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Emergence is a concept that is easy to exhibit, but very hard to formally handle. This paper is about cubic sand grains moving around on nicely packed columns in one dimension (the physical sandpile is two dimensional, but the support of sand columns is one dimensional). The Kadanoff Sandpile Model is a discrete dynamical system describing the evolution of a finite number of stacked grains –as they would fall from an hourglass– to a stable configuration (fixed point). Grains move according to the repeated application of a simple local rule until reaching a fixed point. The main interest of the model relies in the difficulty of understanding its behavior, despite the simplicity of the rule. In this paper we prove the emergence of wave patterns periodically repeated on fixed points. Remarkably, those regular patterns do not cover the entire fixed point, but eventually emerge from a seemingly highly disordered segment. The proof technique we set up associated arguments of linear algebra and combinatorics, which interestingly allow to formally state the emergence of regular patterns without requiring a precise understanding of the chaotic initial segment's dynamic. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove the iterative processes generated by a nonexpansive semigroup and a generalized contraction mapping converge strongly to common fixed point p of the nonexpansive semigroup, and p is the unique solution to some variational inequality. The result extends some theorems in the literature. [ABSTRACT FROM AUTHOR]

The well-known Tarski΄s fixed point theorem asserts that an increasing mapping from a complete lattice into itself has a fixed point. This theorem plays an important role in the development of supermodular games for economic analysis. Let C be a finite lattice consisting of all integer points in an n-dimensional box and f be an increasing mapping from C into itself in terms of lexicographic ordering. It has been shown in the literature that, when f is given as an oracle, a fixed point of f can be found in polynomial time. The problem we consider in this paper is the complexity of determining whether or not f has a unique fixed point. We present a polynomial-time reduction of integer programming to an increasing mapping from C into itself. As a result of this reduction, we prove that, when f is given as an oracle, determining whether or not f has a unique fixed point is Co-NP hard. [ABSTRACT FROM AUTHOR]

In this paper we use fixed point tilings to answer a question posed by Michael Hochman and show that every one-dimensional effectively closed subshift can be implemented by a local rule in two dimensions. The proof uses the fixed-point construction of an aperiodic tile set and its extensions. [ABSTRACT FROM AUTHOR]

In this paper, a reflexive transformation ᵠ on F(S) in the logic system L* is proposed, which is an automorphism of F(S). It is proved that the concepts of truth degree, similarity degree and pseudo-distance are remain under the reflexive transformation ᵠ. Moreover, three different approximate reasoning patterns are introduced in the system L*, and it is also proved that for any A ϵ F(S) and Γ ⊂ F(S), A is an approximate conclusion of Γ if and only if ᵠ(A) is an approximate conclusion of ᵠ(Γ) in the sense of any one of the three patterns. Finally, the properties of reflexive transformation are investigated and relation of fixed points in classical logic system L and those in L* are studied. [ABSTRACT FROM AUTHOR]

In 1975 Chaitin introduced his Ώ number as a concrete example of random real. The real Ώ is defined based on the set of all halting inputs for an optimal prefix-free machine U, which is a universal decoding algorithm used to define the notion of program-size complexity. Chaitin showed Ώ to be random by discovering the property that the first n bits of the base-two expansion of Ώ solve the halting problem of U for all binary inputs of length at most n. In this paper, we introduce a new representation Θ of Chaitin Ώ number. The real Θ is defined based on the set of all compressible strings. We investigate the properties of Θ and show that Θ is random. In addition, we generalize Θ to two directions Θ(T) and ]> with a real T > 0. We then study their properties. In particular, we show that the computability of the real Θ(T) gives a sufficient condition for a real T ϵ (0,1) to be a fixed point on partial randomness, i.e., to satisfy the condition that the compression rate of T equals to T. [ABSTRACT FROM AUTHOR]

It is generally conjectured that the Mann iteration converges faster than the Ishikawa iteration for any operator defined on an arbitrary closed convex subset of a Banach space. The recent result of Babu et al [1] shows that this conjecture can be proved for a class of quasi-contractive operators called the Zamfirescu operators[10]. In this paper it is shown that the proof can indeed be generalised to that of quasicontraction maps. [ABSTRACT FROM AUTHOR]

We present several new observations on the SMS4 block cipher, and discuss their cryptographic significance. The crucial observation is the existence of fixed points and also of simple linear relationships between the bits of the input and output words for each component of the round functions for some input words. This implies that the non-linear function T of SMS4 does not appear random and that the linear transformation provides poor diffusion. Furthermore, the branch number of the linear transformation in the key scheduling algorithm is shown to be less than optimal. The main security implication of these observations is that the round function is not always non-linear. Due to this linearity, it is possible to reduce the number of effective rounds of SMS4 by four. We also investigate the susceptibility of SMS4 to further cryptanalysis. Finally, we demonstrate a successful differential attack on a slightly modified variant of SMS4. These findings raise serious questions on the security provided by SMS4. [ABSTRACT FROM AUTHOR]