We present a new and useful congruence identity satisfied by m-permutable varieties. It has been proved in [L1] that every m-permutable variety satisfies a non-trivial lattice identity (depending only on m). In [L2] we have found another interesting identity: Theorem 1. For m ≥ 3, every m-permutable variety satisfies the congruence identity αβh = αγh, for h = m[ m+1 2 ]− 1 Here, [ ] denotes int...

Brevity is the soul of wit. William Shakespeare A cyclically permutable code is a set of codewords having the property that no codeword is a cyclic shift of another codeword. We study the problem of constructing cyclically permutable codes of large size and low correlation. Cyclically permutable codes are used in code-division multiple-access systems realized by e.g. direct-sequence modulation ...

A cyclically permutable code is a binary code whose codewords are cyclically distinct and have full cyclic order. An important class of these codes are the constant weight cyclically permutable codes. In a code of this class all codewords have the same weight w. These codes have many applications, including in optical code-division multiple-access communication systems and in constructing proto...

The notions of permutable and weak-permutable convergence of a series ∑∞ n=1 an of real numbers are introduced. Classically, these two notions are equivalent, and, by Riemann’s two main theorems on the convergence of series, a convergent series is permutably convergent if and only if it is absolutely convergent. Working within Bishop-style constructive mathematics, we prove that Ishihara’s prin...

The technique of in-situ associative permuting is introduced which is an association of in-situ permuting and in-situ inverting. It is suitable for associatively permutable permutations of {1, 2, . . . , n} where the elements that will be inverted are negative and stored in order relative to each other according to their absolute values. Let K[1 . . . n] be an array of n integer keys each in th...

In this paper I shall extend the formulation of the hole argument to permutable theories. As covariant theories provides a general mathematical framework for classical physics, permutable theories provide the language for quantum physics. This analogy is deeply founded: as covariant theories are defined functorially on the category of manifolds and local diffeomorphisms with values into the cat...