Let x, y, n be integers. I explore x and y for 1 through 15. I use an HP 32Sii to help me with the calculations.
x^2 + y^2 = n^3
2^2 + 2^2 = 2^3
2^2 + 11^2 = 5^3
5^2 + 10^2 = 5^3
Program:
LBL A
x^2
x<>y
x^2
+
3
1/x
y^x
RTN
x^3 + y^3 = n^2
1^3 + 2^3 = 3^2
2^3 + 2^3 = 4^2
8^3 + 4^3 = 24^2
8^3 + 8^3 = 32^2
Program:
LBL B
3
y^x
x<>y
3
y^x
+
√
RTN
x^2 - y^2 = n^3, x > y
15^2 - 10^2 = 5^3
15^2 - 3^2 = 6^3
14^2 - 13^2 = 3^3
10^2 - 6^2 = 4^3
6^2 - 3^2 = 3^3
3^2 - 1^2 = 2^3
Program:
LBL C
x^2
x<>y
x^2
x<>y
-
3
x√y
RTN
x^3 - y^3 = n^2, x > y
14^3 - 7^3 = 49^2
10^3 - 6^3 = 28^2
8^3 - 7^3 = 13^2
Program:
LBL D
3
y^x
x<>y
3
y^x
x<>y
-
√
RTN
If you want to find more integer triplets, happy exploring!
To all the dads out there - Happy Father's Day! To my dad, I am so proud of you!
Until next time,
Eddie
Happy One Week from the Summer Solstice!
This blog is property of Edward Shore. 2013
Sunday, June 16, 2013
Integers: x^2 + y^2 = n^3, x^3 + y^3 = n^2, x^2 - y^2 = n^3, x^3 - y^3 = n^2
Friday, June 7, 2013
HP35S Music Pitch: Next Half-Step Up and Down
HP35S Music Pitch: Next Half-Step Up and Down
Source: "Sourcebook for Programmable Calculators" Texas Instruments (TI-58/59) - 1978 (,I believe, apologize if get the year wrong)
next pitch = original pitch * (twelfth root of 2) ≈ original pitch * 1.05946309436
Pitch is in HZ (Hertz)
256 Hz one octave lower than Middle C
512 Hz Middle C
1024 Hz one octave higher than Middle C
Steps: C C# D D# E F F# G G# A A# B
HP35S Program:
Next half-step up: pitch, XEQ M001
Next half-step down: pitch, XEQ M009
Repeat each step with R/S, stop whenever you like
M001 LBL M
M002 2
M003 12
M004 1/x
M005 y^x
M006 *
M007 R/S
M008 GTO M001
M009 2
M010 12
M011 1/x
M012 y^x
M013 ÷
M014 R/S
M015 GTO M009
Program made on 4/27/2013
Have fun and thanks for your support! Always appreciate the comments and followers! Eddie
This blog is property of Edward Shore. 2013
HP 35S: Sun Altitude, Azimuth, Solar Pond Absorption
HP 35S: Sun Altitude, Azimuth, Solar Pond Absorption
Source: Sun Altitude, Azimuth, Solar Pond Absorption, HP 67/97 Energy Conservation December 1978, Author: HP
Input:
This program asks for:
D = days after March 21 (later will be sun's declination : 23.45 sin (D * .9856°) )
L = latitude given in D.MMSS (degrees minutes seconds) format (avoid ±90°)
T = time before solar noon (12:00 PM), if the time is after noon, enter hours as a negative (example: 3:00 PM → -3)
N = index of refraction of surface/fluid (see below)
Index of Refraction for Common Objects:
Water: 1.33
Ice: 1.309
Glass: 1.52
Diamond: 2.42
Formulas: (Degrees Mode)
Sun Declination
D = 23.45 * sin( days after March 21 * .9856°)
Altitude of the Sun (H):
H = asin (cos L * cos D * cos (15 * T) + sin L * sin D)
Azimuth of the Sun (A): (degree from latitude ground wise north)
A = acos ( (sin H * sin L - sin D) ÷ (cos L * cos H) )
Fraction of the surface penetrated by the sun that hour (T):
T = 2 * n * (x^2 + y^2) * sin H * cos R
Where
R = asin (cos H ÷ n)
x = (cos R + n * sin H)^-1
y = (sin H + n * cos R)^-1
Example 1:
Input:
D = 68 (May 28), L = 46°, T = 4 (8:00 AM), N = 1.33 (water)
Output:
H (altitude) = 35.98991°
A (azimuth) = 84.40835°
F (fraction of coverage) = 0.95943
Example 2:
Input:
D = 90 (June 19), L = 23°, T = -3 (3:00 PM), N = 2.42 (diamond)
Output:
H = 48.81756°
A = 99.85903°
F = 0.82156
Program:
U001 LBL U
U002 INPUT D \\ declination
U003 DEG
U004 0.9856
U005 *
U006 SIN
U007 23.45
U008 *
U009 STO D \\ altitude
U010 COS
U011 INPUT L
U012 HMS→ \\ or → H
U013 STO L
U014 COS
U015 *
U016 INPUT T
U017 15
U018 *
U019 COS
U020 *
U021 RCL L
U022 SIN
U023 RCL D
U024 SIN
U025 *
U026 +
U027 ASIN
U028 STO H
U029 VIEW H \\ azimuth
U030 SIN
U031 RCL L
U032 SIN
U033 *
U034 RCL D
U035 SIN
U036 -
U037 RCL L
U038 COS
U039 RCL H
U040 COS
U041 *
U042 ÷
U043 ACOS
U044 STO A
U045 VIEW A
U046 RCL H \\ fraction
U047 COS
U048 INPUT N
U049 ÷
U050 ASIN
U051 STO R
U052 COS
U053 RCL H
U054 SIN
U055 RCL* N
U056 +
U057 1/x
U058 x^2
U059 RCL H
U060 SIN
U061 RCL R
U062 COS
U063 RCL* N
U064 +
U065 1/x
U066 x^2
U067 +
U068 RCL* N
U069 2
U070 *
U071 RCL H
U072 SIN
U073 *
U074 RCL R
U075 COS
U076 *
U077 STO F
U078 VIEW F
U079 RTN
This blog is property of Edward Shore. 2013
HP35S Fraunhofer Diffraction - Spherical
HP35S Fraunhofer Diffraction - Spherical
Source: HP 67/97 Optics Pac, June 1978
This version takes advantage of the HP's integral function. The trade off is that two labels are required. On the plus side, this program can be typed directly into a 32Sii or 33S.
Labels: F (Main), G (Integral)
Variables:
D = diameter in microns (10^-6 meters)
L = wavelength of light in microns (10^-6 meters)
A = θ, angle of the slit, entered in degrees
Formulas:
X = π D ÷ L
W = X sin A
J = int(cos(T - W sin T) dT, 0, π)/π
I = (X^2 J ÷ W)^2
Output:
Bessel function of the first kind (J) - paused for 2 seconds;
Fraunhofer Intensity (I) - dimensionless - as I understand it, this is how intense the diffraction is
Programs:
LBL F
INPUT D
INPUT L
÷
π
*
STO X
INPUT A
→RAD
STO A
SIN
*
STO W
RAD
0
π
FN= G
∫ FN dT
STO J \\ Bessel store in J for future use, if desired
PSE
PSE
RCL ÷ W
RCL X
x^2
*
x^2
RTN
LBL G
COS(T-W*SIN(T))/π \\ enter as an equation (press [EQN])
RTN
This blog is property of Edward Shore. 2013
Sunday, June 2, 2013
HP 35S: Spherical Triangle
(First draft?)
Formulas:
sin A / sin a = sin B / sin b = sin C / sin c
cos A = - cos B cos C + sin B sin C cos a
cos a = cos b cos c + sin b sin c cos A
A, B, C are angles formed by the great circles (the "lines" of the Spherical triangle). Note that A + B + C > 180°. a, b, and c measure the arc length of great circles as angles measured from the center of the sphere.
Source: http://www.krysstal.com/sphertrig.html
Program: Label S
Calculator: HP 35S
I am not sure if I covered all possible scenarios.
Memory registers B and C are used for temporary purposes.
Given: B, A, b; Goal: a; Label S001
S001 LBL S
S002 SIN
S003 x<>y
S004 SIN
S005 ÷
S006 x<>y
S007 SIN
S008 ×
S009 ASIN
S010 RTN
Given: b, a, B; Goal: A; Label S011
S011 SIN
S012 x<>y
S013 SIN
S014 ×
S015 x<>y
S016 SIN
S017 ÷
S018 ASIN
S019 RTN
Given: a, b, c; Goal: A; Label S020
S020 STO C
S021 COS
S022 x<>y
S023 STO B
S024 COS
S025 ×
S026 x<>y
S027 COS
S028 x<>y
S029 -
S030 RCL B
S031 SIN
S032 RCL C
S033 SIN
S034 ×
S035 ÷
S036 ACOS
S037 RTN
Given: b, A, c; Goal: a; Label S038
S038 STO C
S039 SIN
S040 x<>y
S041 COS
S042 ×
S043 x<>y
S044 STO B
S045 SIN
S046 ×
S047 RCL C
S048 COS
S049 RCL B
S050 COS
S051 ×
S052 +
S053 ACOS
S054 RTN
Given: A, B, C; Find: a; Label S055
S055 STO C
S056 COS
S057 x<>y
S058 STO B
S059 COS
S060 ×
S061 x<>y
S062 COS
S063 +
S064 RCL B
S065 SIN
S066 RCL C
S067 SIN
S068 ×
S069 ÷
S070 ACOS
S071 RTN
Examples:
Given: B = 3.2145°, A = 2.2718°, b = 40°; XEQ S001; Result: a ≈ 65.4058°
Given: b = 60°, a = 40°, B = 4.95°; XEQ S011; Result A ≈ 3.6720°
Given: a = 4.11°, b = 5°, c = 6.03°; XEQ S020; Result A ≈ 42.5439°
Given: b = 3.996°, A = 49°, c = 6.314°; XEQ S038; Result a ≈ 4.7636°
Given: A = 124°, B = 45°, C = 76°; XEQ S055; Result a ≈ 124.4509°
Enjoy - hope this helps and have a great day!
Eddie
This blog is property of Edward Shore. 2013
HP 35S: Planar Triangles
This program is set to solve common problems in plane (regular) triangles.
The programs can be adapted to any side lengths and angles necessary.
Variables of Plane Triangles
Side length a with corresponding angle A,
Side length b with corresponding angle B, and
Side length c with corresponding angle C.
Labels and Stack Set Up: HP 35S
Angle-Angle-Side, Label P001, Stack: B, A, b, Goal: a
Side-Side-Angle, Label P009, Stack: b, a, B, Goal: A
Angle-Side-Angle, Label P015, Stack: b, A, c, Goal: a
Side-Side-Side, Label P033, Stack: a, b, c, Goal: A (angle corresponding to first side length entered)
* If you use a 15C, 32Sii, or other another RPN calculator, you will need to create four labels. The nice thing with the HP 35S is that you can create multiple programs within in one label. Memory registers B and C are temporary.
Program P (Planar Triangles)
\\ Angle-Angle-Side: Law of Sines
\\ Stack: B, A, b; Find: a
P001 LBL P
P002 x<>y
P003 SIN
P004 ×
P005 x<>y
P006 SIN
P007 ÷
P008 RTN
\\ Side-Side-Angle: Law of Sines
\\ Stack: b, a, B; Find: A
P009 SIN
P010 ×
P011 x<>y
P012 ÷
P013 ASIN
P014 RTN
\\ Side-Angle-Side: Law of Cosines
\\ Stack: b, A, c; Find: a
P015 STO C
P016 x<>y
P017 COS
P018 ×
P019 x<>y
P020 STO B
P021 ×
P022 2
P023 ×
P024 +/-
P025 RCL B
P026 x^2
P027 +
P028 RCL C
P029 x^2
P030 +
P031 √
P032 RTN
\\ Side-Side-Side: Law of Cosines
\\ Stack: a, b, c; Find: A
P033 STO C
P034 x^2
P035 x<>y
P036 STO B
P037 x^2
P038 +
P039 x<>y
P040 x^2
P041 -
P042 2
P043 ÷
P044 RCL÷ B
P045 RCL÷ C
P046 ACOS
P047 RTN
Examples (Degrees Mode Used):
AAS: B = 30, A = 40, b = 4; a ≈ 5.1423
SSA: b = 5, a = 4, B = 90°; A ≈ 53.1301°
SAS: b = 8, A= 30°, c = 9; a ≈ 4.5047
SSS: a = 5, b = 4, c = 3; A = 90°
Hope this helps. I plan to post a program regarding spherical triangles.
Take care,
Eddie
This blog is property of Edward Shore. 2013
Sunday, May 19, 2013
HP 35S: Approximate Length of Sunlight During a Day
HP 35S: Length of Sunlight During a Day
Source: Total Daily Amount of Solar Radiation - HP 67/97 Energy Conservation Pac, December 1978, Author: Hewlett Packard
(This is a slight variation instead of a direct port)
Input
You are prompted for D and L where:
D = the number of days from March 21, a 365 day year is assumed
L = latitude (North as positive, South as negative), entered as D.MMSS (degrees-minutes-seconds) format
Output
Approximate number of hours of sunlight, in hours, minutes, seconds
Examples
Los Angeles, April 17: latitude of 34°03' N, 27 days after March 21
D = 27, L = 34.03, answer is approximately 12.573501 (12 hours, 57 minutes, 35.01 seconds)
Rome, September 1: latitude 41°51' N, 164 days after March 21
D = 164, L = 41.52, answer is approximately 12.532269 (12 hours, 53 minutes, 22.69 seconds)
Sydney, June 21: latitude 33°51'31" S, 92 days after March 21
D = 92, L = -33.5131, answer is approximately 9.443922 (9 hours, 44 minutes, 39.22 seconds)
Formulas
This version uses the estimate of sun declination:
D = 23.45 sin(d * 0.9856°)
Since 360/365.25 ≈ 0.985626283368
θ = acos(-tan L × tan D)
L = 24 * θ in radians ÷ π
Program
S001 LBL S
S002 DEG
S003 INPUT D
S004 0.9856
S005 ×
S006 SIN
S007 23.45
S008 ×
S009 INPUT L
S010 HMS→
S011 TAN
S012 x<>y
S013 TAN
S014 ×
S015 +/-
S016 ACOS
S017 ->RAD
S018 24
S019 ×
S020 π
S021 ÷
S022 ->HMS
S023 RTN
This blog is property of Edward Shore. 2013
HP 35S: Air Density & Density Altitude (Metric-US Conversion Factors included)
HP 35S: Air Density & Density Altitude (7 Digit Accuracy*)
*You can use 4 or 5 digits for the HP 15C and other RPN calculators on which each number takes a step. Slightly less accuracy but smaller program.
Source: Shelquist Engineering web page: http://wahiduddin.net/calc/density_altitude.htm
Formulas
Air Density:
D = P ÷ (R * T)
D = density in kg/m^3
P = pressure in Pascals (Pa)
T = temperature in Kelvins
R = specific gas constant = 287.05 J/(kg*°K)
Air Density Altitude: (where the plane/vehicle thinks it is)
H = 44.3307692 - 42.2665143 * D^0.2349695
H = air density in km
U.S.-SI Conversion Factors
Pressure: (convert and store in P)
1 inHg = 2275.5477799 lb/(ft*s^2) = 3386.3881579 Pa
Temperature Conversion Formula: (convert and store in T)
°K = 5/9 * (°F - 32) + 273.15
Length:
1 km = 3280.8398950 ft
Program
D001 LBL D
D002 INPUT P
D003 287.05
D004 INPUT T
D005 *
D006 ÷
D007 STO D
D008 VIEW D \\ display air density in kg/m^3
D009 0.2349696 \\ dimensionless constant
D010 y^x
D011 42.2665143
D012 *
D013 +/-
D014 44.3307692
D015 +
D016 STO H
D017 RTN \\ display density altitude
I chose to use SI units because the constants are much smaller than the formula would be in US units.
This blog is property of Edward Shore. 2013
HP35S: Distance to Horizon
HP35S Distance to Horizon
Calculator: HP 35S
Sources:
* HP 65 Navigation Pac-1 - published in 1974
* Bad Astronomy by Discovery Magazine: http://blogs.discovermagazine.com/badastronomy/2009/01/15/how-far-away-is-the-horizon/#.UYCEhMu9KSM
* Wikihow Article: http://m.wikihow.com/Calculate-the-Distance-to-the-Horizon
Input
Height above sea level, the height includes the height of the land (if any) and the distance to your eyes or eyepiece.
Output
Approximate distance to the horizon in miles.
Conversions
If you use metric, use these conversions before/after running the program:
1 m = 3.28083989501 ft
1 mi = 1.609344 km
Formula
Updated formula based on radius on Earth on average 3,959 miles.
D ≈ √(1.49922*H + H^2)
H is in feet, D is in miles
Example
H = 3 ft, D ≈ 3.67392 miles
Program
H001 LBL H
H002 ENTER
H003 ENTER
H004 1.49922
H005 *
H006 x<>y
H007 x^2
H008 +
H009 √
H010 RTN
This blog is property of Edward Shore. 2013
HP35S Vertical Curve: Elevation at Peak and at End Point
Source: Fundamental of Engineering Supplied-Reference Handbook 8th Ed, 2nd Revision, 2011, NCEES
Calculator: HP 35S
Input
I = initial height
G = grade 1; entering the curve (in decimal form)
H = grade 2; exiting the curve (in decimal form)
L = horizontal length of the curve
G and H are the opposite signs
If G>0 and H<0 a="" and="" curve="" g="" h="" has="" if="" peak.="" similarity="" the="">0, the curve has a valley.
The program listed prompts for the inputs.
Output
1. Point where the curve reaches extreme elevation
2. Press R/S to get the elevation at the extreme point.
3. Press R/S once more to get the elevation at the end of the curve.
None of the outputs are stored.
Formula
A = (G - H)/(2L)
X_extrema = -G/(2A)
Equation of the Curve: y = I + G*x + A*x^2
Examples
Uphill curve:
I = 1,000 ft
G = 7% = 0.07
H = -4% = -0.04
L = 1,368 ft
Point at peak elevation is 870.545 ft into the curve at 1,030.469 ft. The elevation at the end of the curve is 1,020.520 ft.
Downhill curve:
I = 1,580 ft
G = -3% = -0.03
H = 4.2% = 0.042
L = 2.3 mi = 12,144 ft
Point at trough elevation occurs 5,060 ft into the curve at elevation 1,504.1 ft. The elevation at the end of the curve is at 1,652.864 ft.
Program
V001 LBL V
V002 INPUT I
V003 INPUT G
V004 INPUT H
V005 INPUT L
V006 RCL H
V007 RCL - G
V008 2
V009 ÷
V010 RCL ÷ L
V011 STO A
V012 RCL G
V013 x<>y
V014 ÷
V015 2
V016 ÷
V017 +/-
V018 R/S \\ shows point where extreme elevation occurs
V019 XEQ V024
V020 R/S \\ shows extreme elevation
V021 RCL L
V022 XEQ V024 \\ shows ending elevation
V023 RTN
V024 ENTER \\ calculate y subroutine
V025 RCL × A
V026 RCL + G
V027 ×
V028 RCL + I
V029 RTN
This blog is property of Edward Shore. 2013
HP35S: Horizontal Curve - Finding Radius, Chord Length, and Arc Length
HP 35S: Horizontal Curve
Original: HP 33S Surveying Applications, Hewlett Packard, March 1978, pg. 46
Calculator
HP 35S
Input
(see diagram above)
T = Tangent Distance (length of segment from P.C. (Point of Curvature) to P.I. (Point of Tangent Intersection))
A = Central curve in degrees, minutes, seconds
This program prompts for tangent length and central angle
Output
The program gives the following results:
1. Radius of the horizontal curve (R)
2. Press R/S to get the Chord length (C)
3. Press R/S once more to get the arc length of the horizontal curve (L)
The program does not store any results.
Formulas
R = T × (tan(A/2))⁻¹
C = 2 × R × sin(A/2)
L = R × A in radians
Where
T = tangent distance
A = central angle
R = Radius
C = Chord Length
L = Arc Length
Example
Tangent Length: 172.45
Central Angle: 40°22'13" (enter as 40.2213)
Results:
Radius: 469.08079
Chord Length: 323.7172
Arc Length: 330.51163
Program
V001 LBL V
V002 DEG
V003 INPUT T
V004 INPUT A
V005 HMS→\\ sometimes named ->H
V006 STO A
V007 2
V008 ÷
V009 TAN
V010 1/x
V011 ×
V012 R/S \\ display Radius
V013 ENTER
V014 ENTER
V015 2
V016 ×
V017 RCL A
V018 2
V019 ÷
V020 SIN
V021 ×
V022 R/S \\ display Chord Length
V023 x<>y
V024 RCL A
V025 ->RAD
V026 ×
V027 RTN \\ display Arc Length
If you don't have the ->RAD function, you can substitute the following steps:
π, ×, 180, ÷
This blog is property of Edward Shore. 2013
HP15C: Julian Date from Gregorian Date
Thanks to the University of Texas at San Antonio Computer Science Department. The explanation and formulas can be found by clicking on this link.
Note: This program only works for dates on or after October 15, 1582. Thank you Dieter.
Calculator
HP 15C (can be adopted with any RPN keystroke calculator)
Input
Preload the following information into these registers:
R1 = month (1 for January, 12 for December)
R2 = date
R3 = year in four digits (example: 2013)
Output
R0 = Julian Date
Temporary
R4 = a = integer((14-month)/12)
R5 = y = year + 4800 - a
Formulas
(See link above)
a = integer((14-month)/12)
y = year + 4800 - a
m = month + 12a - 3
Julian Date = day + integer((153m+2)/5) + 365y + integer(y/4) - integer(y/100) + integer(y/400) - 32045
Assumptions: 12:00 PM Universal Time is assumed (that's 7:00 AM in Pacific Standard Time or 8:00 AM in Pacific Daylight Savings Time)
Program
LBL A
1
4
RCL - 1
1
2
÷
INT
STO 4
4
8
0
0
RCL + 3
RCL - 4
STO 5
RCL 2
STO 0
1
2
RCL × 4
RCL + 1
3
-
1
5
3
×
2
+
5
÷
INT
STO + 0
3
6
5
RCL × 5
STO + 0
RCL 5
4
÷
INT
STO + 0
RCL 5
1
%
CHS
INT
STO + 0
RCL 5
4
÷
1
%
INT
STO + 0
RCL 0
3
2
0
4
5
-
STO 0
RTN
In HP RPN calculators, a number followed by 1, %, divides said number by 100.
Example:
March 14, 1977 has a Julian Date of 2,443,217
April 28, 2012 has a Julian Date of 2,456,046
December 31, 2015 has a Julian Date of 2,457,388
This blog is property of Edward Shore. 2013
Saturday, May 11, 2013
Greetings from Carpinteria/UCSB - Astronomy
I am in Carpinteria, CA; soon to be headed back home (vacation goes by soooooooo fast!)
Last night was a sight to see. Unfortunately I did not catch the young moon, but I did manage to see Jupiter, and the constellations Ursa Major, Leo, Gemini, and Virgo. I can thank Google Sky for the immense help. Who knew the stars of Cancer were so faint?
My trip to the UCSB Library (Davidson Library) completed a circle for me. It was there where I got the itch to visit the mathematics section of as many libraries I can get to. I first visited the UCSB library last September.
This visit I concentrated more on astronomy than mathematics. Learning about the structure of the Milky Way has become an interest for me.
Thanks to Tycho Brahe, comets are a major part of astronomy. He also brought all the astronomical almanacs to date around 1600, all without a telescope. Telescopes were first used by Galileo Galilei. Brahe's data was eventually inherited by Johannes Kepler, who was both an astrologer and astronomer.
Kepler showed how proportional the orbits of Jupiter (inscribed circle) and Saturn (circumscribed circle) were (well, in general) by drawing two circles around an equilateral triangle. My attempt at drawing is shown above. (Freehand art was never my strength).
Unlike what astronomers believed before, Galileo showed, among other things:
1. Planets are not self-moving.
2. Stars are not close to the Earth, instead they are distant suns.
The Milky Way
Harlow Shapley was the first to locate the Earth's place in the Milky Way Galaxy (around 1919): in the galactic disc two thirds out from the center. Today, astronomers estimate that we are about 24,500 to 27,000 light years away from the center; which could be a good thing. The center of our galaxy is said to be a massive black hole.
During the time of World War II, H.C. van de Hurst suggested trying to detect radio waves emitted by hydrogen atoms. This opened the door to a more detailed map of our celestial sky.
Source: Charles A. Whitney. "The Discovery of our Galaxy" Alfred A. Knoff: New York. 1971
-------
Currently, the center of our Milky Way lies in the constellation Sagittarius (♐). The biggest identifier is a radio source named Sagittarius A*, which can't be seen by the naked eye. It lies near the boarder of Scorpio (Scorpius) and Ophiuchus.
Coordinates of Sagittarius A* (approximately)
RA (α) 17hr 45min 40.04sec (266.416833°)
Dec (δ) -29°00'28.1" (-29.007806°)
I am curious: with the precision of the equinoxes, will Sagittarius A* "move" to Scorpio and/or Ophiuchus in the far future? It stands to reason, since the point (0hr, 0°), the vernal equinox once lied in the constellation Aries 2,000 years ago, now lies in Pisces on its way to Aquarius.
It is also an explanation why the star Polaris in Ursa Minor and the star Vega in Lyra trade the honor of being the North Star approximately every 12,857 years. (Close to 13,000) Source: http://csep10.phys.utk.edu/astr161/lect/time/precession.html
Hipparchus determined that the stars rotated approximately 50 arc seconds (50".3 to 50".4) around the ecliptic pole. Hence, it takes about 25,714 2/7 years for the stars to complete one cycle. (≈ 360°/50".4 = 360°/0.014°)
Until next time, cheers!
Eddie
Thursday, May 9, 2013
Vacation! And some goodies!
Greetings from San Luis Obispo. I am on vacation this week having a great time!
So far I visited Morro Bay (what a sight - despite the presence of a power plant),hung out in downtown SLO, and visited Cal Poly (and it's library - I am going to hit the library at UC Santa Barbara tomorrow).
Some math tips I picked up from my visit:
Fuzzy sets are sets that allow degrees of membership. Instead of having a Yes/No decision of whether an object belongs in a set, degrees of acceptance are allowed.
Let A(x) → [0,1] where A(x) is the degree of acceptance.
The basic properties for normal sets work for fuzzy sets.
From the NIST Handbook of Mathematical Functions (that book is huge!):
Ways to calculate some functions:
Error Function:
erf(z) = 2/√π * ∫(e^(-t^2) dt, 0, z) = 2/√π * Σ((-1)^n * z^(2*n+1) /(n! * (2*n+1)), n=0 to infinity)
Γ(z) = ∫(e^-t * t^(z-1) dt,0,infinity) = (z-1)! = π /(sin (π*z) * Γ(1-z))
Γ(x) ≈ e^-x * x^x * √(2 π/x) * Σ(g_k/x^k, k=0 to infinity) where g_k is from a series. The first few terms are
g_1 = 1
g_2 = 1/12
g_3 = 1/288
g_4 = -139/51840
g_5 = -571/2488320
g_6 = 163879/2090188880
zeta(s) = Σ(n^-s, n=0 to infinity) = 2^(s-1)/Γ(s+1) * ∫(x^s/(sinh x)^2 dx, 0, infinity)
= 1/Γ(s) * ∫(x^(s-1)/(e^x - 1) dx, 0, infinity)
Wishing the best for everyone,
Eddie
