青少年风湿性关节炎新药上市 疗效达85%

r语言quarter函数在R语言中，quarter函数用于将日期向下舍入到指定季度。

它是一种非常常用的时间处理函数，可用于将日期数据进行分组和统计分析。

quarter函数的使用方法非常简单，只需将日期作为参数传入即可。

下面是quarter函数的语法：quarter(x, ...)参数x表示需要处理的日期数据，可以是日期向量、日期字符串或日期对象。

被处理的日期数据可以是单个日期，也可以是一个日期向量。

另外，quarter函数还可以接受其他参数，例如：- "start"参数用于指定每个季度的起始月份，取值范围是1-12，默认值为1。

例如，指定start=1表示每年的第一个月是1月，指定start=3表示每年的第一个月是3月。

- "fiscalyear"参数用于指定是否按照财务年度计算季度，取值为TRUE或FALSE，默认为FALSE。

如果设置为TRUE，则季度将以财务年度为基准计算，财务年度通常不同于日历年度。

下面是一些使用quarter函数的例子，以帮助理解其功能：# 使用日期向量作为参数dates <- c("2021-01-01", "2021-03-15", "2021-06-30", "2021-10-20")quarters <- quarter(dates)print(quarters)# 输出：[1] 1 1 2 4# 将日期向下舍入到对应的季度。

2021-01-01属于第一季度，2021-03-15也属于第一季度，2021-06-30属于第二季度，2021-10-20属于第四季度。

# 使用日期字符串作为参数date <- "2022-07-01"quarter <- quarter(date)print(quarter)# 输出：[1] 3# 将日期向下舍入到对应的季度。

a r X i v :h e p -t h /9512038v 1 6 D e c 1995Supertraces on the Algebras of Observables of the Rational Calogero Model with Harmonic Potential S.E.Konstein and M.A.Vasiliev I.E.Tamm Department of Theoretical Physics,P.N.Lebedev Physical Institute,117924Leninsky Prospect 53,Moscow,Russia.Abstract We define a complete set of supertraces on the algebra SH N (ν),the algebra of observables of the N -body rational Calogero model with harmonic interaction.This result extends the previously known results for the simplest cases of N =1and N =2to arbitrary N .It is shown that SH N (ν)admits q (N )independent supertraces where q (N )is a number of partitions of N into a sum of odd positive integers,so that q (N )>1for N ≥3.Some consequences of the existence of several independent supertraces of SH N (ν)are discussed such as the existence of ideals in associated W ∞-type Lie superalgebras.I IntroductionIn this paper we investigate some properties of the associative algebras which were shown in [1,2,3]to underly the rational Calogero model [4]and were denoted as SH N (ν)in [5].Algebra SH N (ν)is the associative algebra of polynomials constructed from arbitrary ele-ments σof the symmetric group S N and the generating elements a αi obeying the following relations σa αi =a ασ(i )σ,(1) a αi ,a βj =ǫαβA ij ,(2)where i,j =1,...,N ,α,β=0,1,ǫαβ=−ǫβα,ǫ01=1andA ij =δij +ν˜A ij ,˜Aij =δij N l =1K il −K ij .(3)1Here K ij∈S N with i,j=1,...,N,i=j,are the elementary permutations i↔j satisfying the relationsK ij=K ji,K ij K ij=1,K ij K jl=K jl K li=K li K ijfor i=j=l=i andK ij K kl=K kl K ijif i,j,k,l are pairwise different.Note that in this paper repeated Latin indices i,j,k,... do not imply summation.The defining relations(1)-(3)are consistent.In particular,the Jacobi identities[aαi,[aβj,aγk]]+[aβj[aγk,aαi]]+[aγk,[aαi,aβj]]=0(4) are satisfied.An important property of SH N(ν)which allows one to solve the Calogero model[4] is that this algebra possesses inner sl2automorphisms with the generators1Tαβ=the notation SH1).Properties of this algebra are very well studied(see e.g.[13]).Note that since the center of mass coordinates1/N N i=1aαi decouple from everything else in the defining relations(1)-(3),the associative algebra SH N(ν)has the structure SH N(ν)= SH1⊗SH′N(ν)where,by definition,SH′N(ν)is the algebra of elements depending only on the relative coordinates aαi−aαj.The properties of SH′2(ν)are well studied too[14].The algebra SH′2(ν)is defined by the relations[aα,aβ]=ǫαβ(1+2νK),(8) where K is the only nontrivial element of S2while aαare the relative motion oscillators. For the particular case ofν=0one recovers the algebra SH1in the sector of the K independent elements.In[14]it was shown that SH′2(ν)admits a unique supertrace operation defined by the simple formulastr(1)=1,str(K)=−2ν,str(W)=str(W K)=0(9) for any polynomial W∈SH′2of the formW=∞n=1Wα1...αn aα1...aαn(10)with arbitrary totally symmetric multispinors Wα1...αn .For the particular case ofν=0one recovers the supertrace on SH1.Furthermore it was shown in[14]by explicit evaluation of the invariant bilinear form B(x,y)def=str(xy)that forν=l+116(4ν2−1)is an arbitrary constant.In its turn this observation clarified the origin of the ideals of SH′2(ν)atν=l+1Let us note that an attempt to define differently graded traces like,e.g.,an ordinary trace (π≡0)unlikely leads to interesting results.Knowledge of the supertrace operations on SH N(ν)is useful in various respects.One of the most important applications of the supertrace is that it gives rise to n-linear invariant formsstr(a1a2...a n)(14) that allows one to work with the algebra essentially in the same way as with the ordinary finite-dimensional matrix algebras and,for example,construct Lagrangians when work-ing with dynamical theories based on SH N(ν).Another useful property is that since null vectors of any invariant bilinear form span a both-side ideal of the algebra,this gives a powerful device for investigating ideals which decouple from everything under the super-trace operation as it happens in SH2(ν)for half-integerν.It is also worth mentioning that having an explicit form of the trilinear form in one or another basis is practically equivalent to defining a star-product law in the algebra.An important motivation for the analysis of the supertraces of SH N(ν)is due to its deep relationship with the analysis of the representations of this algebra,which in its turn gets applications to the analysis of the wave functions of the Calogero model.For example,given representation of SH N(ν),one can speculate that it induces some super-trace on this algebra as(appropriately regularized)supertrace of(infinite)representation matrices.When the corresponding bilinear form degenerates this would imply that the representation becomes reducible.As we show,the situation for SH N(ν)is very interesting since starting from N=3 it admits more than one independent supertrace in contrast to the cases of N=1and N=2.This fact is in agreement with the results of[5]where it was shown that there exist many inequivalent lowest-weight type representations of SH N(ν)for higher N(these representations are classified according to the representations of S N.)Another important consequence of this phenomenon is that the Lie superalgebras W N,∞(ν)are not simple while appropriate their simple subalgebras possess non-trivial outer automorphisms.The paper is organized as follows.In Section II we analyze consequences of S N and sl2 automorphisms of SH N(ν).In Section III we discuss general properties of the supertraces and consequences of the existence of several independent supertraces.In Section IV we study the restrictions on supertraces of the group algebra of S N considered as a subalgebra of SH N(ν),which follow from the defining relations of SH N(ν).These restrictions are called ground level conditions(GLC).They play a fundamental role in the problem since as we show in Section V every solution of GLC admits a unique extension to some supertrace on SH N(ν).In Appendix A it is shown that the number of independent supertraces on SH N(ν)equals to the number of partitions of N into a sum of odd positive integers.Some technical details of the proof of Section V are collected in Appendices B and C.II Finite-Dimensional Groups of Automorphisms The group algebra of S N is thefinite-dimensional subalgebra of SH N(ν).The elements σ∈S N induce inner automorphisms of SH N(ν).It is well known,that anyσ∈S N can4be expanded into a product of pairwise commuting cyclesσ=c1c2c3...c t,(15) where c w,w=1,...,t,are cyclic permutations acting on distinct subsets of values of indices i.For example,a cycle which acts on thefirst s indices as1→2→...→s→1 has the formc=K12K23...K(s−1)s.(16) We use the notation|c|for the length of the cycle c.For the cycle(16),|c|=s.We take a convention that the cycles of unit length are associated with all values of i such that σ(i)=i,so that the relation w|c w|=N is true.Given permutationσ∈S N,we introduce a new set of basis elements Bσ={b I}instead of{aαi}in the following way.For every cycle c w in the decomposition(15)(w=1,...,t), let usfix some index l w,which belongs to the subset associated with the cycle c w.The basis elements bαw j,j=1,...,|c w|,which realize1-dimensional representations of the commutative cyclic group generated by c w,have the formbαw j=1|c w||c w|k=1(λw)jk aαl(w,k),(17)where l(w,k)=c−k w(l w)andλw=exp(2πi/|c w|).(18) From the definition(17)it follows thatc w bαw j=(λw)j bαw j c w,(19)c w bαn j=bαn j c w,for n=w(20) and thereforeσbαw j=(λw)j bαw jσ.(21) In what follows,instead of writing bαw j we use the notation b I with the label I ac-counting for the full information about the indexα,the index w enumerating cycles in(15),and the index j which enumerates various elements bαw j related to the cycle c w,i.e.I(I=1,...,2N)enumerates all possible triples{α,w,j}.We denote the indexα, the cycle and the eigenvalue in(19)corresponding to somefixed index I asα(I),c(I),andλI=(λw)j,respectively.The notationσ(I)=σ0implies that b I∈Bσ0.B1is the original basis of the generating elements aαi(here1is the unit permutation).Let M(σ)be the matrix which maps B1−→Bσin accordance with(17),b I= i,αM I iα(σ)aαi.(22) Obviously this mapping is ing the matrix notations one can rewrite(21)asσb Iσ−1=2NJ=1ΛI J(σ)b J,∀b I∈Bσ,(23) 5whereΛJ I(σ)=δJ IλI.Every polynomial in SH N(ν)can be expanded into a sum of monomials of the formb I1b I2...b I sσ,(24) where allσ(I k)=σ.Every monomial of this form realizes some one-dimensional repre-sentation of the Abelian group generated by all cyclesc w in the decomposition(15).The commutation relations for the generating elements b I follow from(2)and(3)b I,b J =F IJ=C IJ+νf IJ,(25)whereC IJ=ǫα(I)α(J)δc(I)c(J)δλIλ−1J(26) andf IJ= i,j,α,βM I iα(σ)M J jβ(σ)ǫαβ˜A ij.(27)The indices I,J are raised and lowered with the aid of the symplectic form C IJ µI= J C IJµJ,µI= JµJ C JI; M C IM C MJ=−δJ I.(28)Note that the elements b I are normalized in(17)in such a way that theν-independent part in(25)has the form(26).Another importantfinite-dimensional algebra of inner automorphisms of SH N(ν)is the sl2algebra which acts on the indicesα.It is spanned by the S N-invariant second-order polynomials(5).Evidently,SH N(ν)decomposes into the infinite direct sum of only finite-dimensional irreducible representations of this sl2spanned by various homogeneous polynomials(24).From the defining relations(1)-(3)it follows that SH N(ν)is Z2-graded with respect to the automorphismf(aαj)=−aαj,f(K ij)=K ij(29) which gives rise to the parityπ(13).In applications to higher-spin models,this automor-phism distinguishes between bosons and fermions.The algebra SH N(ν)admits the antiautomorphismρ,ρ(aαk)=iaαk,ρ(K ij)=K ij,(30) which leaves invariant the basic relations(1)-(3)provided that an order of operators is reversed according to the defining property of antiautomorphisms:ρ(AB)=ρ(B)ρ(A). From(15),(16)and(21)it follows thatρ(σ)=σ−1,ρ(b I)=ib J,(31) where J is related to I in such a way thatα(J)=α(I),σ(J)=(σ(I))−1,c(J)=(c(I))−1 andλJ=λ−1I.Note that in higher-spin theories the counterpart ofρdistinguishes between odd and even spins[16].6III General Properties of SupertraceIn this section we summarize some general properties to be respected by any supertrace in SH N(ν).Let A be an arbitrary associative Z2graded algebra with the parity functionπ(x)=0 or1.Suppose that A admits some supertrace operations str p where the label p enumeratesdifferent nontrivial supertraces.We call a supertrace str even(odd)if str(x)=0∀x∈Asuch thatπ(x)=1(0).Let T A be a linear space of supertraces on A.We say that dim T A is the number of supertraces on A.Given parity-preserving(anti)automorphismτand supertrace operation str on A, str(τ(x))is some supertrace as well.For inner automorphismsτ(τ(x)=pxp−1,π(p)=0)it follows from the defining property of the supertrace that str(τ(x))=str(x).Thus,T A forms a representation of the factor-group of the parity preserving automorphisms andantiautomorphisms of A over the normal subgroup of the inner automorphisms of A.Applying this fact to the original parity automorphism(−1)πone concludes that T A can always be decomposed into a direct sum of subspaces of even and odd supertraces,T A=T0A⊕T1A and that T1A=0if the parity automorphism is inner.In the sequel we only consider the case where dim T A<∞and there are no nontrivialodd supertraces.Let A=A1⊗A2with the associative algebras A1and A2endowed with some even supertrace operations t1and t2,respectively.The supertrace on A can bedefined by setting str(a1⊗a2)=t1(a1)t2(a2),∀a1∈A1,∀a2∈A2.As a result,one concludes that T A=T A1⊗T A2.In the case of SH N(ν)one thus can always separate out a contribution of the center of mass coordinates as an overall factor(SH1admits theunique supertrace).If A isfinite-dimensional then the existence of two different supertraces indicates thatA admits non-trivial both-side ideals.Actually,consider the bilinear form B(f,g)=α1str1(fg)+α2str2(fg)with arbitrary parametersα1,α2∈C and elements f,g∈A. The determinant of this bilinear form is some polynomial ofα1andα2.Therefore it vanishes for certain ratiosα1/α2orα2/α1according to the central theorem of algebra. Thus,for these values of the parameters the bilinear formB degenerates and admits non-trivial null vectors x,B(x,g)=0,∀g∈A.It is easy to see that the linear space I of all null vectors x is some both-side ideal of A.For infinite-dimensional algebras the existence of several supertraces does not necessarily imply the existence of ideals.As mentioned in introduction the existence of several supertrace operations may be related to the existence of inequivalent representations.Also it is worth mentioning that for the case of infinite-dimensional algebras and representations under investigation it can be difficult to use the standard(i.e.matrixwise)definition of the supertrace.In this situation the formal definition of the supertraces on the algebra we implement in this paper is the only rigorous one.Let l A be the Lie superalgebra which is isomorphic to A as a linear space and is endowedwith the product law(12).It contains the subalgebra sl A∈l A spanned by elements g such that str p(g)=0for all p.Evidently sl A forms the ideal of l A.The factor algebra t A=l A/sl A is a commutative Lie algebra isomorphic to T∗A as a linear space.Elements of t A different from the unit element of A(which exist if dim T A>1)can induce outer automorphisms of sl A.Let us note that it is this sl A Lie superalgebra which usually has7physical applications.For the case of SH N(ν)under consideration the algebra l SHN(ν)isidentified with the algebra W N,∞(ν)introduced in[8].We therefore conclude that these algebras are not simple for N>2because it is shown below that SH N(ν)admits several supertraces for N>2.Instead one can consider the algebras sW N,∞(ν).Let l A contain some subalgebra L such that A decomposes into a direct sum of irre-ducible representations of L with respect to the adjoint action of L on A via supercom-mutators.Then,only trivial representations of L can contribute to any supertrace on A. Actually,consider some non-trivial irreducible representation R of L.Any r∈R can be represented asr= j[l j,r j},l j∈L,r j∈R(32)since elements of the form(32)span the invariant subspace in R.From(11)it follows then that str(r)=0,∀r∈R.From the definition of the supertrace it follows thatstr(a1a2)+str(a2a1)=0(33) for arbitrary odd elements a1and a2of A.A simple consequence of this relation is that str(a1a2...a n+a2...a n a1+...+a n a1...a n−1)=0(34) is true for an arbitrary even n if all a i are some odd elements of A.Since we assume that the supertrace is even(34)is true for any n.This simple property turns out to be practically useful because,when odd generating elements are subject to some commutation relations with the right hand sides expressed via even generating elements like in(2),it often allows one to reduce evaluation of the supertrace of a degree-n polynomial of a i to supertraces of lower degree polynomials.Another useful property is that in order to show that the characteristic property of the supertrace(11)is true for any x,g∈A,it suffices to show this for a particular case where x is arbitrary while g is an arbitrary generating element of somefixed system of generating elements.Then(11)for general x and g will follow from the properties that A is associative and str is linear.For the particular case of SH N(ν)this means that it is enough to set either g=aαi or g=K ij.Let us now turn to some specific properties of SH N(ν)as a particular realization of A.By identifying L with sl2(5)and taking into account that SH N(ν)decomposes into a direct sum of irreduciblefinite-dimensional representations of sl2,one arrives at the followingLemma1:str(x)can be different from zero only when x is sl2-singlet,i.e.[Tαβ,x]=0. Corollary:Any supertrace on SH N(ν)is even.Analogously one deduces consequences of the S N symmetry.In particular,one proves Lemma2:Given c∈S N such that cF=µF c for some element F and any constant µ=1,str(F)=0.Given monomial F=b I1b I2...b I sσwith b I k∈Bσand a cycle c0in the decomposition(15)ofσone concludes that str(F)=0if k:c(I k)=c0λI k=1where λIkare the eigenvalues(21)of b I k.8IV Ground Level ConditionsLet us analyze restrictions on a form of str(a),a∈S N,which follow from the defining relations of SH N(ν).Firstly,we describe supertraces on the group algebra of S N.Let some permutationσdecomposes into n1cycles of length1,n2cycles of length2,...and n N cycles of length N.The non-negative integers n k satisfy the relationNk=1kn k=N(35) andfixσup to some conjugationσ→τστ−1,τ∈S N.Thusstr(σ)=ϕ(n1,n2,...,n N),(36) whereϕ(n1,n2,...,n N)is an arbitrary function.Obviously the linear space of invariant functions on S N(i.e.such that f(τστ−1)=f(σ))coincides with the linear space of supertraces on the group algebra of S N.Therefore,the dimension of the linear space of supertraces is equal to the number p(N)of independent solutions of(35),the number ofconjugacy classes of S N.One can introduce the generating function for p(N)as P(q)= ∞n=0p(n)q n= ∞k=11Lemma 3:Let c 1and c 2be two distinct cycles in the decomposition (15).Let indices i 1and i 2belong to the subsets of indices associated with the cycles c 1and c 2,respectively.Then the permutation c =c 1c 2K i 1i 2is a cycle of length |c |=|c 1|+|c 2|.Lemma 4:Given cyclic permutation c ∈S N ,let i =j be two indices such that c k (i )=j ,where k is some positive integer,k <|c |.Then cK ij =c 1c 2where c 1,2are some non-coinciding mutually commuting cycles such that |c 1|=k and |c 2|=|c |−k .Using the definition (17),the commutation relations (1)-(3)and Lemmas 3and 4one reduces GLC to the following system of equations:n 2k ϕ(n 1,...,n 2k ,...,n N )=−νn 2k 22k −1 s =k,s =1O s ϕ(n 1,...,n s +1,...,n 2k −s +1,...,n 2k −1,...,n N )+2O k ϕ(n 1,...,n k +2,...,n 2k −1,...,n N )+N s =2k ;s =1sn s ϕ(n 1,...,n s −1,...,n 2k −1,...,n 2k +s +1,...,n N )+2k (n 2k −1)ϕ(n 1,...,n 2k −2,...,n 4k +1,...,n N )(39)where O k =0for k even and O k =1for k odd.Let us note that by virtue of the substitution ϕ(n 1,...,n N )=νE (σ)˜ϕ(n 1,...,n N ),(40)where E (σ)is the number of cycles of even length in the decomposition of σ(15),i.e.E (σ)=n 2+n 4+ (41)one can get rid of the explicit dependence of νfrom GLC (39).As a result,there are two distinguishing cases,ν=0and ν=0.For lower N the conditions (39)take the formϕ(0,1)+2νϕ(2,0)=0(42)for N =2(cf.(9)),ϕ(1,1,0)+2νϕ(3,0,0)+νϕ(0,0,1)=0(43)for N =3andϕ(2,1,0,0)+2νϕ(4,0,0,0)+2νϕ(1,0,1,0)=0ϕ(0,2,0,0)+2νϕ(2,1,0,0)+2νϕ(0,0,0,1)=0ϕ(0,0,0,1)+4νϕ(1,0,1,0)=0for N =4.As a result one finds 1-parametric families of solutions for N =1and N =2and 2-parametric families of solutions for N =3and N =4.Let G N be the number of independent solutions of (39).As we show in the next section G N =dimT SH N (ν)for all ν.In other words all other conditions on the supertrace do not10impose any restrictions on the functionsϕ(n1,...,n N)but merely express supertraces of higher order polynomials of aαi in terms ofϕ(n1,...,n N).In the Appendix A we prove the followingTheorem1:G N=q(N)where q(N)is a number of partitions of N into a sum of odd positive integers,i.e.the number of the solutions of the equation ∞k=0(2k+1)n k=N for non-negative integers n i.One can guess this result from the particular case ofν=0where GLC tell us that ϕ(n1,...,n N)can be nonvanishing(and arbitrary)only when all n2k=0.Interestingly enough,G N remains the same forν=0.V Supertrace for General ElementsIn this section we proveTheorem2:dimT SHN(ν)=G N where G N is the number of independent solutions of theground level conditions(39).The proof of the Theorem2will be given in a constructive way by virtue of the following double induction procedure:(i).Assuming that GLC are true and str{b I,P p(a)σ}=0∀P p(a),σand I provided that b I∈Bσandλ(I)=−1;p≤k orλ(I)=−1,E(σ)≤l,p≤k orλ(I)=−1;p≤k−2,where P p(a)is an arbitrary degree p polynomial of aαi(p is odd)and E(σ)is the number of cycles of even length in the decomposition(15)ofσ,one proves that there exists such a unique extension of the supertrace that the same is true for l→l+1.(ii).Assuming that str{b I,P p(a)σ}=0∀P p(a),σand b I such thatσ(I)=σ,p≤k one proves that there exists such a unique extension of the supertrace that the assumption (i)is true for k→k+2and l=0.As a result this inductive procedure extends uniquely any solution of GLC to some supertrace on the whole SH N(ν).(Let us remind ourselves that the supertrace of any odd element of SH N(ν)is trivially zero by sl2invariance).The inductive proof of the Theorem2is based on the S N covariance of the whole setting and the following importantLemma5:Given permutationσwhich has E(σ)cycles of even length in the decompo-sition(15),the quantity f IJσforσ(I)=σ(J)=σandλI=λJ=−1can be uniquely expanded as f IJσ= qαqσq whereαq are some coefficients and E(σq)=E(σ)−1∀q.Lemma5is a simple consequence of the particular form of the structure coefficients f IJ(27)and Lemmas3and4.The proof is straightforward.Let us stress that it is Lemma5which accounts for the specific properties of the algebra SH N(ν)in the analysis of this section.In practice it is convenient to work with the exponential generating functionsΨσ(µ)=str e Sσ ,S=2N L=1(µL b L),(44)11where σis some fixed element of S N ,b L ∈B σand µL ∈C are independent parameters.By differentiating over µL one can obtain an arbitrary polynomial of b L in front of σ.The exponential form of the generating functions implies that these polynomials are Weyl ordered.In these terms the induction on a degree of polynomials is equivalent to the induction on a degree of homogeneity in µof the power series expansions of Ψσ(µ).As a consequence of the general properties discussed in the preceding sections the generating function Ψσ(µ)must be invariant under the S N similarity transformationsΨτστ−1(µ)=Ψσ(˜µ),(45)where the S N transformed parameters are of the form˜µI = J M (τστ−1)M −1(τ)Λ−1(τ)M (τ)M −1(σ) J I µJ (46)and matrices M (σ)and Λ(σ)are defined in (22)and (23).In accordance with thegeneralargumentof SectionIII the necessary and sufficient conditions for the existence of even supertrace are the S N -covariance conditions (45)and the condition thatstr b L,(expS )σ =0for any σand L.(47)To transform (47)to an appropriate form,let us use the following two general relations which are true for arbitrary operators X and Y and the parameter µ∈C :Xexp (Y +µX )=∂∂µexp (Y +µX )− t 1exp (t 1(Y +µX ))[X,Y ]exp (t 2(Y +µX ))D 1t (49)with the convention that D n −1t =δ(t 1+...+t n −1)θ(t 1)...θ(t n )dt 1...dt n .(50)The relations (48)and (49)can be derived with the aid of the partial integration (e.g.over t 1)and the following formula∂∂µL Ψσ(µ)= (λL t 1−t 2)str exp (t 1S )[b L ,S ]exp (t 2S )σ D 1t.(53)12This condition should be true for anyσand L and plays the central role in the analysis of this section.There are two essentially distinguishing cases,λL=−1andλL=−1.In the latter case,the equation(53)takes the form0= str exp(t1S)[b L,S]exp(t2S)σ D1t,λL=−1.(54) In Appendix B we show by induction that the equations(53)and(54)are consistent in the following sense∂(1+λK)∂µK str exp(t1S)[b L,S]exp(t2S)σ D1t=0,λL=−1.(56) Note that this part of the proof is quite general and does not depend on a concrete form of the commutation relations of aαi in(2).By expanding the exponential e S in(44)into power series inµK(equivalently b K) one concludes that the equation(53)uniquely reconstructs the supertrace of monomials containing b K withλK=−1(from now on called regular polynomials)via supertraces of some lower order polynomials.The consistency conditions(55)and(56)then guarantee that(53)does not impose any additional conditions on the supertraces of lower degree polynomials and allow one to represent the generating function in the formΨσ=Φσ(µ)(57) + L:λL=−1 10µL dτConsider the part of str b I,(expS′)σ which is of order k inµand suppose that E(σ)= l+1.According to(54)the conditions(60)give0= str exp(t1S′)[b I,S′]exp(t2S′)σ D1t.(61) Substituting[b I,S′]=µI+ν M f IMµM,where the quantities f IJ andµI are defined in(25)-(28),one can rewrite the equation(61)in the formµIΦσ(µ)=−ν str exp(t1S′) M f IMµM exp(t2S′)σ D1t.(62)Now we use the inductive hypothesis(i).The right hand side of(62)is a supertrace of at most a degree k−1polynomial of aαi in the sector of degree k polynomials inµ. Therefore one can use the inductive hypothesis(i)to obtainstr exp(t1S′) M f IMµM exp(t2S′)σ D1t= str exp(t2S′)exp(t1S′) M f IMµMσ D1t,where we made use of the simple fact that str(S′Fσ)=−str(FσS′)=str(F S′σ)due to the definition of S′.As a result,the inductive hypothesis allows one to transform(60)to the following formX I≡µIΦσ(µ)+νstr exp(S′) M f IMµMσ =0.(63) By differentiating this equation with respect toµJ one obtains after symmetrization ∂∂µJX I(µ)+∂∂µJ µIΦσ(µ)+(I↔J)=−ν2 L,M(t1−t2)str exp(t1S′)F JLµL exp(t2S′)f IMµMσ D1t+(I↔J).(65) The last term on the right hand side of this expression can be shown to vanish under the supertrace operation due to the factor of(t1−t2),so that one is left with the equationL IJΦσ(µ)=−νwhereR IJ(µ)= M str exp(S′){b J,f IM}µMσ +(I↔J)(67) andL IJ=∂∂µIµJ.(68)The differential operators L IJ satisfy the standard sp(2E(σ))commutation relations [L IJ,L KL]=− C IK L JL+C IL L JK+C JK L IL+C JL L IK .(69) We show by induction in Appendix C that this algebra is consistent with the right-hand side of the basic relation(66)i.e.that[L IJ,R KL]−[L KL,R IJ]=− C IK R JL+C JL R IK+C JK R IL+C IL R JK .(70)Generally,these consistency conditions guarantee that the equations(66)express Φσ(µ)in terms of R IJ in the following wayΦσ(µ)=Φσ(0)+νt(1−t2E(σ))(L IJ R IJ)(tµ),(71)provided thatR IJ(0)=0.(72) The latter condition must hold for the consistency of(66)since its left hand side vanishes atµI=0.In the formula(71)it guarantees that the integral on t converges.In the case under consideration the property(72)is indeed true as a consequence of the definition (67).Taking into account Lemma5and the explicit form of R IJ(67)one concludes that the equation(71)expresses uniquely the supertrace of special polynomials via the supertraces of polynomials of lower degrees or via the supertraces of special polynomials of the same degree with a lower number of cycles of even length provided that theµindependent term Φσ(0)is an arbitrary solution of GLC.This completes the proof of Theorem2. Comment1:The formulae(57)and(71)can be effectively used in practical calculations of supertraces of particular elements of SH N(ν).Comment2:Any supertrace on SH N(ν)is determined unambiguously in terms of its values on the group algebra of S N.Corollary:Any supertrace on SH N(ν)isρ-invariant,str(ρ(x))=str(x)∀x∈SH N(ν), for the antiautomorphismρ(30).This is true due to the Comment2becauseσandσ−1=ρ(σ)belong to the same conjugacy class of S N so that str(ρ(σ))=str(σ).15。

R语言时间序列中文教程R语言是一种广泛应用于统计分析和数据可视化的编程语言。

它提供了丰富的函数和包，使得处理时间序列数据变得非常方便。

本文将为大家介绍R语言中时间序列分析的基础知识和常用方法。

R语言中最常用的时间序列对象是`ts`对象。

通过将数据转换为`ts`对象，可以使用R语言提供的各种函数和方法来分析时间序列数据。

我们可以使用`ts`函数将数据转换为`ts`对象，并指定数据的时间间隔、起始时间等参数。

例如，对于按月份记录的时间序列数据，可以使用以下代码将数据转换为`ts`对象：```Rts_data <- ts(data, start = c(2000, 1), frequency = 12)```在时间序列分析中，常用的一个概念是平稳性。

平稳性表示时间序列的均值和方差在时间上不发生显著变化。

平稳时间序列的特点是，它的自相关函数(ACF)和偏自相关函数(PACF)衰减得很快。

判断时间序列是否平稳可以通过绘制序列的线图和计算序列的自相关函数来进行。

我们可以使用R语言中的`plot`函数和`acf`函数来实现。

例如，对于一个名为`ts_data`的时间序列数据，可以使用以下代码绘制序列的线图和自相关函数图：```Rplot(ts_data)acf(ts_data)```在进行时间序列分析时，经常需要进行模型拟合和预测。

R语言提供了一些常用的函数和包，用于时间序列的模型拟合和预测。

其中，最常用的方法是自回归移动平均模型(ARIMA)。

ARIMA模型是一种广泛应用于时间序列分析的统计模型，它可以描述时间序列数据中的长期趋势、季节性变动和随机波动等特征。

我们可以使用R语言中的`arima`函数来拟合ARIMA模型，并使用`forecast`函数来进行预测。

以下是一个使用ARIMA模型进行时间序列预测的示例代码：```Rmodel <- arima(ts_data, order = c(p, d, q))forecast_result <- forecast(model, h = 12)```以上代码中，`p`、`d`和`q`分别表示ARIMA模型的自回归阶数、差分阶数和移动平均阶数。

分形布朗运动和hurst指数
分形布朗运动是一种随机过程，其特性与布朗运动相似，但具有更复杂的分形结构。

布朗运动是指微观粒子在液体或气体中由于受到分子的不断碰撞而进行的无规则、连续且随机的运动。

而分形布朗运动则是在这种运动过程中引入了分形结构，使得其具有更为复杂的运动模式。

Hurst指数是用来描述分形布朗运动的一个重要参数。

它表示分形布朗运动在时间序列上的长期依赖性或持久性。

Hurst指数的值介于0和1之间，其中0.5表示随机游走，小于0.5表示负持久性，即过去的变化趋势对未来的影响逐渐减弱，而大于0.5则表示正持久性，即过去的变化趋势对未来的影响逐渐增强。

在金融领域中，分形布朗运动和Hurst指数被广泛应用于模拟股票价格等金融时间序列。

由于股票价格具有分形结构和持久性，因此分形布朗运动可以很好地描述股票价格的波动特征。

通过估计Hurst指数，我们可以了解股票价格的波动趋势和未来价格的变化情况。

除了金融领域，分形布朗运动和Hurst指数还在其他领域得到广泛应用。

例如，在地球物理学中，它们被用于模拟地震和海浪等自然现象；在生物学中，它们被用于描述生物种群的增长和变化趋势等。

此外，分形布朗运动和Hurst 指数还被应用于图像处理、信号处理等领域。

总之，分形布朗运动是一种具有复杂分形结构的随机过程，其特性与布朗运动相似但更为复杂。

Hurst指数是描述分形布朗运动的一个重要参数，可以用来估计时间序列的持久性和变化趋势。

在金融、地球物理学、生物学等领域中，分形布朗运动和Hurst指数得到了广泛应用，为我们提供了更准确、更有效的分析方法和工具。

nr中的tbsize计算公式TBSIZE是一种用于计算网络传输速度的公式，它可以帮助我们评估网络的带宽和传输效率。

在计算TBSIZE时，我们需要考虑到传输速度、网络延迟和数据包大小等因素。

下面将详细介绍TBSIZE的计算公式及其在网络传输中的应用。

我们需要了解TBSIZE的定义。

TBSIZE是指传输缓冲区大小（Transmission Buffer Size），它表示在一段时间内可以发送的最大数据量。

传输缓冲区是指在发送数据时，存储待发送数据的内存区域。

通过调整传输缓冲区的大小，可以对网络传输速度进行优化。

TBSIZE的计算公式如下：TBSIZE = (RTT * BDP) / 8其中，RTT表示网络往返时间（Round Trip Time），BDP表示带宽延迟乘积（Bandwidth-Delay Product）。

公式中的8是将单位从比特转换为字节。

带宽延迟乘积（BDP）是指网络带宽和网络延迟的乘积，它反映了网络传输的效率。

网络带宽是指在单位时间内传输的数据量，通常以比特为单位。

网络延迟是指数据从发送端到接收端所需的时间，通常以毫秒为单位。

通过计算TBSIZE，我们可以确定在一段时间内可以发送的最大数据量。

这对于优化网络传输速度和提高用户体验非常重要。

在实际应用中，我们可以根据需求和网络情况来调整传输缓冲区的大小，以达到最佳的传输效果。

在网络传输中，TBSIZE的优化对于提高数据传输速度和减少数据丢失非常重要。

通过合理调整传输缓冲区的大小，可以在保证传输稳定性的同时，提高传输效率。

例如，在视频流传输中，通过增大传输缓冲区的大小，可以减少视频卡顿和数据丢失的情况，提供更流畅的观看体验。

TBSIZE的计算还可以帮助我们评估网络的带宽和延迟状况。

通过监测网络往返时间（RTT）和带宽延迟乘积（BDP），我们可以了解网络的传输效率和瓶颈所在，进而采取相应的优化措施。

总结起来，TBSIZE是一种用于计算网络传输速度的公式，它可以帮助我们评估网络的带宽和传输效率。

hurst指数第一篇：Hurst指数简介及应用领域Hurst指数是一种用于衡量时间序列数据的长期记忆性的统计量，其应用广泛于金融分析、水文学、信号处理等领域。

本文将对Hurst指数进行详细介绍，并探讨其应用领域。

Hurst指数最初是由数学家H.E. Hurst于1951年提出的，其用于衡量时间序列数据的波动性和相关性。

时间序列数据是指一组按时间顺序排列的观测值，例如股票价格、气温记录等。

Hurst指数的取值范围在0到1之间，其中0表示完全反序列相关，1表示完全正序列相关，0.5表示完全随机。

Hurst 指数越接近于0.5，说明时间序列数据的波动性越接近于随机，没有长期记忆性；而越接近于0或1，说明时间序列数据存在较强的趋势性，即具有长期记忆性。

Hurst指数的计算需要借助于重叠子序列的均值计算，具体步骤如下：首先，将时间序列数据分解成不同长度的子序列；然后，计算每个子序列的均值；最后，计算不同子序列长度下的均值之比。

根据计算得到的比值，可得到Hurst指数。

在金融分析中，Hurst指数常被用于衡量股票价格的长期记忆性和预测性。

通过计算Hurst指数，可以评估股票价格的波动性，进而辅助投资者进行风险管理和决策制定。

例如，当股票价格的Hurst指数较高时，说明价格具有较强的趋势性，投资者可以选择更长期的持有策略，以获得更大的收益。

此外，Hurst指数在水文学领域也得到了广泛的应用。

水文学研究常关注各种水文变量的波动性，例如降水量、水位等。

通过计算Hurst指数，可以评估水文变量的长期趋势，进而为水资源管理、洪水预测等提供科学依据。

除金融分析和水文学外，Hurst指数在信号处理、网络分析等领域也有着重要的应用价值。

例如，对于信号处理，Hurst指数可以用于评估信号的分形特性和自相似性，从而指导滤波、数据压缩等算法的设计与优化。

综上所述，Hurst指数是一种用于衡量时间序列数据长期记忆性的统计量，在金融分析、水文学、信号处理等领域有广泛的应用。

hurst 指数python摘要：1.Hurst 指数简介2.Python 在Hurst 指数计算中的应用3.Hurst 指数的计算方法4.Python 代码示例5.总结正文：1.Hurst 指数简介Hurst 指数是一种用来描述时间序列数据的长期记忆特性的指标。

它是由英国统计学家Hurst 在1951 年提出的，被广泛应用于金融、气象、水文等领域。

Hurst 指数的取值范围为0 到1 之间，当指数大于0.5 时，表示时间序列具有正长时相关性，即具有趋势；当指数等于0.5 时，表示时间序列无关；当指数小于0.5 时，表示时间序列具有负长时相关性，即具有反趋势。

2.Python 在Hurst 指数计算中的应用Python 作为一门广泛应用于数据分析和科学计算的语言，拥有丰富的库和工具，可以方便地实现Hurst 指数的计算。

在使用Python 计算Hurst 指数时，常用的库有NumPy、Pandas 和Statsmodels 等。

3.Hurst 指数的计算方法Hurst 指数的计算方法有多种，其中较为常见的有以下几种：- R/S分析法：R/S分析法是Hurst指数计算中最常用的方法，其基本思想是将时间序列数据进行分段，计算各分段的平均值，然后计算各分段平均值之间的相关性。

- 波动率法：波动率法是通过计算时间序列数据的波动率来估计Hurst 指数的方法。

波动率的计算可以采用简单的方差计算，也可以采用更为复杂的GARCH 模型等。

- 功率谱法：功率谱法是通过计算时间序列数据的功率谱来估计Hurst 指数的方法。

功率谱可以反映时间序列在不同时间尺度上的能量分布，从而为Hurst 指数的估计提供依据。

4.Python 代码示例以下是一个使用Python 和Pandas 库计算Hurst 指数的简单示例：```pythonimport pandas as pdfrom scipy import stats# 创建一个简单的时间序列数据data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]# 将数据转换为Pandas 的Series 对象series = pd.Series(data)# 计算R/S分析法的Hurst指数rs_result = stats.rs_one_step(series)hurst_rs = rs_result[1]print("Hurst 指数（R/S 分析法）：", hurst_rs)```5.总结本文介绍了Hurst 指数的计算方法和Python 在Hurst 指数计算中的应用，并通过一个简单的Python 代码示例展示了如何使用Pandas 和Scipy 库计算Hurst 指数。

urwtest参数说明English answer:urwtest is a program that tests the functionality of the Unicode Regular Expressions (URE) library. The library is used by many programs to perform complex text processing tasks, such as finding and replacing text, searching for patterns, and validating input.The urwtest program can be used to test the following aspects of the URE library:Character classes: Character classes are used to match characters that have certain properties, such as being a letter, a digit, or a whitespace character.Anchors: Anchors are used to match characters at the beginning or end of a string, or at the beginning or end of a line.Quantifiers: Quantifiers are used to match characters that occur a certain number of times.Grouping: Grouping is used to group characters together so that they can be treated as a single unit.Backreferences: Backreferences are used to match characters that have been previously matched.The urwtest program can be used to test the URE library by providing a regular expression and a string to match. The program will then output whether the regular expression matches the string.The urwtest program has a number of options that can be used to control its behavior. These options include:-v: Verbose output. This option causes the urwtest program to output more information about the regular expression and the string being matched.-i: Case-insensitive matching. This option causes theurwtest program to ignore the case of the characters in the regular expression and the string being matched.-m: Multiline matching. This option causes the urwtest program to treat the string being matched as a multiline string.-s: Dotall matching. This option causes the urwtest program to treat the dot (.) character in the regular expression as matching any character, including newline characters.-x: Extended syntax. This option causes the urwtest program to allow the use of whitespace characters and comments in the regular expression.The urwtest program can be a useful tool for testing the functionality of the URE library. It can be used to verify that regular expressions are working as expected, and to troubleshoot problems with regular expressions.Here are some examples of how to use the urwtestprogram:$ urwtest 'abc' 'abc'。