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The Algebra of Holonomic Equations

Wolfram Koepf

Konrad-Zuse-Zentrum Berlin,Heilbronner Str.10,D–10711Berlin Abstract.In this article algorithmic methods are presented that have essen-tially been introduced into computer algebra systems like Maple or Mathematica within the last decade.The main ideas are due to Stanley and Zeilberger.Some of them had already been discovered in the last century by Beke,but because of their complexity the underlying algorithms have fallen into oblivion.We give a survey of these techniques,show how they can be used to identify transcen-dental functions,and present implementations of these algorithms in computer algebra systems.

1Algebraic Representation of Transcendental Functions

How can transcendental function be represented by algebraic means?To give this question another?avor:What is the main di?erence between the exponen-tial function f(x)=e x and the function g(x)=e x+|x|/101000,that makes f an elementary function,but not g,although f and g are numerically quite close on a part of the real axis?

Or let’s consider an example of discrete mathematics:Why is the factorial function a n=n!considered to be the most important discrete function,and not b n=n!+n/101000or any other discrete function?

Although these examples refer to the most important continuous and discrete transcendental functions,oddly enough the answers to the above questions are purely algebraic:The exponential function f is characterized by any of the following algebraic properties:

1.f is continuous,f(1)=e,and for all x,y we have f(x+y)=f(x)·f(y);

2.f is di?erentiable,f (x)=f(x)and f(0)=1;

3.f∈C∞,f(x)=∞

n=0a n x n with a0=1,and for all n≥0we have

(n+1)a n+1=a n;

and the factorial function a n is represented by any of the following algebraic properties:

4.a0=1,and for all n≥0we have a n+1=(n+1)a n;

5.the generating function f(x)=∞

n=0a n x n satis?es the di?erential equation

x2f (x)+(x?1)f(x)+1=0with the initial condition f(0)=1.

(Note here that one could argue that property(1.)is not algebraic since the symbol e is needed in the representation.)I do not know any method to represent transcendental functions using functional equations,such as property(1.),but I will show,why and how the other properties can be suitable for this purpose, being mainly concerned with properties(2.)and(4.).In§4we consider,how these representations can be viewed as purely polynomial cases.

Observe that the“generating function”of the factorial function is conver-gent only at the origin,and therefore must be considered as a formal series.In particular,a“closed representation”(whatever that should mean)of the gen-erating function cannot be given.But this is not the main issue here.Rather than working with the generating function itself,it is much better to work with its di?erential equation which is purely algebraic(in fact,it is purely polyno-mial).The same argument applies to the exponential and factorial functions themselves.Rather than working with these transcendental objects,one should represent them by their corresponding di?erential and recurrence equations.

The given properties are structural statements about the corresponding func-tions.Any small modi?cation(even changing the value at a single point)de-stroys this structure.For example,the function g(x)=e x+|x|/101000cannot be characterized by a rule analogous to one of the properties(2.)–(3.).On the other hand,the function h(x)=e x+x/101000can be represented by the di?er-ential equation(x?1)h (x)?x h (x)+h(x)=0with the initial values h(0)=1 and h (0)=1+10?1000.

Therefore,the special(and common)fact about the exponential and fac-torial functions is that they both satisfy a di?erential or recurrence equation, respectively,that is homogeneous,linear,of order one,and has polynomial co-e?cients.

We can generalize this observation[42]:A continuous function of one variable f(x)is holonomic,if it satis?es a homogeneous linear di?erential equation with polynomial coe?cients;we call such a di?erential equation also holonomic.

By linear algebra arguments,Stanley[36]showed that sums and prod-ucts of holonomic functions and the composition with algebraic functions also form holonomic functions.This can be seen as follows:Assume f and g sat-isfy holonomic di?erential equations of order n and m,respectively.We con-sider the linear space L f of functions with rational coe?cients generated by f,f ,f ,...,f(k),....Since f,f ,...,f(n)are linearly dependent by the given holonomic di?erential equation and since by di?erentiation the same conclusion follows for f ,f ,...,f(n+1),and so on inductively,the dimension of L f is≤n. Similarly L g has dimension≤m.We now build the sum L f+L g which is of dimension≤n+m.As f+g,(f+g) ,...,(f+g)(k),...are elements of L f+L g, arbitrary n+m+1many of them are linearly dependent.In particular,f+g satis?es a holonomic di?erential equation of order≤n+m.

Similarly the product and composition cases can be handled.Note that the above proof provides a construction of the resulting holonomic equation by linear algebra techniques.It is remarkable that100years ago,Beke[4]–[5]

already described these algorithms to generate holonomic di?erential equations for the sum and product of f and g from the holonomic di?erential equations of f and g.Hence,he had discovered algorithmic versions of Stanley’s results!

Analogously,a discrete function(sequence)of one variable is called holo-nomic,if it satis?es a homogeneous linear recurrence equation with polynomial coe?cients.Such a recurrence equation is also called holonomic.Sums and products of discrete holonomic functions are again holonomic,and there are similar algorithms to calculate representing holonomic recurrence equations(s.

[34],[25]).

A function

f(x)=

∞ n=0

a n x n

represented by a power series is holonomic if and only if the corresponding power

series coe?cient a n is a holonomic sequence.The holonomic equations for f(x)

and a n can be converted equating coe?cients.

Note that these algorithms were implemented by Salvy and Zimmermann in

the gfun package of Maple’s share library[34].I wrote a Mathematica imple-

mentation,SpecialFunctions,to be obtained by World Wide Web from the ad-

dress ftp://ftp.zib-berlin.de/pub/UserHome/Koepf/SpecialFunctions.

Examples of this implementation will be given later.

2Identi?cation of Transcendental Functions Note that the notion of holonomy provides a normal form for a suitably large

number of transcendental functions,which can then be utilized for identi?cation

purposes.The holonomic equation of lowest order corresponding to a holonomic

function constitutes such a normal form.Once we have calculated the normal

form of a holonomic function,the latter is identi?ed:Two holonomic functions

are identical if and only if they have the same normal form,and satisfy the same

initial conditions.

But also without having access to the lowest order holonomic equations,

one can check whether two holonomic functions agree,since(by linear algebra,

e.g.,)it is easy to see whether two holonomic equations are compatible with

each other.

Therefore,we may ignore that e x,sin x,cos x,arctan x,arcsin x and others

form transcendental functions,and take only their holonomic di?erential equa-

tions f =f,f =?f,f =?f,(1+x2)f +2xf =0,(x2?1)f +xf =0 etc.into account.From these di?erential equations,corresponding di?eren-

tial equations for sums and products can be generated by the above mentioned

technique,using only polynomial arithmetic and linear algebra.For example,

the function f(x)=arcsin2x yields(x2?1)f +3xf +f =0.Note,how-ever,that in the given case one can get even more:The resulting holonomic

di?erential equation is directly equivalent to the holonomic recurrence equa-tion n (1+n )(2+n )a n +2=n 3a n for the coe?cients a n of the Taylor series of arcsin 2x =∞ n =0

a n x n ,and since this holonomic recurrence equation fortunately contains only the two terms a n +2and a n ,it can be solved explicitly,and leads to the representation

arcsin 2x =∞ n =04n n !2

(1+n )(1+2n )!x 2n +2

(compare [18],[41],[20]–[21]).

Note that not only a function like the Airy function Ai (x )(s.e.[1],(10.4))falls under the category of holonomic functions,since it satis?es the simple holonomic di?erential equation f ?xf =0,moreover the classical families of orthogonal polynomials 1and many other special functions form holonomic func-tions [1].These depend on several variables,and we will discuss this situation in §4.

On the other hand,there are functions that are not holonomic,like the tangent function tan x (s.[36],[25]).The identi?cation problem for expressions involving nonholonomic functions can only be treated after preprocessing the input.If,for example,we want to verify the addition formula for the tangent function tan (x +y )=tan x +tan y 1?tan x tan y

by the given method,then we have to replace all occurrences of the tangent function by sines and cosines (which are holonomic)using the rewrite rule tan x =sin x/cos x .We can then generate a polynomial equation by multi-plying both sides by the common denominator.This procedure results in the equivalent representation

(cos x cos y ?sin x sin y )sin (x +y )=(cos y sin x +cos x sin y )cos (x +y )(1)which is easily proved since the algorithms generate the common holonomic di?erential equation f (x )+4f (x )=0with respect to x (or the common holonomic di?erential equation f (y )+4f (y )=0with respect to y )for both sides of (1)where the common initial values are f (0)=cos y sin y ,and f (0)=cos y 2?sin y 2.Assume that for the initial value functions we had obtained di?erent representations (e.g.cos y sin y and sin(2y )/2).These could be veri?ed by the same technique.

In the Mathematica package SpecialFunctions (s.also [22]),the procedure HolonomicDE[f,x]calculates the holonomic di?erential equation of f with re-spect to the variable x using the known holonomic di?erential equations of the primitive functions,and the sum and product algorithms by recursive decent 1As families of orthogonal polynomials they are not polynomials!

through the expression tree.Here we call a function primitive if it is rational, or whenever we use a separate symbol for it and a holonomic di?erential equa-tion is known.Therefore the above mentioned functions(besides the tangent function)are primitive.

The examples given are governed by the following Mathematica session:

In[1]:=<

In[2]:=HolonomicDE[ArcSin[x]^2,x]

(3)

Out[2]=F’[x]+3x F’’[x]+(-1+x)(1+x)F[x]==0

In[3]:=DEtoRE[%,F,x,a,n]

Out[3]=n a[n]-n(1+n)(2+n)a[2+n]==0

In[4]:=Series[ArcSin[x]^2,{x,0}]

k2+2k2

4x k!

Out[4]=Sum[------------------,{k,0,Infinity}]

(1+k)(1+2k)!

In[5]:=HolonomicDE[AiryAi[x],x]

Out[5]=-(x F[x])+F’’[x]==0

In[6]:=HolonomicDE[AiryAi[x]^2,x]

(3)

Out[6]=2F[x]+4x F’[x]-F[x]==0

In[7]:=HolonomicDE[Sin[x+y]*(Sin[x]Sin[y]-Cos[x]Cos[y]),x]

(3)

Out[7]=4F’[x]+F[x]==0

In[8]:=HolonomicDE[Cos[x+y]*(Sin[x]Cos[y]+Cos[x]Sin[y]),x]

(3)

Out[8]=4F’[x]+F[x]==0

In[9]:=HolonomicDE[Cos[y]*Sin[y],y]

(3)

Out[9]=4F’[y]+F[y]==0

In[10]:=HolonomicDE[Sin[2y]/2,y]

Out[10]=4F[y]+F’’[y]==0

One di?culty that may arise with the method described is that in some in-stances the sum and product algorithms will not generate the holonomic di?er-ential equation of lowest order,as in the above example for cos y sin y .In this case,the normal form property is lost.In fact,the sum algorithm calculates a holonomic equation that is valid for any linear combination af +bg rather than the particular

given sum f +g .As a simple example,we consider the sum √1+x +1√1+x satisfying the ?rst order di?erential equation

2(2+x )(1+x )F (x )?xF (x )=0.

This di?erential equation can be found using a method given in [20]–[21],whereas the sum algorithm generates the second order di?erential equation

4(1+x )2

F (x )+4(1+x )F (x )?F (x )=0.

The reason for the existence of a di?erential equation of lower order is due to

the fact that the ratio of the two summands √1+x and 1√1+x forms a rational function.

Similarly,the sum of two consecutive Legendre polynomials P n (x )+P n +1(x )satis?es the second order di?erential equation

(x ?1)(x +1)F (x )+(x +1)F (x )?(n +1)2F (x )=0,

whereas the sum algorithm generates the di?erential equation

0=(x ?1)2(1+x )2F (x )+8(x ?1)x (1+x )F (x )+2 ?2+2n +n 2+6x 2?2n x 2?n 2x 2 F (x )

?4n (2+n )x F (x )+n (1+n )2(2+n )F (x )of fourth order,which is also valid for the di?erence P n (x )?P n +1(x )and for any other linear combination.

For the veri?cation of identities ,this is not an important issue,since the compatibility of two holonomic equations can be easily checked.This situation is similar to proving a rational identity by pure polynomial arithmetic without gcd computations (after having multiplied through by all denominators),and is actually equivalent to a noncommutative polynomial division,see §4.

In the case that the normal form is needed for a particular problem,a fac-torization algorithm can be used,s.§6.

For the discrete functions,the situation is quite similar.We call a function primitive whenever we use a separate symbol for it and a holonomic recurrence

equation is known.To these primitive functions,we add the rational functions and the functions

(mn +b )!,1

(mn +b )!(m ∈Q ),and a n (2)

whose holonomic recurrence equations are known,as primitive functions with respect to the variable n .We consider the factorial function to be equivalent to the Γfunction Γ(a +1)=a !,and declare binomial coe?cients etc.also via factorials.From the holonomic representations of the primitive functions the holonomic equations for all sums and products can be established.E.g.the two equations

(n ?k +1)2F (n +1,k )?(1+n )2F (n,k )=0(3)

and

(k +1)2F (n,k +1)?(n ?k )2F (n,k )=0(4)

for F (n,k )= n

k 2.Whereas these are simple consequences of the representation

of F (n,k )by factorials,the given procedure can be applied,for example,to the

more complicated function F (n,k )=n !+

k !2k to generate the two holonomic equations

nF (n +2,k )?(1+3n +n 2)F (n +1,k )+(1+n )2F (n,k )=0

and

k (2+k )2F (n,k +2)?(1+k )(1+3k +k 2)(3+3k +k 2)F (n,k +1)+k (1+k )3F (n,k )=0.Note that the given approach also covers all kinds of orthogonal polynomials and special functions with respect to their discrete variables,see §4.

In our Mathematica implementation SpecialFunctions ,the procedure HolonomicRE[a,n]calculates the holonomic recurrence equation of a n with re-spect to the variable n taking the known holonomic recurrence equations of the primitive functions into account,and using the sum and product algorithms by recursive decent through the expression tree.The above examples are governed by the following Mathematica session:

In[11]:=HolonomicRE[Binomial[n,k]^2,n]

2Out[11]=(1+n)a[n]-(1-k +n)a[1+n]==0

In[12]:=HolonomicRE[Binomial[n,k]^2,k]

2Out[12]=(-k +n)a[k]-(1+k)a[1+k]==0

In[13]:=HolonomicRE[(n!+k!^2)/k,n]

2Out[13]=(1+n)a[n]+(-1-3n -n )a[1+n]+n a[2+n]==0In[14]:=HolonomicRE[(n!+k!^2)/k,k]

Out[14]=k (1+k)(3+k)a[k]-

22>(1+k)(1+3k +k )(3+3k +k )a[1+k]+

>k (2+k)a[2+k]==0

3Hypergeometric Sums

Rather than having functions given as ?nite sums and products of primitive expressions,an important class of functions,particularly in analysis and com-binatorics,is given by in?nite sums of products of terms of the form (2)

s (n )= k ∈Z

F (n,k ).(5)

Then F (n,k )is an (m,l )-fold hypergeometric term .That is,both F (n +m,k )/F (n,k )and F (n,k +l )/F (n,k )are rational functions with respect to n and k for a certain pair (m,l )∈N 2.For example,by (3)–(4)this is valid for F (n,k )= n k 2with m =l =1.We assume moreover that the sums (5)have ?nite support,i.e.,they are ?nite sums for each particular n ∈N .

A modi?cation [23]of the (fast)Zeilberger algorithm ([43],see also [27],and [31])returns a holonomic recurrence equation valid for s (n ).Zeilberger’s algorithm is based on a decision procedure for inde?nite summation due to Gosper [17].In our example case,Zeilberger’s algorithm ?nds the holonomic recurrence equation (1+n )s (n +1)=2(1+2n )s (n )for s (n )= k ∈Z n k 2=n k =0 n k 2which fortunately has only two terms.Therefore,we are led to the representation

s (n )=n k =0 n

k 2=(2n )!n !.Even though,in general,the resulting recurrence equation has more than two terms,this holonomic equation contains very important structural information

about s(n).This may be used to show that a certain family of polynomials is orthogonal or not[44],and can be an interesting property for numerical purposes (compare[11]–[12]).

In particular,as described in the last section,the generated structural in-formation can be used for the identi?cation of a transcendental function that is given as sum(5).Note that sums of type(5)in general form transcendental functions with respect to the discrete variable n.

For example,to check the identity(compare[37])

n k=0 n

3=n

k=0

2 2k

(6)

which is nontrivial since for n=1it reads1+1=0+2,we need only to show that both sums s(n)satisfy the common recurrence equation

(n+2)2s(n+2)?(16+21n+7n2)s(n+1)?(n+1)2s(n)=0(7) which is the result given by Zeilberger’s algorithm.We also have the same initial values s(0)=1and s(1)=2,so we are done.

In Mathematica these computations are done by

In[15]:=HolonomicRE[Sum[Binomial[n,k]^2,{k,0,n}],n]

Out[15]=-2(1+2n)a[n]+(1+n)a[1+n]==0

In[16]:=HolonomicRE[Sum[Binomial[n,k]^3,{k,0,n}],n]

Out[16]=-8(1+n)a[n]+(-16-21n-7n)a[1+n]+

>(2+n)a[2+n]==0

In[17]:=HolonomicRE[Sum[Binomial[n,k]^2*Binomial[2k,n],{k,0,n}],n]

Out[17]=-8(1+n)a[n]+(-16-21n-7n)a[1+n]+

>(2+n)a[2+n]==0

Note that the example shows that transcendental functions can come in quite di?erent disguises.Might the left or the right hand side of(6)be a preferable representation?This question cannot be answered satisfyingly.A holonomic recurrence equation like(7),de?ning the same transcendental function s(n),is probably the simplest way to describe a function of a discrete variable,since

it postulates how the values of the function can be calculated iteratively.Not only is this a quite e?cient way to calculate the values of s(n),but moreover it is preferable to either of the two representations given in(6),since it gives a unique representation scheme.This is what a normal form is about.

As a further example,we consider the function(α,β,γ∈N0,z,M,d∈R+)

V(α,β,γ)=(?1)α+β+γ·Γ(α+β+γ?d)Γ(d/2?γ)Γ(α+γ?d/2)Γ(β+γ?d/2)Γ(α)Γ(β)Γ(d/2)Γ(α+β+2γ?d)M

·2F1 α+β+γ?d,α+γ?d/2

α+β+2γ?d

(2F1here represents Gau?’s hypergeometric function,see[1],Chapter15),which

plays a role for the computation of Feynman-diagrams[15]2,for which Zeil-berger’s algorithm generates the holonomic recurrence equation

0=(α+β?d+γ)(2α?d+2γ)V(α,β,γ)

+αM(2α+2β?2d+4γ?2z?4αz?2βz+3d z?4γz)V(α+1,β,γ) +2α(1+α)M2(z?1)z V(α+2,β,γ)

and analogous recurrence equations with respect to the variablesβandγ(see [24]).These,in particular,can be used for numerical purposes.

Note that for the application of Zeilberger’s algorithm our Mathematica program uses the Paule-Schorn implementation[31].For the current example,

the output is given by

In[18]:=HolonomicRE[(-1)^(alpha+beta+gamma)*Gamma[alpha+beta+gamma-d]* Gamma[d/2-gamma]*Gamma[alpha+gamma-d/2]*Gamma[beta+gamma-d/2]/

(Gamma[alpha]*Gamma[beta]*Gamma[d/2]*

Gamma[alpha+beta+2*gamma-d]*M^(alpha+beta+gamma-d))*

Hypergeometric2F1[alpha+beta+gamma-d,alpha+gamma-d/2,

alpha+beta+2*gamma-d,z],alpha,V]

Out[18]=(alpha+beta-d+gamma)(2alpha-d+2gamma)V[alpha]+

>alpha M(2alpha+2beta-2d+4gamma-2z-4alpha z-

>2beta z+3d z-4gamma z)V[1+alpha]+

>2alpha(1+alpha)M(-1+z)z V[2+alpha]==0

2I am indebted to Jochem Fleischer who informed me about a misprint in formula(31)of [15].

4Holonomic Systems of Several Variables

In[42],Zeilberger considered the more general situation of functions F of several discrete and continuous variables.If we have d variables,and d(essentially in-dependent)mixed homogeneous linear(partial)di?erence-di?erential equations with polynomial coe?cients in all variables are given for F,then F is called a holonomic system(compare[6]–[8]).In most cases these holonomic equations together with suitably many initial values declare F uniquely.

In particular,we concentrate on the situation,when the given system of holonomic equations is separated,i.e.each of them is either an ordinary di?er-ential equation or a pure recurrence equation.These representing holonomic equations can be generated by the method described in§2whenever F is given in terms of sums and products of primitive functions.

For example,the Legendre polynomials F(n,x)=P n(x)([1],Chapter22) form a holonomic system by their holonomic di?erential equation

(x2?1)F (n,x)+2xF (n,x)?n(1+n)F(n,x)=0(8) and their holonomic recurrence equation

(n+2)F(n+2,x)?(3+2n)xF(n+1,x)+(n+1)F(n,x)=0,(9) together with the initial values

F(0,0)=1,F(1,0)=0,F (0,0)=0,F (1,0)=1.(10) Equations(8)–(10)therefore build a su?cient algebraic,even polynomial struc-ture to represent the functions P n(x)as we shall see now.

If we interpret the(partial)di?erentiations and shifts that occur as opera-tors,and the representing system of holonomic equations as operator equations, then these form a polynomial equations system in a noncommutative polyno-mial ring.For a continuous variable x with di?erential operator D given by DF(n,x)=F (n,x),the product rule implies D(xf)?xDf=f,and hence the commutator rule Dx?xD=1is valid.Similarly for a discrete variable n with the(forward)shift operator N given by NF(n,x)=F(n+1,x),we have N(nF(n,x))?nNF(n,x)=(n+1)F(n+1,x)?nF(n+1,x)=F(n+1,x)= NF(n,x),and therefore the commutator rule Nn?nN=N.Similar rules are valid for all variables involved,whereas all other commutators vanish.

The transformation of a holonomic system given by mixed holonomic di?er-ence-di?erential equations represents an elimination problem in the noncom-mutative polynomial ring considered,that can be solved by noncommutative Gr¨o bner basis methods([3],[16],[19],[42],[45],[38]–[40]),[23]).

Hence,we need the concept of a Gr¨o bner basis.If one applies Gau?’s al-gorithm to a linear system,the variables are eliminated iteratively,resulting in an equivalent system which is simpler in the sense that it contains some equa-tions which are free of some variables involved.Note that connected with an application of Gau?’s algorithm is a certain order of the variables.

The Buchberger algorithm is an elimination process,given a certain term order for the variables(a variable order is no longer su?cient),with which a polynomial system(rather than a linear one)is transformed,resulting in an equivalent system(i.e.,constituting the same ideal)for which the terms that are largest with respect to the term order,are eliminated as far as possible.Note that—in contrast to the linear case—the resulting equivalent system may con-tain more polynomials than the original one.Such a rewritten system is called a Gr¨o bner basis of the ideal generated by the polynomial system given.It turns out that Buchberger’s algorithm can be extended to the noncommutative case that we consider here[19]as long as the rewrite process using the commutator does not increase the variable order.

As an example,we consider F(n,k)= n k in which case we have the Pascal triangle relation F(n+1,k+1)=F(n,k)+F(n,k+1),together with the pure recurrence equation(n+1?k)F(n+1,k)?(n+1)F(n,k)=0with respect to n,say.These equations read as(KN?1?K)F(n,k)=0,and ((n+1?k)N?(n+1))F(n,k)=0in operator notation,K denoting the shift operator with respect to k.Therefore we have the polynomial system

KN?1?K and(n+1?k)N?(n+1).(11) The Gr¨o bner basis of the left ideal generated by(11)with respect to the lexi-cographical term order(k,n,K,N)is given by

(k+1)K+k?n,(n+1?k)N?(n+1),KN?1?K ,

i.e.,the elimination process has generated the pure recurrence equation

(k+1)F(n,k+1)+(k?n)F(n,k)=0

with respect to k.

We used the REDUCE implementation[28]for the noncommutative Gr¨o bner calculations of this article,but I would like to mention that there is also a Maple package Mgfun written by Chyzak[10](to be obtained from http://pauillac.inria.fr/algo/libraries/libraries.html#Mgfun)which can be used for this purpose.

As another example,we consider the Legendre polynomials.In operator notation the holonomic equations(8)–(9)constitute the polynomials

(x2?1)D2+2xD?n(1+n)and(n+2)N2?(3+2n)xN+(n+1).(12) The Gr¨o bner basis of the left ideal generated by(12)with respect to the lexi-cographical term order(D,N,n,x)is given by

(x2?1)D2+2xD?n(1+n),

(1+n)ND?(1+n)xD?(1+n)2,(13)

(x2?1)ND?(1+n)xN+(1+n),(14) (1+n)(x2?1)D?(1+n)2N+x(1+n)2,(15)

(n+2)N2?(3+2n)xN+(n+1) .

After the calculation of the Gr¨o bner basis,for better readability I positioned the operators D and N back to the right,so that the equations can be easily understood as operator equations,again.By the term order chosen,the Gr¨o bner basis contains those equations for which the D-powers are eliminated as far as possible,and(13)–(15)correspond to the relations

P n+1(x)=x P n(x)+(1+n)P n(x),

(x2?1)P n+1(x)=(1+n)(xP n+1(x)?P n(x)),

(x2?1)P n(x)=(1+n)(P n+1(x)?xP n(x))(16) between the Legendre polynomials and their derivatives.

If we are interested in a relation between the Legendre polynomials and their derivatives that is x-free(which is of importance for example for spectral approximation,see[9]),we choose the term order(x,D,N,n)to eliminate x in the?rst place,and obtain a di?erent Gr¨o bner basis containing the x-free polynomial

?(n+2)(n+1)D?(2n+3)(n+2)(n+1)N+(n+2)(n+1)N2D equivalent to the identity

(2n+1)P n(x)=P n+1(x)?P n?1(x)

for the Legendre polynomials(see e.g.[9],formula(2.3.16)).

Here,we present the REDUCE output for the above examples:

1:load ncpoly;

2:nc_setup({D,NN,n,x},{NN*n-n*NN=NN,D*x-x*D=1},left);

3:p1:=(x^2-1)*D^2+2*x*D-n*(1+n)$%differential equation

4:p2:=(n+2)*NN^2-(3+2*n)*x*NN+(n+1)$%recurrence equation

5:nc_groebner({p1,p2});

2222

{d*x-d-2*d*x-n-n,

d*nn*n-d*n*x-d*x-n-n,

d*nn*x-d*nn-nn*n*x-2*nn*x+n+1,

2222

d*n*x-d*n+d*x-d-nn*n+n*x-x,

nn*n-2*nn*n*x-nn*x+n+1}

6:nc_setup({x,D,NN,n},{NN*n-n*NN=NN,D*x-x*D=1},left);

7:result:=nc_groebner({p1,p2});

2222

result:={x*d+2*x*d-d-n-n,

x*d*nn-d*nn*n+d*n+d+2*nn*n+2*nn*n+nn,

x*d*n+x*d-d*nn*n+n+2*n+1,

2*x*nn*n+x*nn-nn*n-n-1,

222232 d*nn*n-d*nn*n-d*n-3*d*n-2*d-2*nn*n-3*nn*n-nn*n} 8:nc_setup({n,x,NN,D},{NN*n-n*NN=NN,D*x-x*D=1},left);

9:nc_compact(part(result,5));

-(2*n+3)*(n+2)*(n+1)*nn+(n+2)*(n+1)*nn*d-(n+2)*(n+1)*d We see,therefore,that by the given procedure new relations(between the bino-

mial coe?cients,and between the derivatives of the Legendre polynomials)can

be discovered.The generation of derivative rules like(16),and the algorithmic work with them is described in[23].

5Holonomic Sums and Integrals

Analogously,with the method in the last section,holonomic recurrence equa-tions for holonomic sums can be generated.Note that the idea to use recurrence equations for the summand to deduce a recurrence equation for the sum is origi-

nally due to Sister Celine Fasenmyer([13]–[14],see[33],Chapter14).Zeilberger [42]brought this into a more general setting.

Consider for example

s(n)=

k=0

F(n,k)=

k=0

P n(x),

then by the product algorithm,we?nd the holonomic recurrence equations

(n?k+1)F(n+1,k)?(1+n)F(n,k)=0

and

(2+k)2F(n,k+2)?(3+2k)(n?k?1)xF(n,k+1)+(n?k)(n?k?1)F(n,k)=0

for the summand F(n,k).The Gr¨o bner basis of the left ideal generated by the corresponding polynomials

(n?k+1)N?(1+n)and(2+k)2K2?(3+2k)(n?k?1)xK+(n?k)(n?k?1) with respect to the lexicographical term order(k,N,n,K)contains the k-free polynomial

(2+n)2K2N2?K(2+n)(3+2n)(K+x)N+(1+n)(2+n)(1+K2+2Kx),(17) which corresponds to a k-free recurrence equation for F(n,k).We use the order (k,N,n,K)because then k-powers are eliminated as far as possible(since we like to?nd a k-free recurrence),and N-powers come next in the elimination process(since the recurrence equation obtained should be of lowest possible order).

Because all shifted sums

s(n)= k∈Z F(n,k)= k∈Z F(n,k+1)= k∈Z F(n,k+2)

generate the same function s(n),and since summing the k-free recurrence equa-tion is equivalent to setting K=1in the corresponding operator equation (check!),the substitution K=1in(17)generates the valid holonomic recur-rence equation

(2+n)s(n+2)?(3+2n)(1+x)s(n+1)+2(1+n)(1+x)s(n)=0

for s(n).

In the general case,we search for a k-free recurrence equation contained in a Gr¨o bner basis of the corresponding left ideal with respect to a suitably chosen weighted[30](or lexicographical(k,N,n,K))term order.For example,the elimination problems described in[45]are automated by this procedure.

On the other hand,it turns out that in many cases the holonomic recurrence equation derived is not of the lowest order.In the next section,we will discuss how this problem can be resolved.

Note that by a similar technique,holonomic integrals can be treated[2].To ?nd a holonomic equation for

I(y):=

F(y,x)dx

for holonomic F(y,x)with respect to the discrete or continuous variable y,cal-culate the Gr¨o bner basis of the left ideal constituted by the holonomic equations of F(y,x)with respect to a suitably chosen weighted or the lexicographical term order(x,D y,y,D x).We search for an x-free holonomic equation E contained in such a Gr¨o bner basis.In case,that F(y,a)=F(y,b)≡0,and enough derivatives of F(y,x)with respect to x vanish at x=a and x=b,by partial integration it follows that the holonomic equation valid for I(y)is given by the substitution D x=0into E(see[2]).

As an example,we consider

I(n):=

∞

?∞

e?x2H n(x)dx,

H n(x)denoting the Hermite polynomials.The method of§2yields the holo-nomic polynomials

2(1+n)+N2?2x N and D2+2(1+n)+2x D

for the integrand.Note that since H n(x)is an odd function for odd n,it is immediately clear that I(n)=0in this case.However,what about even values of n?

The Gr¨o bner basis of the corresponding left ideal contains the two x-free polynomials

N2+N D and N n+n D+D

so that setting D=0we get for I(n)the recurrence equation I(n+1)=0. Indeed,this proves that I(n)=0for n≥1.

As another example,we consider the Abramowitz functions([1],27.5))

A(n,y):=

∞

x n e?x2?y/x dx.

By the method in§2for the integrand F(n,y,x)=x n e?x2?y/x we get the three holonomic polynomials

x?N,?n x+x2D x+2x3?y and1+x D y.

Using the term order(x,D y,y,D x),the di?erential equation

y A (n,y)?(n?1)A (n,y)+2A(n,y)=0,

and using(x,N,n,D),the recurrence equation

2A(n+3,y)?(n+2)A(n+1,y)?y A(n,y)=0

is automatically generated by the given approach(compare[1],(27.5.1),(27.5.3)).

Finally,we mention that similarly an identity like([1],(11.4.28))

∞

0e?a2x2x m?1J n(bx)dx=

Γ(n/2+m/2)b n

2n+1a n+mΓ(n+1)1

n/2+m/2

n+1

?b24a2

(18)

(1F1representing Kummer’s con?uent hypergeometric function)for the Bessel function is proved by the calculation of the common holonomic recurrence equa-tion

0=?(n+3)(n+m)b2I(n)

+2(n+2) 4a2n2+16a2n+12a2?b2m+b2 I(n+2)

+(n+1)(n+4?m)b2I(n+4)

for the left and right hand sides of(18).Note that Zeilberger’s algorithm is not directly applicable to the right hand side,but the extended version of[23]gives the result.

6Noncommutative Factorization and Holonomic Normal Form

Note that neither the sum and product algorithms of§2,nor Zeilberger’s algo-rithm or its extension[23],nor the algorithms for holonomic sums and integrals of§5can guarantee to present the holonomic equation N of lowest order,and therefore the normal form searched for.

In[29]3a Gr¨o bner basis based factorization algorithm was introduced for polynomials in noncommutative polynomial rings given by Lie bracket commu-tator rules.This method is implemented in[28].Given an expression f,and a holonomic equation P of order m of f,one may?nd the normal form N of f using this factorization algorithm by generating the right factors of the non-commutative polynomial p corresponding to P,and checking if any of them,Q, say,(having order l

3Due to a severe bicycle accident of Herbert Melenk,this paper is still un?nished.

To present some examples,we consider Zeilberger’s algorithm?rst.An ex-ample for which Zeilberger’s algorithm does not generate the holonomic recur-rence equation of lowest order is given by the sum(see e.g.[31])

s n:=

k=0

(?1)k n k

for which the holonomic equation

2(2n+3)s n+2+3(5n+7)s n+1+9(n+1)s n=0(19)

is generated.Note that there is an algorithm due to Petkov?s ek[32]to?nd all hypergeometric solutions of holonomic recurrence equations which could be used as next step.However,we may also proceed as follows:The corresponding noncommutative polynomial2(2n+3)N2+3(5n+7)N+9(n+1)is factorized by implementation[28]as

2(2n+3)N2+3(5n+7)N+9(n+1)=((4n+6)N+3(n+1))(N+3).

The right factor N+3corresponds to the holonomic recurrence equation

S n+1+3S n=0,(20)

which,together with the initial value S0=s0=1uniquely de?nes a sequence (S n)n∈N

.Since S1=?3turns out to be compatible with the given sum

s1=

k=0

(?1)k 1k

=?3,

and since(20)implies(19)(right factor!),the sequence s n,which is the unique solution of(19)with s0=1and s1=?3,must equal S n.From(20),however, the closed form s n=(?3)n follows.

Similarly,for any particular d∈N,d≥3,the identity

k=0(?1)k n k

=(?d)n

can be established,for whose left hand side Zeilberger’s algorithm generates a recurrence equation of order d?1(see[31]).

Whereas Petkov?s ek’s algorithm?nds hypergeometric solutions of holonomic recurrence equations as in the example,and therefore not only veri?es identities, but generates closed-form results,our approach is more general in the follow-ing sense.Factorizations with polynomial coe?cients of ordinary holonomic di?erential equations(see[4],[35]for other methods)as well as of any mixed holonomic di?erence-di?erential equation can be calculated.

We give an example of that type for the sum algorithm:Consider the di?er-

ence of successive Gegenbauer polynomials h(x)=C(?1/2)

n+1(x)?C(?1/2)

(x)that

were used in[26].Here the summand f(x):=C(?1/2)

(x)satis?es the holonomic

equation

(x2?1)f (x)+(n?n2)f(x)=0,

and the sum algorithm yields the fourth order equation

(x2?1)2h (x)+4x(x2?1)h (x)?2(n2?1)(x2?1)h (x)+n2(n2?1)h(x)=0 for h(x).The implementation[28]?nds(besides others)the noncommutative

factorization

(x2?1)D2+(1+x)D?n2 (x2?1)D2?(1+x)D+(1?n2) of the corresponding noncommutative polynomial,whose right factor

(x2?1)D2?(1+x)D+(1?n2)

turns out to be compatible with the given function h(x).That is,the corre-sponding di?erential equation and two derivatives thereof are satis?ed by h(x), at x=1.Therefore the holonomic normal form of h(x)is the corresponding di?erential equation

(1?n2)h(x)?(1+x)h (x)+(x2?1)h (x)=0

that was a tool in[26].This result can also be obtained by the method given in[20]–[21].

To evaluate the integrals

I n:=

∞

?∞

x n e?x2H n(x)dx,

we may deduce the holonomic system

N2?2x2N+2(1+n)x2

and

x2D2+2x(x2?n)D+(n+n2+2x2)

of the integrand.The Gr¨o bner basis of this system with respect to the weighted lexicographical order with weights(3,1,0,0)for(x,N,n,D)(i.e.the term x is considered larger than N3,whereas x is smaller than N4,and any power of n and D is smaller than x and N)contains an x-free polynomial,which when evaluated at D=0yields

P(n,N)=(n+5)(n+4)(n+3)N3?(3n+7)(n+5)(n+4)(n+3)N2 +(3n+5)(n+5)(n+4)(n+3)(n+2)N(21)

?(n+5)(n+4)(n+3)(n+2)(n+1)2

corresponding to a recurrence equation of order three.

On the other hand,P(n,N)obviously has the trivial(commutative)factor-ization

P(n,N)=(n+5)(n+4)(n+3) N3?(3n+7)N2+(3n+5)(n+2)N?(n+2)(n+1)2 and the remaining right factor can be represented as

N3?(3n+7)N2+(3n+5)(n+2)N?(n+2)(n+1)2=(N?n?2)(N?n?1)(N?n?1) (note that[28]?nds four di?erent right factors).This leads to the valid recur-rence equation I n+1=(n+1)I n that together with the initial value

I0=

∞

?∞

e?x2dx=

√

gives?nally I n=√

πn!.

Acknowledgement

I would like to thank Prof.Peter Deu?hard for his support and encouragement. I’d also like to thank Herbert Melenk for his advice on Gr¨o bner bases,and for his excellent REDUCE implementation[28].Hopefully,he will have recovered soon!

References

[1]Abramowitz,M.and Stegun,I.A.:Handbook of Mathematical Functions.

Dover Publ.,New York,1964.

[2]Almkvist,G.and Zeilberger,D.:The method of di?erentiation under the

integral sign.J.Symbolic Computation10,571–591(1990).

[3]Becker,Th.and Weispfenning,V.:Gr¨o bner Bases.A Computational Ap-

proach to Commutative Algebra.Springer,New York,1991.

[4]Beke,E.:Die Irreducibilit¨a t der homogenen linearen Di?erentialgleichun-

gen.Math.Ann.45,278–294(1894).

[5]Beke,E.:Die symmetrischen Functionen bei linearen homogenen Di?eren-

tialgleichungen.Math.Ann.45,295–300(1894).

[6]Bernstein,I.N.:Modules over a ring of di?erential operators,study of the

fundamental solutions of equations with constant coe?cients.Functional Anal.Appl.5,1–16(Russian);(89–101)(English)(1971).

Functions of language

Functions of language There are many categorizations of functions of language, not in terms of the concrete specific function functions that language is put to in our daily life, such as to chat, to think, to buy and sell, to read and write, to greet people., but in terms of the more generalized functions language can perform in human communication. There are three main functions:the descriptive function, the expressive function, the social function. The descriptive function, also referred to as the cognitive, or referential, propositional function, is the primary function, which is to convey factual information. e.g. The Sichuan earthquake is the most serious one China has ever suffered. The expressive function, or emotive or attitudinal function is to supply information about the user’s feelings, preferences, prejudices, and values. e.g. I will never go camping with the Simpsons again. The social function, or the interpersonal function, serves to establish and maintain social relations between people. e.g. How can I help you, Sir? In the early 1970s the British linguist M.A.K.Halliday explored in a number of papers the functions of child language. His system contains macro functions:the ideational, the interpersonal ,Textual. Ideational: Language constructs a model of experience and constructs logical relations. When we use language to identify things, to think, or to record information, we use language as a symbolic code to represent the world around us. Playing this function, language serves as a medium that links a person with the world. Interpersonal: Language enacts/represent social relationships. It is concerned with interaction between the addresser and addressee in a discourse situation and the addresser’s attitude toward what he speaks or writes about. For example, the ways in which people address others and refer to themselves such as “Dear Sir “ (or Madam), “Dear professor”, “Johnny”, “yours”, “your obedient servant” indicate the various grade s of interpersonal relations. It is

Language functions

Language functions 1.announcing the beginning ●May I have your attention, please? It’s time for us to start. ●Attention, please. Shall we get down to business? ●Ladies and gentlemen, please be seated. Since we are already a bit late, we should begin the present session immediately. ●It’s a great pleasure for me to open/begin/start this morning’s session. ●I would like to call the session to order. 2.introducing oneself and/or others ●I am Dr. A from ABC University, China, and I am going to be in the chair of this afternoon’s session. ●Let me introduce myself. My name is A from XYZ Institute. It is a privilege for me to chair this session. ●We are honored to have Dr. B from DEF University on my left as co-chairperson. ●On my immediate right is Dr. C from UVW University, and beyond, Professor D from the same university. 3.calling on participants/encouraging questions ●You’re requested to put forward your opinions and views concerning Mr. W’s speech. ●Mr. L, you appear to have some questions to ask regarding this matter, could you please tell us what you think about it? ●Is there anyone else who wants to say something regarding Dr. A’s presentation? ●Any more questions or comments you would like to address to Professor B? ●Well, since nobody seems willing to start the ball rolling, I’ll just like to make a small comment. 4.changing the subject/moving on ●Let’s turn/move on/pass on to the next problem. ●I would like to switch from ABC to DEF. ●We’ll now get off the topic of ABC and move on to the next problem. ●If we could move on to … ●Having looked at …, I’d now like to come to … 5.reminding the speakers/keeping the discussion on the subject ●That’s interesting, but it raises a different point. Could we come back to that later? ●It seems to me that you have just said falls a bit beyond the range of our present topic. Let’s go back to the main theme. ●I’m afraid what you said is not to the point. ●We seem to be getting off the main point. Can I bring the discussion back in the direction of XYZ for a moment? ●Mr. A, you’re getting off the point a bit. I wish you could confine yourself to the point under discussion. 6.moderating the disputes ●Since we do not seem to be able to resolve this difference now, could we move on to the next point? ●As to the result of this discussion, I would say, let’s think it over for two or three minutes.

Windows平台上NCL的安装

图文详解Windows平台上NCL的安装 NCL在Linux下的安装非常容易，只需下载适当版本的文件，设置好环境变量即可使用。 NCL在Windows下的安装则要麻烦一些，需要先安装一个虚拟Linux环境（Cygwin/X）。以下内容详细介绍NCL在Windows平台上的安装过程，希望仅具备Windows基本操作技能的用户也能轻松安装NCL。一、NCL简介二、准备工作三、安装Cygwin/X 四、熟悉Cygwin/X环境五、安装NCL 六、运行NCL范例七、语法高亮显示（此部分供有兴趣的用户参考）八、.hluresfile文件（此部分供有兴趣的用户参考）九、FAQ 十、获取帮助一、NCL简介 NCL（NCAR Command Language）是由NCAR的“Computational & Information Systems Laboratory”开发的。 NCL是一种编程语言，专门用于分析和可视化数据。主要用于以下三个领域：文件输入/输出（File input and output）：资料处理（Data processing）：图形显示（Graphical display）：可生出出版级别的黑白、灰度或彩色图。从5.0起，NCL和NCAR Graphics已经打包在一起发行。2009年3月4日，NCL发布了最新的5.1.0版，该版本更新了地图投影，修正了一些bug，增加了更多的函数及资源。下图为新增的含中国省界的地图（见图1-1）。

二、准备工作 2.1 安装环境安装环境为WinXP Professional SP3，并做如下假定：计算机名：TEAM 用户名：Grissom 安装目录：D:\download 用户在实际安装中，请根据自己系统的信息替换本教程中的计算机名和用户名。特别说明：用户名中不能出现空格，否则会在使用中出现一些问题。 2.2 下载Cygwin/X Cygwin/X=Cygwin+X。通俗地说，Cygwin/X可以在Windows平台上实现命令行+图形的Linux模拟环境。 Cygwin/X的下载与安装非常灵活，用户可根据自己的需求定制。为便于大家的安装，我已下载了安装NCL所需的软件包，包括编译器、编辑器、X Server等，用户可直接从以下地址下载，并解压至D:\download\install 目录下。 Cygwin下载：https://www.wendangku.net/doc/442816200.html,/xglm/2009/2/wnx45afnq7.htm 以下关于Cygwin和Cygwin/X的详细介绍供参考：

Management Functions(作业)

课堂（后）作业 Translate (English to Chinese) expressions Management functions management control Resolve conflicts management by objectives Set a goal management competence Motivate subordinates management inventory form Formulate plans management of trades Previously set goals management principle Controlling function Communication channel管理功能管理控制Resolve conflicts management by objectives 通过解决目标冲突管理设定一个目标管理能力Motivate subordinates management inventory form 激励下属管理库存的形式制定计划，行业管理Previously set goals management principle 先前设定的目标管理原理控制功能Communication channel通信信道 Translate (Chinese to English) 1. Generally speaking, managers performs four basic functions .（履行四项基本职能） Organizations exist to achieve a goal. (是为了实现某一个目标) 3. The textbook on management According to management function arrangement. (是按管理的职能编排的) 4. Management should be Responsible for the organization's goals, establishing an overall strategy. . (负责拟定机构的目标，制定全面策略) 5. The leading function involves in Motivating subordinates, resolve conflicts among members, directing their actions. (激励下属，解决员工间的纠纷，指挥他们的行动) 6. We refer to your application for employment with us（根据你向我们提出的求职申请）and are pleased to offer you the position. 7. Either party may terminate the labor contract（任何一方都可以终止该劳动合同）by giving to the other 1 day’s notice or 1 day’s salary in lieu of notice during probation. 8. If you are guilty of misconduct or any other conduct Damage to the interests of the company or reputation (有损于公司利益或声誉的)，the company can terminate the contract without notice.

ncl源码安装方案

气象项目NCL开发环境配置手册 1建立和测试源码安装ncl所需要的编译环境 1.1安装ncl所需要的编译器源码编译安装ncl需要C编译器和Fortran编译器。C编译器使用gcc即可。安装ncl5.2.1版本时，最好使用gfortran或g95作为Fortran编译器，而不要使用g77。 1.2为外部软件配置环境变量例如设置C编译器环境变量，export CC=gcc；设置Fortran编译器，export FC=gfortran。 2下载安装非可选外部软件注：安装时最好把所有的外部软件都安装到同一根目录下，这样做便于以后告诉ncl编译系统所有的外部软件的安装位置。本文假设所有的外部软件都安装在/usr/local目录下。官网上说如果源码编译安装ncl，则下面的几款软件都是必须安装的： ●JPEG 支持jpeg图形的软件，我下载的jpeg源码安装文件是jpegsrc.v8c.tar.gz。一旦有了源码，执行以下命令进行安装： ./configure --prefix=/usr/local make all install 如果jpeg版本是v6的，则还要额外执行以下命令： make install-lib make install-headers ●zlib 如果想要支持png图形，或者支持grib2数据，则需要下载安装此软件。我下载的zlib源码安装文件是zlib-1.2.5.tar.gz。一旦有了源码，执行以下命令进行安装： ./configure - -prefix=/usr/local make all install ●NetCDF 支持NetCDF数据格式读取的软件包。（如果不需要NetCDF数据读取的话，应该可以不用安装） - 1-

MATLAB的函数及指令Functions and Commands

附录C：索引〖使用说明〗 ●本“索引”列出了在本书叙述文字或算例中所涉及的所有符号、指令、模块和图形对象属性的“关键符（词）”。 ●读者根据“关键符（词）”可从本“索引”找到本书对它的说明或使用算例。 C.1MATLAB的标点及符号算术运算符Arithmetic operators + 加；正号 1.3.3, 2.2.1, 4.3.1 - 减；负号 1.3.3, 2.2.1, 4.3.1 * 矩阵乘 1.3.3, 2.2.1, 4.3.1 .* 数组乘 1.3.3, 2.2.1, 4.3.1 \ 矩阵左除 1.3.3, 2.2.1, 4.5.1, 4.5.3, 5.6.2 / 矩阵右除 1.3.3, 2.2.1, 4.5.1 .\ 数组除 1.3.3, 2.2.1, ./ 数组除 1.3.3, 2.2.1, ^ 矩阵乘方 1.3.3, 2.2.1, 4.4.2 .^ 数组乘方 1.3.3, 2.2.1, ' 共轭转置 2.2.1, 4.3.1 .' 非共轭转置 2.2.1, 4.3.1 关系运算符 Relational operators == 等于 2.2.1, 2.5.1, 4.7.1, 5.1.3 ~= 不等于 2.2.1, 2.5.1, < 小于 2.2.1, 2.5.1, 4.2.3 > 大于 2.2.1, 2.5.1, 4.2.3 <= 小于等于 2.2.1, 2.5.1, 4.10.2 >= 大于等于 2.2.1, 2.5.1, 4.10.2 逻辑运算符 Logical operators & 逻辑与 2.5.2 | 逻辑或 2.5.2, 4.2.3, 4.10.2 ~ 逻辑非 2.5.2 && 先决逻辑与 || 先决逻辑或

wrf手册中文

Chapter 1: Overview Introduction The Advanced Research WRF (ARW) modeling system has been in development for the past few years. The current release is Version 3, available since April 2008. The ARW is designed to be a flexible, state-of-the-art atmospheric simulation system that is portable and efficient on available parallel computing platforms. The ARW is suitable for use in a broad range of applications across scales ranging from meters to thousands of kilometers, including: ?Idealized simulations (e.g. LES, convection, baroclinic waves) ?Parameterization research ?Data assimilation research ?Forecast research ?Real-time NWP ?Coupled-model applications ?Teaching 简介 Advanced Research WRF (ARW)模式系统在过去的数年中得到了发展。最近公布了第三版，从2008年4月开始可供使用。ARW是灵活的，最先进的大气模拟系统，它易移植，并且有效的应用于各种操作系统。ARW适用于从米到成千上万公里尺度的各种天气系统的模拟，它的功能包括： ?理想化模拟（如，LES，对流，斜压波） ?参数化研究 ?数据同化研究 ?预报研究 ?实时数值天气预报 ?耦合模式应用 ?教学 The Mesoscale and Microscale Meteorology Division of NCAR is currently maintaining and supporting a subset of the overall WRF code (Version 3) that includes: ?WRF Software Framework (WSF) ?Advanced Research WRF (ARW) dynamic solver, including one-way, two-way nesting and moving nest. ?The WRF Preprocessing System (WPS) ?WRF Variational Data Assimilation (WRF-Var) system which currently supports 3DVAR capability ?Numerous physics packages contributed by WRF partners and the research community ?Several graphics programs and conversion programs for other graphics tools And these are the subjects of this document. The WRF modeling system software is in the public domain and is freely available for community use.

Functions of advertisement

1.Functions of advertisement: Advertisement is the most important tool, more like a bridge linked consumer with advertiser. The functions of advertisement are multifarious. I will address a few important points here. 1)For Consumer Firstly, advertisements could help consumers understand feature/main function of product, for example Wanglaoji ad shows the product “Afraid lit drink Wanglaoji （怕上火喝王老吉）” clearly; Secondly, advertisements also help consumers familiar with brand image of the advertiser, as we familiar with Mercedes-Benz ads; Thirdly, advertisements help consumers build up style of consumption which just as what Colgate tooth toothpaste does; Finally, advertisements help consumers have better understanding on trends of consumption which like BYD Hybrid advertisements done. Advertisements can educate consumer relevant knowledge 2)For Advertiser On the other hand, advertisement is good friend to advertiser. Advertisements deliver product information to customer, depends on how, where and when that company want to; Advertisements develop interest of customer to product, attract them take out money; Advertisements help advertiser to establish a strong competitive advantage by increasing brand awareness and preference; Advertisements can maintain and increase customer loyalty; 2.Let us have a general look on types of advertisements; 1)Traditional advertisement includes TV ads which we familiar with couple of years; Radio ads as company while driving but please do not text them; newspaper and magazine ads is the main resource that people looking for satisfy demand; outdoor, building, elevator and cars advertisement becomes a city sightline in today’s life. 2)Network based advertisements; such as portal advertisement, ads on SNS, in email, SMS and other types. These low cost advertisements come to be an important part of company promotion channel. 3)On the other hand, advertisement in movie such as Transformer 3 with Lenovo and Shuhua milk, Matrix with Nokia mobile phone and other advertisement in video game, famous with FIFA games. These types of advertisement are flexible and highly target to consumer group. 4)Finally onsite advertisement which mainly use as inside store promotion. Such as flagship store ads and promoter.

《叉车操作手册》

叉车电瓶和充电器使用规程一、电瓶：当叉车配备了其它种类的电瓶及/或充电器时，遵从其制造商的指导。（一）远离火种（爆炸性气体）绝对禁止火焰接近电瓶。电瓶内部会产生爆炸性气体；吸烟、火焰及火花，均会引起电瓶爆炸。（二）小心触电严禁引起短路，电瓶带有高电压和能量。当处理电瓶时，要戴护目镜、穿胶鞋和戴橡皮手套。（三）正确接触严禁将电瓶的正、负极调乱，否则可导致火花、燃烧及/或爆炸。（四）远离工具严禁让工具接近电瓶两极，以免引起火花或短路。（五）勿过量放电严禁让叉车的电量耗至叉车不能移动时，才进行充电。（会引致电池寿命缩短）当电瓶负荷显示器显示无电时，请立刻进行充电。（六）保持清洁保持电瓶上表面干净。严禁使用干布擦电瓶表面，以免引起静电。清洁电瓶要戴护目镜、穿胶鞋和戴橡皮手套。当清洁电瓶后，才能进行充电。（七）穿着安全服为了个人安全，需配戴护目镜、穿胶鞋和戴橡皮手套。（八）小心电瓶电解液。电瓶电解液含有硫酸，严禁让皮肤接触电瓶电解液。（九）急救电瓶电解液含有硫酸，若与之接触可能造成烧伤。发生意外时，请立即进行急救并请医生治理。 ①溅到皮肤上：用水冲洗10-15分钟 ②溅到眼睛上：用水冲洗10-15分钟 ③误咽：喝大量的水和牛奶 ④溅到衣服上：立即脱下衣服如不遵守以上各点，可造成严重的伤亡事故。（十）拧紧电瓶通风盖拧紧电瓶通风盖，以防泄漏电解液。（十一）清洗严禁在叉车上清洗电瓶，以免损坏叉车。（十二）不正常的电瓶如电瓶发生下列情况，与和资格的服务机构或电瓶生产商联系。 ①电瓶发臭 ②电解液变浊 ③电解液减少速度过快 ④电解液温度过高（十三）严禁拆解电瓶不得让电解液耗尽、拆卸或自行维修电瓶。

FME四则运算中数学函数MathFunctions

FME中四则运算中的Math Functions（数学函数）原文： file:///C:/Program%20Files/FME2015.0/help/fme_desktop/FME_Desktop_Help.htm#.. /Subsystems/FME_Transformers/Content/transformer_parameters/math_functions.ht m 四则编辑器支持下列数学函数表达式。下列所有函数的参数为双精度，并返回双精度值，除非它们是值类型转换函数，例如@int(), @double(), 和@real32()。对于所有参数数量不定的函数，参数中包含null,missing或空字符串的值都会被过滤掉。然而，如果变量参数列表只包含null,missing和空字符串，函数返回null。函数接收到非数字型、null、missing或空字符串参数时会返回null，并附加到 fme_expression_warnings列表属性中。对于所有参数固定的函数，返回双精度值，如果参数为NaN，则预计结果为NaN。所以下列提供的描述的前提是函数参数是数值型、non-null，non-infinity 和non-NaN。函数描述 abs(arg) 返回arg 的绝对值 acos(arg) 返回arg 的反余弦值，值的范围[0,pi]。Arg 的值在[-1,1]范围。 add(arg1,[arg2]...) 返回参数的和。 asin(arg) 返回arg的反正弦值，值的范围[-pi/2,pi/2]。Arg 的值在 [-1,1]范围。 atan(arg) 返回arg的反正切值，值的范围在[-pi/2,pi/2]。 atan2(y,x) 返回y/x的反正切值，值的范围在[-pi,pi]。x和y不能同时为 0. average(arg1,[arg2]... ) 输入数字列表并求出平均值。忽略空、missing和null输入，若输入为非数值型数据会导致失败。如果没有输入，返回空字符串。 ceil(arg) 以双精度的方式返回不小于arg的最小整数部分。 cos(arg) 返回arg的余弦值，以弧度为单位。 cosh(arg) 返回arg的双曲余弦。如果发生溢出，返回无穷大。degToRad(arg) 将度转换为弧度。 div(x,y) 计算x/y。如果除数为0，返回无穷大。 double(arg) 以双精度形式返回arg。 exp(arg) 返回arg的指数，以e为底，e的arg次幂，如果发生溢出，返

WRFChem安装及使用说明手册

WRF-Chem安装及使用说明手册 V1.0 编写者：王彬 2015年6月

目录 1. 引言 (3) 1.1 编写目的 (3) 1.2 背景 (3) 1.3 参考资料 (4) 2. 安装过程 (4) 2.1 设定环境变量 (4) 2.2 WRFV3安装 (5) 2.3 WPS与WRFDA的安装 (6) 3. 使用过程 (6) 3.1 WPS数据准备说明 (7) 3.2 气象初始场的准备说明 (8) 3.3 化学数据前处理程序的使用说明 (8) 3.4 化学数据转化程序的使用说明 (9) 附录1 （prep_chem_sources.inp） (14) 附录2（ncl程序） (18) 附录3（convert.f90） (19)

1. 引言 1.1 编写目的本手册为指导WRF-Chem数值模拟的用户，方便轻松掌握WRF-Chem数值模式的安装及使用而编写。希望该手册在大家使用和学习过程中起到引导和帮助的作用。 1.2 背景 WRF-Chem模式是由国家大气研究中心（NCAR）等机构联合开发的新一代大气预报模式，真正实现了气象模式与化学传输模式在时空上的耦合。它不仅能够模拟污染气体和气溶胶的排放、传输以及混合过程，还可用于分析空气质量、云与化学之间的相互作用等。研究表明，WRF-Chem对气象场以及污染物的模拟表现出值得信赖的能力。 WRF-Chem在原有WRF模式的基础上耦合了化学反应过程，它通过在WRF 的辐射方案中引入气溶胶光学厚度、单次散射反照率和不对称因子表现气溶胶的直接辐射效应。此外，还增加了云滴数浓度的计算，这会改变原有WRF辐射方案中云粒子有效半径的计算方式，从而影响云的反照率，体现气溶胶的Twomey 效应。另一方面，Liu et al.提出了以云滴数浓度为阈值的新的云水向雨水自动转换参数化方案，取代了原有的Kessler方案，体现气溶胶的Albrecht效应。

小型机UNIX系统操作指南完整版

小型机UNIX系统操作指南一、 UNIX基本指令 1、Ls –l/-a/-R 列目录 2、Cd 更改目录 3、ps –ef |grep ora_ 看进程 4、kill -9 pid 杀进程 5、more filename 按屏看文本 6、cat filename 看文本 7、strings filename 看文本 8、df 看文件系统 9、du –k 看空间 10、rm 删文件 11、rm –r dir 删目录 12、find . –name filename 查找文件 13、cp *.* /aaa 拷贝文件 14、mv 移动文件 15、tail –f file1 看文件尾，可临控文件变化 16、makedir 建目录 17、id 看用户 18、mount /cdrom 19、>file1 20、set –o vi 可保存历史命令 21、su – user 切换用户 22、passwd 改密码 23、man 查帮助 24、ifconfig 看网卡状态 25、date 看时间

26、chmod 改权限 27、chown 改组/用户 28、vi 编辑器，常用的操作二、 AIX （一）、文件系统 1、VG、PV、LV、LP、PP的概念及关系 2、基本文件系统 root(/) :home用户、usr软件、tmp临时、var控制管理文件（二）、常用指令 1、lsdev –C 看设备 2、info 相关书籍 3、lslpp –l 安装的软件 4、clstat 监控群集状态 5、errpt –a|more 错误日志 6、errclear 0 删除错误日志 7、df –k 看文件系统空间 8、ps –aux 列出当前进程 9、lspv –o 看激活的VG 10、smitty mkgroup\group\rmgroup 对组操作 hacmp 对群集操作 user\mkuser\rmuser 对用户操作 chinet 改IP tcpip 网络配置 11、varyonvg sharedvg 挂VG varyoffvg sharedvgb 卸VG 12、diag 故障诊断 13、cfgmgr –v 重新配置硬件 14、netstat –in 看网卡信息 15、lsvg –p 看VG （二）、系统开、关机

Functions函数-英文试题

Which of the following function prototype is perfectly acceptable? A. int Function(int Tmp = Show()); B. float Function(int Tmp = Show(int, float)); C. Both A and B. D. float = Show(int, float) Function(Tmp); 2. Which of the following statement is correct? A. C++ enables to define functions that take constants as an argument. B. We cannot change the argument of the function that that are declared as constant. C. Both A and B. D. We cannot use the constant while defining the function. 3. Which of the following statement is correct? A. Overloaded functions can have at most one default argument. B. An overloaded function cannot have default argument. C. All arguments of an overloaded function can be default. D. A function if overloaded more than once cannot have default argument 4. Which of the following statement is correct? A. Two functions having same number of argument, order and type of argument can be overloaded if both functions do not have any default argument. B. Overloaded function must have default arguments. C. Overloaded function must have default arguments starting from the left of argument list. D. A function can be overloaded more than once. 5. Which of the following statement will be correct if the function has three arguments passed to it? A. The trailing argument will be the default argument. B. The first argument will be the default argument. C. The middle argument will be the default argument. D. All the argument will be the default argument. 6. Which of the following statement is incorrect? A. Default arguments can be provided for pointers to functions. B. A function can have all its arguments as default. C. Default argument cannot be provided for pointers to functions. D. A default argument cannot be redefined in later declaration. 7. Which of the following statement is correct? A. Constructors can have default parameters. B. Constructors cannot have default parameters. C. Constructors cannot have more than one default parameter. D. Constructors can have at most five default parameters.