# A comparison result for perturbed radial p-laplacians

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 A comaison esult fo etubed adial -Lalacians Raul Manásevich and Guido Swees Diectoy Table of Contents Begin Aticle Coyight c 23 Last Revision Date: Ail 1, 23

2 Table of Contents 1. Intoduction and main esult 1.1.A non-quasimonotone system 1.2.The main condition and the theoem Otimal bounds in the linea case that use one aamete 1.3.Othe examles 2. On the solution oeato 2.1.Elementay oeties of G 2.2.Comaing with a sot of fundamental solution 2.3.Two-sided estimates fo G 2.4.Estimates fo G alied to a singula function 3. Veification of the main condition 4. Main oofs 4.1.Comaison esults fo G. 4.2.A fixed oint agument Poof of Theoem 1.2

3 Section 1: Intoduction and main esult 3 A comaison esult fo etubed adial -Lalacians Raul Manásevich & Guido Swees Abstact Conside the adially symmetic -Lalacian fo 2 unde zeo Diichlet bounday conditions. The main esult of the esent ae is that unde aoiate conditions a solution of a etubed adially symmetic -Lalacian can be comaed with the solution of the unetubed one. As a consequence one obtains a sign eseving esult fo a system of -Lalacians which ae couled in a non-quasimonotone way. 1. Intoduction and main esult One of the goals when studying the Schödinge equation u + V u = f is to find comaison esults, that is, when consideing the oblem fo V 1 and V 2, what ae the conditions such that fo the same f the coesonding solutions u 1 and u 2 can be comaed see [7]. Zhao and collaboatos see [11], [2] and the efeences theein obtained such comaison esults on bounded domains Ω R n, fo u satisfying zeo Diichlet bounday conditions, by estimating the iteated Geen function with the Geen function itself. The difficulties aise both by the singulaity on the diagonal of the solution oeato the Geen function G Ω x, y in dimension n 2 has a singulaity when x = y and as well as by the zeo bounday condition. The main tool in thei oofs ae the Hanack inequalities both in the inteio and at the bounday. With the estimates of Zhao one may even show that fo ε > but small the nonlocal V defined by V u x = ε Ω G Ω x, y u y dy is in this class. As a consequence one obtains a maximum incile fo a system of ellitic equations with a noncooeative couling. In this ae we will show a fist ste in tansfeing such a comaison esult to a nonlinea equation, namely one containing the -Lalacian with the assumtion of adial symmety. Of couse the otential should have the same tye of nonlineaity and that leads us to conside comaison inciles fo { u + λ V u = f in B, 1 u = on B, with B = {x R n ; x < 1}, n 2, 2,, λ > some small aamete and whee V is an oeato having the same homogeneity as the -Lalacian. This oeato may be nonlocal but is assumed to eseve adial symmety. Fo λ = it is well known that f imlies u even in a much moe geneal setting see [1]. Since we will estict ouselves to the adial symmetic case we have following exession fo the -Lalacian: u = 1 n n 1 u 2 u. As a consequence we will find a sign eseving esult fo a system of -Lalace oeatos which ae couled in a noncooeative way. We ecall that the bounday value oblem 1 is called sign eseving if evey solution u is ositive wheneve the souce tem f is ositive. In contay to the oiginal maximum incile, that is, u cannot have a negative minimum, such a sign eseving oety may deend on nonlocal aguments. In the esent ae thee will not be a maximum incile in this oiginal sense but we suoted by Fonda M.A. and Milenio gant-p1-34 Keywods: -lalacian system, ositivity, noncooeative couling AMS-MSC: 35B5, 35J6, 34C1

4 Section 1: Intoduction and main esult 4 will show that fo ositive f solutions u of the etubed λ small and unetubed λ = oblem can be comaed. Hence a sign eseving oety will hold fo the etubed oblem, wheneve λ is small enough, fo all f >. Fo easy efeence we fix the following: Notation 1.1 φ u = u 2 u and its invese is being denoted by φ inv φ inv u = u 2 u ; The solution oeato G fo the adial -Lalacian with Diichlet bounday condition: t s n 1 G f = fsds dt; 2 t f > denotes f fo all [, 1] and f ; f means that thee is c > such that f c1 fo all [, 1] A non-quasimonotone system A secial case that we conside is the following non-quasimonotone if λ > nonlinea ellitic system u = f λφ v in B, v = φ u in B, 3 u = v = on B. Remembe that the system u = F 1 u, v, v = F 2 u, v is quasimonotone iff u F 2 and v F 1, and that in a quasimonotone setting the maximum incile can be used simila as fo one equation. In a linea setting quasimonotone is also known as cooeative. Using the solution oeato G, the Geen oeato defined in 2 fo the adial case, the system coincides with { u + λ φ G φ u = f in B, 4 u = on B, Notice that φ G φ tu = φ t φ G φ u fo all t R and hence satisfies the aoiate homogeneity condition. The 1-dimensional case has been studied in [5]. In a eaction to that ae W. Walte aised the question what would haen in the highe dimensional case. This ae is a fist ste in that diection. The linea case, = 2, of 3 was studied in [6] even fo geneal nonadial functions on smooth domains. A ealie esult fo the ball can be found in [8]. The cucial esult that was used in that ae was the so-called 3G-theoem which oiginates fom Zhao [11]. The nonlinea natue of 3 makes the geneal system much hade. By esticting ouselves to the adial case we ae able to ove a ositivity eseving oety fo this noncooeative system and in doing so we encounte some citical dimensions. Fo the linea case the Geen function becomes unbounded fo n 2. Similaly, the k th iteated Geen function is bounded if and only if 2k > n. Fo the -Lalacian ointwise boundedness of the k th - iteated homogenized Geen oeato, defined by G φ k = G φ G φ k 1 fo k 1, is elated to k > n. These numbes eaea as a estiction in the esults down below. Fo sign eseving esults fo cooeative systems with the -Lalacian we efe to [3]. Positivity eseving oeties of 1 fo = 2 and linea, ossibly nonlocal V, have been studied in [4].

5 Section 1: Intoduction and main esult The main condition and the theoem The basic conditions that we will use to show that a etubation by V does not destoy the ositivity eseving oety fo λ sufficiently small is the following. Condition 1.1 The oeato V is as follows: i. V t u = t V u fo t ; ii. V is continuous fom C 1 [, 1] to C [, 1] and moeove thee is C V > such that V u C[,1] C V u C 1 [,1] fo all u C 1 [, 1]; iii. thee is C V,,n > such that G V G f C V,,n G f fo [, 1] and fo all f C[, 1]. Remak Fo V = φ the thid item in the condition above imlies a nonlinea 3G-tye esult: G φ G f C V,,n G f fo [, 1] and < f C[, 1]. 5 Fo V = φ we ae able to show that iii. is satisfied when both 2 and > 1 2 n hold. See Lemma 3.1 below. Remak Notice that if V and Ṽ satisfy Condition 1.1 then so does V + Ṽ. Only the thid condition needs some eflection. Set v = V G f and ṽ = Ṽ G f and one obtains by φ inv a + b φ inv a + φ inv b fo a, b that t s n 1 G v + ṽ v s + ṽ s ds dt t φ inv G v + G ṽ. Theoem 1.2 the main esult Fix > 2 and suose that the oeato V satisfies Condition 1.1 above. Then thee exists λ such that fo all f C[, 1] with f > and λ [, λ ] the following holds: i. thee exists a solution u C 1, 1 [, 1] of 1 with φ u C 1 [, 1]; ii. evey solution u of 1 satisfies and hence evey solution is ositive. 1 2 G f u 3 2 G f fo [, 1], The oof will be ostoned to the following sections. Remak Notice that we do not state uniqueness of the solution fo the etubed oblem. As can be seen fom the case n = 1 in [5] uniqueness is not obvious in geneal. Fo the non-quasimonotone system 3 we have the following esult.

6 Section 1: Intoduction and main esult 6 Coollay 1.3 If > 1 3n then the adially symmetic case of the non-quasimonotone system in 3 is ositivity eseving fo λ sufficiently small. That is, thee exists λ > such that fo evey λ [, λ ] and f C B with f = f x and f > thee exists a adially symmetic solution u of 3 and evey adially symmetic solution is stictly ositive. Poof. It is sufficient to show that V satisfies Condition 1.1 whee V u = φ G φ u. Indeed Coollay 3.5 imlies that this holds wheneve 2 and > 1 3 n. The aoach of this ae is to get estimates fom above fo the etubation in tems of a function that itself gives a unifom estimate fom below fo the Geen oeato. A stong estiction of this aoach is that one needs to catch the ositive function f in one numbe α f such that, fo some unifom constant C, the following holds: G V f C α f 1 and α f 1 G f. Fo > n we will use α f = G f and fo < n the numbe α f = su [,1] n G f. Needless to say that we do not exect this to give the best ossible esult. The next aagah contains an exlanation fo the case = 2. Otimal bounds in the linea case that use one aamete Fo = 2 a unifom estimate using only one aamete would coincide with obtaining an estimate fom below fo the Geen function G, s by a multile of the oduct g 1 g 2 s whee g 1, g 2 ae ositive functions. Fo the linea Diichlet oblem such an estimate is almost neve otimal since this would mean that the Geen function could be estimated fom below and fom above by multiles of the same oduct. Only fo the 1-dimensional Neumann oblem this is ossible. In the linea case the adial symmetic Geen oeato educes to an integal oeato G 2 f = G, sfsds with the following kenel: G, s = { 1 n 2 sn 1 s 2 n 1 if s, 1 n 2 sn 1 2 n 1 if s <, fo n > 2, G, s = s log max, s fo n = 2. Fo n > 2 these can be estimated in tems of owes of s and and distances to the bounday 1 and 1 s, by 1 c 1 s n 1 min n 2, 1 s 1 s n 2 G, s c 2 s n 1 min n 2, 1 s s n 2, 6 Hence otimal estimates in oduct fom ae fom below and fo the estimate fom above one cannot go beyond G, s c n s n 1 1 s 1, 7 G, s C n s n 1 1 s s n 2 θ 1 1 θ n 2 with θ [, 1]. Otimal two-sided estimates fo the Geen function on geneal domains ae due to Zhao [11]. See also [6] o [9]. As just exlained, the sha exessions that ae used in these aes cannot be of the fom g 1 g 2 s.

7 Section 2: On the solution oeato Othe examles Ou main inteest focuses on the non-quasimonotone system in 3. Othe examles of oeatos V satisfying the main condition Condition 1.1 ae V = φ A with A as follows: i. Au = au + bu, fo > n and a, b C[, 1]. If b = then we may allow > 1 2n. This esult follows fom Lemma s 3.1 and 3.3 and the emak following Condition 1.1. Fo a this examle is not so inteesting since with this local etubation one may oceed by the local aguments of the maximum incile. ii. Au = a, s us sn 1 ds, with aoiate kenel a. Fo the ecise condition see Lemma 3.6. Also kenels like Au = a, s us ds 1 α α o Au = a, t, s us ds 1 α α dt satisfy Condition 1.1 fo aoiate estictions elating α with n and. If we set α = q 1 and a, t, s = χ [t>] χ [s<t] s/t n 1 we find that u is a solution of the non-quasimonotone nonlinea ellitic system 3. Fo = q and n = 1, but with f not necessaily symmetic, this system was studied in [5]. iii. A u = G q φ q. Fo this oeato A, which coesonds with the system u = f λφ v in B, q v = φ q u in B, u = v = on B we find that Condition 1.1 is satisfied when n,, q 2 ae such that n < 2 + q q 1. See Lemma 3.4. A simila condition can be found when using fou diffeent owes as long as the -Lalacians have 2 and the homogeneity fits. 2. On the solution oeato 2.1. Elementay oeties of G The solution oeato fo 1 with λ = is G. Fist note that f C[, 1] imlies that G f C 1, 1 [, 1]. Moeove, if f > set t = inf {t [, 1] ; f t > } and we find that s n 1 φ inv fsds C 1 [, t t, 1] C 1 [, 1]. 9 By integating we find that G f C 2 [, 1] \ {t } C 1, 1 [, 1]. Also9 immediately shows that G f = fo t and G f < fo t < Comaing with a sot of fundamental solution We stat by studying the outcome of this oeato alied on some secial distibutions fo the ight hand side, namely d s = s 1 n δ s, with δ the Diac delta function at s, 1 with weight s 1 n. This weight is the aoiate nomalization fo the adial symmetic natue of the oblem. We will see that = n is citical in the following sense. If n < then and only then the functions G d s ae unifomly bounded with esect to s. 8

8 Section 2: On the solution oeato 8 Lemma 2.1 Set d s = s 1 n δ s. i. If > n, then G d s = n 1 max, s n. ii. If = n, then G d s = log max, s. iii. If < n, then G d s = n max, s n 1. Poof. The esult follows fom a diect comutation. G d s G d s n s s Figue 1: > n esectively n With the solutions fo the delta function we will have the following vesion of a comaison incile. Lemma 2.2 Let u = G f with < f C[, 1]. Then fo evey [, 1] and s, 1 one has u s u G d s s G d s. 1 t σ n 1 t fσdσ dt it is immediate fo f that Poof. Since u = φinv imlies u 1 u 2 and hence 1 holds on [, s]. Fo t [s, 1] we oceed by contadiction. Set v = us G G d ss d s and suose that v τ > u τ fo some τ s, 1. Then thee exist τ 1, τ 2 [s, 1] such that s τ 1 τ τ 2 1 with and with eithe u τ 1 v τ 1 = 1 and u v < 1 fo τ 1, τ 2, τ 2 = 1 o u τ 2 v τ 2 = 1.

9 Section 2: On the solution oeato 9 It follows by an elementay agument that u τ 1 v τ 1 1 u τ 2 v τ The diffeential equations fo u and v on s, 1 give, using φ v <, that u φ n 1 φ u n 1 φ u v = n 1 φ v = n 1 φ v. It follows, afte integating and alying φ inv and by using 11, that fo any τ τ 1, τ 2 1 u τ 2 v τ 2 u τ v τ u τ 1 v τ 1 1. Hence u v on [τ 1, τ 2 ] which imlies u v on [τ 1, τ 2 ], a contadiction Two-sided estimates fo G In the next thee lemmata we will ove a elation between the ue and lowe estimates of G f. Lemma 2.3 If > n, then fo evey f C[, 1] with f > one has n 1 1 G f G f G f fo all [, 1]. 12 Poof. The estimate fom above is obvious by the definition of G. Fo the estimate fom below note that Lemma 2.2 imlies that fo evey ε > Letting ε one finds G f G f G f ε G d ε ε G d ε fo [, 1]. 1 n G f n 1 G f. Lemma 2.4 Suose that = n and let f C[, 1] with f >. Set Then one has Poof. Let be the numbe such that G f θ f = su < 1 1 log. θ f 1 G f θ f 1 log. 13 θ f 1 log = G f. Then G f G f G δ G δ = θ f 1 log log log max, θ 1 log f min log, 1 1 log = θ 1 f min 1 log, log θ f 1.

10 Section 2: On the solution oeato 1 Lemma 2.5 Suose that < n and let f C[, 1] with f >. Set θ f = su n G f. < 1 Then one has θ f min 1, n 1 G f θ f n Remak The numbe θ f is a weighted L -nom fo the function G f. Poof. The estimate fom above follows by the definition of θ f. Fo the estimate fom below conside g = n G f. Since G f C[, 1] we find g = g 1 =. Hence g has an global maximum inside, say in, and θ f = g. By Lemma 2.2 we have G f G f G d G d = θ f n n 1 max, n 1 15 and using that n 1 n 1, we continue 15 by n θ f n 1 = θ f min min n 1 n 1, n 1 n, n n = θ f min θ f min 1, n 1 1 1, n 1. 1 Remak It is cucial in this oof that we ae able to give an estimate indeendent of. Note that any lage exonent, say α > n and θ f = su < 1 α G f fails to give a unifom estimate fom below Estimates fo G alied to a singula function Finally we will show the following two estimates that will be used late. Lemma 2.6 Let α [, n and set g α = α if α > and g = 1 log, then 1 if α <, G g α c n,,α log if α =, α 1 if α >. Poof. Let α, n. Since α < n holds, a staightfowad comutation yields G g α = t s 1 n 1 s α ds dt = t 16

11 Section 3: Veification of the main condition 11 = n α 1 t 1 α dt = c n,,α 1 α if α <, log if α =, α 1 if α >, imlying 16. Fo α = a simila comutation shows that fo some c n, > it holds that G g c n, Veification of the main condition Fist we will show a comaison between the solution oeato G and the iteated G φ G. Lemma 3.1 Suose that 2 satisfies > 1 2n. Let a C[, 1] with a. Then thee is a constant C a,,n such that fo all f C[, 1] : 17 G φ a G f C a,,n G f. 18 Remak The majo estiction hee is > 1 2n. Although we do exect this lemma to hold fo all 2 a oof will be much moe involved. The eason is the following. Fo > 1 2 n we ae able to chaacteize G f by one numbe G f fo > n, see 12, θ f fo = n, see 13 o θ f fo < n, see 14. Indeed this numbe is used to find unifom estimates fom below fo G f and fom above fo G φ G f. Such estimates in tems of one numbe follow fom Lemma 2.6 only if > 1 2 n. Wheneve 2, 1 2 n] such chaacteization by one numbe does not seem to be sufficient and consequently it will be necessay to catue the behavio of G f in a moe elaboate way. Poof. Since G φ a.g f a G φ G f it will be sufficient to conside H f := G φ G f fo < f C[, 1]. Let us denote by 1 the function 1x 1. If > n then the estimates in Lemma 2.3 imly that Fo 1 2 n, n we have H f G f G φ 1 19 t s 1 n 1 G f 1 ds dt t H f G φ G f 1 1 n G f. θ f n t s 1 n 1 θ f s n ds dt t = 1 1 θ f 1 2 n t n+1 dt = max max 1 2 n min 1, 2 n 1, 2 n 1, n 1 2 n θ f 1 G f. θ f 1 2 n

12 Section 3: Veification of the main condition 12 Finally the case = n. It follows that H f G φ θ f 1 log t s 1 n 1 θ f 1 log s n 1 n 1 ds dt t c n θ f t 1 n 1 1 log t dt c n θ f 1 c n G f. Fo such multilication by a it is obvious that it is sufficient to conside the ositive oeato whee a is elaced by a. Fo moe geneal oeatos let us intoduce a slitting in a ositive and a negative at. Lemma 3.2 Suose that V satisfies Condition 1.1. Defining fo f C[, 1] V + G f = max {, V G f } 2 V G f = min {, V G f } 21 we find that V ± G ae continuous fom C[, 1] to C[, 1]. Moeove, thee exist c V, C V > such that V ± G f C[,1] c V f C[,1] fo all f C[, 1]; 22 G V ± G f C V G f fo all f C[, 1]. 23 Poof. The continuity is staightfowad. By Condition 1.1.ii one finds V ± G f V G f c V G f C 1 [,1] c V 2 f. By Condition 1.1.iii G V ± G f G V G f C V,,n G f. Next we addess the etubation by a deivative. Lemma 3.3 Suose that 2 satisfies > n. Let b C[, 1]. Then thee is a constant C b,,n such that fo all f C [, 1] : G φ b. d d G f C b,,n G f. Hence fo all, n, b as above V defined by V u = φ b u satisfies Condition 1.1. Poof. Note that d s n 1 d G f = φ inv f s ds

13 Section 3: Veification of the main condition 13 imlies that V ± G f = φinv φ inv b s n 1 ± f s ds + φ inv b ± s n 1 f s ds s n 1 b f s ds Since > n we may oceed simila as 19 fo f > stating with G V + G f b G f G V + G 1. Fo the couled system we have to deal with G φ G φ G. It will not be much moe touble to have a -Lalacian with anothe exonent say q in the second equation as long as the homogeneity fits. In that case we would have to conside G φ G q φ q G, with G q and φ q defined in Notation 1.1 with the obvious elacement of by q. Lemma 3.4 Suose that n,, q 2 ae such that n < 2+q q 1. Then thee is a constant C,q,n such that fo all f C[, 1] with f : G φ G q φ q G f C,n G f. In othe wods, fo n,, q as above Condition 1.1 is satisfied fo V = φ G q φ q G φ. Coollay 3.5 Suose that > 2 and n < 3. Then V = φ G φ G φ satisfies Condition 1.1. Poof. of Lemma 3.4 Let us denote H,q f = G φ G q φ q G f. If n < then the estimates in Lemma 2.3 imly that Fo n = Lemma 2.6 imlies that H,q f G f G φ G q φ q 1 G f 1 1 n G f. H,q f θ f G φ G q φ q 1 log c n,q θf G φ 1 c n,,q 1. Fo n > we have by Lemma 2.4 and Lemma 2.6 that with H f as in a Lemma 3.1 H f G q φ q c,q,n θ f θ f n = θ f G q q 1 n 1 if q 1 + n >, log if q 1 + n =, 1+ 1 q 1 + n 1 if q 1 + n <. and hence again by Lemma 2.6, deending on the sign of q 1 + n γ,q,n = = q 1 + n 1

14 Section 4: Main oofs 14 then H,q f c,q,n θ f 1 if γ,q,n >, log if γ,q,n =, γ,q,n 1 if γ,q,n <. Since G f c,n θ f 1 we find the esult of the lemma wheneve γ,q,n >, that is, fo n < 2 + q q 1. Lemma 3.6 Let, q 2 and suose that Au = a, s u s sn 1 ds is such that fo some γ < if > n then a, s s n 1 ds C γ 1 if = n then a, s 1 log s s n 1 ds C γ 24 if < n then 1 Then oeato V = φ A satisfies Condition 1.1. n 1 n a, s s ds C γ Poof. Set a, s = a +, s a, s with a +, a and denote g α = α Fist suose that > n. We find by Lemma 2.6 if 1 α < that 1 G V + G f G f G φ a +, s s n 1 ds C G f G φ g α = C G f G g α C,n,α G f 1 The condition 1 α < coincides with α < Fo < n we oceed fo n α < by 1 G V + G f G φ a +, s G f s s n 1 ds θ f G φ a +, s s n s n 1 ds C,n,α θ f Main oofs 4.1. Comaison esults fo G. In this section we comae the Geen oeato fo the etubed and the unetubed ight hand side. Fist we need an elementay estimate: Lemma 4.1 Suose that 2. Fo a, b it holds that φ inv a b φ inv Poof. Fo a b we may use Minkowski s inequality: φ inv a b φ inv a = φ inv a 2 φ inv b. a b φ inv a b + b a b 1 a b 1 + b 1 = b 1.

15 Section 4: Main oofs 15 If a b we oceed by: φ inv a b φ inv a = b a 1 + a 1 b 1 + b 1 = 2 b 1. We will also need the following ode esult: Lemma 4.2 Let g 1, g 2 C[, 1] with g 1 g 2. Then fo all [, 1] : G g 1 G g 2, 25 G g 1 G g Poof. Diectly fom 2: G f = 1 n φ inv sn 1 fsds. Lemma 4.3 Let 2 and let f, g C [, 1] with f. Then fo all [, 1] one finds: G f + g G f 2 G g. 27 If moeove g s θ α fo some α [, n and θ >, then the following estimate holds with C = 2 n α 1 : G f + g G f C θ 1 1 α. 28 Remak Note that G g 1 n Poof. By Lemma 4.2 we find Using that 1 that 1 g L 1 whee g L1 = g s sn 1 ds. G f + g + G f G f + g + G f. Lemma 4.1 shows, 1] imlies a + b 1 a 1 + b 1, we find with a = s n 1 fsds and b = s n 1 gs ds G f + g + G f G g G f + g + G f G f g + G f 2 G g. The estimate of 28 follows by 1 n s α s n 1 ds n = 1 n α n α. Coollay 4.4 Let 2 and let f, g C[, 1] with f. Then fo all [, 1] one finds: G f 2 G g G f + g G f + 2 G g. 29 Poof. The esult follows by 27 and an integation fom = 1.

16 Section 4: Main oofs A fixed oint agument Fo 1 one might obtain a solution when λ is small by the following iteation ocedue. Defining S λ, : C[, 1] C 1 [, 1] C 1 [, 1] by one consides the iteation u = G f, and S λ, f; u := G f λv u. 3 u n+1 = G f λv u n fo n N. Since the esent oblem does not satisfy an ode esevation such an iteation might esult in a sequence that does have a conveging subsequence, but that is not conveging itself. Fo examle it could haen that u 2n u and u 2n+1 u u. The functions u and u do satisfy an 4 th -ode system but do not necessaily satisfy 1. Instead of using such an iteation we will use a fixed oint agument fo existence of a solution to Fo a suvey on fixed oint methods see [1]. u S λ, f; u. Poosition 4.5 Let 2 and λ >. Suose that V satisfies Condition 1.1. Then, with C V > as in Lemma 3.2, the following holds fo all f, g C[, 1] with f > S λ, f; G g G f 2C V,,n λ 1 G g. Poof. By Lemma 4.3 and Lemma 4.2 one finds S λ, f; G g G f = G f λv G g G f and similaly G f + λv G g G f 2 G λv G g S λ, f; G g G f G f λv + G g G f 2 G λv + G g. Condition 1.1 and Lemma 3.2 imly that fo all [, 1] comleting the estimate. G λv G g 2C V λ 1 G g, G λv + G g 2C V λ 1 G g, Coollay 4.6 With, λ, V and C V,,n as in Poosition 4.5, setting it follows that fo all λ that S λ, f; G g satisfies ˆλ = 1 8 V,,n C, 31 [, ˆλ ], f, g C[, 1] with f > and G g 2G f fo all [, 1], < 1 2 G f S λ, f; G g 3 2 G f fo all [, 1].

17 Section 4: Main oofs 17 Poof of Theoem 1.2 An aioi bound. We will fist show that fo λ sufficiently small evey fixed oint of u = S λ, f; u will necessaily lie in [ 1 2 G f, 3 2 G f ]. Indeed, using Coollay 4.4 twice and Condition 1.1 we find that Hence G V u = G V S λ, f; u and fo λ [, λ ] with we get Again using Coollay 4.4 we have = G V G f λv u 2λ 1 C V,,n G f λv u 2λ 1 C V,,n G f + λ V u 2λ 1 C V,,n G f + 4λ 2 C V,,n G V u G V u 2λ 1 C V,,n 1 4λ 2 C V,,n G f λ = 8C V,,n G V u 4λ 1 C V,,n G f. 33 u G f = S λ, f; u G f 2λ 1 G V u. 34 Combining 33 and 34 shows u [ 1 2 G f, 3 2 G f ] fo λ [, λ ]. Existence. Fix f C[, 1] and let us conside D = { u C 1 [, 1] with u C 1 2 G f C 1}. The maing u S λ, f; u fom C 1 [, 1] to itself is comletely continuous and mas D into D. Indeed by Lemma 4.3, the oeties of G and the assumtion on V one finds that Sλ, f; u G f 2 G λv u 4λ 1 V u 1/ 4 c V λ 1 u C 1 [,1], 35 and since S λ, f; u and G f ae zeo in = 1 it follows fo λ [, λ ], whee that λ := 8c V 1, 36 S λ, f; u C 1 [,1] S λ, f; u G f C 1 [,1] + G f C 1 [,1] = S λ, f; u G f + Sλ, f; u G f + G f C 1 [,1] 1 2 G f + 4 c V λ 1 u C 1 [,1] + G f C 1 [,1] 2 G f C 1 [,1]. By Schaude s fixed oint Theoem thee exists u D such that u = S λ, f; u. Conclusion. Thee exists a solution u [ 1 2 G f, 3 2 G f ] wheneve λ [, λ ] with λ = min { λ, λ } defined by 32 and 36.

18 Section 4: Main oofs 18 Refeences [1] H. Amann, Fixed oint equations and nonlinea eigenvalue oblems in odeed Banach saces, SIAM Rev , [2] Kai Lai Chung, and Zhong Xin Zhao, Fom Bownian motion to Schödinge s equation. Gundlehen de Mathematischen Wissenschaften [Fundamental Pinciles of Mathematical Sciences], 312. Singe-Velag, Belin, [3] J. Fleckinge, J. Henandez and F. de Thélin, On maximum inciles and existence of ositive solutions fo some cooeative ellitic systems. Diffeential Integal Equations , no. 1, [4] P. Feitas and G. Swees, Positivity esults fo a nonlocal ellitic equation. Poc. Roy. Soc. Edinbugh Sect. A , no. 4, [5] R. Manásevich and G. Swees, A noncooeative ellitic system with -Lalacians that eseves ositivity. Nonlinea Anal , no. 4, Se. A: Theoy Methods, , 5, 7 [6] E. Mitidiei and G. Swees, Weakly couled ellitic systems and ositivity, Math. Nach , , 6 [7] R. G. Pinsky, Positive hamonic functions and diffusion. Cambidge Studies in Advanced Mathematics, 45. Cambidge Univesity Pess, Cambidge, [8] G. Swees, A stong maximum incile fo a noncooeative ellitic system, SIAM J. Math. Anal , [9] G. Swees, Positivity fo a stongly couled ellitic system by Geen function estimates. J. Geom. Anal , no. 1, [1] J.L. Vázquez, A stong maximum incile fo some quasilinea ellitic equations, Al. Math.Otim , [11] Z. Zhao, Geen functions fo Schödinge oeato and conditioned Feynman-Kac gauge, J. Math. Anal. Al , , 4, 6 Raul Manásevich: Cento de Modelamiento Matemático Univesidad de Chile Av. Blanco Encalada 212 Piso 7 Santiago Chile Guido Swees: Det. of Alied Mathematical Analysis Delft Univesity of Technology P.O. Box GA Delft The Nethelands

### Integer sequences from walks in graphs

otes on umbe Theoy and Discete Mathematics Vol. 9, 3, o. 3, 78 84 Intege seuences fom walks in gahs Enesto Estada, and José A. de la Peña Deatment of Mathematics and Statistics, Univesity of Stathclyde

### MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION

MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe- and subsolution method with

### Chapter 4: Matrix Norms

EE448/58 Vesion.0 John Stensby Chate 4: Matix Noms The analysis of matix-based algoithms often equies use of matix noms. These algoithms need a way to quantify the "size" of a matix o the "distance" between

### Risk Sensitive Portfolio Management With Cox-Ingersoll-Ross Interest Rates: the HJB Equation

Risk Sensitive Potfolio Management With Cox-Ingesoll-Ross Inteest Rates: the HJB Equation Tomasz R. Bielecki Depatment of Mathematics, The Notheasten Illinois Univesity 55 Noth St. Louis Avenue, Chicago,

### Seshadri constants and surfaces of minimal degree

Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth

### 1ST INTERNATIONAL CONFERENCE ON SUPPLY CHAINS

Examination of the inteelation among the ice of the fuel, the cost of tansot feight and the ofit magin Angeliki Paana 1, Aiadni Paana 2, Michael Dagiasis 3, Dimitios Folinas 4, Eaminontas Diamantooulos

### I- LACUNARY GENERALIZED DIFFERENCE CONVERGENT SEQUENCES IN n-normed SPACES

Jounal of Mathematical Analysis ISSN: 17-341, URL: http://www.iliias.com Volume Issue 1(011), Pages 18-4. I- LACUNARY GENERALIZED DIFFERENCE CONVERGENT SEQUENCES IN n-normed SPACES NURTEN FISTIKÇI, MEHMET

### Continuous Compounding and Annualization

Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

### Chapter 5 Review - Part I

Math 17 Chate Review Pat I Page 1 Chate Review - Pat I I. Tyes of Polynomials A. Basic Definitions 1. In the tem b m, b is called the coefficient, is called the vaiable, and m is called the eonent on the

### Capital Investment and Liquidity Management with collateralized debt.

TSE 54 Novembe 14 Capital Investment and Liquidity Management with collatealized debt. Ewan Piee, Stéphane Villeneuve and Xavie Wain 7 Capital Investment and Liquidity Management with collatealized debt.

### UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

### Concept and Experiences on using a Wiki-based System for Software-related Seminar Papers

Concept and Expeiences on using a Wiki-based System fo Softwae-elated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wth-aachen.de,

### Life Insurance Purchasing to Reach a Bequest. Erhan Bayraktar Department of Mathematics, University of Michigan Ann Arbor, Michigan, USA, 48109

Life Insuance Puchasing to Reach a Bequest Ehan Bayakta Depatment of Mathematics, Univesity of Michigan Ann Abo, Michigan, USA, 48109 S. David Pomislow Depatment of Mathematics, Yok Univesity Toonto, Ontaio,

### ROBERTA FILIPPUCCI. div(a( Du )Du) q( x )f(u) in R n,

ENTIRE RADIAL SOLUTIONS OF ELLIPTIC SYSTEMS AND INEQUALITIES OF THE MEAN CURVATURE TYPE ROBERTA FILIPPUCCI Abstact. In this pape we study non existence of adial entie solutions of elliptic systems of the

### Physics 235 Chapter 5. Chapter 5 Gravitation

Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

### Classical Mechanics (CM):

Classical Mechanics (CM): We ought to have some backgound to aeciate that QM eally does just use CM and makes one slight modification that then changes the natue of the oblem we need to solve but much

### The Supply of Loanable Funds: A Comment on the Misconception and Its Implications

JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently Fields-Hat

### ON THE (Q, R) POLICY IN PRODUCTION-INVENTORY SYSTEMS

ON THE R POLICY IN PRODUCTION-INVENTORY SYSTEMS Saifallah Benjaafa and Joon-Seok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poduction-inventoy

### Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

### Nontrivial lower bounds for the least common multiple of some finite sequences of integers

J. Numbe Theoy, 15 (007), p. 393-411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to

### Gauss Law. Physics 231 Lecture 2-1

Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

### LINES AND TANGENTS IN POLAR COORDINATES

LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Pola-coodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and

### The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges

The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee

### Carter-Penrose diagrams and black holes

Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

### Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

### STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts

0.9 Radicals and Equations 0.9 Radicals and Equations In tis section we eview simlifying exessions and solving equations involving adicals. In addition to te oduct, quotient and owe ules stated in Teoem

### Mechanics 1: Motion in a Central Force Field

Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.

### PAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII - SPETO - 1995. pod patronatem. Summary

PCE SEMINIUM Z PODSTW ELEKTOTECHNIKI I TEOII OBWODÓW 8 - TH SEMIN ON FUNDMENTLS OF ELECTOTECHNICS ND CICUIT THEOY ZDENĚK BIOLEK SPŠE OŽNO P.., CZECH EPUBLIC DLIBO BIOLEK MILITY CDEMY, BNO, CZECH EPUBLIC

### On Some Functions Involving the lcm and gcd of Integer Tuples

SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 6, 2 (2014), 91-100. On Some Functions Involving the lcm and gcd of Intege Tuples O. Bagdasa Abstact:

### AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM

AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,

### EXPERIMENT 16 THE MAGNETIC MOMENT OF A BAR MAGNET AND THE HORIZONTAL COMPONENT OF THE EARTH S MAGNETIC FIELD

260 16-1. THEORY EXPERMENT 16 THE MAGNETC MOMENT OF A BAR MAGNET AND THE HORZONTAL COMPONENT OF THE EARTH S MAGNETC FELD The uose of this exeiment is to measue the magnetic moment μ of a ba magnet and

### Top-Down versus Bottom-Up Approaches in Risk Management

To-Down vesus Bottom-U Aoaches in isk Management PETE GUNDKE 1 Univesity of Osnabück, Chai of Banking and Finance Kathainenstaße 7, 49069 Osnabück, Gemany hone: ++49 (0)541 969 4721 fax: ++49 (0)541 969

### 1.4 Phase Line and Bifurcation Diag

Dynamical Systems: Pat 2 2 Bifucation Theoy In pactical applications that involve diffeential equations it vey often happens that the diffeential equation contains paametes and the value of these paametes

### CONCEPTUAL FRAMEWORK FOR DEVELOPING AND VERIFICATION OF ATTRIBUTION MODELS. ARITHMETIC ATTRIBUTION MODELS

CONCEPUAL FAMEOK FO DEVELOPING AND VEIFICAION OF AIBUION MODELS. AIHMEIC AIBUION MODELS Yui K. Shestopaloff, is Diecto of eseach & Deelopment at SegmentSoft Inc. He is a Docto of Sciences and has a Ph.D.

### Revision Guide for Chapter 11

Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams

### Voltage ( = Electric Potential )

V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage

### PACE: Policy-Aware Application Cloud Embedding

PACE: Policy-Awae Alication Cloud Embedding Li Ean Li Vahid Liaghat Hongze Zhao MohammadTaghi Hajiaghayi Dan Li Godon Wilfong Y. Richad Yang Chuanxiong Guo Bell Labs Micosoft Reseach Asia Tsinghua Univesity

### Uniform Rectilinear Motion

Engineeing Mechanics : Dynamics Unifom Rectilinea Motion Fo paticle in unifom ectilinea motion, the acceleation is zeo and the elocity is constant. d d t constant t t 11-1 Engineeing Mechanics : Dynamics

### Housing in the Household Portfolio and Implications for Retirement Saving: Some Initial Finding from SOFIE

Housing in the Household Potfolio and Imlications fo Retiement Saving: Some Initial Finding fom SOFIE Gant Scobie, Tinh Le and John Gibson N EW ZEALAND T REASURY W ORKING P APER 07/04 M ARCH 2007 NZ TREASURY

### Hour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and

Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon

### Skills Needed for Success in Calculus 1

Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

### On Efficiently Updating Singular Value Decomposition Based Reduced Order Models

On Efficiently dating Singula alue Decoosition Based Reduced Ode Models Ralf Zieann GAMM oksho Alied and Nueical Linea Algeba with Secial Ehasis on Model Reduction Been Se..-3. he POD-based ROM aoach.

### 2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

### UPS Virginia District Package Car Fleet Optimization

UPS Viginia Distit Pakage Ca Fleet Otimization Tavis Manning, Divaka Mehta, Stehen Sheae, Malloy Soldne, and Bian Togesen Abstat United Pael Sevie (UPS) is onstantly haged with ealigning its akage a fleet

### Physics 505 Homework No. 5 Solutions S5-1. 1. Angular momentum uncertainty relations. A system is in the lm eigenstate of L 2, L z.

Physics 55 Homewok No. 5 s S5-. Angula momentum uncetainty elations. A system is in the lm eigenstate of L 2, L z. a Show that the expectation values of L ± = L x ± il y, L x, and L y all vanish. ψ lm

### Approximation Algorithms for Data Management in Networks

Appoximation Algoithms fo Data Management in Netwoks Chistof Kick Heinz Nixdof Institute and Depatment of Mathematics & Compute Science adebon Univesity Gemany kueke@upb.de Haald Räcke Heinz Nixdof Institute

### International Monetary Economics Note 1

36-632 Intenational Monetay Economics Note Let me biefly ecap on the dynamics of cuent accounts in small open economies. Conside the poblem of a epesentative consume in a county that is pefectly integated

### Uncertain Version Control in Open Collaborative Editing of Tree-Structured Documents

Uncetain Vesion Contol in Open Collaboative Editing of Tee-Stuctued Documents M. Lamine Ba Institut Mines Télécom; Télécom PaisTech; LTCI Pais, Fance mouhamadou.ba@ telecom-paistech.f Talel Abdessalem

### Lesson 7 Gauss s Law and Electric Fields

Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual

### Edge Detection with Sub-pixel Accuracy Based on Approximation of Edge with Erf Function

56 M. HAGARA, P. KULLA, EDGE DETECTION WITH SUB-PIXEL ACCURACY BASED ON APPROXIMATION OF EDGE Edge Detection with Sub-ixel Accuacy Based on Aoximation of Edge with Ef Function Mioslav HAGARA, Pete KULLA

### 2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

### A r. (Can you see that this just gives the formula we had above?)

24-1 (SJP, Phys 1120) lectic flux, and Gauss' law Finding the lectic field due to a bunch of chages is KY! Once you know, you know the foce on any chage you put down - you can pedict (o contol) motion

### Symmetric polynomials and partitions Eugene Mukhin

Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation

### Chris J. Skinner The probability of identification: applying ideas from forensic statistics to disclosure risk assessment

Chis J. Skinne The pobability of identification: applying ideas fom foensic statistics to disclosue isk assessment Aticle (Accepted vesion) (Refeeed) Oiginal citation: Skinne, Chis J. (2007) The pobability

### Universal Cycles. Yu She. Wirral Grammar School for Girls. Department of Mathematical Sciences. University of Liverpool

Univesal Cycles 2011 Yu She Wial Gamma School fo Gils Depatment of Mathematical Sciences Univesity of Livepool Supeviso: Pofesso P. J. Giblin Contents 1 Intoduction 2 2 De Buijn sequences and Euleian Gaphs

### Seventh Edition DYNAMICS 15Ferdinand P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Texas Tech University

Seenth 15Fedinand P. ee E. Russell Johnston, J. Kinematics of Lectue Notes: J. Walt Ole Texas Tech Uniesity Rigid odies CHPTER VECTOR MECHNICS FOR ENGINEERS: YNMICS 003 The McGaw-Hill Companies, Inc. ll

### The Binomial Distribution

The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between

### 9. Mathematics Practice Paper for Class XII (CBSE) Available Online Tutoring for students of classes 4 to 12 in Physics, Chemistry, Mathematics

Available Online Tutoing fo students of classes 4 to 1 in Physics, 9. Mathematics Class 1 Pactice Pape 1 3 1. Wite the pincipal value of cos.. Wite the ange of the pincipal banch of sec 1 defined on the

### New proofs for the perimeter and area of a circle

New poofs fo the peimete and aea of a cicle K. Raghul Kuma Reseach Schola, Depatment of Physics, Nallamuthu Gounde Mahalingam College, Pollachi, Tamil Nadu 64001, India 1 aghul_physics@yahoo.com aghulkumak5@gmail.com

### Definitions and terminology

I love the Case & Fai textbook but it is out of date with how monetay policy woks today. Please use this handout to supplement the chapte on monetay policy. The textbook assumes that the Fedeal Reseve

### STOCHASTIC MARKOV MODEL APPROACH FOR EFFICIENT VIRTUAL MACHINES SCHEDULING ON PRIVATE CLOUD

Intenational Jounal on Cloud Comuting: Sevices and Achitectue(IJCCSA,Vol., No.3,Novembe 20 STOCHASTIC MARKOV MODEL APPROACH FOR EFFICIENT VIRTUAL MACHINES SCHEDULING ON PRIVATE CLOUD Hsu Mon Kyi and Thinn

### Lecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3

Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each

### Infinite-dimensional Bäcklund transformations between isotropic and anisotropic plasma equilibria.

Infinite-dimensional äcklund tansfomations between isotopic and anisotopic plasma equilibia. Infinite symmeties of anisotopic plasma equilibia. Alexei F. Cheviakov Queen s Univesity at Kingston, 00. Reseach

### Coordinate Systems L. M. Kalnins, March 2009

Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

### Explicit, analytical solution of scaling quantum graphs. Abstract

Explicit, analytical solution of scaling quantum gaphs Yu. Dabaghian and R. Blümel Depatment of Physics, Wesleyan Univesity, Middletown, CT 06459-0155, USA E-mail: ydabaghian@wesleyan.edu (Januay 6, 2003)

### 867 Product Transfer and Resale Report

867 Poduct Tansfe and Resale Repot Functional Goup ID=PT Intoduction: This X12 Tansaction Set contains the fomat and establishes the data contents of the Poduct Tansfe and Resale Repot Tansaction Set (867)

### A PID Tuning Method for Tracking Control of an Underactuated Gantry Crane

Univesal Jounal of Engineeing echanics 1 (13), 45-49 www.aesciences.com A PID Tuning etho fo Tacing Contol of an Uneactuate Ganty Cane. Nazemizaeh Deatment of echanics, Damavan Banch, Islamic Aza Univesity,

### Patent renewals and R&D incentives

RAND Jounal of Economics Vol. 30, No., Summe 999 pp. 97 3 Patent enewals and R&D incentives Fancesca Conelli* and Mak Schankeman** In a model with moal hazad and asymmetic infomation, we show that it can

### The Doppler Effect in Warning Signals: Perceptual Investigations into Aversive Reactions Related to Pitch Changes

The Dopple Effect in Waning Signals: Peceptual Investigations into Avesive Reactions Related to Pitch Changes Clemens Kuhn-Rahloff & Matin Neukom Institute fo Compute Music and Sound Technology (ICST,

### Load Balancing in Processor Sharing Systems

Load Balancing in ocesso Shaing Systems Eitan Altman INRIA Sophia Antipolis 2004, oute des Lucioles 06902 Sophia Antipolis, Fance altman@sophia.inia.f Utzi Ayesta LAAS-CNRS Univesité de Toulouse 7, Avenue

### Load Balancing in Processor Sharing Systems

Load Balancing in ocesso Shaing Systems Eitan Altman INRIA Sophia Antipolis 2004, oute des Lucioles 06902 Sophia Antipolis, Fance altman@sophia.inia.f Utzi Ayesta LAAS-CNRS Univesité de Toulouse 7, Avenue

### Efficient Redundancy Techniques for Latency Reduction in Cloud Systems

Efficient Redundancy Techniques fo Latency Reduction in Cloud Systems 1 Gaui Joshi, Emina Soljanin, and Gegoy Wonell Abstact In cloud computing systems, assigning a task to multiple seves and waiting fo

### Chapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43

Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.

### est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.

9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,

### Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field

### INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS

INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in

### Order-Degree Curves for Hypergeometric Creative Telescoping

Ode-Degee Cuves fo Hyegeometc Ceatve Telescong ABSTRACT Shaosh Chen Deatment of Mathematcs NCSU Ralegh, NC 7695, USA schen@ncsuedu Ceatve telescong aled to a bvaate oe hyegeometc tem oduces lnea ecuence

### Converting knowledge Into Practice

Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading

### AUTOMATIC WHITE BALANCE FOR DIGITAL STILL CAMERA

AUTOMATI WHITE ALANE FOR DIGITAL STILL AMERA Tzan-Sheng hiou ( 邱贊生 ), hiou-shann Fuh ( 傅楸善 ), and Vasha hikane Deatment of omute Science and Infomation Engineeing, National Taiwan Univesity, Taiei, Taiwan.

### VISCOSITY OF BIO-DIESEL FUELS

VISCOSITY OF BIO-DIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use

### UNIVERSIDAD DE CANTABRIA TESIS DOCTORAL

UNIVERSIDAD DE CANABRIA Depatamento de Ingenieía de Comunicaciones ESIS DOCORAL Cyogenic echnology in the Micowave Engineeing: Application to MIC and MMIC Vey Low Noise Amplifie Design Juan Luis Cano de

### Chapter F. Magnetism. Blinn College - Physics Terry Honan

Chapte F Magnetism Blinn College - Physics 46 - Tey Honan F. - Magnetic Dipoles and Magnetic Fields Electomagnetic Duality Thee ae two types of "magnetic chage" o poles, Noth poles N and South poles S.

### Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E

CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known

### Theory and practise of the g-index

Theoy and pactise of the g-index by L. Egghe (*), Univesiteit Hasselt (UHasselt), Campus Diepenbeek, Agoalaan, B-3590 Diepenbeek, Belgium Univesiteit Antwepen (UA), Campus Die Eiken, Univesiteitsplein,

### Week 3-4: Permutations and Combinations

Week 3-4: Pemutations and Combinations Febuay 24, 2016 1 Two Counting Pinciples Addition Pinciple Let S 1, S 2,, S m be disjoint subsets of a finite set S If S S 1 S 2 S m, then S S 1 + S 2 + + S m Multiplication

### Experimentation under Uninsurable Idiosyncratic Risk: An Application to Entrepreneurial Survival

Expeimentation unde Uninsuable Idiosyncatic Risk: An Application to Entepeneuial Suvival Jianjun Miao and Neng Wang May 28, 2007 Abstact We popose an analytically tactable continuous-time model of expeimentation

### Fast FPT-algorithms for cleaning grids

Fast FPT-algoithms fo cleaning gids Josep Diaz Dimitios M. Thilikos Abstact We conside the poblem that given a gaph G and a paamete k asks whethe the edit distance of G and a ectangula gid is at most k.

### CHAT Pre-Calculus Section 10.7. Polar Coordinates

CHAT Pe-Calculus Pola Coodinates Familia: Repesenting gaphs of equations as collections of points (, ) on the ectangula coodinate sstem, whee and epesent the diected distances fom the coodinate aes to

### Timing Synchronization in High Mobility OFDM Systems

Timing Synchonization in High Mobility OFDM Systems Yasamin Mostofi Depatment of Electical Engineeing Stanfod Univesity Stanfod, CA 94305, USA Email: yasi@wieless.stanfod.edu Donald C. Cox Depatment of

### The Lucas Paradox and the Quality of Institutions: Then and Now

Diskussionsbeitäge des Fachbeeichs Witschaftswissenschaft de Feien Univesität Belin Volkswitschaftliche Reihe 2008/3 The Lucas Paadox and the Quality of Institutions: Then and Now Moitz Schulaick und Thomas

### Strength Analysis and Optimization Design about the key parts of the Robot

Intenational Jounal of Reseach in Engineeing and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Pint): 2320-9356 www.ijes.og Volume 3 Issue 3 ǁ Mach 2015 ǁ PP.25-29 Stength Analysis and Optimization Design

### On Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables.

C.Candan EE3/53-METU On Coelation Coefficient The coelation coefficient indicates the degee of linea dependence of two andom vaiables. It is defined as ( )( )} σ σ Popeties: 1. 1. (See appendi fo the poof

### (References to) "Introduction to Electrodynamics" by David J. Griffiths 3rd ed., Prentice-Hall, 1999. ISBN 0-13-805326-X

lectic Fields in Matte page. Capstones in Physics: lectomagnetism. LCTRIC FILDS IN MATTR.. Multipole xpansion A. Multipole expansion of potential B. Dipole moment C. lectic field of dipole D. Dipole in

### Channel selection in e-commerce age: A strategic analysis of co-op advertising models

Jounal of Industial Engineeing and Management JIEM, 013 6(1):89-103 Online ISSN: 013-0953 Pint ISSN: 013-843 http://dx.doi.og/10.396/jiem.664 Channel selection in e-commece age: A stategic analysis of

econsto www.econsto.eu De Open-Access-Publikationsseve de ZBW Leibniz-Infomationszentum Witschaft The Open Access Publication Seve of the ZBW Leibniz Infomation Cente fo Economics Yang, Wei; Feng, Youyi;

### Gravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2

F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,