Functions and functional on finite systems

1992 ◽  
Vol 57 (1) ◽  
pp. 118-130 ◽  
Author(s):  
Libo Lo
Keyword(s):  
Computer Science ◽  
Natural Number ◽  
Number System ◽  
Input Function ◽  
Finite System ◽  
Global Function ◽  
Finite Systems ◽  
Recursive Functions ◽  
The Continuum

The global function on finite systems is a new concept defined by Gurevich in [1] and discussed in [2] and [3]. In the last ten years this concept has become more and more useful in computer science and logic. Gurevich also pointed out the importance of global functionals on finite systems. In this paper we will give a brief introduction to the concepts of global functions and global functionals on finite systems.In studying the natural number system N = 〈N, +,0〉 we often refer to its functions and functionals. There are a lot of books and papers in this area. Kleene in [4] gave a detailed introduction to the recursive functions of N. The functionals of N are normally very difficult to compute because here we need to tell the machine what the input function is, which is not very easy to do. In developing the theory of finite systems the functions and functionals are also very useful. For computing the functionals in finite systems we can take the entire graph of a function as the input, which is not possible in N. We will discuss recursive functions and functionals for finite systems. The definitions of recursive functions are very similar to the case in N, but we will have a very different situation. In N the number of elements is infinite. The number of all possible functions from N to N is the continuum. In a finite system the number of all possible functions is finite. It seems that there is no necessity to define the global functions.

scholarly journals A Concretization of an Approximation Method for Non-Affine Fractal Interpolation Functions

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 767
Author(s):  
Alexandra Băicoianu ◽  
Cristina Maria Păcurar ◽  
Marius Păun
Keyword(s):  
Approximation Method ◽  
Finite System ◽  
Countable System ◽  
Fractal Interpolation ◽  
Interpolation Function ◽  
Finite Systems ◽  
Fractal Interpolation Function ◽  
Fractal Interpolation Functions ◽  
Finite Set ◽  
Interpolation Functions

The present paper concretizes the models proposed by S. Ri and N. Secelean. S. Ri proposed the construction of the fractal interpolation function(FIF) considering finite systems consisting of Rakotch contractions, but produced no concretization of the model. N. Secelean considered countable systems of Banach contractions to produce the fractal interpolation function. Based on the abovementioned results, in this paper, we propose two different algorithms to produce the fractal interpolation functions both in the affine and non-affine cases. The theoretical context we were working in suppose a countable set of starting points and a countable system of Rakotch contractions. Due to the computational restrictions, the algorithms constructed in the applications have the weakness that they use a finite set of starting points and a finite system of Rakotch contractions. In this respect, the attractor obtained is a two-step approximation. The large number of points used in the computations and the graphical results lead us to the conclusion that the attractor obtained is a good approximation of the fractal interpolation function in both cases, affine and non-affine FIFs. In this way, we also provide a concretization of the scheme presented by C.M. Păcurar .

Invers Natural Number System to Maintain User-defined Sequence of Data Records

2021 ◽  
Author(s):  
Seyfettin Öztürk
Keyword(s):  
Natural Number ◽  
Number System

scholarly journals Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers

2019 ◽  
Vol 3 (1) ◽  
pp. 6 ◽  
Author(s):  
Vance Blankers ◽  
Tristan Rendfrey ◽  
Aaron Shukert ◽  
Patrick Shipman
Keyword(s):  
Dynamical Systems ◽  
Natural Number ◽  
Hyperbolic Plane ◽  
Julia Sets ◽  
Number System ◽  
Mandelbrot Set ◽  
Complex Numbers ◽  
Hyperbolic Numbers ◽  
Fractal Sets ◽  
Mandelbrot Sets

Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form x + τ y for x , y ∈ R , and τ 2 = 1 but τ ≠ ± 1 , are the natural number system in which to encode geometric properties of the Minkowski space R 1 , 1 . We show that the hyperbolic analog of the Mandelbrot set parameterizes the connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets.

Large families of incomparable A-isols

1983 ◽  
Vol 48 (2) ◽  
pp. 250-252 ◽  
Author(s):  
William S. Heck
Keyword(s):  
Natural Number ◽  
Natural Partial Order ◽  
List Type ◽  
Type Number ◽  
Closed Set ◽  
Strict Inclusion ◽  
Power Set ◽  
Large Families ◽  
Areas Of Interest ◽  
The Continuum

The set Λ of isols was extensively studied by Dekker and Myhill in [1]. Subsequently, Nerode [3] developed the theory of Λ(A), the set of isols relative to some recursively closed set of functions A.One of the main areas of interest of [1] was the natural partial order ≤ on Λ. In this paper we will examine some of the properties of ≤A on Λ(A). We use the following notations: ∣A∣ is the cardinality of the set A, ⊃ denotes strict inclusion, (a) is the power set of the set a, c is the cardinality of the continuum, and ω = {0, 1, 2, …}. The terms A-isol, A-immune, A-r.e., A-incomparable, etc. all refer to the usual meaning of these words, only taken in the context of the recursively closed set A. ReqA(a) is the A-r.e.t. of which a is a representative. By identifying a finite natural number with the A-r.e.t. consisting of sets of a given finite cardinality we see that ω ⊆ Λ(A); Λ(A) is said to be nontrivial iff ω ⊃ Λ(A). The three results proven in this paper are:Theorem 1. If Λ(A) is nontrivial, then ∣Λ(A)∣ = c.Theorem 2. If∣A∣ < c, then Λ(A) is nontrivial.Theorem 3. If ∣A∣ < c and ∣⊿∣ < c and ⊿ ⊆ Λ(A) − ω, then there is aΓ ⊆ Λ(A) − ω such that:(a) ∣Γ∣ = c.(b) Every member of Γ is A-incomparable with every member of Δ.(c) Any two distinct members of Γ are A-incomparable.

The inconsistency of the natural number system

Kybernetes ◽  
2008 ◽  
Vol 37 (3/4) ◽  
pp. 482-488 ◽  
Author(s):  
Wujia Zhu ◽  
Yi Lin ◽  
Guoping Du ◽  
Ningsheng Gong
Keyword(s):  
Natural Number ◽  
Number System

scholarly journals Counterexample-Guided Partial Bounding for Recursive Function Synthesis

2021 ◽  
pp. 832-855
Author(s):  
Azadeh Farzan ◽  
Victor Nicolet
Keyword(s):  
Recursive Function ◽  
Input Data ◽  
Unbounded Domains ◽  
Program Synthesis ◽  
Input Function ◽  
Synthesis Problem ◽  
Standard Approach ◽  
Data Types ◽  
Recursive Functions

AbstractQuantifier bounding is a standard approach in inductive program synthesis in dealing with unbounded domains. In this paper, we propose one such bounding method for the synthesis of recursive functions over recursive input data types. The synthesis problem is specified by an input reference (recursive) function and a recursion skeleton. The goal is to synthesize a recursive function equivalent to the input function whose recursion strategy is specified by the recursion skeleton. In this context, we illustrate that it is possible to selectively bound a subset of the (recursively typed) parameters, each by a suitable bound. The choices are guided by counterexamples. The evaluation of our strategy on a broad set of benchmarks shows that it succeeds in efficiently synthesizing non-trivial recursive functions where standard across-the-board bounding would fail.

Using Clock Arithmetic to Teach Algebra Concepts

2005 ◽  
Vol 11 (2) ◽  
pp. 104-109
Author(s):  
Elizabeth M. Brown ◽  
Elizabeth Jones
Keyword(s):  
Middle School Students ◽  
Number System ◽  
Abstract Concepts ◽  
Real Numbers ◽  
School Students ◽  
Multiplicative Inverse ◽  
Finite Systems ◽  
Solving Equations ◽  
Original Number ◽  
Addition And Subtraction

Learning algebra concepts can be difficult for middle school students. One reason may be because we work in only one number system, the set of real numbers. Students have only one frame of reference to provide examples of abstract concepts, such as the additive and multiplicative identities, additive and multiplicative inverses, and connections among the operations. These concepts are essential in solving equations. For example, we can think of an equation like 3x + 4 = 7 in the following way: Begin with a number, multiply it by 3, and add 4. If the answer is 7, what number did we start with? To solve this type of equation algebraically, we just undo what was done to the original number, that is, we add -4, the additive inverse of 4, and multiply the result by 1/3, the multiplicative inverse of 3. We can also think of this as subtracting 4 and dividing by 3, because addition and subtraction are inverse operations as are multiplication and division. The ideas of inverse operations and inverse elements are, therefore, central to algebra. Many students understand these ideas well enough to do simple problems like 3x + 4 = 7 but get confused with more difficult problems, such as 3(x + 6) + 2 = 5x + 5, where it is not as easy to see the order in which to undo these operations. We use finite systems to help students understand these key concepts in algebra, including additive and multiplicative identities, additive and multiplicative inverses, closure, and the relationships between addition and subtraction and multiplication and division.

Not all basic number representations are analog: Place coding as a precursor of the natural number system

2008 ◽  
Vol 31 (6) ◽  
pp. 650-651
Author(s):  
Wim Fias ◽  
Tom Verguts
Keyword(s):  
Natural Number ◽  
Number System ◽  
Basic Number ◽  
Number Representation ◽  
Number Representations

AbstractRips et al.'s arguments for rejecting basic number representations as a precursor of the natural number system are exclusively based on analog number coding. We argue that these arguments do not apply to place coding, a type of basic number representation that is not considered by Rips et al.

Nonstandard natural number systems and nonstandard models

1981 ◽  
Vol 46 (2) ◽  
pp. 365-376 ◽  
Author(s):  
Shizuo Kamo
Keyword(s):  
Natural Number ◽  
Ordered Set ◽  
Uniform Space ◽  
Additive Group ◽  
Number System ◽  
Order Type ◽  
The Other ◽  
Number Systems ◽  
Nonstandard Models ◽  
Real Number System

AbstractIt is known (see [1, 3.1.5]) that the order type of the nonstandard natural number system *N has the form ω + (ω* + ω) θ, where θ is a dense order type without first or last element and ω is the order type of N. Concerning this, Zakon [2] examined *N more closely and investigated the nonstandard real number system *R, as an ordered set, as an additive group and as a uniform space. He raised five questions which remained unsolved. These questions are concerned with the cofinality and coinitiality of θ (which depend on the underlying nonstandard universe *U). In this paper, we shall treat nonstandard models where the cofinality and coinitiality of θ coincide with some appropriated cardinals. Using these nonstandard models, we shall give answers to three of these questions and partial answers to the other to questions in [2].

EFFECTS OF REORTHOGONALIZATION: USING RECURSION METHOD IN SOME FINITE SYSTEMS

2004 ◽  
Vol 18 (04) ◽  
pp. 143-148
Author(s):  
JIANGWEI CHEN ◽  
XIANGANG WAN ◽  
JINMING DONG
Keyword(s):  
Density Of States ◽  
Infinite System ◽  
Finite System ◽  
Rounding Errors ◽  
Recursion Method ◽  
Asymptotic Behaviors ◽  
Finite Systems ◽  
Infinite Systems

Using the recursion method in some finite systems, such as C 78, C 82, etc., we find that we cannot obtain sufficient small recursion coefficients bn+1 to cut off the calculation due to the accumulation of rounding errors; and this problem can be avoided conveniently by making the newly gained state orthogonal to the preceding states. Furthermore, our calculations show that, for low symmetrical systems, reorthogonalization is necessary to calculate the correct density of states (DOS). The similar asymptotic behaviors between the finite system and the infinite system indicate that our work may be also useful for applying recursion method in infinite systems.

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