Hypercomplex Numbers

New Approaches to Continuous and Discrete Signals by Using Hypercomplex Numbers

  • The advantage of the use of hypercomplex numbers is due to their structure of commutative and associative algebras of finite order over the field of real numbers. They are used in the representation of continuous and discrete signals. By making use of  some novel algebraic properties, determined by the author, the following results are obtained:
    • extension of the representation of continuous/discrete signals
    • new solutions to some widely used differential/finite differences equations
  • Regarding the discrete signals which are obtained from the uniform sampling of continuous signals, a result which highlights the connections between uniform and discrete signals is published in 1982.
  • The classical integral transforms of Laplace and Fourier type are generalized for the case of hypercomplex numbers, for real valued functions as well as for distributions.
  • These results allow to generalize the notion of transfer function from a complex valued function of complex variable to a hypercomplex valued function of hypercomplex variable.
  • The aforementioned contributions open new directions in the study of dynamical systems, linear and, most important, non-linear.
  • By using the hypercomplex properties of the truncated Walsh series, an exact procedure of parametric identification of the nonlinear analytical dynamical systems is developped.
  • The results are cited in Surfing through Hyperspace: Understanding Higher Universes in Six Easy Lessons, Clifford A. Pickover, Oxford University Press, 2001.

References:

  1. Condurache D., Reprezentari simbolice. Aplicatii in teoria semnalelor si studiul sistemelor dinamice, ISBN 973-97101-8-2, Nord-Est, Iasi, 1996, 218 pag.
  2. Condurache D., Noi procedee simbolice in studiul sistemelor dinamice, Universitatea Tehnica “Gheorghe Asachi Iasi”, 1995.
  3. Condurache D., About some algebraic properties of the Walsh function, Bul.Inst.Polit. Iasi, XXXIII(XXXVII),1-4,sIII, 1987, pp. 19-23.
  4. Condurache D., Integral Transforms on Sobrero Algebras, Bul. Inst. Polit. Iasi, XXXIII (XXXVII), 1-4, sI, 1987, pp. 5-9.
  5. Condurache D., About a 2n+1 Order Algebra, Bul. Inst. Polit. Iasi, XXXI (XXXV), 1-4, sI, 1985, pp. 35-39.
  6. Condurache D., Integral Transforms of Fourier Type on Commutative Algebras, Bul. Inst. Polit. Iasi, XXX (XXXIV), 1-4, sIII, 1984.
  7. Condurache D., Integral Transforms on Algebras, II. Bul. Inst. Polit. Iasi, XXX (XXXIV), 1-4, sIII, 1984, pp. 19-23.
  8. Condurache D., Integral Transforms on Algebras, I. Bul. Inst. Polit. Iasi, XXVIII (XXXII), 1-4, sIII, 1982, pp. 27-36.
  9. Condurache D., Polynomial Algebras and Symbolic Representation of Discrete Signals, Bul. Inst. Polit. Iasi, XXVII (XXXI), 3-4, sI, 1981, pp. 41-46.
  10. Condurache D., On the Symbolic Representation of Discret Signals, Bul. Inst. Polit. Iasi, XXVII (XXXI), 1-2, sI, 1981, pp. 55-59.
  11. Condurache D., Symbolic Representation of Signal on Hyperspaces, part II, Bul.Inst.Polit. Iasi, XXVII(XXXI), 3-4, sI, 1981, pp. 49-56.
  12. Condurache D., Symbolic Representation of Signal on Hyperspaces, part I, Bul.Inst.Polit. Iasi, XXVII (XXXI), 1-2, sI, 1981, pp. 33-42.
  13. Condurache D., Matcovschi M.H., Symbolic representation of continual and discrete signals on finite order algebras, 7-th International Symposium on Automatic Control and Computer Science, Iasi, Romania, 2001.
  14. Condurache D., Poterasu V.F., Active Optimal Loop Control to Reduce the Seismic Response of Nonlinear Isolation System, Transaction of the 9-th International Conference on Structural Mechanics in Reactor Technology, Lausanne, 1987.
  15. Condurache D., Integral Algebraic Approach of Integral Transforms on Finite-Order Algebras, First International Conference on Industrial and Applied Mathematics ICIAM, Paris, 1987.
  16. Condurache D., A New Algebraic Procedure Concerning Symbolic Representation of Signals, First IMOAS-IFAC Symposium on Modeling and Simulation for Control Lumped and Distributed Parameter, Lille-France, 1986.
  17. Condurache D., O metoda directa de integrare in miscarea particulelor in cimp electromagnetic, in volumul «Conceptie, tehnologie si management in constructia de masini», Institutul Politehnic «Gh. Asachi» Iasi, 1992.
  18. Condurache D., Reprezentari simbolice si transformari integrale pe algebre polinomiale, A VI-a Conferinta Nationala de Vibratii in Constructia de Masini, Timisoara, 10-12 dec, 1988.
  19. Condurache D., Proprietati algebrice ale seriilor Walsh trunchiate in identificarea sistemelor mecanice cu memorie, Prima Conferinta nationala de matematici aplicate si mecanica, Cluj-Napoca, 1988.
  20. Condurache D., Lazescu Al., Serii Walsh in identificarea sistemelor neliniare, Sesiunea Jubiliara de comunicari stiintifice “Contributia invatamintului politehnic la dezvoltarea ramurilor de virf ale industriilor din Romania”, sectia de Mecanica si Rezistenta materialelor, Iasi, 1988.
  21. Condurache D., Identificarea exacta a sistemelor neliniare fara memorie utilizind functii Walsh, Sesiunea stiintifica «Creatia tehnica si fiabilitatea in constructia de masini», Iasi, 1985.
  22. Condurache D., Transformari integrale pe algebre comutative, Simpozionul de tehnologie si fiabilitate, Iasi, 1983.
  23. Condurache D., Braier A., Asupra reprezentarii simbolice a unor semnale modulate pe C-algebre, Sesiunea jubiliara Caius Iacob, Bucuresti, 1982.

Comments are closed.