I wrote a Microsoft C# computer program to experimentally calculate poker hand probabilities:
As you can easily discern there are ten probabilities from Royal Flush down to High Card. I am worried about % Error of the Straight case. I used 10 * C(52, 5) = 10 * 2598960 samples to determine the calculated approximate poker hand probabilities.
My uncle was James “Jim” William Jordan who was born on September 5, 1920 and died January 27, 1970 at the relatively young age of 49 years old. He was born in Randolph County Alabama to Charles Webster Jordan and Ida Jane Kenady Jordan. His dad was at the time of his birth a postal clerk and his mother was trained as a school teacher but was a stay at home mom.
In early 1945 my uncle was stationed at Murtha Strip, Mindoro in the Philippines Islands. My uncle was a 2nd Lt. regular bombardier in a Consolidated B-24 Liberator heavy bomber. My uncle was in 1st Lt Robert R. Ferguson’s Aircrew Number 86 in the 530th Bombardment Squadron of the 380th Bombardment Group of the 310th Bomb Wing of the V Bomber Command of the 5th Air Force of the United States Army Air Forces (USAAF). Using the information which I received from the United States Air Force Historical Research Agency (AFHRA) Reel B0369 that contains the Histories of the 380th Bombardment Group for the year 1945 I found mission boards for twelve missions flown by 1st Lt Ferguson in the months April through June 1945. The mission boards are given below:
The following information is from AFHRA Reel A0636A which contains the 530th Bombardment Squadron Histories for June, July and August 1945.
We tried to factor the following numbers with each algorithm: 11^3+2, 2^33+2, 5^15+2, 2^66+2, 2^72+2, 2^81+2, 2^101+2, 2^129+2, and 2^183+2. Shor’s algorithm fully factored all of the numbers. Factoring with cubic integers fully factored all numbers except 2^66+2, 2^71+2, 2^129+2, and 2^183+2.
Typical full output from factoring with cubic integers:
A-Solutions = 973 B-Solutions = 234 Known Eqs = 614 Solutions = 1821 Rows = 1821 Columns = 1701 Kernel rank = 423 Sieved = 326434 Successes0 = 200863 Successes1 = 47073 Successes2 = 2708 Successes3 = 973 Successes4 = 1735 2417851639229258349412354 - 25 DDs 2 p 65537 p 414721 p 44479210368001 p Sets = 189 #Factor Base 1 = 501 #Factor Base 2 = 868 FactB1 time = 00:00:00.000 FactB2 time = 00:00:05.296 Sieve time = 00:00:17.261 Kernel time = 00:00:06.799 Factor time = 00:00:02.327 Total time = 00:00:31.742
A-solutions have no large prime. B-solutions have a large prime between B0 and B1 exclusively which is this case is between 3272 and 50000 exclusively. The known equations are between the rational primes and the cubic primes and their associates of the form p = 6k + 1 that have -2 as a cubic residue. There are 81 rational primes of the form and 243 cubic primes but we keep many other associates of the cubic primes so more a and b pairs are successfully algebraically factored. In out case the algebraic factor base has 868 members. The rational prime factor base also includes the negative unit -1. The kernel rank is the number of independent columns in the matrix. The number of dependent sets is equal to columns – rank which is this case 1701 – 423 = 1278. The number of (a, b) pairs sieved is 326434. Successes0 is the pairs that have gcd(a, b) = 1. Successes1 is the number of (a, b) pairs such that a+b*r is B0-smooth or can be factored by the first 500 primes and the negative unit. r is equal to 2^27. Successes2 is the number of (a, b) pairs whose N[a, b] = a^2-2*b^3 can be factored using the norms of the algebraic primes. Successes3 is the number of A-solutions that are algebraically and rationally smooth. Successes4 is the number of B-solutions without combining to make the count modulo 2 = 0. Successes3 + Successes4 should equal Successes2 provided all proper algebraic primes and their associates are utilized.
Note factoring with cubic integers is very fickle with respect to parameter choice.
The following text is based upon Appendix 5 “Higher Algebraic Number Fields” found in the book “Prime Numbers and Computer Methods for Factorization” by Hans Riesel. Cubic integers are algebraic numbers of the form [a, b, c] which is a short hand for a + b * z + c * z * z where z is the cube root of -2. we have z * z * z = z ^ 3 = -2 and z ^ 4 = -2 * z. The numbers a, b, and c are rational integers in Z. The product of two cubic integers is also a cubic integer (expansion of equation A5.8):
[a, b, c] * [d, e, f] = (a + b * z + c * z * z) * (d + e * z + f * z * z) = a * d + a * e * z + a * f * z * z + b * d * z + b * e * z * z + b * f * z * z * z + c * d * z * z + c * e * z * z * z + c * f * z * z * z * z = a * d – 2 * b * f – 2 * c * e + (a * e + b * d – 2 * c * f) * z + (a * f + b * e + c * d) * z * z = [ a * d – 2 * b * f – 2 * c * e , a * e + b * d – 2 * c * f , a * f + b * e + c * d ]
The defining equation for z is z^3 + 2 = 0. The other two roots of the polynomial f(z) = z^3 + 2 are the complex conjugates z[2] = z * (-1 + i * SQR(3)) / 2 and z[3] = z * (-1 – i * SQR(3)) / 2, where SQR(x) is the square root function such that x = SQR(x) * SQR(x) and i = SQR(-1), the imaginary unit. z[2] and z[3] are complex numbers of the form w = u + i * v, where u is the real part and v is the imaginary part. Let o = (-1 + i * SQR(3)) / 2. z[2] = o * z and z[3] = o * o * z.
Q(z) is a field and the integers of the field form Z(z) which is a ring. The norm of the cubic integer [a, b, c] is (expansion of equation A5.7):
N[a, b, c] = (a + b * z + c * z * z) * (a + b * o * z + c * o * o * z * z) * (a + b * o * o * z + c * o * z * z) = a ^ 3 – 2b ^ 3 + 4 * c ^ 3 + 6 * a * b * c
I wrote a computer application in C# to recreate the results of Appendix 5. Below is a screen shot of the application’s form:
I reproduce the units table of page 300 below:
Positive Units
[ 1, 0, 0, 1] 1
[ 1, 1, 0, 1] -1
[ 1, 2, 1, 1] 1
[ -1, 3, 3, 1] -1
[ -7, 2, 6, 1] 1
[ -19, -5, 8, 1] -1
[ -35, -24, 3, 1] 1
[ -41, -59, -21, 1] -1
[ 1, -100, -80, 1] 1
[ 161, -99, -180, 1] -1
[ 521, 62, -279, 1] 1
[ 1079, 583, -217, 1] -1
[ 1513, 1662, 366, 1] 1
Negative Units
[ -1, 0, 0, -1] -1
[ -1, 1, -1, -1] -1
[ 5, -4, 3, -1] 1
[ -19, 15, -12, -1] -1
[ 73, -58, 46, -1] 1
[ -281, 223, -177, -1] -1
[ 1081, -858, 681, -1] 1
[ -4159, 3301, -2620, -1] -1
[ 16001, -12700, 10080, -1] 1
[ -61561, 48861, -38781, -1] -1
[ 236845, -187984, 149203, -1] 1
[ -911219, 723235, -574032, -1] -1
[ 3505753, -2782518, 2208486, -1] 1
The pertinent factorization of the cubic integer prime numbers less than 300 are given next:
Case 1
[ 1, 0, 1, 5] 5
[ -1, -1, 1, 11] 11
[ 1, -2, 0, 17] 17
[ 3, 0, -1, 23] 23
[ 3, -1, 0, 29] 29
[ 1, -3, 1, 41] 41
[ 3, 1, 1, 47] 47
[ -1, -3, 0, 53] 53
[ 3, 0, 2, 59] 59
[ -1, -2, 2, 71] 71
[ 3, -2, -2, 83] 83
[ 1, -2, 3, 89] 89
[ -3, -4, 0, 101] 101
[ -1, 0, 3, 107] 107
[ 1, -4, 2, 113] 113
[ 3, -3, -1, 131] 131
[ -3, -1, 3, 137] 137
[ 1, -4, -1, 149] 149
[ -3, -3, 2, 167] 167
[ 3, -5, 4, 173] 173
[ 5, -3, 0, 179] 179
[ -1, 2, 4, 191] 191
[ 5, -2, -1, 197] 197
[ 3, 2, 3, 227] 227
[ 5, 0, 3, 233] 233
[ 1, -3, 4, 239] 239
[ 1, -5, 0, 251] 251
[ 1, 0, 4, 257] 257
[ 3, -4, -3, 263] 263
[ 1, -5, 3, 269] 269
[ -1, -1, 4, 281] 281
[ -3, -6, -1, 293] 293
Case 2
[ -1, 0, 2, 31] 31
[ 3, 0, 1, 31] 31
[ -1, -2, 1, 31] 31
[ 1, 1, 2, 43] 43
[ 3, -1, -1, 43] 43
[ 3, -2, 0, 43] 43
[ 1, 0, 3, 109] 109
[ 1, -4, 1, 109] 109
[ 5, 2, 0, 109] 109
[ -1, -4, 0, 127] 127
[ -1, -1, 3, 127] 127
[ 5, -1, 0, 127] 127
[ 5, 1, 1, 157] 157
[ 5, 0, 2, 157] 157
[ 3, -3, -2, 157] 157
[ 3, -4, -1, 223] 223
[ -3, -5, 0, 223] 223
[ 1, -5, 2, 223] 223
[ -1, 1, 4, 229] 229
[ -3, 0, 4, 229] 229
[ 5, -2, -2, 229] 229
[ 1, -5, -1, 277] 277
[ 3, -5, 0, 277] 277
[ -3, -4, 2, 277] 277
[ -1, -5, 1, 283] 283
[ 3, 0, 4, 283] 283
[ 5, 3, 2, 283] 283
Case 3
[ 0, 1, 0, 2] -2
Case 4
[ -1, 1, 0, 3] -3
I reproduce the factorization of the norms of [a, b, 0] found on page 303 are given below:
[ 59, 46, 0, 1] 10707
[ -1, -2, -1, 1] -1
[ -1, 1, 0, 3] -3
[ 1, 1, 2, 43] 43
[ 3, -2, -2, 83] 83
[ 59, 48, 0, 1] -15805
[ -1, 3, 3, 1] -1
[ 1, 0, 1, 5] 5
[ 3, -1, 0, 29] 29
[ 1, -4, 1, 109] 109
[ 61, 48, 0, 1] 5797
[ 1, -3, -3, 1] 1
[ -1, -1, 1, 11] 11
[ 1, -2, 0, 17] 17
[ 3, 0, 1, 31] 31
[ 62, 47, 0, 1] 30682
[ 1, 1, 0, 1] -1
[ 0, 1, 0, 2] -2
[ 3, 0, -1, 23] 23 ^ 2
[ 3, -1, 0, 29] 29
[ 62, 49, 0, 1] 3030
[ 1, -3, -3, 1] 1
[ 0, 1, 0, 2] -2
[ -1, 1, 0, 3] -3
[ 1, 0, 1, 5] 5
[ -3, -4, 0, 101] 101
[ 63, 50, 0, 1] 47
[ 19, 5, -8, 1] 1
[ 3, 1, 1, 47] 47
[ 64, 47, 0, 1] 54498
[ -1, -1, 0, 1] 1
[ 0, 1, 0, 2] -2
[ -1, 1, 0, 3] -3
[ -1, 0, 2, 31] 31
[ -3, -6, -1, 293] 293
[ 65, 53, 0, 1] -23129
[ 1, 1, 0, 1] -1
[ -3, -4, 0, 101] 101
[ -3, 0, 4, 229] 229
[ 66, 53, 0, 1] -10258
[ 1, 2, 1, 1] 1
[ 0, 1, 0, 2] -2
[ 3, 0, -1, 23] 23
[ 3, -4, -1, 223] 223
[ 67, 53, 0, 1] 3009
[ 7, -2, -6, 1] -1
[ -1, 1, 0, 3] -3
[ 1, -2, 0, 17] 17
[ 3, 0, 2, 59] 59
[ 1693, 749, 0, 1] 4012180059
[ -1, -2, -1, 1] -1
[ -1, 1, 0, 3] -3
[ 3, -2, 0, 43] 43
[ 5, 0, 2, 157] 157
[ 7, -3, 0, 397] 397
[ 5, 1, 4, 499] 499
The last factorization is from “The Development of the Number Field Sieve” edited by A. K. Lenstra and H. W. Lenstra, Jr. containing the paper “Factoring with Cubic Integers” by J. M. Pollard page 10.
Shor’s algorithm is a fast method of factoring integers using a quantum computer:
https://en.wikipedia.org/wiki/Shor%27s_algorithm
I have implemented the algorithm classically and performed a few experiments on integers greater than equal 12. I used Floyd’s cycle finding algorithm as implemented with a randomized Pollard’s Rho Factoring method using two second degree polynomials:
https://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm
I factored the seventh Fermat number in less than nine minutes. J. M. Pollard factored this number in 20.1 hours in 1988 using the special number field sieve [see “Lecture Notes in Mathematics 1554 The Development of the Number Field Sieve” edited by A. K. Lenstra and H. W. Lenstra, Jr. The paper by Pollard is entitled “Factoring with Cubic Integers”]. Both Pollard and I used personal computers.
Suppose Moore’s original law is applied to Pollard’s factoring of the seventh Fermat number and also assume computing time is cut in half every two years then we can compute the following table:
A less optimistic table states that the computation time is halved every four years:
I successfully factored 38 of 39 numbers of the form 12, 123, 1234, 12345, … , 1234567890123456789012345678901234567890. The results of my factoring C# Windows form application are displayed below:
I was eventually able to factor the 38-decimal digit number that failed in the above calculation as illustrated below:
The number of bits per number varied from 4-bits to 130-bits. This is still no where near the ability to factor a 3,000-bit Rivest-Shamir-Adleman composite integer so most financial systems remain secure. The link below will take you to the Microsoft C# project directory on my OneDrive account.
https://1drv.ms/f/s!AjoNe9CiN2M9gfMbQ8H_g3GMQEC6iA
The C# source code is at the preceding link in a printable and readable PDF.
I designed and implemented a C# computer language application to model the global greenhouse gas concentrations data found on the NOAA website:
https://www.esrl.noaa.gov/gmd/aggi/aggi.html
I used the latest recommended data for time period 1979 to 2017. The concentrations of three greenhouse gases were modeled: carbon dioxide (CO2), methane (CH4), and nitrous oxide (N2O).
The empirical modeling paradigm I used was simple linear regression. My model goes out to the year 2300. The key formulas used by the model are:
See the website:
https://en.wikipedia.org/wiki/Simple_linear_regression
Some plots of the concentrations in parts per million (PPM) and parts per billion (PPB) are given below.
I designed and implemented a C# computer language application to model the precipitation data found on the NOAA website:
I used the latest recommended data for time period 1895 to 2017. The empirical modeling paradigm I used was simple linear regression. My model goes out to the year 2300. The key formulas used by the model are:
See the website:
https://en.wikipedia.org/wiki/Simple_linear_regression
Some plots of the contiguous U.S. precipitation are shown below. The climate is getting wetter thus some parts of the U.S.maybe more prone to floods.
I designed and implemented a C# computer language application to model the temperature anomaly data found on the NOAA website:
I used the latest recommended data for time period 1895 to 2017. The empirical modeling paradigm I used was simple linear regression. My model goes out to the year 2300. The key formulas used by the model are:
See the website:
https://en.wikipedia.org/wiki/Simple_linear_regression
Below are some plots of the temperature anomaly.
Solve the Laplace equation in the following orthogonal rectilinear coordinate systems:
Solution PDF:
The four methods considered in this study are as follows:
The trapezoidal rule requires (n + 2) function evaluations, n real number increments, and six additional real number arithmetic operations. Simpson’s rule involves (n + 2) function evaluations, n real number increments, and ten additional real number arithmetic operations. Gauss-Legendre quadrature uses n function evaluations, 3 * n real number arithmetic operations, 2 * n index operations, and five additional arithmetic operations. Finally, the Monte Carlo Method requires n function evaluations, n random number generations, 2 * n + 3 additional real number arithmetic operations. The Gauss-Legendre quadrature also involves some complicated orthogonal polynomial operations to determine the abscissas and weights. Below are some results from our test C# application.
We conclude from the preceding dearth of tests that for given n the order of accuracy is generally Gauss-Legendre, Simpson’s, Trapezoidal, and finally Monte Carlo.
Numerical Explorations 