text:
# Educational Qualifications:

B.Sc. Mathematics and Physics, Andhra University 1973 First Class
M.Sc. Mathematics, Andhra University 1975 First Class
Ph.d. Functional Analysis, Indian Statistical Institute 1983

# Honours, Awards and Grants:

Fulbright-Nehru Academic and Professional Excellence Fellowship 2015-16.
Fulbright-Nehru Senior Research Fellowship 2011-12.
Grant to attend ICM-2018.
Grant from Indo-French Institute for Mathematics, for research collaboration with Professor Gilles Godefroy, April 2018.
Research grant from the University of Granada, for research collaboration with Professor M. Martin, April 2018.
Research grant from the University of Memphis, for research collaboration with Professor F. Botelho, May 2018.
A travel grant from National Board for Higher Mathematics (NBHM) for research collaboration with L. Molnar, April 2017.
Travel grants from Indo-US Science and Technology Forum, for the Workshop and Conference Recent Advances in Operator theory and Operator algebras, December 2016' (Professor F. Botelho as the US-PI).
A travel grant from the Outreach Lecturer Fund (OLF) by the Commission for International Exchange of Scholars (CIES) in 2016.
IMU-Simons Foundation Travel grant 2014.
A travel grant from the Occasional Lecturer Fund (OLF) by the Commission for International Exchange of Scholars (CIES) in 2011.
A two month Fellowship from the European Commission, Erasmus Mundus External cooperation program, India4EU, during September and October 2010 at the University of Bologna.
Senior Research Fellowship of the Institute of Advanced Studies, University of Bologna.
India4EU Fellowship extended to June 2011.
A travel grant from the D. S. T and support from the I. S. I to attend the 11th conference on Function theory on infinite dimensional spaces, Madrid, December 2009.
A travel grant from the IMU and a summer research grant from the NSF at University of Iowa (jointly with B. L. Lin) during August 2007.
Full support from the N. B. H. M to attend the I. C. M at Madrid, 2006.
A travel grant from the IMU and a summer research grant from the NSF at S. I. U. E (jointly with K. Jarosz) during 2004.
Principal investigator, DST - NSF project"Banach Algebra techniques and the geometry of Banach spaces" - 2001 -2004. (K. Jarosz as US - PI and P. Bandyopadhyay as co - PI).
Principal investigator of the Indo-French project (IFCPAR) "Studies in geometry of Banach spaces" 2001 - 2004. (G.Godefroy as French PI and D. Li, P. Bandyopadhyay as co-PI's.
(Adjudged as the Best Project by IFCPAR).
Co-PI of the DST-NSF project "Extremal structures in Banach spaces", 2004-07 (V. Indumathi and B.L. Lin as PI).
INSA-PAN nomination for an exchange visit- 1997.
A research grant from the CNRS of Italy and a travel grant from the IMU - 1995.
Indo-EEC Post-doctoral Fellowship - 1991.
Awarded a Gold Medal and Professor S. Meenakshisundaram prize for the performance in the M.Sc final year examinations.
Title of the Phd thesis :
The Alfsen-Effros Structure topology in the theory of complex L^1-preduals.
Thesis advisor : Professor A. K. Roy.
Phd Student: Jayanarayanan, C.R, 2013.

# Publications

## Recent Publications :

1. Geometry of spaces of compact operators, Acta Sci Math (Szeged), To appear.
2. Operators on separable $L^1$-predual space, Journal of Mathematical Analysis and applications, 469 (2019) 252-259. (Jointly with A. K. Roy).
3. Bi-contractive projections on spaces of vector-valued continuous functions, Concrete Operators, to appear (Jointly with F. Botelho).
4. Simultaneous proximinality in spaces of Bochner integrable functions, Numerical Functional Analysis and Optimization, 39 (2018) 1221-1227.
5. Points of strong subdifferentiability in dual spaces, Houston Journal of Mathematics, 41 (2018) 1221-1226.
6. Adjoints of operators as smooth points in spaces of compact operators, Linear and Multilinear Algebra, 66 (2018) 668-670.
7. Weak coproximinality for Banach spaces, J. Nonlinear Funct. Anal., 13, 2017.
8. On a theorem of Abatzoglou for operators on abstract L and M-spaces, Journal of Mathematical Analysis and its applications, 453 (2017) 1000-1004.
9. Smooth points in spaces of operators, Linear Algebra Appl., 517 (2017) 129-133.
10. A Cheney and Wulbert type lifting theorem in optimization, Boll Unione Mat Ital 10 (2017) 585-589.
11. Coproximinality in spaces of Bochner integrable functions, Journal of Convex Analysis, 24 (2017) 955-958.
12. Into isometries that preserve finite dimensional structure of the range, AMS Contemporary Mathematics 687 (2017) 219-224.
13. Coproximinality for quotient spaces, Zeitschrift Fur Analysis und Ihre Anwendungen 36 (2017) 151-157.
14. Simultaneously proximinal subspaces, J. Appl. Anal. , 22 (2016) 115-120.
15. Optimization through dense sets, Bull., Belg., Math. Soc., 23 (2016) 583-594 (jointly with Jayanarayanan, C. R).
16. On small combination of slices in Banach spaces, Extracta Mathematicae, 31 (2016) 1-10 (Jointly with S. Basu).
17. Extremely strict ideals in Banach spaces, Proc. Indian Acad. Sci.(Math. Sci.), 126 (2016) 381-387.
18. On almost isometric ideals in Banach spaces, Monatshefte fur Mathematik, 181 (2016) 169-176.

## From 1981 to 2015:

1. Facial characterizations of complex Lindenstrauss spaces, Trans. Amer. Math. Soc. 268 (1981), 173-186. (Jointly with A. J.Ellis, A. K. Roy and U. Uttersrud).
2. Characterizations of some classes of L^1-preduals by the Alfsen - Effros Structure topology, Israel J.Math. 42 (1982), 20-32.
3. Isometries of A_C(K), Proc. Amer. Math.Soc. 85 (1982), 544- 46.
4. A note on real structure in complex L^1-preduals,Arch. Math., 45 (1985) 267- 269.
5. An intersection property of balls and relations with M-ideals, Math. Z., 197(1988) 277-290. (Jointly with P.Harmand).
6. Intersection properties of balls in tensor products of some Banach spaces, Math. Scand.,64 (1989) 103-118. (Jointlywith A. K. Roy and K. Sundaresan).
7. A note on the R_{n,k} property for L^1 (\mu, E). Canad. Math. Bull., 32 (1989) 74-77.
8. Intersection properties of balls in tensor products of some Banach spaces -II, Indian. J. Pure and Applied Math, 21(1990) 275-729.
9. L(X,C(K)) as a dual space, Proc. Amer.Math. soc., 110 (1990) 727-729.
10. C(K,X) as an M-ideal in WC(K,X), Proc.Indian. Acad. Sci. (Math. Sci) 102 (1992) 217-224.
11. Some remarks on the Schur approximation property, Quaestiones Mathematicae 16 (1993) 149-156.
12. On the injective tensor product of M-embedded spaces, Le Matematiche, 47 (1993) 159-161.
13. Space of compact operators as an M-ideal in its bidual, Extracta Mathematicae, 7 (1993) 114-118.
14. Spaces with the Namioka-Phelps property have trivial L-structure, Archive der Math., 62, (1994) 65-68.
15. L^1 (\mu, X) as a complemented subspace of its bidual, Proc. Indian. Acad. Sci (Math. Sci) 104 (1994) 421-424.
16. Geometry of operator spaces,Michigan Math.J.,41 (1994) 473-490. (Jointly with A. Lima, E. Oja and D. Werner).
17. A note on the extreme points of WC(K,X)_1,J. Ramanujan Math. Soc.,9 (1994) 215-219.
18. On a theorem of Dunford,Pettis and Phillips,Real Analysis Exchange,20 (1994/5) 741-743.
19. M-structure and the space A(K,E),Rendiconti di Matematica 15 (1995),153-160.
20. On the extreme point intersection property,"Function spaces, the second conference",Ed.,K. Jarosz,Lecture notes in pure and applied mathematics No 172, Marcel Dekker 1995,pp 339-346.
21. The space of vector-valued weakly continuous functions as a dual space,"Interaction between functional analysis,harmonic analysis and probability",Ed.,N. J. Kalton,E. Saab,S. Montgomery-Smith, Lecture notes in pure and applied mathematics No 175, Marcel Dekker 1996, pp 387-390.
22. On a new geometric property for Banach spaces,Proc. Indian Acad. Sci(Math. Sci), 107 (1977),35-42.
23. Spaces of operators as continuous function spaces, Extracta Mathematicae , 12 (1997) 251-253.
24. Spaces of Bochner integrable functions and spaces of representable operators as U-ideals,Quart. J. Math. Oxford,48 (1997) 467-478 (Jointly with G. Emmanuele).
25. Some stability results for asymptotic norming properties of Banach spaces, Colloq. Math.,75 (1998) 271-284. (Jointlywith S. Basu).
26. There are no denting points in the unit ball of WC(K,X), Proc. Amer.Math. Soc., 127(1999) 2969-2973.
27. Points of weak*-norm continuity in the unit ball of the space WC(K,X)*,Canad. Math. Bull., 42 (1999) 118-124.
28. Points of weak*-norm continuity in the dual unit ball of injective tensor product spaces, Collectanea Mathematica, 50(1999)269-275.
29. Denting and strongly extreme points in the unit ball of spaces of operators,Proc. Indian Acad.Sci.(Math.Sci),109 (1999) 75-86 .
30. L^1(\mu, X) as a constrained subspace of its bidual, Proc. IndianAcad. Sci.(Math. Sci), 109 (1999) 309-316.
31. Central subspaces of Banach spaces, J. Approx. Theory, 103 (2000)206-222.(Jointly with P. Bandyopadhyay)
32. Boundedness of linear maps,Comm. Math. Univ. Carolinae, 41 (2000) 107-110.
33. Weakly continuous functions of Baire class 1, Extracta Mathematicae, 15(2000) 207-212.
34. Local surjective isometries of function spaces, Expositiones Mathematicae,18 (2000) 285-296.
35. On Ideals in Banach spaces, Rocky Mountain J. Math., 31 (2001) 595-609.
36. On the geometry of higher duals of a Banach space, Illinois J. Math., 45(2001)1389-1392.
37. Diameter-preserving linear bijections of function spaces, J. Aust. Math.Soc., 70 (2001) 323-335 (Jointly with A. K. Roy).
38. Points of weak-norm continuity in the unit ball of Banach spaces, J. Math.Analysis. Appl., 265 (2002) 128-134.
39. Chebyshev centres and centrable sets, Proc. Amer. Math. Soc., 130 (2002)2593-2598.
40. Very smooth points of spaces of operators, Proc. Indian Acad. Sci. (MathSci), 113 (2003) 53-64.
41. Local isometries of function spaces, Math. Z., 243 (2003) 449-469 (jointlywith K. Jarosz) .
42. Proximinal subspaces of $A(K)$ of finite codimension, Int. J. Math. Mathsci.,39 (2003) 2501-2506.
43. Weak* - extreme points of Banach spaces, J. Convex Analysis 10 (2003) 531-539(jointly with S. Dutta).
44. Weak*-extreme points of injective tensor product spaces, Contemporary Mathematics,Amer. Math. Soc., 2003. Vol. 328, pp. 231-237 (Jointly with K. Jarosz).
45. Unitaries in Banach spaces, Illinois J. Math., 48 (2004) 339-351 (Jointlywith P. Bandyopadhyay, K. Jarosz).
46. Renormings and extremal structures, Illinois J. Math., 48 (2004) 1021-1029 (Jointlywith G. Godefroy).
47. Some generalizations of Kadison's theorem: a survey, Extracta Mathematicae,19 (2004) 319-334.
48. Norm-to-weak upper semi-continuity of the preduality map, J. Analysis 12 (2004) 115-124 (Jointly with S. Dutta).
49. Proximinality in generalized direct sums, Int. J. Math. Mathsci, 67(2004) 3663-3670. (Jointly with D. Narayana).
50. Remarks on a result of Khalil and Saleh, Proc. Amer. Math. Soc., 133 (2005) 1721-1722.
51. Local surjective isometries of L(X,C(K)), Proc. Amer. Math. Soc., 133 (2005) 2729-2732.
52. Transitivity of proximality and norm attaining functionals, Colloq.Math., 104(2006) 1-19. (Jointly with D. Narayana).
53. Extending into isometries of K(X,Y), Proc. Amer Math. Soc.,134 (2006) 2079-2082 .
54. Nice surjections on spaces of operators, Proc. Indian Acad. Sci(Math. Sci),116 (2006) 401-409.
55. Some remarks on quasi-Chebyshev subspaces, J. Math. Analysis. Appl., 321 (2006) 193-197. (Jointly with D. Narayana).
56. Some remarks on proximinality in higher dual spaces, J. Math. Analysis. Appl.,328 (2007) 1173-1177.
57. Ball proximinality in Banach spaces, (Jointly with B. L. Lin and P. Bandyopadhyay), Banach spaces and their Applications in Analysis, Conference In Honor of Nigel Kalton's 60th Birthday, Ed.,Randrianantoanina, Beata / Randrianantoanina, Narcisse, de Gruyter Proceedings in Mathematics, 2007, 251-264.
58. Operator versions of the Hahn-Banach theorem, Mathematics Newsletter, 17 (2007) 11-13.
59. Multi-smoothness in Banach spaces, International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 52382, 12 pages, 2007. doi:10.1155/2007/52382. (Jointly with B. L. Lin).
60. A note on the algebraic reflexivity of the group of surjective isometries of K(C(K)), Expositiones Mathematicae, 26 (2008) 79-83.
61. Algebraic reflexivity of some subsets of the isometry group, Linear Algebra and Its Applications, 429 (2008) 1522-1527 (Jointly with S. Dutta).
62. Ball remotal subspaces of Banach spaces, Colloq. Math.,114 (2009)119-133 (Jointly with P. Bandyopadhyay and B. L. Lin).
63. Three space problem for some classes of $L^1$-preduals, J. Math. Anal. Appl. 351 (2009) 311-314.
64. Riesz isomorphisms of tensor products of order unit Banach spaces, Proc. Indian Acad. Sci.(Math. Sci.), 119 (2009) 383-386.
65. On spaces of operators whose duals are isometric to L^1(\mu), Operators and Matrices, 3 (2009) 397-400.
66. Remark on a paper of Sababheh and Khalil, Numerical Functional Analysis and Optimization, 30 (2009) 822-824.
67. A note on a paper of Ercan and Onal, Taiwanese Journal of Mathematics, 14 (2010) 263-266.
68. On remotality for convex sets in Banach spaces, Journal of Approximation Theory, 162 (2010) 392-396 (Jointly with Miguel Martin).
69. Central and almost constrained subspaces of Banach spaces, Operators and Matrices, 6 (2012) 371-383.
70. Intersection Properties of Balls and Projections of Norm One, Mathematics News letter 22 (2012) 203-208.
71. Length inequalities for vectors in normed linear spaces, The Mathematics Student, Vol. 81, Nos. 1-4, (2012) pp.1-12.
72. The One and Half Ball Property in Spaces of Vector-Valued Functions, Journal of Convex Analysis, 20 (2013) 13-23.
73. On intersections of ranges of projections of norm one in Banach spaces, Proc. Amer. Math. Soc, 141 (2013) 3579-3586.
74. On ideals and generalized centers of finite sets in Banach spaces, Journal of Mathematical Analysis and Applications, 398 (2013) 886-888.
75. On a Garkavi type theorem for M-ideals of finite codimension, Numerical Functional Analysis and Optimization, 34 (2013) 1041- 1049.
76. On Silov boundary for function spaces, Topology and its applications, 193 (2015) 175-181 (jointly with A. K. Roy)
77. On algebraic reflexivity of sets of surjective isometries between spaces of weak*-continuous functions, Journal of Mathemaical Analysis and its applications, 432 (2015) 367-373 (jointly with F. Botelho).
78. Best constrained approximation in Banach spaces, Numerical Functional Analysis and Optimization, 36 (2015) 248-255.
79. Approximation properties for spaces of Bochner integrable functions, Journal of Mathematical Analysis and Applications, 423 (2015) 1540-45.
80. Existence sets of best coapproximation and Projections of norm one, Monatshefte fur Mathematik, 176 (2015) 607-614.
81. On intersections of ideals in Banach spaces, Houston Journal of Mathematics, 41 (2015) 589-594.

## Editorials:

1. Member of the Editorial board, Proceedings of the Indian Academy of Sciences, Mathematical Sciences.
2. Member of the Editorial board of Indian Journal of Pure and Applied Mathematics. (2018)
3. Associate Editor, Springer-I. S. I Series, 2018-2022.
4. Member of the Editorial board of 'Operators and Matrices'.
5. Lead Guest Editor, Geometry of Function spaces', Journal of Function Spaces, 2013 (Jointly with M. Martin and H. Hudzik).
6. Guest Editor, Functional Analysis and Operator Theory', ICM2010, Satellite Conference, Operators and Matrices, June 2012. (Jointly with Z. J. Ruan and P. Semrl)
7. Guest Editor, Proc. Indian Acad. Sci. (MathSci), 113 (2003) .
8. Guest Editor, J. Analysis 12 (2004) (Jointly with S. Agrawal and S. H. Kulkarni).
9. Lectures on Operator Theory, R. M. S, Lecture Notes, 2008.
10. Perspectives in Mathematical Sciences, Statistics and Probability, World Scientific, Statistical Science and Interdisciplinary Research - Vol. 7 2009 (Jointly with M. Delampady, B. Rajeev and N. S. N. Sastry).
11. Perspectives in Mathematical Sciences, Pure Mathematics, World Scientific, Statistical Science and Interdisciplinary Research - Vol 8. (2009) (Jointly with M. Delampady, B. Rajeev and N. S. N. Sastry).
12. In The Matrix Mould, Expository articles by Rajendra Bhatia, ((jointly with T. Bhattacharya and Roger A Horn) Hindustan Book Agency, 2012.
13. Problems and Recent Mathods in Operator Theory, AMS CONM 687, 2017. (Jointly with F. Botelho and R. King).

## Preprints:

• Into isometries of Banach spaces, Preprint 2018.
• Approximate Birkhoff-James orthogonality and smoothness in the space of bounded linear operators, Preprint 2018, (Jointly with A. Mal, K. Paul and D. Sain)
• Quotients of tensor product spaces, Preprint 2018 (Jointly with Monika )
• Proximinality properties for quotient spaces, Preprint 2015.
• Relative Chebyshev centres for sums of closed subspaces, Preprint 2016.
• Some remarks on unitaries in Banach spaces, Preprint 2003. (Jointly with S. Dutta).
• On a new class of extreme points, Preprint 2003.
• Some remarks on proximinality in Bochner function spaces, Preprint 1997.
• My Mathematical reviews, A compilation of 80 reviews written for Mathematical Reviews.
• My Zentralblatt reviews, A compilation of 310 Zentralblatt reviews (Also as Taduri S Rao).

## Research Interests :

Geometry of Banach spaces, Vector Measures, Tensor Product spaces, L^1-predual theory.
Choquet theory, Function algebras, Approximation theory.

## Professional and Teaching experience:

• Junior research fellow at the I. S. I from 1975-1977.
• Senior research fellow at the I.S.I from 1977-1981.
• Visiting fellow at the I.S.I during 1981-1982.
• Faculty of I. S. I Bangalore from 1-10-1985 and Professor since June 1995.
• Visiting Assistant Professor at the University of Washington, Seattle, during 1983-84.
• Visiting Assistant Professor at the Texas A and M university, College Station, during 1984-1985.
• C.E.C Post doctoral Fellow at the Mathematical Institute 1, Free University of Berlin during the year1990-91.
• Visited the Agder College, Kristiansand,Norway during July 1991 and the University of Granda, Spain during November1991.
• Visiting Associate Professor at The University of Michigan, Ann Arbor, during Summer term 1994
• Visited the University of Catania,Italy during May 1995,under a bursary from the C.N.R of Italy.
• Nominated by the Indian NationalScience Academy for an Exchange Visit to the Polish Academy of Sciences during September 1997. During this trip I visited the Institute of Mathematics at the Polish Academy of Sciences. Gave talks at the Institute of Mathematics at the University and Technical University of Wroclaw, Adam Mickiewicz University at Poznan and the University at Zielona Göra.
• Visited the Institute of Mathematics 1, Free University of Berlin during the 2nd week of October 97 and May 2002 and gave a talk.
• Visiting Professor, University of Missouri - Columbia from August 99 to May 2000.
• Visiting Professor, University of Iowa, Iowa City, Spring 2005.
• Attended the NSF - CBMS regional conference on Factorization of linear operators and the Geometry of Banach spaces, held at Columbia - Missouri in 1984.
• Attended the Second Conference on Function Spaces at Edwardsville, May 1994.
• Attended the Conference on The Interaction between Functional Analysis, Harmonic Analysis and ProbabilityTheory at Columbia, Missouri in June 1994.
• Attended the conference on Analysis'held at the Ohio State University, Columbus, Ohio during October 1999.
• Invited speaker at the Joint India AMS Mathematics Meeting, December 2003.
• Invited speaker at the 11th Ramanujan Symposium, March 2004.
• Gave the first K. R Unni memorial lectures at the Department of Mathematics, Cochin University of Science and Technology on 17-11-2004.
• Invited speaker at the  International conference on Operator theory, Quantum Probability and Noncommutative Geometry' , I. S.I, Kolkata December 20-23 04.
• Invited speaker at the Mid West conference on Numerical Analysis, 20-22 May 05,University of Iowa, Iowa City.
• Invited speaker at the International conference, Topics in Functional and Numerical Analysis', IIT, Bombay, 7-9th December 2005.
• Invited speaker at the Workshop and International conference on Operator theory and Operator algebras III, 12-22nd December 2005, I. S. I, Bangalore.
• Invited speaker at the Workshop on Functional Analysis and Fuzzy algebras, February 3-4,2006, Siddhartha College of Arts and Science, Vijayawada.
• Invited speaker at the Golden Jubilee Conference', IISc, 20-23 March 2006.
• Convener of the NBHM, Advanced Instructional School on  Harmonic Analysis and Functional Analysis', July 3rd-28th, 2006, Bangalore.
• Attended ICM-2006 at Madrid with full support from the NBHM.
• Invited speaker at the National Seminar on recent advances in Mathematics, Periyar University, Salem, March 12-13 2007.
• Invited speaker at the Symposium on Teaching of Functional Analysis', during the 22nd Annual Conference of the Ramanujan Mathematical Society at NIT, Surathkal, 6th June 2007.
• Invited speaker at the International conference on Functional Analysis and its applications, 28th Nov-1st Dec 2007, at Scott Christian College, Nagercoil.
• Convener of the NBHM workshop on Operator theory, June 11-17, 2007.
• Member NBHM Regional Library committee.
• Coordinator NBHM Nurture Program 2008-10.
• Member, Organizing Committee, A conference on Perspectives in Mathematical Sciences, February 4-8, 2008, ISI Bangalore.
• Convener NBHM, ATML on Functional Analysis, 2-13 June 2008.
• Member, General Assembly Committee, International Congress of Mathematicians (ICM) 2010.
• Chairman, Local organizing committee, 24th Annual conference of the Ramanujan Mathematical Society, May11-13, 2009.
• Convener, NBHM Workshop on Geometry of Banach spaces, June 2009.
• Attended and gave a talk at the 11th conference on Function theory on infinite dimensional spaces, Madrid, 14-17 December 2009.
• Visited the Department of Mathematics, University of Granada, Spain during November-December 2009.
• Member, Organizing committee, ICM Satellite conference on Functional Analysis and Operator theory, August 8th to 12th 2010, ISI Bangalore.
• Attended the International Congress of Mathematicians at Hyderabad, gave a short talk, Chaired a Short talk session.
• Visited the Department of Mathematics of the Università degli Studi of Milano during September 2010 and gave a talk in the SAA Seminar.
• Invited speaker at the 76th Annaul meeting of the Indian Academy of Sciences in Goa, 12th to 14th November 10.
• Gave an invited talk in the special session on Recent advances in Analysis and operator theory' during the 76th Annual Conference of the Indian Mathematical Society in Surat, during December 27th-30th 2010.
• Visited the Department of Mathematics at the University of Memphis in October 2011 and February 2012 and gave talks.
• Visited the Departments of Mathematics, U. S Naval Academy and the University of Maryland, Baltimore county during November 2011 under the Occasional Lecturer Fund (OLF) of the Council for International Exchange of Scholars (CIES) and gave talks.
• Visited the Department of Mathematics at the Texas A&M University, College station, during March 2012 and gave a Colloquioum talk.
• Attended the conference on History, Aspects and Prospects of Mathematics in India, organized by the three academies of Science in India during June 2012 and Chaired a session.
• Invited speaker at the Analysis symposium, during the 27th Annual Meeting of the Ramanujan Mathematical Society, October 2012.
• Member, Organizing committee of the conference, Matrices and Operators, December 2012.
• Convener NBHM, Instructional School for Lecturers in Functional Analysis, 4-15th March 2013.
• Member, Organizing committee, International Workshop on Operator Theory and Applications (IWOTA 2013), December 16-20th, Bangalore.
• Convener NBHM Advanced Instructional School in Operator theory, 2nd to 22nd June 2014.
• Member, Organizing committee, I. S. I Summer school in Analysis and Probability, University of Tripura, 2015.
• Special session on Isometries on Banach spaces, dedicated to the memory of J. Jamison, AMS Southern sectional meeting, University of Memphis, 2015. (Jointly with F. Botelho).
• Member, organizing committee, Workshop and Conference Recent Advances in Operator theory and Operator algebras, I. S. I, Bangalore, December 2016.
• Invited speaker at the conference on Functional Analysis, IIT Bombay, 12-15th October 2017.
• Invited speaker, International Conference on Mathematical Analysis and its Applications in Modeling, Jadavpur University, 9-12 January 2018.
• Plenary Speaker, Recent trends in operator theory, May 3-5th 2018, University of Memphis.
• Invited speaker, International Conference on Mathematics and Statistics, May 7-10th 2018, University of Memphis.
• Invited speaker, Conference and Workshop on Analysis and applications, IIT Madras, June 18-22 2018.
• Reviewer for Mathemaical Reviews.
• Reviewer for Zentral Blatt.
• Member of the American Mathematical Society.
• Member of the European Mathematical Society (EMS).
• Life member Ramanujan Mathematical Society.
• Courses usually taught at the I. S. I, include, advanced courses in Analysis, Linear Algebra and Functional Analysis to the students in the Master of Statistics program of the I. S. I, B. Math, M. Math programs and to Research Scholars in the department. I have been teaching an undergraduate/graduate course and/or a research course every semester for the past 30 years.
• I am a Mentor for the INSPIRE program of the Department of Science and technology (DST) and Volunteer lecturer of the Committee for developing countries (CDC) of the IMU.

# Activities at ISI

Member of the Governing Council of the I. S. I from September 2007 to August 2011, September 2012-August 2014 and November 2015 till date.

Member, Policy, Planning and Evaluation Committe, 2014-18.

Professor-In-Charge, Theoretical Statistics and Mathematics Division, Sept 2012- Aug2014.

Member, Ph.D and D. Sc committee in Mathematics, Indian Statistical Institute, 2008-2014.

Head, I. S. I Bangalore Centre, September 2007 to August 2011.

Member of the Academic Council of the I. S. I.

Member , Research Fellow Advisory Committee, 2014- till date.

Officiating Head, I S I Bangalore Centre, 1st May-July 8th, 2007.

Acting Professor-In-charge, Theoretical Statistics and Mathematics Division, August-September 2007.

Member, Indian Statistical Institute.

Member Executive committee, I. S. I Karnataka Chapter, 2012-16.

Member Executive committee, I. S. I Alumni Association,2012-15.

Head of the Stat-Math Unit at Bangalore, from July 1993 to March 1995.

Member M. Math selection Committee 2003, 2007, 2009.

Member J. R. F selection committee 1989, 90,95,97,04,08,09,10(Convener),11,12, 13 (Chairman) ,14,16.

Member Research Fellow advisory committee at Bangalore, 87-90, 2001-10.

Convener of the Research Fellow advisory committee at Bangalore, 93-95.

Chairman, Computer Committee at Bangalore,2004-2008 (March), Member 2008-10.

Member Leave rule committee-2006.

Convener Platinum Jubilee Lecture Series 2007.

Member Platinum Jubilee PDF Committee 2007.

Age: 64 years.

Born in Gurazala in Andhra Pradesh.

Early education: Mostly at home, the street school (Veedhi badi) in Macherla and Z. P. H. S at Gurazala and Sesha Reddy High school at Bethamcherla .

College education: S.R.R. & C.V.R Government College, Vijayawada.

Marital status: Married with a daughter and a son.

Indian Statistical Institute, Bangalore Centre,

R. V. College Post,

Bangalore-560059,

India.

Phone:

Office:- +91 (80) 26985417.

Home:- +91 (80) 2339 9019.

Fax:

+91 (80) 28484 265

E-mail:

tss@isibang.ac.in ,

srin@fulbrightmail.org


text:
# showing any controllable system can be put in 'controller' form.

I am looking at the proof of the following theorem: Let $\dot{x} =Ax + Bu$ be a controllable single input system, where $\Delta_A:= \det(\lambda I -A) = \lambda^n + a_1\lambda^{n-1} + \ldots + a_{n-1}\lambda + a_n$. Show this system is isomorphic to the system $\dot{x} = \bar{A}x + \bar{B}u$ where: $$\bar{A} = \left( \begin{matrix} 0 & 1 & 0& \dots & 0 & 0\\ 0 &0&1 &\ldots &0&0 & \\ \vdots &\vdots &\vdots & & \vdots &\vdots &\\ 0&0&0& \ldots & 0& 1\\ -a_n& -a_{n-1}& -a_{n-2}& \ldots & -a_2& -a_1 \end{matrix}\right) ,\quad\bar{b}=\left( \begin{matrix} 0\\0\\ \vdots \\0\\0\\1 \end{matrix}\right)$$ This amounts to finding an invertible matrix $S$ such that $\bar{A}=S^{-1}AS$ and $\bar{B} = S^{-1}B$.

The proof however I really cannot follow. This theorem comes from a subject that I am taking for the second time. I distinctly remember begin quite impressed with this theorem last year when the teacher proved it with relative ease. However in the updated lecture notes we are given this year I cannot follow the proof at all.

It starts of with defining the required matrix $S$ via $S = \left(q_1\quad q_2 \quad \ldots \quad q_n\right)$ where \begin{eqnarray} q_n&=&B\\ q_{n-1} &=& AB + a_{1}B\\ q_{n-1} &=& A^2B + a_{1}AB +a_{2}B\\ &\vdots&\\ q_1 &=& A^{n-1}B +a_{1}A^{n-2}B + \ldots +a_{n-1}B \end{eqnarray}

Can anyone show me why this choice of $S$ works? Thanks a lot in advance for any help!

• You need to change the order of the scalars. See the post below. – obareey Dec 6 '13 at 17:45
• Thanks for your comment, I think I made the adequate changes to the definition of the $a_i$'s. – Slugger Dec 6 '13 at 17:56
• Also $\bar{A} = S^{-1} A S$ – obareey Dec 6 '13 at 18:16
• Ah yes thanks for that! – Slugger Dec 6 '13 at 19:36

We need to show that $S \bar{A} = AS$. First look at $S \bar{A}$ (work it out).

$S \bar{A} = \begin{pmatrix} -a_n q_n & -a_{n-1} q_n + q_1 & -a_{n-2} q_n + q_2 & \dots & -a_1 q_n + q_{n-1} \end{pmatrix}$

Now $AS = \begin{pmatrix} A q_1 & A q_2 & \dots & A q_n \end{pmatrix}$. Let us look at each column seperately.

$A q_1 = A^n B + a_1 A^{n-1} B + \dots + a_{n-1} A B = -a_n B = -a_n q_n$.

It follows from the Cayley-Hamilton theorem, because $\Delta_A$ is the characteristic polynomial of $A$. Others will just follow from the definitions

\begin{align} A q_2 &= A^{n-1} B + a_1 A^{n-2} B + \dots + a_{n-2} A B = -a_{n-1} B + q_1 = -a_{n-1} q_n + q_1 \\ A q_3 &= A^{n-2} B + a_1 A^{n-3} B + \dots + a_{n-3} A B = -a_{n-2} B + q_2 = -a_{n-2} q_n + q_2 \\ &\vdots \\ A q_{n-1} &= A^2 B + a_1 AB = -a_2 B + q_{n-2} = -a_2 q_n + q_{n-2} \\ A q_n &= AB = -a_1 B + q_{n-1} = -a_1 q_n + q_{n-1} \\ \end{align}

Hence, $S \bar{A} = AS$. Also, see that $S \bar{B} = q_n = B$.

Edit: Also note that $q_i$ are linearly dependant (ie. $S$ does not have an inverse) if the system is not controllable.

Here is another view (inspired by Luenberger's 1967 paper http://www.stanford.edu/dept/MSandE/cgi-bin/people/faculty/luenberger/pdfs/cfflms.pdf, this is a slightly different solution for a slightly simpler problem):

(Note that even Luenberger wrote that this "is somewhat more difficult to derive".)

Let $P = \begin{bmatrix} b & Ab & \cdots & A^{n-1} b \end{bmatrix}$. Since the system is cc., $P$ is invertible. It is straightforward to show that using the basis $P$, the system $(A,b)$ has the form $\left( \begin{bmatrix} 0 & 0 & 0& \dots & 0 & -a_n\\ 1 &0&0 &\ldots &0& -a_{n-1} & \\ \vdots &\vdots &\vdots & & \vdots &\vdots &\\ 0&0&0& \ldots & 0& -a_2 \\ 0& 0& 0& 0 & 1& -a_1 \end{bmatrix}, e_1 \right)$, where $e_k$ is all zeros except for a one in the $k$th position.

This is close, but we need the transpose instead, and the '$b$' matrix needs to be $e_n$ instead.

The key claim is that the system $(A^T, P^{-T} e_n)$ is cc. Let $W^T = \begin{bmatrix} e_n^T P^{-1} \\ \vdots \\ e_n^T P^{-1} A^{n-1} \end{bmatrix}$ (the transpose of the usual). We need to show that $W$ is invertible. Note that $e_n^T P^{-1} A^{i-1}A^{j-1}b = e_n^T P^{-1} P e_{i+j-1} = \delta_{n,i+j-1}$ for $i,j$ such that $2 \le i+j \le n$. Hence $W^TP =\begin{bmatrix} 0 & 0 & 0& \dots & 0 & 1 \\ 0 &0&0 &\ldots &1& * & \\ \vdots &\vdots &\vdots & & \vdots &\vdots &\\ 0&1& * & \ldots & * & * \\ 1& *& *& \ldots & *& * \end{bmatrix}$, where the $*$ entries do not matter in the context of showing that $W$ is invertible. Since $P$ and the right hand side are invertible, it follows that $W^T$ is invertible and hence $(A^T, P^{-T} e_n)$ is cc. Note that this computation also shows that $W^T P e_1 = e_n$.

Now we note that in the basis $W$, the matrix $W^{-1}A^TW$ has the form given above, hence the matrix $W^T A W^{-T}$ has the desired form. Furthermore, $W^Tb = W^T P e_1 = e_n$, as desired.

The two representations $(A, \mathbf{b})$ and $(\bar{A}, \bar{\mathbf{b}})$ are isomorphic iff there is a bijection $\Phi:\mathbb{R^n} \rightarrow \mathbb{R^n}$ from the original states $\mathbf{x}(t)$ to the states $\mathbf{z}(t)$ in normal form: $\Phi(\mathbf{x}(t)) = W\mathbf{x}(t) = \mathbf{z}(t)$ where $W \in \mathbb{R}^{n \times n}$.

The linear transformation yields the following set of equations (argument $t$ omitted for sake of readability): \begin{align} \mathbf{w}_1^T\mathbf{x} = z_1 \\ \mathbf{w}_1^T\dot{\mathbf{x}} = \mathbf{w}_1^T (A\mathbf{x} + \mathbf{b}u) = \mathbf{w}_1^TA\mathbf{x} + \underbrace{\mathbf{w}_1^T\mathbf{b}}_{=0}u = \dot{z}_1 = z_2 \\ \mathbf{w}_1^TA\dot{\mathbf{x}} = \mathbf{w}_1^TA (A\mathbf{x} + \mathbf{b}u) = \mathbf{w}_1^TA^2\mathbf{x} + \overbrace{\underbrace{\mathbf{w}_1^TA}_{\mathbf{w}_2^T}\mathbf{b}}^{=0}u = \dot{z}_2 = z_3 \\ \vdots \\ \mathbf{w}_1^TA^{n-1}\mathbf{x} = z_n. \end{align} Matrixwise this reads $$\underbrace{ \left[ \begin{array}{c} \mathbf{w}_1^T \\ \mathbf{w}_1^TA \\ \mathbf{w}_1^TA^2 \\ \vdots \\ \mathbf{w}_1^TA^{n-1} \end{array} \right]}_{=W}\mathbf{x} = \mathbf{z}.$$ Where the conditional equation for $\mathbf{w}_1^T$ is $$\left[ \mathbf{w}_1^T\mathbf{b}, \mathbf{w}_2^T\mathbf{b}, ..., \mathbf{w}_n^T\mathbf{b} \right] = \left[ \mathbf{w}_1^T\mathbf{b}, \mathbf{w}_1^TA\mathbf{b}, ..., \mathbf{w}_1^TA^{n-1}\mathbf{b} \right],$$ which, from $\bar{\mathbf{b}} = [0 ... 0 \: 1]^T$, returns $$[0 ... 0 \: 1] = \mathbf{w}_1^T \underbrace{[\mathbf{b} \: A\mathbf{b} \: ... \: A^{n-1}\mathbf{b}]}_{=:S(A, \mathbf{b})}.$$ If $S(A, \mathbf{b})$ is invertible $$\mathbf{w}_1^T = [0 ... 0 \: 1] \: S^{-1}(A, \mathbf{b}),$$ which is a unique solution and a necessary condition for $W$ to be invertible and $\Phi$ to be 1-1 and onto.

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# How do you find the square root of 9/144?

The answer is $\frac{1}{4}$.
$9 = {3}^{2}$ and $144 = {12}^{2}$.
$\sqrt{\frac{9}{144}} = \frac{3}{12}$
Simplify $\frac{\cancel{3}}{\cancel{12}} = \frac{1}{4}$.


text:
# Obeleži sve kategorije koje odgovaraju problemu

### Još detalja - opišite nam problem

Uspešno ste prijavili problem!
Status problema i sve dodatne informacije možete pratiti klikom na link.
Nažalost nismo trenutno u mogućnosti da obradimo vaš zahtev.
Molimo vas da pokušate kasnije.

# 2. Sequences of data and line charts¶

In this lecture we demonstrate:

1. how to navigate through Jupyter notebook;
2. how to represent sequences of data; and
3. how to visualize sequences of data by line charts.

## 2.1. A bit more about Jupyter notebooks¶

Each Jupyter notebook is a sequence of cells, and each cell can contain some text, an expression, or a Python program. Buttons at the top of the page make it possible for you to manipulate cells. We have already used the Run button. Let us now see what other four buttons do:

• Clicking the diskette (first button on the left) saves the notebook.
• Clicking the + adds a new cell below the active cell. (A cell can be activated by clicking on it; an active cell is framed in ***green*** or ***blue***. The difference between the two does not matter at the moment.)
• Up and down arrows move the active cell.

Imagine that, while reading this text, you suddenly felt an unexplicable urge to evaluate $1 + \frac 12 + \frac 13 + \frac 14 + \frac 15 + \frac 16 + \frac 17$. You then have to click this cell (yes, the one you are reading at the moment) and click on the + button. In the new cell that appears below you can enter the expression and Python will evaluate it for you:

In [1]:
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7

Out[1]:
2.5928571428571425

The following eight cells contain the first few verses of a nursery rhyme. The verses are, alas, scrambled. Using the up and down arrows rearrange the cells to get the rhyme right.

Once I caught a fish alive,

Then I let it go again.

Which finger did it bite?

Why did you let it go?

Because it bit my finger so.

This little finger on my right.

One, two, three, four, five,

Six, seven, eight, nine, ten,

## 2.2. Representing sequences of data¶

A sequence of data can be represented as a list: list the numbers within square brackets. For example, in some countries students get marks as letters A, B, C, D and sometimes even F, while in other countries students get marks as numbers 5, 4, 3, 2 and 1. Marks expressed as numbers are more convenient for data analysis, so let these be the marks of a student:

In [2]:
marks = [2, 4, 5, 3, 5]

The system produces no output, of course. It has just registered that the variable marks contains a list of integers. Let us check that everything is as we expect it to be:

In [3]:
marks

Out[3]:
[2, 4, 5, 3, 5]

It is also possible to form lists of strings, like this:

In [4]:
subjects = ["Maths", "English", "Art", "History", "PE"]

Let's check, just in case:

In [5]:
subjects

Out[5]:
['Maths', 'English', 'Art', 'History', 'PE']

The standard function len gives back the length of a list:

In [6]:
len(subjects)

Out[6]:
5

## 2.3. Visualizing sequences of data¶

The following cell contains the data that describe the way the population of our planet has changed in the last millenium. The numbers in the population list are given in billions:

In [7]:
years       = [1000,  1500, 1650, 1750, 1804, 1850, 1900, 1930, 1950, 1960, 1974, 1980, 1987, 1999, 2011, 2020, 2023, 2030, 2037, 2045, 2055, 2100]
population  = [0.275, 0.45, 0.5,  0.7,  1,    1.2,  1.6,  2,    2.55, 3,    4,    4.5,  5,    6,    7,    7.8,  8,    8.5,  9,    9.5,  10,   11.2]

We would like to visualize this data because we, the humans, find the visually represented data as the most appealing.

There are many libraries that come with Python that can help with visualizing data and we will focus on the one called matplotlib.pyplot. The name of the library is very long and complicated, and we shall have to refer to many functions from it, we shall import the entire library at once and at the same time give it a nickname plt.

In [8]:
import matplotlib.pyplot as plt

It is important to stress that one import per notebook suffices! Therefore, we shall import the library once and use it throughout the notebook. On the other hand, in each new notebook we have to import all the libraries we need, but only once per notebook.

One import per notebook!

If you, by accident, import the same library twice -- no worries. Python will not complain, but also will not lose time and resources to import the same library twice.

Back to drawing charts! This is the simplest way to get a chart:

In [9]:
plt.plot(years, population)
plt.show()
plt.close()

The function plot(years, population) tells the system that we want a chart where the horizontal axis (the $x$-axis) represents the numbers listed in years, while the vertical axis (the $y$-axis) represents the numbers listed in population. The function show then does the actual drawing. Finally, the function close cleans up the garbage left after all the computations needed to draw the chart.

The data is represented by a line. This is why such charts are called line charts.

Since this time we imported the entire library under the nickname plt the functions plot, show and close have to be addressed using both their "family name" (the library it comes from) and "first name": plt.plot, plt.show and plt.close.

Next, using the title function we shall give our chart a title:

In [10]:
plt.plot(years, population)
plt.title("The population of the Earth")
plt.show()
plt.close()

We can also put a label along the vertical axis to indicate that the numbers we are displaying are in billions. For this we need the function ylabel ("the label on the $y$-axis""):

In [11]:
plt.plot(years, population)
plt.title("The population of the Earth")
plt.ylabel("(billions)")
plt.show()
plt.close()

In conclusion: functions plot, title and ylabel add various elements to the chart, and only when all the elements have been added to the char we invoke the show function to do the actual drawing. After that we have to clean up by invoking the close function.

## 2.4. Exercises¶

Exercise 1. Look at the following code and then answer the questions:

In [12]:
import matplotlib.pyplot as plt
plt.plot(years, population)
plt.title("The population of the Earth")
plt.ylabel("(billions)")
plt.show()
plt.close()

1. What does import ... as .. do?
2. Why do we have to write plt.plot, and not simply plot?
3. What does plot do?
4. What do title and ylabel do?
5. What do show and close do?

Exercise 2. The following cell contains data about weight and length/height of a boy in the first seven years of his life.

In [13]:
peroid    = ["6 m", "1.5 y", "2.5 y", "3.5 y", "4.5 y", "5.5 y", "6.5 y"]
weightKG  = [5.9,   11.5,    14.8,    20.5,    22.0,    24.2,    29.0   ]
heightCM  = [62.0,  84.0,    97.0,    115.0,   122.5,   131.5,   135.0  ]

Generate two charts: one to display how the weight of the boy changes over time, and the other one to display how the height of the boy changes over time.

Exercise 3. Body mass index, BMI, is the quotient of the weight (in kilograms) of a person and the height (in meters) of the person:

$$\hbox{BMI} = \frac{\hbox{weight in kilograms}}{(\hbox{height in meters})^2}$$

For a boy from Exercise 2 visualize the way his BMI changed over time. The following Python program maight be helpful:

In [14]:
BMI = [0, 0, 0, 0, 0, 0, 0]
for i in range(0, 7):
BMI[i] = weightKG[i] / (heightCM[i] / 100.0)**2

Exercise 4. It is estimated that on July 1st, 2019 the population of China was 1,420,062,022. It is also estimated that the population of China increases by 0.35% per year. Visualize the population of China in the following ten years under the assumtion that these parameters will not change.

In [ ]:


text:
# Complex Numbers¶

## Objective¶

Get comfortable with complex numbers and complex arithmetic. Understand how complex numbers can be visualized and interpreted geometrically.

## Real Numbers¶

"Real numbers" is the formal math name for the set of numbers that we, as software engineers, usually call "floating point" numbers. Some languages call these floats, some call them doubles, some have both terms. Anything that can be written out as a single sequence of digits (even if it has an infinite number of digits) is a real number. They can be positive, negative, or zero. Integers, fractions, irrational numbers like $$\sqrt{2}$$ and even transcendental numbers like $$\pi$$ are all included in the real numbers. Essentially, they are all of the numbers that can be found on a one-dimensional number line like this:

Real numbers can be used to represent the solution to any algebraic equation, with one exception. Equations of this form cannot be solved with them:

$x^2 + y = 0, \qquad y > 0$

To solve this, $$x^2$$ would need to be a negative number. However, since a positive number times a positive number is positive, and a negative number times a negative number is positive, there isn't a real number that produces a negative number when multiplied by itself. Thus, we say that this equation has no real solutions.

## The Imaginary Unit¶

Equations like this come up a lot in physics and quantum mechanics, so we can't just ignore it - we need to solve it somehow. The way we do this is by defining a special number that specifically satisfies this equation:

$x^2 = -1$

We call it the imaginary unit, which is kind of an unfortunate derogatory name given to it a long time ago, when mathematicians didn't like dealing with the square roots of negative numbers. The traditional symbol for the imaginary unit is the letter $$i$$:

$i^2 = -1, \qquad i = \sqrt{-1}$

The imaginary unit is a completely separate entity from the real numbers. It isn't anywhere on the number line. However, we can still do arithmetic with it. For example, consider this equation:

$(x-3)^2 = -16$

As shown before, it doesn't have any real solutions, but we can solve it by using the imaginary unit:

$\displaylines{ (x-3)^2 = -16 \\~\\ x-3 = \sqrt{-16} \\~\\ x = \sqrt{-16} + 3 \\~\\ x = \sqrt{-1 \cdot 16} + 3 \\~\\ x = \sqrt{-1} \cdot \sqrt{16} + 3 \\~\\ x = i \cdot 4 + 3 \\~\\ \boldsymbol{x = 3 + 4i} }$

Because the imaginary unit is a totally separate thing from the real numbers, this solution can't be reduced any further. We have to write it in two parts: a real part (the 3) and an imaginary part (the 4$$i$$). This is an example of what's called a complex number. Complex numbers take this form:

$x = a + bi$

The value $$a$$ is a real number that corresponds to the real part. The value $$b$$ is a real number that corresponds to the imaginary part, and the $$i$$ is used to denote which part is the imaginary one. $$a$$ and $$b$$ can be any real numbers you want, but as a whole, complex numbers have to be written in two parts like this.

Here are some more examples of complex numbers:

$x = 3 + 4i, \qquad y = -2 + 6.3i, \qquad z = \frac{1}{2} - 7i, \qquad w = -4, \qquad v = 7.5i$

Note that if $$b = 0$$, as is the case with the $$w$$ example, then the imaginary part isn't included. It's just a regular old real number. If $$a = 0$$, as in the $$v$$ example, then the real part isn't included. It's just a pure imaginary number.

## The Complex Plane¶

Because complex numbers are represented by two independent real numbers, you can think of a complex number as an array of two floats. Many software libraries will treat them like this, or like an object with two separate fields (a floating-point number for the real component, and a floating-point number for the imaginary component). In other words, complex numbers are two-dimensional entities, where the real component represents one dimension and the imaginary component represents the other.

Naturally, an easy way to visualize a two-dimensional thing is to plot it on a graph. When we do this for complex numbers, we can represent the real component with the X axis and the imaginary component with the Y axis. This specific plot is called the complex plane. To demonstrate, let's graph the five example numbers from above on the complex plane:

In this picture, each complex number is drawn in two ways: the first is a point on the plane, and the second is a vector from the origin to that point.

## Math Properties¶

When adding two complex numbers, you just add the real part to the real part, and the imaginary part to the imaginary part. For example:

$(3 + 4i) + (7 - i) = 10 + 3i$

### Multiplication¶

When multiplying two complex numbers, it's easiest just to follow this formula:

$(a_0 + b_{0}i) \cdot (a_1 + b_{1}i) = (a_{0}a_{1} - b_{0}b_{1}) + i(a_{0}b_{1} + b_{0}a_{1})$

### Absolute Value / Magnitude¶

The magnitude of a complex number is represented in-between straight vertical bars, like this:

$\lvert a + bi \rvert$

It is equal to the length of the vector when the number is plotted on the complex plane. This vector will produce a right triangle where one side is the length of the real component, one side is the length of the imaginary component, and the hypotenuse is the complex number itself. For example, using the point $$x = 3 + 4i$$, the two sides would have lengths 3 and 4:

Following Pythagoras's theorem, $$A^2 + B^2 = C^2$$, the length of the vector is equal to the square root of the sum on both other sides squared:

$C = \sqrt{A^2 + B^2}$

Thus, the magnitude of a complex number is just the real component squared plus the imaginary component squared:

$\lvert a + bi \rvert = \sqrt{a^2 + b^2}$

### Complex Conjugate¶

The complex conjugate is used sometimes when converting a column vector into a row vector (discussed in the next page). It's the same number, but with the sign of the imaginary part flipped. It is represented with an asterisk, like this:

$x = a + bi, \qquad x^* = a - bi$

## Euler's formula¶

One important formula that comes up in quantum computing is Euler's formula:

$e^{i\theta}=\cos(\theta)+i\sin(\theta)$

Any complex number, $$z$$, with absolute value equal to 1 can be written in this form for some $$\theta$$ : $$z=e^{i\theta}$$ . These complex numbers lie on the unit circle in the complex plane and one can think of the angle $$\theta$$ as the angle the complex number, $$e^{i\theta}$$ makes with the $$x$$-axis.

• Khan Academy unit on complex numbers

• What are the complex numbers?
• The complex plane
• Adding and subtracting complex numbers
• Multiplying complex numbers
• Complex conjugates and dividing complex numbers
• Absolute value and angle of complex numbers
• 3Blue1Brown video on complex numbers

• This is a long video, but if offers a clear and deep understanding of complex numbers.

## Knowledge Check¶

### Q1¶

Which arrow represents $$2-3i$$ on the complex plane?

### Q2¶

$(2-3i) + (3+3i) = \; ?$

### Q3¶

$(-5+2i) - (-8+i) = \; ?$

### Q4¶

$4 \cdot (1+5i) = \; ?$

### Q5¶

$-2i \cdot (3-3i) = \; ?$

### Q6¶

Let $$x$$ be a complex number as shown on the complex plane. Which arrow represents $$-i \cdot x$$?

### Q7¶

$(-1-4i) \cdot (4+2i) = \; ?$

### Q8¶

$(3+2i) \cdot (3-2i) = \; ?$

### Q9¶

$|4-3i| = \; ?$

### Q10¶

What is the complex conjugate of $$2+5i$$?

Last update: July 6, 2022


text:
# Possible Typo in Real Analysis Folland Problem 3.5.32 Functions of Bounded Variation

I wanted to do this problem for practice but I believe there may be a typo. The problem is: If $F_1,F_2,\ldots,F\in NBV$ and $F_j\rightarrow F$ pointwise, then $T_F\leq \liminf T_{F_j}$

I believe the corrected version should be: If $F_1,F_2,\ldots,F_n\in NBV$ and $F_j\rightarrow F$ pointwise, then $T_F\leq \liminf T_{F_j}$

I just want to know if these two statements are the same or is there some typo there. Any suggestions or comments is greatly appreciated.

Attempt (not complete) - Let $F_1,F_2,\ldots, F\in NBV$ and $F_j\rightarrow F$ pointwise. Given $\epsilon > 0$ choose points $x_0 < \ldots < x_n = x$ where $x\in\mathbb{R}$ such that $$\sum_{1}^{n}|F(x_j) - F(x_{j-1}| \geq T_F(x) - \epsilon$$ Then $T_F(x_0) \leq \epsilon$. Thus we have \begin{align*} T_F(x) - \epsilon &= \sup\{\sum_{1}^{n}|F(x_j) - F(x_{j-1}|:n\in\mathbb{N},-\infty < x_0 < \ldots < x_n = x\} - \epsilon\\ &\leq \sum_{1}^{n}|F(x_j) - F(x_{j-1}|\\ &\leq \inf\{\sum_{1}^{n}|F(x_j) - F(x_{j-1}|:n\in\mathbb{N},-\infty < x_0 < \ldots < x_n = x\} \end{align*}

I am not sure if what I have done is on the right track or not.

Wolfy, the two statements are correct, but your statement is stronger since it applies also to the case $T_F = \infty$. If $T_F$ is finite, then you can proceed as @Qiyu indicates. Qiyu is assuming implicitly that $T_F$ is finite at the statemnt: "Chose some partion $P = \{x_0,\dots,x_n\}$ such that $T-T(P) < \epsilon$ ...", which cannot be done if $T_F = + \infty$.

Your attempt is in the right track, but there is at lest one typo and it needs some cleaning.

Qiyu's proof can be modified to avoid assuming $T_F < \infty$. Here is how. Choose a partition $P = \{x_0,\dots,x_n\}$ of the interval $[a,b]$ with $a=x_0 < ... < x_n=b$ and choose $\epsilon > 0$.

By the assumption of pointwise convergence, there is $k_0$, that depends on $\epsilon$ and the partition, such that for all $k\ge k_0$ we have $|F_k(x_i) - F(x_i)| < \frac{\epsilon}{2n}$ for $i=0,...,n$.

Then for any $k\ge k_0$, and for $i=1,..,n$ $$|F(x_i) - F(x_{i-1})| \le |F(x_i) - F_k(x_{i})| + |F_k(x_i) - F_k(x_{i-1})| + |F(x_{i-1}) - F_k(x_{i-1})|$$ $$< \frac{\epsilon}{n} + |F_k(x_i) - F_k(x_{i-1})| .$$

Therefore
$$\sum_{1}^{n}|F(x_i) - F(x_{i-1})| < \epsilon + \sum_{1}^{n}|F_k(x_i) - F_k(x_{i-1})| \le \epsilon + T_{F_k}.$$

Now we can take lim inf over $k \ge k_0$ to get:
$$\sum_{1}^{n}|F(x_i) - F(x_{i-1})| \le \epsilon + \liminf_{j\to \infty}T_j$$
and whether the lim-inf is finite or not, since this holds for any $\epsilon > 0$ we get:
$$\sum_{1}^{n}|F(x_i) - F(x_{i-1})| \le \liminf_{j\to \infty}T_j .$$

Since the last inequality holds for any partition $P$, taking supremum over all partitions we get the result.$\square$

$$\dots \dots \dots$$

In particular, this shows that point-wise convergence to $F$ + finite lim-inf implies the limit $F$ is of bounded variation.

Let's work a couple of examples. Let's take one of the standard examples of unbounded variations $$f(x) = \begin{cases} \frac{1}{x}\cos(\frac{\pi}{x}) & \text{ for  x \in (0,1]},\\ 0 & \text{for x=0}. \end{cases}$$ The graph of $f$ oscilates between the lines $y=x$ and $y=-x$ as $x \rightarrow 0^+.$ Take the partition $P=\{ 0, \frac{1}{n},\frac{1}{n-1},.., \frac{1}{2}, 1 \}$. Since $f(\frac{1}{k}) = \frac{(-1)^k}{k}$ for $k=1,2,..$ we get
$$|f(\frac{1}{k}) - f(\frac{1}{k-1}) | = \frac{1}{k} + \frac{1}{k-1}$$
and
$$\sum_{1}^{n}|f(\frac{1}{k}) - f(\frac{1}{k-1})| = 2*ln(n)*(1+ o(1)) ,$$ therefore $T_f = \infty .$

Let $$F_u(x) = \begin{cases} f(x) & \text{ for x \in (\frac{2}{2u+1},1]},\\ 0 & \text{otherwise}. \end{cases}$$ Since $| F_u(x)-f(x)| \le \frac{2}{2u+1}$ for all $x \in [0,1]$, it follows that $F_j\rightarrow f$ uniformly, and pointwise.
$F_u$ is differentiable, except at $x=\frac{2}{2u+1}$, and the derivative is bounded, hence $F_u \in BV([0,1])$. Consider the partition $P=\{ 0, \frac{2}{2u+1}, \frac{1}{u},\frac{1}{u-1},.., \frac{1}{2}, 1 \}$. Because $F_u(0)=F_u(\frac{2}{2u+1}) = 0$, the variation of $F_u$ over this partition is the same sum as above, i.e.: $$\sum_{1}^{u} \frac{1}{k} +\frac{1}{k-1} = 2*ln(u)*(1+ o(1)) ,$$ so $\liminf_{u\to \infty}T_{F_u} \ge \liminf_{u\to \infty}2ln(u) = + \infty$. We have an example where both, $\liminf_{u\to \infty}T_{F_u}$ and $T_f$ are infinite. Your re-statement of the exercise in Folland applies.

$$\dots \dots \dots$$

Let's next take
$$f_u(x)=\begin{cases} f(s)& \text{for x \in (\frac{2}{2u^2+1},\frac{2}{2u+1})},\\ 0 & \text{otherwise}. \end{cases}$$

Since $|f_u(x)| \le \frac{2}{2u+1}$, we have $f_u\rightarrow 0$ uniformly, and in hence pointwise as well. Clearly $T_0 = 0$, so the limit function is of bounded variation. Consider the partition with partion points $$0, \frac{2}{2u^2+1}, \text{\frac{1}{k}, (for k=u+1,..,u^2 )} , \frac{2}{2u+1}, 1 .$$

The variation of $f_u$ over this partition is
$$\sum_{u+1}^{u^2} \frac{1}{k} +\frac{1}{k-1} = 2*ln(u)*(1+ o(1)) ,$$
therefore $\liminf_{u\to \infty}T_{f_u} \ge \liminf_{u\to \infty}2ln(u) = + \infty$. This is an example where $\liminf_{u\to \infty}T_{f_u} = + \infty$ and $T_f$ is finite.

Either I'm missing something or everyone else is. It seems clear to me that if $$F_j\to F$$ pointwise then $$T_F\le\liminf T_{F_j}$$ without any assumption on any of the functions having bounded variation. And there's no epsilons needed either.

If $$x_0<\dots then \begin{aligned}\sum_k|F(x_k)-F(x_{k-1})| &=\lim_j \sum_k|F_j(x_k)-F_j(x_{k-1})| \\&=\liminf_j \sum_k|F_j(x_k)-F_j(x_{k-1})| \\&\le\liminf_jT_{F_j}.\end{aligned}Now taking the sup over all choices of $$x_0<\dots shows that $$T_F \le\liminf T_{F_j}$$.

The problem in the book is well posed. For simplicity, I will denote the sums over partition $P$ by $T_j(P)$ and $T(P)$, and the total variations by $T_j$ and $T$.

Given $\epsilon > 0$, choose some partition $P = \{x_0,\dots,x_n\}$ such that $T-T(P) < \epsilon$. Choose $N$ such that $|F_j(x_k) - F(x_k)| < \epsilon/2n$ for all $j>N$ and $k = 0,1,\dots,n$. Then $$T(P) < T_j(P)+\epsilon \leq T_j + \epsilon$$ for all $j>N$, hence $$T < T(P) + \epsilon \leq \liminf_{j\to \infty}T_j+2\epsilon.$$ Since $\epsilon$ is arbitrary, $T \leq \liminf_{j\to \infty}T_j$.

• By stating it is well-posed. Are you implying that my suggested corrected statement is incorrect? Feb 17, 2016 at 20:21
• @MorganWeiss Well, I do not immediately see what the consequence would be if your $F_{n+1}, \dots, F$ aren't assured to be of bounded variation. Feb 17, 2016 at 22:49
• Is your partition $P$ the same as the points $x_0 < x_1 < \ldots < x_n = x$ for $x\in\mathbb{R}$? Jul 29, 2016 at 1:41
• I think my proof is clear enough on its own, and I don't understand your attempt, if that's what you are trying to relate my proof to. Jul 30, 2016 at 23:03
• Is my attempt incorrect? As in did I make any mistakes? Jul 30, 2016 at 23:12


text:
# Deducing linear combinations of column matrices

I've a $m \times n$ column matrix $A$ that can contain values {0,0.5,1}. i.e $\Omega = \{0,0.5,1\}$

$A$ = $\begin{bmatrix} 0 & 0.5 & 0.5 & 1 & 1 & \cdots & 0 \\ 0.5 & 0 & 1 & 1 & 0 & \cdots & 1 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0.5 & 0.5 & 1 & \cdots & 0 \end{bmatrix}$

let the column matrix $x$ be equal to $\mathbb{R_0^{+ n \times 1}}$ (non-negative real numbers including 0)

$x$ = $\begin{bmatrix} 1 \\ 2 \\ 0 \\ 0.5 \\ 3 \\ \vdots \\ 0.3 \end{bmatrix}$

We know that $Ax = b$. So given $b$ we can find $x$ by $x = A \backslash b$ (performing matrix left division). 99.9% the matrix A is full rank (because it can contain only 3 values and $m$ is very large) but if not I could always use $x$ = pseudo-inverse$(A) \times b$.

My question is now if we are given $b$ and a matrix $A^{\prime}$ such that $A^{\prime} \supseteq A$ where $A^{\prime} \in \mathbb{R^{m \times \textbf{N}}}$ and $\textbf{N} \supseteq n$ and belong to the same domain as $A$ (pardon my math explanation. $A$ and $A^{\prime}$ has same rows but the columns of $A$ are $\subseteq$ of $A^{\prime}$ ), is it possible to find $x$ ? (Note that we do not need $A$, if a column in $A^{\prime}$ is involved in the linear combination it gets a value or else it can just be $0$).

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I wonder why is the down vote ? Does anybody think this is not a valid problem ? It would be helpful for learners if the down voters give a reason for their action so they can avoid repeating it. –  Sunil Feb 24 '11 at 0:56
Use linear programming. –  Yuval Filmus Feb 24 '11 at 20:22


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# Windowing

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

# Windowing Functions

WINDOWING FUNCTIONS

## 1 Spectral Leakage

Suppose we are taking a discrete Fourier transform(DFT) of a signal. When taking the DFT for the signal, we can only sample it for a finite time (the Fourier transform of a function is defined for infinite time). This leads to some unwanted features. Because we are only sampling the signals for a finite time, we need to assume something about the signal outside of this interval. The DFT implicitly assumes that the signal repeats. This causes discontinuities at the edges of the sampled signal. Fourier transforms of discontinuities lead to broad frequency structure. For an example, see figure 1 and 2. Notice the leakage of power in the neighboring spectral bins. This is "spectral leakage".

The time series signals. The top window shows the real time signal in the ’infinite’ limit. The middle window shows that actual sampled wave that we feed into the FFT. The bottom window shows what the FFT thinks the signal is : infinitely repeated sampled signal.

The Fourier transform of the sine wave that does not have sharp discontinuities (top). The FFT of the sampled sine wave, which has sharp discontinuities.

## 2 Windowing

Windowing is a way to minimize spectral leakage by tapering off the ends of the sampled waveform to 0. This is nothing new, and in fact when we take a regular ol’ FFT (the fast Fourier transform algorithm that implements a DFT efficiently), we are in fact multiplying by a windowing function. This windowing function is a square windowing function which is described as being 1 if in the sampled window, and 0 otherwise. Hence, we can write the Fourier transform as

${\displaystyle {\hat {f}}=\int {W(t)f(t)e^{-2\pi {j}\omega {t}}}={\hat {W}}\star {\hat {f}},\,\!}$

where ${\displaystyle \star }$ is the convolution function and ${\displaystyle {\hat {}}}$ denotes the Fourier transform. ${\displaystyle W}$ is the windowing function. Note that the windowing function can either be defined in time space or frequency space. The same term refers to both spaces. The result above follows from the convolution theorem.\

As we noted above, the sharp discontinuities at the ends of the sampled waveform are what cause spectral leakage. To minimize this we can use a different windowing function that tapers off to zero at the ends. This will minimize discontinuities. Some popular windowing functions include the Hanning and Hamming windowing functions, see figure 3 and 4. \

The hanning window and its frequency response. The frequency response is the Fourier transform of the window function on a test tone.

The hamming window and its frequency response. The frequency response is the Fourier transform of the window function on a test tone.

One way to think about spectral leakage is that it is a spreading out of energy/power over spectral bins by the windowing function. The contribution of the spectral bin to its neighbors is the weight of the window function (in frequency space) centered at the given frequency component evaluated at the FFT bin in question. Another way to think about it is to think about the FFT as a series of filters centered on a specific spectral channel. Then the FFT filters response at that spectral channel is given by the windowing function (in frequency space). Therefore, each spectral bin includes signal from all other frequencies in the filters bandwidth.\

## 3 Noise Equivalent Bandwidth

Note that for non square window functions are tapering off at the ends and hence we are in effect throwing away information about the signal. Specifically, we are decreasing signal to noise. One way to quantify this, is the talk about the Noise Equivalent Bandwidth : the bandwidth (of the non rectangular windowed function) that gives us the same amount of noise power as the rectangular window did. Numerically,

${\displaystyle NEB={\frac {B}{B_{eq}}}\,\!}$

There is also a concept of the Signal Equivalent bandwidth. By the radiometer equation, this is just given by the square root of the Noise Equivalent Bandwidth. Hence the NEB is always greater than the SEB. Therefore, there is a crucial trade off for windowing functions. The narrower your window is, the greater your NEB is and hences the lower your signal to noise is going to be. Depending on your needs, you need to pick your windowing function wisely.