In On a theory of computation and complexity over the real numbers ... , Bull. Amer. Math. Soc. 21 (1989), 1-46, Blum, Shub, and Smale investigated computability over the reals and over ordered rings in general. They showed that over the reals, output sets of machines are recursively enumerable (i.e., halting sets of machines). It is asked in the aforementioned paper which ordered rings have th...

Based on the notion of $Q$-sup-lattices (a fuzzy counterpart of complete join-semilattices valuated in a commutative quantale), we present the concept of $Q$-sup-algebras -- $Q$-sup-lattices endowed with a collection of finitary operations compatible with the fuzzy joins. Similarly to the crisp case investigated in cite{zhang-laan}, we characterize their subalgebras and quotients, and following...

Citation Ilker, Efe and A. Nihat Berker. "Odd q-state clock spin-glass models in three dimensions, asymmetric phase diagrams, and multiple algebraically ordered phases. Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The MIT Faculty has made this article openly available. Please sh...

Let P be a set of n points in the finite plane 2 q over the finite field q of q elements, where q is an odd prime power. For any s ∈ q , denote by A(P; s) the number of ordered triangles whose vertices in P having area s. We show that if the cardinality of P is large enough then A(P; s) is close to the expected number |P| 3 /q.

The classical q-analogue of the integers was recently generalized by Morier-Genoud and Ovsienko to give q-analogues rational numbers. Some combinatorial interpretations are already known, namely as rank generating functions for certain partially ordered sets. We a new interpretation, showing that numerators q-rationals count sizes varieties over finite fields, which unions open Schubert cells i...

Setting of this research is Bishop’s constructive mathematics, the mathematics developed on Intuitionistic logic. If (X, =, 6=, θ) is an anti-ordered set, for a coequality q on X we say that it is strongly regular if it is regular and θ ◦ q ⊆ q ◦ θ holds. In this case, θ ◦ q is a quasi-antiorder relation on X such that the relation Θ = π ◦ θ ◦ π−1 on X/q is the maximal anti-order on X/q.

Let S be a thick generalized quadrangle and let G be a group of automorphisms of S. If G acts transitively on the set of non-degenerate ordered pentagons, then S is one of the classical generalized quadrangles W (q), Q(4, q), Q(5, q) or H(3, q2). The possibilities for G in each case are determined. We do not use the classification of the finite simple groups (from which this result also follows).

The mock A N N N I model is generalised in replacing the Ising spins by q-state Potts variables. Performing exact dedecoration transformations for general q and using Monte Carlo techniques in two dimensions for q = 3, a multitude of distinct spatially modulated long-range ordered phases, as well as phases with algebraic order, are found 10 spring from a multiphase point.

In this article, we define the set H of quaternion numbers as the set of all ordered sequences q = 〈x, y,w, z〉 where x,y,w and z are real numbers. The addition, difference and multiplication of the quaternion numbers are also defined. We define the real and imaginary parts of q and denote this by x = R(q), y = I1(q), w = I2(q), z = I3(q). We define the addition, difference, multiplication again...