Order statistics and concentration of l r norms for log-concave vectors

Size: px
Start display at page:

Download "Order statistics and concentration of l r norms for log-concave vectors"

Transcription

1 Jounal of Funcional Analysis Ode saisics and concenaion of l noms fo log-concave vecos Rafał Laała a,b, a Insiue of Mahemaics, Univesiy of Wasaw, Banacha, Waszawa, Poland b Insiue of Mahemaics, Polish Academy of Sciences, ul. Śniadecich 8, Waszawa, Poland Received 30 Novembe 010; acceped 17 Febuay 011 Available online Mach 011 Communicaed by K. Ball Absac We esablish uppe bounds fo ails of ode saisics of isoopic log-concave vecos and apply hem o deive a concenaion of l noms of such vecos. 011 Elsevie Inc. All ighs eseved. Keywods: Log-concave measues; Ode saisics; Concenaion of volume 1. Inoducion and noaion An n-dimensional andom veco is called log-concave if i has a log-concave disibuion, i.e. fo any compac nonempy ses A,B R n and λ 0, 1, P X λa + 1 λb PX A λ PX B 1 λ, whee λa + 1 λb ={λx + 1 λy: x A, y B}. By he esul of Boell [3] a veco X wih full dimensional suppo is log-concave if and only if i has a densiy of he fom e f, whee f : R n, ] is a convex funcion. Log-concave vecos ae fequenly sudied in convex geomey, since by he Bunn Minowsi inequaliy unifom disibuions on convex ses as well as hei lowe dimensional maginals ae log-concave. Reseach paially suppoed by MNiSW Gan no. N N and he Foundaion fo Polish Science. * Addess fo coespondence: Insiue of Mahemaics, Univesiy of Wasaw, Banacha, Waszawa, Poland. addess: laala@mimuw.edu.pl /$ see fon mae 011 Elsevie Inc. All ighs eseved. doi: /j.jfa

2 68 R. Laała / Jounal of Funcional Analysis A andom veco X = X 1,...,X n is isoopic if EX i = 0 and CovX i,x j = δ i,j fo all i, j n. Equivalenly, an n-dimensional andom veco wih mean zeo is isoopic if E,X = fo any R n. Fo any nondegeneae log-concave veco X hee exiss an affine ansfomaion T such ha TXis isoopic. In ecen yeas hee wee deived numeous impoan popeies of log-concave vecos. One of such esuls is he Paouis concenaion of mass [10] ha saes ha fo any isoopic log-concave veco X in R n, P X C n exp n fo 1. 1 One of puposes of his pape is he exension of he Paouis esul o l noms, ha is deiving uppe bounds fo P X, whee x = n x i 1/.Fo [1, his is an easy consequence of 1 and Hölde s inequaliy, howeve he case > equies in ou opinion new ideas. We show ha P X Cn 1/ exp n 1/ fo 1, >, whee C is a consan depending only on see Theoem 8. Ou mehod is based on suiable ail esimaes fo ode saisics of X. Fo an n-dimensional andom veco X by X1 X X n we denoe he noninceasing eaangemen of X 1,..., X n in paicula X1 = max{ X 1,..., X n } and Xn = min{ X 1,..., X n }. Random vaiables X,1 n, ae called ode saisics of X. By 1 we immediaely ge fo isoopic log-concave vecos X, P X exp 1 C fo C n/. The main esul of he pape is Theoem 3 which saes ha he above inequaliy holds fo C logen/ as shows he example of exponenial disibuion his ange of is fo n/ opimal up o a univesal consan. Tail esimaes fo ode saisics can be also applied o povide opimal esimaes fo sup #I=m P I X, whee he supemum is aen ove all subses of {1,...,n} of cadinaliy m [1,n] and P I denoes he coodinaewise pojecion. The deails will be pesened in he fohcoming pape [1]. The oganizaion of he aicle is as follows. In Secion we discuss uppe bounds fo ails of ode saisics and hei connecions wih exponenial concenaion and Paouis esul. Secion 3 is devoed o he deivaion of ail esimaes of l noms fo log-concave vecos. Finally Secion 4 conains a poof of Theoem 4, which is a cucial ool used o deive ou main esul. Thoughou he aicle by C,C 1,... we denoe univesal consans. Values of a consan C may diffe a each occuence. Fo x R n we pu x = x = n x i 1/.. Tail esimaes fo ode saisics If he coodinaes of X ae independen symmeic exponenial andom vaiables wih vaiance one hen i is no had o see ha MedX C 1 logen/ fo any 1 n/. So we may obain a easonable bound fo PX, n/ in he case of isoopic log-concave vecos only fo 1 C logen/. Using he idea ha exponenial andom vecos ae exemal in he

3 R. Laała / Jounal of Funcional Analysis class of uncondiional log-concave vecos i.e. such vecos ha η 1 X 1,...,η n X n has he same disibuion as X fo any choice of signs η i { 1, 1} one may easily deive he following fac. Poposiion 1. If X is a log-concave and uncondiional n-dimensional isoopic andom veco hen P X exp 1C en fo C log. Poof. The esul of Bobov and Nazaov [] implies ha fo any i 1 <i < <i and >0, P X i1,..., X i = PX i1,...,x i exp 1C. Hence P X 1 i 1 < <i n P X i1,..., X i en exp 1C exp 1 C n exp 1C if C logen/. Howeve fo a geneal isoopic log-concave veco wihou uncondiionaliy assumpion we may bound PX i1,...,x i only by exp /C fo C. This suggess ha we should ahe expec bound exp /C han exp /C. If we y o apply he union bound as in he poof of Poposiion 1 i will wo only fo C logen/. Anohe appoach may be based on he exponenial concenaion. We say ha a veco X saisfies exponenial concenaion inequaliy wih a consan α if fo any Boel se A, P X A + αb n 1 1 exp if PX A and >0. Poposiion. If he coodinaes of an n-dimensional veco X have mean zeo and vaiance one and X saisfies exponenial concenaion inequaliy wih a consan α 1 hen P X exp 1 3α en fo 8α log. Poof. Since VaX i = 1wehaveP X i 1/soP X i + exp /αfo >0. Le μ be he disibuion of X. Then he se { A = x R n :# { i: x i } < } has measue μ a leas 1/ fo 4α logen/ indeed we have fo such

4 684 R. Laała / Jounal of Funcional Analysis μ A n n = P 1 { Xi } E 1 { Xi } n exp α n en 1. Le A = A4α logen/. Ifz = x + y A + sb n hen less han / of x i s ae bigge han 4α logen/ and less han / of y i s ae bigge han s, so en P X 4α log + s 1 μ A + sb n exp 1 s. α Fo log-concave vecos i is nown ha exponenial inequaliy is equivalen o seveal ohe funcional inequaliies such as Cheege s and specal gap see [9] fo a deailed discussion and ecen esuls. The song conjecue due o Kannan, Lovász and Simonovis [6] saes ha evey isoopic log-concave veco saisfies Cheege s and heefoe also exponenial inequaliy wih a unifom consan. The conjecue howeve is wide open a ecen esul of Klaag [7] shows ha in he uncondiional case KLS conjecue holds up o log n consan see also [5] fo examples of nonpoduc disibuions ha saisfy specal gap inequaliy wih unifom consans. Bes nown uppe bound fo Cheege s consan fo geneal isoopic log-concave measue is n α fo some α 1/4, 1/ see [9,4]. The main esul of his pape saes ha despie he appoach via he union bound o exponenial concenaion fails he naual esimae fo ode saisics is valid. Namely we have Theoem 3. Le X be an n-dimensional log-concave isoopic veco. Then P X exp 1 en fo C log. C Ou appoach is based on he suiable esimae of momens of he pocess N X, whee N X := n 1 {Xi }, 0. Theoem 4. Fo any isoopic log-concave veco X and p 1 we have E N X p Cp p n fo C log. We pospone a long and bi echnical poof ill he las secion of he pape. Le us only menion a his poin ha i is based on wo ideas. One is he Paouis lage deviaion inequaliy 1 and anohe is an obsevaion ha if we esic a log-concave disibuion o a convex se i is sill log-concave. Poof of Theoem 3. Obseve ha X implies ha N X / on X / and veco X is also isoopic and log-concave. So by Theoem 4 and Chebyshev s inequaliy we ge p

5 R. Laała / Jounal of Funcional Analysis P X p ENX p + EN X p Cp p povided ha C logn /p. So i is enough o ae p = ec 1 and noice ha he esicion on follows by he assumpion ha C logen/. As we aleady noiced one of he main ools in he poof of Theoem 4 is he Paouis concenaion of mass. One may howeve also do he opposie and deive lage deviaions fo he Euclidean nom of X fom ou esimae of momens of N X and he obsevaion ha he disibuion of UX is again log-concave and isoopic fo any oaion U. Moe pecisely he following saemen holds. Poposiion 5. Suppose ha X is a andom veco in R n A 1,A < and any U On, such ha fo some consans E N UX l A1 l l fo A,l n. Then P X n exp 1 n CA 1 fo max{ca 1,A }. Poof. Le us fix A. Hölde s inequaliy gives ha fo any U 1,...,U n On, E l l 1/l N Ui X EN Ui X l A1 l l fo l n. Now le U 1,...,U l be independen andom oaions in On disibued accoding o he Haa measue hen fo l n, A1 l l l E X E U N Ui X = E X EU1 N U1 X l l = EX npy X, Y = n l E X PY X Y1 l, whee Y is a andom veco unifomly disibued on S n 1. Since Y 1 is symmeic, EY 1 = 1/n and EY 4 1 C/n we ge by he Paley Zygmund inequaliy ha PY 1 1 4n 1/C 1 which gives P X n E X C1 P Y X Y1 l C1 A l 1 l. n To conclude he poof i is enough o ae l = ec1 A n. 1 1

6 686 R. Laała / Jounal of Funcional Analysis Concenaion of l noms The aim of his secion is o deive Paouis-ype esimaes fo concenaion of X = n X i 1/. We sa wih pesening wo simple examples. Example 1. Le he coodinaes of X be independen symmeic exponenial.v. s wih vaiance one. Then 1/ = ne X1 1/ 1 C n1/ fo [1,, 1 C log n and p 1/p E X1 p 1/p p C fo p, 1. I is also nown ha in he independen exponenial case wea and song momens ae compaable [8], hence fo, 1/ 1/ = E sup a i X i Theefoe we ge a 1 i 1/ 1/ + C sup E a i X i a 1 i 1/ 1/ + C sup E a i X i 1/ + C. a 1 p 1/p 1/ 1 C n1/ fo p and n C. Example. Fo 1 lex be an isoopic andom veco such ha Y = X X n / n has he exponenial disibuion wih vaiance one. Then by Hölde s inequaliy X n 1/ 1/ Y and i p 1/p n 1/ 1/ Y p 1 C n1/ 1/ p fo p, 1. The examples above show ha he bes we can hope is p 1/p C n 1/ + n 1/ 1/ p fo p, 1, p 1/p C n 1/ + p fo p, [, 3

7 R. Laała / Jounal of Funcional Analysis and p 1/p Clog n + p fo p. 4 O in ems of ails, P X exp 1C n1/ 1/ P X exp 1C fo Cn 1/, [1, ], 5 fo Cn 1/, [, 6 and P X exp 1C fo C log n. 7 Case [1, ] is a simple consequence of he Paouis heoem. Poposiion 6. Esimaes and 5 hold fo all isoopic log-concave vecos X. Poof. We have X n 1/ 1/ X by Hölde s inequaliy, hence and heefoe also 5 immediaely follows by he Paouis esul. Case = is also vey simple. Poposiion 7. Esimaes 4 and 7 hold fo all isoopic log-concave vecos X. Poof. We have P X n P X i n exp /C. Wha is lef is he case << we would lie o obain 6 and 3. We almos ge i excep ha consans explode when appoaches. Theoem 8. Fo any δ>0 hee exis consans C 1 δ, C δ C1 + δ 1/ such ha fo any + δ, P X exp 1 C 1 δ fo C 1 δn 1/ and p 1/p C δ n 1/ + p fo p. The poof of Theoem 8 is based on he following slighly moe pecise esimae.

8 688 R. Laała / Jounal of Funcional Analysis Poposiion 9. Fo > we have P X exp 1 1/ C 1/ fo C n 1/ + log n o in ems of momens p 1/p 1/ C n 1/ + log n + p fo p. Poof. Le s = log n.wehave n X = X i =0 s X. Theoem 3 yields P X en C3 log + exp 1 C fo >0. 8 Obseve ha s log en Cn j j C n. =0 Thus fo 1,..., 0 we ge whee Hence by 8 j=1 s 1/ s P X C n 1/ + P Y =0 =0 Y := X C 3 log en. s 1/ P X C n 1/ + =0 s PY =0 s =0 Fix >0and choose such ha = / / 1/. Then s, =0 exp 1 C 1/.

9 R. Laała / Jounal of Funcional Analysis s = =0 s =0 1 1 C, so we ge 1/ P X C n 1/ + log n + 1 exp 1C. Poof of Theoem 8. Obseve ha 1/ C1 + δ 1/ fo + δ and log n n 1/ and apply Poposiion Poof of Theoem 4 Ou cucial ool will be he following esul. Poposiion 10. Le X be an isoopic log-concave n-dimensional andom veco, A ={X K}, whee K is a convex se in R n such ha 0 < PA 1/e. Then n P A {X i } C 1 PA log PA + ne /C 1 fo C 1. 9 Moeove fo 1 u C, # { i n: P A {X i } e u PA } C u log PA. 10 Poof. Le Y be a andom veco disibued as he veco X condiioned on he se A ha is PY B = PA {X B} PA = PX B K. PX K Noice ha in paicula fo any se B, PX B PAPY B. The veco Y is log-concave, bu no longe isoopic. Since his is only a mae of pemuaion of coodinaes we may assume ha EY 1 EY EY n. Fo α>0le m = mα = # { i: EY i α }. We have EY1 EY m α. Hence by he Paley Zygmund inequaliy, P m Yi 1 αm m P Y i 1 E m Y i 1 4 E m Y i E m Y i 1 C.

10 690 R. Laała / Jounal of Funcional Analysis This implies ha m P Xi 1 αm 1 C PA. Howeve by he esul of Paouis, m P Xi 1 αm exp 1C3 mα fo α C 3. So fo α C 3,exp 1 C 3 mα PA/C and we ge ha We have mα = # { i: EY i α } C 4 α log PA fo α C PA {X i } PA = PY i exp 1 CEYi 1/ and 10 follows by 11. Tae C 3 and le 0 be a nonnegaive inege such ha 0 C Define I 0 = { i: EYi }, I 0 +1 = { i: EYi < 4 0 } and By 11 we ge I j = { i:4 j EY i < 4 1 j }, j = 1,,..., 0. #I j C 4 4 j log PA fo j = 0, 1,..., 0 and obviously #I 0 +1 n. Moeove fo i I j, j 0, Thus Y i PY i P EYi j 1 1/ n 0 +1 PY i = PY i #I 0 + e j=0 i I j 0 C 4 log PA 1 + e j=1 C 1 log PA + ne /C 1. exp 1 1C j j=1 #I j exp 1C j j exp 1C j + ene /C

11 R. Laała / Jounal of Funcional Analysis To finish he poof of 9 i is enough o obseve ha n P A {X i } n = PA PY i. The following wo examples show ha esimae 9 is close o be opimal. Example 1. Tae X 1,X,...,X n o be independen symmeic exponenial andom vaiables wih vaiance 1 and A ={X 1 }. Then PA = 1 e and n P A {X i } n = PA PX i = n 1PA exp /, i= i= heefoe he faco ne /C in 9 is necessay. Example. Tae A ={X 1,...,X } hen n P A {X i } PA. So impovemen of he faco PA log PA in 9 would imply in paicula a bee esimae of PX 1,...,X han exp 1 C and we do no now if such bound is possible o obain. Poof of Theoem 4. We have N X n, so he saemen is obvious if n Cp, in he sequel we will assume ha n 10p. Le C 1 and C be as in Poposiion 10 inceasing C i if necessay we may assume ha PX 1 e /C i fo C i and i = 1,. Le us fix p 1 and C log n, hen p max{c 1, 4C } and ne /C 1 p if C is lage enough. Le l be a posiive inege such ha p l p and l = fo some inege. Since EN X p 1/p EN X l 1/l i is enough o show ha E N X l Cl l. Recall ha by ou assumpion on p,wehave n 5l. To shoen he noaion le B i1,...,i s ={X i1,...,x is } and B = Ω. Define

12 69 R. Laała / Jounal of Funcional Analysis we need o show ha n l ml := EN X l = E 1 {Xi } = n i 1,...,i l =1 PB i1,...,i l, Cl l ml. 1 We divide he sum in ml ino seveal pas. Le j 1 be such inege ha We se and n j1 < log j1 1. I 0 = { i 1,...,i l {1,...,n} l : PB i1,...,i l >e l}, I j = { i 1,...,i l {1,...,n} l : PB i1,...,i l e j l,e j 1 l ]}, 0 <j<j 1 I j1 = { i 1,...,i l {1,...,n} l : PB i1,...,i l e j 1 1 l }. Since {1,...,n} l = j 1 j=0 I j we ge ml = j 1 j=0 m j l, whee m j l := l i 1,...,i l I j PB i1,...,i l fo 0 j j 1. I is easy o bound m j1 l namely since #I j1 n l we have i 1,...,i l I j1 PB i1,...,i l n l e j1 1l l l. To esimae m 0 l we define fis fo I {1,...,n} l and 1 s l, P s I = { i 1,...,i s : i 1,...,i l I fo some i s+1,...,i l }. By Poposiion 10 we ge fo s = 1,...,l 1 PB i1,...,i s+1 i 1,...,i s+1 P s+1 I 0 n i 1,...,i s P s I 0 i s+1 =1 P B i1,...,i s {X is+1 } C 1 PB i1,...,i s log PB i1,...,i s + ne /C 1. i 1,...,i s P s I 0

13 R. Laała / Jounal of Funcional Analysis Obseve ha we have PB i1,...,i s >e l fo i 1,...,i s P s I 0 and ecall ha ne /C 1 p 4l, hence PB i1,...,i s+1 5C 1 l PB i1,...,i s. i 1,...,i s+1 P s+1 I 0 i 1,...,i s P s I 0 So, by easy inducion we obain m 0 l = PB i1,...,i l 5C1 l l 1 PB i1 i 1,...,i l I 0 i 1 P 1 I 0 5C 1 l l 1 ne /C 1 Cl l. Now comes he mos involved pa of he poof esimaing m j l fo 0 <j<j 1. I is based on suiable bounds fo #I j. We will need he following simple combinaoial lemma. Lemma 11. Le l 0 l 1 l s be a fixed sequence of posiive ineges and Then F = { f :{1,,...,l 0 } {0, 1,,...,s}: 1 i s # { : f i } l i }. #F s eli 1 Poof of Lemma 11. Noice ha any funcion f :{1,,...,l 0 } {0, 1,,...,s} is deemined by he ses A i ={: f i} fo i = 0, 1,...,s.Taef F, obviously A 0 ={1,...,l 0 }.If he se A i 1 of cadinaliy a i 1 l i 1 is aleady chosen hen he se A i A i 1 of cadinaliy a mos l i may be chosen in ai ai ai 1 l i li 1 0 l i + li. li li 1 l i li eli 1 l i ways. We come bac o he poof of Theoem 4. Fix 0 <j<j 1,le 1 be a posiive inege such ha 1 < C 1+1. Fo i 1,...,i l I j we define a funcion f i1,...,i l :{1,...,l} {j,j + 1,..., 1 } by he fomula j if PB i1,...,i s exp j+1 PB i1,...,i s 1, f i1,...,i l s = if exp +1 PB i 1,...,is PB i1,...,i s 1 < exp, j < < 1, 1 if PB i1,...,i s <exp 1PB i1,...,i s 1.

14 694 R. Laała / Jounal of Funcional Analysis Noice ha fo all i 1, PX i1 e /C < exp 1PB, so f i1,...,i l 1 = 1 fo all i 1,...,i l. Pu Fo f = f i1,...,i l F j and >j,wehave so F j := { f i1,...,i l : i 1,...,i l I j }. exp j l < PB i1,...,i l <exp # { s: fs }, # { s: fs } j l =: l. 13 Obseve ha he above inequaliy holds also fo = j. Wehavel 1 /l = and 1 =j+1 l l so by Lemma 11 we ge Pu #F j 1 =j+1 el 1 l l e l. Now fix f F j we will esimae he cadinaliy of he se We have I j f := { i 1,...,i l I j : f i1,...,i l = f }. n := # { s {1,...,l}: fs= }, = j,j + 1,..., 1. n j + n j+1 + +n 1 = l, moeove if i 1,...,i s 1 ae fixed and fs= < 1 hen s and by he second pa of Poposiion 10 wih u = +1 /C i s may ae a mos values. Thus 4C log PB i1,...,i s 1 4C +j l 4C l 1 1 #I j f n n 1 =j m n = n n 4C l l n exp Obseve ha by peviously deived esimae 13 we ge n l = j l, exp + j =: m =j + jn.

15 R. Laała / Jounal of Funcional Analysis hence jn j+ l C + j l. =j =j We also have n 1 j 1 l C j 1 l logn /4l j 3 l, whee he las inequaliy holds since C logn /4l and C may be aen abiaily lage. So we ge ha fo any f F j, This shows ha Hence Theefoe Cl l n n1 #I j f exp j 4l l Cl l 3 exp 8 j l. Cl l 3 #I j #F j exp 8 j l Cl l exp + 38 j l. m j l = PB i1,...,i l #I j exp j 1 l Cl l exp j 3 l. i 1,...,i l I j j 1 1 ml = m 0 l + m j1 l + Cl l j=1 m j l l l C l C l exp j 3 l j=1 and 1 holds. Acnowledgmens This wo was done while he auho was aing pa in he Themaic Pogam on Asympoic Geomeic Analysis a he Fields Insiue in Toono. The auho would lie o han Radosław Adamcza and Nicole Tomcza-Jaegemann fo hei helpful commens.

16 696 R. Laała / Jounal of Funcional Analysis Refeences [1] R. Adamcza, R. Laała, A. Liva, A. Pajo, N. Tomcza-Jaegemann, Geomey of log-concave ensembles of andom maices and appoximae econsucion, pepin. [] S.G. Bobov, F.L. Nazaov, On convex bodies and log-concave pobabiliy measues wih uncondiional basis, in: Geomeic Aspecs of Funcional Analysis, in: Lecue Noes in Mah., vol. 1807, Spinge, Belin, 003, pp [3] C. Boell, Convex measues on locally convex spaces, A. Ma [4] O. Guedon, E. Milman, Inepolaing hin-shell and shap lage-deviaion esimaes fo isoopic log-concave measues, pepin, hp://axiv.og/abs/ [5] N. Hue, Specal gap fo some invaian log-concave pobabiliy measues, pepin, hp://axiv.og/abs/ [6] R. Kannan, L. Lovász, M. Simonovis, Isopeimeic poblems fo convex bodies and a localizaion lemma, Discee Compu. Geom [7] B. Klaag, A Bey Esseen ype inequaliy fo convex bodies wih an uncondiional basis, Pobab. Theoy Relaed Fields [8] R. Laała, Tail and momen esimaes fo sums of independen andom vecos wih logaihmically concave ails, Sudia Mah [9] E. Milman, On he ole of convexiy in isopeimey, specal gap and concenaion, Inven. Mah [10] G. Paouis, Concenaion of mass on convex bodies, Geom. Func. Anal

Extremal problems for t-partite and t-colorable hypergraphs

Extremal problems for t-partite and t-colorable hypergraphs Exemal poblems fo -paie and -coloable hypegaphs Dhuv Mubayi John Talbo June, 007 Absac Fix ineges and an -unifom hypegaph F. We pove ha he maximum numbe of edges in a -paie -unifom hypegaph on n veices

More information

ON 3-DIMENSIONAL CONTACT METRIC MANIFOLDS

ON 3-DIMENSIONAL CONTACT METRIC MANIFOLDS Mem. Fac. Inegaed As and Sci., Hioshima Univ., Se. IV, Vol. 8 9-33, Dec. 00 ON 3-DIMENSIONAL CONTACT METRIC MANIFOLDS YOSHIO AGAOKA *, BYUNG HAK KIM ** AND JIN HYUK CHOI ** *Depamen of Mahemaics, Faculy

More information

Combinatorial Approach to M/M/1 Queues. Using Hypergeometric Functions

Combinatorial Approach to M/M/1 Queues. Using Hypergeometric Functions Inenaional Mahemaical Foum, Vol 8, 03, no 0, 463-47 HIKARI Ld, wwwm-hikaicom Combinaoial Appoach o M/M/ Queues Using Hypegeomeic Funcions Jagdish Saan and Kamal Nain Depamen of Saisics, Univesiy of Delhi,

More information

Variance and Covariance Processes

Variance and Covariance Processes Vaiance and Covaiance Pocesses Pakash Balachandan Depamen of Mahemaics Duke Univesiy May 26, 2008 These noes ae based on Due s Sochasic Calculus, Revuz and Yo s Coninuous Maingales and Bownian Moion, Kaazas

More information

336 ERIDANI kfk Lp = sup jf(y) ; f () jj j p p whee he supemum is aken ove all open balls = (a ) inr n, jj is he Lebesgue measue of in R n, () =(), f

336 ERIDANI kfk Lp = sup jf(y) ; f () jj j p p whee he supemum is aken ove all open balls = (a ) inr n, jj is he Lebesgue measue of in R n, () =(), f TAMKANG JOURNAL OF MATHEMATIS Volume 33, Numbe 4, Wine 2002 ON THE OUNDEDNESS OF A GENERALIED FRATIONAL INTEGRAL ON GENERALIED MORREY SPAES ERIDANI Absac. In his pape we exend Nakai's esul on he boundedness

More information

The shortest path between two truths in the real domain passes through the complex domain. J. Hadamard

The shortest path between two truths in the real domain passes through the complex domain. J. Hadamard Complex Analysis R.G. Halbud R.Halbud@ucl.ac.uk Depamen of Mahemaics Univesiy College London 202 The shoes pah beween wo uhs in he eal domain passes hough he complex domain. J. Hadamad Chape The fis fundamenal

More information

[ ] 0. = (2) = a q dimensional vector of observable instrumental variables that are in the information set m constituents of u

[ ] 0. = (2) = a q dimensional vector of observable instrumental variables that are in the information set m constituents of u Genealized Mehods of Momens he genealized mehod momens (GMM) appoach of Hansen (98) can be hough of a geneal pocedue fo esing economics and financial models. he GMM is especially appopiae fo models ha

More information

On Control Problem Described by Infinite System of First-Order Differential Equations

On Control Problem Described by Infinite System of First-Order Differential Equations Ausalian Jounal of Basic and Applied Sciences 5(): 736-74 ISS 99-878 On Conol Poblem Descibed by Infinie Sysem of Fis-Ode Diffeenial Equaions Gafujan Ibagimov and Abbas Badaaya J'afau Insiue fo Mahemaical

More information

7 Wave Equation in Higher Dimensions

7 Wave Equation in Higher Dimensions 7 Wave Equaion in Highe Dimensions We now conside he iniial-value poblem fo he wave equaion in n dimensions, u c u x R n u(x, φ(x u (x, ψ(x whee u n i u x i x i. (7. 7. Mehod of Spheical Means Ref: Evans,

More information

arxiv: v1 [math.co] 4 Apr 2019

arxiv: v1 [math.co] 4 Apr 2019 Dieced dominaion in oiened hypegaphs axiv:1904.02351v1 [mah.co] 4 Ap 2019 Yai Cao Dep. of Mahemaics Univesiy of Haifa-Oanim Tivon 36006, Isael yacao@kvgeva.og.il This pape is dedicaed o Luz Volkmann on

More information

General Non-Arbitrage Model. I. Partial Differential Equation for Pricing A. Traded Underlying Security

General Non-Arbitrage Model. I. Partial Differential Equation for Pricing A. Traded Underlying Security 1 Geneal Non-Abiage Model I. Paial Diffeenial Equaion fo Picing A. aded Undelying Secuiy 1. Dynamics of he Asse Given by: a. ds = µ (S, )d + σ (S, )dz b. he asse can be eihe a sock, o a cuency, an index,

More information

Deviation probability bounds for fractional martingales and related remarks

Deviation probability bounds for fractional martingales and related remarks Deviaion pobabiliy bounds fo facional maingales and elaed emaks Buno Sausseeau Laboaoie de Mahémaiques de Besançon CNRS, UMR 6623 16 Roue de Gay 253 Besançon cedex, Fance Absac In his pape we pove exponenial

More information

POSITIVE SOLUTIONS WITH SPECIFIC ASYMPTOTIC BEHAVIOR FOR A POLYHARMONIC PROBLEM ON R n. Abdelwaheb Dhifli

POSITIVE SOLUTIONS WITH SPECIFIC ASYMPTOTIC BEHAVIOR FOR A POLYHARMONIC PROBLEM ON R n. Abdelwaheb Dhifli Opuscula Mah. 35, no. (205), 5 9 hp://dx.doi.og/0.7494/opmah.205.35..5 Opuscula Mahemaica POSITIVE SOLUTIONS WITH SPECIFIC ASYMPTOTIC BEHAVIOR FOR A POLYHARMONIC PROBLEM ON R n Abdelwaheb Dhifli Communicaed

More information

BMOA estimates and radial growth of B φ functions

BMOA estimates and radial growth of B φ functions c Jounal of echnical Univesiy a Plovdiv Fundamenal Sciences and Applicaions, Vol., 995 Seies A-Pue and Applied Mahemaics Bulgaia, ISSN 3-827 axiv:87.53v [mah.cv] 3 Jul 28 BMOA esimaes and adial gowh of

More information

On The Estimation of Two Missing Values in Randomized Complete Block Designs

On The Estimation of Two Missing Values in Randomized Complete Block Designs Mahemaical Theoy and Modeling ISSN 45804 (Pape ISSN 505 (Online Vol.6, No.7, 06 www.iise.og On The Esimaion of Two Missing Values in Randomized Complee Bloc Designs EFFANGA, EFFANGA OKON AND BASSE, E.

More information

STUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE WEIBULL DISTRIBUTION

STUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE WEIBULL DISTRIBUTION Inenaional Jounal of Science, Technology & Managemen Volume No 04, Special Issue No. 0, Mach 205 ISSN (online): 2394-537 STUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE

More information

A note on characterization related to distributional properties of random translation, contraction and dilation of generalized order statistics

A note on characterization related to distributional properties of random translation, contraction and dilation of generalized order statistics PobSa Foum, Volume 6, July 213, Pages 35 41 ISSN 974-3235 PobSa Foum is an e-jounal. Fo eails please visi www.pobsa.og.in A noe on chaaceizaion elae o isibuional popeies of anom anslaion, conacion an ilaion

More information

Lecture 17: Kinetics of Phase Growth in a Two-component System:

Lecture 17: Kinetics of Phase Growth in a Two-component System: Lecue 17: Kineics of Phase Gowh in a Two-componen Sysem: descipion of diffusion flux acoss he α/ ineface Today s opics Majo asks of oday s Lecue: how o deive he diffusion flux of aoms. Once an incipien

More information

Today - Lecture 13. Today s lecture continue with rotations, torque, Note that chapters 11, 12, 13 all involve rotations

Today - Lecture 13. Today s lecture continue with rotations, torque, Note that chapters 11, 12, 13 all involve rotations Today - Lecue 13 Today s lecue coninue wih oaions, oque, Noe ha chapes 11, 1, 13 all inole oaions slide 1 eiew Roaions Chapes 11 & 1 Viewed fom aboe (+z) Roaional, o angula elociy, gies angenial elociy

More information

Lecture-V Stochastic Processes and the Basic Term-Structure Equation 1 Stochastic Processes Any variable whose value changes over time in an uncertain

Lecture-V Stochastic Processes and the Basic Term-Structure Equation 1 Stochastic Processes Any variable whose value changes over time in an uncertain Lecue-V Sochasic Pocesses and he Basic Tem-Sucue Equaion 1 Sochasic Pocesses Any vaiable whose value changes ove ime in an unceain way is called a Sochasic Pocess. Sochasic Pocesses can be classied as

More information

THE MODULAR INEQUALITIES FOR A CLASS OF CONVOLUTION OPERATORS ON MONOTONE FUNCTIONS

THE MODULAR INEQUALITIES FOR A CLASS OF CONVOLUTION OPERATORS ON MONOTONE FUNCTIONS PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 5, Numbe 8, Augus 997, Pages 93 35 S -9939(973867-7 THE MODULAR INEQUALITIES FOR A CLASS OF CONVOLUTION OPERATORS ON MONOTONE FUNCTIONS JIM QILE

More information

Representing Knowledge. CS 188: Artificial Intelligence Fall Properties of BNs. Independence? Reachability (the Bayes Ball) Example

Representing Knowledge. CS 188: Artificial Intelligence Fall Properties of BNs. Independence? Reachability (the Bayes Ball) Example C 188: Aificial Inelligence Fall 2007 epesening Knowledge ecue 17: ayes Nes III 10/25/2007 an Klein UC ekeley Popeies of Ns Independence? ayes nes: pecify complex join disibuions using simple local condiional

More information

Lecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation

Lecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation Lecue 8: Kineics of Phase Gowh in a Two-componen Sysem: geneal kineics analysis based on he dilue-soluion appoximaion Today s opics: In he las Lecues, we leaned hee diffeen ways o descibe he diffusion

More information

Degree of Approximation of a Class of Function by (C, 1) (E, q) Means of Fourier Series

Degree of Approximation of a Class of Function by (C, 1) (E, q) Means of Fourier Series IAENG Inenaional Jounal of Applied Mahemaic, 4:, IJAM_4 7 Degee of Appoximaion of a Cla of Funcion by C, E, q Mean of Fouie Seie Hae Kihna Nigam and Kuum Shama Abac In hi pape, fo he fi ime, we inoduce

More information

arxiv: v1 [math.ca] 25 Sep 2013

arxiv: v1 [math.ca] 25 Sep 2013 OUNDEDNESS OF INTRINSIC LITTLEWOOD-PALEY FUNCTIONS ON MUSIELAK-ORLICZ MORREY AND CAMPANATO SPACES axiv:39.652v [mah.ca] 25 Sep 23 YIYU LIANG, EIICHI NAKAI 2, DACHUN YANG AND JUNQIANG ZHANG Absac. Le ϕ

More information

MODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE

MODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE Topics MODULE 3 FUNCTION OF A RANDOM VARIABLE AND ITS DISTRIBUTION LECTURES 2-6 3. FUNCTION OF A RANDOM VARIABLE 3.2 PROBABILITY DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE 3.3 EXPECTATION AND MOMENTS

More information

GRADIENT ESTIMATES, POINCARÉ INEQUALITIES, DE GIORGI PROPERTY AND THEIR CONSEQUENCES

GRADIENT ESTIMATES, POINCARÉ INEQUALITIES, DE GIORGI PROPERTY AND THEIR CONSEQUENCES GRADIENT ESTIMATES, POINCARÉ INEQUALITIES, DE GIORGI PROPERTY AND THEIR CONSEQUENCES FRÉDÉRIC ERNICOT, THIERRY COULHON, DOROTHEE FREY Absac. On a doubling meic measue space endowed wih a caé du champ,

More information

@FMI c Kyung Moon Sa Co.

@FMI c Kyung Moon Sa Co. Annals of Fuzzy Mahemaics Infomaics Volume, No. 2, (Apil 20), pp. 9-3 ISSN 2093 930 hp://www.afmi.o.k @FMI c Kyung Moon Sa Co. hp://www.kyungmoon.com On lacunay saisical convegence in inuiionisic fuzzy

More information

Orthotropic Materials

Orthotropic Materials Kapiel 2 Ohoopic Maeials 2. Elasic Sain maix Elasic sains ae elaed o sesses by Hooke's law, as saed below. The sesssain elaionship is in each maeial poin fomulaed in he local caesian coodinae sysem. ε

More information

Computer Propagation Analysis Tools

Computer Propagation Analysis Tools Compue Popagaion Analysis Tools. Compue Popagaion Analysis Tools Inoducion By now you ae pobably geing he idea ha pedicing eceived signal sengh is a eally impoan as in he design of a wieless communicaion

More information

MEEN 617 Handout #11 MODAL ANALYSIS OF MDOF Systems with VISCOUS DAMPING

MEEN 617 Handout #11 MODAL ANALYSIS OF MDOF Systems with VISCOUS DAMPING MEEN 67 Handou # MODAL ANALYSIS OF MDOF Sysems wih VISCOS DAMPING ^ Symmeic Moion of a n-dof linea sysem is descibed by he second ode diffeenial equaions M+C+K=F whee () and F () ae n ows vecos of displacemens

More information

The sudden release of a large amount of energy E into a background fluid of density

The sudden release of a large amount of energy E into a background fluid of density 10 Poin explosion The sudden elease of a lage amoun of enegy E ino a backgound fluid of densiy ceaes a song explosion, chaaceized by a song shock wave (a blas wave ) emanaing fom he poin whee he enegy

More information

Lecture 22 Electromagnetic Waves

Lecture 22 Electromagnetic Waves Lecue Elecomagneic Waves Pogam: 1. Enegy caied by he wave (Poyning veco).. Maxwell s equaions and Bounday condiions a inefaces. 3. Maeials boundaies: eflecion and efacion. Snell s Law. Quesions you should

More information

, on the power of the transmitter P t fed to it, and on the distance R between the antenna and the observation point as. r r t

, on the power of the transmitter P t fed to it, and on the distance R between the antenna and the observation point as. r r t Lecue 6: Fiis Tansmission Equaion and Rada Range Equaion (Fiis equaion. Maximum ange of a wieless link. Rada coss secion. Rada equaion. Maximum ange of a ada. 1. Fiis ansmission equaion Fiis ansmission

More information

Sections 3.1 and 3.4 Exponential Functions (Growth and Decay)

Sections 3.1 and 3.4 Exponential Functions (Growth and Decay) Secions 3.1 and 3.4 Eponenial Funcions (Gowh and Decay) Chape 3. Secions 1 and 4 Page 1 of 5 Wha Would You Rahe Have... $1million, o double you money evey day fo 31 days saing wih 1cen? Day Cens Day Cens

More information

arxiv: v2 [math.pr] 19 Feb 2016

arxiv: v2 [math.pr] 19 Feb 2016 Global Diichle Hea Kenel Esimaes fo Symmeic Lévy Pocesses in Half-space Zhen-Qing Chen and Panki Kim axiv:54.4673v2 [mah.pr] 9 Feb 26 Mach 5, 28 Absac In his pape, we deive explici shap wo-sided esimaes

More information

Monotonic Solutions of a Class of Quadratic Singular Integral Equations of Volterra type

Monotonic Solutions of a Class of Quadratic Singular Integral Equations of Volterra type In. J. Conemp. Mah. Sci., Vol. 2, 27, no. 2, 89-2 Monoonic Soluions of a Class of Quadraic Singular Inegral Equaions of Volerra ype Mahmoud M. El Borai Deparmen of Mahemaics, Faculy of Science, Alexandria

More information

K. G. Malyutin, T. I. Malyutina, I. I. Kozlova ON SUBHARMONIC FUNCTIONS IN THE HALF-PLANE OF INFINITE ORDER WITH RADIALLY DISTRIBUTED MEASURE

K. G. Malyutin, T. I. Malyutina, I. I. Kozlova ON SUBHARMONIC FUNCTIONS IN THE HALF-PLANE OF INFINITE ORDER WITH RADIALLY DISTRIBUTED MEASURE Математичнi Студiї. Т.4, 2 Maemaychni Sudii. V.4, No.2 УДК 57.5 K. G. Malyuin, T. I. Malyuina, I. I. Kozlova ON SUBHARMONIC FUNCTIONS IN THE HALF-PLANE OF INFINITE ORDER WITH RADIALLY DISTRIBUTED MEASURE

More information

Reinforcement learning

Reinforcement learning Lecue 3 Reinfocemen leaning Milos Hauskech milos@cs.pi.edu 539 Senno Squae Reinfocemen leaning We wan o lean he conol policy: : X A We see examples of x (bu oupus a ae no given) Insead of a we ge a feedback

More information

CS 188: Artificial Intelligence Fall Probabilistic Models

CS 188: Artificial Intelligence Fall Probabilistic Models CS 188: Aificial Inelligence Fall 2007 Lecue 15: Bayes Nes 10/18/2007 Dan Klein UC Bekeley Pobabilisic Models A pobabilisic model is a join disibuion ove a se of vaiables Given a join disibuion, we can

More information

arxiv: v2 [stat.me] 13 Jul 2015

arxiv: v2 [stat.me] 13 Jul 2015 One- and wo-sample nonpaameic ess fo he al-o-noise aio based on ecod saisics axiv:1502.05367v2 [sa.me] 13 Jul 2015 Damien Challe 1,2 1 Laboaoie de mahémaiques appliquées aux sysèmes, CenaleSupélec, 92295

More information

LOGARITHMIC ORDER AND TYPE OF INDETERMINATE MOMENT PROBLEMS

LOGARITHMIC ORDER AND TYPE OF INDETERMINATE MOMENT PROBLEMS LOGARITHMIC ORDER AND TYPE OF INDETERMINATE MOMENT PROBLEMS CHRISTIAN BERG AND HENRIK L. PEDERSEN WITH AN APPENDIX BY WALTER HAYMAN We invesigae a efined gowh scale, logaihmic gowh, fo indeeminae momen

More information

Hamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t

Hamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t M ah 5 2 7 Fall 2 0 0 9 L ecure 1 0 O c. 7, 2 0 0 9 Hamilon- J acobi Equaion: Explici Formulas In his lecure we ry o apply he mehod of characerisics o he Hamilon-Jacobi equaion: u + H D u, x = 0 in R n

More information

ENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 9 Solutions [Theorems of Gauss and Stokes]

ENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 9 Solutions [Theorems of Gauss and Stokes] ENGI 44 Avance alculus fo Engineeing Faculy of Engineeing an Applie cience Poblem e 9 oluions [Theoems of Gauss an okes]. A fla aea A is boune by he iangle whose veices ae he poins P(,, ), Q(,, ) an R(,,

More information

Statistical inference versus mean field limit for Hawkes processes

Statistical inference versus mean field limit for Hawkes processes Eleconic Jounal of Saisics Vol. 1 216 1223 1295 ISS: 1935-7524 DOI: 1.1214/16-EJS1142 Saisical infeence vesus mean field limi fo Hawkes pocesses Sylvain Delae Laboaoie de Pobabiliés e Modèles Aléaoies,

More information

On the local convexity of the implied volatility curve in uncorrelated stochastic volatility models

On the local convexity of the implied volatility curve in uncorrelated stochastic volatility models On he local conexiy of he implied olailiy cue in uncoelaed sochasic olailiy models Elisa Alòs Dp. d Economia i Empesa and Bacelona Gaduae School of Economics Uniesia Pompeu Faba c/ramon Tias Fagas, 5-7

More information

Lecture 20: Riccati Equations and Least Squares Feedback Control

Lecture 20: Riccati Equations and Least Squares Feedback Control 34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he

More information

Research Article A Note on Multiplication and Composition Operators in Lorentz Spaces

Research Article A Note on Multiplication and Composition Operators in Lorentz Spaces Hindawi Publishing Copoaion Jounal of Funcion Spaces and Applicaions Volume 22, Aicle ID 29363, pages doi:.55/22/29363 Reseach Aicle A Noe on Muliplicaion and Composiion Opeaos in Loenz Spaces Eddy Kwessi,

More information

IMPROVING ON MINIMUM RISK EQUIVARIANT AND LINEAR MINIMAX ESTIMATORS OF BOUNDED MULTIVARIATE LOCATION PARAMETERS

IMPROVING ON MINIMUM RISK EQUIVARIANT AND LINEAR MINIMAX ESTIMATORS OF BOUNDED MULTIVARIATE LOCATION PARAMETERS REVSTAT Saisical Jounal Volume 8, Numbe 2, Novembe 21, 125 138 IMPROVING ON MINIMUM RISK EQUIVARIANT AND LINEAR MINIMAX ESTIMATORS OF BOUNDED MULTIVARIATE LOCATION PARAMETERS Auhos: Éic Machand Dépaemen

More information

Monochromatic Wave over One and Two Bars

Monochromatic Wave over One and Two Bars Applied Mahemaical Sciences, Vol. 8, 204, no. 6, 307-3025 HIKARI Ld, www.m-hikai.com hp://dx.doi.og/0.2988/ams.204.44245 Monochomaic Wave ove One and Two Bas L.H. Wiyano Faculy of Mahemaics and Naual Sciences,

More information

11!Hí MATHEMATICS : ERDŐS AND ULAM PROC. N. A. S. of decomposiion, properly speaking) conradics he possibiliy of defining a counably addiive real-valu

11!Hí MATHEMATICS : ERDŐS AND ULAM PROC. N. A. S. of decomposiion, properly speaking) conradics he possibiliy of defining a counably addiive real-valu ON EQUATIONS WITH SETS AS UNKNOWNS BY PAUL ERDŐS AND S. ULAM DEPARTMENT OF MATHEMATICS, UNIVERSITY OF COLORADO, BOULDER Communicaed May 27, 1968 We shall presen here a number of resuls in se heory concerning

More information

Ann. Funct. Anal. 2 (2011), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:

Ann. Funct. Anal. 2 (2011), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL: Ann. Func. Anal. 2 2011, no. 2, 34 41 A nnals of F uncional A nalysis ISSN: 2008-8752 elecronic URL: www.emis.de/journals/afa/ CLASSIFICAION OF POSIIVE SOLUIONS OF NONLINEAR SYSEMS OF VOLERRA INEGRAL EQUAIONS

More information

A STOCHASTIC MODELING FOR THE UNSTABLE FINANCIAL MARKETS

A STOCHASTIC MODELING FOR THE UNSTABLE FINANCIAL MARKETS A STOCHASTIC MODELING FOR THE UNSTABLE FINANCIAL MARKETS Assoc. Pof. Romeo Negea Ph. D Poliehnica Univesiy of Timisoaa Depamen of Mahemaics Timisoaa, Romania Assoc. Pof. Cipian Peda Ph. D Wes Univesiy

More information

Risk tolerance and optimal portfolio choice

Risk tolerance and optimal portfolio choice Risk oleance and opimal pofolio choice Maek Musiela BNP Paibas London Copoae and Invesmen Join wok wih T. Zaiphopoulou (UT usin) Invesmens and fowad uiliies Pepin 6 Backwad and fowad dynamic uiliies and

More information

arxiv: v1 [math.ca] 15 Jan 2019

arxiv: v1 [math.ca] 15 Jan 2019 DORRONSORO S THEOREM IN HEISENBERG GROUPS KATRIN FÄSSLER AND TUOMAS ORPONEN axiv:9.4767v [mah.ca] 5 Jan 29 ABSTRACT. A heoem of Doonsoo fom he 98s quanifies he fac ha eal-valued Sobolev funcions on Euclidean

More information

Quantum Algorithms for Matrix Products over Semirings

Quantum Algorithms for Matrix Products over Semirings CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE 2017, Aicle 1, pages 1 25 hp://cjcscsuchicagoedu/ Quanum Algoihms fo Maix Poducs ove Semiings Fançois Le Gall Haumichi Nishimua Received July 24, 2015; Revised

More information

functions on localized Morrey-Campanato spaces over doubling metric measure spaces

functions on localized Morrey-Campanato spaces over doubling metric measure spaces JOURNAL OF FUNCTION SPACES AND APPLICATIONS Volume 9, Numbe 3 2), 245 282 c 2, Scienific Hoizon hp://www.jfsa.ne oundedness of Lusin-aea and gλ funcions on localized Moey-Campanao spaces ove doubling meic

More information

MATHEMATICAL FOUNDATIONS FOR APPROXIMATING PARTICLE BEHAVIOUR AT RADIUS OF THE PLANCK LENGTH

MATHEMATICAL FOUNDATIONS FOR APPROXIMATING PARTICLE BEHAVIOUR AT RADIUS OF THE PLANCK LENGTH Fundamenal Jounal of Mahemaical Phsics Vol 3 Issue 013 Pages 55-6 Published online a hp://wwwfdincom/ MATHEMATICAL FOUNDATIONS FOR APPROXIMATING PARTICLE BEHAVIOUR AT RADIUS OF THE PLANCK LENGTH Univesias

More information

Vehicle Arrival Models : Headway

Vehicle Arrival Models : Headway Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where

More information

New sufficient conditions of robust recovery for low-rank matrices

New sufficient conditions of robust recovery for low-rank matrices New sufficien condiions of ous ecovey fo low-an maices Jianwen Huang a, Jianjun Wang a,, Feng Zhang a, Wendong Wang a a School of Mahemaics and Saisics, Souhwes Univesiy, Chongqing, 400715, China Reseach

More information

r P + '% 2 r v(r) End pressures P 1 (high) and P 2 (low) P 1 , which must be independent of z, so # dz dz = P 2 " P 1 = " #P L L,

r P + '% 2 r v(r) End pressures P 1 (high) and P 2 (low) P 1 , which must be independent of z, so # dz dz = P 2  P 1 =  #P L L, Lecue 36 Pipe Flow and Low-eynolds numbe hydodynamics 36.1 eading fo Lecues 34-35: PKT Chape 12. Will y fo Monday?: new daa shee and daf fomula shee fo final exam. Ou saing poin fo hydodynamics ae wo equaions:

More information

Secure Frameproof Codes Through Biclique Covers

Secure Frameproof Codes Through Biclique Covers Discee Mahemaics and Theoeical Compue Science DMTCS vol. 4:2, 202, 26 270 Secue Famepoof Codes Though Biclique Coves Hossein Hajiabolhassan,2 and Faokhlagha Moazami 3 Depamen of Mahemaical Sciences, Shahid

More information

4 Sequences of measurable functions

4 Sequences of measurable functions 4 Sequences of measurable funcions 1. Le (Ω, A, µ) be a measure space (complee, afer a possible applicaion of he compleion heorem). In his chaper we invesigae relaions beween various (nonequivalen) convergences

More information

Math 10B: Mock Mid II. April 13, 2016

Math 10B: Mock Mid II. April 13, 2016 Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.

More information

Hamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:

Hamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation: M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno

More information

Exponential and Logarithmic Equations and Properties of Logarithms. Properties. Properties. log. Exponential. Logarithmic.

Exponential and Logarithmic Equations and Properties of Logarithms. Properties. Properties. log. Exponential. Logarithmic. Eponenial and Logaihmic Equaions and Popeies of Logaihms Popeies Eponenial a a s = a +s a /a s = a -s (a ) s = a s a b = (ab) Logaihmic log s = log + logs log/s = log - logs log s = s log log a b = loga

More information

The Production of Polarization

The Production of Polarization Physics 36: Waves Lecue 13 3/31/211 The Poducion of Polaizaion Today we will alk abou he poducion of polaized ligh. We aleady inoduced he concep of he polaizaion of ligh, a ansvese EM wave. To biefly eview

More information

The Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales

The Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions

More information

Research Article Weighted Hardy Operators in Complementary Morrey Spaces

Research Article Weighted Hardy Operators in Complementary Morrey Spaces Hindawi Publishing Copoaion Jounal of Funcion Spaces and Applicaions Volume 212, Aicle ID 283285, 19 pages doi:1.1155/212/283285 Reseach Aicle Weighed Hady Opeaos in Complemenay Moey Spaces Dag Lukkassen,

More information

The k-filtering Applied to Wave Electric and Magnetic Field Measurements from Cluster

The k-filtering Applied to Wave Electric and Magnetic Field Measurements from Cluster The -fileing pplied o Wave lecic and Magneic Field Measuemens fom Cluse Jean-Louis PINÇON and ndes TJULIN LPC-CNRS 3 av. de la Recheche Scienifique 4507 Oléans Fance jlpincon@cns-oleans.f OUTLINS The -fileing

More information

Properties of the interface of the symbiotic branching model

Properties of the interface of the symbiotic branching model Popeies of he ineface of he symbioic banching model Jochen Blah 1 and Macel Ogiese 1 (Vesion of 4 Mach 1) Absac The symbioic banching model descibes he evoluion of wo ineacing populaions and if saed wih

More information

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGID ODIES In igid body kinemaics, we use he elaionships govening he displacemen, velociy and acceleaion, bu mus also accoun fo he oaional moion of he body. Descipion of he moion of igid

More information

Kalman Filter: an instance of Bayes Filter. Kalman Filter: an instance of Bayes Filter. Kalman Filter. Linear dynamics with Gaussian noise

Kalman Filter: an instance of Bayes Filter. Kalman Filter: an instance of Bayes Filter. Kalman Filter. Linear dynamics with Gaussian noise COM47 Inoducion o Roboics and Inelligen ysems he alman File alman File: an insance of Bayes File alman File: an insance of Bayes File Linea dynamics wih Gaussian noise alman File Linea dynamics wih Gaussian

More information

Fuzzy Hv-submodules in Γ-Hv-modules Arvind Kumar Sinha 1, Manoj Kumar Dewangan 2 Department of Mathematics NIT Raipur, Chhattisgarh, India

Fuzzy Hv-submodules in Γ-Hv-modules Arvind Kumar Sinha 1, Manoj Kumar Dewangan 2 Department of Mathematics NIT Raipur, Chhattisgarh, India Inenaional Jounal of Engineeing Reseach (IJOER) [Vol-1, Issue-1, Apil.- 2015] Fuzz v-submodules in Γ-v-modules Avind Kuma Sinha 1, Manoj Kuma Dewangan 2 Depamen of Mahemaics NIT Raipu, Chhaisgah, India

More information

Existence of multiple positive periodic solutions for functional differential equations

Existence of multiple positive periodic solutions for functional differential equations J. Mah. Anal. Appl. 325 (27) 1378 1389 www.elsevier.com/locae/jmaa Exisence of muliple posiive periodic soluions for funcional differenial equaions Zhijun Zeng a,b,,libi a, Meng Fan a a School of Mahemaics

More information

Reichenbach and f-generated implications in fuzzy database relations

Reichenbach and f-generated implications in fuzzy database relations INTERNATIONAL JOURNAL O CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 08 Reichenbach and f-geneaed implicaions in fuzzy daabase elaions Nedžad Dukić Dženan Gušić and Nemana Kajmoić Absac Applying a definiion

More information

Notes for Lecture 17-18

Notes for Lecture 17-18 U.C. Berkeley CS278: Compuaional Complexiy Handou N7-8 Professor Luca Trevisan April 3-8, 2008 Noes for Lecure 7-8 In hese wo lecures we prove he firs half of he PCP Theorem, he Amplificaion Lemma, up

More information

The Global Trade and Environment Model: GTEM

The Global Trade and Environment Model: GTEM The Global Tade and Envionmen Model: A pojecion of non-seady sae daa using Ineempoal GTEM Hom Pan, Vivek Tulpulé and Bian S. Fishe Ausalian Bueau of Agiculual and Resouce Economics OBJECTIVES Deive an

More information

A Note on Goldbach Partitions of Large Even Integers

A Note on Goldbach Partitions of Large Even Integers arxiv:47.4688v3 [mah.nt] Jan 25 A Noe on Goldbach Pariions of Large Even Inegers Ljuben Muafchiev American Universiy in Bulgaria, 27 Blagoevgrad, Bulgaria and Insiue of Mahemaics and Informaics of he Bulgarian

More information

Foundations of Statistical Inference. Sufficient statistics. Definition (Sufficiency) Definition (Sufficiency)

Foundations of Statistical Inference. Sufficient statistics. Definition (Sufficiency) Definition (Sufficiency) Foundaions of Saisical Inference Julien Beresycki Lecure 2 - Sufficiency, Facorizaion, Minimal sufficiency Deparmen of Saisics Universiy of Oxford MT 2016 Julien Beresycki (Universiy of Oxford BS2a MT

More information

Existence of positive solution for a third-order three-point BVP with sign-changing Green s function

Existence of positive solution for a third-order three-point BVP with sign-changing Green s function Elecronic Journal of Qualiaive Theory of Differenial Equaions 13, No. 3, 1-11; hp://www.mah.u-szeged.hu/ejqde/ Exisence of posiive soluion for a hird-order hree-poin BVP wih sign-changing Green s funcion

More information

On the Semi-Discrete Davey-Stewartson System with Self-Consistent Sources

On the Semi-Discrete Davey-Stewartson System with Self-Consistent Sources Jounal of Applied Mahemaics and Physics 25 3 478-487 Published Online May 25 in SciRes. hp://www.scip.og/jounal/jamp hp://dx.doi.og/.4236/jamp.25.356 On he Semi-Discee Davey-Sewason Sysem wih Self-Consisen

More information

Convergence of the Neumann series in higher norms

Convergence of the Neumann series in higher norms Convergence of he Neumann series in higher norms Charles L. Epsein Deparmen of Mahemaics, Universiy of Pennsylvania Version 1.0 Augus 1, 003 Absrac Naural condiions on an operaor A are given so ha he Neumann

More information

Multiple Criteria Secretary Problem: A New Approach

Multiple Criteria Secretary Problem: A New Approach J. Stat. Appl. Po. 3, o., 9-38 (04 9 Jounal of Statistics Applications & Pobability An Intenational Jounal http://dx.doi.og/0.785/jsap/0303 Multiple Citeia Secetay Poblem: A ew Appoach Alaka Padhye, and

More information

arxiv: v1 [math.fa] 20 Dec 2018

arxiv: v1 [math.fa] 20 Dec 2018 Diffeeniabiliy of he Evoluion Map and Mackey Coninuiy Maximilian Hanusch axiv:1812.08777v1 mah.fa] 20 Dec 2018 Insiu fü Mahemaik Univesiä Padebon Wabuge Saße 100 33098 Padebon Gemany Decembe 20, 2018 Absac

More information

Supplement for Stochastic Convex Optimization: Faster Local Growth Implies Faster Global Convergence

Supplement for Stochastic Convex Optimization: Faster Local Growth Implies Faster Global Convergence Supplemen for Sochasic Convex Opimizaion: Faser Local Growh Implies Faser Global Convergence Yi Xu Qihang Lin ianbao Yang Proof of heorem heorem Suppose Assumpion holds and F (w) obeys he LGC (6) Given

More information

Probabilistic Models. CS 188: Artificial Intelligence Fall Independence. Example: Independence. Example: Independence? Conditional Independence

Probabilistic Models. CS 188: Artificial Intelligence Fall Independence. Example: Independence. Example: Independence? Conditional Independence C 188: Aificial Inelligence Fall 2007 obabilisic Models A pobabilisic model is a join disibuion ove a se of vaiables Lecue 15: Bayes Nes 10/18/2007 Given a join disibuion, we can eason abou unobseved vaiables

More information

An Automatic Door Sensor Using Image Processing

An Automatic Door Sensor Using Image Processing An Auomaic Doo Senso Using Image Pocessing Depamen o Elecical and Eleconic Engineeing Faculy o Engineeing Tooi Univesiy MENDEL 2004 -Insiue o Auomaion and Compue Science- in BRNO CZECH REPUBLIC 1. Inoducion

More information

Online Completion of Ill-conditioned Low-Rank Matrices

Online Completion of Ill-conditioned Low-Rank Matrices Online Compleion of Ill-condiioned Low-Rank Maices Ryan Kennedy and Camillo J. Taylo Compue and Infomaion Science Univesiy of Pennsylvania Philadelphia, PA, USA keny, cjaylo}@cis.upenn.edu Laua Balzano

More information

Course Notes for EE227C (Spring 2018): Convex Optimization and Approximation

Course Notes for EE227C (Spring 2018): Convex Optimization and Approximation Course Noes for EE7C Spring 018: Convex Opimizaion and Approximaion Insrucor: Moriz Hard Email: hard+ee7c@berkeley.edu Graduae Insrucor: Max Simchowiz Email: msimchow+ee7c@berkeley.edu Ocober 15, 018 3

More information

Stochastic models and their distributions

Stochastic models and their distributions Sochasic models and heir disribuions Couning cusomers Suppose ha n cusomers arrive a a grocery a imes, say T 1,, T n, each of which akes any real number in he inerval (, ) equally likely The values T 1,,

More information

Fig. 1S. The antenna construction: (a) main geometrical parameters, (b) the wire support pillar and (c) the console link between wire and coaxial

Fig. 1S. The antenna construction: (a) main geometrical parameters, (b) the wire support pillar and (c) the console link between wire and coaxial a b c Fig. S. The anenna consucion: (a) ain geoeical paaees, (b) he wie suppo pilla and (c) he console link beween wie and coaial pobe. Fig. S. The anenna coss-secion in he y-z plane. Accoding o [], he

More information

Chapter 2. First Order Scalar Equations

Chapter 2. First Order Scalar Equations Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.

More information

Modelling Dynamic Conditional Correlations in the Volatility of Spot and Forward Oil Price Returns

Modelling Dynamic Conditional Correlations in the Volatility of Spot and Forward Oil Price Returns Modelling Dynamic Condiional Coelaions in he Volailiy of Spo and Fowad Oil Pice Reuns Maeo Manea a, Michael McAlee b and Magheia Gasso c a Depamen of Saisics, Univesiy of Milan-Bicocca and FEEM, Milan,

More information

Math 527 Lecture 6: Hamilton-Jacobi Equation: Explicit Formulas

Math 527 Lecture 6: Hamilton-Jacobi Equation: Explicit Formulas Mah 527 Lecure 6: Hamilon-Jacobi Equaion: Explici Formulas Sep. 23, 2 Mehod of characerisics. We r o appl he mehod of characerisics o he Hamilon-Jacobi equaion: u +Hx, Du = in R n, u = g on R n =. 2 To

More information

Two-dimensional Effects on the CSR Interaction Forces for an Energy-Chirped Bunch. Rui Li, J. Bisognano, R. Legg, and R. Bosch

Two-dimensional Effects on the CSR Interaction Forces for an Energy-Chirped Bunch. Rui Li, J. Bisognano, R. Legg, and R. Bosch Two-dimensional Effecs on he CS Ineacion Foces fo an Enegy-Chiped Bunch ui Li, J. Bisognano,. Legg, and. Bosch Ouline 1. Inoducion 2. Pevious 1D and 2D esuls fo Effecive CS Foce 3. Bunch Disibuion Vaiaion

More information

The Miki-type identity for the Apostol-Bernoulli numbers

The Miki-type identity for the Apostol-Bernoulli numbers Annales Mahemaicae e Informaicae 46 6 pp. 97 4 hp://ami.ef.hu The Mii-ype ideniy for he Aposol-Bernoulli numbers Orli Herscovici, Toufi Mansour Deparmen of Mahemaics, Universiy of Haifa, 3498838 Haifa,

More information

Approximation Algorithms for Unique Games via Orthogonal Separators

Approximation Algorithms for Unique Games via Orthogonal Separators Approximaion Algorihms for Unique Games via Orhogonal Separaors Lecure noes by Konsanin Makarychev. Lecure noes are based on he papers [CMM06a, CMM06b, LM4]. Unique Games In hese lecure noes, we define

More information

This aticle was oiginally published in a jounal published by Elsevie, the attached copy is povided by Elsevie fo the autho s benefit fo the benefit of the autho s institution, fo non-commecial eseach educational

More information

Low-complexity Algorithms for MIMO Multiplexing Systems

Low-complexity Algorithms for MIMO Multiplexing Systems Low-complexiy Algoihms fo MIMO Muliplexing Sysems Ouline Inoducion QRD-M M algoihm Algoihm I: : o educe he numbe of suviving pahs. Algoihm II: : o educe he numbe of candidaes fo each ansmied signal. :

More information