On general fractional abstract Cauchy problem

Eur.Phys.J.Special Topics193,27–47(2011) c EDP Sciences,Springer-Verlag2011DOI:10.1140/epjst/e2011-01379-1T HE E UROPEANP HYSICAL J OURNAL S PECIAL T OPICSReviewA survey on the stability of fractionaldifferential equationsDedicated to Prof.Y.S.Chen on the Occasion of his80th BirthdayC.P.Li1,a and F.R.Zhang1,2,b1Department of Mathematics,Shanghai University,Shanghai200444,PR China2School of Mathematics and Computational Sciences,China University of Petroleum (East China),Dongying257061,PR ChinaReceived01December2010/Received infinal form27January2011Published online4April2011Abstract.Recently,fractional calculus has attracted much attentionsince it plays an important role in manyfields of science and engineer-ing.Especially,the study on stability of fractional differential equationsappears to be very important.In this paper,a brief overview on therecent stability results of fractional differential equations and the ana-lytical methods used are provided.These equations include linear frac-tional differential equations,nonlinear fractional differential equations,fractional differential equations with time-delay.Some conclusions forstability are similar to that of classical integer-order differential equa-tions.However,not all of the stability conditions are parallel to thecorresponding classical integer-order differential equations because ofnon-locality and weak singularities of fractional calculus.Some resultsand remarks are also included.1IntroductionFractional calculus has been300years old history,the development of fractional calculus theory is mainly focused on the pure mathematicalfield.The earliest more or less systematic studies seem to have been made in the19th century by Liouville, Riemann,Leibniz,etc.[1,2].In the last two decades,fractional differential equations (FDEs)have been used to model various stable physical phenomena with anomalous decay,say that are not of exponential type[3].We can refer to[4]for the recent history of fractional calculus.As we all know,many mathematical models of real problems arising in variousfields of science and engineering are either linear systems or nonlinear systems.Nevertheless,most differential systems used to describe physical phenomena are integer-order systems.With the development of fractional calculus, it has been found that the behavior of many systems can be described by using the fractional differential systems[5–9].It is worth mentioning that many physical phenomena having memory and genetic characteristics can be described by using the fractional differential systems.In fact,real world processes generally or most likelya e-mail:lcp@b e-mail:zhangfengrongsong@28The European Physical Journal Special Topicsare fractional order systems[10,11].That is to say,a lot of physical systems showfractional dynamical behavior because of special materials and chemical properties.Recently,the theory of FDEs has been studied and some basic results areobtained including stability theory.The question of stability is of main interest inphysical and biological systems,such as the fractional Duffing oscillator[12,13],frac-tional predator-prey and rabies models[14],etc.It is known that the chaotic sequenceis non-periodic and pseudo-random.Similar to integer-order differential equations,the stability theory of FDEs is also widely applied to chaos and chaos synchroniza-tion[15–19]due to its potential applications in control processing and secure com-munication.However,a few stability results rely on a restrictive modeling of FDEs:the basic hypothesis deals with commensurability,i.e.the fractional derivative ordershave to be an integer multiple of minimal fractional order.Owing to this hypothesis,some stability results are available,based on Matignon’s theorem[20].On the otherhand,the case of incommensurate fractional order can be referred to[21,22].In this paper,we present and discuss some basic results on stability of FDEsincluding linear FDEs,nonlinear FDEs,FDEs with time-delay and the analyticalmethods used.The analysis on stability of FDEs is more complex than that of clas-sical differential equations,since fractional derivatives are nonlocal and have weaklysingular kernels.The earliest study on stability of FDEs started in[23],the authorstudied the case of linear FDEs with Caputo derivative and the same fractional orderα,where0<α≤1.The stability problem comes down to the eigenvalue prob-lem of system matrix.Corresponding to the stability result in[23],Qian et al.[22]recently studied the case of linear FDEs with Riemann-Liouville derivative and thesame fractional orderα,where0<α<1.Then,in[24–26]authors derived thesame conclusion as[23]for the case1<α<2.[21]studied the linear system withmulti-order Caputo derivative and derived a sufficient condition on Lyapunov globalasymptotical stability.In the last decades,many researchers have more interests inthe stability of linear systems and various methods have emerged in succession.Forexample,there are frequency domain methods[27–32],Linear Matrix Inequalities(LMI)methods[25,26,33,34]and conversion methods[10],[35–37].By contrast,the development of stability of nonlinear FDEs is a bit slow.Thestructural stability was studied in[38],where the system with Riemann-Liouvillederivative was considered by using Taylor polynomial.In[39]authors investigatedthe system of nonautonomous FDEs involving Caputo derivative and derived theresult on continuous dependence of solution on initial conditions.The stability inthe sense of Lyapunov has also been studied[40]by using Gronwall’s lemma andSchwartz inequality.We can also refer to[14],[41–47],where the linearization method was considered.But the rigorous theoretical derivation has not been founded.Some researchers weakened the criterion of stability,such as[48]where the L p-stability properties of nonlinear FDEs were investigated.In[49,50],the Mittag-Leffler stability and the fractional Lyapunov’s second method were proposed.At last Deng[51]derived a sufficient stability condition of nonlinear FDEs.In engineering,a kind of system is also very important,namely,the time-delaysystem.In[52–55]authors considered thefinite-time stability of FDEs with time-delay on the basis of real problems.They studied the autonomous and nonau-tonomous fractional differential systems.For the time-delay system stability,we canrefer to[21],[56–58].In[59],Petr´aˇs gave a survey on the methods for stability investigation of a certainclass of fractional differential systems with rational orders and Caputo derivative in aviewpoint of control.To complement the literature,we will review the stability resultsof FDEs with Riemann-Liouville derivative or Caputo derivative and the analyticalmethods.Furthermore,the stability of FDEs with time-delay will also be covered.Thepaper is outlined as follows.In Sec.2,wefirst recall some definitions and propositions.Perspectives on Fractional Dynamics and Control29In Sec.3,some stability conditions of linear FDEs are presented,meanwhile,we give two results.In Sec.4,some stability conditions of nonlinear FDEs are described. Sec.5deals with the stability of FDEs with time-delay.Conclusions and comments are included in Sec.6.2PreliminariesLet us denote by R the set of real numbers,R+the set of positive real numbers and by Z+the set of positive integer numbers,denote by C the set of complex numbers. We denote the real part of complex numberαby Re(α).Two kinds of fractional derivatives,i.e.,the Riemann-Liouville derivative and Caputo derivative,have been often used in fractional differential systems.We briefly introduce these two definitions of fractional derivatives which will be frequently used throughout this paper.Firstly,we introduce the definition of fractional integral[60]. Definition1.The fractional integral(or,the Riemann-Liouville integral)D−αt0,t with fractional orderα∈R+of function x(t)is defined below:D−αt0,t x(t)=1Γ(α)tt0(t−τ)α−1x(τ)dτ,where t=t0is the initial time,Γ(·)is the Euler’s gamma function.Definition2.The Riemann-Liouville derivative with orderαof function x(t)is defined below:RL Dαt0,t x(t)=1Γ(m−α)d mdt mtt0(t−τ)m−α−1x(τ)dτ=d mdt mD−(m−α)t0,t,where m−1≤α<m∈Z+.Definition3.The Caputo derivative with orderαof function x(t)is defined below:C Dαt0,t x(t)=1Γ(m−α)tt0(t−τ)m−α−1x(m)(τ)dτ=D−(m−α)t0,td mdt mx(t),where m−1<α<m∈Z+.There are also two functions that play an important role in the study on stability of FDEs.Definition4.The Mittag-Leffler function is defined byEα(z)=∞k=0z kΓ(kα+1),where Re(α)>0,z∈C.The two-parameter Mittag-Leffler function is defined byEα,β(z)=∞k=0z kΓ(kα+β),where Re(α)>0andβ∈C,z∈C.One can see Eα(z)=Eα,1(z)from the above equations.30The European Physical Journal Special TopicsDefinition 5([61]).The α-exponential function is defined as follows:e λz α=zα−1E α,α(λz α),where z ∈C \0,Re (α)>0,and λ∈C .E α,α(·)is the two-parameter Mittag-Leffler function.Proposition 1.If 0<α<2,βis an arbitrary complex number,then for an arbitrary integer p ≥1the following expansions hold:E α,β(z )=1αz (1−β)/αexp(z 1/α)−p k =11Γ(β−αk )1z k +O 1|z |p +1 ,with |z |→∞,|arg(z )|≤απ2,andE α,β(z )=−pk =11Γ(β−αk )1z k +O 1|z |p +1 ,with |z |→∞,|arg(z )|>απ2.Proof.These results were proved in [60].The following relations for the α-exponential function follow from Definition 5and Proposition 1:Proposition 2[61].If 0<α<2,z ∈C ,then the following asymptotic equivalents for e λz αas |z |reaches infinity are valid:•for |arg(λ)|≤απ2,e λz α∼λ(1−α)/ααexp(λ1/α)z ,•for |arg(λ)|>απ2,e λz α∼−λ−2Γ(−α)1z α+1.In this paper,we will consider the following general type of FDEs involving Caputo derivative or Riemann-Liouville derivativeD ¯αt 0,t x (t )=f (t,x ),(1)with suitable initial values x k =[x k 1,x k 2,...,x kn ]T ∈R n (k =0,1,...,m −1),where x (t )=[x 1(t ),x 2(t ),...,x n (t )]T ∈R n ,¯α=[α1,α2,...,αn ]T ,m −1<αi <m ∈Z +(i =1,2,...,n ),D ¯αt 0,t x (t )=[D α1t 0,t x 1(t ),...,D αn t 0,t x n (t )]T ,f :[t 0,∞)×R n →R n ,D αi t 0,t denotes either C D αi t 0,t or RL D αi t 0,t .In particular,if α1=α2=···=αn =α,then Eq.(1)can be written asD αt 0,t x (t )=f (t,x ).(2)We call Eq.(2)the same order fractional differential system,otherwise,call Eq.(1)multi-order fractional differential system.The following definitions are associated with the stability problem in the paper.Definition 6[49].The constant vector x eq is an equilibrium point of fractional dif-ferential system (1),if and only if f (t,x eq )=D ¯αt 0,t x (t )|x (t )=x eq for all t >t 0.Without loss of generality,let the equilibrium point be x eq =0,we introduce the following definition.Perspectives on Fractional Dynamics and Control31 Definition7.The zero solution of fractional differential system(1)is said to be stable if,for any initial values x k=[x k1,x k2,...,x kn]T∈R n(k=0,1,...,m−1), there exists >0such that any solution x(t)of(1)satisfies x(t) <εfor all t>t0. The zero solution is said to be asymptotically stable if,in addition to being stable, x(t) →0as t→+∞.Definition8.Let1≤p≤∞andΩ⊂[t0,∞),the same order system(2)with Riemann-Liouville derivative is L p(Ω)−stable if the solution x(t)=[x1(t),..., x n(t)]T defined by equationx(t)=x0Γ(α)(t−t0)α−1+1Γ(α)tt0(t−τ)α−1f(τ,x(τ))dτbelongs to L p(Ω).Where0<α<1,x0∈R n is the initial value,f∈C([t0,∞)×R n,R n+)is a continuous positive function.Definition9(Mittag-Leffler Stability)[49].Let B⊂R n be a domain containing the origin.The zero solution of the same order system(2)is said to be Mittag-Leffler stable ifx(t) ≤{m(x(t0))Eα(−λ(t−t0)α)}b,where t0is the initial time,α∈(0,1),λ>0,b>0,m(0)=0,m(x)≥0,and m(x)is locally Lipschitz on x∈B⊂R n with Lipschitz constant L.Definition10.A continuous functionα:[0,∞)→[0,∞)is said to belong to class-K if it is strictly increasing andα(0)=0.Definition11(Generalized Mittag-Leffler Stability).Let B⊂R n be a domain containing the origin.The zero solution of the same order system(2)is said to be generalized Mittag-Leffler stable ifx(t) ≤{m(x(t0))(t−t0)−γEα,1−γ(−λ(t−t0)α)}b,where t0is the initial time,α∈(0,1),−α<γ<1−α,λ≥0,b>0,m(0)=0,m(x)≥0,and m(x)is locally Lipschitz on x∈B⊂R n with Lipschitz constant L.We will also need the following definitions to analyze the case of FDEs with time-delay in Sec.5.First,we introduce the same order fractional differential system with multiple time delays represented by the following differential equation:⎧⎪⎨⎪⎩Dαt0,tx(t)=A0x(t)+mi=1A i x(t−τi)+B0u(t),0≤τ1<τ2<τ3<···<τi<···<τm=Δ(3)with the initial condition x(t0+t)=ψ(t)∈C[−Δ,0].Where0<α<1,Dαi t0,t denoteseither C Dαi t0,t or RL Dαi t0,t.x(t)∈R n is a state vector,u(t)∈R l is a control vector,A i(i=0,1,...,m),B0are constant system matrices of appropriate dimensions,and τi>0(i=1,2,...,m)are pure time delays.Definition12.The same order system(3)(u(t)≡0,∀t)satisfying initial condition x(t0+t)=ψ(t),−Δ≤t≤0,isfinite-time stable w.r.t.{t0,J,δ,ε,δ},δ<εif and only if:ψ c<δ,∀t∈J ,J =[− ,0]∈R32The European Physical Journal Special Topicsimplies:x (t ) <ε,∀t ∈J,where δis a positive real number and ε∈R +,δ<ε, ψ c =max −Δ≤t ≤0 ψ(t ) ,time interval J =[t 0,t 0+T ]⊂R ,quantity T may be either a positive real number or a symbol +∞.Definition 13.System given by (3)satisfying initial condition x (t 0+t )=ψ(t ),−Δ≤t ≤0,is finite-time stable w.r.t.{t 0,J,δ,ε,Δ,αu },δ<εif and only if:ψ c <δ,∀t ∈J ,J =[− ,0]∈Randu (t ) <αu ,∀t ∈Jimplies:x (t ) <ε,∀t ∈J,where δis a positive real number and ε∈R +,δ<ε,αu >0, ψ c =max −Δ≤t ≤0 ψ(t ) ,time interval J =[t 0,t 0+T ]⊂R ,quantity T may be either a positive real number or a symbol +∞.3Stability of linear fractional differential equationsIn this section,we consider the following linear system of FDEsD ¯αt 0,t x (t )=Ax (t ),(4)where x (t )=[x 1(t ),x 2(t ),...,x n (t )]T ∈R n ,matrix A ∈R n ×n ,¯α=[α1,α2,...,αn ]T ,D ¯αt 0,t x (t )=[D α1t 0,t x 1(t ),D α2t 0,t x 2(t ),...,D αn t 0,t x n (t )]T and D αi t 0,t is the Caputo derivativeor Riemann-Liouville derivative of order αi ,where 0<αi ≤2,for i =1,2,···,n .In particular,if α1=α2=···=αn =α,then fractional differential system (4)can be written as the following same order linear systemD αt 0,t x (t )=Ax (t ).(5)3.1The fundamental theoremsFor Eq.(5)and 0<α≤1,Matignon firstly gave a well-known stability result by an algebraic approach combined with the use of asymptotic results,where the necessary and sufficient conditions have been derived,the specific result is as follows [23].Theorem 1.The autonomous same order system (5)with Caputo derivative and initial value x 0=x (0),where 0<α≤1,is•asymptotically stable if and only if |arg(spec (A ))|>απ2.In this case the compo-nents of the state decay towards 0like t −α.•stable if and only if either it is asymptotically stable,or those critical eigenval-ues which satisfy |arg(spec (A ))|=απ2have geometric multiplicity one,spec (A )denotes the eigenvalues of matrix A .Perspectives on Fractional Dynamics and Control 33As we can see,in case α=1,the above stability result shows that roots of the equa-tion det(diag(λ,λ,...,λ)−A)=0lie outside the closed angular sector |arg(λ)|≤απ2,thus generalizing the well-known result for the integer case α=1.For the asymptot-ical stability in Theorem 1,the components of the state are anomalous decay,thus fractional systems have ‘memory’feature and its asymptotical stability is also called t −αstability.Exponential stability cannot be used to characterize the asymptotic stability of fractional differential systems.The case of zero eigenvalues of system matrix A is not included in Theorem 1,since the argument of zero in complex plane can be arbitrary.In view of this situation,Qian et al.recently studied the case of autonomous same order system (5)with Riemann-Liouville derivative by using the asymptotic expansions of Mittag-Leffler function,where 0<α<1.And they stated the zero eigenvalues case of Theorem 1,the corresponding conclusion is as follows [22].Theorem 2.The autonomous same order system (5)with Riemann-Liouville deriv-ative and initial value x 0=RL D α−1t 0,tx (t )|t =t 0,where 0<α<1and t 0=0,is •asymptotically stable if and only if all the non-zero eigenvalues of A satisfy |arg(spec (A ))|>απ2,or A has k -multiple zero eigenvalues corresponding to a Jordan block diag(J 1,J 2,...,J i ),where J l is a Jordan canonical form with order n l , i l =1n l =k ,and n l α<1,1≤l ≤i .•stable if and only if either it is asymptotically stable,or those critical eigenval-ues which satisfy |arg(spec (A ))|=απ2have the same algebraic and geometric multiplicities,or A has k -multiple zero eigenvalues corresponding to a Jordan block matrix diag(J 1,J 2,...,J i ),where J l is a Jordan canonical form with order n l , i l =1n l =k ,and n l α≤1,1≤l ≤i .The proof of Theorem 2is contained in [22].For the autonomous same order system(5)with Riemann-Liouville derivative and 0<α<1,if all the eigenvalues of system matrix A satisfy |arg(spec (A ))|>απ2,then the components of the state decay towards0like t −α−1which is different from the Caputo derivative case.And similar to the proof in [22],for the autonomous same order system (5)with Caputo derivative and 0<α≤1,if all the non-zero eigenvalues of A satisfy |arg(spec (A ))|≥απ2and the critical eigenvalues satisfying |arg(spec (A ))|=απ2have the same algebraic and geometric multiplicities,and the zero eigenvalue of A has the same algebraic and geometric multiplicities,then the zero solution of this system is stable from the representation of the solution.In addition,the zero solution of this system is never asymptotically stable as long as A has zero eigenvalue.The above two theorems dealt with the same order linear fractional differential system.For the multi-order linear fractional differential system (4),Deng et al.firstly studied the case that α,i s are rational numbers between 0and 1,for i =1,2,...,n ,where the following result [21,62]was introduced.Theorem 3.Suppose that α,i s are rational numbers between 0and 1,for i =1,2,...,n .Let M be the lowest common multiple (LCM)of the denominators u i of α,i s ,where αi =v i u i ,(u i ,v i )=1,u i ,v i ∈Z +,i =1,2,...,n ,and set γ=1M .Then the zero solution of system (4)with Caputo derivative and initial value x 0=x (0)is•asymptotically stable if and only if any zero solution of the polynomialdet(diag(λMα1,λMα2,...,λMαn )−A )satisfies |arg(λ)|>γπ/2,the components of the state variable (x 1(t ),x 2(t ),...,x n (t ))T ∈R n decay towards 0like t −α1,t −α2,...,t −αn ,respectively.34The European Physical Journal Special Topics•stable if and only if either it is asymptotically stable or those critical zero solutions λof the above polynomial satisfy |arg(λ)|=γπ/2have geometric multiplicity one.From Theorem 3,suppose α1=α2=···=αn =αare rational numbers between 0and 1,then the result of Theorem 3coincides with the one in Theorem 1.So,Theorem 3is an extension of Theorem 1in respect of rational orders.In [21],the authors used Laplace transform [63]and the final-value theorem to prove Theorem 3.Similarly,Odibat [64]also analyzed the stability of system (4)with 0<α=α1=α2=···=αn ≤1.In [64],the Mittag-Leffler functions and their integer-order derivatives which are analytic functions were used to obtain analytical solutions of the initial value problem (4),then the sufficient stability condition was derived by using the final-value theorem.This result [64]is consistent with the one in Theorem 1and Theorem 3.Matignon’s theorem is in fact the starting point of several results on the stability analysis.For example,Ralti et al.extended Theorem 1to the case 1<α<2[24].Remark 1.Similar to Theorem 2,if we assume that the conditions of Theorem 3hold except replacing C D ¯α0,t and the initial value x 0=x (0)by RL D ¯α0,t and the initial valuex 0=RL D ¯α0,t x (t )|t =0respectively,then the stability result which is an extension ofTheorem 2in respect of rational orders is still available.All the above conclusions are about the case of commensurate fractional order,in addition,some literatures such as [21,22]also involved the case of incommensurate fractional order.If α1,α2,...,αn are not rational numbers but real numbers between 0and 1in system (4),then we have the following result,which was introduced in [21,22]by using the final-value theorem of Laplace transform.Theorem 4.If all the roots of the characteristic equation det(diag(s α1,s α2,...,s αn )−A )=0have negative real parts,then the zero solution of system (4)is asymp-totically stable,where αi is real and lies in (0,1).From the above theorems and Proposition 2,we have the following result.Theorem 5.The autonomous same order system (5)with initial value x 0=x (0)and Riemann-Liouville derivative is asymptotically stable if and only if |arg(spec (A ))|>απ2,where n =2and 0<α≤1.Proof.From [65],this system has unique solution which can be expressed by a gen-eralization of the matrix α-exponential function as follows:x (t )=e At αC =t α−1∞k =0A kt αk Γ[(k +1)α]C.For A ,there exists an invertible matrix P from the algebra such that P −1AP =J,where J is the Jordan canonical form of the matrix A .Here are two cases to discuss:•for J = λ100λ2,x (t )=t α−1∞k =0P J k P −1t αk Γ[(k +1)α]C =P e λ1t α0e λ2t α P −1C.Perspectives on Fractional Dynamics and Control 35•for J =λ10λ,x (t )=t α−1∞k =0P J k P−1t αk Γ[(k +1)α]C =P e λt α∂∂λe λt α0e λt α P −1C.Thus,we have lim t →+∞ x (t ) =0if and only if |arg(spec (A ))|>απ2from Propo-sition 2,and the proof is complete.We can also extend Theorem 5to the case 1<α<2.Theorem 6.The autonomous same order system (5)with Riemann-Liouville deriv-ative and initial values x k =RL D α−k −10,t x (t )|t =0(k =0,1),is asymptotically stable if and only if |arg(spec (A ))|>απ2,where n =2and 1<α<2.Proof.Similar to the above proof,there exists an invertible matrix P such that P −1AP =J ,where J is the Jordan canonical form of matrix A .Let us denote y =P −1x and substitute into Eq.(5),the system can be described asRL D α0,t y (t )=Jy (t ),(6)with initial values RL D α−10,t y (t )|t =0=y 0=P −1x 0and RL D α−20,t y (t )|t =0=y 1=P −1x 1.Here are also two cases to discuss:•for J = λ100λ2,applying Laplace transform to Eq.(6),we have y i (t )=y 0i ·t α−1E α,α(λi t α)+y 1i ·t α−2E α,α−1(λi t α),where y 0i is the i -th component of y 0,y 1i is the i -th component of y 1,i =1,2.According to Proposition 1,y i (t )∼1αy 0i λ(1−α)/αi +y 1i λ(2−α)/αi exp λ1αi t ,with t →∞,|arg(λi )|≤απ2,andy i (t )∼−y 0i λ−2i Γ(−α)t −α−1−y 1i λ−2i Γ(−1−α)t −α−2,with t →∞and |arg(λi )|>απ2.•for J = λ10λ ,similarly,y 1(t )=y 01·t α−1E α,α(λt α)+y 11·t α−2E α,α−1(λt α)+y 02·t 2α−1E (1)α,α(λt α)+y 12·t 2α−2E (1)α,α−1(λt α),y 2(t )=y 02·t α−1E α,α(λt α)+y 12·t α−2E α,α−1(λt α).From Proposition 1,one getsy 02·t 2α−1E (1)α,α(λt α)+y 12·t 2α−2E (1)α,α−1(λt α)∼y 02+y 12α d dσ σ(1−α)/α+σ(2−α)/α exp σ1αt σ=λ,36The European Physical Journal Special Topicswith t→∞,|arg(λ)|≤απ2,andy02·t2α−1E(1)α,α(λtα)+y12·t2α−2E(1)α,α−1(λtα)∼2y02Γ(−α)λ−3t−α−1+2y12Γ(−1−α)λ−3t−α−2,with t→∞and|arg(λ)|>απ2.The proof is complete. Recently,the stability analysis of linear FDEs is much involved.Some new methods are presented,such as the popular Laplace transform methods(i.e.,the frequency domain methods)[27–32],the Linear Matrix Inequalities(LMI)methods[25,26,33, 34]and the conversion methods[10],[35–37].These summaries are as follows.3.2The frequency domain methods for stability analysis of linear FDEsThe stability analysis of linear FDEs is very important in the control area,where the frequency domain method is often used.It is known that the key point of the stability analysis for linear FDEs is to determine the location of the roots of charac-teristic equations in the complex plane.It seems difficult to calculate all the roots of the characteristic equations because of the equations with fractional power options. Therefore,the authors in[27]made the following variable substitution.In general, the characteristic equation of a linear fractional differential equation has the formni=1a i sαi=0,(7)where0<αi≤1,for i=1,2,...,n.Whenαi=u i vi is rational,where(u i,v i)=1,the above equation may be rewritten asni=1a i s i m=0,(8)where m is an integer and m=LCM{v1,v2,...,v n}.Translating this equation into the W-plane yields:ni=1a i W i=0,(9) where W=s1m.The steps for stability analysis are as follows[27,31]:1.For given a i,calculate the roots of Eq.(9)andfind the minimum absolute phaseof all roots|θW min|.2.The condition for stability is|θW min|>π2m,while the condition for oscillation is|θW min|=π2m,otherwise the system is unstable.3.Roots in the primary sheet of the W-plane which have corresponding roots in thes-plane can be obtained byfinding all roots which lie in the region|θW|<πm, then applying the inverse transformation s=W m.Evidently the time response of the system can be easily related to these roots.Recently,Trigeassou et al.[32]has presented a new frequency domain method based on Nyquist’s criterion to test the stability of FDEs.The stability of FDEs with one, two derivatives was investigated,some analytic results for systems with N fractional derivatives were given.The detailed analysis can be referred to[32].Thefirst fre-quency domain method is simple,but it is hard to calculate the roots of Eq.(9). The second frequency domain method is based on Nyquist’s criterion(a graphical approach),so the latter is more intuitive.3.3LMI conditions for stability analysis of linear FDEsIn what follows,we will survey the Linear Matrix Inequalities(LMI)conditions on stability of linear FDEs.LMI has played an important role in control theory since the1960s due to its particular form.The main issue when dealing with LMI is the convexity of the optimization set,however,the stability domain of the same order system described by Eq.(5)with order1≤α<2is a convex set,various LMI methods for defining such a region have already been developed[25,26,33,34].Hence a LMI-based theorem for the stability of fractional differential system(5)with order 1≤α<2can be introduced as follows[25,26,34]:Theorem7.A fractional differential system described by Eq.(5)with order1≤α< 2is asymptotically stable if and only if there exists a matrix P=P T>0,P∈R n×n,such that(A T P+P A)sin(απ2)(A T P−P A)cos(απ2)(P A−A T P)cos(απ2)(A T P+P A)sin(απ2)<0.Unfortunately,the stability domain is not convex when0<α<1.The LMI condi-tions can thus not be directly derived.Here there are three different ideas how one can apply these methods to obtain LMI conditions for the stability of linear FDEs. The following three theorems have also been proven in[25,26,34].Theorem8.A fractional differential system described by Eq.(5)with order0<α< 1is asymptotically stable if there exists a matrix P>0,P∈R n×n,such thatA1α TP+PA1α<0.The derivation of above Theorem8(see also[33])is based on an algebraic transfor-mation of the system(5)combined with Lyapunov’s second method.This method has conservatism(explanation can be seen in[26]),so it is a sufficient condition.In order to avoid the conservatism,a new stability theorem based on a geometric analysis of the stability domain was proposed as follows.Theorem9.A fractional differential system described by Eq.(5)with order0<α< 1is asymptotically stable if and only if there exists a positive definite matrix P∈S, where S denotes the set of symmetric matrices,such that−(−A)12−α TP+P−(−A)12−α<0.However,LMI of Theorem9is not linear in relation to matrix A,thus limiting its use in the more specific control problems.In order to overcome this problem,a third condition is proposed.It based on the fact that instability domain is a convex subset of the complex plane when0<α<1.Theorem10.A fractional differential system described by Eq.(5)with order0<α<1is asymptotically stable if and only if there does not exist any nonnegative rank one matrix Q∈C n×n such that(AQ+QA T)sin(απ2)(AQ−QA T)cos(απ2)(AQ−QA T)cos(απ2)(AQ+QA T)sin(απ2)≥0.The following statement is based on stability domain decomposition[34].。

Hilfer分数阶微分方程解的延拓性孙瑜;顾海波;张艳辉;王仁正【摘要】文章研究了一类带有初值的Hilfer分数阶微分方程.首先应用Schauder 不动点定理,证明了解的局部存在性.然后,在经典微分方程连续性定理的研究思想和方法的基础上,进一步讨论Hilfer分数阶微分方程初值问题延拓定理及分数阶微分方程解的全局存在性.【期刊名称】《新疆师范大学学报（自然科学版）》【年(卷),期】2018(037)001【总页数】9页(P33-41)【关键词】分数阶微分方程;解的存在性;解的延拓;不动点定理【作者】孙瑜;顾海波;张艳辉;王仁正【作者单位】新疆师范大学数学科学学院,新疆乌鲁木齐 830017;新疆师范大学数学科学学院,新疆乌鲁木齐 830017;新疆师范大学数学科学学院,新疆乌鲁木齐830017;巴楚县第二中学,新疆巴楚 843800【正文语种】中文【中图分类】O177.91文章考虑带有初值条件类型的Hilfer分数阶微分方程：其中是Hilfer分数阶微分算子，f∶R+×R→R是给定的连续函数是Riemann-Liouville分数阶积分。

分数阶微积分也被称为广义或任意阶微积分，主要是针对任意阶微分方程进行积分和导数相关理论及应用的研究。

分数阶微积分在科学和工程等多种领域得到实践应用，在控制、系统与信号处理方面尤为突出。

几个世纪以来，研究者已经对整数阶微分方程有了深入的探索，并建立了极为系统和严密的理论体系。

相对于整数阶微分方程的研究，分数阶微分方程相关理论的研究发展缓慢。

然而，近些年来随着分数阶微分方程在众多领域的应用实践，方程模型大量涌出，研究者依据整数阶微分方程的研究思路和方法，对分数阶微分方程有了更为深入的探究。

因此，文章研究了Hilfer分数阶微分方程解的延拓性，全文由三个部分组成。

第一部分列出文章中所需要的定理、引理、注记。

第二部分中应用Schauder不动点定理，证明解的局部存在性。

摘要摘要在近几十年里，分数阶导数越来越引起数学家与物理学家的关注。

分数阶导数的定义有二十种之多，最常被人使用的有：Riemann-Liouville定义，Caputo定义，Jumare’s定义和Conformable定义等。

随着分数阶导数的发展，很多物理工程上的数学模型都可以最终转换成为分数阶微分方程的定解问题，例如：控制论和智能机器人、系统处理和信号识别、热学和光学系统、材料科学及力学和材料系统等。

但是，我们要想找到分数阶微分方程的精确解是相当困难的事情，从而人们转向求分数阶微分方程的近似解析解。

因此，一些逼近方法被应用于求解分数阶微分方程。

目前，在求解分数阶微分方程中比较有效的逼近方法有：同伦摄动法（HPM），同伦分析法（HAM），Adomian分解法(ADM)，变分迭代法（VIM），有限元方法，有限差分方法，线性多步算法和小波分析方法等。

对于上述算法都有其自身的优点与局限性。

在本文中，我们结合了分数阶Sumudu变换和分数阶Elzaki变换，建立了几种新的分数阶微分方程的逼近算法，这些新的算法被成功地应用于求不同类型的分数阶微分方程的近似解析解，通过将新的算法所得逼近解与已有的结果比较，得出我们建立的新的逼近算法具有计算简单、有效、精确度更高等优点。

在本文中我们也成功建立了求解局部分数阶微分方程逼近解的新算法。

本文所建立的四种求分数阶微分方程近似解析解的算法如下：1.分数阶同伦分析变换算法（FHATM）。

分数阶同伦分析变换算法（FHATM）的优点是所求分数阶微分方程的逼近解被辅助参数h所控制，合适的选取h的值将大大加速逼近解的收敛速度，在分数阶同伦分析变换算法中我们加入了分数阶Elzaki变换，使得求解过程简单快捷，通过和传统的经典算法比较可以得出：一些经典的算法可归结为分数阶同伦分析变换算法（FHATM）。

我们使用分数阶同伦分析变换算法（FHATM）成功求解非线性的时间分数阶Fornberg-Whitham方程，二维时间分数阶扩散方程，二维时间分数阶波方程和三维时间分数阶扩散方程。

得分：_______ 南京林业大学研究生课程论文2013 ～2014 学年第 1 学期课程号：PD03088课程名称：工程应用专题题目：分数阶控制理论研究及工程领域的应用学科专业：机械工程学号：********名：***任课教师：**二○一四年一月分数阶控制理论研究及工程领域的应用摘要: 作为控制科学与工程中一个新的研究领域，分数阶控制的研究愈来愈被关注。

本文简要介绍分数阶控制的数学背景和基本知识,对分数阶控制理论及应用(分数阶系统模型、系统分析、分数阶控制器、非线性分数阶系统、系统辨识) 的研究作了总结、评述和展望。

关键词:控制理论；分数阶微积分(FOC)；分数阶系统Fractional Control Theory and EngineeringApplicationsQian Dongxing（Nanjing Forestry University, Nanjing Jiangsu 210037）Abstract: As a new study field of control theory and applications , the fractional order control is attracted much attention recently. In this paper, an overview in this field is surveyed. The historical development and the basic knowledge of fractional-order control are introduced. The latest works of fractional-order control are summarized and reviewed, including mathematical model, system analysis, fractional-order controller, nonlinear fractional order system and identification, etc. Some future trends in its further studies are prospected.Key words: Theory of control ；Fractional order calculus( FOC) ；Fractional order system1 引 言目前,几乎所有的以微分方程描述的控制系统，其微分均考虑为整数阶。

A New Logical Topology Based on Barrel Shifter Network over anAll Optical NetworkN. ChakiUniversity of CalcuttaKolkata, India nabendu@R. ChakiJoint Plant CommitteeKolkata, India rchaki@B. SahaUniversity of CalcuttaKolkata, Indiabimansaha@rediffmail.co mT. Chattopadhyay University of CalcuttaKolkata, India titas@hotpop.co mAbstractThis paper presen ts a n ew logical topology SBS-n et, a Scalable Barrel Shifter n etwork to be used as a logical topology over an all-optical network using WDM. The major emphasis of the presen t work is to improve upon the scalability issue. This SBS-et co ects a y arbitrary numbers of nodes as opposed to the Barrel Shifter, de Bruijn graph a n d Shuffle n et. The average hoppi n g dista ncebetween two nodes using this topology is smaller comparedto that in de Bruijn Graph, Shufflenet & GEM net .1. IntroductionThis paper considers the problem of enhancing the scalability of an optical network [4], [5], [6], [3], by overlaying a new logical topology over a wavelength routed all optical network physical topology. Scalability, indeed, is one of the primary concerns in designing an optical topology due to high prices of components. On an all-optical network physical topology, light paths can be set between any pair of nodes. By carefully selecting lightpaths, a logical topology can be overlaid upon the physical topology of the network. Node pairs that are not directly connected via lightpaths must use a sequence of lightpaths through some intermediate nodes for communication between them. De Bruijn graphs [1], GEM-nets [2] are examples of such multi-hop [7] logical topologies. The present paper compares the features of all different existing logical topologies with a new topology SBS-net.2. SBS-net TopologyThis topology assumes exactly same structure as that of a re-circulating single-stage Barrel-shifter when number of nodes, N=2n , with an undirected link from node c j to c k ,called neighbor of c j , iff c k {c j r 2i mod N}, where i=0,1, …,n-1. Each node has exactly 2n-1 links attached to it. Diameter of the graph is log 2N/2.When 2n-1<N<2n , a node c j’ has neighbors, c j’r 2i mod 2n ,i=0,1,.….,n-1. All neighbors may not be present. So number of links used by a node may be less than 2n-1. We alsoproposed an improvised utilization strategy for unused links. With this improvised strategy the regular interconnection pattern of a node with its neighbors is violated and node c j’can have neighbors other than the nodes c j’r 2i mod 2n , which are called temporary n eighbors . Their addresses are c j’r 2q +2q-1, q {0,1,……, n-1}.3. Routing Strategy for N=2nA region refers to a set of nodes between two neighbors.Whole region:If a region with respect to a node has both itsboundary nodes present, it is said to be a whole region. Broken region:When 2n-1<N<2n , a node may not have all its neighbors. I n this case a region may not have both its boundary nodes, it is said to be a broken region.Property 1: In a region AB, if P is any arbitrary node then if P is closer to B(A), P can be reached from B(A) in fewer number of hops when leaps are to be taken in 2’s power. Property 2: The path to P thru B(A) is the shortest path; P cannot be reached via any other neighbor with fewer number of hops.Routing, here, is very straightforward. Each node keeps a data structure containing information about their neighbors. When a source wants to send data to a destination , it checks to see which region the destination belongs and sends data to the nearer boundary. If boundary node itself is the intended node, i.e., source has a direct link to destination, the job is done. Otherwise we try to find out next region with respect to the node we just reached, which encloses destination. In effect the enclosing region gradually narrows down and eventually destination coincides with a boundary. Maximum possible hopping in this routing is ªlog 2N/2º or ªn/2º.3.1. Problem When 2n-1<N<2nWhen 2n-1<N<2n , all neighbors of a node may not be present. Therefore, a few of the regions may be either not present or broken. Absence of an entire region has no effect over routing. But when there is a broken region, routing via the only existing boundary instead of the nearer one may increase required hopping.4. Revised Routing Strategy for 2n-1<N<2nHere, when a source s wants to send data to any destination d, d could belong to a whole region as before; in this case the routing is same as described in the previous section. Otherwise d belongs to a broken region. In this case s looks for all neighbors of d (d r 2i , i=0,1, …, n-1). It then checks to see which of them (one or more) lie in one of its whole regions. This check ensures both the existence of that neighbor, as well as a viable alternative, called target to send data to. At the most ªlog 22n /2º or ªn/2º hops are required to do so. So maximum number of hops required is ªn/2º+1.Property 3: When 2n-1<N<2n , for a node A in a broken region of another node B, at least one of A’s neighbors fall in a whole region of B.4.1. Problem of Porous RegionsA porous region with respect to a node R is a region such that both of the end nodes of it exists while some of the intermediate nodes are not there. To send data at some existing node in this type of regions, if source chooses the nearer boundary and transfer data, for that boundary or some other subsequent nodes along the path the destination may fall in a broken region.This problem does not exist when N<2n-1+2n-2, as no node can have a region spanning more than 2n-2 nodes.For all such porous regions, one of its terminals would be in 1st & 4th quadrant each. When the destination is c=11a n-3…a 0, i.e., c is in 4th quadrant, routing entails some additional considerations, such as,I . I f the destination is in a whole region, i.e., its terminals are t 1=11a n-3…a 0& t 2=11a n-3…a 0, routing is as usual.II. Otherwise, c is either in porous or broken region; wechoose its farthest neighbor (c r 2n-1), the target, in 2nd quadrant and send data to it. All imaginable enclosing regions (since longest region contains 2n-2nodes) of target must be full. Again maximum number of hops required is ªn/2º+1.5. Unused Links Utilization PolicyConsider a node R. We bifurcate R’s broken region(s) as soon as it(they) is(are) at least half full and attach its unused link with its(their) middle node. This effectively creates the longest possible whole region with length 2i , i=0,1, …, n-2 out of that broken one. Routing to this newly formed region is thus simplified.A node may have half/more than half full region(s), so its unused link(s) is(are) required to connect to its(their) middle node(s).Whereas the node itself may be one such middle node of other node(s). We start from node 0 and moveanticlockwise to utilize unused links at each node, if there is any. Both the origin node and terminal node are considered.5.1. Impact on Neighbor Set of a NodeWhen a node grants link request from elsewhere, it implies that this node is the middle node of a broken region of another node. I ts own neighbor set and consequently its regions are changed.A region of 2l nodes is formed by links 2l & 2l+1. Whenever a link is set up with the middle node of such a region, it gets a new neighbor 2l +2l-1nodes apart. Therefore this will always bifurcate its own 2l -nodes region, be it broken or whole.T he maximum possible hopping for routing in these newly formed regions is ¬n/2¼.6. PerformanceFor a single-stage barrel shifter the minimum number of re-circulations B is upper bounded by B d ª log 2N/2º When 2n-1<N<2n , maximum number of re-circulations required is shown to be ªlog 22n /2º+1 or ªn/2º+1, just one hop greater than the regular one.7. ConclusionSBS-net introduces a higher degree of scalability in multihop WDM optical network, compared to the existing topologies such as de Bruijn Graph and Shufflenet etc. by extending the routing strategy of Barrel Shifter network to connect any number of nodes. A method to utilize unused links when total number of nodes in the modified network is not power 2 is also presented in this paper.8. References[1] Kumar N. Sivarajan, Rajib Ramaswami; Lightwave Networks based on de Bruiju Graphs in IEEE/ACM TRANSACTI ONS ON NETWORKING, vol. 2 no. 1, February, 1994.[2] J. Iness, S. Banerjee, B. Mukherjee; GEMNET : A Generalised, Shuffle exchange based, Regular, Scalable, Modular, Multihop, WDM Lightwave Network in I EEE/ACM TRANSACTI ONS ON NETWORKING vol. 3., no. 4, August, 1995.[3] M. A. Marsan, A. Bianco, E. Leonardi, F. Neri; "A comparison of regular topologies for all optical networks" in Proc. INFOCOM '93, San Francisco, CA, March, 1993.[4] P. E. Green; "The Future of Fiber-optic Computer Networks", IEEE Computer, vol. 24, pp. 78-87, September, 1991.[5] B. Mukherjee : "Optical Communication Networks", Mc-Graw Hill Publishing Compnay, 1st edition.[6] B. Mukherjee, S. Ramamurthy, D. Banerjee and A. Mukherjee : "Some principles for designing a wide-area optical network", Proceedings IEEE INFOCOM '94, 1994.[7] B. Mukherjee : "WDM-based local lightwave networks - Part II : multihop systems", IEEE network magazine,vol 6, no 4, pp 20-32, July 1992.。

一些常见的统计术语翻译Absolute deviation, 绝对离差Absolute number, 绝对数Absolute residuals, 绝对残差Acceleration array, 加速度立体阵Acceleration in an arbitrary direction, 任意方向上的加速度Acceleration normal, 法向加速度Acceleration space dimension, 加速度空间的维数Acceleration tangential, 切向加速度Acceleration vector, 加速度向量Acceptable hypothesis, 可承受假设Accumulation, 累积Accuracy, 准确度Actual frequency, 实际频数Adaptive estimator, 自适应估计量Addition, 相加Addition theorem, 加法定理Additivity, 可加性Adjusted rate, 调整率Adjusted value, 校正值Admissible error, 容许误差Aggregation, 聚集性Alternative hypothesis, 备择假设Among groups, 组间Amounts, 总量Analysis of correlation, 相关分析Analysis of covariance, 协方差分析Analysis of regression, 回归分析Analysis of time series, 时间序列分析Analysis of variance, 方差分析Angular transformation, 角转换ANOVA 〔analysis of variance〕, 方差分析ANOVA Models, 方差分析模型Arcing, 弧/弧旋Arcsine transformation, 反正弦变换Area under the curve, 曲线面积AREG , 评估从一个时间点到下一个时间点回归相关时的误差 ARIMA, 季节和非季节性单变量模型的极大似然估计 Arithmetic grid paper, 算术格纸Arithmetic mean, 算术平均数Arrhenius relation, 艾恩尼斯关系Assessing fit, 拟合的评估Associative laws, 结合律Asymmetric distribution, 非对称分布Asymptotic bias, 渐近偏倚Asymptotic efficiency, 渐近效率Asymptotic variance, 渐近方差Attributable risk, 归因危险度Attribute data, 属性资料Attribution, 属性Autocorrelation, 自相关Autocorrelation of residuals, 残差的自相关Average, 平均数Average confidence interval length, 平均置信区间长度Average growth rate, 平均增长率Bar chart, 条形图Bar graph, 条形图Base period, 基期Bayes' theorem , Bayes定理Bell-shaped curve, 钟形曲线Bernoulli distribution, 伯努力分布Best-trim estimator, 最好切尾估计量Bias, 偏性Binary logistic regression, 二元逻辑斯蒂回归Binomial distribution, 二项分布Bisquare, 双平方Bivariate Correlate, 二变量相关Bivariate normal distribution, 双变量正态分布Bivariate normal population, 双变量正态总体Biweight interval, 双权区间Biweight M-estimator, 双权M估计量Block, 区组/配伍组BMDP(Biomedical puter programs), BMDP统计软件包Bo*plots, 箱线图/箱尾图Breakdown bound, 崩溃界/崩溃点Canonical correlation, 典型相关Caption, 纵标目Case-control study, 病例对照研究Categorical variable, 分类变量Catenary, 悬链线Cauchy distribution, 柯西分布Cause-and-effect relationship, 因果关系Cell, 单元Censoring, 终检Center of symmetry, 对称中心Centering and scaling, 中心化和定标Central tendency, 集中趋势Central value, 中心值CHAID -χ2 Automatic Interaction Detector, 卡方自动交互检测Chance, 机遇Chance error, 随机误差Chance variable, 随机变量Characteristic equation, 特征方程Characteristic root, 特征根Characteristic vector, 特征向量Chebshev criterion of fit, 拟合的切比雪夫准则Chernoff faces, 切尔诺夫脸谱图Chi-square test, 卡方检验/χ2检验Choleskey deposition, 乔洛斯基分解Circle chart, 圆图 Class interval, 组距Class mid-value, 组中值Class upper limit, 组上限Classified variable, 分类变量Cluster analysis, 聚类分析Cluster sampling, 整群抽样Code, 代码Coded data, 编码数据Coding, 编码Coefficient of contingency, 列联系数Coefficient of determination, 决定系数Coefficient of multiple correlation, 多重相关系数Coefficient of partial correlation, 偏相关系数Coefficient of production-moment correlation, 积差相关系数Coefficient of rank correlation, 等级相关系数Coefficient of regression, 回归系数Coefficient of skewness, 偏度系数Coefficient of variation, 变异系数Cohort study, 队列研究Column, 列Column effect, 列效应Column factor, 列因素bination pool, 合并binative table, 组合表mon factor, 共性因子mon regression coefficient, 公共回归系数mon value, 共同值mon variance, 公共方差mon variation, 公共变异munality variance, 共性方差parability, 可比性parison of bathes, 批比拟parison value, 比拟值partment model, 分部模型passion, 伸缩plement of an event, 补事件plete association, 完全正相关plete dissociation, 完全不相关plete statistics, 完备统计量pletely randomized design, 完全随机化设计posite event, 联合事件posite events, 复合事件Concavity, 凹性Conditional e*pectation, 条件期望Conditional likelihood, 条件似然Conditional probability, 条件概率Conditionally linear, 依条件线性Confidence interval, 置信区间Confidence limit, 置信限Confidence lower limit, 置信下限Confidence upper limit, 置信上限Confirmatory Factor Analysis , 验证性因子分析Confirmatory research, 证实性实验研究Confounding factor, 混杂因素Conjoint, 联合分析Consistency, 相合性Consistency check, 一致性检验Consistent asymptotically normal estimate, 相合渐近正态估计Consistent estimate, 相合估计Constrained nonlinear regression, 受约束非线性回归Constraint, 约束Contaminated distribution, 污染分布Contaminated Gausssian, 污染高斯分布Contaminated normal distribution, 污染正态分布Contamination, 污染Contamination model, 污染模型Contingencytable, 列联表Contour, 边界限Contribution rate, 奉献率Control, 对照Controlled e*periments, 对照实验Conventional depth, 常规深度Convolution, 卷积Corrected factor, 校正因子Corrected mean, 校正均值Correction coefficient, 校正系数Correctness, 正确性 Correlation coefficient, 相关系数Correlation inde*, 相关指数Correspondence, 对应Counting, 计数Counts, 计数/频数Covariance, 协方差Covariant, 共变 Co* Regression, Co*回归Criteria for fitting, 拟合准则Criteria of least squares, 最小二乘准则Critical ratio, 临界比Critical region, 拒绝域Critical value, 临界值Cross-over design, 穿插设计Cross-section analysis, 横断面分析Cross-section survey, 横断面调查Crosstabs , 穿插表 Cross-tabulation table, 复合表Cube root, 立方根Cumulative distribution function, 分布函数Cumulative probability, 累计概率Curvature, 曲率/弯曲Curvature, 曲率Curve fit , 曲线拟和 Curve fitting, 曲线拟合Curvilinear regression, 曲线回归Curvilinear relation, 曲线关系Cut-and-try method, 尝试法Cycle, 周期Cyclist, 周期性D test, D检验Data acquisition, 资料收集Data bank, 数据库Data capacity, 数据容量Data deficiencies, 数据缺乏Data handling, 数据处理Data manipulation, 数据处理Data processing, 数据处理Data reduction, 数据缩减Data set, 数据集Data sources, 数据来源Data transformation, 数据变换Data validity, 数据有效性Data-in, 数据输入Data-out, 数据输出Dead time, 停滞期Degree of freedom, 自由度Degree of precision, 精细度Degree of reliability, 可靠性程度Degression, 递减Density function, 密度函数Density of data points, 数据点的密度Dependent variable, 应变量/依变量/因变量Dependent variable, 因变量Depth, 深度Derivative matri*, 导数矩阵Derivative-free methods, 无导数方法Design, 设计Determinacy, 确定性Determinant, 行列式Determinant, 决定因素Deviation, 离差Deviation from average, 离均差Diagnostic plot, 诊断图Dichotomous variable, 二分变量Differential equation, 微分方程Direct standardization, 直接标准化法Discrete variable, 离散型变量DISCRIMINANT, 判断Discriminant analysis, 判别分析Discriminant coefficient, 判别系数Discriminant function, 判别值Dispersion, 散布/分散度Disproportional, 不成比例的Disproportionate sub-class numbers, 不成比例次级组含量Distribution free, 分布无关性/免分布Distribution shape, 分布形状Distribution-free method, 任意分布法Distributive laws, 分配律Disturbance, 随机扰动项Dose response curve, 剂量反响曲线 Double blind method, 双盲法Double blind trial, 双盲试验Double e*ponential distribution, 双指数分布Double logarithmic, 双对数Downward rank, 降秩Dual-space plot, 对偶空间图DUD, 无导数方法Duncan's new multiple range method, 新复极差法/Duncan新法 E-LEffect, 实验效应Eigenvalue, 特征值Eigenvector, 特征向量Ellipse, 椭圆Empirical distribution, 经历分布Empirical probability, 经历概率单位Enumeration data, 计数资料Equal sun-class number, 相等次级组含量 Equally likely, 等可能Equivariance, 同变性Error, 误差/错误Error of estimate, 估计误差Error type I, 第一类错误Error type II, 第二类错误Estimand, 被估量Estimated error mean squares, 估计误差均方Estimated error sum of squares, 估计误差平方和Euclidean distance, 欧式距离Event, 事件Event, 事件E*ceptional data point, 异常数据点E*pectation plane, 期望平面E*pectation surface, 期望曲面E*pected values, 期望值E*periment, 实验E*perimental sampling, 试验抽样E*perimental unit, 试验单位E*planatory variable, 说明变量E*ploratory data analysis, 探索性数据分析E*plore Summarize, 探索-摘要E*ponential curve, 指数曲线E*ponential growth, 指数式增长E*SMOOTH, 指数平滑方法 E*tended fit, 扩大拟合E*tra parameter, 附加参数E*trapolation, 外推法E*treme observation, 末端观测值 E*tremes, 极端值/极值F distribution, F分布F test, F检验Factor, 因素/因子Factor analysis, 因子分析Factor Analysis, 因子分析Factor score, 因子得分Factorial, 阶乘Factorial design, 析因试验设计False negative, 假阴性False negative error, 假阴性错误Family of distributions, 分布族Family of estimators, 估计量族Fanning, 扇面Fatality rate, 病死率Field investigation, 现场调查Field survey, 现场调查Finite population, 有限总体Finite-sample, 有限样本First derivative, 一阶导数 First principal ponent, 第一主成分First quartile, 第一四分位数Fisher information, 费雪信息量Fitted value, 拟合值Fitting a curve, 曲线拟合Fi*ed base, 定基Fluctuation, 随机起伏Forecast, 预测Four fold table, 四格表Fourth, 四分点Fraction blow, 左侧比率Fractional error, 相对误差Frequency, 频率Frequency polygon, 频数多边图Frontier point, 界限点Function relationship, 泛函关系Gamma distribution, 伽玛分布Gauss increment, 高斯增量Gaussian distribution, 高斯分布/正态分布Gauss-Newton increment, 高斯-牛顿增量General census, 全面普查GENLOG (Generalized liner models), 广义线性模型 Geometric mean, 几何平均数Gini's mean difference, 基尼均差GLM (General liner models), 通用线性模型 Goodness of fit, 拟和优度/配合度Gradient of determinant, 行列式的梯度Graeco-Latin square, 希腊拉丁方Grand mean, 总均值Gross errors, 重大错误Gross-error sensitivity, 大错敏感度Group averages, 分组平均Grouped data, 分组资料Guessed mean, 假定平均数Half-life, 半衰期Hampel M-estimators, 汉佩尔M估计量Happenstance, 偶然事件Harmonic mean, 调和均数Hazard function, 风险均数Hazard rate, 风险率Heading, 标目 Heavy-tailed distribution, 重尾分布Hessian array, 海森立体阵Heterogeneity, 不同质Heterogeneity of variance, 方差不齐 Hierarchical classification, 组分组Hierarchical clustering method, 系统聚类法High-leverage point, 高杠杆率点HILOGLINEAR, 多维列联表的层次对数线性模型Hinge, 折叶点Histogram, 直方图Historical cohort study, 历史性队列研究 Holes, 空洞HOMALS, 多重响应分析Homogeneity of variance, 方差齐性Homogeneity test, 齐性检验Huber M-estimators, 休伯M估计量Hyperbola, 双曲线Hypothesis testing, 假设检验Hypothetical universe, 假设总体Impossible event, 不可能事件Independence, 独立性Independent variable, 自变量Inde*, 指标/指数Indirect standardization, 间接标准化法Individual, 个体Inference band, 推断带Infinite population, 无限总体Infinitely great, 无穷大Infinitely small, 无穷小Influence curve, 影响曲线Information capacity, 信息容量Initial condition, 初始条件Initial estimate, 初始估计值Initial level, 最初水平Interaction, 交互作用Interaction terms, 交互作用项Intercept, 截距Interpolation, 插法Interquartile range, 四分位距Interval estimation, 区间估计Intervals of equal probability, 等概率区间Intrinsic curvature, 固有曲率Invariance, 不变性Inverse matri*, 逆矩阵Inverse probability, 逆概率Inverse sine transformation, 反正弦变换Iteration, 迭代Jacobian determinant, 雅可比行列式Joint distribution function, 分布函数Joint probability, 联合概率Joint probability distribution, 联合概率分布K means method, 逐步聚类法Kaplan-Meier, 评估事件的时间长度 Kaplan-Merier chart, Kaplan-Merier图Kendall's rank correlation, Kendall等级相关Kinetic, 动力学Kolmogorov-Smirnove test, 柯尔莫哥洛夫-斯米尔诺夫检验Kruskal and Wallis test, Kruskal及Wallis检验/多样本的秩和检验/H检验Kurtosis, 峰度Lack of fit, 失拟Ladder of powers, 幂阶梯Lag, 滞后Large sample, 大样本Large sample test, 大样本检验Latin square, 拉丁方Latin square design, 拉丁方设计Leakage, 泄漏Least favorable configuration, 最不利构形Least favorable distribution, 最不利分布Least significant difference, 最小显著差法Least square method, 最小二乘法Least-absolute-residuals estimates, 最小绝对残差估计Least-absolute-residuals fit, 最小绝对残差拟合Least-absolute-residuals line, 最小绝对残差线Legend, 图例L-estimator, L估计量L-estimator of location, 位置L 估计量L-estimator of scale, 尺度L估计量Level, 水平Life e*pectance, 预期期望寿命Life table, 寿命表Life table method, 生命表法Light-tailed distribution, 轻尾分布Likelihood function, 似然函数Likelihood ratio, 似然比line graph, 线图Linear correlation, 直线相关Linear equation, 线性方程Linear programming, 线性规划Linear regression, 直线回归Linear Regression, 线性回归Linear trend, 线性趋势Loading, 载荷 Location and scale equivariance, 位置尺度同变性Location equivariance, 位置同变性Location invariance, 位置不变性Location scale family, 位置尺度族Log rank test, 时序检验 Logarithmic curve, 对数曲线Logarithmic normal distribution, 对数正态分布Logarithmic scale, 对数尺度Logarithmic transformation, 对数变换Logic check, 逻辑检查Logistic distribution, 逻辑斯特分布Logit transformation, Logit转换LOGLINEAR, 多维列联表通用模型 Lognormal distribution, 对数正态分布Lost function, 损失函数Low correlation, 低度相关Lower limit, 下限Lowest-attained variance, 最小可达方差LSD, 最小显著差法的简称Lurking variable, 潜在变量M-RMain effect, 主效应Major heading, 主辞标目Marginal density function, 边缘密度函数Marginal probability, 边缘概率Marginal probability distribution, 边缘概率分布Matched data, 配对资料Matched distribution, 匹配过分布Matching of distribution, 分布的匹配Matching of transformation, 变换的匹配Mathematical e*pectation, 数学期望Mathematical model, 数学模型Ma*imum L-estimator, 极大极小L 估计量Ma*imum likelihood method, 最大似然法Mean, 均数Mean squares between groups, 组间均方Mean squares within group, 组均方Means (pare means), 均值-均值比拟Median, 中位数Median effective dose, 半数效量Median lethal dose, 半数致死量Median polish, 中位数平滑Median test, 中位数检验Minimal sufficient statistic, 最小充分统计量Minimum distance estimation, 最小距离估计Minimum effective dose, 最小有效量Minimum lethal dose, 最小致死量Minimum variance estimator, 最小方差估计量MINITAB, 统计软件包Minor heading, 宾词标目Missing data, 缺失值 Model specification, 模型确实定Modeling Statistics , 模型统计Models for outliers, 离群值模型Modifying the model, 模型的修正Modulus of continuity, 连续性模Morbidity, 发病率 Most favorable configuration, 最有利构形Multidimensional Scaling (ASCAL), 多维尺度/多维标度Multinomial Logistic Regression , 多项逻辑斯蒂回归Multiple parison, 多重比拟Multiple correlation , 复相关Multiple covariance, 多元协方差Multiple linear regression, 多元线性回归Multiple response , 多重选项Multiple solutions, 多解Multiplication theorem, 乘法定理Multiresponse, 多元响应Multi-stage sampling, 多阶段抽样Multivariate T distribution, 多元T分布Mutual e*clusive, 互不相容Mutual independence, 互相独立Natural boundary, 自然边界Natural dead, 自然死亡Natural zero, 自然零Negative correlation, 负相关Negative linear correlation, 负线性相关Negatively skewed, 负偏Newman-Keuls method, q检验NK method, q检验No statistical significance, 无统计意义Nominal variable, 名义变量Nonconstancy of variability, 变异的非定常性Nonlinear regression, 非线性相关Nonparametric statistics, 非参数统计Nonparametric test, 非参数检验Nonparametric tests, 非参数检验Normal deviate, 正态离差Normal distribution, 正态分布Normal equation, 正规方程组Normal ranges, 正常围Normal value, 正常值Nuisance parameter, 多余参数/讨厌参数Null hypothesis, 无效假设 Numerical variable, 数值变量Objective function, 目标函数Observation unit, 观察单位Observed value, 观察值One sided test, 单侧检验One-way analysis of variance, 单因素方差分析Oneway ANOVA , 单因素方差分析Open sequential trial, 开放型序贯设计Optrim, 优切尾Optrim efficiency, 优切尾效率Order statistics, 顺序统计量Ordered categories, 有序分类Ordinal logistic regression , 序数逻辑斯蒂回归Ordinal variable, 有序变量Orthogonal basis, 正交基Orthogonal design, 正交试验设计Orthogonality conditions, 正交条件ORTHOPLAN, 正交设计 Outlier cutoffs, 离群值截断点Outliers, 极端值OVERALS , 多组变量的非线性正规相关Overshoot, 迭代过度Paired design, 配对设计Paired sample, 配对样本Pairwise slopes, 成对斜率Parabola, 抛物线Parallel tests, 平行试验Parameter, 参数Parametric statistics, 参数统计Parametric test, 参数检验Partial correlation, 偏相关Partial regression, 偏回归Partial sorting, 偏排序Partials residuals, 偏残差Pattern, 模式Pearson curves, 皮尔逊曲线Peeling, 退层Percent bar graph, 百分条形图Percentage, 百分比Percentile, 百分位数Percentile curves, 百分位曲线Periodicity, 周期性Permutation, 排列P-estimator, P估计量Pie graph, 饼图Pitman estimator, 皮特曼估计量Pivot, 枢轴量Planar, 平坦Planar assumption, 平面的假设PLANCARDS, 生成试验的方案卡Point estimation, 点估计Poisson distribution, 泊松分布Polishing, 平滑Polled standard deviation, 合并标准差Polled variance, 合并方差Polygon, 多边图Polynomial, 多项式Polynomial curve, 多项式曲线Population, 总体Population attributable risk, 人群归因危险度Positive correlation, 正相关Positively skewed, 正偏Posterior distribution, 后验分布Power of a test, 检验效能Precision, 精细度Predicted value, 预测值Preliminary analysis, 预备性分析Principal ponent analysis, 主成分分析Prior distribution, 先验分布Prior probability, 先验概率Probabilistic model, 概率模型probability, 概率Probability density, 概率密度Product moment, 乘积矩/协方差Profile trace, 截面迹图Proportion, 比/构成比Proportion allocation in stratified random sampling, 按比例分层随机抽样Proportionate, 成比例Proportionate sub-class numbers, 成比例次级组含量Prospective study, 前瞻性调查Pro*imities, 亲近性 Pseudo F test, 近似F检验Pseudo model, 近似模型Pseudosigma, 伪标准差Purposive sampling, 有目的抽样QR deposition, QR分解Quadratic appro*imation, 二次近似Qualitative classification, 属性分类Qualitative method, 定性方法Quantile-quantile plot, 分位数-分位数图/Q-Q图Quantitative analysis, 定量分析Quartile, 四分位数Quick Cluster, 快速聚类Radi* sort, 基数排序Random allocation, 随机化分组Random blocks design, 随机区组设计Random event, 随机事件Randomization, 随机化Range, 极差/全距Rank correlation, 等级相关Rank sum test, 秩和检验Rank test, 秩检验Ranked data, 等级资料Rate, 比率Ratio, 比例Raw data, 原始资料Raw residual, 原始残差Rayleigh's test, 雷氏检验Rayleigh's Z, 雷氏Z值 Reciprocal, 倒数Reciprocal transformation, 倒数变换Recording, 记录Redescending estimators, 回降估计量Reducing dimensions, 降维Re-e*pression, 重新表达Reference set, 标准组Region of acceptance, 承受域Regression coefficient, 回归系数Regression sum of square, 回归平方和 Rejection point, 拒绝点Relative dispersion, 相对离散度Relative number, 相对数Reliability, 可靠性Reparametrization, 重新设置参数Replication, 重复Report Summaries, 报告摘要Residual sum of square, 剩余平方和Resistance, 耐抗性Resistant line, 耐抗线Resistant technique, 耐抗技术R-estimator of location, 位置R估计量R-estimator of scale, 尺度R估计量Retrospective study, 回忆性调查Ridge trace, 岭迹Ridit analysis, Ridit分析Rotation, 旋转Rounding, 舍入Row, 行Row effects, 行效应Row factor, 行因素R*C table, R*C表S-ZSample, 样本Sample regression coefficient, 样本回归系数Sample size, 样本量Sample standard deviation, 样本标准差Sampling error, 抽样误差SAS(Statistical analysis system ), SAS统计软件包Scale, 尺度/量表Scatter diagram, 散点图Schematic plot, 示意图/简图Score test, 计分检验Screening, 筛检SEASON, 季节分析 Secondderivative, 二阶导数Second principal ponent, 第二主成分SEM (Structural equation modeling), 构造化方程模型Semi-logarithmic graph, 半对数图Semi-logarithmic paper, 半对数格纸Sensitivity curve, 敏感度曲线Sequential analysis, 贯序分析Sequential data set, 顺序数据集Sequential design, 贯序设计Sequential method, 贯序法Sequential test, 贯序检验法Serial tests, 系列试验Short-cut method, 简捷法 Sigmoid curve, S形曲线Sign function, 正负号函数Sign test, 符号检验Signed rank, 符号秩Significance test, 显著性检验Significant figure, 有效数字Simple cluster sampling, 简单整群抽样Simple correlation, 简单相关Simple random sampling, 简单随机抽样Simple regression, 简单回归simple table, 简单表Sine estimator, 正弦估计量Single-valued estimate, 单值估计Singular matri*, 奇异矩阵Skewed distribution, 偏斜分布Skewness, 偏度Slash distribution, 斜线分布Slope, 斜率Smirnov test, 斯米尔诺夫检验Source of variation, 变异来源Spearman rank correlation, 斯皮尔曼等级相关Specific factor, 特殊因子Specific factor variance, 特殊因子方差Spectra , 频谱Spherical distribution, 球型正态分布Spread, 展布SPSS(Statistical package for the social science), SPSS统计软件包Spurious correlation, 假性相关Square root transformation, 平方根变换Stabilizing variance, 稳定方差Standard deviation, 标准差Standard error, 标准误Standard error of difference, 差异的标准误Standard error of estimate, 标准估计误差Standard error of rate, 率的标准误Standard normal distribution, 标准正态分布Standardization, 标准化Starting value, 起始值Statistic, 统计量Statistical control, 统计控制Statistical graph, 统计图Statistical inference, 统计推断Statistical table, 统计表Steepest descent, 最速下降法Stem and leaf display, 茎叶图Step factor, 步长因子Stepwise regression, 逐步回归Storage, 存Strata, 层〔复数〕Stratified sampling, 分层抽样Stratified sampling, 分层抽样Strength, 强度Stringency, 严密性Structural relationship, 构造关系Studentized residual, 学生化残差/t化残差Sub-class numbers, 次级组含量Subdividing, 分割Sufficient statistic, 充分统计量Sum of products, 积和Sum of squares, 离差平方和Sum of squares about regression, 回归平方和Sum of squares between groups, 组间平方和Sum of squares of partial regression, 偏回归平方和Sure event, 必然事件Survey, 调查Survival, 生存分析Survival rate, 生存率Suspended root gram, 悬吊根图Symmetry, 对称Systematic error, 系统误差Systematic sampling, 系统抽样Tags, 标签Tail area, 尾部面积Tail length, 尾长Tail weight, 尾重Tangent line, 切线Target distribution, 目标分布Taylor series, 泰勒级数Tendency of dispersion, 离散趋势Testing of hypotheses, 假设检验Theoretical frequency, 理论频数Time series, 时间序列Tolerance interval, 容忍区间Tolerance lower limit, 容忍下限Tolerance upper limit, 容忍上限Torsion, 扰率Total sum of square, 总平方和Total variation, 总变异Transformation, 转换Treatment, 处理Trend, 趋势Trend of percentage, 百分比趋势Trial, 试验Trial and error method, 试错法Tuning constant, 细调常数Two sided test, 双向检验Two-stage least squares, 二阶最小平方Two-stage sampling, 二阶段抽样Two-tailed test, 双侧检验Two-way analysis of variance, 双因素方差分析Two-way table, 双向表Type I error, 一类错误/α错误Type II error, 二类错误/β错误UMVU, 方差一致最小无偏估计简称Unbiased estimate, 无偏估计Unconstrained nonlinear regression , 无约束非线性回归Unequal subclass number, 不等次级组含量Ungrouped data, 不分组资料Uniform coordinate, 均匀坐标Uniform distribution, 均匀分布Uniformly minimum variance unbiased estimate, 方差一致最小无偏估计Unit, 单元Unordered categories, 无序分类Upper limit, 上限Upward rank, 升秩Vague concept, 模糊概念Validity, 有效性VARP (Variance ponent estimation), 方差元素估计Variability, 变异性Variable, 变量Variance, 方差Variation, 变异Varima* orthogonal rotation, 方差最大正交旋转Volume of distribution, 容积W test, W检验Weibull distribution, 威布尔分布Weight, 权数Weighted Chi-square test, 加权卡方检验/Cochran检验Weighted linear regression method, 加权直线回归Weighted mean, 加权平均数Weighted mean square, 加权平均方差Weighted sum of square, 加权平方和Weighting coefficient, 权重系数Weighting method, 加权法W-estimation, W估计量W-estimation of location, 位置W估计量Width, 宽度Wilco*on paired test, 威斯康星配对法/配对符号秩和检验Wild point, 野点/狂点Wild value, 野值/狂值Winsorized mean, 缩尾均值Withdraw, 失访 Youden's inde*, 尤登指数Z test, Z检验Zero correlation, 零相关Z-transformation, Z变换。