In 2003 Cohn and Umans introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets S, T and U of a group G satisfying the Triple Product Property (TPP) as a means to bound the exponent ω of the matrix multiplication. We show that S, T and U may be be assumed to contain the identity and be otherwise disjoint. We also give a much shorter proof of the u...

In this note we establish connections between the orthogonality of conjugates of a finite or infinite quasigroup and some strictly transitive subsets of the complete multiplication group of this quasigroup. These connections are used for the investigation of orthogonality of distinct pairs of conjugates for quasigroups (loops) from some classes. For finite quasigroups the quasi-identities corre...

The public goods game is one of the most famous models for studying the evolution of cooperation in sizable groups. The multiplication factor in this game can characterize the investment return from the public good, which may be variable depending on the interactive environment in realistic situations. Instead of using the same universal value, here we consider that the multiplication factor in...

Hardware organized modular multiplication based on division algorithm is one of the effective methods used for RSA encryption/decryption. This paper generalizes the hardware organization of the modular multiplication based on the higher-radix SRT division algorithm, and describes the area/time trade-off of the organization. For the number representation we used the signed-digit number system an...

Addition is an essential operation for the convolution (or FIR filters). In FPGAs, addition should be carried out in a standard way employing ripple-carry adders (rather than carry-save adders), which complicates search for an optimal adder structure as routing order has a substantial influence on the addition cost. Further, complex parameters of addition inputs have been considered e.g. correl...

where v ranges over all places of Q and Qv is the completion of Q at v, denote its TateShafarevich group. As usual, L(E/Q, s) is the complex L-function of E over Q. Since E is now known to be modular, Kolyvagin’s work [11] shows that X(E/Q) is finite if L(E/Q, s) has a zero at s = 1 of order ≤ 1, and that gE/Q is equal to the order of the zero of L(E/Q, s) at s = 1. His proof relies heavily on ...

We describe the structure of coefficient ring $$W^*(pt)=\varOmega_W^*$$ $$c_1$$ -spherical bordism theory for an arbitrary $$SU$$ -bilinear multiplication. prove that any multiplication formal group $$W^*$$ is Landweber exact. Also we show after inverting set $$\mathcal{P}$$ Fermat primes there exists a complex orientation localized $$W^*[\mathcal{P}^{-1}]$$ such coefficients corresponding law ...

For any abelian group G and any function f : G → G we define a commutative binary operation or “multiplication” on G in terms of f . We give necessary and sufficient conditions on f for G to extend to a commutative ring with the new multiplication. In the case where G is an elementary abelian p-group of odd order, we classify those functions which extend G to a ring and show, under an equivalen...

In this paper, we have generalized the definition of vector space by considering the group as a canonical $m$-ary hypergroup, the field as a krasner $(m,n)$-hyperfield and considering the multiplication structure of a vector by a scalar as hyperstructure. Also we will be consider a normed $m$-ary hypervector space and introduce the concept of convergence of sequence on $m$-ary hypernormed space...

In [11], a new approach to subquadratic space complexity multiplication for extended finite fields has recently been proposed for hardware implementation. In this article, we develop the corresponding algorithm for software implementation. Compared to the Karatsuba algorithm, the proposed algorithm has a lower theoretical time complexity when the size of the input is greater than a fixed intege...