There is a wedge product notation that can facilitate the computation of matrix Jacobians. In point of fact, it is never needed at all. The reader who can follow the derivation of the Jacobian in Handout #2 is well equipped to never use wedge products. The notation also expresses the concept of volume on curved surfaces. For advanced readers who truly wish to understand exterior products from a...

We present a quantitative approach to the measurement of shape dissimilarity between two 3D (three-dimensional) curves. Any 3D continuous curve can be digitalized and represented as a 3D discrete curve. Thus, a 3D discrete curve is composed of constant orthogonal straightline segments. In order to represent 3D discrete curves, we use the orthogonal direction change chain code. The chain element...

A double generating functions based method for the discrete-time linear quadratic optimal control problem with fixed and free terminal state, i.e., hard constraint problem and soft constraint problem, is developed in this paper. By this method, the optimal state and input can be expressed by functions only in terms of the boundary conditions and pre-computed coefficients. Accordingly, the whole...

Purpose: This research was conducted with the aim of identifying effective factors on educational reform at the level of national and regional supports for high school mathematics. Methodology: The present study in terms of purpose was applied and in terms of implementation method was qualitative, which was performed in two stages of synthesis research and content analysis. The research enviro...

We study and compare two classes of statistical criteria to assess the significance of exceptional words. Indeed, the Z-score-like criteria, or the normal approximation that is a strict equivalent, suffer from several drawbacks in terms of sensitivity and specificity. Thanks to the combinatorial structure of words, a computation of the exact P-value has been made possible by recent mathematical...

Some years ago we introduced a new topological invariant for foliated manifolds using techniques from noncommutative geometry, in particular the pairing between K-Theory and cyclic cohomology. The motivation came from flat principal G-bundles where the base space is a non simply connected manifold. The computation of this invariant is quite complicated. In this article we try to perform certain...

It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N . Morton proved there were none with N = 4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X1(16), whose rational points had been previously computed. We prove there are none with N = 5. Here the rel...

We give a simple, yet very general definition for distributed protocols. We then define notions of knowledge and common knowledge appropriate for these protocols. We study how changes in the states of knowledge relate to more standard notions of computation. We find that by restricting our formulas to certain sets of global states we can realize different, appropriate definitions of knowledge w...

An analog to intrinsic linking, intrinsic even linking, is explored in the first half of this paper. Four graphs are established to be minor minimal intrinsically even linked, and it is conjectured that they form a complete minor minimal set. Some characterizations are given, using the simplest of the four graphs as an integral part of the arguments, that may be useful in proving the conjecture...