we define a new type of spectrum, called δ-approximate spectrum, of an element a in a complex unital banach algebra a and show that the δ-approximate spectrum σ_δ (a) of a is compact. the relation between the δ-approximate spectrum and the usual spectrum is investigated. also an analogue of the classical gleason-kahane-zelazko theorem is established: for each ε>0, there is δ>0 such that if ϕ is...

If F is an automorphism of Ωn, the n -dimensional spectral unit ball, we show that, in a neighborhood of any cyclic matrix of Ωn, the map F can be written as conjugation by a holomorphically varying non singular matrix. This provides a shorter proof of a theorem of J. Rostand, with a slightly stronger result. 1. Background Let Mn be the set of all n × n complex matrices. For A ∈ Mn denote by sp...

The four lectures follow closely a textbook on Potential Theory in the Complex Plane by T. Ransford, apart from material on harmonic measure which has been borrowed from a lecture course Introduction to Potential Theory with Applications, by C. Kuehn. The material in lecture 5 is borrowed from a survey Logarithmic Potential Theory with Applications to Approximation Theory by E. B. Saff (E-print...

The four lectures follow closely a textbook on Potential Theory in the Complex Plane by T. Ransford, apart from material on harmonic measure which has been borrowed from a lecture course Introduction to Potential Theory with Applications, by C. Kuehn. The material in lecture 5 is borrowed from a survey Logarithmic Potential Theory with Applications to Approximation Theory by E. B. Saff (E-print...

Twenty-six patients with rheumatoid disease affecting the cervical spine underwent surgical treatment for neck pain, neurological deficit, or both. Atlantoaxial subluxation (n=13), subaxial subluxation (n=7) and vertical migration of the odontoid (n=6) were treated. Arthrodesis with autologous bone was augmented with wire, Ransford loop, Hartshill rectangle or Magerl technique. Pain relief occu...

We provide generalisations of two Schwarz-type lemmas — the first a result of Globevnik and the other due to Ransford and White — for holomorphic mappings into the spectral unit ball. The first concerns mappings of the unit disc in C into the spectral unit ball, while the second concerns self-mappings. The aforementioned results apply to holomorphic mappings that map the origin to the origin. W...

In this paper, we consider certain matricial domains that are naturally associated to a given domain of the complex plane. A particular example such is spectral unit ball. We present several results for these domains. Our first result shows — generalizing Ransford–White ball holomorphic automorphism group does not act transitively. also [Formula: see text]-point and Pick–Nevanlinna interpolatio...

We show that, given a non-degenerate, finitely connected domain D, its boundary, and the number of its boundary components, it is possible to compute a conformal mapping of D onto a circular domain without prior knowledge of the circular domain. We do so by computing a suitable bound on the error in the Koebe construction (but, again, without knowing the circular domain in advance). Recent resu...

We define a new type of spectrum, called δ-approximate spectrum, of an element a in a complex unital Banach algebra A and show that the δ-approximate spectrum σ_δ (a) of a is compact. The relation between the δ-approximate spectrum and the usual spectrum is investigated. Also an analogue of the classical Gleason-Kahane-Zelazko theorem is established: For each ε>0, there is δ>0 such that if ϕ is...