scholarly journals The transfer of stylised artistic images in eye movement experiments based on fuzzy differential equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jie Zhou ◽  
Long Li ◽  
Zaiyang Yu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Eye Movement ◽  
Space Environment ◽  
Fuzzy Differential Equation ◽  
Image Study ◽  
The Difference ◽  
Art Research ◽  
Artistic Images ◽  
Artistic Image

Abstract In many aspects of people's production and life, artistic images have been widely used. Because the image has the function of transmitting information, it can provide necessary space environment information for people. However, there are many problems in the design of stylised art images, and hence the usability of images is affected. Due to its unique advantages, the study of artistic eye movement has gradually become a research hotspot. The fuzzy differential equation is an important branch of differential equation theory, which can be used to study eye movement experiments in the field of the art research. In the process of observation, experiment and maintenance, errors cannot be avoided, and the variables and parameters obtained are often fuzzy, incomplete and inaccurate. And fuzzy differential equations can deal with these uncertainties well. At first, this paper studies the migration-image-study-related theory and art image, based on the study of an artistic image that can be divided into instructions image and symbol and image, with the help of eye movement experiment method to investigate the effects of two types of image on people read mechanism. This research mainly uses the fuzzy differential equation for the visual search experimental paradigm to identify the influence of the difference of the effect.

A reduction theory of second order meromorphic differential equations, I

1987 ◽  
Vol 101 (2) ◽  
pp. 323-342
Author(s):  
W. B. Jurkat ◽  
H. J. Zwiesler
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Standard Form ◽  
Reduction Theory ◽  
Fundamental Solution Matrix ◽  
Equivalent Equation ◽  
The Difference ◽  
Solution Matrix ◽  
Meromorphic Differential Equation ◽  
Meromorphic Differential

In this article we investigate the meromorphic differential equation X′(z) = A(z) X(z), often abbreviated by [A], where A(z) is a matrix (all matrices we consider have dimensions 2 × 2) meromorphic at infinity, i.e. holomorphic in a punctured neighbourhood of infinity with at most a pole there. Moreover, X(z) denotes a fundamental solution matrix. Given a matrix T(z) which together with its inverse is meromorphic at infinity (a meromorphic transformation), then the function Y(z) = T−1(z) X(z) solves the differential equation [B] with B = T−1AT − T−1T [1,5]. This introduces an equivalence relation among meromorphic differential equations and leads to the question of finding a simple representative for each equivalence class, which, for example, is of importance for further function-theoretic examinations of the solutions. The first major achievement in this direction is marked by Birkhoff's reduction which shows that it is always possible to obtain an equivalent equation [B] where B(z) is holomorphic in ℂ ¬ {0} (throughout this article A ¬ B denotes the difference of these sets) with at most a singularity of the first kind at 0 [1, 2, 5, 6]. We call this the standard form. The question of how many further simplifications can be made will be answered in the framework of our reduction theory. For this purpose we introduce the notion of a normalized standard equation [A] (NSE) which is defined by the following conditions:(i) , where r ∈ ℕ and Ak are constant matrices, (notation: )(ii) A(z) has trace tr for some c ∈ ℂ,(iii) Ar−1 has different eigenvalues,(iv) the eigenvalues of A−1 are either incongruent modulo 1 or equal,(v) if A−1 = μI, then Ar−1 is diagonal,(vi) Ar−1 and A−1 are triangular in opposite ways,(vii) a12(z) is monic (leading coefficient equals 1) unless a12 ≡ 0; furthermore a21(z) is monic in case that a12 ≡ 0 but a21 ≢ 0.

scholarly journals Existence and uniqueness theorem for a solution of fuzzy differential equations

1999 ◽  
Vol 22 (2) ◽  
pp. 271-279 ◽  
Author(s):  
Jong Yeoul Park ◽  
Hyo Keun Han
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Differential Equations ◽  
Uniqueness Theorem ◽  
Successive Approximation ◽  
Existence And Uniqueness ◽  
Fuzzy Differential Equation ◽  
Existence And Uniqueness Theorem ◽  
Uniqueness Of A Solution

By using the method of successive approximation, we prove the existence and uniqueness of a solution of the fuzzy differential equationx′(t)=f(t,x(t)),x(t0)=x0. We also consider anϵ-approximate solution of the above fuzzy differential equation.

scholarly journals The fundamental importance of differential equations with three singularities in Mathematical Statistics

1985 ◽  
Vol 4 (1) ◽  
pp. 18-24
Author(s):  
H. S. Steyn
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Probability Distributions ◽  
Mathematical Physics ◽  
Second Order ◽  
Discrete Distributions ◽  
Mathematical Statistics ◽  
Multivariate Distributions ◽  
The Difference ◽  
Linear Differential

It is well-known that the solution of a second order linear differential equation with at most five singularities plays a fundamental role in Mathematical Physics. In this paper it is shown that this statement also applies to Mathematical Statistics but with the difference that an equation with three singularities will suffice. Two wide classes of probability distributions are defined as solutions of such a differential equation, one for continuous distributions and one for discrete distributions. These two classes contain as members all the distributions which are normally considered as of importance in Mathematical Statistics. In the continuous case the probability functions are solutions of the relevant second order equation, while in the discrete case the probability generating functions are solutions there-of. By defining appropriate multidimentional extensions corresponding differential equations are obtained for continuous and discrete multivariate distributions.

Climate: A Chain of Identical Reservoirs

1991 ◽  
Author(s):  
James C. G. Walker
Keyword(s):  
Differential Equation ◽  
Energy Balance ◽  
Differential Equations ◽  
Climate Model ◽  
Diffusion Problem ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
The Difference ◽  
A Chain ◽  
The One

One class of important problems involves diffusion in a single spatial dimension, for example, height profiles of reactive constituents in a turbulently mixing atmosphere, profiles of concentration as a function of depth in the ocean or other body of water, diffusion and diagenesis within sediments, and calculation of temperatures as a function of depth or position in a variety of media. The one-dimensional diffusion problem typically yields a chain of interacting reservoirs that exchange the species of interest only with the immediately adjacent reservoirs. In the mathematical formulation of the problem, each differential equation is coupled only to adjacent differential equations and not to more distant ones. Substantial economies of computation can therefore be achieved, making it possible to deal with a larger number of reservoirs and corresponding differential equations. In this chapter I shall explain how to solve a one-dimensional diffusion problem efficiently, performing only the necessary calculations. The example I shall use is the calculation of the zonally averaged temperature of the surface of the Earth (that is, the temperature averaged over all longitudes as a function of latitude). I first present an energy balance climate model that calculates zonally averaged temperatures as a function of latitude in terms of the absorption of solar energy, which is a function of latitude, the emission of long-wave planetary radiation to space, which is a function of temperature, and the transport of heat from one latitude to another. This heat transport is represented as a diffusive process, dependent on the temperature gradient or the difference between temperatures in adjacent latitude bands. I use the energy balance climate model first to calculate annual average temperature as a function of latitude, comparing the calculated results with observed values and tuning the simulation by adjusting the diffusion parameter that describes the transport of energy between latitudes. I then show that most of the elements of the sleq array for this problem are zero. Nonzero elements are present only on the diagonal and immediately adjacent to the diagonal. The array has this property because each differential equation for temperature in a latitude band is coupled only to temperatures in the adjacent latitude bands.

Fuzzy form of Euler Method to Solve Fuzzy Differential Equations

2021 ◽  
Vol 01 ◽  
Author(s):  
Umme Salma Pirzada ◽  
S. Rama Mohan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Riccati Equation ◽  
Initial Value Problems ◽  
Euler Method ◽  
Fuzzy Differential Equation ◽  
Fuzzy Arithmetic ◽  
Local Error ◽  
Initial Value

: This paper proposes fuzzy form of Euler method to solve fuzzy initial value problems. By this method, fuzzy differential equations can be solved directly using fuzzy arithmetic. The solution by this method is readily available in a form of fuzzy-valued function. The method does not require to re-write fuzzy differential equation into system of two crisp ordinary differential equations. Algorithm of the method and local error expression are discussed. An illustration and solution of fuzzy Riccati equation are provided for the applicability of the method.

scholarly journals A stability theory for perturbed differential equations

1979 ◽  
Vol 2 (2) ◽  
pp. 283-297
Author(s):  
Sheldon P. Gordon
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Direct Method ◽  
Mathematical Technique ◽  
General Differential Equation ◽  
Stability And Instability ◽  
The Difference ◽  
Perturbed Differential Equation ◽  
Eventual Stability ◽  
Liapunov's Direct Method

The problem of determining the behavior of the solutions of a perturbed differential equation with respect to the solutions of the original unperturbed differential equation is studied. The general differential equation considered isX′=f(t,X)and the associated perturbed differential equation isY′=f(t,Y)+g(t,Y).The approach used is to examine the difference between the respective solutionsF(t,t0,x0)andG(t,t0,y0)of these two differential equations. Definitions paralleling the usual concepts of stability, asymptotic stability, eventual stability, exponential stability and instability are introduced for the differenceG(t,t0,y0)−F(t,t0,x0)in the case where the initial valuesy0andx0are sufficiently close. The principal mathematical technique employed is a new modification of Liapunov's Direct Method which is applied to the difference of the two solutions. Each of the various stabillty-type properties considered is then shown to be guaranteed by the existence of a Liapunov-type function with appropriate properties.

A study of fuzzy methods for solving system of fuzzy differential equations

2020 ◽  
pp. 1-27
Author(s):  
M. Keshavarz ◽  
T. Allahviranloo ◽  
S. Abbasbandy ◽  
M. H. Modarressi
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Differential Equations ◽  
Harmonic Oscillator ◽  
Fuzzy Differential Equation ◽  
First Order ◽  
Fuzzy Methods ◽  
Effectiveness And Efficiency ◽  
Generalized Hukuhara Differentiability ◽  
Fuzzy Solutions

This paper is devoted to obtain an analytical solution for first-order fuzzy differential equations and system of fuzzy differential equations by different methods by considering the type of generalized Hukuhara differentiability of a solution and without embedding them to crisp equations. Moreover, the fuzzy solutions of a second-order fuzzy differential equation by considering the type of differentiability are obtained using reduction to a system of fuzzy differential equations. The effectiveness and efficiency of the approaches are illustrated by solving several practical examples such as Newton’s law of cooling, the mathematical models for the distribution of a drug in the human body and the fuzzy forced harmonic oscillator problem.

scholarly journals Controllability of Semilinear Neutral Differential Equations with Impulses and Nonlocal Conditions

2021 ◽  
Author(s):  
Oscar Camacho ◽  
Hugo Leiva ◽  
Lenin Riera
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlocal Conditions ◽  
Real Life ◽  
Exact Controllability ◽  
Ordinary Linear Differential Equation ◽  
Neutral Differential Equations ◽  
The Difference ◽  
Linear Differential ◽  
External Forces

When a real-life problem is mathematically modeled by differential equations or another type of equation, there are always intrinsic phenomena that are not taken into account and can affect the behavior of such a model. For example, external forces can abruptly change the model; impulses and delay can cause a breakdown of it. Considering these intrinsic phenomena in the mathematical model makes the difference between a simple differential equation and a differential equation with impulses, delay, and nonlocal conditions. So, in this work, we consider a semilinear nonautonomous neutral differential equation under the influence of impulses, delay, and nonlocal conditions. In this paper we study the controllability of these semilinear neutral differential equations with some of these intrinsic phenomena taking into consideration. Our aim is to prove that the controllability of the associated ordinary linear differential equation is preserved under certain conditions imposed on these new disturbances. In order to achieve our objective, we apply Rothe’s fixed point Theorem to prove the exact controllability of the system. Finally, our method can be extended to the evolution equation in Hilbert spaces with applications to control systems governed by PDE’s equations.

Age Differences in Eye Movements During Reading: Degenerative Problems or Compensatory Strategy?

2019 ◽  
Vol 24 (4) ◽  
pp. 297-311
Author(s):  
José David Moreno ◽  
José A. León ◽  
Lorena A. M. Arnal ◽  
Juan Botella
Keyword(s):  
Eye Movements ◽  
Eye Movement ◽  
Meta Analysis ◽  
Age Groups ◽  
Behavioral Changes ◽  
Aging Effects ◽  
Movement Data ◽  
Sentence Reading ◽  
The Difference ◽  
Eye Movements During Reading

Abstract. We report the results of a meta-analysis of 22 experiments comparing the eye movement data obtained from young ( Mage = 21 years) and old ( Mage = 73 years) readers. The data included six eye movement measures (mean gaze duration, mean fixation duration, total sentence reading time, mean number of fixations, mean number of regressions, and mean length of progressive saccade eye movements). Estimates were obtained of the typified mean difference, d, between the age groups in all six measures. The results showed positive combined effect size estimates in favor of the young adult group (between 0.54 and 3.66 in all measures), although the difference for the mean number of fixations was not significant. Young adults make in a systematic way, shorter gazes, fewer regressions, and shorter saccadic movements during reading than older adults, and they also read faster. The meta-analysis results confirm statistically the most common patterns observed in previous research; therefore, eye movements seem to be a useful tool to measure behavioral changes due to the aging process. Moreover, these results do not allow us to discard either of the two main hypotheses assessed for explaining the observed aging effects, namely neural degenerative problems and the adoption of compensatory strategies.

Normalization Bernstein Basis For Solving Fractional Fredholm-Integro Differential Equation

2018 ◽  
pp. 491
Author(s):  
Abdul Khaleq O. Al-Jubory ◽  
Shaymaa Hussain Salih
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear System ◽  
Differential Equations ◽  
Bernstein Basis ◽  
Traditional Methods ◽  
Integro Differential Equation

In this work, we employ a new normalization Bernstein basis for solving linear Freadholm of fractional integro-differential equations  nonhomogeneous  of the second type (LFFIDEs). We adopt Petrov-Galerkian method (PGM) to approximate solution of the (LFFIDEs) via normalization Bernstein basis that yields linear system. Some examples are given and their results are shown in tables and figures, the Petrov-Galerkian method (PGM) is very effective and convenient and overcome the difficulty of traditional methods. We solve this problem (LFFIDEs) by the assistance of Matlab10.   

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