CC BY
15
УДК 512.554
Вестник СибГАУ Том 17, № 2. С. 324-327
АВТОМОРФИЗМЫ НИЛЬТРЕУГОЛЬНЫХ ПОДКОЛЕЦ АЛГЕБР ШЕВАЛЛЕ ОРТОГОНАЛЬНЫХ ТИПОВ
В. М. Левчук*, А. В. Литаврии
Сибирский федеральный университет Российская Федерация, 660041, г. Красноярск, просп. Свободный, 79 E-mail: vlevchuk@sfu-kras.ru
Алгебра Шевалле над ассоциативно коммутативным кольцом К с единицей характеризуется базисом Ше-валле, который сопоставляют каждой неразложимой системе корней Ф. Все элементы er (r е Ф+) базиса Шевалле дают базис подалгебры МФ(К), называемой нильтреугольной. Автоморфизмы алгебры МФ(К) описали Y. Cao, D. Jiang и D. Wang (J. Algebra, 2007) при К = 2К для лиевых типов Bn, Cn или F4 и при близких ограничениях для других типов. Их описание использует только нестандартный автоморфизм Гиббса; в нашей терминологии это гиперцентральный автоморфизм высоты 2 или 3 (для типа Су). Наша главная цель состоит в описании группы автоморфизмов А кольца Ли МФ(К).
Алгебра МФ(К) лиева типа An-1 представляется алгеброй Ли, ассоциированной с алгеброй NT(n, К) всех нильтреугольных n х n матриц над К. Группы автоморфизмов кольца NT(n, К) и ассоциированного с ним кольца Ли (т. е. A типа An) описал ранее В. М. Левчук (1983). Группу автоморфизмов A для типа Cn недавно описал А. В. Литаврин.
В настоящей работе мы находим нестандартные автоморфизмы алгебр NФ(K) ортогональных типов, когда условие К = 2К нарушается. Оказывается, когда аннулятор элемента 2 в К ненулевой, наибольшая высота гиперцентральных автоморфизмов зависит от лиева ранга. Кроме того, мы находим автоморфизмы алгебры NФ(K) типа Dm которые нестандартны по модулю второго члена нижнего центрального ряда и порождают подгруппу в A, изоморфную определенной подгруппе S в SL(2, К), в частности, S = SL(2, К) при 2К = 0. Стандартные автоморфизмы вместе с построенными нестандартными автоморфизмами порождают всякий автоморфизм алгебры NФ(K). Для всех классических типов лиева ранга > 4 наши результаты показывают, что группа автоморфизмов A является произведением подгрупп центральных и индуцированных кольцевых автоморфизмов и группы автоморфизмов алгебры NФ(K). Используются разработанные ранее методы, в частности, специальное представление алгебр NФ(K) классических типов. Результаты могут быть использованы при разработке криптографических методов.
Ключевые слова: алгебра Шевалле, нильтреугольная подалгебра, автоморфизм кольца Ли, высота гиперцентрального автоморфизма.
Sibirskii Gosudarstvennyi Aerokosmicheskii Universitet imeni Akademika M. F. Reshetneva. Vestnik Vol. 17, No. 2, P. 324-327
AUTOMORPHISMS OF NIL-TRIANGULAR SUBRINGS IN CHEVALLEY ALGEBRA OF ORTHOGONAL TYPE
V. M. Levchuk*, A. V. Litavrin
Siberian Federal University 79, Svobodny Av., Krasnoyarsk, 660041, Russian Federation E-mail: vlevchuk@sfu-kras.ru
Any Chevalley algebra over an associative commutative ring К with the identity is characterized by Chevalley base that correspondents to each indecomposable root system Ф. All elements er (r e Ф+) of Chevalley base give a base of subalgebra NФ(K) which is said to be nil-triangular. Automorphisms of algebras NФ(K) were described by Y. Cao, D. Jiang and D. Wang (J. Algebra, 2007) at К = 2K for Lie type Bw Cn or F4 and under similar restrictions for other types. Their description uses only non-standard Gibbs's automorphisms; in our terminology it is a hypercentral automorphisms of height 2 or 3 (for type Cy). Our main purpose is to describe the automorphism group A of the Lie ring NO(K).
The algebra N0(K) of Lie type An-1 can be represented as Lie algebra which associated to the algebra NT(n, K) of all nil-triangular n x n matrices over K. The automorphism group of the ring NT(n, K) and of its associated Lie ring (i. e., A for the type A,) described earlier V. M. Levchuk (1983). A. V. Litavrin has described the automorphism group A for Lie type Cn recently.
UDC 550.338.2
Sibirskii Gosudarstvennyi Aerokosmicheskii Universitet imeni Akademika M. F. Reshetneva. Vestnik Vol. 17, No. 2, P. 318-323
ON THE ERRORS OF ISOTROPIC APPROXIMATION TO THE GEOMETRIC-OPTICS DESCRIPTION OF IONOSPHERIC RADIO WAVE PROPAGATION
O. A. Laryunin
Institute of Solar-Terrestrial Physics SB RAS 126a, Lermontov Str., Irkutsk, 664033, Russian Federation E-mail: laroleg@inbox.ru
A geometric optics approach is of vital importance when considering high frequency radio wave propagation and space weather effects. Isotropic approximation, i. e. neglect the effect of the Earth's magnetic field, is widely used to simplify the ray-optical description of ionospheric propagation of radio waves. In particular, the advantage of this approach is that for a set of ionospheric models it allows to obtain analytical solutions for the parameters of the ray path. It has been extensively discussed in the literature that the isotropic approximation works quite well for oblique ionospheric sounding being representative of the ordinary mode. Thus, it has been established that with increasing the launch angle (i. e. for more oblique paths), the isotropic approximation error decreases, in fact not being able to distinguish between isotropic and anisotropic media for paths of the order of thousand kilometers long. The efficiency of the approximation is of vital importance for applications where simplification of the software implementation and reduction of the calculation execution time is required. However, this description under certain conditions may produce a significant error. In particular, it is not applicable for vertical and near-vertical sounding in the cases where operating frequency is close to critical frequencies of E and F2 layers. Two-dimensional ray tracing scheme is applied to the initial value problem and to the boundary value problem. The refractive index of the ionosphere is given by the Appleton-Hartree formula (ignoring collisions). Thus, the purpose of the study is to examine the no-field approximation error under different conditions of radio wave propagation on the basis of ray tracing analysis.
Keywords: ionospheric sounding, ray tracing, Earth's magnetic field.
Вестник СибГАУ Том 17, № 2. С. 318-323
О ПОГРЕШНОСТЯХ ИЗОТРОПНОГО ПРИБЛИЖЕНИЯ ПРИ ГЕОМЕТРООПТИЧЕСКОМ ОПИСАНИИ РАСПРОСТРАНЕНИЯ РАДИОВОЛН В ИОНОСФЕРЕ
О. А. Ларюнин
Институт солнечно-земной физики СО РАН Российская Федерация, 664033, г. Иркутск, ул. Лермонтова, 126а E-mail: laroleg@inbox.ru
При прогнозировании коротковолновой радиосвязи и описании эффектов космической погоды особое место занимает лучевой подход (метод геометрической оптики). Изотропное приближение, т. е. пренебрежение эффектами магнитного поля, широко используется для упрощения геометрооптического описания распространения радиоволн в ионосфере. Данное приближение позволяет в некоторых частных случаях получать аналитические выражения для траекторных характеристик. Известным фактом является то, что изотропное приближение может с хорошей точностью описывать обыкновенную компоненту радиосигнала. Так, в настоящей работе, в частности, показано, что погрешность приближения уменьшается с ростом угла излучения (т. е. для более наклонных трасс). Также данное приближение имеет большую практическую значимость при программной реализации метода геометрической оптики, позволяя существенно упростить соответствующий алгоритм. Тем не менее, в определенных условиях погрешность изотропного приближения может быть значительной. Так, пренебрежение эффектами магнитного поля может быть недопустимым при вертикальном и слабонаклонном зондировании на частотах, близких к критическим частотам E- и F2-moee. Выполнен двумерный траекторный синтез в одноточечной и двухточечной постановках задачи. Показатель преломления ионосферы был описан формулой Эпплтона-Хартри.
Ключевые слова: ионосферное зондирование, траекторный синтез, магнитное поле Земли.
1. Introduction
The problem of electromagnetic-wave propagation in anisotropic plasma cannot usually be described by exact analytical solutions even in simplest model cases. One of main approximate methods is a geometric optics which provides an illustrative description of radio wave propagation in terms of ray paths [1-3]. Applicability boundaries of the method (wavelength is smaller than the typical scales of the medium) generally work quite well in the problem of ionospheric propagation of radio waves. An allowance for magneto-ionic effects can be made by using the Appleton-Hartree formula [4]:
n02 = 1 -2u (1 - u
(1)
.2-4 .
2(1 -u)-vsin2 a + ^4(1 -u)2 vsin2 a + v2 sin4 a
where n0 is the ordinary component refractive index;
f2 f2
u =~j22 ; v = ~H ; f 2 is the plasma frequency; f is the
operating frequency; fH is the gyrofrequency.
If f is much greater than fH, the expression for the refractive index of the ordinary wave component can be simplified:
fs_
f
<< 1
2 1 n « 1 - u.
(2)
Relation (2) represents the isotropic (no-field) approximation.
Since the isotropic approximation simplifies the software implementation and reduces the execution time of calculation of the geometric optics method, it is widely used in practical applications. However, the simulation shows that there can be considerable discrepancies between the ray parameters in anisotropic and isotropic cases.
In [5-7], oblique incidence ionograms were synthe-tized from vertical incidence ionograms using the Smith's method [8]. In [6] the Smith's method is used for a long path of Khabarovsk-Tory (ground range D = 2297 km), which seems well founded from this point of view.
The effect of the magnetic field is also ignored in [910] considering paths over 1000 km long. Authors of [11] deal with oblique propagation of radio waves in the iso-tropic medium both for the long paths of Magadan-Tory and Norilsk-Tory as well as for the short path of Usolye-Tory (120 km). However, for example, the widespread Huang-Reinisch method for calculating electron density profiles [12] considers an anisotropic ionosphere and assumes a magnetic field to be constant in magnitude and direction. Authors of [13] also performed numerical ray tracing considering the Earth's magnetic field and gave a specification for the ray paths for near-vertical ionospheric sounding.
Authors of [5] developed a frequency scaling technique, which is based on the use of equivalent operating frequency f = f + Af, which leads to the same results in the absence of the magnetic field, as the actual frequency f in the presence of magnetic field. The additive
correction Af is in particular a function of magnetic dip angle at the ray mid point.
2. Numerical Simulation
For simulations we will use a typical winter mid-latitude plasma frequency profile from the Digisonde DPS-4 sited in Irkutsk (fig. 1), the points were joined using cubic spline interpolation to form a continuous function. The peak of the F2 layer is hm = 242.3 km , the critical frequency of the F2 layer fc = 9.46 MHz, the critical frequency of the E layer fE = 2.19 MHz .
/¿.MHz
Fig. 1. Plasma frequency profile from DPS-4 ionosonde, Irkutsk, December 02, 2011, 06:30 UT
The magnetic field will be considered to be constant in magnitude and direction (as in [12; 13]) with parameters typical for midlatitudes: the gyrofrequency fH = 1.56 MHz, the magnetic inclination I = 70.5°
2.1. Initial value problem (Cauchy problem). Let us state the initial elevation angle 90 = 15° (in relation to the vertical plane) and vary the operating frequency. Compare such characteristics of ray paths as group path and ground range (tab. 1).
As a comparison, we will give characteristic ray paths for a number of frequencies (fig. 2).
The results presented in tab. 1 allow us to plot the relative error versus frequency.
Fig. 3 shows that at frequencies close to critical ones of the E (fe) or F2 (fc) layer (an equivalent of the critical frequencies for oblique propagation according to the
fVIS
relation f0IS = -
[14]), the error increases sharply,
cos 90
whereas, say, at f = 6 MHz the relative error for ground range is equal to 4.4 %.
Consider now the isotropic approximation error for the initial value problem at f = 9 MHz and the alternate elevation angle (tab. 2).
The characteristic ray paths are as follows (fig. 4).
Tab. 2 yields the following dependences (fig. 5).
Raw relative errors at low angles of radiation in fig. 5, b are associated with very short paths (near-vertical sounding). However, local maxima in the vicinity of 8°-9° in both the plots of fig. 5, a, fig. 5, b, call for special consideration.
Table 1
Frequency dependence of group path and ground range for ordinary wave and for isotropic approximation
Frequency, MHz Group path, km (ordinary wave) Group path, km (isotropic approximation) Ground range, km (ordinary wave) Ground range, km (isotropic approximation)
2.5 548.7 484.3 151.2 121.9
3 482.9 477.3 136.8 119.9
3.5 451.0 459.4 125.0 115.3
4 436.4 445.5 118.1 111.8
4.5 437.9 443.2 116.6 111.2
5 443.7 446.2 117.0 111.9
5.5 451.8 452.0 118.3 113.3
6 461.9 460.2 120.5 115.2
6.5 475.8 471.6 123.7 118.0
7 490.7 484.8 127.3 121.2
7.5 507.7 499.9 131.5 124.9
8 528.7 518.3 136.8 129.3
8.5 557.7 542.5 144.1 135.1
9 603.0 578.2 155.6 143.8
9.5 724.8 653.1 185.5 161.8
Fig. 2. Examples of ray paths for ordinary component (solid lines) and isotropic approximation (dashed lines) for the profile shown in fig. 1: a - 3 MHz, b - 6 MHz, c - 9 MHz
Fig. 3. The relative error versus frequency: a - group path; b - ground range
Table 2
Group path and ground range versus elevation angle
Angle, degrees Group path, km (ordinary wave) Group path, km (isotropic approximation) Ground range, km (ordinary wave) Ground range, km (isotropic approximation)
1 629.6 592.8 11.5 9.9
2 629.5 592.5 21.3 19.8
3 629.0 592.0 31.7 29.8
4 628.8 591.3 42.1 40.0
5 626.2 590.4 53.0 49.4
6 624.1 589.4 64.0 59.1
End table 2
Angle, degrees Group path, km (ordinary wave) Group path, km (isotropic approximation) Ground range, km (ordinary wave) Ground range, km (isotropic approximation)
7 621.8 588.2 75.8 69.3
8 622.1 587.0 86.3 78.5
9 619.7 585.7 97.1 87.9
10 617.0 584.3 107.5 97.4
12 611.4 581.4 127.4 116.1
15 603.0 578.2 155.6 143.8
19 595.3 575.3 191.9 180.1
24 590.5 575.2 236.2 225.2
30 593.0 581.9 290.1 280.4
37 610.3 602.0 358.0 349.6
45 650.7 644.3 447.5 440.4
54 734.5 729.9 577.4 571.8
64 930.0 926.9 812.0 807.9
i-.m
u IÜÜ isü jdj
IrV
W»
4
ili'I ifü >ii
Ko Fi: Po
6 i А
3 2 1
Fig. 4. Examples of ray paths: a - <p0 = 4°; b - <p0 = 24°; c - <p0 = 45°. The solid line is the ordinary wave; the dashed line, the isotropic approximation
A>
i+
12 iO 8 6 i 2
II 20 30 4(1 iO 60
/4
\I \
10 30 30 +0 50
Fig. 5. The relative error versus elevation angle for f = 6 MHz: a - group path; b - ground range
It should be noted that at low elevation angles (near-vertical propagation), the ordinary ray path has a typical needle point (see fig. 4, a) [4; 13]; yet the reflection height in near-vertical sounding is equal to the reflection height in vertical incidence and is independent of the elevation angle. The critical angle 90 (relative to the vertical plane), at which the needle point disappears and the reflection heights begin to decrease, is determined by the relation [4; 13]:
sin 90 =
fn
fH + f
sin X,
(3)
where fH is the gyrofrequency; f is the operating frequency; x is the magnetic inclination.
In our case fH = 1.56 MHz, f = 9 MHz, % = 19.5°,
hence (3) yields 90 = 7.4°. The local maxima in the plots of fig. 5, a, b approximate to this angle. Thus, we may conclude that the closer the elevation angle to 90, the
b
a
b
a
larger error given by the isotropic approximation. With increasing the elevation angle (i. e. for more oblique paths), the isotropic approximation error becomes insignificant.
2.2. Two-point boundary value problem. Consider now a two-point boundary value problem in which for each operating frequency we carry out "homing-in" to the
given ground distance (in our case, 120 km) by varying the initial elevation angle of the ray [15]. We will compare the group path and the elevation angle (tab. 3).
In the problem thus stated, characteristic ray paths take the following form (fig. 6).
According to the data from tab. 3, the corresponding curves are as follows (fig. 7).
Table 3
Frequency dependence of group path and elevation angle for ordinary wave and no-field approximation
Frequency, MHz Group path, km (ordinary wave) Group path, km (isotropic approximation) elevation angle, (ordinary wave) elevation angle, (isotropic approximation)
2.5 538.5 481.7 12.14 14.75
3 476.1 477.4 13.27 15.02
3.5 449.2 460.9 14.43 15.57
4 437.0 447.9 15.23 16.04
4.5 438.8 445.6 15.42 16.13
5 444.5 448.1 15.38 16.04
5.5 452.2 453.5 15.21 15.86
6 461.9 461.1 14.94 15.60
6.5 475.2 472.0 14.55 15.24
7 489.6 484.6 14.13 14.85
7.5 506.4 499.3 13.67 14.42
8 527.5 517.3 13.11 13.93
8.5 558.4 542.0 12.37 13.30
9 613.6 580.9 11.24 12.42
Fig. 6. Examples of the ray paths: homing-in to the distance of 120 km: a - 3 MHz; b - 6 MHz; c - 9 MHz
Fig. 7. The frequency dependence of the relative error for the 120-km path: a - group path; b - elevation angle
3. Conclusion
The isotropic approximation is widely used for simplifying the ray-optics description of ionospheric propagation of radio waves. However, this description under certain conditions may produce a significant error. So, we have revealed that when an operating frequency approaches the critical frequencies of the E or F2 layer, the no-field approximation error increases sharply for near-vertical sounding. On the other hand, the simulation has shown that when varying the elevation angle the relative error has a local maximum in the vicinity of the critical angle determined by (3). With increasing the elevation angle (i. e. for more oblique paths), the isotropic approximation error decreases monotonically, practically not being able to distinguish between isotropic and anisot-ropic media for paths over 1000 km long.
Acknowledgments. The study was supported by the RFBR grant No. 14-05-00259a.
References
1. Kravtsov Yu. A., Orlov Yu. I. Geometrical Optics of Inhomogeneous Media. Springer-Verlag Publ., 1990, 312 p.
2. Hunsucker R. D. Radio techniques for probing the terrestrial ionosphere, Berlin; New York, SpringerVerlag, 1991, 293 p.
3. Rawer K. Wave propagation in the ionosphere, Dordrecht, Boston, Kluwer Academic Publ., 1993, 486 p.
4. Ginzburg V. L. Propagation of Electromagnetic Waves in Plasma. Gordon & Breach Science Publ., 1967, 822 p.
5. Bennett J. A., Chen J., Dyson P. L. Analytic calculation of the ordinary (O) and extraordinary (X) mode nose frequencies on oblique ionograms. Journal of Atmospheric and Terrestrial Physics, 1994, Vol. 56, No. 5, P. 631-636.
6. Kotovich G. V., Kim A. G., Mikhailov S. Ya., Grozov V. P., Mikhailov Ya. S. Determining the fo F2 critical frequency at the path midpoint from oblique sounding data based on the Smith method. Geomagnetism andAeronomy, 2006, Vol. 46, No. 4, P. 517-521.
7. Chen J., Bennet J. A., Dyson P. L. Synthesis of oblique ionograms from vertical ionograms using quasi-parabolic segment models of the ionosphere, Journal of Atmospheric and Terrestrial Physics, 1992, Vol. 54, No 3/4, P. 323-331.
8. Smith N. The relation of radio sky-wave transmission to ionosphere measurements. Proc. Inst. Radio Engrs, 1939, Vol. 27, P. 332-347.
9. Balagansky B. A., Sazhin V. I. [Numerical simulation of characteristics of decametric radio waves in the ionosphere with three-dimensional disturbances]. Geomagnetizm i aeronomiya, 2003, Vol. 43, No. 1, P. 92-96 (in Russ.).
10. Bristow W. A., Greenwald R. A. Estimating gravity wave parameters from oblique high-frequency back-scatter: Modeling and analysis. Journal of geophysical research, 1995, Vol. 100, No. A3, P. 3639-3648.
11. Mikhailov S. Ya., Grozov V. P. Recovery of the nonmonotonic altitude profile of the plasma frequency based on the ionospheric oblique sounding data. Radiophysics and Quantum Electronics, 2013, Vol. 56, No. 7, P. 399-412.
12. Reinisch B. W., Huang X. Automatic Calculation of Electron Density Profiles from Digital Ionograms, 3, Processing of Bottomside Ionograms. Radio Science, 1983, Vol. 18, No. 3, P. 477-492.
13. Varshavsky I. I. Issledovaniye Neodnorodnoi Struktury Ionosfery pri Aktivnykh Vozdeistviyakh [Studying Inhomogeneous Structure of the Ionosphere in Active Influences]. Ph. D. Thesis in Physics and Mathematics. Irkutsk, 1986, 154 p.
14. Davies K. Ionospheric Radio Waves. Blaisdell Publ. Co, 1973, 460 p.
15. Coleman C. J. A ray tracing formulation and its application to some problems in over-the-horizon radar, Radio Science, 1998, Vol. 33, No. 4, P. 1187-1197.
Библиографические ссылки
1. Kravtsov Yu. A., Orlov Yu. I. Geometrical Optics of Inhomogeneous Media. Springer-Verlag Publ., 1990. 312 p.
2. Hunsucker R. D. Radio techniques for probing the terrestrial ionosphere. Berlin ; NewYork : SpringerVerlag, 1991. 293 p.
3. Rawer K. Wave propagation in the ionosphere. Dordrecht ; Boston : Kluwer Academic Publ., 1993. 486 p.
4. Ginzburg V. L. Propagation of Electromagnetic Waves in Plasma. Gordon & Breach Science Publ., 1967. 822 p.
5. Bennett J. A., Chen J., Dyson P. L. Analytic calculation of the ordinary (O) and extraordinary (X) mode nose frequencies on oblique ionograms // Journal of Atmospheric and Terrestrial Physics. 1994. Vol. 56, No. 5. P. 631-636.
6. Determining the fo F2 critical frequency at the path midpoint from oblique sounding data based on the Smith method / G. V. Kotovich [et al.] // Geomagnetism and Aeronomy. 2006. Vol. 46, No. 4. P. 517-521.
7. Chen J., Bennet J. A., Dyson P. L. Synthesis of oblique ionograms from vertical ionograms using quasi-parabolic segment models of the ionosphere // Journal of Atmospheric and Terrestrial Physics. 1992. Vol. 54, No 3/4. P. 323-331.
8. Smith N. The relation of radio sky-wave transmission to ionosphere measurements // Proc. Inst. Radio Engrs. 1939. Vol. 27. P. 332-347.
9. Балаганский Б. А., Сажин В. И. Численное моделирование характеристик декаметровых радиоволн в ионосфере с трехмерно-неоднородными возмущениями // Геомагнетизм и аэрономия. 2003. Т. 43, № 1. P. 92-96.
10. Bristow W. A., Greenwald R. A. Estimating gravity wave parameters from oblique high-frequency back-scatter: Modeling and analysis // Journal of geophysical research. 1995. Vol. 100, No. A3. P. 3639-3648.
11. Mikhailov S. Ya., Grozov V. P. Recovery of the nonmonotonic altitude profile of the plasma frequency based on the ionospheric oblique sounding data // Radiophysics and Quantum Electronics. 2013. Vol. 56, No 7. P. 399-412.
12. Reinisch B. W., Huang X. Automatic Calculation of Electron Density Profiles from Digital Ionograms, 3, Processing of Bottomside Ionograms // Radio Science. 1983. Vol. 18, No 3. P. 477-492.
13. Варшавский И. И. Исследование неоднородной структуры ионосферы при активных воздействиях : дис. ... к-та физ.-мат. наук. Иркутск : СибИЗМИР СО АН СССР, 1986. 154 с.
14. Davies K. Ionospheric Radio Waves. Blaisdell Publ. Co, 1973. 460 p.
15. Coleman C. J. A ray tracing formulation and its application to some problems in over-the-horizon radar // Radio Science. 1998. Vol. 33, No 4. P. 1187-1197.
© Laryunin O. A., 2016
