Skip to main content
SpringerLink
Log in

Differential equations in Schatten classes of operators

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

We study a system of differential equations in Schatten classes of operators, \({\mathcal{C}_p(\mathcal{H})\,(1 \leq p < \infty}\)), with \({\mathcal{H}}\) a separable complex Hilbert space. The systems considered are infinite dimensional generalizations of mathematical models of unsupervised learning. In this new setting, we address the usual questions of existence and uniqueness of solutions. Under some restrictions on the spectral properties of the initial conditions, we explicitly solve the system. We also discuss the long-term behavior of solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arrowsmith D., Place C.: An Introduction to Dynamical Systems. Cambridge University Press, New York (1990)

    MATH  Google Scholar 

  2. Arverson W.: A Short Course on Spectral Theory. Cambridge University Press, New York (2000)

    Google Scholar 

  3. Botelho F., Jamison J.: Differential equations in spaces of compact operators. J. Math. Anal. Appl. 308, 105–120 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Botelho, F., Jamison, J.: Qualitative Behavior of Differential equations Associated with Artificial Neural Networks. Preprint (2008)

  5. Chen T., Amari S., Lin Q.: A unified algorithm for principal and minor components extraction. Neural Netw. 11, 385–390 (1998)

    Article  Google Scholar 

  6. Conway J.: A Course in Functional Analysis. Graduate Texts in Mathematics, vol. 96. Springer, Berlin (1990)

    Google Scholar 

  7. Deimling K.: Ordinary Differential Equations in Banach Spaces. Lecture Notes in Mathematics, vol. 596. Springer, Berlin (1977)

    Google Scholar 

  8. Haykin S.: Neural Networks: A Comprehensive Foundation. Macmillan, New York (1994)

    MATH  Google Scholar 

  9. Halmos P.: Introduction to Hilbert Spaces. Chelsea, New York (1957)

    Google Scholar 

  10. Hartman P.: Ordinary Differential Equations. Wiley, New York (1964)

    MATH  Google Scholar 

  11. Hertz, J., Krogh, A., Palmer, R.: Introduction to the Theory of Neural Computation. A Lecture Notes Volume, Santa Fe Institute Studies in the Sciences of Complexity (1991)

  12. Mukherjea A., Pothoven K.: Real and Functional Analysis. Plenum Press, New York (1978)

    MATH  Google Scholar 

  13. Oja E.: A simplified neuron model as a principal component analyzer. J. Math. Biol. 15, 267–273 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  14. Oja E., Karhunen J.: A stochastic approximation of the eigenvectors and eigenvalues of the expectation of a random matrix. J. Math. Anal. Appl. 106, 69–84 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  15. Pavel N.: Differential Equations, Flow Invariance and Applications. Research Notes in Mathematics, vol. 113. Pitman, London (1994)

  16. Schatten R.: Norm Ideals of Completely Continuous Operators. Springer, Berlin (1960)

    MATH  Google Scholar 

  17. Xu L.: Least mean square error recognition principle for self-organizing neural nets. Neural Netw. 6, 627–648 (1993)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, The University of Memphis, Memphis, TN, 38152, USA

    Fernanda Botelho & James E. Jamison

Authors
  1. Fernanda Botelho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James E. Jamison
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fernanda Botelho.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Botelho, F., Jamison, J.E. Differential equations in Schatten classes of operators. Monatsh Math 160, 257–269 (2010). https://doi.org/10.1007/s00605-009-0114-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-009-0114-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation