Basic Results in Functional Analysis

Size: px
Start display at page:

Download "Basic Results in Functional Analysis"

Transcription

1 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 ( ): X Y is ischiz if f ( z) f( ) z ocally if for every, z i a comac se, vs. Globally if for all, z. Derivaive of f() a is f ( z) f( ) f '( ) lim z z f() is differeiable a if derivaive eiss a. he f ( ) C. f() is coiuously differeiable a if derivaive eiss a ad is coiuous. he f ( ) C. f() is uiformly coiuous if i is coiuous ad is derivaive is bouded. ischiz imlies uif. co. Coi. diff. imlies locally ischiz e f() be coi diff o X. he globally ischiz imlies f '( ), for a cos, X Diff imlies co e f( ) f() coi imlies eiss soluio () f() coi ad ischiz imlies eiss a uique soluio () ocally o a comac se, or globally. f() coi diff imlies eiss a uique soluio ()

2 Def. A ier roduc o a liear vecor sace X wih field F is a fucio.,. : X X F such ha for, yz, X, F a., b., iff c., y, y, homogeeous d., yz, y, z, liear e., y y,, comle cojugae Usually.,. : R R R. Def. A orm o R is a fucio.: R R such ha, for, y R, R: a. b. iff c., homogeeiy d. y y, riagle iequaliy Fac. Every orm is a cove fucio. Fac. Every ier roduc defies a orm /, Def. A semiorm or seudoorm does o have roery b. Def. A quasiorm has d. relaced by he milder roery y K y, K Aoher Def. A quasiorm has c. relaced by he milder roery Def. vecor -orm (Holder orms) for i i / R wih comoes i R Fac. k k, for some k i. his meas all vecor orms o R are equivale.

3 Mikowski iequaliy or riagle iequaliy y y Holder s iequaliy. e, y R ad q. he, y y q Cauchy-Schwarz iequaliy is he secial case =q=, y y Sylveser s iequaliy mi ( A) A ma ( A) wih he sigular values of A. Def. Covergece of sequeces. A sequece of vecors { k},, X is said o coverge o a limi vecor if k as k or equivalely, N k N k Def. is a accumulaio oi of sequece { k } if here is a subsequece of { k } ha coverges o. ha is, here is a ifiie subse K of oegaive iegers such ha { k} coverges o. k K Def. A se S X is closed if ad oly if every coverge sequece wih elemes i S has a limi i S. Def. A sequece { k } X is said o be a Cauchy sequece if as k, m k m Fac. Every coverge sequece is Cauchy, bu o vice versa. Def. Baach Sace. A ormed liear sace X is comlee if every Cauchy sequece coverges o a vecor i X. A comlee ormed liear sace is a Baach Sace. Def. A re-hilber Sace is a liear sace X wih a ier roduc. Def. Hilber Sace. A liear sace X wih ier roduc is comlee if every Cauchy sequece coverges o a vecor i X. A comlee liear sace wih ier roduc is a Hilber Sace. Fac. A Hilber sace has a ier roduc, hece a orm, hece is a Baach Sace. 3

4 Fac. X orm is R is a Hilber Sace wih ier roduc. y y,, y R. he associaed / ( ), he vecor orm. Fac. A bouded sequece { k } i R has a leas oe accumulaio oi i Fac. A sequece of real umbers { r k } which is moooically odecreasig ad bouded from above, i.e. rk rk R, R cos coverges o a real umber Fac. A sequece of real umbers { r k } which is moooically oicreasig ad bouded from below, i.e. rk rk R, R cos coverges o a real umber R. Coracio Maig. e S be a closed subse of a Baach sace X ad le be a maig ha mas S io S. Suose ha ( ) ( y) y,, ys, he * * * a. here eiss a uique vecor S such ha ( ). (fied oi) * b. ca be obaied by he mehod of successive aroimaio sarig from ay iiial oi i S. Fac. e f ( ) : R R. he f () d f() d his is acually he riagle iequaliy, for oe ha he ebesgue iegral is So ha b b / f () d lim f( k) k b b / b / f () d lim f( k) lim f( k) k k ier roduc. f, g f ( ) g( ) d D 4

5 ca deoe ime ad D a ime ierval, or ca ake ad iegrae over a regio D R. Def. orm (also deoed orm) of fucio f ( ) : R Y f(.) f( ) d, also deoed as f. D ca deoe ime ad D a ime ierval, or ca ake / ad iegrae over a regio D R. Def. Uiform fucio orm, or suremum orm, or orm, or Chebyshev orm f su f( ) : R Def. f is said o belog o if f (.) is bouded. Def. he ebesque ormed sace { f(.) Y : f(.) } (also deoed ) is If ime iegral, he ier roduc of f ( ), g( ) : R R is f, g f ( ) g( ) d ad orm of fucio f ( ) : R R is / f (.) f ( ) d, If over ime f su f( ) : Defie ier roduc f, g, f ( ) g( ) d ad orm f (.) f ( ) d, Def. he eeded ebesque ormed sace e (also deoed { f(.) Y : f(.), } e / e ) is 5

6 m Mea Value heorem. e f ( ) : R R be differeiable a each oi R, a oe se. e y, wih he lie segme y (, ). he here eiss a oi z of y (, ) such ha f f( y) f( ) y z m Imlici Fucio heorem. e f ( y, ) : RR R be coiuously differeiable a each m oi (, y) R R, a oe se. e (, y) such ha f(, y) ad f Jacobia mari is osigular. (, y) m he here eis eighborhoods U R of ad V R of y such ha y V he equaio f(, y) has a uique soluio U. Moreover he soluio ca be wrie as g( y) where g(.) is coiuously differeiable a y=y. eibiz Formula. () If () f(,) d he () () d ( ) '() f( ( ),,) '( ) f( ( ),,) '( ) f(, ) d d Bellma-Growall emma. e here be coiuous fucios, : R R, ad a coiuous oegaive fucio : R R. If he () () ()() s s ds, ( ) d s () () () s () s e ds, Also, if is a cos he () e, ( ) d If is a cos he ( () e ), () (Growall emma) 6

7 Jese s Iequaliy. Give mari P, scalars ba, ad R a ( ) P ( ) d ( a) ( b) P ( a) ( b) b a b 7

K3 p K2 p Kp 0 p 2 p 3 p

K3 p K2 p Kp 0 p 2 p 3 p Mah 80-00 Mo Ar 0 Chaer 9 Fourier Series ad alicaios o differeial equaios (ad arial differeial equaios) 9.-9. Fourier series defiiio ad covergece. The idea of Fourier series is relaed o he liear algebra

More information

A TAUBERIAN THEOREM FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY

A 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 information

Calculus Limits. Limit of a function.. 1. One-Sided Limits...1. Infinite limits 2. Vertical Asymptotes...3. Calculating Limits Using the Limit Laws.

Calculus 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 information

Prakash Chandra Rautaray 1, Ellipse 2

Prakash Chandra Rautaray 1, Ellipse 2 Prakash Chadra Rauara, Ellise / Ieraioal Joural of Egieerig Research ad Alicaios (IJERA) ISSN: 48-96 www.ijera.com Vol. 3, Issue, Jauar -Februar 3,.36-337 Degree Of Aroimaio Of Fucios B Modified Parial

More information

Lecture 15 First Properties of the Brownian Motion

Lecture 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 information

EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D. S. Palimkar

EXISTENCE 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 information

FIXED FUZZY POINT THEOREMS IN FUZZY METRIC SPACE

FIXED 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 information

Review Exercises for Chapter 9

Review Exercises for Chapter 9 0_090R.qd //0 : PM Page 88 88 CHAPTER 9 Ifiie Series I Eercises ad, wrie a epressio for he h erm of he sequece..,., 5, 0,,,, 0,... 7,... I Eercises, mach he sequece wih is graph. [The graphs are labeled

More information

Comparison between Fourier and Corrected Fourier Series Methods

Comparison between Fourier and Corrected Fourier Series Methods Malaysia Joural of Mahemaical Scieces 7(): 73-8 (13) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Joural homepage: hp://eispem.upm.edu.my/oural Compariso bewee Fourier ad Correced Fourier Series Mehods 1

More information

Lecture 15: Three-tank Mixing and Lead Poisoning

Lecture 15: Three-tank Mixing and Lead Poisoning Lecure 15: Three-ak Miig ad Lead Poisoig Eigevalues ad eigevecors will be used o fid he soluio of a sysem for ukow fucios ha saisfy differeial equaios The ukow fucios will be wrie as a 1 colum vecor [

More information

Common Fixed Point Theorem in Intuitionistic Fuzzy Metric Space via Compatible Mappings of Type (K)

Common Fixed Point Theorem in Intuitionistic Fuzzy Metric Space via Compatible Mappings of Type (K) Ieraioal Joural of ahemaics Treds ad Techology (IJTT) Volume 35 umber 4- July 016 Commo Fixed Poi Theorem i Iuiioisic Fuzzy eric Sace via Comaible aigs of Tye (K) Dr. Ramaa Reddy Assisa Professor De. of

More information

Application of Fixed Point Theorem of Convex-power Operators to Nonlinear Volterra Type Integral Equations

Application of Fixed Point Theorem of Convex-power Operators to Nonlinear Volterra Type Integral Equations Ieraioal Mahemaical Forum, Vol 9, 4, o 9, 47-47 HIKRI Ld, wwwm-hikaricom h://dxdoiorg/988/imf4333 licaio of Fixed Poi Theorem of Covex-ower Oeraors o Noliear Volerra Tye Iegral Equaios Ya Chao-dog Huaiyi

More information

1 Notes on Little s Law (l = λw)

1 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 information

Math-303 Chapter 7 Linear systems of ODE November 16, Chapter 7. Systems of 1 st Order Linear Differential Equations.

Math-303 Chapter 7 Linear systems of ODE November 16, Chapter 7. Systems of 1 st Order Linear Differential Equations. Mah-33 Chaper 7 Liear sysems of ODE November 6, 7 Chaper 7 Sysems of s Order Liear Differeial Equaios saddle poi λ >, λ < Mah-33 Chaper 7 Liear sysems of ODE November 6, 7 Mah-33 Chaper 7 Liear sysems

More information

Some inequalities for q-polygamma function and ζ q -Riemann zeta functions

Some inequalities for q-polygamma function and ζ q -Riemann zeta functions Aales Mahemaicae e Iformaicae 37 (1). 95 1 h://ami.ekf.hu Some iequaliies for q-olygamma fucio ad ζ q -Riema zea fucios Valmir Krasiqi a, Toufik Masour b Armed Sh. Shabai a a Dearme of Mahemaics, Uiversiy

More information

MATH 507a ASSIGNMENT 4 SOLUTIONS FALL 2018 Prof. Alexander. g (x) dx = g(b) g(0) = g(b),

MATH 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 information

Math 6710, Fall 2016 Final Exam Solutions

Math 6710, Fall 2016 Final Exam Solutions Mah 67, Fall 6 Fial Exam Soluios. Firs, a sude poied ou a suble hig: if P (X i p >, he X + + X (X + + X / ( evaluaes o / wih probabiliy p >. This is roublesome because a radom variable is supposed o be

More information

STK4080/9080 Survival and event history analysis

STK4080/9080 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 information

Existence Of Solutions For Nonlinear Fractional Differential Equation With Integral Boundary Conditions

Existence Of Solutions For Nonlinear Fractional Differential Equation With Integral Boundary Conditions Reserch Ivey: Ieriol Jourl Of Egieerig Ad Sciece Vol., Issue (April 3), Pp 8- Iss(e): 78-47, Iss(p):39-6483, Www.Reserchivey.Com Exisece Of Soluios For Nolier Frciol Differeil Equio Wih Iegrl Boudry Codiios,

More information

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS M.A. (Previous) Direcorae of Disace Educaio Maharshi Dayaad Uiversiy ROHTAK 4 Copyrigh 3, Maharshi Dayaad Uiversiy, ROHTAK All Righs Reserved. No par of his publicaio may be reproduced

More information

Final Solutions. 1. (25pts) Define the following terms. Be as precise as you can.

Final Solutions. 1. (25pts) Define the following terms. Be as precise as you can. Mathematics H104 A. Ogus Fall, 004 Fial Solutios 1. (5ts) Defie the followig terms. Be as recise as you ca. (a) (3ts) A ucoutable set. A ucoutable set is a set which ca ot be ut ito bijectio with a fiite

More information

1. Solve by the method of undetermined coefficients and by the method of variation of parameters. (4)

1. Solve by the method of undetermined coefficients and by the method of variation of parameters. (4) 7 Differeial equaios Review Solve by he mehod of udeermied coefficies ad by he mehod of variaio of parameers (4) y y = si Soluio; we firs solve he homogeeous equaio (4) y y = 4 The correspodig characerisic

More information

Solutions to Problems 3, Level 4

Solutions to Problems 3, Level 4 Soluios o Problems 3, Level 4 23 Improve he resul of Quesio 3 whe l. i Use log log o prove ha for real >, log ( {}log + 2 d log+ P ( + P ( d 2. Here P ( is defied i Quesio, ad parial iegraio has bee used.

More information

Extremal graph theory II: K t and K t,t

Extremal 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 information

On Stability of Quintic Functional Equations in Random Normed Spaces

On Stability of Quintic Functional Equations in Random Normed Spaces J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 3, NO.4, 07, COPYRIGHT 07 EUDOXUS PRESS, LLC O Sabiliy of Quiic Fucioal Equaios i Radom Normed Spaces Afrah A.N. Abdou, Y. J. Cho,,, Liaqa A. Kha ad S.

More information

ODEs II, Supplement to Lectures 6 & 7: The Jordan Normal Form: Solving Autonomous, Homogeneous Linear Systems. April 2, 2003

ODEs II, Supplement to Lectures 6 & 7: The Jordan Normal Form: Solving Autonomous, Homogeneous Linear Systems. April 2, 2003 ODEs II, Suppleme o Lecures 6 & 7: The Jorda Normal Form: Solvig Auoomous, Homogeeous Liear Sysems April 2, 23 I his oe, we describe he Jorda ormal form of a marix ad use i o solve a geeral homogeeous

More information

BEST CONSTANTS FOR TWO NON-CONVOLUTION INEQUALITIES

BEST CONSTANTS FOR TWO NON-CONVOLUTION INEQUALITIES BEST CONSTANTS FOR TWO NON-CONVOLUTION INEQUALITIES Michael Chris ad Loukas Grafakos Uiversiy of Califoria, Los Ageles ad Washigo Uiversiy Absrac. The orm of he oeraor which averages f i L ( ) over balls

More information

The Connection between the Basel Problem and a Special Integral

The Connection between the Basel Problem and a Special Integral Applied Mahemaics 4 5 57-584 Published Olie Sepember 4 i SciRes hp://wwwscirporg/joural/am hp://ddoiorg/436/am45646 The Coecio bewee he Basel Problem ad a Special Iegral Haifeg Xu Jiuru Zhou School of

More information

Real Numbers R ) - LUB(B) may or may not belong to B. (Ex; B= { y: y = 1 x, - Note that A B LUB( A) LUB( B)

Real Numbers R ) - LUB(B) may or may not belong to B. (Ex; B= { y: y = 1 x, - Note that A B LUB( A) LUB( B) Real Numbers The least upper boud - Let B be ay subset of R B is bouded above if there is a k R such that x k for all x B - A real umber, k R is a uique least upper boud of B, ie k = LUB(B), if () k is

More information

The Central Limit Theorem

The 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 information

Ideal Amplifier/Attenuator. Memoryless. where k is some real constant. Integrator. System with memory

Ideal Amplifier/Attenuator. Memoryless. where k is some real constant. Integrator. System with memory Liear Time-Ivaria Sysems (LTI Sysems) Oulie Basic Sysem Properies Memoryless ad sysems wih memory (saic or dyamic) Causal ad o-causal sysems (Causaliy) Liear ad o-liear sysems (Lieariy) Sable ad o-sable

More information

VISCOSITY APPROXIMATION TO COMMON FIXED POINTS OF kn- LIPSCHITZIAN NONEXPANSIVE MAPPINGS IN BANACH SPACES

VISCOSITY 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 information

2 Banach spaces and Hilbert spaces

2 Banach spaces and Hilbert spaces 2 Baach spaces ad Hilbert spaces Tryig to do aalysis i the ratioal umbers is difficult for example cosider the set {x Q : x 2 2}. This set is o-empty ad bouded above but does ot have a least upper boud

More information

Solution. 1 Solutions of Homework 6. Sangchul Lee. April 28, Problem 1.1 [Dur10, Exercise ]

Solution. 1 Solutions of Homework 6. Sangchul Lee. April 28, Problem 1.1 [Dur10, Exercise ] Soluio Sagchul Lee April 28, 28 Soluios of Homework 6 Problem. [Dur, Exercise 2.3.2] Le A be a sequece of idepede eves wih PA < for all. Show ha P A = implies PA i.o. =. Proof. Noice ha = P A c = P A c

More information

An interesting result about subset sums. Nitu Kitchloo. Lior Pachter. November 27, Abstract

An 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 information

Research Article A Generalized Nonlinear Sum-Difference Inequality of Product Form

Research 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 information

Moment Generating Function

Moment 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 information

L-functions and Class Numbers

L-functions and Class Numbers L-fucios ad Class Numbers Sude Number Theory Semiar S. M.-C. 4 Sepember 05 We follow Romyar Sharifi s Noes o Iwasawa Theory, wih some help from Neukirch s Algebraic Number Theory. L-fucios of Dirichle

More information

ANSWERS TO MIDTERM EXAM # 2

ANSWERS TO MIDTERM EXAM # 2 MATH 03, FALL 003 ANSWERS TO MIDTERM EXAM # PENN STATE UNIVERSITY Problem 1 (18 pts). State ad prove the Itermediate Value Theorem. Solutio See class otes or Theorem 5.6.1 from our textbook. Problem (18

More information

Supplement for SADAGRAD: Strongly Adaptive Stochastic Gradient Methods"

Supplement for SADAGRAD: Strongly Adaptive Stochastic Gradient Methods Suppleme for SADAGRAD: Srogly Adapive Sochasic Gradie Mehods" Zaiyi Che * 1 Yi Xu * Ehog Che 1 iabao Yag 1. Proof of Proposiio 1 Proposiio 1. Le ɛ > 0 be fixed, H 0 γi, γ g, EF (w 1 ) F (w ) ɛ 0 ad ieraio

More information

Math 2414 Homework Set 7 Solutions 10 Points

Math 2414 Homework Set 7 Solutions 10 Points Mah Homework Se 7 Soluios 0 Pois #. ( ps) Firs verify ha we ca use he iegral es. The erms are clearly posiive (he epoeial is always posiive ad + is posiive if >, which i is i his case). For decreasig we

More information

AN UNCERTAIN CAUCHY PROBLEM OF A NEW CLASS OF FUZZY DIFFERENTIAL EQUATIONS. Alexei Bychkov, Eugene Ivanov, Olha Suprun

AN UNCERTAIN CAUCHY PROBLEM OF A NEW CLASS OF FUZZY DIFFERENTIAL EQUATIONS. Alexei Bychkov, Eugene Ivanov, Olha Suprun Ieraioal Joural "Iformaio Models ad Aalyses" Volume 4, Number 2, 215 13 AN UNCERAIN CAUCHY PROBLEM OF A NEW CLASS OF FUZZY DIFFERENIAL EQUAIONS Alexei Bychkov, Eugee Ivaov, Olha Supru Absrac: he cocep

More information

10.3 Autocorrelation Function of Ergodic RP 10.4 Power Spectral Density of Ergodic RP 10.5 Normal RP (Gaussian RP)

10.3 Autocorrelation Function of Ergodic RP 10.4 Power Spectral Density of Ergodic RP 10.5 Normal RP (Gaussian RP) ENGG450 Probabiliy ad Saisics for Egieers Iroducio 3 Probabiliy 4 Probabiliy disribuios 5 Probabiliy Desiies Orgaizaio ad descripio of daa 6 Samplig disribuios 7 Ifereces cocerig a mea 8 Comparig wo reames

More information

Linear System Theory

Linear System Theory Naioal Tsig Hua Uiversiy Dearme of Power Mechaical Egieerig Mid-Term Eamiaio 3 November 11.5 Hours Liear Sysem Theory (Secio B o Secio E) [11PME 51] This aer coais eigh quesios You may aswer he quesios

More information

Numerical Method for Ordinary Differential Equation

Numerical Method for Ordinary Differential Equation Numerical ehod for Ordiar Differeial Equaio. J. aro ad R. J. Lopez, Numerical Aalsis: A Pracical Approach, 3rd Ed., Wadsworh Publishig Co., Belmo, CA (99): Chap. 8.. Iiial Value Problem (IVP) d (IVP):

More information

A quadratic convergence method for the management equilibrium model

A quadratic convergence method for the management equilibrium model (IJACSA Ieraioal Joural of Advaced Copuer Sciece ad Applicaios Vol No 9 03 A quadraic covergece ehod for he aagee equilibriu odel Jiayi Zhag Feixia School Liyi Uiversiy Feixia Shadog PRChia Absrac i his

More information

Approximately Quasi Inner Generalized Dynamics on Modules. { } t t R

Approximately 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 information

Homework 4. x n x X = f(x n x) +

Homework 4. x n x X = f(x n x) + Homework 4 1. Let X ad Y be ormed spaces, T B(X, Y ) ad {x } a sequece i X. If x x weakly, show that T x T x weakly. Solutio: We eed to show that g(t x) g(t x) g Y. It suffices to do this whe g Y = 1.

More information

TAKA 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.

TAKA 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 information

BE.430 Tutorial: Linear Operator Theory and Eigenfunction Expansion

BE.430 Tutorial: Linear Operator Theory and Eigenfunction Expansion BE.43 Tuorial: Liear Operaor Theory ad Eigefucio Expasio (adaped fro Douglas Lauffeburger) 9//4 Moivaig proble I class, we ecouered parial differeial equaios describig rasie syses wih cheical diffusio.

More information

E will be denoted by n

E will be denoted by n JASEM ISSN 9-8362 All rigs reserved Full-ex Available Olie a p:// wwwbiolieorgbr/ja J Appl Sci Eviro Mg 25 Vol 9 3) 3-36 Corollabiliy ad Null Corollabiliy of Liear Syses * DAVIES, I; 2 JACKREECE, P Depare

More information

Mathematical Statistics. 1 Introduction to the materials to be covered in this course

Mathematical Statistics. 1 Introduction to the materials to be covered in this course Mahemaical Saisics Iroducio o he maerials o be covered i his course. Uivariae & Mulivariae r.v s 2. Borl-Caelli Lemma Large Deviaios. e.g. X,, X are iid r.v s, P ( X + + X where I(A) is a umber depedig

More information

Chapter 2: Time-Domain Representations of Linear Time-Invariant Systems. Chih-Wei Liu

Chapter 2: Time-Domain Representations of Linear Time-Invariant Systems. Chih-Wei Liu Caper : Time-Domai Represeaios of Liear Time-Ivaria Sysems Ci-Wei Liu Oulie Iroucio Te Covoluio Sum Covoluio Sum Evaluaio Proceure Te Covoluio Iegral Covoluio Iegral Evaluaio Proceure Iercoecios of LTI

More information

The analysis of the method on the one variable function s limit Ke Wu

The analysis of the method on the one variable function s limit Ke Wu Ieraioal Coferece o Advaces i Mechaical Egieerig ad Idusrial Iformaics (AMEII 5) The aalysis of he mehod o he oe variable fucio s i Ke Wu Deparme of Mahemaics ad Saisics Zaozhuag Uiversiy Zaozhuag 776

More information

Theorem. Let H be a class of functions from a measurable space T to R. Assume that for every ɛ > 0 there exists a finite set of brackets

Theorem. Let H be a class of functions from a measurable space T to R. Assume that for every ɛ > 0 there exists a finite set of brackets STATISTICAL THEORY SUMMARY ANDREW TULLOCH Theorem Hoeffig s Iequaliy. Suose EX, a a X b. The E e X e b a 8... Uiform Laws of Large Numbers. Bic Coces Defiiio Covergece almos surely. A sequece X, N of raom

More information

N! AND THE GAMMA FUNCTION

N! AND THE GAMMA FUNCTION N! AND THE GAMMA FUNCTION Cosider he produc of he firs posiive iegers- 3 4 5 6 (-) =! Oe calls his produc he facorial ad has ha produc of he firs five iegers equals 5!=0. Direcly relaed o he discree! fucio

More information

ON SOME NEW SEQUENCE SPACES OF NON-ABSOLUTE TYPE RELATED TO THE SPACES l p AND l I. M. Mursaleen and Abdullah K. Noman

ON SOME NEW SEQUENCE SPACES OF NON-ABSOLUTE TYPE RELATED TO THE SPACES l p AND l I. M. Mursaleen and Abdullah K. Noman Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia Available at: htt://www.mf.i.ac.rs/filomat Filomat 25:2 20, 33 5 DOI: 0.2298/FIL02033M ON SOME NEW SEQUENCE SPACES OF NON-ABSOLUTE TYPE RELATED

More information

Math Solutions to homework 6

Math Solutions to homework 6 Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there

More information

Calculus BC 2015 Scoring Guidelines

Calculus BC 2015 Scoring Guidelines AP Calculus BC 5 Scorig Guidelies 5 The College Board. College Board, Advaced Placeme Program, AP, AP Ceral, ad he acor logo are regisered rademarks of he College Board. AP Ceral is he official olie home

More information

2 f(x) dx = 1, 0. 2f(x 1) dx d) 1 4t t6 t. t 2 dt i)

2 f(x) dx = 1, 0. 2f(x 1) dx d) 1 4t t6 t. t 2 dt i) Mah PracTes Be sure o review Lab (ad all labs) There are los of good quesios o i a) Sae he Mea Value Theorem ad draw a graph ha illusraes b) Name a impora heorem where he Mea Value Theorem was used i he

More information

MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES. Boundary Value Problem for the Higher Order Equation with Fractional Derivative

MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES. Boundary Value Problem for the Higher Order Equation with Fractional Derivative Malaysia Joural of Maheaical Scieces 7(): 3-7 (3) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Joural hoepage: hp://eispe.up.edu.y/joural Boudary Value Proble for he Higher Order Equaio wih Fracioal Derivaive

More information

Notes 03 largely plagiarized by %khc

Notes 03 largely plagiarized by %khc 1 1 Discree-Time Covoluio Noes 03 largely plagiarized by %khc Le s begi our discussio of covoluio i discree-ime, sice life is somewha easier i ha domai. We sar wih a sigal x[] ha will be he ipu io our

More information

Boundary-to-Displacement Asymptotic Gains for Wave Systems With Kelvin-Voigt Damping

Boundary-to-Displacement Asymptotic Gains for Wave Systems With Kelvin-Voigt Damping Boudary-o-Displaceme Asympoic Gais for Wave Sysems Wih Kelvi-Voig Dampig Iasso Karafyllis *, Maria Kooriaki ** ad Miroslav Krsic *** * Dep. of Mahemaics, Naioal Techical Uiversiy of Ahes, Zografou Campus,

More information

arxiv:math/ v1 [math.fa] 1 Feb 1994

arxiv:math/ v1 [math.fa] 1 Feb 1994 arxiv:mah/944v [mah.fa] Feb 994 ON THE EMBEDDING OF -CONCAVE ORLICZ SPACES INTO L Care Schü Abrac. I [K S ] i wa how ha Ave ( i a π(i) ) π i equivale o a Orlicz orm whoe Orlicz fucio i -cocave. Here we

More information

ECE534, Spring 2018: Solutions for Problem Set #2

ECE534, Spring 2018: Solutions for Problem Set #2 ECE534, Srig 08: s for roblem Set #. Rademacher Radom Variables ad Symmetrizatio a) Let X be a Rademacher radom variable, i.e., X = ±) = /. Show that E e λx e λ /. E e λx = e λ + e λ = + k= k=0 λ k k k!

More information

5. Matrix exponentials and Von Neumann s theorem The matrix exponential. For an n n matrix X we define

5. Matrix exponentials and Von Neumann s theorem The matrix exponential. For an n n matrix X we define 5. Matrix expoetials ad Vo Neuma s theorem 5.1. The matrix expoetial. For a matrix X we defie e X = exp X = I + X + X2 2! +... = 0 X!. We assume that the etries are complex so that exp is well defied o

More information

MODERN CONTROL SYSTEMS

MODERN CONTROL SYSTEMS MODERN CONTROL SYSTEMS Lecure 9, Sae Space Repreeaio Emam Fahy Deparme of Elecrical ad Corol Egieerig email: emfmz@aa.edu hp://www.aa.edu/cv.php?dip_ui=346&er=6855 Trafer Fucio Limiaio TF = O/P I/P ZIC

More information

On The Generalized Type and Generalized Lower Type of Entire Function in Several Complex Variables With Index Pair (p, q)

On The Generalized Type and Generalized Lower Type of Entire Function in Several Complex Variables With Index Pair (p, q) O he eeralized ye ad eeralized Lower ye of Eire Fucio i Several Comlex Variables Wih Idex Pair, Aima Abdali Jaffar*, Mushaq Shakir A Hussei Dearme of Mahemaics, College of sciece, Al-Musasiriyah Uiversiy,

More information

th m m m m central moment : E[( X X) ] ( X X) ( x X) f ( x)

th 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 information

Advanced Real Analysis

Advanced Real Analysis McGill Uiversity December 26 Faculty of Sciece Fial Exam Advaced Real Aalysis Math 564 December 9, 26 Time: 2PM - 5PM Examier: Dr. J. Galkowski Associate Examier: Prof. D. Jakobso INSTRUCTIONS. Please

More information

Definition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.

Definition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4. 4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad

More information

PAPER : IIT-JAM 2010

PAPER : IIT-JAM 2010 MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure

More information

Actuarial Society of India

Actuarial Society of India Acuarial Sociey of Idia EXAMINAIONS Jue 5 C4 (3) Models oal Marks - 5 Idicaive Soluio Q. (i) a) Le U deoe he process described by 3 ad V deoe he process described by 4. he 5 e 5 PU [ ] PV [ ] ( e ).538!

More information

Fuzzy Dynamic Equations on Time Scales under Generalized Delta Derivative via Contractive-like Mapping Principles

Fuzzy Dynamic Equations on Time Scales under Generalized Delta Derivative via Contractive-like Mapping Principles Idia Joural of Sciece ad echology Vol 9(5) DOI: 7485/ijs/6/v9i5/8533 July 6 ISSN (Pri) : 974-6846 ISSN (Olie) : 974-5645 Fuzzy Dyamic Euaios o ime Scales uder Geeralized Dela Derivaive via Coracive-lie

More information

SUBSERIES CONVERGENCE AND SEQUENCE-EVALUATION CONVERGENCE. Min-Hyung Cho, Hong Taek Hwang and Won Sok Yoo. n t j x j ) = f(x 0 ) f(x j ) < +.

SUBSERIES CONVERGENCE AND SEQUENCE-EVALUATION CONVERGENCE. Min-Hyung Cho, Hong Taek Hwang and Won Sok Yoo. n t j x j ) = f(x 0 ) f(x j ) < +. Kagweo-Kyugki Math. Jour. 6 (1998), No. 2, pp. 331 339 SUBSERIES CONVERGENCE AND SEQUENCE-EVALUATION CONVERGENCE Mi-Hyug Cho, Hog Taek Hwag ad Wo Sok Yoo Abstract. We show a series of improved subseries

More information

Real Analysis Fall 2004 Take Home Test 1 SOLUTIONS. < ε. Hence lim

Real Analysis Fall 2004 Take Home Test 1 SOLUTIONS. < ε. Hence lim Real Aalysis Fall 004 Take Home Test SOLUTIONS. Use the defiitio of a limit to show that (a) lim si = 0 (b) Proof. Let ε > 0 be give. Defie N >, where N is a positive iteger. The for ε > N, si 0 < si

More information

SUMMATION OF INFINITE SERIES REVISITED

SUMMATION 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 information

The Eigen Function of Linear Systems

The Eigen Function of Linear Systems 1/25/211 The Eige Fucio of Liear Sysems.doc 1/7 The Eige Fucio of Liear Sysems Recall ha ha we ca express (expad) a ime-limied sigal wih a weighed summaio of basis fucios: v ( ) a ψ ( ) = where v ( ) =

More information

The Dynamics of a One-Dimensional Parabolic Problem versus the Dynamics of Its Discretization

The Dynamics of a One-Dimensional Parabolic Problem versus the Dynamics of Its Discretization Joural of Differeial Equaios 168, 6792 (2000) doi:10.1006jdeq.2000.3878, available olie a hp:www.idealibrary.com o The Dyamics of a Oe-Dimesioal Parabolic Problem versus he Dyamics of Is Discreizaio Simoe

More information

A note on deviation inequalities on {0, 1} n. by Julio Bernués*

A note on deviation inequalities on {0, 1} n. by Julio Bernués* A oe o deviaio iequaliies o {0, 1}. by Julio Berués* Deparameo de Maemáicas. Faculad de Ciecias Uiversidad de Zaragoza 50009-Zaragoza (Spai) I. Iroducio. Le f: (Ω, Σ, ) IR be a radom variable. Roughly

More information

On stability of first order linear impulsive differential equations

On stability of first order linear impulsive differential equations Ieraioal Joural of aisics ad Applied Mahemaics 218; 3(3): 231-236 IN: 2456-1452 Mahs 218; 3(3): 231-236 218 as & Mahs www.mahsoural.com Received: 18-3-218 Acceped: 22-4-218 IM Esuabaa Deparme of Mahemaics,

More information

Completeness of Random Exponential System in Half-strip

Completeness of Random Exponential System in Half-strip 23-24 Prepri for School of Mahemaical Scieces, Beijig Normal Uiversiy Compleeess of Radom Expoeial Sysem i Half-srip Gao ZhiQiag, Deg GuaTie ad Ke SiYu School of Mahemaical Scieces, Laboraory of Mahemaics

More information

Fractional Lévy Cox-Ingersoll-Ross and Jacobi processes

Fractional Lévy Cox-Ingersoll-Ross and Jacobi processes Holger Fik ad Georg Schlücherma Fracioal Lévy Cox-Igersoll-Ross ad Jacobi rocesses Workig Paer Number 7, 6 Ceer for Quaiaive Risk Aalysis (CEQURA) Dearme of Saisics Uiversiy of Muich h://www.cequra.ui-mueche.de

More information

Introduction to Optimization Techniques

Introduction to Optimization Techniques Itroductio to Optimizatio Techiques Basic Cocepts of Aalysis - Real Aalysis, Fuctioal Aalysis 1 Basic Cocepts of Aalysis Liear Vector Spaces Defiitio: A vector space X is a set of elemets called vectors

More information

Outline. simplest HMM (1) simple HMMs? simplest HMM (2) Parameter estimation for discrete hidden Markov models

Outline. 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 information

Jornal of Kerbala University, Vol. 5 No.4 Scientific.Decembar 2007

Jornal of Kerbala University, Vol. 5 No.4 Scientific.Decembar 2007 Joral of Kerbala Uiversiy, Vol. No. Scieific.Decembar 7 Soluio of Delay Fracioal Differeial Equaios by Usig Liear Mulise Mehod حل الوعادالث التفاضل ت الكسز ت التباطؤ ت باستخذام طز قت هتعذد الخطىاث الخط

More information

Ma 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5

Ma 4121: Introduction to Lebesgue Integration Solutions to Homework Assignment 5 Ma 42: Itroductio to Lebesgue Itegratio Solutios to Homework Assigmet 5 Prof. Wickerhauser Due Thursday, April th, 23 Please retur your solutios to the istructor by the ed of class o the due date. You

More information

UNIT 1: ANALYTICAL METHODS FOR ENGINEERS

UNIT 1: ANALYTICAL METHODS FOR ENGINEERS UNIT : ANALYTICAL METHODS FOR ENGINEERS Ui code: A// QCF Level: Credi vale: OUTCOME TUTORIAL SERIES Ui coe Be able o aalyse ad model egieerig siaios ad solve problems sig algebraic mehods Algebraic mehods:

More information

Integration and Differentiation

Integration and Differentiation ome Clculus bckgroud ou should be fmilir wih, or review, for Mh 404 I will be, for he mos pr, ssumed ou hve our figerips he bsics of (mulivrible) fucios, clculus, d elemer differeil equios If here hs bee

More information

Four equations describe the dynamic solution to RBC model. Consumption-leisure efficiency condition. Consumption-investment efficiency condition

Four equations describe the dynamic solution to RBC model. Consumption-leisure efficiency condition. Consumption-investment efficiency condition LINEAR APPROXIMATION OF THE BASELINE RBC MODEL FEBRUARY, 202 Iroducio For f(, y, z ), mulivariable Taylor liear epasio aroud (, yz, ) f (, y, z) f(, y, z) + f (, y, z)( ) + f (, y, z)( y y) + f (, y, z)(

More information

12 Getting Started With Fourier Analysis

12 Getting Started With Fourier Analysis Commuicaios Egieerig MSc - Prelimiary Readig Geig Sared Wih Fourier Aalysis Fourier aalysis is cocered wih he represeaio of sigals i erms of he sums of sie, cosie or complex oscillaio waveforms. We ll

More information

Averaging of Fuzzy Integral Equations

Averaging of Fuzzy Integral Equations Applied Mahemaics ad Physics, 23, Vol, No 3, 39-44 Available olie a hp://pubssciepubcom/amp//3/ Sciece ad Educaio Publishig DOI:269/amp--3- Averagig of Fuzzy Iegral Equaios Naalia V Skripik * Deparme of

More information

Persistence of Elliptic Lower Dimensional Invariant Tori for Small Perturbation of Degenerate Integrable Hamiltonian Systems

Persistence of Elliptic Lower Dimensional Invariant Tori for Small Perturbation of Degenerate Integrable Hamiltonian Systems JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 08, 37387 997 ARTICLE NO. AY97533 Persisece of Ellipic Lower Dimesioal Ivaria Tori for Small Perurbaio of Degeerae Iegrable Hamiloia Sysems Xu Juxiag Deparme

More information

BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics

BIBECHANA 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 information

MA541 : Real Analysis. Tutorial and Practice Problems - 1 Hints and Solutions

MA541 : Real Analysis. Tutorial and Practice Problems - 1 Hints and Solutions MA54 : Real Aalysis Tutorial ad Practice Problems - Hits ad Solutios. Suppose that S is a oempty subset of real umbers that is bouded (i.e. bouded above as well as below). Prove that if S sup S. What ca

More information

On the Existence and Uniqueness of Solutions for Nonlinear System Modeling Three-Dimensional Viscous Stratified Flows

On the Existence and Uniqueness of Solutions for Nonlinear System Modeling Three-Dimensional Viscous Stratified Flows Joural of Applied Mahemaics ad Physics 58-59 Published Olie Jue i SciRes hp://wwwscirporg/joural/jamp hp://dxdoiorg/6/jamp76 O he Exisece ad Uiqueess of Soluios for oliear Sysem Modelig hree-dimesioal

More information

e x x s 1 dx ( 1) n n!(n + s) + e s n n n=1 n!n s Γ(s) = lim

e x x s 1 dx ( 1) n n!(n + s) + e s n n n=1 n!n s Γ(s) = lim Lecure 3 Impora Special FucioMATH-GA 45. Complex Variable The Euler gamma fucio The Euler gamma fucio i ofe ju called he gamma fucio. I i oe of he mo impora ad ubiquiou pecial fucio i mahemaic, wih applicaio

More information

FUNDAMENTALS OF REAL ANALYSIS by

FUNDAMENTALS OF REAL ANALYSIS by FUNDAMENTALS OF REAL ANALYSIS by Doğa Çömez Backgroud: All of Math 450/1 material. Namely: basic set theory, relatios ad PMI, structure of N, Z, Q ad R, basic properties of (cotiuous ad differetiable)

More information

lim za n n = z lim a n n.

lim za n n = z lim a n n. Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget

More information

Convergence of Solutions for an Equation with State-Dependent Delay

Convergence of Solutions for an Equation with State-Dependent Delay Joural of Mahemaical Aalysis ad Applicaios 254, 4432 2 doi:6jmaa2772, available olie a hp:wwwidealibrarycom o Covergece of Soluios for a Equaio wih Sae-Depede Delay Maria Barha Bolyai Isiue, Uiersiy of

More information