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AbstractLet qbe an odd prime power and suppose that $a,b\in \mathbb {F}_q$ are such that $ab$ and $(1{-}a)(1{-}b)$ are nonzero squares. Let $Q_{a,b} = (\mathbb {F}_q,*)$ be the quasigroup in which the operation is defined by $u*v=u+a(v{-}u)$ if $v-u$ is a square, and $u*v=u+b(v{-}u)$ if $v-u$ is a nonsquare. This quasigroup is called maximally nonassociativeif it satisfies $x*(y*z) = (x*y)*z \Leftrightarrow x=y=z$ . Denote by $\sigma (q)$ the number of $(a,b)$ for which $Q_{a,b}$ is maximally nonassociative. We show that there exist constants $\alpha \approx 0.029\,08$ and $\beta \approx 0.012\,59$ such that if $q\equiv 1 \bmod 4$ , then $\lim \sigma (q)/q^2 = \alpha $ , and if $q \equiv 3 \bmod 4$ , then $\lim \sigma (q)/q^2 = \beta $ .

Let TT, T be a spherical latin bitrade. With each aa1, a2, a3∈Tassociate a set of linear equations EqT, a of the form b1b2b3, where bb1, b2, b3 runs through Ta. Assume a10a2and a31. Then EqT,a has in rational numbers a unique solution \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document} $b_{i}=\bar{b}_{i}$\end{document}. Suppose that \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document} $\bar{b}_{i}\not= \bar{c}_{i}$\end{document}for all b, c∈Tsuch that \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document} $\bar{b}_{i}\not= \bar{c}_{i}$\end{document}and i∈1, 2, 3. We prove that then Tcan be interpreted as a dissection of an equilateral triangle. We also consider group modifications of latin bitrades and show that the methods for generating the dissections can be used for a proof that Tcan be embedded into the operational table of a finite abelian group, for every spherical latin bitrade T. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 1–24, 2010

Every LCC loop Q with Inn Q abelian is nilpotent class two. A loop Q of nilpotency class two is LCC ? L(x, y) = L(y, x) for all x, y ? Q ? I/Z(Mlt Q) is abelian ? [x, y, z] = [x,z,y] for all x, y, z ? Q ? [x, y, z] = [xy, z][x, z]-1for all x, y, z ? Q. All nilpotent LCC loops of order p2are described, and some of their multiplication groups are computed.

If Q is a quasigroup that is free in the class of all quasigroups which are isotopic to an Abelian group, then its multiplication group Mlt Q is a Frobenius group. Conversely, if Mlt Q is a Frobenius group, Q a quasigroup, then Q has to be isotopic to an Abelian group. If Q is, in addition, finite, then it must be a central quasigroup (a T-quasigroup).

We examine two particular constructions that derive from a 2-group G=G(·) another 2-group G(*) for the case when G(·) is one of , , . The constructions (cyclic and dihedral) have the property that x*y=x·y for exactly 3/4 of all pairs (x,y)?G×G.

Let $G(o)$ and $G(*)$ be two groups of finite order $n$, and suppose that each of the sets $\{u\in G;\ uo v=u*v$ for all $v\in G\}$ and $\{v\in G;\ uo v=u*v$ for all $u\in G\}$ has $n/2$ elements. Then $G(*)$ can be obtained from $G(o)$ by one of the two general constructions that are discussed in the paper.

Abstract.:  Several new tools are developed dealing with multiplication groups of loops. One of them is based upon the theorem that the inner mapping group of a loop never acts faithfully on any of its orbits, unless it is trivial. The applications include an analysis of loops whose inner mapping group is nonabelian of order pq. Some important earlier results are proved anew in a shorter way.

Let G(∘) and G(*) be two groups of the finite order n, and let dbe the size of the set {(a, b) ∈G×G;a∘b≠=a*b}. Let Pand Qbe Sylow 2-subgroups of G(∘) and G(*), respectively. If dis less than n2/4, then there exists an isomorphism ϑ: P≃Q, and the normalizers of Pand Qhave the same order.

Without Abstract:

A physical model of directional growth of a1 fluctuations in magnetic field has been put forward. Anisotropic growth of fluctuations along the field direction is due to the tendency towards lower demagnetizing and magnetostatic energy. This tendency is largest in the narrow range of temperatures limited by the Curie temperatures of the a1 and a2 phases.

An = An(*) denotes the unique left distributive binary system on {0, 1,…,2n-1) that satisfies a * 1 = a + 1 mod 2nfor all a ∈ An, and on(a) = k indicates the period 2kof a ∈ An (if b,c ∈ An, then a * b = a * c iff b ‵ c mod 2kand if 0 < b < c < 2kthen a < a * b < a * c). Amongs others, we prove that ot2(2t-2) ≤ t holds for every integer t ≥ 2, the equality taking place iff t is of the form 22*for an integer s ≥ 0.

Let Ak(*) denote the left distributive groupoid defined on [image omitted]  such that [image omitted]  mod2k for every a ∈ Ak For 1≥k

A general-purpose processor cell, called DOP, is presented. The DOP architecture is designed to support efficiently high-level programming languages (HLLs) such as C or Pascal, but still be simple enough to be implemented on one field programmable gate array (FPGA). Special attention is paid to the analysis of HLL requirements on processors. The DOP is designed to be used as a building block (cell) in a FPGA library. Its simplicity allows other microcomputer functional units to be implemented on the same FPGA. The DOP serves as a core for simple solutions using currently available technology.