In 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field surreal numbers was brought to fore by first author and employed provide necessary sufficient conditions for an (ordered -vector space) be isomorphic initial subfield ( -subspace) , i.e. a that is subtree . this sequel, analogous results are established exponential fields, making use slight ...

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

We show that the Weierstrass points of the generic curve of genus g over an algebraically closed field of characteristic 0 generate a group of maximal rank in the Jacobian. The Weierstrass points are a set of distinguished points on curves, which are geometrically intrinsic. In particular, the group these points generate in the Jacobian is a geometric invariant of the curve. A natural question ...

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...

A proof of the Jordan canonical form, suitable for a first course in linear algebra, is given. The proof includes the uniqueness of the number and sizes of the Jordan blocks. The value of the customary procedure for finding the block generators is also questioned. 2000 MSC: 15A21. The Jordan form of linear transformations is an exceeding useful result in all theoretical considerations regarding...

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...