33:
7928:
9172:
2883:
8155:
2701:
919:
7394:
3035:
3896:
In writing, following common mathematical convention, the argument of a differential operator is usually placed on the right side of the operator itself. Sometimes an alternative notation is used: The result of applying the operator to the function on the left side of the operator and on the right
698:
2878:{\displaystyle {\frac {\partial }{\partial z}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}-i{\frac {\partial }{\partial y}}\right)\ ,\quad {\frac {\partial }{\partial {\bar {z}}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right)\ .}
5160:
2438:
1563:
744:
2124:
2927:
6206:
1977:
7923:{\displaystyle {\begin{aligned}L^{*}u&{}=(-1)^{2}D^{2}+(-1)^{1}D+(-1)^{0}(qu)\\&{}=-D^{2}(pu)+D(p'u)+qu\\&{}=-(pu)''+(p'u)'+qu\\&{}=-p''u-2p'u'-pu''+p''u+p'u'+qu\\&{}=-p'u'-pu''+qu\\&{}=-(pu')'+qu\\&{}=Lu\end{aligned}}}
4111:
4886:
3636:
569:
1214:
6501:
3877:
1310:
4488:
1134:
6374:
4023:
5922:
3960:
475:
392:
321:
2285:
4324:
1446:
1870:
3792:
4701:
7234:
is a type of operator on an open subset of a cotangent bundle, as opposed to an open subset of a manifold. It is obtained by extending the notion of a differential operator to the cotangent bundle.
6004:
1710:
5444:
6886:
7399:
2578:
998:
1988:
1407:
2257:
4230:
4356:
7141:
7052:
5740:
3525:
737:
5823:
5684:
3690:
914:{\displaystyle Pf=\sum _{|\alpha |\leq m}a_{\alpha }(x){\frac {\partial ^{|\alpha |}f}{\partial x_{1}^{\alpha _{1}}\partial x_{2}^{\alpha _{2}}\cdots \partial x_{n}^{\alpha _{n}}}}}
1618:
6016:
5602:
3266:
6639:
1896:
6691:
235:
175:
5261:
3357:
2530:
6250:
3061:
204:
531:
3209:
7231:
6987:
5315:
3132:
4029:
3453:
2162:
948:
562:
2205:
3325:
3030:{\displaystyle \nabla =\mathbf {\hat {x}} {\partial \over \partial x}+\mathbf {\hat {y}} {\partial \over \partial y}+\mathbf {\hat {z}} {\partial \over \partial z}.}
5526:
1025:
6270:
3803:
495:
7216:
4782:
4570:
4544:
3547:
3415:
3182:
4730:
4399:
4261:
3293:
1046:
4705:
This formula does not explicitly depend on the definition of the scalar product. It is therefore sometimes chosen as a definition of the adjoint operator. When
8205:
5776:
4152:
3152:
2490:
1430:
141:
121:
3897:
side of the operator, and the difference obtained when applying the differential operator to the functions on both sides, are denoted by arrows as follows:
242:
4937:
1139:
6395:
693:{\displaystyle D^{\alpha }={\frac {\partial ^{|\alpha |}}{\partial x_{1}^{\alpha _{1}}\partial x_{2}^{\alpha _{2}}\cdots \partial x_{n}^{\alpha _{n}}}}}
9061:
1323:. While the total symbol is not intrinsically defined, the principal symbol is intrinsically defined (i.e., it is a function on the cotangent bundle).
8600:
1222:
8605:
4266:
8468:
7274:
8897:
6284:
3714:
2648:
allows for generalizations of differential operators, which do not require the use of calculus. Frequently such generalizations are employed in
8595:
4579:
3966:
5840:
3903:
8724:
2433:{\displaystyle Pf(x)={\frac {1}{(2\pi )^{\frac {d}{2}}}}\int \limits _{\mathbf {R} ^{d}}e^{ix\cdot \xi }p(x,i\xi ){\hat {f}}(\xi )\,d\xi .}
405:
326:
8198:
8887:
8255:
4156:
1558:{\displaystyle Pu(x)=\sum _{|\alpha |=k}P^{\alpha }(x){\frac {\partial ^{\alpha }u}{\partial x^{\alpha }}}+{\text{lower-order terms}}}
9014:
8869:
8368:
2600:
1803:
8845:
8159:
3890:
2604:
8141:
8191:
8412:
8306:
8068:
5931:
1644:
7222:. The notion appears, for instance, in an associative algebra structure on a deformation quantization of a Poisson algebra.
5353:
68:
operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a
8590:
6823:
2119:{\displaystyle (\sigma _{P}(\xi )u)_{\nu }=\sum _{|\alpha |=k}\sum _{\mu }P_{\nu \mu }^{\alpha }(x)\xi _{\alpha }u_{\mu }.}
87:
differential operators, which are the most common type. However, non-linear differential operators also exist, such as the
8500:
8515:
8296:
2535:
964:
1348:
8737:
5539:
is, by contrast, commutative. It can be characterised another way: it consists of the translation-invariant operators.
2210:
3076:
8826:
8717:
8301:
8265:
8090:
8031:
5743:
5471:
of such operators we must assume derivatives of all orders of the coefficients used. Secondly, this ring will not be
4927:
operator is a well-known example of a formal self-adjoint operator. This second-order linear differential operator
2459:,Ο) which satisfy at most polynomial growth conditions in Ο under which this integral is well-behaved comprises the
9096:
8363:
8275:
7315:
4329:
17:
7060:
9206:
9201:
8741:
8331:
7309:
7269:
7151:
5605:
8646:
8245:
6995:
5700:
3472:
2493:
951:
8402:
8892:
8417:
8260:
8250:
8176:
6201:{\displaystyle (D_{i}X_{j}-X_{j}D_{i})-\delta _{i,j},\ \ \ D_{i}D_{j}-D_{j}D_{i},\ \ \ X_{i}X_{j}-X_{j}X_{i}}
2645:
2619:
705:
5785:
5646:
3651:
1972:{\displaystyle P_{\nu \mu }=\sum _{\alpha }P_{\nu \mu }^{\alpha }{\frac {\partial }{\partial x^{\alpha }}}.}
9175:
8948:
8882:
8710:
8270:
8240:
8166:
7299:
3092:
2460:
1575:
5572:
3225:
8912:
8505:
8375:
8336:
8171:
6589:
6650:
209:
149:
9157:
9111:
9035:
8917:
7171:
A differential operator of infinite order is (roughly) a differential operator whose total symbol is a
7146:
This characterization of linear differential operators shows that they are particular mappings between
5198:
5169:
4924:
4375:
3330:
2499:
8341:
6214:
180:
9152:
8968:
8626:
8545:
2886:
2608:
8681:
8651:
8003:
4106:{\displaystyle f{\overleftrightarrow {\partial _{x}}}g=f\cdot \partial _{x}g-g\cdot \partial _{x}f.}
500:
9196:
9004:
8902:
8805:
8540:
6804:
An equivalent, but purely algebraic description of linear differential operators is as follows: an
6792:, which is expressed by saying that differential operators are local. A foundational result is the
4912:
3187:
3075:
The conceptual step of writing a differential operator as something free-standing is attributed to
6894:
5267:
3105:
9101:
8877:
8550:
8520:
8510:
8407:
6537:
5457:
4366:. This definition therefore depends on the definition of the scalar product (or inner product).
2140:
926:
540:
41:
2175:
9132:
9076:
9040:
7998:
4881:{\displaystyle \langle f,P^{*}g\rangle _{L^{2}(\Omega )}=\langle Pf,g\rangle _{L^{2}(\Omega )}}
4382:
3631:{\displaystyle \Delta =\nabla ^{2}=\sum _{k=1}^{n}{\frac {\partial ^{2}}{\partial x_{k}^{2}}}.}
3368:
3298:
2913:
2165:
69:
61:
7357:
5493:
1003:
8839:
8666:
8661:
8555:
8479:
8458:
8326:
8321:
8316:
8311:
8214:
8142:
https://mathoverflow.net/questions/451110/reference-request-inverse-of-differential-operators
8078:
7320:
6513:
6255:
5536:
4745:
3531:
3065:
2917:
2626:
480:
88:
73:
37:
8835:
7186:
4549:
4523:
3421:
3157:
9115:
8560:
8484:
8453:
8050:
8041:
7304:
7147:
5779:
5344:
4708:
4239:
3886:
3271:
2695:
2664:
8702:
8:
9081:
9019:
8733:
8570:
8463:
8357:
7264:
7244:
7155:
6781:
5625:
4117:
2653:
2630:
1761:
permits a local trivialization of the cotangent bundle by the coordinate differentials d
9106:
8973:
8691:
8530:
8525:
8448:
8280:
7964:
7259:
6517:
5752:
5468:
4137:
3380:
3137:
3044:
2649:
2475:
2448:
1415:
126:
106:
9086:
8130:
8086:
8064:
8027:
7968:
7294:
7279:
7254:
6796:
showing that the converse is also true: any (linear) local operator is differential.
6720:
5484:
4501:
4233:
4129:
2657:
2589:
2276:
2272:
1433:
32:
8011:
5155:{\displaystyle Lu=-(pu')'+qu=-(pu''+p'u')+qu=-pu''-p'u'+qu=(-p)D^{2}u+(-p')Du+(q)u.}
2686:, sometimes a complex function is considered to be a function of two real variables
9091:
9009:
8978:
8958:
8943:
8938:
8933:
8676:
8656:
8120:
8056:
8019:
7954:
5472:
3538:
3463:
2905:
2668:
2641:
2615:
2593:
2581:
1727:
1636:
1209:{\displaystyle \xi ^{\alpha }=\xi _{1}^{\alpha _{1}}\cdots \xi _{n}^{\alpha _{n}}.}
77:
45:
8770:
7371:
6496:{\displaystyle X_{1}^{a_{1}}\ldots X_{n}^{a_{n}}D_{1}^{b_{1}}\ldots D_{n}^{b_{n}}}
8953:
8907:
8855:
8850:
8821:
8636:
8565:
8037:
7289:
7284:
6273:
1723:
8780:
8641:
3872:{\displaystyle \Theta =\sum _{k=1}^{n}x_{k}{\frac {\partial }{\partial x_{k}}}.}
9142:
8994:
8795:
8631:
8438:
7249:
6793:
5173:
4359:
3642:
2890:
2634:
1716:
1305:{\displaystyle \sigma (x,\xi )=\sum _{|\alpha |=m}a_{\alpha }(x)\xi ^{\alpha }}
144:
8060:
8023:
5449:
Some care is then required: firstly any function coefficients in the operator
4483:{\displaystyle \langle f,g\rangle =\int _{a}^{b}{\overline {f(x)}}\,g(x)\,dx,}
9190:
9147:
9071:
8800:
8785:
8775:
8134:
6730:
6548:
6525:
5641:
5543:
4363:
3700:
1335:
1129:{\displaystyle p(x,\xi )=\sum _{|\alpha |\leq m}a_{\alpha }(x)\xi ^{\alpha }}
7354:
A Boole
Anthology: Recent and classical studies in the logic of George Boole
950:
is justified (i.e., independent of order of differentiation) because of the
9137:
8790:
8760:
8686:
8621:
8535:
7172:
5560:
5177:
8055:. Grundlehren der mathematischen Wissenschaften. Vol. 269. Springer.
6369:{\displaystyle R\langle D_{1},\ldots ,D_{n},X_{1},\ldots ,X_{n}\rangle /I}
4116:
Such a bidirectional-arrow notation is frequently used for describing the
27:
Typically linear operator defined in terms of differentiation of functions
9066:
9056:
8963:
8765:
8443:
8183:
6384:
5828:
5690:
4379:
4018:{\displaystyle f{\overrightarrow {\partial _{x}}}g=f\cdot \partial _{x}g}
1569:
395:
53:
5917:{\displaystyle R\langle D_{1},\ldots ,D_{n},X_{1},\ldots ,X_{n}\rangle }
5636:â 1. Then the ring of univariate polynomial differential operators over
5343:
with function coefficients is also a differential operator. We may also
3955:{\displaystyle f{\overleftarrow {\partial _{x}}}g=g\cdot \partial _{x}f}
8999:
8831:
8671:
7959:
7942:
6573:
6521:
6276:. Then the ring of multivariate polynomial differential operators over
5554:
5336:
5189:
5165:
This property can be proven using the formal adjoint definition above.
3882:
3088:
3048:
1621:
84:
65:
8433:
8125:
8108:
3052:
470:{\displaystyle |\alpha |=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}}
387:{\displaystyle \alpha =(\alpha _{1},\alpha _{2},\cdots ,\alpha _{n})}
316:{\displaystyle P=\sum _{|\alpha |\leq m}a_{\alpha }(x)D^{\alpha }\ ,}
8473:
8107:
Fedosov, Boris; Schulze, Bert-Wolfgang; Tarkhanov, Nikolai (2002).
6583:). In other words, there exists a linear mapping of vector bundles
4770:
3704:
3040:
3087:
The most common differential operator is the action of taking the
1982:
With this trivialization, the principal symbol can now be written
40:. Harmonic functions are exactly those functions which lie in the
7947:
Proceedings of the Japan
Academy, Series A, Mathematical Sciences
2909:
399:
2614:
In applications to the physical sciences, operators such as the
8154:
6524:-independent description of differential operators between two
3537:
One of the most frequently seen differential operators is the
4319:{\displaystyle \langle Tu,v\rangle =\langle u,T^{*}v\rangle }
7166:
5549:
1865:{\displaystyle (Pu)_{\nu }=\sum _{\mu }P_{\nu \mu }u_{\mu }}
8732:
4749:
operator is an operator equal to its own (formal) adjoint.
3095:
for taking the first derivative with respect to a variable
3787:{\displaystyle \Theta (z^{k})=kz^{k},\quad k=0,1,2,\dots }
1216:
The highest homogeneous component of the symbol, namely,
4696:{\displaystyle T^{*}u=\sum _{k=0}^{n}(-1)^{k}D^{k}\left.}
2897:
8018:, Grundl. Math. Wissenschaft., vol. 256, Springer,
8109:"Analytic index formulas for elliptic corner operators"
8106:
8016:
The analysis of linear partial differential operators I
5999:{\displaystyle D_{1},\ldots ,D_{n},X_{1},\ldots ,X_{n}}
5183:
4732:
is defined according to this formula, it is called the
2271:
and its symbol appear naturally in connection with the
1705:{\displaystyle \sigma _{P}:S^{k}(T^{*}X)\otimes E\to F}
6715:
is the prolongation that associates to any section of
5829:
Ring of multivariate polynomial differential operators
5439:{\displaystyle (D_{1}\circ D_{2})(f)=D_{1}(D_{2}(f)).}
2496:
if its symbol is invertible; that is for each nonzero
7397:
7189:
7063:
6998:
6897:
6881:{\displaystyle f_{0},\ldots ,f_{k}\in C^{\infty }(M)}
6826:
6653:
6592:
6507:
6398:
6287:
6258:
6217:
6019:
5934:
5843:
5788:
5778:(for the standard derivation) can be identified with
5755:
5703:
5649:
5575:
5496:
5356:
5270:
5201:
4940:
4785:
4711:
4582:
4552:
4526:
4402:
4332:
4269:
4242:
4159:
4140:
4032:
3969:
3906:
3806:
3717:
3654:
3550:
3475:
3424:
3383:
3333:
3301:
3274:
3228:
3190:
3160:
3140:
3108:
2930:
2704:
2538:
2502:
2478:
2288:
2213:
2178:
2143:
1991:
1899:
1806:
1647:
1578:
1449:
1418:
1351:
1225:
1142:
1049:
1006:
967:
929:
747:
708:
572:
543:
503:
483:
408:
329:
245:
212:
183:
152:
129:
109:
6387:. Every element can be written in a unique way as a
5693:. Every element can be written in a unique way as a
5555:
Ring of univariate polynomial differential operators
3641:
Another differential operator is the Î operator, or
3466:, who considered differential operators of the form
3219:th order derivatives, the operator may be written:
2573:{\displaystyle \sigma _{P}(\theta ,\dots ,\theta )}
993:{\displaystyle {\frac {\partial }{\partial x_{i}}}}
9062:Spectral theory of ordinary differential equations
7940:
7922:
7210:
7135:
7046:
6981:
6880:
6816:th-order linear differential operator, if for any
6799:
6685:
6633:
6495:
6368:
6264:
6244:
6200:
5998:
5916:
5817:
5770:
5734:
5678:
5596:
5520:
5438:
5309:
5255:
5154:
4880:
4765:a differential operator on Ω, then the adjoint of
4724:
4695:
4564:
4538:
4482:
4350:
4318:
4255:
4224:
4146:
4105:
4017:
3954:
3871:
3786:
3684:
3630:
3519:
3447:
3409:
3351:
3319:
3287:
3260:
3203:
3176:
3146:
3126:
3029:
2877:
2607:, zeros of the principal symbol correspond to the
2572:
2524:
2484:
2432:
2251:
2199:
2156:
2118:
1971:
1864:
1704:
1612:
1557:
1424:
1402:{\displaystyle P:C^{\infty }(E)\to C^{\infty }(F)}
1401:
1304:
1208:
1128:
1019:
992:
942:
913:
731:
692:
556:
525:
489:
469:
386:
315:
229:
198:
169:
135:
115:
8601:List of nonlinear ordinary differential equations
5531:The subring of operators that are polynomials in
4369:
3003:
2973:
2943:
2885:This approach is also used to study functions of
2252:{\displaystyle \operatorname {Hom} (E_{x},F_{x})}
9188:
8606:List of nonlinear partial differential equations
7183:A differential operator acting on two functions
4907:, this defines the adjoint on a dense subset of
4225:{\displaystyle Tu=\sum _{k=0}^{n}a_{k}(x)D^{k}u}
2908:differential operator. It appears frequently in
72:and returns another function (in the style of a
8052:Microdifferential Systems in the Complex Domain
7941:Omori, Hideki; Maeda, Y.; Yoshioka, A. (1992).
7275:Differential calculus over commutative algebras
7154:, allowing the concept to be seen as a part of
6010:the two-sided ideal generated by the elements
3800:variables the homogeneity operator is given by
3374:is sometimes given as either of the following:
1890:is the scalar differential operator defined by
8596:List of linear ordinary differential equations
7943:"Deformation quantization of Poisson algebras"
8718:
8199:
7225:
6391:-linear combination of monomials of the form
5697:-linear combination of monomials of the form
4351:{\displaystyle \langle \cdot ,\cdot \rangle }
7136:{\displaystyle (s)=P(f\cdot s)-f\cdot P(s).}
6355:
6291:
5924:be the non-commutative polynomial ring over
5911:
5847:
5804:
5792:
5665:
5653:
5591:
5579:
5483:. For example we have the relation basic in
4853:
4837:
4809:
4786:
4512:). If one moreover adds the condition that
4415:
4403:
4345:
4333:
4313:
4291:
4285:
4270:
4236:of this operator is defined as the operator
2698:, which are partial differential operators:
2618:play a major role in setting up and solving
7369:
3462:notation's use and creation is credited to
2584:, it follows from the elliptic theory that
8725:
8711:
8213:
8206:
8192:
8083:Differential analysis on complex manifolds
7178:
2279:. Then by the inverse Fourier transform,
2129:In the cotangent space over a fixed point
1793:, respectively, the differential operator
8124:
8010:
8002:
7958:
7167:A differential operator of infinite order
7047:{\displaystyle :\Gamma (E)\to \Gamma (F)}
5735:{\displaystyle X^{a}D^{b}{\text{ mod }}I}
5550:Ring of polynomial differential operators
5542:The differential operators also obey the
4470:
4457:
4123:
3520:{\displaystyle \sum _{k=0}^{n}c_{k}D^{k}}
2420:
2262:
1742:. This symmetric tensor is known as the
186:
9015:Group algebra of a locally compact group
8048:
7981:
7340:
2912:in places like the differential form of
2605:parabolic partial differential equations
31:
4903:. Since smooth functions are dense in
732:{\displaystyle f\in {\mathcal {F}}_{1}}
14:
9189:
5818:{\displaystyle R\langle D,X\rangle /I}
5679:{\displaystyle R\langle D,X\rangle /I}
3685:{\displaystyle \Theta =z{d \over dz}.}
123:linear differential operator is a map
8706:
8187:
8077:
7992:
6570:th-order linear differential operator
4572:, one can also define the adjoint of
4134:Given a linear differential operator
2611:of the partial differential equation.
2451:. A more general class of functions
1765:, which determine fiber coordinates Ο
1613:{\displaystyle P^{\alpha }(x):E\to F}
533:is a function on some open domain in
48:, an important differential operator.
8591:List of named differential equations
5597:{\displaystyle R\langle D,X\rangle }
5460:as many times as the application of
5184:Properties of differential operators
4776:by duality in the analogous manner:
4752:
3891:Euler's homogeneous function theorem
3261:{\displaystyle {d^{n} \over dx^{n}}}
1412:is a differential operator of order
8516:Method of undetermined coefficients
8297:Dependent and independent variables
6760:th-order infinitesimal behavior of
6634:{\displaystyle i_{P}:J^{k}(E)\to F}
5347:differential operators by the rule
5180:) of this operator are considered.
4396:, the scalar product is defined by
24:
8100:
7032:
7017:
6864:
6768:. In particular this implies that
6686:{\displaystyle P=i_{P}\circ j^{k}}
6508:Coordinate-independent description
4870:
4826:
4088:
4066:
4039:
4003:
3976:
3940:
3913:
3850:
3846:
3807:
3718:
3695:This is sometimes also called the
3655:
3604:
3594:
3558:
3551:
3335:
3192:
3015:
3011:
2985:
2981:
2955:
2951:
2931:
2855:
2851:
2834:
2830:
2792:
2788:
2765:
2761:
2744:
2740:
2711:
2707:
1950:
1946:
1531:
1517:
1385:
1363:
974:
970:
883:
855:
830:
806:
718:
662:
634:
609:
589:
230:{\displaystyle {\mathcal {F}}_{2}}
216:
170:{\displaystyle {\mathcal {F}}_{1}}
156:
36:A harmonic function defined on an
25:
9218:
8147:
6820: + 1 smooth functions
6729:This just means that for a given
6520:it is often convenient to have a
5744:Euclidean division of polynomials
5256:{\displaystyle D(f+g)=(Df)+(Dg),}
3352:{\displaystyle \partial _{x}^{n}}
2637:operators have intrinsic meaning.
2525:{\displaystyle \theta \in T^{*}X}
537:-dimensional space. The operator
9171:
9170:
9097:Topological quantum field theory
8413:Carathéodory's existence theorem
8153:
6245:{\displaystyle 1\leq i,j\leq n,}
3000:
2970:
2940:
2347:
1771:. In terms of a basis of frames
199:{\displaystyle \mathbb {R} ^{n}}
7312:(section on symbol of operator)
7270:Invariant differential operator
6800:Relation to commutative algebra
5606:non-commutative polynomial ring
3756:
3077:Louis François Antoine Arbogast
3043:, and is used to calculate the
2785:
961:obtained by replacing partials
7975:
7934:
7882:
7867:
7678:
7663:
7653:
7643:
7616:
7602:
7593:
7584:
7556:
7547:
7538:
7528:
7522:
7516:
7502:
7499:
7487:
7477:
7471:
7465:
7456:
7453:
7434:
7424:
7384:
7363:
7346:
7334:
7205:
7193:
7127:
7121:
7106:
7094:
7085:
7079:
7076:
7064:
7041:
7035:
7029:
7026:
7020:
7011:
6999:
6970:
6967:
6961:
6942:
6936:
6914:
6898:
6875:
6869:
6625:
6622:
6616:
6066:
6020:
5765:
5759:
5430:
5427:
5421:
5408:
5392:
5386:
5383:
5357:
5301:
5292:
5283:
5274:
5247:
5238:
5232:
5223:
5217:
5205:
5143:
5137:
5125:
5111:
5092:
5083:
5023:
4990:
4968:
4953:
4873:
4867:
4829:
4823:
4673:
4667:
4630:
4620:
4556:
4530:
4467:
4461:
4448:
4442:
4370:Formal adjoint in one variable
4206:
4200:
3734:
3721:
3439:
3433:
3400:
3396:
3390:
3384:
2801:
2620:partial differential equations
2567:
2549:
2417:
2411:
2405:
2396:
2381:
2323:
2313:
2301:
2295:
2246:
2220:
2090:
2084:
2044:
2036:
2018:
2011:
2005:
1992:
1817:
1807:
1696:
1687:
1671:
1624:, symmetric on the indices α.
1604:
1595:
1589:
1510:
1504:
1482:
1474:
1462:
1456:
1396:
1390:
1377:
1374:
1368:
1289:
1283:
1261:
1253:
1241:
1229:
1113:
1107:
1085:
1077:
1065:
1053:
952:symmetry of second derivatives
819:
811:
799:
793:
771:
763:
602:
594:
526:{\displaystyle a_{\alpha }(x)}
520:
514:
418:
410:
381:
336:
294:
288:
266:
258:
83:This article considers mainly
13:
1:
8893:Uniform boundedness principle
8113:Annales de l'Institut Fourier
7327:
7054:is defined as the commutator
6536:be two vector bundles over a
5742:. It supports an analogue of
5479:isn't the same in general as
3363:The derivative of a function
3204:{\displaystyle \partial _{x}}
2461:pseudo-differential operators
94:
64:defined as a function of the
8241:Notation for differentiation
7316:MalgrangeâEhrenpreis theorem
7300:Pseudo-differential operator
6982:{\displaystyle \cdots ]]=0.}
5310:{\displaystyle D(af)=a(Df),}
5168:This operator is central to
4677:
4452:
3127:{\displaystyle {d \over dx}}
3082:
3062:chiral differential operator
2592:: it has finite-dimensional
1039:; i.e., the total symbol of
99:Given a nonnegative integer
7:
8337:Exact differential equation
8172:Encyclopedia of Mathematics
7995:Geometry of Dirac operators
7310:AtiyahâSinger index theorem
7237:
7161:
6756:is fully determined by the
4931:can be written in the form
4376:square-integrable functions
4374:In the functional space of
2466:
2157:{\displaystyle \sigma _{P}}
1797:decomposes into components
1342:. Then the linear operator
943:{\displaystyle D^{\alpha }}
557:{\displaystyle D^{\alpha }}
10:
9223:
9036:Invariant subspace problem
7232:microdifferential operator
7226:Microdifferential operator
6572:if it factors through the
5749:Differential modules over
5558:
4918:
4127:
3070:
2896:The differential operator
2472:The differential operator
2200:{\displaystyle T_{x}^{*}X}
206:to another function space
9166:
9125:
9049:
9028:
8987:
8926:
8868:
8814:
8756:
8749:
8647:JĂłzef Maria Hoene-WroĆski
8627:Gottfried Wilhelm Leibniz
8614:
8583:
8493:
8426:
8418:CauchyâKowalevski theorem
8395:
8388:
8350:
8289:
8228:
8221:
8061:10.1007/978-3-642-61665-5
8049:Schapira, Pierre (1985).
8024:10.1007/978-3-642-96750-4
7993:Freed, Daniel S. (1987),
7175:instead of a polynomial.
3320:{\displaystyle D_{x}^{n}}
2887:several complex variables
9005:Spectrum of a C*-algebra
8541:Finite difference method
5521:{\displaystyle Dx-xD=1.}
4913:densely defined operator
3881:As in one variable, the
2275:as follows. Let Æ be a
2267:A differential operator
1738:, and whose codomain is
1020:{\displaystyle \xi _{i}}
237:that can be written as:
9102:Noncommutative geometry
8521:Variation of parameters
8511:Separation of variables
8408:Peano existence theorem
8403:PicardâLindelöf theorem
8290:Attributes of variables
8167:"Differential operator"
7352:James Gasser (editor),
7220:bidifferential operator
7179:Bidifferential operator
6780:) is determined by the
6538:differentiable manifold
6265:{\displaystyle \delta }
3885:of Î are the spaces of
2916:. In three-dimensional
490:{\displaystyle \alpha }
9207:Differential operators
9202:Multivariable calculus
9158:TomitaâTakesaki theory
9133:Approximation property
9077:Calculus of variations
8682:Carl David Tolmé Runge
8256:Differential-algebraic
8215:Differential equations
8160:Differential operators
7924:
7212:
7211:{\displaystyle D(g,f)}
7137:
7048:
6983:
6882:
6687:
6635:
6497:
6370:
6266:
6246:
6202:
6000:
5918:
5819:
5772:
5736:
5680:
5598:
5522:
5440:
5311:
5257:
5170:SturmâLiouville theory
5156:
4882:
4726:
4697:
4619:
4566:
4565:{\displaystyle x\to b}
4540:
4539:{\displaystyle x\to a}
4484:
4352:
4320:
4257:
4226:
4189:
4148:
4124:Adjoint of an operator
4120:of quantum mechanics.
4107:
4019:
3956:
3873:
3833:
3788:
3686:
3632:
3590:
3532:differential equations
3521:
3496:
3449:
3448:{\displaystyle f'(x).}
3411:
3353:
3321:
3289:
3262:
3205:
3178:
3177:{\displaystyle D_{x},}
3148:
3128:
3031:
2879:
2663:In the development of
2574:
2526:
2486:
2434:
2263:Fourier interpretation
2253:
2201:
2166:homogeneous polynomial
2158:
2120:
1973:
1866:
1757:The coordinate system
1706:
1631:order coefficients of
1614:
1559:
1426:
1403:
1306:
1210:
1130:
1021:
994:
944:
915:
733:
694:
558:
527:
491:
471:
388:
317:
231:
200:
171:
137:
117:
49:
9153:BanachâMazur distance
9116:Generalized functions
8667:Augustin-Louis Cauchy
8662:Joseph-Louis Lagrange
8556:Finite element method
8546:CrankâNicolson method
8480:Numerical integration
8459:Exponential stability
8351:Relation to processes
8236:Differential operator
7925:
7321:Hypoelliptic operator
7213:
7138:
7049:
6984:
6883:
6688:
6636:
6514:differential geometry
6498:
6371:
6280:is the quotient ring
6267:
6247:
6203:
6001:
5919:
5820:
5773:
5737:
5681:
5599:
5537:constant coefficients
5523:
5441:
5312:
5258:
5157:
4925:Sturm–Liouville
4883:
4727:
4725:{\displaystyle T^{*}}
4698:
4599:
4567:
4541:
4485:
4353:
4321:
4258:
4256:{\displaystyle T^{*}}
4227:
4169:
4149:
4108:
4020:
3957:
3887:homogeneous functions
3874:
3813:
3789:
3687:
3633:
3570:
3522:
3476:
3450:
3412:
3354:
3322:
3290:
3288:{\displaystyle D^{n}}
3263:
3206:
3179:
3149:
3129:
3032:
2918:Cartesian coordinates
2880:
2696:Wirtinger derivatives
2694:. Use is made of the
2665:holomorphic functions
2627:differential topology
2575:
2527:
2487:
2435:
2254:
2202:
2159:
2121:
1974:
1867:
1707:
1615:
1560:
1427:
1404:
1307:
1211:
1131:
1022:
995:
945:
916:
734:
695:
559:
528:
492:
472:
389:
318:
232:
201:
172:
138:
118:
89:Schwarzian derivative
74:higher-order function
58:differential operator
35:
8898:Kakutani fixed-point
8883:Riesz representation
8561:Finite volume method
8485:Dirac delta function
8454:Asymptotic stability
8396:Existence/uniqueness
8261:Integro-differential
8162:at Wikimedia Commons
7395:
7305:Fundamental solution
7187:
7061:
6996:
6895:
6824:
6651:
6590:
6396:
6285:
6256:
6215:
6017:
5932:
5841:
5786:
5753:
5701:
5647:
5573:
5494:
5354:
5268:
5199:
4938:
4783:
4757:If Ω is a domain in
4709:
4580:
4550:
4524:
4492:where the line over
4400:
4330:
4267:
4240:
4157:
4138:
4030:
3967:
3904:
3804:
3715:
3697:homogeneity operator
3652:
3548:
3473:
3422:
3381:
3331:
3299:
3272:
3226:
3215:When taking higher,
3188:
3158:
3138:
3106:
2928:
2702:
2580:is invertible. On a
2536:
2500:
2476:
2286:
2211:
2176:
2141:
1989:
1897:
1804:
1715:whose domain is the
1645:
1576:
1447:
1416:
1349:
1326:More generally, let
1223:
1140:
1047:
1004:
965:
927:
745:
706:
702:Thus for a function
570:
541:
501:
481:
406:
327:
243:
210:
181:
150:
127:
107:
9082:Functional calculus
9041:Mahler's conjecture
9020:Von Neumann algebra
8734:Functional analysis
8571:Perturbation theory
8551:RungeâKutta methods
8531:Integral transforms
8464:Rate of convergence
8360:(discrete analogue)
8085:, Springer-Verlag,
7265:Fractional calculus
7245:Difference operator
7156:commutative algebra
7152:commutative algebra
6547:-linear mapping of
6492:
6467:
6445:
6420:
5467:requires. To get a
5328:are functions, and
5188:Differentiation is
4435:
4326:where the notation
4118:probability current
3621:
3348:
3316:
3055:of various objects.
2920:, del is defined as
2914:Maxwell's equations
2889:and functions of a
2654:commutative algebra
2644:, the concept of a
2631:exterior derivative
2193:
2083:
1943:
1202:
1177:
907:
879:
854:
686:
658:
633:
564:is interpreted as
9107:Riemann hypothesis
8806:Topological vector
8692:Sofya Kovalevskaya
8526:Integrating factor
8449:Lyapunov stability
8369:Stochastic partial
7960:10.3792/PJAA.68.97
7920:
7918:
7260:Curl (mathematics)
7208:
7133:
7044:
6979:
6878:
6683:
6631:
6518:algebraic geometry
6493:
6471:
6446:
6424:
6399:
6366:
6262:
6242:
6198:
5996:
5914:
5815:
5768:
5732:
5676:
5594:
5518:
5436:
5307:
5253:
5152:
4878:
4722:
4693:
4562:
4536:
4480:
4421:
4348:
4316:
4253:
4222:
4144:
4103:
4015:
3952:
3869:
3784:
3682:
3628:
3607:
3539:Laplacian operator
3517:
3445:
3407:
3349:
3334:
3317:
3302:
3285:
3258:
3201:
3174:
3144:
3124:
3027:
2904:, is an important
2875:
2650:algebraic geometry
2570:
2522:
2482:
2449:Fourier multiplier
2430:
2358:
2249:
2197:
2179:
2154:
2116:
2066:
2065:
2055:
1969:
1926:
1925:
1862:
1838:
1702:
1610:
1555:
1493:
1422:
1399:
1302:
1272:
1206:
1181:
1156:
1126:
1096:
1017:
990:
940:
911:
886:
858:
833:
782:
729:
690:
665:
637:
612:
554:
523:
487:
467:
384:
313:
277:
227:
196:
167:
133:
113:
50:
9184:
9183:
9087:Integral operator
8864:
8863:
8700:
8699:
8579:
8578:
8384:
8383:
8158:Media related to
8070:978-3-642-64904-2
7370:E. W. Weisstein.
7295:Momentum operator
7280:Lagrangian system
7255:Elliptic operator
6992:Here the bracket
6154:
6151:
6148:
6099:
6096:
6093:
5928:in the variables
5771:{\displaystyle R}
5727:
5612:in the variables
5485:quantum mechanics
4753:Several variables
4680:
4502:complex conjugate
4455:
4147:{\displaystyle T}
4130:Hermitian adjoint
4051:
3988:
3925:
3864:
3677:
3623:
3410:{\displaystyle '}
3256:
3147:{\displaystyle D}
3122:
3022:
3006:
2992:
2976:
2962:
2946:
2871:
2862:
2841:
2821:
2808:
2804:
2781:
2772:
2751:
2731:
2718:
2658:Jet (mathematics)
2590:Fredholm operator
2485:{\displaystyle P}
2408:
2340:
2338:
2334:
2277:Schwartz function
2273:Fourier transform
2056:
2030:
1964:
1916:
1829:
1553:
1552:lower-order terms
1545:
1468:
1434:local coordinates
1425:{\displaystyle k}
1247:
1071:
988:
909:
757:
688:
309:
252:
136:{\displaystyle P}
116:{\displaystyle m}
16:(Redirected from
9214:
9174:
9173:
9092:Jones polynomial
9010:Operator algebra
8754:
8753:
8727:
8720:
8713:
8704:
8703:
8677:Phyllis Nicolson
8657:Rudolf Lipschitz
8494:Solution methods
8469:Series solutions
8393:
8392:
8226:
8225:
8208:
8201:
8194:
8185:
8184:
8180:
8157:
8138:
8128:
8126:10.5802/aif.1906
8095:
8074:
8044:
8007:
8006:
7985:
7979:
7973:
7972:
7962:
7938:
7932:
7929:
7927:
7926:
7921:
7919:
7906:
7901:
7888:
7880:
7860:
7855:
7842:
7828:
7820:
7806:
7801:
7788:
7780:
7766:
7755:
7741:
7733:
7716:
7702:
7697:
7684:
7673:
7659:
7636:
7631:
7612:
7583:
7582:
7567:
7562:
7546:
7545:
7515:
7495:
7494:
7452:
7451:
7442:
7441:
7420:
7411:
7410:
7388:
7382:
7381:
7379:
7378:
7372:"Theta Operator"
7367:
7361:
7356:(2000), p. 169;
7350:
7344:
7338:
7217:
7215:
7214:
7209:
7142:
7140:
7139:
7134:
7053:
7051:
7050:
7045:
6988:
6986:
6985:
6980:
6954:
6953:
6932:
6931:
6910:
6909:
6887:
6885:
6884:
6879:
6868:
6867:
6855:
6854:
6836:
6835:
6714:
6692:
6690:
6689:
6684:
6682:
6681:
6669:
6668:
6640:
6638:
6637:
6632:
6615:
6614:
6602:
6601:
6565:is said to be a
6564:
6504:
6502:
6500:
6499:
6494:
6491:
6490:
6489:
6479:
6466:
6465:
6464:
6454:
6444:
6443:
6442:
6432:
6419:
6418:
6417:
6407:
6383:
6377:
6375:
6373:
6372:
6367:
6362:
6354:
6353:
6335:
6334:
6322:
6321:
6303:
6302:
6271:
6269:
6268:
6263:
6251:
6249:
6248:
6243:
6207:
6205:
6204:
6199:
6197:
6196:
6187:
6186:
6174:
6173:
6164:
6163:
6152:
6149:
6146:
6142:
6141:
6132:
6131:
6119:
6118:
6109:
6108:
6097:
6094:
6091:
6087:
6086:
6065:
6064:
6055:
6054:
6042:
6041:
6032:
6031:
6005:
6003:
6002:
5997:
5995:
5994:
5976:
5975:
5963:
5962:
5944:
5943:
5923:
5921:
5920:
5915:
5910:
5909:
5891:
5890:
5878:
5877:
5859:
5858:
5824:
5822:
5821:
5816:
5811:
5777:
5775:
5774:
5769:
5741:
5739:
5738:
5733:
5728:
5725:
5723:
5722:
5713:
5712:
5689:
5685:
5683:
5682:
5677:
5672:
5603:
5601:
5600:
5595:
5527:
5525:
5524:
5519:
5445:
5443:
5442:
5437:
5420:
5419:
5407:
5406:
5382:
5381:
5369:
5368:
5316:
5314:
5313:
5308:
5262:
5260:
5259:
5254:
5161:
5159:
5158:
5153:
5124:
5104:
5103:
5070:
5062:
5051:
5022:
5014:
5003:
4974:
4966:
4887:
4885:
4884:
4879:
4877:
4876:
4866:
4865:
4833:
4832:
4822:
4821:
4804:
4803:
4731:
4729:
4728:
4723:
4721:
4720:
4702:
4700:
4699:
4694:
4689:
4685:
4681:
4676:
4666:
4665:
4655:
4648:
4647:
4638:
4637:
4618:
4613:
4592:
4591:
4571:
4569:
4568:
4563:
4545:
4543:
4542:
4537:
4489:
4487:
4486:
4481:
4456:
4451:
4437:
4434:
4429:
4395:
4358:is used for the
4357:
4355:
4354:
4349:
4325:
4323:
4322:
4317:
4309:
4308:
4262:
4260:
4259:
4254:
4252:
4251:
4231:
4229:
4228:
4223:
4218:
4217:
4199:
4198:
4188:
4183:
4153:
4151:
4150:
4145:
4112:
4110:
4109:
4104:
4096:
4095:
4074:
4073:
4052:
4047:
4046:
4037:
4024:
4022:
4021:
4016:
4011:
4010:
3989:
3984:
3983:
3974:
3961:
3959:
3958:
3953:
3948:
3947:
3926:
3921:
3920:
3911:
3878:
3876:
3875:
3870:
3865:
3863:
3862:
3861:
3845:
3843:
3842:
3832:
3827:
3793:
3791:
3790:
3785:
3752:
3751:
3733:
3732:
3691:
3689:
3688:
3683:
3678:
3676:
3665:
3637:
3635:
3634:
3629:
3624:
3622:
3620:
3615:
3602:
3601:
3592:
3589:
3584:
3566:
3565:
3530:in his study of
3526:
3524:
3523:
3518:
3516:
3515:
3506:
3505:
3495:
3490:
3464:Oliver Heaviside
3454:
3452:
3451:
3446:
3432:
3416:
3414:
3413:
3408:
3406:
3358:
3356:
3355:
3350:
3347:
3342:
3326:
3324:
3323:
3318:
3315:
3310:
3294:
3292:
3291:
3286:
3284:
3283:
3267:
3265:
3264:
3259:
3257:
3255:
3254:
3253:
3240:
3239:
3230:
3210:
3208:
3207:
3202:
3200:
3199:
3183:
3181:
3180:
3175:
3170:
3169:
3153:
3151:
3150:
3145:
3133:
3131:
3130:
3125:
3123:
3121:
3110:
3093:Common notations
3039:Del defines the
3036:
3034:
3033:
3028:
3023:
3021:
3010:
3008:
3007:
2999:
2993:
2991:
2980:
2978:
2977:
2969:
2963:
2961:
2950:
2948:
2947:
2939:
2884:
2882:
2881:
2876:
2869:
2868:
2864:
2863:
2861:
2850:
2842:
2840:
2829:
2822:
2814:
2809:
2807:
2806:
2805:
2797:
2787:
2779:
2778:
2774:
2773:
2771:
2760:
2752:
2750:
2739:
2732:
2724:
2719:
2717:
2706:
2669:complex variable
2642:abstract algebra
2616:Laplace operator
2599:In the study of
2582:compact manifold
2579:
2577:
2576:
2571:
2548:
2547:
2531:
2529:
2528:
2523:
2518:
2517:
2491:
2489:
2488:
2483:
2439:
2437:
2436:
2431:
2410:
2409:
2401:
2377:
2376:
2357:
2356:
2355:
2350:
2339:
2337:
2336:
2335:
2327:
2308:
2258:
2256:
2255:
2250:
2245:
2244:
2232:
2231:
2206:
2204:
2203:
2198:
2192:
2187:
2163:
2161:
2160:
2155:
2153:
2152:
2125:
2123:
2122:
2117:
2112:
2111:
2102:
2101:
2082:
2077:
2064:
2054:
2047:
2039:
2026:
2025:
2004:
2003:
1978:
1976:
1975:
1970:
1965:
1963:
1962:
1961:
1945:
1942:
1937:
1924:
1912:
1911:
1875:on each section
1871:
1869:
1868:
1863:
1861:
1860:
1851:
1850:
1837:
1825:
1824:
1744:principal symbol
1728:cotangent bundle
1711:
1709:
1708:
1703:
1683:
1682:
1670:
1669:
1657:
1656:
1637:symmetric tensor
1619:
1617:
1616:
1611:
1588:
1587:
1568:where, for each
1564:
1562:
1561:
1556:
1554:
1551:
1546:
1544:
1543:
1542:
1529:
1525:
1524:
1514:
1503:
1502:
1492:
1485:
1477:
1431:
1429:
1428:
1423:
1408:
1406:
1405:
1400:
1389:
1388:
1367:
1366:
1338:over a manifold
1317:principal symbol
1311:
1309:
1308:
1303:
1301:
1300:
1282:
1281:
1271:
1264:
1256:
1215:
1213:
1212:
1207:
1201:
1200:
1199:
1189:
1176:
1175:
1174:
1164:
1152:
1151:
1135:
1133:
1132:
1127:
1125:
1124:
1106:
1105:
1095:
1088:
1080:
1026:
1024:
1023:
1018:
1016:
1015:
999:
997:
996:
991:
989:
987:
986:
985:
969:
949:
947:
946:
941:
939:
938:
920:
918:
917:
912:
910:
908:
906:
905:
904:
894:
878:
877:
876:
866:
853:
852:
851:
841:
828:
824:
823:
822:
814:
803:
792:
791:
781:
774:
766:
738:
736:
735:
730:
728:
727:
722:
721:
699:
697:
696:
691:
689:
687:
685:
684:
683:
673:
657:
656:
655:
645:
632:
631:
630:
620:
607:
606:
605:
597:
587:
582:
581:
563:
561:
560:
555:
553:
552:
532:
530:
529:
524:
513:
512:
496:
494:
493:
488:
476:
474:
473:
468:
466:
465:
447:
446:
434:
433:
421:
413:
398:of non-negative
393:
391:
390:
385:
380:
379:
361:
360:
348:
347:
322:
320:
319:
314:
307:
306:
305:
287:
286:
276:
269:
261:
236:
234:
233:
228:
226:
225:
220:
219:
205:
203:
202:
197:
195:
194:
189:
176:
174:
173:
168:
166:
165:
160:
159:
142:
140:
139:
134:
122:
120:
119:
114:
78:computer science
46:Laplace operator
21:
18:Principal symbol
9222:
9221:
9217:
9216:
9215:
9213:
9212:
9211:
9197:Operator theory
9187:
9186:
9185:
9180:
9162:
9126:Advanced topics
9121:
9045:
9024:
8983:
8949:HilbertâSchmidt
8922:
8913:GelfandâNaimark
8860:
8810:
8745:
8731:
8701:
8696:
8637:Jacob Bernoulli
8610:
8575:
8566:Galerkin method
8489:
8427:Solution topics
8422:
8380:
8346:
8285:
8217:
8212:
8165:
8150:
8103:
8101:Further reading
8093:
8071:
8034:
8004:10.1.1.186.8445
7989:
7988:
7984:, § 1.2. § 1.3.
7980:
7976:
7939:
7935:
7917:
7916:
7905:
7899:
7898:
7881:
7873:
7859:
7853:
7852:
7835:
7821:
7813:
7805:
7799:
7798:
7781:
7773:
7759:
7748:
7734:
7726:
7709:
7701:
7695:
7694:
7677:
7666:
7652:
7635:
7629:
7628:
7605:
7578:
7574:
7566:
7560:
7559:
7541:
7537:
7508:
7490:
7486:
7447:
7443:
7437:
7433:
7419:
7415:
7406:
7402:
7398:
7396:
7393:
7392:
7389:
7385:
7376:
7374:
7368:
7364:
7351:
7347:
7339:
7335:
7330:
7325:
7290:Energy operator
7285:Spectral theory
7240:
7228:
7188:
7185:
7184:
7181:
7169:
7164:
7062:
7059:
7058:
6997:
6994:
6993:
6949:
6945:
6921:
6917:
6905:
6901:
6896:
6893:
6892:
6863:
6859:
6850:
6846:
6831:
6827:
6825:
6822:
6821:
6802:
6740:, the value of
6697:
6677:
6673:
6664:
6660:
6652:
6649:
6648:
6610:
6606:
6597:
6593:
6591:
6588:
6587:
6551:
6510:
6485:
6481:
6480:
6475:
6460:
6456:
6455:
6450:
6438:
6434:
6433:
6428:
6413:
6409:
6408:
6403:
6397:
6394:
6393:
6392:
6382:non-commutative
6381:
6358:
6349:
6345:
6330:
6326:
6317:
6313:
6298:
6294:
6286:
6283:
6282:
6281:
6274:Kronecker delta
6257:
6254:
6253:
6216:
6213:
6212:
6192:
6188:
6182:
6178:
6169:
6165:
6159:
6155:
6137:
6133:
6127:
6123:
6114:
6110:
6104:
6100:
6076:
6072:
6060:
6056:
6050:
6046:
6037:
6033:
6027:
6023:
6018:
6015:
6014:
5990:
5986:
5971:
5967:
5958:
5954:
5939:
5935:
5933:
5930:
5929:
5905:
5901:
5886:
5882:
5873:
5869:
5854:
5850:
5842:
5839:
5838:
5837:is a ring, let
5831:
5807:
5787:
5784:
5783:
5754:
5751:
5750:
5726: mod
5724:
5718:
5714:
5708:
5704:
5702:
5699:
5698:
5688:non-commutative
5687:
5668:
5648:
5645:
5644:
5574:
5571:
5570:
5569:is a ring, let
5563:
5557:
5552:
5495:
5492:
5491:
5466:
5455:
5415:
5411:
5402:
5398:
5377:
5373:
5364:
5360:
5355:
5352:
5351:
5332:is a constant.
5269:
5266:
5265:
5200:
5197:
5196:
5186:
5117:
5099:
5095:
5063:
5055:
5044:
5015:
5007:
4996:
4967:
4959:
4939:
4936:
4935:
4921:
4891:for all smooth
4861:
4857:
4856:
4852:
4817:
4813:
4812:
4808:
4799:
4795:
4784:
4781:
4780:
4755:
4716:
4712:
4710:
4707:
4706:
4661:
4657:
4656:
4654:
4653:
4649:
4643:
4639:
4633:
4629:
4614:
4603:
4587:
4583:
4581:
4578:
4577:
4551:
4548:
4547:
4525:
4522:
4521:
4438:
4436:
4430:
4425:
4401:
4398:
4397:
4385:
4372:
4331:
4328:
4327:
4304:
4300:
4268:
4265:
4264:
4247:
4243:
4241:
4238:
4237:
4213:
4209:
4194:
4190:
4184:
4173:
4158:
4155:
4154:
4139:
4136:
4135:
4132:
4126:
4091:
4087:
4069:
4065:
4042:
4038:
4036:
4031:
4028:
4027:
4006:
4002:
3979:
3975:
3973:
3968:
3965:
3964:
3943:
3939:
3916:
3912:
3910:
3905:
3902:
3901:
3857:
3853:
3849:
3844:
3838:
3834:
3828:
3817:
3805:
3802:
3801:
3747:
3743:
3728:
3724:
3716:
3713:
3712:
3669:
3664:
3653:
3650:
3649:
3616:
3611:
3603:
3597:
3593:
3591:
3585:
3574:
3561:
3557:
3549:
3546:
3545:
3511:
3507:
3501:
3497:
3491:
3480:
3474:
3471:
3470:
3425:
3423:
3420:
3419:
3399:
3382:
3379:
3378:
3343:
3338:
3332:
3329:
3328:
3311:
3306:
3300:
3297:
3296:
3279:
3275:
3273:
3270:
3269:
3249:
3245:
3241:
3235:
3231:
3229:
3227:
3224:
3223:
3195:
3191:
3189:
3186:
3185:
3165:
3161:
3159:
3156:
3155:
3139:
3136:
3135:
3114:
3109:
3107:
3104:
3103:
3085:
3073:
3064:. For now, see
3014:
3009:
2998:
2997:
2984:
2979:
2968:
2967:
2954:
2949:
2938:
2937:
2929:
2926:
2925:
2854:
2849:
2833:
2828:
2827:
2823:
2813:
2796:
2795:
2791:
2786:
2764:
2759:
2743:
2738:
2737:
2733:
2723:
2710:
2705:
2703:
2700:
2699:
2609:characteristics
2543:
2539:
2537:
2534:
2533:
2532:the bundle map
2513:
2509:
2501:
2498:
2497:
2477:
2474:
2473:
2469:
2400:
2399:
2363:
2359:
2351:
2346:
2345:
2344:
2326:
2322:
2312:
2307:
2287:
2284:
2283:
2265:
2240:
2236:
2227:
2223:
2212:
2209:
2208:
2207:with values in
2188:
2183:
2177:
2174:
2173:
2148:
2144:
2142:
2139:
2138:
2107:
2103:
2097:
2093:
2078:
2070:
2060:
2043:
2035:
2034:
2021:
2017:
1999:
1995:
1990:
1987:
1986:
1957:
1953:
1949:
1944:
1938:
1930:
1920:
1904:
1900:
1898:
1895:
1894:
1889:
1856:
1852:
1843:
1839:
1833:
1820:
1816:
1805:
1802:
1801:
1784:
1777:
1770:
1724:symmetric power
1678:
1674:
1665:
1661:
1652:
1648:
1646:
1643:
1642:
1635:transform as a
1583:
1579:
1577:
1574:
1573:
1550:
1538:
1534:
1530:
1520:
1516:
1515:
1513:
1498:
1494:
1481:
1473:
1472:
1448:
1445:
1444:
1417:
1414:
1413:
1384:
1380:
1362:
1358:
1350:
1347:
1346:
1296:
1292:
1277:
1273:
1260:
1252:
1251:
1224:
1221:
1220:
1195:
1191:
1190:
1185:
1170:
1166:
1165:
1160:
1147:
1143:
1141:
1138:
1137:
1120:
1116:
1101:
1097:
1084:
1076:
1075:
1048:
1045:
1044:
1011:
1007:
1005:
1002:
1001:
981:
977:
973:
968:
966:
963:
962:
957:The polynomial
934:
930:
928:
925:
924:
900:
896:
895:
890:
872:
868:
867:
862:
847:
843:
842:
837:
829:
818:
810:
809:
805:
804:
802:
787:
783:
770:
762:
761:
746:
743:
742:
723:
717:
716:
715:
707:
704:
703:
679:
675:
674:
669:
651:
647:
646:
641:
626:
622:
621:
616:
608:
601:
593:
592:
588:
586:
577:
573:
571:
568:
567:
548:
544:
542:
539:
538:
508:
504:
502:
499:
498:
482:
479:
478:
477:, and for each
461:
457:
442:
438:
429:
425:
417:
409:
407:
404:
403:
375:
371:
356:
352:
343:
339:
328:
325:
324:
301:
297:
282:
278:
265:
257:
256:
244:
241:
240:
221:
215:
214:
213:
211:
208:
207:
190:
185:
184:
182:
179:
178:
161:
155:
154:
153:
151:
148:
147:
128:
125:
124:
108:
105:
104:
97:
66:differentiation
28:
23:
22:
15:
12:
11:
5:
9220:
9210:
9209:
9204:
9199:
9182:
9181:
9179:
9178:
9167:
9164:
9163:
9161:
9160:
9155:
9150:
9145:
9143:Choquet theory
9140:
9135:
9129:
9127:
9123:
9122:
9120:
9119:
9109:
9104:
9099:
9094:
9089:
9084:
9079:
9074:
9069:
9064:
9059:
9053:
9051:
9047:
9046:
9044:
9043:
9038:
9032:
9030:
9026:
9025:
9023:
9022:
9017:
9012:
9007:
9002:
8997:
8995:Banach algebra
8991:
8989:
8985:
8984:
8982:
8981:
8976:
8971:
8966:
8961:
8956:
8951:
8946:
8941:
8936:
8930:
8928:
8924:
8923:
8921:
8920:
8918:BanachâAlaoglu
8915:
8910:
8905:
8900:
8895:
8890:
8885:
8880:
8874:
8872:
8866:
8865:
8862:
8861:
8859:
8858:
8853:
8848:
8846:Locally convex
8843:
8829:
8824:
8818:
8816:
8812:
8811:
8809:
8808:
8803:
8798:
8793:
8788:
8783:
8778:
8773:
8768:
8763:
8757:
8751:
8747:
8746:
8730:
8729:
8722:
8715:
8707:
8698:
8697:
8695:
8694:
8689:
8684:
8679:
8674:
8669:
8664:
8659:
8654:
8652:Ernst Lindelöf
8649:
8644:
8639:
8634:
8632:Leonhard Euler
8629:
8624:
8618:
8616:
8615:Mathematicians
8612:
8611:
8609:
8608:
8603:
8598:
8593:
8587:
8585:
8581:
8580:
8577:
8576:
8574:
8573:
8568:
8563:
8558:
8553:
8548:
8543:
8538:
8533:
8528:
8523:
8518:
8513:
8508:
8503:
8497:
8495:
8491:
8490:
8488:
8487:
8482:
8477:
8471:
8466:
8461:
8456:
8451:
8446:
8441:
8439:Phase portrait
8436:
8430:
8428:
8424:
8423:
8421:
8420:
8415:
8410:
8405:
8399:
8397:
8390:
8386:
8385:
8382:
8381:
8379:
8378:
8373:
8372:
8371:
8361:
8354:
8352:
8348:
8347:
8345:
8344:
8342:On jet bundles
8339:
8334:
8329:
8324:
8319:
8314:
8309:
8307:Nonhomogeneous
8304:
8299:
8293:
8291:
8287:
8286:
8284:
8283:
8278:
8273:
8268:
8263:
8258:
8253:
8248:
8243:
8238:
8232:
8230:
8223:
8222:Classification
8219:
8218:
8211:
8210:
8203:
8196:
8188:
8182:
8181:
8163:
8149:
8148:External links
8146:
8145:
8144:
8139:
8119:(3): 899â982.
8102:
8099:
8098:
8097:
8091:
8075:
8069:
8046:
8032:
8008:
7987:
7986:
7974:
7933:
7931:
7930:
7915:
7912:
7909:
7904:
7902:
7900:
7897:
7894:
7891:
7887:
7884:
7879:
7876:
7872:
7869:
7866:
7863:
7858:
7856:
7854:
7851:
7848:
7845:
7841:
7838:
7834:
7831:
7827:
7824:
7819:
7816:
7812:
7809:
7804:
7802:
7800:
7797:
7794:
7791:
7787:
7784:
7779:
7776:
7772:
7769:
7765:
7762:
7758:
7754:
7751:
7747:
7744:
7740:
7737:
7732:
7729:
7725:
7722:
7719:
7715:
7712:
7708:
7705:
7700:
7698:
7696:
7693:
7690:
7687:
7683:
7680:
7676:
7672:
7669:
7665:
7662:
7658:
7655:
7651:
7648:
7645:
7642:
7639:
7634:
7632:
7630:
7627:
7624:
7621:
7618:
7615:
7611:
7608:
7604:
7601:
7598:
7595:
7592:
7589:
7586:
7581:
7577:
7573:
7570:
7565:
7563:
7561:
7558:
7555:
7552:
7549:
7544:
7540:
7536:
7533:
7530:
7527:
7524:
7521:
7518:
7514:
7511:
7507:
7504:
7501:
7498:
7493:
7489:
7485:
7482:
7479:
7476:
7473:
7470:
7467:
7464:
7461:
7458:
7455:
7450:
7446:
7440:
7436:
7432:
7429:
7426:
7423:
7418:
7416:
7414:
7409:
7405:
7401:
7400:
7383:
7362:
7345:
7332:
7331:
7329:
7326:
7324:
7323:
7318:
7313:
7307:
7302:
7297:
7292:
7287:
7282:
7277:
7272:
7267:
7262:
7257:
7252:
7250:Delta operator
7247:
7241:
7239:
7236:
7227:
7224:
7207:
7204:
7201:
7198:
7195:
7192:
7180:
7177:
7168:
7165:
7163:
7160:
7144:
7143:
7132:
7129:
7126:
7123:
7120:
7117:
7114:
7111:
7108:
7105:
7102:
7099:
7096:
7093:
7090:
7087:
7084:
7081:
7078:
7075:
7072:
7069:
7066:
7043:
7040:
7037:
7034:
7031:
7028:
7025:
7022:
7019:
7016:
7013:
7010:
7007:
7004:
7001:
6990:
6989:
6978:
6975:
6972:
6969:
6966:
6963:
6960:
6957:
6952:
6948:
6944:
6941:
6938:
6935:
6930:
6927:
6924:
6920:
6916:
6913:
6908:
6904:
6900:
6877:
6874:
6871:
6866:
6862:
6858:
6853:
6849:
6845:
6842:
6839:
6834:
6830:
6801:
6798:
6794:Peetre theorem
6694:
6693:
6680:
6676:
6672:
6667:
6663:
6659:
6656:
6642:
6641:
6630:
6627:
6624:
6621:
6618:
6613:
6609:
6605:
6600:
6596:
6526:vector bundles
6509:
6506:
6488:
6484:
6478:
6474:
6470:
6463:
6459:
6453:
6449:
6441:
6437:
6431:
6427:
6423:
6416:
6412:
6406:
6402:
6365:
6361:
6357:
6352:
6348:
6344:
6341:
6338:
6333:
6329:
6325:
6320:
6316:
6312:
6309:
6306:
6301:
6297:
6293:
6290:
6261:
6241:
6238:
6235:
6232:
6229:
6226:
6223:
6220:
6209:
6208:
6195:
6191:
6185:
6181:
6177:
6172:
6168:
6162:
6158:
6145:
6140:
6136:
6130:
6126:
6122:
6117:
6113:
6107:
6103:
6090:
6085:
6082:
6079:
6075:
6071:
6068:
6063:
6059:
6053:
6049:
6045:
6040:
6036:
6030:
6026:
6022:
5993:
5989:
5985:
5982:
5979:
5974:
5970:
5966:
5961:
5957:
5953:
5950:
5947:
5942:
5938:
5913:
5908:
5904:
5900:
5897:
5894:
5889:
5885:
5881:
5876:
5872:
5868:
5865:
5862:
5857:
5853:
5849:
5846:
5830:
5827:
5814:
5810:
5806:
5803:
5800:
5797:
5794:
5791:
5767:
5764:
5761:
5758:
5731:
5721:
5717:
5711:
5707:
5675:
5671:
5667:
5664:
5661:
5658:
5655:
5652:
5624:the two-sided
5593:
5590:
5587:
5584:
5581:
5578:
5559:Main article:
5556:
5553:
5551:
5548:
5529:
5528:
5517:
5514:
5511:
5508:
5505:
5502:
5499:
5475:: an operator
5464:
5458:differentiable
5453:
5447:
5446:
5435:
5432:
5429:
5426:
5423:
5418:
5414:
5410:
5405:
5401:
5397:
5394:
5391:
5388:
5385:
5380:
5376:
5372:
5367:
5363:
5359:
5318:
5317:
5306:
5303:
5300:
5297:
5294:
5291:
5288:
5285:
5282:
5279:
5276:
5273:
5263:
5252:
5249:
5246:
5243:
5240:
5237:
5234:
5231:
5228:
5225:
5222:
5219:
5216:
5213:
5210:
5207:
5204:
5185:
5182:
5176:(analogues to
5174:eigenfunctions
5163:
5162:
5151:
5148:
5145:
5142:
5139:
5136:
5133:
5130:
5127:
5123:
5120:
5116:
5113:
5110:
5107:
5102:
5098:
5094:
5091:
5088:
5085:
5082:
5079:
5076:
5073:
5069:
5066:
5061:
5058:
5054:
5050:
5047:
5043:
5040:
5037:
5034:
5031:
5028:
5025:
5021:
5018:
5013:
5010:
5006:
5002:
4999:
4995:
4992:
4989:
4986:
4983:
4980:
4977:
4973:
4970:
4965:
4962:
4958:
4955:
4952:
4949:
4946:
4943:
4920:
4917:
4889:
4888:
4875:
4872:
4869:
4864:
4860:
4855:
4851:
4848:
4845:
4842:
4839:
4836:
4831:
4828:
4825:
4820:
4816:
4811:
4807:
4802:
4798:
4794:
4791:
4788:
4769:is defined in
4754:
4751:
4734:formal adjoint
4719:
4715:
4692:
4688:
4684:
4679:
4675:
4672:
4669:
4664:
4660:
4652:
4646:
4642:
4636:
4632:
4628:
4625:
4622:
4617:
4612:
4609:
4606:
4602:
4598:
4595:
4590:
4586:
4561:
4558:
4555:
4535:
4532:
4529:
4500:) denotes the
4479:
4476:
4473:
4469:
4466:
4463:
4460:
4454:
4450:
4447:
4444:
4441:
4433:
4428:
4424:
4420:
4417:
4414:
4411:
4408:
4405:
4371:
4368:
4360:scalar product
4347:
4344:
4341:
4338:
4335:
4315:
4312:
4307:
4303:
4299:
4296:
4293:
4290:
4287:
4284:
4281:
4278:
4275:
4272:
4250:
4246:
4221:
4216:
4212:
4208:
4205:
4202:
4197:
4193:
4187:
4182:
4179:
4176:
4172:
4168:
4165:
4162:
4143:
4125:
4122:
4114:
4113:
4102:
4099:
4094:
4090:
4086:
4083:
4080:
4077:
4072:
4068:
4064:
4061:
4058:
4055:
4050:
4045:
4041:
4035:
4025:
4014:
4009:
4005:
4001:
3998:
3995:
3992:
3987:
3982:
3978:
3972:
3962:
3951:
3946:
3942:
3938:
3935:
3932:
3929:
3924:
3919:
3915:
3909:
3868:
3860:
3856:
3852:
3848:
3841:
3837:
3831:
3826:
3823:
3820:
3816:
3812:
3809:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3759:
3755:
3750:
3746:
3742:
3739:
3736:
3731:
3727:
3723:
3720:
3701:eigenfunctions
3699:, because its
3693:
3692:
3681:
3675:
3672:
3668:
3663:
3660:
3657:
3643:theta operator
3639:
3638:
3627:
3619:
3614:
3610:
3606:
3600:
3596:
3588:
3583:
3580:
3577:
3573:
3569:
3564:
3560:
3556:
3553:
3528:
3527:
3514:
3510:
3504:
3500:
3494:
3489:
3486:
3483:
3479:
3456:
3455:
3444:
3441:
3438:
3435:
3431:
3428:
3417:
3405:
3402:
3398:
3395:
3392:
3389:
3386:
3361:
3360:
3346:
3341:
3337:
3314:
3309:
3305:
3282:
3278:
3252:
3248:
3244:
3238:
3234:
3213:
3212:
3198:
3194:
3173:
3168:
3164:
3143:
3120:
3117:
3113:
3084:
3081:
3072:
3069:
3068:
3067:
3057:
3056:
3037:
3026:
3020:
3017:
3013:
3005:
3002:
2996:
2990:
2987:
2983:
2975:
2972:
2966:
2960:
2957:
2953:
2945:
2942:
2936:
2933:
2922:
2921:
2900:, also called
2894:
2891:motor variable
2874:
2867:
2860:
2857:
2853:
2848:
2845:
2839:
2836:
2832:
2826:
2820:
2817:
2812:
2803:
2800:
2794:
2790:
2784:
2777:
2770:
2767:
2763:
2758:
2755:
2749:
2746:
2742:
2736:
2730:
2727:
2722:
2716:
2713:
2709:
2661:
2638:
2635:Lie derivative
2623:
2612:
2597:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2546:
2542:
2521:
2516:
2512:
2508:
2505:
2481:
2468:
2465:
2443:This exhibits
2441:
2440:
2429:
2426:
2423:
2419:
2416:
2413:
2407:
2404:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2375:
2372:
2369:
2366:
2362:
2354:
2349:
2343:
2333:
2330:
2325:
2321:
2318:
2315:
2311:
2306:
2303:
2300:
2297:
2294:
2291:
2264:
2261:
2248:
2243:
2239:
2235:
2230:
2226:
2222:
2219:
2216:
2196:
2191:
2186:
2182:
2151:
2147:
2127:
2126:
2115:
2110:
2106:
2100:
2096:
2092:
2089:
2086:
2081:
2076:
2073:
2069:
2063:
2059:
2053:
2050:
2046:
2042:
2038:
2033:
2029:
2024:
2020:
2016:
2013:
2010:
2007:
2002:
1998:
1994:
1980:
1979:
1968:
1960:
1956:
1952:
1948:
1941:
1936:
1933:
1929:
1923:
1919:
1915:
1910:
1907:
1903:
1887:
1873:
1872:
1859:
1855:
1849:
1846:
1842:
1836:
1832:
1828:
1823:
1819:
1815:
1812:
1809:
1782:
1775:
1766:
1717:tensor product
1713:
1712:
1701:
1698:
1695:
1692:
1689:
1686:
1681:
1677:
1673:
1668:
1664:
1660:
1655:
1651:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1586:
1582:
1566:
1565:
1549:
1541:
1537:
1533:
1528:
1523:
1519:
1512:
1509:
1506:
1501:
1497:
1491:
1488:
1484:
1480:
1476:
1471:
1467:
1464:
1461:
1458:
1455:
1452:
1421:
1410:
1409:
1398:
1395:
1392:
1387:
1383:
1379:
1376:
1373:
1370:
1365:
1361:
1357:
1354:
1336:vector bundles
1315:is called the
1313:
1312:
1299:
1295:
1291:
1288:
1285:
1280:
1276:
1270:
1267:
1263:
1259:
1255:
1250:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1205:
1198:
1194:
1188:
1184:
1180:
1173:
1169:
1163:
1159:
1155:
1150:
1146:
1123:
1119:
1115:
1112:
1109:
1104:
1100:
1094:
1091:
1087:
1083:
1079:
1074:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1031:is called the
1014:
1010:
984:
980:
976:
972:
937:
933:
903:
899:
893:
889:
885:
882:
875:
871:
865:
861:
857:
850:
846:
840:
836:
832:
827:
821:
817:
813:
808:
801:
798:
795:
790:
786:
780:
777:
773:
769:
765:
760:
756:
753:
750:
726:
720:
714:
711:
682:
678:
672:
668:
664:
661:
654:
650:
644:
640:
636:
629:
625:
619:
615:
611:
604:
600:
596:
591:
585:
580:
576:
551:
547:
522:
519:
516:
511:
507:
486:
464:
460:
456:
453:
450:
445:
441:
437:
432:
428:
424:
420:
416:
412:
383:
378:
374:
370:
367:
364:
359:
355:
351:
346:
342:
338:
335:
332:
312:
304:
300:
296:
293:
290:
285:
281:
275:
272:
268:
264:
260:
255:
251:
248:
224:
218:
193:
188:
164:
158:
145:function space
132:
112:
96:
93:
26:
9:
6:
4:
3:
2:
9219:
9208:
9205:
9203:
9200:
9198:
9195:
9194:
9192:
9177:
9169:
9168:
9165:
9159:
9156:
9154:
9151:
9149:
9148:Weak topology
9146:
9144:
9141:
9139:
9136:
9134:
9131:
9130:
9128:
9124:
9117:
9113:
9110:
9108:
9105:
9103:
9100:
9098:
9095:
9093:
9090:
9088:
9085:
9083:
9080:
9078:
9075:
9073:
9072:Index theorem
9070:
9068:
9065:
9063:
9060:
9058:
9055:
9054:
9052:
9048:
9042:
9039:
9037:
9034:
9033:
9031:
9029:Open problems
9027:
9021:
9018:
9016:
9013:
9011:
9008:
9006:
9003:
9001:
8998:
8996:
8993:
8992:
8990:
8986:
8980:
8977:
8975:
8972:
8970:
8967:
8965:
8962:
8960:
8957:
8955:
8952:
8950:
8947:
8945:
8942:
8940:
8937:
8935:
8932:
8931:
8929:
8925:
8919:
8916:
8914:
8911:
8909:
8906:
8904:
8901:
8899:
8896:
8894:
8891:
8889:
8886:
8884:
8881:
8879:
8876:
8875:
8873:
8871:
8867:
8857:
8854:
8852:
8849:
8847:
8844:
8841:
8837:
8833:
8830:
8828:
8825:
8823:
8820:
8819:
8817:
8813:
8807:
8804:
8802:
8799:
8797:
8794:
8792:
8789:
8787:
8784:
8782:
8779:
8777:
8774:
8772:
8769:
8767:
8764:
8762:
8759:
8758:
8755:
8752:
8748:
8743:
8739:
8735:
8728:
8723:
8721:
8716:
8714:
8709:
8708:
8705:
8693:
8690:
8688:
8685:
8683:
8680:
8678:
8675:
8673:
8670:
8668:
8665:
8663:
8660:
8658:
8655:
8653:
8650:
8648:
8645:
8643:
8640:
8638:
8635:
8633:
8630:
8628:
8625:
8623:
8620:
8619:
8617:
8613:
8607:
8604:
8602:
8599:
8597:
8594:
8592:
8589:
8588:
8586:
8582:
8572:
8569:
8567:
8564:
8562:
8559:
8557:
8554:
8552:
8549:
8547:
8544:
8542:
8539:
8537:
8534:
8532:
8529:
8527:
8524:
8522:
8519:
8517:
8514:
8512:
8509:
8507:
8504:
8502:
8499:
8498:
8496:
8492:
8486:
8483:
8481:
8478:
8475:
8472:
8470:
8467:
8465:
8462:
8460:
8457:
8455:
8452:
8450:
8447:
8445:
8442:
8440:
8437:
8435:
8432:
8431:
8429:
8425:
8419:
8416:
8414:
8411:
8409:
8406:
8404:
8401:
8400:
8398:
8394:
8391:
8387:
8377:
8374:
8370:
8367:
8366:
8365:
8362:
8359:
8356:
8355:
8353:
8349:
8343:
8340:
8338:
8335:
8333:
8330:
8328:
8325:
8323:
8320:
8318:
8315:
8313:
8310:
8308:
8305:
8303:
8300:
8298:
8295:
8294:
8292:
8288:
8282:
8279:
8277:
8274:
8272:
8269:
8267:
8264:
8262:
8259:
8257:
8254:
8252:
8249:
8247:
8244:
8242:
8239:
8237:
8234:
8233:
8231:
8227:
8224:
8220:
8216:
8209:
8204:
8202:
8197:
8195:
8190:
8189:
8186:
8178:
8174:
8173:
8168:
8164:
8161:
8156:
8152:
8151:
8143:
8140:
8136:
8132:
8127:
8122:
8118:
8114:
8110:
8105:
8104:
8094:
8092:0-387-90419-0
8088:
8084:
8080:
8076:
8072:
8066:
8062:
8058:
8054:
8053:
8047:
8043:
8039:
8035:
8033:3-540-12104-8
8029:
8025:
8021:
8017:
8013:
8012:Hörmander, L.
8009:
8005:
8000:
7997:, p. 8,
7996:
7991:
7990:
7983:
7982:Schapira 1985
7978:
7970:
7966:
7961:
7956:
7952:
7948:
7944:
7937:
7913:
7910:
7907:
7903:
7895:
7892:
7889:
7885:
7877:
7874:
7870:
7864:
7861:
7857:
7849:
7846:
7843:
7839:
7836:
7832:
7829:
7825:
7822:
7817:
7814:
7810:
7807:
7803:
7795:
7792:
7789:
7785:
7782:
7777:
7774:
7770:
7767:
7763:
7760:
7756:
7752:
7749:
7745:
7742:
7738:
7735:
7730:
7727:
7723:
7720:
7717:
7713:
7710:
7706:
7703:
7699:
7691:
7688:
7685:
7681:
7674:
7670:
7667:
7660:
7656:
7649:
7646:
7640:
7637:
7633:
7625:
7622:
7619:
7613:
7609:
7606:
7599:
7596:
7590:
7587:
7579:
7575:
7571:
7568:
7564:
7553:
7550:
7542:
7534:
7531:
7525:
7519:
7512:
7509:
7505:
7496:
7491:
7483:
7480:
7474:
7468:
7462:
7459:
7448:
7444:
7438:
7430:
7427:
7421:
7417:
7412:
7407:
7403:
7391:
7390:
7387:
7373:
7366:
7359:
7355:
7349:
7342:
7341:Schapira 1985
7337:
7333:
7322:
7319:
7317:
7314:
7311:
7308:
7306:
7303:
7301:
7298:
7296:
7293:
7291:
7288:
7286:
7283:
7281:
7278:
7276:
7273:
7271:
7268:
7266:
7263:
7261:
7258:
7256:
7253:
7251:
7248:
7246:
7243:
7242:
7235:
7233:
7223:
7221:
7202:
7199:
7196:
7190:
7176:
7174:
7159:
7157:
7153:
7149:
7130:
7124:
7118:
7115:
7112:
7109:
7103:
7100:
7097:
7091:
7088:
7082:
7073:
7070:
7067:
7057:
7056:
7055:
7038:
7023:
7014:
7008:
7005:
7002:
6976:
6973:
6964:
6958:
6955:
6950:
6946:
6939:
6933:
6928:
6925:
6922:
6918:
6911:
6906:
6902:
6891:
6890:
6889:
6872:
6860:
6856:
6851:
6847:
6843:
6840:
6837:
6832:
6828:
6819:
6815:
6811:
6807:
6797:
6795:
6791:
6787:
6783:
6779:
6775:
6771:
6767:
6763:
6759:
6755:
6752: â
6751:
6748:) at a point
6747:
6743:
6739:
6735:
6732:
6727:
6725:
6723:
6718:
6712:
6708:
6704:
6700:
6678:
6674:
6670:
6665:
6661:
6657:
6654:
6647:
6646:
6645:
6628:
6619:
6611:
6607:
6603:
6598:
6594:
6586:
6585:
6584:
6582:
6578:
6575:
6571:
6569:
6562:
6558:
6554:
6550:
6546:
6542:
6539:
6535:
6531:
6527:
6523:
6519:
6515:
6505:
6486:
6482:
6476:
6472:
6468:
6461:
6457:
6451:
6447:
6439:
6435:
6429:
6425:
6421:
6414:
6410:
6404:
6400:
6390:
6386:
6378:
6363:
6359:
6350:
6346:
6342:
6339:
6336:
6331:
6327:
6323:
6318:
6314:
6310:
6307:
6304:
6299:
6295:
6288:
6279:
6275:
6259:
6239:
6236:
6233:
6230:
6227:
6224:
6221:
6218:
6193:
6189:
6183:
6179:
6175:
6170:
6166:
6160:
6156:
6143:
6138:
6134:
6128:
6124:
6120:
6115:
6111:
6105:
6101:
6088:
6083:
6080:
6077:
6073:
6069:
6061:
6057:
6051:
6047:
6043:
6038:
6034:
6028:
6024:
6013:
6012:
6011:
6009:
5991:
5987:
5983:
5980:
5977:
5972:
5968:
5964:
5959:
5955:
5951:
5948:
5945:
5940:
5936:
5927:
5906:
5902:
5898:
5895:
5892:
5887:
5883:
5879:
5874:
5870:
5866:
5863:
5860:
5855:
5851:
5844:
5836:
5826:
5812:
5808:
5801:
5798:
5795:
5789:
5781:
5762:
5756:
5747:
5745:
5729:
5719:
5715:
5709:
5705:
5696:
5692:
5673:
5669:
5662:
5659:
5656:
5650:
5643:
5642:quotient ring
5639:
5635:
5631:
5628:generated by
5627:
5623:
5619:
5615:
5611:
5607:
5588:
5585:
5582:
5576:
5568:
5562:
5547:
5545:
5544:shift theorem
5540:
5538:
5534:
5515:
5512:
5509:
5506:
5503:
5500:
5497:
5490:
5489:
5488:
5486:
5482:
5478:
5474:
5470:
5463:
5459:
5452:
5433:
5424:
5416:
5412:
5403:
5399:
5395:
5389:
5378:
5374:
5370:
5365:
5361:
5350:
5349:
5348:
5346:
5342:
5338:
5333:
5331:
5327:
5323:
5304:
5298:
5295:
5289:
5286:
5280:
5277:
5271:
5264:
5250:
5244:
5241:
5235:
5229:
5226:
5220:
5214:
5211:
5208:
5202:
5195:
5194:
5193:
5191:
5181:
5179:
5175:
5171:
5166:
5149:
5146:
5140:
5134:
5131:
5128:
5121:
5118:
5114:
5108:
5105:
5100:
5096:
5089:
5086:
5080:
5077:
5074:
5071:
5067:
5064:
5059:
5056:
5052:
5048:
5045:
5041:
5038:
5035:
5032:
5029:
5026:
5019:
5016:
5011:
5008:
5004:
5000:
4997:
4993:
4987:
4984:
4981:
4978:
4975:
4971:
4963:
4960:
4956:
4950:
4947:
4944:
4941:
4934:
4933:
4932:
4930:
4926:
4916:
4914:
4910:
4906:
4902:
4898:
4894:
4862:
4858:
4849:
4846:
4843:
4840:
4834:
4818:
4814:
4805:
4800:
4796:
4792:
4789:
4779:
4778:
4777:
4775:
4773:
4768:
4764:
4760:
4750:
4748:
4747:
4743:A (formally)
4741:
4739:
4735:
4717:
4713:
4703:
4690:
4686:
4682:
4670:
4662:
4658:
4650:
4644:
4640:
4634:
4626:
4623:
4615:
4610:
4607:
4604:
4600:
4596:
4593:
4588:
4584:
4575:
4559:
4553:
4533:
4527:
4519:
4515:
4511:
4507:
4503:
4499:
4495:
4490:
4477:
4474:
4471:
4464:
4458:
4445:
4439:
4431:
4426:
4422:
4418:
4412:
4409:
4406:
4393:
4389:
4384:
4381:
4377:
4367:
4365:
4364:inner product
4361:
4342:
4339:
4336:
4310:
4305:
4301:
4297:
4294:
4288:
4282:
4279:
4276:
4273:
4248:
4244:
4235:
4219:
4214:
4210:
4203:
4195:
4191:
4185:
4180:
4177:
4174:
4170:
4166:
4163:
4160:
4141:
4131:
4121:
4119:
4100:
4097:
4092:
4084:
4081:
4078:
4075:
4070:
4062:
4059:
4056:
4053:
4048:
4043:
4033:
4026:
4012:
4007:
3999:
3996:
3993:
3990:
3985:
3980:
3970:
3963:
3949:
3944:
3936:
3933:
3930:
3927:
3922:
3917:
3907:
3900:
3899:
3898:
3894:
3892:
3888:
3884:
3879:
3866:
3858:
3854:
3839:
3835:
3829:
3824:
3821:
3818:
3814:
3810:
3799:
3794:
3781:
3778:
3775:
3772:
3769:
3766:
3763:
3760:
3757:
3753:
3748:
3744:
3740:
3737:
3729:
3725:
3710:
3706:
3702:
3698:
3679:
3673:
3670:
3666:
3661:
3658:
3648:
3647:
3646:
3645:, defined by
3644:
3625:
3617:
3612:
3608:
3598:
3586:
3581:
3578:
3575:
3571:
3567:
3562:
3554:
3544:
3543:
3542:
3541:, defined by
3540:
3535:
3533:
3512:
3508:
3502:
3498:
3492:
3487:
3484:
3481:
3477:
3469:
3468:
3467:
3465:
3461:
3442:
3436:
3429:
3426:
3418:
3403:
3393:
3387:
3377:
3376:
3375:
3373:
3370:
3366:
3344:
3339:
3312:
3307:
3303:
3280:
3276:
3250:
3246:
3242:
3236:
3232:
3222:
3221:
3220:
3218:
3196:
3171:
3166:
3162:
3141:
3118:
3115:
3111:
3102:
3101:
3100:
3098:
3094:
3090:
3080:
3078:
3066:
3063:
3059:
3058:
3054:
3050:
3046:
3042:
3038:
3024:
3018:
2994:
2988:
2964:
2958:
2934:
2924:
2923:
2919:
2915:
2911:
2907:
2903:
2899:
2895:
2892:
2888:
2872:
2865:
2858:
2846:
2843:
2837:
2824:
2818:
2815:
2810:
2798:
2782:
2775:
2768:
2756:
2753:
2747:
2734:
2728:
2725:
2720:
2714:
2697:
2693:
2689:
2685:
2681:
2677:
2673:
2670:
2666:
2662:
2659:
2655:
2651:
2647:
2643:
2639:
2636:
2632:
2628:
2624:
2621:
2617:
2613:
2610:
2606:
2602:
2598:
2596:and cokernel.
2595:
2591:
2587:
2583:
2564:
2561:
2558:
2555:
2552:
2544:
2540:
2519:
2514:
2510:
2506:
2503:
2495:
2479:
2471:
2470:
2464:
2462:
2458:
2454:
2450:
2446:
2427:
2424:
2421:
2414:
2402:
2393:
2390:
2387:
2384:
2378:
2373:
2370:
2367:
2364:
2360:
2352:
2341:
2331:
2328:
2319:
2316:
2309:
2304:
2298:
2292:
2289:
2282:
2281:
2280:
2278:
2274:
2270:
2260:
2241:
2237:
2233:
2228:
2224:
2217:
2214:
2194:
2189:
2184:
2180:
2171:
2167:
2149:
2145:
2137:, the symbol
2136:
2132:
2113:
2108:
2104:
2098:
2094:
2087:
2079:
2074:
2071:
2067:
2061:
2057:
2051:
2048:
2040:
2031:
2027:
2022:
2014:
2008:
2000:
1996:
1985:
1984:
1983:
1966:
1958:
1954:
1939:
1934:
1931:
1927:
1921:
1917:
1913:
1908:
1905:
1901:
1893:
1892:
1891:
1886:
1882:
1878:
1857:
1853:
1847:
1844:
1840:
1834:
1830:
1826:
1821:
1813:
1810:
1800:
1799:
1798:
1796:
1792:
1788:
1781:
1774:
1769:
1764:
1760:
1755:
1753:
1749:
1746:(or just the
1745:
1741:
1737:
1733:
1729:
1725:
1722:
1718:
1699:
1693:
1690:
1684:
1679:
1675:
1666:
1662:
1658:
1653:
1649:
1641:
1640:
1639:
1638:
1634:
1630:
1625:
1623:
1607:
1601:
1598:
1592:
1584:
1580:
1571:
1547:
1539:
1535:
1526:
1521:
1507:
1499:
1495:
1489:
1486:
1478:
1469:
1465:
1459:
1453:
1450:
1443:
1442:
1441:
1439:
1435:
1419:
1393:
1381:
1371:
1359:
1355:
1352:
1345:
1344:
1343:
1341:
1337:
1333:
1329:
1324:
1322:
1318:
1297:
1293:
1286:
1278:
1274:
1268:
1265:
1257:
1248:
1244:
1238:
1235:
1232:
1226:
1219:
1218:
1217:
1203:
1196:
1192:
1186:
1182:
1178:
1171:
1167:
1161:
1157:
1153:
1148:
1144:
1121:
1117:
1110:
1102:
1098:
1092:
1089:
1081:
1072:
1068:
1062:
1059:
1056:
1050:
1042:
1038:
1034:
1030:
1012:
1008:
1000:by variables
982:
978:
960:
955:
953:
935:
931:
923:The notation
921:
901:
897:
891:
887:
880:
873:
869:
863:
859:
848:
844:
838:
834:
825:
815:
796:
788:
784:
778:
775:
767:
758:
754:
751:
748:
740:
724:
712:
709:
700:
680:
676:
670:
666:
659:
652:
648:
642:
638:
627:
623:
617:
613:
598:
583:
578:
574:
565:
549:
545:
536:
517:
509:
505:
484:
462:
458:
454:
451:
448:
443:
439:
435:
430:
426:
422:
414:
401:
397:
376:
372:
368:
365:
362:
357:
353:
349:
344:
340:
333:
330:
310:
302:
298:
291:
283:
279:
273:
270:
262:
253:
249:
246:
238:
222:
191:
162:
146:
130:
110:
102:
92:
90:
86:
81:
79:
75:
71:
67:
63:
59:
55:
47:
43:
39:
34:
30:
19:
9138:Balanced set
9112:Distribution
9050:Applications
8903:KreinâMilman
8888:Closed graph
8687:Martin Kutta
8642:Ămile Picard
8622:Isaac Newton
8536:Euler method
8506:Substitution
8235:
8170:
8116:
8112:
8082:
8051:
8015:
7994:
7977:
7950:
7946:
7936:
7386:
7375:. Retrieved
7365:
7358:Google Books
7353:
7348:
7336:
7229:
7219:
7218:is called a
7182:
7173:power series
7170:
7145:
6991:
6817:
6813:
6809:
6808:-linear map
6805:
6803:
6789:
6785:
6777:
6773:
6769:
6765:
6761:
6757:
6753:
6749:
6745:
6741:
6737:
6733:
6728:
6721:
6716:
6710:
6706:
6702:
6698:
6695:
6643:
6580:
6576:
6567:
6566:
6560:
6556:
6552:
6544:
6540:
6533:
6529:
6511:
6388:
6379:
6277:
6210:
6007:
5925:
5834:
5832:
5748:
5694:
5686:. This is a
5637:
5633:
5629:
5621:
5617:
5613:
5609:
5566:
5564:
5561:Weyl algebra
5541:
5532:
5530:
5480:
5476:
5461:
5450:
5448:
5340:
5334:
5329:
5325:
5321:
5319:
5187:
5178:eigenvectors
5167:
5164:
4928:
4922:
4908:
4904:
4900:
4896:
4892:
4890:
4771:
4766:
4762:
4758:
4756:
4746:self-adjoint
4744:
4742:
4737:
4733:
4704:
4573:
4520:vanishes as
4517:
4513:
4509:
4505:
4497:
4493:
4491:
4391:
4387:
4373:
4133:
4115:
3895:
3880:
3797:
3795:
3708:
3696:
3694:
3640:
3536:
3529:
3459:
3457:
3371:
3364:
3362:
3216:
3214:
3096:
3086:
3074:
2901:
2691:
2687:
2683:
2679:
2675:
2671:
2656:. See also
2585:
2456:
2452:
2444:
2442:
2268:
2266:
2169:
2134:
2130:
2128:
1981:
1884:
1880:
1876:
1874:
1794:
1790:
1786:
1779:
1772:
1767:
1762:
1758:
1756:
1751:
1747:
1743:
1739:
1735:
1731:
1720:
1714:
1632:
1628:
1626:
1567:
1437:
1411:
1339:
1331:
1327:
1325:
1320:
1316:
1314:
1040:
1036:
1033:total symbol
1032:
1028:
958:
956:
922:
741:
701:
566:
534:
239:
100:
98:
82:
57:
51:
29:
9067:Heat kernel
9057:Hardy space
8964:Trace class
8878:HahnâBanach
8840:Topological
8444:Phase space
8302:Homogeneous
8079:Wells, R.O.
6385:simple ring
5691:simple ring
5473:commutative
3883:eigenspaces
1570:multi-index
1440:, we have
396:multi-index
103:, an order-
54:mathematics
9191:Categories
9000:C*-algebra
8815:Properties
8672:John Crank
8501:Inspection
8364:Stochastic
8358:Difference
8332:Autonomous
8276:Non-linear
8266:Fractional
8229:Operations
7377:2009-06-12
7328:References
6644:such that
6574:jet bundle
6555: : Î(
6522:coordinate
6380:This is a
5337:polynomial
5172:where the
4911:: P is a
4895:functions
4263:such that
4128:See also:
3089:derivative
3049:divergence
2646:derivation
2601:hyperbolic
2168:of degree
2164:defines a
1622:bundle map
1043:above is:
95:Definition
8974:Unbounded
8969:Transpose
8927:Operators
8856:Separable
8851:Reflexive
8836:Algebraic
8822:Barrelled
8476:solutions
8434:Wronskian
8389:Solutions
8317:Decoupled
8281:Holonomic
8177:EMS Press
8135:1777-5310
7999:CiteSeerX
7969:119540529
7865:−
7830:−
7811:−
7743:−
7721:−
7707:−
7641:−
7572:−
7532:−
7506:−
7481:−
7460:−
7428:−
7408:∗
7116:⋅
7110:−
7101:⋅
7033:Γ
7030:→
7018:Γ
6965:⋯
6940:⋯
6926:−
6865:∞
6857:∈
6841:…
6671:∘
6626:→
6469:…
6422:…
6356:⟩
6340:…
6308:…
6292:⟨
6260:δ
6234:≤
6222:≤
6176:−
6121:−
6074:δ
6070:−
6044:−
5981:…
5949:…
5912:⟩
5896:…
5864:…
5848:⟨
5805:⟩
5793:⟨
5666:⟩
5654:⟨
5592:⟩
5580:⟨
5504:−
5371:∘
5115:−
5087:−
5053:−
5039:−
4988:−
4951:−
4871:Ω
4854:⟩
4838:⟨
4827:Ω
4810:⟩
4801:∗
4787:⟨
4718:∗
4678:¯
4624:−
4601:∑
4589:∗
4557:→
4531:→
4453:¯
4423:∫
4416:⟩
4404:⟨
4346:⟩
4343:⋅
4337:⋅
4334:⟨
4314:⟩
4306:∗
4292:⟨
4286:⟩
4271:⟨
4249:∗
4171:∑
4089:∂
4085:⋅
4079:−
4067:∂
4063:⋅
4049:↔
4040:∂
4004:∂
4000:⋅
3986:→
3977:∂
3941:∂
3937:⋅
3923:←
3914:∂
3851:∂
3847:∂
3815:∑
3808:Θ
3782:…
3719:Θ
3705:monomials
3656:Θ
3605:∂
3595:∂
3572:∑
3559:∇
3552:Δ
3478:∑
3336:∂
3193:∂
3099:include:
3083:Notations
3079:in 1800.
3053:Laplacian
3016:∂
3012:∂
3004:^
2986:∂
2982:∂
2974:^
2956:∂
2952:∂
2944:^
2932:∇
2856:∂
2852:∂
2835:∂
2831:∂
2802:¯
2793:∂
2789:∂
2766:∂
2762:∂
2754:−
2745:∂
2741:∂
2712:∂
2708:∂
2565:θ
2559:…
2553:θ
2541:σ
2515:∗
2507:∈
2504:θ
2425:ξ
2415:ξ
2406:^
2394:ξ
2374:ξ
2371:⋅
2342:∫
2320:π
2218:
2190:∗
2146:σ
2109:μ
2099:α
2095:ξ
2080:α
2075:μ
2072:ν
2062:μ
2058:∑
2041:α
2032:∑
2023:ν
2009:ξ
1997:σ
1959:α
1951:∂
1947:∂
1940:α
1935:μ
1932:ν
1922:α
1918:∑
1909:μ
1906:ν
1858:μ
1848:μ
1845:ν
1835:μ
1831:∑
1822:ν
1697:→
1691:⊗
1680:∗
1650:σ
1605:→
1585:α
1540:α
1532:∂
1522:α
1518:∂
1500:α
1479:α
1470:∑
1386:∞
1378:→
1364:∞
1298:α
1294:ξ
1279:α
1258:α
1249:∑
1239:ξ
1227:σ
1193:α
1183:ξ
1179:⋯
1168:α
1158:ξ
1149:α
1145:ξ
1122:α
1118:ξ
1103:α
1090:≤
1082:α
1073:∑
1063:ξ
1009:ξ
975:∂
971:∂
936:α
898:α
884:∂
881:⋯
870:α
856:∂
845:α
831:∂
816:α
807:∂
789:α
776:≤
768:α
759:∑
713:∈
677:α
663:∂
660:⋯
649:α
635:∂
624:α
610:∂
599:α
590:∂
579:α
550:α
510:α
485:α
459:α
452:⋯
440:α
427:α
415:α
373:α
366:⋯
354:α
341:α
331:α
303:α
284:α
271:≤
263:α
254:∑
9176:Category
8988:Algebras
8870:Theorems
8827:Complete
8796:Schwartz
8742:glossary
8584:Examples
8474:Integral
8246:Ordinary
8081:(1973),
8014:(1983),
7886:′
7878:′
7840:″
7826:′
7818:′
7786:′
7778:′
7764:″
7753:″
7739:′
7731:′
7714:″
7682:′
7671:′
7657:″
7610:′
7513:′
7238:See also
7162:Variants
6888:we have
6549:sections
6211:for all
5456:must be
5122:′
5068:′
5060:′
5049:″
5020:′
5012:′
5001:″
4972:′
4964:′
4383:interval
3703:are the
3430:′
3404:′
3369:argument
3041:gradient
2494:elliptic
2467:Examples
1883:. Here
400:integers
70:function
62:operator
8979:Unitary
8959:Nuclear
8944:Compact
8939:Bounded
8934:Adjoint
8908:Minâmax
8801:Sobolev
8786:Nuclear
8776:Hilbert
8771:Fréchet
8736: (
8312:Coupled
8251:Partial
8179:, 2001
8042:0717035
7343:, 1.1.7
7150:over a
7148:modules
6731:section
6528:. Let
5780:modules
5640:is the
5604:be the
5345:compose
5192:, i.e.
4919:Example
4234:adjoint
3071:History
2910:physics
2682:
1726:of the
1719:of the
1432:if, in
143:from a
44:of the
38:annulus
8954:Normal
8791:Orlicz
8781:Hölder
8761:Banach
8750:Spaces
8738:topics
8327:Degree
8271:Linear
8133:
8089:
8067:
8040:
8030:
8001:
7967:
6705:) â Î(
6696:where
6559:) â Î(
6252:where
6153:
6150:
6147:
6098:
6095:
6092:
6006:, and
5620:, and
5320:where
5190:linear
4761:, and
3367:of an
3051:, and
2906:vector
2870:
2780:
2629:, the
2594:kernel
1748:symbol
1136:where
323:where
308:
85:linear
60:is an
42:kernel
8766:Besov
8376:Delay
8322:Order
7965:S2CID
7953:(5).
6812:is a
6543:. An
5782:over
5626:ideal
5608:over
5535:with
4378:on a
3327:, or
2902:nabla
2667:of a
2588:is a
2447:as a
1750:) of
1734:with
1620:is a
394:is a
9114:(or
8832:Dual
8131:ISSN
8087:ISBN
8065:ISBN
8028:ISBN
6782:germ
6724:-jet
6719:its
6701:: Î(
6532:and
6516:and
5616:and
5469:ring
5335:Any
5324:and
4923:The
4546:and
4380:real
4232:the
3458:The
3184:and
3045:curl
2690:and
2652:and
2633:and
2603:and
1789:and
1627:The
1330:and
56:, a
8121:doi
8057:doi
8020:doi
7955:doi
6788:in
6784:of
6764:in
6736:of
6512:In
6272:is
5833:If
5565:If
5339:in
4774:(Ω)
4736:of
4576:by
4516:or
4504:of
4362:or
3889:. (
3796:In
3707:in
2898:del
2640:In
2625:In
2492:is
2215:Hom
2172:in
2133:of
1879:of
1785:of
1730:of
1572:α,
1436:on
1334:be
1319:of
1035:of
1027:in
177:on
80:).
76:in
52:In
9193::
8740:â
8175:,
8169:,
8129:.
8117:52
8115:.
8111:.
8063:.
8038:MR
8036:,
8026:,
7963:.
7951:68
7949:.
7945:.
7230:A
7158:.
6977:0.
6776:)(
6726:.
6713:))
5825:.
5746:.
5634:XD
5632:â
5630:DX
5546:.
5516:1.
5487::
5481:Dg
5477:gD
4915:.
4899:,
4740:.
4390:,
3893:)
3711::
3534:.
3295:,
3268:,
3154:,
3134:,
3091:.
3060:A
3047:,
2678:+
2674:=
2463:.
2259:.
1888:ΜΌ
1778:,
1754:.
954:.
739::
497:,
402:,
91:.
9118:)
8842:)
8838:/
8834:(
8744:)
8726:e
8719:t
8712:v
8207:e
8200:t
8193:v
8137:.
8123::
8096:.
8073:.
8059::
8045:.
8022::
7971:.
7957::
7914:u
7911:L
7908:=
7896:u
7893:q
7890:+
7883:)
7875:u
7871:p
7868:(
7862:=
7850:u
7847:q
7844:+
7837:u
7833:p
7823:u
7815:p
7808:=
7796:u
7793:q
7790:+
7783:u
7775:p
7771:+
7768:u
7761:p
7757:+
7750:u
7746:p
7736:u
7728:p
7724:2
7718:u
7711:p
7704:=
7692:u
7689:q
7686:+
7679:)
7675:u
7668:p
7664:(
7661:+
7654:)
7650:u
7647:p
7644:(
7638:=
7626:u
7623:q
7620:+
7617:)
7614:u
7607:p
7603:(
7600:D
7597:+
7594:)
7591:u
7588:p
7585:(
7580:2
7576:D
7569:=
7557:)
7554:u
7551:q
7548:(
7543:0
7539:)
7535:1
7529:(
7526:+
7523:]
7520:u
7517:)
7510:p
7503:(
7500:[
7497:D
7492:1
7488:)
7484:1
7478:(
7475:+
7472:]
7469:u
7466:)
7463:p
7457:(
7454:[
7449:2
7445:D
7439:2
7435:)
7431:1
7425:(
7422:=
7413:u
7404:L
7380:.
7360:.
7206:)
7203:f
7200:,
7197:g
7194:(
7191:D
7131:.
7128:)
7125:s
7122:(
7119:P
7113:f
7107:)
7104:s
7098:f
7095:(
7092:P
7089:=
7086:)
7083:s
7080:(
7077:]
7074:P
7071:,
7068:f
7065:[
7042:)
7039:F
7036:(
7027:)
7024:E
7021:(
7015::
7012:]
7009:P
7006:,
7003:f
7000:[
6974:=
6971:]
6968:]
6962:]
6959:P
6956:,
6951:0
6947:f
6943:[
6937:[
6934:,
6929:1
6923:k
6919:f
6915:[
6912:,
6907:k
6903:f
6899:[
6876:)
6873:M
6870:(
6861:C
6852:k
6848:f
6844:,
6838:,
6833:0
6829:f
6818:k
6814:k
6810:P
6806:R
6790:x
6786:s
6778:x
6774:s
6772:(
6770:P
6766:x
6762:s
6758:k
6754:M
6750:x
6746:s
6744:(
6742:P
6738:E
6734:s
6722:k
6717:E
6711:E
6709:(
6707:J
6703:E
6699:j
6679:k
6675:j
6666:P
6662:i
6658:=
6655:P
6629:F
6623:)
6620:E
6617:(
6612:k
6608:J
6604::
6599:P
6595:i
6581:E
6579:(
6577:J
6568:k
6563:)
6561:F
6557:E
6553:P
6545:R
6541:M
6534:F
6530:E
6503:.
6487:n
6483:b
6477:n
6473:D
6462:1
6458:b
6452:1
6448:D
6440:n
6436:a
6430:n
6426:X
6415:1
6411:a
6405:1
6401:X
6389:R
6376:.
6364:I
6360:/
6351:n
6347:X
6343:,
6337:,
6332:1
6328:X
6324:,
6319:n
6315:D
6311:,
6305:,
6300:1
6296:D
6289:R
6278:R
6240:,
6237:n
6231:j
6228:,
6225:i
6219:1
6194:i
6190:X
6184:j
6180:X
6171:j
6167:X
6161:i
6157:X
6144:,
6139:i
6135:D
6129:j
6125:D
6116:j
6112:D
6106:i
6102:D
6089:,
6084:j
6081:,
6078:i
6067:)
6062:i
6058:D
6052:j
6048:X
6039:j
6035:X
6029:i
6025:D
6021:(
6008:I
5992:n
5988:X
5984:,
5978:,
5973:1
5969:X
5965:,
5960:n
5956:D
5952:,
5946:,
5941:1
5937:D
5926:R
5907:n
5903:X
5899:,
5893:,
5888:1
5884:X
5880:,
5875:n
5871:D
5867:,
5861:,
5856:1
5852:D
5845:R
5835:R
5813:I
5809:/
5802:X
5799:,
5796:D
5790:R
5766:]
5763:X
5760:[
5757:R
5730:I
5720:b
5716:D
5710:a
5706:X
5695:R
5674:I
5670:/
5663:X
5660:,
5657:D
5651:R
5638:R
5622:I
5618:X
5614:D
5610:R
5589:X
5586:,
5583:D
5577:R
5567:R
5533:D
5513:=
5510:D
5507:x
5501:x
5498:D
5465:1
5462:D
5454:2
5451:D
5434:.
5431:)
5428:)
5425:f
5422:(
5417:2
5413:D
5409:(
5404:1
5400:D
5396:=
5393:)
5390:f
5387:(
5384:)
5379:2
5375:D
5366:1
5362:D
5358:(
5341:D
5330:a
5326:g
5322:f
5305:,
5302:)
5299:f
5296:D
5293:(
5290:a
5287:=
5284:)
5281:f
5278:a
5275:(
5272:D
5251:,
5248:)
5245:g
5242:D
5239:(
5236:+
5233:)
5230:f
5227:D
5224:(
5221:=
5218:)
5215:g
5212:+
5209:f
5206:(
5203:D
5150:.
5147:u
5144:)
5141:q
5138:(
5135:+
5132:u
5129:D
5126:)
5119:p
5112:(
5109:+
5106:u
5101:2
5097:D
5093:)
5090:p
5084:(
5081:=
5078:u
5075:q
5072:+
5065:u
5057:p
5046:u
5042:p
5036:=
5033:u
5030:q
5027:+
5024:)
5017:u
5009:p
5005:+
4998:u
4994:p
4991:(
4985:=
4982:u
4979:q
4976:+
4969:)
4961:u
4957:p
4954:(
4948:=
4945:u
4942:L
4929:L
4909:L
4905:L
4901:g
4897:f
4893:L
4874:)
4868:(
4863:2
4859:L
4850:g
4847:,
4844:f
4841:P
4835:=
4830:)
4824:(
4819:2
4815:L
4806:g
4797:P
4793:,
4790:f
4772:L
4767:P
4763:P
4759:R
4738:T
4714:T
4691:.
4687:]
4683:u
4674:)
4671:x
4668:(
4663:k
4659:a
4651:[
4645:k
4641:D
4635:k
4631:)
4627:1
4621:(
4616:n
4611:0
4608:=
4605:k
4597:=
4594:u
4585:T
4574:T
4560:b
4554:x
4534:a
4528:x
4518:g
4514:f
4510:x
4508:(
4506:f
4498:x
4496:(
4494:f
4478:,
4475:x
4472:d
4468:)
4465:x
4462:(
4459:g
4449:)
4446:x
4443:(
4440:f
4432:b
4427:a
4419:=
4413:g
4410:,
4407:f
4394:)
4392:b
4388:a
4386:(
4340:,
4311:v
4302:T
4298:,
4295:u
4289:=
4283:v
4280:,
4277:u
4274:T
4245:T
4220:u
4215:k
4211:D
4207:)
4204:x
4201:(
4196:k
4192:a
4186:n
4181:0
4178:=
4175:k
4167:=
4164:u
4161:T
4142:T
4101:.
4098:f
4093:x
4082:g
4076:g
4071:x
4060:f
4057:=
4054:g
4044:x
4034:f
4013:g
4008:x
3997:f
3994:=
3991:g
3981:x
3971:f
3950:f
3945:x
3934:g
3931:=
3928:g
3918:x
3908:f
3867:.
3859:k
3855:x
3840:k
3836:x
3830:n
3825:1
3822:=
3819:k
3811:=
3798:n
3779:,
3776:2
3773:,
3770:1
3767:,
3764:0
3761:=
3758:k
3754:,
3749:k
3745:z
3741:k
3738:=
3735:)
3730:k
3726:z
3722:(
3709:z
3680:.
3674:z
3671:d
3667:d
3662:z
3659:=
3626:.
3618:2
3613:k
3609:x
3599:2
3587:n
3582:1
3579:=
3576:k
3568:=
3563:2
3555:=
3513:k
3509:D
3503:k
3499:c
3493:n
3488:0
3485:=
3482:k
3460:D
3443:.
3440:)
3437:x
3434:(
3427:f
3401:]
3397:)
3394:x
3391:(
3388:f
3385:[
3372:x
3365:f
3359:.
3345:n
3340:x
3313:n
3308:x
3304:D
3281:n
3277:D
3251:n
3247:x
3243:d
3237:n
3233:d
3217:n
3211:.
3197:x
3172:,
3167:x
3163:D
3142:D
3119:x
3116:d
3112:d
3097:x
3025:.
3019:z
3001:z
2995:+
2989:y
2971:y
2965:+
2959:x
2941:x
2935:=
2893:.
2873:.
2866:)
2859:y
2847:i
2844:+
2838:x
2825:(
2819:2
2816:1
2811:=
2799:z
2783:,
2776:)
2769:y
2757:i
2748:x
2735:(
2729:2
2726:1
2721:=
2715:z
2692:y
2688:x
2684:y
2680:i
2676:x
2672:z
2660:.
2622:.
2586:P
2568:)
2562:,
2556:,
2550:(
2545:P
2520:X
2511:T
2480:P
2457:x
2455:(
2453:p
2445:P
2428:.
2422:d
2418:)
2412:(
2403:f
2397:)
2391:i
2388:,
2385:x
2382:(
2379:p
2368:x
2365:i
2361:e
2353:d
2348:R
2332:2
2329:d
2324:)
2317:2
2314:(
2310:1
2305:=
2302:)
2299:x
2296:(
2293:f
2290:P
2269:P
2247:)
2242:x
2238:F
2234:,
2229:x
2225:E
2221:(
2195:X
2185:x
2181:T
2170:k
2150:P
2135:X
2131:x
2114:.
2105:u
2091:)
2088:x
2085:(
2068:P
2052:k
2049:=
2045:|
2037:|
2028:=
2019:)
2015:u
2012:)
2006:(
2001:P
1993:(
1967:.
1955:x
1928:P
1914:=
1902:P
1885:P
1881:E
1877:u
1854:u
1841:P
1827:=
1818:)
1814:u
1811:P
1808:(
1795:P
1791:F
1787:E
1783:Μ
1780:f
1776:Ό
1773:e
1768:i
1763:x
1759:x
1752:P
1740:F
1736:E
1732:X
1721:k
1700:F
1694:E
1688:)
1685:X
1676:T
1672:(
1667:k
1663:S
1659::
1654:P
1633:P
1629:k
1608:F
1602:E
1599::
1596:)
1593:x
1590:(
1581:P
1548:+
1536:x
1527:u
1511:)
1508:x
1505:(
1496:P
1490:k
1487:=
1483:|
1475:|
1466:=
1463:)
1460:x
1457:(
1454:u
1451:P
1438:X
1420:k
1397:)
1394:F
1391:(
1382:C
1375:)
1372:E
1369:(
1360:C
1356::
1353:P
1340:X
1332:F
1328:E
1321:P
1290:)
1287:x
1284:(
1275:a
1269:m
1266:=
1262:|
1254:|
1245:=
1242:)
1236:,
1233:x
1230:(
1204:.
1197:n
1187:n
1172:1
1162:1
1154:=
1114:)
1111:x
1108:(
1099:a
1093:m
1086:|
1078:|
1069:=
1066:)
1060:,
1057:x
1054:(
1051:p
1041:P
1037:P
1029:P
1013:i
983:i
979:x
959:p
932:D
902:n
892:n
888:x
874:2
864:2
860:x
849:1
839:1
835:x
826:f
820:|
812:|
800:)
797:x
794:(
785:a
779:m
772:|
764:|
755:=
752:f
749:P
725:1
719:F
710:f
681:n
671:n
667:x
653:2
643:2
639:x
628:1
618:1
614:x
603:|
595:|
584:=
575:D
546:D
535:n
521:)
518:x
515:(
506:a
463:n
455:+
449:+
444:2
436:+
431:1
423:=
419:|
411:|
382:)
377:n
369:,
363:,
358:2
350:,
345:1
337:(
334:=
311:,
299:D
295:)
292:x
289:(
280:a
274:m
267:|
259:|
250:=
247:P
223:2
217:F
192:n
187:R
163:1
157:F
131:P
111:m
101:m
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
