This work presents a parametrized family of divergences, namely Alpha-Beta LogDeterminant (Log-Det) divergences, between positive definite unitized trace class operators on a Hilbert space. This is a generalization of the Alpha-Beta Log-Determinant divergences between symmetric, positive definite matrices to the infinite-dimensional setting. The family of Alpha-Beta Log-Det divergences is highl...

Optimal normal bases are special cases of the so-called Gauss periods (Disquisitiones Arithmeticae, Articles 343-366); in particular, optimal normal bases are Gauss periods of type (n,1) for any characteristic and type (n,2) for characteristic 2. We present the multiplication tables and complexities of Gauss periods of type (n, t) for all n and t = 3,4,5 over any finite field and give a slightl...

We show that, for a cw passively mode-locked picosecond Ti:sapphire/DDI laser, the first autocorrelation trace with negligible cw background occurs at a delay time of 20 mu;s, or 1600 round trips from the first relaxationoscillation peak. The trace suggests that the pulse consists of a primary pulse as short as 4.4 ps and of small secondary pulses that form a much wider pedestal of the trace, e...

In this paper, we establish an optimal trace regularity theorem, also known as the hidden regularity theorem [L2], for the anisotropic linear elasticity equation on a bounded domain Ω with Lipschitz boundary. In its simplest form it provides a space-time L estimate for the trace of the normal derivative for the solution. Over the years, such sharp trace regularity theorems have proven to be cru...

The theory of traces was originated by A. Mazurkiewicz to model non-sequential behaviour of a distributed system. The normal form of a trace shows us an optimal way to execute a process occurred on the system. Nevertheless, the independence of actions in many systems is not static and depends on the history of the systems. To describe this fact, D. Kuske and R. Morin proposed a notion of local ...

In this paper, we investigate the relationship between the squared Weil/Tate pairing and the plain Weil/Tate pairing. Along these lines, we first show that the squared pairing for arbitrary chosen point can be transformed into a plain pairing for the trace zero point which has a special form to compute them more efficiently. This transformation requires only a cost of some Frobenius actions. Ad...

A (discrete) dynamical system may have various symmetries and reversing symmetries, which together form its so-called reversing symmetry group. We study the set of 3D trace maps (obtained from two-letter substitution rules) which preserve the Fricke-Vogt invariant I(x, y, z). This set of dynamical systems forms a group G isomorphic with the projective linear (or modular) group PGl(2, Z ). For s...

We study weighted trace automata with weights in strong bimonoids. Traces form a generalization of words that allow to model concurrency; strong bimonoids are algebraic structures that can be regarded as “semirings without distributivity”. A very important example for the latter are bounded lattices, especially non-distributive ones. We show that if both operations of the bimonoid are locally f...

Consider a bipartite quantum system consisting of two subsystems A and B, of dimensions nA and nB , respectively. The overall system has dimension n := nAnB. The state of the total system is represented by an Hermitian n×nmatrix ρ, called the density matrix, which is positive semi-definite and has trace equal to one. Special types of density matrices are product matrices, i.e., matrices ρprod o...

We consider a Hamiltonian system of the form y′(x) = JH(x)y(x), with a locally integrable and nonnegative 2×2-matrix valued Hamiltonian H(x). In the literature dealing with the operator theory of such equations, it is often required in addition that the Hamiltonian H is trace–normed, i.e. satisfies tr H(x) ≡ 1. However, in many examples this property does not hold. The general idea is that one ...