FRACTAL GENESIS OF THE ANGLES OF THE NEUTRINO MIXING MATRIX. Valery Timkov, Serg Timkov, Vladimir Zhukov, Konstantin Afanasiev
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Valery Timkov, Serg Timkov, Vladimir Zhukov, Konstantin Afanasiev. FRACTAL GENESIS OF THE ANGLES OF THE NEUTRINO MIXING MATRIX.. HERALD OF KHMELNYTSKYI NATIONAL UNIVERSITY, KHMELNYTSKYI NATIONAL UNIVERSITY, 2018, 263 (4), pp.263 - 272. hal- 01782443
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Timkov V. F. The Office of National Security and Defense Council of Ukraine Timkov S. V., Zhukov V. A., Afanasiev K.E. Research and Production Enterprise «TZHK»
FRACTAL GENESIS OF THE ANGLES OF THE NEUTRINO MIXING MATRIX.
Annotation
A fractal genesis of the angles of the neutrino mixing matrix hypothesis is being examined. (Pontecorvo – Maki–Nakagawa–Sakata matrix - PMNS matrix). It is assumed that framework for this genesis are gold algebraic fractals mantissas: of the Planck mass, of the Planck time and of the fine-structure constant, namely, sines of angles
12,, 23 13 are gold algebraic fractals: of the fine-structure constant, of the Planck mass and of the Planck time respectively. A gold algebraic fractal mantissa of any real number can be represented as additive gold algebraic fractal – an infinite sum of the power series (more exactly – geometrical) with the basis of the gold ratio large number. Assessed the Higgs boson mass based on additive gold algebraic fractal.
Key words: fractal genesis, neutrino mixing matrix, gold algebraic fractal, Planck constants, fine-structure constant, Higgs boson, spatial and structural levels of matter.
В.Ф. Тимков Аппарат Совета национальной безопасности и обороны Украины С.В. Тимков, В.А. Жуков, К.Е. Афанасьев Научно-производственное предприятие «ТЖК»
ФРАКТАЛЬНЫЙ ГЕНЕЗИС УГЛОВ МАТРИЦЫ СМЕШИВАНИЯ НЕЙТРИНО
Аннотация
Рассматривается гипотеза о фрактальном генезисе углов матрицы смешивания нейтрино (Pontecorvo – Maki–Nakagawa–Sakata matrix - PMNS matrix). Предполагается, что основой этого генезиса являются мантиссы золотых алгебраических фракталов: массы Планка, времени Планка и постоянной тонкой структуры, а именно: синусы углов являются золотыми алгебраическими фракталами соответственно: постоянной тонкой структуры, массы Планка и времени Планка. Мантисса золотого алгебраического фрактала любого вещественного числа может быть представлена в виде аддитивного золотого алгебраического фрактала – бесконечной суммы степенного (более конкретно – геометрического) ряда с основанием в виде большего числа золотой пропорции. Дана оценка массы бозона Хиггса на основе аддитивного золотого алгебраического фрактала.
Ключевые слова: фрактальный генезис, матрица смешивания нейтрино, золотой алгебраический фрактал, константы Планка, постоянная тонкой структуры, бозон Хиггса, пространственные и структурные уровни материи.
1. Introduction.
It’s known that main fundamental physical constants, for example the gravitational constant G, the reduced
Planck constant h, the speed of light in vacuum c, can be determined with Planck constants: the Planck length lp , the
Planck mass mp , the Planck time t p : 32 lp m p l p l p Gc2 ;;.h (1) mp t p t p t p Using the “Planck universal proportions” [1], the Planck constants values [2]: 35 44 8 lp1.616229(38) 10 m ; t p 5.39116(13) 10 s ; m p 2.176470(51) 10 kg ; (2) and the Hubble constant [3]:
H0 66.93 0.62( km / s ) / Mpc , (3) it's possible to evaluate the main spatial characteristics of the observable Universe [4]: mass MU , radius RU ,rotation period TU [5], which is equal to the light signal propagation delay at a distance equal
RU :
MRUU1 17 TtU p t p 4.43153468332610s , (4) mpp l H0 3 TUUU R c T 53 MU m p m p 1.789337036792 10 kg , (5) tpp l G
MTUU 26 RU l p l p cT U 1.328540675427 10m . (6) mtpp As well, in accordance with the law “Planck universal proportions” – for every celestial body of the observable Universe (including the observable Universe), mass which is m, the curvature of space radius is S, which created by gravitational field mass which is with the light signal propagation delay is tdm at distance equal to the gravitational constant G, and Planck force Fp , true: 3 33 lp RSU 11 312 G 22 2 6.673045869 10 m kg s , (7) mp t p M U R U mt dm
lp RSU 44 1 1 2 Fp m p2 M U 2 m 2 1.21048301950 10 kg m s . (8) tp T U t dm
Energy of body E which mass а and also the observable Universe energy EU are: 22 Emc hte dm FSE p;, U Mc U hT e U FR p U (9)
E p where: h – is the quantum of the Planck energy, where E - the Planck energy: E m c2. e p pp t p
It is evident: 22 Ep m p c m p l p 52 1 1 he 3 3.628372528 10 J s . (10) tp t p t p It’s known [6], that charge is function of the moment of the mass. Then for the elementary charge e is true: 11 7 22 e 10 mpp l kg m , (11) where is the fine-structure constant, it’s value is [2]: 7.2973525664 17 103. (12) From formula (11), considering [6]: 107 c2 , (13) 40 4 h2 4 m24 l m l 0 0 p p p p (14) a0 2 7 2 , me e10 m e m p l p t p m e where: a0 the Bohr Radius , me the electron mass, 0 electric constant, it follows:
mpl p a0me a0me mpl p a0me mpl p ; ;m p ;l p ;me . (15) a0me l p mp a0 Considering the formula (11) the Coulomb’s law for two elementary charges in gravitational form is: 2 2 e mp F ke 2 G 2 . (16) r12 r12
In the formula (16) ke is the proportionality factor (Coulomb's constant, or the electric force constant ),
r12 is a distance between two elementary charges. From formulas (11 – 16) it follows then that measurement units of electromagnetic interaction on the basis of constants lp,,, m p t p can be expressed with units of length, of mass and of time[6]. For instance: 11 22 Coulomb: C kC kg m , (17) 11 221 Ampere: A kA kg m s , (18) 13 Volt: 222 (19) V kV kg m s , impedance: 11 (20) k m s , 12 electric capacitance:F kF m s , (21) 1 inductance: H kH m , (22) 11 magnetic induction: 221 (23) Tl kTl kg m s , where: kC,,,,,, k A k V k k F k H k Tl are dimensionless coefficients of proportionality between units of measurements. Given that units of electromagnetic interaction basis is a moving charge that is the function of moment of the mass, then electromagnetic interaction is particular case of gravitational interaction that also is confirmed by formulas (17 – 23). Therefore in certain conditions exists gravitational-electromagnetic resonance [1,7,8]. The parameters of such resonance are defined by constants: lp,,,. m p t p In [9] was shown that between fundamental constants exists fractal connectivity and basic characteristics of a muon are gold algebraic fractals of the Planck mass and of the Planck length. Based on formulas (4 – 16) can be concluded that fundamental physic constants and spatially-energetic characteristics of the observable Universe – are multiplicative gold algebraic fractals of a muon.
Taking into account universal character of constants lp,,,, m p t p for gravitational and electro-magnetic interactions, i.e. for matter macro world, unity and the interrelationship of all spatial and structural levels of matter, it can be assumed that constants are also take part in formation of micro world patterns but given it specific and the influence of scale. It is handier to look for relationship of constants representation on macro and micro levels of matter along the lines of gold algebraic fractals, more particularly – on the basis of gold algebraic fractals mantissas for the physical constants case and mantissas functions for physical processes case.
2. Fractal genesis of the angles of the neutrino mixing matrix.
The angles 12,, 23 13 of neutrino mixing matrix have the following values [10]: 2 sin (12 ) 0.307 0.013, (24) 2 sin (23 )nm 0.51 0.04 (normal mass hierarchy), (25) 2 sin (23 )im 0.50 0.04 (inverted mass hierarchy), (26) 2 2 sin ()13 2.10 0.11 10, (27) or:
sin()12 0.55407580708780 0.114, (28)
sin(23 )nm 0.714142842854284 0.2, (29)
sin()23 im 0.70710 0.2, (30)
sin(13 )0.144913767461894 0.0331662470 9 , (31) then:
12 33.64708224 , 12 34.44990199 , 12 32.83473313 , (32)
23nm 45.572996 , 23 nm 47.86958524 , 23 nm 43.28009362 , (33) 23im45 , 23 im 47.29428287 , 23 im 42.70571713 , (34)
13 8.33228569 , 13 8.54931942 , 13 8.10961446 . (35)
Let’s say that sines of the angles 12,, 23 13 are gold algebraic fractals of constants lp,,,. m p t p Represent the constants lp,,,, m p t p numerical values for whom are defined by the formulas (2, 12), in form of gold algebraic fractals (in GAF form) [4,9,11]: m m 167 mantissa of the Planck length: lp 1.2865859866, then: lp l p f g m, (36) m m 37 mantissa of the Planck mass: mp 1.1756969040, then: mpp m fg kg, (37) m m 208 mantissa of the Planck time: tp 1.588954250534220, then: tp t p f g s, (38) m m 11 mantissa of the fine-structure constant: 1.4522098299, then: fg . (39)
1 fg 2 In formulas(36 – 39) fg - is large number of golden ratio: 2 ,ffgg 1, is equal: ffgg 5 f 0.5 2sin(18 ) 2cos(72 ) 0.61803398874989484820... (40) g 2 Some constants can be used in two positions: as direct or inverse values. For example, the inverse value of the fine-structure constant: 1 137.0359991381545 0, (41) in the form of gold algebraic fractals is:
GAF mantissa of the inverse value of the fine structure constant: 1 mi 1,11418746481018987150,then: mi f 10. (42) g It is obvious that for all constants mantissas and for their inversions, on example of fine-structure constant is true: 1 mi m 1.61803398874989484820... (43) fg Define first five levels of gold algebraic fractals of the constants:
For this sequentially multiply five times the mantissas of this constants by f g . The results are presented in the table 1: Table1
Mantissa name Level1 Level2 Level3 Level4 Level5 Planck` s length 0.79515387 0.49143212 0.30372175 0.1877103 0.11601 Planck' s mass 0.726620 0.44907 0.27754439 0.17153187 0.106012 Planck' s time 0.982028 0.6069265 0.3751012 0.23182529 0.14327590 Fine structure constant 0.89751503 0.55469479 0.3428202 0.21187456 0.13094568
Believing that numbers in the table 1 – are the sines of some angles will present their values in table 2: Table2
Constant name Planck` s length 52.669791 29.4347540 17.681277 10.819194 6.6619637 46.6038340 26.684433 16.11370 9.876896 6.08550 79.1209132 37.3675974 22.03057 13.404558 8.2374534 63.833328 33.68969567 20.048792 12.232228 7.5242428
If the hypothesis under consideration is right then the true values of the angles 12,,, 23 13 the following:
12 33.68969567 fine structure constant , level 2 , (44)
23 46.6038340Planck` s mass , level 1 , (45) 13 8.2374534Planck` s time, level 5. (46) Angles represented in expressions (44-46) are within tolerance limits of the angles ,,, which are given 12 23 13 in the expressions (32 – 35). In [9] is shown that the main characteristics of a muon are gold algebraic fractals of the fundamental constants: of the Planck mass and of the Planck length. Considering: l c p , t p it could be argued that main characteristics of a real elementary particle – muon, underlies the fractal genesis of the angles (44, 45, 46).
In [5] was stated that the angle 13 is close to the integral angle of the Carioles force (which is equal to 9.79°) in the model of rotating observable Universe. According to updated data, this angle is closer to the angle: Planck` s mass , level 4 9.87698 6 . Considering that genesis if the angles and also the integral angle of the Carioles force and of their fractals is determined by Big Bang, it is conceivable that the integral angle of the Carioles force and the angle Planck` s mass, level 4 9.876896 is the one and the same angle. Then contribution of dark matter in general energetic balance of Universe will be not 26.8%, but 26.88% , including contribution of the Carioles force constitutes 6.88%.
Conclusions. On the basis of gold algebraic fractals it is possible to research physical processes at different structural and spatial levels of matter. If the hypothesis under consideration is right then gold algebraic fractals of neutrino mixing matrix are related to the gold algebraic fractals of Planck constants, of a muon main characteristics and of the fine-structure constant.
3. Additive gold algebraic fractals.
In [4,9] were examined simple and multiplicative gold algebraic fractals. On example of the abstract physical constant 0A 1, consider additive gold algebraic fractals. For this represent the constant A in the form of gold algebraic fractal: An A m fg , (47) where: m A GAF mantissa, n integer number, value of structure level of GAF in doing so: 1 1mnA ,0 . (48) fg 1 It is evident that: can be represented as: A 1 mfAi( n 1) , (49) A g where mAi is inversion of the mantissa mA, wherein under the condition of formula (43): 1 mmA Ai . (50) fg Explore the mantissa m A.For this we’ll look at the power series as follows: 1 2 3kk 1 (51) fg f g f g ... f g ... f g . k2 It is obvious that the power series (51) – is geometrical progression given that it’s every element z can be computed under formula: zz11 fg f g f g ,2 z . (52) The series (51) makes sense as much as the following conditions are met: d 1) the necessary test for the convergence: limfdg 0, 1,2,3,... d 2) the sufficient test for the convergence: d 1 f g 1 a) d’Alambert’s: limf g 1, d d f g
d d 1 b) the Cauchy’s radical test: limffgg 1. d 1 1 From the formula (52) follows that the number ffggis denominator of the series (51), because fg 1, than the series (51) – is decreasing geometrical progression, then the sum of the series is: ff11 1 ffk11gg 1,61803398874989484820... (53) gg2 k2 1 fg f g f g For the series: 2 3 4kk 1 (54) fg f g f g ... f g ... f g , k3 amount is: ff22 fk12gg f f 1. (55) g2 g g k3 1 ffgg For the series: 3 4 5kk 1 (56) fg f g f g ... f g ... f g , k4 amount is: 33 ffgg ffk1 0,61803398874989484820... (57) gg2 k4 1 ffgg For series: 4 5 6kk 1 (58) fg f g f g ... f g ... f g , k5 amount is: 44 ffgg ffk12 0,381966011250105151795... (59) gg2 k5 1 ffgg
For series: 5 6 7kk 1 (60) fg f g f g ... f g ... f g , k6 amount is: 55 ffgg ffk13 0,23606797749978969640... (61) gg2 k6 1 ffgg Let: i 1,2,3,... , then in general case for considered series is true: i i1 i 2 k k 1 k 3 (62) fg f g f g ... f g ... f g f g . ki1 For example consequently for k 7,8,9,10,11,12,will get: 4 fg 0,145898033750315455386..., (63) 5 fg 0,09016994374947424102..., (64) 6 fg 0,055728090000841214..., (65) 7 fg 0,0344418537486330267..., (66) 8 fg 0,02128623625220818770..., (67) 9 fg 0,01315561749642483896... (68) It is evident that at k , is true: i i1 i 2 k k 1 k 3 (69) fg f g f g ... f g ... f g f g 0. ki1 GAF mantissa of any physical constant consisted of two parts: first part – is constant, which inherent for all mantissas – this is a number 1, wherein: 02 fg f g f g 1, (70) second part – is constant, let’s define it as: mAc , which inherent only for particular GAF. It’s characteristic number reflecting the specificities of the given fractal: Ac 0,mfg (71) that is GAF m A can be represented as: A Ac2 Ac m1. m fgg f m (72) On the basis of the formulas (51 – 69) it can be argued that an any characteristic number m Ac can be represented with specified accuracy in the form of series sum of the pattern (62). Let S is the series sum of the form (62), representing number with accuracy , i.e.: mSAc . (73) Algorithm of the amount finding S is following: Ac 1) find the amount S1, as series of (62) type, which value as close as possible to the number m , but is less than it on modulus;
2) define the accuracy 1 : Ac mS11 , (74)
3) verify the ratio of accuracy and 1, if: 1, then:
4) define the amount S2 , as series of (62) type, which value as close as possible to the number but is less than it on modulus;
5) define the accuracy 2 : Ac m() S1 S 2 2, (75)
6) verify the ratio of accuracy and 2 , if: 2, then:
7) define the amount S3, as series of (62) type, which value as close as possible to the number 2 , but is less than it on modulus;
8) define the accuracy 3 : Ac m() S1 S 2 S 3 3, (76)
9) verify the ratio of accuracy and 3, and then: 10) do iterations of (74 – 76) type until the condition (73) will be met, then:
SSSS12 ... b , (77) where b is the iterations quantity of (74 – 76) type, required to met the (73) condition. N.B.: if characteristic number of the mantissa m Ac is close to (59 – 68) or multiply to it, then the algorithm of it representation as power series is simplified. From (70, 72, 77) it follows that any physical constant’s mantissa can be represented as additive gold algebraic fractal with any specified accuracy. Let’s see the examples. Represent first ten numbers of a number sequence as additive gold algebraic fractals: 20 1 (fg f g ) f g , (78) 2 3 1 2 (fg f g f g ) f g , (79) 2 4 2 3 (fg f g f g ) f g , (80) 2 4 2 4 (fg 2 f g f g ) f g , (81) 2 5 3 5 (fg f g 2 f g ) f g , (82) 2 7 3 6 (fg 2 f g f g ) f g , (83) 2 8 4 7 (fg f g f g ) f g , (84) 2 4 8 4 8 (fg f g f g f g ) f g , (85) 2 3 6 8 4 9 (fg f g f g f g f g ) f g , (86) 2 6 8 4 10 (fg 2 f g f g f g ) f g , (87) i where fg ,i 1,2,...,8 determined by (57 – 67). Represent as additive gold algebraic fractals – the mass of some elementary particles: leptons (except for neutrino): a electron e, a muon , a tau and bosons: Higgs boson H 0, Z 0 boson, W boson. Wherein will use not the energetic but the gravitational equivalent of a mass. Mass values [12] of the above-mentioned elementary particles: 31 electron mass me : me 9.10938356 11 10kg , (88) 28 muon mass m : m 1.883531594 48 10kg , (89) 27 tau mass m : m 3.16747 29 10kg , (90) Higgs boson mass m : m2.2299 43 1,025 kg (91) H0 H0 boson mass m : m 1.625567 37 1025 kg , (92) Z 0 Z0 boson mass m : m1.43299 27 1025 kg. (93) W W Mass values of elementary particles (88 – 93) in the form of gold algebraic fractals: m m 144 mantissa of the electron mass: e 1.13164233168890, then: meg e f kg, (94) m m 133 mantissa of the muon mass: 1.175788105912,then: m fg kg, (95) m m 127 mantissa of the tau mass: 1.10190162111,then: m fg kg, (96) mantissa of the Higgs boson mass: H 0,m 1.020547272 40, then: m H0,m f 118 kg, (97) H0 g mantissa of the boson mass: Z 0,m 1.20374351254,then: m Z0,m f 119 kg, (98) Z0 g mantissa of the boson mass: W ,m 1.0611412325, then: m W,m f 119 kg. (99) W g Represent mantissas from formulas (94 – 99) as of gold algebraic fractals. Mantissa of the electron mass: is close to multiply value of 9 fg 0.01315561749642483896..., therefore approximately it can be represented: with accuracy 103 : m 29 e fg f g 10 f g , (100) with accuracy 105 : m 2 9 20 23 26 e fg f g 10 f g f g f g f g 1.1316415716769572. (101) Approximation representation with accuracy mantissa of the electron mass: em 1.13164233168890 In the form of additive algebraic gold fractal on the algorithm (74 -77): m 2 5 7 11 13 20 23 26 e fg f g f g f g f g f g f g f g f g 1.1316415716769572. (102) Approximation representation of the electron mass in the form of additive gold algebraic fractal: 2 5 7 11 13 20 23 26 144 mfffffffegggggggggg(). fffkg (103) Mantissa of the muon mass: m 1.175788105912 can be represented approximately as follows: with accuracy :
m 2 ffgg 2 1.175570504585, (104) with accuracy : m 2 4 8 10 16 24 fg+ f g + f g f g + f g f g + f g = 1.175783598349. (105) Approximation representation of the muon mass in the form of additive gold algebraic fractal: 2 4 8 10 16 24 133 m ( fffg + g + g ff g + g ff g + g ) fkg g . (106) Mantissa of the tau mass: m 1.10190162111 can be represented approximately with 105 in the form of additive gold algebraic fractal: m 2 5 10 12 16 21 fg f g f g f g f g f g f g 1.1019001427232. (107) Approximation representation of the tau mass in the form of additive gold algebraic fractal: 2 5 10 12 16 21 127 mffff ().g g g g f g f g ffkg g g (108) Mantissa of the Higgs boson mass: H 0,m 1.02054727240 can be represented in the form of additive algebraic gold fractal: 0,m 2 8 H fg f g f g 1.02128623625220. (109) Value of the Higgs boson mass in energetic equivalent is [12]: m 125.09 0.24GeV . (110) H0 Value of the Higgs boson mass in energetic equivalent with (109) represented in the form of additive algebraic gold fractal is: m ( f f2 f 8) f 118 125.18057589962250GVe , (111) H0 g g g g what is within tolerance on formula (110). Should note that the Higgs boson mass and number 7 (formula (84)) – are one and the same additive algebraic gold. Mantissa of the Z 0 boson mass: Z 0,m 1.20374351254 can be represented approximately with accuracy 106 in the form of additive gold algebraic fractal: 0,m 2 4 6 13 20 22 Z fg f g f g f g f g f g f g 1.203743823360. (112) Approximation representation of the boson mass in the form of additive gold algebraic fractal: m(). ffffff 2 4 6 13 20 ffkg 22 119 (113) Z0 g g g g g g g g Should note that characteristic number inversion of the mantissa of the boson mass 3 7 approximately with accuracy 10 multiply to the value fg by formula (66), therefore inversion of mantissa’s boson mass approximately in the form of additive gold algebraic fractal: 0,mi 2 7 Z fg f g 10 f g . (114) Mantissa of the W boson mass: W ,m 1.0611412325 can be represented approximately with accuracy in the form of additive gold algebraic fractal: ,m 2 6 12 13 17 19 W fg f g f g f g f g f g f g 1.0611400856. (115) Approximate value of the boson mass in the form of gold algebraic: m(). ffffffffkg 2 6 12 13 17 19 119 (116) W g g g g g g g g
Conclusions. Formulas (78 – 87) and (100 – 116) demonstrate that any physical constants which have any numerical value can be represented with specified accuracy in the form of additive gold algebraic fractals. Probably the Higgs boson mass and number 7 are one and the same gold algebraic fractal. In this case the Higgs boson mass can be defined accurately.
References
[1] Timkov, V. F., Timkov, S. V., Zhukov, V. A.,: Planck universal proportions. Gravitational - electromagnetic resonance., International scientific-technical magazine: Measuring and computing devices in technological processes, ISSN 2219-9365, 3 (52), p.p. 7 – 11, 2015. http://journals.khnu.km.ua/vottp/pdf/pdf_full/vottp-2015-3.pdf https://hal.archives-ouvertes.fr/hal-01329094v1 [2] https://physics.nist.gov/cgi-bin/cuu/Value?plkt|search_for=universal_in! [3] Aghanim N., Ashdown M., Aumont J., et al: Planck Collaboration: Planck intermediate results. XLVI. Reduction of large-scale systematic effects in HFI polarization maps and estimation of the reionization optical depth. https://arxiv.org/pdf/1605.02985.pdf [4] Timkov, V. F., Timkov, S. V., Zhukov, V. A.,: Evaluation of the main spatial characteristics of the observable Universe based on the law “Planck Universal Proportions”., Міжнародний науково-технічний журнал: Вимірювальна та обчислювальна техніка в технологічних процесах, ISSN 2219-9365, 1 (54), стор. 144 – 147 , 2016. http://journals.khnu.km.ua/vottp/pdf/pdf_full/2016/vottp-2016-1.pdf https://hal.archives-ouvertes.fr/hal-01330333v3 [5] Timkov, V. F., Timkov, S. V.,: Rotating space of the Universe, as the source of dark energy and dark matter., International scientific-technical magazine: Measuring and computing devices in technological processes, ISSN 2219-9365, 3 (52), p.p. 200 – 204, 2015. http://journals.khnu.km.ua/vottp/pdf/pdf_full/vottp-2015-3.pdf https://hal.archives-ouvertes.fr/hal-01329145v1 [6] Timkov, V. F., Timkov, S. V., Zhukov, V. A.,: Electric charge as a function of the moment of mass. Gravitational form of Coulomb`s law., International scientific-technical magazine: Measuring and computing devices in technological processes, ISSN 2219-9365, 3 (56), pp.27 – 32, 2016. http://journals.khnu.km.ua/vottp/pdf/pdf_full/2016/vottp-2016-3.pdf https://hal.archives-ouvertes.fr/hal-01374611v1 [7] Timkov, V. F., Timkov, S. V., Zhukov, V. A.,: Gravitational-electromagnetic resonance of the Sun as one of the possible sources of auroral radio emission of the planets in the kilometer range., International scientific-technical magazine: Measuring and computing devices in technological processes, ISSN 2219-9365, 4 (53), p.p. 23 – 32, 2015. http://journals.khnu.km.ua/vottp/pdf/pdf_full/vottp-2015-4.pdf https://hal.archives-ouvertes.fr/hal-01232287v1 [8] Timkov, V. F., Timkov, S. V., Zhukov, V. A.,: Gravitational-electromagnetic resonance of the Sun in the low- frequency of the radio spectrum of the Jupiter., International scientific-technical magazine: Measuring and computing devices in technological processes, ISSN 2219-9365, 2 (55), p.p. 198 – 203, 2016. http://journals.khnu.km.ua/vottp/pdf/pdf_full/2016/vottp-2016-2.pdf https://hal.archives-ouvertes.fr/hal-01326265v1 [9] Timkov, V. F., Timkov, S. V., Zhukov, V. A., Afanasiev, K.E.,: Fractal structure of the fundamental constants. Numerical evaluation of the values of some of fundamental constants with use of the major characteristics of muon., International scientific-technical magazine: Measuring and computing devices in technological processes, ISSN 2219-9365, 3 (59), p.p. 188 – 194, 2017. http://fetronics.ho.com.ua/pdf/pdf_full/2017/vottp-2017-3.pdf https://hal.archives-ouvertes.fr/hal-01581300v1 [10] http://pdg.lbl.gov/2017/tables/rpp2017-sum-leptons.pdf [11] Timkov, V. F., Timkov, S. V., Zhukov, V. A.,: Fractal structure of the Universe., International scientific-technical magazine: Measuring and computing devices in technological processes, ISSN 2219-9365, 2 (55), p.p. 190 – 197, 2016. http://journals.khnu.km.ua/vottp/pdf/pdf_full/2016/vottp-2016-2.pdf https://hal.archives-ouvertes.fr/hal-01330337v1 [12] http://pdglive.lbl.gov/Viewer.action
Литература
[1] Timkov, V. F., Timkov, S. V., Zhukov, V. A.,: Planck universal proportions. Gravitational - electromagnetic resonance.,Міжнародний науково-технічний журнал: Вимірювальна та обчислювальна техніка в технологічних процесах, ISSN 2219-9365, 3 (52), стор. 7 – 11, 2015. http://journals.khnu.km.ua/vottp/pdf/pdf_full/vottp-2015-3.pdf https://hal.archives-ouvertes.fr/hal-01329094v1 [2] https://physics.nist.gov/cgi-bin/cuu/Value?plkt|search_for=universal_in! [3] Aghanim N., Ashdown M., Aumont J., et al: Planck Collaboration: Planck intermediate results. XLVI. Reduction of large-scale systematic effects in HFI polarization maps and estimation of the reionization optical depth. https://arxiv.org/pdf/1605.02985.pdf [4] Timkov, V. F., Timkov, S. V., Zhukov, V. A.,: Evaluation of the main spatial characteristics of the observable Universe based on the law “Planck Universal Proportions”., Міжнародний науково-технічний журнал: Вимірювальна та обчислювальна техніка в технологічних процесах, ISSN 2219-9365, 1 (54), стор. 144 – 147 , 2016. http://journals.khnu.km.ua/vottp/pdf/pdf_full/2016/vottp-2016-1.pdf https://hal.archives-ouvertes.fr/hal-01330333v3 [5] Timkov, V. F., Timkov, S. V.,: Rotating space of the Universe, as the source of dark energy and dark matter., Міжнародний науково-технічний журнал: Вимірювальна та обчислювальна техніка в технологічних процесах, ISSN 2219-9365, 3 (52), стор. 200 – 204, 2015. http://journals.khnu.km.ua/vottp/pdf/pdf_full/vottp-2015-3.pdf https://hal.archives-ouvertes.fr/hal-01329145v1 [6] Timkov, V. F., Timkov, S. V., Zhukov, V. A.,: Electric charge as a function of the moment of mass. Gravitational form of Coulomb`s law., Міжнародний науково-технічний журнал: Вимірювальна та обчислювальна техніка в технологічних процесах, ISSN 2219-9365, 3 (56), стор. 27 – 32, 2016. http://journals.khnu.km.ua/vottp/pdf/pdf_full/2016/vottp-2016-3.pdf https://hal.archives-ouvertes.fr/hal-01374611v1 [7] Timkov, V. F., Timkov, S. V., Zhukov, V. A.,: Gravitational-electromagnetic resonance of the Sun as one of the possible sources of auroral radio emission of the planets in the kilometer range., Міжнародний науково-технічний журнал: Вимірювальна та обчислювальна техніка в технологічних процесах, ISSN 2219-9365, 4 (53), стор. 23 – 32, 2015. http://journals.khnu.km.ua/vottp/pdf/pdf_full/vottp-2015-4.pdf https://hal.archives-ouvertes.fr/hal-01232287v1 [8] Timkov, V. F., Timkov, S. V., Zhukov, V. A.,: Gravitational-electromagnetic resonance of the Sun in the low- frequency of the radio spectrum of the Jupiter., Міжнародний науково-технічний журнал: Вимірювальна та обчислювальна техніка в технологічних процесах, ISSN 2219-9365, 2 (55), стор. 198 – 203, 2016. http://journals.khnu.km.ua/vottp/pdf/pdf_full/2016/vottp-2016-2.pdf https://hal.archives-ouvertes.fr/hal-01326265v1 [9] Timkov, V. F., Timkov, S. V., Zhukov, V. A., Afanasiev, K.E.,: Fractal structure of the fundamental constants. Numerical evaluation of the values of some of fundamental constants with use of the major characteristics of muon., Міжнародний науково-технічний журнал: Вимірювальна та обчислювальна техніка в технологічних процесах, ISSN 2219-9365, 3 (59), p.p. 188 – 194, 2017. http://fetronics.ho.com.ua/pdf/pdf_full/2017/vottp-2017-3.pdf https://hal.archives-ouvertes.fr/hal-01581300v1 [10] http://pdg.lbl.gov/2017/tables/rpp2017-sum-leptons.pdf [11] Timkov, V. F., Timkov, S. V., Zhukov, V. A.,: Fractal structure of the Universe., International scientific-technical magazine: Міжнародний науково-технічний журнал: Вимірювальна та обчислювальна техніка в технологічних процесах, ISSN 2219-9365, 2 (55), стор. 190 – 197, 2016. http://journals.khnu.km.ua/vottp/pdf/pdf_full/2016/vottp-2016-2.pdf https://hal.archives-ouvertes.fr/hal-01330337v1 [12] http://pdglive.lbl.gov/Viewer.action