Data Dependence of New Iterative Schemes

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "Data Dependence of New Iterative Schemes"

Transcript

1 Mathematics Volume : 4 Issue : 6 Jue 4 ISSN X Data Depedece of New Iterative Schemes KEYWORDS CR Iteratio Data Depedece New Multistep Iteratio Quasi Cotractive * Aarti Kadia Assistat Professor Departmet of mathematics Deshbadhu College Delhi Uiversity New Delhi-9 * Correspodig Author ABSTRACT operators I this paper we prove data depedece of ew multistep iterative scheme for quasi cotractive operators that is by usig a approximate quasi -cotractive operator we approximate the fixed poit of the give INTRODUCTION Fixed poit theory is ivolved i various types of issues such as to fid the fixed poit existece ad uiqueess of fixed poit etc Data depedece of fixed poit is oe of these issues ad has become a subject of research iterest from some time ow The data depedece of various iterative schemes has bee studied by various authors like Rus ad Muresa [8] Rus et al [6 7] Beride[4] Espiola ad Petrusel [7] Marki [] Chifu ad Petrusel [4] Olatiwo [] Soltuz [9 ] Soltuz ad Grosa [] Chugh ad Kumar [5] Gursoy et al[5] ad several refereces thereof Our mai iterest i this paper is to show data depedece of ew multi step [] ad CR [6] iterative schemes usig quasi cotractive operators For the backgroud of our expositio we first metio some cotractive mappigs Zemfirescu [] established a ice geeralizatio of the Baach fixed poit theorem as follows: Let (X d be a complete metric space ad T: X X mappigs for which there exists real umbers αβγsatisfyig α < β < γ < respectively such that for each xy E at least oe of the followig is true: ( z d Ty αd( x y ( z d Ty β[ d( x Tx + d( y Ty] ( z d Ty γ[ d( x Ty + d( y Tx] 3 ( The the mappig T satisfyig ( is called Zamfirescu cotractio Remark Mappig which satisfy (z is called a Kaa mappig while the mappig satisfyig (z 3 is called Chatterjea operator The cotractive coditio ( implies Tx Ty δ x Tx + δ x y ad Tx Ty δ x Ty + δ x y β γ where δ max { α δ < β γ for all xy E Osilike ad Udomee [4] defied a ew geeral defiitio of quasi cotractive operator as follows: Tx Ty δ x y + L x Tx L δ [] (3 xy E ad some After that a more geeral defiitio was itroduced by Imoru ad Olatiwo [3] as follows: if there exists a costat δ < ad a mootoically icreasig ad cotiuous fuctio :[ [ such that for xy E Tx Ty δ x y + ϕ( x Tx ϕ with ϕ ( (4 Now we recapitulate some iterative schemes i the literature of fixed poit theory Let X be a Baach space ad E be a closed covex subset of X If T: X X a mappigs o E the F { p X : Tp p deotes the set of fixed poits of T For x { x is defied as x α x αty ( + + y ( β x + β Tx T E Ishikawa Iteratio [8] where { α ad { are real sequeces i [ ] satisfyig β α Observe that if β for each the the Ishikawa iteratio process (5 reduces to the Ma iteratio scheme For x E the Noor three step iterative scheme [3] { x is defied as x ( α x + + Ty y ( β x + Tz z ( γ x + Tx where { α { β ad { γ are real sequeces i [ ] satisfyig α If γ the Noor iteratio process (6 reduces to Ishikawa Iteratio scheme (5 I 7 Agarwal et al defied the Agarwal et al iterative scheme [] as x ( - α Tx + + α Ty y ( - β x + β Tx (7 INDIAN JOURNAL OF APPLIED RESEARCH X 335

2 where { α ad { β are sequeces of positive umbers i [ ] with α Quite recetly Phuegrattaa ad Suatai [5] itroduced SP iterative scheme as x ( α y + + αty y ( β z + β Tz z ( γ x + γ Tx (8 where { α { β ad { γ are sequeces of positive umbers i [ ] satisfyig α We shall use the followig iterative schemes: (a New multi step iterative scheme x+ ( α y + αty y ( β Tx + βty i k p k k y ( β x + β Tx (9 ad u+ ( α v + αtv v ( β Tu + βtv i k p k k v ( β u + β Tu where { α i { β i k k be the real sequeces of positive umbers i [ ] satisfyig α (b CR iterative scheme x E x+ ( α y + αty y ( β Tx + βtz z ( γ x + γtx ad u+ ( α v αtv + v ( β Tu + β Tw w ( γ u + γtu ( where{ α { β ad { γ are sequeces of positive umbers i [ ] satisfyig α due to Chugh ad Kumar [6] Now to prove our mai results we eed followig results i sequel Defiitio 3 [3] Let T T be two operators We say T is approximate operator of T if for all x X ad for a fixed TxTx ε ε > we have Volume : 4 Issue : 6 Jue 4 ISSN X Lemma 4[] Let { be a oegative sequece for α which oe suppose there exists it satisfies the followig iequality: α ( λ α + λσ where λ ( N λ The lim supα lim sup σ I such that for all ad σ N Theorem 5 [7]: Let T: E E be a mappig satisfyig (4 ad { be defied by ( with real x sequece { α { β ad { γ i [ satisfyig α The the sequece { x coverges to uique fixed poit of T Theorem 6[9]: Let T: E E be a mappig satisfyig (4 ad { be defied by (9 with real x sequece { α i { β i k i [ satisfyig α The the sequece { x coverges to uique fixed poit of T Mai Results Theorem Let T : E E be a mappig satisfyig (4 Let T be a approximate operator of T as i Defiitioad{ x { u be two CR iterative schemes defied by ( associated to T ad T respectively where { α { β ad { γ are real ( i α( δ sequeces i [ satisfyig Let ( ii α p Tp ad q Tq the we have the followig estimate: 4ε pq δ Proof: Usig (4 ad ( we have the followig estimates: 336 X INDIAN JOURNAL OF APPLIED RESEARCH

3 x u ( α ( y v + α ( Ty Tv + + ( α y v + α Ty Tv ( α y v + α Ty Tv + Tv Tv { { ( ( ( α y v + α Ty Tv + Tv Tv ( α y v + α δ y v + ϕ T y y + ε ( α ( δ y v + αϕ Ty y + αε y v ( β Tu + β Tw ( β Tx Tu + β Tz Tw ( ( β Tx Tu + Tu Tu + β Tz Tw + Tw T w ( β { Tx Tu + Tu Tu + β{ Tz Tw + Tw Tw ( β { δ x u + ϕ x + ε + β{ δ z w + ϕ z + ε ( β δ x u + ( β ϕ x + ( β ε+ βδ z w + βϕ z + βε ( ad ( γ ( z w ( γ x u + Tx Tu ( γ x u + γ Tx Tu { ( ( ( γ x u + γ δ x u + ϕ Tx x + ε ( γ ( δ x u + γϕ Tx x + γε (3 Combiig ( ( ad (3 we have x ( β δ x u u ( α ( δ + ( β ϕ + ( β ε x α δ { βδ z w βϕ z βε ( Ty y αε ( ( αϕ + ( α ( δ( β δ x u + ( α ( δ β δ z w x z ( Ty y αε + ( α ( δ( β ϕ + ( α ( δ( β ε + ( α ( δ βϕ + ( α ( δ βε + αϕ + Volume : 4 Issue : 6 Jue 4 ISSN X ( α( δ( β δ x u + ( α ( δ βδ ( γ ( δ + γϕ + γε { x u x x z ( Ty y + ( α( δ( β ϕ + ( α( δ( β ε + ( α ( δ βϕ + ( α ( δ βε+ αϕ + αε {( α( δ( β( δ( γ( δ x u δ( α( δ βγϕ x ( α( δ βδγ ε ( α( δ( β ϕ x ( α( δ( β ε z ( Ty y ( ( α ( δ βϕ + ( α ( δ βε+ αϕ + αε It may be oted that for { α { β { [ ad δ < the followig γ iequalities hold: ( α δ < ( α ( β( δ( γ( δ < αβδ < α It follows from assumptio (i that ( α( δ < α( δ α I Now usig (5 ad (6 i (4 we get ( α ( δ αϕ( z ( Ty y x u x u + Tx x αε+ αϕ + αϕ which further implies x u ( α ( δ x u + + (5 (6 x + + z + ( Ty y {ϕ 4 ε ϕ ϕ + α( δ ( δ (7 Let us deote a x u r ( α δ ad x + + z + ( Ty y {ϕ 4 ε ϕ ϕ σ ( δ INDIAN JOURNAL OF APPLIED RESEARCH X 337

4 Now from Theorem we have lim x p Also T satisfies coditio (4 ad Tp p F hece lim x Tx lim y Ty lim z Tz T Volume : 4 Issue : 6 Jue 4 ISSN X u+ ( α v + αtv v Tu T v i k p p p v ( β u + β Tu ( β + β Because{ x { y { z poit of T coverges to fixed lim ϕ( x Tx lim ϕ( y Ty lim ϕ( z Tz Sice ϕ is cotiuous hece usig Lemma (7 yields 4ε pq ( δ Theorem Let T : E E be a mappig satisfyig (4 Let T be a approximate operator of T as i Defiitio ad { x { u be two iterative schemes defied by (9 associated to T ad T respectively where { α ad { β i i k are real sequeces i[ satisfyig i ( i o β α < i k Let p Tp ad q Tq ( ii α the we have the followig estimate kε pq δ Proof: For a give x E ad u E we cosider the followig iterative schemes for T ad T I this paper we use followig iterative scheme x+ ( α y + αty y ( β Tx + βty i k p k k y ( β x + β Tx (9 the usig (4 (8 ad (9 yield the followig estimates: x u ( α ( y v + α ( Ty Tv + + ( α y v + α Ty Tv ( α y v + α Ty Tv + Tv Tv ( α y v + α Ty Tv + α Tv Tv ( α ( δ y v + αϕ( y Ty + αε y v ( β Tu + β ( Ty Tv ( ( β Tx Tu + Tu Tu + β Ty Tv + Tv Tv { Tx Tu Tu Tu β { Ty Tv Tv Tv { x u x { y v ( Ty y ε ( β ( β δ + ϕ + ε + β δ + ϕ + ( β δ + β ϕ ( x u ( Tx x ( + ( β ε+ βδ y v + βϕ Ty y + βε ad ( y v ( β ( x u + β ( Ty Tv 3 3 ( β δ x u + βδ y v + βϕ( y Ty x + βε+ ( β ϕ + ( β ε Combiig ( ( ad ( we have ( x u ( α ( δ y v + αϕ( y Ty + αε + + α δ β δ x u + α δ β δ β x u ( ( ( ( ( ( + ( α ( δ βδ β y v + ( α ( δ β δβ ϕ( y Ty ( α( δ βδβε + ( α( δ βδ ( β ϕ x ( Ty y x + α δ β δ β ε + α δ β ϕ ( ( ( ( ( ( + ( α ( δ( β ε + ( α ( δ β ϕ + ( α ( δ βε+ αϕ( y Ty + αε Thus iductively we get (3 ad (8 338 X INDIAN JOURNAL OF APPLIED RESEARCH

5 ( β δ x+ u+ ( α( δ x k 3 k 3 u + βδ ( β + + δ ββ ( β + ( α( δ δ ββ β y v k k k k 3 + [ αϕ ( y Ty + ( α( δ βϕ ( Ty y + ( α( δ( β ϕ x + ( α( δ βδ ( β ϕ x + ( α ( δ β δβ ϕ( y Ty ] + [ αε 3 3 ( ( ( ( ( + α δ β ε + α δ β ε ( α( δ βδ ( β ε ( α( δ βδβε ] (4 Usig (4 ad (9 y v ( β ( x u + β Tu k k k k ( β x u + β Tx Tu k k ( β ( δ x u + β ϕ( x Tx + β ε k k k (5 Now by combiig (4 ad (5 x u x u ( β δ + + ( α( δ k 3 k 3 + βδ ( β + + δ ββ ( β x x ( + [ αϕ( y Ty + ( α ( δ βϕ Ty y ( α( δ( β ϕ + ( α( δ βδ ( β ϕ ( α ( δ β δβ ϕ( y Ty k ( ( ( x Tx ] [ + α δ β δβ β ϕ + αε ( ( ( ( ( + α δ β ε + α δ β ε ( ( ( + α δ βδ β ε k ( α ( δ βδβε ( α ( δ βδβ β ε ] Volume : 4 Issue : 6 Jue 4 ISSN X x u [ ( α ( δ] x u + αϕ( y Ty + αϕ( y Ty αϕ ( y Ty + + αϕ ( x Tx + kε [ ( α ( δ] x u ϕ ( y Ty + ϕ( y Ty + + αϕ ( x Tx + kε + α( δ ( δ (9 Let us deote a x u r α ( δ ad { ϕ( y Ty + ϕ( y Ty + + ϕ( x Tx + kε σ : ( δ Now from Theorem we have lim x p Also T satisfies coditio (4 ad Tp p F usig the similar argumet as i Theorem 3 we get lim x Tx lim y Ty lim y Ty lim y Ty k k Sice ϕ is cotiuous we have lim ϕ( x Tx lim ϕ( y Ty lim ϕ( y Ty lim ϕ( y Ty k k Hece usig Lemma (9 yields kε pq δ T which further implies x u α δ β δ β δ x u k + + ( ( ( ( ( ( + αϕ( y Ty + δα β ϕ( y Ty + δαββϕ( y Ty + + δ α β β ϕ( x Tx k k + αε+ δαβε+ + δ αβ β ε p k ( α δ < ( α < k ( β( δ( β ( δ k k δ αβ β < α It follows from assumptio (i that ( α( δ < α( δ α I (6 (7 (8 Hece usig (7 ad (8 i (6 we get INDIAN JOURNAL OF APPLIED RESEARCH X 339

6 Volume : 4 Issue : 6 Jue 4 ISSN X REFERENCE [] Agarwal RP O Rega D ad Sahu DR: Iterative costructio of fixed poits of early asymptotically oexpasive mappigs Joural of Noliear ad Covex Aalysis 8( ( [] BE Rhoades SM Soltuz The equivalece betwee Ma-Ishikawa iteratios ad multi-step iteratio Noliear Aalysis 58(4 9-8 [3] CO Imoru MO Olatiwo O the stability of Picard ad Ma iteratio processes Carpathia Joural of Mathematics 9(3 o 55-6 [4] Chifu G Petru sel Existece ad Data Depedece of Fixed Poits ad Strict Fixed Poits for Cotractive- Type Multivalued Operators Fixed Poit Theory ad Applicatios 7(7 Article ID pages [5] F Gursoy et al Data depedece results of ew multi-step ad s-iterative schemes for cotractive-like operators Fixed Poit Theory ad Applicatio 86/ [6] IA Rus A Petru sel A Sˆıtamaria Data depedece of the fixed poits set of multivalued weakly Picard operators Stud Uiv Babes-Bolyai Math 46 ( ( - [7] IA Rus A Petru sel A Sˆıtamaria Data depedece of the fixed poit set of some multivalued weakly Picard operators Noliear Aalysis: Theory Methods & Applicatios 5 ( [8] IA Rus S Muresa Data depedece of the fixed poits set of weakly Picard operators Stud Uiv Babes-Bolyai 43 ( [9] JO Olaleru H Akewe O multistep iterative scheme for approximatig the commo fixed poits of cotractive-like operators It Joural of Mathematics ad mathematical scieces Volume Article ID pages [] JT Marki Cotiuous depedece of fixed poit sets Proc AMS 38 ( [] M O Olatiwo O the cotiuous depedece of the fixed poits for ( -cotractive-type operators Kragujevac Joural of Mathematics 34( 9- [] M O Olatiwo Some results o the cotiuous depedece of the fixed poits i ormed liear space Fixed Poit Theory (9 o 5-57 [3] MA Noor New approximatio schemes for geeral variatioal iequalities Joural of Mathematical Aalysis ad Applicatios 5( o 7-9 [4] MO Osilike A Udomee Short proofs of stability results for fixed poit iteratio procedures for a class of cotractive-type mappigs Idia Joural of Pure ad Applied Mathematics 3(999 o 9-34 [5] R Chugh V Kumar Data depedece of Noor ad SP iterative schemes whe dealig with quasi-cotractive operators Iteratioal Joural of Computer Applicatios 3( o5 [6] R Chugh V Kumar S Kumar Strog covergece of a ew three step iterative scheme i Baach spaces America Joural of Computatioal Mathematics ( [7] R Esp ıola A Petru sel Existece ad data depedece of fixed poits for multivalued operators o gauge spaces J Math Aal Appl 39 ( [8] S Ishikawa Fixed poits by a ew iteratio method Proc Amer Math Soc 44( [9] SM Soltuz Data depedece for Ishikawa iteratio Lecturas Mathematicas 5(4 o [] SM Soltuz Data depedece for Ma iteratio Octogo Math Magazie 9( [] SM Soltuz T Grosa Data depedece for Ishikawa iteratio whe dealig with cotractive like operators Fixed Poit Theory ad Applicatios 8(8 Article ID pages [] T Zamfirescu Fixed poit theorems i metric spaces Archiv der Mathematik 3(97 o 9-98 [3] V Beride Iterative Approximatio of Fixed Poits Spriger Berli (7 [4] V Beride O the approximatio of fixed poits of weak cotractive mappigs Carpathia J Math 9(3 o 7- [5] W Phuegrattaa S Suatai O the rate of covergece of Ma Ishikawa Noor ad SP iteratios for cotiuous fuctios o a arbitrary iterval Joural of Computatioal ad Applied Mathematics 35( [6] W Takahashi Iterative methods for approximatio of fixed poits ad their applicatios Joural of the Operatios Research Society of Japa 43( o 87-8 [7] WR Ma Mea value methods i iteratios Proc Amer Math Soc 4( [8] Xu MA Noor Ishikawa ad Ma iteratio process with errors for oliear strogly accretive operator equatios J Math Aal Appl 4( X INDIAN JOURNAL OF APPLIED RESEARCH

COMMON RANDOM FIXED POINT THEOREMS IN SYMMETRIC SPACES

COMMON RANDOM FIXED POINT THEOREMS IN SYMMETRIC SPACES Iteratioal Joural of Avacemets i Research & Techology, Volume, Issue, Jauary-03 ISSN 78-7763 COMMON RANDOM FIXED POINT THEOREMS IN SYMMETRIC SPACES Dr Neetu Vishwakarma a Dr M S Chauha Sagar Istitute of

Διαβάστε περισσότερα

On Inclusion Relation of Absolute Summability

On Inclusion Relation of Absolute Summability It. J. Cotemp. Math. Scieces, Vol. 5, 2010, o. 53, 2641-2646 O Iclusio Relatio of Absolute Summability Aradhaa Dutt Jauhari A/66 Suresh Sharma Nagar Bareilly UP) Idia-243006 aditya jauhari@rediffmail.com

Διαβάστε περισσότερα

1. For each of the following power series, find the interval of convergence and the radius of convergence:

1. For each of the following power series, find the interval of convergence and the radius of convergence: Math 6 Practice Problems Solutios Power Series ad Taylor Series 1. For each of the followig power series, fid the iterval of covergece ad the radius of covergece: (a ( 1 x Notice that = ( 1 +1 ( x +1.

Διαβάστε περισσότερα

Research Article Finite-Step Relaxed Hybrid Steepest-Descent Methods for Variational Inequalities

Research Article Finite-Step Relaxed Hybrid Steepest-Descent Methods for Variational Inequalities Hidawi Publishig Corporatio Joural of Iequalities ad Applicatios Volume 2008, Article ID 598632, 13 pages doi:10.1155/2008/598632 Research Article Fiite-Step Relaxed Hybrid Steepest-Descet Methods for

Διαβάστε περισσότερα

On Generating Relations of Some Triple. Hypergeometric Functions

On Generating Relations of Some Triple. Hypergeometric Functions It. Joural of Math. Aalysis, Vol. 5,, o., 5 - O Geeratig Relatios of Some Triple Hypergeometric Fuctios Fadhle B. F. Mohse ad Gamal A. Qashash Departmet of Mathematics, Faculty of Educatio Zigibar Ade

Διαβάστε περισσότερα

Introduction of Numerical Analysis #03 TAGAMI, Daisuke (IMI, Kyushu University)

Introduction of Numerical Analysis #03 TAGAMI, Daisuke (IMI, Kyushu University) Itroductio of Numerical Aalysis #03 TAGAMI, Daisuke (IMI, Kyushu Uiversity) web page of the lecture: http://www2.imi.kyushu-u.ac.jp/~tagami/lec/ Strategy of Numerical Simulatios Pheomea Error modelize

Διαβάστε περισσότερα

On Certain Subclass of λ-bazilevič Functions of Type α + iµ

On Certain Subclass of λ-bazilevič Functions of Type α + iµ Tamsui Oxford Joural of Mathematical Scieces 23(2 (27 141-153 Aletheia Uiversity O Certai Subclass of λ-bailevič Fuctios of Type α + iµ Zhi-Gag Wag, Chu-Yi Gao, ad Shao-Mou Yua College of Mathematics ad

Διαβάστε περισσότερα

A study on generalized absolute summability factors for a triangular matrix

A study on generalized absolute summability factors for a triangular matrix Proceedigs of the Estoia Acadey of Scieces, 20, 60, 2, 5 20 doi: 0.376/proc.20.2.06 Available olie at www.eap.ee/proceedigs A study o geeralized absolute suability factors for a triagular atrix Ere Savaş

Διαβάστε περισσότερα

Στα επόμενα θεωρούμε ότι όλα συμβαίνουν σε ένα χώρο πιθανότητας ( Ω,,P) Modes of convergence: Οι τρόποι σύγκλισης μιας ακολουθίας τ.μ.

Στα επόμενα θεωρούμε ότι όλα συμβαίνουν σε ένα χώρο πιθανότητας ( Ω,,P) Modes of convergence: Οι τρόποι σύγκλισης μιας ακολουθίας τ.μ. Στα πόμνα θωρούμ ότι όλα συμβαίνουν σ ένα χώρο πιθανότητας ( Ω,,). Modes of covergece: Οι τρόποι σύγκλισης μιας ακολουθίας τ.μ. { } ίναι οι ξής: σ μια τ.μ.. Ισχυρή σύγκλιση strog covergece { } lim = =.

Διαβάστε περισσότερα

Homework for 1/27 Due 2/5

Homework for 1/27 Due 2/5 Name: ID: Homework for /7 Due /5. [ 8-3] I Example D of Sectio 8.4, the pdf of the populatio distributio is + αx x f(x α) =, α, otherwise ad the method of momets estimate was foud to be ˆα = 3X (where

Διαβάστε περισσότερα

SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018

SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018 Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals

Διαβάστε περισσότερα

Μια εισαγωγή στα Μαθηματικά για Οικονομολόγους

Μια εισαγωγή στα Μαθηματικά για Οικονομολόγους Μια εισαγωγή στα Μαθηματικά για Οικονομολόγους Μαθηματικά Ικανές και αναγκαίες συνθήκες Έστω δυο προτάσεις Α και Β «Α είναι αναγκαία συνθήκη για την Β» «Α είναι ικανή συνθήκη για την Β» Α is ecessary for

Διαβάστε περισσότερα

SUPPLEMENT TO ROBUSTNESS, INFINITESIMAL NEIGHBORHOODS, AND MOMENT RESTRICTIONS (Econometrica, Vol. 81, No. 3, May 2013, )

SUPPLEMENT TO ROBUSTNESS, INFINITESIMAL NEIGHBORHOODS, AND MOMENT RESTRICTIONS (Econometrica, Vol. 81, No. 3, May 2013, ) Ecoometrica Supplemetary Material SUPPLEMENT TO ROBUSTNESS, INFINITESIMAL NEIGHBORHOODS, AND MOMENT RESTRICTIONS (Ecoometrica, Vol. 81, No. 3, May 213, 1185 121) BY YUICHI KITAMURA,TAISUKE OTSU, ANDKIRILL

Διαβάστε περισσότερα

A New Class of Analytic p-valent Functions with Negative Coefficients and Fractional Calculus Operators

A New Class of Analytic p-valent Functions with Negative Coefficients and Fractional Calculus Operators Tamsui Oxford Joural of Mathematical Scieces 20(2) (2004) 175-186 Aletheia Uiversity A New Class of Aalytic -Valet Fuctios with Negative Coefficiets ad Fractioal Calculus Oerators S. P. Goyal Deartmet

Διαβάστε περισσότερα

L.K.Gupta (Mathematic Classes) www.pioeermathematics.com MOBILE: 985577, 4677 + {JEE Mai 04} Sept 0 Name: Batch (Day) Phoe No. IT IS NOT ENOUGH TO HAVE A GOOD MIND, THE MAIN THING IS TO USE IT WELL Marks:

Διαβάστε περισσότερα

Homomorphism in Intuitionistic Fuzzy Automata

Homomorphism in Intuitionistic Fuzzy Automata International Journal of Fuzzy Mathematics Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 39-45 Research India Publications http://www.ripublication.com/ijfms.htm Homomorphism in Intuitionistic

Διαβάστε περισσότερα

2 Composition. Invertible Mappings

2 Composition. Invertible Mappings Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,

Διαβάστε περισσότερα

Binet Type Formula For The Sequence of Tetranacci Numbers by Alternate Methods

Binet Type Formula For The Sequence of Tetranacci Numbers by Alternate Methods DOI: 545/mjis764 Biet Type Formula For The Sequece of Tetraacci Numbers by Alterate Methods GAUTAMS HATHIWALA AND DEVBHADRA V SHAH CK Pithawala College of Eigeerig & Techology, Surat Departmet of Mathematics,

Διαβάστε περισσότερα

LAD Estimation for Time Series Models With Finite and Infinite Variance

LAD Estimation for Time Series Models With Finite and Infinite Variance LAD Estimatio for Time Series Moels With Fiite a Ifiite Variace Richar A. Davis Colorao State Uiversity William Dusmuir Uiversity of New South Wales 1 LAD Estimatio for ARMA Moels fiite variace ifiite

Διαβάστε περισσότερα

n r f ( n-r ) () x g () r () x (1.1) = Σ g() x = Σ n f < -n+ r> g () r -n + r dx r dx n + ( -n,m) dx -n n+1 1 -n -1 + ( -n,n+1)

n r f ( n-r ) () x g () r () x (1.1) = Σ g() x = Σ n f < -n+ r> g () r -n + r dx r dx n + ( -n,m) dx -n n+1 1 -n -1 + ( -n,n+1) 8 Higher Derivative of the Product of Two Fuctios 8. Leibiz Rule about the Higher Order Differetiatio Theorem 8.. (Leibiz) Whe fuctios f ad g f g are times differetiable, the followig epressio holds. r

Διαβάστε περισσότερα

The Neutrix Product of the Distributions r. x λ

The Neutrix Product of the Distributions r. x λ ULLETIN u. Maaysia Math. Soc. Secod Seies 22 999 - of the MALAYSIAN MATHEMATICAL SOCIETY The Neuti Poduct of the Distibutios ad RIAN FISHER AND 2 FATMA AL-SIREHY Depatet of Matheatics ad Copute Sciece

Διαβάστε περισσότερα

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal

Διαβάστε περισσότερα

Last Lecture. Biostatistics Statistical Inference Lecture 19 Likelihood Ratio Test. Example of Hypothesis Testing.

Last Lecture. Biostatistics Statistical Inference Lecture 19 Likelihood Ratio Test. Example of Hypothesis Testing. Last Lecture Biostatistics 602 - Statistical Iferece Lecture 19 Likelihood Ratio Test Hyu Mi Kag March 26th, 2013 Describe the followig cocepts i your ow words Hypothesis Null Hypothesis Alterative Hypothesis

Διαβάστε περισσότερα

Biorthogonal Wavelets and Filter Banks via PFFS. Multiresolution Analysis (MRA) subspaces V j, and wavelet subspaces W j. f X n f, τ n φ τ n φ.

Biorthogonal Wavelets and Filter Banks via PFFS. Multiresolution Analysis (MRA) subspaces V j, and wavelet subspaces W j. f X n f, τ n φ τ n φ. Chapter 3. Biorthogoal Wavelets ad Filter Baks via PFFS 3.0 PFFS applied to shift-ivariat subspaces Defiitio: X is a shift-ivariat subspace if h X h( ) τ h X. Ex: Multiresolutio Aalysis (MRA) subspaces

Διαβάστε περισσότερα

MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS

MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS FUMIE NAKAOKA AND NOBUYUKI ODA Received 20 December 2005; Revised 28 May 2006; Accepted 6 August 2006 Some properties of minimal closed sets and maximal closed

Διαβάστε περισσότερα

Coefficient Inequalities for a New Subclass of K-uniformly Convex Functions

Coefficient Inequalities for a New Subclass of K-uniformly Convex Functions International Journal of Computational Science and Mathematics. ISSN 0974-89 Volume, Number (00), pp. 67--75 International Research Publication House http://www.irphouse.com Coefficient Inequalities for

Διαβάστε περισσότερα

1. Introduction and Preliminaries.

1. Introduction and Preliminaries. Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.yu/filomat Filomat 22:1 (2008), 97 106 ON δ SETS IN γ SPACES V. Renuka Devi and D. Sivaraj Abstract We

Διαβάστε περισσότερα

SOME PROPERTIES OF FUZZY REAL NUMBERS

SOME PROPERTIES OF FUZZY REAL NUMBERS Sahand Communications in Mathematical Analysis (SCMA) Vol. 3 No. 1 (2016), 21-27 http://scma.maragheh.ac.ir SOME PROPERTIES OF FUZZY REAL NUMBERS BAYAZ DARABY 1 AND JAVAD JAFARI 2 Abstract. In the mathematical

Διαβάστε περισσότερα

IIT JEE (2013) (Trigonomtery 1) Solutions

IIT JEE (2013) (Trigonomtery 1) Solutions L.K. Gupta (Mathematic Classes) www.pioeermathematics.com MOBILE: 985577, 677 (+) PAPER B IIT JEE (0) (Trigoomtery ) Solutios TOWARDS IIT JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL SURVIVE

Διαβάστε περισσότερα

SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES. Reading: QM course packet Ch 5 up to 5.6

SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES. Reading: QM course packet Ch 5 up to 5.6 SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES Readig: QM course packet Ch 5 up to 5. 1 ϕ (x) = E = π m( a) =1,,3,4,5 for xa (x) = πx si L L * = πx L si L.5 ϕ' -.5 z 1 (x) = L si

Διαβάστε περισσότερα

Statistical Inference I Locally most powerful tests

Statistical Inference I Locally most powerful tests Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided

Διαβάστε περισσότερα

Commutative Monoids in Intuitionistic Fuzzy Sets

Commutative Monoids in Intuitionistic Fuzzy Sets Commutative Monoids in Intuitionistic Fuzzy Sets S K Mala #1, Dr. MM Shanmugapriya *2 1 PhD Scholar in Mathematics, Karpagam University, Coimbatore, Tamilnadu- 641021 Assistant Professor of Mathematics,

Διαβάστε περισσότερα

ANOTHER EXTENSION OF VAN DER CORPUT S INEQUALITY. Gabriel STAN 1

ANOTHER EXTENSION OF VAN DER CORPUT S INEQUALITY. Gabriel STAN 1 Bulleti of the Trasilvaia Uiversity of Braşov Vol 5) - 00 Series III: Mathematics, Iformatics, Physics, -4 ANOTHER EXTENSION OF VAN DER CORPUT S INEQUALITY Gabriel STAN Abstract A extesio ad a refiemet

Διαβάστε περισσότερα

4.6 Autoregressive Moving Average Model ARMA(1,1)

4.6 Autoregressive Moving Average Model ARMA(1,1) 84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this

Διαβάστε περισσότερα

Homework 4.1 Solutions Math 5110/6830

Homework 4.1 Solutions Math 5110/6830 Homework 4. Solutios Math 5/683. a) For p + = αp γ α)p γ α)p + γ b) Let Equilibria poits satisfy: p = p = OR = γ α)p ) γ α)p + γ = α γ α)p ) γ α)p + γ α = p ) p + = p ) = The, we have equilibria poits

Διαβάστε περισσότερα

Homomorphism of Intuitionistic Fuzzy Groups

Homomorphism of Intuitionistic Fuzzy Groups International Mathematical Forum, Vol. 6, 20, no. 64, 369-378 Homomorphism o Intuitionistic Fuzz Groups P. K. Sharma Department o Mathematics, D..V. College Jalandhar Cit, Punjab, India pksharma@davjalandhar.com

Διαβάστε περισσότερα

Every set of first-order formulas is equivalent to an independent set

Every set of first-order formulas is equivalent to an independent set Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent

Διαβάστε περισσότερα

Solve the difference equation

Solve the difference equation Solve the differece equatio Solutio: y + 3 3y + + y 0 give tat y 0 4, y 0 ad y 8. Let Z{y()} F() Taig Z-trasform o both sides i (), we get y + 3 3y + + y 0 () Z y + 3 3y + + y Z 0 Z y + 3 3Z y + + Z y

Διαβάστε περισσότερα

Supplemental Material: Scaling Up Sparse Support Vector Machines by Simultaneous Feature and Sample Reduction

Supplemental Material: Scaling Up Sparse Support Vector Machines by Simultaneous Feature and Sample Reduction Supplemetal Material: Scalig Up Sparse Support Vector Machies by Simultaeous Feature ad Sample Reductio Weizhog Zhag * 2 Bi Hog * 3 Wei Liu 2 Jiepig Ye 3 Deg Cai Xiaofei He Jie Wag 3 State Key Lab of CAD&CG,

Διαβάστε περισσότερα

J. of Math. (PRC) Shannon-McMillan, , McMillan [2] Breiman [3] , Algoet Cover [10] AEP. P (X n m = x n m) = p m,n (x n m) > 0, x i X, 0 m i n. (1.

J. of Math. (PRC) Shannon-McMillan, , McMillan [2] Breiman [3] , Algoet Cover [10] AEP. P (X n m = x n m) = p m,n (x n m) > 0, x i X, 0 m i n. (1. Vol. 35 ( 205 ) No. 4 J. of Math. (PRC), (, 243002) : a.s. Marov Borel-Catelli. : Marov ; Borel-Catelli ; ; ; MR(200) : 60F5 : O2.4; O236 : A : 0255-7797(205)04-0969-08 Shao-McMilla,. Shao 948 [],, McMilla

Διαβάστε περισσότερα

On a four-dimensional hyperbolic manifold with finite volume

On a four-dimensional hyperbolic manifold with finite volume BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In

Διαβάστε περισσότερα

Congruence Classes of Invertible Matrices of Order 3 over F 2

Congruence Classes of Invertible Matrices of Order 3 over F 2 International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and

Διαβάστε περισσότερα

The Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle Chemistry 460 Sprig 015 Dr. Jea M. Stadard March, 015 The Heiseberg Ucertaity Priciple A policema pulls Werer Heiseberg over o the Autobah for speedig. Policema: Sir, do you kow how fast you were goig?

Διαβάστε περισσότερα

PROPERTIES OF CERTAIN INTEGRAL OPERATORS. a n z n (1.1)

PROPERTIES OF CERTAIN INTEGRAL OPERATORS. a n z n (1.1) GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 5, 995, 535-545 PROPERTIES OF CERTAIN INTEGRAL OPERATORS SHIGEYOSHI OWA Abstract. Two integral operators P α and Q α for analytic functions in the open unit disk

Διαβάστε περισσότερα

A General Note on δ-quasi Monotone and Increasing Sequence

A General Note on δ-quasi Monotone and Increasing Sequence International Mathematical Forum, 4, 2009, no. 3, 143-149 A General Note on δ-quasi Monotone and Increasing Sequence Santosh Kr. Saxena H. N. 419, Jawaharpuri, Badaun, U.P., India Presently working in

Διαβάστε περισσότερα

THE SECOND ISOMORPHISM THEOREM ON ORDERED SET UNDER ANTIORDERS. Daniel A. Romano

THE SECOND ISOMORPHISM THEOREM ON ORDERED SET UNDER ANTIORDERS. Daniel A. Romano 235 Kragujevac J. Math. 30 (2007) 235 242. THE SECOND ISOMORPHISM THEOREM ON ORDERED SET UNDER ANTIORDERS Daniel A. Romano Department of Mathematics and Informatics, Banja Luka University, Mladena Stojanovića

Διαβάστε περισσότερα

2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)

2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p) Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

Διαβάστε περισσότερα

C.S. 430 Assignment 6, Sample Solutions

C.S. 430 Assignment 6, Sample Solutions C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order

Διαβάστε περισσότερα

A Note on Intuitionistic Fuzzy. Equivalence Relation

A Note on Intuitionistic Fuzzy. Equivalence Relation International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com

Διαβάστε περισσότερα

Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequality for metrics: Let (X, d) be a metric space and let x, y, z X.

Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequality for metrics: Let (X, d) be a metric space and let x, y, z X. Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequalit for metrics: Let (X, d) be a metric space and let x,, z X. Prove that d(x, z) d(, z) d(x, ). (ii): Reverse triangle inequalit for norms:

Διαβάστε περισσότερα

derivation of the Laplacian from rectangular to spherical coordinates

derivation of the Laplacian from rectangular to spherical coordinates derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used

Διαβάστε περισσότερα

Solutions: Homework 3

Solutions: Homework 3 Solutios: Homework 3 Suppose that the radom variables Y,, Y satisfy Y i = βx i + ε i : i,, where x,, x R are fixed values ad ε,, ε Normal0, σ ) with σ R + kow Fid ˆβ = MLEβ) IND Solutio: Observe that Y

Διαβάστε περισσότερα

A Decomposition Algorithm for the Solution of Fractional Quadratic Riccati Differential Equations with Caputo Derivatives

A Decomposition Algorithm for the Solution of Fractional Quadratic Riccati Differential Equations with Caputo Derivatives America Joural of Computatioal ad Applied Mathematics 01, (3): 83-91 DOI: 10.593/j.ajcam.01003.03 A Decompositio Algorithm for the Solutio of Fractioal Quadratic Riccati Differetial Equatios with Caputo

Διαβάστε περισσότερα

Fractional Colorings and Zykov Products of graphs

Fractional Colorings and Zykov Products of graphs Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is

Διαβάστε περισσότερα

Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in

Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that

Διαβάστε περισσότερα

Other Test Constructions: Likelihood Ratio & Bayes Tests

Other Test Constructions: Likelihood Ratio & Bayes Tests Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :

Διαβάστε περισσότερα

1. Matrix Algebra and Linear Economic Models

1. Matrix Algebra and Linear Economic Models Matrix Algebra ad Liear Ecoomic Models Refereces Ch 3 (Turkigto); Ch 4 5 (Klei) [] Motivatio Oe market equilibrium Model Assume perfectly competitive market: Both buyers ad sellers are price-takers Demad:

Διαβάστε περισσότερα

Some new generalized topologies via hereditary classes. Key Words:hereditary generalized topological space, A κ(h,µ)-sets, κµ -topology.

Some new generalized topologies via hereditary classes. Key Words:hereditary generalized topological space, A κ(h,µ)-sets, κµ -topology. Bol. Soc. Paran. Mat. (3s.) v. 30 2 (2012): 71 77. c SPM ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v30i2.13793 Some new generalized topologies via hereditary

Διαβάστε περισσότερα

Certain Sequences Involving Product of k-bessel Function

Certain Sequences Involving Product of k-bessel Function It. J. Appl. Coput. Math 018 4:101 https://doi.org/10.1007/s40819-018-053-8 ORIGINAL PAPER Certai Sequeces Ivolvig Product of k-bessel Fuctio M. Chad 1 P. Agarwal Z. Haouch 3 Spriger Idia Private Ltd.

Διαβάστε περισσότερα

J. of Math. (PRC) Banach, , X = N(T ) R(T + ), Y = R(T ) N(T + ). Vol. 37 ( 2017 ) No. 5

J. of Math. (PRC) Banach, , X = N(T ) R(T + ), Y = R(T ) N(T + ). Vol. 37 ( 2017 ) No. 5 Vol. 37 ( 2017 ) No. 5 J. of Math. (PRC) 1,2, 1, 1 (1., 225002) (2., 225009) :. I +AT +, T + = T + (I +AT + ) 1, T +. Banach Hilbert Moore-Penrose.. : ; ; Moore-Penrose ; ; MR(2010) : 47L05; 46A32 : O177.2

Διαβάστε περισσότερα

Bessel function for complex variable

Bessel function for complex variable Besse fuctio for compex variabe Kauhito Miuyama May 4, 7 Besse fuctio The Besse fuctio Z ν () is the fuctio wich satisfies + ) ( + ν Z ν () =. () Three kids of the soutios of this equatio are give by {

Διαβάστε περισσότερα

ORDINAL ARITHMETIC JULIAN J. SCHLÖDER

ORDINAL ARITHMETIC JULIAN J. SCHLÖDER ORDINAL ARITHMETIC JULIAN J. SCHLÖDER Abstract. We define ordinal arithmetic and show laws of Left- Monotonicity, Associativity, Distributivity, some minor related properties and the Cantor Normal Form.

Διαβάστε περισσότερα

Matrices and Determinants

Matrices and Determinants Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z

Διαβάστε περισσότερα

F A S C I C U L I M A T H E M A T I C I

F A S C I C U L I M A T H E M A T I C I F A S C I C U L I M A T H E M A T I C I Nr 46 2011 C. Carpintero, N. Rajesh and E. Rosas ON A CLASS OF (γ, γ )-PREOPEN SETS IN A TOPOLOGICAL SPACE Abstract. In this paper we have introduced the concept

Διαβάστε περισσότερα

Example Sheet 3 Solutions

Example Sheet 3 Solutions Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note

Διαβάστε περισσότερα

3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β

3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β 3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle

Διαβάστε περισσότερα

Section 8.3 Trigonometric Equations

Section 8.3 Trigonometric Equations 99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.

Διαβάστε περισσότερα

The Number of Zeros of a Polynomial in a Disk as a Consequence of Restrictions on the Coefficients

The Number of Zeros of a Polynomial in a Disk as a Consequence of Restrictions on the Coefficients The Number of Zeros of a Polynomial in a Disk as a Consequence of Restrictions on the Coefficients Robert Gardner Brett Shields Department of Mathematics and Statistics Department of Mathematics and Statistics

Διαβάστε περισσότερα

Lecture 17: Minimum Variance Unbiased (MVUB) Estimators

Lecture 17: Minimum Variance Unbiased (MVUB) Estimators ECE 830 Fall 2011 Statistical Sigal Processig istructor: R. Nowak, scribe: Iseok Heo Lecture 17: Miimum Variace Ubiased (MVUB Estimators Ultimately, we would like to be able to argue that a give estimator

Διαβάστε περισσότερα

Intuitionistic Fuzzy Ideals of Near Rings

Intuitionistic Fuzzy Ideals of Near Rings International Mathematical Forum, Vol. 7, 202, no. 6, 769-776 Intuitionistic Fuzzy Ideals of Near Rings P. K. Sharma P.G. Department of Mathematics D.A.V. College Jalandhar city, Punjab, India pksharma@davjalandhar.com

Διαβάστε περισσότερα

Generating Set of the Complete Semigroups of Binary Relations

Generating Set of the Complete Semigroups of Binary Relations Applied Mathematics 06 7 98-07 Published Online January 06 in SciRes http://wwwscirporg/journal/am http://dxdoiorg/036/am067009 Generating Set of the Complete Semigroups of Binary Relations Yasha iasamidze

Διαβάστε περισσότερα

Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------

Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- ----------------- Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin

Διαβάστε περισσότερα

Ψηφιακή Επεξεργασία Εικόνας

Ψηφιακή Επεξεργασία Εικόνας ΠΑΝΕΠΙΣΤΗΜΙΟ ΙΩΑΝΝΙΝΩΝ ΑΝΟΙΚΤΑ ΑΚΑΔΗΜΑΪΚΑ ΜΑΘΗΜΑΤΑ Ψηφιακή Επεξεργασία Εικόνας Φιλτράρισμα στο πεδίο των συχνοτήτων Διδάσκων : Αναπληρωτής Καθηγητής Νίκου Χριστόφορος Άδειες Χρήσης Το παρόν εκπαιδευτικό

Διαβάστε περισσότερα

12. Radon-Nikodym Theorem

12. Radon-Nikodym Theorem Tutorial 12: Radon-Nikodym Theorem 1 12. Radon-Nikodym Theorem In the following, (Ω, F) is an arbitrary measurable space. Definition 96 Let μ and ν be two (possibly complex) measures on (Ω, F). We say

Διαβάστε περισσότερα

2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.

2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits. EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.

Διαβάστε περισσότερα

Uniform Convergence of Fourier Series Michael Taylor

Uniform Convergence of Fourier Series Michael Taylor Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula

Διαβάστε περισσότερα

Supplement to A theoretical framework for Bayesian nonparametric regression: random series and rates of contraction

Supplement to A theoretical framework for Bayesian nonparametric regression: random series and rates of contraction Supplemet to A theoretical framework for Bayesia oparametric regressio: radom series ad rates of cotractio A Proof of Theorem 31 Proof of Theorem 31 First defie the followig quatity: ɛ = 3 t α, δ = α α

Διαβάστε περισσότερα

Homomorphism and Cartesian Product on Fuzzy Translation and Fuzzy Multiplication of PS-algebras

Homomorphism and Cartesian Product on Fuzzy Translation and Fuzzy Multiplication of PS-algebras Annals of Pure and Applied athematics Vol. 8, No. 1, 2014, 93-104 ISSN: 2279-087X (P), 2279-0888(online) Published on 11 November 2014 www.researchmathsci.org Annals of Homomorphism and Cartesian Product

Διαβάστε περισσότερα

Reminders: linear functions

Reminders: linear functions Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U

Διαβάστε περισσότερα

Jordan Journal of Mathematics and Statistics (JJMS) 4(2), 2011, pp

Jordan Journal of Mathematics and Statistics (JJMS) 4(2), 2011, pp Jordan Journal of Mathematics and Statistics (JJMS) 4(2), 2011, pp.115-126. α, β, γ ORTHOGONALITY ABDALLA TALLAFHA Abstract. Orthogonality in inner product spaces can be expresed using the notion of norms.

Διαβάστε περισσότερα

Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.

Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ. Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action

Διαβάστε περισσότερα

ST5224: Advanced Statistical Theory II

ST5224: Advanced Statistical Theory II ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known

Διαβάστε περισσότερα

On Quasi - f -Power Increasing Sequences

On Quasi - f -Power Increasing Sequences Ieaioal Maheaical Fou Vol 8 203 o 8 377-386 Quasi - f -owe Iceasig Sequeces Maheda Misa G Deae of Maheaics NC College (Auooous) Jaju disha Mahedaisa2007@gailco B adhy Rolad Isiue of echoy Golahaa-76008

Διαβάστε περισσότερα

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)

Διαβάστε περισσότερα

Uniform Estimates for Distributions of the Sum of i.i.d. Random Variables with Fat Tail in the Threshold Case

Uniform Estimates for Distributions of the Sum of i.i.d. Random Variables with Fat Tail in the Threshold Case J. Math. Sci. Uiv. Tokyo 8 (2, 397 427. Uiform Estimates for Distributios of the Sum of i.i.d. om Variables with Fat Tail i the Threshold Case By Keji Nakahara Abstract. We show uiform estimates for distributios

Διαβάστε περισσότερα

5. Choice under Uncertainty

5. Choice under Uncertainty 5. Choice under Uncertainty Daisuke Oyama Microeconomics I May 23, 2018 Formulations von Neumann-Morgenstern (1944/1947) X: Set of prizes Π: Set of probability distributions on X : Preference relation

Διαβάστε περισσότερα

Homework 3 Solutions

Homework 3 Solutions Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For

Διαβάστε περισσότερα

Approximation of distance between locations on earth given by latitude and longitude

Approximation of distance between locations on earth given by latitude and longitude Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth

Διαβάστε περισσότερα

On New Subclasses of Analytic Functions with Respect to Conjugate and Symmetric Conjugate Points

On New Subclasses of Analytic Functions with Respect to Conjugate and Symmetric Conjugate Points Global Journal of Pure Applied Mathematics. ISSN 0973-768 Volume, Number 3 06, pp. 849 865 Research India Publications http://www.ripublication.com/gjpam.htm On New Subclasses of Analytic Functions with

Διαβάστε περισσότερα

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Chapter 6: Systems of Linear Differential. be continuous functions on the interval Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations

Διαβάστε περισσότερα

Second Order Partial Differential Equations

Second Order Partial Differential Equations Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y

Διαβάστε περισσότερα

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required) Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts

Διαβάστε περισσότερα

arxiv: v1 [math.nt] 17 Sep 2016

arxiv: v1 [math.nt] 17 Sep 2016 arxiv:609.057v [math.nt] 7 Sep 06 Covolutio idetities for Tetraacci umbers Ruse Li School of Mathematics ad Statistics Wuha Uiversity Wuha 43007 Chia limajiashe@whu.edu.c Abstract We give covolutio idetities

Διαβάστε περισσότερα

Proof of Lemmas Lemma 1 Consider ξ nt = r

Proof of Lemmas Lemma 1 Consider ξ nt = r Supplemetary Material to "GMM Estimatio of Spatial Pael Data Models with Commo Factors ad Geeral Space-Time Filter" (Not for publicatio) Wei Wag & Lug-fei Lee April 207 Proof of Lemmas Lemma Cosider =

Διαβάστε περισσότερα

Degenerate Perturbation Theory

Degenerate Perturbation Theory R.G. Griffi BioNMR School page 1 Degeerate Perturbatio Theory 1.1 Geeral Whe cosiderig the CROSS EFFECT it is ecessary to deal with degeerate eergy levels ad therefore degeerate perturbatio theory. The

Διαβάστε περισσότερα

Homework 8 Model Solution Section

Homework 8 Model Solution Section MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx

Διαβάστε περισσότερα

CHAPTER 103 EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES

CHAPTER 103 EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES CHAPTER 3 EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES EXERCISE 364 Page 76. Determie the Fourier series for the fuctio defied by: f(x), x, x, x which is periodic outside of this rage of period.

Διαβάστε περισσότερα

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1 Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the

Διαβάστε περισσότερα

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all

Διαβάστε περισσότερα

k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +

k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R + Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b

Διαβάστε περισσότερα