A sharp Trudinger-Moser type inequality for unbounded domains in R n

Size: px
Start display at page:

Download "A sharp Trudinger-Moser type inequality for unbounded domains in R n"

Transcription

1 A sharp Trudiger-Moser type iequality for ubouded domais i R Yuxiag Li ad Berhard Ruf Abstract The Trudiger-Moser iequality states that for fuctios u H, 0 (Ω) (Ω R a bouded domai) with Ω u dx oe has Ω (eα u )dx c Ω, with c idepedet of u. Recetly, the secod author has show that for = 2 the boud c Ω may be replaced by a uiform costat d idepedet of Ω if the Dirichlet orm is replaced by the Sobolev orm, i.e. requirig Ω ( u + u )dx. We exted here this result to arbitrary dimesios > 2. Also, we prove that for Ω = R the supremum of (e α u R fuctios is attaied. The proof is based o a blow-up procedure. )dx over all such Keywords: Trudiger-Moser iequality, blow-up, best costat, ubouded domai. Mathematics subject classificatio (2000): 35J50, 46E35 Itroductio Let H,p 0 (Ω), Ω R, be the usual Sobolev space, i.e. the completio of C0 ( ) u H,p (Ω) = ( u p + u p p )dx. Ω It is well-ow that H,p p 0 (Ω) L p (Ω) if p < H,p 0 (Ω) L (Ω) if < p The case p = is the it case of these embeddigs ad it is ow that H, 0 (Ω) L q (Ω) for q < +. (Ω) with the orm Whe Ω is a bouded domai, we usually use the Dirichlet orm u D = ( u dx) i place of H,. I this case, we have the famous Trudiger-Moser iequality (see [P], [T], [M]) for the it case p = which states that { sup (e α u < + whe α α )dx = c(ω, α) (.) u D Ω = + whe α > α where α = ω, ad ω is the measure of the uit sphere i R. The Trudiger-Moser result has bee exteded to Sobolev spaces of higher order ad Soboleve spaces over compact

2 maifolds (see [A], [Fo]). Moreover, for ay bouded Ω, the costat c(ω, α ) ca be attaied. For the attaiability, we refer to [C-C], [F], [Li], [L], [L2], [d-d-r], [L3]. Aother iterestig extesio of (.) is to costruct Trudiger-Moser type iequalities o ubouded domais. Whe = 2, this has bee doe by B. Ruf i [R]. O the other had, for a ubouded domai i R, S. Adachi ad K. Taaa ([A-T]) get a weaer result. Let S. Adachi ad K. Taaa s result says that: 2 Φ(t) = e t t j j!. Theorem A For ay α (0, α ) there is a costat C(α) such that j= u u Φ(α( ) L )dx C(α) (R ) R u L (R ) u L (R ) I this paper, we shall discuss the critical case α = α. followig:, for u H, (R ) \ {0}. (.2) More precisely, we prove the Theorem.. There exists a costat d > 0, s.t. for ay domai Ω R, sup Φ(α u )dx d. (.3) u H, (Ω), u H, (Ω) Ω The iequality is sharp: for ay α > α, the supremum is +. We set Further, we will prove S = Theorem.2. S is attaied. u H, (R ) =, s.t. sup u H, (R ), u H, (R ) R Φ(α u )dx. I other words, we ca fid a fuctio u H, (R ), with S = R Φ(α u )dx. The secod part of Theorem. is trivial: Give ay fixed α > α, we tae β (α, α). By (.) we ca fid a positive sequece {u } i {u H, 0 (B ) : u dx = }, B such that + e βu B = +. By Lio s Lemma, we get u 0. The by the compact embeddig theorem, we may assume u L p (B ) 0 for ay p >. The, R ( u + u )dx, ad u α( u H, ) 2 > βu

3 whe are sufficietly large. So, we get u Φ(α( + R u H, ) )dx + (e βu B )dx = +. The first part of Theorem. ad Theorem.2 will be proved by blow up aalysis. We will use the ideas from [L] ad [L2] (see also [A-M] ad [A-D]). However, i the ubouded case we do ot obtai the strog covergece of u i L (R ), ad so we eed more techiques. Cocretely, we will fid positive ad symmetric fuctios u H, 0 (B R ) which satisfy B R ( u + u )dx = ad Φ(β u )dx = sup B R B ( v + v )=, v H, R 0 (B R ) B R Φ(β v )dx. Here, β is a icreasig sequece tedig to α, ad R is a icreasig sequece tedig to +. Furthermore, u satisfies the followig equatio: div u 2 u + u = u Φ (β u ), λ where λ is a Lagrage multiplier. The, there are two possibilities. If c = max u is bouded from above, the it is easy to see that + R (Φ(β u where u is the wea it of u. R Φ(α u )dx, or ) β u ( )! )dx = (Φ(α u α ) u R ( )! )dx It the follows that either R Φ(β u S α ( )!. If c is ot bouded, the ey poit of the proof is to show that β c (u (r x) c ) log( + c r ), locally for a suitably chose sequece r (ad with c = ( ω ) ), ad that c u G, )dx coverges to o ay Ω R \ {0}, where G is some Gree fuctio. This will be doe i sectio 3. The, we will get i sectio 4 the followig 3

4 Propositio.3. If S ca ot be attaied, the where A = r 0 (G(r) + α log r ). S mi{ α ( )!, ω eαa++/2+ +/() }, So, to prove the attaiability, we oly eed to show that S > mi{ α ( )!, ω eαa++/2+ +/() }. I sectio 5, we will costruct a fuctio sequece u ɛ such that Φ(α uɛ )dx > ω eαa++/2+ +/() R whe ɛ is sufficietly small. Ad i the last sectio we will costruct, for each > 2, a fuctio sequece u ɛ such that for ɛ sufficietly small Φ(α uɛ )dx > α R ( )!. Thus, together with Ruf s result of attaiability i [R] for the case = 2, we will get Theorem.2. 2 The maximizig sequece Let {R } be a icreasig sequece which diverges to ifiity, ad {β } a icreasig sequece which coverges to α. By compactess, we ca fid positive fuctios u H, 0 (B R ) with B R ( u + u )dx = such that Φ(β u )dx = sup Φ(β v )dx. B R B ( v + v )=, v H, B R R Moreover, we may assume that R Φ(β u 0 (B R ) )dx = B R Φ(β u Lemma 2.. Let u as above. The a) u is a maximizig sequece for S; b) u may be chose to be radially symmetric ad decreasig. )dx is icreasig. Proof. a) Let η be a cut-off fuctio which is o B ad 0 o R \ B 2. The give ay ϕ H, (R ) with R ( ϕ + ϕ )dx =, we have τ (L) := ( η( x R L )ϕ + η( x L )ϕ )dx, as L +. Hece for a fixed L ad R > 2L ϕ Φ(β τ(l) )dx B 2L Φ(β η( x L )ϕ τ(l) )dx 4 B R Φ(β u )dx

5 By the Levi Lemma, we the have ϕ Φ(α τ(l) )dx + The, lettig L +, we get Hece, we get + R Φ(α ϕ )dx + Φ(β u )dx = sup R R Φ(β u R Φ(β u R ( v + v )=, v H, (R ) )dx. )dx. R Φ(α v )dx. b) Let u be the radial rearragemet of u, the we have τ := ( u + u )dx ( u + u )dx =. B R B R It is well-ow that τ = iff u is radial. Sice Φ(β u )dx = Φ(β u B R B R we have Φ(β ( u ) )dx B R τ B R Φ(β u ad = holds iff τ =. Hece τ = ad Φ(β u )dx = sup B R B ( v + v )=, v H, R 0 (B R ) )dx, )dx, B R Φ(β v )dx. So, we ca assume u = u ( x ), ad u (r) is decreasig. Assume ow u u. The, to prove Theorem. ad.2, we oly eed to show that Φ(β u + )dx = Φ(α u )dx. R R 3 Blow up aalysis By the defiitio of u we have the equatio div u 2 u + u where λ is the costat satisfyig λ = First, we eed to prove the followig: u B R = u Φ (β u Φ (β u ), (3.) λ )dx. 5

6 Lemma 3.. if λ > 0. Proof. Assume λ 0. The u dx C R u R Φ (β u )dx Cλ 0. Sice u ( x ) is decreasig, we have u (L) u, ad the Set ɛ = u (L) ω L. (3.2) ω L. The u (x) ɛ for ay x /, ad hece we have, usig the form of Φ, that Φ(β u )dx C u dx Cλ 0. R \ R \ Ad o, sice u 0 i L q ( ) for ay q >, we have by Lebesgue [ Φ(β u + )dx Cu + Φ (β u )dx + Cλ + Φ(0)dx + = 0. {x :u (x) } Φ(β u )dx ] This is impossible. We deote c = max u = u (0). The we have Lemma 3.2. If sup c < +, the i) Theorem. holds; ii) if S is ot attaied, the S α ( )!. Proof. If sup c < +, the u u i C loc (R ). By (3.2), we are able to fid L s.t. u (x) ɛ for x /. The R \ (Φ(β u Lettig ɛ 0, we get Hece + + ) β u ( )! )dx C R (Φ(β u Φ(β u ) = R 2 u R \ ) β u ( )! )dx = R Φ(α u )dx + 6 dx Cɛ 2 R u (Φ(α u α ) u R ( )! )dx. α ( )! + 2 dx Cɛ. R (u u )dx. (3.3)

7 Whe u = 0, we ca deduce from (3.3) that Now, we assume u 0. Set τ = By the Levi Lemma, we have τ. Let ũ = u( x τ ). The, we have ũ dx = R ad The ũ dx = τ R R ( ũ + ũ )dx S + α ( )!. R u dx R u dx = + u dx R. u dx R + + R u dx, R u dx. ( u + u )dx =. R Hece, we have by (3.3) S Φ(α ũ )dx R = τ Φ(α u )dx R [ = Φ(α u )dx + (τ α ] ) R R ( )! u dx = Φ(β u + )dx + (τ ) (Φ(α u ) R R = S + (τ ) (Φ(α u α ) R ( )! u )dx + (τ ) α ( )! u )dx R (Φ(α u ) α ( )! u )dx Sice Φ(α u ) α ()! u > 0, we have τ =, ad the So, u is a extremal fuctio. S = R Φ(α u )dx. From ow o, we assume c +. We perform a blow-up procedure: We defie r = c λ e β c. 7

8 By (3.2) we ca fid a sufficietly large L such that u o R \. The (u u (L)) + dx ad hece, by (.), we have e α[(u u (L))+ ] C(L). Clearly, for ay p < α we ca fid a costat C(p), s.t. pu α [(u u (L)) + ] + C(p), ad the we get Hece, λ e β 2 c = e β 2 c C [ e pu u dx e R \ u R \ β 2 c + dx < C = C(L, p). Φ (β u )dx + u e β 2 u u dx. Φ (β u )dx ] Sice u coverges strogly i L q ( ) for ay q >, we get λ Ce β 2 c Now, we set r Ce β 2 c. v (x) = u (r x), w (x) = β c (v c ),, ad hece where v ad w are defied o Ω = {x R : r x B }. Usig the defiitio of r ad (3.) we have div w 2 w = v ( β ) e β (v c ) + O(r c ). c By Theorem 7 i [S], we ow that osc BR ω C(R) for ay R > 0. The from the result i [T] (or [D]), it follows that w C,δ (B R ) < C(R). Therefore w coverges i Cloc ad v c 0 i Cloc. Sice we get β (v v c = c ( + v c ) c ) w i Cloc 0, ad so we have = c ( + v c + O( c c 2 )), div w 2 w = ( α ) e w, (3.4) 8

9 with w(0) = 0 = max w. Sice ω is radially symmetric ad decreasig, it is easy to see that (3.4) has oly oe solutio. We ca chec that w(x) = log( + c x ), ad e w dx =, R where c = ( ω ). The, L + + u e β u r λ dx = L + e w dx =. (3.5) For A >, let u A = mi{u, c A }. We have Lemma 3.3. For ay A >, there holds sup ( u A + u A )dx + R A. (3.6) Proof. Sice {x : u c A } c A {u c A } u, we ca fid a sequece ρ 0 s.t. {x : u c A } B ρ. Sice u coverges i L p (B ) for ay p >, we have u A + {u > c p dx A } + u p {u > c dx = 0, A } ad + R (u c A )+ u p dx = 0 for ay p > 0. Hece, testig equatio (3.) with (u c A ) +, we have R ( (u c A )+ + (u c ) A )+ u dx = (u c R A )+ u λ e β u dx + o() (u c u r A )+ e β u dx + o() λ v c /A = ( v c + ) e w +o() dx + o(). c c Hece if + R ( (u c A )+ + (u c ) A )+ u dx A e w dx. A 9

10 Lettig L +, we get if + R Now observe that ( u A + u A )dx = R ( (u c A )+ + (u c ) A )+ u dx A A. + R ( (u c A )+ + (u c ) A )+ u dx (u c R A )+ u dx ( A ) + o(). u {u > c dx + A } u A {u > c dx A } Hece, we get this Lemma. Corollary 3.4. We have + R \B δ ( u + u )dx = 0, for ay δ > 0, ad the u = 0. Proof. Lettig A +, the for ay costat c, we have ( u + u )dx 0. {u c} So we get this Corollary. Lemma 3.5. We have Φ(β u + )dx R ad cosequetly Proof. We have Φ(β u dx) R L + + λ c +, ad sup {u c A } (e β u r Φ(β u )dx + R Φ(β (u A ) )dx + A )dx = sup c λ, (3.7) c < +. (3.8) λ {u > c A } c λ Φ (β u u R λ )dx Φ (β u )dx. 0

11 Applyig (3.2), we ca fid L such that u o R \. The by Corollary 3.4 ad the form of Φ, we have Φ(pβ (u A ) )dx C(p) u dx = 0 (3.9) + R \ R \ for ay p > 0. Sice by Lemma 3.3 sup R ( u A + + u A )dx A < whe A >, it follows from (.) that sup dx < + for ay p < A. Sice for ay p < p we have p(u A ) sup e p β ((ua u (L))+ ) p ((u A u (L)) + ) + C(p, p ), Φ(pβ (u A ) )dx < + (3.0) for ay p < A. The o BL, by the wea compactess of Baach space, we get + Φ(β (u A ) )dx = Φ(0)dx = 0. Hece we have + Φ(β u )dx R A L + = A + c λ c λ + Cɛ. u λ Φ (β u )dx + Cɛ As A ad ɛ 0 we obtai (3.7). If λ c was bouded or sup c λ = +, it would follow from (3.7) that sup R ( v + v )dx=,v H, (R ) R Φ(α v )dx = 0, which is impossible. λ u Lemma 3.6. We have that c we have + Φ (β u ) coverges to δ 0 wealy, i.e. for ay ϕ D(R ) R ϕ c u λ Φ (β u )dx = ϕ(0).

12 Proof. Suppose supp ϕ B ρ. We split the itegral B ρ ϕ c u λ Φ (β u )dx + {u c A }\r + r {u < c A } We have I A ϕ C 0 u R \r λ = I + I 2 + I 3. Φ (β u )dx = A ϕ C 0( e w +o() dx), ad I 2 = ϕ(r x) c (c + (v c )) B L c By (3.9) ad (3.0) we have e w+o() dx = ϕ(0) R Φ(pβ u A )dx < C for ay p < A. We set q + p =. The we get by (3.8) e w dx + o() = ϕ(0) + o(). I 3 = {u c A } ϕ c u λ Φ (β u )dx c λ ϕ C 0 u L q (R ) e β u A L p (R ) 0. Lettig L +, we deduce ow that + R ϕ c u λ Φ (β u )dx = ϕ(0). Propositio 3.7. O ay Ω R \{0}, we have that c G C,α loc (R \ {0}) satisfies the followig equatio: Proof. We set U = c u coverges to G i C (Ω), where div G 2 G + G = δ 0. (3.) u, which satisfy by (3.) the equatios: div U 2 U + U For our purpose, we eed to prove that = c u λ B R U q dx C(q, R), 2 Φ (β u ). (3.2)

13 where C(q, R) does ot deped o. We use the idea i [St] to prove this statemet. Set Ω t = {0 U t}, U t = mi{u, t}. The we have ( U t + U t )dx Ω t ( U t U + U t U ) = R R U t c u λ Φ (β u )dx 2t. Let η be a radially symmetric cut-off fuctio which is o B R ad 0 o B c 2R. The, B 2R ηu t dx C (R) + C 2 (R)t. The, whe t is bigger tha C (R) C 2 (R), we have B 2R ηu t dx 2C 2 (R) t. Set ρ such that U (ρ) = t. The we have { } if v dx : v H, 0 (B 2R ) ad v Bρ = t 2C 2 (R) t. B 2R O the other had, the if is achieved by t log x 2R / log 2R ρ. By a direct computatio, we have ad hece for ay t > C (R) C 2 (R) ω t (log 2R ρ ) 2C 2(R), {x B 2R : U t} = B ρ C 3 (R)e A(R)t, where A(R) is a costat oly depedig o R. The, for ay δ < A, B R e δu dx µ({m U m + })e δ(m+) m=0 e (A δ)m e δ C. m=0 The, testig the equatio (3.2) with the fuctio log +2(U U (R)) + +(U U (R)) + U ( + U U (R))( + 2U 2U (R)) dx B R log 2 B R c u λ Φ (β u )dx, we get U log + 2(U U (R)) B R + (U U (R)) dx C. Give q <, by Youg s Iequality, we have [ U q U dx B R B R ( + U U (R))( + 2U 2U (R)) + (( + U )( + 2U )) [ U ] ( + U U (R))( + 2U 2U (R)) + CeδU dx. B R 3 q ] dx

14 Hece, we are able to assume that U coverges to a fuctio G wealy i H,p (B R ) for ay R ad p <. Applyig Lemma 3.6, we get (3.). Hece U is bouded i L q (Ω) for ay q > 0. By Corollary 3.4 ad Theorem A, e β u is also bouded i L q (Ω) for ay q > 0. The, applyig Theorem 2.8 i [S], ad the mai result i [T] (or [D]), we get U C,α (Ω) C. So, U coverges to G i C (Ω). For the Gree fuctio G we have the followig results: Lemma 3.8. G C,α loc (R \ {0}) ad ear 0 we ca write G = α log r + A + O(r log r) ; (3.3) here, A is a costat. Moreover, for ay δ > 0, we have ( c + u + (c u ) )dx = ( G + G )dx R \B δ R \B δ = G(δ)( G dx). B δ Proof. Slightly modifyig the proof i [K-L], we ca prove G = α log r + A + o(). Oe ca refer to [L2] for details. Further, testig the equatio (3.2) with, we get ω G (r) r 2 G = G B r = G dx = + O(r log r). B r The, we get (3.3). We have u R \B δ Φ (β u )dx C Recall that U H, 0 (B R ). By equatio (3.2) we get R \B δ ( U + U )dx = c By (3.4) ad (3.8) we the get + λ u R \B δ Φ (β u u dx 0. (3.4) R \B δ ( U + U )dx = R \B δ + = G(δ) )dx B δ U U 2 U ds. B δ U U 2 U ds G 2 ds = G(δ)( G dx). B δ 4 B δ G

15 We are ow i the positio to complete the proof of Theorem.: We have see i (3.9) that Φ(β u )dx C. R \B R So, we oly eed to prove o B R, e β u B R dx < C The classical Trudiger-Moser iequality implies that dx < C = C(R). By Propositio 3.7, u (R) = O( B R e β ((u u (R))+ ) c ), ad hece we have u ((u u (R)) + + u (R)) ((u u (R)) + ) + C, The, we get e β u B R C. 4 The proof of Propositio.3 We will use a result of Carleso ad Chag (see [C-C]): Lemma 4.. Let B be the uit ball i R. Assume that u is a sequece i H, 0 (B) with B u dx =. If u 0, the sup (e α u )dx B e +/2+ +/(). + Proof of Propositio.3: Set u (x) = (u (x) u (δ)) + u L (Bδ ) of Carleso ad Chag, we have sup + By Lemma 3.8, we have ( c R \B δ B e β u B δ u + (c which is i H, 0 (B δ ). The by the result B δ ( + e +/2+ +/() ). u ) )dx G(δ)( B δ G dx), 5

16 ad therefore we get u dx = B δ ( u + u )dx R \B δ u dx = G(δ) + ɛ (δ), (4.) B δ c where ɛ (δ) = 0. δ 0 + By (3.9) i Lemma 3.5 we have L + + e β u B ρ\r dx = B ρ, for ay ρ < δ. Furthermore, o B ρ we have by (4.) (u ) u ( G(δ)+ɛ (δ) c ) = u ( + G(δ) + ɛ (δ) + O( c c 2 = u + G(δ)(u ) + O(c c ) )) u log δ ( )α. The we have L + + e β u B ρ\r dx O(δ ) L + + e β u B ρ\r dx B ρ O(δ ). Sice u 0 o B δ \ B ρ, we get (e βu + B δ \B ρ the Lettig ρ 0, we get 0 L + + L + + B δ \r (e β u B δ \r (e β u )dx = 0, )dx B ρ O(δ ). )dx = 0. So, we have L + + r (e β u )dx e +/2+ +/() B δ. 6

17 Now, we fix a L. The for ay x r, we have u β u = β ( u ) L ( u (Bδ ) dx) B δ = β (u + u (δ) u ) L ( u (Bδ ) dx) B δ ( usig that u (δ) = O( ) ad u L (B δ ) = + O( = β ( u + u (δ) + O( + c = β u ( + u (δ) + O( u [ c = β u It is easy to chec that c ) ) ( B δ u dx ) 2 + u (δ) u ) ) ( G(δ)+ɛ (δ) c ] G(δ)+ɛ (δ) c + O( 2 c ) u (r x) c, ad ( u (r x) ) u (δ) G(δ). ). c ) ) So, we get L + + (e β u r )dx = L + + eαg(δ) r (e β u )dx e αg(δ) δ ω e +/2+ +/() = e α( α log δ +A+O(δ log δ)) δ ω e +/2+ +/(). Lettig δ 0, the the above iequality together with Lemma 3.2 imply Propositio.3. 5 The test fuctio I this sectio, we will costruct a fuctio sequece {u ɛ } H, (R ) with u ɛ H, = which satisfies Φ(α u ɛ ω )dx > eαa++/2+ +/(), R for ɛ > 0 sufficietly small. Let C () log(+c x ɛ )+Λ ɛ x Lɛ u ɛ = α C G( x ) C x > Lɛ, where Λ ɛ, C ad L are fuctios of ɛ (which will be defied later, by ( 5.), (5.2), (5.5) ) which satisfy 7

18 ad i) L +, C +, ad Lɛ 0, as ɛ 0 ; ii) C iii) log L C () log(+cl )+Λ ɛ α C 0, as ɛ 0. = G(Lɛ) C ; We use the ormalizatio of u ɛ to obtai iformatio o Λ ɛ, C ad L. We have ( u ɛ + u ( ɛ )dx = R \ɛ C G dx + G dx ) Bc Lɛ BLɛ c 2 G = C G(Lɛ) G ɛ ds G(Lɛ) G(Lɛ) G dx B = Lɛ. ɛ u ɛ dx = = α C α C cl 0 cl 0 C u ( + u) du (( + u) ) ( + u) du where we used the fact = α C + α C 2 =0 C ( ) log( + c L ) + O( L ) C = ( ) + /2 + /3 + + /( ) α C + α C log( + c L ) + O( L ), C 2 =0 C ( ) = It is easy to chec that ad thus we get R ( u ɛ + u ɛ )dx = ɛ u ɛ dx = O((Lɛ) C log L), { ( ) ( + /2 + + /( ) ) + α α C A } +( ) log( + c L ) log(lɛ) + φ, 8

19 where ) φ = O ((Lɛ) C log L + (Lɛ) log Lɛ + L. Settig R ( u ɛ + u ɛ )dx =, we obtai α C = ( ) ( + /2 + + /( ) ) + α A + log (+cl ) L log ɛ + φ By ii) we have ad hece this implies that = ( ) ( + /2 + + /( ) ) + α A + log ω log ɛ + φ. α C ( ) log( + c L ) + Λɛ = αg(lɛ) ( ) ( + /2 + + /( ) ) + α A log (Lɛ) + φ + Λ ɛ = αg(lɛ) ; Next, we compute (5.) Λ ɛ = ( )( + /2 + + /( )) + φ. (5.2) e α uɛ ɛ dx. Clearly, ϕ(t) = t + t is icreasig whe 0 t ad decreasig whe t 0, the t t, whe t <. Thus we have by ii), for ay x ɛ The we have α uɛ = α C ( ) log( + c x ɛ ) + Λɛ α C α C ( ) log( + c x ( ɛ ) + Λɛ ). α C ɛ e α uɛ dx e αc log(+c x ɛ ) Λɛ ɛ = e αc Λɛ ɛ ( + c x dx ) = e αc Λɛ ( )ɛ cl = e αc Λɛ ( )ɛ cl 0 0 = e αc Λɛ ɛ ( + O(L )) u 2 ( + u) du ((u + ) ) 2 ( + u) du (5.3) = ω eαa++/2+ +/() ) + O ((Lɛ) C log L + L + (Lɛ) log Lɛ. 9

20 Here, we used the fact The ɛ Φ(α u ɛ m =0 ( ) m m + C m = m +. )dx ω ) eαa++/2+ +/() +O ((Lɛ) C log L + L + (Lɛ) log Lɛ. Moreover, o R \ ɛ we have the estimate Φ(α uɛ )dx α R \ɛ ( )! R \ɛ G(x) C dx, ad thus we get Φ(α uɛ )dx ω R + α G(x) ()! R \ɛ C = ω eαa++/2+ +/() [ + α ()! C R \ɛ eαa++/2+ +/() ) dx + O ((Lɛ) C log L + L + (Lɛ) log Lɛ G(x) dx + O ((Lɛ) C + C log L + L ) ] + C (Lɛ) log Lɛ (5.4) We ow set the Lɛ 0 as ɛ 0. equatio (5.). We set L = log ɛ ; (5.5) We the eed to prove that there exists a C = C(ɛ) which solves Sice f(t) = α t ( )( + /2 + + /( )) + α A + log ω log ɛ + φ, for ɛ small, ad for ɛ small, f has a zero i satisfies Therefore, as ɛ 0, we have f(( 2 α log ɛ ) ) = log ɛ + o() + φ < 0 f(( log ɛ ) ) = 2α 2 log ɛ + o() + φ > 0 ( ( 2α log ɛ ), ( 2 α log ɛ ) α C = log ɛ + O(). log L C 0, ). Thus, we defied C, ad it 20

21 ad the (Lɛ) C + log L + C L + C (Lɛ) log Lɛ 0. Therefore, i), ii), iii) hold ad we ca coclude from (5.4) that for ɛ > 0 sufficietly small Φ(α uɛ )dx > ω eαa++/2+ +/(). R 6 The test fuctio 2 I this sectio we costruct, for > 2, fuctios u ɛ such that u ɛ Φ(α ( ) α )dx > R u ɛ H, ( )!, for ɛ > 0 sufficietly small. Let ɛ = e αc, ad u ɛ = c log x L α c x < Lɛ Lɛ x L 0 L x, where L is a fuctio of ɛ which will be defied later. ad We have The u ɛ Φ(α ( R u ɛ H, u ɛ dx = ω R ) α )dx ( )! R u ɛ =, c (Lɛ) + ω L α c R u ɛ dx + R u ɛ dx + α! ɛ r log rdr. 2 R \ɛ uɛ ( + R u ɛ dx) dx = α ( )! α ( )! + ω c (Lɛ) + ω L α c ɛ r log rdr + α! ( 2 ω L /c () 2 ( α ) 2 + ω c (Lɛ) + ω L α c ɛ r log 2 r ɛ r log rdr ) We ow as that L satisfies c 0, as ɛ 0. (6.) L 2

22 The, for sufficietly small ɛ, we have α ( )! + α! ( + ω c (Lɛ) + ω L α c 2 ω L /c () 2 ( α ) 2 + ω c (Lɛ) + ω L α c 2 c B L B2 L = c L ( = c L (B 2 2 c L ( 2) c B 2 ) B 2 ), ɛ r log rdr + ɛ r log 2 r ɛ r log rdr ) where B, B 2 are positive costats. Whe > 2, we may choose L = b c 2 ; the, for b sufficietly large, we have L ( 2) B B c 2 = B b ( 2) B 2 > 0, ad (6.) holds. Thus, we have proved that for ɛ > 0 sufficietly small u ɛ Φ(α ( R u ɛ H, (R ) ) )dx > α ( )!. Refereces [A-T] S. Adachi ad K. Taaa: Trudiger type iequalites i R ad their best expoets. Proc. AMS, 28: ,999. [A] D. R. Adams: A sharp iequality of J. Moser for higher order derivatives. Aals of Math., 28: , 988. [A-D] Adimurthi ad O.Druet, Blow up aalysis i dimesio 2 ad a sharp form of Trudiger Moser iequality, Comm. PDE 29 No -2, (2004), [A-M] Adimurthi ad M. Struwe: Global Compactess properties of semiliear elliptic equatios with critical expoetial growth. J. Fuct. Aal., 75, o :25 67, [C-C] L. Carleso ad S. Y. A. Chag: o the existece of a extremal fuctio for a iequality of J.Moser. Bull. Sc. Math., 0, 3-27, 986. [d-d-r] D.G. de Figueiredo, J.M. do O, B.Ruf: O a iequality by N.Trudiger ad J.Moser ad related elliptic equatios. Comm. Pure. Appl. Math., 55, 35-52, [D] E. DiBeedetto: C,α local regularity of wea solutio of degeerate elliptic equatios. Noliear Aalysis. 7 o.8, ,

23 [F] M. Flucher: Extremal fuctios for Trudiger-Moser iequality i 2 dimesios. Comm. Math. Helv., 67,47-497, 992. [Fo] L. Fotaa: Sharp borderlie Sobolev iequalities o compact Riemaia maifolds. Comm. Math. Helv., 68,45-454, 993. [K-L] S. Kicheassamy ad L. Vero: Sigular solutios of the p-laplace equatio. Math. A., 275,599-65, 986. [L] Y. Li: Moser-Trudiger iequality o maifold of dimesio two. J. Partial Differetial Equatios 4, o. 2, 63 92, 200. [L2] Y. Li: The extremal fuctios for Moser-Trudiger iequality o compact Riemaia maifolds, To appear i Sci. Chiese, series A. [L3] Y. Li: Remars o the Extremal Fuctios for the Moser-Trudiger Iequalities. To appear i Acta Mathematica Siica. [L-L] Y. Li ad P. Liu: A Moser-Trudiger iequality o the boudary of a Riemma Surface. Math. Z., 250, o.2, , [Li] K.C. Li: Extremal fuctios for Moser s iequality. Tras. Amer. Math. Sco., 348, , 996. [M] J. Moser: A sharp form of a Iequality by N.Trudiger. Id. Uiv. Math. J., 20,077-09, 97. [P] S. I. Pohozaev: The Sobolev embeddig i the case pl =. Proceedigs of the Techical Scietific Coferece o Advaces of Scietific Research Mathematics Sectio, 58-70, Mosov. Eerget. Ist., Moscow, 965 [R] B. Ruf: A sharp Trudiger-Moser type iequality for ubouded domais i R 2. J. Fuct. Aal. 29, o. 2, , [T] P. Tolsdorf: Regularity for a more geeral class of qusiliear elliptic equatios. J.D.E., 5: [S] J. Serri: Local behavior of solutios of qusai-liear equatios. Acta. Math.,, , 964. [St] M. Struwe: Positive solutio of critical semiliear elliptic equatios o o-cotractible plaar domai. J. Eur. Math. Soc., 2(4): , [Tr] N. S. Trudiger: O embeddig ito Orlicz space ad some applicatios, J. Math. Mech. 7: , 967. Yuxiag Li ICTP, Mathematics Sectio, Strada Costiera, I-3404 Trieste, Italy address: liy@ictp.it Berhard Ruf Dipartimeto di Matematica, Uiversità di Milao, via Saldii 50, 2033 Mila, Italy address: Berhard.Ruf@mat.uimi.it 23

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

More information

THE ABRACADABRA PROBLEM

THE ABRACADABRA PROBLEM THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected

More information

Class Meeting # 16: The Fourier Transform on R n

Class Meeting # 16: The Fourier Transform on R n MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,

More information

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009) 18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

More information

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series 8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series

More information

Asymptotic Growth of Functions

Asymptotic Growth of Functions CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

More information

Infinite Sequences and Series

Infinite Sequences and Series CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

More information

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

More information

A note on the boundary behavior for a modified Green function in the upper-half space

A note on the boundary behavior for a modified Green function in the upper-half space Zhag ad Pisarev Boudary Value Problems (015) 015:114 DOI 10.1186/s13661-015-0363-z RESEARCH Ope Access A ote o the boudary behavior for a modified Gree fuctio i the upper-half space Yulia Zhag1 ad Valery

More information

ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE

ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE by Maria E Gageoea ad Silvia Moldoveau

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

More information

Factors of sums of powers of binomial coefficients

Factors of sums of powers of binomial coefficients ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the

More information

ON THE DENSE TRAJECTORY OF LASOTA EQUATION

ON THE DENSE TRAJECTORY OF LASOTA EQUATION UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLIII 2005 ON THE DENSE TRAJECTORY OF LASOTA EQUATION by Atoi Leo Dawidowicz ad Najemedi Haribash Abstract. I preseted paper the dese trajectory

More information

Convexity, Inequalities, and Norms

Convexity, Inequalities, and Norms Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for

More information

Irreducible polynomials with consecutive zero coefficients

Irreducible polynomials with consecutive zero coefficients Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK

THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E MCCARTHY, SANDRA POTT, AND BRETT D WICK Abstract We provide a ew proof of Volberg s Theorem characterizig thi iterpolatig sequeces as those for

More information

Sequences and Series

Sequences and Series CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

More information

Department of Computer Science, University of Otago

Department of Computer Science, University of Otago Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

More information

Chapter 5: Inner Product Spaces

Chapter 5: Inner Product Spaces Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

More information

A probabilistic proof of a binomial identity

A probabilistic proof of a binomial identity A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

More information

Lecture 5: Span, linear independence, bases, and dimension

Lecture 5: Span, linear independence, bases, and dimension Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;

More information

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

More information

4.3. The Integral and Comparison Tests

4.3. The Integral and Comparison Tests 4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece

More information

On the L p -conjecture for locally compact groups

On the L p -conjecture for locally compact groups Arch. Math. 89 (2007), 237 242 c 2007 Birkhäuser Verlag Basel/Switzerlad 0003/889X/030237-6, ublished olie 2007-08-0 DOI 0.007/s0003-007-993-x Archiv der Mathematik O the L -cojecture for locally comact

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Research Article Sign Data Derivative Recovery

Research Article Sign Data Derivative Recovery Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov

More information

THE HEIGHT OF q-binary SEARCH TREES

THE HEIGHT OF q-binary SEARCH TREES THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average

More information

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL. Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory

More information

INFINITE SERIES KEITH CONRAD

INFINITE SERIES KEITH CONRAD INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal

More information

AP Calculus BC 2003 Scoring Guidelines Form B

AP Calculus BC 2003 Scoring Guidelines Form B AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet

More information

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

More information

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if

More information

A Note on Sums of Greatest (Least) Prime Factors

A Note on Sums of Greatest (Least) Prime Factors It. J. Cotemp. Math. Scieces, Vol. 8, 203, o. 9, 423-432 HIKARI Ltd, www.m-hikari.com A Note o Sums of Greatest (Least Prime Factors Rafael Jakimczuk Divisio Matemática, Uiversidad Nacioal de Luá Bueos

More information

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

More information

Entropy of bi-capacities

Entropy of bi-capacities Etropy of bi-capacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc 44306 Nates, Frace iva.kojadiovic@uiv-ates.fr Jea-Luc Marichal Applied Mathematics

More information

MARTINGALES AND A BASIC APPLICATION

MARTINGALES AND A BASIC APPLICATION MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this

More information

Analysis Notes (only a draft, and the first one!)

Analysis Notes (only a draft, and the first one!) Aalysis Notes (oly a draft, ad the first oe!) Ali Nesi Mathematics Departmet Istabul Bilgi Uiversity Kuştepe Şişli Istabul Turkey aesi@bilgi.edu.tr Jue 22, 2004 2 Cotets 1 Prelimiaries 9 1.1 Biary Operatio...........................

More information

Lecture 4: Cheeger s Inequality

Lecture 4: Cheeger s Inequality Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a d-regular

More information

Overview of some probability distributions.

Overview of some probability distributions. Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

More information

Elementary Theory of Russian Roulette

Elementary Theory of Russian Roulette Elemetary Theory of Russia Roulette -iterestig patters of fractios- Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some

More information

Building Blocks Problem Related to Harmonic Series

Building Blocks Problem Related to Harmonic Series TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite

More information

The Gompertz Makeham coupling as a Dynamic Life Table. Abraham Zaks. Technion I.I.T. Haifa ISRAEL. Abstract

The Gompertz Makeham coupling as a Dynamic Life Table. Abraham Zaks. Technion I.I.T. Haifa ISRAEL. Abstract The Gompertz Makeham couplig as a Dyamic Life Table By Abraham Zaks Techio I.I.T. Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 32000, Haifa, Israel Abstract A very famous

More information

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics

More information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

More information

A Faster Clause-Shortening Algorithm for SAT with No Restriction on Clause Length

A Faster Clause-Shortening Algorithm for SAT with No Restriction on Clause Length Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 49-60 A Faster Clause-Shorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer Sciece

More information

Acta Acad. Paed. Agriensis, Sectio Mathematicae 29 (2002) 77 87. ALMOST SURE FUNCTIONAL LIMIT THEOREMS IN L p( ]0, 1[ ), WHERE 1 p <

Acta Acad. Paed. Agriensis, Sectio Mathematicae 29 (2002) 77 87. ALMOST SURE FUNCTIONAL LIMIT THEOREMS IN L p( ]0, 1[ ), WHERE 1 p < Acta Acad. Paed. Agriesis, Sectio Mathematicae 29 22) 77 87 ALMOST SUR FUNCTIONAL LIMIT THORMS IN L ], [ ), WHR < József Túri Nyíregyháza, Hugary) Dedicated to the memory of Professor Péter Kiss Abstract.

More information

Theorems About Power Series

Theorems About Power Series Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

More information

1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION 1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

More information

Lipschitz maps and nets in Euclidean space

Lipschitz maps and nets in Euclidean space Lipschitz maps ad ets i Euclidea space Curtis T. McMulle 1 April, 1997 1 Itroductio I this paper we discuss the followig three questios. 1. Give a real-valued fuctio f L (R ) with if f(x) > 0, is there

More information

Degree of Approximation of Continuous Functions by (E, q) (C, δ) Means

Degree of Approximation of Continuous Functions by (E, q) (C, δ) Means Ge. Math. Notes, Vol. 11, No. 2, August 2012, pp. 12-19 ISSN 2219-7184; Copyright ICSRS Publicatio, 2012 www.i-csrs.org Available free olie at http://www.gema.i Degree of Approximatio of Cotiuous Fuctios

More information

1 Computing the Standard Deviation of Sample Means

1 Computing the Standard Deviation of Sample Means Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

More information

Running Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis

Running Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.

More information

A Recursive Formula for Moments of a Binomial Distribution

A Recursive Formula for Moments of a Binomial Distribution A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,

More information

Modified Line Search Method for Global Optimization

Modified Line Search Method for Global Optimization Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o

More information

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas: Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries

More information

3. Greatest Common Divisor - Least Common Multiple

3. Greatest Common Divisor - Least Common Multiple 3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

An Efficient Polynomial Approximation of the Normal Distribution Function & Its Inverse Function

An Efficient Polynomial Approximation of the Normal Distribution Function & Its Inverse Function A Efficiet Polyomial Approximatio of the Normal Distributio Fuctio & Its Iverse Fuctio Wisto A. Richards, 1 Robi Atoie, * 1 Asho Sahai, ad 3 M. Raghuadh Acharya 1 Departmet of Mathematics & Computer Sciece;

More information

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

More information

1 Correlation and Regression Analysis

1 Correlation and Regression Analysis 1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

More information

CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homework 4 Solutions CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

More information

Metric, Normed, and Topological Spaces

Metric, Normed, and Topological Spaces Chapter 13 Metric, Normed, ad Topological Spaces A metric space is a set X that has a otio of the distace d(x, y) betwee every pair of poits x, y X. A fudametal example is R with the absolute-value metric

More information

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5 Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

More information

. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2

. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2 4. Basic feasible solutios ad vertices of polyhedra Due to the fudametal theorem of Liear Programmig, to solve ay LP it suffices to cosider the vertices (fiitely may) of the polyhedro P of the feasible

More information

Lecture 3. denote the orthogonal complement of S k. Then. 1 x S k. n. 2 x T Ax = ( ) λ x. with x = 1, we have. i = λ k x 2 = λ k.

Lecture 3. denote the orthogonal complement of S k. Then. 1 x S k. n. 2 x T Ax = ( ) λ x. with x = 1, we have. i = λ k x 2 = λ k. 18.409 A Algorithmist s Toolkit September 17, 009 Lecture 3 Lecturer: Joatha Keler Scribe: Adre Wibisoo 1 Outlie Today s lecture covers three mai parts: Courat-Fischer formula ad Rayleigh quotiets The

More information

Hypothesis testing. Null and alternative hypotheses

Hypothesis testing. Null and alternative hypotheses Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

More information

Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT

Vladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee

More information

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8 CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive

More information

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Cosider a legth- sequece x[ with a -poit DFT X[ where Represet the idices ad as +, +, Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Usig these

More information

A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design

A Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA 16804-0030 haupt@ieee.org Abstract:

More information

Math 113 HW #11 Solutions

Math 113 HW #11 Solutions Math 3 HW # Solutios 5. 4. (a) Estimate the area uder the graph of f(x) = x from x = to x = 4 usig four approximatig rectagles ad right edpoits. Sketch the graph ad the rectagles. Is your estimate a uderestimate

More information

Heavy Traffic Analysis of a Simple Closed Loop Supply Chain

Heavy Traffic Analysis of a Simple Closed Loop Supply Chain Heavy Traffic Aalysis of a Simple Closed Loop Supply Chai Arka Ghosh, Sarah M. Rya, Lizhi Wag, ad Aada Weerasighe April 8, 2 Abstract We cosider a closed loop supply chai where ew products are produced

More information

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006 Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam

More information

ABOUT A DEFICIT IN LOW ORDER CONVERGENCE RATES ON THE EXAMPLE OF AUTOCONVOLUTION

ABOUT A DEFICIT IN LOW ORDER CONVERGENCE RATES ON THE EXAMPLE OF AUTOCONVOLUTION ABOUT A DEFICIT IN LOW ORDER CONVERGENCE RATES ON THE EXAMPLE OF AUTOCONVOLUTION STEVEN BÜRGER AND BERND HOFMANN Abstract. We revisit i L 2 -spaces the autocovolutio equatio x x = y with solutios which

More information

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13 EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

More information

Journal of Combinatorial Theory, Series A

Journal of Combinatorial Theory, Series A Joural of Combiatorial Theory, Series A 118 011 319 345 Cotets lists available at ScieceDirect Joural of Combiatorial Theory, Series A www.elsevier.com/locate/jcta Geeratig all subsets of a fiite set with

More information

Incremental calculation of weighted mean and variance

Incremental calculation of weighted mean and variance Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

More information

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The

More information

Convex Bodies of Minimal Volume, Surface Area and Mean Width with Respect to Thin Shells

Convex Bodies of Minimal Volume, Surface Area and Mean Width with Respect to Thin Shells Caad. J. Math. Vol. 60 (1), 2008 pp. 3 32 Covex Bodies of Miimal Volume, Surface Area ad Mea Width with Respect to Thi Shells Károly Böröczky, Károly J. Böröczky, Carste Schütt, ad Gergely Witsche Abstract.

More information

BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets

BENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets BENEIT-CST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal - Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts

More information

Lehmer s problem for polynomials with odd coefficients

Lehmer s problem for polynomials with odd coefficients Aals of Mathematics, 166 (2007), 347 366 Lehmer s problem for polyomials with odd coefficiets By Peter Borwei, Edward Dobrowolski, ad Michael J. Mossighoff* Abstract We prove that if f(x) = 1 k=0 a kx

More information

2. Degree Sequences. 2.1 Degree Sequences

2. Degree Sequences. 2.1 Degree Sequences 2. Degree Sequeces The cocept of degrees i graphs has provided a framewor for the study of various structural properties of graphs ad has therefore attracted the attetio of may graph theorists. Here we

More information

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).

Example 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here). BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly

More information

How To Solve The Homewor Problem Beautifully

How To Solve The Homewor Problem Beautifully Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log

More information

NOTES ON PROBABILITY Greg Lawler Last Updated: March 21, 2016

NOTES ON PROBABILITY Greg Lawler Last Updated: March 21, 2016 NOTES ON PROBBILITY Greg Lawler Last Updated: March 21, 2016 Overview This is a itroductio to the mathematical foudatios of probability theory. It is iteded as a supplemet or follow-up to a graduate course

More information

Chapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity

More information

THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE

THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE JAVIER CILLERUELO Abstract. We obtai, for ay irreducible quadratic olyomial f(x = ax 2 + bx + c, the asymtotic estimate log l.c.m. {f(1,..., f(} log. Whe

More information

2-3 The Remainder and Factor Theorems

2-3 The Remainder and Factor Theorems - The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x

More information

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is 0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

More information

Partial Di erential Equations

Partial Di erential Equations Partial Di eretial Equatios Partial Di eretial Equatios Much of moder sciece, egieerig, ad mathematics is based o the study of partial di eretial equatios, where a partial di eretial equatio is a equatio

More information

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required.

S. Tanny MAT 344 Spring 1999. be the minimum number of moves required. S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,

More information

Rényi Divergence and L p -affine surface area for convex bodies

Rényi Divergence and L p -affine surface area for convex bodies Réyi Divergece ad L p -affie surface area for covex bodies Elisabeth M. Werer Abstract We show that the fudametal objects of the L p -Bru-Mikowski theory, amely the L p -affie surface areas for a covex

More information

arxiv:1506.03481v1 [stat.me] 10 Jun 2015

arxiv:1506.03481v1 [stat.me] 10 Jun 2015 BEHAVIOUR OF ABC FOR BIG DATA By Wetao Li ad Paul Fearhead Lacaster Uiversity arxiv:1506.03481v1 [stat.me] 10 Ju 2015 May statistical applicatios ivolve models that it is difficult to evaluate the likelihood,

More information

LECTURE 13: Cross-validation

LECTURE 13: Cross-validation LECTURE 3: Cross-validatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Three-way data partitioi Itroductio to Patter Aalysis Ricardo Gutierrez-Osua Texas A&M

More information

Solutions to Selected Problems In: Pattern Classification by Duda, Hart, Stork

Solutions to Selected Problems In: Pattern Classification by Duda, Hart, Stork Solutios to Selected Problems I: Patter Classificatio by Duda, Hart, Stork Joh L. Weatherwax February 4, 008 Problem Solutios Chapter Bayesia Decisio Theory Problem radomized rules Part a: Let Rx be the

More information

5 Boolean Decision Trees (February 11)

5 Boolean Decision Trees (February 11) 5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected

More information

Chapter 14 Nonparametric Statistics

Chapter 14 Nonparametric Statistics Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they

More information

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized? 5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso

More information