Jordan Canonical Form ( JCF) is one of the most important, and useful, concepts in linear algebra. In this book we develop JCF and show how to apply it to solving systems of differential equations. We first develop JCF, including the concepts involved in it–eigenvalues, eigenvectors, and chains of generalized eigenvectors.We begin with the diagonalizable case and then proceed to the general cas...

In this paper we discuss algorithmic aspects of the computation of the Jordan canonical form. Inspired by the Golub & Wilkinson paper 9] on the computation of the Jordan canonical form, an O(n 3) algorithm was developed by Beelen & Van Dooren 3] for computing the Kronecker structure of an arbitrary pencil B ? A. Here we show how the ideas of this algorithm lead to a special algorithm for recons...

A common problem in frequent graph mining is the size of the output, which can easily exceed the size of the database to analyze. In the application area of molecular fragment mining a promising approach to tackle this problem is to treat certain substructures as a unit. Among such structures, rings are most prominent, and by requiring that either a ring is present as a whole in a fragment, or ...

In frequent subgraph mining one tries to find all subgraphs that occur with a user-specified minimum frequency in a given graph database. The basic approach is to grow subgraphs, adding an edge and maybe a node in each step, to count the number of database graphs containing them, and to eliminate infrequent subgraphs. The predominant method to avoid redundant search (the same subgraph can be gr...

We consider cubic surfaces with rational coefficients that contain a rational point and satisfy a certain condition regarding their coefficients. Each such cubic surface is shown to be birationally equivalent to a surface of the form z2 = f(x, y), where f(x, y) is a polynomial of degree at most 4. Our method is similar to that which Tate and Silverman used in [2] to put cubic curves into Weiers...

R. Feng, and H. Wu recently established a certain mean-value formula for the x-coordinates of the n-division points on an elliptic curve given in Weierstrass form (A mean value formula for elliptic curves, 2010, available at http://eprint.iacr.org/2009/586.pdf ). We prove a similar result for both the x and y-coordinates on a twisted Edwards elliptic curve.

Let E/Q be an elliptic curve with discriminant ∆, and let P ∈ E(Q). The standard method for computing the canonical height ĥ(P ) is as a sum of local heights ĥ(P ) = λ̂∞(P ) + ∑ p λ̂p(P ). There are well-known series for computing the archimedean height λ̂∞(P ), and the non-archimedean heights λ̂p(P ) are easily computed as soon as all prime factors of ∆ have been determined. However, for curves wi...

A canonical form for circuit diagrams with a designated start state is presented. The form is based upon nite automata minimization and Shannon's canonical form for boolean expressions.

Given a linear time-invariant control system, it is well known that the transmission zeroes are the generalized eigenvalues of a matrix pencil. Adding outputs to place additional zeroes is equivalent to appending rows to this pencil to place new generalized eigenvalues. Adding inputs is likewise equivalent to appending columns. Since both problems are dual to each other, we only show in this pa...