Affine moments of a random vector

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1 Affie momets of a radom vector Erwi Lutwak, Sogju Lv, Deae Yag, ad Gaoyog Zhag Abstract A affie ivariat -th momet measure is defied for a radom vector ad used to rove shar momet-etroy iequalities that are more geeral ad stroger tha stadard momet-etroy iequalities Idex Terms Momet, affie momet, iformatio measure, iformatio theory, etroy, Réyi etroy, Shao etroy, Shao theory, etroy iequality, Gaussia, geeralized Gaussia I INTRODUCTION The momet-etroy iequality states that a cotiuous radom variable with give secod momet has maximal Shao etroy if ad oly if it is a Gaussia see, for examle, 5], Theorem 965) The Fisher iformatio iequality states that a cotiuous radom variable with give Fisher iformatio has miimal Shao etroy if ad oly if it is a Gaussia see 4]) I 32] these classical iequalities were exteded to shar iequalities ivolvig the Réyi etroy, -th momet, ad geeralized Fisher iformatio of a radom variable, where the extremal distributios are o loger Gaussias but are fat-tailed distributios that the authors call geeralized Gaussias There are differet ways to exted the defiitio of a -th momet to radom vectors i R The most obvious oe is to defie it usig the Euclidea orm of the radom vector This, however, assumes that the stadard ier roduct o R gives the right ivariat scalar measure of error or oise i the radom vector It is more aroriate to seek a defiitio of momet that does ot rely o the stadard ier roduct For examle, if the momet of a radom vector is defied usig the stadard Euclidea orm, the the extremals for the corresodig classical momet-etroy iequality are Gaussias with covariace matrix equal to a multile of the idetity It is more desirable to defie a ivariat momet measure, where ay Gaussia or geeralized Gaussia) is a extremal distributio ad ot just the oe whose covariace matrix is a multile of the idetity Oe aroach for defiig a ivariat momet measure, take i 35], leads to the defiitio of the -th momet matrix ad the corresodig shar momet-etroy iequalities Here, we itroduce a differet aroach, where a scalar affie momet measure is defied by averagig -dimesioal -th momets obtaied by rojectig a -dimesioal radom vector alog each ossible directio i R with resect to a give robability measure, ad the otimizig this average over all robability measures with give -th momet A E Lutwak elutwak@olyedu), D Yag dyag@olyedu), ad G Zhag gzhag@olyedu) are with the Deartmet of Mathematics, Polytechic Istitute of New York Uiversity, ad were suorted i art by NSF Grat DMS S Lv lvsogju@26com) is with the College of Mathematics ad Comuter Sciece, Chogqig Normal Uiversity, ad was suorted i art by Chiese NSF Grat 84 exlicit itegral formula for this momet is derived We the establish shar iformatio iequalities givig a shar lower boud of Réyi etroy i terms of the affie momet These ew affie momet-etroy iequalities imly the mometetroy iequalities obtaied i 32], 35] I 26] a aroach similar to the oe take here is used to defie a affie ivariat versio of Fisher iformatio, ad a corresodig shar Fisher iformatio iequality is established It is worth otig that the affie momet measure itroduced here, as well as the affie Fisher iformatio studied i 26], is closely related to aalogous affie ivariat geeralizatios of the surface area of a covex body see, for examle, 9], 27], 3]) I fact, the results reseted here are art of a ogoig effort by the authors to exlore coectios betwee iformatio theory ad geometry Previous results o this iclude a coectio betwee the etroy ower iequality i iformatio theory ad the Bru- Mikowski iequality i covex geometry first demostrated by Lieb 23] ad also discussed by Costa ad Cover 2] This was develoed further by Cover, Dembo, ad Thomas 4] also, see 5]) Also, see ] for geeralizatios of various etroy ad Fisher iformatio iequalities related to mass trasortatio, ad 4] 8], 37], 38] for a ew coectio betwee affie iformatio iequalities ad log-cocavity I view of this, the authors of 9] bega to systematically exlore coectios betwee iformatio theory ad covex geometry The goals are to both establish iformatio-theoretic iequalities that are the couterarts of geometric iequalities ad ivestigate ossible alicatios of ideas i iformatio theory to covex geometry This has led to aers i both iformatio theory see 9], 26], 3], 32], 35]) ad covex geometry see 28], 29]) I the ext sectio we follow the suggestio of a referee ad rovide a brief survey of this ogoig effort II INFORMATION THEORETIC AND GEOMETRIC INEQUALITIES The usual way of associatig a radom vector X i R with a comact covex set K i R is to defie X as the uiform radom vector i K I 9], a differet costructio of radom vectors associated to a star-shaed set was itroduced It was also show that the iformatio theoretic ivariats of the distributios costructed, called cotoured distributios are equivalet to geometric ivariats of the covex set K This rovides a direct lik betwee shar iformatio theoretic iequalities satisfied by cotoured distributios ad shar geometric iequalities satisfied by covex sets Let K be a bouded star-shaed set i R about the origi Its gauge fuctio, g K : R, ), is defied by g K x) if{t > : x tk}

2 2 The gauge fuctio is homogeeous of degree A radom vector X has a cotoured distributio if its robability desity fuctio is give by f X x) ψg K x x )), where K is a bouded star-shaed set with resect to the oit x, ad ψ is a -dimesioal fuctio that we call the radial rofile If ψ is mootoe, the the level sets of f X are dilates of K with resect to x A straightforward calculatio shows that the etroy hx) of X is give by hx) c ψ, )V K), where c ψ, ) is a costat deedig oly o the radial rofile ψ ad the dimesio ad V K) is the -dimesioal volume ie, -dimesioal Lebesgue measure) of K Aother calculatio shows that the Fisher iformatio JX) of X is give by JX) c ψ, )S 2 K), where c ψ, ) is aother costat deedig o ψ ad ad dsx) S 2 K) x νx) K is called the L 2 surface area of K, where the outer uit ormal vector νx) exists for almost every x K with resect to the )-dimesioal Hausdorff measure ds The L 2 surface area S 2 K) is a geometric ivariat i the L Bru-Mikowski theory i covex geometry The usual surface area is viewed as the L surface area i the L Bru-Mikowski theory see 25]) The formula of the Fisher iformatio ad the L 2 surface area imlies that the classical iformatio theory is closely associated to the L 2 Bru-Mikowski theory Let K be a covex body comact covex set with oemty iterior) ad X be a radom vector i R Deote by B the -dimesioal uit ball ad by Z the -dimesioal stadard Gaussia radom vector The followig variatioal formula, lim t V K 2 tb) V K) t 2 S 2K), where 2 is the L 2 Mikowski additio see 25]), ad the de Bruij s idetity, hx tz) hx) lim t t 2 JX), further illustrate the close coectio of iformatio theory ad geometry These coectios lead to the defiitio of a ew ellisoid i the L 2 Bru-Mikowski theory which is the couterart of the Fisher iformatio matrix see 28]), ad thus a Cramer- Raó iequality for star-shaed sets was roved i 29] The results were, i tur, alied to iformatio theory to show ew iformatio-theoretic iequalities see 9]) It is the atural to ivestigate the couterart of the L Bru-Mikowski theory i iformatio theory The ew ellisoid discovered was show i 33] to be the L 2 case of a family of ellisoids, called L Joh ellisoids, while the classical Joh ellisoid the ellisoid of maximal volume iside a covex body) is the L case This ew result i covex geometry suggests that it is atural to defie a cocet of L Fisher iformatio matrix as a corresodig object of the L Joh ellisoid The usual Fisher iformatio matrix is the L 2 case This was doe i the recet aer 26] A extesio of the covariace matrix to L covariace matrix also called -momet matrix) was give earlier i the aer 35], which corresods to aother family of ellisoids i geometry that cotais the classical Legedre ellisoid ad is cocetually dual to the family of L Joh ellisoids Affie isoerimetric iequalities are cetral i the L Bru- Mikowski theory The authors have bee exlorig their corresodig affie iformatio-theoretic iequalities See the survey aers 24] ad 42] o affie isoerimetric iequalities i covex geometry For, λ >, ad ideedet radom vectors X, Y i R, the followig momet-etroy iequality was roved i 3], E X Y ) c N λ X) N λ Y ), ) where N λ deotes the λ-réyi etroy ower, ad c is the best costat that is attaied whe X, Y are certai geeralized Gaussia radom vectors The affie isoerimetric iequality behid this momet-etroy iequality is a L extesio of the well-kow Blaschke-Sataló iequality i geometry see 36]) The Shao etroy hx) ad the λ-réyi etroy ower N λ X) of a cotiuous radom vector X i R are affie ivariats, that is, they are ivariat uder liear trasformatios of radom vectors To establish affie iformatiotheoretic iequalities as couterarts of affie isoerimetric iequalities, affie otios of Fisher iformatio ad momets as corresodig otios of affie surface areas are eeded I 26], the authors itroduced the otio of affie, λ)- Fisher iformatio Ψ,λ X) of a radom vector X i R, which is a aalogue of the L itegral affie surface area of a covex body It was show that the followig affie Fisher iformatio ad etroy iequality holds: Ψ,λ X)N λ X) λ )) c, 2) where <, λ >, ad c is the best costat that is attaied whe X is a geeralized Gaussia radom vector This iequality is roved by usig a L affie Sobolev iequality established i 3] see also 4]) The L affie Sobolev iequality is stroger tha the classical L Sobolev iequality ad comes from the L Petty rojectio iequality established i 27] which is a imortat affie isoerimetric iequality i the L Bru-Mikowski theory of covex geometry It is oe of the uroses this aer to itroduce the otio of affie -th momet M X) of a radom vector X i R, ad to establish a affie momet-etroy iequality We shall rove the followig theorem Theorem : If <, λ >, ad X is a radom vector i R with fiite λ-réyi etroy ad -th momet, the M X) c N λ X), 3)

3 3 where c is the best costat that is attaied oly whe X is a geeralized Gaussia radom vector The iequality 3) is roved by usig ) Thus, the affie isoerimetric iequality behid the affie momet-etroy iequality 3) is the L extesio of the Blaschke-Sataló iequality III PRELIMINARIES Let X be a radom vector i R with robability desity fuctio f X If A is a ivertible matrix, the f AX y) A f X A y), 4) where A is the absolute value of the determiat of A A The -th momet of a radom vector For, ), the -th momet of X is defied to be E X ) x f X x)dx, R where deotes the stadard Euclidea orm ad dx the stadard Lebesgue measure i R B Etroy ower The Shao etroy of X is give by hx) f X x) log f X x) dx R For λ >, the λ-réyi etroy ower of X is defied to be f X x) λ λ) dx if λ, N λ X) R e hx) if λ, ad the λ-réyi etroy is By 4), for ay ivertible matrix A A Defiitio h λ X) log N λ X) N λ AX) A Nλ X), 5) IV GENERALIZED GAUSSIANS If > ad β < /, the corresodig geeralized stadard Gaussia radom vector Z R has desity fuctio a f Z x),β β x β if β, a, e x / if β, where t max{t, } for t R, β Γ 2 ) π 2 B, β ) if β <, Γ 2 a,β ) π 2 Γ ) if β, β ) Γ 2 ) π 2 B, β ) if β >, Γ ) deotes the gamma fuctio, ad B, ) deotes the beta fuctio Ay radom vector that ca be writte as W AZ, for a ivertible matrix A, is called a geeralized Gaussia If 2 ad β, the Z is the stadard Gaussia radom vector with mea ad variace matrix equal to the idetity The fuctios i the geeralized stadard Gaussia distributios, which are also called Bareblatt fuctios, have bee foud to arise aturally as extremals for Sobolev tye iequalities ad shar iequalities relatig momet, Réyi etroy, ad geeralized Fisher iformatio see, for examle, ], 3], ], ], 3], 6], 7], 2], 26], 3], 32], 34], 35]) The usual Gaussias ad Studet distributios are geeralized Gaussias The whole class of geeralized Gaussia distributios i this form were first studied i 3] as extremal distributios of momet-etroy iequalities May authors have studied secial cases of geeralized Gaussias see, for examle, ] 3], 3], 6], 8], 2], 22], 39]) B Iformatio measures of geeralized Gaussias Give < < ad λ > / ), set the arameters ad β of the stadard geeralized Gaussia Z so that, β λ, 6) ad β whe λ We assume throughout this aer that, λ, ad the arameters ad β of the stadard geeralized Gaussia satisfy these equatios The -th momet of Z is give by E Z ), 7) ad its Réyi etroy ower by a,β β λ) if λ, N λ Z) a, e if λ See 35] for similar formulas The followig are sketches of calculatio For β < < λ < ), by olar coordiates ad chage of variable β r ) t, we have E Z ) a,β x β β x dx R a,β ω β β r r dr a,β ω β ) a,β π 2 Γ 2 ) β, ) t β t) dt B β, ) 8)

4 4 ad N λ Z) a λ,β a,β R ω β x ) λ β ) a ω,β a ω,β a,β β β ) β r β β ) λ), where a,β a λ ),β dx r dr λ) t β t) dt B β, ) ] λ) ] λ) ] λ) For < β < λ > ), by olar coordiates ad chage of variable β r t, we have ad E Z ) a,β x β β x dx R a,β ω β β r r dr ) a,β ω β a,βπ 2 ) Γ 2 ) β, N λ Z) a λ,β a,β R ω a ω,β a ω,β β a,β B t t) β dt β, ) β x ) λ β ) β β ) β r β dx r dr λ) ] λ) t t β dt ] λ) ) B β, λ) ) ] λ) For β λ ), by olar coordiates ad chage of variable r t, we have ad thus, E Z ) a, R x e x dx a, ω e r r dr a, ω e t t dt a, π 2 Γ 2 ) Γ ), hz) f Z x) log f Z x) dx R f Z x) log a, e x ) dx R ) f Z x) log a, f Z x) x dx R log a, E Z ) log a,, N Z) a, e V NOTIONS OF AN AFFINE MOMENT If G is the stadard Gaussia radom vector i R with mea ad variace matrix equal to the idetity, the the classical momet-etroy iequality see, for examle, 5]) states that for a radom vector X i R, E X 2 ) N X) 2 E G 2 ) N G) 2, 9) with equality if ad oly if X tg, for some t R\{} I 3], 32], 35], this was exteded to the followig iequality for the λ-réyi etroy ad -th momet Theorem 2: If, ), λ > / ), ad X R is a radom vector such that N λ X), E X ) <, the E X ) N λ X) E Z ) N λ Z), with equality if ad oly if there exists t > such that X tz, where Z is the stadard geeralized Gaussia with arameters ad β satisfyig 6) I 35], a affie -th momet of a radom vector X is defied by m X) if{e AX ) : A SL)}, ad the followig affie momet-etroy iequality was show Theorem 3: If, ), λ > / ), ad X is a radom vector i R satisfyig N λ X), E X ) <, the m X) N λ X) E Z ) N λ Z),

5 5 with equality if ad oly if X T Z for the stadard geeralized Gaussia Z ad some T GL) Theorem 3 is formally stroger tha Theorem 2, but the two are equivalet Theorem 3 is a affie formulatio of Theorem 2 The affie momet m X) has o exlicit formula i terms of the desity of X This makes the calculatio difficult Is there a otio of affie momets that has a exlicit formula ad also gives a essetially stroger momet-etroy iequality tha those i Theorems 2 ad 3? The defiitio of m X) ca be formulated differetly as follows: Let F be the class of orms o R that is give by F { A : A SL)}, where A is defied by x A Ax, x R The m X) if A F E X A ) We shall use a larger class of orms tha F to defie the affie momet M X) The orms are geerated by the -cosie trasforms of desity fuctios We shall give a exlicit formula for the affie momet M X) ad establish the affie momet-etroy iequality i Theorem which is essetially stroger tha Theorems 2 ad 3 A similar aroach was used i 26] to defie a otio of affie Fisher iformatio VI THE AFFINE -TH MOMENT OF A RANDOM VECTOR A Defiitio A radom vector X i R with desity g is said to have fiite -momet for >, if R x gx) dx < The -cosie trasform of a radom vector Y with fiite - th momet ad desity g defies the followig orm o R, x Y, x y gy) dy, x R ) R If > ad X is a radom vector, the we defie the affie -th momet of X to be M X) if E X Y, ), ) N λ Y )c where each radom vector Y is ideedet of X ad has fiite -th momet ad λ-réyi etroy ower equal to a costat c which will be chose aroriately later The defiitio above aears to deed o the arameter λ, but by Theorem 5 ad 4), which are stated i Sectios VI-B ad VI-D, the value of M X) is i fact ideedet of λ whe the costat c is roerly chose We also show below that the ifimum i the defiitio above is achieved, ad the affie -th momet M X) is ivariat uder volumereservig liear trasformatios of the radom vector X B A itegral reresetatio for affie momets The followig is a secial case of the dual Mikowski iequality for radom vectors established i 3], Lemma 4 Lemma 4: If >, λ >, ad X ad Y are ideedet radom vectors i R with fiite -th momet, the E y X )gy) dy R ) N λ Y ) a E u X ) du, S where g is the desity of Y, S is the uit shere i R, du deotes the Lebesgue measure o S, a B, λ ) if λ <, e ) a Γ ) if λ, ) a B, λ if λ >, λ ad a λ ) λ λ λ λ ) Equality is attaied, if the desity of Y is give by b a y λ if λ <, gy) be a y if λ, b a y λ if λ >, 2) for a, b >, where the orm is give by y E y X ) for each y R The followig is the itegral reresetatio for the affie -th momet Theorem 5: If > ad X is a radom vector with fiite -th momet ad desity f, the M X) c E u X ) du, S where c a c Proof: If Y is a radom vector such that N λ Y ) c, the by the Fubii theorem ad Lemma 4, E X Y, ) x y gy) dy fx) dx R R x y fx) dx gy) dy R R E y X )gy) dy R c E u X ) du S Moreover, by the equality coditio of Lemma 4, equality is attaied if Y is a radom vector whose desity is give by

6 6 2), ormalized so that N λ Y ) c Therefore, M X) if E X Y, ) N λ Y )c c E u X ) du S C Affie ivariace of the affie -th momet Theorem 6: If X is a radom vector i R with fiite -th momet, the M AX) M X), for each A SL) Proof: If A GL), the it follows by ) that E AX Y, ) Ax y gy) dy fx) dx R R x A t y gy) dy fx) dx R R E X A t Y ) Thus, if A SL), the by 5), M AX) if N λ Y )c E AX Y, ) if N λ Y )c E X A t Y, ) if N λ A t Y )c E X A t Y, ) M X) D The affie -th momet of a sherically cotoured radom vector Deote the volume of the uit ball i R by ad observe that Let ω π 2 Γ 2 ), S du ω 3) ω, u e du 2π S c ω ω 2π 2 2 Γ 2 ) Γ 2 ), where e is a fixed uit vector, for examle, e,,, ) We choose the costat c so that the costat c a c is give by ) ω, Γ 2 ) Γ 2 )Γ 2 ) ) 4) π 2 Γ 2 ) A radom vector X is called sherically cotoured if its desity f ca be writte as fx) F x ), x R, where F :, ), ) Lemma 7: If X is a sherically cotoured radom vector with fiite -th orm ad desity give by fx) F x ), x R, the M X) ω F ρ)ρ dρ 5) Proof: For each u S, E u X ) u x F x ) dx R u v dv F ρ)ρ dρ S ω, F ρ)ρ dρ By Theorem 5, M X) c E u X ) du S E Affie versus Euclidea ω F ρ)ρ dρ Lemma 8: If > ad X is a radom vector i R, the M X) E X ) 6) Equality holds if ad oly if the fuctio v E v X ) is costat for v S I articular, equality holds if X is sherically cotoured Proof: If f is the desity of X ad u S, the E u X ) du S u x fx) dx du S R ) u x R S x du x fx) dx ω, E X ) Therefore, by Theorem 5, 3), ad Hölder s iequality, ω c ) M X) E u X ) du ω S E u X ) du ω S ω, E X ) ω The equality coditio follows by the equality coditio of Hölder s iequality By the theorem above ad 7), we get the followig Corollary 9: If Z is the stadard geeralized Gaussia, the M Z)

7 7 VII AFFINE -TH MOMENT-ENTROPY INEQUALITIES A Proof of Theorem The followig biliear momet-etroy iequality is established i 3] Theorem : Let, λ > / ) There exists a costat c > such that if X ad Y are ideedet radom vectors i R with fiite -th momets, the E X Y ) c N λ X) N λ Y ), with equality holdig if ad oly if X ad Y are certai geeralized Gaussias We use this theorem to establish the followig affie momet-etroy iequality, which is Theorem Theorem : If, ), λ / ), ), ad X is a radom vector i R with fiite λ-réyi etroy ad -th momet, the M X) N λ X) M Z) N λ Z), 7) with equality if ad oly if X is a geeralized Gaussia Proof: If Y is ideedet of X ad has fiite -th momet ad Réyi etroy ower N λ Y ) c, the by ) ad Theorem, E X Y, ) E X Y ) c c N λx) The desired iequality 7) ow follows by the defiitio ) of M X) The equality coditio follows by the equality coditios of Lemma 4 ad Theorem or 7) ad Corollary 9) B Affie imlies Euclidea Proositio 2: The affie momet-etroy iequality i Theorem is stroger tha the Euclidea momet-etroy iequality i Theorem 2 Proof: Observe that equality holds i Lemma 8 for a stadard geeralized Gaussia radom vector Z because it is sherically cotoured By Lemma 8, Theorem, ad the equality coditio of Lemma 8, E X ) N λ X) M X) N λ X) M Z) N λ Z) E Z ) N λ Z) Therefore, the Euclidea momet-etroy iequality i Theorem 2 is weaker tha the affie momet-etroy iequality i Theorem ACKNOWLEDGMENT The authors would like to thak the referees for their helful commets REFERENCES ] S Amari, Differetial-geometrical methods i statistics, ser Lecture Notes i Statistics New York: Sriger-Verlag, 985, vol 28 2] A Adai, O the geometry of geeralized Gaussia distributios, J Multivariate Aal, vol, o 4, , 29 3] E Arika, A iequality o guessig ad its alicatio to sequetial decodig, IEEE Tras Iform Theory, vol 42, 99 5, 996 4] A D Barbour, O Johso, I Kotoyiais, ad M Madima, Comoud Poisso aroximatio via iformatio fuctioals, Electro J Probab, vol 5, , 2 5] S Bobkov ad M Madima, Cocetratio of 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8 8 28] E Lutwak, D Yag, ad G Zhag, A ew ellisoid associated with covex bodies, Duke Math J vol 4, , 2 29] E Lutwak, D Yag, ad G Zhag, The Cramer-Rao iequality for star bodies, Duke Math J vol 2, 59 8, 22 3] E Lutwak, D Yag, ad G Zhag, Shar affie L Sobolev iequalities, J Differetial Geom, vol 62, 7 38, 22 3] E Lutwak, D Yag, ad G Zhag, Momet etroy iequalities, Aals of Probability, vol 32, , 24 32] E Lutwak, D Yag, ad G Zhag, Cramer-Raó ad momet-etroy iequalities for Reyi etroy ad geeralized Fisher iformatio, IEEE Tras Iform Theory, vol 5, , 25 33] E Lutwak, D Yag, ad G Zhag, L Joh ellisoids, Proc Lodo Math Soc vol 9, , 25 34] E Lutwak, D Yag, ad G Zhag, Otimal Sobolev orms ad the L Mikowski roblem, It Math Res Not, 26, Art ID 62987, 2 35] E Lutwak, D Yag, ad G Zhag, Momet-Etroy Iequalities for a Radom Vector, IEEE Tras Iform Theory, vol 53, 63 67, 27 36] E Lutwak ad G Zhag, Blaschke-Sataló iequalities, J Differetial Geom, vol 47, 6, ] M Madima ad A Barro, Geeralized etroy ower iequalities ad mootoicity roerties of iformatio, IEEE Tras Iform Theory, vol 53, o 7, , 27 38] M Madima ad P Tetali, Iformatio iequalities for joit distributios, with iterretatios ad alicatios, IEEE Tras Iform Theory, vol 56, o 6, , 2 39] S Nadarajah, TheKotz-tye distributio with alicatios, Statistics, vol 37, o 4, , 23 4] A J Stam, Some iequalities satisfied by the quatities of iformatio of Fisher ad Shao, Iformatio ad Cotrol, vol 2, 2, 959 4] G Zhag, The affie Sobolev iequality, J Differetial Geom, vol 53, 83 22, ] G Zhag, New affie isoerimetric iequalities, i: ICCM 27, vol II, Erwi Lutwak is Professor of Mathematics at Polytechic Istitute of NYU He received his BS ad MS i Mathematics from this istitutio whe it was amed Polytechic Istitute of Brookly ad his PhD i Mathematics whe it was amed Polytechic Istitute of New York Sogju Lv is Associate Professor of Mathematics at Chogqig Normal Uiversity He received his BS ad MS i Mathematics from Hua Normal Uiversity, ad his PhD i Mathematics from Shaghai Uiversity Deae Yag is Professor of Mathematics at Polytechic Istitute of New York Uiversity He received his BA i Mathematics ad Physics from Uiversity of Pesylvaia ad PhD i Mathematics from Harvard Uiversity Gaoyog Zhag is Professor of Mathematics at Polytechic Istitute of New York Uiversity He received his BS i Mathematics from Wuha Uiversity of Sciece ad Techology, MS i Mathematics from Wuha Uiversity, ad PhD i Mathematics from Temle Uiversity