Infimum Properties Differ in the Weak Truth-table Degrees and the Turing Degrees

Abstract We prove that there are non-recursive r.e. sets A and C with A < _T C such that for every set . Both authors are supported by “863” and the National Science Foundation of China     

How to build a hypercomputer

We claim that the theoretical hypercomputation problem has already been solved, and that what remains is an engineering problem. We review our construction of the Halting Function (the function that settles the Halting Problem) and then sketch possible blueprints for an actual hypercomputer.     

On the Hardness of Computing Intersection,Union and Minkowski Sum of Polytopes

For polytopes P ₁,P ₂⊂ℝ ^d , we consider the intersection P ₁∩P ₂, the convex hull of the union CH(P ₁∪P ₂), and the Minkowski sum P ₁+P ₂. For the Minkowski sum, we prove that enumerating the facets of P ₁+P ₂ is NP-hard if P ₁ and P ₂ are specified by facets, or if P ₁ is specified by vertices and P ₂ is a polyhedral cone specified by facets. For the intersection, we prove that computing the facets or the vertices of the intersection of two polytopes is NP-hard if one of them is given by vertices and the other by facets. Also, computing the vertices of the intersection of two polytopes given by vertices is shown to be NP-hard. Analogous results for computing the convex hull of the union of two polytopes follow from polar duality. All of the hardness results are established by showing that the appropriate decision version, for each of these problems, is NP-complete.     

On effectively computable realizations of choice functions: Dedicated to Professors Kenneth J. Arrow and Anil Nerode

Let X be a compact, convex subset of R_n, and let $\text{〈R(X),}\text{F}{}_{\mathrm{R}}\text{〉}$ be a recursive space of alternatives, where R(X) is the image of X in a recursive metric space, and $\text{F}{}_{\mathrm{R}}$ is the family of all recursive subsets of R(X). If $\text{C:}\text{F}{}_{\mathrm{R}}\text{→}\text{F}{}_{\mathrm{R}}$ is a non-trivial recursively representable choice function that is rational in the sense of Richter, we prove that C has no recursive realization within Church's Thesis. Our proof is not a diagonalization argument and uses no paradoxical statements from formal systems. Instead, the proof is a Kleene-Post reduction style argument and uses the Turing equivalence between mechanical devices of computation and the recursive functions of Gödel and Kleene.     

Group classification of systems of non-linear reaction-diffusion equations with general diffusion matrix. II. Generalized Turing systems

Group classification of systems of two coupled non-linear reaction-diffusion equation with a diagonal diffusion matrix is carried out. Symmetries of diffusion systems with singular diffusion matrix and additional first order derivative terms are described.     

Chemical waves

In our paper we try to describe the basic concepts of chemical waves and spatial pattern formation in a simple way. We pay particular attention to self-organisation phenomena in extended excitable systems. These result in the appearance of travelling waves, spiral waves, target patterns, Turing structures or more complicated structures called scroll waves, which are three-dimensional systems. We describe the most famous oscillating reaction, the Belousov-Zhabotinsky (BZ) reaction, in greater detail. This is because it is of great interest in both physical chemistry and in studies on the evolution and sustenance of self-organising biological systems.