trusciencetrutechnology@blogspot.com Volume 2019, Issue No.B, 19 January 2019, Time:05:54:41 AM
14 Dr.Biresh Guha St. Kolkata-700017
The Indian Science Congress Association 106th
Mathematics & Statistics Session held in January 3-7, 2019.
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CYCLIC AND V4 KLEIN GROUP ANALYSIS TO DESCRIBE VARIED MASS TERMS
Professor Dr. Kotcherlakota . L. N.
trusciencetrutechnology@blogspot.com
17-11-10, Narasimha Ashram, Maharanipeta.P.O. Visakhapatnam-530002.AP.
Mobile No. 09491902867
Paper presented on 5 January 2019 at LPU, Mathematics session, ISC106.
ABSTRACT
The fourth order groups, the Cyclic, and the V4 Klein groups have been analyzed to obtain the mass-dependent Eigenvalues. The Cyclic group reveals both the real and imaginary Eigenvalues. The V4 Klein group, on the other hand, presents only real Eigenvalues. Both the positive and anti-positive mass terms are used adopting that the antiparticles have negative masses, like the electron-positron formulation. The different sets of mass values, adopting that the mass m2=-m1 positive masses and m4=-m3 for the anti-positive masses are adopted. The mass values m1 and m3 were varied from 0.1, 0.3, 0.4, 0.5, 0.6, 0.7, 0.9 and the respective Eigenvalues obtained.
INTRODUCTION
Two different sets of Klein and the ordinary 4th order-matrices were used to generate the distinct groups. The ordinary 4th order-matrices gave distinct roots suggesting the complex and real roots of considerable complexity. All the roots possible are enlisted and complex terms indicated. Unlike this, the Klein set gave plain straight forward roots and several sets of roots are listed. They are found to be real and no complex behavior evident. The calculations have thus brought out the distinctive features of the two different sets of Klein and ordinary 4th order-matrices which were presented in this paper.
The fourth order groups, the Cyclic, and the V4 Klein groups have been analyzed to obtain the mass-dependent Eigenvalues. The Cyclic group reveals both the real and imaginary Eigenvalues. The V4 Klein group, on the other hand, presents only real Eigenvalues. Both the positive and anti-positive mass terms are used adopting that the antiparticles have negative masses, like the electron-positron formulation. The different sets of mass values, adopting that the mass m2=-m1 positive masses and m4=-m3 for the anti-positive masses are adopted. The mass values m1 and m3 were varied from 0.1, 0.2, 0.4, 0.5, 0.6, 0.7, 0.9 and the respective Eigenvalues obtained.
Cyclic group
m1:=0.5;m3:=0.5;l4+6.3222534.l3-114.9917341.l2-160.437934.l+2862.351717;
x1:=3.8888;x2:=-2.3308;x3:=-2.779+i*0.295864;x4:= -2.779-i*0.295864;
m1:=0.4; m3:=0.6;l4+6.207556.l3-115.644.l2-151.1417089.l+2901.352736;
x1:=63.49464;x2:=52.294641; x3:=-60.996419+i*5.295;x4:= -60.996419-i*5.295;
m1:=0.2; m3:=0.8; l4+5.823121.l3-120.5275984.l2-123.7828232.l+3186447232;
x1:=4.01739;x2:=-2.08390;x3:=-2.965+i*0.7;x4:= -2.965-i*0.7;
m1:=0.8;m3:=0.2;l4+6.578668.l3-113.7629972.l2-183.2204372.l+2774.711933;
x1:=63.35153295;x2:=51.27793295; x3:=-60.60406705+i*4.2396;x4:= -60.60406705-i*4.2396;
m1:=0.6;m3:=0.4; l4+6.417112198.l3-114.4915552.l2-168.5062189.l+2830.142332;
x1:=3.8989;x2:=-4.3803; x3:=-1.7593+i*0.1942;x4:= -1.7593-i*0.1942;
m1:=0.7;m3:=0.3; l4-4.l3-3.85778.l2+1.271112.l-0.1060436733;
x1:=3.92885;x2:=-4.44195;x3:=-1.74343+i*0.148628;x4:= -1.74343-i*0.148628;
V4 or Klein Group
Klein set of Matrices: The Vierergruppe is the Abelian abstract group on four elements that is isomorphic to the finite group c2xc2 and the dihedral group D2. The masses m1, m2, m3, m4 are set, such that m2=-m1 and m4=-m3. Later m1=m3 was used to simplify the matrix for diagonalization.
A:=<<e-m1-l,a,b,c|a,e-m2-l,c,b|b,c,e-m3-l|c,b,a,e-m4-l>>;
Also a=e=1 was chosen setting b=1/2, c=1/3.
The roots set as x1, x2, x3 and x4 for each set of m1 and m3 values.
m1:=0.5; m3:=0.5; l4-4.l3-3.77778.l2+1.1112.l-0.09875;
y3-4.88889.y2+7.97223.y-4.34028;
x1:=3.88884;x2:=-2.33076;x3:=-2.77904+i*0.29586;x4:= -2.77904-i*0.29586;
m1:=0.2; m3:=0.8;l4-4.l3-3.95778.l2+1.471112.l-0.11065
y3-4.97889.y2+8.332223.y-4.72338;
x1:=3.98289; x2:=-2.129898; x3:=-2.926496+I*0.7; x4:= -2.926496-I*0.7;
m1:=0.6;m3:=0.4; l4-4.l3-3.79778.l2+1.151112l-1.0087707;
y3-4.89889.y2+8.012223.y-4.382047;
x1:=3.8988; x2:=-3.37576; x3:=-2.26145+I*0.81478; x4:= -2.26145-I*0.81478;
m1:=0.7;m3:=0.3; l 4-4. l 3-3.85778. l 2+1.271112.l-0.1064367;
y3-4.92889.y2+8.132223.y-4.508547;
x1:=3.928891; x2:=-2.211821; x3:=-2.858535+I*0.5257028; x4:= -2.858535-I*0.5257028;
m1:=0.1;m3:=0.9;l4-4.l3-4.09778.l2+1.751112.l-0.10871;
y3-5.04889.y2+8.612223.y-5.03255;
x1:=4.0488902; x2:=-2.0316578; x3:=-3.0086162+I*0.882414; x4:= -3.0086162-I*0.882414;
REFERENCES
2. Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990.
CITE THIS AS Weisstein, Eric W. "Cyclic Group C_4." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CyclicGroupC4.html
Acknowledgment:
The author is greatly indebted to Late Prof K. R. Rao D.Sc.(Madras), D.Sc.(London) for his guidance and encouragement, to publish newer thoughts in Science & Technology. I was with him at Andhra University, from 1959-1972, and learned from him, many research adventures. I am also grateful to Late Mrs. Peramma Rangadhama Rao for her support and interest in my works. Submitted on 23 July 2018.
The author is greatly indebted to Late Prof K. R. Rao D.Sc.(Madras), D.Sc.(London) for his guidance and encouragement, to publish newer thoughts in Science & Technology. I was with him at Andhra University, from 1959-1972, and learned from him, many research adventures. I am also grateful to Late Mrs. Peramma Rangadhama Rao for her support and interest in my works. Submitted on 23 July 2018.
Paper Presentation Certificate at LPU, Jalandhar
on 5th Jan. 2019 Cyclic And V4 Klein...Varied mass terms.
Abstract of my Paper in Maths Proc. 2019
Kotcherlakota L N and Page.86
Abstract details noted by Chairman at LPU Maths session
L. N. Kotcherlakota
Prof. Nandadulal Bairagi
Professor and Coordinator,
Centre for Mathematical Biology and Ecology,
Department of Mathematics, Jadavpur University,
Kolkata-700 032; Mobile : 09433110781 / 08240364057;
E-mail : nbairagi@yahoo.co.in / nbairagi@math.jdvu.ac.in
Professor and Coordinator,
Centre for Mathematical Biology and Ecology,
Department of Mathematics, Jadavpur University,
Kolkata-700 032; Mobile : 09433110781 / 08240364057;
E-mail : nbairagi@yahoo.co.in / nbairagi@math.jdvu.ac.in
He was very cordial with me and showed me the inclusion of my paper in the proceedings, but since haven't received the acceptance or inclusion of my paper in the ABSTRACTS. I presented it by the postal display.
Fig.1a
Fig.1b
Fig.1c
Fig.1d
Fig.2a
Fig.2b
Fig.3 Office room at LPU 41-1011 my accommodation
The staff at the Office were very cordial and both the ladies looked after me well. on 10 floors and staff at downstairs.
Fig.4a In the autorickshaw battery driven at
New Delhi seated S V Chalam on 9 Jan 2016.
S V Chalam photographs taken in Visakhapatnam.
Fig.4b S V Chalam at Narasimha Ashram Puja Room
Fig.4c S V Chalam at Narasimha Ashram
in my office room.
in my office room.
Fig.4d S V Chalam at Narasimha Ashram
I was received by S V Chalam at New Delhi Airport on 1 January 2019, and by his son's car took me to his house in Dwaraka Colony, where I met his wife and the house I found quite cozy. After the lunch, around 2:30PM he took me to the railway station, by a taxi, who deceived us and hardly 15 minutes time left to reach the other side of the railway station. I have to run on the long upstairs bridge to reach the train destination platform. I am amazed at the AMRITSAR SHATABDI. I left at the scheduled time and they were serving profusely on the train.
On return 8 Jan. 2019, At New Delhi station I was received by S V Chalam, around 11:30PM and took me to his house by around 12:15AM or so. Next morning I had IDLI breakfast, around 9AM. Lunch I had around 2PM typical Visakhapatnam preparations. His wife hails from near Bhimadole village has mastered the preparations and she is a typical housewife.
With S V Chalam on 9 Jan. 2019 I gave all the material of ISC106, to him. He said he would intimate his brother in the US, Dr. Pavanram who specifically asked for paper presentations made by me, both in Maths and Physics sessions. Around 2PM we had our lunch and left around 3PM to take the Metro train. He left me at the New Delhi station, after purchasing his platform ticket (that took nearly a half hour) and when I boarded the train GT express to Vijayawada, I told him, to go home as it was already 6PM.
With S V Chalam on 9 Jan. 2019 I gave all the material of ISC106, to him. He said he would intimate his brother in the US, Dr. Pavanram who specifically asked for paper presentations made by me, both in Maths and Physics sessions. Around 2PM we had our lunch and left around 3PM to take the Metro train. He left me at the New Delhi station, after purchasing his platform ticket (that took nearly a half hour) and when I boarded the train GT express to Vijayawada, I told him, to go home as it was already 6PM.
Fig.5a Dr. S. Juneja's son in the car to drop at
ISC106 Venue on 2 Jan 2019
ISC106 Venue on 2 Jan 2019
Fig.5b Mrs. Juneja at their hospital in Jalandhar,
on 2 Jan. 2019.
on 2 Jan. 2019.
Fig.5c With Mrs. Dr. Juneja opposite
their fabulous Hospital in Jalandhar on 2 Jan 2019.
their fabulous Hospital in Jalandhar on 2 Jan 2019.
Received around 9PM by Dr. S. Juneja and his son a Medical student at Amritsar, and took me to their house, in their car, three-storied amazing hospital premises, and I slept immediately, couldn't talk much. Morning, I was entertained with Wheat Upma, specially prepared by her loving wife, also a Doctor, and their son took me by car to the venue of the ISC106, by a roundabout route, in view of the police protection at the main gate. He didn't have his breakfast, and his mother called him to return home.
Fig.6 ISC106 office room with staff
and my room 41 1011 on the 10th floor.
and my room 41 1011 on the 10th floor.
I have registered at the ISC106 Office and they gave me a splendid room on the 10th floor of a building with a magnificent view of the venue of ISC106. Office staff were quite cordial and very accommodative to see my stay at the 10 floors was quite comfortable.
One office boy gave me the room heater that kept me alert and comfortable and helped me to fetch an Ola taxi to go to the Jalandhar city station at a cost of Rs.350/- at 3pm on the 8 January 2019.
One office boy gave me the room heater that kept me alert and comfortable and helped me to fetch an Ola taxi to go to the Jalandhar city station at a cost of Rs.350/- at 3pm on the 8 January 2019.
Acknowledgment
The author Professor Dr. K. Lakshmi Narayana is indebted to Late Prof. K. R. Rao D.Sc.(Madras) D.Sc.(London) for his guidance and inspiration to present these articles on my website. He is grateful to his mother Late Mrs. Peramma Rangadhama Rao for her sustained interest in my ventures.
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