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In this chapter, we consider the representation of vector functions (often referred to as “vector fields”) with low-order (constant and linear) polynomial basis functions on simple cells, such as triangles or quadrilateral cells in two dimensions or tetrahedrons and bricks in three dimensions. As will soon be apparent, there are multiple ways of defining vector basis functions, and therefore the approach requires some consideration. The proper representation of a function depends on what will be done with it-do we need to compute the curl of the function, for instance? If so, the representation might be different than if we need to compute the divergence of the function. We use the term curl conforming to denote the space of vector functions that maintain first-order tangential-vector continuity throughout the domain and can be differentiated via the curl operation, without producing unbounded or generalized functions (Dirac delta functions) in the process. The term divergence conforming is used to denote the complementary space of vector functions that maintain first-order normal-vector continuity throughout the domain and can therefore be differentiated via the divergence operation. (First order or C_{0} continuity is continuity of the function itself, but not necessarily continuity of its first derivatives.) The simple low-order polynomial vector basis functions in widespread use are either curl conforming or divergence conforming; seldom we will use functions that maintain complete continuity and belong to both the curl-conforming and divergence-conforming spaces, although it is possible to define such functions.

This paper describes an efficient implementation of a form of linear semi-infinite programming (LSIP). We look at maximizing (minimizing) a linear function over a set of constraints formed by positive trigonometric polynomials. Previous studies about LSIP are formulated using semi-definite programming (SDP), this is typically done by using the Kalman Yakubovich Popov (KYP) lemma or using a trace operation involving a Grammian matrix, which can be computationally expensive. The proposed algorithm is based on simplex method that directly solves the LSIP without any parameterization. Numerical results show that the proposed LISP algorithm is significantly more efficient than existing SDP solvers using KYP lemma and Grammian matrix, in both execution time and memory.

Given a graph G = (V, E) with a set W ⊆ V of vertices, we enumerate colorings to W such that for every two enumerated colorings c and c' the corresponding colored graphs (G, c) and (G, c') are not isomorphic. This problem has an important application in the study of isomers of chemical graphs such as generation of benzen isomers from a tree-like chemical graph structure. The number of such colorings can be computed efficiently based on Polya's theorem. However, enumerating each from the set of these colorings without using a large space is a challenging problem in general. In this paper, we propose a method for enumerating these colorings when the automorphisms of G are determined by two axial symmetries, and show that our algorithm can be implemented to run in polynomial delay and polynomial space.

Word search is a classical puzzle to search for all given words on a given assignment of letters to a rectangular grid (matrix). This problem is clearly in P. The inverse of this problem is more difficult, which asks to assign letters in a given alphabet to a matrix of given size so that every word in a given wordset can be found horizontally, vertically, or diagonally. This problem is in NP; it admits a trivial polynomial-size certificate. We prove its NP-hardness. It turns out to be so even under the following restrictions: 1) the alphabet size is 2 (binary) and 2) all the words to be found are of length at most 2. These results are optimal in the sense that decreasing these bounds 2 to 1 makes the problem be trivially in P.

Polynomial parahermitian matrices can accurately and elegantly capture the space-time covariance in broadband array problems. To factorise such matrices, a number of polynomial EVD (PEVD) algorithms have been suggested. At every step, these algorithms move various amounts of off-diagonal energy onto the diagonal, to eventually reach an approximate diagonalisation. In practical experiments, we have found that the relative performance of these algorithms depends quite significantly on the type of parahermitian matrix that is to be factorised. This paper aims to explore this performance space, and to provide some insight into the characteristics of PEVD algorithms.