Upper triangular matrix operators with diagonal (T_1,T_2), T_2 k-nilpotent

If \(T=\left(\begin{array}{clcr}T_1&\quad C\\ 0&\quad T_2\end{array}\right) \in B(\mathcal{X }_1\oplus \mathcal{X }_2)\) is a Banach space upper triangular operator matrix with diagonal \((T_1, T_2)\) such that \(T_2\) is \(k\)-nilpotent for some integer \(k\ge 1\), then \(T\) inherits a number of its spectral properties, such as SVEP, Bishop’s property \((\beta )\) and the equality of Browder and Weyl spectrum, from those of \(T_1\). This paper studies such operators. The conclusions are then applied to provide a general framework for results pertaining (for example) to Browder, Weyl type theorems and supercyclicity for classes of Hilbert space operators, such as \(k\)-quasi hyponormal, \(k\)-quasi isometric and \(k\)-quasi paranormal operators, defined by a positivity condition.     

Unavoidable Traces Of Set Systems

Sauer, Shelah, Vapnik and Chervonenkis proved that if a set system on n vertices contains many sets, then the set system has full trace on a large set. Although the restriction on the size of the groundset cannot be lifted, Frankl and Pach found a trace structure that is guaranteed to occur in uniform set systems even if we do not bound the size of the groundset. In this note we shall give three sequences of structures such that every set system consisting of sufficiently many sets contains at least one of these structures with many sets.     

Plasma parameters within the cathode spot of the vacuum arc

The composition of the vacuum arc plasma for five elements (Cd, Mg, Al, Ni, and Mo) is calculated by the Saha equation, which assumes local thermodynamic equilibrium conditions within the ionization region of the cathode spot(s). The lowering of the ionization potential due to the high density of charged particles is considered. By matching the computed and the measured plasma ionic composition, the electron density and the temperature are estimated. The experimental plasma compositions can be approximated only at a high electron density (10¹⁹-10 ²¹ cm^-3) and at electron temperatures in the range of a few electronvolts     

Numerical differentiation of photomechanical data using smoothing spline functions with generalized boundary conditions

The smoothing spline function has been used for differentiation of photomechanical data in experimental stress analysis. However, since the available software packages use the algorithm with the so-called natural boundary conditions which are not necessarily suitable in many circumstances, errors tend to occur in both the first and second derivatives. In this paper we have modified the algorithm so that the user can have more freedom in selecting boundary conditions which suit the problem best and therefore, as a result, improve the accuracy of the analysis. Two examples are given to show the improvement achievable using the new scheme.     

A classical no-interaction theorem for bosonic strings and its quantum significance

We show that, in dimensions equal or greater than three, classical strings do not join or split. This is part of a no-interaction theorem that is also proved. The quantum theory is considered.