Scouting the Mandelbrot set with memory

An exploratory study is made on the dynamics of the map defining the Mandelbrot set endowed with memory (m) of past iterations, that is, , . © 2014 Wiley Periodicals, Inc. Complexity 21: 84–96, 2016     

The area of the Mandelbrot set

Summary We obtain upper bounds for the area of the Mandelbrot set. An effective procedure is given for computing the coefficients of the conformal mapping from the exterior of the unit circle onto the exterior of the Mandelbrot set. The upper bound is obtained by computing finitely many of these coefficient and applying Green's Theorem. The error in such calculations is estimated by deriving explicit formulas for infinitely many of the coefficients and comparing.Partially supported by a grant from the National Science Foundation     

On the complement of the Mandelbrot set

We study the uniformization function of the Mandelbrot set via the behavior of multipliers of periodic orbits     

Homeomorphisms between limbs of the Mandelbrot set

Using a family of higher degree polynomials as a bridge, together with complex surgery techniques, we construct a homeomorphism between any two limbs of the Mandelbrot set of equal denominator. Induced by these homeomorphisms and complex conjugation, we obtain an involution between each limb and itself, whose fixed points form a topological arc. All these maps have counterparts at the combinatorial level relating corresponding external arguments. Assuming local connectivity of the Mandelbrot set we may conclude that the constructed homeomorphisms between limbs are compatible with the embeddings of the limbs in the plane. As usual we plough in the dynamical planes and harvest in the parameter space.     

On the Julia set of the perturbed Mandelbrot map

In this work, we present numerical results which support the smooth decomposition method of the generalized Julia set by Peintge et al., in the case of other perturbations of the Mandelbrot map studied in our previous work (Argyris J, Andreadis I, Karakasidis T. Chaos, Solitons & Fractals 1999). We also calculate the generalized Julia set of a Mandelbrot map subject to noise. Hence, we are in a position to examining numerically the stability of this set under small noise.     

On the lower semicontinuity of the set of efficient solutions to parametric generalized systems

In this paper, we investigate weak vector solutions and global vector solutions to a parametric generalized system (PGS). By using a weaker assumption, we discuss the lower semicontinuity of solution maps for PGS under the case that the $f$ -efficient solution mapping may not be single-valued. These results extend the recent ones in the literature. Some examples are given to illustrate our results.     

An efficient algorithm for the parametric resource allocation problem

The parametric resource allocation problem asks to minimize the sum of separable single-variable convex functions containing a parameter λ, Σ_{i = 1}ⁿ (ƒ_i(x_i + λg_i(x_i)), under simple constraints Σ_{i = 1}ⁿ x_i = M, l_i≤x_i≤u_i and x_i: nonnegative integers for i = 1, 2, …, n, where M is a given positive integer, and l_i and u_i are given lower and upper bounds on x_i. This paper presents an efficient algorithm for computing the sequence of all optimal solutions when λ is continuously changed from 0 to ∞. The required time is O(G√Mlog² n + n log n + n log(M/n)), where G = Σ_{i = 1}ⁿ u_i − Σ_{i = 1n} l_i and an evaluation of ƒ_i(·) or g_i(·) is assumed to be done in constant time.     

Connectivity of the Mandelbrot set for the family of renormalization transformations

We show that the Mandelbrot set for the family of renormalization transformations of 2-dimensional diamond-like hierachical Potts models in statistical mechanics is connected. We also give an upper bound for the Hausdorff dimension of Julia set when it is a quasi-circle.     

On the boundaries of the parametric resonance domain

The problem of stability for a system of linear differential equations with coefficients which are periodic in time and depend on the parameters is considered. The singularities of the general position arising at the boundaries of the stability and instability (parametric resonance) domains in the case of two and three parameters are listed. A constructive approach is proposed which enables one, in the first approximation, to determine the stability domain in the neighbourhood of a point of the boundary (regular or singular) from the information at this point. This approach enables one to eliminate a tedious numerical analysis of the stability region in the neighbourhood of the boundary point and can be employed to construct the boundaries of parametric resonance domains. As an example, the problem of the stability of the oscillations of an articulated pipe conveying fluid with a pulsating velocity is considered. In the space of three parameters (the average fluid velocity and the amplitude and frequency of pulsations) a singularity of the boundary of the stability domain of the “dihedral angle” type is obtained and the tangential cone to the stability domain is calculated.     

An efficient algorithm for the complete set partitioning problem

The complete set partitioning problem is the well known set partitioning problem with all possible nonzero binary columns in the constraint matrix. A highly specialized enumerative algorithm, which never requires the explicit maintenance of the constraint matrix, is presented. Computational results, with data reflecting a particular corporate tax payment scenario that can be modelled as a complete set partitioning problem, is also given.     

An efficient decision making approach in incomplete soft set

Zou et al. (2008) [21] presented weighted-average of all possible choice values approach of soft sets under incomplete information system in decision making. However, the approach is hard to understand and involves a great amount of computation. In order to simplify the approach, we present the simplified probability to directly instead of the incomplete information, and demonstrate the equivalence between the weighted-average of all possible choice values approach and the simplified probability approach. Finally, comparison results show that the proposed approach involves relatively less computation and is easier to implement and understand as compared with the weighted-average of all possible choice values approach.     

An O(n 2) active set algorithm for the solution of a parametric quadratic program

In this paper, an O(n ²) active set method is presented for minimizing the parametric quadratic function (1/2)x′Dx-a′x + λmax(c - γ x,0) subject to l ⩽ x ⩽ b, for all nonnegative values of the parameter γ. Here, D is a positive diagonal n x n matrix, a and γ are arbitrary N-vectors, c is an arbitrary scalar, l and b are arbitrary n-vectors, such thatl ⩽ b. An extension of this algorithm is presented for minimizing the parametric function (1/2)x′Dx-a ^′ x + λ |γ′x - c| subject to l ⩽ x ⩽ b. It is also shown that these problems arise naturally in a tax programming problem. This revised version was published online in August 2006 with corrections to the Cover Date.     

An efficient cubic trigonometric B-spline collocation scheme for the time-fractional telegraph equation

In this paper, a proficient numerical technique for the time-fractional telegraph equation(TFTE) is proposed. The chief aim of this paper is to utilize a relatively new type of B-spline called the cubic trigonometric B-spline for the proposed scheme. This technique is based on finite difference formulation for the Caputo time-fractional derivative and cubic trigonometric B-splines based technique for the derivatives in space. A stability analysis of the scheme is presented to confirm that the errors do not amplify. A convergence analysis is also presented. Computational experiments are carried out in addition to verify the theoretical analysis. Numerical results are contrasted with a few present techniques and it is concluded that the presented scheme is progressively right and more compelling.     

Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

A fundamental theme in holomorphic dynamics is that the local geometry of parameter space (e.g. the Mandelbrot set) near a parameter reflects the geometry of the Julia set, hence ultimately the dynamical properties, of the corresponding dynamical system. We establish a new instance of this phenomenon in terms of entropy.     

逃逸时间法生成M集中控制变量B的确定

逃逸时间算法是生成Mandelbrot集(简称M集)最常用的算法,本文针对非线性复映射f(z)=zm c为迭代函数的情形进行讨论.首先,根据逃逸时间算法的基本原理给出相应的算法步骤;然后,对迭代函数f(z)=zm c进行了详细研究,从而合理地确定了算法中需要控制的变量B(参数值c0的取值范围)的取值,这样就大大地减少了迭代次数,从而提高了算法的运算效率.     

An all-linear programming relaxation algorithm for optimizing over the efficient set

The problem (P) of optimizing a linear function over the efficient set of a multiple objective linear program has many important applications in multiple criteria decision making. Since the efficient set is in general a nonconvex set, problem (P) can be classified as a global optimization problem. Perhaps due to its inherent difficulty, it appears that no precisely-delineated implementable algorithm exists for solving problem (P) globally. In this paper a relaxation algorithm is presented for finding a globally optimal solution for problem (P). The algorithm finds an exact optimal solution to the problem after a finite number of iterations. A detailed discussion is included of how to implement the algorithm using only linear programming methods. Convergence of the algorithm is proven, and a sample problem is solved.Research supported by a grant from the College of Business Administration, University of Florida, Gainesville, Florida, U.S.A.     

On the approximation of plane curves by parametric cubic splines

A plane curveC can be approximated by a parametric cubic spline as follows. Points (x _i ,y _i ) are chosen in order alongC and a monotonically increasing variable is assigned values _i at the points (x _i ,y _i ): _i = the cumulative chordal distance from (x ₁ ,y ₁ ). The points ( _i ,x _i ) and ( _i ,y _i ) are then fitted separately by cubic splinesx() andy(), to obtain : (x(),y()). This paper establishes estimates for the errors involved in approximatingC by . It is found that the error in position betweenC and decreases likeh ³, whereh is the maximum length of arc between consecutive knots onC. For first derivatives, the error behaves likeh ²; for second derivatives, likeh.     

The set of periodic scalar differential equations with cubic nonlinearities

We study the structure induced by the number of periodic solutions on the set of differential equations x^′=f(t,x) where f∈C³(R²) is T-periodic in t, fx³(t,x)<0 for every (t,x)∈R², and f(t,x)→?∞ as x→∞, uniformly on t. We find that the set of differential equations with a singular periodic solution is a codimension-one submanifold, which divides the space into two components: equations with one periodic solution and equations with three periodic solutions. Moreover, the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension-two submanifold.