Abstract: It will be shown that the stabilizer clone of a transformation monoid is either trivial, i.e., it is generated by the monoid itself, or it contains an essentially binary function.

Abstract: We present some results extending d.c. duality theory to the generalized convex setting.Given a binary operation ∗ on the extended real line, we consider the problem of minimizingƒ∗–hwithƒ and h being functions defined on an arbitrary set. For this problem. we obtain results which encompass, as particular cases, some well known duality theorems for the infimum of the difference of two functions and the infimum of the maximum of two functions, as well as some well known formulae for the conjugate of the difference of two functions and the conjugate of type Lau of the maximum of two functions. Instead of considering the Fenchel conjugation operator, our results are expressed with the aid of dualities which are associated to the operation ∗.We also study. for such dualities, the corresponding generalized subdifferential

Abstract: The paper proposes a new operation for a class of slope-monotone closed curves. This operation is closely related to the Minkowski sum in that the Minkowski sum can be considered as the binary operation for figures bounded by those closed curves. In this sense, the proposed algebra is a generalization of the Minkowski sum. Conventionally, the inverse of the Minkowski sum can be defined naturally for convex figures, but the generalization of the inverse to nonconvex figures results in a complicated algebraic structure. This paper, on the other hand, gives a natural definition of the inverse for nonconvex figures, and thus constructs a simple algebraic structure.

Abstract: We describe the free objects in the variety of algebras involving several mutually distributive binary operations. Also, we show how an associative operation can be constructed on such systems in good cases, thus obtaining a two way correspondence between LD-monoids (sets with a left self-distributive and a compatible associative operation) and multi-LD-systems (sets with a family of mutually distributive operations).

Abstract: Cannon’s algorithm is a memory-efficient matrix multiplication technique for parallel computers with toroidal mesh interconnections. This algorithm assumes that input matrices are block distributed, but it is not clear how it can deal with block-cyclic distributed matrices. This paper generalizes Cannon’s algorithm for the case when input matrices are blockcyclic distributed across a two-dimensional processor array with an arbitrary number of processors and toroidal mesh interconnections. An efficient scheduling technique is proposed so that the number of communication steps is reduced to the least common multiple of P and Q for a given P x Q processor array. In addition, a partitioning and communication scheme is proposed to reduce the number of page faults for the case when matrices are too large to fit into main memory. Performance analysis shows that the proposed generalized Cannon’s algorithm (GCA) requires fewer page faults than a previously proposed algorithm (SUMMA). Experimental results on Intel Paragon show that GCA performs better than SUMMA when blocks of size larger than about (65 x 65) are used. However, GCA performance degrades if the block size is relatively small while SUMMA maintains the same performance. It is also shown that GCA maintains higher performance for large matrices than SUMMA