Abstract: A new improvement of author's techniques of trilinear aggregating, uniting and canceling, is presented and applied to accelerate multiplication of matrices of moderate sizes. This results in the Exact Computing algorithms for n × n matrix multiplication in only ( n +2)[1.75( n +2)+( n 2 + 4 n +3)/3] essential multiplication steps for arbitrary even n . Also the new techniques allow us to simplify the design of the fastest known APA-algorithms for matrix multiplication.

Abstract: The axiomatization of algebras of functions or partial functions under various operations has been studied by several authors. Menger [2] deals with the problem of axiomatizing the algebraic properties of 1-ary functions from the reals to the reals. He considers algebras with one of three binary operations corresponding to addition, multiplication or composition. This study is furthered by Schweizer and Sklar in [3]–[6]. In addition to the operations mentioned above they also introduce a partial ordering which corresponds to restriction of functions. In the second of these papers they give a set of axioms such that any system satisfying these axioms is order isomorphic to a concrete system of partial functions under composition and ordered by restriction. In [7] Schweizer and Sklar extend their work to the algebra of multiplace vector-valued functions. In [8] Whitlock studies abstract multiplaced function systems given by super-associative laws and shows that these systems are isomorphically embeddable in a concrete system of multiplaced functions where the operation on the functions is substitution of an m -ary function in an n -ary function. In this paper we consider the set of functions from U α , the set of α-sequences, to U where a is an infinite ordinal. As opposed to the composition operations studied in the above works, we consider the composition operations * κ , for κ A , * κ , V κ › κ where V κ are the projection (selector) functions on the κth place. In particular, the polynomials over an algebra form such an algebra called a polynomial substitution algebra. In §6 we show that a first-order axiom system and a condition of local finiteness, given by Pinter in a talk at Berkeley in 1972 to characterize term substitution abgebras, also characterize these polynomial substitution algebras.

Abstract: The problem of deciding whether a partial binary operation, a "bin", can be embedded in a semigroup is the associativity problem (for general bins). It is known that it is equivalent to the word problem for (semi)groups and thus unsolvable, even for the class of finite bins. This paper establishes a close association between bins and their wordchains and 3-connected 3-regular planar graphs, or, equivalently convex 3-regular polyhedral nets (skeletons). This permits a constructive approach revealing the combinatorial depth of the associativity problem in detail and leads to a naturally enumerable hierarchy of standard wordchain patterns, of universal bins, and of associative laws. Each bin is a superposition of homomorphic images, i.e. "colourings" of edges, of universal bins. One side result is a purely algebraic equivalent of the 4-colour-theorem. The obtained results open further ways for an efficient search by computer for simplest non-associativity contradictions. It is hoped that they lead to solutions of the associativity problem for further subclasses of bins, further insight into the structure of partial binary operations and of polyhedra and will yield precise measures of presentations for associative systems and their classifications.

Abstract: We give below a rather concrete and intuitively motivated definition and construction of a free algebra which make the latter more comprehendible than when it is defined and constructed according to the manner usually given in the literature. Let A = {a, b} and M = {m, n, p} be algebras each with one binary operation, say, multiplication (which, as usual, we omit to denote) defined by their respective multiplication tables :

Abstract: Let t be a disjoint sum of tensors associated to matrix multiplication. The rank of the tensorial powers of t is bounded by an expression involving the elements of t and an exponent for matrix multiplication. This relation leads to a trascendental equation defining a new exponent for matrix multiplication.