Never give up!
BROAD AREA(S) OF INTEREST--FLUID MECHANICS, ANALYTIC NUMBER THEORY, TOPOLOGY
MathScinet author ID 778677

#### Research Accomplishment

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

A. 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. IF2010~1.089, SCI indexed Journal; MathSciNet Link Cited in: EPJ, Microgravity Sci. Tech, IOSR 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 Convection, Phys. Fluids (AIP), 21, 6, (2009), 064105-16. SCI indexed Journal Cited in:JMMM, Transp. Por. Med., Acta Mech., Mincrogravity Sci. Tech.,Phys. Fluids, Int. Jour. of Heat and Mass Transfer., Ph.D.Thesis , JMMM, Aerospace Sci. and Tech., Phys. Rev. E, Microgravity Sci. Tech, Aerospace 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 in: Phys. 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
BAnalytic 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 convolution, J. 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), 41-53. MathSciNet 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 #A242225, The On-Line Encyclopedia of Integer Sequences.
7. Jitender Singh, Sequence #A241885, The On-Line Encyclopedia of Integer Sequences.
C. General Mathematics
1. (New)Jitender Singh. A Short Proof that Lebesgue Outer Measure of an Interval is Its Length. American Mathematical Monthly (2017) Accepted for publication
2. (New)Jitender Singh. A Noninductive Proof of de Moivre's Formula. American Mathematical Monthly (2017) Accepted for publication
3. Jitender Singh. Another Proof of the Binomial Theorem. American Mathematical Monthly, 124, (2017), 658.
4. Jitender Singh and Harpreet K. Grover. Limit and Continuity revisited via Convergence (Unpublished)

#### Impact Factors of some Journals

 Journal Impact factor 2014 Impact Factor 2015-16
 American Mathematical Monthly 0.349 International Journal of Number Theory 0.318 0.463 Journal of Integer Sequences 0.55
 Physical Review E 2.288 2.252 Physics of Fluids 2.031 2.017 Fluid Dynamics Research 0.99 0.846 Meccanica 1.949 1.982 Applied Mathematics and Computation 1.551 1.345 ANZIAM Journal 1.025 0.957 Journal of magnetism and magnetic materials 1.97 2.357 International Journal of Thermal Sciences 2.769 3.615

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

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."Touring the Calculus Gallery", AMM
6. K. Janich. "Topology''
7. N. Bourbaki, "Elements of Mathematics-General Topology"(English Translation)
8. Armstrong, "Basic Topology"
9. J. Conway. "Functional Analysis"
10. Landau Lifshitz, "Fluid Dynamics"
11. T. M. Apostol, "Introduction to Analytic Number Theory"
12. Dummit and Foote, "Modern Algebra"
13. J. Munkres, "Topology"
14. J. Munkres, "Elements of Algebraic Topology"
15. S. Morris, "Topology without Tears"
16. W. Rudin, "Principles of Mathematical Analysis"
17. Ahlfors, "Complex Analysis"
18. J. Stillwell, "Mathematics and its History"
19. K. S. Sarkaria,"Topological Work of Henri Poincare"
20. K. S. Sarkaria, "A topological paradox of Motion"
21. S. Chandraskhar, "Hydrodynamic and Hydromagnetic Stability"
22. M. Spivak, "Calculus on Manifolds"
23. L. S. Pontryagin, "Selected Works: Topological Groups Vol. 2"
24. M. Farkas, "Periodic Motions"
25. T. Tao, "Analysis I & II"
26. M. Ahmed, "How many squares are there Mr. Franklin ?"
27. W. H. Gustafson,"What is the probability that two group elements commute ?"

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