Polar Integers and Polar Integer Optimization
Abstract
This is a pioneering work, introducing a special class of complex numbers, wherein their absolute values and arguments given in a Polar coordinate system are integers, which when considered within the complex plane, constitute Unicentered Radial Lattice and similarly for quaternions and Euclidean R2 and R3 Spaces. The corresponding Optimization Problems are introduced as well.References
M. Agrawal, N. Kayal and N. Saxena, PRIMES is in P, Annals of Mathematics, vol. 160, pp. 781–793, 2004.
S. Bernstein, U. Kahler, I. Sabadini and F. Sommen, Hypercomplex Analysis: New Perspectives and Applications, Birkhauser, 2014.
L. M. B. C. Campos, Complex Analysis with Applications to Flows and Fields, CRC Press, 2011.
J. Coolidge, The Origin of Polar Coordinates, American Mathematical Monthly, Mathematical Association of America, vol. 59, no. 2, pp. 78–85, 2018.
T. Cormen, C. Leiserson, R. Rivest and C. Stein, Introduction To Algorithms, The MIT Press, Cambridge, 2009.
C. A. Floudas and P. M. Pardalos, Encyclopedia of Optimization, Springer, New York, 2009.
I. Frenkel and M. Libine, Quaternionic analysis, representation theory and physics, Advances in Mathematics, vol. 218, pp. 1806–1877, 2008.
B. Green and T. Tao, Linear equations in primes, Annals of Mathematics, vol.171, pp. 1753–1850, 2010.
R. Hemmecke, M. Koppe, J. Lee and R. Weismantel, ¨ Nonlinear Integer Programming, in 50 Years of Integer Programming 1958–2008: The Early Years and State-of-the-Art Surveys (eds. M. Junger, T. Liebling, D. Naddef, W. Pulleyblank, W. Reinelt, G. Rinaldi and Wolsey), Springer-Verlag, Berlin, pp. 561–618, 2010.
G. James, Modern Engineering Mathematics, Trans-Atlantic Pubns Inc., 2015.
I. Kleiner, From Numbers to Rings: The Early History of Ring Theory, Elem. Math, Birkhauser, Basel, vol. 53, pp. 18–35, 1998.
E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons Inc., 2011.
P. Moon, D. E Spencer, Circular-Cylinder Coordinates (r, ψ, z). Field Theory Hand-book, Including Coordinate Systems, Differential Equations, and Their Solutions, (corrected 2nd ed.). New York City: Springer-Verlag., pp. 12–17, 1989.
P. Moon, D. E Spencer, Spherical Coordinates (r, θ, ψ). Field Theory Hand-book, Including Coordinate Systems, Differential Equations, and Their Solutions, (corrected 2nd ed., 3rd print ed.). New York City: Springer-Verlag., pp. 24–27, 1988.
V. Neale, Closing the Gap: The Quest to Understand Prime Numbers, Oxford University Press, 2017.
V. Scheidemann, Introduction to complex analysis in several variables, Birkhauser, 2005.
W. T. Shaw, Complex Analysis with Mathematica, Cambridge, 2006.
Y. Shipilevsky, Complex and quaternionic optimization, Numerical Algebra, Control and Optimization, . vol. 10, no. 3, pp. 249–255, 2020.
Y. Shipilevsky, Prime Optimization, Statistics, Optimization, & Information Computing, vol. 9, no. 2, pp. 453–458, 2021.
V. Scheidemann, Introduction to complex analysis in several variables, Birkh¨auser, 2005.
W. T. Shaw, Complex Analysis with Mathematica, Cambridge, 2006.
Y. Shipilevsky, Complex and quaternionic optimization, Numerical Algebra, Control and Optimization, . vol. 10, no. 3, pp. 249–255, 2020.
Y. Shipilevsky, Prime Optimization, Statistics, Optimization, & Information Computing, vol. 9, no. 2, pp. 453–458, 2021.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).