Algebraic structures in which the property of commutativity is substituted by mediality are introduced. We consider (associative) graded algebras and instead almost (generalized or $\varepsilon$-commutativity) we introduce ("commutativity-to-mediality" ansatz). Higher twisted products "deforming" brackets (being medial analog Lie brackets) defined. Toyoda's theorem connects (universal) with abe...

Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ elements $x_1,\ldots,x_n$ are equal to each other. The $\mathcal{F}^n_1$ said quasitrivial and those $\mathcal{F}^n_n$ idempotent. We show that $\mathcal{F}^n_1=\cdots =\mathcal{F}^n_{n-2}\subseteq\mathc...

The necessary and sufficient conditions under which a Menger algebra can be isomorphically represented by medial n-ary operations are proposed. Since hypercompositional regarded as generalization of algebra, for this reason, the situation hyperoperations is further examined representation theorem algebras such concepts proved.

We design 4-ary codebooks using Reed Muller codes for multiple-input multiple-output (MIMO) beamforming systems. An upperbound on the maximum correlation magnitude metric of a 4-ary codebook is derived from the minimum Hamming distance of the corresponding binary linear block code. The 4-ary MIMO beamforming codebooks for the number of transmit antennas equal to 4, 8, 16, and 32 are tabulated. ...

Factoring a Synchronous Context-Free Grammar into an equivalent grammar with a smaller number of nonterminals in each rule enables more efficient strategies for synchronous parsing. We present an algorithm for factoring an n-ary SCFG into a k-ary grammar in time O(kn). We also show how to efficiently compute the exact number of k-ary parsable permutations of length n, and discuss asymptotic beh...

Fundamentally binary theories are nonsignaling theories in which measurements of many outcomes are constructed by selecting from binary measurements. They constitute a sensible alternative to quantum theory and have never been directly falsified by any experiment. Here we solve two open problems related to them raised in Phys. Rev. Lett. 117, 150401 (2016). We first show that fundamentally bina...