Area distortion of quasiconformal mappings


 Osborne Beasley
 2 years ago
 Views:
Transcription
1 Area dstorton of quasconformal mappngs K. Astala 1 Introducton A homeomorphsm f : Ω Ω between planar domans Ω and Ω s called Kquasconformal f t s contaned n the Sobolev class W2,loc 1 (Ω) and ts drectonal dervatves satsfy max α α f(x) Kmn α α f(x) a.e. x Ω. In recent years quasconformal mappngs have been an effcent tool n the study of dynamcal systems of the complex plane. We show here that, n turn, methods or deas from dynamcal systems can be used to solve a number of open questons n the theory of planar quasconformal mappngs. It has been known snce the work of Ahlfors [A] and Mor [Mo] that Kquasconformal mappngs are locally Hölder contnuous wth exponent 1/K. The functon f 0 (z) = z z 1 K 1 (1) shows that ths exponent s the best possble. In addton to dstance, quasconformal mappngs dstort also the area by a power dependng only on K, as shown frst by ojarsk [j]. Snce f 0 (r) = π 1 1 K (r) 1 K, where (r) = {z C : z < r}, t s natural to expect that the optmal exponent n area dstorton s smlarly 1/K. In ths paper we gve a postve answer to ths problem and prove the followng result whch was conjectured and formulated n ths precse form by Gehrng and Rech [GR]. We shall denote by the open unt dsk and by E the area of the planar set E. Theorem 1.1 Suppose f : s a Kquasconformal mappng wth f(0) = 0. Then we have fe M E 1/K (2) for all orel measurable sets E. Moreover, the constant M = M(K) depends only on K wth M(K) = 1 + O(K 1). For the proof of (2) we consder famles { } n 1 of dsjont dsks = (λ) whch depend holomorphcally on the parameter λ (n a sense to be defned later). After an approxmaton (2) now becomes equvalent to ( n ) 1 λ (λ) C (0) 1+ λ, (3) 1
2 where C depends only on λ. Furthermore, teratng the confguraton one s led to measures on Cantor sets and there we shall apply the Ruelleowen thermodynamc formalsm [w]; f we wrte (3) n terms of the topologcal pressure, then the proof comes out n a transparent way. The functon f 0 s extremal n the dstorton of area as well as dstance, and therefore t s natural to ask [I, 9.2] f for quasconformal mappngs the Hölder contnuty alone, rather than the dlataton, mples the nequalty (2). However, ths turns out to be false, as shown recently by P.Koskela [K]. As s well known the optmal control of area dstorton answers several questons n ths feld. For example, n general domans Ω one can nterpret (2) n terms of the local ntegrablty of the Jacoban J f of the quasconformal mappng f. Ths leads to a soluton of the well known problem [LV], [Ge] on the value of the constant p(k) = sup{p : J f L p loc (Ω) for each K quasconformal f on Ω}. Corollary 1.2 In every planar doman Ω, p(k) = K K 1. In other words, for each Kquasconformal f : Ω Ω, f W 1 p,loc(ω), p < 2K K 1. The example (1) shows that ths s false for p 2K K 1. Theorem 1.1 governs also the dstorton of the Hausdorff dmenson dm(e) of a subset E. Corollary 1.3 Let f : Ω Ω be Kquasconformal and suppose E Ω s compact. Then 2 K dm(e) dm(fe) 2 + (K 1) dm(e). (4) Ths nequalty, as well, s the best possble. Theorem 1.4 For each 0 < t < 2 and K 1 there s a set E C wth dm(e) = t and a Kquasconformal mappng f of C such that dm(fe) = 2 K dm(e) 2 + (K 1) dm(e). The estmate (4) was suggested by Gehrng and Väsälä [GV]. It can also be formulated [IM2] n the symmetrc form 1 ( 1 K dm(e) 1 ) 1 2 dm(fe) 1 ( 2 K 1 dm(e) 1 ). (5) 2 The results 1.3 and 1.4 are closely related to the removablty propertes of quasregular mappngs, snce n plane domans these can be represented as compostons of analytc functons and quasconformal mappngs. The strongest removablty conjecture, due to Iwanec and Martn [IM1], suggest that sets of Hausdorff d measure zero, d = n K+1, are removable for bounded quasregular mappngs n Rn. Here we obtan the followng. 2
3 Corollary 1.5 In planar domans sets E of Hausdorff dmenson dm(e) < 2 K + 1 are removable for bounded quasregular mappngs. Conversely, for each K 1 and t > 2/(K + 1) there s a t dmensonal set E C whch s not removable for some bounded K quasregular mappngs. In addton to [IM1] removablty questons have recently been studed for nstance n [JV], [KM] and [R]. Fnally, we menton the applcatons to the regularty results of quasregular mappngs. Recall that a mappng f W 1 q,loc(ω), 1 < q < 2, s sad to be weakly quasregular, f J f 0 almost everywhere and max Df(x)h K mn Df(x)h a.e. x Ω. h =1 h =1 Then f s K quasregular n the usual sense f f W2,loc 1 (Ω),.e. f J f s locally ntegrable. We can now consder the number q(k), the nfmum of the q s such that every weakly K quasregular mappng f Wq,loc 1 (Ω) s actually K quasregular. Corollary 1.6 q(k) = K 2K + 1. Indeed, Lehto and Vrtanen [LV] have proven that the precse estmate on the L p ntegrablty, Corollary 1.2, mples that q(k) 2K 2K K+1. The opposte nequalty q(k) K+1 was shown by Iwanec and Martn n [IM1]. Quasconformal mappngs are also the homeomorphc solutons of the ellptc dfferental equatons f(z) = µ(z) f(z); here µ s the complex dlataton or the eltram coeffcent of f wth µ = K 1 K+1 < 1. Hence there are close connectons to the sngular ntegral operators and especally to the eurlngahlfors operator Sω(z) = 1 π C ω(ζ) dm(ζ) (ζ z) 2, see [I],[IM1],[IK] for example. In fact, ths operator was the man tool n the work of ojarsk [j] and GehrngRech [GR], c.f. also [IM2]. elow we shall use mostly dfferent approaches and the role of the S operator remans mplct. Stll the area dstorton nequalty has a number of mplcatons on the propertes of S. In partcular, we have Corollary 1.7 There s a constant α 1 such that for any measurable set E, Sχ E dm E log α E. (6) It s for ths consequence that we must show the asymptotc estmate M(K) = 1 + O(K 1) and then, actually, Theorem 1.1 s equvalent to 1.7, cf. [GR]. 1 1 Davd Hamlton and Tadeusz Iwanec have ponted out that now (6) holds wth α = eπ. 3
4 If we consder general functons ω L ( ) then the nequalty (6) mples the correct exponental decay for {z : R Sω > t} when t. As a consequence, for each δ > 1 there s a constant M δ < such that Sv dm δ v log(1 + M δ v v ) dm, v L log L( ). (7) Here v = 1 π v dm s the ntegral mean of v. It s a natural queston whether (7) holds at δ = 1 as n (6) wth characterstc functons. Ths would also mply the IwanecMartn removablty conjecture n the planar case. However, n the last secton we show, agan by consderng the nequaltes arsng from the thermodynamc formalsm, that n fact (7) fals when δ = 1. Further results equvalent to the GehrngRech conjecture have been gven by Iwanec and Koseck [IK]. These nclude applcatons to the L 1 theory of analytc functons, quadratc dfferentals and crtcal values of harmonc functons. Moreover, by results of Lavrentev, ers and others the solutons to the ellptc dfferental equatons A(x) u = 0 can be nterpreted n terms of quasregular mappngs f. Therefore Corollary 1.2 yelds sharp exponents of ntegrablty on the gradent u; note that the dlataton of f and so necessarly the optmal ntegrablty exponent depends n a complcated manner on all the entres of the matrx A rather than just on ts ellptcty coeffcent. Acknowledgements Durng the preparaton of ths manuscrpt a number of people found smplfcatons n the frst prelmnary notes. Especally I would lke to thank Alexander Eremenko and Jose Fernandez who both ponted out Corollary 2.4. Also, I would thank Tadeusz Iwanec and Mchel Znsmester for mportant dscussons and comments on the topcs n ths paper and many people, ncludng Alexander Eremenko, Fred Gehrng, Seppo Rckman, Juha Henonen and Pekka Koskela for readng and makng correctons to the frst draft. 2 Holomorphc deformatons of Cantor sets Let us frst consder a famly { } n of nonntersectng subdsks of. We shall study the quasconformal deformaton of such famles and, n partcular, estmate sums r( ) t, t R, (8) where r( ) denotes the radus of. Lookng for the extremal phenomena we can terate the confguraton { } n and are thus led to Cantor sets. There one needs measures µ whch reflect n a natural manner the propertes of the sums (8). It turns out that such measures can, ndeed, be found by usng the thermodynamc formalsm ntroduced by Ruelle and owen, c.f. [w], [W]. To descrbe ths n more detal suppose hence that we are gven smlartes γ, 1 n, for whch = γ. Snce the γ are contractons, there s a unque compact 4
5 subset J of the unt dsk for whch n γ (J) = J. Thus J s selfsmlar n the termnology of Hutchnson [H]. We can also reverse ths pcture and defne the mappng g : by g = γ 1. Now J = g k, k=0 and g s a nto1 expandng mappng wth J completely nvarant, J = gj = g 1 J. Furthermore, g represents the shft on J; n a natural manner we can dentfy the pont x J wth the sequence (j k ) k=0 {1,..., n}n by defnng j k = f g k (x). Then, n ths dentfcaton, g : (j k ) k=0 (j k+1) k=0. In the sequel we use the notaton J = J(g) for our Cantor set and say also that t s generated by the smlartes γ. Next, let s = dm(j(g)), the Hausdorff dmenson of J(g). Then the Hausdorff s measure s nonzero and fnte on J(g) and after a normalzaton t defnes a probablty measure µ s whch s nvarant under the shft g,.e. µ s (g 1 E) = µ s (E) for all borelan E J(g). A general and systematc way to produce further nvarant measures s provded by the Ruelleowen formalsm: Gven a Hölder contnuous and real valued functon ψ on J(g) there s a unque shftnvarant probablty measure µ = µ ψ, called the Gbbs measure of ψ, for whch the supremum P (ψ) = sup { h µ (g) + J(g) ψ dµ : µ s g nvarant} (9) s attaned, see [w] or [W]. Here h µ (g) denotes the entropy of µ and the quantty P (ψ) s called the topologcal pressure of ψ. Let us then look for the Gbbs measures that are related to the sums (8). Recall that s = dm(j(g)) s the unque soluton of P ( s log g ) = 0, and ths suggest the choces ψ t = t log g. It then readly follows from [w, Lemma I.1.20], that ( P ( t log g n ) = log γ t) ( n = log r( ) t). (10) 5
6 In fact, the functons ψ = ψ t are n our stuaton locally constant and therefore t can be shown that the system g : (J(g), µ ψ ) (J(g), µ ψ ) s ernoull. In other words, the numbers p = µ ψ (J ) satsfy n p = 1 and on J(g) = {1,..., n} N µ ψ s the product measure determned by the probablty dstrbuton {p } n of {1,..., n}. Ths enables one to make the dynamcal approach more elementary, as ponted to us by Alexander Eremenko. We are grateful to hm for lettng us to nclude ths smplfcaton here. For the readers convenence let us recall the proof of the varatonal prncple, the counterpart of (9), n the elementary settng of product measures. Then also the entropy of µ = µ ψ attans the smple form h µ (g) = p log p. 1 Lemma 2.1 Let ν be a product measure on J(g) determned by the probablty dstrbuton {q } n. Then for each t R, wth equalty f and only f h ν (g) t log g dν log( r( ) t ) J(g) q = r( ) t n r( ) t, 1 n. Proof: Snce the logarthm s concave on R +, h ν (g) t J(g) log g dν = q log γ t q = q log r( ) t q log( r( ) t ) where the equalty holds f and only f q r( ) t 1 n. has the same value for each Remark. In the elementary settng, one can use (10) as the defnton of the pressure P ( t log g ). Note also that f s = dm(j(g)), then n r( ) s = 1, or P ( s log g ) = 0, and the extremal measure n Lemma 2.1 s agan the normalzed Hausdorff s measure. We shall next consder holomorphc famles of Cantor sets or pars (g λ, J(g λ )), λ. y ths we mean that each set J(g λ ) s generated as above by smlartes γ,λ (z) = a (λ)z + b (λ), 1 n, where the coeffcents a (λ) 0, b (λ) now depend holomorphcally on the parameter λ. On the other hand, we can also consder the (λ) = γ,λ and say that { (λ)} n 1 s a holomorphc famly of dsjont dsks n. oth of these confguratons can be descrbed as holomorphc motons; recall that a functon Φ : A C s called a holomorphc moton of a set A C f 6
7 () for any fxed a A, the map λ Φ(λ, a) s holomorphc n () for any fxed λ, the map a Φ λ (a) = Φ(λ, a) s an njecton, and () the mappng Φ 0 s the dentty on A. In fact, (global) quasconformal mappngs and holomorphc motons are just dfferent expressons of the same geometrc quantty. For nstance, accordng to Slodkowsk s generalzed λ lemma ([Sl], see also [AM], 3.3) the correspondence γ,0 (z) γ,λ (z) for z and 1 n, extends to a quasconformal mappng Φ λ : C C wth K(Φ λ ) 1+ λ 1 λ. Therefore the estmate ( n ) 1 λ 1+ λ (λ) C (0) (11) s a specal case of the GehrngRech conjecture. ut after smplfyng arguments, gven later n Lemmas 3.1 and 3.3, we wll see that the conjecture s n fact equvalent to (11). Expressng ths nequalty now n terms of the topologcal pressure (10) we end up wth the followng formulaton. Theorem 2.2 Suppose that (g λ, J(g λ )) depends holomorphcally on the parameter λ. Then 1 + λ 1 λ P ( 2 log g 0 ) P ( 2 log g λ ) 1 λ 1 + λ P ( 2 log g 0 ). Proof: y the varatonal nequalty 2.1 for each λ there s a unque (product) measure µ λ such that P ( 2 log g λ ) = h µλ (g λ ) 2 log g λ (z) dµ λ (z) (12) and clearly log g λ (z) s harmonc n λ. To use Harnack s nequalty we freeze the measure µ λ. In other words, gven a probablty dstrbuton {p } n 1 on {1,..., n}, defne for each λ a product measure µ λ on J(g λ ) by the condton µ λ (J(g λ ) (λ)) = p ; ths s possble snce the dsks (λ) reman dsjont. y the constructon, h µλ (g λ ) s also constant n λ. Moreover, we have that P ( 2 log g λ ) < 0, snce P ( s log g λ ) s strctly decreasng n s and t vanshes for s = dm(j(g)) < 2. Alternatvely, we may also use here the dentty (10) to P ( 2 log g λ ) = log( n r( (λ)) 2 ) < 0. If now the numbers {p } are so chosen that µ 0 = µ 0 (the maxmzng measure n (12) when the parameter λ = 0), then Harnack s nequalty wth 2.1 mples that J(g λ ) 1 + λ 1 λ P ( 2 log g 0 ) = 1 + λ ( h µ0 (g 0 ) 2 1 λ h µλ (g λ ) 2 J(g λ ) J(g 0 ) log g λ dµ λ P ( 2 log g λ ) 7 log g 0 dµ 0 )
8 whch proves the frst of the requred nequaltes. symmetry n λ and 0. The second follows smlarly by When t > 2 the same nequaltes hold for P ( t log g λ ) as well. However, smaller exponents must change wth λ and we shall later see how ths reflects n the precse dstorton of Hausdorff dmenson under quasconformal mappngs. Corollary 2.3 If (g λ, J(g λ )) s as above and 0 < t 2, set t(λ) = Then 1 t(λ) P ( t(λ) log g λ ) 1 λ λ t P ( t log g 0 ). t(1 + λ ) (1 λ + t λ ). Proof: If {µ λ } λ s a famly of product measures on J(g λ ), all defned by a fxed probablty dstrbuton {p } n 1 lke n the prevous theorem, then by t(λ) h µ λ (g λ ) log g λ dµ λ = h µλ (g λ )( 1 t(λ) 1 2 ) h µ λ (g λ ) log g λ dµ λ J(g λ ) 1 λ ( h µ0 (g 0 )( λ t 1 2 ) h µ 0 (g 0 ) 1 λ λ t P ( t log g 0 ) J(g 0 ) J(g λ ) log g 0 dµ 0 ) and takng the supremum over the product measures on J(g λ ) proves the clam. The above estmates for the topologcal pressure hold actually n a much greater generalty. We can consder, for nstance, polynomallke mappngs of Douady and Hubbard [DH]. More precsely, suppose we have a famly of holomorphc functons f λ defned on the open sets U λ, λ, such that U λ f λ U λ. We need to assume that J(f λ ) = f n λ U λ n=0 s a mxng repeller for f λ. That s, f λ 0 for z J(f λ) and J(f λ ) s compact n C wth no proper f λ nvarant relatvely open subsets. Then the f λ are expandng on J(f λ ) and the thermodynamc formalsm extends to f λ : J(f λ ) J(f λ ), see [w] or [Ru]. To consder the dependence on the parameter, let U λ depend contnuously on λ and let (λ, a) f λ (a) be holomorphc whenever defned. ecause the functons are expandng, we have a holomorphc moton of the perodc ponts ([MSS], p.198). Snce these are dense n the repeller J(f λ ), by the λ lemma of Mañé, Sad and Sullvan we obtan a holomorphc moton Φ of J(f 0 ) such that J(f λ ) = Φ λ J(f 0 ) and f λ Φ λ = Φ λ f 0. Combnng these facts we conclude that 1 t(λ) P ( t(λ) log f λ ) 1 λ λ t P ( t log f 0 ). (13) 8
9 Namely, snce the varatonal prncple generalzes to ths settng, the proof of (13) s as n Lemma 2.3. In ths case to show that P ( t(λ) log f λ ) < 0 we may use Mannng s formula [M] h µ (f λ ) dm(µ) = log f λ dµ J(f λ ) and the fact [Su] that dm(µ) nf{dm(e) : µ(e) = 1} dm(j(f λ )) < 2. These hold for any ergodc f λ nvarant measure on J(f λ ). Especally, startng from a measure µ on J(f 0 ) we can take the mages µ λ = Φ λ µ under the holomorphc moton, and snce the entropy s an somorphsm nvarant, (13) follows. On the other hand, f one looks for the mnmal approach to the quasconformal area dstorton, then the above leads also to a proof for (11) that avods the thermodynamc formalsm. In fact, ths was shown to us by A. Eremenko and J.Fernandez, who ndependently ponted out the followng result on the (nonharmonc!) functon log f(z). Corollary 2.4 Let n = {z C n : z < 1}. If f : n s a holomorphc mappng such that all of ts coordnate functons f are everywhere nonzero, then 1 + z 1 z log f(0) log f(z) log f(0). 1 z 1 + z Proof: If f = (f 1,..., f n ), consder numbers p > 0 wth n 1 p = 1 and set p u(z) = p log f (z) 2. Then u(z) s harmonc and by Jensen s nequalty, e u(z) n f 1 p (z) 2 p < 1, u s also postve. Hence usng the concavty of the logarthm and the Harnack s nequalty we may deduce log f(z) 2 p log f (z) 2 p 1 + z 1 z p log f (0) 2 p. proves the frst nequalty. The second follows by symme Choosng fnally p = f (0) 2 f(0) 2 try. 3 Dstorton of area We shall reduce the proof of the area dstorton estmate fe M E 1/K nto two dstnct specal cases. In the frst, where we use the nequaltes of the prevous secton, let us assume that E s a fnte unon of nonntersectng dsks = (z, r ), 1 n. 9
10 Lemma 3.1 Suppose that f : s K quasconformal wth f(0) = 0. If f s conformal n E = n 1, then ( n ) 1 f C(K) K, where the constant C(K) depends only on K. Moreover, C(K) = 1 + O(K 1). Proof: Extend f frst to C by a reflecton across S 1 and assume wthout loss of generalty that f(1) = 1. Then we can embed f to a holomorphc famly of quasconformal mappngs of C. However, n order to control the dstorton as K we need to modfy f near. Thus, f µ s the eltram coeffcent of (the extended) f, defne new dlatatons by { λ K+1 µ λ (z) = K 1 µ(z), z 2, 0, z > 2. y the measurable Remann mappng theorem there are unque µ λ quasconformal mappngs f λ : C C normalzed by the condton f λ (z) z = O( 1 ) as z. z Then f λ s conformal n E, f λ (z) and ts dervatves (when z E) depend holomorphcally on λ [A, Theorem 3], f 0 (z) z and f λ 0 = K 1 K+1, then f λ0 = Φ f, (14) where Φ s conformal n f(0, 2). To apply Theorem 2.2 note that by Koebe s 1/4 theorem ( D (λ) f λ (z ), r ) 4 f λ(z ) f λ (z, r ) and smlarly f λ (0, 2) (0, 8). Also D (λ) = ψ λ, D (0), where ψ,λ (z) = f λ(z ) (z z ) + f λ (z ), and thus {D (λ)} n 1 s a holomorphc famly of dsjont dsks contaned n (0, 8). Therefore we need only choose extra smlartes φ : (0, 8) D (0), 1 n, set γ,λ = ψ,λ φ and note that these generate a holomorphc famly of Cantor sets J(g λ ) (0, 8). y Theorem 2.2 P ( 2 log g λ 1 λ ) 1+ λ P ( 2 log g 0 ) or, n other words, by (11) 4 λ ( f λ(z ) 2 r 2 1+ λ n ) 1 λ 1+ λ 32. r 2 The Lemma wll then be completed by smple approxmaton arguments. Snce the mages of crcles under global quasconformal mappngs have bounded dstorton, f λ π max z f λ (z) f λ (z ) 2 πc 0 ( λ ) mn z f λ (z) f λ (z ) 2 10
11 πc 0 ( λ ) f λ(z ) 2 r 2, where the last estmate follows from the Schwarz lemma. Moreover, the correct expresson for the constant C 0 ( λ ), see [L, p.16], shows that C 0 ( λ ) = 1+O( λ ). If we choose λ 0 = K 1 K+1, t then follows that f λ 0 E C 1 (K) E 1 K wth C 1 (K) = 1 + O(K 1). It remans to show that the functon Φ n (14) satsfes Φ (z) C 2 (K) = (1 + O(K 1)) 1 for all z. Frst, snce the dameter of f λ0 (0, 2) s at least four [P, 11.1], the basc bounds on the crcular dstorton, see [L, I.2.5], mply that (f λ0 (0), ρ(k)) f λ0 = Φ for a ρ(k) > 0 dependng only on K. As above, Φ (0) ρ(k) by the Schwarz lemma. Yet another applcaton of the Schwarz lemma, ths tme to the functon λ f λ (z) z, gves f λ (z) z 10 λ, z (0, 2). Ths shows that we may choose ρ(k) = (1 + O(K 1)) 1. Furthermore, as fs 1 = S 1 and f 1 s unformly Hölder contnuous wth constants dependng only on K, f (0, 2) (0, R) for an R = R(K) > 1. Then Koebe s dstorton theorem combned wth Lehto s majorant prncple [L, II.3.5] proves that and the requred estmates follow. ( Φ (z) Φ (1 z R (0) )3 ) K 1 K z R, z, Remark 3.2 The above proof gves us the followng varatonal prncple for planar quasconformal mappngs: Suppose we are gven numbers p > 0 wth n p = 1 and dsjont dsks. Then for each K quasconformal mappng f : for whch f(0) = 0 and we have the nequalty f n 1 s conformal, (15) p log f p 1 K p log p + C(K) (16) where C(K) = O(K 1) depends only on K. n ) In fact, choosng p = f /( f shows that (16) generalzes Lemma 3.1. Somewhat curously, the varatonal nequalty (16) s not true for general quasconformal mappngs, for mappngs whch do not satsfy (15). We shall return to ths n Secton 5 where t wll have mplcatons on the estmates of the L log L norm of the eurlngahlfors operator. To prove the complementary case n the area dstorton nequalty we use therefore a dfferent method. We shall apply here the approach due to Gehrng and Rech [GR] based on a parametrc representaton. 11
12 Lemma 3.3 Let f : be K quasconformal wth f(0) = 0. If E s closed and f s conformal outsde E, then fe b(k) E where b(k) = 1 + O(K 1) depends only on K. Proof: As n [GR] defne the eltram coeffcents ν t (z) = sgn(µ(z)) tanh( t T arctanh µ(z) ), t R +, where µ s the complex dlataton of f, T = log K and sgn(w) = w w f w 0 wth sgn(0) = 0. y the measurable Remann mappng theorem we can fnd ν t quasconformal h t : wth h t (0) = 0. If A(t) = h t E, then Gehrng and Rech show that d dt A(t) = φs(χ ) dxdy + c(t) h ht E te (17) where S s the eurlngahlfors operator and c(t) s unformly bounded. The functon φ depends only on the famly {h t }, not on E, and from [GR, (2.6) and (3.6)] we conclude that φ 1 and that φ(w) = 0 whenever µ(h 1 t (w)) = 0. Suppose now that f s conformal outsde the compact subset E. Then µ 0 n \E and, n partcular, we obtan φ(z) χ ht E (z). ut S : L 2 L 2 s an sometry and therefore for any set F C, ( ) Sχ F dxdy F 1 2 Sχ F 2 1 dxdy 2 = F. (18) Thus F C d dt A(t) C 0A(t), 0 < t <, and an ntegraton gves h t E = A(t) e C 0t A(0) = e C 0t E. Takng t = log K shows that fe e C 0 log K E = b(k) E, where b(k) = 1 + O(K 1). Remark. On the other hand, as kndly ponted out by the referee, f one consders n C the normal solutons f λ (z) = z + O( z 1 ) of the eltram equaton f z = µf z wth µ supported on E, then n that stuaton the followng argument gves a drect proof for a very precse estmate f(e) K E. Namely, for ω = f z we have f z = 1 + Sω wth ω = µ(1 + Sµ + SµSµ +...). In vew of S 2 = 1 we obtan f(e) = 1 + Sω 2 ω 2 E + 2R Sω, E E 12
13 where for the k th terate E SµSµ... Sµ µ k E as n (18). Thus f(e) E + 2 µ E + 2 µ 2 E +... = E ( µ ) = K E. The area dstorton nequalty s now an mmedate corollary of the two prevous lemmas 3.1 and 3.3. Proof of Theorem 1.1: Suppose that f : s K quasconformal and f(0) = 0. In provng the estmate fe M E 1/K t suffces to study sets of the type E = n 1, where the are subdsks of wth parwse dsjont closures. The general case follows then from Vtal s coverng theorem. To factor f we fnd by the measurable Remann mappng theorem a K quasconformal mappng g :, g(0) = 0, wth complex dlataton µ g = χ \E µ f. Then g s conformal n E and f = h g, where h : s also K quasconformal, h(0) = 0, but now h s conformal outsde ge. Snce quasconformal mappngs preserve sets of zero area, h( ge) = ge = 0, and then Lemmas 3.1 and 3.3 mply fe = h(ge) b(k) ge b(k)c(k) E 1 K, where M(K) b(k)c(k) = 1 + O(K 1) as requred. 2 One of the equvalent formulatons of Theorem 1.1 s the statement that for a K quasconformal f the Jacoban J f belongs to the class weakl p ( ), p = K K 1. Corollary 3.4 If f : s K quasconformal, f(0) = 0, then for all s > 0, ( M {z : J f (z) s} s ) K K 1, where M depends only on K. Moreover the exponent p = K K 1 Proof: If E s = {z : J f (z) s}, then by Theorem 1.1 s E s J f dm = fe s M(K) E s E s No p larger than 1 K. s the best possble. K K 1 wll do, snce E s = π(ks) K 1 K 1 for f(z) = z z K 1. Proof of Corollary 1.2: If D s a compact dsk n the doman Ω and f : Ω Ω s K quasconformal, choose conformal ψ, φ whch map negbourhoods of D and f D, respectvely, onto the unt dsk. As ψ D and φ fd are blpschtz, applyng Corollary 3.4 to φ f ψ 1 proves that J f L p K loc (Ω) for all p < K 1. 2 Davd Hamlton has nformed us that the same methods can also be used to obtan good bounds for the constant M n fe M E 1/K f one consders nstead of the case f : those mappngs f whch are conformal outsde wth f(z) z = O(1/ z ). 13
14 4 Dstorton of dmenson In the prevous secton we determned the quasconformal area dstorton from the propertes of the pressure P ( 2 log g λ ). Smlarly Corollary 2.3, or the varatonal nequalty (16) wth a sutable choce of the probabltes p, also admts a geometrc nterpretaton: If f : s K quasconformal wth f(0) = 0 and f, n addton, f s conformal n the unon of the dsks, 1 n, then f tk 1+t(K 1) ( C(K) t) 1 1+t(K 1), 0 < t 1, where the constant C(K) depends only on K. Snce the complementary lemma 3.3 fals for exponents t < 2, n the general case we content wth slghtly weaker nequaltes. Lemma 4.1 If 0 < t < 1, f : s K quasconformal, f(0) = 0 and { } n 1 are parwse dsjont sets n, then whenever (1 + t(k 1)) 1 tk < p 1. ( f p C K (t, p) t) 1 1+t(K 1) Proof: We use the ntegrablty of the Jacoban J f as n [GV]. Snce p(1 + t(k 1)) > tk we can choose an exponent 1 < p 0 < K K 1 such that 1 K q 1 + t(k 1) 0 < p, (19) tk where q 0 = p 0 s the conjugate exponent. Then usng Hölder s nequalty twce one p 0 1 obtans f p = ( ) p ( ) p p J f dm p J 0 p q f dm 0 0 ( ) p p J 0 f dm p 0 ( p p 0 ) q 0 p 0 p p 0 p p 0 ( ) p p J 0 f dm p 0 ( p p 0 ) q 0 p 0 p p 0 p p 0. K 1 On the other hand, as p 0 K < 1 < p 1+t(K 1) tk, t follows that > 1 + t(k 1). Combnng ths wth Corollary 3.4 (or 1.2) yelds ( f p M p q 0 (1+t(K 1))) 1 1+t(K 1), p 0 p 0 p where M depends only on p 0 and K. Snce by (19) also t < p q 0 (1 + t(k 1)), the clam follows. 14
15 Proof of Corollary 1.3: If f : Ω Ω s K quasconformal, let E Ω be a compact subset wth dm(e) < 2. Choose also a number 1 2dm(E) < t 1 and cover E by squares wth parwse dsjont nterors. Accordng to [LV], Theorems III.8.1 and III.9.1, da(f ) 2 C 0 f, where the constant C 0 depends only on K, E and Ω. Hence we conclude from Lemma 4.1 that ( da(f ) δ C 1 da( ) 2t) 1 1+t(K 1), δ > 2tK 1 + t(k 1). Wth a proper choce of the coverng { } the sum on the rght hand sde can be made arbtrarly small and thus dm(f E) δ. Consequently, dm(fe) 2 K dm(e) 2 + (K 1) dm(e) (20) whch proves the corollary. In the specal case of K quascrcles Γ, the mages of S 1 under global K quasconformal mappngs, Corollary 1.3 reads as dm(γ) 1 + ( K 1 ) = 2 2 K + 1 K + 1. Ths sharpens recent results due to JonesMakarov [JM] and eckerpommerenke [P]. On the other hand, ecker and Pommerenke showed that f the dlataton K 1, then ) 2 ( ( K 1 K+1 dm(γ) K 1 2. K+1) These results suggest the followng Queston 4.2 If Γ s a K quascrcle, s t true that dm(γ) 1 + In the postve case, s the bound sharp? ( K 1 ) 2. K + 1 Let us next show that the equalty can occur n (20) for any value of K and dm(e). Note frst that n terms of the holomorphc motons Corollary 1.3 obtans the followng form. Corollary 4.3. Let Φ : E C be a holomorphc moton of a set E C and wrte d(λ) = dm(φ λ (E)). Then d(λ) 2d(0) (2 d(0)) 1 λ (21) 1+ λ + d(0). Proof: y Slodkowsk s extended λ lemma Φ λ s a restrcton of a K quasconformal mappng of C, K 1+ λ 1 λ, and hence the clam follows from 1.3. For the converse, we start by constructng holomorphc motons of Cantor sets such that the equalty holds n (21) up to a gven ε > 0. Thus for each, say, n 10 fnd 15
16 dsjont dsks (z, r) all of the same radus r = r n, such that 1 2 nr2 1. If 0 < t < 2, let also β(t) = log(n 1/t r). (22) For n large enough, β(t) > 0 and t 2 log r log r β(t) t + ɛ. (23) 2 Set then a t (λ) = exp( β(t) 1 λ 1+λ ). Clearly a t s holomorphc n wth a t ( ) = \{0}. Therefore we can consder the holomorphc famly of smlartes γ,λ (z) = ra t (λ)z + z. Snce the dsks γ,λ (z, r) are dsjont, the smlartes γ,λ generate Cantor sets (g λ, J(g λ )) as n Secton 2. Furthermore, the dervatves γ,λ do not depend on and so the dmenson d(λ) = dm(j(g λ )) s determned from the equaton n(r a t (λ) ) d(λ) = 1. y (22) n(r a t (0) ) t = 1 and therefore d(0) = t. Smlarly, f 0 < λ < 1, t follows from (23) that d(0) d(λ) = log(r a t(λ) ) log(r a t (0) ) d(0) 2 + ( 1 d(0) 2 = log r β(t) 1 λ 1+λ log r β(t) ) 1 λ + ε. (24) 1 + λ Proof of Theorem 1.4: Choose a countable collecton { k } 1 of parwse dsjont subdsks of and defne, usng the argument above, n each dsk k a holomorphc moton Φ of a Cantor set J k wth Φ λ (J k ) k. If d(λ) = dm(φ λ (J k )), we may assume that d(0) = t and that for each k (24) holds wth ε = 1 k. Clearly ths constructon determnes a holomorphc moton Ψ of the unon J = k J k. Wrtng stll d(λ) = dm(ψ λ (J)) we have d(λ) = 2d(0) (2 d(0)) 1 λ 0 λ < 1. 1+λ + d(0), Now Slodkowsk s generalzed λ lemma apples and Ψ extends to a K quasconformal mappng f of C, where K = 1+λ 1 λ, 0 λ < 1. In other words, f E = J, then dm(e) = t and dm(fe) = (2Kdm(E))/(2 + (K 1)dm(E)). Fnally, Corollary 1.5 s an mmedate consequence of 1.3 and 1.4 snce K quasregular mappngs f can be factored as f = φ g, where φ s holomorphc and g K quasconformal; for holomorphc φ sets E wth dm(e) < 1 are removable by Panlevé s theorem whle those wth dm(e) > 1 are never removable [Ga, III. 4.5]. 16
17 Therefore n consderng the removablty questons for K quasregular mappngs, the dmenson d K = 2 K+1 s the borderlne case and there we have the IwanecMartn conjecture that all sets of zero Hausdorff d K measure are removable. More generally, t s natural to ask whether the precse bound on the dmenson dm(fe) 2 K dm(e) 2+(K 1) dm(e) gven by Corollary 1.3 s stll correct on the level of measures. 2Kτ 2+τ(K 1). If f s a planar K quas Queston 4.4 Let 0 < τ < 2 and δ = δ K (τ) = conformal mappng, s t true that H τ (E) = 0 H δ (fe) = 0. If not, what s the optmal Hausdorff measure H h or measure functon h such that f H h H τ? 5 Estmates for the eurlngahlfors operator As we saw earler quasconformal mappngs have mportant connectons to the sngular ntegrals and n partcular to the eurlngahlfors operator, the complex Hlbert transform Sω(z) = 1 ω(ζ) dm(ζ) π (ζ z) 2. C There are even hgher dmensonal counterparts, see [IM1] and the references there. In fact, many propertes of the S operator can be reduced to the dstorton results of quasconformal mappngs. We shall here consder only the operaton of S on the functon space L log L and refer to the work of Iwanec and Koseck [IK] for further results. In case of the characterstc functons ω = χ E we have then by Corollary 1.7 that Sχ E dm E log α E for all orel subsets E of a dsk C; the constant α does not depend on E or. Ths translates also to the L settng: Corollary 5.1 Let C be a dsk. If ω s a measurable functon such that ω(z) χ (z) a.e. then {z : R Sω(z) > t} 2α e t. (26) Proof: Let E + = {z : R Sω > t}. Snce S has a symmetrc kernel, t E + R Sω dm = R ωsχ E+ E + dm E + log α E + by (25). Thus E + α e t and snce by the same argument E = {z : R Sω < t} satsfes E α e t, the nequalty (26) follows. (25) 17
18 The estmate (26) s sharp snce for ω = (z/z)χ (z) we have Sω = (1 + 2 log z )χ (z). For the modulus Sω Iwanec and Koseck [IK, proposton 12] have shown that (25) mples {z : Sω(z) > t} α(1 + 19t) e t. (27) It remans open f the lnear term 19t can be replaced by a constant. Corollary 5.2 For each δ > 1 there s a constant M(δ) < such that Sv dm δ whenever the rght hand sde s fnte. v(z) log (1 + M(δ) v(z) ) dm(z) v Proof: Let ω be a functon, unmodular n and vanshng n C\, such that Sv dm = ωsv dm = v Sω v dm. v We apply then the elementary nequalty ab a log(1+a)+exp(b) 1. Snce accordng to (27) e Sω /δ α 19δ 1 dm (1 + δ 1 δ 1 ) = M 1(δ), t follows that Sv dm M 1 (δ) v dm + δ v log (1 + δ v ) dm. v Defne now E 0 = {z : v(z) < 1 e v }. As t t log 1 t s ncreasng on (0, 1 e ), e E 0 v log( v v ) dm v E 0 v dm, where we use the conventon 0 log 0 = 0. Thus ( v ) v log (1 + δ v ) dm v \E 0 v log v (e + δ) dm + where M 2 = e 2 + eδ. In concluson, f M = M 2 exp(m 1 (δ)), Sv dm δ whch completes the estmaton. E 0 v log(1 + δ e ) dm v ) v log (M 2 dm, (28) v v log (M v ) dm δ v v log (1 + M v ) dm, v 18
19 Snce the varatonal nequalty (16), p log f p 1 K p log p + C(K) wth C(K) = O(K 1) and f conformal, was the key n the area dstorton Theorem 1.1 t s of nterest to know whether the nequalty s vald wthout any conformalty assumptons. Another natural queston s whether Corollary 5.2 stll holds at δ = 1; for characterstc functons ths s true and (25) wth [IK, proposton 19] mples that for nonnegatve functons v, \E Sv dm E v(z) log (1 + α v(z) ) dm(z), f supp(v) E. v Indeed, t can be shown that these two questons are equvalent (f v 0 n Corollary 5.2). However, t turns out that the answer to them s the negatve. We omt here the proof of the equvalence; nstead we gve frst a smple counterexample to the general varatonal nequalty and then show how ths reflects n the L log L estmates of the complex Hlbert transform. Example 5.3 Choose 0 < ρ < 1 and for 1 n consder the dsjont dsks = (ρ, aρ ) where 0 < a < 1 ρ 1+ρ. Let also p = 1 n and f 0(z) = z z 1 K 1. Then whle p log p = (n + 1) log ρ + log n + log πa 2 p log f 0 n + 1 p K log ρ + log n + C 0 = 1 K p log p + K 1 K log n + C 1, where C 0, C 1 depend only on K and a. Lettng n shows that the varatonal nequalty fals for f 0. Proposton 5.4 For each M < there s an ε > 0 and a nonnegatve functon v L log L( ) such that Sv dm > (1 + ε) v(z) log (1 + M v(z) ) dm(z). v Proof: y nequalty (28) t suffces to show that for no M < does Sv dm hold for all nonnegatve functons v L log L( ). v(z) log (M v(z) ) dm(z) (29) v 19
20 We argue by contradcton. Hence consder frst the mappng f(z) = z z K 1 and mbedd t to a one parameter famly of quasconformal mappngs h t :, as n the proof of Lemma 3.3. Thus for t = log K, h t = f. Suppose next that we have dsjont open sets {D } n 1 and numbers p > 0 wth n1 p = 1. Set then p v t (z) = h t D χ h t D (z). Clearly v t dm = 1 and f ψ(t) = v t log(mv t ) dm then by Jensen s nequalty ψ(t) 0 for M π. Furthermore, we can deduce from the GehrngRech dentty (17) that ψ p d (t) = h t D dt h td = φsv dm c(t) where c(t) s unformly bounded. Thus f (29) holds, then ψ (t) ψ(t) c(t) and after ntegraton ψ(t) e t ψ(0) e t t 0 c(s)e s ds = e t ψ(0) + c 1 (t). Takng t = log K we obtan ( p ) ( p ) p log K p log + C(K), fd D where C(K) = (K 1) log M + c 1 (log K). Fnally, f, p are as n the prevous example wth f 1 (z) = f 0 (z) = z z 1 K 1, we can choose D = f 0. ut ths would mean that p log f 0 p 1 K p log p C(K) K, contradctng Example 5.3. Therefore (29) cannot hold and so the estmate of Corollary 5.2 s sharp. REFERENCES [A] Ahlfors, L., On quasconformal mappngs. J. Analyse Math., 3 (1954), [A] Ahlfors, L. & ers L., Remann s mappng theorem for varable metrcs. Ann. Math., 72 (1960), [AM] Astala, K. & Martn, G., Holomorphc Motons. Preprnt, [P] ecker, J. & Pommerenke Chr., On the Hausdorff Dmenson of Quascrcles. Ann. Acad. Sc. Fenn. Ser. A I Math., 12 (1987), [j] ojarsk,., Generalzed solutons of a system of dfferental equatons of frst order and ellptc type wth dscontnuous coeffcents. Math. Sb., 85 (1957), [w] owen, R., Equlbrum States and the Ergodc Theory of Anosov Dffeomorphsms. Lecture Notes n Math., 470. SprngerVerlag, New YorkHedelberg, [DH] Douady, A. & Hubbard J., On the dynamcs of polynomallke mappngs. Ann. Sc. Ec. Norm. Sup., 18 (1985),
Recurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More information1 Example 1: Axisaligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationWe are now ready to answer the question: What are the possible cardinalities for finite fields?
Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the
More informationREGULAR MULTILINEAR OPERATORS ON C(K) SPACES
REGULAR MULTILINEAR OPERATORS ON C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. The purpose of ths paper s to characterze the class of regular contnuous multlnear operators on a product of
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationgreatest common divisor
4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no
More informationNew bounds in BalogSzemerédiGowers theorem
New bounds n BalogSzemerédGowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A
More informationA Probabilistic Theory of Coherence
A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want
More informationn + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)
MATH 16T Exam 1 : Part I (InClass) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total
More informationFormula of Total Probability, Bayes Rule, and Applications
1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σalgebra: a set
More informationOn Lockett pairs and Lockett conjecture for πsoluble Fitting classes
On Lockett pars and Lockett conjecture for πsoluble Fttng classes Lujn Zhu Department of Mathematcs, Yangzhou Unversty, Yangzhou 225002, P.R. Chna Emal: ljzhu@yzu.edu.cn Nanyng Yang School of Mathematcs
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More information+ + +   This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationRing structure of splines on triangulations
www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAMReport 201448 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
More informationLoop Parallelization
  Loop Parallelzaton C52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I,J]+B[I,J] ED FOR ED FOR analyze
More informationNONCONSTANT SUM REDANDBLACK GAMES WITH BETDEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia
To appear n Journal o Appled Probablty June 2007 OCOSTAT SUM REDADBLACK GAMES WITH BETDEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate
More informationThe eigenvalue derivatives of linear damped systems
Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by YeongJeu Sun Department of Electrcal Engneerng IShou Unversty Kaohsung, Tawan 840, R.O.C emal: yjsun@su.edu.tw
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationEmbedding lattices in the Kleene degrees
F U N D A M E N T A MATHEMATICAE 62 (999) Embeddng lattces n the Kleene degrees by Hsato M u r a k (Nagoya) Abstract. Under ZFC+CH, we prove that some lattces whose cardnaltes do not exceed ℵ can be embedded
More informationProductForm Stationary Distributions for Deficiency Zero Chemical Reaction Networks
Bulletn of Mathematcal Bology (21 DOI 1.17/s11538195174 ORIGINAL ARTICLE ProductForm Statonary Dstrbutons for Defcency Zero Chemcal Reacton Networks Davd F. Anderson, Gheorghe Cracun, Thomas G. Kurtz
More informationwhere the coordinates are related to those in the old frame as follows.
Chapter 2  Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of noncoplanar vectors Scalar product
More information6. EIGENVALUES AND EIGENVECTORS 3 = 3 2
EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a nonzero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :
More informationCHAPTER 7 VECTOR BUNDLES
CHAPTER 7 VECTOR BUNDLES We next begn addressng the queston: how do we assemble the tangent spaces at varous ponts of a manfold nto a coherent whole? In order to gude the decson, consder the case of U
More informationLogistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification
Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson
More information2.4 Bivariate distributions
page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together
More informationOPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL. Thomas S. Ferguson and C. Zachary Gilstein UCLA and Bell Communications May 1985, revised 2004
OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL Thomas S. Ferguson and C. Zachary Glsten UCLA and Bell Communcatons May 985, revsed 2004 Abstract. Optmal nvestment polces for maxmzng the expected
More informationPricing Overage and Underage Penalties for Inventory with Continuous Replenishment and Compound Renewal Demand via Martingale Methods
Prcng Overage and Underage Penaltes for Inventory wth Contnuous Replenshment and Compound Renewal emand va Martngale Methods RAF Jun3  comments welcome, do not cte or dstrbute wthout permsson Junmn
More informationCOLLOQUIUM MATHEMATICUM
COLLOQUIUM MATHEMATICUM VOL. 74 997 NO. TRANSFERENCE THEORY ON HARDY AND SOBOLEV SPACES BY MARIA J. CARRO AND JAVIER SORIA (BARCELONA) We show that the transference method of Cofman and Wess can be extended
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More informationx f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60
BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More informationSection 2 Introduction to Statistical Mechanics
Secton 2 Introducton to Statstcal Mechancs 2.1 Introducng entropy 2.1.1 Boltzmann s formula A very mportant thermodynamc concept s that of entropy S. Entropy s a functon of state, lke the nternal energy.
More informationOn the Interaction between Load Balancing and Speed Scaling
On the Interacton between Load Balancng and Speed Scalng Ljun Chen, Na L and Steven H. Low Engneerng & Appled Scence Dvson, Calforna Insttute of Technology, USA Abstract Speed scalng has been wdely adopted
More informationA Lyapunov Optimization Approach to Repeated Stochastic Games
PROC. ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, OCT. 2013 1 A Lyapunov Optmzaton Approach to Repeated Stochastc Games Mchael J. Neely Unversty of Southern Calforna http://wwwbcf.usc.edu/
More informationz(t) = z 1 (t) + t(z 2 z 1 ) z(t) = 1 + i + t( 2 3i (1 + i)) z(t) = 1 + i + t( 3 4i); 0 t 1
(4.): ontours. Fnd an admssble parametrzaton. (a). the lne segment from z + to z 3. z(t) z (t) + t(z z ) z(t) + + t( 3 ( + )) z(t) + + t( 3 4); t (b). the crcle jz j 4 traversed once clockwse startng at
More informationEE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN
EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson  3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson  6 Hrs.) Voltage
More informationFisher Markets and Convex Programs
Fsher Markets and Convex Programs Nkhl R. Devanur 1 Introducton Convex programmng dualty s usually stated n ts most general form, wth convex objectve functons and convex constrants. (The book by Boyd and
More informationThe descriptive complexity of the family of Banach spaces with the πproperty
Arab. J. Math. (2015) 4:35 39 DOI 10.1007/s4006501401163 Araban Journal of Mathematcs Ghadeer Ghawadrah The descrptve complexty of the famly of Banach spaces wth the πproperty Receved: 25 March 2014
More informationRealistic Image Synthesis
Realstc Image Synthess  Combned Samplng and Path Tracng  Phlpp Slusallek Karol Myszkowsk Vncent Pegoraro Overvew: Today Combned Samplng (Multple Importance Samplng) Renderng and Measurng Equaton Random
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationFinite Math Chapter 10: Study Guide and Solution to Problems
Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationGeneralizing the degree sequence problem
Mddlebury College March 2009 Arzona State Unversty Dscrete Mathematcs Semnar The degree sequence problem Problem: Gven an nteger sequence d = (d 1,...,d n ) determne f there exsts a graph G wth d as ts
More informationLecture 3: Force of Interest, Real Interest Rate, Annuity
Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annutymmedate, and ts present value Study annutydue, and
More informationCautiousness and Measuring An Investor s Tendency to Buy Options
Cautousness and Measurng An Investor s Tendency to Buy Optons James Huang October 18, 2005 Abstract As s well known, ArrowPratt measure of rsk averson explans a ratonal nvestor s behavor n stock markets
More informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 19982016. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng
More informationQUANTUM MECHANICS, BRAS AND KETS
PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented
More informationNMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 7. Root Dynamcs 7.2 Intro to Root Dynamcs We now look at the forces requred to cause moton of the root.e. dynamcs!!
More informationLecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.
Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook
More informationProduction. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem.
Producer Theory Producton ASSUMPTION 2.1 Propertes of the Producton Set The producton set Y satsfes the followng propertes 1. Y s nonempty If Y s empty, we have nothng to talk about 2. Y s closed A set
More informationUsing Series to Analyze Financial Situations: Present Value
2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated
More informationLinear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits
Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.
More informationOn Competitive Nonlinear Pricing
On Compettve Nonlnear Prcng Andrea Attar Thomas Marott Franços Salané February 27, 2013 Abstract A buyer of a dvsble good faces several dentcal sellers. The buyer s preferences are her prvate nformaton,
More informationThe Greedy Method. Introduction. 0/1 Knapsack Problem
The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton
More informationProject Networks With MixedTime Constraints
Project Networs Wth MxedTme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More informationSolution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.
Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces
More informationNondegenerate Hilbert Cubes in Random Sets
Journal de Théore des Nombres de Bordeaux 00 (XXXX), 000 000 Nondegenerate Hlbert Cubes n Random Sets par Csaba Sándor Résumé. Une légère modfcaton de la démonstraton du lemme des cubes de Szemeréd donne
More informationDo Hidden Variables. Improve Quantum Mechanics?
Radboud Unverstet Njmegen Do Hdden Varables Improve Quantum Mechancs? Bachelor Thess Author: Denns Hendrkx Begeleder: Prof. dr. Klaas Landsman Abstract Snce the dawn of quantum mechancs physcst have contemplated
More informationL10: Linear discriminants analysis
L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss
More informationBandwdth Packng E. G. Coman, Jr. and A. L. Stolyar Bell Labs, Lucent Technologes Murray Hll, NJ 07974 fegc,stolyarg@research.belllabs.com Abstract We model a server that allocates varyng amounts of bandwdth
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationPassive Filters. References: Barbow (pp 265275), Hayes & Horowitz (pp 3260), Rizzoni (Chap. 6)
Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationINTERPRETING TRUE ARITHMETIC IN THE LOCAL STRUCTURE OF THE ENUMERATION DEGREES.
INTERPRETING TRUE ARITHMETIC IN THE LOCAL STRUCTURE OF THE ENUMERATION DEGREES. HRISTO GANCHEV AND MARIYA SOSKOVA 1. Introducton Degree theory studes mathematcal structures, whch arse from a formal noton
More informationStability, observer design and control of networks using Lyapunov methods
Stablty, observer desgn and control of networks usng Lyapunov methods von Lars Naujok Dssertaton zur Erlangung des Grades enes Doktors der Naturwssenschaften  Dr. rer. nat.  Vorgelegt m Fachberech 3
More informationCS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering
Lecture 7a Clusterng Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Clusterng Groups together smlar nstances n the data sample Basc clusterng problem: dstrbute data nto k dfferent groups such that
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More information) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance
Calbraton Method Instances of the Cell class (one nstance for each FMS cell) contan ADC raw data and methods assocated wth each partcular FMS cell. The calbraton method ncludes event selecton (Class Cell
More informationThe Noether Theorems: from Noether to Ševera
14th Internatonal Summer School n Global Analyss and Mathematcal Physcs Satellte Meetng of the XVI Internatonal Congress on Mathematcal Physcs *** Lectures of Yvette KosmannSchwarzbach Centre de Mathématques
More informationErrorPropagation.nb 1. Error Propagation
ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then
More informationEfficient Project Portfolio as a tool for Enterprise Risk Management
Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse
More informationOn the Solution of Indefinite Systems Arising in Nonlinear Optimization
On the Soluton of Indefnte Systems Arsng n Nonlnear Optmzaton Slva Bonettn, Valera Ruggero and Federca Tnt Dpartmento d Matematca, Unverstà d Ferrara Abstract We consder the applcaton of the precondtoned
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationHedging InterestRate Risk with Duration
FIXEDINCOME SECURITIES Chapter 5 Hedgng InterestRate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cashflows Interest rate rsk Hedgng prncples DuratonBased Hedgng Technques Defnton of duraton
More informationAPPLICATIONS OF VARIATIONAL PRINCIPLES TO DYNAMICS AND CONSERVATION LAWS IN PHYSICS
APPLICATIONS OF VAIATIONAL PINCIPLES TO DYNAMICS AND CONSEVATION LAWS IN PHYSICS DANIEL J OLDE Abstract. Much of physcs can be condensed and smplfed usng the prncple of least acton from the calculus of
More informationUniform topologies on types
Theoretcal Economcs 5 (00), 445 478 555756/000445 Unform topologes on types YChun Chen Department of Economcs, Natonal Unversty of Sngapore Alfredo D Tllo IGIER and Department of Economcs, Unverstà Lug
More informationGeneral Auction Mechanism for Search Advertising
General Aucton Mechansm for Search Advertsng Gagan Aggarwal S. Muthukrshnan Dávd Pál Martn Pál Keywords game theory, onlne auctons, stable matchngs ABSTRACT Internet search advertsng s often sold by an
More informationFINITE HILBERT STABILITY OF (BI)CANONICAL CURVES
FINITE HILBERT STABILITY OF (BICANONICAL CURVES JAROD ALPER, MAKSYM FEDORCHUK, AND DAVID ISHII SMYTH* To Joe Harrs on hs sxteth brthday Abstract. We prove that a generc canoncally or bcanoncally embedded
More informationOn Leonid Gurvits s proof for permanents
On Leond Gurvts s proof for permanents Monque Laurent and Alexander Schrver Abstract We gve a concse exposton of the elegant proof gven recently by Leond Gurvts for several lower bounds on permanents.
More informationQuantization Effects in Digital Filters
Quantzaton Effects n Dgtal Flters Dstrbuton of Truncaton Errors In two's complement representaton an exact number would have nfntely many bts (n general). When we lmt the number of bts to some fnte value
More informationANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING
ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 6105194390,
More informationSection C2: BJT Structure and Operational Modes
Secton 2: JT Structure and Operatonal Modes Recall that the semconductor dode s smply a pn juncton. Dependng on how the juncton s based, current may easly flow between the dode termnals (forward bas, v
More informationThe Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets
. The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely
More informationNatural hpbem for the electric field integral equation with singular solutions
Natural hpbem for the electrc feld ntegral equaton wth sngular solutons Alexe Bespalov Norbert Heuer Abstract We apply the hpverson of the boundary element method (BEM) for the numercal soluton of the
More informationSection 5.4 Annuities, Present Value, and Amortization
Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today
More informationTo Fill or not to Fill: The Gas Station Problem
To Fll or not to Fll: The Gas Staton Problem Samr Khuller Azarakhsh Malekan Julán Mestre Abstract In ths paper we study several routng problems that generalze shortest paths and the Travelng Salesman Problem.
More information