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Register a new userProperties of Carry Value Transformation
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Added by
Sudhakar Sahoo
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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The notion of Carry Value Transformation (CVT) is a model of Discrete Deterministic Dynamical System. In this paper, we have studied some interesting properties of CVT and proved that (1) the addition of any two non-negative integers is same as the sum of their CVT and XOR values. (2) While performing the repeated addition of CVT and XOR of two non-negative integers a and b (where a >= b), the number of iterations required to get either CVT=0 or XOR=0 is at most the length of a when both are expressed as binary strings. A similar process of addition of Modified Carry Value Transformation (MCVT) and XOR requires a maximum of two iterations for MCVT to be zero. (3) An equivalence relation is defined in the set (Z x Z) which divides the CV table into disjoint equivalence classes.
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CVT and XOR are two binary operations together used to calculate the sum of two non-negative integers on using a recursive mechanism. In this present study the convergence behaviors of this recursive mechanism has been captured through a tree like structure named as CVT-XOR Tree. We have analyzed how to identify the parent nodes, leaf nodes and internal nodes in the CVT-XOR Tree. We also provide the parent information, depth information and the number of children of a node in different CVT-XOR Trees on defining three different matrices. Lastly, one observation is made towards very old Mathematical problem of Goldbach Conjecture.
In this paper the theory of Carry Value Transformation (CVT) is designed and developed on a pair of n-bit strings and is used to produce many interesting patterns. One of them is found to be a self-similar fractal whose dimension is same as the dimension of the Sierpinski triangle. Different construction procedures like L-system, Cellular Automata rule, Tilling for this fractal are obtained which signifies that like other tools CVT can also be used for the formation of self-similar fractals. It is shown that CVT can be used for the production of periodic as well as chaotic patterns. Also, the analytical and algebraic properties of CVT are discussed. The definition of CVT in two-dimension is slightly modified and its mathematical properties are highlighted. Finally, the extension of CVT and modified CVT (MCVT) are done in higher dimensions.
Here the Integral Value Transformations (IVTs) are considered to be Discrete Dynamical System map in the spacemathbb{N}_(0). In this paper, the dynamics of IVTs is deciphered through the light of Topological Dynamics.
We provide formulas to compute the coefficients entering the affine scaling needed to get a non-degenerate function for the asymptotic distribution of the maxima of some kind of observable computed along the orbit of a randomly perturbed dynamical system. This will give information on the local geometrical properties of the stationary measure. We will consider systems perturbed with additive noise and with observational noise. Moreover we will apply our techniques to chaotic systems and to contractive systems, showing that both share the same qualitative behavior when perturbed.
In this paper, the notion of Integral Value Transformations (IVTs), a class of Discrete Dynamical Maps has been introduced. Then notion of Affine Discrete Dynamical System (ADDS) in the light of IVTs is defined and some rudimentary mathematical properties of the system are depicted. Collatz Conjecture is one of the most enigmatic problems in 20th Century. The Conjecture was posed by German Mathematician L. Collatz in 1937. There are much advancement in generalizing and defining analogous conjectures, but even to the date, there is no fruitful result for the advancement for the settlement of the conjecture. We have made an effort to make a Collatz type problem in the domain of IVTs and we have been able to solve the problem in 2011 [1]. Here mainly, we have focused and inquired on Collatz-like ADDS. Finally, we have designed the Optimal Distributed and Parallel Environment (ODPE) in the light of ADDS.
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