bluuebunny/crossref_metadata_2025_split huggingface dataset:
On rationally convex hulls
Authors: Basener Richard F. 
Published: July 2010

For a compact set

X
⊆

C

n

X \subseteq {{\mathbf {C}}^n}

, let

h
r

(
X
)

{h_r}(X)

denote the rationally convex hull of X; let

Δ

{\mathbf {\Delta }}

denote the closed unit disk in C; and, following Wermer, for a compact set S such that

∂

Δ

⊆
S
⊆

Δ

\partial {\mathbf {\Delta }} \subseteq S \subseteq {\mathbf {\Delta }}

let

X
S

=
S
×
S
∩
∂

Δ

2

{X_S} = S \times S \cap \partial {{\mathbf {\Delta }}^2}

. It is shown that
\[

h
r

(

X
S

)
=
{
(
z
,
w
)
∈
S
×
S

|

u
S

(
z

)
+

u
S

(
w
)
≤
1
}

{h_r}({X_S}) = \{ (z,w) \in S \times S|{u_S}{(z)^ + }{u_S}(w) \leq 1\}

\]
where

u
S

{u_S}

is a function on S which, in the case when S is smoothly bounded, is specified by requiring

u
S

|

∂

Δ

=
0
,

u
S

|

∂
S
∖
∂

Δ

=
1

{u_S}{|_{\partial {\mathbf {\Delta }}}} = 0,{u_S}{|_{\partial S\backslash \partial {\mathbf {\Delta }}}} = 1

and

u
S

|

int
⁡
S

{u_S}{|_{\operatorname {int} S}}

harmonic. In particular this provides a precise description of

h
r

(
X
)

{h_r}(X)

for certain sets

X
⊆

C

2

X \subseteq {{\mathbf {C}}^2}

with the property that

h
r

(
X
)
≠
X

{h_r}(X) \ne X

, but

h
r

(
X
)

{h_r}(X)

does not contain analytic structure (as Wermer demonstrated, there are S for which

X
=

X
S

X = {X_S}

has these properties). Furthermore, it follows that whenever

h
r

(

X
S

)
≠

X
S

{h_r}({X_S}) \ne {X_S}

then there is a Gleason part of

h
r

(

X
S

)

{h_r}({X_S})

for the algebra

R
(

X
S

)

R({X_S})

with positive four-dimensional measure. In fact, the Gleason part of any point

(
z
,
w
)
∈

h
r

(

X
S

)
∩
int
⁡

Δ

2

(z,w) \in {h_r}({X_S}) \cap \operatorname {int} {{\mathbf {\Delta }}^2}

such that

u
S

(
z
)
+

u
S

(
w
)
>
1

{u_S}(z) + {u_S}(w) > 1

has positive four-dimensional measure. A similar idea is then used to construct a compact rationally convex set

Y
⊆

C

2

Y \subseteq {{\mathbf {C}}^2}

such that each point of Y is a peak point for

R
(
Y
)

R(Y)

even though

R
(
Y
)
≠
C
(
Y
)

R(Y) \ne C(Y)

; namely,

Y
=

X
~

T

=
{
(
z
,
w
)
∈

C

2

|

z
∈
T
,

|

w

|

=

1
−

|

z

|

2

}

Y = {\tilde X_T} = \{ (z,w) \in {{\mathbf {C}}^2}|z \in T,|w| = \sqrt {1 - |z{|^2}} \}

where T is any compact subset of

int
⁡

Δ

\operatorname {int} {\mathbf {\Delta }}

having the property that

R
(
T
)
≠
C
(
T
)

R(T) \ne C(T)

even though there are no nontrivial Jensen measures for

R
(
T
)

R(T)

. This example is more concrete than the original example of such a uniform algebra which was discovered by Cole. It is possible to show, for instance, that

R
(

X
~

T

)

R({\tilde X_T})

is not even in general locally dense in

C
(

X
~

T

)

C({\tilde X_T})

, a possibility which had been suggested by Stuart Sidney. Finally, smooth examples (3-spheres in

C

6

{{\mathbf {C}}^6}

) with the same pathological properties are obtained from

X
S

{X_S}

and

X
~

T

{\tilde X_T}

.
The Riemann problem for general 2×2 conservation laws
Authors: Liu Tai Ping 
Published: July 2010

The Riemann Problem for a system of hyperbolic conservation laws of form
\[

(
1
)

u
t

+
f

(
u
,
υ
)

x

=
0
,

υ
t

+
g

(
u
,
υ
)

x

=
0

(1)\quad \begin {array}{*{20}{c}} {{u_t} + f{{(u,\upsilon )}_x} = 0,} \\ {{\upsilon _t} + g{{(u,\upsilon )}_x} = 0} \\ \end {array}

\]
with arbitrary initial constant states
\[

(
2
)

(

u
0

(
x
)
,

v
0

(
x
)
)
=

{

(

u
l

,

v
l

)
,

x
>
0
,

(

u
r

,

v
r

)
,

x
>
0
,

(2)\quad ({u_0}(x),{v_0}(x)) = \left \{ {\begin {array}{*{20}{c}} {({u_l},{v_l}),\quad x > 0,} \\ {({u_r},{v_r}),\quad x > 0,} \\ \end {array} } \right .

\]
is considered. We assume that

f
υ

>
0
,

g
u

>
0

{f_\upsilon } > 0,{g_u} > 0

. Let

l
i

(

r
i

)

{l_i}({r_i})

be the left (right) eigenvectors of

d
F
≡
d
(
f
,
g
)

dF \equiv d(f,g)

for eigenvalues

λ
1

>

λ
2

{\lambda _1} > {\lambda _2}

. Instead of assuming the usual convexity condition

d

λ
i

(

r
i

)
≠
0
,
i
=
1
,
2

d{\lambda _i}({r_i}) \ne 0,i = 1,2

we assume that

d

λ
i

(

r
i

)
=
0

d{\lambda _i}({r_i}) = 0

on disjoint union of

1
1

-dim manifolds in the

(
u
,
υ
)

(u,\upsilon )

plane. Oleinik’s condition (E) for single equation is extended to system (1); again call this new condition (E). Our condition (E) implies Lax’s shock inequalities and, in case

d

λ
i

(

r
i

)
≠
0

d{\lambda _i}({r_i}) \ne 0

, the two are equivalent. We then prove that there exists a unique solution to the Riemann Problem (1) and (2) in the class of shocks, rarefaction waves and contact discontinuities which satisfies condition (E).
Power residues and nonresidues in arithmetic progressions
Authors: Hudson Richard H. 
Published: July 2010

Let k be an integer

≥
2

\geq 2

and p a prime such that

v
k

(
p
)
=
(
k
,
p
−
1
)
>
1

{v_k}(p) = (k,p - 1) > 1

. Let

b
n
+
c
(
n
=
0
,
1
,
…
;
b
≥
2
,
1
≤
c
>
b
,
(
b
,
p
)
=
(
c
,
p
)
=
1
)

bn + c(n = 0,1, \ldots ;b \geq 2,1 \leq c > b,(b,p) = (c,p) = 1)

be an arithmetic progression. We denote the smallest kth power nonresidue in the progression

b
n
+
c

bn + c

by

g
(
p
,
k
,
b
,
c
)

g(p,k,b,c)

, the smallest quadratic residue in the progression

b
n
+
c

bn + c

by

r
2

(
p
,
b
,
c
)

{r_2}(p,b,c)

, and the nth smallest prime kth power nonresidue by

g
n

(
p
,
k
)
,
n
=
0
,
1
,
2
,
…

{g_n}(p,k),n = 0,1,2, \ldots

. If

C
(
p
)

C(p)

is the multiplicative group consisting of the residue classes

mod

p

\bmod \;p

, then the kth powers

mod

p

\bmod \;p

form a multiplicative subgroup,

C
k

(
p
)

{C_k}(p)

. Among the

v
k

(
p
)

{v_k}(p)

cosets of

C
k

(
p
)

{C_k}(p)

denote by T the coset to which c belongs (where c is the first term in the progression

b
n
+
c
)

bn + c)

, and let

h
(
p
,
k
,
b
,
c
)

h(p,k,b,c)

denote the smallest number in the progression

b
n
+
c

bn + c

which does not belong to T so that

h
(
p
,
k
,
b
,
c
)

h(p,k,b,c)

is a natural generalization of

g
(
p
,
k
,
b
,
c
)

g(p,k,b,c)

. We prove by purely elementary methods that

h
(
p
,
k
,
b
,
c
)

h(p,k,b,c)

is bounded above by

2

7

/

4

b

5

/

2

p

2

/

5

+
3

b
3

p

1

/

5

+

b
2

{2^{7/4}}{b^{5/2}}{p^{2/5}} + 3{b^3}{p^{1/5}} + {b^2}

if p is a prime for which either b or

p
−
1

p - 1

is a kth power nonresidue. The restriction on b and

p
−
1

p - 1

may be lifted if

p
>

(

g
1

(
p
,
k
)

)

7.5

p > {({g_1}(p,k))^{7.5}}

. We further obtain a similar bound for

r
2

(
p
,
b
,
c
)

{r_2}(p,b,c)

for every prime p, without exception, and we apply our results to obtain a bound of the order of

p

2

/

5

{p^{2/5}}

for the nth smallest prime kth power nonresidue of primes which are large relative to

Π

j
=
1

n
−
1

g
j

(
p
,
k
)

\Pi _{j = 1}^{n - 1}{g_j}(p,k)

.
Weighted 𝐿² approximation of entire functions
Authors: Wohlgelernter Devora 
Published: July 2010

Let

S
S

be the space of entire functions

f
(
z
)

f(z)

such that

|

|

f
(
z
)

|

|

2

=
∫
∫

|

f
(
z
)

|

2

d
m
(
z
)

||f(z)|{|^2} = \smallint \smallint |f(z){|^2}dm(z)

, where

m
m

is a positive measure defined on the Borel sets of the complex plane. Write

d
m
(
z
)
=
K
(
z
)
d

A
z

=
K
(
r
,
θ
)
d
A
z

dm(z) = K(z)d{A_z} = K(r,\theta )dAz

. Theorem 1. If

ln
⁡

inf
θ

K
(
r
,
θ
)

\ln {\inf _\theta }K(r,\theta )

is asymptotic to

ln
⁡

sup
θ

K
(
r
,
θ
)

\ln {\sup _\theta } K(r,\theta )

(together with other mild restrictions) then polynomials are dense in

S
S

. Theorem 2. Let

K
(
z
)
=

e

−
ϕ
(
z
)

K(z) = {e^{ - \phi (z)}}

where

ϕ
(
z
)

\phi (z)

is a convex function of

z
z

such that all exponentials belong to

S
S

. Then polynomials are dense in

S
S

.
Product integral techniques for abstract hyperbolic partial differential equations
Authors: Spellmann J. W. 
Published: July 2010

Explicit and implicit product integral techniques are used to represent a solution

U
U

to the abstract system:

U

12

(
x
,
y
)
=
A
U
(
x
,
y
)
;
U
(
x
,
0
)
=
p
=
U
(
0
,
y
)

{U_{12}}(x,y) = AU(x,y);U(x,0) = p = U(0,y)

. The coefficient

A
A

is a closed linear transformation defined on a dense subspace

D
(
A
)

D(A)

of the Banach space

X
X

and the point

p
p

in

D
(
A
)

D(A)

satisfies the condition that

|

|

A
i

p

|

|

>

S
i

(
i
!

)

3

/

2

||{A^i}p|| > {S^i}{(i!)^{3/2}}

for all integers

i
⩾
0

i \geqslant 0

and some

S
>
0

S > 0

. The implicit technique is developed under the additional assumption that

A
A

generates a strongly continuous semigroup of bounded linear transformations on

X
X

. Both methods provide representations for the

J
0

{J_0}

Bessel function.
The Boolean space of orderings of a field
Authors: Craven Thomas C. 
Published: July 2010

It has been pointed out by Knebusch, Rosenberg and Ware that the set

X
X

of all orderings on a formally real field can be topologized to make a Boolean space (compact, Hausdorff and totally disconnected). They have called the sets of orderings

W
(
a
)
=
{
>

in

X

|

a
>
0
}

W(a) = \{ > {\text { in }}X|a > 0\}

the Harrison subbasis of

X
X

. This subbasis is closed under symmetric difference and complementation. In this paper it is proved that, given any Boolean space

X
X

, there exists a formally real field

F
F

such that

X
X

is homeomorphic to the space of orderings on

F
F

. Also, an example is given of a Boolean space and a basis of clopen sets closed under symmetric difference and complementation which cannot be the Harrison subbasis of any formally real field.
On 𝐽-convexity and some ergodic super-properties of Banach spaces
Authors: Brunel Antoine , Sucheston Louis 
Published: July 2010

Given two Banach spaces

F

|

|

F||

and

X

|

|

|

|

X||\,||

, write

F

fr

X

iff

F{\text { fr }}X{\text { iff}}

for each finite-dimensional subspace

F
′

F’

of

F
F

and each number

ε
>
0

\varepsilon > 0

, there is an isomorphism

V
V

of

F
′

F’

into

X
X

such that

|

|

x

|

−

|

|

V
x

|

|

|

≤
ε

||x| - ||Vx||| \leq \varepsilon

for each

x
x

in the unit ball of

F
′

F’

. Given a property

P

{\mathbf {P}}

of Banach spaces,

X
X

is called super-

P

iff

F

fr

X

{\mathbf {P}}{\text { iff }}F{\text { fr }}X

implies

F
F

is

P

{\mathbf {P}}

. Ergodicity and stability were defined in our articles On

B
B

-convex Banach spaces, Math. Systems Theory 7 (1974), 294-299, and C. R. Acad. Sci. Paris Ser. A 275 (1972), 993, where it is shown that super-ergodicity and super-stability are equivalent to super-reflexivity introduced by R. C. James [Canad. J. Math. 24 (1972), 896-904].

Q
Q

-ergodicity is defined, and it is proved that super-

Q
Q

-ergodicity is another property equivalent with super-reflexivity. A new proof is given of the theorem that

J
J

-spaces are reflexive [Schaffer-Sundaresan, Math. Ann. 184 (1970), 163-168]. It is shown that if a Banach space

X
X

is

B
B

-convex, then each bounded sequence in

X
X

contains a subsequence

(

y
n

)

({y_n})

such that the Cesàro averages of

(
−
1

)
i

y
i

{( - 1)^i}{y_i}

converge to zero.
On the generation of one-relator groups
Authors: Pride Stephen J. 
Published: July 2010

This paper is concerned with obtaining information about the Nielsen equivalence classes and

T
T

-systems of certain two-generator HNN groups, and in particular of certain two-generator one-relator groups. The theorems presented here extend results of the author appearing in the Proceedings of the Second International Conference on the Theory of Groups. In particular it is shown here that if

G
=
⟨
a
,
t
;

a

α
1

E
r

−
1

a

β
1

E
r

⋯

a

a
s

E
r

−
1

a

B
s

E
r

)

G = \langle a,t;{a^{{\alpha _1}}}E_r^{ - 1}{a^{{\beta _1}}}{E_r} \cdots {a^{{a_s}}}E_r^{ - 1}{a^{{B_s}}}{E_r})

where the

α
j

{\alpha _j}

are positive, the

β
i

{\beta _i}

are nonzero,

E
r

{E_r}

has the form

[

a

∈
1

,
[

a

∈
2

,
[
⋯
,
[

a

∈
r

,
t
]
⋯
]
]
]

[{a^{{ \in _1}}},[{a^{{ \in _2}}},[ \cdots ,[{a^{{ \in _r}}},t] \cdots ]]]

with

|

∈
1

|

=

|

∈
2

|

=
⋯
=

|

∈
r

|

=
1

|{ \in _1}| = |{ \in _2}| = \cdots = |{ \in _r}| = 1

, then in a large number of cases

G
G

has one Nielsen equivalence class. Similar results are also obtained for certain groups with more than one relator. A fair proportion of the paper is given to developing a method for reducing pairs of elements in HNN groups. This method has some of the features of Nielsen’s reduction theorem for free groups. One other interesting result obtained here is that a one-relator group with torsion which has one

T
T

-system is Hopfian. The early part of the paper is discursive. It contains most of the known results concerning

T
T

-systems of one-relator groups, and highlights several open problems, some of which have been raised by other authors.
On differential rings of entire functions
Authors: Cayford A. H. , Straus E. G. 
Published: July 2010

Consider an entire function

f
f

which is a solution of the differential equation
\[

[

c
0

(
z
)
+

c
1

(
z
)
D
+
…
+

c
m

(
z
)

D
m

]
(

f
n

)
=
P
(
f
,

f
′

,
…
,

f

(
k
)

)

[{c_0}(z) + {c_1}(z)D + \ldots + {c_m}(z){D^m}]({f^n}) = P(f,f’, \ldots ,{f^{(k)}})

\]
where

c
i

(
z
)

{c_i}(z)

are entire functions in a differential ring

R
R

and

P
P

is a polynomial in a differential field related to

R
R

. We prove the following THEOREM. If

f
f

satisfies the equation above then

f
f

is of finite type in case

R
=

C

R = {\mathbf {C}}

and of finite exponential order in case

R
=

C

[
z
]

R = {\mathbf {C}}[z]

. We use this result to prove a conjecture made in [2] that entire functions of order

ρ
>
s

\rho > s

, all of whose derivatives at

s
s

points are integers in an imaginary quadratic number field, must be solutions of linear differential equations with constant coefficients and therefore of order

⩽
1

\leqslant 1

.
Interpolation and uniqueness results for entire functions
Authors: Child James D. 
Published: July 2010

Let

K
[
Ω
]

K[\Omega ]

denote the collection of entire functions of exponential type whose Borel transforms are analytic on

Ω
c

{\Omega ^c}

(the complement of the simply connected domain taken relative to the sphere). Let

f
f

be in

K
[
Ω
]

K[\Omega ]

and set

L
n

(
f
)
=

(
2
π
i

)

−
1

∫
Γ

g
n

(
λ
)
F
(
λ
)
d
λ
(
n
=
0
,
1
,
…
)

{L_n}(f) = {(2\pi i)^{ - 1}}\smallint \Gamma {g_n}(\lambda )F(\lambda )d\lambda (n = 0,1, \ldots )

where

F
F

is the Borel transform of

f
,
Γ
⊂
Ω

f,\Gamma \subset \Omega

is a simple closed contour chosen so that

F
F

is analytic outside and on

Γ
\Gamma

and each

g
n

{g_n}

is in

H
(
Ω
)

H(\Omega )

(the collection of functions analytic on

Ω
\Omega

). In what follows read ’the sequence of linear functionals

{

L
n

(
f
)
}

\{ {L_n}(f)\}

’ wherever the sequence of functions ’

{

g
n

}

\{ {g_n}\}

’ appears. Let

T
T

denote a continuous linear operator from

H
(
Ω
)

H(\Omega )

to

H
(
Λ
)

H(\Lambda )

where

Λ
\Lambda

is also a simply connected domain. The topologies on

H
(
Ω
)

H(\Omega )

and

H
(
Λ
)

H(\Lambda )

are those of uniform convergence on compact subsets of

Ω
\Omega

(resp.

Λ
\Lambda

). The purpose of this paper is to consider uniqueness preserving operators, i.e., operators

T
T

which have the property that

K
[
Λ
]

K[\Lambda ]

is a uniqueness class for

{
T
(

g
n

)
}

\{ T({g_n})\}

whenever

K
[
Ω
]

K[\Omega ]

is a uniqueness class for

{

g
n

}

\{ {g_n}\}

, and to examine interpolation preserving operators, i.e., operators

T
T

which have the property that

K
[
Λ
]

K[\Lambda ]

interpolates the sequence of complex numbers

{

b
n

}

\{ {b_n}\}

relative to

{
T
(

g
n

)
}

\{ T({g_n})\}

whenever

K
[
Ω
]

K[\Omega ]

interpolates

{

b
n

}

\{ {b_n}\}

relatives to

{

g
n

}

\{ {g_n}\}

. Once some classes of uniqueness preserving operators and some classes of interpolation preserving operators have been found, we proceed to obtain new uniqueness and interpolation results from our knowledge of these operators and from previously known uniqueness and interpolation results. Operators which multiply by analytic functions and some differential operators are considered. Composition operators are studied and the results are used to extend the interpolation results for sequences of functions of the form

{

[
W
(
ζ
)

]
n

}

\{ {[W(\zeta )]^n}\}

where

W
W

is analytic and univalent on a simply connected domain.
Boundary value problems for second order differential equations in convex subsets of a Banach space
Authors: Schmitt Klaus , Volkmann Peter 
Published: July 2010

Let E be a real Banach space, C a closed, convex subset of E and

f
:
[
0
,
1
]
×
E
×
E
→
E

f:[0,1] \times E \times E \to E

be continuous. Let

u
0

,

u
1

∈
C

{u_0},{u_1} \in C

and consider the boundary value problem

(

∗

)

u
=
f
(
t
,
u
,

u
′

)
,

u
(
0
)
=

u
0

,

u
(
1
)
=

u
1

.

\begin{equation}\tag {$\ast $} u = f(t,u,u’),\quad u(0) = {u_0},\quad u(1) = {u_1}.\end{equation}

We establish sufficient conditions in order that

(
∗
)

(\ast )

have a solution

u
:
[
0
,
1
]
→
C

u:[0,1] \to C

.
On bounded functions satisfying averaging conditions. II
Authors: Cahill Rotraut Goubau 
Published: July 2010

Let

S
(
f
)

S(f)

denote the subspace of

L
∞

(

R
n

)

{L^\infty }({R^n})

consisting of those real valued functions f for which
\[

lim

r
→
0

1

|

B
(
x
,
r
)

|

∫

B
(
x
,
r
)

f
(
y
)
d
y
=
f
(
x
)

\lim \limits _{r \to 0} \frac {1}{{|B(x,r)|}} {\int } _{B(x,r)}f(y)dy = f(x)

\]
for all x in

R
n

{R^n}

and let

L
(
f
)

L(f)

be the subspace of

S
(
f
)

S(f)

consisting of the approximately continuous functions. A number of results concerning the existence of functions in

S
(
f
)

S(f)

and

L
(
f
)

L(f)

with special properties are obtained. The extreme points of the unit balls of both spaces are characterized and it is shown that

L
(
f
)

L(f)

is not a dual space. As a preliminary step, it is shown that if E is a

G
δ

{G_\delta }

set of measure 0 in

R
n

{R^n}

, then the complement of E can be decomposed into a collection of closed sets in a particularly useful way.
Sobolev inequalities for weight spaces and supercontractivity
Authors: Rosen Jay 
Published: July 2010

For

ϕ
∈

C
2

(

R

n

)

\phi \in {C^2}({{\mathbf {R}}^n})

with

ϕ
(
x
)
=
a

|

x

|

1
+
s

\phi (x) = a|x{|^{1 + s}}

for

|

x

|

⩾

x
0

,
a
,
s
>
0

|x| \geqslant {x_0},a,s > 0

, define the measure

d
μ
=
exp
⁡
(
−
2
ϕ
)

d
n

x

d\mu = \exp ( - 2\phi ){d^n}x

on

R

n

{{\mathbf {R}}^n}

. We show that for any

k
∈

Z

+

k \in {{\mathbf {Z}}^ + }

\[

∫

|

f

|

2

|

lg
⁡
(

|

f

|

)

|

2
s
k

/

(
s
+
1
)

d
μ

⩽
c

{

∑

|

α

|

=
0

k

‖

D
α

f
‖

L
2

(
d
μ
)

2

+

‖
f
‖

L
2

(
d
μ
)

2

∙

|

lg
⁡
(

‖
f
‖

L
2

(
d
μ
)

)

|

2
s
k

/

(
s
+
1
)

}

\begin {array}{*{20}{c}} {\int {|f{|^2}|\lg (|f|){|^{2sk/(s + 1)}}d\mu } } \hfill \\ { \leqslant c\left \{ {\sum \limits _{|\alpha | = 0}^k {\left \|{D^\alpha }f\right \|_{{L_2}(d\mu )}^2 + \left \|f\right \|_{{L_2}(d\mu )}^2 \bullet |\lg (\left \|f\right \|_{{L_2}(d\mu )}){|^{2sk/(s + 1)}}} } \right \}} \hfill \\ \end {array}

\]
As a consequence we prove

e

−
t

∇
∗

⋅
∇

:

L
q

(

R

n

,
d
μ
)
→

L
p

(

R

n

,
d
μ
)
,
p
,
q
≠
1
,
∞

{e^{ - t{\nabla ^\ast } \cdot \nabla }}:{L_q}({{\mathbf {R}}^n},d\mu ) \to {L_p}({{\mathbf {R}}^n},d\mu ),p,q \ne 1,\infty

, is bounded for all

t
>
0

t > 0

.
On bounded univalent functions whose ranges contain a fixed disk
Authors: Barnard Roger W. 
Published: July 2010

Let

S

\mathcal {S}

denote the standard normalized class of regular, univalent functions in

K
=

K
1

=
{
z
:

|

z

|

>
1
}

K = {K_1} = \{ z:|z| > 1\}

. Let

F

\mathcal {F}

be a given compact subclass of

S

\mathcal {S}

. We consider the following two problems. Problem 1. Find

max

|

a
2

|

\max |{a_2}|

for

f
∈

F

f \in \mathcal {F}

. Problem 2. For

|

z

|

=
r
>
1

|z| = r > 1

, find the

max

(
min
)

|

f
(
z
)

|

\max \;(\min )|f(z)|

for

f
∈

F

f \in \mathcal {F}

. In this paper we are concerned with the subclass

S

d
∗

(
M
)
=
{
f
∈

S

:

K
d

⊂
f
(
K
)
⊂

K
M

}

\mathcal {S}_d^\ast (M) = \{ f \in \mathcal {S}:{K_d} \subset f(K) \subset {K_M}\}

. Through the use of the Julia variational formula and the Loewner theory we determine the extremal functions for Problems 1 and 2 for the class

S

d
∗

(
M
)

\mathcal {S}_d^\ast (M)

, for all d, M such that

1
4

⩽
d
⩽
1
⩽
M
⩽
∞

\tfrac {1}{4} \leqslant d \leqslant 1 \leqslant M \leqslant \infty

.
The topological structure of 4-manifolds with effective torus actions. I
Authors: Pao Peter Sie 
Published: July 2010

Torus actions on orientable 4-manifolds have been studied by F. Raymond and P. Orlik [8] and [9]. The equivariant classification problem has been completely answered there. The problem then arose as to what can be said about the topological classification of these manifolds. Specifically, when are two manifolds homeomorphic if they are not equivariantly homeomorphic? In some cases this problem was answered in the above mentioned papers. For example, if the only nontrivial stability groups are finite cyclic, then the manifolds are essentially classified by their fundamental groups. In the presence of fixed points, a connected sum decomposition in terms of

S
4

,

S
2

×

S
2

,
C

P
2

,
−
C

P

−
2

,

S
1

×

S
3

{S^4},{S^2} \times {S^2},C{P^2}, - C{P^{ - 2}},{S^1} \times {S^3}

, and three families of elementary 4-manifolds,

R
(
m
,
n
)
,
T
(
m
,
n
;

m
′

,

n
′

)
,
L
(
n
;
p
,
q
;
m
)

R(m,n),T(m,n;m’,n’),L(n;p,q;m)

has been obtained (where m, n,

m
′

,

n
′

m’,n’

, p, and q are integers). In addition, a stable homeomorphic relation for the manifolds

R
(
m
,
n
)

R(m,n)

and

T
(
m
,
n
;

m
′

,

n
′

)

T(m,n;m’,n’)

can also be found in [9]. But the topological classification of R’s, T’s, and especially L’s were still unsolved problems. Furthermore, the connected sum decomposition of a manifold with fixed points, even in the simply connected case, is not unique. In this paper, we completely classify the manifolds with fixed points. For the manifolds R’s and T’s, the above mentioned stable homeomorphic relation is proved to be the topological classification. The manifolds

L
(
n
;
p
,
q
;
m
)

L(n;p,q;m)

form a very interesting family of 4-manifolds. They behave similarly to lens spaces. For example, the fundamental group of

L
(
n
;
p
,
q
;
m
)

L(n;p,q;m)

is finite cyclic of order n. And it is proved that

π
1

(
L
(
n
;
p
,
q
;
m
)
)

{\pi _1}(L(n;p,q;m))

and

π
1

(
L
(
n
;

p
′

,

q
′

;

m
′

)
)

{\pi _1}(L(n;p’,q’;m’))

act identically on the second cohomology of their common universal covering space,

(

S
2

×

S
2

)
#
⋯
#

(

S
2

×

S
2

)

({S^2} \times {S^2})\# \cdots \# \;({S^2} \times {S^2})

(

n
−
1

n - 1

copies), even though

L
(
n
;
p
,
q
;
m
)

L(n;p,q;m)

and

L
(
n
;

p
′

,

q
′

;

m
′

)

L(n;p’,q’;m’)

are not homotopically equivalent for some

(
n
;

p
′

,

q
′

;

m
′

)

(n;p’,q’;m’)

’s. This family of manifolds is explicitly constructed and completely classified. In addition, a normal form is imposed on a connected sum decomposition mentioned above. These nromal forms are unique. Most of the material of this paper appeared first in the author’s doctoral dissertation. The author would like to thank his thesis advisor Professor F. Raymond for his help and encouragement.