Spiral waves in CIMA model and its LBGK simulation

Introduction In 1952, Turing set up a formalism capable of describing formation of stationary concentration patterns by symmetry breaking in'reaction-diffusion systemsl']. This type of spatiallydistributed stationary concentration pattern is called Turing pattern. Another type of interesting structure is concentration spiral wave which is also involved in reaction-diffusion systems.In a special case, called the activator-inhibitor system, the following chemicajs are involved:an activator capa…     

Restrictions on the degree spectra of algebraic structures

We construct the degree b ≤ 0″ admitting no algebraic structure with degree spectrum {x: x ? b}. Moreover, we solve Miller’s problem of distinguishing incomparable degrees by the spectra of linear orderings.     

Increasing η ‐representable degrees

In this paper we prove that any Δ₃⁰ degree has an increasing η ‐representation. Therefore, there is an increasing η ‐representable set without a strong η ‐representation (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strong jump-traceability I: The computably enumerable case

Recent investigations in algorithmic randomness have lead to the discovery and analysis of the fundamental class K of reals called the K-trivial reals, defined as those whose initial segment complexity is identical with that of the sequence of all 1's. There remain many important open questions concerning this class, such as whether there is a combinatorial characterization of the class and whether it coincides with possibly smaller subclasses, such as the class of reals which are not sufficiently powerful as oracles to cup a Turing incomplete Martin-Löf random real to the halting problem. Hidden here is the question of whether there exist proper natural subclasses of K. We show that the combinatorial class of computably enumerable, strongly jump-traceable reals, defined via the jump operator by Figueira, Nies and Stephan [Santiago Figueira, André Nies, Frank Stephan, Lowness properties and approximations of the jump, Electr. Notes Theor. Comput. Sci. 143 (2006) 45-57], is such a class, and show that like K, it is an ideal in the computably enumerable degrees. This is the first example of a class of reals defined by a “cost function” construction which forms a proper subclass of K. Further, we show that every c.e., strongly jump-traceable set is not Martin-Löf cuppable, thus giving a combinatorial property which implies non-ML cuppability.     

A hierarchy of Turing degrees of divergence bounded computable real numbers

A real number x is f-bounded computable (f-bc, for short) for a function f if there is a computable sequence (x_s) of rational numbers which converges to x f-bounded effectively in the sense that, for any natural number n, the sequence (x_s) has at most f(n) non-overlapping jumps of size larger than 2^-n. f-bc reals are called divergence bounded computable if f is computable. In this paper we give a hierarchy theorem for Turing degrees of different classes of f-bc reals. More precisely, we will show that, for any computable functions f and g, if there exists a constant γ>1 such that, for any constant c, f(nγ)+n+cg(n) holds for almost all n, then the classes of Turing degrees given by f-bc and g-bc reals are different. As a corollary this implies immediately the result of [R. Rettinger, X. Zheng, On the Turing degrees of the divergence bounded computable reals, in: CiE 2005, June 8–15, Amsterdam, The Netherlands, Lecture Notes in Computer Science, vol. 3526, 2005, Springer, Berlin, pp. 418–428.] that the classes of Turing degrees of d-c.e. reals and divergence bounded computable reals are different.     

Note on a universal quantum Turing machine

In this Letter, we construct a novel model of universal quantum Turing machine (QTM) by means of a property of Riemann zeta function, which is free from the specific time for an input data and efficiently simulates each step of a given QTM.     

Stability of synchronized dynamics and pattern formation in coupled systems: Review of some recent results

In arbitrarily coupled dynamical systems (maps or ordinary differential equations), the stability of synchronized states (including equilibrium point, periodic orbit or chaotic attractor) and the formation of patterns from loss of stability of the synchronized states are two problems of current research interest. These two problems are often treated separately in the literature. Here, we present a unified framework in which we show that the eigenvalues of the coupling matrix determine the stability of the synchronized state, while the eigenvectors correspond to patterns emerging from desynchronization. Based on this simple framework three results are derived: First, general approaches are developed that yield constraints directly on the coupling strengths which ensure the stability of synchronized dynamics. Second, when the synchronized state becomes unstable spatial patterns can be selectively realized by varying the coupling strengths. Distinct temporal evolution of the spatial pattern can be obtained depending on the bifurcating synchronized state. Third, given a desired spatiotemporal pattern, one is able to design coupling schemes which give rise to that pattern as the coupled system evolves. Systems with specific coupling schemes are used as examples to illustrate the general methods.