Research Article A Statistical Cohomogeneity One Metric on...

Research ArticleA Statistical Cohomogeneity One Metric onthe Upper Plane with Constant Negative Curvature

Limei Cao1 Didong Li2 Erchuan Zhang2 Zhenning Zhang3 and Huafei Sun2

1School of Mathematics and Physics University of Science and Technology Beijing Beijing 100083 China2School of Mathematics and Statistics Beijing Institute of Technology Beijing 100081 China3College of Applied Sciences Beijing University of Technology Beijing 100124 China

Correspondence should be addressed to Huafei Sun huafeisunbiteducn

Received 24 September 2014 Revised 15 December 2014 Accepted 15 December 2014 Published 30 December 2014

Academic Editor Carlo Cattani

Copyright copy 2014 Limei Cao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

we analyze the geometrical structures of statistical manifold S consisting of all the wrapped Cauchy distributionsWe prove that S isa simply connected manifold with constant negative curvature119870 = minus2 However it is not isometric to the hyperbolic space becauseS is noncomplete In fact S is approved to be a cohomogeneity one manifold Finally we use several tricks to get the geodesics andexplore the divergence performance of them by investigating the Jacobi vector field

1 Introduction

So far more and more geometric approaches have beenapplied to various fields such as in statistics physics andcontrol [1ndash6] by analyzing their complexity characterizationsor giving their solutions

Entropic dynamics is a theoretical framework con-structed on statistical manifolds to explore the possibilityof laws of physics [7] Until now researchers have triedto study a variety of entropic dynamic models from theviewpoint of information geometry [4ndash6 8] In [8] twoentropic dynamical models are considered in which the sta-bility of the Jacobi vector fields on Riemannian manifolds isinvestigated In [4] authors explored the geometric structuresand instability of entropic dynamical models Informationgeometric characterization of fractional Brownian motionswas considered in [5] Hamiltonian dynamics are studiedfrom the viewpoint of geometry in [6]

In probability theory and directional statistics [9] awrapped probability distribution is a continuous probabilitydistribution whose state can be considered as lying on a unit119899-sphere 119878

119899 Examples include the Bingham distribution [10]the Kent distribution (or the Fisher-Bingham distribution)[11] and the vonMises-Fisher distribution [9 12] Particularlyfor one-dimensional cases a wrapped distribution consists

of points on the unit circle For instance a wrapped Cauchydistribution [9 13 14]

119901 (119909 120583 120574) =1

2120587

sinh 120574

cosh 120574 minus cos (119909 minus 120583)(1)

is a wrapped probability distribution that results from theldquowrappingrdquo of the Cauchy distribution around the unit circlewhere apparently 119909 isin [minus120587 120587) which is parameterized fromthe unit circle 120574 isin 119877 and 120583 gt 0 The functions sinhand cosh are respectively the hyperbolic sine function andthe hyperbolic cosine function As the random variable 119909 isin

[minus120587 120587] the wrapped probability distribution can be con-sidered as an angular probability to measure the directionalprobabilityTherefore it has beenwidely applied in the field ofrandomwalk for studying animalmovements [15 16] such asidentifying fishing trip behavior and estimating fishing effort[17] and ringing recoveries of pied flycatchers [18]

Assume that all information relevant to dynamical evolu-tion of the model can be obtained from the associated proba-bility distribution which is the wrapped Cauchy distributionin our consideration We will call this model the WCED(wrapped Cauchy entropic dynamical) model for simplicity

The remainder of the paper is organized as follows InSection 2 the geometric structures including the Fishermet-ric and the sectional curvature of the WCED are calculated

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2014 Article ID 832683 6 pageshttpdxdoiorg1011552014832683

2 Advances in Mathematical Physics

In Section 3 this manifold is proved to be simply connectedbut not complete hence not isometric to the hyperbolic spaceHowever it is proved to be a cohomogeneity one manifoldby Theorem 14 In Section 4 several tricks are used basedon some special properties to get geodesics and the Jacobivector fields The geodesics are the sets of fixed points ofisometric transformations which include symmetries to linesparalleled to 119910-axis Meanwhile the Jacobi vector fields arecalculated Then the instability of the geodesic spreads isanalyzed via the behaviors of the the Jacobi vector field

Remark 1 It is well known that the upper plane R2+admits a

hyperbolic metric 119892119867with constant negative curvature That

is

119892119867

=

[[[

[

1

11991020

01

1199102

]]]

]

(2)

However in this paper we give another metric on R2+

with constant negative curvature but not isometric to thehyperbolic space

2 Geometric Structures of the WCED Model

From the theory of information geometry one can definean 119898-dimensional statistical manifold 119872 over unit sphereswhich is a set of probability densities namely

119872 = 119901 (119909 120579) (3)

The parameter 120579 = (1205791

1205792

120579119898

) plays the role of coordi-nate systems The Fisher information matrix 119892 acting as aRiemannian metric is then given by the expectation

(119892119894119895) = (119864 [(120597

119894119897) (120597119895119897)]) (4)

where 119897(119909 120579) = ln119901(119909 120579) and 120597119894

= 120597120597120579119894 On a statistical

manifold the Riemannian connection coefficient Γ119894119895119896

is (seeeg [1 2 19ndash21])

Γ119894119895119896

= 119864 [(120597119894120597119895119897) (120597119896119897)] +

1 minus 120572

2119864 [(120597119894119897) (120597119895119897) (120597119896119897)] (5)

which can be expressed as a classical formula Γ119894119895119896

=

(12)(120597119894119892119895119896

+ 120597119895119892119896119894

minus 120597119896119892119894119895) equivalently

If (1205791

1205792

120579119898

) is a local coordinate of 119872 then forany vector field 119883 on 119872 one has 119883 = 119883

119894

120597119894 where 119883

119894

are smooth functions with respect to (1205791

1205792

120579119898

) TheEinstein summation convention is used here and all throughthe paper With the Riemannian connection the curvaturetensor is defined by [19 20]

119877 (119883 119884) 119885 = nabla119883

nabla119884119885 minus nabla

119884nabla119883

119885 minus nabla[119883119884]

119885 (6)

and the Riemannian curvature tensor is

119877 (119883 119884 119885 119882) = 119892 (119877 (119885 119882) 119883 119884) (7)

where 119883 119884 119885 119882 are smooth vector fields on 119872 and [sdot sdot]

represents the Lie bracket namely [119883 119884] = 119883 ∘ 119884 minus 119884 ∘ 119883The local representation of the curvature tensor satisfies

119877119898119895119896119897

= (120597119898

Γ119905

119895119896minus 120597119895Γ119905

119898119896) 119892119905119897

+ (Γ119898119905119897

Γ119905

119895119896minus Γ119895119905119897

Γ119905

119898119896) (8)

where Γ119896

119898119895= Γ119898119895119905

119892119905119896 and (119892

119905119896

) is the inverse of the metricmatrix (119892

119905119896)

The Ricci curvature and the sectional curvature aredefined by

119877119898119896

= 119877119898119895119897119896

119892119895119897

(9)

119870119894119895

= minus

119877119894119895119894119895

119892119894119894119892119895119895

minus 1198922

119894119895

(10)

respectivelyA curve 120585(119905) on 119872 is said to be a geodesic if its tangent

vector 120585(119905) is displaced parallel along the curve 120585(119905) that is

nabla 120585(119905)

120585 (119905) = 0 (11)

and it has the following local form

d2120579119896

d1199052+ Γ119896

119895119898

d120579119895

d119905

d120579119898

d119905= 0 (12)

Suppose that the solution of (12) is 120579(119905 120573) where 120573 is a vectorwhose components are the integration constants

Considering the wrapped Cauchy entropic dynamicalmodel the corresponding statistical manifold is

119878 = 119901 (119909 120579) | 120579 = (1205791

1205792

) isin R2

+ (13)

where 119901(119909 120579) is given by (1) with parameters 1205791

= 120583 and 1205792

=

120574 The Fisher information matrix can be easily calculated as

119892 = diag (1

2sinh21205792

1

2sinh21205792) (14)

Proposition 2 Thenonzero components of the Ricci curvatureare

11987711

= 11987722

= minus1

sinh21205792 (15)

Proof A direct calculation of (5) gives the nonvanishingRiemannian connection coefficients

Γ112

= minusΓ121

= minusΓ211

= minusΓ222

=cosh 120579

2

2sinh31205792 (16)

From the relation Γ119896

119898119895= Γ119898119895119905

119892119905119896 one can get another type of

connection coefficients

Γ2

11= minusΓ1

12= minusΓ1

21= minusΓ2

22=cosh 120579

2

sinh 1205792 (17)

From (8) the following nonzero component of the curvaturetensor is obtained

1198771212

=1

2sinh41205792 (18)

Therefore the proof can be finished by a simple substitutionof (18) into (9)

Advances in Mathematical Physics 3

The following theorem is obtained by substituting (18)and (14) into (10)

Theorem 3 119878 is a manifold with constant negative curvature119870 = minus2

3 Topological and Geometric Properties of 119878

Lemma 4 119878 is diffeomorphic to R2+and hence simply con-

nected

Proof The topology as well as the smooth structure of 119878 isinduced by those onR2

+ hence 119878 is diffeomorphic toR2

+ The

fact that R2+is simply connected implies that 119878 is also simply

connected

If 119878 were also complete it would be a space formwith constant negative curvature which was isometric tothe hyperbolic space 119867

2

(minus2) However 119878 is not completeWe begin with a necessary and sufficient condition of thecompleteness

Definition 5 (see [19]) Let (119872 119892) be a RiemannianmanifoldA smooth curve 120574 [0 +infin) rarr 119872 is called a divergent curveif for any compact subset 119870 sub 119872 there exists 119905

0gt 0 st

120574(1199050) notin 119870 If 120574 is a divergent curve its length is defined by

119871 (120574) = lim119904rarr+infin

int

119904

0

100381610038161003816100381610038161205741015840

(119905)10038161003816100381610038161003816d119905 (19)

Lemma 6 (see [22]) (119872 119892 119889) is a connected Riemannianmanifold then 119872 is complete if and only if every divergentcurve has infinite length

Proof (1) Suppose that 119872 is complete and 120574 is a divergentcurve Consider

119881119899

= 119901 isin 119872 | 119889 (119901 120574 (0)) le 119899 forall119899 isin N (20)

where for each 119899 119881119899is bounded and closed hence compact

The divergence of 120574 implies that for all 119899 isin Nlowast exist119905119899

ge 0 st120574(119905119899) notin 119881119899 As a result

int

119905119899

0

100381610038161003816100381610038161205741015840

(119905)10038161003816100381610038161003816d119905 ge 119889 (120574 (0) 120574 (119905

119899)) ge 119899 (21)

hence 119871(120574) = +infin(2) Suppose every divergent curve has infinite length

Assume that 119872 is noncomplete which implies that exist119901 isin 119872existV0

isin 119879119901119872 and 120574(119905) = exp

119901(119905V0) is only defined on 119905 isin [0 1)

119871(120574) lt |V0| lt +infin means 120574 is not divergent that is there

exists a compact subset 119870 sub 119872 st 120574(119905) isin 119870 for all 119905 Choosesequence 119905

119899+infin

119899=1 st 119905

119899rarr 1 and 119905

119899lt 119905119899+1

for all 119899 isin N It isobvious that 119889(120574(119905

119899) 120574(119905119898

)) rarr 0 when 119899 119898 are sufficientlylarge Hence 120574(119905

119899) is a Cauchy sequence The completeness

of 119872 implies that there exists 119902 isin 119870 st 120574(119905119899) rarr 119902

Then 120574(1) = 119902 is a extension of 120574 from [0 1) to [0 1]

which contradicts to the assumption which implies that119872 isin fact complete

Theorem 7 119878 is noncomplete

Proof Suppose that 120574(119905) = (0 119905) isin R2+ 119905 isin [1 +infin) is a curve

along the 119910-axis It is obvious that 120574 is divergent Howeverthe length of 120574 satisfies

119871 (120574) = int

+infin

1

100381610038161003816100381610038161205741015840

(119905)10038161003816100381610038161003816d119905

= int

+infin

1

1

radic2 sinh (119905)

d119905

=1

radic2

ln(tanh(119905

2))

1003816100381610038161003816100381610038161003816

+infin

1

=ln (119890 + 1) minus ln (119890 minus 1)

radic2

lt +infin

(22)

This implies 119878 is noncomplete by Lemma 6

Although 119878 is neither isometric to the hyperbolic spacenor a symmetric space it is ldquopartly symmetricrdquo which ispresented in Lemma 10

Definition 8 Let 119868(119878) be the group of isometric transforma-tions on 119878 A family of transformations on 119878 are defined as

120593120572

119878 997888rarr 119878 (1205791

1205792

) 997891997888rarr (2120572 minus 1205791

1205792

) forall120572 isin R

(23)

Remark 9 120593120572is just the reflection about the line 120579

1

= 120572which is a line parallel to the 119910-axis

Lemma 10 120593120572

isin 119868(119878) for all 120572 isin R

Proof For any fixed 120572 isin R for all 119901 = (1205791

1205792

) isin 119878 119902 =

120593120572(119901) = (2120572 minus 120579

1

1205792

)Consider

1205741

(119905) = (1205791

+ 119905 1205792

) 1205742

(119905) = (1205791

1205792

+ 119905)

(120593120572)lowast119901

(120597

1205971205791) =

dd119905

10038161003816100381610038161003816100381610038161003816119905=0

(120593120572

∘ 1205741

(119905))

=dd119905

10038161003816100381610038161003816100381610038161003816119905=0

(2120572 minus 1205791

minus 119905 1205792

) = minus120597

1205971205791

10038161003816100381610038161003816100381610038161003816119902

(120593120572)lowast119901

(120597

1205971205792) =

dd119905

10038161003816100381610038161003816100381610038161003816119905=0

(120593120572

∘ 1205742

(119905))

=dd119905

10038161003816100381610038161003816100381610038161003816119905=0

(2120572 minus 1205791

1205792

+ 119905) =120597

1205971205792

10038161003816100381610038161003816100381610038161003816119902

119892119902

((120593120572)lowast119901

(120597

1205971205791) (120593120572)lowast119901

(120597

1205971205791))

= 119892119902

(minus120597

1205971205791 minus

120597

1205971205791) =

1

2sinh21205792= 119892119901

(120597

1205971205791

120597

1205971205791)


119892119902

((120593120572)lowast119901

(120597

1205971205791) (120593120572)lowast119901

(120597

1205971205792))

= 119892119902

(minus120597

1205971205791

120597

1205971205792) = 0 = 119892

119901(

120597

1205971205791

120597

1205971205792)

119892119902

((120593120572)lowast119901

(120597

1205971205792) (120593120572)lowast119901

(120597

1205971205792))

= 119892119902

(120597

1205971205792

120597

1205971205792) =

1

2sinh21205792= 119892119901

(120597

1205971205792

120597

1205971205792)

(24)

Hence 120593120572is an isometry

As a result 119866 = 120593120572

| 120572 isin R is a subgroup of 119868(119878)Lemma 10 implies that 119878 is ldquoalmostrdquo homogeneous althoughnot truly homogeneous In fact 119878 is a cohomogeneity onemanifold defined in Definition 11

Definition 11 Let (119872 119892) be a Riemannianmanifold and 119868(119872)

is its group of isometric transformations Then 119868(119872) has anatural action Φ on 119872

Φ 119868 (119872) times 119872 997888rarr 119872 (120593 119901) 997891997888rarr 120593 (119901)

forall120593 isin 119868 (119872) 119901 isin 119872

(25)

119872 is a cohomogeneity one manifold if the codimension oforbit space of this action is 1

Remark 12 The codimension of the orbit space is closelyrelated to the homogeneous properties For example if 119872 isa homogeneous space then the orbit space is whole manifold119872 and hence its codimension is 0The less that codimensionis the more homogeneous the manifold is In other wordsa cohomogeneity one manifold is ldquothe most homogeneousmanifoldrdquo except for the real ones

Actually cohomogeneity one manifolds are natural gen-eralizations of homogeneousmanifoldsThe systematic studyof cohomogeneity onemanifolds was started by Bergery whosuccessfully constructed new invariant Einstein metrics oncohomogeneity one manifolds In addition Bryant and Sala-mon constructed special metrics with exceptional holonomygroups 119866

2and Spin(7) on these manifolds [23ndash30]

We state a well known result without proof beforeTheorem 14

Lemma 13 Every homogeneous space is complete

Theorem 14 119878 is a cohomogeneity one manifold

Proof Let 119901 = (1205791

1205792

) isin 119878 and 119868(119878)119901

= 120593(119901) | 120593 isin 119868(119878)

be its orbit space If dim(119868(119878)119901) = 2 119878 will be homogeneous

Lemma 13 implies that 119878 is complete which contradicts toTheorem 7 This contradiction implies dim(119868(119878)

119901) le 1

On the other hand

119866119901

= 120593 (119901) | 120593 isin 119866 = 120593120572

(119901) | 120572 isin R

= (2120572 minus 1205791

1205792

) | 120572 isin R = (119909 1205792

) | 119909 isin R

(26)

Since 119866 is a subgroup of 119868(119878) 119866119901

sub 119868(119878)119901 This implies that

dim (119868 (119878)119901) ge dim (119866

119901) = 1 (27)

Based on the two inequalities we conclude that dim(119868(119878)119901) =

1 hence its codimension is 1 which means that 119878 is acohomogeneity one manifold

4 Instability Analysis

In this section we will calculate some geodesics and investi-gate the instability of the geodesics by the Jacobi vector fields

Combining (12) and (17) the geodesic equations of theWCED model are given by the following differential system

d21205791

d1199052minus

2 cosh 1205792

sinh 1205792

d1205791

d119905

d1205792

d119905= 0

d21205792

d1199052+cosh 120579

2

sinh 1205792(d1205791

d119905)

2

minuscosh 120579

2

sinh 1205792(d1205792

d119905)

2

= 0

(28)

Generally these equations are quite hard to be solvedhowever we can get some special geodesics based on thecohomogeneity one property

Lemma 15 All lines parallel to the y-axis are geodesics

Proof It is well known that the fixed-points set of anyisometry is a geodesic Since 120593

120572is an isometry and its fixed-

points set is the line 119909 = 1205791 the lemma follows

According to Lemma 15 the equation of such geodesicsatisfies

1205791

= 1205791

0 (29)

Hence (28) can be reduced to

d1205791

d119905= 0

d21205792

d1199052minuscosh 120579

2

sinh 1205792(d1205792

d119905)

2

= 0

(30)

Then solution is expressed as

1205791

= 1198863

1205792

= ln100381610038161003816100381610038161003816100381610038161003816

1198862exp 119886

1119905 + 1

1198862exp 119886

1119905 minus 1

100381610038161003816100381610038161003816100381610038161003816

(31)

Consider the parameter family of geodesics

F (119862) = 120579119897

(119905 119862)119897=12

(32)

where 120579119897 is a geodesic satisfying (28) and 119862 is an integration

constant vectorThe length of geodesics inF(119862) is defined as

Θ (119905 120573) = int (119892119894119895d120579119894d120579119895

)12

(33)


Therefore the length of geodesics on WCED model isobtained as

Θ (119905 119862) = int

119905

0

radic11989211

(d1205791

d119905)

2

+ 11989222

(d1205792

d119905)

2

d120591 =radic21198621

2119905

(34)

where 1198621is constant In order to investigate the behavior of

two neighboring geodesics labeled by the parameter 1198621 we

consider the following difference

ΔΘ = (

radic2 (1198621

+ 1205751198621)

2minus

radic21198621

2) 119905 (35)

It is clear that ΔΘ diverges that is the length of twoneighboring geodesics with slightly different parameters 119862

1

and 1198621

+ 1205751198621differs in a remarkable way as 119905 rarr infin

The stability of the geodesics is completely determinedby the curvature of manifold Studying the stability ofdynamics means determining the evolution of perturbationsof geodesics For isotropic manifolds the geodesic spreadis unstable only if their constant sectional curvatures arenegative As long as the curvatures are negative the geodesicspread is unstable even if the manifold is no longer isotropic[19] However it is more attractive to know the divergentdegree of a geodesic

The Jacobi vector field 119869 that is the evolution perturba-tion vector satisfies the following (Jacobi) equation

nabla 120585(119905)

nabla 120585(119905)

119869 = 119877 ( 120585 (119905) 119869) 120585 (119905) (36)

where 120585(119905) is a geodesic on manifold 119872 It is also called thegeodesic derivation equation as it has close connections withthe geodesic In general the Jacobi equations are difficult tosolve but in particular this manifold has a constant negativecurvature which is of great help Given a normal geodesic 120585

and supposing 119869 is a normal Jacobi vector field along 120585(119905) then

119877 ( 120585 (119905) 119869) 120585 (119905) = 2 (119892 ( 120585 (119905) 120585 (119905)) 119869 minus 119892 ( 120585 (119905) 119869) 120585 (119905))

= 2119869

(37)

Thus the Jacobi equation is reduced to

d2119869 (119905)

d1199052minus 2119869 (119905) = 0 (38)

Choose a unit orthogonal frame 119890119894(119905) st 119890

2(119905) = 120585(119905) and

suppose

119869 (119905) = 119869119894

(119905) 119890119894(119905) (39)

Then the Jacobi equation satisfies

d21198691 (119905)d1199052

minus 21198691

(119905) = 0

1198692

(119905) = 0

(40)

The solutions are

1198691

(119905) = 1198621119890radic2119905

+ 1198622119890minusradic2119905

1198692

(119905) = 0

(41)

This means that the geodesic spread onmanifold is describedby means of an exponential order divergent Jacobi vectorfield

5 Conclusion and Remarks

In this paper we investigate the manifold of wrapped Cauchydistributions By considering the geometric structures of theWCED we conclude that the WCED is a constant negativecurvature space with119870 = minus2 By a series of lemmas and theo-rems we prove that it is a cohomogeneity one manifold Thisexample is interesting because of its statistical backgroundwhile some other examples are deliberately constructed Inaddition we calculate the geodesics and the Jacobi vector fieldby some tricks based on some special properties of 119878 As aresult the divergent behavior of geodesics can be describedas an exponential order divergent Jacobi vector field

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Authorsrsquo Contribution

The authors declare that the study was realized in collab-oration with the same responsibility All authors read andapproved the final paper

Acknowledgments

This work is supported by the National Natural ScienceFoundations of China (nos 11126161 and NSFC61440058)Beijing Higher Education Young Elite Teacher Project (noYETP 0388) and the Fundamental Research Funds for theCentral Universities (no FRF-BR-12-005)

References

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[2] S Amari and H Nagaoka Methods of Information Geo metryOxford University Press Oxford UK 2000

[3] T Li L Peng and H Sun ldquoThe geometric structure of theinverse gamma distributionrdquo Beitrage zur Algebra und Geome-trie vol 49 no 1 pp 217ndash225 2008

[4] L Peng H Sun D Sun and J Yi ldquoThe geometric structuresand instability of entropic dynamical modelsrdquo Advances inMathematics vol 227 no 1 pp 459ndash471 2011

[5] L Peng H Sun and G Xu ldquoInformation geometric charac-terization of the complexity of fractional Brownian motionsrdquoJournal of Mathematical Physics vol 53 no 12 Article ID123305 12 pages 2012


[6] L Peng H Sun and X Sun ldquoGeometric structure of Hamilto-nian dynamics with conformal Eisenhart metricrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2011Article ID 710274 26 pages 2011

[7] A Caticha ldquoEntropic dynamicsrdquo in Bayesian Inference andMaximum Entropy Methods in Science and Engineering R LFry Ed AIP Conference Proceedings pp 617ndash302 2002

[8] C Cafaro and S A Ali ldquoJacobi fields on statistical manifolds ofnegative curvaturerdquo Physica D Nonlinear Phenomena vol 234no 1 pp 70ndash80 2007

[9] K V Mardia and P E Jupp Directional Statistics JohnWiley ampSons New York NY USA 1999

[10] C Bingham ldquoAn antipodally symmetric distribution on thesphererdquoThe Annals of Statistics vol 2 pp 1201ndash1225 1974

[11] J T Kent ldquoThe Fisher-Bingham distribution on the sphererdquoJournal of the Royal Statistical Society vol 44 pp 71ndash80 1982

[12] R A Fisher ldquoDispersion on a sphererdquo Proceedings of the RoyalSociety of London A vol 217 pp 295ndash305 1953

[13] N I Fisher Statistical Analysis of Circular Data CambridgeUniversity Press Cambridge UK 1996

[14] G Borradaile Statistics of Earth Science Data Springer NewYork NY USA 2003

[15] K A Keating and S Cherry ldquoModeling utilization distributionsin space and timerdquo Ecology vol 90 no 7 pp 1971ndash1980 2009

[16] H-I Wu B-L Li T A Springer and W H Neill ldquoModellinganimal movement as a persistent random walk in two dimen-sions expected magnitude of net displacementrdquo EcologicalModelling vol 132 no 1-2 pp 115ndash124 2000

[17] Y Vermard E Rivot S Mahevas P Marchal and D GascuelldquoIdentifying fishing trip behaviour and estimating fishing effortfrom VMS data using Bayesian Hidden Markov ModelsrdquoEcological Modelling vol 221 no 15 pp 1757ndash1769 2010

[18] K Thorup J Raboslashl and J J Madsen ldquoCan clock-and-compassexplain the distribution of ringing recoveries of pied flycatch-ersrdquo Animal Behaviour vol 60 no 2 pp F3ndashF8 2000

[19] M P do Carmo Riemannian Geometry Birkhauser BostonMass USA 1992

[20] P Petersen Riemannian Geometry Springer New York NYUSA 2nd edition 2006

[21] C R Rao ldquoInformation and the accuracy attainable in theestimation of statistical parametersrdquo Bulletin of the CalcuttaMathematical Society vol 37 pp 81ndash91 1945

[22] W Chen and X Li Introduction to Riemannian GeometryPeking University Press Beijing China 2012

[23] D Alekseevsky ldquoRiemannian manifolds of cohomogeneityonerdquo in Proceedings of the Colloquium on Differential Geometryvol 56 of Colloeq Math Soc J Bolyai pp 9ndash22 1989

[24] A Alekseevsky and D Alekseevsky ldquoG-manifold with one-dimensional orbit spacerdquo Advances in Soviet Mathematics vol8 pp 1ndash31 1992

[25] A Besse Einstein Manifolds Springer Berlin Germany 1987[26] R L Bryant ldquoMetrics with holonomy 119866

2or spin (7)rdquo in Arbeit-

stagung Bonn 1984 vol 1111 of Lecture Notes in Mathematics pp269ndash277 Springer Berlin Germany 1985

[27] R L Bryant ldquoA survey of Riemannian metrics with specialholonomy groupsrdquo in Proceedings of the International Congressof Mathematicians vol 1 pp 505ndash514 AMS Berkeley CalifUSA 1987

[28] R L Bryant ldquoMetrics with exceptional holonomyrdquo Annals ofMathematics Second Series vol 126 no 3 pp 525ndash576 1987

[29] R Bryant and F Harvery ldquoSome remarks on the geometry ofmanifolds with exceptional holonomyrdquo preprint 1994

[30] R L Bryant and S M Salamon ldquoOn the construction ofsome complete metrics with exceptional holonomyrdquo DukeMathematical Journal vol 58 no 3 pp 829ndash850 1989

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

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OptimizationJournal of


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International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


In Section 3 this manifold is proved to be simply connectedbut not complete hence not isometric to the hyperbolic spaceHowever it is proved to be a cohomogeneity one manifoldby Theorem 14 In Section 4 several tricks are used basedon some special properties to get geodesics and the Jacobivector fields The geodesics are the sets of fixed points ofisometric transformations which include symmetries to linesparalleled to 119910-axis Meanwhile the Jacobi vector fields arecalculated Then the instability of the geodesic spreads isanalyzed via the behaviors of the the Jacobi vector field

Remark 1 It is well known that the upper plane R2+admits a

hyperbolic metric 119892119867with constant negative curvature That

is

119892119867

=

[[[

[

1

11991020

01

1199102

]]]

]

(2)

However in this paper we give another metric on R2+

with constant negative curvature but not isometric to thehyperbolic space

2 Geometric Structures of the WCED Model

From the theory of information geometry one can definean 119898-dimensional statistical manifold 119872 over unit sphereswhich is a set of probability densities namely

119872 = 119901 (119909 120579) (3)

The parameter 120579 = (1205791

1205792

120579119898

) plays the role of coordi-nate systems The Fisher information matrix 119892 acting as aRiemannian metric is then given by the expectation

(119892119894119895) = (119864 [(120597

119894119897) (120597119895119897)]) (4)

where 119897(119909 120579) = ln119901(119909 120579) and 120597119894

= 120597120597120579119894 On a statistical

manifold the Riemannian connection coefficient Γ119894119895119896

is (seeeg [1 2 19ndash21])

Γ119894119895119896

= 119864 [(120597119894120597119895119897) (120597119896119897)] +

1 minus 120572

2119864 [(120597119894119897) (120597119895119897) (120597119896119897)] (5)

which can be expressed as a classical formula Γ119894119895119896

=

(12)(120597119894119892119895119896

+ 120597119895119892119896119894

minus 120597119896119892119894119895) equivalently

If (1205791

1205792

120579119898

) is a local coordinate of 119872 then forany vector field 119883 on 119872 one has 119883 = 119883

119894

120597119894 where 119883

119894

are smooth functions with respect to (1205791

1205792

120579119898

) TheEinstein summation convention is used here and all throughthe paper With the Riemannian connection the curvaturetensor is defined by [19 20]

119877 (119883 119884) 119885 = nabla119883

nabla119884119885 minus nabla

119884nabla119883

119885 minus nabla[119883119884]

119885 (6)

and the Riemannian curvature tensor is

119877 (119883 119884 119885 119882) = 119892 (119877 (119885 119882) 119883 119884) (7)

where 119883 119884 119885 119882 are smooth vector fields on 119872 and [sdot sdot]

represents the Lie bracket namely [119883 119884] = 119883 ∘ 119884 minus 119884 ∘ 119883The local representation of the curvature tensor satisfies

119877119898119895119896119897

= (120597119898

Γ119905

119895119896minus 120597119895Γ119905

119898119896) 119892119905119897

+ (Γ119898119905119897

Γ119905

119895119896minus Γ119895119905119897

Γ119905

119898119896) (8)

where Γ119896

119898119895= Γ119898119895119905

119892119905119896 and (119892

119905119896

) is the inverse of the metricmatrix (119892

119905119896)

The Ricci curvature and the sectional curvature aredefined by

119877119898119896

= 119877119898119895119897119896

119892119895119897

(9)

119870119894119895

= minus

119877119894119895119894119895

119892119894119894119892119895119895

minus 1198922

119894119895

(10)

respectivelyA curve 120585(119905) on 119872 is said to be a geodesic if its tangent

vector 120585(119905) is displaced parallel along the curve 120585(119905) that is

nabla 120585(119905)

120585 (119905) = 0 (11)

and it has the following local form

d2120579119896

d1199052+ Γ119896

119895119898

d120579119895

d119905

d120579119898

d119905= 0 (12)

Suppose that the solution of (12) is 120579(119905 120573) where 120573 is a vectorwhose components are the integration constants

Considering the wrapped Cauchy entropic dynamicalmodel the corresponding statistical manifold is

119878 = 119901 (119909 120579) | 120579 = (1205791

1205792

) isin R2

+ (13)

where 119901(119909 120579) is given by (1) with parameters 1205791

= 120583 and 1205792

=

120574 The Fisher information matrix can be easily calculated as

119892 = diag (1

2sinh21205792

1

2sinh21205792) (14)

Proposition 2 Thenonzero components of the Ricci curvatureare

11987711

= 11987722

= minus1

sinh21205792 (15)

Proof A direct calculation of (5) gives the nonvanishingRiemannian connection coefficients

Γ112

= minusΓ121

= minusΓ211

= minusΓ222

=cosh 120579

2

2sinh31205792 (16)

From the relation Γ119896

119898119895= Γ119898119895119905

119892119905119896 one can get another type of

connection coefficients

Γ2

11= minusΓ1

12= minusΓ1

21= minusΓ2

22=cosh 120579

2

sinh 1205792 (17)

From (8) the following nonzero component of the curvaturetensor is obtained

1198771212

=1

2sinh41205792 (18)

Therefore the proof can be finished by a simple substitutionof (18) into (9)






nected



+ The


connected


2



0gt 0 st


119871 (120574) = lim119904rarr+infin

int

119904

0

100381610038161003816100381610038161205741015840

(119905)10038161003816100381610038161003816d119905 (19)



119881119899





int

119905119899

0

100381610038161003816100381610038161205741015840

(119905)10038161003816100381610038161003816d119905 ge 119889 (120574 (0) 120574 (119905

119899)) ge 119899 (21)



isin 119879119901119872 and 120574(119905) = exp




119899+infin

119899=1 st 119905

119899rarr 1 and 119905

119899lt 119905119899+1


119899) 120574(119905119898









119871 (120574) = int

+infin

1

100381610038161003816100381610038161205741015840

(119905)10038161003816100381610038161003816d119905

= int

+infin

1

1


d119905

=1

radic2

ln(tanh(119905

2))

1003816100381610038161003816100381610038161003816

+infin

1

=ln (119890 + 1) minus ln (119890 minus 1)

radic2

lt +infin

(22)




120593120572

119878 997888rarr 119878 (1205791

1205792

) 997891997888rarr (2120572 minus 1205791

1205792


(23)


1


Lemma 10 120593120572



1205792

) isin 119878 119902 =

120593120572(119901) = (2120572 minus 120579

1

1205792

)Consider

1205741

(119905) = (1205791

+ 119905 1205792

) 1205742

(119905) = (1205791

1205792

+ 119905)

(120593120572)lowast119901

(120597

1205971205791) =

dd119905

10038161003816100381610038161003816100381610038161003816119905=0

(120593120572

∘ 1205741

(119905))

=dd119905

10038161003816100381610038161003816100381610038161003816119905=0

(2120572 minus 1205791

minus 119905 1205792

) = minus120597

1205971205791

10038161003816100381610038161003816100381610038161003816119902

(120593120572)lowast119901

(120597

1205971205792) =

dd119905

10038161003816100381610038161003816100381610038161003816119905=0

(120593120572

∘ 1205742

(119905))

=dd119905

10038161003816100381610038161003816100381610038161003816119905=0

(2120572 minus 1205791

1205792

+ 119905) =120597

1205971205792

10038161003816100381610038161003816100381610038161003816119902

119892119902

((120593120572)lowast119901

(120597

1205971205791) (120593120572)lowast119901

(120597

1205971205791))

= 119892119902

(minus120597

1205971205791 minus

120597

1205971205791) =

1

2sinh21205792= 119892119901

(120597

1205971205791

120597

1205971205791)


119892119902

((120593120572)lowast119901

(120597

1205971205791) (120593120572)lowast119901

(120597

1205971205792))

= 119892119902

(minus120597

1205971205791

120597

1205971205792) = 0 = 119892

119901(

120597

1205971205791

120597

1205971205792)

119892119902

((120593120572)lowast119901

(120597

1205971205792) (120593120572)lowast119901

(120597

1205971205792))

= 119892119902

(120597

1205971205792

120597

1205971205792) =

1

2sinh21205792= 119892119901

(120597

1205971205792

120597

1205971205792)

(24)


As a result 119866 = 120593120572




Φ 119868 (119872) times 119872 997888rarr 119872 (120593 119901) 997891997888rarr 120593 (119901)

forall120593 isin 119868 (119872) 119901 isin 119872

(25)








Proof Let 119901 = (1205791

1205792

) isin 119878 and 119868(119878)119901

= 120593(119901) | 120593 isin 119868(119878)



119901) le 1

On the other hand

119866119901

= 120593 (119901) | 120593 isin 119866 = 120593120572

(119901) | 120572 isin R

= (2120572 minus 1205791

1205792

) | 120572 isin R = (119909 1205792

) | 119909 isin R

(26)



dim (119868 (119878)119901) ge dim (119866

119901) = 1 (27)






d21205791

d1199052minus

2 cosh 1205792

sinh 1205792

d1205791

d119905

d1205792

d119905= 0

d21205792

d1199052+cosh 120579

2

sinh 1205792(d1205791

d119905)

2

minuscosh 120579

2

sinh 1205792(d1205792

d119905)

2

= 0

(28)







1205791

= 1205791

0 (29)


d1205791

d119905= 0

d21205792


2

sinh 1205792(d1205792

d119905)

2

= 0

(30)


1205791

= 1198863

1205792

= ln100381610038161003816100381610038161003816100381610038161003816

1198862exp 119886

1119905 + 1

1198862exp 119886

1119905 minus 1

100381610038161003816100381610038161003816100381610038161003816

(31)


F (119862) = 120579119897

(119905 119862)119897=12

(32)



Θ (119905 120573) = int (119892119894119895d120579119894d120579119895

)12

(33)



Θ (119905 119862) = int

119905

0

radic11989211

(d1205791

d119905)

2

+ 11989222

(d1205792

d119905)

2

d120591 =radic21198621

2119905

(34)




ΔΘ = (

radic2 (1198621

+ 1205751198621)

2minus

radic21198621

2) 119905 (35)


1

and 1198621




nabla 120585(119905)

nabla 120585(119905)

119869 = 119877 ( 120585 (119905) 119869) 120585 (119905) (36)



119877 ( 120585 (119905) 119869) 120585 (119905) = 2 (119892 ( 120585 (119905) 120585 (119905)) 119869 minus 119892 ( 120585 (119905) 119869) 120585 (119905))

= 2119869

(37)


d2119869 (119905)

d1199052minus 2119869 (119905) = 0 (38)


2(119905) = 120585(119905) and

suppose

119869 (119905) = 119869119894

(119905) 119890119894(119905) (39)


d21198691 (119905)d1199052

minus 21198691

(119905) = 0

1198692

(119905) = 0

(40)

The solutions are

1198691

(119905) = 1198621119890radic2119905

+ 1198622119890minusradic2119905

1198692

(119905) = 0

(41)








Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














nected



+ The


connected


2



0gt 0 st


119871 (120574) = lim119904rarr+infin

int

119904

0

100381610038161003816100381610038161205741015840

(119905)10038161003816100381610038161003816d119905 (19)



119881119899





int

119905119899

0

100381610038161003816100381610038161205741015840

(119905)10038161003816100381610038161003816d119905 ge 119889 (120574 (0) 120574 (119905

119899)) ge 119899 (21)



isin 119879119901119872 and 120574(119905) = exp




119899+infin

119899=1 st 119905

119899rarr 1 and 119905

119899lt 119905119899+1


119899) 120574(119905119898









119871 (120574) = int

+infin

1

100381610038161003816100381610038161205741015840

(119905)10038161003816100381610038161003816d119905

= int

+infin

1

1


d119905

=1

radic2

ln(tanh(119905

2))

1003816100381610038161003816100381610038161003816

+infin

1

=ln (119890 + 1) minus ln (119890 minus 1)

radic2

lt +infin

(22)




120593120572

119878 997888rarr 119878 (1205791

1205792

) 997891997888rarr (2120572 minus 1205791

1205792


(23)


1


Lemma 10 120593120572



1205792

) isin 119878 119902 =

120593120572(119901) = (2120572 minus 120579

1

1205792

)Consider

1205741

(119905) = (1205791

+ 119905 1205792

) 1205742

(119905) = (1205791

1205792

+ 119905)

(120593120572)lowast119901

(120597

1205971205791) =

dd119905

10038161003816100381610038161003816100381610038161003816119905=0

(120593120572

∘ 1205741

(119905))

=dd119905

10038161003816100381610038161003816100381610038161003816119905=0

(2120572 minus 1205791

minus 119905 1205792

) = minus120597

1205971205791

10038161003816100381610038161003816100381610038161003816119902

(120593120572)lowast119901

(120597

1205971205792) =

dd119905

10038161003816100381610038161003816100381610038161003816119905=0

(120593120572

∘ 1205742

(119905))

=dd119905

10038161003816100381610038161003816100381610038161003816119905=0

(2120572 minus 1205791

1205792

+ 119905) =120597

1205971205792

10038161003816100381610038161003816100381610038161003816119902

119892119902

((120593120572)lowast119901

(120597

1205971205791) (120593120572)lowast119901

(120597

1205971205791))

= 119892119902

(minus120597

1205971205791 minus

120597

1205971205791) =

1

2sinh21205792= 119892119901

(120597

1205971205791

120597

1205971205791)


119892119902

((120593120572)lowast119901

(120597

1205971205791) (120593120572)lowast119901

(120597

1205971205792))

= 119892119902

(minus120597

1205971205791

120597

1205971205792) = 0 = 119892

119901(

120597

1205971205791

120597

1205971205792)

119892119902

((120593120572)lowast119901

(120597

1205971205792) (120593120572)lowast119901

(120597

1205971205792))

= 119892119902

(120597

1205971205792

120597

1205971205792) =

1

2sinh21205792= 119892119901

(120597

1205971205792

120597

1205971205792)

(24)


As a result 119866 = 120593120572




Φ 119868 (119872) times 119872 997888rarr 119872 (120593 119901) 997891997888rarr 120593 (119901)

forall120593 isin 119868 (119872) 119901 isin 119872

(25)








Proof Let 119901 = (1205791

1205792

) isin 119878 and 119868(119878)119901

= 120593(119901) | 120593 isin 119868(119878)



119901) le 1

On the other hand

119866119901

= 120593 (119901) | 120593 isin 119866 = 120593120572

(119901) | 120572 isin R

= (2120572 minus 1205791

1205792

) | 120572 isin R = (119909 1205792

) | 119909 isin R

(26)



dim (119868 (119878)119901) ge dim (119866

119901) = 1 (27)






d21205791

d1199052minus

2 cosh 1205792

sinh 1205792

d1205791

d119905

d1205792

d119905= 0

d21205792

d1199052+cosh 120579

2

sinh 1205792(d1205791

d119905)

2

minuscosh 120579

2

sinh 1205792(d1205792

d119905)

2

= 0

(28)







1205791

= 1205791

0 (29)


d1205791

d119905= 0

d21205792


2

sinh 1205792(d1205792

d119905)

2

= 0

(30)


1205791

= 1198863

1205792

= ln100381610038161003816100381610038161003816100381610038161003816

1198862exp 119886

1119905 + 1

1198862exp 119886

1119905 minus 1

100381610038161003816100381610038161003816100381610038161003816

(31)


F (119862) = 120579119897

(119905 119862)119897=12

(32)



Θ (119905 120573) = int (119892119894119895d120579119894d120579119895

)12

(33)



Θ (119905 119862) = int

119905

0

radic11989211

(d1205791

d119905)

2

+ 11989222

(d1205792

d119905)

2

d120591 =radic21198621

2119905

(34)




ΔΘ = (

radic2 (1198621

+ 1205751198621)

2minus

radic21198621

2) 119905 (35)


1

and 1198621




nabla 120585(119905)

nabla 120585(119905)

119869 = 119877 ( 120585 (119905) 119869) 120585 (119905) (36)



119877 ( 120585 (119905) 119869) 120585 (119905) = 2 (119892 ( 120585 (119905) 120585 (119905)) 119869 minus 119892 ( 120585 (119905) 119869) 120585 (119905))

= 2119869

(37)


d2119869 (119905)

d1199052minus 2119869 (119905) = 0 (38)


2(119905) = 120585(119905) and

suppose

119869 (119905) = 119869119894

(119905) 119890119894(119905) (39)


d21198691 (119905)d1199052

minus 21198691

(119905) = 0

1198692

(119905) = 0

(40)

The solutions are

1198691

(119905) = 1198621119890radic2119905

+ 1198622119890minusradic2119905

1198692

(119905) = 0

(41)








Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119892119902

((120593120572)lowast119901

(120597

1205971205791) (120593120572)lowast119901

(120597

1205971205792))

= 119892119902

(minus120597

1205971205791

120597

1205971205792) = 0 = 119892

119901(

120597

1205971205791

120597

1205971205792)

119892119902

((120593120572)lowast119901

(120597

1205971205792) (120593120572)lowast119901

(120597

1205971205792))

= 119892119902

(120597

1205971205792

120597

1205971205792) =

1

2sinh21205792= 119892119901

(120597

1205971205792

120597

1205971205792)

(24)


As a result 119866 = 120593120572




Φ 119868 (119872) times 119872 997888rarr 119872 (120593 119901) 997891997888rarr 120593 (119901)

forall120593 isin 119868 (119872) 119901 isin 119872

(25)








Proof Let 119901 = (1205791

1205792

) isin 119878 and 119868(119878)119901

= 120593(119901) | 120593 isin 119868(119878)



119901) le 1

On the other hand

119866119901

= 120593 (119901) | 120593 isin 119866 = 120593120572

(119901) | 120572 isin R

= (2120572 minus 1205791

1205792

) | 120572 isin R = (119909 1205792

) | 119909 isin R

(26)



dim (119868 (119878)119901) ge dim (119866

119901) = 1 (27)






d21205791

d1199052minus

2 cosh 1205792

sinh 1205792

d1205791

d119905

d1205792

d119905= 0

d21205792

d1199052+cosh 120579

2

sinh 1205792(d1205791

d119905)

2

minuscosh 120579

2

sinh 1205792(d1205792

d119905)

2

= 0

(28)







1205791

= 1205791

0 (29)


d1205791

d119905= 0

d21205792


2

sinh 1205792(d1205792

d119905)

2

= 0

(30)


1205791

= 1198863

1205792

= ln100381610038161003816100381610038161003816100381610038161003816

1198862exp 119886

1119905 + 1

1198862exp 119886

1119905 minus 1

100381610038161003816100381610038161003816100381610038161003816

(31)


F (119862) = 120579119897

(119905 119862)119897=12

(32)



Θ (119905 120573) = int (119892119894119895d120579119894d120579119895

)12

(33)



Θ (119905 119862) = int

119905

0

radic11989211

(d1205791

d119905)

2

+ 11989222

(d1205792

d119905)

2

d120591 =radic21198621

2119905

(34)




ΔΘ = (

radic2 (1198621

+ 1205751198621)

2minus

radic21198621

2) 119905 (35)


1

and 1198621




nabla 120585(119905)

nabla 120585(119905)

119869 = 119877 ( 120585 (119905) 119869) 120585 (119905) (36)



119877 ( 120585 (119905) 119869) 120585 (119905) = 2 (119892 ( 120585 (119905) 120585 (119905)) 119869 minus 119892 ( 120585 (119905) 119869) 120585 (119905))

= 2119869

(37)


d2119869 (119905)

d1199052minus 2119869 (119905) = 0 (38)


2(119905) = 120585(119905) and

suppose

119869 (119905) = 119869119894

(119905) 119890119894(119905) (39)


d21198691 (119905)d1199052

minus 21198691

(119905) = 0

1198692

(119905) = 0

(40)

The solutions are

1198691

(119905) = 1198621119890radic2119905

+ 1198622119890minusradic2119905

1198692

(119905) = 0

(41)








Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Θ (119905 119862) = int

119905

0

radic11989211

(d1205791

d119905)

2

+ 11989222

(d1205792

d119905)

2

d120591 =radic21198621

2119905

(34)




ΔΘ = (

radic2 (1198621

+ 1205751198621)

2minus

radic21198621

2) 119905 (35)


1

and 1198621




nabla 120585(119905)

nabla 120585(119905)

119869 = 119877 ( 120585 (119905) 119869) 120585 (119905) (36)



119877 ( 120585 (119905) 119869) 120585 (119905) = 2 (119892 ( 120585 (119905) 120585 (119905)) 119869 minus 119892 ( 120585 (119905) 119869) 120585 (119905))

= 2119869

(37)


d2119869 (119905)

d1199052minus 2119869 (119905) = 0 (38)


2(119905) = 120585(119905) and

suppose

119869 (119905) = 119869119894

(119905) 119890119894(119905) (39)


d21198691 (119905)d1199052

minus 21198691

(119905) = 0

1198692

(119905) = 0

(40)

The solutions are

1198691

(119905) = 1198621119890radic2119905

+ 1198622119890minusradic2119905

1198692

(119905) = 0

(41)








Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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