In the unit on rings, I explained category theory and general rings at the same time. Then I talked mostly about commutative rings. In the unit on modules, I again mixed category theory into the basic notions and progressed to the structure theorem for finitely generated modules over PID’s. Jordan canonical forms were used as an application. The uniqueness part of the structure theorem was put ...

In this paper we establish that the causal order determined by an Ol’shanski semigroup on the corresponding homogeneous space is globally hyperbolic. Using this fact, we present sufficient conditions for a special class of Lie semigroups to admit a canonical “triple decomposition,” namely those for which the Lie algebra is of Cayley type. This theory applies in particular to semigroups which ar...

It is an easy consequence of the Jordan canonical form that a matrix A ∈Mn×n(C) can be decomposed into a sum A = DA + NA where DA is a diagonalizable matrix, NA a nilpotent matrix, and such that DANA = NADA. It is clear that both DA and NA also commute with A. This decomposition is often referred to as the Jordan decomposition and has found many applications throughout the years. For example, i...

Given an affine subspace of square matrices, we consider the problem of minimizing the spectral abscissa (the largest real part of an eigenvalue). We give an example whose optimal solution has Jordan form consisting of a single Jordan block, and we show, using nonlipschitz variational analysis, that this behaviour persists under arbitrary small perturbations to the example. Thus although matric...

It has been conjectured [Blair] that every definite clause logic program containing exactly one rule is canonical. This note presents a simple proof

We study the ground-state properties of hard-core bosons trapped by arbitrary confining potentials on one-dimensional optical lattices. A recently developed exact approach based on the Jordan-Wigner transformation is used. We analyze the large distance behavior of the one-particle density matrix, the momentum distribution function, and the lowest natural orbitals. In addition, the low-density l...

Among the various canonical forms which were proposed for constant linear systems, the one due to Brunovsky [1] certainly is the most profound. It characterizes a dynamics modulo the group of static state feedbacks by a finite set of pure integrators. Its proof, which is quite computational, has been improved in various ways, and can be found in several textbooks (see, e.g., [12, 13, 20, 21] an...

The Rabi Hamiltonian, describing the coupling of a two-level system to a single quantized boson mode, is studied in the BargmannFock representation. The corresponding system of differential equations is transformed into a canonical form in which all regular singularities between zero and infinity have been removed. The canonical or Birkhoff-transformed equations give rise to a two-dimensional e...

We introduce RS-vector machines (RS-VMs) as a canonical form of vector machines. They are based on vector operations called repeat and stretch. Repeat enlarges a vector (a1 , a2 , ..., am) to (a1 , a2 , ..., am , a1 , a2, ..., am) and stretch enlarges (a1, a2, ..., am) to (a1, a1, a2, a2, ..., am, am), when the expansion factor d(m)=2. It is shown that we can change the power of RS-VMs dependin...

We carefully perform a Hamiltonian Dirac's constraint analysis of the $\ensuremath{\omega}=\ensuremath{-}\frac{3}{2}$ Brans-Dicke theory with Gibbons-Hawking-York boundary term. The Poisson brackets are computed via functional derivatives. After brief summary results for $\ensuremath{\omega}\ensuremath{\ne}\ensuremath{-}\frac{3}{2}$ case [G. Gionti S. J., Canonical addresses inequivalence betwe...