How To Prove That A Multplcty Map Is A Natural Map
|
|
- Vernon Sanders
- 3 years ago
- Views:
Transcription
1 ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS CAILLO DE LELLIS AND EANUELE SPADARO Abstract. We prove several results on Almgren s multple valued functons and ther lnks to ntegral currents. In partcular, we gve a smple proof of the fact that a Lpschtz multple valued map naturally defnes an nteger rectfable current; we derve explct formulae for the boundary, the mass and the frst varatons along certan specfc vectorfelds; and explot ths connecton to derve a delcate reparametrzaton property for multple valued functons. These results play a crucal role n our new proof of the partal regularty of area mnmzng currents [5, 6, 7]. 0. Introducton It s known snce the poneerng work of Federer and Flemng [10] that one can naturally assocate an nteger rectfable current to the graph of a Lpschtz functon n the Eucldean space, ntegratng forms over the correspondng submanfold, endowed wth ts natural orentaton. It s then possble to derve formulae for the boundary of the current, ts mass and ts frst varatons along smooth vector-felds. oreover, all these formulae have mportant Taylor expansons when the current s suffcently flat. In ths paper we provde elementary proofs for the correspondng facts n the case of Almgren s multple valued functons (see [4] for the relevant defntons). The connecton between multple valued functons and ntegral currents s crucal n the analyss of the regularty of area mnmzng currents for two reasons. On the one hand, t provdes the necessary tools for the approxmaton of currents wth graphs of multple valued functon. Ths s a fundamental dea for the study of the regularty of mnmzng currents n the classcal sngle-vaued case, and t also plays a fundamental role n the proof of Almgren s partal regularty result (cf. [1, 5]). In ths perspectve, explct expressons for the mass and the frst varatons are necessary to derve the rght estmates on the man geometrc quanttes nvolved n the regularty theory (cf. [5, 6, 7]). On the other hand, the connecton can be exploted to nfer nterestng conclusons about the multple valued functons themselves. Ths pont of vew has been taken frutfully n many problems for the case of classcal functons (see, for nstance, [11, 12] and the references theren), and has been recently exploted n the multple valued settng n [3, 14]. The prototypcal example of nterest here s the followng: let f : R m Ω R n be a Lpschtz map and Gr(f) ts graph. If the Lpschtz constant of f s small and we change coordnates n R m+n wth an orthogonal transformaton close to the dentty, then the set Gr(f) s the graph of a Lpschtz functon f over some doman Ω also n the new system of coordnates. In fact t s easy to see that there exst sutable maps Ψ and Φ such that f(x) = Ψ ( x, f(φ(x)) ). In the multple valued 1
2 2 CAILLO DE LELLIS AND EANUELE SPADARO case, t remans stll true that Gr(f) s the graph of a new Lpschtz map f n the new system of coordnates, but we are not aware of any elementary proof of such statement, whch has to be much more subtle because smple relatons as the one above cannot hold. It turns out that the structure of Gr(f) as ntegral current gves a smple approach to ths and smlar ssues. Several natural estmates can then be proved for f, although more nvolved and much harder. The last secton of the paper s dedcated to these questons; more careful estmates obtaned n the same ven wll also be gven n [6], where they play a crucal role. ost of the conclusons of ths paper are already establshed, or have a counterpart, n Almgren s monograph [1], but we are not always able to pont out precse references to statements theren. However, also when ths s possble, our proofs have an ndependent nterest and are n our opnon much smpler. ore precsely, the materal of Sectons 1 and 2 s covered by [1, Sectons ], where Almgren deals wth general flat chans. Ths s more than what s needed n [5, 6, 7], and for ths reason we have chosen to treat only the case of nteger rectfable currents. Our approach s anyway smpler and, nstead of relyng, as Almgren does, on the ntersecton theory of flat chans, we use rather elementary tools. For the theorems of Secton 3 we cannot pont out precse references, but Taylor expansons for the area functonal are ubqutous n [1, Chapters 3 and 4]. The theorems of Secton 4 do not appear n [1], as Almgren seems to consder only some partcular classes of deformatons (the squeeze and squash, see [1, Chapter 5]), whle we derve farly general formulas. Fnally, t s very lkely that the conclusons of Secton 5 appear n some form n the constructon of the center manfold of [1, Chapter 4], but we cannot follow the ntrcate arguments and notaton of that chapter. In any case, our approach to reparametrzons of multple valued maps seems more flexble and powerful, capable of further applcatons, because, as t was frst realzed n [4], we can use tools from metrc analyss and metrc geometry developed n the last 20 years. Acknowledgments The research of Camllo De Lells has been supported by the ERC grant agreement RA (Regularty for Area nmzng currents), ERC The authors are warmly thankful to Bll Allard for several enlghtenng conversatons and hs constant enthusastc encouragement; and very grateful to Luca Spolaor and atteo Focard for carefully readng a prelmnary verson of the paper and for ther very useful comments. Camllo De Lells s also very thankful to the Unversty of Prnceton, where he has spent most of hs sabbatcal completng ths and the papers [5, 6, 7]. 1. Q-valued push-forwards We use the notaton, for: the eucldean scalar product, the naturally nduced nner products on p-vectors and p-covectors and the dualty parng of p-vectors and p-covectors; we nstead restrct the use of the symbol to matrx products. Gven a C 1 m-dmensonal submanfold Σ R N, a functon f : Σ R k and a vector feld X tangent to Σ, we denote by D X f the dervatve of f along X, that s D X f(p) = (f γ) (0) whenever γ s a smooth curve on Σ wth γ(0) = p and γ (0) = X(p). When k = 1, we denote by f the vector feld tangent to Σ such that f, X = D X f for every tangent vector feld X. For general
3 ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 3 k, Df x : T x Σ R k wll be the lnear operator such that Df x X(x) = D X f(x) for any tangent vector feld X. We wrte Df for the map x Df x and sometmes we wll also use the notaton Df(x) n place of Df x. Havng fxed an orthonormal base e 1,... e m on T x Σ and lettng (f 1,..., f k ) be the components of f, we can wrte f = m =1 a e and Df for the usual Hlbert-Schmdt norm: m k Df 2 = D e f 2 = f 2 = a 2. =1 =, All the notaton above s extended to the dfferental of Lpschtz multple valued functons at ponts where they are dfferentable n the sense of [4, Defnton 1.19]: although the defnton n there s for eucldean domans, ts extenson to C 1 submanfolds Σ R N s done, as usual, usng coordnate charts. We wll keep the same notaton also when f = Y s a vector feld,.e. takes values n R N, the same Eucldean space where Σ s embedded. In that case we defne addtonally dv Σ Y := D e Y, e. oreover, when Y s tangent to Σ, we ntroduce the covarant dervatve D Σ Y x,.e. a lnear map from T x Σ nto tself whch gves the tangental component of D X Y. Thus, f we denote by p x : R N T x Σ the orthogonal proecton onto T x Σ, we have D Σ Y x = p x DY (x). It follows that D Σ Y X = X Y, where we use for the connecton (or covarant dfferentaton) on Σ compatble wth ts structure as Remannan submanfold of R N. Such covarant dfferentaton s then extended n the usual way to general tensors on Σ. When dealng wth C 2 submanfolds Σ of R N we wll denote by A the followng tensor: A x as a blnear map on T x Σ T x Σ takng values on T x Σ (the orthogonal complement of T x Σ) and f X and Y are vector felds tangent to Σ, then A(X, Y ) s the normal component of D X Y, whch we wll denote by DX Y. A s called second fundamental form by some authors (cf. [13, Secton 7], where the tensor s denoted by B) and we wll use the same termnology, although n dfferental geometry t s more customary to call A shape operator and to use second fundamental form for scalar products A(X, Y ), η wth a fxed normal vector feld (cf. [8, Chapter 6, Secton 2] and [15, Vol. 3, Chapter 1]). In addton, H wll denote the trace of A (.e. H = A(e, e ) where e 1,..., e m s an orthonormal frame tangent to Σ) and wll be called mean curvature Push-forward through multple valued functons of C 1 submanfolds. In what follows we consder an m-dmensonal C 1 submanfold Σ of R N and use the word measurable for those subsets of whch are H m -measurable. Any tme we wrte an ntegral over (a measurable subset of) Σ we understand that ths ntegral s taken wth respect to the H m measure. We recall the followng lemma whch, even f not stated explctely n [4], s contaned n several arguments theren. Lemma 1.1 (Decomposton). Let Σ be measurable and F : A Q (R n ) Lpschtz. Then there are a countable partton of n bounded measurable subsets ( N) and Lpschtz functons f : R n ( {1,..., Q}) such that (a) F = Q =1 f for every N and Lp(f ) Lp(F ), ;
4 4 CAILLO DE LELLIS AND EANUELE SPADARO (b) N and, {1,..., Q}, ether f f or f (x) f (x) x ; (c) we have DF (x) = Q =1 Df (x) for a.e. x. Proof. The proof s by nducton on Q. For Q = 1 t s obvous. Assume the statement for any Q < Q and fx F and. Note that, wthout loss of generalty, we can assume that s bounded. We set 0 := {x : y = y(x) R n wth F (x) = Q y }. Clearly, 0 s measurable because t s the countermage of a closed subset of A Q (R n ). oreover, y : 0 R n s Lpschtz. We then set f 0 = y for every {1,..., Q}. Next, consder x 0. By [4, Proposton 1.6] there exst a ball B x, two postve numbers Q 1 and Q 2, and two Lpschtz Q l -valued functons g l : B x A Ql (R n ) for l = 1, 2 such that F Bx = g 1 + g 2. We can apply the nductve hypothess to g 1 and g 2, and conclude that F Bx can be reduced to the form as n (a) and (b) when restrcted to a (sutably chosen) countable partton of B x nto measurable sets. Snce Σ s paracompact, we can fnd a countable cover {B x } of \ 0, from whch (a) and (b) follow. The last statement can be easly verfed at every Lebesgue pont x where F and all the f s are dfferentable. When F : Σ R n s a proper Lpschtz functon and Σ R N s orented, the current S = F n R n s gven by S(ω) = ω(f (x)), DF (x) e(x) dh m (x) ω D m (R n ), where e(x) = e 1 (x)... e m (x) s the orentng m-vector of Σ and DF (x) e = (DF x e 1 )... (DF x e m ), (cf. [13, Remark 26.21(3)]; as usual D m (Ω) denotes the space of smooth m-forms compactly supported n Ω). Usng the Decomposton Lemma 1.1 t s possble to extend ths defnton to multple valued functons. To ths purpose, we gve the defnton of proper multple valued functons. Defnton 1.2 (Proper Q-valued maps). A measurable F : A Q (R n ) s called proper f there s a measurable selecton F 1,..., F Q as n [4, Defnton 1.1] (.e. F = F ) such that (F ) 1 (K) s compact for every compact K R n. It s then obvous that f there exsts such a selecton, then every measurable selecton shares the same property. We warn the reader that the termnology mght be slghtly msleadng, as the condton above s effectvely stronger than the usual properness of maps takng values n the metrc space (A Q (R n ), G), even when F s contnuous: the standard noton of properness would not ensure the well-defnton of the multple-valued push-forward. Defnton 1.3 (Q-valued push-forward). Let Σ R N be a C 1 orented manfold, Σ a measurable subset and F : A Q (R n ) a proper Lpschtz map. Then, we defne the push-forward of through F as the current T F =, (f ), where and f are
5 ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 5 as n Lemma 1.1: that s, T F (ω) := Q ω(f (x)), Df (x) e(x) dh m (x) N =1 } {{ } T (ω) ω D m (R n ). (1.1) We frst want to show that T s well-defned. Snce F s proper, we easly deduce that T (ω) Lp(F ) ω H m ((f ) 1 )(spt(ω)) <. On the other hand, upon settng F (x) := f (x) for x, we have (f ) 1 (spt(ω)) = (F ) 1 (spt(ω)) and (f ) 1 (spt(ω)) (f ) 1 (spt(ω)) = for, thus leadng to Q T (ω) Lp(F ) ω H m ((F ) 1 (spt(ω))) < +., =1 Therefore, we can pass the sum nsde the ntegral n (1.1) and, by Lemma 1.1, get Q T F (ω) = ω(f l (x)), DF l (x) e(x) dh m (x) ω D m (R n ). (1.2) l=1 In partcular, recallng the standard theory of rectfable currents (cf. [13, Secton 27]) and the area formula (cf. [13, Secton 8]), we have acheved the followng proposton. Proposton 1.4 (Representaton of the push-forward). The defnton of the acton of T F n (1.1) does not depend on the chosen partton nor on the chosen decomposton {f }, (1.2) holds and, hence, T F s a (well-defned) nteger rectfable current gven by T F = (Im(F ), Θ, τ) where: (R1) Im(F ) = x spt(f (x)) = Q N =1 f ( ) s an m-dmensonal rectfable set; (R2) τ s a Borel untary m-vector orentng Im(F ); moreover, for H m -a.e. p Im(F ), we have Df (x) e(x) 0 for every,, x wth f (x) = p and τ(p) = ± Df (x) e(x) Df (x) e(x) ; (1.3) (R3) for H m -a.e. p Im(F ), the (Borel) multplcty functon Θ equals Df Θ(p) := τ, (x) e(x) Df (x). e(x),,x:f (x)=p 1.2. Push-forward of Lpschtz submanfolds. As for the classcal push-forward, Defnton 1.3 can be extended to domans Σ whch are Lpschtz submanfolds usng the fact that such Σ can be chopped nto C 1 peces. Recall ndeed the followng fact. Theorem 1.5 ([13, Theorem 5.3]). If Σ s a Lpschtz m-dmensonal orented submanfold, then there are countably many C 1 m-dmensonal orented submanfolds Σ whch cover H m -a.s. Σ and such that the orentatons of Σ and Σ concde on ther ntersecton.
6 6 CAILLO DE LELLIS AND EANUELE SPADARO Defnton 1.6 (Q-valued push-forward of Lpschtz submanfolds). Let Σ R N be a Lpschtz orented submanfold, Σ a measurable subset and F : A Q (R n ) a proper Lpschtz map. Consder the {Σ } of Theorem 1.5 and set F := F Σ. Then, we defne the push-forward of through F as the nteger rectfable current T F := T F. The aboved defnton can be extended to Q-valued pushforwards of general rectfable currents n a straghtforward way: however ths wll never be used n the papers [5, 6, 7] and thus goes beyond the scope of our work. The followng concluson s a smple consequence of Theorem 1.5 and classcal arguments n geometrc measure theory (cf. [13, Secton 27]). Lemma 1.7. Let, Σ and F be as n Defnton 1.6 and consder a Borel untary m- vector e orentng Σ. Then T F s a well-defned nteger rectfable current for whch all the conclusons of Proposton 1.4 hold. As for the classcal push-forward, T F s nvarant under blpschtz change of varables. Lemma 1.8 (Blpschtz nvarance). Let F : Σ A Q (R n ) be a Lpschtz and proper map, Φ : Σ Σ a blpschtz homeomorphsm and G := F Φ. Then, T F = T G. Proof. The lemma follows trvally from the correspondng result for classcal push-forwards (see [9, & ]), the Decomposton Lemma 1.1 and the defnton of Q-valued pushforward. We wll next use the area formula to compute explctely the mass of T F. Followng standard notaton, we wll denote by JF (x) the Jacoban determnant of DF,.e. the number DF (x) e = det((df (x)) T DF (x)) Lemma 1.9 (Q-valued area formula). Let Σ, and F = F be as n Defnton 1.6. Then, for any bounded Borel functon h : R n [0, [, we have h(p) d T F (p) h(f (x)) JF (x) dh m (x). (1.4) Equalty holds n (1.4) f there s a set of full measure for whch DF (x) e(x), DF (y) e(y) 0 x, y and, wth F (x) = F (y). (1.5) If (1.5) holds the formula s vald also for bounded real-valued Borel h wth compact support. Proof. Let h : R n [0, [ be a Borel functon. Consder a decomposton as n the Decomposton Lemma 1.1 and the nteger rectfable currents T of (1.1). By the classcal area formula, see [13, Remark 27.2], we have h(p) d T (p) h(f (x))jf (x) dhm (x). (1.6) Summng ths nequalty over and and usng Lemma 1.1(c), we easly conclude (1.4). When (1.5) holds, we can choose τ of Proposton 1.4 such that the dentty (1.3) has always the + sgn. Defne Θ (p) := H 0 ({x : f (x) = p}. We then conclude from Proposton
7 ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 7 1.4(R3) that Θ(p) =, Θ (p) for H m -a.e. p Im(F ). On the other hand, agan by [13, Remark 27.2], equalty holds n (1.6) and, moreover, we have the denttes T = Θ H m Im(f ), T F = ΘH m Im(F ). Ths easly mples the second part of the lemma and hence completes the proof. A partcular class of push-forwards are gven by graphs. Defnton 1.10 (Q-graphs). Let Σ, and f = f be as n Defnton 1.6. Defne the map F : A Q (R N+n ) as F (x) := Q =1 (x, f (x)). T F s the current assocated to the graph Gr(f) and wll be denoted by G f. Observe that, f Σ, f and F are as n Defnton 1.10, then the condton (1.5) s always trvally satsfed. oreover, when Σ = R m the well-known Cauchy-Bnet formula gves m (JF ) 2 = 1 + (det A) 2, k=1 A k (DF ) where k (B) denotes the set of all k k mnors of the matrx B. Lemma 1.9 gves then the followng corollary n the case of Q-graphs Corollary 1.11 (Area formula for Q-graphs). Let Σ = R m, R m and f be as n Defnton Then, for any bounded compactly supported Borel h : R m+n R, we have ( m ) 1 h(p) d G f (p) = h(x, f (x)) 1 + (det A) 2 2 dx. (1.7) 2. Boundares k=1 A k (DF ) In the classcal theory of currents, when Σ s a Lpschtz manfold wth Lpschtz boundary and F : Σ R N s Lpschtz and proper, then (F Σ ) = F Σ (see [9, ]). Ths result can be extended to multple-valued functons. Theorem 2.1 (Boundary of the push-forward). Let Σ be a Lpschtz submanfold of R N wth Lpschtz boundary, F : Σ A Q (R n ) a proper Lpschtz functon and f = F Σ. Then, T F = T f. The man buldng block s the followng small varant of [4, Homotopy Lemma 1.8]. Lemma 2.2. There s c(q, m) > 0 such that, for every closed cube C R m centered at x 0 and every F Lp(C, A Q (R n )), we can fnd G Lp(C, A Q (R n )) satsfyng: () G C = F C =: f, Lp(G) c Lp(F ) and G(F, G) L c Lp(F ) dam(c); () there are Lpschtz mult-valued maps G and f (wth {1,..., J}) such that G = J =1 G, f = J =1 f and G G = (x 0, a ) G f for some a R n. Proof. The proof of () s contaned n [4, Lemma 1.8]. Concernng (), the proof s contaned n the nductve argument of [4, Lemma 1.8], t suffces to complement the arguments there wth the followng fact: f C = [ 1, 1] m, u Lp( C, A Q (R n )) and
8 8 CAILLO DE LELLIS AND EANUELE SPADARO G(x) = ( ) x x u s the cone-lke extenson of u to C (where x = sup x x ), then G G = 0 G u. The proof of ths clam s a smple consequence of the Decomposton Lemma 1.1 and the very defnton of G u. Consder, ndeed, a countable measurable partton = C and Lpschtz functons u wth u = u. Accordng to our defntons, G u =, (u ) =:, T. Consder now for each the set R := {λx : x, λ ]0, 1]} and defne G (λx) := λu (x) for every x and λ ]0, 1]. The sets R are a measurable decomposton of C \ {0} and we have G = G R. Therefore, settng S := (G ) R, we have G G =, S. On the other hand, by the classcal theory of currents S = 0 T (see [9, Secton ]). Snce ((S ) + (T )) <, the desred clam follows. Proof of Theorem 2.1. The proof s by nducton on the dmenson m. Snce every Lpschtz manfold can be trangulated and the statement s nvarant under blpschtz homemorphsms, t suffces to prove the theorem when Σ = [0, 1] m. Next, gven a classcal Lpschtz map Φ : R N R k, let Φ F be the multple-valued map Φ(F ) (cf. [4, Secton 1.3.1]). If F s a classcal Lpschtz map, then T Φ F = Φ F Σ = Φ T F (cf. [9, ]). The same dentty holds for Q-valued map, as the Decomposton Lemma 1.1 easly reduces t to the sngle-valued case. Then, f p : R m R m+n R m+n s the orthogonal proecton on the second components, we have p G F = T F. Gven the classcal commutaton of boundary and (sngle-valued) push-forward (see [9, Secton ]) we are then reduced to provng he dentty G F = G f. We turn therefore to the case G F. The startng step m = 1 s an obvous corollary of the Lpschtz selecton prncple [4, Proposton 1.2]. Indeed, for F Lp([0, 1], A Q (R n )), there exst functons F Lp([0, 1], R n ) such that F = F. Therefore, T F = T F and T F = T F = ( F (1) F (0) ) = T f. For the nductve argument, consder the dyadc decomposton at scale 2 l of [0, 1] m : [0, 1] m = Q k,l, wth Q k,l = 2 l (k + [0, 1] m ). k {0,...,2 l 1} m In each Q k,l, let u k,l be the cone-lke extenson gven by Lemma 2.2 of f k,l := F Qk,l. Denote by u l the Q- functon on [0, 1] m whch concdes wth u k,l on each Q k,l. Obvously the u l s are equ-lpschtz and converge unformly to F by Lemma 2.2 (). Set T l := G ul = k G u k,l. By the nductve hypothess G fk,l = 0. Snce ( p T ) = T p T (see [13, Secton 26]), Lemma 2.2 mples G uk,l = G fk,l. Consderng that the boundary faces common to adacent cubes come wth opposte orentatons, we conclude T l = G f. By Corollary 1.11, lm sup l ((T l ) + ( T l )) < and so the compactness theorem for ntegral currents (see [13, Theorem 27.3]) guarantees the exstence of an ntegral current T whch s the weak lmt of a subsequence of {T l } (not relabeled). It suffces therefore to show that:
9 ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 9 (C) f Ω R m s an open set and u l s a sequence of Lpschtz Q-valued maps on Ω such that u l converge unformly to some F and T l := G ul converge to an ntegral current T, then T = G F. We wll prove (C) by nducton over Q: the case Q = 1 s classcal (see for nstance [11, Theorem 2, Secton 3.1 n Chapter 3] and [11, Proposton 2, Secton 2.1 n Chapter 3]). We assume (C) holds for every Q < Q and want to prove t for Q. Fx a sequence as n (C). Clearly T s supported n the rectfable set Gr(F ). Fx an orthonormal bass e 1,..., e m of R m and extend t to an orthonormal bass of R m+n wth postve orentaton. Set e = e 1... e m. Thanks to the Lpschtz regularty of F, Gr(F ) can be orented by m-planes τ wth the property that τ, e c > 0, where the constant c depends on Lp(F ). We have T = (Gr(F ), τ, Θ) and G F = (Gr(F ), τ, Θ): we ust need to show that Θ = Θ H m -a.e. on Gr(F ). As observed n Lemma 1.1 there s a closed set 0 and a Lpschtz functon f 0 such that: F (x) = Q f 0 (x) for every x 0 ; F splts locally on Ω = Ω\ 0 nto (Lpschz) functons takng less than Q values. Usng the nducton hypothess, t s trval to verfy that T Ω R n = G F Ω R n. Thus we ust need to show that Θ(x, f 0 (x)) = Θ(x, f 0 (x)) for H m -a.e. x 0. Consder the orthogonal proecton p : R m+n R m. By the well-known formula for the pusforward of currents (see [9, Lemma ]), we have p T = Θ Ω and p G F = Θ Ω, where Θ (x) = Θ(x, y) and Θ (x) = Θ(x, y). (x,y) Gr(F ) (x,y) Gr(F ) Therefore Θ (x) = Θ(x, f 0 (x)) and Θ (x) = Θ(x, f 0 (x)) for H m -a.e. x 0. On the other hand, by the defnton of G F and the very same formula for the push-forward (.e. [9, Lemma ]) t s easy to see that p G F = Q Ω = p T l. Snce p T l converges to p T, we conclude that Θ Q Θ H m -a.e. on Ω, whch n turn mples Θ(x, f 0 (x)) = Θ(x, f 0 (x)) for a.e. x 0. Ths completes the proof of the nductve step. 3. Taylor expanson of the area functonal In ths secton we compute the Taylor expanson of the area functonal n several forms. To ths am, we fx the followng notaton and hypotheses. Assumpton 3.1. We consder the followng: () an open submanfold R m+n of dmenson m wth H m () <, whch s the graph of a functon ϕ : R m Ω R n wth ϕ C 3 c; A and H wll denote, respectvely, the second fundamental form and the mean curvature of ; (U) a regular tubular neghborhood U of,.e. the set of ponts {x + y : x, y T x, y < c 0 }, where the thckness c 0 s suffcently small so that the nearest pont proecton p : U s well defned and C 2 ; the thckness s supposed to be larger than a fxed geometrc constant;
10 10 CAILLO DE LELLIS AND EANUELE SPADARO (N) a Q-valued map F : A Q (R m+n ) of the form Q Q F (x) = x + N (x), =1 =1 where N : A Q (R m+n ) satsfes x + N (x) U, N (x) T x for every x and Lp(N) c. We recall the notaton η F := 1 Q F, for every multple valued functon F = F. Theorem 3.2 (Expanson of (T F )). If, F and N are as n Assumpton 3.1 and c s smaller than a geometrc constant, then (T F ) = Q H m () Q η N + H, 1 DN 2 2 ( ) + P 2 (x, N ) + P 3 (x, N, DN ) + R 4 (x, DN ), (3.1) where P 2, P 3 and R 4 are C 1 functons wth the followng propertes: () n P 2 (x, n) s a quadratc form on the normal bundle of satsfyng P 2 (x, n) C A(x) 2 n 2 x, n T x ; (3.2) () P 3 (x, n, D) = L (x, n)q (x, D), where n L (x, n) are lnear forms on the normal bundle of and D Q (x, D) are quadratc forms on the space of (m + n) (m + n)-matrces, satsfyng L (x, n) Q (x, D) C A(x) n D 2 x, n T x, D ; () R 4 (x, D) = D 3 L(x, D), for some functon L wth Lp(L) C, whch satsfes L(x, 0) = 0 for every x and s ndependent of x when A 0. oreover, for any Borel functon h : R m+n R, ( ) h d T F h F C A h F N + h ( DN 2 + A N 2 ), (3.3) and, f h(p) = g(p(p)) for some g, we have h d T F Q (1 H, η N DN 2 ) g C ( A 2 N 2 + DN 4) g. (3.4) In partcular, as a smple corollary of the theorem above, we have the followng. Corollary 3.3 (Expanson of (G f )). Assume Ω R m s an open set wth bounded measure and f : Ω A Q (R n ) a Lpschtz map wth Lp(f) c. Then, (G f ) = Q Ω + 1 Df 2 + R 4 (Df ), (3.5) 2 where R 4 C 1 satsfes R 4 (D) = D 3 L(D) for L wth Lp( L) C and L(0) = 0. Ω Ω
11 ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 11 Proof. The corollary s reduced to Theorem 3.2 by smply settng = Ω {0}, N = N (x) := (0, f (x)) and F (x) = F (x) = (x, f (x)). Snce n ths case A vanshes, (3.1) gves precsely (3.5). Proof of Theorem 3.2. We wll n fact prove the statement for (T F V ), where V s any Borel subset of. Under ths generalty, by the decomposton Lemma 1.1, t s enough to consder the case F V = G, where each G = F V = x + N V s a (one-valued!) Lpschtz map. Next observe that (1.5) obvously holds f c s suffcently small. Therefore, (T F V ) = ((F ) V ), and, snce η N = 1 Q N, the formula (3.1) follows from summng the correspondng denttes ((F ) V ) = H m (V ) + H, N + 1 DN 2 V 2 V ( ) + P 2 (x, N ) + P 3 (x, N, DN ) + R 4 (x, DN ). (3.6) V To smplfy the notaton we drop the subscrpt n the proof of (3.6). Usng the area formula, we have that (F V ) = DF ξ dh m, where ξ = ξ 1... ξ m s the smple m-vector assocated to an orthonormal frame on T. By smple multlnear algebra DF ξ = det, where s the m m matrx gven by k = DF ξ, DF ξ k = ξ + DN ξ, ξ k + DN ξ k V = δ k + DN ξ, ξ k + DN ξ k, ξ + DN ξ, DN ξ k. (3.7) } {{ } } {{ } a k b k Set a = (a k ), b = (b k ) and denote by 2 (a + b) and 3 (a + b), respectvely, the sum of all 2 2 and that of all 3 3 mnors of the matrx (a + b); smlarly denote by R(a + b) the sum of all k k mnors wth k 4. Then, det = 1 + tr (a + b) + 2 (a + b) + 3 (a + b) + R(a + b). (3.8) Observe that the entres of a are lnear n DN and those of b are quadratc. Thus, 2 (a + b) = 2 (a) + 2 (b) + C 2 (a, b), (3.9) 3 (a + b) = 3 (a) + C 4 (a, b), (3.10) where C 2 (a, b) s a lnear combnaton of terms of the form a k b lm and C 4 (a, b) s a polynomal n the entres of DN satsfyng the nequalty C 4 (a, b) C DN 4. Recall the
12 12 CAILLO DE LELLIS AND EANUELE SPADARO Taylor expanson 1 + τ = 1 + τ τ 2 + τ 3 + g(τ), where g s an analytc functon wth g(τ) τ 4. Wth the ad of (3.8), (3.9) and (3.10) we reach the followng concluson: DF ξ tr (a + b) + 2 (a) + C 2 (a, b) + 3 (a) = (tr a)2 + 2 tr a tr b + 2 tr a 2 (a) (tr a)3 + + R 4, (3.11) 8 16 where R 4 s an analytc functon of the entres of DN whch satsfes R 4 (DN) C DN 4. Observe next that tr b = k DN ξ k, DN ξ k = DN 2. oreover, DN ξ, ξ k = ξ ( N, ξ k ) N, ξ ξ k = N, A(ξ, ξ k ). Thus, by the symmetry of the second fundamental form, we have a k = 2 A(ξ, ξ k ), N and tr a = 2 H, N. We then can rewrte DF ξ DN 2 = 1 H, N + + 2(a) (tr a)2 2 } 2 {{ 8 } P C 2(a, b) + 3 (a) tr a (tr b + 2(a)) (tr a)3 + } 2 {{ 4 16 } +R 4. (3.12) P 3 Integratng (3.12) we reach (3.6). It remans to show that P 2, P 3 and R 4 satsfy (), () and (). If A = 0, then s flat and the frame ξ 1,..., ξ m can be chosen constant, so that R 4 wll not depend on x. Next, each b k s a quadratc polynomal n the entres of DN, wth coeffcents whch are C 2 functons of x. Instead each a k can be seen as a lnear functon n DN wth coeffcents whch are C 2 functons of x, but also as a lnear functon L k of N, wth a C 1 dependence on x. In the latter case we have the bound L k (x, n) A(x) n. Therefore the clams n () and () follow easly. Fnally, snce R 4 s an analytc functon of the entres of DN satsfyng R 4 (DN) C DN 4, the representaton n () follows from the elementary consderaton that R 4(D) s a Lpschtz functon vanshng at the orgn. D 3 Fnally, observe that the argument above mples (3.4) when g s the ndcator functon of any measurable set and the general case follows from standard measure theory. The dentty (3.3) follows easly from the same formulas for DF ξ, usng ndeed cruder estmates Taylor expanson for the excess n a cylnder. The last results of ths secton concerns estmates of the excess n dfferent systems of coordnates, n partcular wth respect to tlted planes and curvlnear coordnates. Proposton 3.4 (Expanson of a curvlnear excess). There exst a dmensonal constant C > 0 such that, f, F and N are as n Assumpton 3.1 wth c small enough, then T F (x) (p(x)) 2 d T F (x) DN 2 C ( A 2 N 2 + DN 4 ), (3.13)
13 ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 13 where T F and are the unt m-vectors orentng T F and T, respectvely. Proof. Let p and defne (p) = ξ 1... ξ m for some orthonormal frame ξ 1,..., ξ m for T and T F (F (p)) = ζ ζ wth ζ = (ξ 1 + DN p ξ 1 )... (ξ m + DN p ξ m ). Our assumptons mply p(f (p)) = p. Usng the Q-valued area formula and obvous computatons we get T F p 2 d T F (x) = ζ ζ 2 ( ζ = 2 ζ ) ζ,. As already computed n the proof of Theorem 3.2, ζ = Q Q H, η N + DN O( A 2 N 2 + DN 4 ). If we next defne Bk := ξ, ξ k + DN ξ k = δ k N, A(ξ, ξ k ), we then get ζ, = det B = Q Q H, η N + O( A 2 N 2 ). Hence the clamed formula follows easly. Next we compute the excess of a Lpschtz graph wth respect to a tlted plane. Theorem 3.5 (Expanson of a cylndrcal excess). There exst dmensonal constants C, c > 0 wth the followng property. Let f : R m A Q (R n ) be a Lpschtz map wth Lp (f) c. For any 0 < s, set A := B s D(η f) and denote by τ the orented untary m-dmensonal smple vector to the graph of the lnear map y A y. Then, we have G f τ 2 d G f G(Df, Q A ) 2 C Df B s B 4. (3.14) s C s Proof of Theorem 3.5. Argung as n the prevous proofs, thanks to Lemma 1.1, we can wrte f = f and process local computatons (when needed) as f each f were Lpschtz. oreover, we have that τ = ξ ξ wth ξ = (e 1 + A e 1 )... (e m + A e m ). Here and for the rest of ths proof, we dentfy R m and R n wth the subspaces R m {0} and {0} R n of R m+n, respectvely: ths ustfes the notaton e + A e for e R m and A e R n. Next, we recall that ξ = ξ, ξ = det(δ + A e, A e ) = A 2 + O( A 4 ).
14 14 CAILLO DE LELLIS AND EANUELE SPADARO By Corollary 1.11 we also have E := G f τ 2 d G f = 2 (G f ) 2 G f, τ d G f C s = 2 Q B s + ( Df 2 + O( Df 4 )) 2 (e 1 + Df e 1 )... (e m + Df e m ), τ. B s On the other hand A e, e k = 0 = Df e, e k. Therefore, (e 1 + Df e 1 )... (e m + Df e m ), τ = ξ 1 det(δ k + Df e, A e k ) ) 1 = (1 + A ( O( A 4 ) 1 + Df : A + O( Df 2 A 2 ) ). Recallng that A Cs m Df C ( s m Df 4) 1 4, we then conclude ( ) E = Df 2 + Q B s A 2 2 Df : A + O Df 4 B s B s B s ( ) ( ) = Df A 2 + O Df 4 = G(Df, Q A ) 2 + O Df 4. B s B s B s B s 4. Frst varatons In ths secton we compute the frst varatons of the currents nduced by multple valued maps. These formulae are ultmately the lnk between the statonarty of area mnmzng currents and the partal dfferental equatons satsfed by sutable approxmatons. We use here the followng standard notaton: gven a current T n R N and a vector feld X C 1 (R N, R N ), we denote the frst varaton of T along X by δt (X) := d dt t=0 (Φ t T ), where Φ :] η, η[ U R N s any C 1 sotopy of a neghborhood U of spt(t ) wth Φ(0, x) = x for any x U and d dε ε=0 Φ ε = X (n what follows we wll often use Φ ε for the map x Φ(ε, x)). It would be more approprate to use the notaton δt (Φ) (see, for nstance, [9, Secton 5.1.7]), but snce the currents consdered n ths paper are rectfable, t s well known that the frst varaton depends only on X and s gven by the formula δt (X) = dv T X d T, (4.1) where dv T X = D e X, e for any orthonormal frame e 1,..., e m wth e 1... e m = T (see [9, 5.1.8] and cf. [13, Secton 2.9]). We begn wth the expanson for the frst varaton of graphs. In what follows, A : B wll denote the usual Hlbert Schmdt scalar product of two k matrces.
15 ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 15 Theorem 4.1 (Expanson of δg f (X)). Let Ω R m be a bounded open set and f : Ω A Q (R n ) a map wth Lp(f) c. Consder a functon ζ C 1 (Ω R m, R n ) and the correspondng vector feld χ C 1 (Ω R n, R m+n ) gven by χ(x, y) = (0, ζ(x, y)). Then, δg ( ) f(χ) Dx ζ(x, f ) + D y ζ(x, f ) Df : Df C Dζ Df 3. (4.2) Ω The next two theorems deal wth general T F as n Assumpton 3.1. However we restrct our attenton to outer and nner varatons, where we borrow our termnology from the elastcty theory and the lterature on harmonc maps. Outer varatons result from deformatons of the normal bundle of whch are the dentty on and map each fber nto tself, whereas nner varatons result from composng the map F wth sotopes of. Theorem 4.2 (Expanson of outer varatons). Let, U, p and F be as n Assumpton 3.1 wth c suffcently small. If ϕ C 1 c () and X(p) := ϕ(p(p))(p p(p)), then δt F (X) = where Err 3 C ( ϕ DN 2 + ) (N Dϕ) : DN Q ϕ H, η N + } {{ } Err 1 Err 2 C Ω 3 Err (4.3) =2 ϕ A 2 N 2 (4.4) ( ϕ ( DN 2 N A + DN 4) + Dϕ ( DN 3 N + DN N 2 A )). (4.5) Let Y be a C 1 vector feld on T wth compact support and defne X on U settng X(p) = Y (p(p)). Let {Ψ ε } ε ] η,η[ be any sotopy wth Ψ 0 = d and d dε ε=0 Ψ ε = Y and defne the followng sotopy of U: Φ ε (p) = Ψ ε (p(p)) + (p p(p)). Clearly X = d dε ε=0 Φ ε. Theorem 4.3 (Expanson of nner varatons). Let, U and F be as n Assumpton 3.1 wth c suffcently small. If X s as above, then ( DN 2 δt F (X) = dv Y ) 3 DN : (DN D Y ) + Err, (4.6) 2 =1 where Err 3 C ( Err 1 = Q H, η N dv Y + D Y H, η N ), (4.7) Err 2 C A ( 2 DY N 2 + Y N DN ), (4.8) ( Y A DN 2( N + DN ) + DY ( A N 2 DN + DN 4)). (4.9)
16 16 CAILLO DE LELLIS AND EANUELE SPADARO 4.1. Proof of Theorem 4.1. Set Φ ε (x, y) := (x, y + ε ζ(x, y)). For ε suffcently small Φ ε s a dffeomorphsm of Ω R n d nto ntself. oreover, Φ dε ε ε=0 = χ. Let f ε = f + ε ζ(x, f ). Snce (Φ ε ) G f = G fε, we can apply Corollary 3.3 to compute δg f (χ) = d (G fε ) (3.5) = d 1 ( D(f + ε ζ) 2 + dε ε=0 dε ε=0 2 R 4 (D(f + ε ζ)) ) ( ) d = Q Dx ζ(x, f ) + D y ζ(x, f ) Df : Df + R4 (Df + εdζ). dε ε=0 Snce R 4 () = 3 L() for some Lpschtz L wth L(0) = 0, we can estmate as follows: d R4 ( + εζ) dε ε=0 CL() 2 Dζ + C 3 Lp(L) Dζ C 3 Dζ, thus concludng the proof Proof of Theorem 4.2. Consder the map Φ ε (p) = p + εx(p). If ε s suffcently small, Φ ε maps U dffeomorphcally n a neghborhood of and we obvously have δt F (X) = d ((Φ dε ε) T F ) ε=0. Next set F ε (x) = x + N (x)(1 + ε ϕ(x)) and observe that (Φ ε ) T F = T Fε. Thus we can apply Theorem 3.2 to get: ( δt F (X) = ϕ DN 2 + ) (N Dϕ) : DN Q ϕ H, η N } {{ } =:Err 1 d + P 2 (x, N (1 + εϕ)) dε ε=0 } {{ } =:Err 2 d ( + P3 (x, N (1 + εϕ), D(N (1 + εϕ))) + R 4 (x, D(N (1 + εϕ))) ). dε ε=0 } {{ } =:Err 3 Snce n P 2 (x, n) s a quadratc form, we have P 2 (x, N (1 + εϕ)) = (1 + εϕ) 2 P 2 (x, N ) and thus (4.4) follows from (3.2). Next, by Theorem 3.2(), we have the bound d P 3 (x, N (1 + εϕ), D(N (1 + εϕ))) dε ε=0 C A(x) ( Dϕ N 2 DN + ϕ N DN 2). Fnally, takng nto account Theorem 3.2(): d R 4 (x, D(N (1 + εϕ))) dε ε=0 C ( DN 3 + DN 3 Lp(L) ) ( N Dϕ + DN ϕ). Puttng together the last two nequaltes we get (4.5).
17 ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS Proof of Theorem 4.3. Set F ε (x) = x + N (Ψ 1 ε (x)). Clearly, Φ ε T F = T Fε. Fx an orthonormal frame e 1,..., e m on T and let e = e 1... e m. By Lemma 1.9, δt F (X) = d (T Fε ) = d (DF ε, ) e. dε ε=0 dε ε=0 Fx {1,..., Q}. Usng the chan rule [4, Proposton 1.12], we have: (DF ε, ) e = w 1 (ε, x)... w m (ε, x) =: w(ε, x), where w (ε, x) = e (x) + DN Ψ 1 ε (x) DΨ 1 ε x e (x). Set v (ε, x) = w (ε, Ψ ε (x)). Snce Ψ 0 s the dentty, we obvously have v(0, ) = DF e. If we denote by JΨ ε (x) the Jacoban determnant of the transformaton Ψ ε, we can change varable n the ntegral to conclude: d (DF ε, ) e = d v(ε, x) JΨ ε (x) dε ε=0 dε ε=0 = (DF ) e d JΨ ε + v(0, x) 1 ε v(0, x), v(0, x) dε ε=0 = (DF ) e dv Y + ε v(0, x), (DF ) e } {{ } } {{ } I,1 I,2 + ε v(0, x), (DF ) e ( DF e 1 1 ). } {{ } I,3 Thus, δt F (X) = I,1 + I,2 + I,3 =: I 1 + I 2 + I 3 and we wll next estmate these three terms separately. Step 1. Estmate on I 1. By the Q-valued area formula of Lemma 1.9 and (3.4) n Theorem 3.2, I 1 = Q dv Y + 1 DN 2 dv Y Q H, η N dv Y + Err 2 where Err C ( A 2 N 2 + DN 4 ) dv Y. Snce dv Y = 0 (recall that Y Cc 1 ()), we easly conclude that I 1 = 1 3 DN 2 dv Y Q H, η N dv Y + Err, (4.10) 2 where the Err s satsfy the estmates (4.8) and (4.9). Step 2. Estmate on I 2. Set ζ (x) := ε v(0, x), (DF ) e = ε v(0, x), v(0, x) = 1 2 =2 d dε v(ε, x) 2. ε=0
18 18 CAILLO DE LELLIS AND EANUELE SPADARO Snce v(ε, x) 2 s ndependent of the orthonormal frame chosen, havng fxed a pont x, we can mpose D e = 0 at x. By multlnearty ε v(0, x) = v 1 (0, x)... ε v (0, x)... v m (0, x). (4.11) We next compute ε v (0, x) = ( ( ) ) e (Ψ ε (x)) + DN ε x DΨ 1 ε ε=0 ε e ε=0 Ψε(x) (Ψ ε (x)) = D Y e (x) + DN x [Y, e ](x), (4.12) where [Y, e ] s the Le bracket. On the other hand, snce D e (x) = 0, we have D Y e (x) = A(e, Y ) and [Y, e ](x) = e Y (x). Recall that v (0, ) = e + DN e. By the usual computatons n multlnear algebra, t turns out that ζ = det, where the entres of the m m matrx are gven by: αβ = e α + DN e α, e β + DN e β = δ αβ + O( A N ) + O( DN 2 ) for β, α = e α + DN e α, A(e, Y ) DN e Y. (The entres for α are computed as n the proof of Theorem 3.2). Denote by n α the (m 1) (m 1) mnor whch s obtaned by deletng the α row and the column. We then easly get the followng estmates: n C( DN 2 + A N ) for α, (4.13) oreover, observe that α n = 1 + O( DN 2 + A N ). (4.14) α = DN e α, DN e Y e α, DN e Y + A(e α, Y ), DN e = DN e α, DN e Y + A(e α, e Y ), N + A(e α, Y ), DN e. (4.15) We therefore conclude from (4.13), (4.14) and (4.15) that ζ (x) = det = ( 1) +α α n α α = ( DN e, DN e Y + A(e, e Y ), N + A(e, Y ), DN e ) ( + O DY ( DN 4 + A 2 N 2) ) + Y ( A DN 3 + A 2 N DN ). (4.16) Summng over and ntegratng, we then acheve I 2 = DN : (DN D Y ) + J 2 + Err 2 + Err 3, (4.17)
19 ULTIPLE VALUED FUNCTIONS AND INTEGRAL CURRENTS 19 where Err 2, Err 3 are estmated as n (4.8), (4.9), and ( J 2 = Q A(e, e Y ), η N + A(e, Y ), D e η N ). In order to treat ths last term, we consder the vector feld Z = A(e, Y ), η N e. Z s ndependent of the choce of the orthonormal frame e : therefore, to compute ts dvergence at a specfc pont x we can assume D e = 0. We then get dv Z = ( ) A(e, Y ), D e η N + De A(e, Y ), η N + A(e, e Y ), η N, where the tensor DX A(U, Y ) s defned as (D X (A(U, Y ))) A( X U, Y ) A(U, X Y ), (recall that (D X W ) denotes the normal component of D X W ). The Codazz-anard equatons (cf. [16, Chapter 7.C, Corollary 15] mply the symmetry of D A. Thus, De A(e, Y ), η N = DY A(e, e ), η N = DY H, η N. (4.18) Summarzng (and recallng that η N s normal to ), dv Z = ( A(e, Y ), D e η N + A(e, e Y ), η N ) + D Y H, η N. (4.19) Snce Z s compactly supported n, ntegratng (4.19) and usng the dvergence theorem we conclude 0 = Q 1 J 2 + D Y H, η N. We thus get I 2 = DN : (DN D Y ) Q D Y H, η N + Err 2 + Err 3. Step 3. Estmate on I 3. From the proof of Theorem 3.2, (cf. (3.11) and (3.12)) we conclude 1 (DF ) e C ( DN 2 + A N ). To show that I 3 can be estmated wth Err 2 and Err 3 observe that, by (4.16) we have ε v(0, x), (DF ) e = ζ (x) C DN 2 DY + C A DY N + C A DN Y. 5. Reparametrzng multple valued graphs In ths secton we explot the lnk between currents and multple valued functons n the opposte drecton, n order to gve condtons under whch Q-valued graphs can be sutably reparametrzed and to establsh relevant estmates on the parametrzaton. We fx the short-hand notaton e = e 1... e m+n, e m = e 1... e m and e n = e m+1... e m+n, where e 1,..., e m, e m+1,..., e m+n s the standard bass of R m R n. We wll often use the notaton π 0 and π 0 for R m {0} and {0} R n.
20 20 CAILLO DE LELLIS AND EANUELE SPADARO Theorem 5.1 (Q-valued parametrzatons). Let Q, m, n N and s < r < 1. Then, there are constants c 0, C > 0 (dependng on Q, m, n and r ) wth the followng property. Let ϕ, s and U be as n Assumpton 3.1 wth Ω = B s and let f : B r A Q (R n ) be such that ϕ C 2 + Lp(f) c 0 and ϕ C 0 + f C 0 c 0 r. (5.1) Set Φ(x) := (x, ϕ(x)). Then, there are maps F and N as n Assumpton 3.1(N) such that T F = G f U and Lp(N) C ( D 2 ϕ C 0 N C 0 + Dϕ C 0 + Lp(f) ), (5.2) 1 2 Q N(Φ(p)) G(f(p), Q ϕ(p) ) 2 Q N(Φ(p)) p B s, (5.3) η N(Φ(p)) C η f(p) ϕ(p) + CLp(f) Dϕ(p) N(Φ(p)) p B s. (5.4) Fnally, assume p B s and (p, η f(p)) = ξ + q for some ξ and q T ξ. Then, G(N(ξ), Q q ) 2 Q G(f(p), Q η f(p) ). (5.5) For further reference, we state the followng mmedate corollary of Theorem 5.1, correspondng to the case of a lnear ϕ. Proposton 5.2 (Q-valued graphcal reparametrzaton). Let Q, m, n N and s < r < 1. There exst postve constants c, C (dependng only on Q, m, n and r ) wth the followng s property. Let π 0 and π be m-planes wth π π 0 c and f : B r (π 0 ) A Q (π0 ) wth Lp(f) c and f cr. Then, there s a Lpschtz map g : B s (π) A Q (π ) wth G g = G f C s (π) and such that the followng estmates hold on B s (π): g C 0 Cr π π 0 + C f C 0, (5.6) Lp(g) C π π 0 + CLp(f). (5.7) In fact the proof of Theorem 5.1 wll gve a more precse nformaton about the map F, namely ts pontwse values can be determned wth a natural geometrc algorthm. Defnton 5.3 (ultplcty n Q-valued maps). Gven a Q-valued map F, we say that a pont p has multplcty k n F (x) f we can wrte F (x) = k p + Q k =1 p where p p for every,.e. f p has multplcty k when treatng F (x) as a 0-dmensonal ntegral current. Lemma 5.4 (Geometrc reparametrzaton). The values of F n Theorem 5.1 can be determned at any pont p as follows. Let κ be the orthogonal complement of T p. Then, Gr(f) (p + κ) s nonempty, conssts of at most Q ponts and every q Gr(f) (p + κ) has n F (p) the same multplcty of p π 0 (q) n f(p π0 (q)) Exstence of the parametrzaton. The next lemma s a natural outcome of the Ambroso-Krchhem approach to the theory of currents [2]. Followng [9, Secton 4.3], f T s a flat m-dmensonal current n U and h : U R k a Lpschtz map wth k m, we denote by T, h, y the slce of T wth respect to h at the pont y (well-defned for a.e. y R k ). Snce we deal wth normal currents, the equvalence of the classcal Federer- Flemng theory and the modern Ambroso-Krchhem theory (cf. [2, Theorem 11.1]) allows us to use all the results of the paper [2].
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 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 informationBERNSTEIN POLYNOMIALS
On-Lne 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 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 informationHow To Assemble The Tangent Spaces Of A Manfold Nto A Coherent Whole
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 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 information1 Example 1: Axis-aligned 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 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 non-coplanar vectors Scalar product
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 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 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 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 σ-algebra: a set
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 annuty-mmedate, and ts present value Study annuty-due, and
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 informationRing structure of splines on triangulations
www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAM-Report 2014-48 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 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 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 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 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 (In-Class) 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 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 informationPSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12
14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) 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 informationHow To Calculate The Accountng Perod Of Nequalty
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 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 informationOn Lockett pairs and Lockett conjecture for π-soluble Fitting classes
On Lockett pars and Lockett conjecture for π-soluble Fttng classes Lujn Zhu Department of Mathematcs, Yangzhou Unversty, Yangzhou 225002, P.R. Chna E-mal: ljzhu@yzu.edu.cn Nanyng Yang School of Mathematcs
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 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 informationNatural hp-bem for the electric field integral equation with singular solutions
Natural hp-bem for the electrc feld ntegral equaton wth sngular solutons Alexe Bespalov Norbert Heuer Abstract We apply the hp-verson of the boundary element method (BEM) for the numercal soluton of the
More informationLoop Parallelization
- - Loop Parallelzaton C-52 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 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 information21 Vectors: The Cross Product & Torque
21 Vectors: The Cross Product & Torque Do not use our left hand when applng ether the rght-hand rule for the cross product of two vectors dscussed n ths chapter or the rght-hand rule for somethng curl
More informationAN EFFECTIVE MATRIX GEOMETRIC MEAN SATISFYING THE ANDO LI MATHIAS PROPERTIES
MATHEMATICS OF COMPUTATION Volume, Number, Pages S 5-578(XX)- AN EFFECTIVE MATRIX GEOMETRIC MEAN SATISFYING THE ANDO LI MATHIAS PROPERTIES DARIO A. BINI, BEATRICE MEINI AND FEDERICO POLONI Abstract. We
More informationArea distortion of quasiconformal mappings
Area dstorton of quasconformal mappngs K. Astala 1 Introducton A homeomorphsm f : Ω Ω between planar domans Ω and Ω s called K-quasconformal f t s contaned n the Sobolev class W2,loc 1 (Ω) and ts drectonal
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 informationProduct-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks
Bulletn of Mathematcal Bology (21 DOI 1.17/s11538-1-9517-4 ORIGINAL ARTICLE Product-Form Statonary Dstrbutons for Defcency Zero Chemcal Reacton Networks Davd F. Anderson, Gheorghe Cracun, Thomas G. Kurtz
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 informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy S-curve Regresson Cheng-Wu Chen, Morrs H. L. Wang and Tng-Ya Hseh Department of Cvl Engneerng, Natonal Central Unversty,
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 non-empty If Y s empty, we have nothng to talk about 2. Y s closed A set
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 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 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 Kosmann-Schwarzbach Centre de Mathématques
More informationThe descriptive complexity of the family of Banach spaces with the π-property
Arab. J. Math. (2015) 4:35 39 DOI 10.1007/s40065-014-0116-3 Araban Journal of Mathematcs Ghadeer Ghawadrah The descrptve complexty of the famly of Banach spaces wth the π-property Receved: 25 March 2014
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 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 informationThe Development of Web Log Mining Based on Improve-K-Means Clustering Analysis
The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
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 informationHedging Interest-Rate Risk with Duration
FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton
More informationNonbinary Quantum Error-Correcting Codes from Algebraic Curves
Nonbnary Quantum Error-Correctng Codes from Algebrac Curves Jon-Lark Km and Judy Walker Department of Mathematcs Unversty of Nebraska-Lncoln, Lncoln, NE 68588-0130 USA e-mal: {jlkm, jwalker}@math.unl.edu
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 informationComplete Fairness in Secure Two-Party Computation
Complete Farness n Secure Two-Party Computaton S. Dov Gordon Carmt Hazay Jonathan Katz Yehuda Lndell Abstract In the settng of secure two-party computaton, two mutually dstrustng partes wsh to compute
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
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 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 informationPower-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts
Power-of-wo Polces for Sngle- Warehouse Mult-Retaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)
More informationCan Auto Liability Insurance Purchases Signal Risk Attitude?
Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang
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 informationNON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia
To appear n Journal o Appled Probablty June 2007 O-COSTAT SUM RED-AD-BLACK GAMES WITH BET-DEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate
More informationRotation Kinematics, Moment of Inertia, and Torque
Rotaton Knematcs, Moment of Inerta, and Torque Mathematcally, rotaton of a rgd body about a fxed axs s analogous to a lnear moton n one dmenson. Although the physcal quanttes nvolved n rotaton are qute
More informationFace Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching)
Face Recognton Problem Face Verfcaton Problem Face Verfcaton (1:1 matchng) Querymage face query Face Recognton (1:N matchng) database Applcaton: Access Control www.vsage.com www.vsoncs.com Bometrc Authentcaton
More informationPolitecnico di Torino. Porto Institutional Repository
Poltecnco d orno Porto Insttutonal Repostory [Proceedng] rbt dynamcs and knematcs wth full quaternons rgnal Ctaton: Andres D; Canuto E. (5). rbt dynamcs and knematcs wth full quaternons. In: 16th IFAC
More informationSeries Solutions of ODEs 2 the Frobenius method. The basic idea of the Frobenius method is to look for solutions of the form 3
Royal Holloway Unversty of London Department of Physs Seres Solutons of ODEs the Frobenus method Introduton to the Methodology The smple seres expanson method works for dfferental equatons whose solutons
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 information7.5. Present Value of an Annuity. Investigate
7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationInertial Field Energy
Adv. Studes Theor. Phys., Vol. 3, 009, no. 3, 131-140 Inertal Feld Energy C. Johan Masrelez 309 W Lk Sammamsh Pkwy NE Redmond, WA 9805, USA jmasrelez@estfound.org Abstract The phenomenon of Inerta may
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 informationForecasting the Direction and Strength of Stock Market Movement
Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems
More informationTENSOR GAUGE FIELDS OF DEGREE THREE
TENSOR GAUGE FIELDS OF DEGREE THREE E.M. CIOROIANU Department of Physcs, Unversty of Craova, A. I. Cuza 13, 2585, Craova, Romana, EU E-mal: manache@central.ucv.ro Receved February 2, 213 Startng from a
More informationThe Distribution of Eigenvalues of Covariance Matrices of Residuals in Analysis of Variance
JOURNAL OF RESEARCH of the Natonal Bureau of Standards - B. Mathem atca l Scence s Vol. 74B, No.3, July-September 1970 The Dstrbuton of Egenvalues of Covarance Matrces of Resduals n Analyss of Varance
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, Arrow-Pratt measure of rsk averson explans a ratonal nvestor s behavor n stock markets
More informationConversion between the vector and raster data structures using Fuzzy Geographical Entities
Converson between the vector and raster data structures usng Fuzzy Geographcal Enttes Cdála Fonte Department of Mathematcs Faculty of Scences and Technology Unversty of Combra, Apartado 38, 3 454 Combra,
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 informationJoe Pimbley, unpublished, 2005. Yield Curve Calculations
Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward
More informationImplied (risk neutral) probabilities, betting odds and prediction markets
Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT - We show that the well known euvalence between the "fundamental theorem of
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 informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2016. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng
More informationGlobal stability of Cohen-Grossberg neural network with both time-varying and continuous distributed delays
Global stablty of Cohen-Grossberg neural network wth both tme-varyng and contnuous dstrbuted delays José J. Olvera Departamento de Matemátca e Aplcações and CMAT, Escola de Cêncas, Unversdade do Mnho,
More informationMatrix Multiplication I
Matrx Multplcaton I Yuval Flmus February 2, 2012 These notes are based on a lecture gven at the Toronto Student Semnar on February 2, 2012. The materal s taen mostly from the boo Algebrac Complexty Theory
More informationarxiv:1311.2444v1 [cs.dc] 11 Nov 2013
FLEXIBLE PARALLEL ALGORITHMS FOR BIG DATA OPTIMIZATION Francsco Facchne 1, Smone Sagratella 1, Gesualdo Scutar 2 1 Dpt. of Computer, Control, and Management Eng., Unversty of Rome La Sapenza", Roma, Italy.
More informationSimple Interest Loans (Section 5.1) :
Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part
More informationTexas Instruments 30X IIS Calculator
Texas Instruments 30X IIS Calculator Keystrokes for the TI-30X IIS are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the
More informationHÜCKEL MOLECULAR ORBITAL THEORY
1 HÜCKEL MOLECULAR ORBITAL THEORY In general, the vast maorty polyatomc molecules can be thought of as consstng of a collecton of two electron bonds between pars of atoms. So the qualtatve pcture of σ
More informationTHE HIT PROBLEM FOR THE DICKSON ALGEBRA
TRNSCTIONS OF THE MERICN MTHEMTICL SOCIETY Volume 353 Number 12 Pages 5029 5040 S 0002-9947(01)02705-2 rtcle electroncally publshed on May 22 2001 THE HIT PROBLEM FOR THE DICKSON LGEBR NGUY ÊN H. V. HUNG
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 informationUpper Bounds on the Cross-Sectional Volumes of Cubes and Other Problems
Upper Bounds on the Cross-Sectonal Volumes of Cubes and Other Problems Ben Pooley March 01 1 Contents 1 Prelmnares 1 11 Introducton 1 1 Basc Concepts and Notaton Cross-Sectonal Volumes of Cubes (Hyperplane
More informationF-Rational Rings and the Integral Closures of Ideals
Mchgan Math. J. 49 (2001) F-Ratonal Rngs and the Integral Closures of Ideals Ian M. Aberbach & Crag Huneke 1. Introducton The hstory of the Brançon Skoda theorem and ts ensung avatars n commutatve algebra
More informationHow To Understand The Results Of The German Meris Cloud And Water Vapour Product
Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPP-ATBD-ClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller
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 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.
More informationBandwdth Packng E. G. Coman, Jr. and A. L. Stolyar Bell Labs, Lucent Technologes Murray Hll, NJ 07974 fegc,stolyarg@research.bell-labs.com Abstract We model a server that allocates varyng amounts of bandwdth
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 -Jun-3 - comments welcome, do not cte or dstrbute wthout permsson Junmn
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, 610-519-4390,
More informationAn Overview of Financial Mathematics
An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take
More information1.1 The University may award Higher Doctorate degrees as specified from time-to-time in UPR AS11 1.
HIGHER DOCTORATE DEGREES SUMMARY OF PRINCIPAL CHANGES General changes None Secton 3.2 Refer to text (Amendments to verson 03.0, UPR AS02 are shown n talcs.) 1 INTRODUCTION 1.1 The Unversty may award Hgher
More informationRESEARCH DISCUSSION PAPER
Reserve Bank of Australa RESEARCH DISCUSSION PAPER Competton Between Payment Systems George Gardner and Andrew Stone RDP 2009-02 COMPETITION BETWEEN PAYMENT SYSTEMS George Gardner and Andrew Stone Research
More informationTHE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES
The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered
More informationPoint cloud to point cloud rigid transformations. Minimizing Rigid Registration Errors
Pont cloud to pont cloud rgd transformatons Russell Taylor 600.445 1 600.445 Fall 000-014 Copyrght R. H. Taylor Mnmzng Rgd Regstraton Errors Typcally, gven a set of ponts {a } n one coordnate system and
More informationForecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network
700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School
More informationEnergies of Network Nastsemble
Supplementary materal: Assessng the relevance of node features for network structure Gnestra Bancon, 1 Paolo Pn,, 3 and Matteo Marsl 1 1 The Abdus Salam Internatonal Center for Theoretcal Physcs, Strada
More information