A discrete chaotic multimodel based on 2D Hénon maps

Size: px
Start display at page:

Download "A discrete chaotic multimodel based on 2D Hénon maps"

Transcription

1 A discrete chotic multimodel sed on D Hénon mps Ameni Dridi, Rni Lind Filli, Mohmed Benreje 3 LA.R.A, Automtique, ENIT, BP 37, Le Belvédère, 00 Tunis, Tunisi. meni.dridi.ing@gmil.com rni_lind@hotmil.fr 3 mohmed.enreje@enit.rnu.tn Astrct In this pper, we propose to design new fmily of discrete-time chotic systems sed on the use of the multimodel pproch. This pproch is used to represent wy to generte new fmily of discrete-time chotic systems, from p discrete time chotic sis models. Our results re illustrted on the specific cse of new discrete-time chotic multimodel sed on two chotic D Hénon mps with two different sets of prmeters leding to different ehviors. The chos chrcteriztion of otined multimodels is performed using ifurction digrms. Inde Terms Chos, multimodels, discrete-time, Hénon mp, ifurction digrm I. INTRODUCTION In the recent yers, there is growing interest to the use of chos-sed techniques in the secure communiction field. Chotic systems proved tht they re efficient to uild roust cryptosystems due to their severl fetures especilly the noiselike time series nd the sensitive dependence on initil conditions [-4]. In order to hve n efficient cryptosystem, some rules, detiled in [5], need to e pplied where the compleity of the chotic used system is considered s fundmentl issue for ll types of cryptosystems. In prllel, glol pproch sed on multiple Liner Time Invrint (LTI) models defined round different operting point hs received significnt ttention. This multimodel pproch is conve polytopic representtion tht cn e otined y the interpoltion of LTI models. Every model represents vlid operting rnge. Three techniques re used to otin the mutimodel either y identifiction [6-9] when input nd output dt re ville or y lineriztion round different operting points or y polytopic trnsformtion[6-9], if we hve the nlytic model. Numerous works ws pulished concerning the mutimodel pproch nd its stility study [6-9]. In this pper, the mutimodel pproch is used to uild new clss of discrete-time chotic systems which constitutes n etension of previous results of continuous chotic processes using the mutimodel pproch. In [0] Cherrier nd Boutye hve proposed to use the definition of multimodel to interpolte continuous chotic susystems. It s proven tht the resulting system hs comple chotic ehvior. The pper is orgnized s follows: Section II presents the wy to design multimodel sed on p discrete time chotic susystems hving different prmeters nd interpolte them using the pproprite ctivtion function. The specific cse of discrete time multimodels sed on two D Hénon mps is presented in section III. It is lso tested in this section; the chotic ehviors through ifurction digrms nd concluding remrks re given. In section IV, cse of interpoltion of chotic nd non-chotic D Hénon mps is considered. II. BUILDING A NEW CHAOTIC DISCRETE-TIME MULTIMODEL: PROBLEM STATEMENT Consider the n-dimensionl discrete-time in Lurie systems s follows ( k ) A f ( ), i,,..., p () i i kt is the discrete-time, T smpling time, vector A i, i,,..., p, re n n n R is the stte constnt mtrices nd f i ( ), i,,..., p, nonliner vector. The mutimodel pproch proposed in [0] is etended to the cse of discrete-time chotic systems to interpolte p susystems hving different ehviors. The new multimodel, resulting from the interpoltion of systems () with different sets of prmeters, is descried s following p ( k ) ( )( A f ( ) i i i i y C ()

2 where y is the output vector, C constnt mtri with n pproprite size nd i, i,,..., p ctivtion functions modeling the weighting of the su-model i, chrcterized in the glol model, y A i, i,,..., p such us The prmeters chosen for the two sic models (4) nd (5) re such s =.4, = 0.3, =.5, =0.4 with initil vlues 0 0, 0, 0 [0-4]. p i i 0 i i, p (3) Since the multimodel is uilt in order to e integrted in cryptosystem nd for the purpose of incresing security, the ctivtion functions hve to e chosen such s they ensure kind of miing etween the different su-models. It doesn t hve to fvor model, ut llows rel trnsition etween them. This llows in one hnd, to enhnce the compleity of the system nd, secondly, to ensure continuous synchroniztion, in the sense tht there is no loss of synchroniztion [0]. In the net section, re proposed two multimodels corresponding to () uilt from two susystems hving two different ehviors using n pproprite ctivtion function. The first mutimodel is comintion of two chotic systems nd the second comintion of chotic nd non-chotic system. Bifurction digrms of the otined multimodels re used to show if they re chotic or not. Fig. The chotic ttrctor of the Hénon mp for nd 00,0 =.4, = 0.3 III. IMPLEMENTATION OF THE CHAOTIC D HÉNON MAPS In this section, for this first emple, we hve chosen s se models two systems of D Hénon mps, with two different sets of prmeters. Considered first discrete-time D Hénon susystem, which is descried s follows [0-4] where k, k ( k ) ( k ) (4) is the stte vector nd nd re ifurction prmeters of Hénon mp. To uild the multimmodel, the system (4) is interpolted with the following Hénon mp using two different sets of prmeters chrcterizing y two different chotic ehviors. ( k ) ( k ) The corresponding ttrctors re found respectively in Fig. nd Fig.. (5) Fig. Chotic ttrctor of Hénon mp for 0 0, 0 =.5, =0.4 nd Once cn note tht the first susystem Hénon mp does not hve strnge ttrctor for ll vlues of the prmeters nd. For emple, y keeping fied t 0.3, the ifurction digrm of Fig. 3 shows tht for 0.4. the Hénon mp hs stle periodic orit. Besides, s presented in Fig. 4, for 0.4 nd, the second susystem discrete-time Hénon mp (5) hs chotic ehvior, illustrted y the ifurction digrm of Fig. 4. The chosen ctivtion function is descried s following [0] ( ) ( tnh( )) / (6)

3 where is prmeter set so tht the μ function performs rel trnsition etween the two Hénon susystems. The resulting multi-model simultions re shown in Fig. 5, ws set t the vlue of ,.. The mutlimodel otined from the comintion of two chotic systems (4) nd (5) gives us lrger intervl of prmeters vlues which is dvntgeous to the security of the encrypting scheme [5]. Fig 3. The ifurction digrmmen of the Hénon mpfor =0.3, nd 0 0, 0 vrile, Fig 5. The chotic ttrctor of the multimodel for =.5, = 0.4 nd 0 0, 0 =.4, =0.3, Fig 4. Bifurction digrmme Hénon mp for vrile, 0 0, 0 =0.4 nd Bifurction digrm Fig. 6 shows tht the chotic ehvior of the multimodel is otined for 0.9,.4, such s 0.3, =0.4 nd.5 while, for the sme fied vlues, the chotic ehvior of (4) is otined for.5,.4 s shown in Fig.3. The intervl size of the multimodel s vlues originting chos is lrger thn those of system (4). The sme pplies is otined for the multimodel y vrying the prmeter nd for the sme fied vlues. In fct, s it is shown in Fig.6 the chotic ehvior of the multimodel is otined for.,.. While the chotic ehvior of (5) is otined for Fig 6.Bifurction digrm of the discrete-time multimodel for =0.3, =.5, =0.4 nd 0 0, 0 Fig 6.Bifurction digrm of the discrete-time multimodel for =.4, = 0.3, =0.4 nd 0 0, 0 vrile, vrile,

4 IV. INTERPOLATION OF A CHAOTIC AND A NON- CHAOTIC D HÉNON MAPS In this section, for this second emple, we hve chosen s se models two systems: chotic D Hénon susystem (4) with fied prmeters Hénon susystem (5). =.4, For the set prmeter 0.3, =0.3, nd non-chotic D 0.9 with initil vlues 0 0, 0, the susystem doesn t present chotic ehvior s it is shown in Fig. 7. In fct, the figure doesn t illustrte strnge ttrctor nd the ifurction digrm Fig. 8 shows tht chosen prmeters doesn t led to chos. However, ifurction digrm of Fig. 0 shows tht for the chosen prmeter the multimodel doesn t hve chotic ehvior. Figure 9. Attrctor of the otined multimodel for 0.3, 0.9 nd 0 0, 0 =.4, = 0.3, Fig 7. Attrctor of the non chotic Hénon mp for 0.3, nd 0 0, Figure 0. Bifurction digrm of the multimodel for 0.3, 0.9 nd 0 0, 0 vrile, = 0.3, V. CONCLUSION Fig 8. Bifurction digrm of the Hénon mp for vrile, 0.9 nd 0 0, 0 The simultion results of Fig. 9 don t give cler illustrtion of the multimodel ttrctor otined from the interpoltion of the first susystem (4) chrcterized y =.4, = 0.3 nd the second susystem (5) chrcterized y 0.3, 0.9 nd for the chosen ctivtion function (6). The multimodel pproch is used in this pper to uild discrete-time chotic multimodels. It hs een shown, y the use of discrete time Hénon mp D, tht the interpoltion of two chotic systems cn enhnce the compleity of the chos however miing non-chotic system with chotic one doesn t led necessry to chos. Bifurction digrms illustrte the rnge of possile prmeters, giving to the multimodel, chotic ehvior. REFERENCES [] A. V. Oppenheim, K. M. Cuomo nd S. H. Strogtz, Synchroniztion of Lorenz-sed chotic circuits with pplictions to communictions, IEEE Trns. on Circ. Syst. II, vol.40, no.0, pp , 993.

5 [] H. Dedieu, M. P. Kennedy nd M. Hsler, Chos shift keying: modultion nd demodultion of chotic crrier using selfsynchronizing Chu s circuits, IEEE Trns. on Circ. Syst. I: vol. 40, no.0, pp , 993 [3] U. Prlitz, L. O. Chu, L. Kocrev K. S. Hlle nd A. Shng, Trnsmission of digitl signls y chotic synchroniztion. Int. J. of Bifurct. nd Chos, vol., 993. [4] G. Millériou nd C. Mir, Coding scheme sed on chos synchroniztion from noninvertile mps Int. J. of Bifurct. nd Chos, vol.8, no.0, pp.09 09, 998. [5] G. Alvrez nd S. Li, Some sic cryptogrphic requirements for chos-sed cryptosystems, Int. J. of Bifurct. nd Chos, vol.6, no.8, pp. 9 5, 006. [6] M. Chdli, J. Rgot, nd D. Mquin, Multiqurdrtic stility nd stiliztion of continuous time multiple-model, th IFAC Symposium on Automtion in Mining, Minerl nd Metl processing, Nncy, Frnce, 004 [7] N. Elfelly, J-Y. Dieulot, M. Benreje nd P. Borne, A new pproch for multimodel identifiction of comple systems sed on oth neurl nd fuzzy clustering lgorithms, Eng. App. of Artificil Intelligence, vol. 3, n 7, pp , 00 [0] E. Cherrier, M. Boutye J. Rgot, M.A. Aziz-Aloui, Chotic multimodels: ppliction to oserver-sed synchroniztion, IEEE Trns. on Cir. Syst. II. [] G. Bier, nd M. Klein, Mimum hyperchos in generlized Hénon circuit, Phys. Lett, A., vol.5, no.67, pp. 8 84, 990. [8] N. Elfelly, J.-Y. Dieulot, M. Benreje nd P. Borne, Multimodel control design using unsupervised clssifiers, Studies in Informtics nd Control, ISSN 0-766, vol., no., pp. 0-08, 0. [9] N. Elfelly, J-Y. Dieulot, M. Benreje nd P. Borne, A Neurl Approch of Multimodel Representtion of Comple Processes, Int. J. of Computers, Communictions & Control, vol. 3, pp.49-60, 008. [] RL. Filli, M Benreje, P. Borne, On oserver-sed secure communiction design using discrete-time hyperchotic systems, Communictions in Nonliner Science nd Numericl Simultion, vol. 9, no.5, pp.44-43, 04 [3] RL. Filli, S. Hmmmi, M. Benreje, On Synchroniztion, Anti- Synchroniztion nd Hyrid Synchroniztion of 3D Discrete Generlized Hénon Mp, Nonliner Dynmics nd Systems Theory, vol., no., pp. 8-95, 0 [4] E. Li, G. Li, G. Wen nd H. Wng, Hopf, Bifurction of the thirdorder Hénon system sed on n eplicit criterion Nonliner Anlysis: Theory, Methods & Applictions, vol.70, no.9, pp , 009..

ADAPTIVE CONTROL AND SYNCHRONIZATION OF RÖSSLER PROTOTYPE-4 SYSTEM

ADAPTIVE CONTROL AND SYNCHRONIZATION OF RÖSSLER PROTOTYPE-4 SYSTEM ADAPTIVE CONTROL AND SYNCHRONIZATION OF RÖSSLER PROTOTYPE-4 SYSTEM Sundrpndin Vidynthn 1 1 Reserch nd Development Centre, Vel Tech Dr. RR & Dr. SR Technicl University Avdi, Chenni-600 06, Tmil Ndu, INDIA

More information

Lecture Solution of a System of Linear Equation

Lecture Solution of a System of Linear Equation ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville - D. Keffer, 5/9/98 (updted /) Lecture 8- - Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner

More information

QUADRATURE is an old-fashioned word that refers to

QUADRATURE is an old-fashioned word that refers to World Acdemy of Science Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Vol:5 No:7 011 A New Qudrture Rule Derived from Spline Interpoltion with Error Anlysis Hdi Tghvfrd

More information

BIFURCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS

BIFURCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS BIFRCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS FRANCESCA AICARDI In this lesson we will study the simplest dynmicl systems. We will see, however, tht even in this cse the scenrio of different possible

More information

A New Receiver for Chaotic Digital Transmissions: The Symbolic Matching Approach

A New Receiver for Chaotic Digital Transmissions: The Symbolic Matching Approach A New Receiver for Chotic Digitl Trnsmissions: The Symbolic Mtching Approch Gilles BUREL nd Stéphne AZOU LEST, Université de Bretgne Occidentle CS 93837, 29238 BREST cede 3, Frnce Abstrct : Chotic digitl

More information

10.2 The Ellipse and the Hyperbola

10.2 The Ellipse and the Hyperbola CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point

More information

Creating A New Planck s Formula of Spectral Density of Black-body Radiation by Means of AF(A) Diagram

Creating A New Planck s Formula of Spectral Density of Black-body Radiation by Means of AF(A) Diagram nd Jogj Interntionl Physics Conference Enhncing Network nd Collortion Developing Reserch nd Eduction in Physics nd Nucler Energy Septemer 6-9, 007, Yogykrt-Indonesi Creting A New Plnck s Formul of Spectrl

More information

NUMERICAL STUDY OF COEXISTING ATTRACTORS FOR THE HÉNON MAP

NUMERICAL STUDY OF COEXISTING ATTRACTORS FOR THE HÉNON MAP NUMERICAL STUDY OF COEXISTING ATTRACTORS FOR THE HÉNON MAP ZBIGNIEW GALIAS Deprtment of Electricl Engineering, AGH University of Science nd Technology, Mickiewicz 30, 30 059 Krków, Polnd glis@gh.edu.pl

More information

Neuro-Fuzzy Modeling of Superheating System. of a Steam Power Plant

Neuro-Fuzzy Modeling of Superheating System. of a Steam Power Plant Applied Mthemticl Sciences, Vol. 1, 2007, no. 42, 2091-2099 Neuro-Fuzzy Modeling of Superheting System of Stem Power Plnt Mortez Mohmmdzheri, Ali Mirsephi, Orng Asef-fshr nd Hmidrez Koohi The University

More information

INEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV FUNCTIONAL. Mohammad Masjed-Jamei

INEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV FUNCTIONAL. Mohammad Masjed-Jamei Fculty of Sciences nd Mthemtics University of Niš Seri Aville t: http://www.pmf.ni.c.rs/filomt Filomt 25:4 20) 53 63 DOI: 0.2298/FIL0453M INEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV

More information

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations. Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one

More information

Bases for Vector Spaces

Bases for Vector Spaces Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything

More information

arxiv:math/ v1 [math.ds] 23 Aug 2006

arxiv:math/ v1 [math.ds] 23 Aug 2006 rxiv:mth/0608568v1 [mth.ds] Aug 006 Anlysis of D chotic system Gheorghe Tign, Dumitru Opriş Astrct A D nonliner chotic system, clled the T system, is nlyzed in this pper. Horseshoe chos is investigted

More information

New Expansion and Infinite Series

New Expansion and Infinite Series Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

More information

Model Reduction of Finite State Machines by Contraction

Model Reduction of Finite State Machines by Contraction Model Reduction of Finite Stte Mchines y Contrction Alessndro Giu Dip. di Ingegneri Elettric ed Elettronic, Università di Cgliri, Pizz d Armi, 09123 Cgliri, Itly Phone: +39-070-675-5892 Fx: +39-070-675-5900

More information

LINEAR ALGEBRA APPLIED

LINEAR ALGEBRA APPLIED 5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order

More information

Note 12. Introduction to Digital Control Systems

Note 12. Introduction to Digital Control Systems Note Introduction to Digitl Control Systems Deprtment of Mechnicl Engineering, University Of Ssktchewn, 57 Cmpus Drive, Ssktoon, SK S7N 5A9, Cnd . Introduction A digitl control system is one in which the

More information

Linear and Non-linear Feedback Control Strategies for a 4D Hyperchaotic System

Linear and Non-linear Feedback Control Strategies for a 4D Hyperchaotic System Pure nd Applied Mthemtics Journl 017; 6(1): 5-13 http://www.sciencepublishinggroup.com/j/pmj doi: 10.11648/j.pmj.0170601.1 ISSN: 36-9790 (Print); ISSN: 36-981 (Online) Liner nd Non-liner Feedbck Control

More information

Calculus of variations with fractional derivatives and fractional integrals

Calculus of variations with fractional derivatives and fractional integrals Anis do CNMAC v.2 ISSN 1984-820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810-193 Aveiro, Portugl

More information

Introduction to Electronic Circuits. DC Circuit Analysis: Transient Response of RC Circuits

Introduction to Electronic Circuits. DC Circuit Analysis: Transient Response of RC Circuits Introduction to Electronic ircuits D ircuit Anlysis: Trnsient esponse of ircuits Up until this point, we hve een looking t the Stedy Stte response of D circuits. StedyStte implies tht nothing hs chnged

More information

Parse trees, ambiguity, and Chomsky normal form

Parse trees, ambiguity, and Chomsky normal form Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs

More information

ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTOR

ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTOR ADAPTIVE CONTROL AND SYNCHRONIZATION OF A HIGHLY CHAOTIC ATTRACTOR Sundrpndin Vidynthn 1 1 Reserch nd Development Centre, Vel Tech Dr. RR & Dr. SR Technicl University Avdi, Chenni-600 062, Tmil Ndu, INDIA

More information

Fundamental Theorem of Calculus

Fundamental Theorem of Calculus Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under

More information

Using Pisarenko Harmonic Decomposition for the design of 2-D IIR Notch filters

Using Pisarenko Harmonic Decomposition for the design of 2-D IIR Notch filters WSEAS RANSACIONS on SIGNAL PROCESSING Using Prenko Hrmonic Decomposition for the design of -D IIR Notch filters Astrct In th pper, the Prenko Hrmonic Decomposition used for the design of -D (wo- Dimensionl

More information

ADAPTIVE CONTROL AND SYNCHRONIZATION OF LIU S FOUR-WING CHAOTIC SYSTEM WITH CUBIC NONLINEARITY

ADAPTIVE CONTROL AND SYNCHRONIZATION OF LIU S FOUR-WING CHAOTIC SYSTEM WITH CUBIC NONLINEARITY ADAPTIVE CONTROL AND SYNCHRONIZATION OF LIU S FOUR-WING CHAOTIC SYSTEM WITH CUBIC NONLINEARITY Sunrpnin Viynthn 1 1 Reserch n Development Centre, Vel Tech Dr. RR Dr. SR Technicl University Avi, Chenni-600

More information

Review of Gaussian Quadrature method

Review of Gaussian Quadrature method Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

More information

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3 2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

More information

Application of Exact Discretization for Logistic Differential Equations to the Design of a Discrete-Time State-Observer

Application of Exact Discretization for Logistic Differential Equations to the Design of a Discrete-Time State-Observer 5 Proceedings of the Interntionl Conference on Informtion nd Automtion, December 58, 5, Colombo, Sri Ln. Appliction of Exct Discretiztion for Logistic Differentil Equtions to the Design of Discrete-ime

More information

How to simulate Turing machines by invertible one-dimensional cellular automata

How to simulate Turing machines by invertible one-dimensional cellular automata How to simulte Turing mchines by invertible one-dimensionl cellulr utomt Jen-Christophe Dubcq Déprtement de Mthémtiques et d Informtique, École Normle Supérieure de Lyon, 46, llée d Itlie, 69364 Lyon Cedex

More information

Topics Covered AP Calculus AB

Topics Covered AP Calculus AB Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.

More information

Homework Solution - Set 5 Due: Friday 10/03/08

Homework Solution - Set 5 Due: Friday 10/03/08 CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.

More information

Preview 11/1/2017. Greedy Algorithms. Coin Change. Coin Change. Coin Change. Coin Change. Greedy algorithms. Greedy Algorithms

Preview 11/1/2017. Greedy Algorithms. Coin Change. Coin Change. Coin Change. Coin Change. Greedy algorithms. Greedy Algorithms Preview Greed Algorithms Greed Algorithms Coin Chnge Huffmn Code Greed lgorithms end to e simple nd strightforwrd. Are often used to solve optimiztion prolems. Alws mke the choice tht looks est t the moment,

More information

FUZZY HOMOTOPY CONTINUATION METHOD FOR SOLVING FUZZY NONLINEAR EQUATIONS

FUZZY HOMOTOPY CONTINUATION METHOD FOR SOLVING FUZZY NONLINEAR EQUATIONS VOL NO 6 AUGUST 6 ISSN 89-668 6-6 Asin Reserch Publishing Networ (ARPN) All rights reserved wwwrpnjournlscom FUZZY HOMOTOPY CONTINUATION METHOD FOR SOLVING FUZZY NONLINEAR EQUATIONS Muhmmd Zini Ahmd Nor

More information

Torsion in Groups of Integral Triangles

Torsion in Groups of Integral Triangles Advnces in Pure Mthemtics, 01,, 116-10 http://dxdoiorg/1046/pm011015 Pulished Online Jnury 01 (http://wwwscirporg/journl/pm) Torsion in Groups of Integrl Tringles Will Murry Deprtment of Mthemtics nd Sttistics,

More information

CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation

CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation CS2N: The Coming Revolution in Computer Architecture Lortory 2 Preprtion Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes

More information

SOME INTEGRAL INEQUALITIES OF GRÜSS TYPE

SOME INTEGRAL INEQUALITIES OF GRÜSS TYPE RGMIA Reserch Report Collection, Vol., No., 998 http://sci.vut.edu.u/ rgmi SOME INTEGRAL INEQUALITIES OF GRÜSS TYPE S.S. DRAGOMIR Astrct. Some clssicl nd new integrl inequlities of Grüss type re presented.

More information

AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS. I. Fedotov and S. S. Dragomir

AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS. I. Fedotov and S. S. Dragomir RGMIA Reserch Report Collection, Vol., No., 999 http://sci.vu.edu.u/ rgmi AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS I. Fedotov nd S. S. Drgomir Astrct. An

More information

Chapter 6 Techniques of Integration

Chapter 6 Techniques of Integration MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln

More information

CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010

CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010 CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w

More information

arxiv:solv-int/ v1 4 Aug 1997

arxiv:solv-int/ v1 4 Aug 1997 Two-dimensionl soliton cellulr utomton of deutonomized Tod-type A. Ngi 1,2, T. Tokihiro 1, J. Stsum 1, R. Willox 1,3 nd K. Kiwr 4 1 Grdute School of Mthemticl Sciences, University of Tokyo, rxiv:solv-int/9708001v1

More information

Designing finite automata II

Designing finite automata II Designing finite utomt II Prolem: Design DFA A such tht L(A) consists of ll strings of nd which re of length 3n, for n = 0, 1, 2, (1) Determine wht to rememer out the input string Assign stte to ech of

More information

The Trapezoidal Rule

The Trapezoidal Rule _.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

More information

Research Article Analytical Solution of the Fractional Fredholm Integrodifferential Equation Using the Fractional Residual Power Series Method

Research Article Analytical Solution of the Fractional Fredholm Integrodifferential Equation Using the Fractional Residual Power Series Method Hindwi Compleity Volume 7, Article ID 457589, 6 pges https://doi.org/.55/7/457589 Reserch Article Anlyticl Solution of the Frctionl Fredholm Integrodifferentil Eqution Using the Frctionl Residul Power

More information

M344 - ADVANCED ENGINEERING MATHEMATICS

M344 - ADVANCED ENGINEERING MATHEMATICS M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If

More information

Characterization of Impact Test Response of PCCP with System Identification Approaches

Characterization of Impact Test Response of PCCP with System Identification Approaches 7th World Conference on Nondestructive Testing, 25-28 Oct 2008, Shnghi, Chin Chrcteriztion of Impct Test Response of PCCP with System Identifiction Approches Astrct Zheng LIU, Alex WANG, nd Dennis KRYS

More information

CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)

CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.) CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts

More information

DYNAMICAL ANALYSIS OF AN NON-ŠIL NIKOV CHAOTIC SYSTEM

DYNAMICAL ANALYSIS OF AN NON-ŠIL NIKOV CHAOTIC SYSTEM THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY Series A OF THE ROMANIAN ACADEMY Volume 9 Numer /8 pp. 8 6 DYNAMICAL ANALYSIS OF AN NON-ŠIL NIKOV CHAOTIC SYSTEM Kui Bio DENG Xio Bo WEI Yi Yn KONG

More information

Matching patterns of line segments by eigenvector decomposition

Matching patterns of line segments by eigenvector decomposition Title Mtching ptterns of line segments y eigenvector decomposition Author(s) Chn, BHB; Hung, YS Cittion The 5th IEEE Southwest Symposium on Imge Anlysis nd Interprettion Proceedings, Snte Fe, NM., 7-9

More information

AUTOMATA AND LANGUAGES. Definition 1.5: Finite Automaton

AUTOMATA AND LANGUAGES. Definition 1.5: Finite Automaton 25. Finite Automt AUTOMATA AND LANGUAGES A system of computtion tht only hs finite numer of possile sttes cn e modeled using finite utomton A finite utomton is often illustrted s stte digrm d d d. d q

More information

Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem

Application Chebyshev Polynomials for Determining the Eigenvalues of Sturm-Liouville Problem Applied nd Computtionl Mthemtics 5; 4(5): 369-373 Pulished online Septemer, 5 (http://www.sciencepulishinggroup.com//cm) doi:.648/.cm.545.6 ISSN: 38-565 (Print); ISSN: 38-563 (Online) Appliction Cheyshev

More information

Coalgebra, Lecture 15: Equations for Deterministic Automata

Coalgebra, Lecture 15: Equations for Deterministic Automata Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined

More information

1 Nondeterministic Finite Automata

1 Nondeterministic Finite Automata 1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you

More information

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O 1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

More information

dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.

dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω. Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd

More information

Quadratic Forms. Quadratic Forms

Quadratic Forms. Quadratic Forms Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte

More information

Network Analysis and Synthesis. Chapter 5 Two port networks

Network Analysis and Synthesis. Chapter 5 Two port networks Network Anlsis nd Snthesis hpter 5 Two port networks . ntroduction A one port network is completel specified when the voltge current reltionship t the terminls of the port is given. A generl two port on

More information

SOLVING SYSTEMS OF EQUATIONS, ITERATIVE METHODS

SOLVING SYSTEMS OF EQUATIONS, ITERATIVE METHODS ELM Numericl Anlysis Dr Muhrrem Mercimek SOLVING SYSTEMS OF EQUATIONS, ITERATIVE METHODS ELM Numericl Anlysis Some of the contents re dopted from Lurene V. Fusett, Applied Numericl Anlysis using MATLAB.

More information

1B40 Practical Skills

1B40 Practical Skills B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need

More information

APPROXIMATE LIMIT CYCLES FOR THE RAYLEIGH MODEL

APPROXIMATE LIMIT CYCLES FOR THE RAYLEIGH MODEL ROMAI J, 4, 228, 73 8 APPROXIMATE LIMIT CYCLES FOR THE RAYLEIGH MODEL Adelin Georgescu, Petre Băzăvn, Mihel Sterpu Acdemy of Romnin Scientists, Buchrest Deprtment of Mthemtics nd Computer Science, University

More information

Interpreting Integrals and the Fundamental Theorem

Interpreting Integrals and the Fundamental Theorem Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of

More information

MT Integral equations

MT Integral equations MT58 - Integrl equtions Introduction Integrl equtions occur in vriety of pplictions, often eing otined from differentil eqution. The reson for doing this is tht it my mke solution of the prolem esier or,

More information

Math 131. Numerical Integration Larson Section 4.6

Math 131. Numerical Integration Larson Section 4.6 Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n

More information

Section 4: Integration ECO4112F 2011

Section 4: Integration ECO4112F 2011 Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

More information

The practical version

The practical version Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht

More information

Improved Results on Stability of Time-delay Systems using Wirtinger-based Inequality

Improved Results on Stability of Time-delay Systems using Wirtinger-based Inequality Preprints of the 9th World Congress he Interntionl Federtion of Automtic Control Improved Results on Stbility of ime-dely Systems using Wirtinger-bsed Inequlity e H. Lee Ju H. Prk H.Y. Jung O.M. Kwon S.M.

More information

Fig. 1. Open-Loop and Closed-Loop Systems with Plant Variations

Fig. 1. Open-Loop and Closed-Loop Systems with Plant Variations ME 3600 Control ystems Chrcteristics of Open-Loop nd Closed-Loop ystems Importnt Control ystem Chrcteristics o ensitivity of system response to prmetric vritions cn be reduced o rnsient nd stedy-stte responses

More information

Boolean algebra.

Boolean algebra. http://en.wikipedi.org/wiki/elementry_boolen_lger Boolen lger www.tudorgir.com Computer science is not out computers, it is out computtion nd informtion. computtion informtion computer informtion Turing

More information

CS 188: Artificial Intelligence Fall Announcements

CS 188: Artificial Intelligence Fall Announcements CS 188: Artificil Intelligence Fll 2009 Lecture 20: Prticle Filtering 11/5/2009 Dn Klein UC Berkeley Announcements Written 3 out: due 10/12 Project 4 out: due 10/19 Written 4 proly xed, Project 5 moving

More information

The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.

The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests. ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

Lecture 3: Curves in Calculus. Table of contents

Lecture 3: Curves in Calculus. Table of contents Mth 348 Fll 7 Lecture 3: Curves in Clculus Disclimer. As we hve textook, this lecture note is for guidnce nd supplement only. It should not e relied on when prepring for exms. In this lecture we set up

More information

Things to Memorize: A Partial List. January 27, 2017

Things to Memorize: A Partial List. January 27, 2017 Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved

More information

Tutorial Automata and formal Languages

Tutorial Automata and formal Languages Tutoril Automt nd forml Lnguges Notes for to the tutoril in the summer term 2017 Sestin Küpper, Christine Mik 8. August 2017 1 Introduction: Nottions nd sic Definitions At the eginning of the tutoril we

More information

Safety Controller Synthesis for Switched Systems using Multiscale Symbolic Models

Safety Controller Synthesis for Switched Systems using Multiscale Symbolic Models Sfety Controller Synthesis for Switched Systems using Multiscle Symolic Models Antoine Girrd Lortoire des Signux et Systèmes Gif sur Yvette, Frnce Séminire du LAAS Toulouse, 29 Juin, 2016 A. Girrd (L2S-CNRS)

More information

Adaptive Stabilizing Control of Power System through Series Voltage Control of a Unified Power Flow Controller

Adaptive Stabilizing Control of Power System through Series Voltage Control of a Unified Power Flow Controller Adptive Stilizing Control of Power System through Series Voltge Control of Unified Power Flow Controller AHMA Rhim SA Al-Biyt Deprtment of Electricl Engineering King Fhd University of Petroleum & Minerls

More information

Exploring parametric representation with the TI-84 Plus CE graphing calculator

Exploring parametric representation with the TI-84 Plus CE graphing calculator Exploring prmetric representtion with the TI-84 Plus CE grphing clcultor Richrd Prr Executive Director Rice University School Mthemtics Project rprr@rice.edu Alice Fisher Director of Director of Technology

More information

Application of Exp-Function Method to. a Huxley Equation with Variable Coefficient *

Application of Exp-Function Method to. a Huxley Equation with Variable Coefficient * Interntionl Mthemticl Forum, 4, 9, no., 7-3 Appliction of Exp-Function Method to Huxley Eqution with Vrible Coefficient * Li Yo, Lin Wng nd Xin-Wei Zhou. Deprtment of Mthemtics, Kunming College Kunming,Yunnn,

More information

Minimal DFA. minimal DFA for L starting from any other

Minimal DFA. minimal DFA for L starting from any other Miniml DFA Among the mny DFAs ccepting the sme regulr lnguge L, there is exctly one (up to renming of sttes) which hs the smllest possile numer of sttes. Moreover, it is possile to otin tht miniml DFA

More information

September 13 Homework Solutions

September 13 Homework Solutions College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are

More information

Domino Recognizability of Triangular Picture Languages

Domino Recognizability of Triangular Picture Languages Interntionl Journl of Computer Applictions (0975 8887) Volume 57 No.5 Novemer 0 Domino Recognizility of ringulr icture Lnguges V. Devi Rjselvi Reserch Scholr Sthym University Chenni 600 9. Klyni Hed of

More information

Bifurcation structures in maps of Hénon type

Bifurcation structures in maps of Hénon type Nonlinerity 11 (1998) 1233 1261. Printed in the UK PII: S0951-7715(98)84171-6 Bifurction structures in mps of Hénon type Ki T Hnsen nd Predrg Cvitnović NORDITA, Blegdmsvej 17, DK-2100 Copenhgen Ø, Denmrk

More information

Nonlinear element of a chaotic generator

Nonlinear element of a chaotic generator INVESTIGACIÓN REVISTA MEXICANA DE FÍSICA 53 (3) 159 163 JUNIO 2007 Nonliner element of chotic genertor E Cmpos-Cntón JS Murguí I Cmpos Cntón b nd M Chvir-Rodríguez Deprtmento de Físico Mtemátics b Fc de

More information

Solutions of Klein - Gordan equations, using Finite Fourier Sine Transform

Solutions of Klein - Gordan equations, using Finite Fourier Sine Transform IOSR Journl of Mthemtics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 13, Issue 6 Ver. IV (Nov. - Dec. 2017), PP 19-24 www.iosrjournls.org Solutions of Klein - Gordn equtions, using Finite Fourier

More information

Ordinary differential equations

Ordinary differential equations Ordinry differentil equtions Introduction to Synthetic Biology E Nvrro A Montgud P Fernndez de Cordob JF Urchueguí Overview Introduction-Modelling Bsic concepts to understnd n ODE. Description nd properties

More information

Resistive Network Analysis

Resistive Network Analysis C H A P T E R 3 Resistive Network Anlysis his chpter will illustrte the fundmentl techniques for the nlysis of resistive circuits. The methods introduced re sed on the circuit lws presented in Chpter 2:

More information

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS. THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem

More information

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

More information

Undergraduate Research

Undergraduate Research Undergrdute Reserch A Trigonometric Simpson s Rule By Ctherine Cusimno Kirby nd Sony Stnley Biogrphicl Sketch Ctherine Cusimno Kirby is the dughter of Donn nd Sm Cusimno. Originlly from Vestvi Hills, Albm,

More information

Safety Controller Synthesis for Switched Systems using Multiscale Symbolic Models

Safety Controller Synthesis for Switched Systems using Multiscale Symbolic Models Sfety Controller Synthesis for Switched Systems using Multiscle Symolic Models Antoine Girrd Lortoire des Signux et Systèmes Gif sur Yvette, Frnce Workshop on switching dynmics & verifiction Pris, Jnury

More information

Precalculus Due Tuesday/Wednesday, Sept. 12/13th Mr. Zawolo with questions.

Precalculus Due Tuesday/Wednesday, Sept. 12/13th  Mr. Zawolo with questions. Preclculus Due Tuesd/Wednesd, Sept. /th Emil Mr. Zwolo (isc.zwolo@psv.us) with questions. 6 Sketch the grph of f : 7! nd its inverse function f (). FUNCTIONS (Chpter ) 6 7 Show tht f : 7! hs n inverse

More information

Chaos in drive systems

Chaos in drive systems Applied nd Computtionl Mechnics 1 (2007) 121-126 Chos in drive systems Ctird Krtochvíl, Mrtin Houfek, Josef Koláčný b, Romn Kříž b, Lubomír Houfek,*, Jiří Krejs Institute of Thermomechnics, brnch Brno,

More information

Module 6: LINEAR TRANSFORMATIONS

Module 6: LINEAR TRANSFORMATIONS Module 6: LINEAR TRANSFORMATIONS. Trnsformtions nd mtrices Trnsformtions re generliztions of functions. A vector x in some set S n is mpped into m nother vector y T( x). A trnsformtion is liner if, for

More information

The Islamic University of Gaza Faculty of Engineering Civil Engineering Department. Numerical Analysis ECIV Chapter 11

The Islamic University of Gaza Faculty of Engineering Civil Engineering Department. Numerical Analysis ECIV Chapter 11 The Islmic University of Gz Fculty of Engineering Civil Engineering Deprtment Numericl Anlysis ECIV 6 Chpter Specil Mtrices nd Guss-Siedel Associte Prof Mzen Abultyef Civil Engineering Deprtment, The Islmic

More information

0 N. S. BARNETT AND S. S. DRAGOMIR Using Gruss' integrl inequlity, the following pertured trpezoid inequlity in terms of the upper nd lower ounds of t

0 N. S. BARNETT AND S. S. DRAGOMIR Using Gruss' integrl inequlity, the following pertured trpezoid inequlity in terms of the upper nd lower ounds of t TAMKANG JOURNAL OF MATHEMATICS Volume 33, Numer, Summer 00 ON THE PERTURBED TRAPEZOID FORMULA N. S. BARNETT AND S. S. DRAGOMIR Astrct. Some inequlities relted to the pertured trpezoid formul re given.

More information

Extended tan-cot method for the solitons solutions to the (3+1)-dimensional Kadomtsev-Petviashvili equation

Extended tan-cot method for the solitons solutions to the (3+1)-dimensional Kadomtsev-Petviashvili equation Interntionl Jornl of Mthemticl Anlysis nd Applictions ; (): 9-9 Plished online Mrch, (http://www.scit.org/jornl/ijm) Extended tn-cot method for the solitons soltions to the (+)-dimensionl Kdomtsev-Petvishvili

More information

Fredholm Integral Equations of the First Kind Solved by Using the Homotopy Perturbation Method

Fredholm Integral Equations of the First Kind Solved by Using the Homotopy Perturbation Method Int. Journl of Mth. Anlysis, Vol. 5, 211, no. 19, 935-94 Fredholm Integrl Equtions of the First Kind Solved by Using the Homotopy Perturbtion Method Seyyed Mhmood Mirzei Deprtment of Mthemtics, Fculty

More information

p-adic Egyptian Fractions

p-adic Egyptian Fractions p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction

More information

Adaptive STATCOM Control for a Multi-machine Power System

Adaptive STATCOM Control for a Multi-machine Power System Adptive SACOM Control for Multi-mchine Power System AHMARhim nd MBer As King Fhd University of Petroleum & Minerls, Dhhrn, Sudi Ari; Sudi Electricity Compny, Dmmm, Sudi Ari Astrct Synchronous sttic compenstor

More information

4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTORS. 4.0 Introduction. Objectives. Activity 1 4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

More information