+BS[n])/(C+dS[n], where the values of the constants
A,B,C, and D depend on the number of terms used in a
continued fraction expansion. Integrals involving Legendre
polynomials are easily evaluated using the basic computer
codes for rem and quo in MAPLE. Go **HERE** for the details.

**July 26, 2017-What is the Witch of Agnesi Curve?**

This is a 2D mathematical curve which in its simplest
form reads y(x)=1/(1+x^{2}). It is named after Maria
Agnesi (1718-1799) a polymath from Milan and professor at
the University of Bologna. Fluent in six languages and
conversant in philosophy and theology, she was the first
person to study the curve in greater detail although Fermat
had already looked at it a century earlier. The term Witch
became attached to the curve do to an incorrect translation
of the Latin-Italian word averisera into English. After
deriving the parametric form of the curve from a geometrical
viewpoint we proceed in determining its various properties.
We also look at several related curves. Go **HERE** for the details.

**August 23, 2017-What is a Solar Eclipse and some of
it's Properties?**

An eclipse of the Sun by the Moon occurs whenever the
Moon finds itself between the Sun and Earth along a straight
line lying in the ecliptic plane. It is a relatively rare
occurrence with yesterday's eclipse being the first one I
have seen here in Florida since 1970.It is our purpose in
this short article to explain some of the characteristics of
a solar eclipse using diagrams and some simple trigonometric
relations. Among other things we show why the path of
totality in yesterday's eclipse was along a 23.5 deg tilted
diagonal and had a width of about 50 km. Also the angles
subtended by the Sun and the Moon as seen from Earth are
determined. Go **HERE** for the details.

**September 1, 2017-Why do fixed Time-Increments become
Subjectively Shorter as one Ages?**

It is well known that individuals judge fixed time
increments, such as the length of a year, as becoming
shorter as they age. We explain here why this is so and in
the process come up the tentative law that -"Subjective
measurements of time intervals decrease as one ages as the
reciprocal of the logarithm of the observer's age". We
motivate this as yet unproven law via various examples such
as why one's eighteen year old freshman students seem to be
getting younger every year and why the judged time by a four
year old to his next birthday seems to be about three times
as long as that judged by an advanced age senior citizen for
the same time interval. Go **HERE**** **for the
discussion.

**September
16, 2017-What are the Properties of n sided
regular Pyramids?**

We discuss the properties of regular pyramids including their volume and surface area. Cones as pyramids with a regular polygon base with an** i**nfinite number of sides are also
examined. The volume of all such pyramids equals one
third the product of their base area multiplied by the
height. A graphic of a cone in a cylinder is also
presented. Go**
****HERE**** **for
the discussion.

September 21, 2017-What is a Void Fraction and its values for bundled Regular Polygons?

We examine the voids produced when bundling polygons in
either a square centered or close packed arrangement.
Starting with the limiting case of circles we then proceed
to polygons with either 4x2^{n} or 6x2^{n }sides.
Both cases produce periodic arrangements of voids. The size
of these voids have importance in certain practical
applications such as the cooling superconductors and
providing conduits in oscillatory heat transfer. Go
**HERE
**for the details.

**October 1, 2017-What is the Largest Rectangle which can
be placed into an Oblique Triangle?**

We solve a math puzzle posed in this week's Wall Street
Journal (Sept.30th). The question is to find the largest
rectangle which just fits into an oblique triangle. First
solving the problem in its most general form we show that no
matter what the side-length of the triangle are, the
triangle area is exactly twice the optimized rectangular
area. For the special case of a triangle with sides 9-19-17
we find the triangle area to be 36 so that the optimized
rectangle area becomes 18. Go **HERE** for the details.

**October 8, 2017-What is ASCII and how may it be used to
convert Numbers and Letters to Binary Form?**

We show how ASCII (American Standard Code for
Information Interchange) converts both numbers and letters
to binary form suitable for electronic communication. The
idea behind the method is to first construct a 3x26 table in
which the first row contains all integers 1 through
26, the second row gives the binary form for these numbers
and the third row contains the 26 letters A through Z.One
uses an eight bit code for each number and letter. The first
few bits on the left read 0011 for numbers and 010 for
capital letters. The terms on the right of a group are the
bits in binary form of the particular letter or number
in question. Thus, for example, the letter K reads 01001011
in binary. In addition to numbers and letters the 8 bit code
offers forms for punctuation marks, brackets and spacings
plus mathematical symbols and lower case letters. Go **HERE** for details of the
discussion.

**October 12, 2017-How can one use Irrational Numbers to
code Messages?**

We generate large random numbers using the product of
irrationals. Such products N act as very long passwords for
encrypted messages. The Ns can be expressed in compact
form for easy transmission. An example of one such
number and its compact construction form is
N=2LPSG=544147299248829936628591544616211725396215111691.
Only the sender and his friendly receiver have the same list
of Ns in their repertoire but no one else does. This means
they can easily decipher the product M(N) but an adversary
can't since he doesn't know the N being used and is highly
unlikely recover it by a trail and error approach when N is
fifty or more digits long. Go **HERE** for the details.

**October 16, 2017-How does one quickly distinguish between
Prime and Composite Numbers?**

We derive a new prime number function defined as
F(N)=1/[Nf(N^{2})] to distinguish between prime and
composite numbers. Here f(N^{2}) is the number
fraction. Whenever F(N)=1 we have a prime number and when
F(N)<1 it is composite. We obtain graphs for F(N) versus
N and show how one can quickly calculate the number of
primes present in any given range of N. Comparison is also
made with the prime density function of Legendre and Gauss.
Go **HERE** for the details
of the discussion.

**October 24, 2017-What are Twin-Primes and how are they
Generated?**

We re-look at twin primes such as [5,7], [11,13], and
[59,61] by use of a hexagonal integer spiral. These primes ,
when greater than three, have the property that their
average value equals a multiple of six. That is" The condition for the existence of a
twin prime is that both 6n$\pm $1
are primes and that N=6n has N mod(6)=0". We also
use the number-fraction f(N) to generate some interesting
patterns were the sequence of increasing N goes as
prime-supercomposite-prime. We have named this the PSP
pattern. Go **HERE**
for the details of the discussion.

**November 1, 2017-How do Nodes, Connectors, and Sub-Areas
relate to each other in Graph Theory?**

It is well known that any polyhedron has its vertexes V
related to its edges E and its faces F by the Euler Formula
V+F-E=2. There is an exact 2D analogue to this formula which
states the number of nodes plus inside areas minus the sum
of connecting lines equal unity. It is our purpose here to
explore this second possibility further from the viewpoint
of graph theory. We start with a collection of N points
(nodes) lying in the x-y plane and then connect them to each
other by straight lines(collectors C) producing a number of
sub-areas (A). By looking at several different examples , we
arrive at the formula N+A-C=1. This continues to hold
for any distribution of points lying in a plane. Go **HERE** for the details of
the discussion.

**December 21, 2017- What is the relation between
Temperature and Time for Cooling a Cup Of Coffee?**

We examine the classic problem of the cooling of a cup
of coffee both from an experimental and analytic point of
view to show that the process follows essentially a Newton
Cooling Law. The cases of plain water, black coffee, and
coffee mixed with cream are considered. They all follow the
same law where the non-dimensional temperature goes as
$\theta $(t)=exp(-kt)
with k an experimentally determined constant. It is found
that one need not add the cream to hot coffee when first
served in order for the mixture to stay warm as long as
possible. Go **HERE** for the
discussion.

**December 25, 2017-What is the Area of a Spherical Cap
drawn on a Sphere by a Compass?**

**2018**

We answer a question appearing in last Sunday's Wall Street Journal puzzle page. The question is to see how the area of a spherical cap created by a compass tracing out a circle of radius L on the surface of a sphere varies with sphere radius.We approach the problem from both an elementary viewpoint and then also via more rigorous mathematics. In both case one finds the spherical cap area generated is always equal to A=$\pi $L^{2}
regardless of the sphere size. Go **HERE** for the details
of the discussion.

**January 5, 2018-How does one determine Price Trends for
any Commodity including Stocks?**

** **

It is well known that any commodity will have periods of long uptrends and long downtrends during its existence.If these trends can accurately identified early after the onset of a new trend large profits can be made.The secret for positive financial returns is to first identify the trend one is in and then stay with the trend until there is a reversal. Whether one acts on the long or short side of a market is immaterial as long as the price stays above a lag curve during bull markets and below a lag curve for bear markets. Go**HERE** to see the details
of our discussion.

**January 30, 2018-How does one convert Infinite Series to
Infinite Products?**

It is known that any polynomial f(x) may be
expressed as a product function involving all roots of such
a polynomial. This fact continuous to hold for certain
periodic functions with an infinite number of distinct
zeros so that an infinite series can be expressed as
its equivalent form as an infinite product..We show in this
article how this may be accomplished by looking at the
specific functions sin(x)/x, cos(x). cos(x)^{2}, and
tan(x)/x. Often such equalities can produce very interesting
results such as Leonard Euler's famous identity Pi^{2}/6=1+1/4+1/9+1/16+...
. Go **HERE
**for details of the discussion.

**February 10, 2018-How can one use Sticks to construct
any Polygon?**

We show how one can construct any N sided polygon by the
concatenation of straight sticks of specified length L
connected to their neighbor by a specified angle $\theta $. The
conditions that the resultant figure close on itself are
that the sum of the external angles add up to

2$\pi $ and that the sum of the x and y components of the stick arrangement vanish. The oblique triangle and non-symmetric quadrangle are discussed in detail with specific examples given. Also the interior area for any regular polygon is derived. Go**HERE** for the details.

**March 1, 2018-How can one evaluate any Trigonometric
Function using only the Values of Tan(a) in
0<a<45deg?**

We use a new approach involving higher order Legendre
Polynomials to obtain twenty digit accurate approximations
for the tangent function tan(a) over the limited range
of 0<a<45deg. It is then shown that this
information may be used to determine the values of any of
the other trigonometric functions at any 'a' including
points outside the region. Also the location of poles of the
tangent function are well estimated by setting the
polynomial in the denominator of its approximation to zero.
Go **HERE** for details of the
discussion.

**March 6, 2018-When is the Time to Buy and the Time to
Sell a Stock or other Commodity?**

Applying the aphorisms (1) For every season there is a
time to sow and a time to reap and (2) Know when to
hold them and know when to fold 'em to stock and commodity
investments, we explore the financial adage Buy Cheap and
Sell Dear. That is, we discuss what procedure one should use
to be on the right side of market moves using both long term
price charts and especially designed lag curves. Analysing
the S&P500 Stock Index, we indicate the points where one
should buy and be long and the points in time where one
should be short. A long term following of the outlined
procedure has produced excellent returns for me over what is
now a time span of over sixty years. Go **HERE** for the details.

**March 11, 2018-What are the basic Principles of
Calculus?**

Calculus is that field of mathematics to which students are exposed right after analytic geometry and trigonometry. It deals with continuous functions defined over given intervals. The two parts of the subject are known as Differential Calculus dealing with derivatives of functions and Integral Calculus involving the determination of areas under specified curves. We present in this article a very much abbreviated version of calculus which is intended to be used as a handy supplement to existing calculus books often exceeding five-hundred pages in length or so. We call the present description Calculus in a Nutshell. Once you master it you will br well on the way to understanding of the entire subject. Go**HERE** for the details
of the discussion.

**March 22, 2018-What was the famous grave marker placed at
the now lost grave of Archimedes of Syracuse?**

It is well known that the famous ancient Greek
mathematician, physicist, and inventor Archimedes
(287-212BC) had a column topped by a sphere in a cylinder
placed above his now lost grave in Syracuse, Italy. During
his researches he had found that the volume of a sphere to
that of a circumscribing cylinder is precisely 3/2. We
modify this problem and show via some elementary calculus
that the volume ratio of an ellipsoid relative to an
inscribed and circumscribed cylinder goes as sqrt(3)
and 3/2 respectively, regardless of the aspect
ratio of the ellipsoid. Go **HERE** for details of
the calculations.

**March 26, 2018-How does one construct Large Parabolic
Mirrors using a Spin Technique?**

We examine a method of spin generating parabolic mirrors
ideal for solar experiments requiring solar concentrations
of several hundred suns. With an added secondary
concentrator (axicon) at the primary mirror focus one is
able to achieve about a thousand sun concentration making
such a two reflector configuration ideal for the pumping of
small cylindrical liquid cooled lasers. The
construction technique was first developed by a group of us
working on a NASA contract in the early 1980s . It involves
revolving a preformed near parabolic surface of a stabilized
foam pad at constant angular velocity about a vertical axis
while poring liquid epoxy onto this rotating
platfom. After several hours of spinning the epoxy
hardens to form a nearly perfect parabolic surface attached
to the underlying foam layer. After coating the resultant
mirror with reflective mylar, a large diameter D and long
focal length F mirror will have been produced. Go **HERE**
for details of the discussion.

**April 5, 2018-How good are the Correlations between
various US stock Indexes?**

We look at the correlation in price and time of various
US stock Indexes including the S&P500 and the Nasdaq
Composite.The correlations between the indexes are found to
be excellent on both a long term (20year) and short term (3
month) basis allowing one to make accurate predictions of
when an index and hence it underlying stocks are in an
uptrend (bull market) or a downtrend (bear market).The more
speculative stocks present in the Nasdaq index tend to rise
more during bull markets and fall more during bear markets
relative the the stocks present in the S&P500. Go **HERE** for the details
of the discussion.

**April 9, 2018-What are ETFs and how may they be used
during Bull and Bear Markets?**

ETFs are a new type of investment instrument which
relies on the underlying value of a collection of stocks.
They have become very popular with investors during the last
decade who recognized that it is almost impossible to
improve on investment returns exceeding a good market
average such as the S&P500. We discuss in this article
some of the characteristics of ETFS ( Exchange Traded Funds)
including what type of ETF to use during bull and bear
markets. The best known and also largest product of price
times daily volume ETFs are SPY and QQQ applicable during
up-markets. Their down-market counterparts are SDS and QID,
respectively. Go **HERE** for details of our
discussions.

**April 24, 2018-What are the Characteristics of
Projectiles associated with Hand-Held Weapons?**

We examine the properties of projectiles ranging from
thrown rocks to bullets from sniper rifles. The important
quantities of velocity, mass, momentum and kinetic energy
for these** **different projectiles are examined and
presented in MKS units in both graphic and table form. It is
found that all of the five different projectiles considered
have momenta slightly above 2 kg-m/s and all can be lethal
at different ranges. Go **HERE** for details of the
discussion.

**May 10, 2018-How may Lag Curves be used to predict
Uptrends an Downtrends in Equities?**

We show how using only lag curves for both long and
short term time windows one can accurately indicate at what
stage an equity finds itself and more importantly whether it
is in an uptrend or downtrend. Using only two types of price
history graphs, one based on a twenty year time period, and
the other on a six month time scale, we can accurately
predict the type of market one is in be it a Bull Market
(uptrend in prices) or a Bear Market(downturn in prices).
The case of GE is discussed in some detail showing how this
widely held stock , although in a long term downtrend, still
offers opportunities on the long side at shorter time
scales. Go **HERE**** **to see the
rest of the discussion.

**May 25, 2018-What is a Caustic and how may it be
generated using Ray Tracing?**

We use geometrical optics to generate the familiar light
pattern often seen at the bottom of coffee or tea cups.
Using ray tracing we first find the formula for all straight
lines representing reflected light rays of off a circle by
an incoming parallel light array. After plotting a
family of these curves, a typical cusp pattern associated
with a standard caustic emerges. Comparison of this caustic
with the classic nephroid are also made. Go **HERE
f**or details of the discussion.

**May 26, 2018-What was Operation Hydra?**

Operation Hydra was the code name for a 596 heavy
British bomber attack on Peenemuende along the Baltic Sea in
northern Germany. The raid took place seventy-five years ago
on the night of August 17-18, 1943. The British had
just discovered the top secret rocket development facility
there and with greatest urgency devised a plan in early
1943 to eliminate as many of the scientists and
engineers and their families by saturation bombing of
the housing facilities in Peenemuende. I happened to
be living there at the time as a six year old boy so that my
family and I were on the receiving end of this bomb attack.
At the end of the raid we had escaped with our lives twice
that night. Go** **to** ****OPERATION-HYDRA** to see
further details of the story and how it influenced my later
life.

**June 3, 2018-How do Areas, Connectors, and Vertexes vary
for N sided Regular Polygons subject to superimposed
Diagonals?**

The 2D version of Euler's famous topological formula
reads A+V-C=1, where V are the number of points (nodes)
connected by C lines to produce A sub-areas. In this article
we will be concerned with applying this result to N sided
regular polygons connected by D diagonals connecting
opposite vertexes. In particular we are interested in how
many sub-areas A are formed by connecting the vertexes
of a polygon by the maximum number of possible allowed
diagonals. Treating polygons N=4,5,6,7, and 8,we obtain some
interesting patterns which when colored make for interesting
computer art. Go **HERE
**for the details.

**June 6, 2018-What are the Rust Belt Cities, how did They
get that Way, and what is the Route to Their Recovery?**

Starting in the late 1970s many cities in a swath about
the Great Lakes including Detroit, Flint,Youngstown,
Cleveland, Pittsburgh , Buffalo and Erie found that their
major industries could no longer compete with foreign
manufacturers on quality and price. As a result the
manufacturers in these cities went out of business producing
massive unemployment , an increase of poverty levels, and a
mass exodus. These cities soon became to be known as the
Rust Belt Cities. Most even today still have not been able
to recover. I detail my own observations of one of these. In
addition we give a recipe for eventual recovery of these
cities following the Pittsburgh model of jobs created
through scientific and technological innovation
produced by a highly educated work force. We show why
tariffs are useless for such recoveries while
financial support for industries and services having
favorable trade balances with the rest of the world needs to
be encouraged and subsidized by the government. Go to **RUST-BELT-CITIES
**for further details.

**June 17, 2018-Where is the Danube River located and what
are some of the more important Cities lying along its
Banks?**

We give a brief travelogue on the River
Danube starting at its source in the Black Forest to its
exit into the Black Sea in Romania. In particular we discuss
the history of some of the better known cities lying along
its banks and also present some of our own
observations acquired through many years of travel in the
region. Starting at the source at Donaueschingen in Swabia
to its end in Romania at the Black Sea we give some of the
historical details of the better known cities along the way
including Ulm, Neuburg, Ingolstadt, Regensburg, Passau,
Linz, Vienna, Bratislava, Budapest, and Belgrade. Also
we have added some photos taken by us or found on the
internet to accompany the dialogue. Go **DANUBE
**for the details.

**June 30, 2018-What are the Properties of Regular n-sided
Polygons?**

We examine the properties of all n-sided regular
polygons including their external and internal vertex
angles, their area and the length of their diagonals. A
generic form for the area of all n-sided polygons is derived
and the limit of this area to that of a circle is shown as n
approaches infinity. In addition to deriving the Golden
Ratio for pentagon diagonals, an infinite number of other
irrational numbers are obtained with some of these
expressible as functions of the roots of integers. An upper
and lower bound for Pi is also derived using polygons with
large n. Go **HERE**
for the details of the discussion.

**JULY 7, 2018-What is meant by Compound Interest?**

The return on capital by compound interest is examined
in some detail. Beginning with compounding on a yearly
bases, we reduce the payment increments to ever shorter time
intervals until the point of continuous compounding is
reached where the capital C(n) increases exponentially
as C(n)=C_{o} exp(in) , with n being the years
held, i the yearly interest rate and C_{0} the
initial capital invested. A discussion of the Federal
Reserve's misguided inflation policy which has reduced
the true value of the dollar by a factor of over seven
in the last hundred years is also discussed. In addition
we emphasize the fact that (in a paper economy)
successful investment requires that the yearly return
exceeds losses due to inflation and taxes. Click on **COMPOUND-INTEREST** to
see more details.

**July 27, 2018-What are the Properties of Integers?**

We discuss the properties of all positive integers, how
they first developed plus their even-odd character and prime
or composite forms.

After introducing the concept of number fraction f(N), we clearly show that primes correspond to f(N)=0 and super-composites to f(N)>1.4. Mersenne, Fermat, and Perfect Numbers are also discussed. In addition, logarithms of integers to various bases are examined. Go to**ALL-ABOUT-INTEGERS.pdf**** **
for further details.

**August 14, 2018-How is the Hexagonal Integer Spiral
generated and used to classify Prime Numbers?**

It is shown how from a class discussion on the point
function (1+i)^{n} we developed a new type of
integer spiral for which all

primes fall along just two radial lines 6n+1 and 6n-1 provided the primes are greater than three. The vertexes of the resultant Hexagonal Integer Spiral locate all positive integers starting from 1 through N. Mersenne and Fermat primes are shown to be sub-classes of the 6n+1 and 6n-1 primes, respectively. We also introduce a more general prime number generator T[a,b,c]=a^{c}+b. Go to **GENERATING-A-HEXAGONAL-SPIRAL****.pdf**** **for further details.

**August 25, 2018-What is the complete form of the Goldbach
Conjecture?**

It is well known that the main part of the
Goldbach Conjecture in number theory is that all even
numbers can be represented by the sum of at most two
primes. What is less often mentioned is his extension to odd
numbers as also conveyed in 1742 letter to Leonard Euler.
The second part of Goldbach's Conjecture is that any odd
number N can be represented by the sum of at most three
primes. We look here in more detail at this second part
proving that it is also valid. The proof depends on the fact
that all the odd numbers we considered have multiple triplet
expansions with the number increasing monotonically with
increasing N. Since only one triplet is required to confirm
the conjecture, it is clear that it also holds. Go **HERE** for further
details.

**September 2, 2018-What are some Additional Features of
the Five Platonic Solids?**

In an earlier note we discussed the mathematics of the
Platonic Solids. In this article we extend these discussions
to include

certain additional facts less well known about these five solids, such as the fact that they all make fair die for any gambling game. After deriving their surface area and volume and their corresponding surface to volume ratio,we show how they may be constructed using metal, wood, or cardboard. In particular we concentrate on the dodacahedron which apparently held religious significance to first and second century AD Romans. Go to**MORE-PLATONIC-SOLIDS**
for further details of the discussion.

**September 10, 2018-What are some of the 2D Patterns which
fold into closed 3D Structures?**

We examine the 2D patterns which can be used to form the
Platonic Solids when folded. Also other patterns capable of
producing other 3D structures of not necessaroily the same
faces when folded are considered. In all cases the resultant
closed polyhedra satisfy the Euler condition that V+F-E=2 is
satisfied. Here V are the number of vertices of the 3D
structure while E represents the edges and F the number of
faces. Go to **FOLDING-2D-PATTERNS.pdf**
for further details.

**September 16, 2018-What is the Twin Prime Conjecture and
how can it be Confirmed?**

** **We look at the well known conjecture that the
number of twin-primes are infinite. Although no rigorous
mathematical proof exists that this is so, a simple
extrapolation of the twin prime sum for integers out to
about N=20,000 allows us to come up with a power law which
states that the twin-prime sum S(N) relates to the integers
N=6n as S(N)=0.19248 N ^{0.7532}. Thus
as N goes to infinity so does S(N) and hence there must be
an infinite number of semi-primes. Go to **TWIN-PRIME-SUM.pdf **for
further details.

**September 20, 2018-What are the times of Sunrise, Sunset,
and Local Noon at any point in the Northern Hemisphere?**

We examine the equations for the Celestial Triangle to
determine local noon and the times of sunrise and sunset at
any latitude between LAT=0 and LAT=67.5deg and any
longitude. The precise values for here in Gainesville,
Florida (LAT=29.65N, LONG=82.45W) are calculated for several
different dates throughout the year especially at the
Equinoxes and Solstices.. The time between sunrise and
sunset as a function of latitude are also discussed. A graph
is created to determine the departure in minutes from a mean
value for local noon due to the earth's not quite circular
orbit about the Sun. Go to **SUNRISE-SUNSET** for
further details.

**September 26, 2018-What was the Roman Limes?**

We describe the location and characteristics of
the ancient Roman defense wall across in southern Germany
known as the Limes Germanicus. This was a border wall
between the Roman occupied territories west of the Rhine and
south of the Danube extending from present day Cologne on
the Rhine to Regensburg on the Danube. Its purpose was to
prevent the northern Barbarians from entering Roman occupied
territory. It existed from 83AD to 260AD, was about 500km
long and studded by forts and watch towers at 5km
intervals. The main legions assigned to enforce the
border wall were located at Mainz (Moguntiacum) and
Regensburg (Castra Regina). Click on **ROMAN-LIMES **for
further details. Also we point out that limes is pronounced
' leemess' and not as the citrus fruit.

**October 21, 2018-What is the Trump Market and what
produced its Rise and Fall?**

We examine the behavior of the Standard and Poors 500
Stock Index over the past two years starting in November of
2016 through today of October 21, 2018.The reason for
its sharp rise until January of this year was mainly due to
the election of Donald Trump as president of the United
States. Some of the more onerous restrictions placed on the
economy by the previous administration and the promise of
more of the same had Hillary Clinton been elected made most
investors extremely optimistic that she wasn't. By January
of 2018, however, the Trump up Market had risen too far
compared to historical standards and the new imposition of
tariffs promised to restrict profits and hence US stock
dividends. Although there was a brief reprieve from a
further downturn by the summer of this year, the sell off
continued starting in early October of this year and we are
probably now seeing the fall of the Trump Market . Go **HERE** for further
details including reasons for the market actions.

**Nov.11, 2018-How does one Construct a regular Icosahedron
?**

Of all of the five platonic solids, the twenty faced
icosahedron is the most complicated and difficult to
analyse. We show in this note

how one can use elementary mathematics to locate all twelve of its vertices and from this information construct an icosahedron by tiling with equilateral triangles. Using polar coordinates, the vertex points are readily calculated. This allows for the construction for a wire-frame map in the shape of an icosahedron. Once this frame is covered by tiles, the desired figure will result. We also show how one can use a 2D Duerer net to produce this structure by a simple folding procedure. Go** ****HERE**
for details of the discussion.

**Nov.15, 2018-How does one easily obtain Accurate Values
for Tangent and other Trigonometric Functions?**

About a decade ago we discovered a new way to rapidly
obtain estimates for certain functions using integrals
involving the Legendre polynomials P(n,x). We want in this
note to extend these earlier discussions, concentrating on
the two integrals int(P(2n,x)cos(ax),x=0..1) and
int(P(2n+1,x)sin(ax),x=0..1). After finding quotient
approximations for tan(a) to various degrees of accuracy, we
produce a table of tangent values accurate to at least ten
digits in the range 0<a<Pi/4. Once this has been
accomplished we then use this information to find values for
the other trigonometric functions to the same order of
accuracy. Go to **MORE-ON-TAN **for
details of the discussion.

**Dec.3, 2018-How do Number Sequences relate to Finite
Difference Equations?**

We examine numerous number sequences S(n) and show how
they relate to finite difference equations subjected to
certain

starting conditions. Among the sequences considered are the power sequence S(n)=sum(k^{p},k=1..n) and the
Fibonacci Sequence and its variations. We also look at
f[n+3]=f[[n+2]+f[n+1]+f[n] , with f[1]=1, [2]=2 and
f[3]=3. Here the resultant sequence reads
S(n)=1,2,3,6,11,20,37,68,125,... and the ratio of the
elements approach the unique value of 1.839286755...
as n gets large. Go **HERE
**

for details of the discussion.

**Dec.21, 2018-How does one optimize the return on the
Exchange Traded Fund SPY?**

We use the long term price history of the popular
exchange traded fund SPY versus a drawn by eye lag curve to
show how one can trade this fund successfully in both bull
and bear markets. A simple buy and hold strategy over the 24
year time period of 1994-2018 would have increased one's
investment by a factor of 6x. On the other hand, our lag
curve approach would have achieved a far superior
multiplication of 22x. Go **HERE**
for details of how the procedure works.

**Dec.27, 2018-How can one ride long term Price Waves of
the S&P 500 Index for consistent Profits?**

We show how the use of long term price data for the
S&P 500 Stock Index exhibits a wave like behavior which
when used in conjunction with a lag curve can yield
consistent profits on invested capital on both the long and
short sides of the US stock market. Using a thirty year
historic chart of the S&P 500 Index, one can clearly
distinguish when the stock market is in an uptrend or
a downtrend. Following these trends produces profits
far superior to what can be accomplished by a buy and hold
approach. We also show how applying the present wave method
to the 1920-1940 time period would have gotten one out of
any long position unscathed some two month prior to the
October 1929 market crash. Go** ****HERE**
for details of the discussion.

**DEC.31, 2018-How can one use Lag-Curves and Cross-Over
Points to determine longer term Trends for any Stock?**

We show how uptrends and downtrends can be found for any
stock by using long term price data with a running Lag-Curve
to find the cross-over points where long or short
transactions should be taken. After a detailed discussion of
GE, we examine the stocks AMZN, MCD, AAPL and HD plus the
global Dow Jones Average $DJWO. In all cases these stocks
and the global average have been in a downtrend since
September of this year after impressive earlier gains.
According to our technical rules one should never hold such
formally active up moving stocks in a downtrending market.
We also discuss the behavior of one of my earliest ill-timed
stock purchases in 1959. We show how , as a naive investor
some sixty years ago, I set myself up for some losses by
buying during the down trending phase of RTN at $50/sh after
it had dropped from a maximum of $72 earlier in the year.
Needless to say this action was never repeated by me at any
later date for any stock. Go **HERE** for further
details of our overall trend discussion.

**2019**

**JAN.4, 2019-What are the Area Ratios between Generations
of Nested Regular Polygons?**

We look at the problem of nesting regular polygons such
that the (N+1) generation has its vertices just touch each
of the centers of the sides of the previous Nth
generation. After looking at the specific cases of
equilateral triangles, squares, and hexagons, we develop a
general formula for the area of the nth generation of any
regular polygon. Although the found generic formulas for the
areas of the Nth and (N+1) th generation are quite
complicated, the area ratio R assumes the simple form of
R=1/ [cos(Pi/n)]^{2} , where n equals the number of
sides of the polygon. Go **HERE** for details of
the discussion.

**JAN.7, 2019-Are all Prime Numbers greater than Three of
the Form 6n±1?**

We show that all primes five or greater have the form
6n+1 or 6n-1 without exception. Thus the prime N=14566243 is
equivalent to 6(2427707)+1 and N=107507=6(17918)-1. That
primes satisfy this form is a neccessary but not sufficient
condition since there are also many composite numbers
which have the from 6n±1. We also show that all primes
fall along just two radial lines in a hexagonal integer
spiral representation. Go** HERE
**for the details of the discussion.

**FEB.9, 2019-How did Eratosthenes make the first accurate
measure of the Earth's Circumference?**

The first accurate measurement of the Earth's
circumference C was carried out by Eratosthenes of Cyrene in
about 246BC. He used the Sun's angle difference at local
noon during the Summer Solstice at both Alexdandria
(Lat-31.2N) and Syene (Lat-24N). The angle difference was
7.2 degrees. From it, simple geometry showed that the
Earth's circumference is C=(360/7.2) S=250,000 stadia, with
S being the known distance of 5000 stadia between the two
cities. This estimate is very close to the actual Earth
circumference although some questions remain about exactly
what value of the stadia Eratosthenes was using. We also
discuss two simpler alternate measurement routes he could
have taken to measure C, but didn't, involving the
Alexandria Light-House. Go **HERE** for the details of
the discussion.

**FEB.15, 2019-What is Parallax?**

When looking at an object by eye one receives two slightly different images on the retina for each eye. The brain merges these images into a single one lying at distance D where the object is located. Parallax represents the angle $\alpha $=arctan(d/D), where 2d is the distance between the eyes** **and D the
distance to the object. The concept of parallax explains not
only stereosopic vision but is also of great use in finding
distances between stellar bodies. We describe its use in
determining the earth-sun distance and also the 4.2
light-year distance to Proxima Centauri. Go **HERE
**for details of the discussion.

**FEB. 20, 2019-What are the Properties of a Binary Abacus?**

We show how to construct and manipulate an abacus
based on the binary number system. After presenting a
schematic of such a digital device, we show how the basic
operations of addition, subtraction, multiplication, and
division are achieved. It is shown that any number in binary
form when multiplied by 2^{n} simply adds n
zeros to its sequence. All numbers equal to 2^{n}-1
are represented in binary by n ones while a number 2^{n}+1
has the form 1 followed by n-1 zeros and finally ending in a
one. Go **HERE
**for details of the discussion.

**MARCH 9, 2019- How does one determine the Mass Centers
and Centroids of Bodies?**

After deriving the basic equations for mass centers and
centroids of bodies, we apply these formulas to carry out
detailed calculations making use of symmetry and elementary
calculus. Among other problems, we determine the center of
mass of a cone with density which varies linearly with
height and the centroid for a standard pyramid. Also
we show how different calculations are simplest to
carry out with choice of an appropriate coordinate system.
Go **HERE**
for details of the discussion.

**MARCH 11, 2019-How can one eliminate the biannual need
for changeover between Standard Time-Daylight Savings Time
?**

The changeover from standard to daylight savings time (DST) and visa versa has become a real burden to not only homeowners who are required to often reset over a dozen clocks a year but also for the travel agencies and especially the airlines. This morning I saw an interesting article in the local newspaper(the Sun) about Marco Rubio(R-FL who is proposing doing away with standard time entirely and just staying on DST year round. He is on the right track but is meeting opposition from teachers who worry about their young students having to line up for their schools bus while it is still dark. We propsse here an alternative approach which would eliminate both standard and daylight savings time as they are today and instead add one half hour to the local noon as found in the middle of the four time zones within the US. Go**HERE** for details of the
discussion.

**APRIL 13,2019-How can one use Lag Curves and Price
History to determine Market Tends?**

We explain how one can construct and use Lag Curves in
conjunction with Price History to determine when a commodity
is in an up or down trend. It is shown, through numerous
examples, how one may use such signals to obtain
profitable returns on any stock or stock index. The
approach we are using is purely technical and ignores
fundamentals and the need for predictions when a trend will
end. The points for buy (B) and sell (S) occur when the
price curve P(t) and lag curve become equal to each other.
Such points are referred to as cross-over points. Go **HERE** for details of
the method.

**APRIL 15, 2019-What are the Properties of Electromagnetic
Waves?**

We examine the properties of the electromagnetic waves
predicted by Maxwell's equations and show how frequency
varies inversely with wavelength for waves ranging from
radio to gamma rays.Also we show that only certain
frequencies are transparent to the atmosphere. After
classifying things according to wavelength and frequency,
their role in wi-fi and cell phone technology is discussed.
In addition we look at possible health effects produced on
living cells for both non-polarizing and polarizing forms of
this em radiation. Go **HERE
**for details of the discussion.

**May 7, 2019-How were the Moai moved from their Quarry to
the Shoreline of Easter Island?**

We examine the megalithic structures termed Moai
on Easter Island by discussing their features, physical
properties, quarry and transport techniques. These statues,
consisting mainly of large stylized heads equal to half
their height, where hewn out of soft volcanic tuff and
transported for miles to reach their final position on
top of a wall facing with their backs to the ocean. It is
shown that the most logical means of transport available to
the natives ( unfamiliar with the wheel and metal working )
was dragging the Moai horizontally. With enough men, some
lubrication, and good ropes this should have been a
relatively simple task. Go **HERE
**for details of the discussion.

**May 20. 2019-How are the Binomial Expansion, Pascal's
Triangle, Coin Flipping and the Gaussian related to each
other?**

We show that the Binomial Coefficient C[n,k) for large n
approaches a standard Gaussian. Furthermore a normalized
Pascal Triangle can be used to measure the probability of
any coin flipping procedure. Also, the time dependent
temperature in the vicinity of a local hot spot can be be
nicely represented bv a widening but height decreasing
Gaussian. The IQ disribution of the US population also has
the form of a Gaussian with a mean of 100 and a standard
deviation of 15. This fact allows one to directly measure
the fraction of the US population which has an IQ above a
specified value. Go
**HERE** for details
of the discussion.

**June1, 2019-How can Price Time Windows be used in
conjunction with Lag Curves to detect Equity Trends?**

We show how with the use of different length price
windows taken together with sketched in lag curves one can
predict price changes into the immediate future. On does not
know how long a new trend will last but nevertheless can act
accordingly until the trend changes. For the case of
exchange traded funds (ETFs) such as SPY and QQQ we
use three time windows of twenty-five, five and one
year duration to indicate whether markets are in an up or
down trend. Points where trends change are marked by B or S
and indicate what action should be taken. Go **HERE** for details of the
discussion.

**June 12, 2019-How does one generate and manipulate
Numbers in different Bases?**

We examine numbers N expressed in different bases with
emphasis on binary and hexadecimal forms. A list of the
first 40 decimal numbers are tabulated in their equivalent
binary forms given. Observation of how such numbers are
generated are discussed in detail. We also show
manipulations with a binary abacus. Several additional
calculations involving a binary systems are presented.
The hexadecimal system is shown to produce extremely simple
presentations for 16 taken to any positive integer power. Go
**HERE**
for the details of our discussion.

**July 2, 2019-What is a Frieze and how is it Constructed?**

A frieze is any base pattern which is repeated
continually in a given direction.Good examples of such
figures are found in architecture, rug borders, floor
mosaics and web page separators. We show how they are
constructed using Heaviside Step Functions, Examples derived
include square pulses, waves, and arrow friezes. A second
method of construction using copy and paste in the microsoft
paint program is also discussed, Go **HERE** for the details.

**August 4, 2019-How does one trade ETFs such as SPY using
the S&P500 Stock Index and its Lag Curve?**

We show how one can use a broad stock market average
such as the S&P500 Index with its time-dependent price
P(t) and its constructed lag curve $\lambda $(t) to
successfully trade and invest in certain Exchange Traded
Funds such as SPY and QQQ. Although the return using this
approach will not match the possibly higher yield investing
in individual stocks, the method is much safer and offers
excellent liquidity when dealing with high
price-volume ETFs. We present a long term historical price
chart extending back 25 years to help determine the type of
stock market one is in at present. The basic rule to adhere
to at all times is to be long only if the price P(t)>$\lambda $(t)
and short when P(t)<$\lambda $(t).
Go **HERE**** **for
details of the discussion.

**August 27, 2019-What are the four main uses of Nuclear
Energy?**

We examine in some detail the four basic uses of nuclear
energy. These are bombs, nuclear reactors, isotopes in
medicine , and nuclear propulsion. Schematics of both
fission and thermonuclear bombs are presented and
their various components discussed. Basic nuclear reactors
are shown two consist of two loops which exchange heat
occurring at a common heat exchanger. The use of cyclotrons
to create radioisotopes is also discussed as well as a
potential nuclear powered cruise missile. Go **HERE** for details of
the discussion.

**September 5, 2019-How does one distinguish Prime Numbers
from Composite Numbers?**

We divide all numbers into either composite numbers
characterized by multiple divisors and primes which
are only divisible by themselves and one. This separation is
most easily accomplished by evaluating the number fraction
f(N)=[sigma(N)-N-1]/N, where sigma(N) is the divisor
function of number theory. For all prime numbers the value
of f(N)=0 while for composites f(N)>0. We also show from
a graph of N versus f(N) that all primes greater than three
have the form 6n+1 or 6n-1 without exception. Twin primes
and semi-primes are also examined using points located
along a hexagonal integer spiral. Go **HERE** for
details of the discussion.

**September 11, 2019-What is the KTL Method and how can it
be used to yield very accurate estimates for Tan(x)?**

We discuss a new mathematical technique based on the use
of even Legendre polynomials to produce highly accurate
approximations for all trigonometric functions. The
procedure was first introduced on our MATHFUNC web page in
2012. It has since that time been named the KTL method in
the literature. By evaluating the integral
int[cos(ax)P(2n,x),x=0..1], where P(2n.x) are the 2nth order
even Legendre Polynomials, one finds a quotient
approximation for tan(a) accurate to any desired number of
digits provided one makes n large enough. Since tan(a)
relates directly to sin(a) and cos(a), we are able to easily
generate a table with elements accurate to 50 places in
0<a<Pi/4. Go **HERE** for details of
the discussion.

**September 14, 2019-How can dual Price Charts distinguish
between Bull Markets and Bear Markets?**

We create dual price charts for a variety of different
commodities including stocks, bonds, and home prices. These
allow us to clearly distinguish between Bull(uptrend) and
Bear (downtrend) Markets leading to profitable transactions
on both the long and short side of markets. The dual
curves present both price and inverse price as a
function of time typically over ten year time frames. Go **HERE** for details of
the procedure.

**October 3, 2019-How does one construct Fractal Curves
starting with a Unit Square?**

We examine the construction of several fractal curves
using different base elements and starting with a unit
square. The results for the first few iterations allow one
to predict both the enclosed area A(n) and the boundary
length L(n) for each generation. One finds that A(n) remains
one for all iterations n (for the particular generating
elements chosen) while the bounding curve increases
continually in length following the rule L(n)=4(b/a)^{n}.
Here 'b' is the the number of length increments making up
the generating element while 'a' is the direct distance in
increments between the ends of the generating element. That
is, the fractal curve will have a Hausdorff Dimension equal
to d=ln(b)/ln(a). Go **HERE** for details of the
discussion.

**November 11, 2019-How does one determine when a Stock
is a Buy or a Sale?**

We show that the basic idea of buying low and selling
high can be applied to stocks by use of long term historical
price charts coupled to running lag curves. As a result of
such a technical approach one can better understand when any
equity is expensive or inexpensive and thus when a
transaction should take place. Numerous examples of past and
future buying and selling opportunities for better known US
stocks are analyzed and clear buy(B) and sell(S) signals
given. At the moment many of these exhibit upward moving
trends indicating a Bull Market.Go **HERE** for details of the
discussion.

**November 18, 2019-What were the buy and sell points
for the Dow Jones Industrial Average over the past 100
years?**

We use five long term(20 year) price charts together with lag curves to determine when one is in a longer term uptrend (bull market) or longer term downtrend (bear market). The beginning of such trends are designated by B for being long the market and S for being short the market.Typically in a twenty year range one has about three up markets and three down markets. By heeding the B and S signals one will always remain on the right side of a trend. Detailed discussions of each of the five long term graphs show how one would have been out of any long position several months before the 1929 market crash and been able to take advantage of the factor of ten rise in stocks during the period 1980-2000. Also one would have gotten out of any long positions shortly before the beginning of the dot.com bubble collapse(2000) and the great recession(2008). Go**HERE** for details of the
discussion.

**November 28, 2019-How are all the Positive****
Integers categorized into their various Sub-Groups?**

We show how all positive integers N may be classified
into subgroups of even, odd, prime, semi-prime,twin-prime,
composite,and super-composite. Both of our earlier found
concepts of hexagonal integral spiral and number fraction
f(N) are employed to do this. Results show that f(N)=0
produces primes, 0<f(N)<1 yields composites, and
f(N)>1 produces

super-composites with numerous divisors.Twin primes exist only if their mean valueequals 6n, with n=1,2,3,. .Go**HERE** for details of
the discussion.

**December 12, 2019-How do Microwave Ovens work?**

Microwave ovens, their configuration, power source, and
cooking ability are discussed in detail. We first present a
schematic of a typical microwave oven, then look at the
magnetron

power source, and next discuss how the generated microwaves produce heating of the water molecules in foods. The microwaves used are typically in the 2450MHz range

meaning the wavelength is about 12 cm. The microwaves used are non-ionizing radiation since their photon energies are orders ofmagnitude smaller than that required

for tearing molecules apart as occurs with x-rays. Go** HERE**
for details of the discussion.

**
2020**

**March 1, 2020-How Does a Vapor Compression Refrigerator
work?**

A standard vapor compression refrigerator functions
by withdrawing heat from a cold reservoir and
depositing it at the hot side of the device. The typical
liquid-gas mixture

used in this heat exchanger is tetrahflouromethane alsoknown as R-314a. A typical refrigerator has just four basic components. These are the compressor, the condenser, the

expansion-valve, and the evaporator. Such machines are extremely reliable and no longer use toxic or other harmful gases.Go**HERE** for details
of the discussion.

Let's begin-

**2010**

This implies an energy consumption rate of-

[474x10^18J]/[365x24x3600]s=1,50x10^13W=15TeraWatts

The sources for this energy are oil, coal, gas, hydro, nuclear, and renewables in descending order. Go HERE for further discussions.

November 1, 2010- How did early stone carvers construct massive cyclopean walls using boulders of irregular shape and how where they able to produce extremely tight fits between neigboring stones without the use of mortar?

December 17, 2010-What is the origin of the Decimal Numbering System?

With only a few exceptions, the base ten number system is the dominant one used throughout the world for commerce and measurements. Why is that? Clearly it stems from the fact that early man used his ten fingers to count objects and that such finger counting was soon replaced by symbols such as sticks and by marks recorded on surfaces. The Roman number system of R={I, II, III, IV, V, VI, VII, VIII, IX, X}clearly hints at such stick figures as does the earlier Chinese method of representing numbers by bamboo sticks. We want HERE to discuss the probable origin of the Decimal Numbering System and indicate how the basic mathematical operations of addition, subtraction, multiplication, and division are accomplished. A few other number systems are also discussed.

December 22, 2010-Why are the most prosperous and technologically advanced and creative countries around the world located within two relatively narrow temperature bands ranging from yearly lows no lower than -10degC and highs no higher than +30degC ?

If one looks at the most technologically advanced countries around the world it is clear that these lie within two temperature bands which I call the creativity(or JUST RIGHT) bands. These zones have produced and are producing the majority of the literature, art, science, and technology and have been the home of such creative individuals as Plato, Confucius, Michelangelo, daVinci, Shakespeare, Newton, Rembrandt, Bach, Tolstoy, Beethoven,Voltaire, Edison, Picasso, and Einstein.The northern hemisphere band includes western Europe, the western part of Russia , Iran, northern India, China, Korea, Japan, the southern part of Canada, and the United States. The southern hemisphere band includes the southern and eastern parts of Australia, New-Zealand, Chile, the northern parts of Argentina, the southern part of Brazil, and South Africa. We discuss HERE why it is that within these bands most of the advancements in the arts, literature, and science have been and are being made and why most regions outside of these bands remain underdeveloped exibiting very little in the form of advanced human intellectual activity or creativity.

January 22, 2011- What is the big deal with Binary and how does one manipulate things mathematically in this Number System?

Although most of us were brought up in school counting things in a base ten (decimal) number system it is clear that other number systems and especially the Binary System are becoming more and more important. Most computer manipulations, data storage, and the transmission of writing and music are now pretty much handled in binary because of the conveneince of a base requiring just two digits instead of the ten associated with decimal. We dicuss HERE some of the properties of the Binary Number System, look at several problems which are well handled in binary, and show how one can mechanically perform basic mathematical operations using a binary abacus.

March 1, 2011-What is relative time and how is it measured?

Time in an absolute sense does not exist as made clear by numerous philosophers(Kant, Heidegger) and by modern technical analysis including relativity. It can however be measured in a relative sense by comparing the interval beteen two events to that of a known time interval such as the period of one earth orbit about the sun. We discuss HERE how relative time measurements were first developed using astronomical observations and later employing atomic clocks. We use a modified Deborah number to measure relative time. An example of a digital combination of date and time is also presented.

March 27,2011- What are the Economics of Solar Energy Conversion using Photovoltaic Methods?

The recent events in

May 5, 2011- What is the BMI and how is it Measured?

The ever increasing problem of obesity in this country has increased interest in having a quick way to estimate the fat content of the human body. Clearly too much fat is unhealthy and indicates a calorie food intake above that required for the individuals energy output. An excellent approximate measure of fat content is the Body Mass Index(BMI) first proposed by the Belgium polymath A. Quetelet in which one looks at the ratio of a persons weight W ( kg)divided by the square of the heigh H( meters). Population statistics show that the normal range lies between 20 and 25, with values above 25 considered overweight up to 30 and obese above 30. Today nearly 70% of the United States population has a BMI above 25. Go HERE to see our discussion on this index and how one can construct a simple circular sliderule to quickly determine ones BMI.

MAY 17, 2011- What is Symmetry and where is it Encountered?

Most recognize symmetry in an object , be it in the human face, in architecture, or mathematics , but often find it difficult to pinpoint exactly what makes the object so. We discuss HERE some of the characteristics of line, plane, and rotational symmetry by analyzing several different examples. Bilateral symmetry is demonstrated by looking at mirror images of the face of Charlize Theron and of the Taj Mahal. Next hexagons, circles, ellipsoids, cubes , the Laplace equation , and the five petal Rhodonea are examined for their symmetry.

July 1, 2011-Do Stock Indexes World-Wide Correlate?

We have been aware for a long time that stock averages throughout the world have a tendency to correlate . This has been especially true in recent years probably do to the increasing speeds of electronic communications and the use of computer arbitrage. We present HERE a graph of the excellent correlation observed between the S&P500, the German DAX Index, the DOW World Index, the Hang Sang Index, and the ILF ETF over a ten year period. Also a discussion of why such correlations should exist is presented.

July 12, 2011-What is RSA Cryptography and its Connection to Large Prime Numbers?

One of the best and most used cryptography techniques is the RSA approach which uses both public and private keys. It involves a public key which relies on the use of two large prime numbers p and q. The product N=pq is essentially impossible to factor with even the fastest electronic computers in any finite amount of time. Go HERE to see our explanation of the RSA method in simple terms, view an unbreakable public key which I constructed in just a few minutes, and find an example of sending a short encoded message from party B to party A.

July 26, 2011-How does an Energy Balance relate to Weight Control?

It is well known that a large fraction of the American public are overweight according to BMI measurements. This fact has become more prevalent in recent years and has been attributed to overeating accompanied by insufficient energy expenditure due to sedentary working conditions. Mathematically one can say that DW/dt=I-E=B, where W is a person's weight, I is the food intake, E the energy output and B the energy balance factor. Go HERE for a discussion of this problem and how it can most effectively be solved. Decrease in calorie intake and increase in physical activity are the well known factors making for weight loss ( B<0). Weight gain occurs when B>0.

August 1, 2011-What is meant by Exponential Growth and Decay and what are the Properties of an Exponential Function?

Although one often hears and reads about exponential growth in the literature, it is not always clear that the authors know precisely what they are talking about. In discussing true exponential behaviour, it is first necessary to understand what is meant by an exponential function, how it arises and what its properties are. We discuss HERE some of these. Both population growth and radioactive decay are described and we show how one can quickly estimate the value of an exponential function via a quotient of two polynomials.

August 4, 2011-Could the US experience Hyperinflation?

Actions by the Federal Reserve in recent years has essentially tripled the monetary base to about 2.5trillion dollars by simply printing money without any hard assets backing up these dollars. This is a recipe for hyperinflation as witnessed in Weimar during the 1920s and more recently in Zimbabwe. The reason we have not yet seen any major spike in inflation is due to the fact that banks are reluctant to loan out this new money as they have a risk free way to make money directly of the government via government bonds. Once house prices reverse their downtrend and the velocity of this extra money increases, large increases in inflation can be expected. Go HERE for our discussion of some of the histrorical aspects of hyperinflation and the possibility that it could happen in this country.

August 27, 2011-What are Self-Similar Patterns and how are they Generated?

Recent increased interest in fractals and how they may be used to explain many natural phenomena has got me to thinking more about how intricate 2D patterns can be generated from some very simple laws of replication. In particular, we look HERE at self-similar patterns which are generated from elementary geometrical figures such as squares, triangles and hexagons. The problem is treated as one of biological generation.

September 1, 2011-Can one create an easy to understand 3D Fractal based on a simple Reproduction Law?

Still thinking about fractals from our August 27th note, we recently posed ourselves the question is there an elementary three dimensional fractal whose basic initiator structure is simple and one where this structure is replicated forever through an infinite number of generations? We answer this question in the affirmative using a simple cube to generate smaller and smaller self-similar cubes. Go HERE for the details. We have not seen this type of 3d fractal before although some other more complicated 3D fractals have received attention earlier. It is hoped that this type of discussion might lead to a future increased emphasis on 3D fractal structures with the anticipation that they might prove helpful in describing the geometry of certain viruses. We also show how a simply cubic fractal model can be constructed from wood with the right tools and a lot of patience.

October 3, 2011-What is a Pelekinon and how is it used to measure Date and Time?

Before the advent of mechanical and electronic clocks, people depended on sundials to determine the time of day and used elaborate structures (such as Stonehenge) to measure the time of year when the solstices occur. The ancient Greeks used one of the simplest sundials called the Pelekinon. It was capable of telling both the hour of the day and the day of the year. The Pelekinon consists esentially of a vertical pole(the gnomon) and a flat plane on which they empirically observed and marked down the locus of the shadow produced by the pole throughout the year. We want here to discusss the mathematics behind the Pelekinon and show how one can use spherical trigonometry involving the astronomical triangle to accurately predict the movement of the shadow. Go HERE for the discussion.

October 23, 2011-What is the impact speed of a bullet fired straight up upon its return to earth?

Recent vidios showing Libyan freedom fighters firing their guns randomly into the air during victory celebrations, has brought the question to my attention of whether or not such acts pose dangers to individuals nearby. We analyse the problem of the returning bullets by first calculating what height H a bullet , fired from a standard Kalashnikov AK-47, will reach and then determine the impact speed V

December 12, 2011-What is Geometric Art what are the Basic Elements used in its Creation?

It is possible to create appealing abstract art work based on the superposition of simple geometrical shapes. I term this Geometric Art and give examples arising from religion, philosopy, and other movements. In addition I show how this type of art , involving straight lines, circles, squares and other simple 2D geometric figures, can readily be created by computer graphics or via handicraft approaches involving, for example, wood work. Go HERE for the details.

February 1, 2012-How can an Abacus be used to Increase one's Ability to quickly Add and Multiply Numbers?

The standard method of adding, subtrtacting, multiplying and dividing numbers by the standard techniques taught in schools is very often not the quickest way to obtain an answer. Rather using an Abacus which requires a decimal concept of numbers can produce much faster results. We discuss HERE how this is accomplished by looking at how elementary mathematical operations are handled with an Abacus. Several different shortcuts involving multiplication based on recognizable powers of intergers and expresing numbers as differences of simpler numbers are discussed.

February 3, 2012-What are Logarithms and how may they be Used to Multiply and Divide Numbers?

Prior to the advent of hand calculators and electronic computers, the way to handle multiplication and divisions arising in complex mathematical expressions was to first convert the numbers contained therein to logarithms , perform the logarithmic operations, and then invert to get the final answer. We discuss HERE what is meant by a logarithm, how one manipulates them, and how slide rules can be used to quickly calculate approximate answers .

February 26, 2012-How does one express Large and Small Numbers?

Although most readers will be familiar with the exponetial notation for number such as 1000 being equvalent to 10

March 20, 2012-How does Facial Recognition by Eletronic Computer work and what steps might be taken to improve the Procedure?

One of the more important and growing tasks in digital recognition is how to quickly identify a face from a collection of millions of other faces electronially and do so with one hundred percent accuracy. So far such approaches have been only partially successful and will fail when simple changes in hairstyle, aging, or facial orientation are introduced. This seems strange in view of the fact our brain allows facial identification in split seconds by an as yet incompletely understood process. We suggest here that improved computer recognition in shorter times might be accomplished by comparing a given face with the norm of a human face and then identifying things by only the differences. Go HERE for our thoughts on such a process.

May 17, 2012-How does the Concept Life and Evolution not violate the Law of Entropy?

The concept of life and its accompanyting evolution seems to be in contradiction to the Law of Entropy which requires that all systems move in time from an ordered state (low Entropy) to one of disorder (high Entropy). After first discussing the very slow rate of change produced by evolution over thousands of generations , we show how life and accompanying evolution is indeed possible without a violation of the Entropy Law by considering the entire environment . Go HERE for a discussion of the details.

May 27, 2012-What is the Difference between a Mean and a Median?

When reading articles in the popular press concerning population trends, wealth distribution , and weather records one quite often encounters the terms mean and median without the articles explaining the difference . This can lead to confusion on part of the readers and in addition is sometimes used to hide certain facts such as the true unemployment rate and the great wealth disparity existing between diffenert segments of the US population. We discuss HERE these two concepts in greater detail and show how highly skewed input data can lead to large differences between the median and mean of a data set. The AGM method for integral evaluation and the Pareto Curve used by economists is also discussed.

June 17, 2012-What are Pareto Curves and how are they be used to discuss Wealth Distribution?

Over a hundred years ago the Italian engineer and economist Vilfredo Pareto noticed that some 20% of any population typically controls about 80% of the wealth. His statistical analysis led to the power law curve y=x

July 4, 2012-What is the Relation between Dip-Stick Readings and the Volume of Gasoline left in a Storage Tank?

Several years ago one of our undergraduate students who was working part-time at a local gas station, asked me to explain to him the relation between dip-stick readings and the volume of gasoline remaing in an underground storage tank. In looking at the problem it became clear that one should expect a non-linear relation between fluid level and fluid volume. The relation depends very much on the type of tank cross-section one is dealing with. Go HERE to see some detailed calculations for both cylindrical cross-section and spherical cross-section tanks.

July 17, 2012-How were the Pyramids at Giza constructed using only 2500 BC Technology?

One of the most impressive sights in the world are the three pyramids on the Giza Plateau just west of Cairo, Egypt. People have speculated for years how a civilization having not yet invented the wheel nor familiar with iron tools and other simple machines such as pulleys could possibly have been able to construct such massive monuments. We show

We examine the laws of conservation of energy and of momentum to discuss the properties of an ICBM trajectory. Using a minimum of mathematics , we obtain explicit formulas for the height H reached by the missile and the time to impact . An important parameter entering the analysis is the non-dimensional parameter alpha=2gR/V

An interesting problem in 2D geometry is what type of tiles can be used to cover a flat surface (such as a floor) without leaving empty spaces between the tiles. Square and rectangular tiles are obvious examples which can do this. But as we will discuss

When dealing with geometrical figures such as circles and spheres one is often interested in the maximum distance from the figure edge to a chord running beteween two points A and B also located on the edge. This quantity is known as the Sagitta (from the Latin word sagittarius for arrow) and has important applications in architecture and line of sight radar among other areas. Go

November 26, 2012-How does one distinguish between a Composite and a Prime Number when the Number under consideration becomes large?

Most of you where taught in middle or
high school how to distinguish between a prime number
which is divisible only by itself and one and a composite
number which has multiple additional factors. What is
often not clear is how does one know when a number
is prime or composite when the number N becomes very
large. We show you **HERE**
how one can quickly
distinguish between these two groups by using our recently
established fact that all primes above N=3 are of the form
Q=6n+1 or Q=6n-1. Also we introduce a new parameter termed
the Number Fraction which can rapidly distinguish between
the two groups of numbers using a simple computer
routine. Also the Sieve of Eratosthenes is
discussed and a table for the first few Q primes is
presented.

Although the validity of the theory of global warming is still in some dispute, there is little question that ocean levels have varied from much below the present level during the ice ages and above the present during the times of the dinosauers. We look

December 28, 2012- What are the four basic Temperature Scales and how does one convert between them?

We
look at the four commonly used temperature
scales, their history and callibration points.
Starting with the Fahrenheit Scale based on
the freezing of a brine mixture(0 deg F) and
the boiling of water(212 deg F). This is
followed by the Celsius Scale using
freezing water as 0 deg C and boiling water as
100 deg C. Finally the Kelvin and Rankine
Scales are discussed and how their zero points
correspond to absolute zero. It is shown how
the conversion formulas F=(9/5)C+32=R-459.67
and K=273.15+C are derived. Go **HERE** for
the details.

**2013**

January 4, 2013-What is a Complex Number and what are some of its uses?

We present an elementary
discussion of complex numbers and how they arise
and how they may be used to find certain
trignometric formulas. After demonstrating how
the use of such numbers arose in connection with
trying to understaning the meaning of numbers
such as sqrt(-2), we show how one takes roots of the complex number z=a+ib** **and how
to use it to evaluate certain
integrals. Go HERE
for the discussion.

**January
11, 2013-How did the Wheel
develop?**

One of man 's earliest and most significant inventions is the wheel. It developed over thousands of years starting with simple wheels constructed from wooden planks to the latest 541 ft diameter Singapore Flyer Ferris wheel using tension spokes. Go HERE for the details of the historical development. Among the discussion is a treatment of rolling friction of wheels and how bearings are used to minimize frictional resistance at a wheel hub.

**February
16, 2013- What
is the Impact
Speed of an
Asteroid
hitting the
Earth?**

In the last few days the news has been saturated with reports of a near miss of the earth by a football field sized asteroid and also the direct impact of a much smaller asteroid near Chelyabinsk, Russia which caused sonic boom damage. We examine the trajectories of such asteroids and in particular calculate the impact speed an asteroid will have when originating from the asteroid belt. Go HERE for the details.

January 4, 2013-What is a Complex Number and what are some of its uses?

One of man 's earliest and most significant inventions is the wheel. It developed over thousands of years starting with simple wheels constructed from wooden planks to the latest 541 ft diameter Singapore Flyer Ferris wheel using tension spokes. Go HERE for the details of the historical development. Among the discussion is a treatment of rolling friction of wheels and how bearings are used to minimize frictional resistance at a wheel hub.

In the last few days the news has been saturated with reports of a near miss of the earth by a football field sized asteroid and also the direct impact of a much smaller asteroid near Chelyabinsk, Russia which caused sonic boom damage. We examine the trajectories of such asteroids and in particular calculate the impact speed an asteroid will have when originating from the asteroid belt. Go HERE for the details.

One often encounters situations where the ability to make quick mental math calculations without the aid of pocket calculators or pen and paper come in handy. For example , calculating the amount of an 18% tip on a dinner at a restaurant or the miles per gallon one's car has achieved during a recent trip. We discuss HERE how such simple mathematical calculations involving just addition, subtraction, multiplication, and division are performed mentally. Also we show how quick approximations to certain mathematical questions can be obtained without outside help or before someone pulls out an iphone to calculate or google things.

Those believing in Keynesian Economivcs claim that when a country is in recession or depression the way to get out of it is to devalue the currency by running the printing presses. Ben Bernanke and the Federal Reserve have been doing this since 2008 at an ever increasing rate with little to show for it except a dilution of the currencty by over 300%, the stoking of a stock market mania , and an artificial increase in bank earnings. Their policies have not helped unemployment and have had the direct effect of depriving retirees of a reasonable return on their life savings by running an essentially zero interest rate policy. It is time for the Fed to change course and to do so before international currency wars erupt and the dollar looses the little remaining value it has left. The best solution would be to immediately clean house at the Fedreal Reserve and in particular remove Chairman Bernanke before further damage is done. Go HERE for the details of the discussion.

If you ask individuals how is the earth's tilt axis is related to seasonal changes and how one locates the north and south poles on the Celestial Sphere, most will not be able to answer. We give HERE a brief answer to these questions including how to determine the local latitude by measuing the altitude of the celestial poles above the horizon. A time lapse photo showing the circular paths stars take in the night sky about the south celestial pole is also presented.

There are five basic Platonic Convex Polyhedra known as the Tetrahedron, the Hexahedron, the Octahedron, the Dodecahedron, and the Icosahedron. They were discovered by the ancient Greeks and form the starting point for most most discussions on 3D geometry. We show how these solids can be constructed via computer by using a guiding sphere on which all vertices of these polyhedra are located. Go HERE for the details. We also show how one can use cardboard cut-outs and wood polygons to construct these polyhedra .

We use simple geometry and elementary calculus to determine the volume and surface areas of pyramids consisting of a regular polygon base and a vertex placed at height H above the centroid of the base. The results show that all of these pyramids have a volume equal to the product of the base area times the height divided by three. The properties of the square base Great Pyramid of Cheops at Giza are also discussed including the volume of such pyramids during construction. Go HERE for the details.

It has now been several weeks since the Guardian Newspaper's revalation of massive warantless spying by the National Security Agency(NSA) on all Americans . A heavy defense of these actions has been mounted by the mass media and the Executive, Legislative and Judicial Branches of the US Government. The essence of their arguments being that such surveilance is neseccary to protect against acts of terrorism. We are expected to believe the recent testimony by General Keith Alexander that this program has indeed protected us from over fifty attacks although he could not tell us what they were because its 'secret'. There seems to be only a minority including myself who recognize that there has been a violation of the 4th Amenment to the US Constitution and the

The Nicaraguan government has recently agreed to the building of a canal across southern Nicaragua by a Hong Kong based company. We discuss HERE the ecomomics of such a proposal and compare it with the cost of modernizing the Panama Canal. The difficulties with the construction of a trans-Nicaraguan canal, which would be about twice the length of the Panama Canal, would be the need to cut through higher terrain than in Panama and also be subject to potential earthquake damage, not to mention border disputes with Coasta Rica. We conclude that a more economic approach for all involved would be to not undertake the construction of such a canal but rather increase the size of existing locks of the Panama Canal in order to accomidate larger ships. The financial support for such an undertaking could come from a world wide consortium of companies, not unlike the AirBus consortium, if Panama can be made to agree.

July 20, 2013-At what height H above the equator must a satellite be placed in order to be Geosynchronous?

A geosynchronous satellite is one whose angular velocity matches that of the earth's rotation rate of

ω=2π /(365.25x24x3600) r/s. Using Newton's second law one finds that such a match occurs when the satellite height H above the equator is H=R{-1+[g/ωR]

One of the more important concepts students encounter in analytic geometry are the Conic Sections. These are 2D curves formed by the intersection of a plane intersecting a cone at different angles. They may be circles(e=0), ellipses(e<1), parabolas(e=1), or hyperbolas(e>1) depending upon the eccentricity e. We show HERE how to use some simple mathematics to derive these curves. Applications for each of the figures are also briefly discussed.

Number Theory is that branch of mathematics which deals with the relation between the positive integers 1,2,3,4,5.. . It has become of more practical importance in recent years with the advent of high speed electronic computers and the use of digital encryption. Basically the integers break up into two groups- the prime numbers such as 3,5,7,11,13.. and the composite numbers 4,6,8,9,10,.. We state some of the known properties for these groups and introduce some new concepts such as the number fraction f(n) and the plotting of Q primes along a hexagonal spiral. Go HERE for the details.

October 3, 2013-What is the Future of the US National Debt?

Perhaps the most critical economic problem facing this country at the present time is a run-away national debt which has increased by a factor of over fifty in the last half century and at the present time stands at 17.065 trillion dollars and accelerating. We make a linear extrapolation of this debt into the near future and show that in about seven years the US Treasury will be paying well over a trillion dollars a year just to service this debt. Clearly this is unsustainable and will lead to a default of the dollar. Printing of money by Ben Bernanke will work only as long as the lenders don't recognize the futility of his actions and recognize that their dollar holdings are being depreciated on a daily basis. A calculation based on the assumption that the US monetary system will collapse when the debt reaches 130% of GDP, suggests this point will be reached shortly before 2020. Go

In the history of mathematics and especially in number theory investigators have established many mathematical principles based on the generalization of observations based on special cases. In many instances such generalizartion have proven to be valid for all integers and have led to the establishment of certain universal principles while in other cases the generalizations have failed. We want here to discuss some of the more important results obtained by generalization in number theory and then add a few thoughts on our own generalizations involving prime numbers. Go

November 28, 2013 (Thanksgiving)-How does one use Vector Operations to determine Angles between Straight Lines in Space?

It is known that any straight line connecting points [x

December 6, 2013-How does one determine the Volume of an N sided Polyhedron?

anuary 1, 2014-What is the Relationship

As many of you are aware, Sudoku is a very popular and addictive number puzzle in which one is asked to find all integers in a square array given a certain number of starting values. It is governed by several simple rules and is related to Euler's Latin Square.We show that the solution procedure is simply an extensive manipulation of elements in an n x n square matrix and involves finding the compliment to known numbers found in the row, column and sub-matrix which contains the element a

February 1, 2014-How did the Ancient Egyptians measure Slopes during Pyramid Construction?

February 20, 2014-What are Magic Squares and how are they Constructed?

March 1, 2014-What is the relation between Grade and Angle of an Inclined Surface?

March 9, 2014-What is a Temperature Well and how can its properties be used to build a better Heat Exchanger?

May 27, 2014-What is the Area of any N sided Irregular Polygon?

JUNE 11, 2014-What are Sequences and how are they Generated?

Sudoku is a math game in which one is given a few numbers in a square array and asked to find the remaining ones in order to generate a complete Sudoku Square. These squares have the property that all rows and columns in the n x n array contain each of the n numbers only once. The n sub-matrices of the square also contain each of the n symbols only once. It is our purpose here to look at the reverse (and much easier) problem of first generating some generic Sudoku Squares, examine their properties, and then use them to construct Sudoku Puzzles. Go

July 6, 2014-How does one determine Latitude and Date using only Astronomical Observations?

A Reuleaux Triangle is constructed from an equilateral triangle of side-length s by drawing circular arcs of this radius from each of the three vertexes. When such a triangle has its centroid follow a closed path about a fixed rotation axis, the triangle can sweep out a near perfect square. This allows one to use such triangles to drill holes of nearly square cross-section. We show

August 12, 2014-How does one determine the Curvature of a Function f(x) and its corresponding Evolute?

The Neolithic monument Stonehenge near Salisbury England was built about 4500 years ago. It's reason for being have fluctuated over the years from that of a primitive astronomical observatory to a place of worship and burials. We look here at the basic equations for determining the azimuth at sunrise and sunset during the solstices, to show that the orientation of the monument strongly supports its role as an instrument for measuring the time of the Summer and Winter Solstice, but nothing more elaborate than that. We find. among other things, that the hour angle (HA) along sun's path during the winter solstice is very close to one radian (3hrs 49min) after local noon. Go

**September
25, 2014-What are the Properties of Lenses?
**

December 18, 2014-What are the Dimensions of the Great Pyramid of Cheops?

January 28, 2015- How much does a Football's Pressure drop with Decrease in Air Temperature?

January 31, 2015-How does one Construct 2D Figures using only Straight Edge and Compass?

February 7, 2015-What is the Gaussian, its Properties and better known Applications?

February 9, 2015-What is an Integer Spiral and what are some of its more important Properties?

February 23, 2015-What are some of the Areas which can be created by the Intersection of Circles and Straight Lines?

We examine the concatenation of line increments L with
specified orientation theta to form both open ended and
closed curves in two dimensions. After discussing
how a piece of land areas can be defined by the length
of its sides measured in paces and the compass direction
these sides take, we proceed to construct curves such as
the rectangular pulse, regular polygons, and spirals. Go **HERE** for the details.

We look at the four basic mathematical manipulations of addition, subtraction, multiplication, and subtraction and show how these operations can be performed quickly and accurately in one's mind without use of pencil and paper or electronic calculator. Also we indicate how to take powers and roots of integers plus calculate percentages of numbers. Go

May 13, 2015-How does one find the Focal Points of Parabolic and Elliptic Reflectors?

We construct some two dimensional curves formed by the concatenation of straight line segments connected to each other by specified angles. Figures like a Swiss cross, a multiple pulse function, a magnified and upside down pentagon, and a hexagonal spriral are generated. Closed curves are formed when the sum of the connection angles after m elements add up to 2 pi. Go

August 1, 2015-How can one use a Square to generate intricate Fractal Patterns?

August 21, 2015-How does one generate complicated 2D curves via a concatination of straight line elements?

October 1, 2015-How does one determine the Area of an N sided Polygon?

October 15, 2015-What is a Cruise Missile, its development History, and future Directions?

December 12, 2015-How can one find Super-Composites?

December 20, 2015-What is the Black Snowflake and Cellular Automata generated by it?

January 15, 2016-What are the Seven Rules for Successful Stock Market Investing?

We discuss the basic investment rules I follow for successful investment in equity markets. These rules have evolved over a fifty year period ever since my first stock purchases as a teenager in the mid fiftees. The basic idea is to be long stocks only in bull markets and short stocks only during bear markets. The type of market one is in is determined by looking at the longer term price trend of the market averages when compared to a running lag curve. Investment risks are lowered by not going on margin and dealing with ETFs as opposed to individual stocks. Go

circumscribing and inscribing a triangle are derived. Next the possibility of tiling with oblique triagular tiles is demonstrated . Also a tile flooring involving the simultaneous use of equilateral and isosceles triangles are shown. Finally we discuss how oblique triangles are important in the functioning of trusses and tripods. Go

May 31,2016-How does one determine the Properties of an Equilateral Triangle using only Intersecting Interior Lines?

June 6, 2016-How do the Pascal Triangle and the Gaussian relate to Games off Chance?

z[n+1}=(a+ib)

We solve the latest puzzle posed in the mathematical recreation page of the Nov.19, 2016 issue of the Wall Street Journal and then consider several variations thereof including the spacing left by a bundle of seven circles tightly bound by a regular hexagon. Among other problems considered are Heron's Formula for any oblique triangle whose sides are tangent to a single radius R circle placed within the triangle. Also we show how the Golden Ratio follows naturally from the radius ratio of two different circles intersecting a pentagon. Go

We show that all primes five or greater have the form 6n$\pm $1. This fact makes it possible to locate all but the primes 2 and 3 along two radial lines 6n+1 and 6n+5 which cross the vertexes of a hexagonal spiral. This geometrical picture of all positive integers allows one to quickly distinguish between prime and composite numbers. The combination of hexagonal spiral and radial line is reminiscent of a spider web and is the reason we often refer to this pattern as such. Any odd number of the form 6n+3 and all even numbers must always be composite numbers no matter how large N becomes. Go

January 20, 2017-What are the Properties of Squares?

February 6, 2017-What are the Laws of Geometrical Optics?

We introduce two new functions alpha(N)={sigma(N)-N-1}/N and beta(N)={tau(N)-2}/N

One of the more intriguing problems in economics is how to distinguish longer term price trends compared to short term price fluctuations. One way to accomplish this is to first look at a longer term price history and use this to filter out the higher frequency components. Once this has been done one can then draw a type of long term moving average referred by us as a lag curve, When the lag curve lies below the smoothed price one has a buy signal and when the lag curve lies above the smoothed price one has a sell condition. As we will show

constants to 100 place accuracy?

We discuss the properties of regular pyramids including their volume and surface area. Cones as pyramids with a regular polygon base with an

September 21, 2017-What is a Void Fraction and its values for bundled Regular Polygons?

We answer a question appearing in last Sunday's Wall Street Journal puzzle page. The question is to see how the area of a spherical cap created by a compass tracing out a circle of radius L on the surface of a sphere varies with sphere radius.We approach the problem from both an elementary viewpoint and then also via more rigorous mathematics. In both case one finds the spherical cap area generated is always equal to A=$\pi $L

It is well known that any commodity will have periods of long uptrends and long downtrends during its existence.If these trends can accurately identified early after the onset of a new trend large profits can be made.The secret for positive financial returns is to first identify the trend one is in and then stay with the trend until there is a reversal. Whether one acts on the long or short side of a market is immaterial as long as the price stays above a lag curve during bull markets and below a lag curve for bear markets. Go

2$\pi $ and that the sum of the x and y components of the stick arrangement vanish. The oblique triangle and non-symmetric quadrangle are discussed in detail with specific examples given. Also the interior area for any regular polygon is derived. Go

Calculus is that field of mathematics to which students are exposed right after analytic geometry and trigonometry. It deals with continuous functions defined over given intervals. The two parts of the subject are known as Differential Calculus dealing with derivatives of functions and Integral Calculus involving the determination of areas under specified curves. We present in this article a very much abbreviated version of calculus which is intended to be used as a handy supplement to existing calculus books often exceeding five-hundred pages in length or so. We call the present description Calculus in a Nutshell. Once you master it you will br well on the way to understanding of the entire subject. Go

After introducing the concept of number fraction f(N), we clearly show that primes correspond to f(N)=0 and super-composites to f(N)>1.4. Mersenne, Fermat, and Perfect Numbers are also discussed. In addition, logarithms of integers to various bases are examined. Go to

primes fall along just two radial lines 6n+1 and 6n-1 provided the primes are greater than three. The vertexes of the resultant Hexagonal Integer Spiral locate all positive integers starting from 1 through N. Mersenne and Fermat primes are shown to be sub-classes of the 6n+1 and 6n-1 primes, respectively. We also introduce a more general prime number generator T[a,b,c]=a

certain additional facts less well known about these five solids, such as the fact that they all make fair die for any gambling game. After deriving their surface area and volume and their corresponding surface to volume ratio,we show how they may be constructed using metal, wood, or cardboard. In particular we concentrate on the dodacahedron which apparently held religious significance to first and second century AD Romans. Go to

how one can use elementary mathematics to locate all twelve of its vertices and from this information construct an icosahedron by tiling with equilateral triangles. Using polar coordinates, the vertex points are readily calculated. This allows for the construction for a wire-frame map in the shape of an icosahedron. Once this frame is covered by tiles, the desired figure will result. We also show how one can use a 2D Duerer net to produce this structure by a simple folding procedure. Go

starting conditions. Among the sequences considered are the power sequence S(n)=sum(k

for details of the discussion.

When looking at an object by eye one receives two slightly different images on the retina for each eye. The brain merges these images into a single one lying at distance D where the object is located. Parallax represents the angle $\alpha $=arctan(d/D), where 2d is the distance between the eyes

The changeover from standard to daylight savings time (DST) and visa versa has become a real burden to not only homeowners who are required to often reset over a dozen clocks a year but also for the travel agencies and especially the airlines. This morning I saw an interesting article in the local newspaper(the Sun) about Marco Rubio(R-FL who is proposing doing away with standard time entirely and just staying on DST year round. He is on the right track but is meeting opposition from teachers who worry about their young students having to line up for their schools bus while it is still dark. We propsse here an alternative approach which would eliminate both standard and daylight savings time as they are today and instead add one half hour to the local noon as found in the middle of the four time zones within the US. Go

We use five long term(20 year) price charts together with lag curves to determine when one is in a longer term uptrend (bull market) or longer term downtrend (bear market). The beginning of such trends are designated by B for being long the market and S for being short the market.Typically in a twenty year range one has about three up markets and three down markets. By heeding the B and S signals one will always remain on the right side of a trend. Detailed discussions of each of the five long term graphs show how one would have been out of any long position several months before the 1929 market crash and been able to take advantage of the factor of ten rise in stocks during the period 1980-2000. Also one would have gotten out of any long positions shortly before the beginning of the dot.com bubble collapse(2000) and the great recession(2008). Go

super-composites with numerous divisors.Twin primes exist only if their mean valueequals 6n, with n=1,2,3,. .Go

power source, and next discuss how the generated microwaves produce heating of the water molecules in foods. The microwaves used are typically in the 2450MHz range

meaning the wavelength is about 12 cm. The microwaves used are non-ionizing radiation since their photon energies are orders ofmagnitude smaller than that required

for tearing molecules apart as occurs with x-rays. Go

used in this heat exchanger is tetrahflouromethane alsoknown as R-314a. A typical refrigerator has just four basic components. These are the compressor, the condenser, the

expansion-valve, and the evaporator. Such machines are extremely reliable and no longer use toxic or other harmful gases.Go

We
discuss the
Corona
Pandemic
and
show how
existing data
can be used to
make
predictions
about the
number of
infections and
deaths to be
expected
during the

coming months. We also show what makes this virus such an effective killer by keeping the infection rate low but he means of virus transmission easy. Go

**HERE** for
details of the
discussion.

coming months. We also show what makes this virus such an effective killer by keeping the infection rate low but he means of virus transmission easy. Go

We
use existing
data on the
total number
of Corona
Virus deaths
to extrapolate
these numbers
thirty days
into the
future. A
straight line
extrapolation
curve relating
the logarithm
of the total
number of
deaths D(t) to
the number of
days t since
January 22,
allows us to
do this. We
expect the
world wide
deaths to
reach 2.5
million a
month from
now. Go **HERE** for details of the discussion.

We
look at the
latest data
(t=95) on
world-wide
deaths D(t)
caused by the
Corona Virus
and also
present an
asymptote for
the number of
expected
deaths over
the next few
months. The
linear
asymptote
reads
D(t)=-465,200
+7010 t , so
we can expect
a quarter
million deaths
world-wide
when t=102
(May 2). This
is a week from
now. We will
be studying
this latest
death trend
very carefully
looking for
the point
where D(t)
falls
significantly
below the
asymptote.
Also some new
observations
on the effect
of mitigations
on fatalities
will be
discussed. The
largest death
rates are
expected for
those
individuals in
confined
spaces under
continued
exposure to
corona viruses
from infected
and
asymptomatic
neighbors. Go** ****HERE**
for details of
the
discussion.

We
examine the
latest chart
of both the
cumulative
worldwide
number of
fatalities
D(t) due to
the
CoronaVirus
and the daily
changes
D(t)-D(t-1)
starting from
Jan.22 of this
year(t=1)
through
today(t=116).
It was already
shown earlier
on this web
page that D(t)
has an
exponential
behavior
for the first
t=70 days or
so, then it
transitions
into a linear
behavior
mode, and
finally gives
an indication
of a
leveling off
ending near
D(t) of one
half million
fatalities
worldwide by
July
20.(t=180).
The
corresponding
final numbers
for the USA at
t=180 will be
around
143,000.Go **HERE**
for details of
the
discussion.

**JUNE
1,2020- What
did Nathan
Rothshild mean
by the saying
"Buy Sheep and
Sell Deer"?**

Back in
the early 18
hundreds a
reporter asked
the banker and
speculator
Nathan
Rothshild how
he became so
rich. His
reply, in his
heavy foreign
accent,
was to buy
sheep and sell
dear. What he
meant was that
one should buy
things when
they are
inexpensive
and sell them
when they have
become
overvalued.
This is in
essence the
fundamental
law of
capitalism
easy to state
but extremely
hard to apply
successfully.
It is our
purpose here
to show how
one can follow
this rule with
considerable
success by
studying long
term graphs of
any commodity
of interest
and then
marking
observed
turning points
on such graphs
by buy(B) and
sell(S)
signals. Go**
HERE**
for details of
the
discussion.

We look at the latest (t=162) world-wide deaths due to the Corona-Virus. The deaths continue to rise with the US now responsible for 25% of the world total. After three months into the pandemic the cumulative trend has gone from an exponential form to a linear phase represented by D(t)=-244,868+4,709 t. Until the D(t) data falls significantly below this upward linear trend the hope for a pandemic cessation seems unlikely. Even after a departure point is reached the final worldwide totals will equal about twice the value at this yet unknown departure point. Go

JULY 29, 2020-What is the Latest on Factoring Large Semi-Primes?

August 1, 2020-HOW DOES ONE GENERATE LARGE PRIMES USING STRINGS PRODUCED BY FINITE SERIES?

August 18, 2020-What are the latest Facts concerning the CoronaPandemic?

September 23, 2020-How does one define the Roman Numerals and how are they Manipulated?

September 26, 2020-What does the Covid 19 data look like on Day 246 into the Corona Pandemic?

We look at the latest statistics on the Corona Virus and summarize the on-line data into a single graph giving the world cumulative death rate D(t) as a function of days t since January 22, 2020. After day 60 one notices that the D(t) trend is going upward in a nearly linear manner approximated by D(t)=-320000+5300t. So far there seems to be no slowing down of theD(t) trend. Using the given linear approximation formula, one can make predictions about the near future. A table giving some of these future D(t) values for the US population only is presented. Go HERE for further details of the discussion.

October 25, 2020-What are the Wave Characteristics of Stock Prices?

November 9, 2020-How does one produce 2D Curves by solving certain first order Differential Equations ?

November 12, 2020-What is Potosi and Cerro Rico?

November 14, 2020-What are SBS Patterns for Stock Prices?

December 9th, 2020-How do things look on DAY 321 of the Corona Virus Pandemic?

December 28, 2020-How does one construct modified Pascal Triangles?

n!. These trianglar arrays can be constructed using summations involving earlier elements. One of the new modified Pascal Triangles reads 1;2 2;3 4 3;4 6 6 4;

. Another one reads 1;1 1;1 6 1;1 16 16 1; . Go HERE for further aspects of the discussion.

January 15, 2021-How does the Corona 19 Pandemic look on Day 358?

January 19, 2021-How does one factor any Semi-Prime using the S Function?

January 22,2021-What are the Properties of the S Function and its role in factoring Semi-Primes?

We examine the point function S=[sigma(N)-N-1]/2=(p+q)/2 used in factoring of a semi-prime N=pq into its two components p and q. Two approaches are described and then applied to the special case of N=26578351 to yield the prime factors p=3797 and q=5683. When N is smaller than about 10^40 one's home PC can be used directly to find the sigma function of sigma(N)=21587832. For still larger Ns above 10^40 one needs an alternate approach starting with a guess for alpha=p/sqrt(N). Go HERE for details of the discussion.

January 23, 2021-What is Scientific Notation and the Meaning of Greek Prefixes in Number Designation?

February 11, 2021-How does one factor

We examine the formula [q,p]=S$\pm $sqrt(S^2-N) which has the ability to factor any semi-prime N=pq into its two prime components p and q. Here S=[sigma(N)-N-1]/2 with sigma(N) being the standard sigma function of number theory. For the small value of N=77 we have sigma(N)=96 and S=9. So that [p,q]=[7,11]. For values as high as N=10^40, values for sigma(N) can be obtained in split seconds on my PC and hence N=pq is factored. It is also shown that the forty digit semi-prime

N=1912492750926191821089996842096354214449 factors into its components [p,q]=

Febuary 18, 2021- How are the two Basic Laws of Trigonometry derived?

February 21, 2021-What is an SBS Stock Channel?

March 5, 2021-What are the Stock Channels for the Golden Five?

GFI=0.4828 GOOG+4.2868 MSFT+1.4309 TSLA+7.9453 AAPL +0.3130 AMZN

March 8, 2021-How does one generate large Primes using combinations of Infinite Series?

March 17, 2021-What are some further Properties of Number Fractions and Hexagonal Integer Spirals?

April 1, 2021-How are the Roots of Quadratic and Cubic Algebraic Equations found using their Depressed Form?

April 5, 2021-How does one find the Roots to a Complex Function f(z)?

April 12, 2021-How Effective are the new Corona 19 Virus Vaccines?

April 16, 2021-How does one get Bounds on Pi using the Archimedes Method of inner and outer Polygons for a unit Circle?

We show how the formula [p,q]=S±sqrt(S^2-N) can be used to factor large semi-primes N=pq into their components. Here S=[sigma(N)-N-1]/2=[(1+alpha^2)/(2 alpha)]sqrt(N) and p=alpha sqrt(N) with q=(1/alpha) sqrt. A second variable appearing in the calculations is epsilon which must be adjusted to make R=sqrt(S^2-N) an integer. Several specific examples for semi-primes as high as eight digits are looked at in detail. Go HERE for further details.

April 30, 2021-How did Newton manipulate the Binomial Theroem to generate numerous additional Identities?

May 4, 2021-How can one use the Diophantine Equation to obtain highly accurate Expansions for the Square Roots of Positive Integers?

are also presented. Go HERE for the details of the discussion.

May 28, 2021-What is a simple way to construct a Hexagonal-Integer-Spiral?

June 5, 2021-What is a Formula for detecting Weight Trends in Individuals?

June 8, 2021-What are the Properties of Semi-Primes?

June 13, 2021-What is the LCDS food diet?

July 1,2021-What are some Further Properties of Semi-Primes?

July 9, 2021-What are the basic manipulations involved in Mental Arithmetic?

With a little practice one can carry out many mathematical operations of summation, subtraction, multiplication and division without the need for external tools. We demonstrate this in his article for several different cases. Among other results we show how to find powers and roots of numbers. Also we show how capital grows under compound interest and how to quickly calculate tips mentally. Go HERE for details of the discussion.

July 15, 2021-What are the SBS Channels for the S&P 500 and the Dow Jones Industrial Average ?

JULY 30, 2021-What are the Properties of a new Modified Pascal Triangle?

Augut 10, 2021-What is the Sigma Function and some of its Related Forms?

August 12, 2021-What does one mean by B followed by p>lambda and S followed by p<lambda?

August 15, 2021-What is meant by the Double-Kite Structure of the Human Face?

August 20, 2021-What is the Silk Road?

August 23, 2021-What are the better known Spirals?

August 27, 2021-How does one construct Angles using only Compass and Ruler?

September 2, 2021-What Angles have their Trigonometric Forms expressed as the Square Roots of 3, 4,or 5?

sqrt[(1/2+(1/4)sqrt((5+sqrt(5))/2))] = 0.9876883.. . This means 9 degrees has trig functions involving only roots of five. We find no such root solutions when n is an odd number seven or greater. Go HERE for futher details of the discussion.

September 7, 2021-How does one apply the L'Hospital Rule to evaluate certain Quotients at limiting values of x?

We examine quotients F(x)= f(x)/g(x) in the limit where the quotient has either value 0/0 or infinity/infinity. Specific examples considered are F(x)=(x^3-2x+1)/(x-1)^2 and F(x)=sin(x)^2/x^3 at x=1 and x=0, respectively. Also we look at the value of (1+1/n)^n as n approaches infinity. Go HERE for further details of the discussion.

Today is my 85th Birthday and an appropriate time to formally end

Kurzwegs's TECH BLOG

Many of the ideas discussed on this Web Page over the past eleven years can be expanded further specifically those involving semi-prime factoring, market movements, hexagonal integer spirals, the modified PASCAL triangle, KTL method for approximating any trigonometric function, and generation of large primes using products of irrational numbers.

Once in a while I will be coming up with some new articles falling under the auspices of this Tech Blog Page. These will come at a slower and more irregular rate since my original September 16, 2021 sign-off. A few word descriptions and their URL addresses follow-

(1)-Relaton between Polynomials and Infinite Sequences found at-

https://mae.ufl.edu/~uhk/POLY-SEQ.pdf

(2)-Appoximating Trig Functions using the KTL Method-

https://mae.ufl.edu/~uhk/KTL-METHOD.pdf

(3)-More on the number fraction f(N)-

https://mae.ufl.edu/~uhk/NUMBER-FRACTION-2021.pdf

https://mae.ufl.edu/~uhk/PRIMES-FROM-IRRATIONALS.pdf

(5)-Barcharts and Barchart Readers-

(7)-Five year ETF stock price windows-

https://mae.ufl.edu/~uhk/Five-Year-Stock-Windows.pdf

(8)-Solving the Brahmagupta-Pell Equation-

https://mae.ufl.edu/~uhk/brahma-pell.pdf

(9)-Some Equations involving n!-

https://mae.ufl.edu/~uhk/FACTORIAL-FORMULAS.pdf

(10)-Determining Market Trends-

https://mae.ufl.edu/~uhk/MARKET-TRENDS.pdf

(11)-Deriving Four Trigonometric Formulas-

https://mae.ufl.edu/~uhk/FOUR-TRIG-FORMULAS.pdf

(12)-Location of Primes and Semi-Primes along a Hexagonal Integer Spiral

https://mae.ufl.edu/~uhk/HEX-PRIME-SEMI-PRIME.pdf

(13)-Tessellations using Oblique Triangles-

https://mae.ufl.edu/~uhk/TRIANGLE-TESSELLATION.pdf

https://mae.ufl.edu/~uhk/STOCK-PRICE-YIELD.pdf

https://mae.ufl.edu/~uhk/LATEST-NUMBER-FRACTION.pdf

(16)-Properties of a Modified Pascal Triangle-

(17)-Leibniz Rule for Differentiating under an Integral Sign-

(18)-Solving simultaneos Diophantine Equations-

https://mae.ufl.edu/~uhk/DIOPHANTINE-NEW.pdf

(19)-How does a Thermobaric Bomb work?

https://mae.ufl.edu/~uhk/THERMOBARIC.pdf

(20)-Constructing an Integer Sandwich

https://mae.ufl.edu/~uhk/SANDWICH.pdf

(21)-Bull and Bear View-

https://mae.ufl.edu/~uhk/BULL-AND-BEAR-VIEW.pdf

(22)-Prime Generation by a Quadratic Formula-

https://mae.ufl.edu/~uhk/PRIME-QUADRATIC-FORMULA.pdf

(23)-Iteration Methods

https://mae.ufl.edu/~uhk/ITERATION-METHODS.pdf

(24)-Manipulating Logarithms-

https://mae.ufl.edu/~uhk/MANIP-LOGS.pdf

(25)-Christmas-Tree-Patterns-

https://mae.ufl.edu/~uhk/CHRISTMAS-TREE-PATTERNS.pdf

(26)-Heron Formula Derivation-

https://mae.ufl.edu/~uhk/HERON-FORMULA-DERIVATION.pdf

(27)-Two Circles-

https://mae.ufl.edu/~uhk/TWO-CIRCLES.pdf

(28)-Centroid Irrational Polygons-

https://mae.ufl.edu/~uhk/CENTROIDS-IRREGULAR-POLYGONS.pdf

(29)-Inflation History-

https://mae.ufl.edu/~uhk/INFLATION-HISTORY.pdf

(30)-TCT Scans-

https://mae.ufl.edu/~uhk/TCT-SCAN.pdf

(31)-Factoring using Diophantine-

https://mae.ufl.edu/~uhk/FACTORING-USING-DIOPHANTINE.pdf

(32)-Ntuple Algebraic Formulas-

https://mae.ufl.edu/~uhk/NTUPLE-ALGEBRAIC-FORMULAS.pdf

(33)-Christmas Tree Construction-

https://mae.ufl.edu/~uhk/CHRISTMAS-TREE-CONSTRUCTION.pdf

(34)-Number Sandwich-

https://mae.edu.ufl/~uhk/NUMBER-SANDWICH.pdf

(35)-Financial Bubbles-

https://mae.ufl.edu/~uhk/FINANCIAL-BUBBLES.pdf

(36)-Number Systems for Base b-

https://mae.ufl.edu/~uhk//NUMBER-SYSTEMS-b.pdf

(37)-Sigma Function Variations-

https://mae.ufl.edu/~uhk/SIGMA-VARIATIONS.pdf

(38)-Sub-Area Determination-

https://mae.ufl.edu/~uhk/AREA-DETERMINATION.pdf

(39)-Hole through Sphere-

https://mae.ufl.edu/~uhk/VOID-FRACTION.pdf

(40)-CT Scans for SPY and QQQ-

https://mae.ufl.edu/~uhk/CTSCANS-ETFS.pdf

(41)-Large Twin Primes-

https://mae.ufl.edu/~uhk/LARGE-TWIN-PRIMES.pdf

(42)-More Hex Spiral-

https://mae.ufl.edu/~uhk/NEW-FACTORIZATION-FORMULA.pdf

(45)-Factoring without Sigma

https://mae.ufl.edu/~uhk/FACTOR-LARGE-NEW.pdf

(46)-Asymptotes for Continuous Functions

https://mae.ufl.edu/~uhk/ASYMPTOTES.pd

(47)-Correlation of Stock Averages

https://mae.ufl.edu/~uhk/CORRELATION-STOCK-AVERAGES.pdf

(48)-Latest on the Integer Spiral

https://mae.ufl.edu/~uhk/LATEST-HEX-SPIRAL.pdf

(49)-Solving the H(x) Equation

https://mae.ufl.edu/~uhk/H(X)EQ.pdf

(50)-numberfraction-versus-sigma

https://mae.ufl.edu/~uhk/f-versus-sigma.pdf

(51)-All About Triangles

https://mae.ufl.edu/~uhk/ALL-ABOUT-TRIANGLES.pdf

(52)-Number-Fraction-New

https://mae.ufl.edu/~uhk/NUM-FRAC-NEW.pdf

(53)-Evaluate-Product-Function

https://mae.ufl.edu/~uhk/EVAL-PRODUCT.pdf

(54)-Area of Regular Polygons

https://mae.ufl.edu/~uhk/POLYGON-AREA-NEW.pdf

(55)-Latest-on Polygons

https://mae.ufl.edu/~uhk/LATEST-ON-POLYGONS.pdf

(56)-More-on-Mersenne-Primes

https://mae.ufl.edu/~uhk/MORE-ON-MERSENNE.pdf

(57)-Solving N(n,a)=(2^n) + a

https://mae.ufl.edu/~uhk/N(n,a).pdf

(58)-Buy-Low-Sell-High

https://mae.ufl.edu//~uhk/BUY-LOW-SELL-HIGH.pdf

(59)-Areas of Polygons via Sub-Triangles

https://mae.ufl.edu/~uhk/POLYGON-AREAS.pdf

(60)-Unconvential Methods of Factoring N=pq

https://mae.ufl.edu/~uhk/UNCONVENTIONAL-FACTORING.pdf

(61)-Generating 2D Curves

https://mae.ufl.edu/~uhk/GENERATING-2D-CURVES.pdf

(62)-Tau and Sigma-Functions

https://mae.ufl.edu/~uhk/TAU-FUNCTION.pdf

(63)-Trends in Equities

https://mae.ufl.edu/~uhk/TRENDS.pdf

(64)-Latest-N-Factoring

https://mae.ufl.edu/~uhk/LATEST-FACTORING.pdf

(65)-Quad Numbers and Semi-Primes

https://mae.ufl.edu/~uhk/QUAD-NUMBERS.pdf

(66)-New-N-Factor

https://mae.ufl.edu/~uhk/NEW-N-FACTOR.pdf

(67)-Solving-Diophantine

https://mae.ufl.edu/~uhk/SOLVING-DIOPHANTINE.pdf

(68)-Stock Windows

https://mae.ufl.edu/~uhk/STOCK-WINDOWS.pdf

(69)-More Diophantine

https://mae.ufl.edu/~uhk/MORE-DIOPHANTINE.pdf

(70)-Iteration-Procedures

https://mae.ufl.edu/~uhk/ITERATION-PROCEDURES.pdf

(71)-Neck-Pains

https://mae.ufl.edu/~uhk/NECK-PAINS.pdf

(72)-Pythagorean-Theorem

https://mae.ufl.edu/~uhk/PYTHAGOREAN-THEOREM.pdf

(73)-Tricks for Integral Evaluations

https://mae.ufl.edu/~uhk/TRICKS-WITH-INTEGRALS.pdf

(74)-Latest-On-Fibonacci

https://mae.ufl.edu/~uhk/LATEST-ON-FIBONACCI.pdf

(75)-Functional-Equations

https://mae.ufl.edu/~uhk/FUNCTIONAL-EQUATIONS.pdf

(76)-Pressure at 12500 ft

https:/mae.ufl.edu/~uhk/PRESSURE-AT-12500.pdf

(77)-Factoring Using Sigma

https://mae.ufl.edu/~uhk/FACTORING-USING-SIGMA.pdf

(78)-More on Sigma

https://mae.ufl.edu/~uhk/MORE-ON-SIGMA.pdf

(79)-Triple-Primes

https://mae.ufl.edu/~uhk/TRIPLE-PRIMES.pdf

(80)-Golden-Ratio

https://mae.ufl.edu/~uhk/GOLDEN-RATIO.pdf

(81)-AVERAGE-VERSUS-BS

(82)-Complex-Spiral

https://mae.ufl.edu/~uhk/COMPLEX-SPIRAL.pdf

(83)-Divisor-Function

https://mae.ufl.edu/~uhk/DIVISOR-FUNCTION.pdf

(84)-Gaussian-Latest

https://mae.ufl.edu/~uhk/GAUSSIAN-LATEST.pdf

(85)-Temperature-Scales

https://mae.ufl.edu/~uhk/TEMPERATURE-SCALES.pdf

(87)-Number-Triplets

https://mae.ufl.edu/~uhk/NUMBER-TRIPLETS.pdf

(88)-Integration by Parts

https://mae.ufl.edu/~uhk//INTEGRATION-BY-PARTS.pdf

(89)-Number-Fraction-New

https://mae.ufl.edu/~uhk/NUMBER-FRACTION-NEW.pdf

(90)-Using a Diophantine Equation to factor a Semi-Prime

https://mae.ufl.edu/~uhk/DIOPHANTINE-SEMI-PRIME.pdf

(91)-Properties of a modified Pascal Triangle

https://mae.ufl.edu/~uhk/modified-pascal.pdf

(92)-Mathematical Induction

https://mae.ufl.edu/~uhk/MATHEMATICAL-INDUCTION.pdf

(93)-Semi-Prime Factoring using F(x)

https://mae.ufl.edu/~uhk/F(x).pdf

(94)-Summing-Finite-Series

https://mae.ufl.edu/~uhk/SUMMING-FINITE-SERIES.pdf

(95)-Exponential-Function

https://mae.ufl.edu/~uhk/EXPONENTIAL-FUNCTION.pdf

(96)-Telescoping

https://mae.ufl.edu/~uhk/TELESCOPING.pdf

(97)-Euler-Variation

https://mae.ufl.edu/~uhk/EULER-VARIATION.pdf

(98)-Use of Windows to Predict Price Futures

https://mae.ufl.edu/~uhk/STOCK-WINDOWS.pdf

(99)-Properties of the Gamma Function

https://mae.ufl.edu/~uhk/GAMMA-FUNCTION.pdf

(100)-Evaluating-f(N)

https://mae.ufl.edu/~uhk/EVALUATING-f(N).pdf

(101)-Sigma, Tau, and Number Fraction for p^n

https://mae.ufl.edu/~uhk/VALUES-PN.pdf

(102)-Price in the Future

https://mae.ufl.edu/~uhk/PRICE-IN-FUTURE.pdf

(103)-Pyramid-of-Khufu

https://mae.ufl.edu/~uhk/PYRAMID-OF-KHUFU.pdf

(104)-Function Approximation

https://mae.ufl.edu/~uhk/FUNCTION-APPROXIMATION.pdf

(105)-Factoring-of-N

https://mae.ufl.edu/~uhk/FACTORING-OF-N.pdf

(106)-Prime-Via-Hex

https://mae.ufl.edu/~uhk/PRIME-VIA-HEX.pdf

(107)-Hyperbolic Aproximation

https://mae.ufl.edu/~uhk/HYPERBOLIC-FUNCTION-APPROXIMATIONS.pdf

(108)-Properties of Legendre Polynomials

https://mae.ufl.edu/~uhk/LEGENDRE-PROPERTIES.pdf

(109)-Properties-Modified-Pascal

https://mae.ufl.edu/~uhk/PROPERTIES-MODIFIED-PASCAL.pdf

(110)-Latest-Fac-N

https://mae.ufl.edu/~uhk/LATEST-FAC-N.pdf

(111)-Number Symmetry for N=pq

https://mae.ufl.edu/~uhk/NUMBER-SYMMETRY.pdf

(112)-Price Filtering

https://mae.ufl.edu/~uhk/PRICE-FILTER.pdf

(113)-Necessary and Sufficient Condtion for Primes

https://mae.ufl.edu/~uhk/necessary-sufficient-primes.pdf

(114)-Golden-Ratio-Diophantine

https://mae.ufl.edu/~uhk/GOLDEN-RATIO-DIOPHANTINE.pdf

(115)-Finding Primes

https://mae.ufl.edu/~uhk/FINDING-PRIMES.pdf

(116)-Stock Futures using Past Time Windows

(117)-Factoring N=1122973

https://mae.ufl.edu/~uhk/FACTORING-1122973.pdf

(118)-Evaluating N Using Sigma

https://mae.ufl.edu/~uhk/EVALUATING-N-USING-SIGMA.pdf

(119)-A Necessary and Sufficient Condition That a Number is Prime

https://mae.ufl.edu/~uhk/NESESSARY-AND-SUFFICIENT.pdf

(120)-Modular Arithmetic

https://mae.ufl.edu/~uhk/MODULAR-ARITHMETIC.pdf

(121)-Solution-by-Lambert

https://mae.ufl.edu/~uhk/SOLUTION-BY-LAMBERT.pdf

(122)-Inverse Functions

https://mae.ufl.edu/~uhk/INVERSE-FUNCTIONS.pdf

Das Ende