Trapezoidal Maps Research Articles

Extended Fuzzy Rule Suram for Coffee Drying System

Extended fuzzy rule Suram is an algorithm developed to control a drying system using diesel as an energy source by modifying the value of the fuzzy membership function {0.5,1]. For a coffee drying room control system with solar energy, the bleak rule is based on fuzzy logic with variables of weather, air condition and wind speed. This algorithm is applied to the coffee drying process. The state variable membership function is represented in error values ​​and changes in error with a typical triangular and trapezoidal map for weather variables, air conditions, while wind speed is expressed in terms of wind speed. The results of the analysis of experiments with 16 fuzzy rules to control system output according to weather conditions obtained optimization of the use of solar energy by minimizing the use of electrical energy by heating. This algorithm also adjusts the coffee drying schedule by controlling the chamber, namely through temperature and humidity control. The results of the application of this algorithm show that the efficiency of electrical energy reaches 40,86%.

Conjugacy between trapezoid maps

Conjugacy between trapezoid maps

Reducing computational requirement of stability analysis of milling by partial averaging

The trapezoidal rule and Taylor theorem are used to establish a novel reformulation technique called Partial Averaging (PA). PA is seen to reduce computational time (CT) of computer central processing unit (CPU) with- out lose of accuracy. Two types of PA are proposed, one involves use of trapezoidal rule to give a map called Full Trapezoidal Rule Map with Partial Averaging (FTRMPA) while the other is achieved through exact analysis to give a map called Partial Trapezoidal Rule Map with Partial Averaging (PTRMPA). The latter is demonstrated to be more accurate and more time conserving in stability analysis of milling process (both one and two degree of freedom systems were considered) and delayed damped Mathieu equation. With all other things being equal the results of stability analysis of fully-immersed milling process using FTRMPA and PTRMPA are seen to be identical with those of full-discretization by Ding et al. (1) further validating the presented reformulation. PA is applied on the Second Order Least Squares Approximated Full-discretization method of (18) to illustrate its usefulness for reducing the cumbersome analysis and CT of the full-discretization method when analysis gets more complicated by higher order interpolation/ approximation theory.

On Minimum Link Monotone Path Problems

The problem of finding monotone paths between two given points has useful applications in path planning, and in particular, it is useful to look for minimum link paths. We are given a simple polygon P or a polygonal domain D with n vertices and a triplet of input parameters: (s, t, d), where s and t are two points in the plane and d is any direction. We show how to answer a query for the existence of a d-monotone path between s and t inside P (or D) in logarithmic time after preprocessing P in O(En) time, or D in O(En + ERlogR) time, where E is the size of the visibility graph of P (or D), and R is the number of reflex vertices in D. Our approach is based on the novel idea utilizing the dual graph of the trapezoidal map of P (or D). For polygonal domains, our approach uses a trapezoidal map associated with each visibility edge of D, and we show how to compute this large set of trapezoidal maps efficiently. Furthermore, we show how to output a minimum linkd-monotone path between points s and t, for an arbitrary input triplet (s, t, d).

Periodic unique beta-expansions: the Sharkovskiĭ ordering

AbstractLetβ∈(1,2). Eachx∈[0,1/(β−1)] can be represented in the formwhere εk∈{0,1} for allk(aβ-expansion ofx). If$\beta >{(1+\sqrt 5)}/{2}$, then, as is well known, there always existx∈(0,1/(β−1)) which have a uniqueβ-expansion. We study (purely) periodic uniqueβ-expansions and show that for eachn≥2 there exists$\beta _n\in [{(1+\sqrt 5)}/{2},2)$such that there are no unique periodicβ-expansions of smallest periodnforβ≤βnand at least one such expansion forβ>βn. Furthermore, we prove thatβk<βmif and only ifkis less thanmin the sense of the Sharkovskiĭ ordering. We give two proofs of this result, one of which is independent, and the other one links it to the dynamics of a family of trapezoidal maps.

Markov Incremental Constructions

A classic result asserts that many geometric structures can be constructed optimally by successively inserting their constituent parts in random order. These randomized incremental constructions (RICs) still work with imperfect randomness: the dynamic operations need only be “locally” random. Much attention has been given recently to inputs generated by Markov sources. These are particularly interesting to study in the framework of RICs, because Markov chains provide highly nonlocal randomness, which incapacitates virtually all known RIC technology. We generalize Mulmuley’s theory of Θ-series and prove that Markov incremental constructions with bounded spectral gap are optimal within polylog factors for trapezoidal maps, segment intersections, and convex hulls in any fixed dimension. The main contribution of this work is threefold: (i) extending the theory of abstract configuration spaces to the Markov setting; (ii) proving Clarkson–Shor-type bounds for this new model; (iii) applying the results to classical geometric problems. We hope that this work will pioneer a new approach to randomized analysis in computational geometry.

A Dynamic Dictionary for Priced Information with Application

In this paper we design a dynamic dictionary for the priced information model initiated by Charikar et al. Assume that a set S consisting of n elements is given such that each element has an associated price, a positive real number. The cost of performing an operation on elements of S is a function of their prices. The cost of an algorithm is the sum of the costs of all operations it performs. The objective is to design algorithms which incur low cost. In this model we propose a dynamic dictionary, supporting search, insert and delete, for keys drawn from a linearly ordered set. As an application we show that the dictionary can be used in computing the trapezoidal map of a set of line segments, a fundamental problem in computational geometry.

THE SHUFFLING BUFFER

The complexity of randomized incremental algorithms is analyzed with the assumption of a random order of the input. To guarantee this hypothesis, the n data have to be known in advance in order to be mixed what contradicts with the on-line nature of the algorithm. We present the shuffling buffer technique to introduce sufficient randomness to guarantee an improvement on the worst case complexity by knowing only k data in advance. Typically, an algorithm with O(n2) worst-case complexity and O(n) or O(n log n) randomized complexity has an [Formula: see text] complexity for the shuffling buffer. We illustrate this with binary search trees, the number of Delaunay triangles or the number of trapezoids in a trapezoidal map created during an incremental construction.

Union and split operations on dynamic trapezoidal maps

We propose algorithms to perform two new operations on an arrangement of line segments in the plane, represented by a trapezoidal map: the split of the map along a given vertical line D, and the union of two trapezoidal maps computed in two vertical slabs of the plane that are adjacent through a vertical line D. The data structure we use is a modified Influence Graph, still allowing dynamic insertions and deletions of line segments in the map. The algorithms for both operations run in O(s D logn+ log 2n) time, where n is the number of line segments in the map, and s D is the number of line segments intersected by D.

Superconvergence of Period-Doubling Cascade in Trapezoid Maps: Its Rigorous Proof and Superconvergence of the Period-Doubling Cascade Starting from a Period p Solution

In the symmetric and the asymmetric trapezoid maps, as a slope a of the trapezoid is increased, a period-doubling cascade occurs. The symbolic sequence of periodic points in this cascade is the Metropolis-Stein-Stein sequence R*m, and the convergence of the onset points am of the period 2m solutions to the accumulation point ac is exponentially fast. In a previous paper, we proved these results. In this paper, we give a detailed description of proofs regarding the results. Further, we study the period-doubling cascade starting from the period p (≥3) solution and show that the convergence of the onset points of the period p ×2m solutions is also exponentially fast.

Quadratic Convergence in Period Doubling to Chaos for Trapezoid Maps

The trapezoid mapge(x) is defined for fixede∈(0,1) byge(x)=x/eforx∈[0,e],ge(x)=1 forx∈(e,2−e), andge(x)=(2−x)/eforx∈[2−e,2]. For a giveneand the associated one-parameter family {λge(x): 1<λ<2}, letting λn(e) be the smallest value of λ>1 for which a fixedx∈(e,2−e), sayxc, is a periodic point of period 2n, Beyer and Stein conjectured in 1982 that, for anye<1, the parameter sequence {λn(e)}∞1is quadratically convergent. In this paper the conjecture is proved. Further, the quadratic convergence is generalized to nonisosceles trapezoid maps.

On the Bifurcation of Trapezoid Maps: An Example of a Class of One-Dimensional Maps with a Superconvergent Period Doubling Cascade

We investigate the period doubling cascade in symmetric and asymmetric trapezoid maps. We prove that as the slope a of the trapezoid is increased, there occurs a cascade of period doubling bifurcations. Further we prove that the symbolic sequence is the Metropolis­ Stein-Stein sequence R*m and that the ratio 8m of the interval of a, where the 2m-cycle is stable, to the interval where the 2 m + 1 _cycle is stable scales as 8m ~ ,(-1)'/3(ac,2/3)2', where ac is the accumulation point of the period doubling cascade, and , is the ratio of the two slopes of the trapezoid. That is, convergence of the onset point of the 2 m -cycle to ac is extremely rapid, and the Feigenbaum constant 8 == limm~oo 8m is infinite. Further, we give approximate expressions for the accumulation point as functions of b, the length of the smaller side of the trapezoid. The agreement between theoretical results and numerically obtained data is excellent.

Transfinite Function Iteration and Surreal Numbers

Louck has developed a relation between surreal numbers up to the first transfinite ordinal ω and aspects of iterated trapezoid maps. In this paper, we present a simple connection between transfinite iterates of the inverse of the tent map and the class of all the surreal numbers. This connection extends Louck's work to all surreal numbers. In particular, one can define the arithmetic operations of addition, multiplication, division, square roots, etc., of transfinite iterates by conversion of them to surreal numbers. The extension is done by transfinite induction. Inverses of other unimodal onto maps of a real interval could be considered and then the possibility exists of obtaining different structures for surreal numbers.

The digital tent map and the trapezoidal map

Much insight into nonlinear dynamics has come from studying simple maps such as the logistic map. In this paper we discuss the tent map and the trapezoidal map and introduce the digital tent map. These maps and the logistic map are all convex unimodal one-dimensional maps. All these maps have the same symbolic dynamics. Algebraic manipulation of high order iterations of nonlinear maps is in general unprofitable since they generate high order nonlinearities. This does not hold for the piecewise linear maps we discuss. The digital tent map is particularly easy to study, and its relation to the trapezoidal map makes possible explicit calculation of the parameter value for every bifurcation and supercycle: the onset of chaos occurs at a parameter value related to the Morse-Thue constant. Every bifurcation of the trapezoidal map corresponds to a periodic attractor of a digital tent map: this permits a numerical classification of all cycles of convex unimodal one-dimensional maps. This is compared with the ordering of trajectories in symbolic dynamics. It is shown how to calculate the parameter value for each bifurcation of the trapezoidal map: all bifurcations and supercycles of this map occur at rational values of the height of the trapezium. The trapezium height for every supercycle is a terminating binary fraction. All cycles with period up to 30 have been identified and counted, and widths of the associated windows calculated. There are 17 896 679 distinct cycles of period 30, so a full tabulation is not practicable. For the trapezoidal map the sum of the widths of all periodic windows is equal to the maximum height of the trapezium. Hence for this map the measure of parameter values at which chaos occurs is zero. There are however infinitely many transcendental parameter values at which the dynamics is not periodic.

On Periodic Orbits of Trapezoid Maps

Iterations of a special one-dimensional map (trapezoid map) which is a piecewise linear map with a flat top are studied. Using a symbolic dynamical method, it is shown that the stable periodic orbits of the map are characterized as finite shift maximal sequences in the sense extended by Sun and Helmberg and the parameter regions where the map produces a stable periodic orbit with any period are determined. The method used in the present paper is a simple method which converts a bifurcation problem of trapezoid maps to an initial value problem of a tent (triangle) map. This method is applicable to various chaotic maps which have a flat segment.

Monotonicity properties of the family of trapezoidal maps

Trapezoidal maps which are everywhere expanding out of their plateau form a three parameter familyT, up to affine changes of coordinates. We show that splittingT according to the various possible dynamical “behaviors” (we make this word precise in the process), yields a codimension one foliation. Some consequences of our result in terms of the monotonicity along simple one parameter families inT are then drawn. All together, aperiodic behavior is rare both from the topological and the measure theoretical point of view inT.

Parallel triangulation of a polygon in two calls to the trapezoidal map

Parallel triangulation of a polygon in two calls to the trapezoidal map

On the universal sequence generated by a class of unimodal functions

The universality of the Metropolis, Stein, and Stein (MSS). sequence ( J. Combin. Theory 15 (1973), 25–44) is established for a wide class of unimodal functions. The standard value of an LR-sequence is defined and a computational formula for it is established. An order on all finite LR-sequences is defined. It is shown that this order is equivalent to the order of Collet and Eckman (CE) (“Iterated Maps on the Interval as Dynamcal Systems,” Birkhauser, Boston, 1980), Louck and Metropolis (“Symbolic Dynamics of Trapezoidal Maps,” Reidel-Kluwer, Hingham, Ma, 1986) and Beyer, Mauldin, and Stein (BMS), ( J. Math.Anal. Appl. 115 (1986), 305–362) . The contiguity of harmonics is proved for any finite LR-sequence: Finally using an important result of BMS, it is shown that a pattern is legal if and only if it is a pattern associated with a positive solution λ of , the sequence of equations [ λf] k ( yo)= y 0 ( k= 1, 2,…).

Universality, multiplicity, and the effect of iron impurities in the Belousov–Zhabotinskii reaction

In experiments on the Belousov–Zhabotinskii reaction in a flow reactor we have observed dynamical behavior that is described well by one-dimensional maps with a single maximum. A sequence of period doubling bifurcations was observed as a parameter was varied, and beyond the accumulation point for the period doubling sequence there was a sequence of periodic states that has the same symbolic dynamics as the states of the U (universal) sequence of Metropolis, Stein, and Stein (1973). However, in another experiment with malonic acid from a different vendor, we found that some states with particular symbol sequences occurred in three different parameter ranges rather than in one range as in the U sequence. Analysis of the effect of impurities in the reagents showed that some impurities (e.g., Fe3+ and esters of malonic acid) at concentrations of only a few ppm produced dramatic changes in the dynamics; such impurities are contained in commercially available malonic acid. Experiments with purified malonic acid indicate that the Fe2+/Fe3+ and Cu+/Cu2+ redox couples act as co-catalysts of the Belousov–Zhabotinskii reaction; their effect can be explained by a free radical mechanism. Other metal ions (e.g., Ni2+ and Co2+) at low concentrations have negligible effect on the dynamics. Finally, both the sequences with universal ordering and the sequences with multiplicity are described well by indented trapezoid maps of the type analyzed by Beyer, Mauldin, and Stein (1986).