Thabet Abdeljawad 1. Çankaya Üniversitesi Fen-Edebiyat Fakültesi, Journal of Arts and Sciences Say : 9 / May s 2008
|
|
- John Walker
- 5 years ago
- Views:
Transcription
1 Çaaya Üiversiesi Fe-Edebiya Faülesi, Jural Ars ad Scieces Say : 9 / May s 008 A Ne e Cai Rule ime Scales abe Abdeljawad Absrac I is w, i eeral, a e cai rule eeral ime scale derivaives des beave well as i e case usual derivaive Hwever, we discuss sme special cases were e ime scale derivaive as e usual cai rule e resuls are aalyzed r b e dela ad abla ime scales derivaives Key wrds: ime scales derivaive, -ime scale, H-ime scale, Frward jump perar, Bacward jump perar,rd-ciuus, ld-ciuus, Cai rule Irduci ad Prelimiaries e calculus ime scales was irduced [], [] uiy e ciuus ad discree aalysis As usually expeced we we eeralize sme ery we lse sme ice prperies Oe e basic cceps a researcers as care abu we deal wi diereial euais, variaial calculus ad s, ime scales is cai rule Fr a reas we ry ere summarize ad rmulae sme cases were e cai rule beys e rder, s a i mi be useul use i ex wrs i e uure Fr e ery dela derivaive ime scales we reer [] ad r e e r abla derivaive we reer [] A ime scale is a arbirary empy clsed subse e real lie us real umbers,, ad aural umbers, are examples ime scales ruu is aricle ad llwi [], e ime scale will be deed by Here are dw sme examples ime scales wic we will sudy e cai rule i is aricle Çaaya Üiversiesi, Fe-Edebiya Faülesi, Maemai-Bilisayar Bölümü, Aara abe@caayaedur
2 A Ne O e Cai Rule O ime Scales Examples : N 0 0 Fr, le We 0, e ery - : N calculus is baied [-sur] wi 0 a imprper accumulai pi, were e bacward derivaive is usually deied We e ime scale will ave as a imprper pi, were e rward derivaive is usually deied i :, N 0 Fr ad, le i0 i :, N i0 e rward jump perar : is deied by i s : s wile e bacward jump perar : is deied by sups : s, 3 were, i sup ie i as maximum ad sup i ie i as 0 miimum I is be ed a i e ad 0 Als i, e ad A pi is called ri scaered i, le-scaered i ad islaed i I ceci e rward ad bacward raiess ucis, : 0, are deied, respecively, by ad I rder deie e rward ad bacward ime scale derivaive, we eed e ses ad wic are derived rm e ime scale as llws: I as a le-scaered maximum M, M ad erwise I as a ri-scaered miimum m, e e m ad erwise Assume : ad e e rward ime-scale derivaive is e umber prvided i exiss wi e prpery a ive ay 0, ere exiss a eibrd U ie U, r sme 0 suc a s s s, s U 4 Mrever, we say a is dela diereiable prvided a r all Similarly, e bacward ime-scale derivaive is e umber, prvided i exiss wi e prpery a ive ay 0, ere exiss a eibrd U ie U, r sme 0 suc a s s s, s U, 5
3 abe ABDELJAWAD ad mrever, we say a is abla diereiable prvided a r all e llwi w erems are valid r e rward ad bacward ime-scale derivaives: erem Assume : is a uci ad le e we ave e llwi: i I is dela diereiable a, e is ciuus a ii I is ciuus a ad is ri-scaered, e is dela diereiable a wi 6 iii I is ri-dese ie e is dela diereiable a i ad ly i e s limi lim exiss as a iie umber I is case s s s lim s s 7 iv I is dela diereiable a, e 8 erem Assume : is a uci ad le e we ave e llwi: i I is abla diereiable a, e is ciuus a ii I is ciuus a ad is le-scaered, e is abla diereiable a wi iii I is le-dese ie e is abla diereiable a i ad ly i e s limi, lim exiss as a iie umber I is case s s s lim s s 9 iv I is abla diereiable a, e 0 e llwi prduc rmulas are useul: I case e ime scales ad ive abve i examples ad all e des are islaed excep r e limi pis ad e jump perars ad are iverses eac er ad ece e llwi relais bewee e dela derivaive ad e abla derivaive ld a b Fr a uci :, will mea e dela derivaive a ad a similar ai ca be assied als r e abla derivaive 3
4 A Ne O e Cai Rule O ime Scales Deiii A uci : is called rd-ciuus prvided i is ciuus a ridese pis i ad is le-sided limis exis iie a le-dese pis i, ad is called ld-ciuus prvided i is ciuus a le-dese pis i ad is ri-sided limis exis iie a ri-dese pis i Ciuus ucis are clearly rd-ciuus ad ld-ciuus Als i is easy see a is a example a rd-ciuus uci wic is ciuus ad is a example a ld-ciuus uci wic is ciuus is is, curse, rue i e ime scale cais le-dese ri-scaered des ad ri-dese le-scaered des, respecively erem 3 [] Exisece dela aiderivaives Every rd-ciuus uci as dela aiderivaive I paricular, i, e F deied by 4 0 F : s s, r 3 0 is a dela aideivaive erem 4 Exisece abla aiderivaives Every ld-ciuus uci as abla aiderivaive I paricular, i 0, e F deied by F : s s, 0 r 4 is a abla aideivaive e Cai Rule ad Special Cases Recall a i ad are diereiable real-valued ucis deied, e e cai rule saes : d 5 d Quesi: Fr wic ucis :, : ad ime scales e llwi relai is valid: r, 6 were Clearly rm e usual cai rule e abve relai is rue r e ime scale r ay w direiable ucis ad wi Bu i yu le yur ime scale be e aurals N, e : ad :, ad is case yu ca id may examples seueces ad r wic e abve relai is valid Hwever, i ae ay seuece ad le r example e e abve relai i e uesi is valid r e dela derivaive Fr e sae cmpleeess we sae e eeral ime scale dela cai rule erem 5 [] Assume : is ciuus, : is dela diereiable, ad : is ciuusly diereiable e ere exiss a umber c, wi c 7 Deiii [4] A ime scale is said ave e H-prpery i is rward jumpi perar as e rm:, r sme ad We call suc a ime sacle a H-ime scale
5 abe ABDELJAWAD 5 I is easy see a N,,, ad are all examples H-ime scales Prpsii Le be a H-ime scale, : is w imes dela diereiable uci ad e, 8 Pr By 8 we ave e leads 9 ad applyi 8 aai r e uci we e 0 e e resul llws by i a Similarly, by e elp 0 ad we ca prve Prpsii Le be a H-ime scale, : is w imes abla diereiable uci ad e, Remar I ac, Prpsii ad Prpsii are valid r ime scales wse rward jump perar ad bacward jump perar are dela ad abla diereiable, respecively I paricular, e are valid r e ime scales ad Lemma Le be a a H-ime scale e, r all N ad We w eeralize Prpsii ad Prpsii, r e H-ime scale wi meas e cmpsii imes ad, respecively Prpsii 4 Le be a H-ime scale, : is w imes dela diereiable uci ad, r N e, Pr We llw by iduci e case is rue by Prpsii Assume e resul is rue r e, 8 applied implies a, 3 ad ece by we bai 4 Aai 8 implies a = 5 Frm e iduci sep, i llws a 6
6 A Ne O e Cai Rule O ime Scales By Lemma ad a is csa we cclude a, 7 ad e pr is cmplee Similarly ad by e elp 0 we ca als sae a cai rule resul r e abla derivaive Prpsii 5 Le be a H-ime scale, : is w imes dela diereiable uci ad, r N e, Fr e purpse cmpariss we sae e llwi cai rule i e ery -calculus, wic is Lemma [3] Prpsii 6 Le x cx, were c ad are csas ad ay uci e D x D x D x 8 were D x meas e abla derivaive e ime scale, 0 As direc cseueces Prpsii 4 ad Prpsii 5, we sae e llwi w useul crllaries Crllary Le be a H-ime scale ad : be a rd-ciuus uci I, ad F s s, N, N e F 9 Crllary Le be a H-ime scale ad : be a ld-ciuus uci I, ad F s s, N, N e F 30 REFERENCES Ber M, Peers A, Dyamic Euais ime Scales: A irduci wi applicais Birauser, Bs, 00 Ber M, Peers A, Advaces i Dyamic Euais ime Scales, Birauser, Bs, 00 Ers, e isry -calculus ad a ew med Liceiae esis, UUDM Repr 000: 6; p://www mauuse/mas/licspd Rui A C Ferreira ad Delim F M rres, Hier-Order Calculus Variais ime ScalesarXiv: v [maoc] 30 Sep 007 6
7 abe ABDELJAWAD 7
Research & Reviews: Journal of Statistics and Mathematical Sciences
Research & Reviews: Jural f Saisics ad Mahemaical Scieces iuus Depedece f he Slui f A Schasic Differeial Equai Wih Nlcal diis El-Sayed AMA, Abd-El-Rahma RO, El-Gedy M Faculy f Sciece, Alexadria Uiversiy,
More informationA L A BA M A L A W R E V IE W
A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N
More informationOn two general nonlocal differential equations problems of fractional orders
Malaya Journal of Maemaik, Vol. 6, No. 3, 478-482, 28 ps://doi.org/.26637/mjm63/3 On wo general nonlocal differenial equaions problems of fracional orders Abd El-Salam S. A. * and Gaafar F. M.2 Absrac
More informationMethod For Solving Fuzzy Integro-Differential Equation By Using Fuzzy Laplace Transformation
INERNAIONAL JOURNAL OF SCIENIFIC & ECHNOLOGY RESEARCH VOLUME 3 ISSUE 5 May 4 ISSN 77-866 Meod For Solving Fuzzy Inegro-Differenial Equaion By Using Fuzzy Laplace ransformaion Manmoan Das Danji alukdar
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 15 10/30/2013. Ito integral for simple processes
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.65/15.7J Fall 13 Lecure 15 1/3/13 I inegral fr simple prcesses Cnen. 1. Simple prcesses. I ismery. Firs 3 seps in cnsrucing I inegral fr general prcesses 1 I inegral
More informationFuzzy Laplace Transforms for Derivatives of Higher Orders
Maemaical Teory and Modeling ISSN -58 (Paper) ISSN 5-5 (Online) Vol, No, 1 wwwiiseorg Fuzzy Laplace Transforms for Derivaives of Higer Orders Absrac Amal K Haydar 1 *and Hawrra F Moammad Ali 1 College
More informationON DIFFERENTIABILITY OF ABSOLUTELY MONOTONE SET-VALUED FUNCTIONS
Folia Maemaica Vol. 16, No. 1, pp. 25 30 Aca Universiais Lodziensis c 2009 for Universiy of Lódź Press ON DIFFERENTIABILITY OF ABSOLUTELY MONOTONE SET-VALUED FUNCTIONS ANDRZEJ SMAJDOR Absrac. We prove
More informationMATH 507a ASSIGNMENT 4 SOLUTIONS FALL 2018 Prof. Alexander. g (x) dx = g(b) g(0) = g(b),
MATH 57a ASSIGNMENT 4 SOLUTIONS FALL 28 Prof. Alexader (2.3.8)(a) Le g(x) = x/( + x) for x. The g (x) = /( + x) 2 is decreasig, so for a, b, g(a + b) g(a) = a+b a g (x) dx b so g(a + b) g(a) + g(b). Sice
More informationThe Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales
Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions
More informationOutline. simplest HMM (1) simple HMMs? simplest HMM (2) Parameter estimation for discrete hidden Markov models
Oulie Parameer esimaio for discree idde Markov models Juko Murakami () ad Tomas Taylor (2). Vicoria Uiversiy of Welligo 2. Arizoa Sae Uiversiy Descripio of simple idde Markov models Maximum likeliood esimae
More informationSingle Platform Emitter Location
Sigle Plarm Emier Lcai AOADF FOA Ierermeery TOA SBI LBI Emier Lcai is Tw Esimai Prblems i Oe: Esimae Sigal Parameers a Deed Emier s Lcai: a Time--Arrival TOA Pulses b Pase Ierermeery: Pase is measured
More informationSome Properties of Semi-E-Convex Function and Semi-E-Convex Programming*
The Eighh Ieraioal Symposium o Operaios esearch ad Is Applicaios (ISOA 9) Zhagjiajie Chia Sepember 2 22 29 Copyrigh 29 OSC & APOC pp 33 39 Some Properies of Semi-E-Covex Fucio ad Semi-E-Covex Programmig*
More informationEXTENDED BÉZOUT IDENTITIES
_ XTNDD ÉZOUT IDNTITIS A Quadra Schl f Mahemaics Uiversiy f Leeds Leeds LS2 9T Uied Kigdm fax: 00 44 133 233 5145 e-mail: quadraamsaleedsacuk Keywrds: Mulidimesial sysems primeess exeded ézu ideiies -flaess
More informationOscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode,
More informationNeutron Slowing Down Distances and Times in Hydrogenous Materials. Erin Boyd May 10, 2005
Neu Slwig Dw Disaces ad Times i Hydgeus Maeials i Byd May 0 005 Oulie Backgud / Lecue Maeial Neu Slwig Dw quai Flux behavi i hydgeus medium Femi eame f calculaig slwig dw disaces ad imes. Bief deivai f
More information1 Notes on Little s Law (l = λw)
Copyrigh c 26 by Karl Sigma Noes o Lile s Law (l λw) We cosider here a famous ad very useful law i queueig heory called Lile s Law, also kow as l λw, which assers ha he ime average umber of cusomers i
More informationC o r p o r a t e l i f e i n A n c i e n t I n d i a e x p r e s s e d i t s e l f
C H A P T E R I G E N E S I S A N D GROWTH OF G U IL D S C o r p o r a t e l i f e i n A n c i e n t I n d i a e x p r e s s e d i t s e l f i n a v a r i e t y o f f o r m s - s o c i a l, r e l i g i
More informationS n. = n. Sum of first n terms of an A. P is
PROGREION I his secio we discuss hree impora series amely ) Arihmeic Progressio (A.P), ) Geomeric Progressio (G.P), ad 3) Harmoic Progressio (H.P) Which are very widely used i biological scieces ad humaiies.
More informationP a g e 3 6 of R e p o r t P B 4 / 0 9
P a g e 3 6 of R e p o r t P B 4 / 0 9 p r o t e c t h um a n h e a l t h a n d p r o p e r t y fr om t h e d a n g e rs i n h e r e n t i n m i n i n g o p e r a t i o n s s u c h a s a q u a r r y. J
More informationAbstract Interpretation and the Heap
Absrac Ierpreai ad he Heap Cmpuer Sciece ad Arificial Ielligece Labrary MIT Wih slides ad eamples by Mly Sagiv. Used wih permissi. Nv 18, 2015 Nvember 18, 2015 1 Recap: Cllecig Semaics Cmpue fr each prgram
More informationStochastic Reliability Analysis of Two Identical Cold Standby Units with Geometric Failure & Repair Rates
DOI: 0.545/mjis.07.500 Socasic Reliabiliy Analysis of Two Idenical Cold Sandby Unis wi Geomeric Failure & Repair Raes NITIN BHARDWAJ AND BHUPENDER PARASHAR Email: niinbardwaj@jssaen.ac.in; parasar_b@jssaen.ac.in
More informationA TAUBERIAN THEOREM FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY
U.P.B. Sci. Bull., Series A, Vol. 78, Iss. 2, 206 ISSN 223-7027 A TAUBERIAN THEOREM FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY İbrahim Çaak I his paper we obai a Tauberia codiio i erms of he weighed classical
More informationarxiv: v1 [math.nt] 13 Feb 2013
APOSTOL-EULER POLYNOMIALS ARISING FROM UMBRAL CALCULUS TAEKYUN KIM, TOUFIK MANSOUR, SEOG-HOON RIM, AND SANG-HUN LEE arxiv:130.3104v1 [mah.nt] 13 Feb 013 Absrac. In his paper, by using he orhogonaliy ype
More information7. Discrete Fourier Transform (DFT)
7 Discrete ourier Trasor (DT) 7 Deiitio ad soe properties Discrete ourier series ivolves to seueces o ubers aely the aliased coeiciets ĉ ad the saples (T) It relates the aliased coeiciets to the saples
More informationHigher Order Difference Schemes for Heat Equation
Available a p://pvau.edu/aa Appl. Appl. Ma. ISSN: 9-966 Vol., Issue (Deceber 009), pp. 6 7 (Previously, Vol., No. ) Applicaions and Applied Maeaics: An Inernaional Journal (AAM) Higer Order Difference
More informationBasic Results in Functional Analysis
Preared by: F.. ewis Udaed: Suday, Augus 7, 4 Basic Resuls i Fucioal Aalysis f ( ): X Y is coiuous o X if X, (, ) z f( z) f( ) f ( ): X Y is uiformly coiuous o X if i is coiuous ad ( ) does o deed o. f
More informationMoment Generating Function
1 Mome Geeraig Fucio m h mome m m m E[ ] x f ( x) dx m h ceral mome m m m E[( ) ] ( ) ( x ) f ( x) dx Mome Geeraig Fucio For a real, M () E[ e ] e k x k e p ( x ) discree x k e f ( x) dx coiuous Example
More informationBIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics
Biod Prasad Dhaal / BIBCHANA 9 (3 5-58 : BMHSS,.5 (Olie Publicaio: Nov., BIBCHANA A Mulidisciliary Joural of Sciece, Techology ad Mahemaics ISSN 9-76 (olie Joural homeage: h://ejol.ifo/idex.h/bibchana
More informationAn interesting result about subset sums. Nitu Kitchloo. Lior Pachter. November 27, Abstract
A ieresig resul abou subse sums Niu Kichloo Lior Pacher November 27, 1993 Absrac We cosider he problem of deermiig he umber of subses B f1; 2; : : :; g such ha P b2b b k mod, where k is a residue class
More informationAN EXTENSION OF A RESULT ABOUT THE ORDER OF CONVERGENCE
Bulleti o Mathematical Aalysis ad Applicatios ISSN: 8-9, URL: http://www.bmathaa.or Volume 3 Issue 3), Paes 5-34. AN EXTENSION OF A RESULT ABOUT THE ORDER OF CONVERGENCE COMMUNICATED BY HAJRUDIN FEJZIC)
More informationA Generalization of Hermite Polynomials
Ieraioal Mahemaical Forum, Vol. 8, 213, o. 15, 71-76 HIKARI Ld, www.m-hikari.com A Geeralizaio of Hermie Polyomials G. M. Habibullah Naioal College of Busiess Admiisraio & Ecoomics Gulberg-III, Lahore,
More informationChapter 1 Fundamental Concepts
Chaper 1 Fundamenal Conceps 1 Signals A signal is a paern of variaion of a physical quaniy, ofen as a funcion of ime (bu also space, disance, posiion, ec). These quaniies are usually he independen variables
More informationResearch Article On a Class of q-bernoulli, q-euler, and q-genocchi Polynomials
Absrac ad Applied Aalysis Volume 04, Aricle ID 696454, 0 pages hp://dx.doi.org/0.55/04/696454 Research Aricle O a Class of -Beroulli, -Euler, ad -Geocchi Polyomials N. I. Mahmudov ad M. Momezadeh Easer
More informationThe Quantum Theory of Atoms and Molecules: The Schrodinger equation. Hilary Term 2008 Dr Grant Ritchie
e Quanum eory of Aoms and Molecules: e Scrodinger equaion Hilary erm 008 Dr Gran Ricie An equaion for maer waves? De Broglie posulaed a every paricles as an associaed wave of waveleng: / p Wave naure of
More informationUndetermined coefficients for local fractional differential equations
Available online a www.isr-publicaions.com/jmcs J. Mah. Compuer Sci. 16 (2016), 140 146 Research Aricle Undeermined coefficiens for local fracional differenial equaions Roshdi Khalil a,, Mohammed Al Horani
More informationFermat Numbers in Multinomial Coefficients
1 3 47 6 3 11 Joural of Ieger Sequeces, Vol. 17 (014, Aricle 14.3. Ferma Numbers i Muliomial Coefficies Shae Cher Deparme of Mahemaics Zhejiag Uiversiy Hagzhou, 31007 Chia chexiaohag9@gmail.com Absrac
More informationModeling Micromixing Effects in a CSTR
delig irixig Effes i a STR STR, f all well behaved rears, has he wides RTD i.e. This eas ha large differees i perfrae a exis bewee segregaed flw ad perais a axiu ixedess diis. The easies hig rea is he
More informationApproximating the Powers with Large Exponents and Bases Close to Unit, and the Associated Sequence of Nested Limits
In. J. Conemp. Ma. Sciences Vol. 6 211 no. 43 2135-2145 Approximaing e Powers wi Large Exponens and Bases Close o Uni and e Associaed Sequence of Nesed Limis Vio Lampre Universiy of Ljubljana Slovenia
More informationIntroduction to Algorithms
Itroductio to Algorithms 6.046J/8.40J LECTURE 9 Radomly built biary search trees Epected ode depth Aalyzig height Coveity lemma Jese s iequality Epoetial height Post mortem Pro. Eri Demaie October 7, 2005
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 4 9/16/2013. Applications of the large deviation technique
MASSACHUSETTS ISTITUTE OF TECHOLOGY 6.265/5.070J Fall 203 Lecure 4 9/6/203 Applicaios of he large deviaio echique Coe.. Isurace problem 2. Queueig problem 3. Buffer overflow probabiliy Safey capial for
More informationDensity estimation. Density estimations. CS 2750 Machine Learning. Lecture 5. Milos Hauskrecht 5329 Sennott Square
Lecure 5 esy esmao Mlos Hauskrec mlos@cs..edu 539 Seo Square esy esmaos ocs: esy esmao: Mamum lkelood ML Bayesa arameer esmaes M Beroull dsrbuo. Bomal dsrbuo Mulomal dsrbuo Normal dsrbuo Eoeal famly Noaramerc
More informationLecture 3: Resistive forces, and Energy
Lecure 3: Resisive frces, and Energy Las ie we fund he velciy f a prjecile ving wih air resisance: g g vx ( ) = vx, e vy ( ) = + v + e One re inegrain gives us he psiin as a funcin f ie: dx dy g g = vx,
More informationMATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence
MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,
More informationû s L u t 0 s a ; i.e., û s 0
Te Hille-Yosida Teorem We ave seen a wen e absrac IVP is uniquely solvable en e soluion operaor defines a semigroup of bounded operaors. We ave no ye discussed e condiions under wic e IVP is uniquely solvable.
More informationEXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D. S. Palimkar
Ieraioal Joural of Scieific ad Research Publicaios, Volue 2, Issue 7, July 22 ISSN 225-353 EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D S Palikar Depare of Maheaics, Vasarao Naik College, Naded
More informationTHE GENERATION OF THE CURVED SPUR GEARS TOOTHING
5 INTERNATIONAL MEETING OF THE CARPATHIAN REGION SPECIALISTS IN THE FIELD OF GEARS THE GENERATION OF THE CURVED SPUR GEARS TOOTHING Boja Şefa, Sucală Felicia, Căilă Aurica, Tăaru Ovidiu Uiversiaea Teică
More informationSUMMATION OF INFINITE SERIES REVISITED
SUMMATION OF INFINITE SERIES REVISITED I several aricles over he las decade o his web page we have show how o sum cerai iiie series icludig he geomeric series. We wa here o eed his discussio o he geeral
More informationAdvection! Discontinuous! solutions shocks! Shock Speed! ! f. !t + U!f. ! t! x. dx dt = U; t = 0
p://www.d.edu/~gryggva/cfd-course/ Advecio Discoiuous soluios socks Gréar Tryggvaso Sprig Discoiuous Soluios Cosider e liear Advecio Equaio + U = Te aalyic soluio is obaied by caracerisics d d = U; d d
More informationTAKA KUSANO. laculty of Science Hrosh tlnlersty 1982) (n-l) + + Pn(t)x 0, (n-l) + + Pn(t)Y f(t,y), XR R are continuous functions.
Iera. J. Mah. & Mah. Si. Vol. 6 No. 3 (1983) 559-566 559 ASYMPTOTIC RELATIOHIPS BETWEEN TWO HIGHER ORDER ORDINARY DIFFERENTIAL EQUATIONS TAKA KUSANO laculy of Sciece Hrosh llersy 1982) ABSTRACT. Some asympoic
More informationSTK4080/9080 Survival and event history analysis
STK48/98 Survival ad eve hisory aalysis Marigales i discree ime Cosider a sochasic process The process M is a marigale if Lecure 3: Marigales ad oher sochasic processes i discree ime (recap) where (formally
More informationCalculus Limits. Limit of a function.. 1. One-Sided Limits...1. Infinite limits 2. Vertical Asymptotes...3. Calculating Limits Using the Limit Laws.
Limi of a fucio.. Oe-Sided..... Ifiie limis Verical Asympoes... Calculaig Usig he Limi Laws.5 The Squeeze Theorem.6 The Precise Defiiio of a Limi......7 Coiuiy.8 Iermediae Value Theorem..9 Refereces..
More informationln y t 2 t c where c is an arbitrary real constant
SOLUTION TO THE PROBLEM.A y y subjec o condiion y 0 8 We recognize is as a linear firs order differenial equaion wi consan coefficiens. Firs we sall find e general soluion, and en we sall find one a saisfies
More informationLecture 15 First Properties of the Brownian Motion
Lecure 15: Firs Properies 1 of 8 Course: Theory of Probabiliy II Term: Sprig 2015 Isrucor: Gorda Zikovic Lecure 15 Firs Properies of he Browia Moio This lecure deals wih some of he more immediae properies
More informationFIXED FUZZY POINT THEOREMS IN FUZZY METRIC SPACE
Mohia & Samaa, Vol. 1, No. II, December, 016, pp 34-49. ORIGINAL RESEARCH ARTICLE OPEN ACCESS FIED FUZZY POINT THEOREMS IN FUZZY METRIC SPACE 1 Mohia S. *, Samaa T. K. 1 Deparme of Mahemaics, Sudhir Memorial
More informationI M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o
I M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o u l d a l w a y s b e t a k e n, i n c l u d f o l
More informationMulti-objective Programming Approach for. Fuzzy Linear Programming Problems
Applied Mathematical Scieces Vl. 7 03. 37 8-87 HIKARI Ltd www.m-hikari.cm Multi-bective Prgrammig Apprach fr Fuzzy Liear Prgrammig Prblems P. Padia Departmet f Mathematics Schl f Advaced Scieces VIT Uiversity
More informationAN EXTENSION OF LUCAS THEOREM. Hong Hu and Zhi-Wei Sun. (Communicated by David E. Rohrlich)
Proc. Amer. Mah. Soc. 19(001, o. 1, 3471 3478. AN EXTENSION OF LUCAS THEOREM Hog Hu ad Zhi-Wei Su (Commuicaed by David E. Rohrlich Absrac. Le p be a prime. A famous heorem of Lucas saes ha p+s p+ ( s (mod
More informationExcursions of Max-Weight Dynamics
Excursios o Max-Weigh Dyamics Joh N. Tsisiklis (wih Arsala Shariassab ad Jamal Golesai, Shari U. Workshop o The Nex Wave i Neworkig Research i hoor o Jea Walrad Simos Isiue, Berkeley Sepember 27 Sepember
More informationChapter 3.1: Polynomial Functions
Ntes 3.1: Ply Fucs Chapter 3.1: Plymial Fuctis I Algebra I ad Algebra II, yu ecutered sme very famus plymial fuctis. I this secti, yu will meet may ther members f the plymial family, what sets them apart
More informationA new Type of Fuzzy Functions in Fuzzy Topological Spaces
IOSR Jurnal f Mathematics (IOSR-JM e-issn: 78-578, p-issn: 39-765X Vlume, Issue 5 Ver I (Sep - Oct06, PP 8-4 wwwisrjurnalsrg A new Type f Fuzzy Functins in Fuzzy Tplgical Spaces Assist Prf Dr Munir Abdul
More information1 Solutions to selected problems
1 Soluions o seleced problems 1. Le A B R n. Show ha in A in B bu in general bd A bd B. Soluion. Le x in A. Then here is ɛ > 0 such ha B ɛ (x) A B. This shows x in B. If A = [0, 1] and B = [0, 2], hen
More informationAn Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
More informationT Promotion. Residential. February 15 May 31 LUTRON. NEW for 2019
M NEW fr 2019 A e yer brigs fres skig ruiy fr Lur L reverse- dimmers sé sluis, iludig e rdus. Ple rder, e ll el drive sles rug i-sre merdisig rr smlig, el yu mee yur 2019 gls. Mesr L PRO dimmer Our s flexible
More informationFinish reading Chapter 2 of Spivak, rereading earlier sections as necessary. handout and fill in some missing details!
MAT 257, Handou 6: Ocober 7-2, 20. I. Assignmen. Finish reading Chaper 2 of Spiva, rereading earlier secions as necessary. handou and fill in some missing deails! II. Higher derivaives. Also, read his
More informationDynamic h-index: the Hirsch index in function of time
Dyamic h-idex: he Hirsch idex i fucio of ime by L. Egghe Uiversiei Hassel (UHassel), Campus Diepebeek, Agoralaa, B-3590 Diepebeek, Belgium ad Uiversiei Awerpe (UA), Campus Drie Eike, Uiversieisplei, B-260
More informationExtended Laguerre Polynomials
I J Coemp Mah Scieces, Vol 7, 1, o, 189 194 Exeded Laguerre Polyomials Ada Kha Naioal College of Busiess Admiisraio ad Ecoomics Gulberg-III, Lahore, Pakisa adakhaariq@gmailcom G M Habibullah Naioal College
More informationA Note on Random k-sat for Moderately Growing k
A Noe o Radom k-sat for Moderaely Growig k Ju Liu LMIB ad School of Mahemaics ad Sysems Sciece, Beihag Uiversiy, Beijig, 100191, P.R. Chia juliu@smss.buaa.edu.c Zogsheg Gao LMIB ad School of Mahemaics
More informationDensity estimation III.
Lecure 6 esy esmao III. Mlos Hausrec mlos@cs..eu 539 Seo Square Oule Oule: esy esmao: Bomal srbuo Mulomal srbuo ormal srbuo Eoeal famly aa: esy esmao {.. } a vecor of arbue values Objecve: ry o esmae e
More informationIntegrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number
MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios
More informationSampling Example. ( ) δ ( f 1) (1/2)cos(12πt), T 0 = 1
Samplig Example Le x = cos( 4π)cos( π). The fudameal frequecy of cos 4π fudameal frequecy of cos π is Hz. The ( f ) = ( / ) δ ( f 7) + δ ( f + 7) / δ ( f ) + δ ( f + ). ( f ) = ( / 4) δ ( f 8) + δ ( f
More informationApproximately Quasi Inner Generalized Dynamics on Modules. { } t t R
Joural of Scieces, Islamic epublic of Ira 23(3): 245-25 (22) Uiversiy of Tehra, ISSN 6-4 hp://jscieces.u.ac.ir Approximaely Quasi Ier Geeralized Dyamics o Modules M. Mosadeq, M. Hassai, ad A. Nikam Deparme
More information2007 Spring VLSI Design Mid-term Exam 2:20-4:20pm, 2007/05/11
7 ri VLI esi Mid-erm xam :-4:m, 7/5/11 efieτ R, where R ad deoe he chael resisace ad he ae caaciace of a ui MO ( W / L μm 1μm ), resecively., he chael resisace of a ui PMO, is wo R P imes R. i.e., R R.
More informationThe Central Limit Theorem
The Ceral Limi Theorem The ceral i heorem is oe of he mos impora heorems i probabiliy heory. While here a variey of forms of he ceral i heorem, he mos geeral form saes ha give a sufficiely large umber,
More informationDensity estimation III.
Lecure 4 esy esmao III. Mlos Hauskrec mlos@cs..edu 539 Seo Square Oule Oule: esy esmao: Mamum lkelood ML Bayesa arameer esmaes MP Beroull dsrbuo. Bomal dsrbuo Mulomal dsrbuo Normal dsrbuo Eoeal famly Eoeal
More informationTopic 9 - Taylor and MacLaurin Series
Topic 9 - Taylor ad MacLauri Series A. Taylors Theorem. The use o power series is very commo i uctioal aalysis i act may useul ad commoly used uctios ca be writte as a power series ad this remarkable result
More informationConditional Probability and Conditional Expectation
Hadou #8 for B902308 prig 2002 lecure dae: 3/06/2002 Codiioal Probabiliy ad Codiioal Epecaio uppose X ad Y are wo radom variables The codiioal probabiliy of Y y give X is } { }, { } { X P X y Y P X y Y
More informationDistribution of Mass and Energy in Five General Cosmic Models
Inerninl Jurnl f Asrnmy nd Asrpysics 05 5 0-7 Publised Online Mrc 05 in SciRes p://wwwscirprg/jurnl/ij p://dxdirg/0436/ij055004 Disribuin f Mss nd Energy in Five Generl Csmic Mdels Fdel A Bukri Deprmen
More informationTAYLOR AND MACLAURIN SERIES
Calculus TAYLOR AND MACLAURIN SERIES Give a uctio ( ad a poit a, we wish to approimate ( i the eighborhood o a by a polyomial o degree. c c ( a c( a c( a P ( c ( a We have coeiciets to choose. We require
More informationECE 340 Lecture 15 and 16: Diffusion of Carriers Class Outline:
ECE 340 Lecure 5 ad 6: iffusio of Carriers Class Oulie: iffusio rocesses iffusio ad rif of Carriers Thigs you should kow whe you leave Key Quesios Why do carriers diffuse? Wha haes whe we add a elecric
More informationSome Newton s Type Inequalities for Geometrically Relative Convex Functions ABSTRACT. 1. Introduction
Malaysia Joural of Mahemaical Scieces 9(): 49-5 (5) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Joural homepage: hp://eispem.upm.edu.my/joural Some Newo s Type Ieualiies for Geomerically Relaive Covex Fucios
More informationA SHORT INTRODUCTION TO BANACH LATTICES AND
CHAPTER A SHORT INTRODUCTION TO BANACH LATTICES AND POSITIVE OPERATORS In tis capter we give a brief introduction to Banac lattices and positive operators. Most results of tis capter can be found, e.g.,
More informationFresnel Dragging Explained
Fresel Draggig Explaied 07/05/008 Decla Traill Decla@espace.e.au The Fresel Draggig Coefficie required o explai he resul of he Fizeau experime ca be easily explaied by usig he priciples of Eergy Field
More informationF (u) du. or f(t) = t
8.3 Topic 9: Impulses and dela funcions. Auor: Jeremy Orloff Reading: EP 4.6 SN CG.3-4 pp.2-5. Warmup discussion abou inpu Consider e rae equaion d + k = f(). To be specific, assume is in unis of d kilograms.
More informationAdditional Exercises for Chapter What is the slope-intercept form of the equation of the line given by 3x + 5y + 2 = 0?
ddiional Eercises for Caper 5 bou Lines, Slopes, and Tangen Lines 39. Find an equaion for e line roug e wo poins (, 7) and (5, ). 4. Wa is e slope-inercep form of e equaion of e line given by 3 + 5y +
More informationChapter 6 - Work and Energy
Caper 6 - Work ad Eergy Rosedo Pysics 1-B Eploraory Aciviy Usig your book or e iere aswer e ollowig quesios: How is work doe? Deie work, joule, eergy, poeial ad kieic eergy. How does e work doe o a objec
More informationInstitute for Mathematical Methods in Economics. University of Technology Vienna. Singapore, May Manfred Deistler
MULTIVARIATE TIME SERIES ANALYSIS AND FORECASTING Manfred Deisler E O S Economerics and Sysems Theory Insiue for Mahemaical Mehods in Economics Universiy of Technology Vienna Singapore, May 2004 Inroducion
More informationVISCOSITY APPROXIMATION TO COMMON FIXED POINTS OF kn- LIPSCHITZIAN NONEXPANSIVE MAPPINGS IN BANACH SPACES
Joral o Maheaical Scieces: Advaces ad Alicaios Vole Nber 9 Pages -35 VISCOSIY APPROXIMAION O COMMON FIXED POINS OF - LIPSCHIZIAN NONEXPANSIVE MAPPINGS IN BANACH SPACES HONGLIANG ZUO ad MIN YANG Deare o
More informationDistribution of Mass and Energy in Closed Model of the Universe
Ieraial Jural f Asrmy ad Asrysics, 05, 5, 9-0 Publised Olie December 05 i SciRes ://wwwscirrg/jural/ijaa ://dxdirg/046/ijaa05540 Disribui f Mass ad Eergy i Clsed Mdel f e Uiverse Fadel A Bukari Dearme
More informationTime discretization of quadratic and superquadratic Markovian BSDEs with unbounded terminal conditions
Time discreizaion of quadraic and superquadraic Markovian BSDEs wih unbounded erminal condiions Adrien Richou Universié Bordeaux 1, INRIA équipe ALEA Oxford framework Le (Ω, F, P) be a probabiliy space,
More informationth m m m m central moment : E[( X X) ] ( X X) ( x X) f ( x)
1 Trasform Techiques h m m m m mome : E[ ] x f ( x) dx h m m m m ceral mome : E[( ) ] ( ) ( x) f ( x) dx A coveie wa of fidig he momes of a radom variable is he mome geeraig fucio (MGF). Oher rasform echiques
More informationThe Components of Vector B. The Components of Vector B. Vector Components. Component Method of Vector Addition. Vector Components
Upcming eens in PY05 Due ASAP: PY05 prees n WebCT. Submiing i ges yu pin ward yur 5-pin Lecure grade. Please ake i seriusly, bu wha cuns is wheher r n yu submi i, n wheher yu ge hings righ r wrng. Due
More informationP a g e 5 1 of R e p o r t P B 4 / 0 9
P a g e 5 1 of R e p o r t P B 4 / 0 9 J A R T a l s o c o n c l u d e d t h a t a l t h o u g h t h e i n t e n t o f N e l s o n s r e h a b i l i t a t i o n p l a n i s t o e n h a n c e c o n n e
More informationResearch Article A Generalized Nonlinear Sum-Difference Inequality of Product Form
Joural of Applied Mahemaics Volume 03, Aricle ID 47585, 7 pages hp://dx.doi.org/0.55/03/47585 Research Aricle A Geeralized Noliear Sum-Differece Iequaliy of Produc Form YogZhou Qi ad Wu-Sheg Wag School
More informationSolution. The straightforward approach is surprisingly difficult because one has to be careful about the limits.
ose ad Varably Homewor # (8), aswers Q: Power spera of some smple oses A Posso ose A Posso ose () s a sequee of dela-fuo pulses, eah ourrg depedely, a some rae r (More formally, s a sum of pulses of wdh
More informationCalculus I Homework: The Derivative as a Function Page 1
Calculus I Homework: Te Derivative as a Function Page 1 Example (2.9.16) Make a careful sketc of te grap of f(x) = sin x and below it sketc te grap of f (x). Try to guess te formula of f (x) from its grap.
More information176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s
A g la di ou s F. L. 462 E l ec tr on ic D ev el op me nt A i ng er A.W.S. 371 C. A. M. A l ex an de r 236 A d mi ni st ra ti on R. H. (M rs ) A n dr ew s P. V. 326 O p ti ca l Tr an sm is si on A p ps
More informationExtremal graph theory II: K t and K t,t
Exremal graph heory II: K ad K, Lecure Graph Theory 06 EPFL Frak de Zeeuw I his lecure, we geeralize he wo mai heorems from he las lecure, from riagles K 3 o complee graphs K, ad from squares K, o complee
More informationInternational Journal of Mathematical Archive-3(1), 2012, Page: Available online through
eril Jrl f Mhemicl rchive-3 Pge: 33-39 vilble lie hrgh wwwijmif NTE N UNFRM MTRX SUMMBLTY Shym Ll Mrdl Veer Sigh d Srbh Prwl 3* Deprme f Mhemics Fcly f Sciece Brs Hid Uiversiy Vrsi UP - ND E-mil: shym_ll@rediffmilcm
More informationCopyright Paul Tobin 63
DT, Kevin t. lectric Circuit Thery DT87/ Tw-Prt netwrk parameters ummary We have seen previusly that a tw-prt netwrk has a pair f input terminals and a pair f utput terminals figure. These circuits were
More informationOrientation. Connections between network coding and stochastic network theory. Outline. Bruce Hajek. Multicast with lost packets
Connecions beween nework coding and sochasic nework heory Bruce Hajek Orienaion On Thursday, Ralf Koeer discussed nework coding: coding wihin he nework Absrac: Randomly generaed coded informaion blocks
More information