5183:
15525:
12957:
4873:
3903:
4483:
17896:
2024:
15157:
12649:
5178:{\displaystyle {\begin{aligned}{\frac {(\psi \circ f)_{\overline {z}}}{(\psi \circ f)_{z}}}&={\frac {(\psi _{z}\circ f)f_{\overline {z}}+(\psi _{\overline {z}}\circ f){\overline {f_{\overline {z}}}}}{(\psi _{z}\circ f)f_{z}+(\psi _{\overline {z}}\circ f){\overline {f_{z}}}}}={\frac {f_{\overline {z}}}{f_{z}}}=\mu (z).\end{aligned}}}
17321:≤ |z| ≤ 1 onto the closure of Ω, such that after a conformal change the induced metric on the annulus can be continued smoothly by reflection in both boundaries. The annulus is a fundamental domain for the group generated by the two reflections, which reverse orientation. The images of the fundamental domain under the group fill out
6850:
14852:
2472:
3497:
4119:
17690:
16360:
If the support is not compact the same trick used in the smooth case can be used to construct a solution in terms of two homeomorphisms associated to compactly supported
Beltrami coefficients. Note that, because of the assumptions on the Beltrami coefficient, a Möbius transformation of the extended
3259:
16369:
There are several proofs of the uniqueness of solutions of the
Beltrami equation with a given Beltrami coefficient. Since applying a Möbius transformation of the complex plane to any solution gives another solution, solutions can be normalised so that they fix 0, 1 and ∞. The method of solution of
17306:
The method used to prove the smooth
Riemann mapping theorem can be generalized to multiply connected planar regions with smooth boundary. The Beltrami coefficient in these cases is smooth on an open set, the complement of which has measure zero. The theory of the Beltrami equation with measurable
1712:
9314:
14066:
will be considered—adequate for most applications—namely those functions which are smooth an open set Ω (the regular set) with complement Λ a closed set of measure zero (the singular set). Thus Λ is a closed set that is contained in open sets of arbitrarily small area. For measurable
Beltrami
10313:
Conversely
Beltrami coefficients defined on the closures of the upper halfplane or unit disk which satisfy these conditions on the boundary can be "reflected" using the formulas above. If the extended functions are smooth the preceding theory can be applied. Otherwise the extensions will be
1508:
11114:
13714:
16356:
of homeomorphic solutions of
Beltrami's equation in the case of Beltrami coefficients of compact support. It also shows that the inverse homeomorphisms and composed homeomorphisms satisfy Beltrami equations and that all computations can be performed by restricting to regular sets.
11266:
18484:
15520:{\displaystyle {\begin{aligned}A(f_{n}(U))&=\iint _{U}J(f_{n})\,dx\,dy=\iint _{U}|\partial _{z}f_{n}|^{2}-|\partial _{\overline {z}}f_{n}|^{2}\,dx\,dy\\&\leq \iint _{U}|\partial _{z}f_{n}|^{2}\,dx\,dy\leq \|\partial _{z}f_{n}|_{U}\|_{p}^{2}\,A(U)^{1-2/p}.\end{aligned}}}
1812:
17333:, by the L theory is a homeomorphism. It is smooth on away from 0 by elliptic regularity. By uniqueness it preserves the unit circle, together with its interior and exterior. Uniqueness of the solution also implies that reflection there is a conjugate Möbius transformation
12952:{\displaystyle {\begin{aligned}Pf(w)&=Cf(w)+\pi ^{-1}\iint _{\mathbf {C} }{f(z) \over z}\,dx\,dy\\&=-{1 \over \pi }\iint _{\mathbf {C} }f(z)\left({1 \over z-w}-{1 \over z}\right)\,dx\,dy=-{1 \over \pi }\iint _{\mathbf {C} }{f(z)w \over z(z-w)}\,dx\,dy.\end{aligned}}}
17270:
16293:
16125:
has weak distributional derivatives in 1 + L and L. Pairing with smooth functions of compact support in Ω, these derivatives coincide with the actual derivatives at points of Ω. Since Λ has measure zero, the distributional derivatives equal the actual derivatives in
19186:
10874:
6571:
14683:
10474:
17430:
open disks removed, onto the closure of Ω. It can be assumed that 0 lies in the interior of the domain. Again after a modification of the diffeomorphism and conformal change near the boundary, the metric can be assumed to be compatible with reflection. Let
11652:
6479:
4766:
10208:
18010:
12304:
18977:
9630:
7743:
14720:
1096:
16093:
15980:
14533:
6319:
8919:
958:
3898:{\displaystyle \displaystyle \mu (z)={\frac {(u_{x}^{2}+v_{x}^{2})-(u_{y}+v_{y})^{2}+2i(u_{x}u_{y}+v_{x}v_{y})}{(u_{x}^{2}+v_{x}^{2})+(u_{y}^{2}+v_{y})^{2}+2{\sqrt {(u_{x}^{2}+v_{x}^{2})(u_{y}^{2}+v_{y}^{2})-(u_{x}u_{y}+v_{x}v_{y})^{2}}}}},}
18745:
2274:
8531:
4478:{\displaystyle \displaystyle \mu (z)={\frac {{\bigl (}(u_{x}+iu_{y})+i(v_{x}+iv_{y}){\bigr )}{\bigl (}(u_{x}+iu_{y})-i(v_{x}+iv_{y}){\bigr )}}{{\bigl (}(u_{x}+v_{y})+i(v_{x}-u_{y}){\bigr )}{\bigl (}(u_{x}+v_{y})-i(v_{x}-u_{y}){\bigr )}}}.}
19838:
9507:
115:
13342:
6990:
6076:
2904:
17891:{\displaystyle \displaystyle {\widehat {\mu }}(z)=\mu (z)\,\,(z\in \mathbf {H} ),\,\,{\widehat {\mu }}(z)=0\,\,(z\in \mathbf {R} ),\,\,{\widehat {\mu }}(z)={\overline {\mu ({\overline {z}})}}\,\,({\overline {z}}\in \mathbf {H} ).}
15087:
16458:
17122:
9739:
5603:
2756:
11389:
3091:
13823:
640:
10989:
2259:
10305:
1519:
9165:
788:
14253:
7992:
4047:
12456:
10348:
extending smoothly to the boundary and the identity on a neighbourhood of 0. Suppose that in addition the induced metric on the closure of the unit disk can be reflected in the unit circle to define a smooth metric on
10010:
15740:
4605:
proved the existence of isothermal coordinates locally in the analytic case by reducing the
Beltrami to an ordinary differential equation in the complex domain. Here is a cookbook presentation of Gauss's technique.
13065:
1265:
7250:
11000:
9010:
16996:
14164:
13489:
10085:
5881:
2140:
543:
13442:
11481:
11152:
16876:
6871:
In the simplest cases the
Beltrami equation can be solved using only Hilbert space techniques and the Fourier transform. The method of proof is the prototype for the general solution using L spaces, although
2019:{\displaystyle \displaystyle \partial _{z}={1 \over 2}(\partial _{x}-i\partial _{y}),\,\,\partial _{\overline {z}}={1 \over 2}(\partial _{x}+i\partial _{y});\,\,\,\,dz=dx+i\,dy,\,\,d{\overline {z}}=dx-i\,dy.}
19688:
18256:
16348:. Pairing with smooth functions of compact support in Ω ∩ Ω*, these agree with the usual derivatives. So the distributional derivatives are given by the usual derivatives off Λ ∪ Λ*, a set of measure zero.
9905:
10549:
16599:
8380:
18583:
12064:
16707:
15162:
12654:
9119:
8715:
8605:
8211:
7821:
7508:
17131:
satisfying
Beltrami's equation and with distributional derivatives locally in L, it can be assumed after applying a Möbius transformation that 0 is not in the singular set of the Beltrami coefficient
14016:
8057:
7063:
1226:
20013:
16548:
12522:
10611:
7370:
17161:
16170:
16775:
7174:
9395:
19611:
4878:
8145:
226:
13150:
12611:
6845:{\displaystyle \displaystyle {h_{\overline {z}} \over h_{z}}={\frac {(a^{2}+b^{2})E\;(E-G+2iF)}{(a^{2}+b^{2})E\;(E+G+2{\sqrt {EG-F^{2}}})}}={\frac {E-G+2iF}{E+G+2{\sqrt {EG-F^{2}}}}}=\mu (z).}
20055:, pp. 314–317, which is pp. 455–460 in the first or second edition, but note that there is a typo in equation (**) on page 315 or 457. The right-hand side, given as −β/α, should be −α/β.
4551:
3478:
18988:
10729:
8254:
The elliptic regularity argument to prove smoothness, however, is the same everywhere and uses the theory of L Sobolev spaces on the torus. Let ψ be a smooth function of compact support on
14550:
7878:
16933:
10363:
7310:
7114:
11524:
7434:
3363:
12126:
7570:
6325:
4635:
10107:
6529:
2576:
2526:
20857:
Papadopoulos, Athanase, ed. (2012), Handbook of Teichmüller theory. Vol. III, IRMA Lectures in
Mathematics and Theoretical Physics, 19, European Mathematical Society (EMS), Zürich,
6162:
5376:
20832:
Papadopoulos, Athanase, ed. (2009), Handbook of Teichmüller theory. Vol. II, IRMA Lectures in Mathematics and Theoretical Physics, 13, European Mathematical Society (EMS), Zürich,
17907:
12163:
4865:
4111:
20807:
Papadopoulos, Athanase, ed. (2007), Handbook of Teichmüller theory. Vol. I, IRMA Lectures in Mathematics and Theoretical Physics, 11, European Mathematical Society (EMS), Zürich,
18779:
17282:
results in a solution of Beltrami's equation with a Beltrami coefficient of compact support. The directional derivatives are still locally in L. The coefficient ν depends only on
3413:
188:
14847:{\displaystyle \displaystyle {\iint u\cdot \psi =\lim \iint u_{n}\cdot \psi =-\lim \iint f_{n}\cdot \partial _{\overline {z}}\psi =-\iint f\cdot \partial _{\overline {z}}\psi ,}}
9534:
6563:
7607:
11896:
5938:
434:
396:
17498:, or equivalently the reflected domain by the Schottky group, fill out the regular set for the Schottky group. It acts properly discontinuously there. The complement is the
969:
5455:
5258:
5212:
308:
for the equation can be used to show that the uniformizing map from the unit disk to the domain extends to a C function from the closed disk to the closure of the domain.
15997:
15884:
14435:
6170:
2467:{\displaystyle \displaystyle ds^{2}=|f_{z}\,dz+f_{\overline {z}}d{\overline {z}}|^{2}=|f_{z}|^{2}\,\left|dz+{f_{\overline {z}} \over f_{z}}\,d{\overline {z}}\right|^{2}.}
8761:
799:
16361:
complex plane can be applied to make the singular set of the Beltrami coefficient compact. In that case one of the homeomorphisms can be chosen to be a diffeomorphism.
18594:
5417:
4593:
8423:
2625:
3029:
2999:
2970:
2941:
19746:
9406:
5475:
5278:
4833:
4809:
3083:
3056:
47:
5711:
13185:
6906:
5962:
436:
is nearly square, not just nearly rectangular. The Beltrami equation is the equation that has to be solved in order to construct isothermal coordinate systems.
5762:
5232:
4789:
2596:
2764:
14976:
3254:{\displaystyle \displaystyle u_{y}={\frac {Fu_{x}-{\sqrt {EG-F^{2}}}\,v_{x}}{E}}\quad {\text{and}}\quad v_{y}={\frac {{\sqrt {EG-F^{2}}}\,u_{x}+Fv_{x}}{E}}.}
16380:
9933:
has extra symmetry properties; since the two regions are related by a Möbius transformation (the Cayley transform), the two cases are essentially the same.
17043:
9661:
2637:
244:. Various techniques have been developed for solving the equation. The most powerful, developed in the 1950s, provides global solutions of the equation on
11281:
19721:
give a survey of various methods of extension, including variants of the Ahlfors-Beurling extension which are smooth or analytic in the open unit disk.
17585:
defines a smooth diffeomorphism of the unit disc with the interior of these circles removed onto the closure of Ω, which is holomorphic in the interior.
13737:
10683:
equal to the identity on a neighbourhood of 0 and with an explicit form on a tubular neighbourhood of the unit circle. In fact taking polar coordinates (
1707:{\displaystyle \displaystyle {\begin{pmatrix}u_{x}^{2}+v_{x}^{2}&u_{x}u_{y}+v_{x}v_{y}\\u_{x}u_{y}+v_{x}v_{y}&u_{y}^{2}+v_{y}^{2}\end{pmatrix}}.}
569:
10893:
9309:{\displaystyle \displaystyle {\mu _{g\circ f^{-1}}\circ f={f_{z} \over {\overline {f_{z}}}}{\mu _{g}-\mu _{f} \over 1-{\overline {\mu _{f}}}\mu _{g}}.}}
2146:
11957:
strictly less than one. Their approach involved the theory of quasiconformal mappings to establish directly the solutions of Beltrami's equation when
10223:
5498:
686:
17513:
extends by reflection to the regular set. The corresponding Beltrami coefficient is invariant for the reflection group generated by the reflections
14192:
7889:
9159:
is a quasiconformal conformal diffeomorphism. These form a group and their Beltrami coefficients can be computed according to the following rule:
3923:
18184:). It leaves invariant the Jordan curve and acts properly discontinuously on each of the two components. The quotients of the two components by Γ
12355:
11977:
17661:
implies that the homeomorphisms can be adjusted near the edges and the vertices of the triangulation to produce diffeomorphisms. The metric on
9950:
1503:{\displaystyle \displaystyle ds^{2}=du^{2}+dv^{2}=(u_{x}^{2}+v_{x}^{2})\,dx^{2}+2(u_{x}u_{y}+v_{x}v_{y})\,dx\,dy+(u_{y}^{2}+v_{y}^{2})\,dy^{2},}
15681:
11109:{\displaystyle \displaystyle \kappa (\theta )=y^{\prime \prime }(\theta )x^{\prime }(\theta )-x^{\prime \prime }(\theta )y^{\prime }(\theta )}
20697:
13709:{\displaystyle \displaystyle |f(w_{1})-f(w_{2})|\leq {K_{p} \over 1-\|\mu \|_{\infty }C_{p}}\|\mu \|_{p}|w_{1}-w_{2}|^{1-2/p}+|w_{1}-w_{2}|.}
12984:
136:
7195:
4488:
The right-hand factors in the numerator and denominator are equal and, since the Jacobian is positive, their common value can't be zero; so
11666:
are smooth functions on the circle. These coefficients can be defined by recurrence so that the transformed metric only has even powers of
11261:{\displaystyle \displaystyle \alpha ={1 \over 2\pi }\int _{0}^{2\pi }\kappa (\theta )\,d\theta ,\,\,\,h(\theta )=\kappa (\theta )-\alpha .}
8947:
16944:
14104:
10025:
5802:
2035:
18479:{\displaystyle \displaystyle {{{|f(z_{1})-f(z_{2})| \over |f(z_{1})-f(z_{3})|}\leq a\cdot {|z_{1}-z_{2}|^{b} \over |z_{1}-z_{3}|^{b}}}.}}
453:
17458:. Its fundamental domain consists of the original fundamental domain with its reflection in the unit circle added. If the reflection is
13353:
11404:
16790:
9914:
is the quasiconformal diffeomorphism of the sphere fixing 0 and ∞ with coefficient λ, then the transformation formulas above show that
19622:
9765:
5431:
axis of the original coordinates and is parameterized in the same way. All other isothermal coordinate systems are then of the form
11929:
variable. It follows that, after composing with this diffeomorphism, the extension of the metric obtained by reflecting in the line
10493:
16559:
8323:
18495:
12014:
21067:
16646:
9033:
8656:
8549:
8170:
7762:
10314:
continuous but with a jump in the derivatives at the boundary. In that case the more general theory for measurable coefficients
7460:
17265:{\displaystyle \displaystyle \nu (g(z))={g_{z} \over {\overline {g_{z}}}}\,{\mu -\lambda \over 1-{\overline {\lambda }}\mu }.}
16288:{\displaystyle \displaystyle \nu (f(z))={f_{z} \over {\overline {f_{z}}}}\,{\mu ^{*}-\mu \over 1-{\overline {\mu }}\mu ^{*}},}
13859:
8007:
7009:
1163:
19957:
19470:
with the additional properties that it commutes with the action of SU(1,1) by Möbius transformations and is quasiconformal if
16496:
12471:
20956:
20870:
20845:
20820:
20630:
20574:
20428:
10560:
7321:
6876:
has indicated a method for handling the general case using only Hilbert spaces: the method relies on the classical theory of
16718:
16370:
the Beltrami equation using the Beurling transform also provides a proof of uniqueness for coefficients of compact support
10554:
preserves the unit circle together with its interior and exterior. From the composition formulas for Beltrami coefficients
7125:
21019:
17573:
is the reflection in another circle in the unit disk. Looking at fixed points, the circles arising this way for different
9329:
19496:
17594:
17526:≥ 0. Since the limit set has measure zero, the Beltrami coefficient extends uniquely to a bounded measurable function on
17294:, so any two solutions of the original equation will produce solutions near 0 with distributional derivatives locally in
293:
16340:. The derivatives are uniformly bounded in 1 + L and L, so as before weak limits give the distributional derivatives of
9929:
The solutions of Beltrami's equation restrict to diffeomorphisms of the upper halfplane or unit disk if the coefficient
8080:
193:
20029:
19181:{\displaystyle \displaystyle {(z_{1},z_{2};z_{3},z_{4})={(z_{1}-z_{3})(z_{2}-z_{4}) \over (z_{2}-z_{3})(z_{1}-z_{4})}}}
13084:
12545:
10869:{\displaystyle \displaystyle G(r,\theta )=(x(\theta )-(1-r)y^{\prime }(\theta ),y(\theta )+(1-r)x^{\prime }(\theta )).}
8217:
289:
14678:{\displaystyle \displaystyle {\|u_{n}\|_{p},\,\,\|v_{n}\|_{p}\leq {\|\mu _{n}\|_{p} \over 1-\|\mu _{n}\|_{\infty }}.}}
4491:
3418:
20929:
20686:
20659:
20600:
20486:
17298:
and the same Beltrami coefficient. They are therefore equal. Hence the solutions of the original equation are equal.
11120:
10469:{\displaystyle \displaystyle \mu (z)={z^{2} \over {\overline {z}}^{2}}{\overline {\mu ({\overline {z}}^{-1})}}^{-1}.}
11647:{\displaystyle \displaystyle f(\theta ,t)=\theta +f_{1}(\theta )t+f_{2}(\theta )t^{2}+\cdots =\theta +g(\theta ,t)}
7832:
16891:
16608:
is weakly holomorphic. Applying Weyl's lemma it is possible to conclude that there exists a holomorphic function
15132:
carries closed sets of measure zero onto closed sets of measure zero. It suffices to check this for a compact set
14079:, the solution of the Beltrami equation can be obtained as a limit of solutions for smooth Beltrami coefficients.
10336:
be an open simply connected domain in the complex plane with smooth boundary containing 0 in its interior and let
7273:
18156:
which is holomorphic on the lower half plane and smooth on the upper half plane. The image of the real axis is a
7081:
6474:{\displaystyle \displaystyle h_{\overline {z}}=((aE-bF-a{\sqrt {EG-F^{2}}}\,)+i(bE+aF-b{\sqrt {EG-F^{2}}}\,))/2.}
4761:{\displaystyle \displaystyle {f_{\overline {z}} \over f_{z}}=\mu (z)={\frac {E-G+2iF}{E+G+2{\sqrt {EG-F^{2}}}}}.}
1154:
10203:{\displaystyle \displaystyle \mu (z)={z^{2} \over {\overline {z}}^{2}}{\overline {\mu ({\overline {z}}^{-1})}},}
7389:
3295:
20876:
Partyka, Dariusz; Sakan, Ken-Ichi; Zając, Józef (1999), "The harmonic and quasiconformal extension operators",
20439:"Saggio di interpretazione della geometria non euclidea (Essay on the interpretation of noneuclidean geometry)"
12079:
8624:) with the same property. It is the counterpart on the torus of the Beurling transform. The standard theory of
7523:
18005:{\displaystyle \displaystyle {\widehat {\mu }}(g(z))={{\overline {g_{z}}} \over g_{z}}\,{\widehat {\mu }}(z),}
12299:{\displaystyle \displaystyle {h=g_{z}=T(g_{\overline {z}})=T(f_{\overline {z}})=T(\mu f_{z})=T(\mu h)+T\mu .}}
5234:
to give one isothermal coordinate system, then all of the possible isothermal coordinate systems are given by
20948:
18972:{\displaystyle \displaystyle {|(f(z_{1}),f(z_{2});f(z_{3}),f(z_{4}))|\leq c|(z_{1},z_{2};z_{3},z_{4})|^{d},}}
17494:
are the reflections in the interior circles in the original domain. The images of the original domain by the
6487:
2534:
2484:
39:
20:
20516:
Douady, Adrien; Earle, Clifford J. (1986), "Conformally natural extension of homeomorphisms of the circle",
9625:{\displaystyle \displaystyle {{\widetilde {\mu }}(w)=(w^{\prime }/{\overline {w^{\prime }}})\cdot \mu (z).}}
6084:
5298:
20995:
Tukia, Pekka (1985), "Quasiconformal extension of quasisymmetric mappings compatible with a Möbius group",
7738:{\displaystyle \displaystyle h=g_{z}=T(g_{\overline {z}})=T(f_{\overline {z}})=T(\mu f_{z})=T(\mu h)+T\mu }
4838:
4056:
14317:
9140:
must be onto by connectedness of the sphere, since its image is an open and closed subset; but then, as a
3372:
162:
9148:
must cover each point of the sphere the same number of times. Since only ∞ is sent to ∞, it follows that
6534:
20695:
Morrey, Charles B. (1936), "On the solutions of quasi-linear elliptic partial differential equations.",
17657:. The homeomorphisms can be chosen to be piecewise linear on corresponding triangulations. A result of
19467:
18229:
12632:
11680:
257:
1091:{\displaystyle \displaystyle |\mu |^{2}={E+G-2{\sqrt {EG-F^{2}}} \over E+G+2{\sqrt {EG-F^{2}}}}<1.}
304:
for simply connected bounded open domains in the complex plane. When the domain has smooth boundary,
21072:
18027:, preserving the real axis and the upper and lower half planes. Conjugation of the group elements by
16463:
for smooth functions ψ of compact support are also valid in the distributional sense for L functions
401:
363:
16088:{\displaystyle \displaystyle {\mu ^{\prime }(w)=-{{\overline {g_{w}}} \over g_{w}}\cdot \mu (g(w))}}
15975:{\displaystyle \displaystyle {\mu ^{\prime }(f(z))=-{f_{z} \over {\overline {f_{z}}}}\cdot \mu (z)}}
14528:{\displaystyle \displaystyle {u_{n}=\partial _{\overline {z}}f_{n},\,\,v_{n}=\partial _{z}f_{n}-1.}}
14094:≤ 1, equal to 1 on a neighborhood of Λ and 0 off a slightly larger neighbourhood, shrinking to Λ as
6314:{\displaystyle \displaystyle h_{z}=((aE+bF+a{\sqrt {EG-F^{2}}}\,)+i(bE-aF+b{\sqrt {EG-F^{2}}}\,))/2}
5948:) is then tangent to the solution curve of the differential equation that passes through the point (
21077:
19938:| > 1 onto the inside and outside of a Jordan curve. They extend continuously to homeomorphisms
17654:
11981:
8914:{\displaystyle \displaystyle {J(f)=|f_{z}|^{2}-|f_{\overline {z}}|^{2}=|f_{z}|^{2}(1-|\mu |^{2}).}}
953:{\displaystyle \displaystyle (E-G+2iF)(E-G-2iF)=(E+G+2{\sqrt {EG-F^{2}}})(E+G-2{\sqrt {EG-F^{2}}})}
20672:, Die Grundlehren der mathematischen Wissenschaften, vol. 126 (2nd ed.), Springer-Verlag
5434:
5237:
5191:
18740:{\displaystyle \displaystyle {a^{-1}\leq {|f(z_{1})-f(z_{2})| \over |f(z_{1})-f(z_{3})|}\leq a.}}
18100:
17155:
can be calculated directly using the change of variables formula for distributional derivatives:
10659:
with the properties above, it can be assumed after an affine transformation that the boundary of
10645:
10327:
5889:
301:
19264:
is a quasi-Möbius homeomorphism then it is also quasisymmetric. Indeed, it is immediate that if
8526:{\displaystyle \displaystyle {Ue_{m}={{\overline {m}} \over m}e_{m}\,\,(m\neq 0),Ue_{0}=e_{0}.}}
21082:
20034:
20024:
6877:
1755:
557:
345:
285:
281:
237:
19833:{\displaystyle \displaystyle {\|\mu \|_{\infty }<1,\,\,\,\mu (z)=F_{\overline {z}}/F_{z}.}}
9502:{\displaystyle \displaystyle {\mu _{g}(z)={z^{2} \over {\overline {z}}^{2}}\mu _{f}(z^{-1}).}}
9019:
can be solved by the same methods as above giving a solution tending to 0 at ∞. By uniqueness
5381:
4566:
110:{\displaystyle {\partial w \over \partial {\overline {z}}}=\mu {\partial w \over \partial z}.}
20898:
Sibner, Robert J. (1965), "Uniformization of symmetric Riemann surfaces by Schottky groups",
20735:
20545:, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, pp. 307–324
15116:, so to check weak convergence to 0 it enough to check inner products with a dense subset of
14691:
Passing to a subsequence if necessary, it can be assumed that the sequences have weak limits
2601:
1803:
356:
orthogonally and, in addition, the parameter spacing is the same — that is, for small enough
229:
17365:. Composing with a Möbius transformation that fixes the unit circle it can be assumed that
13483:
yield the following uniform estimates for the normalized solution of the Beltrami equation:
13337:{\displaystyle \displaystyle |Pf(w_{1})-Pf(w_{2})|\leq K_{p}\|f\|_{p}|w_{1}-w_{2}|^{1-2/p},}
6985:{\displaystyle \displaystyle {\widehat {Tf}}(z)={{\overline {z}} \over z}{\widehat {f}}(z).}
6071:{\displaystyle \displaystyle dh=c(x,y){\bigl (}E\,dx+(F+i{\sqrt {EG-F^{2}}}\,)\,dy{\bigr )}}
3004:
2974:
2945:
2916:
300:
can also be derived using the Beltrami equation. One of the simplest applications is to the
277:
20966:
20852:
20827:
20640:
20584:
19947:
of the unit circle onto the Jordan curve on the boundary. By construction they satisfy the
18173:
7072:
6565:, which is the complex conjugate of the denominator. Simplifying the result, we find that
5460:
5263:
4818:
4812:
4794:
3061:
3034:
20769:
20728:
20457:
19277:
13176:
13168:
11962:
5687:
8:
10668:
8150:
2899:{\displaystyle =((u_{x}-v_{y})+i(v_{x}+u_{y})){\bigl (}E+G+2{\sqrt {EG-F^{2}}}{\bigr )}.}
317:
305:
14058:
The theory of the Beltrami equation can be extended to measurable Beltrami coefficients
12635:. On L its image is contained in Hölder continuous functions with Hölder exponent 1 − 2
11945:
theory to prove the existence and uniqueness of solutions when the Beltrami coefficient
21038:
20797:
20757:
18239:
15082:{\displaystyle \displaystyle {\mu v-\mu _{n}v_{n}=\mu (v-v_{n})+(\mu -\mu _{n})v_{n}.}}
12620:> 2 implies convergence in L, so these formulas are compatible with the L theory if
6889:
5747:
5217:
4774:
2581:
253:
20912:
18052:
16453:{\displaystyle \displaystyle (P\psi )_{\overline {z}}=\psi ,\,\,\,(P\psi )_{z}=T\psi }
11980:
the Beurling transform and its inverse are known to be continuous for the L norm. The
1721:
both preserves orientation and induces a metric that differs from the original metric
21042:
20952:
20925:
20866:
20841:
20816:
20716:
20682:
20655:
20626:
20596:
20570:
20482:
20424:
18225:
17646:
17117:{\displaystyle \displaystyle {f_{z}-1,\,\,f_{\overline {z}}\in L^{p}(\mathbf {C} ).}}
16309:
are given by the actual derivatives on Ω ∩ Ω*. In fact this follows by approximating
9734:{\displaystyle \displaystyle {\mu =\psi \mu +(1-\psi )\mu =\mu _{0}+\mu _{\infty }.}}
9516:, a Beltrami coefficient is not a function. Under a holomorphic change of coordinate
8625:
7184:
5713:
along the solution path of that differential equation that passes through the point (
2751:{\displaystyle \displaystyle ((u_{x}+v_{y})+i(v_{x}-u_{y})){\bigl (}E-G+2iF{\bigr )}}
297:
241:
20738:(1938), "On the Solutions of Quasi-Linear Elliptic Partial Differential Equations",
20711:
17325:
with 0 removed and the Beltrami coefficient is smooth there. The canonical solution
11384:{\displaystyle \displaystyle ds^{2}=(1+t\psi (t)h(\theta ))^{2}d\theta ^{2}+dt^{2},}
9748:
be the quasiconformal diffeomorphism of the sphere fixing 0 and ∞ with coefficient
21028:
21004:
20982:
20907:
20885:
20858:
20833:
20808:
20789:
20765:
20747:
20724:
20706:
20616:
20525:
20505:
20453:
18059:. It must therefore be given by multiplying by a positive smooth function that is Γ
13818:{\displaystyle \displaystyle \|\mu \|_{p}\leq (\pi R^{2})^{1/p}\|\mu \|_{\infty }.}
11902:
635:{\displaystyle \displaystyle g(x,y)={\begin{pmatrix}E&F\\F&G\end{pmatrix}}}
273:
125:
35:
20777:
20479:
Partial differential equations, with supplements by Lars Gårding and A. N. Milgram
19616:
where ψ is a smooth function with values in , equal to 0 near 0 and 1 near 1, and
19478:: Tukia's approach has the advantage of also applying in higher dimensions.) When
11965:. In the L approach Hölder continuity follows automatically from operator theory.
11670:
in the coefficients. This condition is imposed by demanding that no odd powers of
10984:{\displaystyle \displaystyle ds^{2}=(1+t\kappa (\theta ))^{2}d\theta ^{2}+dt^{2},}
4617:) = (0, 0), is given by the real and imaginary parts of a complex-valued function
3920:) has dropped out and the expression inside the square root is the perfect square
20962:
20942:
20849:
20824:
20636:
20610:
20580:
20564:
20412:
19483:
9922:
is a quasiconformal diffeomorphism of the sphere fixing 0 and ∞ with coefficient
9513:
2254:{\displaystyle \displaystyle f_{\overline {z}}=((u_{x}-v_{y})+i(v_{x}+u_{y}))/2,}
146:
20481:, Lectures in Applied Mathematics, vol. 3A, American Mathematical Society,
20421:
Elliptic partial differential equations and quasiconformal mappings in the plane
18202:. This identification is compatible with the natural complex structures on both
10300:{\displaystyle \displaystyle {f(z)={\overline {f({\overline {z}}^{-1})}}^{-1},}}
5598:{\displaystyle \displaystyle q'(t)={\frac {-F-i{\sqrt {EG-F^{2}}}}{E}}(q(t),t),}
20938:
20647:
20566:
Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1
18750:
Conversely if this condition is satisfied for all such triples of points, then
17538:
onto itself preserving the unit circle, its exterior and interior. Necessarily
17530:. smooth on the regular set. The normalised solution of the Beltrami equation
17455:
16374:
and for which the distributional derivatives are in 1 + L and L. The relations
14301:
12462:
7998:
5295:
are real analytic, Gauss constructed a particular isothermal coordinate system
783:{\displaystyle \displaystyle \mu (x,y)={E-G+2iF \over E+G+2{\sqrt {EG-F^{2}}}}}
14248:{\displaystyle \displaystyle {\|\mu _{n}\|_{\infty }\leq \|\mu \|_{\infty }.}}
21061:
20973:
Tukia, P.; Väisälä, J. (1980), "Quasisymmetric embeddings of metric spaces",
20720:
20538:
19474:
is quasisymmetric. (A less elementary method was also found independently by
11922:
9640:
8611:
7987:{\displaystyle \displaystyle {h=(I-A)^{-1}T\mu ,\,\,\,T^{*}h=(I-B)^{-1}\mu }}
6873:
20987:
14346:. The limits will satisfy the same Hölder estimates and be holomorphic for |
4042:{\displaystyle u_{x}^{2}v_{y}^{2}-2u_{x}v_{x}u_{y}v_{y}+v_{x}^{2}u_{y}^{2}.}
20509:
20438:
20423:, Princeton Mathematical Series, vol. 48, Princeton University Press,
20416:
20399:
18157:
14082:
In fact in this case the singular set Λ is compact. Take smooth functions φ
12451:{\displaystyle \displaystyle h=(I-A)^{-1}T\mu ,\,\,\,T^{-1}=(I-B)^{-1}\mu }
11905:, there is a diffeomorphism defined in a neighbourhood of the unit circle,
11511:
11275:
coordinate and a conformal change in the metric, the metric takes the form
9141:
20890:
20778:"Obstructions to the smoothing of piecewise-differentiable homeomorphisms"
15120:. The inner products with functions of compact support in Ω are zero for
10005:{\displaystyle \displaystyle \mu (z)={\overline {\mu ({\overline {z}})}},}
19192:
17411:
17313:
If Ω is a doubly connected planar region, then there is a diffeomorphism
15735:{\displaystyle \displaystyle {\partial _{\overline {z}}(f\circ g_{n})=0}}
11124:
10636:
which is holomorphic on the interior. Thus, if a suitable diffeomorphism
27:
20621:
20550:
Glutsyuk, Alexey A. (2008), "Simple proofs of uniformization theorems",
18043:
then yield zero as the Beltrami coefficient. Thus the metric induced on
14045:. So the inverse diffeomorphisms also satisfy uniform Hölder estimates.
21033:
21009:
20801:
20761:
20530:
17466:
17143:
with Beltrami coefficient λ supported near 0, the Beltrami coefficient
13060:{\displaystyle \displaystyle (Pf)_{\overline {z}}=f,\,\,\,(Pf)_{z}=Tf.}
5956:). Because we are assuming the metric to be analytic, it follows that
4053:
must preserve orientation to give isothermal coordinates, the Jacobian
793:
This coefficient has modulus strictly less than one since the identity
19466:, generalizing earlier results of Ahlfors and Beurling, produced such
17301:
12616:
Convergence of functions with fixed compact support in the L norm for
10353:. The corresponding Beltrami coefficient is then a smooth function on
7245:{\displaystyle \displaystyle {\partial _{\overline {z}}(E\star f)=f.}}
4815:
and whose derivative is nowhere zero. Since any holomorphic function
4609:
An isothermal coordinate system, say in a neighborhood of the origin (
19351:, so the estimate follows by applying the quasi-Möbius inequality to
17499:
14320:, passing to a subsequence if necessary, it can be assumed that both
9635:
Defining a smooth Beltrami coefficient on the sphere in this way, if
9005:{\displaystyle \displaystyle {k_{\overline {z}}=\mu k_{z}+\mu _{z}.}}
6880:
to establish Hölder estimates that are automatic in the L theory for
5188:
Thus, the coordinate system given by the real and imaginary parts of
269:
20793:
20752:
17037:(∞) = ∞, the conditions that the derivatives are locally in L force
16991:{\displaystyle \displaystyle {f_{\overline {z}}=g_{\overline {z}}.}}
14159:{\displaystyle \displaystyle {\mu _{n}=(1-\varphi _{n})\cdot \mu .}}
10080:{\displaystyle \displaystyle f(z)={\overline {f({\overline {z}})}},}
5876:{\displaystyle \displaystyle E\,dx+(F+i{\sqrt {EG-F^{2}}}\,)\,dy=0.}
5772:. More generally, suppose that we move by an infinitesimal vector (
2135:{\displaystyle \displaystyle f_{z}=((u_{x}+v_{y})+i(v_{x}-u_{y}))/2}
538:{\displaystyle \displaystyle ds^{2}=E\,dx^{2}+2F\,dx\,dy+G\,dy^{2}}
249:
19371:
is quasisymmetric, it suffices to find a uniform upper bound for |
18063:-invariant. Any such function corresponds to a smooth function on
18023:
of the corresponding Beltrami equation defines a homeomorphism of
17435:
be the group generated by reflections in the boundary circles of Ω
13437:{\displaystyle \displaystyle K_{p}=\|z^{-1}(z-1)^{-1}\|_{q}/\pi .}
11476:{\displaystyle \displaystyle \psi (t)=1+a_{1}t+a_{2}t^{2}+\cdots }
10887:
as parameter, the induced metric near the unit circle is given by
8540:
is a unitary and on trigonometric polynomials or smooth functions
6862:
20681:, Graduate Texts in Mathematics, vol. 109, Springer-Verlag,
19462:
of the closed unit disk which is diffeomorphism on its interior.
16871:{\displaystyle \displaystyle {f_{z}-g_{z}=T(\mu (f_{z}-g_{z})).}}
10090:
so leaves the real axis and hence the upper halfplane invariant.
8297:
can be regarded as a distribution and smooth function on a torus
332:
on that surface typically don't intersect the curves of constant
20862:
20837:
20812:
19683:{\displaystyle \displaystyle {f(e^{i\theta })=e^{ig(\theta )},}}
18103:, i.e. is uniformized and inherits a natural complex structure.
14048:
7315:
is locally square integrable. The above equation can be written
20615:, Memoirs of the American Mathematical Society, vol. 191,
18077:
by this function results in a conformally equivalent metric on
15095:
The first term tends weakly to 0, while the second term equals
11976:
is smooth of compact support proceeds as in the L case. By the
9900:{\displaystyle \displaystyle \lambda (z)=\left\circ g^{-1}(z).}
20922:
A comprehensive introduction to differential geometry. Vol. IV
20543:
Le théorème d'intégrabilité des structures presque complexes.
19215:: this follows by multiplying the first inequality above for (
17009:
is both holomorphic and antiholomorphic, so a constant. Since
16134:
satisfies Beltrami's equation since the actual derivatives do.
10544:{\displaystyle \displaystyle {h_{\overline {z}}=\mu (z)h_{z}}}
10318:
is required and is most directly handled within the L theory.
8924:
This Jacobian is nowhere vanishing by a classical argument of
8258:, identically equal to 1 on a neighbourhood of the support of
4597:
2913:
The real and imaginary parts of this identity linearly relate
20496:
Bers, Lipman (1961), "Uniformization by Beltrami equations",
18039:). Composing the original diffeomorphism with the inverse of
17637:
are homeomorphic to a fixed quotient of the upper half plane
17454:
consisting of orientation-preserving mappings is a classical
17278:
can be chosen so that ν vanishes near zero. Applying the map
16594:{\displaystyle \displaystyle {\partial _{\overline {z}}F=0.}}
10310:
so leaves the unit circle and hence the unit disk invariant.
8375:{\displaystyle \displaystyle F_{\overline {z}}=\mu F_{z}+GF,}
5492:
that satisfies the following ordinary differential equation:
4602:
233:
19866:
be the corresponding solution of the Beltrami equation. Let
18578:{\displaystyle \displaystyle {|z_{1}-z_{2}|=|z_{1}-z_{3}|,}}
12059:{\displaystyle \displaystyle {f_{\overline {z}}=\mu f_{z},}}
5427:
axis of his isothermal coordinate system coincides with the
4555:
Thus, the local coordinate system given by a diffeomorphism
16702:{\displaystyle \displaystyle {P(f_{\overline {z}})=f(z)+z}}
15642:. This follows from the elliptic regularity results in the
12333:, then their operator norms are strictly less than 1 and (
9114:{\displaystyle \displaystyle {J(f)=(1-|\mu |^{2})|e^{2k}|}}
8710:{\displaystyle \displaystyle {\partial _{z}(I-U\mu )F=UG.}}
8600:{\displaystyle \displaystyle {U(P_{\overline {z}})=P_{z}.}}
8206:{\displaystyle \displaystyle \partial _{\overline {z}}f=0,}
7816:{\displaystyle \displaystyle {AF=T\mu F,\,\,\,\,BF=\mu TF}}
4811:
be a complex-valued function of a complex variable that is
20947:, History of Mathematics, vol. 10, Providence, R.I.:
17400:| ≤ 1 onto the closure of Ω, holomorphic in the interior.
14885:
satisfies the Beltrami equation with Beltrami coefficient
12639:
once a suitable constant is added. In fact for a function
9755:. Let λ be the Beltrami coefficient of compact support on
7503:{\displaystyle \displaystyle f_{\overline {z}}=\mu f_{z},}
19724:
In the case of a diffeomorphism, the Alexander extension
15875:
satisfies the Beltrami equation with Beltrami coefficient
14011:{\displaystyle \displaystyle |f(z)|\leq \left\cdot R=CR,}
12631:
is not continuous on L except as a map into functions of
8650:
are invertible on each Sobolev space. On the other hand,
8052:{\displaystyle \displaystyle {g_{\overline {z}}=T^{*}h,}}
7068:
where the subscripts denote complex partial derivatives.
7058:{\displaystyle \displaystyle T(f_{\overline {z}})=f_{z},}
1221:{\displaystyle \displaystyle u_{x}v_{y}-v_{x}u_{y}>0.}
20008:{\displaystyle \displaystyle {f=f_{1}^{-1}\circ f_{2}.}}
19268:
is quasi-Möbius so is its inverse. It then follows that
16543:{\displaystyle \displaystyle {F=f-P(f_{\overline {z}})}}
12517:{\displaystyle \displaystyle g_{\overline {z}}=T^{-1}h,}
3491:
that does give us isothermal coordinates. We then have
10606:{\displaystyle \displaystyle {\mu _{F\circ h^{-1}}=0,}}
7826:
then their operator norms are strictly less than 1 and
7365:{\displaystyle \displaystyle {(Cf)_{\overline {z}}=f.}}
7255:
The same holds for distributions of compact support on
5668:. Gauss then defines his isothermal coordinate system
19458:
of the unit circle can be extended to a homeomorphism
11072:
11028:
9655:
can be written as a sum of two Beltrami coefficients:
5652:, this differential equation determines the values of
1529:
600:
20477:
Bers, Lipman; John, Fritz; Schechter, Martin (1979),
19961:
19960:
19750:
19749:
19626:
19625:
19500:
19499:
19315:. To prove a Hölder estimate, it can be assumed that
19199:
is quasisymmetric then it is also quasi-Möbius, with
18992:
18991:
18783:
18782:
18598:
18597:
18499:
18498:
18260:
18259:
17911:
17910:
17694:
17693:
17165:
17164:
17047:
17046:
16948:
16947:
16895:
16894:
16794:
16793:
16770:{\displaystyle \displaystyle {f_{z}=T(\mu f_{z})+1.}}
16722:
16721:
16650:
16649:
16563:
16562:
16500:
16499:
16384:
16383:
16174:
16173:
16001:
16000:
15888:
15887:
15685:
15684:
15160:
14980:
14979:
14724:
14723:
14554:
14553:
14439:
14438:
14300:
satisfy uniform Hölder estimates. They are therefore
14196:
14195:
14108:
14107:
13863:
13862:
13741:
13740:
13493:
13492:
13357:
13356:
13189:
13188:
13088:
13087:
12988:
12987:
12652:
12549:
12548:
12475:
12474:
12359:
12358:
12167:
12166:
12083:
12082:
12018:
12017:
11683:
11528:
11527:
11408:
11407:
11285:
11284:
11156:
11155:
11004:
11003:
10897:
10896:
10733:
10732:
10564:
10563:
10497:
10496:
10367:
10366:
10227:
10226:
10111:
10110:
10029:
10028:
9954:
9953:
9769:
9768:
9665:
9664:
9538:
9537:
9410:
9409:
9333:
9332:
9169:
9168:
9037:
9036:
8951:
8950:
8765:
8764:
8660:
8659:
8553:
8552:
8427:
8426:
8327:
8326:
8174:
8173:
8084:
8083:
8011:
8010:
7893:
7892:
7836:
7835:
7766:
7765:
7611:
7610:
7527:
7526:
7464:
7463:
7393:
7392:
7325:
7324:
7277:
7276:
7199:
7198:
7169:{\displaystyle \displaystyle {E(z)={1 \over \pi z},}}
7129:
7128:
7084:
7013:
7012:
6910:
6909:
6575:
6574:
6537:
6490:
6329:
6328:
6174:
6173:
6087:
5966:
5965:
5892:
5806:
5805:
5750:
5690:
5502:
5501:
5488:) be some complex-valued function of a real variable
5463:
5437:
5384:
5301:
5266:
5240:
5220:
5194:
4876:
4841:
4821:
4797:
4777:
4639:
4638:
4569:
4494:
4123:
4122:
4059:
3926:
3501:
3500:
3421:
3375:
3298:
3095:
3094:
3064:
3037:
3007:
2977:
2948:
2919:
2767:
2641:
2640:
2604:
2584:
2537:
2487:
2278:
2277:
2150:
2149:
2039:
2038:
1816:
1815:
1523:
1522:
1269:
1268:
1167:
1166:
973:
972:
803:
802:
690:
689:
573:
572:
457:
456:
404:
366:
232:. Classically this differential equation was used by
196:
165:
50:
11941:
Douady and others have indicated ways to extend the
9390:{\displaystyle \displaystyle {g(z)=f(z^{-1})^{-1},}}
7443:
is given on Fourier transforms as multiplication by
6896:) defined on the Fourier transform of an L function
268:< ∞. The same method applies equally well on the
20410:
20323:
20205:
20067:
19606:{\displaystyle \displaystyle {F(r,\theta )=r\exp,}}
17302:
Uniformization of multiply connected planar domains
15622:tends uniformly to the identity on compacta. Hence
11961:is smooth with fixed compact support are uniformly
10628:is a smooth diffeomorphism between the closures of
9132:∪ ∞ into itself which is locally a diffeomorphism,
1766:as a complex-valued function of a complex variable
328:) coordinate system on it. The curves of constant
20007:
19832:
19682:
19605:
19419:|, uniformly small. In this case there is a point
19180:
18971:
18757:An apparently weaker condition on a homeomorphism
18739:
18577:
18478:
18004:
17890:
17264:
17116:
16990:
16927:
16885:μ has operator norm on L less than 1, this forces
16870:
16769:
16701:
16593:
16542:
16452:
16287:
16087:
15974:
15734:
15519:
15081:
14846:
14699:in L. These are the distributional derivatives of
14677:
14527:
14247:
14158:
14010:
13817:
13708:
13436:
13336:
13144:
13059:
12951:
12605:
12516:
12450:
12298:
12120:
12058:
11890:
11646:
11475:
11383:
11260:
11108:
10983:
10868:
10605:
10543:
10468:
10299:
10202:
10079:
10004:
9899:
9733:
9624:
9501:
9389:
9308:
9113:
9004:
8913:
8709:
8599:
8525:
8374:
8205:
8140:{\displaystyle \displaystyle {f(z)=CT^{*}h(z)+z.}}
8139:
8051:
7986:
7872:
7815:
7737:
7564:
7502:
7428:
7364:
7304:
7244:
7168:
7108:
7057:
6984:
6844:
6557:
6523:
6473:
6313:
6156:
6070:
5932:
5875:
5756:
5705:
5597:
5469:
5449:
5411:
5370:
5272:
5252:
5226:
5206:
5177:
4859:
4827:
4803:
4783:
4760:
4587:
4545:
4477:
4105:
4041:
3897:
3472:
3407:
3357:
3253:
3077:
3050:
3023:
2993:
2964:
2935:
2898:
2750:
2619:
2590:
2570:
2520:
2466:
2253:
2134:
2018:
1725:only by a positive, smoothly varying scale factor
1706:
1502:
1220:
1090:
952:
782:
634:
537:
428:
390:
221:{\displaystyle \partial /\partial {\overline {z}}}
220:
182:
109:
20875:
20740:Transactions of the American Mathematical Society
20590:
20476:
20394:, Ann. Acad. Sci. Fenn. Ser. A. I., vol. 206
20293:
20165:
20130:
20103:
20072:
19718:
17406:For regions with a higher degree of connectivity
14868:using the fact that Beltrami coefficients of the
13145:{\displaystyle \displaystyle f(z)=PT^{-1}h(z)+z.}
12606:{\displaystyle \displaystyle f(z)=CT^{-1}h(z)+z.}
11936:
10321:
21059:
19843:This is also true for the other extensions when
15652:has non-vanishing Jacobian there. In particular
14769:
14741:
8289:, so, identifying opposite sides of the square,
7263:is an L function with compact support, then its
4546:{\displaystyle \mu (z)=f_{\overline {z}}/f_{z}.}
3473:{\displaystyle f_{\overline {z}}/f_{z}=\mu (z),}
288:can be proved using the equation, including the
20392:Conformality with respect to Riemannian metrics
18152:of the Beltrami equation is a homeomorphism of
17602:showed that two compact Riemannian 2-manifolds
17534:is a smooth diffeomorphism of the closure of Ω
11394:where ψ is an analytic real-valued function of
6859:thus gives the desired isothermal coordinates.
6531:and then multiply numerator and denominator by
1762:To determine when this happens, we reinterpret
20349:
20347:
20345:
19284:be the set of cube roots of unity, so that if
17588:
17509:. It has measure zero. The induced metric on Ω
12461:where the right hand sides can be expanded as
7997:where the right hand sides can be expanded as
20746:(1), American Mathematical Society: 126–166,
20698:Bulletin of the American Mathematical Society
20667:
20608:
20303:
17676:be the corresponding Beltrami coefficient on
16616:almost everywhere. Abusing notation redefine
14291:of the Beltrami equations and their inverses
14049:Solution for measurable Beltrami coefficients
11674:appear in the formal power series expansion:
7873:{\displaystyle \displaystyle {(I-A)h=T\mu .}}
6062:
5996:
4463:
4392:
4385:
4314:
4305:
4228:
4221:
4144:
2888:
2843:
2742:
2714:
21017:Väisälä, Jussi (1984), "Quasi-Möbius maps",
20972:
20364:
20333:
20331:
19759:
19752:
19331:are greater than a fixed distance away from
17641:by a discrete cocompact subgroup Γ of PSL(2,
16928:{\displaystyle \displaystyle {f_{z}=g_{z}.}}
16477:is a solution of the Beltrami equation with
15462:
15426:
14658:
14644:
14627:
14613:
14598:
14584:
14570:
14556:
14231:
14224:
14212:
14198:
13962:
13955:
13938:
13931:
13802:
13795:
13749:
13742:
13603:
13596:
13574:
13567:
13413:
13371:
13267:
13260:
12974:by a constant, it follows exactly as in the
9639:is such a coefficient then, taking a smooth
7305:{\displaystyle \displaystyle {Cf=E\star f,}}
311:
20515:
20342:
19463:
16301:defined on Ω ∩ Ω*. The weak derivatives of
16105:(Ω), with corresponding singular set Λ ' =
12535:. Hence, up to the addition of a constant,
9128:induces a smooth map of the Riemann sphere
7109:{\displaystyle D=\partial _{\overline {z}}}
4598:Isothermal coordinates for analytic metrics
20679:Univalent functions and Teichmüller spaces
20537:
20308:
20273:
20249:
20237:
20224:
20182:
20087:
19854:to a Beltrami coefficient on the whole of
19435:. Applying the quasi-Möbius inequality to
18136:. It defines a Beltrami coefficient λ omn
17620:> 1 can be simultaneously uniformized.
17447:. Moreover, the index two normal subgroup
17392:is a smooth diffeomorphism of the annulus
13155:On the other hand, the integrand defining
12073:a smooth function of compact support, set
8628:shows that the operators corresponding to
8610:Similarly it extends to a unitary on each
7517:a smooth function of compact support, set
7429:{\displaystyle \displaystyle (Cf)_{z}=Tf.}
6705:
6643:
5378:the one that he chose being the one with
3358:{\displaystyle r=(u_{x}^{2}+v_{x}^{2})/E,}
21032:
21008:
20986:
20937:
20911:
20889:
20751:
20710:
20620:
20529:
20462:
20328:
19779:
19778:
19777:
18084:which agrees with the Poincaré metric on
17976:
17859:
17858:
17805:
17804:
17783:
17782:
17754:
17753:
17732:
17731:
17221:
17069:
17068:
16420:
16419:
16418:
16230:
15476:
15416:
15409:
15345:
15338:
15236:
15229:
14711:, since if ψ is smooth of compact support
14583:
14582:
14541:These lie in L and are uniformly bounded:
14483:
14482:
14062:. For simplicity only a special class of
13024:
13023:
13022:
12935:
12928:
12852:
12845:
12750:
12743:
12402:
12401:
12400:
12131:and assume that the first derivatives of
12121:{\displaystyle \displaystyle g(z)=f(z)-z}
11873:
11808:
11220:
11219:
11218:
11208:
9512:This formula reflects the fact that on a
8473:
8472:
7937:
7936:
7935:
7792:
7791:
7790:
7789:
7575:and assume that the first derivatives of
7565:{\displaystyle \displaystyle g(z)=f(z)-z}
6455:
6398:
6295:
6238:
6081:for some smooth, complex-valued function
6053:
6049:
6004:
5859:
5855:
5810:
3365:which is positive, while the Jacobian of
3214:
3151:
2435:
2390:
2310:
2005:
1976:
1975:
1965:
1943:
1942:
1941:
1940:
1876:
1875:
1513:a metric whose first fundamental form is
1482:
1433:
1426:
1357:
520:
507:
500:
477:
20646:
20609:Iwaniec, Tadeusz; Martin, Gaven (2008),
20549:
20436:
20187:
20108:
20082:
18234:An orientation-preserving homeomorphism
15148:denotes the Jacobian of a function, then
14905:follows by continuity from the relation
12962:Note that the constant is added so that
4113:is the positive square root; so we have
2481:of this induced metric is defined to be
336:orthogonally. A new coordinate system (
21016:
20775:
20562:
20398:
20389:
20375:
20288:
20261:
20219:
20192:
20159:
20144:
20124:
20077:
19482:is a diffeomorphism of the circle, the
17658:
16145:are solutions constructed as above for
15795:. Since it is locally a homeomorphism,
14282:The corresponding normalised solutions
8925:
6524:{\displaystyle h_{\overline {z}}/h_{z}}
5764:, since the starting condition is then
5214:is also isothermal. Indeed, if we fix
2571:{\displaystyle f_{\overline {z}}/f_{z}}
2521:{\displaystyle f_{\overline {z}}/f_{z}}
21060:
20919:
20897:
20734:
20694:
20353:
20052:
19740:> 1. Accordingly, in the unit disc
14426:are weakly differentiable. In fact let
10357:vanishing near 0 and ∞ and satisfying
8755:into itself, the Jacobian is given by
6157:{\displaystyle c(x,y)=a(x,y)+ib(x,y).}
5371:{\displaystyle h(x,y)=u(x,y)+iv(x,y),}
3487:Conversely, consider a diffeomorphism
3264:It follows that the metric induced by
21048:
20994:
20676:
20593:An Introduction to Teichmüller spaces
20298:
19728:can be continued to any larger disk |
19475:
18164:into two components. Conjugation of Γ
18106:With this conformal change in metric
17901:It satisfies the invariance property
15112:. The terms are uniformly bounded in
14877:are supported in a fixed closed disk.
14864:. A similar argument applies for the
14025:> 0 depends only on the L norm of
13479:) converge. The Hölder estimates for
13463:<1. Hence the Neumann series for (
11909:= 0, for which the formal expression
10340:be a diffeomorphism of the unit disk
9528:), the coefficient is transformed to
4860:{\displaystyle \psi _{\overline {z}}}
4106:{\displaystyle u_{x}v_{y}-v_{x}u_{y}}
20975:Ann. Acad. Sci. Fenn. Ser. A I Math.
20670:Quasiconformal mappings in the plane
20668:Lehto, Olli; Virtanen, K.I. (1973),
20591:Imayoshi, Y.; Taniguchi, M. (1992),
20495:
20467:
20337:
18219:
17599:
17418:. There is a smooth diffeomorphism
17307:coefficients is therefore required.
16938:But then from the Beltrami equation
16467:since they can be written as L of ψ
16117:satisfies the Beltrami equation for
14178:are smooth with compact support in |
14041:. It has the same L norm as that of
3408:{\displaystyle r{\sqrt {EG-F^{2}}},}
1153:preserves orientation just when its
352:do intersect the curves of constant
183:{\displaystyle \partial /\partial z}
135:, with derivatives that are locally
20924:(3nd ed.), Publish or Perish,
20404:Lectures on quasiconformal mappings
19930:are univalent holomorphic maps of |
17595:Simultaneous uniformization theorem
16161:satisfies the Beltrami equation for
10217:of the Beltrami equation satisfies
10019:of the Beltrami equation satisfies
7003:with partial derivatives in L then
6558:{\displaystyle {\overline {h_{z}}}}
3480:the new coordinate system given by
294:simultaneous uniformization theorem
13:
20030:Measurable Riemann mapping theorem
19763:
19486:provides another way of extending
19423:at a distance greater than 1 from
18121:. The diffeomorphism between onto
17577:must be disjoint. It follows that
17029:is holomorphic off the support of
16566:
16008:
15895:
15823:. it follows that the Jacobian of
15688:
15638:is smooth on the regular set Ω of
15430:
15378:
15302:
15262:
14822:
14789:
14662:
14498:
14455:
14338:converge uniformly on compacta to
14235:
14216:
13966:
13942:
13806:
13578:
11800:
11271:After a change of variable in the
11091:
11069:
11047:
11025:
10845:
10793:
10479:The quasiconformal diffeomorphism
9819:
9721:
9643:ψ equal to 0 near 0, equal 1 for |
9589:
9572:
9136:must be a diffeomorphism. In fact
8663:
8389:is smooth. On the canonical basis
8176:
7451:as multiplication by its inverse.
7202:
7092:
3415:which is also positive. So, when
1925:
1909:
1878:
1860:
1844:
1818:
661:> 0) that varies smoothly with
290:measurable Riemann mapping theorem
236:to prove the existence locally of
205:
197:
174:
166:
155:) of norm less than 1, called the
95:
87:
62:
54:
14:
21094:
20913:10.1090/s0002-9947-1965-0188431-2
20324:Astala, Iwaniec & Martin 2009
20206:Astala, Iwaniec & Martin 2009
20068:Astala, Iwaniec & Martin 2009
19719:Partyka, Sakan & Zając (1999)
19451:yields the required upper bound.
19403:)| in the case of a triple with |
15140:is a bounded open set containing
14308:; they are even holomorphic for |
14029:. So the Beltrami coefficient of
11891:{\displaystyle \left\cdot \left.}
11486:A formal diffeomorphism sending (
10671:produces a smooth diffeomorphism
10663:has length 2π and that 0 lies in
7179:a locally integrable function on
5664:either less than or greater than
4563:solves the Beltrami equation for
1137:)) be a smooth diffeomorphism of
18242:if there are positive constants
18031:gives a new cocompact subgroup Γ
17877:
17794:
17743:
17645:). Γ can be identified with the
17102:
13451:> 2 sufficiently close to 2,
12881:
12786:
12716:
10640:can be constructed, the mapping
10213:then by uniqueness the solution
10015:then by uniqueness the solution
8747:is smooth. Regarded as a map of
6995:It is a unitary operator and if
6867:for smooth Beltrami coefficients
2598:equals the Beltrami coefficient
276:and plays a fundamental role in
20712:10.1090/S0002-9904-1936-06297-X
20569:, Matrix Editions, Ithaca, NY,
20369:
20358:
20316:
20279:
20267:
20255:
20243:
20231:
20166:Bers, John & Schechter 1979
20104:Bers, John & Schechter 1979
20073:Bers, John & Schechter 1979
18140:which this time is extended to
17410:+ 1, the result is essentially
14967:. The difference can be written
11949:is bounded and measurable with
8153:can now be used to deduce that
5640:). If we specify the value of
3174:
3168:
429:{\displaystyle b\leq v\leq b+h}
391:{\displaystyle a\leq u\leq a+h}
145:is a given complex function in
21068:Partial differential equations
21051:Generalized analytic functions
21020:Journal d'Analyse Mathématique
20944:Sources of hyperbolic geometry
20210:
20199:
20173:
20150:
20138:
20115:
20094:
20058:
20046:
19858:by setting it equal to 0 for |
19789:
19783:
19670:
19664:
19647:
19631:
19595:
19589:
19586:
19580:
19568:
19559:
19553:
19547:
19541:
19532:
19517:
19505:
19323:is uniformly small. Then both
19170:
19144:
19141:
19115:
19110:
19084:
19081:
19055:
19046:
18994:
18954:
18949:
18897:
18893:
18882:
18878:
18875:
18862:
18853:
18840:
18831:
18818:
18809:
18796:
18790:
18786:
18765:, that is there are constants
18719:
18715:
18702:
18693:
18680:
18673:
18666:
18662:
18649:
18640:
18627:
18620:
18566:
18538:
18530:
18502:
18457:
18428:
18415:
18386:
18366:
18362:
18349:
18340:
18327:
18320:
18313:
18309:
18296:
18287:
18274:
18267:
18188:are naturally identified with
17995:
17989:
17939:
17936:
17930:
17924:
17881:
17860:
17849:
17836:
17824:
17818:
17798:
17784:
17773:
17767:
17747:
17733:
17728:
17722:
17713:
17707:
17426:, given by the unit disk with
17184:
17181:
17175:
17169:
17106:
17098:
16860:
16857:
16831:
16825:
16756:
16740:
16688:
16682:
16673:
16655:
16535:
16517:
16431:
16421:
16395:
16385:
16193:
16190:
16184:
16178:
16080:
16077:
16071:
16065:
16019:
16013:
15967:
15961:
15915:
15912:
15906:
15900:
15721:
15702:
15487:
15480:
15451:
15399:
15373:
15328:
15297:
15283:
15257:
15226:
15213:
15190:
15187:
15181:
15168:
15061:
15042:
15036:
15017:
14142:
14123:
13882:
13878:
13872:
13865:
13778:
13761:
13698:
13670:
13642:
13613:
13541:
13537:
13524:
13515:
13502:
13495:
13400:
13387:
13306:
13277:
13243:
13239:
13226:
13214:
13201:
13191:
13129:
13123:
13098:
13092:
13035:
13025:
12999:
12989:
12922:
12910:
12899:
12893:
12801:
12795:
12734:
12728:
12691:
12685:
12669:
12663:
12590:
12584:
12559:
12553:
12432:
12419:
12379:
12366:
12279:
12270:
12261:
12245:
12236:
12218:
12209:
12191:
12108:
12102:
12093:
12087:
11982:Riesz–Thorin convexity theorem
11937:Hölder continuity of solutions
11805:
11786:
11742:
11736:
11707:
11701:
11640:
11628:
11597:
11591:
11572:
11566:
11544:
11532:
11418:
11412:
11339:
11335:
11329:
11323:
11317:
11302:
11245:
11239:
11230:
11224:
11205:
11199:
11102:
11096:
11083:
11077:
11058:
11052:
11039:
11033:
11014:
11008:
10939:
10935:
10929:
10914:
10859:
10856:
10850:
10837:
10825:
10819:
10813:
10804:
10798:
10785:
10773:
10767:
10761:
10755:
10749:
10737:
10715:in ) be a parametrization of ∂
10526:
10520:
10444:
10421:
10377:
10371:
10322:Smooth Riemann mapping theorem
10274:
10251:
10238:
10232:
10187:
10164:
10121:
10115:
10064:
10051:
10039:
10033:
9989:
9976:
9964:
9958:
9890:
9884:
9779:
9773:
9694:
9682:
9614:
9608:
9599:
9564:
9558:
9552:
9491:
9475:
9428:
9422:
9370:
9353:
9344:
9338:
9105:
9087:
9083:
9073:
9064:
9054:
9048:
9042:
8903:
8893:
8884:
8874:
8864:
8848:
8834:
8813:
8799:
8783:
8776:
8770:
8687:
8672:
8576:
8558:
8486:
8474:
8123:
8117:
8095:
8089:
7967:
7954:
7914:
7901:
7850:
7838:
7722:
7713:
7704:
7688:
7679:
7661:
7652:
7634:
7552:
7546:
7537:
7531:
7404:
7394:
7337:
7327:
7228:
7216:
7140:
7134:
7035:
7017:
6999:is a tempered distribution on
6975:
6969:
6934:
6928:
6900:as a multiplication operator:
6835:
6829:
6747:
6706:
6699:
6673:
6668:
6644:
6637:
6611:
6459:
6456:
6408:
6399:
6351:
6348:
6299:
6296:
6248:
6239:
6191:
6188:
6148:
6136:
6124:
6112:
6103:
6091:
6050:
6014:
5991:
5979:
5907:
5901:
5856:
5820:
5700:
5694:
5588:
5579:
5573:
5567:
5517:
5511:
5400:
5388:
5362:
5350:
5338:
5326:
5317:
5305:
5165:
5159:
5098:
5074:
5058:
5039:
5012:
4988:
4967:
4948:
4926:
4913:
4897:
4884:
4681:
4675:
4579:
4573:
4504:
4498:
4458:
4432:
4423:
4397:
4380:
4354:
4345:
4319:
4300:
4271:
4262:
4233:
4216:
4187:
4178:
4149:
4133:
4127:
3877:
3830:
3824:
3788:
3785:
3749:
3732:
3700:
3694:
3658:
3653:
3607:
3589:
3562:
3556:
3520:
3511:
3505:
3464:
3458:
3341:
3305:
2838:
2835:
2809:
2800:
2774:
2771:
2709:
2706:
2680:
2671:
2645:
2642:
2614:
2608:
2380:
2364:
2350:
2296:
2236:
2233:
2207:
2198:
2172:
2169:
2120:
2117:
2091:
2082:
2056:
2053:
1934:
1905:
1869:
1840:
1479:
1443:
1423:
1377:
1354:
1318:
1239:the standard Euclidean metric
984:
975:
946:
905:
902:
861:
855:
831:
828:
804:
706:
694:
589:
577:
124:a complex distribution of the
1:
20949:American Mathematical Society
20383:
20294:Imayoshi & Taniguchi 1992
20131:Imayoshi & Taniguchi 1992
17369:is a reflection in a circle |
16364:
14088:of compact support with 0 ≤ φ
12317:are the operators defined by
11510:) can be defined as a formal
10093:Similarly for the unit disc |
7756:are the operators defined by
7454:Now in the Beltrami equation
7119:is given by the distribution
40:partial differential equation
16:Partial differential equation
20563:Hubbard, John Hamal (2006),
19804:
18761:of the circle is that it be
18238:of the circle is said to be
17959:
17868:
17853:
17844:
17443:iz a fundamental domain for
17329:of the Beltrami equation on
17247:
17214:
17079:
16977:
16959:
16667:
16574:
16529:
16403:
16263:
16223:
16042:
15948:
15696:
15579:), for applying the inverse
15310:
14830:
14797:
14463:
14053:
13007:
12485:
12230:
12203:
12029:
11988:are continuous functions of
10667:. The smooth version of the
10655:To produce a diffeomorphism
10508:
10448:
10430:
10401:
10278:
10260:
10191:
10173:
10145:
10068:
10059:
9993:
9984:
9856:
9824:
9647:| > 1 and satisfying 0 ≤
9594:
9452:
9283:
9230:
9124:is nowhere vanishing. Since
8962:
8826:
8570:
8452:
8337:
8184:
8160:In fact, off the support of
8022:
7673:
7646:
7474:
7345:
7210:
7100:
7029:
6947:
6587:
6550:
6500:
6339:
5450:{\displaystyle \psi \circ h}
5260:for the various holomorphic
5253:{\displaystyle \psi \circ f}
5207:{\displaystyle \psi \circ f}
5135:
5113:
5086:
5032:
5026:
5000:
4979:
4905:
4851:
4791:be such a function, and let
4651:
4519:
3431:
2547:
2497:
2444:
2417:
2343:
2329:
2160:
1985:
1886:
348:when the curves of constant
213:
70:
7:
20018:
18588:then the condition becomes
17672:which is Γ-invariant. Let
17589:Simultaneous uniformization
17404:Multiply connected domains.
17139:is a smooth diffeomorphism
14941:. It suffices to show that
14033:is smooth and supported in
13179:with an explicit estimate:
9936:For the upper halfplane Im
8928:. In fact formally writing
8743:Thus the original function
5933:{\displaystyle q'(t)=dx/dy}
1802:) so that we can apply the
645:is a positive real matrix (
439:To see how this works, let
240:on a surface with analytic
10:
21099:
20437:Beltrami, Eugenio (1867),
18223:
18144:by defining λ to be 0 off
18132:which is invariant under Γ
18128:induces another metric on
17592:
13846:= 0, it follows that for |
12643:of compact support define
12633:vanishing mean oscillation
10325:
7375:Moreover, still regarding
4867:identically zero, we have
258:singular integral operator
18:
20390:Ahlfors, Lars V. (1955),
20304:Lehto & Virtanen 1973
19464:Douady & Earle (1986)
18070:. Dividing the metric on
17381:< 1. It follows that
17311:Doubly connected domains.
16784:is another solution then
16101:on the regular set Ω ' =
14304:on any compact subset of
14071:with compact support in |
12966:(0) = 0. Since
12008:In the Beltrami equation
8247:tends to 0 uniformly as |
8223:is even holomorphic for |
5477:with nonzero derivative.
5280:with nonzero derivative.
360:, the little region with
316:Consider a 2-dimensional
312:Metrics on planar domains
21:Laplace–Beltrami equation
20920:Spivak, Michael (1999),
20365:Tukia & Väisälä 1980
20040:
19847:is only quasisymmetric.
17680:. It can be extended to
17655:universal covering space
14390:imply that in the limit
12527:has the same support as
8313:). As a distribution on
8277:lies in a large square |
8062:has the same support as
5780:) away from some point (
5412:{\displaystyle h(x,0)=x}
4588:{\displaystyle \mu (z).}
4559:is isothermal just when
19:Not to be confused with
20988:10.5186/aasfm.1980.0531
20900:Trans. Amer. Math. Soc.
20776:Munkres, James (1960),
20461:English translation in
20446:Giornale di Mathematica
18174:quasi-Fuchsian subgroup
18113:can be identified with
18101:compact Riemann surface
17357:denotes reflection in |
16712:and so differentiating
16624:'(z) − 1 lies in L and
15558:)). On the other hand
14889:. In fact the relation
13078:to produce a solution:
13074:can be used instead of
11984:implies that the norms
11978:Calderón–Zygmund theory
11657:where the coefficients
10646:Riemann mapping theorem
10328:Riemann mapping theorem
6878:quasiconformal mappings
5648:) for some start value
3908:where the scale factor
2627:of the original metric
2620:{\displaystyle \mu (z)}
1737:), the new coordinates
302:Riemann mapping theorem
286:uniformization theorems
282:quasiconformal mappings
20736:Morrey, Charles B. Jr.
20510:10.1002/cpa.3160140304
20498:Comm. Pure Appl. Math.
20309:Douady & Buff 2000
20274:Douady & Buff 2000
20250:Douady & Buff 2000
20238:Douady & Buff 2000
20225:Douady & Buff 2000
20183:Douady & Buff 2000
20088:Douady & Buff 2000
20035:Isothermal coordinates
20025:Quasiconformal mapping
20009:
19834:
19684:
19607:
19182:
18973:
18741:
18579:
18480:
18230:Douady–Earle extension
18006:
17892:
17623:As topological spaces
17414:generalization of the
17266:
17118:
16992:
16929:
16872:
16771:
16703:
16595:
16544:
16454:
16289:
16089:
15976:
15736:
15661:≠ 0 on Ω. In fact for
15571:) eventually contains
15521:
15083:
14848:
14679:
14529:
14249:
14160:
14012:
13819:
13710:
13438:
13338:
13146:
13061:
12953:
12607:
12518:
12452:
12300:
12122:
12060:
11892:
11648:
11477:
11385:
11262:
11110:
10985:
10870:
10607:
10545:
10470:
10301:
10204:
10081:
10006:
9901:
9735:
9626:
9503:
9391:
9310:
9115:
9006:
8915:
8724:is smooth, so too is (
8711:
8601:
8527:
8376:
8207:
8141:
8053:
7988:
7874:
7817:
7739:
7566:
7504:
7430:
7366:
7306:
7246:
7170:
7110:
7059:
6986:
6846:
6559:
6525:
6475:
6315:
6158:
6072:
5934:
5877:
5758:
5707:
5620:are here evaluated at
5599:
5471:
5451:
5413:
5372:
5274:
5254:
5228:
5208:
5179:
4861:
4829:
4805:
4785:
4762:
4589:
4547:
4479:
4107:
4043:
3899:
3474:
3409:
3359:
3255:
3079:
3052:
3025:
3024:{\displaystyle v_{y},}
2995:
2994:{\displaystyle v_{x},}
2966:
2965:{\displaystyle u_{y},}
2937:
2936:{\displaystyle u_{x},}
2900:
2752:
2621:
2592:
2572:
2531:The Beltrami quotient
2522:
2468:
2264:the metric induced by
2255:
2136:
2020:
1756:isothermal coordinates
1708:
1504:
1222:
1141:onto another open set
1092:
954:
784:
636:
558:first fundamental form
539:
430:
392:
238:isothermal coordinates
222:
184:
111:
21049:Vekua, I. N. (1962),
20891:10.4064/-48-1-141-177
20652:Differential Geometry
20612:The Beltrami equation
20468:Bers, Lipman (1958),
20010:
19835:
19685:
19608:
19183:
18974:
18742:
18580:
18481:
18051:and conformal to the
18007:
17893:
17649:of the manifolds and
17267:
17119:
17017:(0), it follows that
16993:
16930:
16873:
16772:
16704:
16596:
16545:
16455:
16352:This establishes the
16290:
16090:
15977:
15737:
15522:
15084:
14849:
14680:
14530:
14318:Arzelà–Ascoli theorem
14250:
14161:
14013:
13820:
13711:
13439:
13339:
13147:
13062:
12954:
12627:The Cauchy transform
12608:
12519:
12453:
12301:
12123:
12061:
11893:
11649:
11478:
11386:
11263:
11111:
10986:
10871:
10608:
10546:
10471:
10302:
10205:
10082:
10007:
9902:
9736:
9627:
9504:
9392:
9311:
9116:
9007:
8916:
8712:
8602:
8528:
8377:
8208:
8142:
8054:
7989:
7875:
7818:
7740:
7567:
7505:
7439:Indeed, the operator
7431:
7367:
7307:
7247:
7171:
7111:
7060:
6987:
6847:
6560:
6526:
6484:We form the quotient
6476:
6316:
6159:
6073:
5935:
5878:
5759:
5708:
5600:
5472:
5470:{\displaystyle \psi }
5452:
5414:
5373:
5275:
5273:{\displaystyle \psi }
5255:
5229:
5209:
5180:
4862:
4830:
4828:{\displaystyle \psi }
4806:
4804:{\displaystyle \psi }
4786:
4763:
4590:
4548:
4480:
4108:
4044:
3900:
3475:
3410:
3360:
3256:
3080:
3078:{\displaystyle v_{y}}
3053:
3051:{\displaystyle u_{y}}
3026:
2996:
2967:
2938:
2901:
2753:
2622:
2593:
2573:
2523:
2469:
2256:
2137:
2021:
1804:Wirtinger derivatives
1709:
1505:
1223:
1093:
955:
785:
637:
540:
431:
393:
230:Wirtinger derivatives
223:
185:
112:
20677:Lehto, Olli (1987),
20552:Fields Inst. Commun.
19958:
19747:
19623:
19497:
18989:
18780:
18595:
18496:
18257:
18047:is invariant under Γ
17908:
17691:
17668:induces a metric on
17416:retrosection theorem
17162:
17044:
16945:
16892:
16791:
16719:
16647:
16560:
16497:
16381:
16171:
15998:
15885:
15834:. On the other hand
15788:is holomorphic near
15682:
15592:eventually contains
15158:
15136:of measure zero. If
14977:
14721:
14551:
14436:
14414:are homeomorphisms.
14193:
14105:
13860:
13738:
13490:
13354:
13186:
13085:
12985:
12650:
12546:
12472:
12356:
12164:
12080:
12015:
11681:
11525:
11405:
11282:
11153:
11001:
10894:
10730:
10679:onto the closure of
10675:from the closure of
10561:
10494:
10364:
10224:
10108:
10026:
9951:
9766:
9662:
9535:
9407:
9330:
9166:
9034:
8948:
8762:
8657:
8550:
8424:
8324:
8171:
8081:
8008:
7890:
7833:
7763:
7608:
7524:
7461:
7390:
7322:
7274:
7196:
7126:
7082:
7073:fundamental solution
7010:
6907:
6572:
6535:
6488:
6326:
6171:
6085:
5963:
5890:
5803:
5748:
5706:{\displaystyle q(0)}
5688:
5499:
5461:
5435:
5382:
5299:
5264:
5238:
5218:
5192:
4874:
4839:
4819:
4795:
4775:
4636:
4567:
4492:
4120:
4057:
3924:
3498:
3419:
3373:
3296:
3092:
3062:
3035:
3005:
2975:
2946:
2917:
2765:
2638:
2602:
2582:
2535:
2485:
2275:
2147:
2036:
1813:
1520:
1266:
1255:induces a metric on
1164:
970:
800:
687:
674:Beltrami coefficient
570:
454:
402:
364:
194:
163:
157:Beltrami coefficient
48:
20878:Banach Center Publ.
20595:, Springer-Verlag,
20541:; Buff, X. (2000),
20472:, Courant Institute
19986:
19484:Alexander extension
18754:is quasisymmetric.
18019:in Γ. The solution
17439:. The interior of Ω
15630:) has measure zero.
15475:
15124:sufficiently large.
14021:where the constant
11195:
10669:Schoenflies theorem
8151:Elliptic regularity
7259:. In particular if
4035:
4020:
3956:
3941:
3823:
3805:
3784:
3766:
3717:
3693:
3675:
3555:
3537:
3340:
3322:
1691:
1673:
1564:
1546:
1478:
1460:
1353:
1335:
548:be a smooth metric
318:Riemannian manifold
306:elliptic regularity
296:. The existence of
21034:10.1007/bf02790198
21010:10.1007/bf02392471
20531:10.1007/bf02392590
20311:, pp. 321–322
20252:, pp. 319–320
20240:, pp. 319–320
20005:
20004:
19969:
19830:
19829:
19680:
19679:
19603:
19602:
19280:. To see this let
19178:
19177:
18969:
18968:
18737:
18736:
18575:
18574:
18476:
18475:
18002:
18001:
17888:
17887:
17262:
17261:
17114:
17113:
17025:. Note that since
16988:
16987:
16925:
16924:
16868:
16867:
16767:
16766:
16699:
16698:
16591:
16590:
16540:
16539:
16450:
16449:
16285:
16284:
16085:
16084:
15972:
15971:
15732:
15731:
15541:) is small, so is
15517:
15515:
15461:
15079:
15078:
14860:and similarly for
14844:
14843:
14675:
14674:
14525:
14524:
14245:
14244:
14156:
14155:
14008:
14007:
13815:
13814:
13706:
13705:
13434:
13433:
13334:
13333:
13142:
13141:
13057:
13056:
12970:only differs from
12949:
12947:
12603:
12602:
12514:
12513:
12465:. It follows that
12448:
12447:
12296:
12295:
12118:
12117:
12056:
12055:
11888:
11838:
11730:
11644:
11643:
11473:
11472:
11381:
11380:
11258:
11257:
11178:
11106:
11105:
10981:
10980:
10866:
10865:
10644:proves the smooth
10603:
10602:
10541:
10540:
10466:
10465:
10297:
10296:
10200:
10199:
10077:
10076:
10002:
10001:
9897:
9896:
9731:
9730:
9622:
9621:
9499:
9498:
9387:
9386:
9306:
9305:
9111:
9110:
9015:This equation for
9002:
9001:
8941:, it follows that
8911:
8910:
8707:
8706:
8626:Fredholm operators
8597:
8596:
8523:
8522:
8372:
8371:
8301:. By construction
8243:, it follows that
8203:
8202:
8137:
8136:
8049:
8048:
8001:. It follows that
7984:
7983:
7870:
7869:
7813:
7812:
7735:
7734:
7562:
7561:
7500:
7499:
7426:
7425:
7383:as distributions,
7362:
7361:
7302:
7301:
7242:
7241:
7185:Schwartz functions
7166:
7165:
7106:
7055:
7054:
6982:
6981:
6890:Beurling transform
6842:
6841:
6555:
6521:
6471:
6470:
6311:
6310:
6154:
6068:
6067:
5930:
5873:
5872:
5754:
5703:
5595:
5594:
5467:
5457:for a holomorphic
5447:
5409:
5368:
5270:
5250:
5224:
5204:
5175:
5173:
4857:
4825:
4801:
4781:
4758:
4757:
4585:
4543:
4475:
4474:
4103:
4039:
4021:
4006:
3942:
3927:
3895:
3894:
3809:
3791:
3770:
3752:
3703:
3679:
3661:
3541:
3523:
3470:
3405:
3355:
3326:
3308:
3251:
3250:
3075:
3048:
3021:
2991:
2962:
2933:
2896:
2748:
2747:
2617:
2588:
2568:
2518:
2464:
2463:
2251:
2250:
2132:
2131:
2016:
2015:
1704:
1703:
1694:
1677:
1659:
1550:
1532:
1500:
1499:
1464:
1446:
1339:
1321:
1218:
1217:
1088:
1087:
950:
949:
780:
779:
632:
631:
625:
535:
534:
443:be an open set in
426:
388:
298:conformal weldings
280:and the theory of
278:Teichmüller theory
254:Beurling transform
248:and relies on the
218:
180:
107:
20958:978-0-8218-0529-9
20871:978-3-03719-103-3
20846:978-3-03719-055-5
20821:978-3-03719-029-6
20632:978-0-8218-4045-0
20622:10.1090/memo/0893
20576:978-0-9715766-2-9
20430:978-0-691-13777-3
19949:conformal welding
19807:
19278:Hölder continuous
19174:
18773:> 0 such that
18724:
18468:
18371:
18226:Conformal welding
18220:Conformal welding
17986:
17974:
17962:
17921:
17871:
17856:
17847:
17815:
17764:
17704:
17647:fundamental group
17422:of the region Ω
17256:
17250:
17219:
17217:
17082:
16980:
16962:
16670:
16620:. The conditions
16612:that is equal to
16577:
16532:
16406:
16279:
16266:
16228:
16226:
16121:′. In fact
16057:
16045:
15953:
15951:
15699:
15313:
14833:
14800:
14668:
14466:
13982:
13723:is supported in |
13594:
13177:Hölder continuous
13169:Hölder inequality
13010:
12926:
12873:
12838:
12825:
12778:
12741:
12488:
12233:
12206:
12032:
11963:Hölder continuous
11925:expansion in the
11823:
11715:
11176:
10511:
10451:
10433:
10412:
10404:
10281:
10263:
10194:
10176:
10156:
10148:
10071:
10062:
9996:
9987:
9861:
9859:
9830:
9827:
9597:
9549:
9463:
9455:
9299:
9286:
9235:
9233:
8965:
8829:
8573:
8460:
8455:
8340:
8273:. The support of
8187:
8025:
7676:
7649:
7477:
7348:
7213:
7159:
7103:
7032:
6966:
6955:
6950:
6925:
6855:Gauss's function
6821:
6818:
6751:
6745:
6603:
6590:
6553:
6503:
6453:
6396:
6342:
6293:
6236:
6047:
5853:
5757:{\displaystyle x}
5565:
5559:
5227:{\displaystyle f}
5151:
5138:
5119:
5116:
5089:
5035:
5029:
5003:
4982:
4936:
4908:
4854:
4784:{\displaystyle f}
4752:
4749:
4667:
4654:
4629:) that satisfies
4522:
4469:
3889:
3886:
3434:
3400:
3245:
3212:
3172:
3166:
3149:
2884:
2591:{\displaystyle f}
2550:
2500:
2479:Beltrami quotient
2447:
2433:
2420:
2346:
2332:
2163:
1988:
1903:
1889:
1838:
1079:
1076:
1036:
944:
900:
777:
774:
680:is defined to be
264:) for all 1 <
242:Riemannian metric
216:
131:in some open set
102:
76:
73:
32:Beltrami equation
21090:
21073:Complex analysis
21054:
21053:, Pergamon Press
21045:
21036:
21013:
21012:
21003:(3–4): 153–193,
20991:
20990:
20969:
20934:
20916:
20915:
20894:
20893:
20804:
20772:
20755:
20731:
20714:
20691:
20673:
20664:
20643:
20624:
20605:
20587:
20559:
20546:
20534:
20533:
20512:
20491:
20473:
20470:Riemann surfaces
20463:Stillwell (1996)
20460:
20443:
20433:
20413:Iwaniec, Tadeusz
20407:
20400:Ahlfors, Lars V.
20395:
20378:
20373:
20367:
20362:
20356:
20351:
20340:
20335:
20326:
20320:
20314:
20283:
20277:
20271:
20265:
20264:, pp. 97–98
20259:
20253:
20247:
20241:
20235:
20229:
20214:
20208:
20203:
20197:
20177:
20171:
20154:
20148:
20142:
20136:
20119:
20113:
20098:
20092:
20062:
20056:
20050:
20014:
20012:
20011:
20006:
20003:
19999:
19998:
19985:
19977:
19839:
19837:
19836:
19831:
19828:
19824:
19823:
19814:
19809:
19808:
19800:
19767:
19766:
19716:
19704:
19689:
19687:
19686:
19681:
19678:
19674:
19673:
19646:
19645:
19612:
19610:
19609:
19604:
19601:
19454:A homeomorphism
19367:. To check that
19314:
19313:
19307:
19187:
19185:
19184:
19179:
19176:
19175:
19173:
19169:
19168:
19156:
19155:
19140:
19139:
19127:
19126:
19113:
19109:
19108:
19096:
19095:
19080:
19079:
19067:
19066:
19053:
19045:
19044:
19032:
19031:
19019:
19018:
19006:
19005:
18978:
18976:
18975:
18970:
18967:
18963:
18962:
18957:
18948:
18947:
18935:
18934:
18922:
18921:
18909:
18908:
18896:
18885:
18874:
18873:
18852:
18851:
18830:
18829:
18808:
18807:
18789:
18746:
18744:
18743:
18738:
18735:
18725:
18723:
18722:
18714:
18713:
18692:
18691:
18676:
18670:
18669:
18661:
18660:
18639:
18638:
18623:
18617:
18612:
18611:
18584:
18582:
18581:
18576:
18573:
18569:
18564:
18563:
18551:
18550:
18541:
18533:
18528:
18527:
18515:
18514:
18505:
18485:
18483:
18482:
18477:
18474:
18470:
18469:
18467:
18466:
18465:
18460:
18454:
18453:
18441:
18440:
18431:
18425:
18424:
18423:
18418:
18412:
18411:
18399:
18398:
18389:
18383:
18372:
18370:
18369:
18361:
18360:
18339:
18338:
18323:
18317:
18316:
18308:
18307:
18286:
18285:
18270:
18264:
18011:
18009:
18008:
18003:
17988:
17987:
17979:
17975:
17973:
17972:
17963:
17958:
17957:
17948:
17946:
17923:
17922:
17914:
17897:
17895:
17894:
17889:
17880:
17872:
17864:
17857:
17852:
17848:
17840:
17831:
17817:
17816:
17808:
17797:
17766:
17765:
17757:
17746:
17706:
17705:
17697:
17469:with generators
17271:
17269:
17268:
17263:
17257:
17255:
17251:
17243:
17234:
17223:
17220:
17218:
17213:
17212:
17203:
17201:
17200:
17191:
17123:
17121:
17120:
17115:
17112:
17105:
17097:
17096:
17084:
17083:
17075:
17058:
17057:
16997:
16995:
16994:
16989:
16986:
16982:
16981:
16973:
16964:
16963:
16955:
16934:
16932:
16931:
16926:
16923:
16919:
16918:
16906:
16905:
16877:
16875:
16874:
16869:
16866:
16856:
16855:
16843:
16842:
16818:
16817:
16805:
16804:
16776:
16774:
16773:
16768:
16765:
16755:
16754:
16733:
16732:
16708:
16706:
16705:
16700:
16697:
16672:
16671:
16663:
16600:
16598:
16597:
16592:
16589:
16579:
16578:
16570:
16549:
16547:
16546:
16541:
16538:
16534:
16533:
16525:
16459:
16457:
16456:
16451:
16439:
16438:
16408:
16407:
16399:
16294:
16292:
16291:
16286:
16280:
16278:
16277:
16276:
16267:
16259:
16250:
16243:
16242:
16232:
16229:
16227:
16222:
16221:
16212:
16210:
16209:
16200:
16094:
16092:
16091:
16086:
16083:
16058:
16056:
16055:
16046:
16041:
16040:
16031:
16029:
16012:
16011:
15981:
15979:
15978:
15973:
15970:
15954:
15952:
15947:
15946:
15937:
15935:
15934:
15925:
15899:
15898:
15851:| (1 − |μ|), so
15741:
15739:
15738:
15733:
15730:
15720:
15719:
15701:
15700:
15692:
15526:
15524:
15523:
15518:
15516:
15509:
15508:
15504:
15474:
15469:
15460:
15459:
15454:
15448:
15447:
15438:
15437:
15408:
15407:
15402:
15396:
15395:
15386:
15385:
15376:
15371:
15370:
15355:
15337:
15336:
15331:
15325:
15324:
15315:
15314:
15306:
15300:
15292:
15291:
15286:
15280:
15279:
15270:
15269:
15260:
15255:
15254:
15225:
15224:
15209:
15208:
15180:
15179:
15088:
15086:
15085:
15080:
15077:
15073:
15072:
15060:
15059:
15035:
15034:
15010:
15009:
15000:
14999:
14959:tends weakly to
14853:
14851:
14850:
14845:
14842:
14835:
14834:
14826:
14802:
14801:
14793:
14784:
14783:
14756:
14755:
14684:
14682:
14681:
14676:
14673:
14669:
14667:
14666:
14665:
14656:
14655:
14636:
14635:
14634:
14625:
14624:
14611:
14606:
14605:
14596:
14595:
14578:
14577:
14568:
14567:
14534:
14532:
14531:
14526:
14523:
14516:
14515:
14506:
14505:
14493:
14492:
14478:
14477:
14468:
14467:
14459:
14450:
14449:
14354:. The relations
14254:
14252:
14251:
14246:
14243:
14239:
14238:
14220:
14219:
14210:
14209:
14165:
14163:
14162:
14157:
14154:
14141:
14140:
14119:
14118:
14017:
14015:
14014:
14009:
13988:
13984:
13983:
13981:
13980:
13979:
13970:
13969:
13947:
13946:
13945:
13930:
13929:
13925:
13912:
13911:
13901:
13885:
13868:
13824:
13822:
13821:
13816:
13810:
13809:
13794:
13793:
13789:
13776:
13775:
13757:
13756:
13715:
13713:
13712:
13707:
13701:
13696:
13695:
13683:
13682:
13673:
13665:
13664:
13660:
13645:
13639:
13638:
13626:
13625:
13616:
13611:
13610:
13595:
13593:
13592:
13591:
13582:
13581:
13559:
13558:
13549:
13544:
13536:
13535:
13514:
13513:
13498:
13443:
13441:
13440:
13435:
13426:
13421:
13420:
13411:
13410:
13386:
13385:
13367:
13366:
13343:
13341:
13340:
13335:
13329:
13328:
13324:
13309:
13303:
13302:
13290:
13289:
13280:
13275:
13274:
13259:
13258:
13246:
13238:
13237:
13213:
13212:
13194:
13151:
13149:
13148:
13143:
13119:
13118:
13066:
13064:
13063:
13058:
13043:
13042:
13012:
13011:
13003:
12958:
12956:
12955:
12950:
12948:
12927:
12925:
12905:
12888:
12886:
12885:
12884:
12874:
12866:
12844:
12840:
12839:
12831:
12826:
12824:
12810:
12791:
12790:
12789:
12779:
12771:
12760:
12742:
12737:
12723:
12721:
12720:
12719:
12709:
12708:
12612:
12610:
12609:
12604:
12580:
12579:
12523:
12521:
12520:
12515:
12506:
12505:
12490:
12489:
12481:
12457:
12455:
12454:
12449:
12443:
12442:
12415:
12414:
12390:
12389:
12305:
12303:
12302:
12297:
12294:
12260:
12259:
12235:
12234:
12226:
12208:
12207:
12199:
12184:
12183:
12127:
12125:
12124:
12119:
12065:
12063:
12062:
12057:
12054:
12050:
12049:
12034:
12033:
12025:
12001:tends to 1 when
11992:. In particular
11897:
11895:
11894:
11889:
11884:
11880:
11872:
11868:
11867:
11866:
11851:
11850:
11837:
11804:
11803:
11777:
11773:
11772:
11768:
11761:
11756:
11755:
11746:
11745:
11729:
11653:
11651:
11650:
11645:
11609:
11608:
11590:
11589:
11565:
11564:
11482:
11480:
11479:
11474:
11465:
11464:
11455:
11454:
11439:
11438:
11390:
11388:
11387:
11382:
11376:
11375:
11360:
11359:
11347:
11346:
11298:
11297:
11267:
11265:
11264:
11259:
11194:
11186:
11177:
11175:
11164:
11115:
11113:
11112:
11107:
11095:
11094:
11076:
11075:
11051:
11050:
11032:
11031:
10990:
10988:
10987:
10982:
10976:
10975:
10960:
10959:
10947:
10946:
10910:
10909:
10875:
10873:
10872:
10867:
10849:
10848:
10797:
10796:
10612:
10610:
10609:
10604:
10601:
10591:
10590:
10589:
10588:
10550:
10548:
10547:
10542:
10539:
10538:
10537:
10513:
10512:
10504:
10475:
10473:
10472:
10467:
10461:
10460:
10452:
10447:
10443:
10442:
10434:
10426:
10416:
10413:
10411:
10410:
10405:
10397:
10394:
10393:
10384:
10306:
10304:
10303:
10298:
10295:
10291:
10290:
10282:
10277:
10273:
10272:
10264:
10256:
10246:
10209:
10207:
10206:
10201:
10195:
10190:
10186:
10185:
10177:
10169:
10159:
10157:
10155:
10154:
10149:
10141:
10138:
10137:
10128:
10086:
10084:
10083:
10078:
10072:
10067:
10063:
10055:
10046:
10011:
10009:
10008:
10003:
9997:
9992:
9988:
9980:
9971:
9906:
9904:
9903:
9898:
9883:
9882:
9867:
9863:
9862:
9860:
9855:
9854:
9845:
9843:
9842:
9833:
9831:
9829:
9828:
9823:
9822:
9813:
9801:
9800:
9791:
9740:
9738:
9737:
9732:
9729:
9725:
9724:
9712:
9711:
9631:
9629:
9628:
9623:
9620:
9598:
9593:
9592:
9583:
9581:
9576:
9575:
9551:
9550:
9542:
9508:
9506:
9505:
9500:
9497:
9490:
9489:
9474:
9473:
9464:
9462:
9461:
9456:
9448:
9445:
9444:
9435:
9421:
9420:
9396:
9394:
9393:
9388:
9385:
9381:
9380:
9368:
9367:
9315:
9313:
9312:
9307:
9304:
9300:
9298:
9297:
9296:
9287:
9282:
9281:
9272:
9263:
9262:
9261:
9249:
9248:
9238:
9236:
9234:
9229:
9228:
9219:
9217:
9216:
9207:
9196:
9195:
9194:
9193:
9120:
9118:
9117:
9112:
9109:
9108:
9103:
9102:
9090:
9082:
9081:
9076:
9067:
9011:
9009:
9008:
9003:
9000:
8996:
8995:
8983:
8982:
8967:
8966:
8958:
8920:
8918:
8917:
8912:
8909:
8902:
8901:
8896:
8887:
8873:
8872:
8867:
8861:
8860:
8851:
8843:
8842:
8837:
8831:
8830:
8822:
8816:
8808:
8807:
8802:
8796:
8795:
8786:
8716:
8714:
8713:
8708:
8705:
8671:
8670:
8606:
8604:
8603:
8598:
8595:
8591:
8590:
8575:
8574:
8566:
8532:
8530:
8529:
8524:
8521:
8517:
8516:
8504:
8503:
8471:
8470:
8461:
8456:
8448:
8446:
8441:
8440:
8381:
8379:
8378:
8373:
8358:
8357:
8342:
8341:
8333:
8212:
8210:
8209:
8204:
8189:
8188:
8180:
8146:
8144:
8143:
8138:
8135:
8113:
8112:
8058:
8056:
8055:
8050:
8047:
8040:
8039:
8027:
8026:
8018:
7993:
7991:
7990:
7985:
7982:
7978:
7977:
7947:
7946:
7925:
7924:
7879:
7877:
7876:
7871:
7868:
7822:
7820:
7819:
7814:
7811:
7744:
7742:
7741:
7736:
7703:
7702:
7678:
7677:
7669:
7651:
7650:
7642:
7627:
7626:
7571:
7569:
7568:
7563:
7509:
7507:
7506:
7501:
7495:
7494:
7479:
7478:
7470:
7435:
7433:
7432:
7427:
7412:
7411:
7371:
7369:
7368:
7363:
7360:
7350:
7349:
7341:
7311:
7309:
7308:
7303:
7300:
7265:Cauchy transform
7251:
7249:
7248:
7243:
7240:
7215:
7214:
7206:
7175:
7173:
7172:
7167:
7164:
7160:
7158:
7147:
7115:
7113:
7112:
7107:
7105:
7104:
7096:
7075:of the operator
7064:
7062:
7061:
7056:
7050:
7049:
7034:
7033:
7025:
6991:
6989:
6988:
6983:
6968:
6967:
6959:
6956:
6951:
6943:
6941:
6927:
6926:
6921:
6913:
6851:
6849:
6848:
6843:
6822:
6820:
6819:
6817:
6816:
6798:
6780:
6757:
6752:
6750:
6746:
6744:
6743:
6725:
6698:
6697:
6685:
6684:
6671:
6636:
6635:
6623:
6622:
6609:
6604:
6602:
6601:
6592:
6591:
6583:
6577:
6564:
6562:
6561:
6556:
6554:
6549:
6548:
6539:
6530:
6528:
6527:
6522:
6520:
6519:
6510:
6505:
6504:
6496:
6480:
6478:
6477:
6472:
6466:
6454:
6452:
6451:
6433:
6397:
6395:
6394:
6376:
6344:
6343:
6335:
6320:
6318:
6317:
6312:
6306:
6294:
6292:
6291:
6273:
6237:
6235:
6234:
6216:
6184:
6183:
6163:
6161:
6160:
6155:
6077:
6075:
6074:
6069:
6066:
6065:
6048:
6046:
6045:
6027:
6000:
5999:
5939:
5937:
5936:
5931:
5923:
5900:
5882:
5880:
5879:
5874:
5854:
5852:
5851:
5833:
5763:
5761:
5760:
5755:
5721:), and thus has
5712:
5710:
5709:
5704:
5604:
5602:
5601:
5596:
5566:
5561:
5560:
5558:
5557:
5539:
5524:
5510:
5476:
5474:
5473:
5468:
5456:
5454:
5453:
5448:
5418:
5416:
5415:
5410:
5377:
5375:
5374:
5369:
5279:
5277:
5276:
5271:
5259:
5257:
5256:
5251:
5233:
5231:
5230:
5225:
5213:
5211:
5210:
5205:
5184:
5182:
5181:
5176:
5174:
5152:
5150:
5149:
5140:
5139:
5131:
5125:
5120:
5118:
5117:
5112:
5111:
5102:
5091:
5090:
5082:
5070:
5069:
5051:
5050:
5037:
5036:
5031:
5030:
5022:
5016:
5005:
5004:
4996:
4984:
4983:
4975:
4960:
4959:
4946:
4937:
4935:
4934:
4933:
4911:
4910:
4909:
4901:
4882:
4866:
4864:
4863:
4858:
4856:
4855:
4847:
4834:
4832:
4831:
4826:
4810:
4808:
4807:
4802:
4790:
4788:
4787:
4782:
4767:
4765:
4764:
4759:
4753:
4751:
4750:
4748:
4747:
4729:
4711:
4688:
4668:
4666:
4665:
4656:
4655:
4647:
4641:
4594:
4592:
4591:
4586:
4552:
4550:
4549:
4544:
4539:
4538:
4529:
4524:
4523:
4515:
4484:
4482:
4481:
4476:
4470:
4468:
4467:
4466:
4457:
4456:
4444:
4443:
4422:
4421:
4409:
4408:
4396:
4395:
4389:
4388:
4379:
4378:
4366:
4365:
4344:
4343:
4331:
4330:
4318:
4317:
4310:
4309:
4308:
4299:
4298:
4283:
4282:
4261:
4260:
4245:
4244:
4232:
4231:
4225:
4224:
4215:
4214:
4199:
4198:
4177:
4176:
4161:
4160:
4148:
4147:
4140:
4112:
4110:
4109:
4104:
4102:
4101:
4092:
4091:
4079:
4078:
4069:
4068:
4048:
4046:
4045:
4040:
4034:
4029:
4019:
4014:
4002:
4001:
3992:
3991:
3982:
3981:
3972:
3971:
3955:
3950:
3940:
3935:
3904:
3902:
3901:
3896:
3890:
3888:
3887:
3885:
3884:
3875:
3874:
3865:
3864:
3852:
3851:
3842:
3841:
3822:
3817:
3804:
3799:
3783:
3778:
3765:
3760:
3748:
3740:
3739:
3730:
3729:
3716:
3711:
3692:
3687:
3674:
3669:
3656:
3652:
3651:
3642:
3641:
3629:
3628:
3619:
3618:
3597:
3596:
3587:
3586:
3574:
3573:
3554:
3549:
3536:
3531:
3518:
3479:
3477:
3476:
3471:
3451:
3450:
3441:
3436:
3435:
3427:
3414:
3412:
3411:
3406:
3401:
3399:
3398:
3380:
3364:
3362:
3361:
3356:
3348:
3339:
3334:
3321:
3316:
3260:
3258:
3257:
3252:
3246:
3241:
3240:
3239:
3224:
3223:
3213:
3211:
3210:
3192:
3189:
3184:
3183:
3173:
3170:
3167:
3162:
3161:
3160:
3150:
3148:
3147:
3129:
3124:
3123:
3110:
3105:
3104:
3084:
3082:
3081:
3076:
3074:
3073:
3057:
3055:
3054:
3049:
3047:
3046:
3031:and solving for
3030:
3028:
3027:
3022:
3017:
3016:
3000:
2998:
2997:
2992:
2987:
2986:
2971:
2969:
2968:
2963:
2958:
2957:
2942:
2940:
2939:
2934:
2929:
2928:
2905:
2903:
2902:
2897:
2892:
2891:
2885:
2883:
2882:
2864:
2847:
2846:
2834:
2833:
2821:
2820:
2799:
2798:
2786:
2785:
2757:
2755:
2754:
2749:
2746:
2745:
2718:
2717:
2705:
2704:
2692:
2691:
2670:
2669:
2657:
2656:
2626:
2624:
2623:
2618:
2597:
2595:
2594:
2589:
2577:
2575:
2574:
2569:
2567:
2566:
2557:
2552:
2551:
2543:
2527:
2525:
2524:
2519:
2517:
2516:
2507:
2502:
2501:
2493:
2473:
2471:
2470:
2465:
2459:
2458:
2453:
2449:
2448:
2440:
2434:
2432:
2431:
2422:
2421:
2413:
2407:
2389:
2388:
2383:
2377:
2376:
2367:
2359:
2358:
2353:
2347:
2339:
2334:
2333:
2325:
2309:
2308:
2299:
2291:
2290:
2260:
2258:
2257:
2252:
2243:
2232:
2231:
2219:
2218:
2197:
2196:
2184:
2183:
2165:
2164:
2156:
2141:
2139:
2138:
2133:
2127:
2116:
2115:
2103:
2102:
2081:
2080:
2068:
2067:
2049:
2048:
2025:
2023:
2022:
2017:
1989:
1981:
1933:
1932:
1917:
1916:
1904:
1896:
1891:
1890:
1882:
1868:
1867:
1852:
1851:
1839:
1831:
1826:
1825:
1713:
1711:
1710:
1705:
1699:
1698:
1690:
1685:
1672:
1667:
1656:
1655:
1646:
1645:
1633:
1632:
1623:
1622:
1609:
1608:
1599:
1598:
1586:
1585:
1576:
1575:
1563:
1558:
1545:
1540:
1509:
1507:
1506:
1501:
1495:
1494:
1477:
1472:
1459:
1454:
1422:
1421:
1412:
1411:
1399:
1398:
1389:
1388:
1370:
1369:
1352:
1347:
1334:
1329:
1314:
1313:
1298:
1297:
1282:
1281:
1235:to pull back to
1227:
1225:
1224:
1219:
1210:
1209:
1200:
1199:
1187:
1186:
1177:
1176:
1097:
1095:
1094:
1089:
1080:
1078:
1077:
1075:
1074:
1056:
1038:
1037:
1035:
1034:
1016:
998:
993:
992:
987:
978:
959:
957:
956:
951:
945:
943:
942:
924:
901:
899:
898:
880:
789:
787:
786:
781:
778:
776:
775:
773:
772:
754:
736:
713:
641:
639:
638:
633:
630:
629:
544:
542:
541:
536:
533:
532:
490:
489:
470:
469:
435:
433:
432:
427:
397:
395:
394:
389:
274:upper half plane
227:
225:
224:
219:
217:
209:
204:
189:
187:
186:
181:
173:
126:complex variable
116:
114:
113:
108:
103:
101:
93:
85:
77:
75:
74:
66:
60:
52:
36:Eugenio Beltrami
21098:
21097:
21093:
21092:
21091:
21089:
21088:
21087:
21078:Operator theory
21058:
21057:
20959:
20939:Stillwell, John
20932:
20794:10.2307/1970228
20753:10.2307/1989904
20689:
20662:
20648:Kreyszig, Erwin
20633:
20603:
20577:
20489:
20441:
20431:
20386:
20381:
20374:
20370:
20363:
20359:
20352:
20343:
20336:
20329:
20321:
20317:
20284:
20280:
20272:
20268:
20260:
20256:
20248:
20244:
20236:
20232:
20215:
20211:
20204:
20200:
20178:
20174:
20155:
20151:
20143:
20139:
20120:
20116:
20099:
20095:
20063:
20059:
20051:
20047:
20043:
20021:
19994:
19990:
19978:
19973:
19962:
19959:
19956:
19955:
19946:
19929:
19922:
19899:
19872:
19819:
19815:
19810:
19799:
19795:
19762:
19758:
19751:
19748:
19745:
19744:
19714:
19702:
19657:
19653:
19638:
19634:
19627:
19624:
19621:
19620:
19501:
19498:
19495:
19494:
19311:
19309:
19305:
19256:
19249:
19242:
19235:
19228:
19221:
19164:
19160:
19151:
19147:
19135:
19131:
19122:
19118:
19114:
19104:
19100:
19091:
19087:
19075:
19071:
19062:
19058:
19054:
19052:
19040:
19036:
19027:
19023:
19014:
19010:
19001:
18997:
18993:
18990:
18987:
18986:
18958:
18953:
18952:
18943:
18939:
18930:
18926:
18917:
18913:
18904:
18900:
18892:
18881:
18869:
18865:
18847:
18843:
18825:
18821:
18803:
18799:
18785:
18784:
18781:
18778:
18777:
18718:
18709:
18705:
18687:
18683:
18672:
18671:
18665:
18656:
18652:
18634:
18630:
18619:
18618:
18616:
18604:
18600:
18599:
18596:
18593:
18592:
18565:
18559:
18555:
18546:
18542:
18537:
18529:
18523:
18519:
18510:
18506:
18501:
18500:
18497:
18494:
18493:
18461:
18456:
18455:
18449:
18445:
18436:
18432:
18427:
18426:
18419:
18414:
18413:
18407:
18403:
18394:
18390:
18385:
18384:
18382:
18365:
18356:
18352:
18334:
18330:
18319:
18318:
18312:
18303:
18299:
18281:
18277:
18266:
18265:
18263:
18262:
18261:
18258:
18255:
18254:
18232:
18222:
18215:
18208:
18201:
18194:
18187:
18179:
18167:
18148:. The solution
18135:
18127:
18120:
18112:
18098:
18091:
18083:
18076:
18069:
18062:
18053:Poincaré metric
18050:
18034:
17978:
17977:
17968:
17964:
17953:
17949:
17947:
17945:
17913:
17912:
17909:
17906:
17905:
17876:
17863:
17839:
17832:
17830:
17807:
17806:
17793:
17756:
17755:
17742:
17696:
17695:
17692:
17689:
17688:
17667:
17636:
17629:
17615:
17608:
17597:
17591:
17572:
17559:
17550:
17537:
17521:
17512:
17508:
17493:
17484:
17477:
17464:
17453:
17442:
17438:
17425:
17391:
17304:
17242:
17235:
17224:
17222:
17208:
17204:
17202:
17196:
17192:
17190:
17163:
17160:
17159:
17101:
17092:
17088:
17074:
17070:
17053:
17049:
17048:
17045:
17042:
17041:
16972:
16968:
16954:
16950:
16949:
16946:
16943:
16942:
16914:
16910:
16901:
16897:
16896:
16893:
16890:
16889:
16851:
16847:
16838:
16834:
16813:
16809:
16800:
16796:
16795:
16792:
16789:
16788:
16750:
16746:
16728:
16724:
16723:
16720:
16717:
16716:
16662:
16658:
16651:
16648:
16645:
16644:
16569:
16565:
16564:
16561:
16558:
16557:
16524:
16520:
16501:
16498:
16495:
16494:
16489:
16472:
16434:
16430:
16398:
16394:
16382:
16379:
16378:
16367:
16339:
16330:
16272:
16268:
16258:
16251:
16238:
16234:
16233:
16231:
16217:
16213:
16211:
16205:
16201:
16199:
16172:
16169:
16168:
16051:
16047:
16036:
16032:
16030:
16028:
16007:
16003:
16002:
15999:
15996:
15995:
15988:or equivalently
15942:
15938:
15936:
15930:
15926:
15924:
15894:
15890:
15889:
15886:
15883:
15882:
15866:
15859:
15850:
15833:
15827:is non-zero at
15822:
15805:
15794:
15787:
15770:
15763:
15754:
15715:
15711:
15691:
15687:
15686:
15683:
15680:
15679:
15672:is large enough
15667:
15660:
15617:
15600:
15587:
15566:
15553:
15514:
15513:
15500:
15490:
15486:
15470:
15465:
15455:
15450:
15449:
15443:
15439:
15433:
15429:
15403:
15398:
15397:
15391:
15387:
15381:
15377:
15372:
15366:
15362:
15353:
15352:
15332:
15327:
15326:
15320:
15316:
15305:
15301:
15296:
15287:
15282:
15281:
15275:
15271:
15265:
15261:
15256:
15250:
15246:
15220:
15216:
15204:
15200:
15193:
15175:
15171:
15161:
15159:
15156:
15155:
15111:
15103:
15068:
15064:
15055:
15051:
15030:
15026:
15005:
15001:
14995:
14991:
14981:
14978:
14975:
14974:
14958:
14949:
14940:
14931:
14922:
14913:
14876:
14825:
14821:
14792:
14788:
14779:
14775:
14751:
14747:
14725:
14722:
14719:
14718:
14661:
14657:
14651:
14647:
14637:
14630:
14626:
14620:
14616:
14612:
14610:
14601:
14597:
14591:
14587:
14573:
14569:
14563:
14559:
14555:
14552:
14549:
14548:
14511:
14507:
14501:
14497:
14488:
14484:
14473:
14469:
14458:
14454:
14445:
14441:
14440:
14437:
14434:
14433:
14389:
14380:
14371:
14362:
14337:
14328:
14299:
14290:
14266:
14234:
14230:
14215:
14211:
14205:
14201:
14197:
14194:
14191:
14190:
14177:
14136:
14132:
14114:
14110:
14109:
14106:
14103:
14102:
14098:increases. Set
14093:
14087:
14056:
14051:
13975:
13971:
13965:
13961:
13948:
13941:
13937:
13921:
13917:
13913:
13907:
13903:
13902:
13900:
13893:
13889:
13881:
13864:
13861:
13858:
13857:
13845:
13834:
13805:
13801:
13785:
13781:
13777:
13771:
13767:
13752:
13748:
13739:
13736:
13735:
13697:
13691:
13687:
13678:
13674:
13669:
13656:
13646:
13641:
13640:
13634:
13630:
13621:
13617:
13612:
13606:
13602:
13587:
13583:
13577:
13573:
13560:
13554:
13550:
13548:
13540:
13531:
13527:
13509:
13505:
13494:
13491:
13488:
13487:
13459:
13422:
13416:
13412:
13403:
13399:
13378:
13374:
13362:
13358:
13355:
13352:
13351:
13320:
13310:
13305:
13304:
13298:
13294:
13285:
13281:
13276:
13270:
13266:
13254:
13250:
13242:
13233:
13229:
13208:
13204:
13190:
13187:
13184:
13183:
13111:
13107:
13086:
13083:
13082:
13038:
13034:
13002:
12998:
12986:
12983:
12982:
12946:
12945:
12906:
12889:
12887:
12880:
12879:
12875:
12865:
12830:
12814:
12809:
12808:
12804:
12785:
12784:
12780:
12770:
12758:
12757:
12724:
12722:
12715:
12714:
12710:
12701:
12697:
12672:
12653:
12651:
12648:
12647:
12572:
12568:
12547:
12544:
12543:
12498:
12494:
12480:
12476:
12473:
12470:
12469:
12435:
12431:
12407:
12403:
12382:
12378:
12357:
12354:
12353:
12255:
12251:
12225:
12221:
12198:
12194:
12179:
12175:
12168:
12165:
12162:
12161:
12156:
12147:
12081:
12078:
12077:
12045:
12041:
12024:
12020:
12019:
12016:
12013:
12012:
12000:
11939:
11933:= 0 is smooth.
11856:
11852:
11846:
11842:
11827:
11822:
11818:
11799:
11795:
11785:
11781:
11757:
11751:
11747:
11735:
11731:
11719:
11714:
11710:
11688:
11684:
11682:
11679:
11678:
11665:
11604:
11600:
11585:
11581:
11560:
11556:
11526:
11523:
11522:
11460:
11456:
11450:
11446:
11434:
11430:
11406:
11403:
11402:
11371:
11367:
11355:
11351:
11342:
11338:
11293:
11289:
11283:
11280:
11279:
11187:
11182:
11168:
11163:
11154:
11151:
11150:
11090:
11086:
11068:
11064:
11046:
11042:
11024:
11020:
11002:
10999:
10998:
10971:
10967:
10955:
10951:
10942:
10938:
10905:
10901:
10895:
10892:
10891:
10844:
10840:
10792:
10788:
10731:
10728:
10727:
10648:for the domain
10581:
10577:
10570:
10566:
10565:
10562:
10559:
10558:
10533:
10529:
10503:
10499:
10498:
10495:
10492:
10491:
10453:
10435:
10425:
10424:
10417:
10415:
10414:
10406:
10396:
10395:
10389:
10385:
10383:
10365:
10362:
10361:
10330:
10324:
10283:
10265:
10255:
10254:
10247:
10245:
10244:
10228:
10225:
10222:
10221:
10178:
10168:
10167:
10160:
10158:
10150:
10140:
10139:
10133:
10129:
10127:
10109:
10106:
10105:
10054:
10047:
10045:
10027:
10024:
10023:
9979:
9972:
9970:
9952:
9949:
9948:
9875:
9871:
9850:
9846:
9844:
9838:
9834:
9832:
9818:
9814:
9812:
9802:
9796:
9792:
9790:
9789:
9785:
9767:
9764:
9763:
9754:
9720:
9716:
9707:
9703:
9666:
9663:
9660:
9659:
9588:
9584:
9582:
9577:
9571:
9567:
9541:
9540:
9539:
9536:
9533:
9532:
9514:Riemann surface
9482:
9478:
9469:
9465:
9457:
9447:
9446:
9440:
9436:
9434:
9416:
9412:
9411:
9408:
9405:
9404:
9373:
9369:
9360:
9356:
9334:
9331:
9328:
9327:
9292:
9288:
9277:
9273:
9271:
9264:
9257:
9253:
9244:
9240:
9239:
9237:
9224:
9220:
9218:
9212:
9208:
9206:
9186:
9182:
9175:
9171:
9170:
9167:
9164:
9163:
9152:is one-to-one.
9104:
9095:
9091:
9086:
9077:
9072:
9071:
9063:
9038:
9035:
9032:
9031:
8991:
8987:
8978:
8974:
8957:
8953:
8952:
8949:
8946:
8945:
8936:
8897:
8892:
8891:
8883:
8868:
8863:
8862:
8856:
8852:
8847:
8838:
8833:
8832:
8821:
8817:
8812:
8803:
8798:
8797:
8791:
8787:
8782:
8766:
8763:
8760:
8759:
8736:and hence also
8666:
8662:
8661:
8658:
8655:
8654:
8619:
8586:
8582:
8565:
8561:
8554:
8551:
8548:
8547:
8512:
8508:
8499:
8495:
8466:
8462:
8447:
8445:
8436:
8432:
8428:
8425:
8422:
8421:
8397:
8353:
8349:
8332:
8328:
8325:
8322:
8321:
8179:
8175:
8172:
8169:
8168:
8108:
8104:
8085:
8082:
8079:
8078:
8035:
8031:
8017:
8013:
8012:
8009:
8006:
8005:
7970:
7966:
7942:
7938:
7917:
7913:
7894:
7891:
7888:
7887:
7837:
7834:
7831:
7830:
7767:
7764:
7761:
7760:
7698:
7694:
7668:
7664:
7641:
7637:
7622:
7618:
7609:
7606:
7605:
7600:
7591:
7525:
7522:
7521:
7490:
7486:
7469:
7465:
7462:
7459:
7458:
7407:
7403:
7391:
7388:
7387:
7340:
7336:
7326:
7323:
7320:
7319:
7278:
7275:
7272:
7271:
7205:
7201:
7200:
7197:
7194:
7193:
7151:
7146:
7130:
7127:
7124:
7123:
7095:
7091:
7083:
7080:
7079:
7045:
7041:
7024:
7020:
7011:
7008:
7007:
6958:
6957:
6942:
6940:
6914:
6912:
6911:
6908:
6905:
6904:
6869:
6812:
6808:
6797:
6781:
6758:
6756:
6739:
6735:
6724:
6693:
6689:
6680:
6676:
6672:
6631:
6627:
6618:
6614:
6610:
6608:
6597:
6593:
6582:
6578:
6576:
6573:
6570:
6569:
6544:
6540:
6538:
6536:
6533:
6532:
6515:
6511:
6506:
6495:
6491:
6489:
6486:
6485:
6462:
6447:
6443:
6432:
6390:
6386:
6375:
6334:
6330:
6327:
6324:
6323:
6302:
6287:
6283:
6272:
6230:
6226:
6215:
6179:
6175:
6172:
6169:
6168:
6086:
6083:
6082:
6061:
6060:
6041:
6037:
6026:
5995:
5994:
5964:
5961:
5960:
5919:
5893:
5891:
5888:
5887:
5847:
5843:
5832:
5804:
5801:
5800:
5749:
5746:
5745:
5736:This rule sets
5689:
5686:
5685:
5553:
5549:
5538:
5525:
5523:
5503:
5500:
5497:
5496:
5462:
5459:
5458:
5436:
5433:
5432:
5383:
5380:
5379:
5300:
5297:
5296:
5265:
5262:
5261:
5239:
5236:
5235:
5219:
5216:
5215:
5193:
5190:
5189:
5172:
5171:
5145:
5141:
5130:
5126:
5124:
5107:
5103:
5101:
5081:
5077:
5065:
5061:
5046:
5042:
5038:
5021:
5017:
5015:
4995:
4991:
4974:
4970:
4955:
4951:
4947:
4945:
4938:
4929:
4925:
4912:
4900:
4896:
4883:
4881:
4877:
4875:
4872:
4871:
4846:
4842:
4840:
4837:
4836:
4820:
4817:
4816:
4796:
4793:
4792:
4776:
4773:
4772:
4743:
4739:
4728:
4712:
4689:
4687:
4661:
4657:
4646:
4642:
4640:
4637:
4634:
4633:
4600:
4568:
4565:
4564:
4534:
4530:
4525:
4514:
4510:
4493:
4490:
4489:
4462:
4461:
4452:
4448:
4439:
4435:
4417:
4413:
4404:
4400:
4391:
4390:
4384:
4383:
4374:
4370:
4361:
4357:
4339:
4335:
4326:
4322:
4313:
4312:
4311:
4304:
4303:
4294:
4290:
4278:
4274:
4256:
4252:
4240:
4236:
4227:
4226:
4220:
4219:
4210:
4206:
4194:
4190:
4172:
4168:
4156:
4152:
4143:
4142:
4141:
4139:
4121:
4118:
4117:
4097:
4093:
4087:
4083:
4074:
4070:
4064:
4060:
4058:
4055:
4054:
4030:
4025:
4015:
4010:
3997:
3993:
3987:
3983:
3977:
3973:
3967:
3963:
3951:
3946:
3936:
3931:
3925:
3922:
3921:
3880:
3876:
3870:
3866:
3860:
3856:
3847:
3843:
3837:
3833:
3818:
3813:
3800:
3795:
3779:
3774:
3761:
3756:
3747:
3735:
3731:
3725:
3721:
3712:
3707:
3688:
3683:
3670:
3665:
3657:
3647:
3643:
3637:
3633:
3624:
3620:
3614:
3610:
3592:
3588:
3582:
3578:
3569:
3565:
3550:
3545:
3532:
3527:
3519:
3517:
3499:
3496:
3495:
3484:is isothermal.
3446:
3442:
3437:
3426:
3422:
3420:
3417:
3416:
3394:
3390:
3379:
3374:
3371:
3370:
3344:
3335:
3330:
3317:
3312:
3297:
3294:
3293:
3235:
3231:
3219:
3215:
3206:
3202:
3191:
3190:
3188:
3179:
3175:
3169:
3156:
3152:
3143:
3139:
3128:
3119:
3115:
3111:
3109:
3100:
3096:
3093:
3090:
3089:
3069:
3065:
3063:
3060:
3059:
3042:
3038:
3036:
3033:
3032:
3012:
3008:
3006:
3003:
3002:
2982:
2978:
2976:
2973:
2972:
2953:
2949:
2947:
2944:
2943:
2924:
2920:
2918:
2915:
2914:
2887:
2886:
2878:
2874:
2863:
2842:
2841:
2829:
2825:
2816:
2812:
2794:
2790:
2781:
2777:
2766:
2763:
2762:
2741:
2740:
2713:
2712:
2700:
2696:
2687:
2683:
2665:
2661:
2652:
2648:
2639:
2636:
2635:
2603:
2600:
2599:
2583:
2580:
2579:
2562:
2558:
2553:
2542:
2538:
2536:
2533:
2532:
2512:
2508:
2503:
2492:
2488:
2486:
2483:
2482:
2454:
2439:
2427:
2423:
2412:
2408:
2406:
2396:
2392:
2391:
2384:
2379:
2378:
2372:
2368:
2363:
2354:
2349:
2348:
2338:
2324:
2320:
2304:
2300:
2295:
2286:
2282:
2276:
2273:
2272:
2239:
2227:
2223:
2214:
2210:
2192:
2188:
2179:
2175:
2155:
2151:
2148:
2145:
2144:
2123:
2111:
2107:
2098:
2094:
2076:
2072:
2063:
2059:
2044:
2040:
2037:
2034:
2033:
1980:
1928:
1924:
1912:
1908:
1895:
1881:
1877:
1863:
1859:
1847:
1843:
1830:
1821:
1817:
1814:
1811:
1810:
1693:
1692:
1686:
1681:
1668:
1663:
1657:
1651:
1647:
1641:
1637:
1628:
1624:
1618:
1614:
1611:
1610:
1604:
1600:
1594:
1590:
1581:
1577:
1571:
1567:
1565:
1559:
1554:
1541:
1536:
1525:
1524:
1521:
1518:
1517:
1490:
1486:
1473:
1468:
1455:
1450:
1417:
1413:
1407:
1403:
1394:
1390:
1384:
1380:
1365:
1361:
1348:
1343:
1330:
1325:
1309:
1305:
1293:
1289:
1277:
1273:
1267:
1264:
1263:
1205:
1201:
1195:
1191:
1182:
1178:
1172:
1168:
1165:
1162:
1161:
1070:
1066:
1055:
1039:
1030:
1026:
1015:
999:
997:
988:
983:
982:
974:
971:
968:
967:
938:
934:
923:
894:
890:
879:
801:
798:
797:
768:
764:
753:
737:
714:
712:
688:
685:
684:
624:
623:
618:
612:
611:
606:
596:
595:
571:
568:
567:
528:
524:
485:
481:
465:
461:
455:
452:
451:
403:
400:
399:
365:
362:
361:
320:, say with an (
314:
208:
200:
195:
192:
191:
169:
164:
161:
160:
94:
86:
84:
65:
61:
53:
51:
49:
46:
45:
24:
17:
12:
11:
5:
21096:
21086:
21085:
21080:
21075:
21070:
21056:
21055:
21046:
21014:
20992:
20970:
20957:
20935:
20930:
20917:
20895:
20873:
20855:
20830:
20805:
20788:(3): 521–554,
20773:
20732:
20692:
20687:
20674:
20665:
20660:
20644:
20631:
20606:
20601:
20588:
20575:
20560:
20547:
20539:Douady, Adrien
20535:
20513:
20504:(3): 215–228,
20493:
20487:
20474:
20465:
20448:(in Italian),
20434:
20429:
20411:Astala, Kari;
20408:
20406:, Van Nostrand
20396:
20385:
20382:
20380:
20379:
20368:
20357:
20341:
20327:
20315:
20313:
20312:
20306:
20301:
20296:
20291:
20278:
20266:
20254:
20242:
20230:
20228:
20227:
20222:
20209:
20198:
20196:
20195:
20190:
20185:
20172:
20170:
20169:
20163:
20149:
20137:
20135:
20134:
20128:
20114:
20112:
20111:
20106:
20093:
20091:
20090:
20085:
20080:
20075:
20070:
20057:
20044:
20042:
20039:
20038:
20037:
20032:
20027:
20020:
20017:
20016:
20015:
20002:
19997:
19993:
19989:
19984:
19981:
19976:
19972:
19968:
19965:
19942:
19934:| < 1 and |
19927:
19920:
19897:
19870:
19841:
19840:
19827:
19822:
19818:
19813:
19806:
19803:
19798:
19794:
19791:
19788:
19785:
19782:
19776:
19773:
19770:
19765:
19761:
19757:
19754:
19691:
19690:
19677:
19672:
19669:
19666:
19663:
19660:
19656:
19652:
19649:
19644:
19641:
19637:
19633:
19630:
19614:
19613:
19600:
19597:
19594:
19591:
19588:
19585:
19582:
19579:
19576:
19573:
19570:
19567:
19564:
19561:
19558:
19555:
19552:
19549:
19546:
19543:
19540:
19537:
19534:
19531:
19528:
19525:
19522:
19519:
19516:
19513:
19510:
19507:
19504:
19260:Conversely if
19254:
19247:
19240:
19233:
19226:
19219:
19189:
19188:
19172:
19167:
19163:
19159:
19154:
19150:
19146:
19143:
19138:
19134:
19130:
19125:
19121:
19117:
19112:
19107:
19103:
19099:
19094:
19090:
19086:
19083:
19078:
19074:
19070:
19065:
19061:
19057:
19051:
19048:
19043:
19039:
19035:
19030:
19026:
19022:
19017:
19013:
19009:
19004:
19000:
18996:
18980:
18979:
18966:
18961:
18956:
18951:
18946:
18942:
18938:
18933:
18929:
18925:
18920:
18916:
18912:
18907:
18903:
18899:
18895:
18891:
18888:
18884:
18880:
18877:
18872:
18868:
18864:
18861:
18858:
18855:
18850:
18846:
18842:
18839:
18836:
18833:
18828:
18824:
18820:
18817:
18814:
18811:
18806:
18802:
18798:
18795:
18792:
18788:
18748:
18747:
18734:
18731:
18728:
18721:
18717:
18712:
18708:
18704:
18701:
18698:
18695:
18690:
18686:
18682:
18679:
18675:
18668:
18664:
18659:
18655:
18651:
18648:
18645:
18642:
18637:
18633:
18629:
18626:
18622:
18615:
18610:
18607:
18603:
18586:
18585:
18572:
18568:
18562:
18558:
18554:
18549:
18545:
18540:
18536:
18532:
18526:
18522:
18518:
18513:
18509:
18504:
18487:
18486:
18473:
18464:
18459:
18452:
18448:
18444:
18439:
18435:
18430:
18422:
18417:
18410:
18406:
18402:
18397:
18393:
18388:
18381:
18378:
18375:
18368:
18364:
18359:
18355:
18351:
18348:
18345:
18342:
18337:
18333:
18329:
18326:
18322:
18315:
18311:
18306:
18302:
18298:
18295:
18292:
18289:
18284:
18280:
18276:
18273:
18269:
18240:quasisymmetric
18221:
18218:
18213:
18206:
18199:
18192:
18185:
18177:
18165:
18133:
18125:
18118:
18110:
18096:
18092:. In this way
18089:
18081:
18074:
18067:
18060:
18048:
18032:
18013:
18012:
18000:
17997:
17994:
17991:
17985:
17982:
17971:
17967:
17961:
17956:
17952:
17944:
17941:
17938:
17935:
17932:
17929:
17926:
17920:
17917:
17899:
17898:
17886:
17883:
17879:
17875:
17870:
17867:
17862:
17855:
17851:
17846:
17843:
17838:
17835:
17829:
17826:
17823:
17820:
17814:
17811:
17803:
17800:
17796:
17792:
17789:
17786:
17781:
17778:
17775:
17772:
17769:
17763:
17760:
17752:
17749:
17745:
17741:
17738:
17735:
17730:
17727:
17724:
17721:
17718:
17715:
17712:
17709:
17703:
17700:
17684:by reflection
17665:
17659:Munkres (1960)
17634:
17627:
17613:
17606:
17590:
17587:
17568:
17555:
17546:
17535:
17517:
17510:
17506:
17489:
17482:
17473:
17462:
17456:Schottky group
17451:
17440:
17436:
17423:
17389:
17317:of an annulus
17303:
17300:
17273:
17272:
17260:
17254:
17249:
17246:
17241:
17238:
17233:
17230:
17227:
17216:
17211:
17207:
17199:
17195:
17189:
17186:
17183:
17180:
17177:
17174:
17171:
17168:
17127:For a general
17125:
17124:
17111:
17108:
17104:
17100:
17095:
17091:
17087:
17081:
17078:
17073:
17067:
17064:
17061:
17056:
17052:
16999:
16998:
16985:
16979:
16976:
16971:
16967:
16961:
16958:
16953:
16936:
16935:
16922:
16917:
16913:
16909:
16904:
16900:
16879:
16878:
16865:
16862:
16859:
16854:
16850:
16846:
16841:
16837:
16833:
16830:
16827:
16824:
16821:
16816:
16812:
16808:
16803:
16799:
16778:
16777:
16764:
16761:
16758:
16753:
16749:
16745:
16742:
16739:
16736:
16731:
16727:
16710:
16709:
16696:
16693:
16690:
16687:
16684:
16681:
16678:
16675:
16669:
16666:
16661:
16657:
16654:
16628:(0) = 0 force
16602:
16601:
16588:
16585:
16582:
16576:
16573:
16568:
16551:
16550:
16537:
16531:
16528:
16523:
16519:
16516:
16513:
16510:
16507:
16504:
16490:- 1 in L then
16485:
16468:
16461:
16460:
16448:
16445:
16442:
16437:
16433:
16429:
16426:
16423:
16417:
16414:
16411:
16405:
16402:
16397:
16393:
16390:
16387:
16366:
16363:
16350:
16349:
16335:
16326:
16298:
16297:
16296:
16295:
16283:
16275:
16271:
16265:
16262:
16257:
16254:
16249:
16246:
16241:
16237:
16225:
16220:
16216:
16208:
16204:
16198:
16195:
16192:
16189:
16186:
16183:
16180:
16177:
16163:
16162:
16135:
16111:
16110:
16098:
16097:
16096:
16095:
16082:
16079:
16076:
16073:
16070:
16067:
16064:
16061:
16054:
16050:
16044:
16039:
16035:
16027:
16024:
16021:
16018:
16015:
16010:
16006:
15990:
15989:
15985:
15984:
15983:
15982:
15969:
15966:
15963:
15960:
15957:
15950:
15945:
15941:
15933:
15929:
15923:
15920:
15917:
15914:
15911:
15908:
15905:
15902:
15897:
15893:
15877:
15876:
15869:
15868:
15864:
15855:
15846:
15831:
15818:
15803:
15792:
15783:
15768:
15759:
15752:
15745:
15744:
15743:
15742:
15729:
15726:
15723:
15718:
15714:
15710:
15707:
15704:
15698:
15695:
15690:
15674:
15673:
15665:
15656:
15647:
15632:
15631:
15613:
15596:
15583:
15562:
15549:
15530:
15529:
15528:
15527:
15512:
15507:
15503:
15499:
15496:
15493:
15489:
15485:
15482:
15479:
15473:
15468:
15464:
15458:
15453:
15446:
15442:
15436:
15432:
15428:
15425:
15422:
15419:
15415:
15412:
15406:
15401:
15394:
15390:
15384:
15380:
15375:
15369:
15365:
15361:
15358:
15356:
15354:
15351:
15348:
15344:
15341:
15335:
15330:
15323:
15319:
15312:
15309:
15304:
15299:
15295:
15290:
15285:
15278:
15274:
15268:
15264:
15259:
15253:
15249:
15245:
15242:
15239:
15235:
15232:
15228:
15223:
15219:
15215:
15212:
15207:
15203:
15199:
15196:
15194:
15192:
15189:
15186:
15183:
15178:
15174:
15170:
15167:
15164:
15163:
15150:
15149:
15126:
15125:
15107:
15099:
15092:
15091:
15090:
15089:
15076:
15071:
15067:
15063:
15058:
15054:
15050:
15047:
15044:
15041:
15038:
15033:
15029:
15025:
15022:
15019:
15016:
15013:
15008:
15004:
14998:
14994:
14990:
14987:
14984:
14969:
14968:
14954:
14945:
14936:
14927:
14918:
14909:
14879:
14878:
14872:
14857:
14856:
14855:
14854:
14841:
14838:
14832:
14829:
14824:
14820:
14817:
14814:
14811:
14808:
14805:
14799:
14796:
14791:
14787:
14782:
14778:
14774:
14771:
14768:
14765:
14762:
14759:
14754:
14750:
14746:
14743:
14740:
14737:
14734:
14731:
14728:
14713:
14712:
14688:
14687:
14686:
14685:
14672:
14664:
14660:
14654:
14650:
14646:
14643:
14640:
14633:
14629:
14623:
14619:
14615:
14609:
14604:
14600:
14594:
14590:
14586:
14581:
14576:
14572:
14566:
14562:
14558:
14543:
14542:
14538:
14537:
14536:
14535:
14522:
14519:
14514:
14510:
14504:
14500:
14496:
14491:
14487:
14481:
14476:
14472:
14465:
14462:
14457:
14453:
14448:
14444:
14428:
14427:
14385:
14376:
14367:
14358:
14333:
14324:
14302:equicontinuous
14295:
14286:
14262:
14256:
14255:
14242:
14237:
14233:
14229:
14226:
14223:
14218:
14214:
14208:
14204:
14200:
14173:
14167:
14166:
14153:
14150:
14147:
14144:
14139:
14135:
14131:
14128:
14125:
14122:
14117:
14113:
14089:
14083:
14055:
14052:
14050:
14047:
14019:
14018:
14006:
14003:
14000:
13997:
13994:
13991:
13987:
13978:
13974:
13968:
13964:
13960:
13957:
13954:
13951:
13944:
13940:
13936:
13933:
13928:
13924:
13920:
13916:
13910:
13906:
13899:
13896:
13892:
13888:
13884:
13880:
13877:
13874:
13871:
13867:
13843:
13832:
13826:
13825:
13813:
13808:
13804:
13800:
13797:
13792:
13788:
13784:
13780:
13774:
13770:
13766:
13763:
13760:
13755:
13751:
13747:
13744:
13717:
13716:
13704:
13700:
13694:
13690:
13686:
13681:
13677:
13672:
13668:
13663:
13659:
13655:
13652:
13649:
13644:
13637:
13633:
13629:
13624:
13620:
13615:
13609:
13605:
13601:
13598:
13590:
13586:
13580:
13576:
13572:
13569:
13566:
13563:
13557:
13553:
13547:
13543:
13539:
13534:
13530:
13526:
13523:
13520:
13517:
13512:
13508:
13504:
13501:
13497:
13455:
13445:
13444:
13432:
13429:
13425:
13419:
13415:
13409:
13406:
13402:
13398:
13395:
13392:
13389:
13384:
13381:
13377:
13373:
13370:
13365:
13361:
13345:
13344:
13332:
13327:
13323:
13319:
13316:
13313:
13308:
13301:
13297:
13293:
13288:
13284:
13279:
13273:
13269:
13265:
13262:
13257:
13253:
13249:
13245:
13241:
13236:
13232:
13228:
13225:
13222:
13219:
13216:
13211:
13207:
13203:
13200:
13197:
13193:
13153:
13152:
13140:
13137:
13134:
13131:
13128:
13125:
13122:
13117:
13114:
13110:
13106:
13103:
13100:
13097:
13094:
13091:
13068:
13067:
13055:
13052:
13049:
13046:
13041:
13037:
13033:
13030:
13027:
13021:
13018:
13015:
13009:
13006:
13001:
12997:
12994:
12991:
12960:
12959:
12944:
12941:
12938:
12934:
12931:
12924:
12921:
12918:
12915:
12912:
12909:
12904:
12901:
12898:
12895:
12892:
12883:
12878:
12872:
12869:
12864:
12861:
12858:
12855:
12851:
12848:
12843:
12837:
12834:
12829:
12823:
12820:
12817:
12813:
12807:
12803:
12800:
12797:
12794:
12788:
12783:
12777:
12774:
12769:
12766:
12763:
12761:
12759:
12756:
12753:
12749:
12746:
12740:
12736:
12733:
12730:
12727:
12718:
12713:
12707:
12704:
12700:
12696:
12693:
12690:
12687:
12684:
12681:
12678:
12675:
12673:
12671:
12668:
12665:
12662:
12659:
12656:
12655:
12614:
12613:
12601:
12598:
12595:
12592:
12589:
12586:
12583:
12578:
12575:
12571:
12567:
12564:
12561:
12558:
12555:
12552:
12525:
12524:
12512:
12509:
12504:
12501:
12497:
12493:
12487:
12484:
12479:
12463:Neumann series
12459:
12458:
12446:
12441:
12438:
12434:
12430:
12427:
12424:
12421:
12418:
12413:
12410:
12406:
12399:
12396:
12393:
12388:
12385:
12381:
12377:
12374:
12371:
12368:
12365:
12362:
12307:
12306:
12293:
12290:
12287:
12284:
12281:
12278:
12275:
12272:
12269:
12266:
12263:
12258:
12254:
12250:
12247:
12244:
12241:
12238:
12232:
12229:
12224:
12220:
12217:
12214:
12211:
12205:
12202:
12197:
12193:
12190:
12187:
12182:
12178:
12174:
12171:
12152:
12143:
12129:
12128:
12116:
12113:
12110:
12107:
12104:
12101:
12098:
12095:
12092:
12089:
12086:
12067:
12066:
12053:
12048:
12044:
12040:
12037:
12031:
12028:
12023:
11996:
11938:
11935:
11899:
11898:
11887:
11883:
11879:
11876:
11871:
11865:
11862:
11859:
11855:
11849:
11845:
11841:
11836:
11833:
11830:
11826:
11821:
11817:
11814:
11811:
11807:
11802:
11798:
11794:
11791:
11788:
11784:
11780:
11776:
11771:
11767:
11764:
11760:
11754:
11750:
11744:
11741:
11738:
11734:
11728:
11725:
11722:
11718:
11713:
11709:
11706:
11703:
11700:
11697:
11694:
11691:
11687:
11661:
11655:
11654:
11642:
11639:
11636:
11633:
11630:
11627:
11624:
11621:
11618:
11615:
11612:
11607:
11603:
11599:
11596:
11593:
11588:
11584:
11580:
11577:
11574:
11571:
11568:
11563:
11559:
11555:
11552:
11549:
11546:
11543:
11540:
11537:
11534:
11531:
11484:
11483:
11471:
11468:
11463:
11459:
11453:
11449:
11445:
11442:
11437:
11433:
11429:
11426:
11423:
11420:
11417:
11414:
11411:
11392:
11391:
11379:
11374:
11370:
11366:
11363:
11358:
11354:
11350:
11345:
11341:
11337:
11334:
11331:
11328:
11325:
11322:
11319:
11316:
11313:
11310:
11307:
11304:
11301:
11296:
11292:
11288:
11269:
11268:
11256:
11253:
11250:
11247:
11244:
11241:
11238:
11235:
11232:
11229:
11226:
11223:
11217:
11214:
11211:
11207:
11204:
11201:
11198:
11193:
11190:
11185:
11181:
11174:
11171:
11167:
11162:
11159:
11117:
11116:
11104:
11101:
11098:
11093:
11089:
11085:
11082:
11079:
11074:
11071:
11067:
11063:
11060:
11057:
11054:
11049:
11045:
11041:
11038:
11035:
11030:
11027:
11023:
11019:
11016:
11013:
11010:
11007:
10992:
10991:
10979:
10974:
10970:
10966:
10963:
10958:
10954:
10950:
10945:
10941:
10937:
10934:
10931:
10928:
10925:
10922:
10919:
10916:
10913:
10908:
10904:
10900:
10877:
10876:
10864:
10861:
10858:
10855:
10852:
10847:
10843:
10839:
10836:
10833:
10830:
10827:
10824:
10821:
10818:
10815:
10812:
10809:
10806:
10803:
10800:
10795:
10791:
10787:
10784:
10781:
10778:
10775:
10772:
10769:
10766:
10763:
10760:
10757:
10754:
10751:
10748:
10745:
10742:
10739:
10736:
10719:by arclength,
10614:
10613:
10600:
10597:
10594:
10587:
10584:
10580:
10576:
10573:
10569:
10552:
10551:
10536:
10532:
10528:
10525:
10522:
10519:
10516:
10510:
10507:
10502:
10477:
10476:
10464:
10459:
10456:
10450:
10446:
10441:
10438:
10432:
10429:
10423:
10420:
10409:
10403:
10400:
10392:
10388:
10382:
10379:
10376:
10373:
10370:
10323:
10320:
10308:
10307:
10294:
10289:
10286:
10280:
10276:
10271:
10268:
10262:
10259:
10253:
10250:
10243:
10240:
10237:
10234:
10231:
10211:
10210:
10198:
10193:
10189:
10184:
10181:
10175:
10172:
10166:
10163:
10153:
10147:
10144:
10136:
10132:
10126:
10123:
10120:
10117:
10114:
10088:
10087:
10075:
10070:
10066:
10061:
10058:
10053:
10050:
10044:
10041:
10038:
10035:
10032:
10013:
10012:
10000:
9995:
9991:
9986:
9983:
9978:
9975:
9969:
9966:
9963:
9960:
9957:
9908:
9907:
9895:
9892:
9889:
9886:
9881:
9878:
9874:
9870:
9866:
9858:
9853:
9849:
9841:
9837:
9826:
9821:
9817:
9811:
9808:
9805:
9799:
9795:
9788:
9784:
9781:
9778:
9775:
9772:
9752:
9742:
9741:
9728:
9723:
9719:
9715:
9710:
9706:
9702:
9699:
9696:
9693:
9690:
9687:
9684:
9681:
9678:
9675:
9672:
9669:
9633:
9632:
9619:
9616:
9613:
9610:
9607:
9604:
9601:
9596:
9591:
9587:
9580:
9574:
9570:
9566:
9563:
9560:
9557:
9554:
9548:
9545:
9510:
9509:
9496:
9493:
9488:
9485:
9481:
9477:
9472:
9468:
9460:
9454:
9451:
9443:
9439:
9433:
9430:
9427:
9424:
9419:
9415:
9398:
9397:
9384:
9379:
9376:
9372:
9366:
9363:
9359:
9355:
9352:
9349:
9346:
9343:
9340:
9337:
9317:
9316:
9303:
9295:
9291:
9285:
9280:
9276:
9270:
9267:
9260:
9256:
9252:
9247:
9243:
9232:
9227:
9223:
9215:
9211:
9205:
9202:
9199:
9192:
9189:
9185:
9181:
9178:
9174:
9122:
9121:
9107:
9101:
9098:
9094:
9089:
9085:
9080:
9075:
9070:
9066:
9062:
9059:
9056:
9053:
9050:
9047:
9044:
9041:
9013:
9012:
8999:
8994:
8990:
8986:
8981:
8977:
8973:
8970:
8964:
8961:
8956:
8932:
8926:Ahlfors (1966)
8922:
8921:
8908:
8905:
8900:
8895:
8890:
8886:
8882:
8879:
8876:
8871:
8866:
8859:
8855:
8850:
8846:
8841:
8836:
8828:
8825:
8820:
8815:
8811:
8806:
8801:
8794:
8790:
8785:
8781:
8778:
8775:
8772:
8769:
8718:
8717:
8704:
8701:
8698:
8695:
8692:
8689:
8686:
8683:
8680:
8677:
8674:
8669:
8665:
8615:
8608:
8607:
8594:
8589:
8585:
8581:
8578:
8572:
8569:
8564:
8560:
8557:
8534:
8533:
8520:
8515:
8511:
8507:
8502:
8498:
8494:
8491:
8488:
8485:
8482:
8479:
8476:
8469:
8465:
8459:
8454:
8451:
8444:
8439:
8435:
8431:
8393:
8383:
8382:
8370:
8367:
8364:
8361:
8356:
8352:
8348:
8345:
8339:
8336:
8331:
8251:| tends to ∞.
8214:
8213:
8201:
8198:
8195:
8192:
8186:
8183:
8178:
8148:
8147:
8134:
8131:
8128:
8125:
8122:
8119:
8116:
8111:
8107:
8103:
8100:
8097:
8094:
8091:
8088:
8060:
8059:
8046:
8043:
8038:
8034:
8030:
8024:
8021:
8016:
7999:Neumann series
7995:
7994:
7981:
7976:
7973:
7969:
7965:
7962:
7959:
7956:
7953:
7950:
7945:
7941:
7934:
7931:
7928:
7923:
7920:
7916:
7912:
7909:
7906:
7903:
7900:
7897:
7881:
7880:
7867:
7864:
7861:
7858:
7855:
7852:
7849:
7846:
7843:
7840:
7824:
7823:
7810:
7807:
7804:
7801:
7798:
7795:
7788:
7785:
7782:
7779:
7776:
7773:
7770:
7746:
7745:
7733:
7730:
7727:
7724:
7721:
7718:
7715:
7712:
7709:
7706:
7701:
7697:
7693:
7690:
7687:
7684:
7681:
7675:
7672:
7667:
7663:
7660:
7657:
7654:
7648:
7645:
7640:
7636:
7633:
7630:
7625:
7621:
7617:
7614:
7596:
7587:
7573:
7572:
7560:
7557:
7554:
7551:
7548:
7545:
7542:
7539:
7536:
7533:
7530:
7511:
7510:
7498:
7493:
7489:
7485:
7482:
7476:
7473:
7468:
7437:
7436:
7424:
7421:
7418:
7415:
7410:
7406:
7402:
7399:
7396:
7373:
7372:
7359:
7356:
7353:
7347:
7344:
7339:
7335:
7332:
7329:
7313:
7312:
7299:
7296:
7293:
7290:
7287:
7284:
7281:
7253:
7252:
7239:
7236:
7233:
7230:
7227:
7224:
7221:
7218:
7212:
7209:
7204:
7177:
7176:
7163:
7157:
7154:
7150:
7145:
7142:
7139:
7136:
7133:
7117:
7116:
7102:
7099:
7094:
7090:
7087:
7066:
7065:
7053:
7048:
7044:
7040:
7037:
7031:
7028:
7023:
7019:
7016:
6993:
6992:
6980:
6977:
6974:
6971:
6965:
6962:
6954:
6949:
6946:
6939:
6936:
6933:
6930:
6924:
6920:
6917:
6868:
6861:
6853:
6852:
6840:
6837:
6834:
6831:
6828:
6825:
6815:
6811:
6807:
6804:
6801:
6796:
6793:
6790:
6787:
6784:
6779:
6776:
6773:
6770:
6767:
6764:
6761:
6755:
6749:
6742:
6738:
6734:
6731:
6728:
6723:
6720:
6717:
6714:
6711:
6708:
6704:
6701:
6696:
6692:
6688:
6683:
6679:
6675:
6670:
6667:
6664:
6661:
6658:
6655:
6652:
6649:
6646:
6642:
6639:
6634:
6630:
6626:
6621:
6617:
6613:
6607:
6600:
6596:
6589:
6586:
6581:
6552:
6547:
6543:
6518:
6514:
6509:
6502:
6499:
6494:
6482:
6481:
6469:
6465:
6461:
6458:
6450:
6446:
6442:
6439:
6436:
6431:
6428:
6425:
6422:
6419:
6416:
6413:
6410:
6407:
6404:
6401:
6393:
6389:
6385:
6382:
6379:
6374:
6371:
6368:
6365:
6362:
6359:
6356:
6353:
6350:
6347:
6341:
6338:
6333:
6321:
6309:
6305:
6301:
6298:
6290:
6286:
6282:
6279:
6276:
6271:
6268:
6265:
6262:
6259:
6256:
6253:
6250:
6247:
6244:
6241:
6233:
6229:
6225:
6222:
6219:
6214:
6211:
6208:
6205:
6202:
6199:
6196:
6193:
6190:
6187:
6182:
6178:
6153:
6150:
6147:
6144:
6141:
6138:
6135:
6132:
6129:
6126:
6123:
6120:
6117:
6114:
6111:
6108:
6105:
6102:
6099:
6096:
6093:
6090:
6079:
6078:
6064:
6059:
6056:
6052:
6044:
6040:
6036:
6033:
6030:
6025:
6022:
6019:
6016:
6013:
6010:
6007:
6003:
5998:
5993:
5990:
5987:
5984:
5981:
5978:
5975:
5972:
5969:
5940:, the vector (
5929:
5926:
5922:
5918:
5915:
5912:
5909:
5906:
5903:
5899:
5896:
5884:
5883:
5871:
5868:
5865:
5862:
5858:
5850:
5846:
5842:
5839:
5836:
5831:
5828:
5825:
5822:
5819:
5816:
5813:
5809:
5753:
5702:
5699:
5696:
5693:
5606:
5605:
5593:
5590:
5587:
5584:
5581:
5578:
5575:
5572:
5569:
5564:
5556:
5552:
5548:
5545:
5542:
5537:
5534:
5531:
5528:
5522:
5519:
5516:
5513:
5509:
5506:
5466:
5446:
5443:
5440:
5408:
5405:
5402:
5399:
5396:
5393:
5390:
5387:
5367:
5364:
5361:
5358:
5355:
5352:
5349:
5346:
5343:
5340:
5337:
5334:
5331:
5328:
5325:
5322:
5319:
5316:
5313:
5310:
5307:
5304:
5269:
5249:
5246:
5243:
5223:
5203:
5200:
5197:
5186:
5185:
5170:
5167:
5164:
5161:
5158:
5155:
5148:
5144:
5137:
5134:
5129:
5123:
5115:
5110:
5106:
5100:
5097:
5094:
5088:
5085:
5080:
5076:
5073:
5068:
5064:
5060:
5057:
5054:
5049:
5045:
5041:
5034:
5028:
5025:
5020:
5014:
5011:
5008:
5002:
4999:
4994:
4990:
4987:
4981:
4978:
4973:
4969:
4966:
4963:
4958:
4954:
4950:
4944:
4941:
4939:
4932:
4928:
4924:
4921:
4918:
4915:
4907:
4904:
4899:
4895:
4892:
4889:
4886:
4880:
4879:
4853:
4850:
4845:
4824:
4800:
4780:
4769:
4768:
4756:
4746:
4742:
4738:
4735:
4732:
4727:
4724:
4721:
4718:
4715:
4710:
4707:
4704:
4701:
4698:
4695:
4692:
4686:
4683:
4680:
4677:
4674:
4671:
4664:
4660:
4653:
4650:
4645:
4599:
4596:
4584:
4581:
4578:
4575:
4572:
4542:
4537:
4533:
4528:
4521:
4518:
4513:
4509:
4506:
4503:
4500:
4497:
4486:
4485:
4473:
4465:
4460:
4455:
4451:
4447:
4442:
4438:
4434:
4431:
4428:
4425:
4420:
4416:
4412:
4407:
4403:
4399:
4394:
4387:
4382:
4377:
4373:
4369:
4364:
4360:
4356:
4353:
4350:
4347:
4342:
4338:
4334:
4329:
4325:
4321:
4316:
4307:
4302:
4297:
4293:
4289:
4286:
4281:
4277:
4273:
4270:
4267:
4264:
4259:
4255:
4251:
4248:
4243:
4239:
4235:
4230:
4223:
4218:
4213:
4209:
4205:
4202:
4197:
4193:
4189:
4186:
4183:
4180:
4175:
4171:
4167:
4164:
4159:
4155:
4151:
4146:
4138:
4135:
4132:
4129:
4126:
4100:
4096:
4090:
4086:
4082:
4077:
4073:
4067:
4063:
4038:
4033:
4028:
4024:
4018:
4013:
4009:
4005:
4000:
3996:
3990:
3986:
3980:
3976:
3970:
3966:
3962:
3959:
3954:
3949:
3945:
3939:
3934:
3930:
3906:
3905:
3893:
3883:
3879:
3873:
3869:
3863:
3859:
3855:
3850:
3846:
3840:
3836:
3832:
3829:
3826:
3821:
3816:
3812:
3808:
3803:
3798:
3794:
3790:
3787:
3782:
3777:
3773:
3769:
3764:
3759:
3755:
3751:
3746:
3743:
3738:
3734:
3728:
3724:
3720:
3715:
3710:
3706:
3702:
3699:
3696:
3691:
3686:
3682:
3678:
3673:
3668:
3664:
3660:
3655:
3650:
3646:
3640:
3636:
3632:
3627:
3623:
3617:
3613:
3609:
3606:
3603:
3600:
3595:
3591:
3585:
3581:
3577:
3572:
3568:
3564:
3561:
3558:
3553:
3548:
3544:
3540:
3535:
3530:
3526:
3522:
3516:
3513:
3510:
3507:
3504:
3469:
3466:
3463:
3460:
3457:
3454:
3449:
3445:
3440:
3433:
3430:
3425:
3404:
3397:
3393:
3389:
3386:
3383:
3378:
3354:
3351:
3347:
3343:
3338:
3333:
3329:
3325:
3320:
3315:
3311:
3307:
3304:
3301:
3262:
3261:
3249:
3244:
3238:
3234:
3230:
3227:
3222:
3218:
3209:
3205:
3201:
3198:
3195:
3187:
3182:
3178:
3165:
3159:
3155:
3146:
3142:
3138:
3135:
3132:
3127:
3122:
3118:
3114:
3108:
3103:
3099:
3072:
3068:
3045:
3041:
3020:
3015:
3011:
2990:
2985:
2981:
2961:
2956:
2952:
2932:
2927:
2923:
2911:
2910:
2909:
2908:
2907:
2906:
2895:
2890:
2881:
2877:
2873:
2870:
2867:
2862:
2859:
2856:
2853:
2850:
2845:
2840:
2837:
2832:
2828:
2824:
2819:
2815:
2811:
2808:
2805:
2802:
2797:
2793:
2789:
2784:
2780:
2776:
2773:
2770:
2744:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2716:
2711:
2708:
2703:
2699:
2695:
2690:
2686:
2682:
2679:
2676:
2673:
2668:
2664:
2660:
2655:
2651:
2647:
2644:
2616:
2613:
2610:
2607:
2587:
2565:
2561:
2556:
2549:
2546:
2541:
2515:
2511:
2506:
2499:
2496:
2491:
2475:
2474:
2462:
2457:
2452:
2446:
2443:
2438:
2430:
2426:
2419:
2416:
2411:
2405:
2402:
2399:
2395:
2387:
2382:
2375:
2371:
2366:
2362:
2357:
2352:
2345:
2342:
2337:
2331:
2328:
2323:
2319:
2316:
2313:
2307:
2303:
2298:
2294:
2289:
2285:
2281:
2262:
2261:
2249:
2246:
2242:
2238:
2235:
2230:
2226:
2222:
2217:
2213:
2209:
2206:
2203:
2200:
2195:
2191:
2187:
2182:
2178:
2174:
2171:
2168:
2162:
2159:
2154:
2142:
2130:
2126:
2122:
2119:
2114:
2110:
2106:
2101:
2097:
2093:
2090:
2087:
2084:
2079:
2075:
2071:
2066:
2062:
2058:
2055:
2052:
2047:
2043:
2027:
2026:
2014:
2011:
2008:
2004:
2001:
1998:
1995:
1992:
1987:
1984:
1979:
1974:
1971:
1968:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1939:
1936:
1931:
1927:
1923:
1920:
1915:
1911:
1907:
1902:
1899:
1894:
1888:
1885:
1880:
1874:
1871:
1866:
1862:
1858:
1855:
1850:
1846:
1842:
1837:
1834:
1829:
1824:
1820:
1715:
1714:
1702:
1697:
1689:
1684:
1680:
1676:
1671:
1666:
1662:
1658:
1654:
1650:
1644:
1640:
1636:
1631:
1627:
1621:
1617:
1613:
1612:
1607:
1603:
1597:
1593:
1589:
1584:
1580:
1574:
1570:
1566:
1562:
1557:
1553:
1549:
1544:
1539:
1535:
1531:
1530:
1528:
1511:
1510:
1498:
1493:
1489:
1485:
1481:
1476:
1471:
1467:
1463:
1458:
1453:
1449:
1445:
1442:
1439:
1436:
1432:
1429:
1425:
1420:
1416:
1410:
1406:
1402:
1397:
1393:
1387:
1383:
1379:
1376:
1373:
1368:
1364:
1360:
1356:
1351:
1346:
1342:
1338:
1333:
1328:
1324:
1320:
1317:
1312:
1308:
1304:
1301:
1296:
1292:
1288:
1285:
1280:
1276:
1272:
1229:
1228:
1216:
1213:
1208:
1204:
1198:
1194:
1190:
1185:
1181:
1175:
1171:
1099:
1098:
1086:
1083:
1073:
1069:
1065:
1062:
1059:
1054:
1051:
1048:
1045:
1042:
1033:
1029:
1025:
1022:
1019:
1014:
1011:
1008:
1005:
1002:
996:
991:
986:
981:
977:
961:
960:
948:
941:
937:
933:
930:
927:
922:
919:
916:
913:
910:
907:
904:
897:
893:
889:
886:
883:
878:
875:
872:
869:
866:
863:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
830:
827:
824:
821:
818:
815:
812:
809:
806:
791:
790:
771:
767:
763:
760:
757:
752:
749:
746:
743:
740:
735:
732:
729:
726:
723:
720:
717:
711:
708:
705:
702:
699:
696:
693:
676:of the metric
643:
642:
628:
622:
619:
617:
614:
613:
610:
607:
605:
602:
601:
599:
594:
591:
588:
585:
582:
579:
576:
546:
545:
531:
527:
523:
519:
516:
513:
510:
506:
503:
499:
496:
493:
488:
484:
480:
476:
473:
468:
464:
460:
425:
422:
419:
416:
413:
410:
407:
387:
384:
381:
378:
375:
372:
369:
313:
310:
252:theory of the
215:
212:
207:
203:
199:
179:
176:
172:
168:
118:
117:
106:
100:
97:
92:
89:
83:
80:
72:
69:
64:
59:
56:
34:, named after
15:
9:
6:
4:
3:
2:
21095:
21084:
21083:Moduli theory
21081:
21079:
21076:
21074:
21071:
21069:
21066:
21065:
21063:
21052:
21047:
21044:
21040:
21035:
21030:
21026:
21022:
21021:
21015:
21011:
21006:
21002:
20998:
20993:
20989:
20984:
20980:
20976:
20971:
20968:
20964:
20960:
20954:
20950:
20946:
20945:
20940:
20936:
20933:
20931:0-914098-70-5
20927:
20923:
20918:
20914:
20909:
20905:
20901:
20896:
20892:
20887:
20883:
20879:
20874:
20872:
20868:
20864:
20860:
20856:
20854:
20851:
20847:
20843:
20839:
20835:
20831:
20829:
20826:
20822:
20818:
20814:
20810:
20806:
20803:
20799:
20795:
20791:
20787:
20783:
20782:Ann. of Math.
20779:
20774:
20771:
20767:
20763:
20759:
20754:
20749:
20745:
20741:
20737:
20733:
20730:
20726:
20722:
20718:
20713:
20708:
20704:
20700:
20699:
20693:
20690:
20688:0-387-96310-3
20684:
20680:
20675:
20671:
20666:
20663:
20661:0-486-66721-9
20657:
20653:
20649:
20645:
20642:
20638:
20634:
20628:
20623:
20618:
20614:
20613:
20607:
20604:
20602:0-387-70088-9
20598:
20594:
20589:
20586:
20582:
20578:
20572:
20568:
20567:
20561:
20557:
20553:
20548:
20544:
20540:
20536:
20532:
20527:
20523:
20519:
20514:
20511:
20507:
20503:
20499:
20494:
20492:, Chapter VI.
20490:
20488:0-8218-0049-3
20484:
20480:
20475:
20471:
20466:
20464:
20459:
20455:
20451:
20447:
20440:
20435:
20432:
20426:
20422:
20418:
20417:Martin, Gaven
20414:
20409:
20405:
20401:
20397:
20393:
20388:
20387:
20377:
20372:
20366:
20361:
20355:
20350:
20348:
20346:
20339:
20334:
20332:
20325:
20319:
20310:
20307:
20305:
20302:
20300:
20297:
20295:
20292:
20290:
20287:
20286:
20282:
20276:, p. 321
20275:
20270:
20263:
20258:
20251:
20246:
20239:
20234:
20226:
20223:
20221:
20218:
20217:
20213:
20207:
20202:
20194:
20191:
20189:
20188:Glutsyuk 2008
20186:
20184:
20181:
20180:
20176:
20168:, p. 277
20167:
20164:
20161:
20158:
20157:
20153:
20146:
20141:
20132:
20129:
20126:
20123:
20122:
20118:
20110:
20109:Glutsyuk 2008
20107:
20105:
20102:
20101:
20097:
20089:
20086:
20084:
20083:Glutsyuk 2008
20081:
20079:
20076:
20074:
20071:
20069:
20066:
20065:
20061:
20054:
20049:
20045:
20036:
20033:
20031:
20028:
20026:
20023:
20022:
20000:
19995:
19991:
19987:
19982:
19979:
19974:
19970:
19966:
19963:
19954:
19953:
19952:
19950:
19945:
19941:
19937:
19933:
19926:
19919:
19915:
19911:
19907:
19903:
19896:
19892:
19888:
19884:
19880:
19876:
19869:
19865:
19861:
19857:
19853:
19848:
19846:
19825:
19820:
19816:
19811:
19801:
19796:
19792:
19786:
19780:
19774:
19771:
19768:
19755:
19743:
19742:
19741:
19739:
19735:
19731:
19727:
19722:
19720:
19712:
19708:
19700:
19696:
19675:
19667:
19661:
19658:
19654:
19650:
19642:
19639:
19635:
19628:
19619:
19618:
19617:
19598:
19592:
19583:
19577:
19574:
19571:
19565:
19562:
19556:
19550:
19544:
19538:
19535:
19529:
19526:
19523:
19520:
19514:
19511:
19508:
19502:
19493:
19492:
19491:
19489:
19485:
19481:
19477:
19473:
19469:
19465:
19461:
19457:
19452:
19450:
19446:
19442:
19438:
19434:
19430:
19426:
19422:
19418:
19414:
19410:
19406:
19402:
19398:
19394:
19390:
19386:
19382:
19378:
19374:
19370:
19366:
19362:
19358:
19354:
19350:
19346:
19342:
19338:
19334:
19330:
19326:
19322:
19318:
19303:
19299:
19295:
19291:
19287:
19283:
19279:
19275:
19271:
19267:
19263:
19258:
19253:
19246:
19239:
19232:
19225:
19218:
19214:
19210:
19206:
19202:
19198:
19195:. In fact if
19194:
19165:
19161:
19157:
19152:
19148:
19136:
19132:
19128:
19123:
19119:
19105:
19101:
19097:
19092:
19088:
19076:
19072:
19068:
19063:
19059:
19049:
19041:
19037:
19033:
19028:
19024:
19020:
19015:
19011:
19007:
19002:
18998:
18985:
18984:
18983:
18964:
18959:
18944:
18940:
18936:
18931:
18927:
18923:
18918:
18914:
18910:
18905:
18901:
18889:
18886:
18870:
18866:
18859:
18856:
18848:
18844:
18837:
18834:
18826:
18822:
18815:
18812:
18804:
18800:
18793:
18776:
18775:
18774:
18772:
18768:
18764:
18760:
18755:
18753:
18732:
18729:
18726:
18710:
18706:
18699:
18696:
18688:
18684:
18677:
18657:
18653:
18646:
18643:
18635:
18631:
18624:
18613:
18608:
18605:
18601:
18591:
18590:
18589:
18570:
18560:
18556:
18552:
18547:
18543:
18534:
18524:
18520:
18516:
18511:
18507:
18492:
18491:
18490:
18471:
18462:
18450:
18446:
18442:
18437:
18433:
18420:
18408:
18404:
18400:
18395:
18391:
18379:
18376:
18373:
18357:
18353:
18346:
18343:
18335:
18331:
18324:
18304:
18300:
18293:
18290:
18282:
18278:
18271:
18253:
18252:
18251:
18249:
18245:
18241:
18237:
18231:
18227:
18217:
18212:
18205:
18198:
18191:
18183:
18175:
18171:
18163:
18159:
18155:
18151:
18147:
18143:
18139:
18131:
18124:
18116:
18109:
18104:
18102:
18095:
18087:
18080:
18073:
18066:
18058:
18054:
18046:
18042:
18038:
18030:
18026:
18022:
18018:
17998:
17992:
17983:
17980:
17969:
17965:
17954:
17950:
17942:
17933:
17927:
17918:
17915:
17904:
17903:
17902:
17884:
17873:
17865:
17841:
17833:
17827:
17821:
17812:
17809:
17801:
17790:
17787:
17779:
17776:
17770:
17761:
17758:
17750:
17739:
17736:
17725:
17719:
17716:
17710:
17701:
17698:
17687:
17686:
17685:
17683:
17679:
17675:
17671:
17664:
17660:
17656:
17652:
17648:
17644:
17640:
17633:
17626:
17621:
17619:
17612:
17605:
17601:
17596:
17586:
17584:
17580:
17576:
17571:
17567:
17563:
17558:
17554:
17549:
17545:
17541:
17533:
17529:
17525:
17520:
17516:
17505:
17501:
17497:
17492:
17488:
17481:
17476:
17472:
17468:
17461:
17457:
17450:
17446:
17434:
17429:
17421:
17417:
17413:
17409:
17405:
17401:
17399:
17395:
17388:
17384:
17380:
17376:
17372:
17368:
17364:
17360:
17356:
17352:
17348:
17344:
17340:
17336:
17332:
17328:
17324:
17320:
17316:
17312:
17308:
17299:
17297:
17293:
17289:
17285:
17281:
17277:
17258:
17252:
17244:
17239:
17236:
17231:
17228:
17225:
17209:
17205:
17197:
17193:
17187:
17178:
17172:
17166:
17158:
17157:
17156:
17154:
17150:
17146:
17142:
17138:
17134:
17130:
17109:
17093:
17089:
17085:
17076:
17071:
17065:
17062:
17059:
17054:
17050:
17040:
17039:
17038:
17036:
17032:
17028:
17024:
17020:
17016:
17012:
17008:
17004:
16983:
16974:
16969:
16965:
16956:
16951:
16941:
16940:
16939:
16920:
16915:
16911:
16907:
16902:
16898:
16888:
16887:
16886:
16884:
16863:
16852:
16848:
16844:
16839:
16835:
16828:
16822:
16819:
16814:
16810:
16806:
16801:
16797:
16787:
16786:
16785:
16783:
16762:
16759:
16751:
16747:
16743:
16737:
16734:
16729:
16725:
16715:
16714:
16713:
16694:
16691:
16685:
16679:
16676:
16664:
16659:
16652:
16643:
16642:
16641:
16639:
16635:
16631:
16627:
16623:
16619:
16615:
16611:
16607:
16586:
16583:
16580:
16571:
16556:
16555:
16554:
16526:
16521:
16514:
16511:
16508:
16505:
16502:
16493:
16492:
16491:
16488:
16484:
16480:
16476:
16471:
16466:
16446:
16443:
16440:
16435:
16427:
16424:
16415:
16412:
16409:
16400:
16391:
16388:
16377:
16376:
16375:
16373:
16362:
16358:
16355:
16347:
16343:
16338:
16334:
16329:
16324:
16320:
16316:
16312:
16308:
16304:
16300:
16299:
16281:
16273:
16269:
16260:
16255:
16252:
16247:
16244:
16239:
16235:
16218:
16214:
16206:
16202:
16196:
16187:
16181:
16175:
16167:
16166:
16165:
16164:
16160:
16156:
16152:
16148:
16144:
16140:
16136:
16133:
16129:
16124:
16120:
16116:
16113:
16112:
16108:
16104:
16100:
16099:
16074:
16068:
16062:
16059:
16052:
16048:
16037:
16033:
16025:
16022:
16016:
16004:
15994:
15993:
15992:
15991:
15987:
15986:
15964:
15958:
15955:
15943:
15939:
15931:
15927:
15921:
15918:
15909:
15903:
15891:
15881:
15880:
15879:
15878:
15874:
15871:
15870:
15863:
15858:
15854:
15849:
15845:
15841:
15837:
15830:
15826:
15821:
15817:
15813:
15809:
15806:) ≠ 0. Since
15802:
15798:
15791:
15786:
15782:
15778:
15774:
15767:
15762:
15758:
15751:
15747:
15746:
15727:
15724:
15716:
15712:
15708:
15705:
15693:
15678:
15677:
15676:
15675:
15671:
15664:
15659:
15655:
15651:
15648:
15645:
15641:
15637:
15634:
15633:
15629:
15625:
15621:
15616:
15612:
15608:
15604:
15599:
15595:
15591:
15586:
15582:
15578:
15574:
15570:
15565:
15561:
15557:
15552:
15548:
15544:
15540:
15536:
15532:
15531:
15510:
15505:
15501:
15497:
15494:
15491:
15483:
15477:
15471:
15466:
15456:
15444:
15440:
15434:
15423:
15420:
15417:
15413:
15410:
15404:
15392:
15388:
15382:
15367:
15363:
15359:
15357:
15349:
15346:
15342:
15339:
15333:
15321:
15317:
15307:
15293:
15288:
15276:
15272:
15266:
15251:
15247:
15243:
15240:
15237:
15233:
15230:
15221:
15217:
15210:
15205:
15201:
15197:
15195:
15184:
15176:
15172:
15165:
15154:
15153:
15152:
15151:
15147:
15143:
15139:
15135:
15131:
15128:
15127:
15123:
15119:
15115:
15110:
15106:
15102:
15098:
15094:
15093:
15074:
15069:
15065:
15056:
15052:
15048:
15045:
15039:
15031:
15027:
15023:
15020:
15014:
15011:
15006:
15002:
14996:
14992:
14988:
14985:
14982:
14973:
14972:
14971:
14970:
14966:
14962:
14957:
14953:
14948:
14944:
14939:
14935:
14930:
14926:
14921:
14917:
14912:
14908:
14904:
14900:
14896:
14892:
14888:
14884:
14881:
14880:
14875:
14871:
14867:
14863:
14859:
14858:
14839:
14836:
14827:
14818:
14815:
14812:
14809:
14806:
14803:
14794:
14785:
14780:
14776:
14772:
14766:
14763:
14760:
14757:
14752:
14748:
14744:
14738:
14735:
14732:
14729:
14726:
14717:
14716:
14715:
14714:
14710:
14706:
14702:
14698:
14694:
14690:
14689:
14670:
14652:
14648:
14641:
14638:
14631:
14621:
14617:
14607:
14602:
14592:
14588:
14579:
14574:
14564:
14560:
14547:
14546:
14545:
14544:
14540:
14539:
14520:
14517:
14512:
14508:
14502:
14494:
14489:
14485:
14479:
14474:
14470:
14460:
14451:
14446:
14442:
14432:
14431:
14430:
14429:
14425:
14421:
14417:
14416:
14415:
14413:
14409:
14405:
14401:
14397:
14393:
14388:
14384:
14379:
14375:
14370:
14366:
14361:
14357:
14353:
14349:
14345:
14341:
14336:
14332:
14327:
14323:
14319:
14315:
14311:
14307:
14303:
14298:
14294:
14289:
14285:
14280:
14278:
14274:
14270:
14265:
14261:
14240:
14227:
14221:
14206:
14202:
14189:
14188:
14187:
14185:
14181:
14176:
14172:
14151:
14148:
14145:
14137:
14133:
14129:
14126:
14120:
14115:
14111:
14101:
14100:
14099:
14097:
14092:
14086:
14080:
14078:
14074:
14070:
14067:coefficients
14065:
14061:
14046:
14044:
14040:
14036:
14032:
14028:
14024:
14004:
14001:
13998:
13995:
13992:
13989:
13985:
13976:
13972:
13958:
13952:
13949:
13934:
13926:
13922:
13918:
13914:
13908:
13904:
13897:
13894:
13890:
13886:
13875:
13869:
13856:
13855:
13854:
13853:
13849:
13842:
13838:
13831:
13811:
13798:
13790:
13786:
13782:
13772:
13768:
13764:
13758:
13753:
13745:
13734:
13733:
13732:
13730:
13726:
13722:
13702:
13692:
13688:
13684:
13679:
13675:
13666:
13661:
13657:
13653:
13650:
13647:
13635:
13631:
13627:
13622:
13618:
13607:
13599:
13588:
13584:
13570:
13564:
13561:
13555:
13551:
13545:
13532:
13528:
13521:
13518:
13510:
13506:
13499:
13486:
13485:
13484:
13482:
13478:
13474:
13470:
13466:
13462:
13458:
13454:
13450:
13430:
13427:
13423:
13417:
13407:
13404:
13396:
13393:
13390:
13382:
13379:
13375:
13368:
13363:
13359:
13350:
13349:
13348:
13330:
13325:
13321:
13317:
13314:
13311:
13299:
13295:
13291:
13286:
13282:
13271:
13263:
13255:
13251:
13247:
13234:
13230:
13223:
13220:
13217:
13209:
13205:
13198:
13195:
13182:
13181:
13180:
13178:
13174:
13171:implies that
13170:
13166:
13162:
13158:
13138:
13135:
13132:
13126:
13120:
13115:
13112:
13108:
13104:
13101:
13095:
13089:
13081:
13080:
13079:
13077:
13073:
13053:
13050:
13047:
13044:
13039:
13031:
13028:
13019:
13016:
13013:
13004:
12995:
12992:
12981:
12980:
12979:
12977:
12973:
12969:
12965:
12942:
12939:
12936:
12932:
12929:
12919:
12916:
12913:
12907:
12902:
12896:
12890:
12876:
12870:
12867:
12862:
12859:
12856:
12853:
12849:
12846:
12841:
12835:
12832:
12827:
12821:
12818:
12815:
12811:
12805:
12798:
12792:
12781:
12775:
12772:
12767:
12764:
12762:
12754:
12751:
12747:
12744:
12738:
12731:
12725:
12711:
12705:
12702:
12698:
12694:
12688:
12682:
12679:
12676:
12674:
12666:
12660:
12657:
12646:
12645:
12644:
12642:
12638:
12634:
12630:
12625:
12623:
12619:
12599:
12596:
12593:
12587:
12581:
12576:
12573:
12569:
12565:
12562:
12556:
12550:
12542:
12541:
12540:
12538:
12534:
12530:
12510:
12507:
12502:
12499:
12495:
12491:
12482:
12477:
12468:
12467:
12466:
12464:
12444:
12439:
12436:
12428:
12425:
12422:
12416:
12411:
12408:
12404:
12397:
12394:
12391:
12386:
12383:
12375:
12372:
12369:
12363:
12360:
12352:
12351:
12350:
12348:
12344:
12340:
12336:
12332:
12328:
12324:
12320:
12316:
12312:
12291:
12288:
12285:
12282:
12276:
12273:
12267:
12264:
12256:
12252:
12248:
12242:
12239:
12227:
12222:
12215:
12212:
12200:
12195:
12188:
12185:
12180:
12176:
12172:
12169:
12160:
12159:
12158:
12155:
12151:
12146:
12142:
12138:
12134:
12114:
12111:
12105:
12099:
12096:
12090:
12084:
12076:
12075:
12074:
12072:
12051:
12046:
12042:
12038:
12035:
12026:
12021:
12011:
12010:
12009:
12006:
12004:
11999:
11995:
11991:
11987:
11983:
11979:
11975:
11971:
11966:
11964:
11960:
11956:
11952:
11948:
11944:
11934:
11932:
11928:
11924:
11923:Taylor series
11920:
11916:
11912:
11908:
11904:
11903:Borel's lemma
11885:
11881:
11877:
11874:
11869:
11863:
11860:
11857:
11853:
11847:
11843:
11839:
11834:
11831:
11828:
11824:
11819:
11815:
11812:
11809:
11796:
11792:
11789:
11782:
11778:
11774:
11769:
11765:
11762:
11758:
11752:
11748:
11739:
11732:
11726:
11723:
11720:
11716:
11711:
11704:
11698:
11695:
11692:
11689:
11685:
11677:
11676:
11675:
11673:
11669:
11664:
11660:
11637:
11634:
11631:
11625:
11622:
11619:
11616:
11613:
11610:
11605:
11601:
11594:
11586:
11582:
11578:
11575:
11569:
11561:
11557:
11553:
11550:
11547:
11541:
11538:
11535:
11529:
11521:
11520:
11519:
11517:
11513:
11509:
11505:
11501:
11497:
11493:
11489:
11469:
11466:
11461:
11457:
11451:
11447:
11443:
11440:
11435:
11431:
11427:
11424:
11421:
11415:
11409:
11401:
11400:
11399:
11397:
11377:
11372:
11368:
11364:
11361:
11356:
11352:
11348:
11343:
11332:
11326:
11320:
11314:
11311:
11308:
11305:
11299:
11294:
11290:
11286:
11278:
11277:
11276:
11274:
11254:
11251:
11248:
11242:
11236:
11233:
11227:
11221:
11215:
11212:
11209:
11202:
11196:
11191:
11188:
11183:
11179:
11172:
11169:
11165:
11160:
11157:
11149:
11148:
11147:
11144:
11142:
11138:
11134:
11130:
11126:
11122:
11099:
11087:
11080:
11065:
11061:
11055:
11043:
11036:
11021:
11017:
11011:
11005:
10997:
10996:
10995:
10977:
10972:
10968:
10964:
10961:
10956:
10952:
10948:
10943:
10932:
10926:
10923:
10920:
10917:
10911:
10906:
10902:
10898:
10890:
10889:
10888:
10886:
10882:
10862:
10853:
10841:
10834:
10831:
10828:
10822:
10816:
10810:
10807:
10801:
10789:
10782:
10779:
10776:
10770:
10764:
10758:
10752:
10746:
10743:
10740:
10734:
10726:
10725:
10724:
10723:has the form
10722:
10718:
10714:
10710:
10706:
10702:
10698:
10695:and letting (
10694:
10690:
10686:
10682:
10678:
10674:
10670:
10666:
10662:
10658:
10653:
10651:
10647:
10643:
10639:
10635:
10631:
10627:
10623:
10619:
10598:
10595:
10592:
10585:
10582:
10578:
10574:
10571:
10567:
10557:
10556:
10555:
10534:
10530:
10523:
10517:
10514:
10505:
10500:
10490:
10489:
10488:
10486:
10482:
10462:
10457:
10454:
10439:
10436:
10427:
10418:
10407:
10398:
10390:
10386:
10380:
10374:
10368:
10360:
10359:
10358:
10356:
10352:
10347:
10343:
10339:
10335:
10329:
10319:
10317:
10311:
10292:
10287:
10284:
10269:
10266:
10257:
10248:
10241:
10235:
10229:
10220:
10219:
10218:
10216:
10196:
10182:
10179:
10170:
10161:
10151:
10142:
10134:
10130:
10124:
10118:
10112:
10104:
10103:
10102:
10100:
10097:| < 1, if
10096:
10091:
10073:
10056:
10048:
10042:
10036:
10030:
10022:
10021:
10020:
10018:
9998:
9981:
9973:
9967:
9961:
9955:
9947:
9946:
9945:
9943:
9939:
9934:
9932:
9927:
9925:
9921:
9917:
9913:
9893:
9887:
9879:
9876:
9872:
9868:
9864:
9851:
9847:
9839:
9835:
9815:
9809:
9806:
9803:
9797:
9793:
9786:
9782:
9776:
9770:
9762:
9761:
9760:
9758:
9751:
9747:
9726:
9717:
9713:
9708:
9704:
9700:
9697:
9691:
9688:
9685:
9679:
9676:
9673:
9670:
9667:
9658:
9657:
9656:
9654:
9650:
9646:
9642:
9641:bump function
9638:
9617:
9611:
9605:
9602:
9585:
9578:
9568:
9561:
9555:
9546:
9543:
9531:
9530:
9529:
9527:
9523:
9519:
9515:
9494:
9486:
9483:
9479:
9470:
9466:
9458:
9449:
9441:
9437:
9431:
9425:
9417:
9413:
9403:
9402:
9401:
9382:
9377:
9374:
9364:
9361:
9357:
9350:
9347:
9341:
9335:
9326:
9325:
9324:
9322:
9319:Moreover, if
9301:
9293:
9289:
9278:
9274:
9268:
9265:
9258:
9254:
9250:
9245:
9241:
9225:
9221:
9213:
9209:
9203:
9200:
9197:
9190:
9187:
9183:
9179:
9176:
9172:
9162:
9161:
9160:
9158:
9155:The solution
9153:
9151:
9147:
9143:
9139:
9135:
9131:
9127:
9099:
9096:
9092:
9078:
9068:
9060:
9057:
9051:
9045:
9039:
9030:
9029:
9028:
9026:
9022:
9018:
8997:
8992:
8988:
8984:
8979:
8975:
8971:
8968:
8959:
8954:
8944:
8943:
8942:
8940:
8935:
8931:
8927:
8906:
8898:
8888:
8880:
8877:
8869:
8857:
8853:
8844:
8839:
8823:
8818:
8809:
8804:
8792:
8788:
8779:
8773:
8767:
8758:
8757:
8756:
8754:
8750:
8746:
8741:
8739:
8735:
8731:
8727:
8723:
8702:
8699:
8696:
8693:
8690:
8684:
8681:
8678:
8675:
8667:
8653:
8652:
8651:
8649:
8646:
8642:
8638:
8635:
8631:
8627:
8623:
8618:
8613:
8612:Sobolev space
8592:
8587:
8583:
8579:
8567:
8562:
8555:
8546:
8545:
8544:
8543:
8539:
8518:
8513:
8509:
8505:
8500:
8496:
8492:
8489:
8483:
8480:
8477:
8467:
8463:
8457:
8449:
8442:
8437:
8433:
8429:
8420:
8419:
8418:
8416:
8413:
8409:
8405:
8401:
8396:
8392:
8388:
8368:
8365:
8362:
8359:
8354:
8350:
8346:
8343:
8334:
8329:
8320:
8319:
8318:
8317:it satisfies
8316:
8312:
8308:
8304:
8300:
8296:
8292:
8288:
8284:
8280:
8276:
8272:
8269:
8265:
8261:
8257:
8252:
8250:
8246:
8242:
8238:
8234:
8230:
8226:
8222:
8219:
8199:
8196:
8193:
8190:
8181:
8167:
8166:
8165:
8163:
8158:
8156:
8152:
8132:
8129:
8126:
8120:
8114:
8109:
8105:
8101:
8098:
8092:
8086:
8077:
8076:
8075:
8073:
8069:
8065:
8044:
8041:
8036:
8032:
8028:
8019:
8014:
8004:
8003:
8002:
8000:
7979:
7974:
7971:
7963:
7960:
7957:
7951:
7948:
7943:
7939:
7932:
7929:
7926:
7921:
7918:
7910:
7907:
7904:
7898:
7895:
7886:
7885:
7884:
7865:
7862:
7859:
7856:
7853:
7847:
7844:
7841:
7829:
7828:
7827:
7808:
7805:
7802:
7799:
7796:
7793:
7786:
7783:
7780:
7777:
7774:
7771:
7768:
7759:
7758:
7757:
7755:
7751:
7731:
7728:
7725:
7719:
7716:
7710:
7707:
7699:
7695:
7691:
7685:
7682:
7670:
7665:
7658:
7655:
7643:
7638:
7631:
7628:
7623:
7619:
7615:
7612:
7604:
7603:
7602:
7599:
7595:
7590:
7586:
7582:
7578:
7558:
7555:
7549:
7543:
7540:
7534:
7528:
7520:
7519:
7518:
7516:
7496:
7491:
7487:
7483:
7480:
7471:
7466:
7457:
7456:
7455:
7452:
7450:
7446:
7442:
7422:
7419:
7416:
7413:
7408:
7400:
7397:
7386:
7385:
7384:
7382:
7378:
7357:
7354:
7351:
7342:
7333:
7330:
7318:
7317:
7316:
7297:
7294:
7291:
7288:
7285:
7282:
7279:
7270:
7269:
7268:
7267:, defined as
7266:
7262:
7258:
7237:
7234:
7231:
7225:
7222:
7219:
7207:
7192:
7191:
7190:
7189:
7186:
7182:
7161:
7155:
7152:
7148:
7143:
7137:
7131:
7122:
7121:
7120:
7097:
7088:
7085:
7078:
7077:
7076:
7074:
7069:
7051:
7046:
7042:
7038:
7026:
7021:
7014:
7006:
7005:
7004:
7002:
6998:
6978:
6972:
6963:
6960:
6952:
6944:
6937:
6931:
6922:
6918:
6915:
6903:
6902:
6901:
6899:
6895:
6891:
6887:
6883:
6879:
6875:
6874:Adrien Douady
6866:
6860:
6858:
6838:
6832:
6826:
6823:
6813:
6809:
6805:
6802:
6799:
6794:
6791:
6788:
6785:
6782:
6777:
6774:
6771:
6768:
6765:
6762:
6759:
6753:
6740:
6736:
6732:
6729:
6726:
6721:
6718:
6715:
6712:
6709:
6702:
6694:
6690:
6686:
6681:
6677:
6665:
6662:
6659:
6656:
6653:
6650:
6647:
6640:
6632:
6628:
6624:
6619:
6615:
6605:
6598:
6594:
6584:
6579:
6568:
6567:
6566:
6545:
6541:
6516:
6512:
6507:
6497:
6492:
6467:
6463:
6448:
6444:
6440:
6437:
6434:
6429:
6426:
6423:
6420:
6417:
6414:
6411:
6405:
6402:
6391:
6387:
6383:
6380:
6377:
6372:
6369:
6366:
6363:
6360:
6357:
6354:
6345:
6336:
6331:
6322:
6307:
6303:
6288:
6284:
6280:
6277:
6274:
6269:
6266:
6263:
6260:
6257:
6254:
6251:
6245:
6242:
6231:
6227:
6223:
6220:
6217:
6212:
6209:
6206:
6203:
6200:
6197:
6194:
6185:
6180:
6176:
6167:
6166:
6165:
6164:We thus have
6151:
6145:
6142:
6139:
6133:
6130:
6127:
6121:
6118:
6115:
6109:
6106:
6100:
6097:
6094:
6088:
6057:
6054:
6042:
6038:
6034:
6031:
6028:
6023:
6020:
6017:
6011:
6008:
6005:
6001:
5988:
5985:
5982:
5976:
5973:
5970:
5967:
5959:
5958:
5957:
5955:
5951:
5947:
5943:
5927:
5924:
5920:
5916:
5913:
5910:
5904:
5897:
5894:
5869:
5866:
5863:
5860:
5848:
5844:
5840:
5837:
5834:
5829:
5826:
5823:
5817:
5814:
5811:
5807:
5799:
5798:
5797:
5795:
5791:
5787:
5783:
5779:
5775:
5771:
5767:
5751:
5743:
5739:
5734:
5732:
5728:
5724:
5720:
5716:
5697:
5691:
5683:
5679:
5675:
5671:
5667:
5663:
5659:
5655:
5651:
5647:
5643:
5639:
5635:
5631:
5627:
5623:
5619:
5615:
5611:
5591:
5585:
5582:
5576:
5570:
5562:
5554:
5550:
5546:
5543:
5540:
5535:
5532:
5529:
5526:
5520:
5514:
5507:
5504:
5495:
5494:
5493:
5491:
5487:
5483:
5478:
5464:
5444:
5441:
5438:
5430:
5426:
5422:
5406:
5403:
5397:
5394:
5391:
5385:
5365:
5359:
5356:
5353:
5347:
5344:
5341:
5335:
5332:
5329:
5323:
5320:
5314:
5311:
5308:
5302:
5294:
5290:
5286:
5281:
5267:
5247:
5244:
5241:
5221:
5201:
5198:
5195:
5168:
5162:
5156:
5153:
5146:
5142:
5132:
5127:
5121:
5108:
5104:
5095:
5092:
5083:
5078:
5071:
5066:
5062:
5055:
5052:
5047:
5043:
5023:
5018:
5009:
5006:
4997:
4992:
4985:
4976:
4971:
4964:
4961:
4956:
4952:
4942:
4940:
4930:
4922:
4919:
4916:
4902:
4893:
4890:
4887:
4870:
4869:
4868:
4848:
4843:
4822:
4814:
4798:
4778:
4754:
4744:
4740:
4736:
4733:
4730:
4725:
4722:
4719:
4716:
4713:
4708:
4705:
4702:
4699:
4696:
4693:
4690:
4684:
4678:
4672:
4669:
4662:
4658:
4648:
4643:
4632:
4631:
4630:
4628:
4624:
4620:
4616:
4612:
4607:
4604:
4595:
4582:
4576:
4570:
4562:
4558:
4553:
4540:
4535:
4531:
4526:
4516:
4511:
4507:
4501:
4495:
4471:
4453:
4449:
4445:
4440:
4436:
4429:
4426:
4418:
4414:
4410:
4405:
4401:
4375:
4371:
4367:
4362:
4358:
4351:
4348:
4340:
4336:
4332:
4327:
4323:
4295:
4291:
4287:
4284:
4279:
4275:
4268:
4265:
4257:
4253:
4249:
4246:
4241:
4237:
4211:
4207:
4203:
4200:
4195:
4191:
4184:
4181:
4173:
4169:
4165:
4162:
4157:
4153:
4136:
4130:
4124:
4116:
4115:
4114:
4098:
4094:
4088:
4084:
4080:
4075:
4071:
4065:
4061:
4052:
4036:
4031:
4026:
4022:
4016:
4011:
4007:
4003:
3998:
3994:
3988:
3984:
3978:
3974:
3968:
3964:
3960:
3957:
3952:
3947:
3943:
3937:
3932:
3928:
3919:
3915:
3911:
3891:
3881:
3871:
3867:
3861:
3857:
3853:
3848:
3844:
3838:
3834:
3827:
3819:
3814:
3810:
3806:
3801:
3796:
3792:
3780:
3775:
3771:
3767:
3762:
3757:
3753:
3744:
3741:
3736:
3726:
3722:
3718:
3713:
3708:
3704:
3697:
3689:
3684:
3680:
3676:
3671:
3666:
3662:
3648:
3644:
3638:
3634:
3630:
3625:
3621:
3615:
3611:
3604:
3601:
3598:
3593:
3583:
3579:
3575:
3570:
3566:
3559:
3551:
3546:
3542:
3538:
3533:
3528:
3524:
3514:
3508:
3502:
3494:
3493:
3492:
3490:
3485:
3483:
3467:
3461:
3455:
3452:
3447:
3443:
3438:
3428:
3423:
3402:
3395:
3391:
3387:
3384:
3381:
3376:
3368:
3352:
3349:
3345:
3336:
3331:
3327:
3323:
3318:
3313:
3309:
3302:
3299:
3291:
3287:
3283:
3279:
3275:
3271:
3267:
3247:
3242:
3236:
3232:
3228:
3225:
3220:
3216:
3207:
3203:
3199:
3196:
3193:
3185:
3180:
3176:
3163:
3157:
3153:
3144:
3140:
3136:
3133:
3130:
3125:
3120:
3116:
3112:
3106:
3101:
3097:
3088:
3087:
3086:
3070:
3066:
3043:
3039:
3018:
3013:
3009:
2988:
2983:
2979:
2959:
2954:
2950:
2930:
2925:
2921:
2893:
2879:
2875:
2871:
2868:
2865:
2860:
2857:
2854:
2851:
2848:
2830:
2826:
2822:
2817:
2813:
2806:
2803:
2795:
2791:
2787:
2782:
2778:
2768:
2761:
2760:
2759:
2758:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2701:
2697:
2693:
2688:
2684:
2677:
2674:
2666:
2662:
2658:
2653:
2649:
2634:
2633:
2632:
2630:
2611:
2605:
2585:
2563:
2559:
2554:
2544:
2539:
2529:
2513:
2509:
2504:
2494:
2489:
2480:
2460:
2455:
2450:
2441:
2436:
2428:
2424:
2414:
2409:
2403:
2400:
2397:
2393:
2385:
2373:
2369:
2360:
2355:
2340:
2335:
2326:
2321:
2317:
2314:
2311:
2305:
2301:
2292:
2287:
2283:
2279:
2271:
2270:
2269:
2267:
2247:
2244:
2240:
2228:
2224:
2220:
2215:
2211:
2204:
2201:
2193:
2189:
2185:
2180:
2176:
2166:
2157:
2152:
2143:
2128:
2124:
2112:
2108:
2104:
2099:
2095:
2088:
2085:
2077:
2073:
2069:
2064:
2060:
2050:
2045:
2041:
2032:
2031:
2030:
2012:
2009:
2006:
2002:
1999:
1996:
1993:
1990:
1982:
1977:
1972:
1969:
1966:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1937:
1929:
1921:
1918:
1913:
1900:
1897:
1892:
1883:
1872:
1864:
1856:
1853:
1848:
1835:
1832:
1827:
1822:
1809:
1808:
1807:
1805:
1801:
1797:
1793:
1789:
1785:
1781:
1777:
1773:
1769:
1765:
1760:
1758:
1757:
1752:
1748:
1744:
1740:
1736:
1732:
1728:
1724:
1720:
1700:
1695:
1687:
1682:
1678:
1674:
1669:
1664:
1660:
1652:
1648:
1642:
1638:
1634:
1629:
1625:
1619:
1615:
1605:
1601:
1595:
1591:
1587:
1582:
1578:
1572:
1568:
1560:
1555:
1551:
1547:
1542:
1537:
1533:
1526:
1516:
1515:
1514:
1496:
1491:
1487:
1483:
1474:
1469:
1465:
1461:
1456:
1451:
1447:
1440:
1437:
1434:
1430:
1427:
1418:
1414:
1408:
1404:
1400:
1395:
1391:
1385:
1381:
1374:
1371:
1366:
1362:
1358:
1349:
1344:
1340:
1336:
1331:
1326:
1322:
1315:
1310:
1306:
1302:
1299:
1294:
1290:
1286:
1283:
1278:
1274:
1270:
1262:
1261:
1260:
1258:
1254:
1250:
1246:
1242:
1238:
1234:
1214:
1211:
1206:
1202:
1196:
1192:
1188:
1183:
1179:
1173:
1169:
1160:
1159:
1158:
1157:is positive:
1156:
1152:
1148:
1144:
1140:
1136:
1132:
1128:
1124:
1120:
1116:
1112:
1108:
1104:
1084:
1081:
1071:
1067:
1063:
1060:
1057:
1052:
1049:
1046:
1043:
1040:
1031:
1027:
1023:
1020:
1017:
1012:
1009:
1006:
1003:
1000:
994:
989:
979:
966:
965:
964:
963:implies that
939:
935:
931:
928:
925:
920:
917:
914:
911:
908:
895:
891:
887:
884:
881:
876:
873:
870:
867:
864:
858:
852:
849:
846:
843:
840:
837:
834:
825:
822:
819:
816:
813:
810:
807:
796:
795:
794:
769:
765:
761:
758:
755:
750:
747:
744:
741:
738:
733:
730:
727:
724:
721:
718:
715:
709:
703:
700:
697:
691:
683:
682:
681:
679:
675:
670:
668:
664:
660:
656:
652:
648:
626:
620:
615:
608:
603:
597:
592:
586:
583:
580:
574:
566:
565:
564:
563:
559:
555:
551:
529:
525:
521:
517:
514:
511:
508:
504:
501:
497:
494:
491:
486:
482:
478:
474:
471:
466:
462:
458:
450:
449:
448:
446:
442:
437:
423:
420:
417:
414:
411:
408:
405:
385:
382:
379:
376:
373:
370:
367:
359:
355:
351:
347:
343:
339:
335:
331:
327:
323:
319:
309:
307:
303:
299:
295:
291:
287:
283:
279:
275:
271:
267:
263:
260:defined on L(
259:
255:
251:
247:
243:
239:
235:
231:
210:
201:
177:
170:
158:
154:
150:
149:
144:
140:
139:
134:
130:
127:
123:
104:
98:
90:
81:
78:
67:
57:
44:
43:
42:
41:
37:
33:
29:
22:
21050:
21024:
21018:
21000:
20996:
20978:
20974:
20943:
20921:
20903:
20899:
20881:
20877:
20785:
20781:
20743:
20739:
20702:
20696:
20678:
20669:
20651:
20611:
20592:
20565:
20555:
20551:
20542:
20521:
20517:
20501:
20497:
20478:
20469:
20449:
20445:
20420:
20403:
20391:
20376:Väisälä 1984
20371:
20360:
20318:
20289:Ahlfors 1966
20281:
20269:
20262:Ahlfors 1966
20257:
20245:
20233:
20220:Ahlfors 1966
20212:
20201:
20193:Ahlfors 1966
20175:
20162:, p. 99
20160:Ahlfors 1966
20152:
20147:, p. 98
20145:Ahlfors 1966
20140:
20133:, p. 88
20125:Ahlfors 1966
20117:
20096:
20078:Ahlfors 1966
20060:
20048:
19948:
19943:
19939:
19935:
19931:
19924:
19917:
19916:| ≥ 1. Thus
19913:
19909:
19905:
19901:
19894:
19890:
19886:
19882:
19878:
19874:
19867:
19863:
19859:
19855:
19851:
19849:
19844:
19842:
19737:
19733:
19729:
19725:
19723:
19710:
19706:
19698:
19694:
19692:
19615:
19487:
19479:
19476:Tukia (1985)
19471:
19468:an extension
19459:
19455:
19453:
19448:
19444:
19440:
19436:
19432:
19428:
19424:
19420:
19416:
19412:
19408:
19404:
19400:
19396:
19392:
19388:
19384:
19380:
19376:
19372:
19368:
19364:
19360:
19356:
19352:
19348:
19344:
19340:
19336:
19332:
19328:
19324:
19320:
19316:
19301:
19297:
19293:
19289:
19285:
19281:
19273:
19269:
19265:
19261:
19259:
19251:
19244:
19237:
19230:
19223:
19216:
19212:
19208:
19204:
19200:
19196:
19191:denotes the
19190:
18981:
18770:
18766:
18763:quasi-Möbius
18762:
18758:
18756:
18751:
18749:
18587:
18488:
18247:
18243:
18235:
18233:
18210:
18203:
18196:
18189:
18181:
18169:
18161:
18158:Jordan curve
18153:
18149:
18145:
18141:
18137:
18129:
18122:
18114:
18107:
18105:
18093:
18085:
18078:
18071:
18064:
18056:
18044:
18040:
18036:
18028:
18024:
18020:
18016:
18014:
17900:
17681:
17677:
17673:
17669:
17662:
17650:
17642:
17638:
17631:
17624:
17622:
17617:
17610:
17603:
17598:
17582:
17578:
17574:
17569:
17565:
17561:
17556:
17552:
17547:
17543:
17539:
17531:
17527:
17523:
17518:
17514:
17503:
17495:
17490:
17486:
17479:
17474:
17470:
17459:
17448:
17444:
17432:
17427:
17419:
17415:
17407:
17403:
17402:
17397:
17393:
17386:
17382:
17378:
17374:
17370:
17366:
17362:
17358:
17354:
17350:
17346:
17342:
17338:
17334:
17330:
17326:
17322:
17318:
17314:
17310:
17309:
17305:
17295:
17291:
17287:
17283:
17279:
17275:
17274:
17152:
17148:
17144:
17140:
17136:
17132:
17128:
17126:
17034:
17030:
17026:
17022:
17018:
17014:
17010:
17006:
17002:
17000:
16937:
16882:
16880:
16781:
16779:
16711:
16637:
16633:
16629:
16625:
16621:
16617:
16613:
16609:
16605:
16603:
16552:
16486:
16482:
16481:(0) = 0 and
16478:
16474:
16469:
16464:
16462:
16371:
16368:
16359:
16353:
16351:
16345:
16341:
16336:
16332:
16327:
16322:
16318:
16314:
16310:
16306:
16302:
16158:
16154:
16150:
16146:
16142:
16138:
16131:
16127:
16122:
16118:
16114:
16106:
16102:
15872:
15861:
15856:
15852:
15847:
15843:
15839:
15835:
15828:
15824:
15819:
15815:
15811:
15807:
15800:
15796:
15789:
15784:
15780:
15776:
15772:
15765:
15760:
15756:
15749:
15669:
15662:
15657:
15653:
15649:
15643:
15639:
15635:
15627:
15623:
15619:
15614:
15610:
15606:
15602:
15597:
15593:
15589:
15584:
15580:
15576:
15572:
15568:
15563:
15559:
15555:
15550:
15546:
15542:
15538:
15534:
15145:
15141:
15137:
15133:
15129:
15121:
15117:
15113:
15108:
15104:
15100:
15096:
14964:
14960:
14955:
14951:
14946:
14942:
14937:
14933:
14928:
14924:
14919:
14915:
14910:
14906:
14902:
14898:
14894:
14890:
14886:
14882:
14873:
14869:
14865:
14861:
14708:
14704:
14700:
14696:
14692:
14423:
14419:
14411:
14407:
14403:
14399:
14395:
14391:
14386:
14382:
14377:
14373:
14368:
14364:
14359:
14355:
14351:
14347:
14343:
14339:
14334:
14330:
14325:
14321:
14316:. So by the
14313:
14309:
14305:
14296:
14292:
14287:
14283:
14281:
14276:
14272:
14268:
14263:
14259:
14257:
14183:
14179:
14174:
14170:
14168:
14095:
14090:
14084:
14081:
14076:
14072:
14068:
14063:
14059:
14057:
14042:
14038:
14034:
14030:
14026:
14022:
14020:
13851:
13847:
13840:
13836:
13829:
13827:
13728:
13724:
13720:
13718:
13480:
13476:
13472:
13468:
13464:
13460:
13456:
13452:
13448:
13446:
13346:
13172:
13164:
13160:
13156:
13154:
13075:
13071:
13069:
12978:theory that
12975:
12971:
12967:
12963:
12961:
12640:
12636:
12628:
12626:
12621:
12617:
12615:
12539:is given by
12536:
12532:
12528:
12526:
12460:
12346:
12342:
12338:
12334:
12330:
12326:
12322:
12318:
12314:
12310:
12308:
12153:
12149:
12144:
12140:
12136:
12132:
12130:
12070:
12068:
12007:
12005:tends to 2.
12002:
11997:
11993:
11989:
11985:
11973:
11972:theory when
11969:
11967:
11958:
11954:
11950:
11946:
11942:
11940:
11930:
11926:
11918:
11914:
11910:
11906:
11900:
11671:
11667:
11662:
11658:
11656:
11515:
11512:power series
11507:
11503:
11499:
11495:
11491:
11487:
11485:
11395:
11393:
11272:
11270:
11145:
11140:
11136:
11132:
11128:
11118:
10993:
10884:
10880:
10878:
10720:
10716:
10712:
10708:
10704:
10700:
10696:
10692:
10688:
10684:
10680:
10676:
10672:
10664:
10660:
10656:
10654:
10649:
10641:
10637:
10633:
10629:
10625:
10621:
10617:
10615:
10553:
10484:
10480:
10478:
10354:
10350:
10345:
10341:
10337:
10333:
10331:
10315:
10312:
10309:
10214:
10212:
10098:
10094:
10092:
10089:
10016:
10014:
9941:
9937:
9935:
9930:
9928:
9923:
9919:
9915:
9911:
9909:
9756:
9749:
9745:
9743:
9652:
9648:
9644:
9636:
9634:
9525:
9521:
9517:
9511:
9399:
9323:(0) = 0 and
9320:
9318:
9156:
9154:
9149:
9145:
9142:covering map
9137:
9133:
9129:
9125:
9123:
9024:
9020:
9016:
9014:
8938:
8933:
8929:
8923:
8752:
8748:
8744:
8742:
8737:
8733:
8729:
8725:
8721:
8719:
8647:
8644:
8640:
8636:
8633:
8629:
8621:
8616:
8609:
8541:
8537:
8535:
8414:
8411:
8407:
8403:
8399:
8394:
8390:
8386:
8384:
8314:
8310:
8306:
8302:
8298:
8294:
8290:
8286:
8282:
8278:
8274:
8270:
8267:
8263:
8259:
8255:
8253:
8248:
8244:
8240:
8236:
8232:
8228:
8224:
8220:
8218:Weyl's lemma
8215:
8161:
8159:
8154:
8149:
8074:is given by
8071:
8067:
8063:
8061:
7996:
7882:
7825:
7753:
7749:
7747:
7597:
7593:
7588:
7584:
7580:
7576:
7574:
7514:
7512:
7453:
7448:
7444:
7440:
7438:
7380:
7376:
7374:
7314:
7264:
7260:
7256:
7254:
7187:
7180:
7178:
7118:
7070:
7067:
7000:
6996:
6994:
6897:
6893:
6885:
6884:> 2. Let
6881:
6870:
6864:
6863:Solution in
6856:
6854:
6483:
6080:
5953:
5949:
5945:
5941:
5885:
5793:
5789:
5785:
5781:
5777:
5773:
5769:
5765:
5741:
5737:
5735:
5730:
5726:
5722:
5718:
5714:
5681:
5677:
5673:
5669:
5665:
5661:
5657:
5653:
5649:
5645:
5641:
5637:
5633:
5629:
5625:
5621:
5617:
5613:
5609:
5607:
5489:
5485:
5481:
5479:
5428:
5424:
5420:
5292:
5288:
5284:
5282:
5187:
4770:
4626:
4622:
4618:
4614:
4610:
4608:
4601:
4560:
4556:
4554:
4487:
4050:
3917:
3913:
3909:
3907:
3488:
3486:
3481:
3366:
3289:
3285:
3281:
3277:
3273:
3269:
3265:
3263:
2912:
2628:
2530:
2478:
2476:
2268:is given by
2265:
2263:
2028:
1799:
1795:
1791:
1787:
1783:
1779:
1775:
1771:
1767:
1763:
1761:
1754:
1750:
1746:
1742:
1738:
1734:
1730:
1726:
1722:
1718:
1716:
1512:
1256:
1252:
1248:
1244:
1240:
1236:
1232:
1230:
1150:
1146:
1142:
1138:
1134:
1130:
1126:
1122:
1118:
1114:
1110:
1106:
1102:
1100:
962:
792:
677:
673:
671:
666:
662:
658:
654:
650:
646:
644:
561:
553:
549:
547:
444:
440:
438:
357:
353:
349:
344:) is called
341:
337:
333:
329:
325:
321:
315:
265:
261:
245:
159:, and where
156:
152:
147:
142:
141:, and where
137:
132:
128:
121:
119:
31:
25:
21027:: 218–234,
20884:: 141–177,
20863:10.4171/103
20838:10.4171/055
20813:10.4171/029
20354:Sibner 1965
20127:, p. 9
20053:Spivak 1999
19951:condition:
19862:| ≥ 1. Let
19850:Now extend
19272:(and hence
19193:cross-ratio
17600:Bers (1961)
14418:The limits
13159:is in L if
12135:are L. Let
11125:plane curve
10487:satisfying
9940:> 0, if
9759:defined by
8157:is smooth.
7579:are L. Let
5744:, 0) to be
5672:by setting
5480:Gauss lets
4813:holomorphic
1753:are called
1745:defined on
1149:. The map
28:mathematics
21062:Categories
20997:Acta Math.
20981:: 97–114,
20770:0018.40501
20729:62.0565.02
20705:(5): 316,
20518:Acta Math.
20458:01.0275.02
20384:References
20299:Lehto 1987
19893:| ≤ 1 and
19304:| = 2 sin
18250:such that
18224:See also:
18099:becomes a
17593:See also:
17467:free group
17465:, it is a
17337:such that
17013:(0) = 0 =
16553:satisfies
16365:Uniqueness
14406:, so that
14275:norm with
13070:Moreover,
12157:– 1. Then
10326:See also:
10101:satisfies
9944:satisfies
7601:– 1. Then
7183:. Thus on
5423:. So the
2631:just when
1231:And using
346:isothermal
284:. Various
21043:189767039
20906:: 79–85,
20721:0002-9904
20654:, Dover,
20558:: 125–143
20524:: 23–48,
20338:Bers 1961
19988:∘
19980:−
19805:¯
19781:μ
19764:∞
19760:‖
19756:μ
19753:‖
19668:θ
19643:θ
19593:θ
19578:ψ
19575:−
19557:θ
19539:ψ
19530:
19515:θ
19158:−
19129:−
19098:−
19069:−
18887:≤
18727:≤
18697:−
18644:−
18614:≤
18606:−
18553:−
18517:−
18443:−
18401:−
18380:⋅
18374:≤
18344:−
18291:−
18180:of PSL(2,
18160:dividing
18035:of PSL(2,
17984:^
17981:μ
17960:¯
17919:^
17916:μ
17874:∈
17869:¯
17854:¯
17845:¯
17834:μ
17813:^
17810:μ
17791:∈
17762:^
17759:μ
17740:∈
17720:μ
17702:^
17699:μ
17616:of genus
17500:limit set
17253:μ
17248:¯
17245:λ
17240:−
17232:λ
17229:−
17226:μ
17215:¯
17167:ν
17086:∈
17080:¯
17060:−
16978:¯
16960:¯
16845:−
16829:μ
16807:−
16744:μ
16668:¯
16575:¯
16567:∂
16530:¯
16512:−
16447:ψ
16428:ψ
16413:ψ
16404:¯
16392:ψ
16354:existence
16274:∗
16270:μ
16264:¯
16261:μ
16256:−
16248:μ
16245:−
16240:∗
16236:μ
16224:¯
16176:ν
16063:μ
16060:⋅
16043:¯
16026:−
16009:′
16005:μ
15959:μ
15956:⋅
15949:¯
15922:−
15896:′
15892:μ
15709:∘
15697:¯
15689:∂
15668:in Ω, if
15495:−
15463:‖
15431:∂
15427:‖
15424:≤
15379:∂
15364:∬
15360:≤
15311:¯
15303:∂
15294:−
15263:∂
15248:∬
15202:∬
15053:μ
15049:−
15046:μ
15024:−
15015:μ
14993:μ
14989:−
14983:μ
14837:ψ
14831:¯
14823:∂
14819:⋅
14813:∬
14810:−
14804:ψ
14798:¯
14790:∂
14786:⋅
14773:∬
14767:−
14761:ψ
14758:⋅
14745:∬
14736:ψ
14733:⋅
14727:∬
14663:∞
14659:‖
14649:μ
14645:‖
14642:−
14628:‖
14618:μ
14614:‖
14608:≤
14599:‖
14585:‖
14571:‖
14557:‖
14518:−
14499:∂
14464:¯
14456:∂
14236:∞
14232:‖
14228:μ
14225:‖
14222:≤
14217:∞
14213:‖
14203:μ
14199:‖
14149:μ
14146:⋅
14134:φ
14130:−
14112:μ
14054:Existence
13990:⋅
13967:∞
13963:‖
13959:μ
13956:‖
13953:−
13943:∞
13939:‖
13935:μ
13932:‖
13915:π
13887:≤
13807:∞
13803:‖
13799:μ
13796:‖
13765:π
13759:≤
13750:‖
13746:μ
13743:‖
13685:−
13651:−
13628:−
13604:‖
13600:μ
13597:‖
13579:∞
13575:‖
13571:μ
13568:‖
13565:−
13546:≤
13519:−
13428:π
13414:‖
13405:−
13394:−
13380:−
13372:‖
13315:−
13292:−
13268:‖
13261:‖
13248:≤
13218:−
13113:−
13008:¯
12917:−
12877:∬
12871:π
12863:−
12828:−
12819:−
12782:∬
12776:π
12768:−
12712:∬
12703:−
12699:π
12574:−
12500:−
12486:¯
12445:μ
12437:−
12426:−
12409:−
12395:μ
12384:−
12373:−
12349:μ. Hence
12289:μ
12274:μ
12249:μ
12231:¯
12204:¯
12112:−
12039:μ
12030:¯
11921:) is the
11861:−
11832:≥
11825:∑
11813:θ
11801:′
11779:⋅
11724:≥
11717:∑
11699:ψ
11632:θ
11620:θ
11614:⋯
11595:θ
11570:θ
11551:θ
11536:θ
11470:⋯
11410:ψ
11353:θ
11333:θ
11315:ψ
11252:α
11249:−
11243:θ
11237:κ
11228:θ
11213:θ
11203:θ
11197:κ
11192:π
11180:∫
11173:π
11158:α
11121:curvature
11100:θ
11092:′
11081:θ
11073:′
11070:′
11062:−
11056:θ
11048:′
11037:θ
11029:′
11026:′
11012:θ
11006:κ
10953:θ
10933:θ
10927:κ
10854:θ
10846:′
10832:−
10817:θ
10802:θ
10794:′
10780:−
10771:−
10765:θ
10747:θ
10583:−
10575:∘
10568:μ
10518:μ
10509:¯
10455:−
10449:¯
10437:−
10431:¯
10419:μ
10402:¯
10369:μ
10285:−
10279:¯
10267:−
10261:¯
10192:¯
10180:−
10174:¯
10162:μ
10146:¯
10113:μ
10069:¯
10060:¯
9994:¯
9985:¯
9974:μ
9956:μ
9877:−
9869:∘
9857:¯
9825:¯
9820:∞
9816:μ
9810:μ
9807:−
9794:μ
9771:λ
9722:∞
9718:μ
9705:μ
9698:μ
9692:ψ
9689:−
9677:μ
9674:ψ
9668:μ
9606:μ
9603:⋅
9595:¯
9590:′
9573:′
9547:~
9544:μ
9484:−
9467:μ
9453:¯
9414:μ
9375:−
9362:−
9290:μ
9284:¯
9275:μ
9269:−
9255:μ
9251:−
9242:μ
9231:¯
9198:∘
9188:−
9180:∘
9173:μ
9069:μ
9061:−
8989:μ
8972:μ
8963:¯
8889:μ
8881:−
8827:¯
8810:−
8685:μ
8679:−
8664:∂
8571:¯
8481:≠
8453:¯
8417:, define
8347:μ
8338:¯
8185:¯
8177:∂
8110:∗
8037:∗
8023:¯
7980:μ
7972:−
7961:−
7944:∗
7930:μ
7919:−
7908:−
7863:μ
7845:−
7803:μ
7781:μ
7732:μ
7717:μ
7692:μ
7674:¯
7647:¯
7556:−
7484:μ
7475:¯
7346:¯
7292:⋆
7223:⋆
7211:¯
7203:∂
7153:π
7101:¯
7093:∂
7030:¯
6964:^
6948:¯
6923:^
6827:μ
6806:−
6763:−
6733:−
6651:−
6588:¯
6551:¯
6501:¯
6441:−
6427:−
6384:−
6370:−
6361:−
6340:¯
6281:−
6258:−
6224:−
6035:−
5841:−
5788:), where
5547:−
5533:−
5527:−
5465:ψ
5442:∘
5439:ψ
5268:ψ
5245:∘
5242:ψ
5199:∘
5196:ψ
5157:μ
5136:¯
5114:¯
5093:∘
5087:¯
5079:ψ
5053:∘
5044:ψ
5033:¯
5027:¯
5007:∘
5001:¯
4993:ψ
4980:¯
4962:∘
4953:ψ
4920:∘
4917:ψ
4906:¯
4891:∘
4888:ψ
4852:¯
4844:ψ
4823:ψ
4799:ψ
4737:−
4694:−
4673:μ
4652:¯
4571:μ
4520:¯
4496:μ
4446:−
4427:−
4368:−
4266:−
4125:μ
4081:−
3958:−
3828:−
3560:−
3503:μ
3456:μ
3432:¯
3388:−
3292:), where
3200:−
3137:−
3126:−
2872:−
2788:−
2723:−
2694:−
2606:μ
2548:¯
2498:¯
2445:¯
2418:¯
2344:¯
2330:¯
2186:−
2161:¯
2105:−
2000:−
1986:¯
1926:∂
1910:∂
1887:¯
1879:∂
1861:∂
1854:−
1845:∂
1819:∂
1259:given by
1189:−
1064:−
1024:−
1010:−
980:μ
932:−
918:−
888:−
844:−
838:−
811:−
762:−
719:−
692:μ
415:≤
409:≤
377:≤
371:≤
270:unit disk
214:¯
206:∂
198:∂
175:∂
167:∂
96:∂
88:∂
82:μ
71:¯
63:∂
55:∂
38:, is the
20941:(1996),
20650:(1991),
20419:(2009),
20402:(1966),
20019:See also
19296:, then |
18172:gives a
17564:. where
16640:. Hence
15609:) since
15533:Thus if
14279:< ∞.
14267:tend to
13828:Setting
13447:For any
12624:> 2.
10616:so that
9027:so that
8262:and set
8231:. Since
8070:. Hence
5898:′
5796:satisfy
5684:) to be
5508:′
5419:for all
3369:is then
3268:is then
1155:Jacobian
653:> 0,
649:> 0,
447:and let
292:and the
20967:1402697
20853:2524085
20828:2284826
20802:1970228
20762:1989904
20641:2377904
20585:2245223
19912:) for |
19889:) for |
19732:| <
19310:√
19236:) and (
16473:'s. If
16130:. Thus
15860:≠ 0 at
15646:theory.
14398:= id =
14372:= id =
14350:| >
14312:| >
14271:in any
14182:| <
14075:| <
13731:, then
13471:) and (
11123:of the
11119:is the
10879:Taking
8402:) with
8227:| >
7447:/2 and
6888:be the
556:. The
21041:
20965:
20955:
20928:
20869:
20844:
20819:
20800:
20768:
20760:
20727:
20719:
20685:
20658:
20639:
20629:
20599:
20583:
20573:
20485:
20456:
20427:
19387:)| / |
18982:where
17485:where
17353:where
17001:Hence
16881:Since
16313:* and
16149:* and
16141:* and
15771:). So
13347:where
13167:. The
13163:= 1 −
11494:) to (
10994:where
10883:= 1 −
9023:+ 1 =
8720:Since
8385:where
8305:is in
8216:so by
7883:Hence
5886:Since
5660:) for
5616:, and
5608:where
5291:, and
4049:Since
3085:gives
2029:Since
30:, the
21039:S2CID
20798:JSTOR
20758:JSTOR
20442:(PDF)
20285:See:
20216:See:
20179:See:
20121:See:
20100:See:
20064:See:
20041:Notes
19736:with
19713:) + 2
19693:with
19411:| = |
19343:with
19308:/3 =
19276:) is
17653:is a
17412:Bers'
17377:with
17135:. If
16153:then
15842:) = |
15748:near
13850:| ≤
12069:with
11953:norm
10691:) in
10344:onto
9651:≤ 1,
9400:then
8536:Thus
8398:of L(
7513:with
6892:on L(
5283:When
4603:Gauss
1790:) + i
1717:When
234:Gauss
20953:ISBN
20926:ISBN
20867:ISBN
20842:ISBN
20817:ISBN
20717:ISSN
20683:ISBN
20656:ISBN
20627:ISBN
20597:ISBN
20571:ISBN
20483:ISBN
20425:ISBN
20156:See
19923:and
19904:) =
19877:) =
19769:<
19705:) =
19447:and
19431:and
19395:) −
19379:) −
19327:and
19207:and
18246:and
18228:and
18209:and
18195:and
18117:/ Γ
18088:/ Γ
18015:for
17630:and
17522:for
17373:| =
17361:| =
17290:and
17147:for
17033:and
16636:) =
16618:F:=G
16344:* ∘
16331:and
16305:* ∘
16157:* ∘
16109:(Λ).
15144:and
14707:) –
14695:and
14422:and
14410:and
14342:and
14329:and
14258:The
14186:and
14169:The
14037:| ≤
13839:and
13727:| ≤
12531:and
12325:and
12313:and
11968:The
11146:Let
11143:)).
10711:)) (
10632:and
10332:Let
9744:Let
8639:and
8293:and
8285:| ≤
8281:|, |
8237:CT*h
8066:and
7752:and
7379:and
7071:The
5792:and
5768:(0)=
5729:) =
5628:and
4835:has
4771:Let
3058:and
3001:and
2477:The
1778:) =
1741:and
1212:>
1113:) =(
1101:Let
1082:<
672:The
665:and
398:and
272:and
256:, a
228:are
190:and
120:for
21029:doi
21005:doi
21001:154
20983:doi
20908:doi
20904:116
20886:doi
20859:doi
20834:doi
20809:doi
20790:doi
20766:Zbl
20748:doi
20725:JFM
20707:doi
20617:doi
20526:doi
20522:157
20506:doi
20454:JFM
19701:+ 2
19527:exp
19339:in
19292:in
19257:).
18489:If
18168:by
18055:on
17502:of
17396:≤ |
16780:If
16604:So
16321:by
16137:If
15799:' (
15097:μ φ
14770:lim
14742:lim
13719:If
13175:is
12331:μTF
12323:TμF
12309:If
11901:By
11514:in
10483:of
9910:If
8406:in
7748:If
3171:and
2578:of
1749:by
1251:on
1145:in
560:of
552:on
26:In
21064::
21037:,
21025:44
21023:,
20999:,
20977:,
20963:MR
20961:,
20951:,
20902:,
20882:48
20880:,
20865:,
20850:MR
20848:,
20840:,
20825:MR
20823:,
20815:,
20796:,
20786:72
20784:,
20780:,
20764:,
20756:,
20744:43
20742:,
20723:,
20715:,
20703:42
20701:,
20637:MR
20635:,
20625:,
20581:MR
20579:,
20556:53
20554:,
20520:,
20502:14
20500:,
20452:,
20444:,
20415:;
20344:^
20330:^
19881:∘
19717:.
19490::
19443:,
19439:,
19427:,
19415:−
19407:−
19363:,
19359:,
19355:,
19347:≠
19335:,
19319:–
19300:−
19288:≠
19211:=
19203:=
18769:,
18216:.
17609:,
17581:∘
17560:∘
17551:=
17542:∘
17390:−1
17385:∘
17349:∘
17345:=
17341:∘
17286:,
17151:∘
17021:=
17005:−
16763:1.
16587:0.
16317:=
15814:∘
15779:∘
15775:=
15755:=
15588:,
14963:⋅
14950:⋅
14932:+
14923:⋅
14914:=
14901:+
14897:⋅
14893:=
14521:1.
14039:CR
13835:=
13475:−
13467:−
13173:Pf
13157:Pf
12972:Cf
12968:Pf
12964:Pf
12345:=
12337:−
12329:=
12327:BF
12321:=
12319:AF
12148:=
12139:=
11518::
11506:),
11398::
11135:),
10703:),
10652:.
10624:∘
10620:=
9926:.
9918:∘
9520:=
9144:,
8937:=
8751:=
8740:.
8730:μU
8728:–
8722:UG
8643:–
8632:–
8410:+
8266:=
8239:+
8235:=
8164:,
7592:=
7583:=
7445:iz
7381:Cf
6468:2.
5952:,
5946:dy
5944:,
5942:dx
5870:0.
5794:dy
5790:dx
5784:,
5778:dy
5776:,
5774:dx
5733:.
5717:,
5680:,
5632:=
5624:=
5612:,
5287:,
4625:,
4613:,
3916:,
3280:)
3276:,
2528:.
1806::
1798:+i
1786:+i
1774:+i
1759:.
1733:,
1249:dv
1247:+
1245:du
1243:=
1241:ds
1215:0.
1125:),
1085:1.
669:.
657:−
655:EG
340:,
324:,
21031::
21007::
20985::
20979:5
20910::
20888::
20861::
20836::
20811::
20792::
20750::
20709::
20619::
20528::
20508::
20450:6
20322:*
20001:.
19996:2
19992:f
19983:1
19975:1
19971:f
19967:=
19964:f
19944:i
19940:f
19936:z
19932:z
19928:2
19925:F
19921:1
19918:F
19914:z
19910:z
19908:(
19906:G
19902:z
19900:(
19898:2
19895:F
19891:z
19887:z
19885:(
19883:F
19879:G
19875:z
19873:(
19871:1
19868:F
19864:G
19860:z
19856:C
19852:μ
19845:f
19826:.
19821:z
19817:F
19812:/
19802:z
19797:F
19793:=
19790:)
19787:z
19784:(
19775:,
19772:1
19738:R
19734:R
19730:z
19726:F
19715:π
19711:θ
19709:(
19707:g
19703:π
19699:θ
19697:(
19695:g
19676:,
19671:)
19665:(
19662:g
19659:i
19655:e
19651:=
19648:)
19640:i
19636:e
19632:(
19629:f
19599:,
19596:]
19590:)
19587:)
19584:r
19581:(
19572:1
19569:(
19566:i
19563:+
19560:)
19554:(
19551:g
19548:)
19545:r
19542:(
19536:i
19533:[
19524:r
19521:=
19518:)
19512:,
19509:r
19506:(
19503:F
19488:f
19480:f
19472:f
19460:F
19456:f
19449:z
19445:y
19441:w
19437:x
19433:z
19429:y
19425:x
19421:w
19417:y
19413:x
19409:z
19405:x
19401:z
19399:(
19397:f
19393:x
19391:(
19389:f
19385:y
19383:(
19381:f
19377:x
19375:(
19373:f
19369:f
19365:b
19361:y
19357:a
19353:x
19349:b
19345:a
19341:S
19337:b
19333:a
19329:y
19325:x
19321:y
19317:x
19312:3
19306:π
19302:b
19298:a
19294:S
19290:b
19286:a
19282:S
19274:f
19270:f
19266:f
19262:f
19255:3
19252:z
19250:,
19248:4
19245:z
19243:,
19241:2
19238:z
19234:4
19231:z
19229:,
19227:3
19224:z
19222:,
19220:1
19217:z
19213:b
19209:d
19205:a
19201:c
19197:f
19171:)
19166:4
19162:z
19153:1
19149:z
19145:(
19142:)
19137:3
19133:z
19124:2
19120:z
19116:(
19111:)
19106:4
19102:z
19093:2
19089:z
19085:(
19082:)
19077:3
19073:z
19064:1
19060:z
19056:(
19050:=
19047:)
19042:4
19038:z
19034:,
19029:3
19025:z
19021:;
19016:2
19012:z
19008:,
19003:1
18999:z
18995:(
18965:,
18960:d
18955:|
18950:)
18945:4
18941:z
18937:,
18932:3
18928:z
18924:;
18919:2
18915:z
18911:,
18906:1
18902:z
18898:(
18894:|
18890:c
18883:|
18879:)
18876:)
18871:4
18867:z
18863:(
18860:f
18857:,
18854:)
18849:3
18845:z
18841:(
18838:f
18835:;
18832:)
18827:2
18823:z
18819:(
18816:f
18813:,
18810:)
18805:1
18801:z
18797:(
18794:f
18791:(
18787:|
18771:d
18767:c
18759:f
18752:f
18733:.
18730:a
18720:|
18716:)
18711:3
18707:z
18703:(
18700:f
18694:)
18689:1
18685:z
18681:(
18678:f
18674:|
18667:|
18663:)
18658:2
18654:z
18650:(
18647:f
18641:)
18636:1
18632:z
18628:(
18625:f
18621:|
18609:1
18602:a
18571:,
18567:|
18561:3
18557:z
18548:1
18544:z
18539:|
18535:=
18531:|
18525:2
18521:z
18512:1
18508:z
18503:|
18472:.
18463:b
18458:|
18451:3
18447:z
18438:1
18434:z
18429:|
18421:b
18416:|
18409:2
18405:z
18396:1
18392:z
18387:|
18377:a
18367:|
18363:)
18358:3
18354:z
18350:(
18347:f
18341:)
18336:1
18332:z
18328:(
18325:f
18321:|
18314:|
18310:)
18305:2
18301:z
18297:(
18294:f
18288:)
18283:1
18279:z
18275:(
18272:f
18268:|
18248:b
18244:a
18236:f
18214:2
18211:M
18207:1
18204:M
18200:2
18197:M
18193:1
18190:M
18186:2
18182:C
18178:2
18176:Γ
18170:h
18166:1
18162:C
18154:C
18150:h
18146:H
18142:C
18138:H
18134:1
18130:H
18126:2
18123:M
18119:1
18115:H
18111:1
18108:M
18097:1
18094:M
18090:1
18086:H
18082:1
18079:M
18075:1
18072:M
18068:1
18065:M
18061:1
18057:H
18049:1
18045:H
18041:f
18037:R
18033:1
18029:f
18025:C
18021:f
18017:g
17999:,
17996:)
17993:z
17990:(
17970:z
17966:g
17955:z
17951:g
17943:=
17940:)
17937:)
17934:z
17931:(
17928:g
17925:(
17885:.
17882:)
17878:H
17866:z
17861:(
17850:)
17842:z
17837:(
17828:=
17825:)
17822:z
17819:(
17802:,
17799:)
17795:R
17788:z
17785:(
17780:0
17777:=
17774:)
17771:z
17768:(
17751:,
17748:)
17744:H
17737:z
17734:(
17729:)
17726:z
17723:(
17717:=
17714:)
17711:z
17708:(
17682:C
17678:H
17674:μ
17670:H
17666:1
17663:M
17651:H
17643:R
17639:H
17635:2
17632:M
17628:1
17625:M
17618:g
17614:2
17611:M
17607:1
17604:M
17583:h
17579:F
17575:i
17570:i
17566:S
17562:h
17557:i
17553:S
17548:i
17544:R
17540:h
17536:1
17532:h
17528:C
17524:i
17519:i
17515:R
17511:1
17507:0
17504:G
17496:G
17491:i
17487:R
17483:0
17480:R
17478:∘
17475:i
17471:R
17463:0
17460:R
17452:0
17449:G
17445:G
17441:1
17437:1
17433:G
17428:k
17424:1
17420:F
17408:k
17398:z
17394:s
17387:h
17383:F
17379:s
17375:s
17371:z
17367:g
17363:r
17359:z
17355:R
17351:h
17347:g
17343:R
17339:h
17335:g
17331:C
17327:h
17323:C
17319:r
17315:F
17296:L
17292:g
17288:λ
17284:μ
17280:z
17276:λ
17259:.
17237:1
17210:z
17206:g
17198:z
17194:g
17188:=
17185:)
17182:)
17179:z
17176:(
17173:g
17170:(
17153:g
17149:f
17145:ν
17141:g
17137:g
17133:μ
17129:f
17110:.
17107:)
17103:C
17099:(
17094:p
17090:L
17077:z
17072:f
17066:,
17063:1
17055:z
17051:f
17035:f
17031:μ
17027:f
17023:g
17019:f
17015:g
17011:f
17007:g
17003:f
16984:.
16975:z
16970:g
16966:=
16957:z
16952:f
16921:.
16916:z
16912:g
16908:=
16903:z
16899:f
16883:T
16864:.
16861:)
16858:)
16853:z
16849:g
16840:z
16836:f
16832:(
16826:(
16823:T
16820:=
16815:z
16811:g
16802:z
16798:f
16782:g
16760:+
16757:)
16752:z
16748:f
16741:(
16738:T
16735:=
16730:z
16726:f
16695:z
16692:+
16689:)
16686:z
16683:(
16680:f
16677:=
16674:)
16665:z
16660:f
16656:(
16653:P
16638:z
16634:z
16632:(
16630:F
16626:F
16622:F
16614:F
16610:G
16606:F
16584:=
16581:F
16572:z
16536:)
16527:z
16522:f
16518:(
16515:P
16509:f
16506:=
16503:F
16487:z
16483:f
16479:f
16475:f
16470:n
16465:h
16444:T
16441:=
16436:z
16432:)
16425:P
16422:(
16416:,
16410:=
16401:z
16396:)
16389:P
16386:(
16372:μ
16346:f
16342:f
16337:n
16333:g
16328:n
16325:*
16323:f
16319:f
16315:g
16311:f
16307:f
16303:f
16282:,
16253:1
16219:z
16215:f
16207:z
16203:f
16197:=
16194:)
16191:)
16188:z
16185:(
16182:f
16179:(
16159:f
16155:f
16151:μ
16147:μ
16143:f
16139:f
16132:g
16128:L
16123:g
16119:μ
16115:g
16107:f
16103:f
16081:)
16078:)
16075:w
16072:(
16069:g
16066:(
16053:w
16049:g
16038:w
16034:g
16023:=
16020:)
16017:w
16014:(
15968:)
15965:z
15962:(
15944:z
15940:f
15932:z
15928:f
15919:=
15916:)
15913:)
15910:z
15907:(
15904:f
15901:(
15873:g
15867:.
15865:0
15862:z
15857:z
15853:f
15848:z
15844:f
15840:f
15838:(
15836:J
15832:0
15829:z
15825:f
15820:n
15816:f
15812:h
15810:=
15808:f
15804:1
15801:z
15797:h
15793:1
15790:z
15785:n
15781:g
15777:f
15773:h
15769:0
15766:z
15764:(
15761:n
15757:f
15753:1
15750:z
15728:0
15725:=
15722:)
15717:n
15713:g
15706:f
15703:(
15694:z
15670:n
15666:0
15663:z
15658:z
15654:f
15650:f
15644:L
15640:μ
15636:f
15628:K
15626:(
15624:f
15620:f
15618:∘
15615:n
15611:g
15607:K
15605:(
15603:f
15601:∘
15598:n
15594:g
15590:U
15585:n
15581:g
15577:K
15575:(
15573:f
15569:U
15567:(
15564:n
15560:f
15556:U
15554:(
15551:n
15547:f
15545:(
15543:A
15539:U
15537:(
15535:A
15511:.
15506:p
15502:/
15498:2
15492:1
15488:)
15484:U
15481:(
15478:A
15472:2
15467:p
15457:U
15452:|
15445:n
15441:f
15435:z
15421:y
15418:d
15414:x
15411:d
15405:2
15400:|
15393:n
15389:f
15383:z
15374:|
15368:U
15350:y
15347:d
15343:x
15340:d
15334:2
15329:|
15322:n
15318:f
15308:z
15298:|
15289:2
15284:|
15277:n
15273:f
15267:z
15258:|
15252:U
15244:=
15241:y
15238:d
15234:x
15231:d
15227:)
15222:n
15218:f
15214:(
15211:J
15206:U
15198:=
15191:)
15188:)
15185:U
15182:(
15177:n
15173:f
15169:(
15166:A
15146:J
15142:K
15138:U
15134:K
15130:f
15122:n
15118:L
15114:L
15109:n
15105:v
15101:n
15075:.
15070:n
15066:v
15062:)
15057:n
15043:(
15040:+
15037:)
15032:n
15028:v
15021:v
15018:(
15012:=
15007:n
15003:v
14997:n
14986:v
14965:v
14961:μ
14956:n
14952:v
14947:n
14943:μ
14938:n
14934:μ
14929:n
14925:v
14920:n
14916:μ
14911:n
14907:u
14903:μ
14899:v
14895:μ
14891:u
14887:μ
14883:f
14874:n
14870:g
14866:g
14862:v
14840:,
14828:z
14816:f
14807:=
14795:z
14781:n
14777:f
14764:=
14753:n
14749:u
14739:=
14730:u
14709:z
14705:z
14703:(
14701:f
14697:v
14693:u
14671:.
14653:n
14639:1
14632:p
14622:n
14603:p
14593:n
14589:v
14580:,
14575:p
14565:n
14561:u
14513:n
14509:f
14503:z
14495:=
14490:n
14486:v
14480:,
14475:n
14471:f
14461:z
14452:=
14447:n
14443:u
14424:g
14420:f
14412:g
14408:f
14404:f
14402:∘
14400:g
14396:g
14394:∘
14392:f
14387:n
14383:f
14381:∘
14378:n
14374:g
14369:n
14365:g
14363:∘
14360:n
14356:f
14352:R
14348:z
14344:g
14340:f
14335:n
14331:g
14326:n
14322:f
14314:R
14310:z
14306:C
14297:n
14293:g
14288:n
14284:f
14277:p
14273:L
14269:μ
14264:n
14260:μ
14241:.
14207:n
14184:R
14180:z
14175:n
14171:μ
14152:.
14143:)
14138:n
14127:1
14124:(
14121:=
14116:n
14096:n
14091:n
14085:n
14077:R
14073:z
14069:μ
14064:μ
14060:μ
14043:f
14035:z
14031:f
14027:μ
14023:C
14005:,
14002:R
13999:C
13996:=
13993:R
13986:]
13977:p
13973:C
13950:1
13927:p
13923:/
13919:1
13909:p
13905:K
13898:+
13895:1
13891:[
13883:|
13879:)
13876:z
13873:(
13870:f
13866:|
13852:R
13848:z
13844:2
13841:w
13837:z
13833:1
13830:w
13812:.
13791:p
13787:/
13783:1
13779:)
13773:2
13769:R
13762:(
13754:p
13729:R
13725:z
13721:μ
13703:.
13699:|
13693:2
13689:w
13680:1
13676:w
13671:|
13667:+
13662:p
13658:/
13654:2
13648:1
13643:|
13636:2
13632:w
13623:1
13619:w
13614:|
13608:p
13589:p
13585:C
13562:1
13556:p
13552:K
13542:|
13538:)
13533:2
13529:w
13525:(
13522:f
13516:)
13511:1
13507:w
13503:(
13500:f
13496:|
13481:P
13477:B
13473:I
13469:A
13465:I
13461:k
13457:p
13453:C
13449:p
13431:.
13424:/
13418:q
13408:1
13401:)
13397:1
13391:z
13388:(
13383:1
13376:z
13369:=
13364:p
13360:K
13331:,
13326:p
13322:/
13318:2
13312:1
13307:|
13300:2
13296:w
13287:1
13283:w
13278:|
13272:p
13264:f
13256:p
13252:K
13244:|
13240:)
13235:2
13231:w
13227:(
13224:f
13221:P
13215:)
13210:1
13206:w
13202:(
13199:f
13196:P
13192:|
13165:p
13161:q
13139:.
13136:z
13133:+
13130:)
13127:z
13124:(
13121:h
13116:1
13109:T
13105:P
13102:=
13099:)
13096:z
13093:(
13090:f
13076:C
13072:P
13054:.
13051:f
13048:T
13045:=
13040:z
13036:)
13032:f
13029:P
13026:(
13020:,
13017:f
13014:=
13005:z
13000:)
12996:f
12993:P
12990:(
12976:L
12943:.
12940:y
12937:d
12933:x
12930:d
12923:)
12920:w
12914:z
12911:(
12908:z
12903:w
12900:)
12897:z
12894:(
12891:f
12882:C
12868:1
12860:=
12857:y
12854:d
12850:x
12847:d
12842:)
12836:z
12833:1
12822:w
12816:z
12812:1
12806:(
12802:)
12799:z
12796:(
12793:f
12787:C
12773:1
12765:=
12755:y
12752:d
12748:x
12745:d
12739:z
12735:)
12732:z
12729:(
12726:f
12717:C
12706:1
12695:+
12692:)
12689:w
12686:(
12683:f
12680:C
12677:=
12670:)
12667:w
12664:(
12661:f
12658:P
12641:f
12637:p
12629:C
12622:p
12618:p
12600:.
12597:z
12594:+
12591:)
12588:z
12585:(
12582:h
12577:1
12570:T
12566:C
12563:=
12560:)
12557:z
12554:(
12551:f
12537:f
12533:g
12529:μ
12511:,
12508:h
12503:1
12496:T
12492:=
12483:z
12478:g
12440:1
12433:)
12429:B
12423:I
12420:(
12417:=
12412:1
12405:T
12398:,
12392:T
12387:1
12380:)
12376:A
12370:I
12367:(
12364:=
12361:h
12347:T
12343:h
12341:)
12339:A
12335:I
12315:B
12311:A
12292:.
12286:T
12283:+
12280:)
12277:h
12271:(
12268:T
12265:=
12262:)
12257:z
12253:f
12246:(
12243:T
12240:=
12237:)
12228:z
12223:f
12219:(
12216:T
12213:=
12210:)
12201:z
12196:g
12192:(
12189:T
12186:=
12181:z
12177:g
12173:=
12170:h
12154:z
12150:f
12145:z
12141:g
12137:h
12133:g
12115:z
12109:)
12106:z
12103:(
12100:f
12097:=
12094:)
12091:z
12088:(
12085:g
12071:μ
12052:,
12047:z
12043:f
12036:=
12027:z
12022:f
12003:p
11998:p
11994:C
11990:p
11986:C
11974:μ
11970:L
11959:μ
11955:k
11951:L
11947:μ
11943:L
11931:t
11927:t
11919:t
11917:,
11915:θ
11913:(
11911:f
11907:t
11886:.
11882:]
11878:t
11875:d
11870:)
11864:1
11858:n
11854:t
11848:n
11844:f
11840:n
11835:1
11829:n
11820:(
11816:+
11810:d
11806:)
11797:g
11793:+
11790:1
11787:(
11783:[
11775:]
11770:)
11766:!
11763:n
11759:/
11753:n
11749:g
11743:)
11740:n
11737:(
11733:h
11727:0
11721:n
11712:(
11708:)
11705:t
11702:(
11696:t
11693:+
11690:1
11686:[
11672:t
11668:t
11663:n
11659:f
11641:)
11638:t
11635:,
11629:(
11626:g
11623:+
11617:=
11611:+
11606:2
11602:t
11598:)
11592:(
11587:2
11583:f
11579:+
11576:t
11573:)
11567:(
11562:1
11558:f
11554:+
11548:=
11545:)
11542:t
11539:,
11533:(
11530:f
11516:t
11508:t
11504:t
11502:,
11500:θ
11498:(
11496:f
11492:t
11490:,
11488:θ
11467:+
11462:2
11458:t
11452:2
11448:a
11444:+
11441:t
11436:1
11432:a
11428:+
11425:1
11422:=
11419:)
11416:t
11413:(
11396:t
11378:,
11373:2
11369:t
11365:d
11362:+
11357:2
11349:d
11344:2
11340:)
11336:)
11330:(
11327:h
11324:)
11321:t
11318:(
11312:t
11309:+
11306:1
11303:(
11300:=
11295:2
11291:s
11287:d
11273:t
11255:.
11246:)
11240:(
11234:=
11231:)
11225:(
11222:h
11216:,
11210:d
11206:)
11200:(
11189:2
11184:0
11170:2
11166:1
11161:=
11141:θ
11139:(
11137:y
11133:θ
11131:(
11129:x
11127:(
11103:)
11097:(
11088:y
11084:)
11078:(
11066:x
11059:)
11053:(
11044:x
11040:)
11034:(
11022:y
11018:=
11015:)
11009:(
10978:,
10973:2
10969:t
10965:d
10962:+
10957:2
10949:d
10944:2
10940:)
10936:)
10930:(
10924:t
10921:+
10918:1
10915:(
10912:=
10907:2
10903:s
10899:d
10885:r
10881:t
10863:.
10860:)
10857:)
10851:(
10842:x
10838:)
10835:r
10829:1
10826:(
10823:+
10820:)
10814:(
10811:y
10808:,
10805:)
10799:(
10790:y
10786:)
10783:r
10777:1
10774:(
10768:)
10762:(
10759:x
10756:(
10753:=
10750:)
10744:,
10741:r
10738:(
10735:G
10721:G
10717:U
10713:θ
10709:θ
10707:(
10705:y
10701:θ
10699:(
10697:x
10693:R
10689:θ
10687:,
10685:r
10681:u
10677:D
10673:G
10665:U
10661:U
10657:F
10650:U
10642:f
10638:F
10634:U
10630:D
10626:h
10622:F
10618:f
10599:,
10596:0
10593:=
10586:1
10579:h
10572:F
10535:z
10531:h
10527:)
10524:z
10521:(
10515:=
10506:z
10501:h
10485:C
10481:h
10463:.
10458:1
10445:)
10440:1
10428:z
10422:(
10408:2
10399:z
10391:2
10387:z
10381:=
10378:)
10375:z
10372:(
10355:C
10351:C
10346:U
10342:D
10338:F
10334:U
10316:μ
10293:,
10288:1
10275:)
10270:1
10258:z
10252:(
10249:f
10242:=
10239:)
10236:z
10233:(
10230:f
10215:f
10197:,
10188:)
10183:1
10171:z
10165:(
10152:2
10143:z
10135:2
10131:z
10125:=
10122:)
10119:z
10116:(
10099:μ
10095:z
10074:,
10065:)
10057:z
10052:(
10049:f
10043:=
10040:)
10037:z
10034:(
10031:f
10017:f
9999:,
9990:)
9982:z
9977:(
9968:=
9965:)
9962:z
9959:(
9942:μ
9938:z
9931:μ
9924:μ
9920:g
9916:f
9912:f
9894:.
9891:)
9888:z
9885:(
9880:1
9873:g
9865:]
9852:z
9848:g
9840:z
9836:g
9804:1
9798:0
9787:[
9783:=
9780:)
9777:z
9774:(
9757:C
9753:∞
9750:μ
9746:g
9727:.
9714:+
9709:0
9701:=
9695:)
9686:1
9683:(
9680:+
9671:=
9653:μ
9649:ψ
9645:z
9637:μ
9618:.
9615:)
9612:z
9609:(
9600:)
9586:w
9579:/
9569:w
9565:(
9562:=
9559:)
9556:w
9553:(
9526:z
9524:(
9522:w
9518:w
9495:.
9492:)
9487:1
9480:z
9476:(
9471:f
9459:2
9450:z
9442:2
9438:z
9432:=
9429:)
9426:z
9423:(
9418:g
9383:,
9378:1
9371:)
9365:1
9358:z
9354:(
9351:f
9348:=
9345:)
9342:z
9339:(
9336:g
9321:f
9302:.
9294:g
9279:f
9266:1
9259:f
9246:g
9226:z
9222:f
9214:z
9210:f
9204:=
9201:f
9191:1
9184:f
9177:g
9157:f
9150:f
9146:f
9138:f
9134:f
9130:C
9126:f
9106:|
9100:k
9097:2
9093:e
9088:|
9084:)
9079:2
9074:|
9065:|
9058:1
9055:(
9052:=
9049:)
9046:f
9043:(
9040:J
9025:e
9021:h
9017:k
8998:.
8993:z
8985:+
8980:z
8976:k
8969:=
8960:z
8955:k
8939:e
8934:z
8930:f
8907:.
8904:)
8899:2
8894:|
8885:|
8878:1
8875:(
8870:2
8865:|
8858:z
8854:f
8849:|
8845:=
8840:2
8835:|
8824:z
8819:f
8814:|
8805:2
8800:|
8793:z
8789:f
8784:|
8780:=
8777:)
8774:f
8771:(
8768:J
8753:R
8749:C
8745:f
8738:F
8734:F
8732:)
8726:I
8703:.
8700:G
8697:U
8694:=
8691:F
8688:)
8682:U
8676:I
8673:(
8668:z
8648:μ
8645:U
8641:I
8637:U
8634:μ
8630:I
8622:T
8620:(
8617:k
8614:H
8593:.
8588:z
8584:P
8580:=
8577:)
8568:z
8563:P
8559:(
8556:U
8542:P
8538:U
8519:.
8514:0
8510:e
8506:=
8501:0
8497:e
8493:U
8490:,
8487:)
8484:0
8478:m
8475:(
8468:m
8464:e
8458:m
8450:m
8443:=
8438:m
8434:e
8430:U
8415:Z
8412:i
8408:Z
8404:m
8400:T
8395:m
8391:e
8387:G
8369:,
8366:F
8363:G
8360:+
8355:z
8351:F
8344:=
8335:z
8330:F
8315:T
8311:T
8309:(
8307:L
8303:F
8299:T
8295:μ
8291:F
8287:R
8283:y
8279:x
8275:F
8271:f
8268:ψ
8264:F
8260:μ
8256:C
8249:z
8245:f
8241:z
8233:f
8229:R
8225:z
8221:f
8200:,
8197:0
8194:=
8191:f
8182:z
8162:μ
8155:f
8133:.
8130:z
8127:+
8124:)
8121:z
8118:(
8115:h
8106:T
8102:C
8099:=
8096:)
8093:z
8090:(
8087:f
8072:f
8068:g
8064:μ
8045:,
8042:h
8033:T
8029:=
8020:z
8015:g
7975:1
7968:)
7964:B
7958:I
7955:(
7952:=
7949:h
7940:T
7933:,
7927:T
7922:1
7915:)
7911:A
7905:I
7902:(
7899:=
7896:h
7866:.
7860:T
7857:=
7854:h
7851:)
7848:A
7842:I
7839:(
7809:F
7806:T
7800:=
7797:F
7794:B
7787:,
7784:F
7778:T
7775:=
7772:F
7769:A
7754:B
7750:A
7729:T
7726:+
7723:)
7720:h
7714:(
7711:T
7708:=
7705:)
7700:z
7696:f
7689:(
7686:T
7683:=
7680:)
7671:z
7666:f
7662:(
7659:T
7656:=
7653:)
7644:z
7639:g
7635:(
7632:T
7629:=
7624:z
7620:g
7616:=
7613:h
7598:z
7594:f
7589:z
7585:g
7581:h
7577:g
7559:z
7553:)
7550:z
7547:(
7544:f
7541:=
7538:)
7535:z
7532:(
7529:g
7515:μ
7497:,
7492:z
7488:f
7481:=
7472:z
7467:f
7449:C
7441:D
7423:.
7420:f
7417:T
7414:=
7409:z
7405:)
7401:f
7398:C
7395:(
7377:f
7358:.
7355:f
7352:=
7343:z
7338:)
7334:f
7331:C
7328:(
7298:,
7295:f
7289:E
7286:=
7283:f
7280:C
7261:f
7257:C
7238:.
7235:f
7232:=
7229:)
7226:f
7220:E
7217:(
7208:z
7188:f
7181:C
7162:,
7156:z
7149:1
7144:=
7141:)
7138:z
7135:(
7132:E
7098:z
7089:=
7086:D
7052:,
7047:z
7043:f
7039:=
7036:)
7027:z
7022:f
7018:(
7015:T
7001:C
6997:f
6979:.
6976:)
6973:z
6970:(
6961:f
6953:z
6945:z
6938:=
6935:)
6932:z
6929:(
6919:f
6916:T
6898:f
6894:C
6886:T
6882:p
6865:L
6857:h
6839:.
6836:)
6833:z
6830:(
6824:=
6814:2
6810:F
6803:G
6800:E
6795:2
6792:+
6789:G
6786:+
6783:E
6778:F
6775:i
6772:2
6769:+
6766:G
6760:E
6754:=
6748:)
6741:2
6737:F
6730:G
6727:E
6722:2
6719:+
6716:G
6713:+
6710:E
6707:(
6703:E
6700:)
6695:2
6691:b
6687:+
6682:2
6678:a
6674:(
6669:)
6666:F
6663:i
6660:2
6657:+
6654:G
6648:E
6645:(
6641:E
6638:)
6633:2
6629:b
6625:+
6620:2
6616:a
6612:(
6606:=
6599:z
6595:h
6585:z
6580:h
6546:z
6542:h
6517:z
6513:h
6508:/
6498:z
6493:h
6464:/
6460:)
6457:)
6449:2
6445:F
6438:G
6435:E
6430:b
6424:F
6421:a
6418:+
6415:E
6412:b
6409:(
6406:i
6403:+
6400:)
6392:2
6388:F
6381:G
6378:E
6373:a
6367:F
6364:b
6358:E
6355:a
6352:(
6349:(
6346:=
6337:z
6332:h
6308:2
6304:/
6300:)
6297:)
6289:2
6285:F
6278:G
6275:E
6270:b
6267:+
6264:F
6261:a
6255:E
6252:b
6249:(
6246:i
6243:+
6240:)
6232:2
6228:F
6221:G
6218:E
6213:a
6210:+
6207:F
6204:b
6201:+
6198:E
6195:a
6192:(
6189:(
6186:=
6181:z
6177:h
6152:.
6149:)
6146:y
6143:,
6140:x
6137:(
6134:b
6131:i
6128:+
6125:)
6122:y
6119:,
6116:x
6113:(
6110:a
6107:=
6104:)
6101:y
6098:,
6095:x
6092:(
6089:c
6063:)
6058:y
6055:d
6051:)
6043:2
6039:F
6032:G
6029:E
6024:i
6021:+
6018:F
6015:(
6012:+
6009:x
6006:d
6002:E
5997:(
5992:)
5989:y
5986:,
5983:x
5980:(
5977:c
5974:=
5971:h
5968:d
5954:y
5950:x
5928:y
5925:d
5921:/
5917:x
5914:d
5911:=
5908:)
5905:t
5902:(
5895:q
5867:=
5864:y
5861:d
5857:)
5849:2
5845:F
5838:G
5835:E
5830:i
5827:+
5824:F
5821:(
5818:+
5815:x
5812:d
5808:E
5786:y
5782:x
5770:x
5766:q
5752:x
5742:x
5740:(
5738:h
5731:x
5727:y
5725:(
5723:q
5719:y
5715:x
5701:)
5698:0
5695:(
5692:q
5682:y
5678:x
5676:(
5674:h
5670:h
5666:s
5662:t
5658:t
5656:(
5654:q
5650:s
5646:s
5644:(
5642:q
5638:t
5636:(
5634:q
5630:x
5626:t
5622:y
5618:G
5614:F
5610:E
5592:,
5589:)
5586:t
5583:,
5580:)
5577:t
5574:(
5571:q
5568:(
5563:E
5555:2
5551:F
5544:G
5541:E
5536:i
5530:F
5521:=
5518:)
5515:t
5512:(
5505:q
5490:t
5486:t
5484:(
5482:q
5445:h
5429:x
5425:u
5421:x
5407:x
5404:=
5401:)
5398:0
5395:,
5392:x
5389:(
5386:h
5366:,
5363:)
5360:y
5357:,
5354:x
5351:(
5348:v
5345:i
5342:+
5339:)
5336:y
5333:,
5330:x
5327:(
5324:u
5321:=
5318:)
5315:y
5312:,
5309:x
5306:(
5303:h
5293:G
5289:F
5285:E
5248:f
5222:f
5202:f
5169:.
5166:)
5163:z
5160:(
5154:=
5147:z
5143:f
5133:z
5128:f
5122:=
5109:z
5105:f
5099:)
5096:f
5084:z
5075:(
5072:+
5067:z
5063:f
5059:)
5056:f
5048:z
5040:(
5024:z
5019:f
5013:)
5010:f
4998:z
4989:(
4986:+
4977:z
4972:f
4968:)
4965:f
4957:z
4949:(
4943:=
4931:z
4927:)
4923:f
4914:(
4903:z
4898:)
4894:f
4885:(
4849:z
4779:f
4755:.
4745:2
4741:F
4734:G
4731:E
4726:2
4723:+
4720:G
4717:+
4714:E
4709:F
4706:i
4703:2
4700:+
4697:G
4691:E
4685:=
4682:)
4679:z
4676:(
4670:=
4663:z
4659:f
4649:z
4644:f
4627:y
4623:x
4621:(
4619:f
4615:y
4611:x
4583:.
4580:)
4577:z
4574:(
4561:f
4557:f
4541:.
4536:z
4532:f
4527:/
4517:z
4512:f
4508:=
4505:)
4502:z
4499:(
4472:.
4464:)
4459:)
4454:y
4450:u
4441:x
4437:v
4433:(
4430:i
4424:)
4419:y
4415:v
4411:+
4406:x
4402:u
4398:(
4393:(
4386:)
4381:)
4376:y
4372:u
4363:x
4359:v
4355:(
4352:i
4349:+
4346:)
4341:y
4337:v
4333:+
4328:x
4324:u
4320:(
4315:(
4306:)
4301:)
4296:y
4292:v
4288:i
4285:+
4280:x
4276:v
4272:(
4269:i
4263:)
4258:y
4254:u
4250:i
4247:+
4242:x
4238:u
4234:(
4229:(
4222:)
4217:)
4212:y
4208:v
4204:i
4201:+
4196:x
4192:v
4188:(
4185:i
4182:+
4179:)
4174:y
4170:u
4166:i
4163:+
4158:x
4154:u
4150:(
4145:(
4137:=
4134:)
4131:z
4128:(
4099:y
4095:u
4089:x
4085:v
4076:y
4072:v
4066:x
4062:u
4051:f
4037:.
4032:2
4027:y
4023:u
4017:2
4012:x
4008:v
4004:+
3999:y
3995:v
3989:y
3985:u
3979:x
3975:v
3969:x
3965:u
3961:2
3953:2
3948:y
3944:v
3938:2
3933:x
3929:u
3918:y
3914:x
3912:(
3910:r
3892:,
3882:2
3878:)
3872:y
3868:v
3862:x
3858:v
3854:+
3849:y
3845:u
3839:x
3835:u
3831:(
3825:)
3820:2
3815:y
3811:v
3807:+
3802:2
3797:y
3793:u
3789:(
3786:)
3781:2
3776:x
3772:v
3768:+
3763:2
3758:x
3754:u
3750:(
3745:2
3742:+
3737:2
3733:)
3727:y
3723:v
3719:+
3714:2
3709:y
3705:u
3701:(
3698:+
3695:)
3690:2
3685:x
3681:v
3677:+
3672:2
3667:x
3663:u
3659:(
3654:)
3649:y
3645:v
3639:x
3635:v
3631:+
3626:y
3622:u
3616:x
3612:u
3608:(
3605:i
3602:2
3599:+
3594:2
3590:)
3584:y
3580:v
3576:+
3571:y
3567:u
3563:(
3557:)
3552:2
3547:x
3543:v
3539:+
3534:2
3529:x
3525:u
3521:(
3515:=
3512:)
3509:z
3506:(
3489:f
3482:f
3468:,
3465:)
3462:z
3459:(
3453:=
3448:z
3444:f
3439:/
3429:z
3424:f
3403:,
3396:2
3392:F
3385:G
3382:E
3377:r
3367:f
3353:,
3350:E
3346:/
3342:)
3337:2
3332:x
3328:v
3324:+
3319:2
3314:x
3310:u
3306:(
3303:=
3300:r
3290:y
3288:,
3286:x
3284:(
3282:g
3278:y
3274:x
3272:(
3270:r
3266:f
3248:.
3243:E
3237:x
3233:v
3229:F
3226:+
3221:x
3217:u
3208:2
3204:F
3197:G
3194:E
3186:=
3181:y
3177:v
3164:E
3158:x
3154:v
3145:2
3141:F
3134:G
3131:E
3121:x
3117:u
3113:F
3107:=
3102:y
3098:u
3071:y
3067:v
3044:y
3040:u
3019:,
3014:y
3010:v
2989:,
2984:x
2980:v
2960:,
2955:y
2951:u
2931:,
2926:x
2922:u
2894:.
2889:)
2880:2
2876:F
2869:G
2866:E
2861:2
2858:+
2855:G
2852:+
2849:E
2844:(
2839:)
2836:)
2831:y
2827:u
2823:+
2818:x
2814:v
2810:(
2807:i
2804:+
2801:)
2796:y
2792:v
2783:x
2779:u
2775:(
2772:(
2769:=
2743:)
2738:F
2735:i
2732:2
2729:+
2726:G
2720:E
2715:(
2710:)
2707:)
2702:y
2698:u
2689:x
2685:v
2681:(
2678:i
2675:+
2672:)
2667:y
2663:v
2659:+
2654:x
2650:u
2646:(
2643:(
2629:g
2615:)
2612:z
2609:(
2586:f
2564:z
2560:f
2555:/
2545:z
2540:f
2514:z
2510:f
2505:/
2495:z
2490:f
2461:.
2456:2
2451:|
2442:z
2437:d
2429:z
2425:f
2415:z
2410:f
2404:+
2401:z
2398:d
2394:|
2386:2
2381:|
2374:z
2370:f
2365:|
2361:=
2356:2
2351:|
2341:z
2336:d
2327:z
2322:f
2318:+
2315:z
2312:d
2306:z
2302:f
2297:|
2293:=
2288:2
2284:s
2280:d
2266:f
2248:,
2245:2
2241:/
2237:)
2234:)
2229:y
2225:u
2221:+
2216:x
2212:v
2208:(
2205:i
2202:+
2199:)
2194:y
2190:v
2181:x
2177:u
2173:(
2170:(
2167:=
2158:z
2153:f
2129:2
2125:/
2121:)
2118:)
2113:y
2109:u
2100:x
2096:v
2092:(
2089:i
2086:+
2083:)
2078:y
2074:v
2070:+
2065:x
2061:u
2057:(
2054:(
2051:=
2046:z
2042:f
2013:.
2010:y
2007:d
2003:i
1997:x
1994:d
1991:=
1983:z
1978:d
1973:,
1970:y
1967:d
1963:i
1960:+
1957:x
1954:d
1951:=
1948:z
1945:d
1938:;
1935:)
1930:y
1922:i
1919:+
1914:x
1906:(
1901:2
1898:1
1893:=
1884:z
1873:,
1870:)
1865:y
1857:i
1849:x
1841:(
1836:2
1833:1
1828:=
1823:z
1800:y
1796:x
1794:(
1792:v
1788:y
1784:x
1782:(
1780:u
1776:y
1772:x
1770:(
1768:f
1764:f
1751:f
1747:S
1743:v
1739:u
1735:y
1731:x
1729:(
1727:r
1723:g
1719:f
1701:.
1696:)
1688:2
1683:y
1679:v
1675:+
1670:2
1665:y
1661:u
1653:y
1649:v
1643:x
1639:v
1635:+
1630:y
1626:u
1620:x
1616:u
1606:y
1602:v
1596:x
1592:v
1588:+
1583:y
1579:u
1573:x
1569:u
1561:2
1556:x
1552:v
1548:+
1543:2
1538:x
1534:u
1527:(
1497:,
1492:2
1488:y
1484:d
1480:)
1475:2
1470:y
1466:v
1462:+
1457:2
1452:y
1448:u
1444:(
1441:+
1438:y
1435:d
1431:x
1428:d
1424:)
1419:y
1415:v
1409:x
1405:v
1401:+
1396:y
1392:u
1386:x
1382:u
1378:(
1375:2
1372:+
1367:2
1363:x
1359:d
1355:)
1350:2
1345:x
1341:v
1337:+
1332:2
1327:x
1323:u
1319:(
1316:=
1311:2
1307:v
1303:d
1300:+
1295:2
1291:u
1287:d
1284:=
1279:2
1275:s
1271:d
1257:S
1253:T
1237:S
1233:f
1207:y
1203:u
1197:x
1193:v
1184:y
1180:v
1174:x
1170:u
1151:f
1147:C
1143:T
1139:S
1135:y
1133:,
1131:x
1129:(
1127:v
1123:y
1121:,
1119:x
1117:(
1115:u
1111:y
1109:,
1107:x
1105:(
1103:f
1072:2
1068:F
1061:G
1058:E
1053:2
1050:+
1047:G
1044:+
1041:E
1032:2
1028:F
1021:G
1018:E
1013:2
1007:G
1004:+
1001:E
995:=
990:2
985:|
976:|
947:)
940:2
936:F
929:G
926:E
921:2
915:G
912:+
909:E
906:(
903:)
896:2
892:F
885:G
882:E
877:2
874:+
871:G
868:+
865:E
862:(
859:=
856:)
853:F
850:i
847:2
841:G
835:E
832:(
829:)
826:F
823:i
820:2
817:+
814:G
808:E
805:(
770:2
766:F
759:G
756:E
751:2
748:+
745:G
742:+
739:E
734:F
731:i
728:2
725:+
722:G
716:E
710:=
707:)
704:y
701:,
698:x
695:(
678:g
667:y
663:x
659:F
651:G
647:E
627:)
621:G
616:F
609:F
604:E
598:(
593:=
590:)
587:y
584:,
581:x
578:(
575:g
562:g
554:S
550:g
530:2
526:y
522:d
518:G
515:+
512:y
509:d
505:x
502:d
498:F
495:2
492:+
487:2
483:x
479:d
475:E
472:=
467:2
463:s
459:d
445:C
441:S
424:h
421:+
418:b
412:v
406:b
386:h
383:+
380:a
374:u
368:a
358:h
354:v
350:u
342:v
338:u
334:y
330:x
326:y
322:x
266:p
262:C
250:L
246:C
211:z
202:/
178:z
171:/
153:U
151:(
148:L
143:μ
138:L
133:U
129:z
122:w
105:.
99:z
91:w
79:=
68:z
58:w
23:.
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