Ramanujan’s partition congruences can be proved by first showing that the coefficients in the expansions of (q; q)∞ satisfy certain divisibility properties when r = 4, 6 and 10. We show that much more is true. For these and other values of r , the coefficients in the expansions of (q; q)∞ satisfy arithmetic relations, and these arithmetic relations imply the divisibility properties referred to ...

We address the problem of refactoring existing, closely related products into product line representations. Our approach is based on comparing and matching artifacts of these existing products and merging those deemed similar while explicating those that vary. Our work focuses on formal specification of a product line refactoring operator called merge-in that puts individual products together i...

1: Properties of the shuffle product in positive characteristic suggest to consider a p−homogeneous form σ : Fp〈〈X1, . . . ,Xk〉〉 −→ Fp〈〈X1, . . . ,Xk〉〉 on the vector space Fp〈〈X1, . . . ,Xk〉〉 of formal power series in k free noncommuting variables. The form σ preserves rational elements in Fp〈〈X1, . . . ,Xk〉〉, algebraic series of Fp[[X]] and induces a bijection on the affine subspace 1+m of for...

A {it topological index} of a graph is a real number related to the graph; it does not depend on labeling or pictorial representation of a graph. In this paper, we present the upper bounds for the product version of reciprocal degree distance of the tensor product, join and strong product of two graphs in terms of other graph invariants including the Harary index and Zagreb indices.

In this note first by considering the notion of general fuzzy automata (for simplicity GFA), we define the notions of direct product, restricted direct product and join of two GFA. Also, we introduce some operations on (Fuzzy) sets and then prove some related theorems. Finally we construct the general fuzzy recognizers and recognizable sets and give the notion of (trim) reversal of a given GFA....