Normal form for travelling kinks in discrete Klein-Gordon lattices

We study travelling kinks in the spatial discretizations of the nonlinear Klein--Gordon equation, which include the discrete $\phi^4$ lattice and the discrete sine--Gordon lattice. The differential advance-delay equation for travelling kinks is reduced to the normal form, a scalar fourth-order differential equation, near the quadruple zero eigenvalue. We show numerically non-existence of monotonic kinks (heteroclinic orbits between adjacent equilibrium points) in the fourth-order equation. Making generic assumptions on the reduced fourth-order equation, we prove the persistence of bounded solutions (heteroclinic connections between periodic solutions near adjacent equilibrium points) in the full differential advanced-delay equation with the technique of center manifold reduction. Existence and persistence of multiple kinks in the discrete sine--Gordon equation are discussed in connection to recent numerical results of \cite{ACR03} and results of our normal form analysis

Secants of Lagrangian Grassmannians

We study the dimensions of secant varieties of the Grassmannian of Lagrangian subspaces in a symplectic vector space. We calculate these dimensions for third and fourth secant varieties. Our result is obtained by providing a normal form for four general points on such a Grassmannian and by explicitly calculating the tangent spaces at these four points

Perbandingan antara Dua Pendekatan Normalisasi Data Hingga Kondisi Fourth Normal Form (4nf)

Data normalization is a technique used to create databases that are able to store data in an effective and efficient manner. This, in turn, will ensure the integrity of the data contained within those databases. The process of normalizing a database from completely unnormalized to its ultimate form is a gradual one, and in practical terms this is commonly thought of as necessitating at least four steps, each of which is marked with a specific "normalized form", usually annotated as 1NF, 2NF, 3NF, and 4NF. This paper compares two different approaches to achieve 4NF. The first method comprises the four steps usually described in books and articles, while the second approach needs only two-and-a-half of those steps to attain 4NF

Performance Criteria for Relational Database Normalization

The fourth normal form where data redundancy is eliminated is a more efficient construct for storage and user access