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limite-et-continuité.pdf

p.f(a) + q.f(b)=(p + q).f(c). Exercice 19 [ 01808 ] [Correction]. Notre objectif dans cet exercice est d'établir la proposition :.





31Image formation by Mirrors and Lenses - Physics Courses

pqf = 1?1 qf p Real =fp =(10)(30) ?=15cmpf30 ?10 =? q=? 15=?0 5p30 InvertedReduced Example An object is placed 30 cm in front of a diverginglens with a focal length of -10 cm Find the image distance and magnification + 1=1 pqf = 1?1F qf p10 cm fp( ?10)(30)30 cm == =?7 5cmp?f30 ??( 10)Virtual image =? =? ?=q7 5M 0 25 30 Upright imagereduced



Generalized Fibonacci Sequences and Its Properties - Journal

F pF qF k with k k k t 12 F a F b 012 This was introduced by Gupta Panwar and Sikhwal We shall use the Induction method and Binet’s formula for derivation 1 Introduction It is well-known that the Fibonacci sequence is most prominent examples of recursive sequence The Fibonacci sequence is famous for possessing wonderful



d p x q (notation means f = (pf qf - University of Arizona

around using integration by parts For any two functions f;g in C1 0 [a;b] hLf;gi= Z b a d dx p(x) df dx g(x) + q(x)f(x)g(x)dx = Z b a p(x) df dx dg dx + q(x)f(x)g(x)dx = Z b a d dx p(x) dg dx f(x) + q(x)f(x)g(x)dx = hf;Lgi: Thus S-L operators are self-adjoint on C1 0 [a;b]



Rt The iff - Purdue University

Probability measures P and Q on S coincide if Pf Qf t f e BCS Roof of Theorem 1 2 Suppose that for any closed AES we are able to find a function f e BC S such that I



Notes on Convergence of Probability Measures by Billingsly

gives Qf QF Stringing these together gives PF QF akingT to zero gives PF QF(like in the last theorem QF !QF) Of course the argument works equally well to show QF PF so we get the desired result De nition 7 We say that a probability measure P on S is tight for every >0 there exists a compact set Kso that PK>1 This notion of tight is a

Metric

Space ST L universe metric Four

Stipulations

l f S Rt 2 The y 0 iff a y 3

Tayfly

a 4

16,2Elley

thy z

Examples

of

Metric

Spaces

it

Euclidean

plane

IRwith

Kay Titty

Function

space C a b flay max att yell tea b Iii

Sequence

space l R Tn n 1 195
E Cato easy sup 191
971
For themetric space Se S denote the Borel a field Fora probability measure P onS S wewrite Pf to mean ffdP S for bounded continuous realvalued f For probability measures Pand P we say Pn converges weakly to P and write Pn y p if Pnf Pf for f e BC s where BC s bounded continuous real f on S

Theoremtt

Every probability measure Pon S S is regular that is for each At S I closed F E Aand open G

Asuchthat

P G F L E E o arbitrary Proof of Thm 1.1 Lets identify a class f of sets in S that satisfies the prop C Put the closed sets in

Suppose

A is closed

ChooseFA

and define Gs resl nA is Then P G F P GS PCF

However

if Str o then poi d E F andhence P AGS I PLF as Sto Note Sis open and closed Ci closed under complements

Suppose

A E S satisfies the assertion Then A doesaswell iiclosed under countable additivity

Suppose

An ES satisfies the assertion thatis for en o I Fn closed and an open so that Fn E An E Gn Al and PGnFn E En

Choose

En 92
1 Hence UFn E VAN E U Gn n n L V not nec closed open Find no so that P UF UF E Men and noticethat UF E U An E Un Gn n Eno n Hence G is a o algebra containing the open sets

Conclude

G IS A subclass G C S is called a separating class if a probability measure is completely decided by itsvalueson s that is if PLAY Q A H A'ES then PA QA Y A E S

Theorem

1 I implies that the closedsets in S form a separating class that is they completely decidethe probability measure

Theorem

1.2

Probability

measures P and Q on S coincide if Pf Qf t f e BCS Roof of

Theorem1.2

Suppose

that for any closed AES we areable to find a function f e BC S such that I E f E Ia where A res La A se Then notice PLA PIE PIE

QfÉQA

Since Q A I QEA as e to above implies PCA E Q A if A is closed

Similarly

QAKRA IA 1 it flat l l a A A E

Notice

that IA E fa t Ine A probability measure P on S S is tight if for each e o F a compact set K suchthat PKI E A family Tl of probability measures onS S is tight if each measure inquotesdbs_dbs35.pdfusesText_40
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