Eigenvalues of Words in Two Positive Definite Letters

Preprint English OPEN
Hillar, Christopher J ; Johnson, Charles R (2005)
  • Subject: Mathematics - Operator Algebras | Mathematics - Rings and Algebras | 15A24, 15A57 | 15A18, 15A90
    arxiv: Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) | Computer Science::Formal Languages and Automata Theory

The question of whether all words in two real positive definite letters have only positive eigenvalues is addressed and settled (negatively). This question was raised some time ago in connection with a long-standing problem in theoretical physics. A large class of words that do guarantee positive eigenvalues is identified, and considerable evidence is given for the conjecture that no other words do. In the process, a fundamental question about solvability of symmetric word equations is encountered.
  • References (6)

    F (p, q) = F (p, q) = g1(q)ap + g2(q)bp + g3(q)cp, h1(p)rq + h2(p)sq + h3(p)tq, F (p, q) = F (p, q) = F (p, q) < 0; F (p, q) < 10 · F (p − 1, q); and F (p, q) < 10 · F (p, q − 1).

    [BMV] D. Bessis, P. Moussa, and M. Villani, Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics, J. Math. Phys., 16 (1975), pp. 2318-2325.

    [HJ] R. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985.

    [K] R. Kemp, On the number of words in the language {w ∈ Σ∗ | w = wR}2, Discrete Math., 40 (1982), pp. 225-234.

    [S] I. Spitkovsky, private communication, Williamsburg, VA, 1999.

    Mathematics Department, College of William and Mary, Williamsburg, VA 23187- 8795 E-mail address: crjohnso@math.wm.edu

  • Similar Research Results (2)
  • Metrics
    No metrics available
Share - Bookmark