# Temperatures and Altitudes of Isothermal
Layers of Earth's, Titan's and Jupiter's Atmospheres in a Simultaneous Proportion of the Golden Ratio Constant, 1.6180339 Introduction The golden ratio constant ?=1.6180339?is a constant of the ancient times. The only constant older than ? is ? (Rhind's mathematical papyrus, XIX century BC). First concrete application of this constant is noted in the proportions of the famous Parthenon (constructed in 447 BC and completed in 432 BC). Known algebra form of the constant ? was obtained on the basis of the Euclidean definition and it sums to: ... .618 1 2 5 1 = + Constant ? has a unique unit fraction:
... . ...
61803398 1 1 1 1 1 1 1 1 = + + + +
Value of the constant is also found in the convergence condition for consecutive order members F n =F n?1 +F n?2 with arbitrary initial numbers, of which certainly the best known one is the Fibonacci sequence with initial numbers 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 134, ... Although most of the current literature that deals with the topic of golden ratio constant is still of a predominantly sensationalistic character, last 20 years were marked by the production of a significant number of scientific papers in which the golden ratio constant had a central role. Systematization of these papers is not yet possible to perform, because the golden ratio constant is becoming a phenomenon found simultaneously in numerous areas. From quantum mechanics (El Nashie, 1994), chemical reactions (Heyrovska, 2006;Heyrovska, 2007;Yablonsky et al., 2010) in DNA structures (Yamagishi et al., 2008;Perez, 2010), the human brain functions (Conte et al., 2009;Pletzer et al., 2010), the structure of the human heart (Henein et al., 2011), the human facial features and facial attractiveness ratings (Schmid et al., 2008;Mizumoto et al., 2009) and structure of plants (Mathai and Davis, 1974;Ridley, 1982;Lanling and Wang, 2009). From micro and meta structures, the special mathematical phenomenon has multiple application in mathematics, science and engineering (Stakhov, 1989, Stakhov 2005), realistic networks (Estrada, 2007), the production of textiles (Gao et al., 2007), Special relativity theory (Sigalotti and Mejias, 2006) and cosmology (Stakhov and Rozin, 2007).
Golden ratio constant still has no significance in applied mathematics in a level of the constants e and ?. Therefore, the established relation of the hyperbolic functions and formations of Fibonacci and Lucas sequences is of particular importance (Stakhov and Rozin, 2005) as well as relations of constants e and ? with ? (Tanackov et al., 2011). In addition to the above listed and many not mentioned scientific papers related to the golden ratio constant, it should be noted that the existence of golden ratio constantis established in studies of the Earth's atmosphere (Willoughby, 2011).
II. Golden Ratio in the Earth's Atmosphere 99% of the Earth's atmosphe reconsists of two chemical elements, 78% nitrogen and 21% oxygen. The troposphere extends upwards from right above the boundary layer, and ranges in height from an average of 9 km (5.6 mi; 30000 ft) at the poles, to 17 km (11 mi; 56,000 ft) at the Equator (Gettelman et al., 2002).
In the absence of inversions and not considering moisture, the temperature lapse rate for this layer is 6.5 C/km, on average, according to the U.S. Standard Atmosphere. Boundary layer of troposphere and stratosphere is the tropopause. Isothermal phenomenon of the tropopause is, on average, around ?60 C.
With increasing height, temperature rises through the stratosphere at a negative lapse rate until the next inversion, which is achieved through stratopause. The stratopause is the level of the atmosphere which is the boundary between stratosphere and mesosphere.
On Earth, the stratopause is 50 to 55 kilometers. Altitude of temperature inversions in the stratopause may vary, there may be significant fluctuations in the isothermal zone of the stratosphere with several peaks (Prakash Raju et al., 2011) of medium temperatures around ~?8 C or ~?5 C at altitudes of 46 km or 53.5 km respectively, in tropical regions, or significantly lower temperatures that in polar regions generate the stratopause temperature up to ?20 C (France et al., 2012) . Temperature variations of the stratopause during the year can be found in a wide range of ?20 C to +2 C at altitudes of 55 km to 42 km respectively (Alexander et al., 2011).
Further altitude increase is characterized by positive lapse rate until the next inversion, which is realized through mesopause. Intervals of minimum temperature of about ?90 C at altitudes of 85 km to100 km were measured at the latitude of 23 S (Clemesha et al., 2011). Detailed analysis of temperatures and altitudes of mesopause in the tropical zone (13.5 N79.2 E) next to the clear influence of the season, introduces °°°°°°°°°°°N otes also the function of latitude and emphasizes the mesopause altitude of 100 km (Ratnam et al., 2010). In the polar area (80°S and 223°E), established altitudes of mesopause were 92 km (Sheese et al., 2011).
The same mesopause altitudes of 92 km were established at lowerl atitudes (68°N and 21°E). Mesopause is the coldest part of the atmosphere, with fascinating temperatures that may be lower than100 K. However, such low temperatures are characteristic for the specific conditions in polar areas-polar mesospheric cloud season (Schmidlin, 1992) Conventional average representation of the relations between mean temperatures of mean altitudes of atmosphere layers is shown in Figure 1. Typical values of temperatures and altitudes have an assumed simultaneous proportion of the golden ratio.
Fig. 1: Mean temperatures and mean altitudes of Earth's atmosphere layers III. Golden Ratio in the Atmosphere of Titan After Jupiter's satellite Ganymede, Saturn's satellite Titan is the second largest satellite in the solar system and the only known moon with a dense atmosphere, with an average surface pressure of 146.7 kPa, with nitrogen (98.4%) and methane (1.6%) structure.
# °°°°°°°°°°°N otes
According to the data for 79.8°N, temperature on the surface of Titan, at 0.0 km altitude is 90.15°K. A drop in temperature to 48 km altitude achieves the isothermal minimum of 66.45°K. Growth of temperature to 80km reaches isothermal maximum of 119.39°C, and at 100 km the isothermal minimum of 109.23°K (Schinder et al., 2012).
Relations in altitudes of 0 km, 48 km, 80 km and 100 km where are the established isothermal minima and maxima of temperatures of the Titan's atmosphere, have departures from the golden ratio constant of 3.00% (1) and 0.43% (2), respectively:
03005 0 666666 1 666 1 0 48 0 80 . , . = ? ? ? â??" = ? ? (1)00432 0 625 1 625 1 48 100 80 100 . , . = ? ? ? â??" = ? ? (2)
The temperature values of characteristic altitudes T(0 km)=90.15°K, T(48 km)=66.45°K, T(80 km)=119.3°K and T(100 km)=109.23°K have numerical coincidences with values of linear combinations of the golden ratio constant. All these numerical coincidences result in deviations of less than 0.5% (3), ( 4), ( 5
. ? = ? ? ? ? ? ? + ? ? ? ? ? ? ? + ? ? â??" = ? ? (3). ? = ? ? ? ? ? â??" = ? ?(5)
Relations of altitudes and temperatures of isothermal layers of the Titan's atmosphere, have a larger number of numerical coincidences with the golden ratio constant or its linear combinations (Figure 2.). Jupiter's atmosphere has a dominant composition of hydrogen (86%) and helium (14%). Atmospheric pressure on Jupiteris is 70 kPa. Analogue to terrestrial terminology, Jupiter?s stratosphere extends from the tropopause at 28km (atmospheric pressure of 280 mbar) to the mesopause at ?350 km (atmospheric pressure of ?0001 mbar). Temperatures of Jupiter's upper stratosphere have been established with a Galileo Atmosphere Structure Instrument (ASI) (Young et al., 2005).
In regard to the overall temperature range of Jupiter's atmosphere, in the region between 90 km and 290 km, the mean temperature is essentially isothermal. At three different altitudes ?95 km, ?117 km and ?264 km, three extreme temperature values of the stratosphere were established, two temperature maxima and one temperature minimum. Data were obtained from two sensors (z 1 and z 2 ) at altitude of 6.5° North in December 1995. A detailed description of the resultsis available in the literature (Young et al., 2005).
Temperature of ?161°K at ?95 km altitude is the first extreme value of the isothermal zone of Jupiter's stratosphere. Average temperature value of Jupiter's stratosphere is 158.1°K and is close to the first extreme value.
The other extreme temperature value of Jupiter's stratosphere is ?148°K at ?117 km altitude which is also the established minimum temperature of Jupiter's stratosphere. Close value, but not the lowest, of 149.8°K at 262.736 km altitude (measured with sensor z 1 ) and 149.9°K at 260.785 km altitude (measured with sensor z 2 ) is still about 2°K higher. Altitude of ?117 km is reported to have a lower value of temperature fluctuations of Jupiter's stratosphere from ??0.07, while the temperature fluctuation at altitude of ?261 km is lower and is around ? ?0.06.
Third extreme temperature value of Jupiter's stratosphere is ?170°K at altitude of ?164 km. This extreme value of temperature is the very maximum of the entire stratosphere. The following table distinguishes characteristic accelerometer data for sensors z 1 and z 2. (Table 1.). Hypothesis on the temperature and altitude proportion of the golden ratio in the upper stratosphere of Jupiter can be confirmed by the ANOVA test. Hypotheses on the proportion in characteristic values of temperatures and latitudes of Jupiter's stratosphere, is proposed with the calculation of the third value based on choosing two values and function context of the constant ?=1.6180339.
In accordance with the assumption on a "golden" ratio of altitude differences and the characteristic values of temperatures of isothermal layers of the Earth's atmosphere as well as numerical coincidences of altitude and temperature differences in isothermal layers in the atmosphere of Titan, a hypothesis is set on the simultaneous proportions of altitude and temperature differences of isothermal layers in the atmosphere of Jupiter.
The possibility of verifying the hypothesis by the ANOVA test was designed due to the existence of variance in the minimum statistical set of values of altitudes and temperatures. The minimum statistical set was formed with measurements from two sensors of the mission Galileo ASI (Young et al., 2005).
The basis of the hypothesis has a predictive character. It is estimated that based on the two selected values of temperatures or altitudes of isothermal layers of the Jupiter's atmosphere, it is possible to establish a functional context of the constant ?=1.6180339 and make a prediction of altitude and temperature of the following isothermal layer (Figure3). Based on the maximum values of temperatures measured with sensor z 1 (169.6°K and 160.6°K), there is an estimation of the quotient of ? value of differences in maximum temperatures and minimum temperature of the isothermal layers of the Jupiter's atmosphere. From this assumption, replacing the value of the maximum, the predictive temperature of the minimum is calculated and it amounts to (6):
â??" ? ? ? = ? â??" ? = ? ? ) z ( P ) z ( P ) z ( P ) z ( P ) z ( P ) z ( P ) z ( P ) z ( P 1 2 1 3 1 2 1 1 1 2 1 3 1 2 1 1 * ) z ( P , ) z ( P ) z ( P ) z ( P 1 2 1 2 1 1 1 3 2 037 146 = â??" = ? ? ? â??"(6)
Analogously, for maximal values of temperature measured with z 2 sensor (170.2°K and 161.4°K), the second predictive value of the golden ratio proportion is obtained from (7):
161 147 = â??" = ? ? ? â??" ? = ? ?(7)
Hypothesis on the equality of predictive and empirical values was verified by the Duncan test. Duncan test was chosen because of the liberality towards the error of the first kind and the introduction of the mean value factor. Data were analyzed with Microsoft Statistica 4.5.
Predictive value is consistent with the empirical measured value. With significance p=0.1337>0.05 it is accepted that mean values of predictive (146.59±0.7945°K) and empirical (148.00±0.1414°K) values of characteristic temperatures in isothermal layers of Jupiter's stratosphere are congruent. Proportion of the golden ratio constant in the relations of extreme temperature of Jupiter's stratosphere is significant (Table 2.).
# 147,900
For altitude, the following linear ratio was elected for values measured with z 1 sensor:
â??" + ? = ? ? â??" + ? = ? ? 2 3 2 2 3 1 3 1 2 1 3 1 1 1 3 1 2 1 31[ ] â??" ? + ? = ? â??" ) z ( P ) z ( P ) z ( P ) z ( P 1 3 1 2 1 3 1 1 2 3 2 â??" + ? = ? + ? + â??" ) z ( P ) z ( P ) z ( P 1 2 1 3 1 1 2 3 2 2 2 3 2 ) z ( P ) ( ) z ( P ) ( ) z ( P 1 2 1 3 1 1 3 2 1 2 2 + ? = + ? + â??"(8)
The first predictive value of isothermal layer altitude of the Jupiter s atmosphere is (9):
* * ) z ( P , ) z ( P ) ( ) z ( P ) ( ) z ( P 1 2 1 2 2 1 3 1 1 900 117 3 2 1 2 2 = â??" = + ? + ? +(9)
Analogously, for the value of altitude measured withsensor z 2, the second predictive value of isothermal layer altitude of the Jupiter's atmosphere is (10)
# Conclusion and Recommendations
The assumption about the part of the golden ratio constant ?=1.6180339... in the Earth's atmosphere, numerical coincidence is in the atmosphere of Titan and the prediction significance in the atmosphere of Jupiter, may represent an important incentive for further research in numerical models of the atmosphere. Though the part of golden ratio constants in natural phenomena has a primarily sensationalistic character, two basic mathematical fields should be emphasized, that have an important role in studies of the atmosphere, and where the golden ratio constant can have an exact mathematical importance.
The first mathematical field refers to the phenomena of fractals. Fractal phenomena have been multiply noted in the atmosphere of the Earth (Baryshnikova et al., 1989;Collins, and Rastogi, 1989) and the atmospheres of Titan and Jupiter (West and Smith, 1991;Cabane et al., 1993;Friedson et al., 2002;Rannou et al, 2003). Temperature amplitudes (Peusse et al., 2006), mean global potential energy distribution for vertical wavelengths (Fröhlich et al., 2007), air velocity in the atmospheric layers (Wrasse et al., 2006), etc. are largely self-similar and have properties of stochastic fractals. The final attractor does not exist because the system dynamics of the atmosphere is under the constant influence of cyclical phenomena (time of day, season, solar activity,etc.) or impulse phenomena (volcanic eruptions, etc.). The potential role of golden ratio constant is in the prognosis of equilibrium status of altitudes and temperatures of isothermal layers of the atmosphere. Based on the results presented in this paper, the final attractor can be determined in the proportion of the golden ratio constant, which is in the fractal concept simultaneously the initial fractal.
Another field of mathematics concerns the application of hyperbolic functions. General role of hyperbolic functions in studies of fluid dynamics is known, and specific application in atmospheric research (Kraginsky and Oparin, 2003), as well as the synthesis of hyperbolic and fractal mathematical concepts (Harlander and Maas, 2007), Binet's formulas for the calculation of Fibonacci and Lucas sequence members in a continuous domain have an exact mathematical relation with Riccati hyperbolic functions (Stakhov and Rozin, 2005). The spherical form of the atmosphere and the atmospheric layers is a space dominated by non-Euclidean geometry, i.e. hyperbolic geometry, which is based on the golden ratio constant (Stakhov, 2006). Substitution of classic analytic forms of hyperbolic function with classes of hyperbolic function based on the golden ratio constant, presents a possible way of introducing golden ratio constant
![. Thermal structure at the midlatitude of mesopause (41°N, 105°W) has the established minimum temperatures from?100 C (172 K at summer solstice) and a high of?61 C (212 K at nearly one month following winter solstice) (She et al., 1993), or from ?100 C (165 K) to ?59 C (214 K) in Wuppertal (51 N,7 E) ( Höppner and Bittner, 2007). These temperatures are established at altitudes from 87 km to 99 km. Overall, altitudes and temperatures of the tropopause, stratopause and mesopause depend on the season, latitude, local weather (Clemesha et al., 2011), volcanic eruptions (Hampson et al., 2006), planetary and gravity waves (Limpasuvan et al., 2012), solar cycles (She et al., 2002), etc.](image-2.png "")
2![Fig. 2: Numerical coincidences with the golden ratio in Titan's atmosphere IV. Golden Ratio in the Atmosphere of Jupiter](image-3.png "NotesFig. 2 :")
3![Fig. 3: Three wave trains in the Galileo ASI data, and numerical values of altitudes with extreme values of temperature in the stratosphere of Jupiter with a graphic proportion of golden mean](image-4.png "Fig. 3 :")
SensorsMeasured temperatures (°K)Measured Altitudes (km)P 1 (z 1 ) P 1 (z 2 )169,6 170,2Average P 1a 169.9165,147 163,376Average P 1a 164,261P 2 (z 1 ) P 2 (z 2 )148,1 147,9Average P 2a 148.0116,513 117,907Average P 2a 117,21NotesP 3 (z 1 ) P 3 (z 2 )160,6 161,4Average P 3a 161.095,594 94,665Average P 3a 95,130
11
3 P P 1( (z z 2 2) )P 2 P 2( (z z 2 2) )23 P(z2)P 1(z2)P 2(z2).P 2(z2* )NotesYear 20191 45ersion I VIIIssueVolume XIX( F )Frontier Researchof ScienceGlobal Journal© 2019 Global Journals
2Jupiter's stratosphereTemepratureValuesMeans and standard deviations( ( 3 P P 2 3 2 ? ?z 1 z 2) )? ?( ( 1 P ? P 1 ?z 1 z 2) )= =P 2 P 2( (z 1 z 2* ) * )146,037 147,161146.59±0.7945z P 1 ( 2 z ( P 2 2) )148,100148.00±0,1414
3Year 20191 46ersion I VIIIssueVolume XIX( F )Frontier Researchof ScienceGlobal Journal2P 1(z2( ) + 2) 1 3 + + ? 2 2 ? ()P 3(z2)=P 2(z2)â??". 116702=P 2(z2* * )(10)© 2019 Global Journals
*
SPAlexander
ARKlekociuk
DJMurphy
2011
Rayleigh lidar observations of gravity wave activity in the winter upper stratosphere and lower mesosphere above
*
Davis
Journal of Geophysical Research
116
16
Atmospheres
*
Fractal analysis of the pre-hurricane atmosphere from satellite data
YSBaryshnikova
GMZaslavsky
EALupyan
SSMoiseyev
EASharkov
Advances in Space Research
9
1989
*
MCabane
ERannou
GChassefière
Israel
Fractal aggregates in Titan's atmosphere
1993
41
*
Sodium lidar measurements of mesopause region temperatures at Advances in Space Research
BClemesha
DSimonich
PBatista
2011
47
*
Fractal analysis of gravity wave spectra in the middle atmosphere
RLCollins
PKRastogi
Journal of Atmospheric and Terrestrial Physics
51
1989
*
Fractal fluctuations and quantum-like chaos in the brain by analysis of variability of brain waves: A new method based on a fractal variance function and random matrix theory: A link with El Naschie fractal Cantorian space-time and V. Weiss and H. Weiss golden ratio in brain
EConte
AKhrennikov
AFederici
JPZbilut
Chaos, Solitons& Fractals
41
2009
*
Is quantum space a random cantor set with a golden mean dimension at the core?
MSEl Nashie
Chaos, Solitons & Fractals
4
1994
*
Graphs (networks) with golden spectral ratio
EEstrada
Chaos, Solitons and Fractals
33
2007
*
2012: A climatology of stratopause temperature and height in the polar vortex and anticyclones
JAFrance
VLHarvey
CERandall
MHHitchman
MJSchwartz
Journal of Geophysical research
117
27
Atmospheres
*
AJFriedson
ASWong
YLYung
Models for Polar Haze Formation in Jupiter's Stratosphere
2002
158
*
The global distribution of gravity wave energy in the lower stratosphere derived from GPS data and gravity wave modelling: Attempt and challenges
KFröhlich
TSchmidt
MErn
PPreusse
ADe La Torre
JWickert
CJacobi
Journal of Atmospheric and Solar-Terrestrial Physics
69
2007
*
Golden mean and fractal dimension of goose down
JGao
NPan
WYu
International Journal of Nonlinear Sciences and Numerical Simulation
8
2007
*
Distribution and influence of convection in the tropical tropopause region
AGettelman
MLSalby
FSassi
Journal of Geophysical Research: Atmospheres
107
2002
*
The dynamical influence of the Pinatubo eruption in the subtropical stratosphere
JHampson
CClaud
PKeckhut
AHauchecorne
Journal of Atmospheric and Solar-Terrestrial Physics
68
2006
*
Internal boundary layers in a well-mixed equatorial atmosphere/ocean
UHarlander
LR MMaas
Dynamics of Atmospheres and Oceans
44
2007
*
The human heart: application of the golden ratio and angle
MYHenein
YZhao
RNicoll
LSun
AWKhir
KFranklin
PLindqvist
International Journal of Cardiology
150
2011
*
Dependence of ion-water distances on covalent radii, ionic radii in water and distances of oxygen and hydrogen of water from ion/water boundaries
RHeyrovska
Chemical Physics Letters
429
2006
*
Dependences of molar volumes in solids, partial molal and hydrated ionic volumes of alkali halides on covalent and ionic radii and the golden ratio
RHeyrovska
Chemical Physics Letters
436
2007
*
Evidence for solar signals in the mesopause temperature variability?
KHöppner
MBittner
Journal of Atmospheric and Solar-Terrestrial Physics
69
2007
*
Parallel framework for numerical modeling of the problems described by hyperbolic equations with applications in atmosphere flows modeling
LMKraginsky
AMOparin
Second MIT Conference on Computational Fluid and Solid Mechanics
KJBathe
2003. June 17-20
*
Modeling golden section in plants
ZLLanling
GWang
Progress in Natural Science
19
2009
*
The roles of planetary and gravity waves during a major stratospheric sudden warming as characterized in WACCM
VLimpasuvan
JHRichter
YJOrsolini
FStordal
OKKvissel
Journal of Atmospheric and Solar-Terrestrial Physics
2012
*
Constructing the sunflower head
AMMathai
ADDavis
Mathematical Biosciences
20
1974
*
Assessment of facial golden proportions among young Japanese women
YMizumoto
DSDeguchi
WC FKelvin
American Journal of Orthodontics and Dentofacial Orthopedics
136
2009
*
Codon populations in single-stranded whole human genome DNA Are fractal and fine-tuned by the Golden Ratio 1.618
JCPerez
Interdisciplinary Sciences: Computational Life Sciences
2010
2
*
When frequencies never synchronize: The golden mean and the resting EEG
BPletzer
HKerschbaum
WKlimesch
Brain Research
1335
2010
*
UJPrakash Raju
MKrishnaiah
YBKumar
DKothandan
PKeckhut
Tropical Stratopause features -Rayleigh lidar observations over Gadanki (13.5°N, 79.2°E)
India
2011
2
Advances in Applied Science Research
*
PPreusse
EManfred
SDEckermann
CDWarner
RHPicard
PKnieling
MKrebsbach
MJames
JMRussell
MGMlynczak
CJMertens
MRiese
Tropopause to mesopause gravity waves in August: Measurement and modeling
2006
68
*
Tropical mesopause: Is it always close to 100 km
MVRatnam
AKPatra
BVKrishna Murthy
Journal of Geophysical Research: Atmospheres
115
27
2010
*
A model of Titan's haze of fractal aerosols constrained by multiple observations
PRannou
CPMc Kay
RDLorenz
Planetary and Space Science
51
2003
*
Packing efficiency in sunflower heads
JNRidley
Mathematical Biosciences
58
1982
*
PJSchinder
FMFlasar
EAMarouf
RGFrench
CAMc Ghee
AJKliore
NJRappaport
EBarbinis
DFleischman
AAnabtawi
The Structure of Titan's Atmosphere from Cassini Radio Occultations: Occultations from the Prime and Equinox Missions
2012
221
*
Computation of a face attractiveness index based on neoclassical canons, symmetry, and golden ratios
KSchmid
DMarx
ASamal
Pattern Recognition
41
2008
*
First observation of mesopause temperatures lower than 100 K
FJSchmidlin
Geophysical Research Letters
19
1992
°Notes
*
Observed thermal structure of a midlatitude mesopause
CYShe
JRYu
HChen
Geophysical Research Letters
20
1993
*
Evidence of solar cycle effect in the mesopause region: Observed temperatures in 1999 and 2000 at 98.5km over Fort Collins, CO (41°N, 105°W)
CYShe
JSherman
JDVance
TYuan
ZLHu
BPWilliams
KArnold
PAcott
DAKrueger
Journal of Atmospheric and Solar-Terrestrial Physics
64
2002
*
Mesopause temperatures during the polar mesospheric cloud season
PESheese
EJLewellyn
RLGattinger
AEBourassa
DADegenstein
NDLloyd
ICMc Dade
10.1029/2011GL047437
Geophysical Research Letters
38
2011
*
The golden ratio in special relativity
LD GSigalotti
AMejias
Chaos, Solitons and Fractals
30
2006
*
The golden section in measurement theory
AStakhov
Computers & Mathematics
17
1989
*
The Generalized Principle of the Golden Section and its applications in mathematics, science, and engineering
AStakhov
Chaos, Solitons and Fractals
26
2005
*
AStakhov
BRozin
On a new class of hyperbolic functions
2005
23
*
AStakhov
Fundamentals of a new kind of mathematics based on the Golden Section
2006
27
*
The "golden" hyperbolic models of Universe
AStakhov
BRozin
Chaos, Solitons and Fractals
34
2007
*
The Golden Ratio in Probablistic and Artificial Intelligence, Tehni?ki vjesnik
ITanackov
JTepi?
MKostelac
2011
18
*
GSYablonsky
DConstales
GBMarin
Coincidences in chemical kinetics: Surprising news about simple reactions
2010
65
*
Nucleotide Frequencies in Human Genome and Fibonacci Numbers
ME BYamagishi
AIShimabukuro
Bulletin of Mathematical Biology
70
2008
*
LAYoung
RVYelle
RYoung
ASeiff
DBKirk
Gravity waves in Jupiter's stratosphere, as measured by the Galileo ASI experiment
2005
173
*
The Golden Radius in Balanced Atmospheric Flows
HEWilloughby
Monthly Weather Review
139
2011
*
ARWest
HPSmith
Evidence for aggregate particles in the atmospheres of Titan and Jupiter
90
*
CMWrasse
TNakamura
TTsuda
HTakahashi
AFMedeiros
MJTaylor
DGobbi
ASSuratno
EAchmad
AGAdmiranto
Reverse ray tracing of the mesospheric gravity waves observed at 23°S (Brazil) and 7°S
Indonesia) in
2006