Third chains are chords that built from a concatenation of minor or major thirds. A triad 0, 4, 7 is third chain, concatenating 4 with 7. important in harmony, and we shall demonstrate how they applied to the management harmonic contents general chords. This approach for analysis musical compositions.

Predicate classes are a new linguistic construct designed to complement normal classes in objectoriented languages. Like a normal class, a predicate class has a set of superclasses, methods, and instance variables. However, unlike a normal class, an object is automatically an instance of a predicate class whenever it satisfies a predicate expression associated with the predicate class. The pred...

These are lecture notes for a series of five lectures I gave to other graduate students about characteristic classes through UT Austin’s summer minicourse program (see https://www.ma.utexas.edu/users/ richard.wong/Minicourses.html for more details). Beware of potential typos. In these notes I cover the basic theory of Stiefel-Whitney, Wu, Chern, Pontrjagin, and Euler classes, introducing some i...

a ring $r$ is a strongly clean ring if every element in $r$ is the sum of an idempotent and a unit that commutate. we construct some classes of strongly clean rings which have stable range one. it is shown that such cleanness of $2 imes 2$ matrices over commutative local rings is completely determined in terms of solvability of quadratic equations.

‎in this paper‎, ‎we give a necessary and sufficient condition for the equality of two symmetrized decomposable polynomials‎. ‎then‎, ‎we study some algebraic and geometric properties of the induced operators over symmetry classes of polynomials in the case of linear characters‎.

Reducing the cost of construction is a primary concern for owners, designers, and builders of facilities. Today, project teams often perform constructability reviews during the design phase to identify design constraints that limit a constructor’s ability to perform construction operations effectively and increase construction cost. However, the current manual process is timeconsuming and prone...

Episodic memory involves a mechanism that binds information into a coherent representation structure. Especially, more complex memory structures are required when there are more overlapping elements among different episodes (Humphreys, Bain & Pike, 1989), and the ability to use memory structures of different complexities increases throughout development (Yim, Dennis, & Sloutsky, 2013). Although...

An r l -system is an r-uniform hypergraph in which every set of l vertices lies in at most one edge. Let mk r l be the minimum number of edges in an r l -system that is not k-colorable. Using probabilistic techniques, we prove that ar l kr−1 ln k l/ l−1 ≤ mk r l ≤ br l kr−1 ln k l/ l−1 where br l is explicitly defined and ar l is sufficiently small. We also give a different argument proving (fo...