Showing papers on "Equivalence class published in 1967"

Cell growth problems

Abstract: The square lattice is the set of all points of the plane whose Cartesian coordinates are integers. A cell of the square lattice is a point-set consisting of the boundary and interior points of a unit square having its vertices at lattice points. An n-omino is a union of n cells which is connected and has no finite cut set. The set of all n-ominoes, Rn is an infinite set for each n; however, we are interested in the elements of two finite sets of equivalence classes, Sn and Tn , which are defined on the elements of Rn as follows: Two elements of Rn belong to the same equivalence class (i) in Sn , or (ii) in Tn , if one can be transformed into the other by (i) a translation or (ii) by a translation, rotation, and reflection of the plane.

A Tabular Method for the Synthesis of Multithreshold Threshold Elements

Abstract: This paper presents a tabular method for synthesizing Boolean functions having four or less variables with multithreshold threshold elements. The method is similar to that used for conventional single-threshold threshold elements. All 224 functions of four variables are divided into 221 equivalence classes by variable complementations and/or permutations and/or function complementation. Each equivalence class is characterized by a subset of its corresponding Rademacher-Walsh coefficients, the size of the subset being determined by the number of thresholds required to realize that equivalence class. An arbitrary Boolean function of four or less variables is synthesized by systematically calculating subsets of its Rademacher-Walsh coefficients until, through simple equivalence operations, the equivalence class of the function is found in a table of the 221 equivalence classes. The table indicates a multithreshold realization of the given function. The table shows that any 4-variable function can be realized with at most five thresholds, or by a network of conventional, or single-threshold, threshold elements with at most three gates in which each gate has the identical weight vector for the four input variables.

A Lie product for the cohomology of subalgebras with coefficients in the quotient

Abstract: The product operation in B, and similarly in A, induces a graded Lie algebra (GLA) structure (here called the cup structure) on H*(B, B) and H*(B, A) (cf., e.g., Gerstenhaber [2], Nijenhuis and Richardson [6]), and i* is known to be a homomorphism of these structures. The cup structure on H*(B, B) is abelian; cf. [2]. I t is also known that H*(B, B) has another GLA structure (here called the comp structure) with respect to the reduced grading (elements of H(B, B) have reduced degree w — 1; cf. [2], [7]). The following theorem supplements this information.