# Research Area(s)

- Quantum Harmonic Analysis
- Applications of Mathematics in Quantum Technologies 2.0
- Post-Quantum Cryptography

# Research Topics

- Quantum Systems: theory, modelling and simulation; Complex Networks; Operator Equations
- Harmonic analysis of signals; Riemann's zeta and Broadband Redundancy; fast-transform bases; nonlinear oscillators
- Data Encryption

# Current and Recent Graduate Students

*Elias Hassani Sangani (*Ph.D. student, co-supervised by Dr. Francis Bui and Dr. Jonathan Norton) is conducting research in cryptography and cybersecurity.*Madeline Berezowski (*Ph.D. student) is exploring new ways of analyzing quantum observables and their dynamics.*Mandana Bidarvand*(Ph.D. student, co-supervised by Dr. Shahedul Khan) is investigating the mathematics of infinite arrays of qubits.*Elias Hassani Sangani*successfully defended his M.Sc. thesis "An Information Security System for Image Encryption Applications: Architecture and Performance Evaluation" on 21 August 2023. Elias' research work has been co-supervised by Dr. Francis Bui (College of Engineering) and Dr. Jonathan Norton (College of Medicine).*Elias w*ill continue as a PhD student at the College of Engineering at USask, and will work under the joint supervision of these three supervisors.*Qi Guo*successfully defended his M.Sc. thesis "Logistic Operator Equation and the Induced Stochastic Process for Complex System Modelling" on 21 August 2018.*Izabela Vlahu*successfully defended her PhD thesis "The Orthogonal Band Decomposition of the Finite Dirichlet Matrix and Its Applications" on 13 December 2017. Izabela's research work was co-supervised by Prof. Paul Babyn (then @ Department of Medical Imaging, U of S).*Arash Shamloo*successfully defended his PhD thesis "Mathematical Modelling of Nano-Electronic Systems" on 11 December 2014.

# Current and Recent Teaching

Math 436/818 Mathematical Modelling II (T2 2023-24), a course focused on quantum system modelling

Math 818: The Q+ Seminar (T1, 2023-24), a graduate research seminar focused on quantum harmonic analysis

Math 438.3/838.3 Methods of Applied Mathematics II, (T2 2022-23), a course focused on operator theory

Math 436.3/818.3 Mathematical Modelling II (T2 2021-22), a course focused on quantum system modelling

# Recent preprints

- Analysing matrix operators via harmonic analysis on [the group of positive rationals]

# Peer-Reviewed Publications

- (with Jonas Fransson) Solving the Bose-Hubbard model in new ways, Quantum 6, 728 (2022)
- Toral Diffeomorphisms Induce Quantum Superoperators via TAQS: Kilgour, D.M., Kunze, H., Makarov, R., Melnik, R., Xu Wang, S. (Eds.), Recent Developments in Mathematical, Statistical and Computational Sciences; The V AMMCS International Conference, Waterloo, Canada, August 18-23, 2019; Springer Proceedings in Mathematics and Statistics 2021
- (with Qi Guo) An almost solvable model of complex network dynamics, Russian Journal of Mathematical Physics, Vol. 27 (2020), 469-477
- (with Patrick Navez and Alexandre Zagoskin) Entangling continuous variables with a qubit array, Physical Review B, Vol. 100 (2019) 144506
- (with A.M. Zagoskin) An exactly solvable quantum-metamaterial type model, Journal of Physics A: Mathematical and Theoretical (Institute of Physics), Vol. 52 (2019) 395304
- A nonlocal transform to map and track quantum dynamics, Journal of Physics A: Mathematical and Theoretical (Institute of Physics), Vol. 52 (2019) 305301
- Riemann's zeta function and the broadband structure of pure harmonics, The IMA Journal of Applied Mathematics (Oxford Academic), Vol. 82 (2017), 1238-1252
- On the Dirichlet matrix operators in sequence spaces, Applicationes Mathematicae (Institute of Mathematics, Polish Academy of Sciences), 44,2 (2017), 185-196
- Image processing via simulated quantum dynamics, Random Operators and Stochastic Equations (De Gruyter), Vol. 25, No. 1 (2017), 27-39
- (with M. J. Everitt, J. H. Samson, S. E. Savel'ev, A. M. Zagoskin, S. Heidel, J. C. Zúñiga-Anaya) Recursive simulation of quantum annealing, Journal of Physics A: Mathematical and Theoretical, Vol. 48 (2015) 415301.
- Interacting Bose gas, the logistic law, and complex networks, Russian Journal of Mathematical Physics, Vol. 22, No 1 (2015), 98-111
- (with Y. Guo and G.-F. Zheng) Existence and Asymptotic Behavior of Solutions for Nonlinear Maxwell Equations Arising in Mesoscopic Electromagnetism, Journal of Nonlinear Analysis B: Real World Applications, 20C (2014), 99-111, DOI: 10.1016/j.nonrwa.2014.05.002
- (with A. Shamloo) A numerically efficient approach to the modelling of double-Qdot channels, Nanoscale Systems: Mathematical Modeling, Theory and Applications ("NanoSystems MMTA"). Volume 2, 145-156, ISSN (Online) 2299-3290, DOI: 10.2478/nsmmt-2013-0009, September 2013
- The Dirichlet ring and unconditional bases in L2[0,2pi], Functional Analysis and Its Applications, Vol. 47, No. 3, 227-232, 2013; Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 47, No. 3, pp. 75-81, 2013
- Signals generated in memristive circuits, Nanoscale Systems: Mathematical Modeling, Theory and Applications ("NanoSystems MMTA"). Volume 1, pp. 48-57, ISSN (Online) 2299-3219, DOI: 10.2478/nsmmt-2012-0004, November 2012
- Factorizing matrices by Dirichlet multiplication, Linear Algebra and its Applications, Vol. 438, No. 5 (2013), 2385-2393
- On an eigenvalue problem with a reciprocal-linear term, Waves in Random and Complex Media, Vol. 22, No. 2 (2012), 186-206, doi: 10.1080/17455030.2011.636085
- Spectra of Nonlocally Bound Quantum Systems, Russian Journal of Mathematical Physics, Vol. 18, No.2 (2011), 227-241, doi:10.1134/S1061920811020117
- A fast-transform basis with hysteretic features, IEEE Conference Proceedings: Electrical and Computer Engineering (CCECE), 2011 24th Canadian Conference on, 8-11 May 2011, pp.000253-000257, doi: 10.1109/CCECE.2011.6030449
- Evolving states in nonlocal type dynamics of composite systems, Journal of Geometry and Physics, Vol. 61 (2011), 26-36, doi:10.1016/j.geomphys.2010.09.006
- The Problem of Extrapolation of Quantum States, Advanced Studies in Theoretical Physics, Vol. 4, 2010, No. 12, 575 - 585
- Exploiting the Paradigm of Quantum Information in Nanosystem Modeling, AIP Conference Proceedings Vol. 1148 (August 2009), 233-236
- Stationary states in nonlocal type dynamics of composite systems, Journal of Geometry and Physics, Vol. 59 (2009), 1604-1612, doi:10.1016/j.geomphys.2009.07.015
- A model for nonlocal bonding in bipartite systems, Journal of Modern Optics, Vol. 56, Issue 12 (2009), 1363-1368
- Quantum entanglement in composite systems, Theoretical and Mathematical Physics, Vol. 159 (2009) 654-666
- Modeling a quantum Hall system via elliptic equations, Advances in Mathematical Physics, Vol. 2009, Article ID 514081, 9 pages
- Parametric models of the Hall effect in the strongly correlated regime, Waves in Random and Complex Media, Vol. 19, No. 1 (2009), 5-10
- Fractional Quantization of Hall Resistance as a Consequence of Mesoscopic Feedback, Russian Journal of Mathematical Physics, Vol. 15, No.1 (2008), 122-127
- Sensitivity to lattice structure in the mesoscopic-loop models of planar systems, Advanced Studies in Theoretical Physics, Vol. 1 (2007), 433-448
- Integrability in the mesoscopic dynamics, Journal of Geometry and Physics, Vol. 55, No.1 (2005), 1-18
- Mesoscopic mechanics, Journal of Physics and Chemistry of Solids, Vol. 65/8-9 (2004), 1507-1515, Special issue: Inhomogeneous and strongly correlated materials and the study of matter at extreme conditions
- Novel nondiffusive flows with applications to image enhancement, International Journal of Computational and Numerical Analysis and Applications, Vol. 3, No. 3 (2003), 325-349
- The (fully) Nonlinear Maxwell Theory delineated, Journal of Geometry and Physics, Vol. 45, No. 1-2 (2003), 54-74
- Nonlinear Maxwell Theory and Electrons in Two Dimensions, Communications in Mathematical Physics, Vol. 226 (2002), 559-566
- (with R. R. Coifman) New methods of controlled total variation reduction for digital functions, SIAM Journal on Numerical Analysis, Vol. 39, No. 2 (2001), 480-498
- (with R. R. Coifman) Combining the calculus of variations and wavelets for image enhancement, Applied and Computational Harmonic Analysis, Vol. 9 (2000), 1-18
- On an equation arising from the geometry of Riemannian submersions, Journal fur die reine und angewandte Mathematik, Vol. 514 (1999), 1-8
- High Accuracy Multiwavelets with Short Supports, IEEE Transactions on Signal Processing, Vol. 46, No. 12 (1998), 3420-3422
- Magnetic Oscillations and Maxwell Theory, Physics Letters A, Vol. 228 (1997), 347-350

# Other publications

- Qualitative Analysis of Quantum Signals, CMS Notes, Vol. 49, No. 5 / November 2017
- The Quantum Sieve of Eratosthenes, J Phys Math 7:180. doi:10.4172/2090-0902.1000180
- (with Alexandre Zagoskin) Can We See or Compute our Way out of Plato’s Cave?, J Appl Computat Math 3: 192. doi:10.4172/2168-9679.1000192
- Complexity in Mathematics and Complexity of a Physical Process, J Appl Computat Math 2: e134. doi:10.4172/2168-9679.1000e134

# Patents

- Method of Enhancement of the Visual Display of Images and Other Visual Data Records, US Patent Number 7194143, issue date 3/20/07.
*The patent can be looked up here.*

# Recent Invited Talks and Conference Presentations

- Analysing operators that evade traditional computational approaches, Advanced Research & Technology Forum, MathWorks (Boston, MA, USA), 10 January 2024 (remotely)
- The physics of an array of boson sites expressed via the Dirichlet series, Friday Physics seminar, Department of Physics and Astronomy, Uppsala University, 8 December 2023
- Applying group duality in the analysis of operators, Department of Mathematics, Uppsala University, The PDEs and Applications seminar, 5 December 2023
- New applications of harmonic analysis to quantum theory and engineering (version 2), Analysis of Complex Physical Systems Seminar, Department of Mathematics, University of Texas at Austin, 6 April 2022 (remotely)
- New applications of harmonic analysis to quantum theory and engineering, Department of Physics and Engineering Physics, University of Saskatchewan, 25 January 2022
- Quantum applications of harmonic analysis on the group of positive rationals, 75th+1 CMS Summer Meeting, Ottawa (remote platform), 7-11 June 2021
- (P. Navez, A. Sowa, A. Zagoskin) Entangling continuous variables with a qubit array, The fifth AMMCS International Conference (AMMCS2019), Waterloo, Ontario, 18-23 August 2019
- Perturbing Hamiltonians via automorphism of the torus, The fifth AMMCS International Conference (AMMCS2019), Waterloo, Ontario, 18-23 August 2019
- Newer nonlocal transforms and their applications in quantum engineering, Saskatoon Summer Symposium: Quantum Engineering Meets Harmonic Analysis, Saskatoon, Saskatchewan, 12-16 August 2019
- Qubits, wavelets, fractals, bands, CMS 2019 Regina, Regina, Saskatchewan, 7-10 June 2019
- Mathematics and Industrial Enterprises...Activities Joined at the Hip (?), Keynote@ Canadian Undergraduate Mathematics Conference, CUMC 2018, Saskatoon, Saskatchewan, 11-15 July 2018
- The Relevance of Harmonic Analysis in the Analysis of Quantum Systems, The Second International Conference on Mathematical Characterization, Control, Analysis and Applications of Complex Information and Systems (2nd CMCAA 2018), Beijing, China, 30 June-1 July 2018
- Harmonic Analysis in Quantum Engineering 2.0, School of Mathematics, Beijing Institute of Technology, Beijing, China, 27 June 2018
- The Role of Harmonic Analysis in Quantum Engineering, the Landau Seminar at the Department of Physics, Loughborough University, Loughborough, Leicestershire, England, 21 June 2018
- Deploying Harmonic Analysis in Quantum and Nano Engineering, The Fields Institute, Focus Program on Nanoscale Systems and Coupled Phenomena, Mathematical Analysis, Modelling, and Applications, Toronto, Ontario, 14 May 2018; watch it here
- The Broadband Redundancy and Its Applications, the Landau Seminar at the Department of Physics, Loughborough University, Loughborough, Leicestershire, England, 19 October, 2017
- (A. Sowa, A. Zagoskin, and A. Balanov) A Nonlinear Master Equation Model for Quantum Metamaterials, The fourth AMMCS International Conference (AMMCS2017), Waterloo, Ontario, 20-25 August 2017
- (B-Z. Li, A. Sowa) Numerical Methods for Analysis and Representation of Quantum States Associated with Integral Transform, The fourth AMMCS International Conference (AMMCS2017), Waterloo, Ontario, 20-25 August 2017
- (S.A. Melli, A.P. Sowa, K.A. Wahid) Image Denoising via Redundant Quantum Channels, The fourth AMMCS International Conference (AMMCS2017), Waterloo, Ontario, 20-25 August 2017
- (Q. Guo, A. Sowa) Examples of Quantum-dynamical Nonlaws and Their Properties, The fourth AMMCS International Conference (AMMCS2017), Waterloo, Ontario, 20-25 August 2017
- Qualitative Analysis of Quantum States and Observables, the Landau Seminar at the Department of Physics, Loughborough University, Loughborough, Leicestershire, England, 28 June, 2017
- Harmonic Analysis of Quantum States and Observables, EMN Quantum 2017, Vienna, Austria, 18-22 June, 2017
- Searching for a spread-spectrum approach to some inverse problems, 10th International Conference on Scientific Computing and Applications (ICSCA 2016), The Fields Institute, Toronto, Ontario, 6-10 June 2016
- Towards Quantum Image Processing, Saskatoon mini-Symposium: Applications of Quantum Engineering to Magnetic Resonance Imaging, Saskatoon, Saskatchewan, 18-19 August 2014
- Linear Operators for the Analysis of Nonlinear Systems, 11th World Congress of Computaitonal Mechanics, 5th European Conference on Computaitonal Mechanics, Barcelona, Spain, 20-25 July 2014
- Applying simulated quantum dynamics, Canadian Applied and Industrial Mathematics Society 2014 Annual Meeting, Saskatoon, Saskatchewan, 22-26 June 2014
- Emerging applications of simulated quantum dynamics, International Focus Workshop on Metamaterials and Quantum Criticality, Loughborough, Leicestershire, UK, March 25, 2014
- From nonlinear eigenvalue problems to fast transforms, number theoretic operators, and special quantum states, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland, November 21, 2013
- Nonlinear oscillators and number theoretic quantum states, the Landau Seminar at the Department of Physics, Loughborough University, Loughborough, Leicestershire, England, October 30, 2013
- Mathematical analogies between the eigenmodes of memristive nanosystems and some cryptography-specialized quantum states, BIT's 3rd Annual World Congress of Nanoscience and Nanotechnology-2013, Xi'an, China, September 28, 2013
- Mathematical Models of Complex Phenomena, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, China, September 24, 2013
- Operators resulting from the Dirichlet series, 2012 Canadian Mathematical Society Summer Meeting, Regina, Saskatchewan, June 2-4, 2012
- Modelling with the Logistic Operator Equation, 3rd International Conference on Econophysics, Loughborough, England, September 23-29, 2011
- A Fast-Transform Basis with Hysteretic Features, 24th Canadian Conference on Electrical and Computer Engineering, Niagara Falls, Ontario, May 8-11, 2011
- Nonlinear eigenvalue problems with applications to quantum systems, 2010 Canadian Mathematical Society Winter Meeting, Vancouver, British Columbia, December 4-6, 2010
- (F. Sage, A. Sowa), Numerical Quantum Chaos in Nonlocal Quantum Dynamics, presented (by FS) at the 1st Meeting of the Canadian Prairie Theoretical Physics Network, Lethbridge, August 25, 2010
- Nonlinear quantum spectroscopy, OTAMP 2010, Fifth International Conference on Operator Theory Analysis and Mathematical Physics, Bedlewo, Poland, August 5 - 12, 2010
- Nonlinear eigenvalue problems and quantum systems, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland, July 27, 2010
- Quantum spectroscopy, nonlinear operator equations, etc., Fourth Annual Meeting of the Prairie Network for Research in Mathematical Sciences, Manitoba, April 30-May 2, 2010
- On the topological characteristic of the spectra of some composite quantum systems, 2009 Canadian Mathematical Society Winter Meeting, Windsor, Ontario, December 5-7, 2009
- On the noncommutative Schrödinger equation, Third Annual Meeting of the Prairie Network for Research in Mathematical Sciences, Saskatchewan, April 29-May 1, 2009
- On the Stationary Solutions of the Noncommutative Schrödinger Equation, The Sixth IMACS International Conference on: Nonlinear Evolution Equations and Wave Phenomena, Georgia, USA, March 23-26, 2009
- Exploiting the paradigm of quantum information in nanosystem modeling, The Sixth International Conference on Computational Methods of Science and Engineering, ICCMSE 2008, Greece, Sept. 25-30, 2008