Doing mathematics is a long-term thing. I've had graduate students who said, "OK", I'm doing my PhD, and at the end of the four years, I'll have learnt everything I need to know, and I'll be a leader in the field". It doesn't work that way! You have to work through undergraduate, and through graduate, and even after you finish, there is still a lot more to learn. Mathematics is huge. You have to keep pushing yourself and not be content with doing just one or two things and sit in this niche of mathematics and never venture out of it, if you want to really progress. I'd describe it as like running a marathon. You can't just sprint right through it. You have to keep learning, and really enjoy doing mathematics. If you don't enjoy it, you won't have the stamina to keep at it. But it is very rewarding if you keep at it.    ----Terence Tao
Science is merely an illusion-A. Einstein
                                            Questions Of the Month...!                     

                                                                                                                                                                                                                        

                                                                                                                                                                                                                        

     BROAD AREA(S) OF INTEREST--FLUID MECHANICS, ANALYTIC                      NUMBER THEORY, TOPOLOGY
MathScinet author ID 778677
                    
        
Research 
     Hydrodynamic stability
My specialization is in numerical computing in Hydrodynamical systems, such as temperature-modulated Rayleigh-Benard convection(TMRBC), Taylor-Couette flow in ferrofluids, Dean instability in ferrofluids etc. We explored TMRBC for bicritical states using Fourier-Floquet analysis. Some nonlinear calculations on patterns of TMRBC have also been done using FEM by modeling the problem in a finite rectangular box of aspect ratio 10[See Kaur and Singh]. Energy stability analysis of transient convection in ferrofluids has also been done by us. Currently, we are exploring on temperature-modulated convection in a vertical fluid layer for bicritical states, using the methods developed for dealing with the TMRBC. 

     Analytic Number Theory

Motivated by the marvelous work of the Mathematical incarnate Leonhard Euler during the eighteenth century, I have worked on investigating the properties of Power Sums of integers via defining them in a more general setting. The motivation behind this work is the following simple identity $1+2+\ldots=\frac{n(n+1)}{2}$. Some new interesting analytical results are obtained. Some power sums of the Euler Phi function are also studied.  

    
 General Mathematics

Besides, we have contributed in General Mathematics, on new proofs of well known fundamental theorems such as binomial theorem, de Moivre's formula, Cauchy type product, subgroups of real line etc.

Contributions

     Hydrodynamic stability:

  1. Puneet Kaur and  Jitender Singh. Heat transfer in a thermally modulated fluid layer. Inter. Jour. Ther. Sci. (Elsevier), 114C (2017), 35-43. Link
  2. Puneet Kaur, Jitender Singh, Renu bajaj. “RayleighB\'enard convection with two frequency temperature modulation”, Phys. Rev. E, 93, (2016), 043111.
  3. Jitender Singh and Renu Bajaj. “Dean Instabilityin ferrofluids”, Meccanica, 51 (2016), 835-847
  4. Jitender Singh and Renu Bajaj. Bicritical states in temperature modulated Rayleigh Benard convection. Phys.  Rev. E, 92 (2015), 013005.
  5. Jitender Singh and S. S. Singh. Instability in modulated Rotating Rayleigh-Benard convection. Fluid Dyn. Res.(IOP) 46(1) (2014),  015504.
  6. Jitender Singh. A nonlinear shooting method and its application to nonlinear Rayleigh-B\'enard convection, ISRN Mathematical Physics (2013), 6502081-9.     
  7. Singh, S. S. and Jitender Singh. Effect of Corrugation on Incident qSV-waves in pre-stressed elastic half-spaces, International Journal of Applied Mathematics and Mechanics, 9(9), (2013), 92-106 
  8. Jitender Singh, E. L. Hines, and Daciana Iliescu. Global stability results for temperature modulated convection in ferrofluids. Appl. Math. Comp. 219 (2013), 6204-6211.  
  9. Jitender Singh and Renu Bajaj. Convective instability in a ferrofluid layer with temperature modulated rigid boundaries. Fluid Dyn. Res. (IOP), 43(2011), 025502. IF2010~1.089SCI indexed Journal; MathSciNet  Link  Cited in: EPJ, Microgravity Sci. TechIOSR Journal of Applied Physics
  10. Jitender Singh. Energy-relaxation for transient convection in ferrofluids, Phys. Rev. E (APS), 82, 2 (2010), 026311-9 IF2009~2.400, IF2010~2.352 SCI indexed Journal Cited in: Proc. of Royal Soc. A (2012) ,  Phys. Rev. E (2012)
  11. Jitender Singh and Renu Bajaj. Temperature Modulation in Ferrofluid ConvectionPhys. Fluids (AIP),  21, 6, (2009), 064105-16. SCI indexed Journal Cited in:JMMMTransp. Por. Med., Acta Mech., Mincrogravity  Sci. Tech.,Phys. FluidsInt. Jour. of Heat and Mass Transfer.Ph.D.Thesis , JMMMAerospace Sci. and Tech., Phys. Rev. EMicrogravity Sci. TechAerospace Sci. Tech.IOSR Journal of Applied Physics
  12. Jitender Singh and Renu Bajaj. Thermal modulation in Rayleigh-Benard convection,  The ANZIAM J. (Cambridge University Press)50, 2, (2008), 231-245. IF2008~0.380, IF2009~0.286, IF2010~0.414 MathSciNet  Link 
  13. Jitender Singh and Renu Bajaj. Parametric modulation in the Taylor-Couette ferrofluid flow, Fluid Dyn. Res.(IOP), 40, (2008), 737-752. MathSciNet Link SCI indexed Journal (Poster: ICIAM 2007)Cited in: Physics Prodcedia
  14. Jitender Singh and Renu Bajaj. Nonaxisymmetric modes of Couette-Taylor instability in ferrofluids with radial flow, Magnetohydrodynamics, 42, (2006), 57-68. IF~0.457
  15. Jitender Singh and Renu Bajaj. Stability of ferrofluid flow in rotating porous cylinders with radial flow, Magnetohydrodynamics, 42, (2006), 41-56. IF~0.457
  16. Jitender Singh and Renu Bajaj. Stability of nonaxisymmetric ferrofluid flow in rotating cylinders with magnetic field, Int. J. Math. Math. Sci.23 (2005) 3727-3737.  MathSciNet Link; Cited inPhys. Rev. (2009)Phys. Rev. (2010)
  17. Jitender Singh and Renu Bajaj. Couette flow in ferrofluids with magnetic field, J. Magn.  Magn. Mat., 294 (2005), 53-62 IF2008~1.283, IF2009~1.204, IF2010~1.689 SCI indexed Journal Cited in: Phys. Rev E (2009) ; Phys. Rev. E (2010) ; Physics Prodcedia (2010)Physical Review E (2013), Open Journal of Fluid Dynamics
             
     Analytic Number Theory
  1. Jitender Singh. Sums of products of power sums, J. Integer Sequences, 19 (2016) Article 16.1.2. 
  2. Jitender Singh. On an arithmetic convolutionJ. Integer Sequences, 17 (2014) Article 17.6.7. 
  3. Jitender Singh. Sums of products involving power sums of $\varphi(n)$ integers. J. numbers (2014) pp 1-6

  4. Jitender Singh. Defining Power sums of n and φ(n) Integers, Int. Jour. Num. Th., 5, (1), (2009), 41-53MathSciNet Link.  ZENTRABLATMath Link, IF2008~0.473, IF2010~0.395 (Presented in the International Congress of Mathematicians (ICM-2010) held in Hyderabad (HICC) on August 21, 2010). Cited in: International Journal of Number theory(2012), The Ramanujan Journal (2012), arXive(2013)arXive2013 , Preprint(2013), arXiv:1306.5848v1 2013, Advances in Applied Mathematics(2013)
  5. Jitender Singh. Subgroups of the additive group of real line. arXiv1312.7067v1 (2013), pp 6.

  6. Jitender Singh, Sequence #A242225The On-Line Encyclopedia of Integer Sequences.

  7. Jitender Singh, Sequence #A241885The On-Line Encyclopedia of Integer Sequences.

           
    
 General Mathematics
  1. (New)Jitender Singh. A Noninductive Proof of de Moivre's Formula. American  Mathematical Monthly (2017Accepted for publication
  2. (New)Jitender Singh. Another Proof of the Binomial TheoremAmerican Mathematical Monthly, 124, (2017), 658.

  3. (New)Jitender Singh and Harpreet K. Grover. Limit and Continuity revisited via Convergence (Unpublished)

                  Impact Factors of some Journals
            (I will have to use this info for promotion purpose!): 
                                                                                                         (2014)    (2015/16)      
                    Physical Review E----------------------------------2.288         2.252
                    Physics of Fluids-----------------------------------2.031           2.017
                    Fluid Dynamics Research-------------------------0.990         0.846
                    Meccanica-------------------------------------------1.949          1.828
                    Applied Mathematics and Computation-------- 1.551           1.345
                    International Journal of Number Theory-------0.318          0.463
                    Journal of Integer Sequences---------------------0.000        0.550
                    Journal of magnetism and magnetic materials-1.970          2.357
                    ANZIAM-------------------------------------------- 1.025          0.957
                    International Journal of Thermal Sciences------------------ 2.769
                    American Mathematical Monthly---------------------------- 0.349


I have extreme interest in learning Topology, and in future, I wish to contribute to this area of Mathematics. 

                                                                                                                                                                                                                        

I wish good luck and happy learning to you (visitor) and I request you to wish the same for me!
      Recommended readings 
           Here is a list of some of my favorite books/articles I often love to read:
  1. W. P. Johnson. The curious history of Faa di Bruno's Formula
  2. E. Bombier. Problem of the millennium: The Riemann Hypothesis.
  3. Jonathan. Sondow Ramanujan, Robin, Riemann Hypothesis and Recent Results
  4. W. Dunham. "Euler: The Master of Us All". (My Favorite Author!)
  5. W. Dunham. "The Calculus Gallery" 
  6. W. Dunham."Touring the Calculus Gallery", AMM
  7. K. Janich. "Topology'' 
  8. N. Bourbaki, "Elements of Mathematics-General Topology"(English Translation)
  9. Armstrong, "Basic Topology"
  10. J. Conway. "Functional Analysis"
  11. Landau Lifshitz, "Fluid Dynamics"
  12. T. M. Apostol, "Introduction to Analytic Number Theory"
  13. Dummit and Foote, "Modern Algebra"
  14. J. Munkres, "Topology"
  15. J. Munkres, "Elements of Algebraic Topology"
  16. S. Morris, "Topology without Tears"
  17. W. Rudin, "Principles of Mathematical Analysis"
  18. Ahlfors, "Complex Analysis"
  19. R. Bird, "Transport Phenomena"
  20. A. Einstein, "The Meaning of Relativity"
  21. J. Stillwell, "Mathematics and its History"
  22. K. S. Sarkaria,"Topological Work of Henri Poincare"
  23. K. S. Sarkaria, "A topological paradox of Motion"
  24. S. Chandraskhar, "Hydrodynamic and Hydromagnetic Stability"
  25. M. Spivak, "Calculus on Manifolds"
  26. L. S. Pontryagin, "Selected Works: Topological Groups Vol. 2"  
  27. M. Farkas, "Periodic Motions"
  28. T. Tao, "Analysis I & II"
  29. M. Ahmed, "How many squares are there Mr. Franklin ?"
  30. W. H. Gustafson,"What is the probability that two group elements commute ?"