Please I want some clue about the bibliography (mathematical,of course)of supersymmetry.Anything,from groundlevel to the state-of-the-art.Thanks very much. cututo@data54.com alex
Mircea Eugen SELARIU | (194.102.63.2) | Thursday, 21 November 2002 7:10:05 PM
"POLITEHNICAL" UNIVERSITY OF TIMISOARA
Mechanical Faculty. Manufacturing Engineering Department
THE SUPERMATHEMATICA
A little group of teaching staff from the Politechnical University of Timisoara (in the past called Polytechnic Institute "Traian Vuia") having a tremendous love for mathematics, informatics and for important areas of science, gathered in the "ECCENTRIC CLUB" and wish to offer you their know-how service of implementation of the SUPERMATHEMATICS in your know-ledge and computers.
Supermathematics was born from a millennial and desperate effort of the human kind for building the world such as it is : non-linear and complex instead of linear and simple. It is the materialisation of the mathematicians dream, displayed in 1872 by Felix Klein in "The program from Erlangen" in the need of having a infinity of mathematics and the need of operating with them.
This dream become reality in 1978 through the work “ECCENTRIC CIRCULAR FUNCTIONS”, which showed that to each point on a surface corresponds a mathematics: to the origin O, in other words the center correspond the mathematics we are obliged to name CENTRIC and to the other points E (e,epsilon) corresponds the other mathematics newly named ECCENTRIC.
The eccentric trigonometry uses the point E, named ECCENTER (because it was expelled from the center ) as a pole for a straight line, of direction and whose crossing points M1,2 with a trigonometric circle C(O, 1) -having the center O and a unit radius- generates the new functions : x =cecc t or cex t - eccentric cosines, y = secc t or sex t - eccentric sinus, y/x = tex t -eccentric tangent and so on; emphasis the new elemental functions of greater importance in science and technic rex t - eccentric radius and dex - eccentric derivative, which in centric ( e = 0 ) corresponds to the Euler-Cotes functions rad U = Exp[i.u] respectively der u =i.Exp[i.u] . Therefore, the centric trigonometry results as a particular case of the zero eccentricity ( e = 0 ) , from the eccentric trigonometry.
The discovery of pass over from the centric to eccentric in mathematics is similar to the pass over from geocentric to heliocentric in cosmology; bath areas enjoying the high level from one to infinite !
By replacing in the parametric equations of different known curves-let's name it centric- sin , cos with cex and sex , for each point E, we obtain another form of curve of the same kind. Thus for the same radius there is an infinity of circles for the same values a and b, a infinity of ellipses and as well parabolas, hyperbolas, spirals, and the new curves.
For an eccenter E mobile point, on a plane curve with a certain mowing law, we obtain another new mathematical forms. Some of them resembling to art. This new mathematics is a necessary instrument, waited for a long time the proof being the number and the great diversity of periodical functions necessary in science and technic and the complicated way to get to them. To obtain new periodical functions, Prof. Dr. Math. Valeriu Alaci (of Timisoara) in 1940, replaced the trigonometric circle with a
square and next with the rhomb and so he defined squarable trigonometrical functions and rhombical trigonometrical function . Prof. N. O. Enculescu (of Timisoara), in 1940, replace the circle with a polygon with n sides and so he defined a polygonal trigonometric functions. The Romanian scientist Eng. George (Gogu) Constantinescu (London) replaced the circle with thinvolute and he have defined the rumanian sinus Sir and romanian cosines Cor . Marcusevici replaced the circle with lemniscate, with the ellipse and other curves and he defined lemniscate trigonometrical functions and generalisation trigonometrical functions. In 1877 Dr. Biehringer replaced the right-angled triangle with acute triangle and he defined the
oblique trigonometrical functions. At 3 Nov. 1823 Janos Bolyay, in Timisoara, when he has been discovering the formula a the first non-Euclidean geometry (ctn ), he said "From nothing I have created a new world" . Nevertheless they not finding the simples solution to achieve a new mathematics world. The Supermathematics has achieved this new mathematics world by moving a single point (the pole E) from the center O(0,0) of trigonometrical circle anywhere in the plane of the circle. "We must be grateful to God for this building of the world so that all what is simple (easy) is also true and all what is complicated ( difficult) is false!".
Typical for discoveries up to now, declared Capitza, is that their value is well known after 20 - 30 years; in Rumania this period being much longer. We have waited for 25 years, in the mean time the supermathematics got reached with the ELEVATED functions ( cel , sel , etc. ) and with the EXOTIC functions ( cexo , sexo , etc.) and they have been multiplied like functions of centric independent variable. The supermathematics function, with centric independent variable, have the advantage to be even than the numerical eccentricity is e > 1. All this supermathematics functions survey the spaces of all scientific objects and produce, at the their basement, a tzunami ( solitons ) that brake the boundaries amount them and are defying the differences between non-linear and linear.
We are considering us the cripples that hobble on a good way and we are convinced that we shall overcome the trotters that frisk (skip) on a wrong way. Nevertheless we don't consider it a fair play, to wait for the end of the competition, we are decided to show this secrete way. This is the reason that makes us writing you. We hope that you haven't lost your sense of humour and understand us. So we suggest you to remind the story of Napoleon and Robert Fulton. Being convinced that you have the sense of reality, we looking forward with confidence, for an answer regarding a possible future co-operation.
Eng. Mircea Eugen Selariu
< http://eng.utt.ro/~mselariu > , < E-mail : mselariu@icmct.uvt.ro or mselariu@eng.utt.ro >
President of "Supermathematics-Eccentric Club"
Mircea Eugen SELARIU | (194.102.63.2) | Thursday, 21 November 2002 9:20:41 PM
NEW PLANE CURVES
Note 2 : ELLIPTIC ECCENTRICS WITH MOBILE ECCENTRE
Mircea Eugen SELARIU
0. Abstract
In the paper are presented the parametric equations and the graphics of some new plane curves called elliptic eccentrics ( because the chosen generic curve it is an ellipsis ), for the case in which the eccentre, that defines the eccentric circular functions, it is a mobile point - in the case of the fixed orthogonal projections - and , respectively, a fixed point - in the variable projection case. By combining the cases one can be obtained the elliptic eccentrics, with mobile eccentre and with variable projections.
1. Introduction
In this note it is extended the problem of the elliptic eccentrics, defined in note 1 [1 ] and treated for the case in which the E ( e,EPS) eccentre is a fixed point, to the case when e and / or are variable, i.e. the E (e,EPS) eccentre is a mobile point.
In the group of these new plane curves are included two large sets. The first, is the elliptic eccentric set with orthogonal projections ( fixed : horizontal and vertical ) and their importance is due to their possibility of representing the characteristic ( or the movement trajectory ) in the phase plane for non-lineal free systems, without amortisation with a single degree of freedom.
The second set is that of the elliptic eccentrics with variable projections. In it are included the Jukowski profile and the spirals with more arms, like the galaxy arms, and all these show the importance of these new plane curves.
2. Elliptic Eccentrics With Mobile Eccentre
and Fixed Orthogonal Projections
These are closed plane curves and their parametric equations are obtained from the parametric equations of the ellipsis ( the generic curve ) with the a and b semi-axes :
( 1 ) M
in which the centric circular functions cos and sin are substituted by the eccentric circular functions cext and sext ( eccentric cosines and eccentric sinus of , defined in [ 2 ] and [ 3 ] ) ,
for an Ev (e, ) mobile eccentre on a certain curve, consequently e = e ( ) , = ( ), t being time the time or an other certain parameter.
Thus, the parametric equations of these elliptic eccentrics are:
( 2 ) M
Figure 1. The elliptic eccentric with Figure 2. The Jukowski profile obteined
variable Ey eccentre and as an elliptic eccentric with fixed
Ex fixed in the origin eccenter and with variable projections.
In Figure 1 is presented, for example, an elliptic eccentric at which the Ex (0, 0 ) eccentre is fixed in the origin and the Ey ( ey, y) eccentre is mobile on the x axis according to the law :
( 3 ) Ey
in which the e0 is the oscillation amplitude of the Ey eccentre on the x axis.
The parametric equations of this eccentric are :
( 4 ) M
All the elliptic eccentrics from this set have maximum dimension in the x axis direction of 2.a and in the y axis direction 2.b.
As in the case of the eccentrics with fixed eccentre, if the Ex and Ey eccentres are on the x axis, this axis becomes also axis of symmetry of the eccentric ( see Figure 1 ).
The eccentric graphic results as it follows. From the points Ex and Ey occupy at a certain moment in the plane are drawn two parallel directions to each other, of a variable angle with the x axis, which intersects the circles with the a and b radii in Ma ( xa, ya ) and Mb ( xb, yb ). The M (x, y ) point on the vertical lowered from Ma with the horizontal one drawn from Mb, as show the arrows in Figure 1.
If from the Ma and Mb points are drawn two variable directions according to a certain law, one can obtained the eccentrics with variable projections.
3. Elliptic Eccentrics with Fixed Eccentre
and Variable Projections
Such an eccentric it is the symmetric or asymmetric Jukowski profile, obtained as eccentric, shown in Figure 2
In this case also has been applied the method of the Ex and Ey eccentre overlap in the origin of the (x,y) co-ordinate axes so that the two circles Ca with the a radius and Cb with the b radius have the centres in the distinct Oa and Ob points, a method described in [ 1 ].
The graphic construction of this eccentric is not at all different from the graphic construction of the Jukowski profile which it represents . From the common eccentre E Ea Eb O are drawn two symmetrical radial directions vs. the x axis : one of the direction and the other of the direction. The intersection of these directions with the Ca and Cb circles are the Ma points, respectively, Mb of polar co-ordinates :
( 5 ) M a
( 6 ) M b
in which the rex function is the eccentric circular function radial, eccentric of , numerically equal with the distance ratio from the eccentre and from the circle's center to the point M 1,2 from the circle, for a certain direction, i.e.
( 7 ) rex1,2 ( t ,E ) = -e. cos (t - eps)+Sqrt[1-(e.sin t)^2]
In the ( 5 ) and ( 6 ) relations have been considered the main determinations ( 1 ) of the radical sign is plus, in this case there is no more necessary to specify the index, considering the trigonometric circle intersection ( R = 1 ) with positive half-line drawn from E.
The and angles, which determine the relative positions of the Oa and Ob centres of the two circles, are those from the literature : the Oa and Ob points are disposed on two symmetric directions vs. the y axis, the directions passing throughout the O origin, being the same line that passes also through the point of tangency of the two Ca and Cb circles.
The M point on the eccentric-the Jukowski profile is determined at the intersection of the straight lines drawn from Ma with the direction b = - and from Mb of the size a = .
With ( 7 ) one can determine the vector radii of the Ma and M b points on the two circles Ca and Cb respectively vs. the pole E O. These are :
( 8 )
and they represent the absolute value of the vectors OMa and OMb respective.
The sum of these vectors is the OM vector with a module, i.e.
( 9 ) the angle between and being -2. .
Considering the circle with variable radius r = , with the center in O ,and Mb being the variable eccentre on the Cb circle, the M point co-ordinates are the eccentric cosines and eccentre sinus of the angle with the mentioned variable parameters i.e.:
( 10 )M
The relation ( 10 ) can be written more simply by using the eccentric circular function expressions cex . and sex . ( see [2] and [ 3 ] )
The same result is got if Ma would be considered as mobile eccentre on the Ca circle, in which case e = and , the circle radius also being variable ( r = ) and the positive half-line direction is - .
An elliptic eccentric with variable projections, having the form of a spiral with four arms, like the galaxy, is shown in Figure 3.
Figure 3. Elliptic eccentric with variable projections and with variable eccentre,
the form being a four arm spiral
The big circle ( Ca ) with the radius a has the eccentricity ea = 0, i.e. Ea , and the Cb circle with smaller radius ( b ) has the eccentre on the y axis. At the half-lines intersection drawn radial from O Ea Eb on the direction vs. the x axis with the C a and Cb circles are obtained the Ma and , respectively, Mb points. From here are drawn variable projection to determine, at their intersections, the M point of the eccentric. These oblique projections on the variable directions, from Ma, and b from Mb, have been chosen as it follows:
( 11 )
The first arm, in dial1, starts from the point 0( which is not the co-ordinate system origin ) and tends asymptotically to a parallel with the +y axis. The point for is at y ( . Each of the four arms exists only in one single dial, tending each asymptotically to infinite. None of the four arms is identical with the other and the new plane curves has no points in the O origin and close vicinity of it.
4. Elliptic Eccentrics with Variable Eccentre
and with Variable Oblique Projections
They can be obtained by combining the two circular eccentric types presented before.
5. Bibliography
[1] Selariu Mircea -Eugen, NEW PLANE CURVES. Note 1: Definition of the Elliptic
Eccentric with Fixed Eccentre , The V-th National
Conference on Mechanical Vibrations in the Manufacturing
Engineering, Timisoara / Romania, 1985, p.175...182
[2] Selariu Mircea -Eugen, CIRCULAR ECCENTRIC FUNCTIONS, The II-th National
Conference on Mechanical Vibrations in the Manufacturing
Engineering, Timisoara / Romania, 1978,Vol.I, p.101...108
[ 3] Selariu Mircea -Eugen, CIRCULAR ECCENTRIC FUNCTIONS and THIS
EXTENSIONS, Bul.St.Tech. IP"TV"Timisoara, Seria Mecanica,
Tom 25 (39), Fasc.1-1980, p.189 ...196.
Mircea Eugen SELARIU | (194.102.63.2) | Thursday, 21 November 2002 9:28:04 PM
THE DEFINITION OF THE ELLIPTIC ECCENTRIC
WITH FIXED ECCENTER
Mircea Eugen SELARIU
0. Abstract
In the paper are defined the elliptic eccentrics. They are included in the new plane curve set, called eccentric vs. the known plane curves called centric, which are used as generic curves ( of genesis ) of the new ones. The elliptic eccentric has as generic curve an ellipsis, and from which it is obtained by substituting the centric circular functions with the eccentric ones in their parametric equations.
Here are presented the perametric equations of the elliptic eccentric, using a fixed eccenter plan and the graphics of these new plane curves.
1. Introduction
The eccentrics are plane curves and their introduction in mathematics was possible after the introduction of the eccentric circular functions defined in the paper [ 1 ] and [ 2 ]. Among the eccentric circular functions important, for the definition of the eccentrics, are the eccentric cosinus ( cex t) and eccentric sinus ( sex t) functions, wich have a lot of mathematical applications and technic applications [ 3 ].
Let it be the C trigonometric cercle, with the O center in the origin of a straight rectangular axes system ( x, y ) and a point E(e, eps) in the cercle plane, with polare coordinates. The half-straight line with the origin in O and with direction with the x axis intersects C in P ( x P, y P ) and a semi-straight line parallel with the first but with the origin in the point E, called eccenter, intersects C in the M point ( x, y ) . If the P point coordinates are centric circular functions xP = cos t and y P = sin t, the point M coordinates are eccentric circular functions.
( 1 ) M (x, y )
Because M belongs to C there exists a half-straight line, drawn from O which intersects C in M , and which direction is written . So the coordinates of M can be simultaneously expressed with the centric circular functions as functions of and with those eccentric ones as functions of . The difference between the angles is called phase shifting angle .
For e > 1, E is exterior to C, the half-straight line from E intersects C in two points M1 and M 2 ( or M 1,2 ) and so the eccentric circular functions will have two determinations : a main one, denoted with index 1, for which the M 1 point rotates on the C cercle in the increasing sense of , i.e. in the trigonometric sens and a second determination, denoted withh the index 2, for which M 2 rotates on C in inverted sense of the trigonometric sense.
For e < 1, E is interior to C and the two determinations can appear by the intersection of C with the half-straight lines : the positive half-straight line for the main determination 1 and the negative one for the secondary 2 one ( the minus sign in front of the radical 2 ).
Thus, the eccentric cosinus and sinus function expression with E eccenter (e,eps ) are :
( 2 ) M 1,2
2. The Eccentric Definition
The eccentric is a plane curve which parametric equqtions are obtained by replacing the centric circular functions ( sin , cos ) , with eccentric ones which have the same variable ( sex , cex ), in the parametric equations of curves as the cercle, the ellipssis, the hyperbola, the parabola, the spyral etc.
Because, for e = 0 , the eccentric cex and sexv functions degenerate in centric circular cos and , respectivly, sin functions, it results that the cercle, the ellipsis, the hyperbola etc are particular cases of the parabolic, hyprbolic, elliptic circular excentrics, obtained for e = 0 ( when the E ecceter returns in the cercle center and in the origin of the O coordinate axes ).
If for example, for a value of the r radius exists a single cercle with the center in the O origin, for the same there exist an infinity of circular eccentrics corresponding to the infinity of positions the E eccentrer can occupy in the cercle plane.
One can distinguish circular eccentrics with fixed eccenter and circular eccentrics with mobile eccentrer, the eccenter moving on any known curve, according to a given law, in the first case e and are constant, and in the second case [ 4 ] , [ 5 ] they are variable function of .
To each eccentric correspond two eccentrers Ex ( ex , ) and Ey ( e y, ) corresponding to the two coordinates x and y of a M (x, y) point on the eccentric. If the numeric eccentricities are equal between them (e x=e y ) and the eccenters Ex and E y are situated on the same straight line, the eccentric degenerates again in the generic curve from which it derived.
3. Elliptic Eccentrics
The elliptic eccentric provides from the ellipsis and has following parametric equations :
( 3 ) M
Here, 2a is the major axis of the ellipsis and 2b is the minor axis, E x and E y are the two eccenters corresponding to the x abscissa, respectively, to the y ordinate of the eccentric.
The graphic construction of the elliptic eccentric is very similar to that of the ellipsis : from the two eccenters E x and E y are drawn two half-straight lines with the variable direction and are determined the two intersection points of them with the two cercles with the radii a and b ( M a ( x a , y a ) and , respectively M b ( x b, y b ) ). The coordinats of two points are :
( 4 ) M a
( 5 ) M b
Because the M point coordinate ( x, y ) are those given by the equations
( 3 ) in comparison with ( 4 ) and ( 5 ) it results that :
( 6 ) M
and consequently the point M on the eccentric is obtained at the vertical intersection drawn from the points M a with the horizontals drawn from the points M b ( Figure 1 ) .
As it was shown for ex = e ax / a and e bx / b equal and is obtained again an ellipsis ( Figure 2 ) . It is seen that it this case, the M a and M b points are disposed on the same radial direction, i. e. .
If the two eccenters are disposed on the x axis then the eccentric has the x as a symetry axis. If these eccentric points E x and E y are disposed an the y axis, than the elliptic eccentric admits y as symetry axis.
The eccentrics have no symmetry center, only when they are degenerated in the curves from which they came from and if they at their turn, have such a symmetry center.
If the two points E x and E y are identical ( Figure 3 ) the numeric eccentricities e x and e y are of different values, while the e ax and e by eccentricities are equal.
Because the y a and x b coordinates do not determine M ( it being determined by x = x a and y = y b ) and coinsequently do not influence the elliptic eccentric form, the C a cercle with the origin in O a= 0 and with the eccenter E x E a may suffer an undefinite translation in the y axis direction, so that x a should not modify and consequently it should not modify the eccentric form. Similarly the C b cercle with the center in O b = 0 and with the eccenter E b , may be translated along the x axis without that y b should modify the elliptic eccentric form.
` The two C a and C b cercles can be displaced, each along the admitted direction ( which does not modify the eccentric form ) , until the two E x and E y eccenters are overlapped in a E eccenter which is common to the two displaced cercles ( Figure 4 ) . Thus, the two M a and M b points, necessary to the eccentric construction., can be obtained drawing one single radial directions whith angle from E ( in the place of the two parallel radial directions drawn from E x and , respectively, E y ) to determine the same points M a and M b ,necessary to the graphic determination of the same eccentrics. Briefly the situation of two concentric cercles, i.e. with common center, can be transformed in the situation of two eccentric cercles with common eccenter , or the eccentric eccenters can be overlapped by the corresponding displacement of the cercles.
The elliptic eccenter forms have shapes. Some of them have symmetric aerodynamic profiles ( Figure 5 ) and others have asymmetric aerodynamic profiles ( Figure 3, 6, 7 and 8 ) . These eccentrics have been obtained graphically by the common eccenter method.
In this set are also included the eccentric Lissajous figures ( curves ) [ 6 ] .
4. Bibliography
1 Selariu M.E. FUNC[II CIRCULARE EXCENTRICE
( Eccentric Circular Functions ) A II-a Conf. VCM ,Timi}oara, 1978, Vol.I, p. 101...108
2 Selariu M.E. FUNC[II CIRCULARE EXCENTRICE {I EXTENSIA LOR Bul. St.&Tehn. IP"TV"T, Tom. 25 ( 39 ), Fasc.I-1980,p. 189 ... 196
3 Selariu M.E. APLICATII TEHNICE ale FUNCTIILOR CIRCULARE EXCENTRICE.Nota I rex si dex Com. IV Conf. PUPR , Timi}oara, 1981, p. 142... 150
4 Selariu M.E. CURBE PLANE NOI. Nota II :
ELLIPTIC ECCENTRICS with MOBILE ECCENTRICS Luc. V-a Conf. Nat. VCM , Timisoara, 1985, p.183 ... 188
5 Selariu M.E. CURBE PLANE NOI. Nota III :
CIRCULAR ECCENTRICS and HYPERBOLIC ECCENTRICS Luc. V-a Conf. Nat. VCM , Timisoara, 1985, p.189 ... 194
6 Selariu M.E. CURBE PLANE NOI . Nota IV :
ECCENTRIC LISSAJOUS FIGURES Luc. V-a Conf. Nat. VCM , Timi}oara, 1985, p.195 ... 202
Mircea Eugen SELARIU | (194.102.63.2) | Thursday, 21 November 2002 9:31:34 PM
BIBLIOGRAFIE
IN DOMENIUL S U P E R M A T E M A T I C I I
1 Selariu Mircea FUNCTII CIRCULARE EXCENTRICE Com. I Conferinta Nationala de Vibratii in Constructia de Masini , Timisoara , 1978, pag.101...108.
2 Selariu Mircea FUNCTII CIRCULARE EXCENTRICE si EXTENSIA LOR. Bul .St.si Tehn. al I.P. ”TV” Timisoara, Seria Mecanica, Tomul 25(39), Fasc. 1-1980, pag. 189...196
3 Selariu Mircea STUDIUL VIBRATIILOR LIBERE ale UNUI SISTEM NELINIAR, CONSERVATIV cu AJUTORUL FUNCTIILOR CIRCULARE EXCENTRICE Com. I Conf. Nat. Vibr.in C.M. Timisoara,1978, pag. 95...100
4 Selariu Mircea APLICATII TEHNICE ale FUNCTIILOR CIRCULARE EXCENTRICE Com.a IV-a Conf. PUPR, Timisoara, 1981, Vol.1. pag. 142...150
5 Selariu Mircea THE DEFINITION of the ELLIPTIC ECCENTRIC with FIXED ECCENTER A V-a Conf. Nat. de Vibr. in Constr. de Masini,Timisoara, 1985, pag. 175...182
6 Selariu Mircea ELLIPTIC ECCENTRICS with MOBILE ECCENTER IDEM pag. 183...188
7 Selariu Mircea CIRCULAR ECCENTRICS and HYPERBOLICS ECCENTRICS Com. a V-a Conf. Nat. V. C. M. Timisoara, 1985, pag. 189...194.
8 Selariu Mircea ECCENTRIC LISSAJOUS FIGURES IDEM, pag. 195...202
9 Selariu Mircea FUNCTIILE SUPERMATEMATICE CEX si SEX- SOLUTIILE UNOR SISTEME MECANICE NELINIARE Com. a VII-a Conf.Nat. V.C.M., Timisoara,1993, pag. 275...284.
10 Selariu Mircea SUPERMATEMATICA Com.VII Conf. Internat. de Ing. Manag. si Tehn.,TEHNO’95 Timisoara, 1995, Vol. 9: Matematica Aplicata,. pag.41...64
11 Selariu Mircea FORMA TRIGONOMETRICA a SUMEI si a DIFERENTEI NUMERELOR COMPLEXE Com.VII Conf. Internat. de Ing. Manag. si Tehn., TEHNO’95 Timisoara, 1995, Vol. 9: Matematica Aplicata,.,pag. 65...72
12 Selariu Mircea MISCAREA CIRCULARA EXCENTRICA Com.VII Conf. Internat. de Ing. Manag. si Tehn. TEHNO’95., Timisoara, 1995 Vol.7: Mecatronica, Dispozitive si Rob.Ind.,pag. 85...102
13 Selariu Mircea RIGIDITATEA DINAMICA EXPRIMATA
CU FUNCTII SUPERMATEMATICE Com.VII Conf. Internat. de Ing. Manag. si Tehn., TEHNO’95 Timisoara, 1995 Vol.7: Mecatronica, Dispoz. si Rob.Ind.,pag. 185...194
14 Selariu Mircea DETERMINAREA ORICAT DE EXACTA A RELATIRI DE CALCUL A INTEGRALEI ELIPTICE COMPLETE DE SPETA INTAIA Bul. VIII-a Conf. de Vibr. Mec., Timisoara,1996, Vol III,
pag.15 ... 24.
15 Petrisor Emilia ON THE DYNAMICS OF THE DEFORMED STANDARD MAP Workshop Dynamicas Days'94, Budapest, si Analele Univ.din Timisoara, Vol.XXXIII, Fasc.1-1995, Seria Mat.-Inf.,pag. 91...105
16 Petrisor Emilia SISTEME DINAMICE HAOTICE Seria Monografii matematice, Tipografia Univ. de Vest din Timisoara, 1992
17 Cioara Romeo FORME CLASICE PENTRU FUNCTII CIRCULARE EXCENTRICE Proceedings of the Scientific Communications Meetings of "Aurel Vlaicu" University, Third Edition, Arad, 1996, pg.61 ...65
18 Preda Horea REPREZENTAREA ASISTATA A TRAIECTORILOR IN PLANUL FAZELOR A VIBRATIILOR NELINIARE Com. VI-a Conf.Nat.Vibr. in C.M. Timisoara, 1993, pag.
19 Selariu Mircea
Ajiduah Cristoph
Bozantan Emil (USA)
Filipescu Avram INTEGRALELE UNOR FUNCTII SUPERMATEMATICE Com. VII Conf.Intern.de Ing.Manag.si Tehn. TEHNO’95 Timisoara. 1995,Vol.IX: Matem.Aplic. pag.73...82
20 Selariu Mircea
Fritz Georg (G)
Meszaros A.(G) ANALIZA CALITATII MISCARILOR PROGRAMATE cu FUNCTII SUPERMATEMATICE IDEM, Vol.7: Mecatronica, Dispozitive si Rob.Ind.,
pag. 163...184
21 Selariu Mircea
Szekely Barna
( Ungaria ) ALTALANOS SIKMECHANIZMUSOK FORDULATSZAMAINAK ATVITELI FUGGVENYEI MAGASFOKU MATEMATIKAVAL Bul.St al Lucr. Prem.,Universitatea din Budapesta, nov. 1992
22 Selariu Mircea
Popovici Maria A FELSOFOKU MATEMATIKA ALKALMAZASAI Bul.St al Lucr. Prem., Universitatea din Budapesta, nov. 1994
23 Konig Mariana
Selariu Mircea PROGRAMAREA MISCARII DE CONTURARE A ROBOTILOR INDUSTRIALI cu AJUTORUL FUNCTIILOR TRIGONOMETRICE CIRCULARE EXCENTRICE MEROTEHNICA, Al V-lea Simp. Nat.de Rob.Ind.cu Part .Internat. Bucuresti, 1985
pag.419...425
24 Konig Mariana
Selariu Mircea PROGRAMAREA MISCARII de CONTURARE ale R I cu AJUTORUL FUNCTIILOR TRIGONOMETRICE CIRCULARE EXCENTRICE, Merotehnica, V-lea Simp. Nat.de RI cu participare internationala, Buc.,1985, pag. 419 ... 425.
25 Konig Mariana
Selariu Mircea THE STUDY OF THE UNIVERSAL PLUNGER IN CONSOLE USING THE ECCENTRIC CIRCULAR FUNCTIONS Com. V-a Conf. PUPR, Timisoara, 1986, pag.37...42
26 Staicu Florentiu
Selariu Mircea CICLOIDELE EXPRIMATE CU AJUTORUL FUNCTIEI SUPERMATEMATICE REX Com. VII Conf. Internationala de Ing.Manag. si Tehn ,Timisoara “TEHNO’95”pag.195-204
27 Gheorghiu Em. Octav
Selariu Mircea
Bozantan Emil FUNCTII CIRCULARE EXCENTRICE DE SUMA DE ARCE Ses.de com.st.stud.,Sectia Matematica,Timisoara, Premiul II pe 1983
28 Gheorghiu Emilian Octav
Selariu Mircea
Cojerean Ovidiu FUNCTII CIRCULARE EXCENTRICE. DEFINItII, PROPRIETATI, APLICATII TEHNICE. Ses. de com.st.stud. Sectia Matematica, premiul II pe 1985.
29 Filipescu Avram APLICAREA FUNCTIILOR ( ExPH ) EXCENTRICE PSEUDOHIPERBOLICE IN TEHNICA Com.VII-a Conf. Internat.de Ing. Manag. si Tehn. TEHNO'95, Timisoara, Vol. 9. Matematica aplicata., pag. 181 ... 185
30 Dragomir Lucian
(Toronto
- Canada ) UTILIZAREA FUNCTIILOR SUPERMATEMATICE IN CAD / CAM : SM-CAD / CAM. Nota I-a: REPREZENTARE IN 2D Com.VII-a Conf. Internat.de Ing. Manag. si Tehn. TEHNO'95, Timisoara, Vol. 9. Matematica aplicata., pag. 83 ... 90
31 Selariu Serban UTILIZAREA FUNCTIILOR SUPERMATEMATICE IN CAD / CAM : SM-CAD / CAM. Nota I I -a: REPREZENTARE IN 3D Com.VII-a Conf. Internat.de Ing. Manag. si Tehn. TEHNO'95, Timisoara, Vol. 9. Matematica aplicata., pag. 91 ... 96
32 Staicu Florentiu DISPOZITIVE UNIVERSALE de PRELUCRARE a SUPRAFETELOR COMPLEXE de TIPUL EXCENTRICELOR ELIPTICE Com. Ses. anuale de com.st. Oradea ,1994
33 George LeMac The eccentric trigonometric functions:
an extention of classical trigonometric functions.
The University of Western Ontario, London, Ontario, Canada
Depertment of Applied Mathematics
May 18, 2001
34 Selariu Mircea PROIECTAREA DISPOZITIVELOR DE PRELUCRARE, Cap. 17 din PROIECTAREA DISPOZITIVELOR Editura Didactica si Pedagogica, Bucuresti, 1982, pag. 474 ... 543
Mircea Eugen SELARIU | (194.102.63.2) | Thursday, 21 November 2002 10:36:18 PM
DETERMINAREA ORICAT DE EXACTA A RELATIEI DE CALCUL
A INTEGRALEI ELIPTICE COMPLETE DE SPETA INTAIA
M i r c e a S e l a r i u , Universitatea “POLITEHNICA” din Timisoara,
Bd. Mihai Viteazul , 1 C.P. 625, 1900- Timisoara/ Romania
0. Abstract.
THE CALCULUS RELATION DETERMINATION, WITH WHATEVER PRECISION,
OF COMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND.
These papers show a calculus relation ( 50 ) of complete elliptic integral K(k) with minimum 9 precise decimals and the possibility to obtain a more precisely relation.. It results by application Landen’s method, of geometrical-arithmetical average, not for obtain a numerical value but to obtain a compute algebraically relation after 5 steps of a geometrical transformation, called “CENTERED PROCESS”.
0. Rezumat
Frecventa este marimea fizica care ,astazi, se poate masura cu cea mai mare precizie. De aceea , definitia uni-tatii de lungime ( metrul etalon de la Sèvres-Paris ) a fost inlocuita, in 1983, cu multiplii lungimii de unda a unei oscilatii ( radiatia kriptonului 86 ), iar unitatea de timp a fost redefinita prin multiplii de perioade ale unei anumite radiatii. Calculul frecventelor diverselor sisteme tehnice , in special neliniare, nu s-a ridicat ,insa, la acelasi nivel dorit de precizie .
Integrala eliptica completa de speta intaia K(k) poate oferi solutia determinarii cu precizie a frecventelor unor sisteme neliniare, dar seria de puteri ( 6 ), prin care ea se exprima, este slab convergenta. De aceea, au aparut metode numerice, ca metoda Landen sau a mediei aritmetico-geometrice, care ofera cu precizie valoarea (numerica ) a lui K pentru un modul k dat, valori prezentate tabelar, cu diverse zecimale exacte- 9 in Abramovitz [ 20 ] s.a.[19].
Ideea autorului a fost de a obtine nu valoarea numerica a lui K(k), ci o expresie algebrica ( relatie de calcul ) din care sa rezulte cu o precizie impusa ( oricat de ridicata se doreste ) valoarea integralei pentru oricare valoare k, si nu numai pentru cele existente in tabele, evitandu-se , astfel, interpolarile uneori necesare. Pentru precizii nelimitate, aceasta relatie de calcul este K(k) = /2R( k ) si, pentru minimum 9 zecimale exacte, s-a constatat ca functia RN(k) necesita doar 5 pasi , astfel ca R5( k ) este patratul perfect
R5 ( k ) = =
cu notatiile G = = = = si A = = =
Algoritmul de calcul prezentat in lucrare, ce constituie, totodata, si o transformare geometrica noua, denumita “centrare” - pentru ca la N cercul trigonometric excentric, cu excentricitatea numerica k , se transforma in cercul cu excentricitate numerica nula (kN = 0), deci centric- stabileste transformarile din pas in pas , peste doi, trei sau patru pasi si poate stabili,in continuare, si peste mai multi pasi. De exemplu, relatia anterioara R5 da dependenta dintre marimile din pasul 5 cu cele obtinute dupa pasul intai, adica peste patru pasi. Se poate obtine R9 cu o relatie asemanatoare in care si , dar preciziile astfel obtinute ar depasi cu mult cerintele practice.