Multiplication tables for non-prime odd numbers The untouched tables which will give us a better understanding of the distribution of prime odd numbers

The sieve method is used to separate prime numbers from non-prime numbers. If the set of prime odd numbers cannot be written as multiplication tables, the set of non-prime odd numbers can be written as multiplication tables. Thus, each odd number that does not appear in these multiplication tables is certainly a prime odd number. Based on these tables, it was proved by the opposite method that the series of prime numbers are random series. Although they are random, they can be easily tracked using the opposite method. The counter-example is used to proof that it is not possible to write whole multiplication tables of prime odd numbers on formula of [(𝑎×𝑏)+𝑐] or [(𝑎×𝑏)−𝑐]. Instead, partial multiplication tables can be used. It also was proved that the number 1 is a prime odd number.

Acceleration of Sparse Matrix-Vector Multiplication by Region Traversal

Sparse matrix-vector multiplication (shortly SpM×V) is one of most common subroutines in numerical linear algebra. The problem is that the memory access patterns during SpM×V are irregular, and utilization of the cache can suffer from low spatial or temporal locality. Approaches to improve the performance of SpM×V are based on matrix reordering and register blocking. These matrix transformations are designed to handle randomly occurring dense blocks in a sparse matrix. The efficiency of these transformations depends strongly on the presence of suitable blocks. The overhead of reorganization of a matrix from one format to another is often of the order of tens of executions ofSpM×V. For this reason, such a reorganization pays off only if the same matrix A is multiplied by multiple different vectors, e.g., in iterative linear solvers.This paper introduces an unusual approach to accelerate SpM×V. This approach can be combined with other acceleration approaches andconsists of three steps:1) dividing matrix A into non-empty regions,2) choosing an efficient way to traverse these regions (in other words, choosing an efficient ordering of partial multiplications),3) choosing the optimal type of storage for each region.All these three steps are tightly coupled. The first step divides the whole matrix into smaller parts (regions) that can fit in the cache. The second step improves the locality during multiplication due to better utilization of distant references. The last step maximizes the machine computation performance of the partial multiplication for each region.In this paper, we describe aspects of these 3 steps in more detail (including fast and time-inexpensive algorithms for all steps). Ourmeasurements prove that our approach gives a significant speedup for almost all matrices arising from various technical areas.

Equivalence elementaire et decidabilite pour des structures du type groupe agissant sur un groupe abelien

We prove an Ax-Kochen-Ershov like transfer principle for groups acting on groups. The simplest case is the following: let B be a soluble group acting on an abelian group G so that G is a torsion-free divisible module over the group ring ℤ[B], then the theory of B determines the one of the two-sorted structure 〈G,B,*〉, where * is the action of B on C. More generally, we show a similar principle for structures 〈G,B,*〉, where G is a torsion-free divisible module over the quotient of ℤ[B] by the annulator of G.Two applications come immediately from this result:First, for not necessarily commutative domains, where we consider the action of a subgroup of the invertible elements on the additive group. We obtain then the decidability of a weakened structure of ring, with partial multiplication.The second application is to pure groups. The semi-direct product of G by B is bi-interpretable with our structure 〈G,B,*〉. Thus, we obtain stable decidable groups that are not linear over a field.