Fractal Behavior in the Clarification Process of Cane … Behavior in the Clarification Process of Cane Sugar Production XiaomoYu,1 JunkeYe,2 JingHu,3,4 XiaopingLiao,2 andJianboGao3,4

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 868313 9 pageshttpdxdoiorg1011552013868313

Research ArticleFractal Behavior in the Clarification Process ofCane Sugar Production

Xiaomo Yu1 Junke Ye2 Jing Hu34 Xiaoping Liao2 and Jianbo Gao34

1 College of Light Industry and Food Engineering Guangxi University 100 Daxue Road Nanning Guangxi 530005 China2 College of Mechanical Engineering Guangxi University 100 Daxue Road Nanning Guangxi 530005 China3 Institute of Complexity Science and Big Data Technology Guangxi University 100 Daxue Road Nanning Guangxi 530005 China4 PMB Intelligence LLC Sunnyvale CA 94087 USA

Correspondence should be addressed to Jianbo Gao jbgaopmbgmailcom

Received 2 September 2013 Accepted 6 October 2013

Academic Editor Ahmed El Wakil

Copyright copy 2013 Xiaomo Yu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Cane sugar production is an important industrial process One of the most important steps in cane sugar production is theclarification process which provides high-quality concentrated sugar syrup crystal for further processing To gain fundamentalunderstanding of the physical and chemical processes associated with the clarification process and help design better approachesto improve the clarification of the mixed juice we explore the fractal behavior of the variables pertinent to the clarification processWe show that the major variables in this key process all show persistent long-range correlations for time scales up to at least a fewdays Persistent long-range correlations amount to unilateral deviations from a preset target This means that when the process isin a desired mode such that the target variables color of the produced sugar and its clarity degree both satisfy preset conditionsthey will remain so for a long period of time However adversity could happen in the sense that when they do not satisfy therequirements the adverse situation may last quite long These findings have to be explicitly accounted for when designing activecontrolling strategies to improve the quality of the produced sugar

1 Introduction

For centuries sugar has been a highly regarded and widelytraded commodity for the simple reason that sweetness is anessential ingredient of life Before 1990 sugar beet accountedfor about 40 of the total sugar production in the worldSince then cane sugar market has been rapidly expandingdue to the low cost in sugarcane production According tothe US Department of Agriculture the global cane sugarproduction reached 1175 million tons in 2009 amountingto 7833 of the total sugar production Among the majorcane sugar producer and consumer China ranks the 3rdaccounting for 7 of the world totalproduction

In general cane sugar manufacturing consists of sixphases milling clarification evaporation crystallizationcentrifuging and drying so as to obtain the white sugarbrown sugar and other products In particular the juice fromthe milling workshop is called mixed juice

Besides water sugarcane and reduced sugar the mixedjuice also contains many organic and inorganic nonsugarcomponents such as colloidal substances inorganic salts(iron magnesium aluminum calcium etc) and pigmentsWhile these nonsugar components are residual nutrients inthe sugarcane they are detrimental to the sugar productionFor example flavonoids multiacid and organic-acid canmake the mixed juice appear dark brown Since the nonsugarcomponents affect not only the appearance and color but alsothe concentration of the sugar (ie reflected as the sweetnessof sugar by a consumer) they have to be carefully removedThe purpose of the clarification process is to remove as manynonsugar components as possible improve the purity of thejuice and reduce its viscosity and color values This is criticalfor providing high-quality concentrated sugar syrup crystalto the boiling stage Therefore the clarification stage is a keyprocess in cane sugar production

2 Mathematical Problems in Engineering

There have been efforts to improve the clarification ofthe mixed juice by using neural network-based predictionschemes [1 2] Such approaches amount to treating theclarification process as a black or gray box and thus do notyield any understanding of the basic physics and chemistryinvolved in clarification To help gain more fundamentalunderstanding and guide the design of better approachesto improve clarification in this paper we carry out fractalanalysis of the key variables pertinent to the clarificationprocess

Among the types of activity that characterize complexsystems the most ubiquitous and puzzling is perhaps theappearance of 1119891120572 noise a form of temporal or spatialfluctuation characterized by a power-law decaying powerspectral density A subclass of such processes denoted as11198912119867+1 is called processes with long-range correlations

(or long memories) characterized by a Hurst parameter 119867Depending on whether 0 lt 119867 lt 12 119867 = 12 or12 lt 119867 lt 1 [3] they are said to have antipersistent corr-elations memoryless or only short-range correlations orpersistent long-range correlations Prominent examples ofsuch processes include vision [4] finance [5] DNA sequences[6ndash10] human cognition [11] and coordination [12] posture[13] cardiac dynamics [14ndash17] and the distribution of primenumbers [18] to name but a few In this paper we wish toexplore whether fractal theory can shed new light on theclarification process of cane sugar production

The remainder of the paper is organized as follows InSection 2 we provide some details about the cane sugarclarification process explain the various variables to beanalyzed here and discuss the challenges of analyzing thosevariables In Section 3 we carry out fractal analysis of thesevariables using the key concept in random fractal theory theHurst parameter which characterizes the basic correlationstructure of the signals Concluding discussions are presentedin Section 4

2 Clarification of Mixed Juice inCane Sugar Production

Sugar clarification is a key process in cane sugar productionIt is characterized by two important indices called color valueand clarity degree Whether the color value and the claritydegree can achieve the desired or preset values can criticallyaffect the quality of the cane sugar and the revenue of afactory

Many methods have been developed for clarifying themixed juiceThe two major ones are the carbonation methodand the sulphitationmethodThe former uses themilk of limeand carbon dioxide as main detergent on the mixed juiceBesides high cost the accompanying pollution by carbonatedmud is hard to deal with Therefore its usage has beensubstantially reduced over the years

Unlike the carbonate method the sulphitation methodadds clarifying agent SO

2in addition to lime and phosphoric

acid Due to the absorption effect of calcium sulfite (whichis the product of lime and SO

2) together with the inhibition

of pigment formation due to SO2 the sulphitation method

is significantly better than the carbonate method In thefollowing we will focus on the sulphitation method for theclarification of the mixed juice

The sulphitation method for clarification is schematicallyshown in Figure 1 It is a complicated physical chemicalprocess involving solvents such as SO

2 milk of lime and

phosphoric acid Roughly the process can be divided intofour stages cane juice phosphorus-preliming first heatingSO2sulphitation-neutralization and settlement after 2nd-

heatingMore specifically in the preliming stage after the firstheating to 55sim65∘C milk of lime and phosphoric acid areadded to themixed sugar cane juice In the next sulphur fumi-gation process SO

2gas enters the chemical process pipeline

At the end of the pipeline milk of lime is further addedfor achieving neutralization since phosphoric acid sulfitereactswith calciumhydroxide to form calciumphosphate andcalcium sulfate Then the 2nd heating is carried out so thatsugar cane colloids ofmixed juice can not only fully condensein the precipitator but also (1) accelerate precipitation (2)decrease juice viscosity and (3) facilitate the precipitatedparticles to sink fast Then the clarified juice can come outof the settler from the top of each later while the mud juice isdischarged into the vacuum suction filter by gravity Filteredjuice can then be directly incorporated into the clear juiceheating device and sent to the evaporation device to condenseinto syrup Therefore controlling the process parameterssteadily is the key to improve the quality of clarified juice andsugar

In this paper time series of the 11 variables indicated inFigure 1 will be analyzed using random fractal theory Thedata were measured in an advanced cane sugar productionfactory in Guangxi Zhuang Autonomous Region of ChinaGuangxi is the largest sugar producer in China amounting tomore than 60of the total sugar in the countryThedatawerecollected with a time interval of 2 hours and lasted for thewhole season (about 4 months) These variables are groupedinto three groups with the last two variables 119909

10 color value

and11990911 the clarity degree being the target variablesThe time

series data of these 11 variables are shown in Figures 2 3 and4 As can be seen the major challenge in analyzing these datais nonstationarity that is the statistical moments includingmean and variance vary with time Understandably this maybe a salient feature for variables that are measured over a verylong period of time (which is about 4 months here)

3 Fractal Analysis of the Clarification Process

In this section we first overview the basics of fractal processeswith long memory and then briefly describe two methodsfor detecting fractal structures from data with trends One isthe seminal detrended fluctuation analysis (DFA) [19] Theother is adaptive fractal analysis (AFA) [20ndash23] Since AFAprovides additional advantages over DFA to deal with signalswith arbitrary trends we only present results of AFA for thevariables pertinent to the clarification process of cane sugarproduction here

31 Basics of Fractal Processes with LongMemory As we havementioned fractal processes with long memory are charac-terized by the Hurst parameter 119867 To better understand the

Mathematical Problems in Engineering 3

Mixed juice ofsugar cane

Milk of lime

Milk of lime

Phosphoricacid

Sulfur

Clarified juice

Clear juice

Mud juice

Vacuum suctionfilter

Extractedsample

neutralizedjuice

Filtered mud

Extractedsample

clarified juice

Corrected brix(x1)

Correctedpolarimeter (x2)

pH value of mixedjuice (x3)

Total dosage of

Neutral pH value(x5)

Sulphur fumigationintensity (x6)

Corrected brix(x7)


pH value ofclarified juice (x9)

Color value (x10)

Clarity degree(x11)

Evaporation

Filter

1st-heating (55sim65∘C)

Extractedsample

mixed juice

2nd-heating (100sim103∘C)

Sedimentationsettler (35sim40min)

Purified juice-heating

Sulphur fumigation

Preliming (pH 64sim72)

Sulphur fumigation-neutralization (pH 68sim74)

(pH 68sim74)

intensity (15sim18mL)

(300sim400mgkg juice)

dioxide

P2O5 P2O5 (x4)

Figure 1 Schematic of the clarification stage of cane sugar production Variables 119909119894 119894 = 1 2 11 are analyzed here

18

16

140 500 1000 1500 2000 2500

Time (hours)

x1

(a)

0 500 1000 1500 2000 2500

Time (hours)

16

14

12

x2

(b)

0 500 1000 1500 2000 2500

Time (hours)

8

6

x3

(c)

0 500 1000 1500 2000 2500

Time (hours)

600

400

200

x4

(d)

Figure 2 Signals 1199091to 1199094

meaning of119867 it is useful to mathematically be more preciseLet 119909

1 1199092 119909

119899 be a stationary stochastic process with

mean 119909 and autocorrelation function of the type

119903 (119896) sim 1198962119867minus2

as 119896 997888rarr infin (1)

where 0 lt 119867 lt 1 is the Hurst parameter When 12 lt 119867 lt1 sum119896119903(119896) = infin leading to the term long range correlation

1199091 1199092 119909

119899 is often called an increment (or noise) process

Its power spectral density (PSD) is 11198912119867minus1 Its integration

119906 (119894) =

119894

sum

119896=1

(119909119896minus 119909) 119894 = 1 2 119899 (2)

is called a random walk process having PSD 11198912119867+1 Being1119891 processes they cannot be aptly modeled by Markov


0 500 1000 1500 2000 2500

Time (hours)

9

8

7

6

x5

(a)

0 500 1000 1500 2000 2500

Time (hours)

25

20

15

10

5

x6

(b)

Figure 3 Signals 1199095and 119909

6

0 500 1000 1500 2000 2500

Time (hours)

20

15

10

x7

(a)

0 500 1000 1500 2000 2500

Time (hours)

20

15

10

x8

(b)

0 500 1000 1500 2000 2500

Time (hours)

8

7

x9

(c)

0 500 1000 1500 2000 2500

Time (hours)

3000

2000

1000

x10

(d)

0 500 1000 1500 2000 2500

Time (hours)

80

60

100

x11

(e)


processes or ARIMA models [24] since the PSD for thoseprocesses is distinctly different from 1119891 To adequatelymodel 1119891 processes fractional order processes have to beused The most popular is the fractional Brownian motionmodel [3]

To deepen our understanding of the Hurst parameter letus smooth 119909

1 1199092 119909

119899 using nonoverlapping windows to

yield a new time series

119883(119898)

119905=(119909119905119898minus119898+1

+ sdot sdot sdot + 119909119905119898)

119898 119905 ge 1 (3)

It can be proven that the variance of the new time series isgiven by [25]

var (119883(119898)) = 12059021198982119867minus2 (4)

where 1205902 is the variance of original stochastic process1199091 1199092 119909

119899 Equation (4) offers an excellent means of

understanding 119867 For example if 119867 = 050 119898 = 100 thenvar(119883(119898)) = 1205902100 When 119867 = 075 in order to havevar(119883(119898)) = 1205902100 then we need 119898 = 104 which is much

larger than 119898 = 100 for the case of 119867 = 050 On the otherhand when 119867 = 025 if we still want var(119883(119898)) = 1205902100then 119898 asymp 215 much smaller than 119898 = 100 the case of119867 = 050 An interesting lesson from such a simple discussionis that if a time series is short while its 119867 is close to 1 thensmoothing is not a viable option for reducing the variationsthere

There are many excellent methods for estimating 119867 [2526] In the next two subsections we describe the methodsthat are most promising for detecting fractal variations invariables pertinent to the clarification process of cane sugarproduction

32 Detrended Fluctuation Analysis (DFA) Denote a timeseries of interest by 119909(1) 119909(2) 119909(3) DFA works as follows[19] First divide a given time series of length 119873 into lfloor119873119897rfloornonoverlapping segments each containing 119897 points Thendefine the local trend in each segment to be the ordinate of alinear least-squares fit or best polynomial fit of the time seriesin that segment this is schematically shown in Figure 5 usinga linear trend as an example Finally compute the ldquodetrended


2000

1000

0

minus1000

minus2000

0 200 400 600 800 1000

EEG

dat

ax(n)

Index n

Figure 5 A schematic showing local detrending in the DFAmethod

walkrdquo denoted by 119909119897(119899) as the difference between the original

ldquowalkrdquo 119909(119899) and the local trend One then examines thefollowing scaling behavior

119865119889(119897) = ⟨

119897

sum

119894=1

119909119897(119894)2

⟩ sim 1198972119867

(5)

where the angle brackets denote ensemble averages of all thesegments

Note that the best linear or polynomial fits of DFA mayhave large discontinuities at the boundaries of adjacent seg-ments This is clearly shown in Figure 5 Such discontinuitiesprevent DFA from effectively removing a complicated trendsuch as the 11-year cycle in sunspot numbers [27] As a resultDFA alone may not be effective enough to detect fractalvariations from data with trends

33 Adaptive Fractal Analysis (AFA) AFA is built upon theadaptive detrending algorithm [28 29] It fixes the potentialproblem of DFA by finding a globally smooth trend signalBecause of this AFA has additional advantages over DFA[20 21] For example AFA can deal with arbitrary strongnonlinear trends while DFA cannot [23 27] AFA has betterresolution of fractal scaling behavior for short time series[22] AFA has a direct interpretation in terms of spectralenergy while DFA does not [23] and there is a simple proofof why AFA yields the correct 119867 while such a proof is notavailable for DFA

To find the global trend themethod first partitions a timeseries into segments (or windows) of length 119908 = 2119899 + 1points where neighboring segments overlap by 119899 + 1 pointsand thus introducing a time scale of ((119908 + 1)2)120591 = (119899 + 1)120591where 120591 is the sampling time For each segment we fit a bestpolynomial of order119872 Note that119872 = 0 and 1 correspondto piecewise constant and linear fitting respectively Denotethe fitted polynomial for the 119894th and (119894 + 1)th segments by119910(119894)

(1198971) 119910(119894+1)(119897

2) 1198971 1198972= 1 2119899 + 1 respectively Note that

the length of the last segment may be smaller than 2119899+ 1 Wedefine the fitting for the overlapped region as

119910(119888)

(119897) = 1199081119910(119894)

(119897 + 119899) + 1199082119910(119894+1)

(119897) 119897 = 1 2 119899 + 1

(6)

where 1199081= (1 minus (119897 minus 1)119899) 119908

2= (119897 minus 1)119899 can be written as

(1 minus 119889119895)119899 119895 = 1 2 where 119889

119895denotes the distances between

the point and the centers of 119910(119894) and 119910(119894+1) respectively Thismeans that the weights decrease linearly with the distancebetween the point and the center of the segment Such aweighting ensures symmetry and effectively eliminates anyjumps or discontinuities around the boundaries of neigh-boring segments In fact the scheme ensures that the fittingis continuous everywhere is smooth at the nonboundarypoints and has the right- and left-derivatives at the boundaryThe method can effectively determine any kind of trendsignalThose for the 11 variables studies here are shown as redcurves in Figures 2ndash4

Now we can describe AFA If we start from an incrementprocess 119909(1) 119909(2) Similar to DFA we first construct arandom walk process using (2) If the original data can beconsidered as a random walk-like process which is true forEEG [25 30 31] and sea clutter radar returns [26 32 33] thenthis step is not necessary However for ideal fractal processesthere is no penalty if this is done even though the process isalready a random walk process

Next for a window size 119908 we determine for the randomwalk process 119906(119894) (or the original process if it is already arandom walk process) a global trend V(119894) 119894 = 1 2 119873Here 119873 is the length of the random walk process Theresidual 119906(119894) minus V(119894) characterizes fluctuations around theglobal trend and its variance yields the Hurst parameter 119867[23]

119865(2)

(119908) = [1

119873

119873

sum

119894=1

(119906 (119894) minus V (119894))2]

12

sim 119908119867

(7)

34 Fractal Variations in the Clarification Process of CaneSugar Production Let us first focus on the group 1 variables119909119894 119894 = 1 2 3 4 which represent corrected brix corrected

polarimeter pH value of themixed juice and the total dosageof P2O5 respectively Their AFA curves are shown in Figures

6(a)ndash6(d) We observe that there are two scaling regionsfor 1199091 on time scales shorter than 25 times 2 = 64 hours

the Hurst parameter is 074 On time scales greater than 64hours the Hurst parameter becomes 142 This means thatfor time scales smaller than about 3 days the variation in 119909

1

has persistent long-range correlations On time scales longerthan about 3 days the signal becomes very non-stationarymdash119867 gt 1 is often associated with non-stationarity [25 26] Thebehavior of 119909

2is very similar to that of 119909

1 Indeed one can

clearly observe that the signal of 1199092looks very similar to that

of 1199091 However the behaviors of 119909

3and 119909

4are quite different

from those of 1199091and 119909

2 In fact they have better fractal

scaling behavior (ie straighter linear relations in log2119865(119908)

versus log2119908 plots) and stronger persistent correlations

Next we consider the variables 1199095and 119909

6 the neutral pH

value and the sulphur fumigation intensityTheir AFA curves


6

6

4

4

2

2

0

minus2

8 10

H = 142

H = 074

minus4

log2 w

log2F(w)

(a)

642 8 10

6

4

2

0

minus2

-4

H = 135

H = 074

log2 w

log2F(w)

(b)

642 8 10

H = 078

2

0

minus2

log 2F(w)

log2 w

(c)

642 8 10

H = 077

H = 087

10

8

6

4

log2F(w)

log2 w

(d)

Figure 6 Adaptive fractal analysis (AFA) of signals 1199091to 1199094 log2119865(119908) versus log

2119908 for (a) 119909

1 (b) 119909

2 (c) 119909

3 and (d) 119909

4

642 8 10

H = 088

H = 074

4

2

0

minus2

log2F(w)

log2 w

(a)

642 8 10

H = 109

H = 084

8

6

4

2

0

minus2

log2F(w)

log2 w

(b)

Figure 7 Adaptive fractal analysis (AFA) of signals (a) 1199095and (b) 119909

6


642 8 10

6

4

2

0

minus2

H = 109

H = 085

log2F(w)

log2 w

(a)

642 8 10

H = 107

H = 080

6

4

2

0

minus2

log2F(w)

log2 w

(b)

642 8 10

2

0

minus2

minus4

H = 102

H = 066

log2F(w)

log2 w

(c)

H = 101

H = 087

14

12

10

8

6642 8 10

log2F(w)

log2 w

(d)

H = 063

H = 073

7

5

3

1642 8 10

log2F(w)

log2 w

(e)

Figure 8 Adaptive fractal analysis (AFA) of signals (a) 1199097 (b) 119909

8 (c) 119909

9 (d) 119909

10 and (e) 119909

11


are shown in Figure 7 They also have two scaling regionsand persistent correlations on short time scales (about 64hours for 119909

5and 128 hours for 119909

6) On longer time scales

the correlation in 1199095becomes more persistent (119867 changes

from 074 to 088) while 1199096becomes non-stationary since119867

becomes larger than 1Third we consider 119909

119894 119894 = 7 8 11 which represents

the corrected brix corrected polarimeter the pH value forthe clarified juice after the 2nd heating the color valueand the clarity degree respectively Their AFA curves areshown in Figure 8 We first examine 119909

119894 119894 = 7 8 9 It is

instructive to compare the curves in Figures 8(a)ndash8(c) withthose in Figures 6(a)ndash6(c) In particular we observe thatfor the brix and polarimeter on shorter time scales thecorrelations become more consistent since now 119867 becomesbigger However on longer time scales 119867 becomes onlyslightly larger than 1Therefore the degree of nonstationarityhas decreased The behavior of the pH value becomes morecomplicated than that in Figure 6 since there are two scalingregions in Figure 8(c) but there is only one scaling behaviorin Figure 6(c)

Finally we examine the AFA curves for the two mostimportant variables color value 119909

10 and clarity degree 119909

11

which are shown in Figures 8(d) and 8(e) We observe thatfor 11990910 the first scaling with 119867 = 087 is up to a time scale

27

times 2 = 256 hours Beyond that time scale 119867 becomes 1therefore the signal becomes marginally nonstationary Theclarify degree on the other hand has an 119867 = 073 for timescales up to about 64 hours and 119867 = 063 for longer timescales

In summary we have observed that persistent correla-tions up to a few days are salient features of the 11 variablesthat are most important for the clarification process of canesugar production

4 Concluding Discussions

Cane sugar production is an important industrial processOne of the most important steps in cane sugar productionis the clarification process which provides high-quality con-centrated sugar syrup crystal for further processing To gainfundamental understanding of the physical and chemicalprocesses associated with the clarification process and helpdesign better approaches to improve the clarification of themixed juice in this paper we have examined the fractalbehavior of the 11 variables pertinent to the clarificationprocess We have shown that they all show persistent long-range correlations for time scales up to at least a fewdays Persistent long-range correlations amount to unilateraldeviations from a preset target This means that when theprocess is in a desired mode such that the target variablescolor of the produced sugar and its clarity degree bothsatisfy preset conditions they will remain so for a longperiod of time However adversity could happen in thesense that when they do not satisfy the requirements theadverse situation may last quite long These findings have tobe explicitly accounted for when designing active controllingstrategies to improve the quality of the produced sugar

Conflict of Interests

The authors declare no conflict of interests

Acknowledgment

This work (X Yu and X Liao) was partially supported bythe Chinese Natural Science Foundation project number50965003

References

[1] S Kamat V Diwanji J G Smith and K P Madhavan ldquoModel-ing of pH process using recurrent neural network and wavenetrdquoin Proceedings of the IEEE International Conference on Compu-tational Intelligence for Measurement Systems and Applications(CIMSA rsquo05) pp 209ndash214 Giardini Naxos Italy July 2005

[2] X Lin J Yang H Liu S Song and C Song ldquoAn improvedmethod of DHP for optimal control in the clarifying process ofsugar cane juicerdquo in Proceedings of the International JointConference on Neural Networks (IJCNN rsquo09) pp 1814ndash1819Atlanta Ga USA June 2009

[3] B B Mandelbrot The Fractal Geometry of Nature W H Free-man San Francisco Calif USA 1982

[4] J B Gao V A Billock I Merk et al ldquoInertia and memory inambiguous visual perceptionrdquo Cognitive Processing vol 7 no 2pp 105ndash112 2006

[5] J Gao J HuW Tung and Y Zheng ldquoMultiscale analysis of eco-nomic time series by scale-dependent lyapunov exponentrdquoQuantitative Finance vol 13 no 2 pp 265ndash274 2013

[6] W Li and K Kaneko ldquoLong-range correlation and partial 1119891120572spectrum in a noncoding DNA sequencerdquo Europhysics Lettersvol 17 no 7 pp 655ndash660 1992

[7] R F Voss ldquoEvolution of long-range fractal correlations and 1fnoise in DNA base sequencesrdquo Physical Review Letters vol 68pp 3805ndash3808 1992

[8] C-K Peng S V Buldyrev A L Goldberger et al ldquoLong-rangecorrelations in nucleotide sequencesrdquoNature vol 356 no 6365pp 168ndash170 1992

[9] J Gao Y Qi Y Cao andW-W Tung ldquoProtein coding sequenceidentification by simultaneously characterizing the periodicand random features of DNA sequencesrdquo Journal of Biomedicineand Biotechnology vol 2005 no 2 pp 139ndash146 2005

[10] J Hu J-B Gao Y Cao E Bottinger andW Zhang ldquoExploitingnoise in array CGH data to improve detection of DNA copynumber changerdquo Nucleic Acids Research vol 35 no 5 articlee35 2007

[11] D L Gilden T Thornton and M W Mallon ldquo1f noise inhuman cognitionrdquo Science vol 267 no 5205 pp 1837ndash18391995

[12] Y Chen M Ding and J A Scott Kelso ldquoLong memory pro-cesses (1119891120572 type) in human coordinationrdquo Physical ReviewLetters vol 79 no 22 pp 4501ndash4504 1997

[13] J J Collins and C J de Luca ldquoRandom walking during quietstandingrdquo Physical Review Letters vol 73 no 5 pp 764ndash7671994

[14] P C Ivanov M G Rosenblum C-K Peng et al ldquoScaling be-haviour of heartbeat intervals obtained by wavelet-based time-series analysisrdquo Nature vol 383 no 6598 pp 323ndash327 1996

[15] L A Nunes Amaral A L Goldberger P C Ivanov and HEugene Stanley ldquoScale-independent measures and pathologic


cardiac dynamicsrdquo Physical Review Letters vol 81 no 11 pp2388ndash2391 1998

[16] P C Ivanov L A Nunes Amaral A L Goldberger et al ldquoMulti-fractality in human heartbeat dynamicsrdquo Nature vol 399 no6735 pp 461ndash465 1999

[17] P Bernaola-Galvan F C Ivanov L A Nunes Amaral and H EStanley ldquoScale invariance in the nonstationarity of human heartraterdquo Physical Review Letters vol 87 no 16 Article ID 168105 4pages 2001

[18] M Wolf ldquo1f noise in the distribution of prime numbersrdquo Phy-sica A vol 241 no 3-4 pp 493ndash499 1997

[19] C-K Peng S V Buldyrev S Havlin M Simons H E Stanleyand A L Goldberger ldquoMosaic organization of DNA nucle-otidesrdquo Physical Review E vol 49 no 2 pp 1685ndash1689 1994

[20] M A Riley N Kuznetsov S Bonnette S Wallot and J B GaoldquoA tutorial introduction to adaptive fractal analysisrdquo Frontiers inFractal Physiology 2012

[21] N Kuznetsov S Bonnette J B Gao and M A Riley ldquoAdaptivefractal analysis reveals limits to fractal scaling in center ofpressure trajectoriesrdquo Annals of Biomedical Engineering vol 41no 8 pp 1646ndash1660 2013

[22] J B Gao J Hu X Mao and M Perc ldquoCulturomics meets ran-dom fractal theory insights into long-range correlations ofsocial and natural phenomena over the past two centuriesrdquoJournal of the Royal Society Interface vol 9 no 73 pp 1956ndash1964 2012

[23] J Gao J Hu andW-W Tung ldquoFacilitating joint chaos and frac-tal analysis of biosignals through nonlinear adaptive filteringrdquoPLoS ONE vol 6 no 9 Article ID e24331 2011

[24] G E P Box and G M Jenkins Time Series Analysis Forecastingand Control Holden-Day San Francisco Calif USA 1976

[25] J Gao Y Cao W-w Tung and J Hu Multiscale Analysis ofComplex Time Series Integration of Chaos and Random FractalTheory and BeyonD Wiley-Interscience New York NY USA2007

[26] J Gao J Hu W-W Tung Y Cao N Sarshar and V P Roycho-wdhury ldquoAssessment of long-range correlation in time serieshow to avoid pitfallsrdquo Physical Review E vol 73 no 1 ArticleID 016117 2006

[27] JHu J Gao andXWang ldquoMultifractal analysis of sunspot timeseries the effects of the 11-year cycle and fourier truncationrdquoJournal of Statistical Mechanics vol 2009 no 2 Article IDP02066 2009

[28] J Gao H Sultan J Hu and W-W Tung ldquoDenoising nonlineartime series by adaptive filtering and wavelet shrinkage acomparisonrdquo IEEE Signal Processing Letters vol 17 no 3 pp237ndash240 2010

[29] W W Tung J B Gao J Hu and L Yang ldquoRecovering chaoticsignals in heavy noise environmentsrdquo Physical Review E vol 83no 4 Article ID 046210 9 pages 2011

[30] R CHwa andT C Ferree ldquoScaling properties of fluctuations inthe human electroencephalogramrdquo Physical Review E vol 66no 2 Article ID 021901 8 pages 2002

[31] P A Robinson ldquoInterpretation of scaling properties of electro-encephalographic fluctuations via spectral analysis and under-lying physiologyrdquo Physical Review E vol 67 no 3 Article ID032902 pp 0329021ndash0329024 2003

[32] J Hu J Gao F L Posner Y I Zheng and W-W Tung ldquoTargetdetection within sea clutter a comparative study by fractalscaling analysesrdquo Fractals vol 14 no 3 pp 187ndash204 2006

[33] J HuW-W Tung and J Gap ldquoDetection of low observable tar-gets within sea clutter by structure function based multifractalanalysisrdquo IEEE Transactions on Antennas and Propagation vol54 no 1 pp 136ndash143 2006

Submit your manuscripts athttpwwwhindawicom

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013


Mathematical Problems in Engineering

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2013

ISRN Applied Mathematics


Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

International Journal of

Combinatorics


Journal of Function Spaces and Applications

International Journal of Mathematics and Mathematical Sciences


ISRN Geometry


Discrete Dynamics in Nature and Society



Volume 2013

Advances in

Mathematical Physics

ISRN Algebra


ProbabilityandStatistics

Journal of


ISRN Mathematical Analysis


Journal ofApplied Mathematics


Advances in

DecisionSciences



Stochastic AnalysisInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal


ISRN Discrete Mathematics


Differential EquationsInternational Journal of

Volume 2013


There have been efforts to improve the clarification ofthe mixed juice by using neural network-based predictionschemes [1 2] Such approaches amount to treating theclarification process as a black or gray box and thus do notyield any understanding of the basic physics and chemistryinvolved in clarification To help gain more fundamentalunderstanding and guide the design of better approachesto improve clarification in this paper we carry out fractalanalysis of the key variables pertinent to the clarificationprocess

Among the types of activity that characterize complexsystems the most ubiquitous and puzzling is perhaps theappearance of 1119891120572 noise a form of temporal or spatialfluctuation characterized by a power-law decaying powerspectral density A subclass of such processes denoted as11198912119867+1 is called processes with long-range correlations

(or long memories) characterized by a Hurst parameter 119867Depending on whether 0 lt 119867 lt 12 119867 = 12 or12 lt 119867 lt 1 [3] they are said to have antipersistent corr-elations memoryless or only short-range correlations orpersistent long-range correlations Prominent examples ofsuch processes include vision [4] finance [5] DNA sequences[6ndash10] human cognition [11] and coordination [12] posture[13] cardiac dynamics [14ndash17] and the distribution of primenumbers [18] to name but a few In this paper we wish toexplore whether fractal theory can shed new light on theclarification process of cane sugar production

The remainder of the paper is organized as follows InSection 2 we provide some details about the cane sugarclarification process explain the various variables to beanalyzed here and discuss the challenges of analyzing thosevariables In Section 3 we carry out fractal analysis of thesevariables using the key concept in random fractal theory theHurst parameter which characterizes the basic correlationstructure of the signals Concluding discussions are presentedin Section 4

2 Clarification of Mixed Juice inCane Sugar Production

Sugar clarification is a key process in cane sugar productionIt is characterized by two important indices called color valueand clarity degree Whether the color value and the claritydegree can achieve the desired or preset values can criticallyaffect the quality of the cane sugar and the revenue of afactory

Many methods have been developed for clarifying themixed juiceThe two major ones are the carbonation methodand the sulphitationmethodThe former uses themilk of limeand carbon dioxide as main detergent on the mixed juiceBesides high cost the accompanying pollution by carbonatedmud is hard to deal with Therefore its usage has beensubstantially reduced over the years

Unlike the carbonate method the sulphitation methodadds clarifying agent SO

2in addition to lime and phosphoric

acid Due to the absorption effect of calcium sulfite (whichis the product of lime and SO

2) together with the inhibition

of pigment formation due to SO2 the sulphitation method

is significantly better than the carbonate method In thefollowing we will focus on the sulphitation method for theclarification of the mixed juice

The sulphitation method for clarification is schematicallyshown in Figure 1 It is a complicated physical chemicalprocess involving solvents such as SO

2 milk of lime and

phosphoric acid Roughly the process can be divided intofour stages cane juice phosphorus-preliming first heatingSO2sulphitation-neutralization and settlement after 2nd-

heatingMore specifically in the preliming stage after the firstheating to 55sim65∘C milk of lime and phosphoric acid areadded to themixed sugar cane juice In the next sulphur fumi-gation process SO

2gas enters the chemical process pipeline

At the end of the pipeline milk of lime is further addedfor achieving neutralization since phosphoric acid sulfitereactswith calciumhydroxide to form calciumphosphate andcalcium sulfate Then the 2nd heating is carried out so thatsugar cane colloids ofmixed juice can not only fully condensein the precipitator but also (1) accelerate precipitation (2)decrease juice viscosity and (3) facilitate the precipitatedparticles to sink fast Then the clarified juice can come outof the settler from the top of each later while the mud juice isdischarged into the vacuum suction filter by gravity Filteredjuice can then be directly incorporated into the clear juiceheating device and sent to the evaporation device to condenseinto syrup Therefore controlling the process parameterssteadily is the key to improve the quality of clarified juice andsugar

In this paper time series of the 11 variables indicated inFigure 1 will be analyzed using random fractal theory Thedata were measured in an advanced cane sugar productionfactory in Guangxi Zhuang Autonomous Region of ChinaGuangxi is the largest sugar producer in China amounting tomore than 60of the total sugar in the countryThedatawerecollected with a time interval of 2 hours and lasted for thewhole season (about 4 months) These variables are groupedinto three groups with the last two variables 119909

10 color value

and11990911 the clarity degree being the target variablesThe time

series data of these 11 variables are shown in Figures 2 3 and4 As can be seen the major challenge in analyzing these datais nonstationarity that is the statistical moments includingmean and variance vary with time Understandably this maybe a salient feature for variables that are measured over a verylong period of time (which is about 4 months here)

3 Fractal Analysis of the Clarification Process

In this section we first overview the basics of fractal processeswith long memory and then briefly describe two methodsfor detecting fractal structures from data with trends One isthe seminal detrended fluctuation analysis (DFA) [19] Theother is adaptive fractal analysis (AFA) [20ndash23] Since AFAprovides additional advantages over DFA to deal with signalswith arbitrary trends we only present results of AFA for thevariables pertinent to the clarification process of cane sugarproduction here

31 Basics of Fractal Processes with LongMemory As we havementioned fractal processes with long memory are charac-terized by the Hurst parameter 119867 To better understand the



Milk of lime

Milk of lime

Phosphoricacid

Sulfur

Clarified juice

Clear juice

Mud juice


Extractedsample

neutralizedjuice

Filtered mud

Extractedsample

clarified juice

Corrected brix(x1)



Total dosage of



Corrected brix(x7)



Color value (x10)

Clarity degree(x11)

Evaporation

Filter


Extractedsample

mixed juice




Sulphur fumigation



(pH 68sim74)



dioxide

P2O5 P2O5 (x4)


18

16

140 500 1000 1500 2000 2500

Time (hours)

x1

(a)

0 500 1000 1500 2000 2500

Time (hours)

16

14

12

x2

(b)

0 500 1000 1500 2000 2500

Time (hours)

8

6

x3

(c)

0 500 1000 1500 2000 2500

Time (hours)

600

400

200

x4

(d)



1 1199092 119909



119903 (119896) sim 1198962119867minus2

as 119896 997888rarr infin (1)


1199091 1199092 119909



119906 (119894) =

119894

sum

119896=1

(119909119896minus 119909) 119894 = 1 2 119899 (2)



0 500 1000 1500 2000 2500

Time (hours)

9

8

7

6

x5

(a)

0 500 1000 1500 2000 2500

Time (hours)

25

20

15

10

5

x6

(b)


6

0 500 1000 1500 2000 2500

Time (hours)

20

15

10

x7

(a)

0 500 1000 1500 2000 2500

Time (hours)

20

15

10

x8

(b)

0 500 1000 1500 2000 2500

Time (hours)

8

7

x9

(c)

0 500 1000 1500 2000 2500

Time (hours)

3000

2000

1000

x10

(d)

0 500 1000 1500 2000 2500

Time (hours)

80

60

100

x11

(e)




1 1199092 119909



119883(119898)

119905=(119909119905119898minus119898+1

+ sdot sdot sdot + 119909119905119898)

119898 119905 ge 1 (3)


var (119883(119898)) = 12059021198982119867minus2 (4)








2000

1000

0

minus1000

minus2000

0 200 400 600 800 1000

EEG

dat

ax(n)

Index n




119865119889(119897) = ⟨

119897

sum

119894=1

119909119897(119894)2

⟩ sim 1198972119867

(5)





(1198971) 119910(119894+1)(119897



119910(119888)

(119897) = 1199081119910(119894)

(119897 + 119899) + 1199082119910(119894+1)

(119897) 119897 = 1 2 119899 + 1

(6)



(1 minus 119889119895)119899 119895 = 1 2 where 119889





119865(2)

(119908) = [1

119873

119873

sum

119894=1

(119906 (119894) minus V (119894))2]

12

sim 119908119867

(7)





1



1 Indeed one can



3and 119909







6 the neutral pH



6

6

4

4

2

2

0

minus2

8 10

H = 142

H = 074

minus4

log2 w

log2F(w)

(a)

642 8 10

6

4

2

0

minus2

-4

H = 135

H = 074

log2 w

log2F(w)

(b)

642 8 10

H = 078

2

0

minus2

log 2F(w)

log2 w

(c)

642 8 10

H = 077

H = 087

10

8

6

4

log2F(w)

log2 w

(d)


2119908 for (a) 119909

1 (b) 119909

2 (c) 119909

3 and (d) 119909

4

642 8 10

H = 088

H = 074

4

2

0

minus2

log2F(w)

log2 w

(a)

642 8 10

H = 109

H = 084

8

6

4

2

0

minus2

log2F(w)

log2 w

(b)


6


642 8 10

6

4

2

0

minus2

H = 109

H = 085

log2F(w)

log2 w

(a)

642 8 10

H = 107

H = 080

6

4

2

0

minus2

log2F(w)

log2 w

(b)

642 8 10

2

0

minus2

minus4

H = 102

H = 066

log2F(w)

log2 w

(c)

H = 101

H = 087

14

12

10

8

6642 8 10

log2F(w)

log2 w

(d)

H = 063

H = 073

7

5

3

1642 8 10

log2F(w)

log2 w

(e)


8 (c) 119909

9 (d) 119909

10 and (e) 119909

11










119894 119894 = 7 8 9 It is




11


27







Acknowledgment


References





































OperationsResearch

Advances in








Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences










Volume 2013



Milk of lime

Milk of lime

Phosphoricacid

Sulfur

Clarified juice

Clear juice

Mud juice


Extractedsample

neutralizedjuice

Filtered mud

Extractedsample

clarified juice

Corrected brix(x1)



Total dosage of



Corrected brix(x7)



Color value (x10)

Clarity degree(x11)

Evaporation

Filter


Extractedsample

mixed juice




Sulphur fumigation



(pH 68sim74)



dioxide

P2O5 P2O5 (x4)


18

16

140 500 1000 1500 2000 2500

Time (hours)

x1

(a)

0 500 1000 1500 2000 2500

Time (hours)

16

14

12

x2

(b)

0 500 1000 1500 2000 2500

Time (hours)

8

6

x3

(c)

0 500 1000 1500 2000 2500

Time (hours)

600

400

200

x4

(d)



1 1199092 119909



119903 (119896) sim 1198962119867minus2

as 119896 997888rarr infin (1)


1199091 1199092 119909



119906 (119894) =

119894

sum

119896=1

(119909119896minus 119909) 119894 = 1 2 119899 (2)



0 500 1000 1500 2000 2500

Time (hours)

9

8

7

6

x5

(a)

0 500 1000 1500 2000 2500

Time (hours)

25

20

15

10

5

x6

(b)


6

0 500 1000 1500 2000 2500

Time (hours)

20

15

10

x7

(a)

0 500 1000 1500 2000 2500

Time (hours)

20

15

10

x8

(b)

0 500 1000 1500 2000 2500

Time (hours)

8

7

x9

(c)

0 500 1000 1500 2000 2500

Time (hours)

3000

2000

1000

x10

(d)

0 500 1000 1500 2000 2500

Time (hours)

80

60

100

x11

(e)




1 1199092 119909



119883(119898)

119905=(119909119905119898minus119898+1

+ sdot sdot sdot + 119909119905119898)

119898 119905 ge 1 (3)


var (119883(119898)) = 12059021198982119867minus2 (4)








2000

1000

0

minus1000

minus2000

0 200 400 600 800 1000

EEG

dat

ax(n)

Index n




119865119889(119897) = ⟨

119897

sum

119894=1

119909119897(119894)2

⟩ sim 1198972119867

(5)





(1198971) 119910(119894+1)(119897



119910(119888)

(119897) = 1199081119910(119894)

(119897 + 119899) + 1199082119910(119894+1)

(119897) 119897 = 1 2 119899 + 1

(6)



(1 minus 119889119895)119899 119895 = 1 2 where 119889





119865(2)

(119908) = [1

119873

119873

sum

119894=1

(119906 (119894) minus V (119894))2]

12

sim 119908119867

(7)





1



1 Indeed one can



3and 119909







6 the neutral pH



6

6

4

4

2

2

0

minus2

8 10

H = 142

H = 074

minus4

log2 w

log2F(w)

(a)

642 8 10

6

4

2

0

minus2

-4

H = 135

H = 074

log2 w

log2F(w)

(b)

642 8 10

H = 078

2

0

minus2

log 2F(w)

log2 w

(c)

642 8 10

H = 077

H = 087

10

8

6

4

log2F(w)

log2 w

(d)


2119908 for (a) 119909

1 (b) 119909

2 (c) 119909

3 and (d) 119909

4

642 8 10

H = 088

H = 074

4

2

0

minus2

log2F(w)

log2 w

(a)

642 8 10

H = 109

H = 084

8

6

4

2

0

minus2

log2F(w)

log2 w

(b)


6


642 8 10

6

4

2

0

minus2

H = 109

H = 085

log2F(w)

log2 w

(a)

642 8 10

H = 107

H = 080

6

4

2

0

minus2

log2F(w)

log2 w

(b)

642 8 10

2

0

minus2

minus4

H = 102

H = 066

log2F(w)

log2 w

(c)

H = 101

H = 087

14

12

10

8

6642 8 10

log2F(w)

log2 w

(d)

H = 063

H = 073

7

5

3

1642 8 10

log2F(w)

log2 w

(e)


8 (c) 119909

9 (d) 119909

10 and (e) 119909

11










119894 119894 = 7 8 9 It is




11


27







Acknowledgment


References





































OperationsResearch

Advances in








Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences










Volume 2013


0 500 1000 1500 2000 2500

Time (hours)

9

8

7

6

x5

(a)

0 500 1000 1500 2000 2500

Time (hours)

25

20

15

10

5

x6

(b)


6

0 500 1000 1500 2000 2500

Time (hours)

20

15

10

x7

(a)

0 500 1000 1500 2000 2500

Time (hours)

20

15

10

x8

(b)

0 500 1000 1500 2000 2500

Time (hours)

8

7

x9

(c)

0 500 1000 1500 2000 2500

Time (hours)

3000

2000

1000

x10

(d)

0 500 1000 1500 2000 2500

Time (hours)

80

60

100

x11

(e)




1 1199092 119909



119883(119898)

119905=(119909119905119898minus119898+1

+ sdot sdot sdot + 119909119905119898)

119898 119905 ge 1 (3)


var (119883(119898)) = 12059021198982119867minus2 (4)








2000

1000

0

minus1000

minus2000

0 200 400 600 800 1000

EEG

dat

ax(n)

Index n




119865119889(119897) = ⟨

119897

sum

119894=1

119909119897(119894)2

⟩ sim 1198972119867

(5)





(1198971) 119910(119894+1)(119897



119910(119888)

(119897) = 1199081119910(119894)

(119897 + 119899) + 1199082119910(119894+1)

(119897) 119897 = 1 2 119899 + 1

(6)



(1 minus 119889119895)119899 119895 = 1 2 where 119889





119865(2)

(119908) = [1

119873

119873

sum

119894=1

(119906 (119894) minus V (119894))2]

12

sim 119908119867

(7)





1



1 Indeed one can



3and 119909







6 the neutral pH



6

6

4

4

2

2

0

minus2

8 10

H = 142

H = 074

minus4

log2 w

log2F(w)

(a)

642 8 10

6

4

2

0

minus2

-4

H = 135

H = 074

log2 w

log2F(w)

(b)

642 8 10

H = 078

2

0

minus2

log 2F(w)

log2 w

(c)

642 8 10

H = 077

H = 087

10

8

6

4

log2F(w)

log2 w

(d)


2119908 for (a) 119909

1 (b) 119909

2 (c) 119909

3 and (d) 119909

4

642 8 10

H = 088

H = 074

4

2

0

minus2

log2F(w)

log2 w

(a)

642 8 10

H = 109

H = 084

8

6

4

2

0

minus2

log2F(w)

log2 w

(b)


6


642 8 10

6

4

2

0

minus2

H = 109

H = 085

log2F(w)

log2 w

(a)

642 8 10

H = 107

H = 080

6

4

2

0

minus2

log2F(w)

log2 w

(b)

642 8 10

2

0

minus2

minus4

H = 102

H = 066

log2F(w)

log2 w

(c)

H = 101

H = 087

14

12

10

8

6642 8 10

log2F(w)

log2 w

(d)

H = 063

H = 073

7

5

3

1642 8 10

log2F(w)

log2 w

(e)


8 (c) 119909

9 (d) 119909

10 and (e) 119909

11










119894 119894 = 7 8 9 It is




11


27







Acknowledgment


References





































OperationsResearch

Advances in








Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences










Volume 2013


2000

1000

0

minus1000

minus2000

0 200 400 600 800 1000

EEG

dat

ax(n)

Index n




119865119889(119897) = ⟨

119897

sum

119894=1

119909119897(119894)2

⟩ sim 1198972119867

(5)





(1198971) 119910(119894+1)(119897



119910(119888)

(119897) = 1199081119910(119894)

(119897 + 119899) + 1199082119910(119894+1)

(119897) 119897 = 1 2 119899 + 1

(6)



(1 minus 119889119895)119899 119895 = 1 2 where 119889





119865(2)

(119908) = [1

119873

119873

sum

119894=1

(119906 (119894) minus V (119894))2]

12

sim 119908119867

(7)





1



1 Indeed one can



3and 119909







6 the neutral pH



6

6

4

4

2

2

0

minus2

8 10

H = 142

H = 074

minus4

log2 w

log2F(w)

(a)

642 8 10

6

4

2

0

minus2

-4

H = 135

H = 074

log2 w

log2F(w)

(b)

642 8 10

H = 078

2

0

minus2

log 2F(w)

log2 w

(c)

642 8 10

H = 077

H = 087

10

8

6

4

log2F(w)

log2 w

(d)


2119908 for (a) 119909

1 (b) 119909

2 (c) 119909

3 and (d) 119909

4

642 8 10

H = 088

H = 074

4

2

0

minus2

log2F(w)

log2 w

(a)

642 8 10

H = 109

H = 084

8

6

4

2

0

minus2

log2F(w)

log2 w

(b)


6


642 8 10

6

4

2

0

minus2

H = 109

H = 085

log2F(w)

log2 w

(a)

642 8 10

H = 107

H = 080

6

4

2

0

minus2

log2F(w)

log2 w

(b)

642 8 10

2

0

minus2

minus4

H = 102

H = 066

log2F(w)

log2 w

(c)

H = 101

H = 087

14

12

10

8

6642 8 10

log2F(w)

log2 w

(d)

H = 063

H = 073

7

5

3

1642 8 10

log2F(w)

log2 w

(e)


8 (c) 119909

9 (d) 119909

10 and (e) 119909

11










119894 119894 = 7 8 9 It is




11


27







Acknowledgment


References





































OperationsResearch

Advances in








Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences










Volume 2013


6

6

4

4

2

2

0

minus2

8 10

H = 142

H = 074

minus4

log2 w

log2F(w)

(a)

642 8 10

6

4

2

0

minus2

-4

H = 135

H = 074

log2 w

log2F(w)

(b)

642 8 10

H = 078

2

0

minus2

log 2F(w)

log2 w

(c)

642 8 10

H = 077

H = 087

10

8

6

4

log2F(w)

log2 w

(d)


2119908 for (a) 119909

1 (b) 119909

2 (c) 119909

3 and (d) 119909

4

642 8 10

H = 088

H = 074

4

2

0

minus2

log2F(w)

log2 w

(a)

642 8 10

H = 109

H = 084

8

6

4

2

0

minus2

log2F(w)

log2 w

(b)


6


642 8 10

6

4

2

0

minus2

H = 109

H = 085

log2F(w)

log2 w

(a)

642 8 10

H = 107

H = 080

6

4

2

0

minus2

log2F(w)

log2 w

(b)

642 8 10

2

0

minus2

minus4

H = 102

H = 066

log2F(w)

log2 w

(c)

H = 101

H = 087

14

12

10

8

6642 8 10

log2F(w)

log2 w

(d)

H = 063

H = 073

7

5

3

1642 8 10

log2F(w)

log2 w

(e)


8 (c) 119909

9 (d) 119909

10 and (e) 119909

11










119894 119894 = 7 8 9 It is




11


27







Acknowledgment


References





































OperationsResearch

Advances in








Volume 2013


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ISRN Geometry





Volume 2013

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Volume 2013


642 8 10

6

4

2

0

minus2

H = 109

H = 085

log2F(w)

log2 w

(a)

642 8 10

H = 107

H = 080

6

4

2

0

minus2

log2F(w)

log2 w

(b)

642 8 10

2

0

minus2

minus4

H = 102

H = 066

log2F(w)

log2 w

(c)

H = 101

H = 087

14

12

10

8

6642 8 10

log2F(w)

log2 w

(d)

H = 063

H = 073

7

5

3

1642 8 10

log2F(w)

log2 w

(e)


8 (c) 119909

9 (d) 119909

10 and (e) 119909

11










119894 119894 = 7 8 9 It is




11


27







Acknowledgment


References





































OperationsResearch

Advances in








Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences










Volume 2013










119894 119894 = 7 8 9 It is




11


27







Acknowledgment


References





































OperationsResearch

Advances in








Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences










Volume 2013






















OperationsResearch

Advances in








Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences










Volume 2013


OperationsResearch

Advances in








Volume 2013


Combinatorics





ISRN Geometry





Volume 2013

Advances in


ISRN Algebra



Journal of






Advances in

DecisionSciences










Volume 2013

Documents

Fractal Behavior in the Clarification Process of Cane … Behavior in the Clarification Process of Cane Sugar Production XiaomoYu,1 JunkeYe,2 JingHu,3,4 XiaopingLiao,2 andJianboGao3,4