A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE PART V: FRACTALS EVERYWHERE

2005 ◽  
Vol 15 (12) ◽  
pp. 3701-3849 ◽  
Author(s):  
LEON O. CHUA ◽  
VALERY I. SBITNEV ◽  
SOOK YOON
Keyword(s):  
Characteristic Function ◽  
Characteristic Functions ◽  
Global Dynamics ◽  
Local Rule ◽  
Digital To Analog Converter ◽  
Potential Applications ◽  
Depth Study ◽  
Global Representation ◽  
Synaptic Coefficient ◽  
The Right

This fifth installment is devoted to an in-depth study of CA Characteristic Functions, a unified global representation for all 256 one-dimensional Cellular Automata local rules. Except for eight rather special local rules whose global dynamics are described by an affine (mod 1) function of only one binary cell state variable, all characteristic functions exhibit a fractal geometry where self-similar two-dimensional substructures manifest themselves, ad infinitum, as the number of cells (I + 1) → ∞. In addition to a complete gallery of time-1 characteristic functions for all 256 local rules, an accompanying table of explicit formulas is given for generating these characteristic functions directly from binary bit-strings, as in a digital-to-analog converter. To illustrate the potential applications of these fundamental formulas, we prove rigorously that the "right-copycat" local rule [Formula: see text] is equivalent globally to the classic "left-shift" Bernoulli map. Similarly, we prove the "left-copycat" local rule [Formula: see text] is equivalent globally to the "right-shift" inverse Bernoulli map. Various geometrical and analytical properties have been identified from each characteristic function and explained rigorously. In particular, two-level stratified subpatterns found in most characteristic functions are shown to emerge if, and only if, b1 ≠ 0, where b1 is the "synaptic coefficient" associated with the cell differential equation developed in Part I. Gardens of Eden are derived from the decimal range of the characteristic function of each local rule and tabulated. Each of these binary strings has no predecessors (pre-image) and has therefore no past, but only the present and the future. Even more fascinating, many local rules are endowed with binary configurations which not only have no predecessors, but are also fixed points of the characteristic functions. To dramatize that such points have no past, and no future, they are henceforth christened "Isles of Eden". They too have been identified and tabulated.

EXTRACTION OF COGNITIVE CHARACTERISTIC FUNCTIONS ACCORDING TO VARIOUS PARAMETER CHANGES OF VIBRATORY STIMULATION

2019 ◽  
Vol 19 (08) ◽  
pp. 1940052
Author(s):  
MI-HYUN CHOI ◽  
SOON-CHEOL CHUNG
Keyword(s):  
Characteristic Function ◽  
Subjective Assessment ◽  
Characteristic Functions ◽  
Vibratory Stimulation ◽  
Cognitive Characteristics ◽  
Cognitive Characteristic ◽  
Normal Cognitive Function ◽  
Four Levels ◽  
Vibration Simulation ◽  
The Right

In this study, vibratory stimuli with various intensities (four levels) and frequencies (10, 50, 100, 150, 200, 225, 250 and 300[Formula: see text]Hz) were applied to the right index finger to extract cognitive characteristic functions for the intensity and frequency. For subjective assessment, an experiment was conducted with 30 healthy adults in their twenties who were right-handed and had normal cognitive function. One trial of the experiment was composed of a vibration phase (0.1 s) and a rest phase (10 s). After vibration simulation was applied once for each intensity and frequency, the participants filled out a subjective assessment form. After extracting the score of each word, the cognitive characteristic function was derived through discriminant analysis and regression analysis according to the frequency and intensity. Through the cognitive characteristic function, the level of cognitive characteristics of each word can be investigated according to the influence of the frequency and intensity. The results observed through the cognitive characteristics function of each word showed that the words affected by the frequency and intensity were “light,” “thick,” “heavy,” “blunt,” “vibrating,” “fast” and “weak.” “Itchy” and “slow” were cognitive characteristics affected only by the frequency, and “strong” and “push” were affected only by the intensity. Through the vibratory stimulation presentation parameters, i.e., frequency and intensity, the results capable of predicting the degrees of various cognitive characteristics were presented.

SOME EXTENSIONS OF A LEMMA OF KOTLARSKI

2012 ◽  
Vol 28 (4) ◽  
pp. 925-932 ◽  
Author(s):  
Kirill Evdokimov ◽  
Halbert White
Keyword(s):  
Characteristic Function ◽  
Characteristic Functions ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Weaker Conditions

This note demonstrates that the conditions of Kotlarski’s (1967, Pacific Journal of Mathematics 20(1), 69–76) lemma can be substantially relaxed. In particular, the condition that the characteristic functions of M, U1, and U2 are nonvanishing can be replaced with much weaker conditions: The characteristic function of U1 can be allowed to have real zeros, as long as the derivative of its characteristic function at those points is not also zero; that of U2 can have an isolated number of zeros; and that of M need satisfy no restrictions on its zeros. We also show that Kotlarski’s lemma holds when the tails of U1 are no thicker than exponential, regardless of the zeros of the characteristic functions of U1, U2, or M.

Certain Fourier Transforms of Distributions: II

1954 ◽  
Vol 6 ◽  
pp. 186-189 ◽  
Author(s):  
Eugene Lukacs ◽  
Otto Szász
Keyword(s):  
Characteristic Function ◽  
Fourier Transforms ◽  
Necessary Conditions ◽  
Laplace Transforms ◽  
Characteristic Functions ◽  
Necessary Condition ◽  
Multiple Roots ◽  
Not Given ◽  
Class Of Functions

In an earlier paper (1), published in this journal, a necessary condition was given which the reciprocal of a polynomial without multiple roots must satisfy in order to be a characteristic function. This condition is, however, valid for a wider class of functions since it can be shown (2, theorem 2 and corollary to theorem 3) that it holds for all analytic characteristic functions. The proof given in (1) is elementary and has some methodological interest since it avoids the use of theorems on singularities of Laplace transforms. Moreover the method used in (1) yields some additional necessary conditions which were not given in (1) and which do not seem to follow easily from the properties of analytic characteristic functions.

A new weighting scheme for arc based circle cone-beam CT reconstruction

2021 ◽  
pp. 1-19
Author(s):  
Wei Wang ◽  
Xiang-Gen Xia ◽  
Chuanjiang He ◽  
Zemin Ren ◽  
Jian Lu
Keyword(s):  
Characteristic Function ◽  
Weighting Function ◽  
Cone Beam Ct ◽  
Reconstruction Algorithm ◽  
Cone Beam ◽  
Characteristic Functions ◽  
Projection Data ◽  
Ct Reconstruction ◽  
Scanning Angle ◽  
Beam Geometry

In this paper, we present an arc based fan-beam computed tomography (CT) reconstruction algorithm by applying Katsevich’s helical CT image reconstruction formula to 2D fan-beam CT scanning data. Specifically, we propose a new weighting function to deal with the redundant data. Our weighting function ϖ ( x _ , λ ) is an average of two characteristic functions, where each characteristic function indicates whether the projection data of the scanning angle contributes to the intensity of the pixel x _ . In fact, for every pixel x _ , our method uses the projection data of two scanning angle intervals to reconstruct its intensity, where one interval contains the starting angle and another contains the end angle. Each interval corresponds to a characteristic function. By extending the fan-beam algorithm to the circle cone-beam geometry, we also obtain a new circle cone-beam CT reconstruction algorithm. To verify the effectiveness of our method, the simulated experiments are performed for 2D fan-beam geometry with straight line detectors and 3D circle cone-beam geometry with flat-plan detectors, where the simulated sinograms are generated by the open-source software “ASTRA toolbox.” We compare our method with the other existing algorithms. Our experimental results show that our new method yields the lowest root-mean-square-error (RMSE) and the highest structural-similarity (SSIM) for both reconstructed 2D and 3D fan-beam CT images.

Integral Limit Laws for Additive Functions

1973 ◽  
Vol 25 (1) ◽  
pp. 194-203
Author(s):  
J. Galambos
Keyword(s):  
Characteristic Function ◽  
Fourier Transforms ◽  
Mean Value ◽  
Characteristic Functions ◽  
Asymptotic Formulas ◽  
Mean Value Theorem ◽  
Multiplicative Functions ◽  
Additive Functions ◽  
Limit Laws ◽  
Integral Limit

In the present paper a general form of integral limit laws for additive functions is obtained. Our limit law contains Kubilius’ results [5] on his class H. In the proof we make use of characteristic functions (Fourier transforms), which reduces our problem to finding asymptotic formulas for sums of multiplicative functions. This requires an extension of previous results in order to enable us to take into consideration the parameter of the characteristic function in question. We call this extension a parametric mean value theorem for multiplicative functions and its proof is analytic on the line of [4].

scholarly journals Rectifying Performance of Heterojunction Based on α-Borophene Nanoribbons with Edge Passivation

2020 ◽  
Vol 15 (1) ◽  
Author(s):  
Guoliang Yu ◽  
Wence Ding ◽  
Xianbo Xiao ◽  
Xiaobo Li ◽  
Guanghui Zhou
Keyword(s):  
Electronic Transport ◽  
First Principles ◽  
First Principles Calculations ◽  
Zigzag Edge ◽  
Hydrogen Atoms ◽  
Rectification Effect ◽  
Structural Details ◽  
Potential Applications ◽  
Matching Degree ◽  
The Right

Abstract We propose a planar model heterojunction based on α-borophene nanoribbons and study its electronic transport properties. We respectively consider three types of heterojunctions. Each type consists of two zigzag-edge α-borophene nanoribbons (Z αBNR), one is metallic with unpassivated or passivated edges by a hydrogen atom (1H-Z αBNR) and the other is semiconducting with the edge passivated by two hydrogen atoms (2H-Z αBNR) or a single nitrogen atom (N-Z αBNR). Using the first-principles calculations combined with the nonequilibrium Green’s function, we observe that the rectifying performance depends strongly on the atomic structural details of a junction. Specifically, the rectification ratio of the junction is almost unchanged when its left metallic ribbon changes from ZBNR to 1H-Z αBNR. However, its ratio increases from 120 to 240 when the right semiconducting one varies from 2H-Z αBNR to N-Z αBNR. This rectification effect can be explained microscopically by the matching degree the electronic bands between two parts of a junction. Our findings imply that the borophene-based heterojunctions may have potential applications in rectification nano-devices.

The Characteristic Function, a Method-Specific Alternative to the Horwitz Function

2012 ◽  
Vol 95 (6) ◽  
pp. 1803-1806 ◽  
Author(s):  
Michael Thompson
Keyword(s):  
Detection Limit ◽  
Characteristic Function ◽  
Characteristic Functions ◽  
Mathematical Form ◽  
Essential Difference ◽  
Food Sector ◽  
Wide Range ◽  
Specific Alternative ◽  
Additional Role

Abstract The Horwitz function is compared with the characteristic function as a descriptor of the precision of individual analytical methods. The Horwitz function describes the trend of reproducibility SDs observed in collaborative trials in the food sector over a wide range of concentrations of the analyte. However, it is imperfectly adaptable for describing the precision of individual methods, which is the role of the characteristic function. An essential difference between the two functions is that the characteristic function can accommodate a detection limit. This makes it a useful alternative when the precision of a method down to a detection limit is of interest. Many characteristic functions have a simple mathematical form, the parameters of which can be estimated with the usual resources. The Horwitz function serves an additional role as a fitness-for-purpose criterion in the form of the Horwitz ratio (HorRat). This use also has some shortcomings. The functional form of the characteristic function (with suitable prescribed parameters) is better adapted to this task.

The optical imagery of curved surfaces

1957 ◽  
Vol 240 (1223) ◽  
pp. 458-461
Keyword(s):  
Characteristic Function ◽  
Functional Equations ◽  
Individual Characteristic ◽  
Optical Systems ◽  
Characteristic Functions ◽  
Curved Surfaces ◽  
Angular Characteristic ◽  
Optical Imagery ◽  
The Individual ◽  
Angular Characteristic Function

The form of Hamilton’s angular characteristic function for the aberrationless imagery of one surface of rotation on another, and the connexions between the coefficients of the surface and functional equations, are found. When several optical systems of the type considered are arranged in succession the relations between the coefficients of the individual characteristic functions and those of the combination are obtained. These connexions enable all aberrations to be computed without resorting to ray tracing.

Market Exclusivity and the Right to Health in South Africa: A Primer for Tackling the Paradox

2020 ◽  
Vol 64 (1) ◽  
pp. 53-80
Author(s):  
Collins C Ajibo
Keyword(s):  
South Africa ◽  
Healthcare Services ◽  
Patent Law ◽  
Global Dynamics ◽  
Protective Measures ◽  
Universal Healthcare ◽  
Future Direction ◽  
The Right ◽  
Rule Making ◽  
Universal Healthcare Coverage

AbstractSouth Africa remains confronted with challenges to the realization of the right to healthcare services, shaped by both national and global dynamics. The proliferation of exclusivity regimes in intellectual property (IP) rule-making poses a threat to affordable healthcare services. Although South Africa is not a signatory to any of these enhanced IP norms, it may still be at risk through transposition, given that the current norm-setting constitutes the future direction of rule-making. These global dynamics are compounded by overly protective measures in South Africa's patent law as well as non-IP factors, particularly the prevalent weak health infrastructure. Although South Africa's IP Policy Phase I incorporates a raft of changes to address the situation, capacity constraints could thwart effective outcomes. Consequently, beyond the current patent law reform, there should be a roadmap for how to manage global IP norm-setting as well as non-IP factors, to foster universal healthcare coverage in South Africa.

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