Let F/Q be a number field closed under complex conjugation. Denote by L0(F (t)) the Witt group of hermitian forms over F (t). We find full invariants for detecting non–zero elements in L0(F (t))⊗Q, this group plays an important role in topology in the work done by Casson and Gordon. 1. L-groups and signatures Let R be a ring with (possibly trivial) involution. An –hermitian ( = ±1) form is a se...

A Hermitian matrix X is called a g-inverse of a Hermitian matrix A, denoted by A, if it satisfies AXA = A. In this paper, a group of explicit formulas are established for calculating the global maximum and minimum ranks and inertias of the difference A − PNP , where both A and N are Hermitian g-inverses of two Hermitian matrices A and N , respectively. As a consequence, necessary and sufficient...

We give a necessary and sufficient condition for the reality of the spectrum of a non-Hermitian Hamiltonian admitting a complete set of biorthonormal eigenvectors. Recently, we have explored in [1] the basic mathematical structure underlying the spectral properties of PT -symmetric Hamiltonians [2]. In particular, we have shown that these properties are associated with a class of more general (...

For a given standard Hamiltonian H with arbitrary complex scalar and vector potentials in one-dimension, we construct an invertible antilinear operator τ such that H is τ -anti-pseudo-Hermitian, i.e., H = τHτ−1. We use this result to give the explicit form of a linear Hermitian invertible operator with respect to which any standard PT symmetric Hamiltonian with a real degree of freedom is pseud...

Let M be a mixed graph and [Formula: see text] be its Hermitian-adjacency matrix. If we add a Randić weight to every edge and arc in M, then we can get a new weighted Hermitian-adjacency matrix. What are the properties of this new matrix? Motivated by this, we define the Hermitian-Randić matrix [Formula: see text] of a mixed graph M, where [Formula: see text] ([Formula: see text]) if [Formula: ...

The hermitian level of composition algebras with involution over a ring is studied. In particular, it is shown that the hermitian level of a composition algebra with standard involution over a semilocal ring, where two is invertible, is always a power of two when finite. Furthermore, any power of two can occur as the hermitian level of a composition algebra with nonstandard involution. Some bou...

This article contains a short summary of an oral presentation in the 2nd International Workshop on “Pseudo-Hermitian Hamiltonians in Quantum Physics” (14.-16.6.2004, Villa Lanna, Prague, Czech Republic). The purpose of the presentation has been to introduce a non-Hermitian generalization of pseudo-Hermitian Quantum Theory (QT) allowing to reconcile the orthogonal concepts of causality, Poincaré...

Let K be a number field and OK its ring of integers. Let E be an OK-module of rank N + 1 in P(E), the projective space representing lines in E. For all closed subvarieties X ⊆ P(E K) of dimension d let deg(X) be its degree with respect to the canonical line bundle O(1) of P(E K). If E is endowed with the structure of hermitian vector bundle over Spec(OK) we can define the Arakelov degree d̂eg ( ...

We consider several families of categories. The first are quotients of Andersen’s tilting module categories for quantum groups of Lie type B at odd roots of unity. The second consists of categories of type BC constructed from idempotents in BMW -algebras. Our main result is to show that these families coincide as braided tensor categories using a recent theorem of Tuba and Wenzl. By appealing t...

We introduce an additive but not F4-linear map S from Fn4 to F 4 and exhibit some of its interesting structural properties. If C is a linear [n, k, d]4-code, then S(C) is an additive (2n, 2 , 2d)4-code. If C is an additive cyclic code then S(C) is an additive quasi-cyclic code of index 2. Moreover, if C is a module θ-cyclic code, a recently introduced type of code which will be explained below,...