bluuebunny/crossref_metadata_2025_split huggingface dataset:
The regularity of the locally integrable and continuous solutions of nonlinear functional equations
Authors: Światak Halina 
Published: June 2010

The purpose of this paper is to show a general method which allows one to find all the continuous (and sometimes also all the locally integrable) solutions of functional equations by considering solutions of class

C
m

{C^m}

. One can do it if one is assured that all the continuous (or all the locally integrable) solutions of a given equation are functions of class

C
m

{C^m}

or

C
∞

{C^\infty }

. Such a property is characteristic for the solutions

f
:

R
n

→
R

f:{R^n} \to R

of the equations

(

∗

)

∑

i
=
1

k

a
i

(
x
,
t
)
f
(

ϕ
i

(
x
,
t
)
)
=
F
(
x
,
f
(

λ
1

(
x
)
)
,
…
,
f
(

λ
s

(
x
)
)
)
+
b
(
x
,
t
)
,

\begin{equation}\tag {$\ast $} \sum \limits _{i = 1}^k {{a_i}(x,t)f({\phi _i}(x,t)) = F(x,f({\lambda _1}(x)), \ldots ,f({\lambda _s}(x))) + b(x,t),} \end{equation}

where

x
∈

R
n

,
t
∈

R
r

,
n
⩾
1
,
r
⩾
1

x \in {R^n},t \in {R^r},n \geqslant 1,r \geqslant 1

and where the functions

ϕ
i

:

R

n
+
r

→

R
n

,

λ
j

:

R
n

→

R
n

,

a
i

:

R

n
+
r

→
R
,
b
:

R

n
+
r

→
R
,
F
:

R

n
+
s

→
R

{\phi _i}:{R^{n + r}} \to {R^n},{\lambda _j}:{R^n} \to {R^n},{a_i}:{R^{n + r}} \to R,b:{R^{n + r}} \to R,F:{R^{n + s}} \to R

satisfy some regularity assumptions and the assumptions which guarantee that an equation obtained by differentiating

(
∗
)

(\ast )

and fixing t is of constant strength, hypoelliptic at a point

x
0

{x_0}

. A general theorem, concerning the regularity of the continuous and locally integrable solutions f of

(
∗
)

(\ast )

, is formulated and proved by the reduction to the corresponding problem for the distributional solutions of linear partial differential equations.
Compact perturbations of certain von Neumann algebras
Authors: Plastiras Joan K. 
Published: June 2010

Let

E

\mathcal {E}

be a sequence of mutually orthogonal, finite dimensional projections whose sum is the identity on a Hilbert space

H

\mathcal {H}

. If we denote the commutant of

E

\mathcal {E}

by

D

(

E

)

\mathcal {D}(\mathcal {E})

and the ideal of compact operators on

H

\mathcal {H}

by

C

(

H

)

\mathcal {C}(\mathcal {H})

, then it is easily verified that

D

(

E

)
+

C

(

H

)
=
{
T
+
K
:
T
∈

D

(

E

)
,
K
∈

C

(

H

)
}

\mathcal {D}(\mathcal {E}) + \mathcal {C}(\mathcal {H}) = \{ T + K:T \in \mathcal {D}(\mathcal {E}),K \in \mathcal {C}(\mathcal {H})\}

is a

C
∗

{C^\ast }

-algebra. In this paper we classify all such algebras up to

∗

^\ast

-isomorphism and characterize them by examining their relationship to certain quasidiagonal and quasitriangular algebras.
Group presentations corresponding to spines of 3-manifolds. II
Authors: Osborne R. P. , Stevens R. S. 
Published: June 2010

Let

ϕ
=
⟨

a
1

,
…
,

a
n

|

R
1

,
…
,

R
m

⟩

\phi = \langle {a_1}, \ldots ,{a_n}|{R_1}, \ldots ,{R_m}\rangle

denote a group presentation. Let

K
ϕ

{K_\phi }

denote the corresponding 2-complex. It is well known that every compact 3-manifold has a spine of the form

K
ϕ

{K_\phi }

for some

ϕ
\phi

, but that not every

K
ϕ

{K_\phi }

is a spine of a compact 3-manifold. Neuwirth’s algorithm (Proc. Cambridge Philos. Soc. 64 (1968), 603-613) decides whether

K
ϕ

{K_\phi }

can be a spine of a compact 3-manifold. However, it is impractical for presentations of moderate length. In this paper a simple planar graph-like object, called a RR-system (railroad system), is defined. To each RR-system corresponds a whole family of compact orientable 3-manifolds with spines of the form

K
ϕ

{K_\phi }

, where

ϕ
\phi

has a particular form (e.g.,

⟨
a
,
b

a
m

b
n

a
p

b
n

,

a
m

b
n

a
m

b
q

⟩

\langle a,b{a^m}{b^n}{a^p}{b^n},{a^m}{b^n}{a^m}{b^q}\rangle

), subject only to certain requirements of relative primeness of certain pairs of exponents. Conversely, every

K
ϕ

{K_\phi }

which is a spine of some compact orientable 3-manifold can be obtained in this way. An equivalence relation on RR-systems is defined so that equivalent RR-systems determine the same family of manifolds. Results of Zieschang are applied to show that the simplest spine of 3-manifolds arises from a particularly simple kind of RR-system called a reduced RR-system. Following Neuwirth, it is shown how to determine when a RR-system gives rise to a collection of closed 3-manifolds.
The sharpness of Lorentz’s theorem on incomplete polynomials
Authors: Saff E. B. , Varga R. S. 
Published: June 2010

For any fixed

θ
\theta

with

0
>
θ
>
1

0 > \theta > 1

, G. G. Lorentz recently showed that bounded sequences

{

Σ

θ

n
i

⩽
k
⩽

n
i

a
k

(
i
)

(
1
+
t
)

k

}

i
=
1

∞

\{\Sigma _{\theta {n_i} \leqslant k \leqslant {n_i}} {{a_k}(i){{(1 + t)}^k}\} _{i = 1}^\infty }

of incomplete polynomials on

[
−
1
,
+
1
]

[ - 1, + 1]

tend uniformly to zero on closed intervals of

[
−
1
,
Δ
(
θ
)
)

[ - 1,\Delta (\theta ))

, where

2

θ
2

−
1
⩽
Δ
(
θ
)
>
2
θ
−
1

2{\theta ^2} - 1 \leqslant \Delta (\theta ) > 2\theta - 1

. In this paper, we show that

Δ
(
θ
)
=
2

θ
2

−
1

\Delta (\theta ) = 2{\theta ^2} - 1

is best possible, and that the geometric convergence to zero of such sequences on closed intervals

[

t
0

,

t
1

]

[{t_0},{t_1}]

can be precisely bounded above as a function of

t
j

{t_j}

and

θ
\theta

. Extensions of these results to the complex plane are also included.
The free boundary for elastic-plastic torsion problems
Authors: Friedman Avner , Pozzi Gianni A. 
Published: June 2010

Consider the variational inequality: Find

u

∈

K

u\, \in \, K

such that

∫
Q

∇
u
⋅

∇

(
υ

−

u
)

⩾

μ

∫
Q

(
υ

−

u
)

(
μ

>

0
)

\int _Q {\nabla u \cdot \, \nabla \, (\upsilon \, -\, u)\, \geqslant \, \mu \, \int _Q\, {(\upsilon \, -\, u)\,(\mu \, >\, 0)} }

for any

υ

∈

K

\upsilon \, \in \, K

, where

K

=

{
w

∈

H
0
1

(
Q
)
;

|

∇

w

|

⩽

1
}

K\, =\, \{ w\, \in \, H_0^1(Q);\,\left | {\nabla \, w} \right |\, \leqslant \, 1\}

and Q is a 2-dimensional simply connected domain in

R
2

{R^2}

with piecewise

C
3

{C^3}

boundary. The solution u represents the stress function in a torsion problem of an elastic-plastic bar with cross section Q. The sets

E

=

{
x

∈

Q
;

|

∇

u
(
x
)

|

>

1
}

E\, =\, \{ x\, \in \, Q;\,\left | {\nabla \, u(x)} \right |\, >\, 1\}

,

P

=

{
x

∈

Q
;

|

∇

u
(
x
)

|

=

1
}

P\, =\, \{ x\, \in \, Q;\,\left | {\nabla \, u(x)} \right |\, =\, 1\}

are the elastic and plastic sets respectively. The purpose of this paper is to study the free boundary

∂
E

∩

Q

\partial E\, \cap \, Q

; more specifically, an estimate is derived on the number of points of local maximum of the free boundary.
Matrix-valued special functions and representation theory of the conformal group. I. The generalized gamma function
Authors: Gross Kenneth I. , Holman Wayne J. 
Published: June 2010

This article examines in detail the matrix-valued gamma function
\[

Γ

λ
0

(
α
)

=

∫
P

e

−

tr

(
r
)

λ
0

(
r
,

r
¯

)

(
det

r
)

α
−
2

d
r

{\Gamma ^{{\lambda ^0}}}\,(\alpha )\, = \,\int _P {{e^{ - {\text {tr}}(r)}}{\lambda ^0}(r,\,\bar r){{(\det \,r)}^{\alpha - 2}}\,} dr

\]
associated to the conformal group

G

=

U
(
2
,

2
)

G\, = \,U(2,\,2)

. Here,

α
\alpha

is a continuous complex parameter,

λ
0

{\lambda ^0}

runs through a family of “weights” of

K

=

U
(
2
)

×

U
(
2
)

K\, = \,U(2)\, \times \,U(2)

, P is the cone of

2

×

2

2\, \times \,2

positive-definite Hermitian matrices, and the integral is well known to converge absolutely for

Re

(
α
)

>

1

{\text {Re}}(\alpha )\, > \,1

. However, until now very little has been known about the analytic continuation for the general weight

λ
0

{\lambda ^0}

. The results of this paper include the following: The complete analytic continuation of

Γ

λ
0

{\Gamma ^{{\lambda ^0}}}

is determined for all weights

λ
0

{\lambda ^0}

. In analogy to the case of the classical gamma function it is proved that for any weight

λ
0

{\lambda ^0}

the mapping

α

→

Γ

λ
0

(
α

)

−
1

\alpha \, \to \,{\Gamma ^{{\lambda ^0}}}\,{(\alpha )^{ - 1}}

is entire. A new integral formula is given for the inverse of the gamma function. An explicit calculation is given for the normalized variant of the gamma matrix that arises in the reproducing kernel for the spaces in which the holomorphic discrete series of G is realized, and one observes that the behavior of the analytic continuation for weights “in general position” is markedly different from the special cases in which the gamma function “is scalar". The full analytic continuation of the holomorphic discrete series for G is determined. The gamma function for the forward light cone (the boundary orbit) is found, and the associated Hardy space of vector-valued holomorphic functions is described. Analogs are given for some of the well-known formulas for the classical gamma function. As an epilogue, applications of the matrix-valued gamma function, such as generalizations to

2

×

2

2\, \times \,2

matrix space of the classical binomial theorem, are announced. These applications require a detailed understanding of the (generalized) Bessel functions associated to the conformal group that will be treated in the sequel to this paper.
Baire category principle and uniqueness theorem
Authors: Hwang J. S. 
Published: June 2010

Applying a theorem of Bagemihl and Seidel (1953), we prove that if

S
2

{S_2}

is a set of second category in

(
α
,
β
)

(\alpha ,\beta )

, where

0
⩽
α
>
β
⩽
2
π

0 \leqslant \alpha > \beta \leqslant 2\pi

, and if

f
(
z
)

f(z)

is a function meromorphic in the sector

Δ
(
α
,
β
)
=
{
z
:
0
>

|
z
|

>
∞
,
α
>
arg
⁡
z
>
β
}

\Delta (\alpha ,\beta ) = \{ z:0 > \left | z \right | > \infty ,\alpha > \arg z > \beta \}

for which

lim

_

r
→
∞

|

f
(
r

e

i
θ

)

|

>
0

{\underline {{\operatorname {lim}}} _{r \to \infty }}\left | {f(r{e^{i\theta }})} \right | > 0

, for all

θ
∈

S
2

\theta \in {S_2}

, then there exists a sector

Δ
(

α
′

,

β
′

)
⊆
Δ
(
α
,
β
)

\Delta (\alpha ’,\beta ’) \subseteq \Delta (\alpha ,\beta )

such that

(

α
′

,

β
′

)
⊆

S
¯

2

,

S
2

(\alpha ’,\beta ’) \subseteq {\bar S_2},{S_2}

is second category in

(

α
′

,

β
′

)

(\alpha ’,\beta ’)

, and

f
(
z
)

f(z)

has no zero in

Δ
(

α
′

,

β
′

)

\Delta (\alpha ’,\beta ’)

. Based on this property, we prove several uniqueness theorems.
Markov processes with identical hitting probabilities
Authors: Glover Joseph 
Published: June 2010

Let

(
X
(
t
)
,

P
x

)

(X(t),{P^x})

and

(
Y
(
t
)
,

Q
x

)

(Y(t),{Q^x})

be transient Hunt processes on a state space

E
E

satisfying the hypothesis of absolute continuity (Meyer’s hypothesis (L)). Let

T
(
K
)

T(K)

be the first entrance time into a set

K
K

, and assume

P
x

(
T
(
K
)
>
∞
)
=

Q
x

(
T
(
K
)
>
∞
)

{P^x}(T(K) > \infty ) = {Q^x}(T(K) > \infty )

for all compact sets

K
⊆
E

K \subseteq E

. There exists a strictly increasing continuous additive functional of

X
(
t
)
,
A
(
t
)

X(t),A(t)

, so that if

T
(
t
)
=

inf

{
s
:
A
(
s
)
>
t
}

T(t) = {\text {inf}}\{s:A(s) > t\}

, then

(
X
(
T
(
t
)
)
,

P
x

)

(X(T(t)),{P^x})

and

(
Y
(
t
)
,

Q
x

)

(Y(t),{Q^x})

have the same joint distributions. An analogous result is stated if

X
X

and

Y
Y

are right processes (with an additional hypothesis). These theorems generalize the Blumenthal-Getoor-McKean Theorem and have interpretations in terms of potential theory.
Normal subgroups of 𝐷𝑖𝑓𝑓^{Ω}(𝑅ⁿ)
Authors: Mascaró Francisca 
Published: June 2010

Let

Ω
\Omega

be a volume element on

R

n

{{\mathbf {R}}^n}

.

Dif

f

Ω

(

R

n

)

{\text {Dif}}{{\text {f}}^\Omega }({{\mathbf {R}}^n})

is the group of

Ω
\Omega

-preserving diffeomorphisms of

R

n

{{\mathbf {R}}^n}

.

Diff

W
Ω

(

R

n

)

{\text {Diff}}_W^\Omega ({{\mathbf {R}}^n})

is the subgroup of all elements whose set of nonfixed points has finite

Ω
\Omega

-volume.

Diff

f
Ω

(

R

n

)

{\text {Diff}}_f^\Omega ({{\mathbf {R}}^n})

is the subgroup of all elements whose support has finite

Ω
\Omega

-volume.

Diff

c
Ω

(

R

n

)

{\text {Diff}}_c^\Omega ({{\mathbf {R}}^n})

is the subgroup of all elements with compact support.

Diff

co

Ω

(

R

n

)

{\text {Diff}}_{{\text {co}}}^\Omega ({{\mathbf {R}}^n})

is the subgroup of all elements compactly

Ω
\Omega

-isotopic to the identity. We prove, in the case

vo

l

Ω

R

n

>
∞

{\text {vo}}{{\text {l}}_{\Omega }}{{\mathbf {R}}^n} > \infty

and for

n

⩾

3

{\text {n}} \geqslant {\text {3}}

that any subgroup of

Dif

f

Ω

(

R

n

)

{\text {Dif}}{{\text {f}}^\Omega }({{\mathbf {R}}^n})

,

N
N

, is normal if and only if

Diff

co

Ω

(

R

n

)
⊂
N
⊂

Diff

c
Ω

(

R

n

)

{\text {Diff}}_{{\text {co}}}^\Omega ({{\mathbf {R}}^n}) \subset N \subset {\text {Diff}}_c^\Omega ({{\mathbf {R}}^n})

. If

vo

l

Ω

R

n

=
∞

{\text {vo}}{{\text {l}}_{\Omega }}{{\mathbf {R}}^n} = \infty

, any subgroup of

Dif

f

Ω

(

R

n

)

{\text {Dif}}{{\text {f}}^\Omega }({{\mathbf {R}}^n})

,

N
N

, satisfying

Diff

co

Ω

(

R

n

)
⊂
N
⊂

Diff

c
Ω

(

R

n

)

{\text {Diff}}_{{\text {co}}}^\Omega ({{\mathbf {R}}^n}) \subset N \subset {\text {Diff}}_c^\Omega ({{\mathbf {R}}^n})

is normal, for

n
⩾

3

n \geqslant {\text {3}}

, there are no normal subgroups between

Diff

W
Ω

(

R

n

)

{\text {Diff}}_W^\Omega ({{\mathbf {R}}^n})

and

Dif

f

Ω

(

R

n

)

{\text {Dif}}{{\text {f}}^\Omega }({{\mathbf {R}}^n})

and for

n
⩾
4

n \geqslant 4

there are no normal subgroups between

Diff

c
Ω

(

R

n

)

{\text {Diff}}_c^\Omega ({{\mathbf {R}}^n})

and

Diff

f
Ω

(

R

n

)

{\text {Diff}}_f^\Omega ({{\mathbf {R}}^n})

.
Arithmetic equivalent of essential simplicity of zeta zeros
Authors: Mueller Julia 
Published: June 2010

Let

R
(
x
)

R(x)

and

S
(
t
)

S(t)

be the remainder terms in the prime number theorem and the Riemann-von Mangoldt formula respectively, that is

ψ
(
x
)
=
x
+
R
(
x
)

\psi (x) = x + R(x)

and

N
(
t
)
=
(
1

/

2
π
)

∫
0
t

log
⁡
(
τ

/

2
π
)

d
τ
+
S
(
t
)
+
7

/

8
+
O
(
1

/

t
)

N(t) = (1/2\pi )\int _0^t {\log (\tau /2\pi )\,d\tau + S(t) + 7/8 + O(1/t)}

. We are interested in the following integrals:

J
(
T
,
β
)
=

∫
1

T
β

(
R
(
x
+
x

/

T
)
−
R
(
x
)
)

2

d
x

/

x
2

J(T,\beta ) = \int _1^{{T^\beta }} {{{(R(x + x/T) - R(x))}^2}dx/{x^2}}

and

I
(
T
,
α
)
=

∫
1
T

(
S
(
t
+
α

/

L
)
−
S
(
t
)
)

2

d
t

I(T,\alpha ) = \int _1^T {{{(S(t + \alpha /L) - S(t))}^2}dt}

, where

L
=

(
2
π

)

−
1

log

T

L = {(2\pi )^{ - 1}}\log \,T

. Furthermore, denote by

N
(
T
,
α
)
(

N

∗

(
T
)
)

N(T,\alpha )(N^{\ast }(T))

the number of pairs of zeros

1
2

+
i
Υ
,

1
2

+
i

Υ
′

\frac {1} {2} + i\Upsilon ,\frac {1} {2} + i\Upsilon ’

with

0
>
Υ
⩽
T

0 > \Upsilon \leqslant T

and

0
>
(

Υ
′

−
Υ
)
L
⩽
α

(
(

Υ
′

−
Υ
)
L
=
0
)

0 > (\Upsilon ’ - \Upsilon )L \leqslant \alpha \,((\Upsilon ’ - \Upsilon )L = 0)

—i.e., off-diagonal and diagonal pairs. Theorem. Assume the Riemann hypothesis. The following three hypotheses (A), (B) and

(

C

1

,

C

2

)

({{\text {C}}_1},{{\text {C}}_2})

are equivalent: for

β
→
∞

\beta \to \infty

and

α
→
0

\alpha \to 0

as

T
→
∞

T \to \infty

we have (A)

J
(
T
,
β
)
∼
β

T

−
1

log
2

T

J(T,\beta ) \sim \beta {T^{ - 1}}{\log ^2}T

, (B)

I
(
T
,
α
)
∼
α
T

I(T,\alpha ) \sim \alpha T

and

(

C

1

)

N

∗

(
T
)
∼
T
L
,
(

C

2

)
N
(
T
,
α
)
=
o
(
T
L
)

({{\text {C}}_1})\;N^{\ast }(T) \sim TL,({{\text {C}}_2})N(T,\alpha ) = o(TL)

. Hypothesis

(

C

1

,

C

2

)

({{\text {C}}_1},{{\text {C}}_2})

is called the essential simplicity hypothesis.
A necessary and sufficient condition for the asymptotic version of Ahlfors’ distortion property
Authors: Rodin Burton , Warschawski S. E. 
Published: June 2010

Let

f
f

be a conformal map of

R
=
{
w
=
u
+
i
v
∈

C

|

φ
0

(
u
)
>
v
>

φ
1

(
u
)
}

R = \{w = u + iv \in {\mathbf {C}}|{\varphi _0}(u) > v > {\varphi _1}(u)\}

onto

S
=
{
z
=
x
+
i
y
∈

C

|

0
>
y
>
1
}

S = \{z = x + iy \in {\mathbf {C}}|0 > y > 1\}

where the

φ
j

∈

C
0

(
−
∞
,
∞
)

{\varphi _j} \in {C^0}( - \infty ,\infty )

and

Re
⁡
f
(
w
)
→
±
∞

\operatorname {Re} f(w) \to \pm \infty

as

Re
⁡
w
→
±
∞

\operatorname {Re} w \to \pm \infty

. There are well-known results giving conditions on

R
R

sufficient for the distortion property

Re
⁡
f
(
u
+
i
v
)
=

∫
0
u

(

φ
1

−

φ
0

)

−
1

d
u
+

const

.
+
o
(
1
)

\operatorname {Re} f(u + iv) = \int _0^u ({\varphi _1} - {\varphi _0})^{- 1}du + {\text {const}}. + o(1)

, where

o
(
1
)
→
0

o(1) \to 0

as

u
→
+
∞

u \to + \infty

. In this paper the authors give a condition on

R
R

which is both necessary and sufficient for

f
f

to have this property.
Stop rule inequalities for uniformly bounded sequences of random variables
Authors: Hill Theodore P. , Kertz Robert P. 
Published: June 2010

If

X

0

,

X
1

,
…

{X_{0}},{X_1},\ldots

is an arbitrarily-dependent sequence of random variables taking values in

[
0
,
1
]

[0,1]

and if

V
(

X
0

,

X
1

,
…
)

V({X_0},{X_1},\ldots )

is the supremum, over stop rules

t
t

, of

E

X
t

E{X_t}

, then the set of ordered pairs

{
(
x
,
y
)
:
x
=
V
(

X
0

,

X
1

,
…
,

X
n

)

\{ (x,y):x = V({X_0},{X_1},\ldots ,{X_n})

and

y
=
E
(

max

j

⩽

n

X
j

)

y = E({\max _{j\, \leqslant \,n}}{X_j})

for some

X
0

,
…
,

X
n

}

{X_0},\ldots ,{X_n}\}

is precisely the set
\[

C
n

=
{
(
x
,
y
)
:
x
⩽
y
⩽
x

(

1
+
n

(
1
−

x

1

/

n

)

)
;
0
⩽
x
⩽
1
}
;

{C_n} = \{ (x,y):x \leqslant y \leqslant x\,( {1 + n\,(1 - {x^{1/n}})} );0 \leqslant x \leqslant 1\} ;

\]
and the set of ordered pairs

{
(
x
,
y
)
:
x
=
V
(

X

0

,

X
1

,
…
)

\{ (x,y):x = V({X_{0}},{X_1},\ldots )

and

y
=
E
(

sup
n

X
n

)

y = E({\sup _n}\;{X_n})

for some

X
0

,

X
1

,
…
}

{X_0},{X_1},\ldots \}

is precisely the set
\[

C
=

⋃

n
=
1

∞

C
n

.

C = \bigcup \limits _{n = 1}^\infty {{C_n}} .

\]
As a special case, if

X
0

,

X
1

,
…

{X_0},{X_1},\ldots

is a martingale with

E

X
0

=
x

E{X_0} = x

, then

E
(

max

j
⩽
n

X
)
⩽
x
+
n
x
(
1
−

x

1

/

n

)

E({\max _{j \leqslant n}} X) \leqslant x + nx(1 - {x^{1\,/\,n}})

and

E
(

sup
n

X
n

)
⩽
x
−
x
ln

x

E({\sup _n}\;{X_n}) \leqslant x - x\ln \;x

, and both inequalities are sharp.
Estimates for eigenfunctions and eigenvalues of nonlinear elliptic problems
Authors: Cosner Chris 
Published: June 2010

We consider solutions to the nonlinear eigenvalue problem
\[

(
∗
)

A
(
x
,

u
→

)

u
→

+
λ
f
(
x
,

u
→

)
=
0

in

Ω
,

u
→

=
0

on

∂
Ω
,

u
→

=

0
,

on

∂
Ω
,

u
→

=
0
,

(*)\quad A(x,\vec u)\vec u + \lambda f(x,\vec u) = 0\:\quad {\text {in}}\,\Omega ,\quad \vec u = 0\:\quad {\text {on}}\,\partial \Omega ,\quad \vec u{\text { = }}0,\quad {\text {on}}\partial \Omega ,\quad \vec {u} = 0,

\]
where (*) is a quasilinear strongly coupled second order elliptic system of partial differential equations and

Ω
⊆

R

n

\Omega \subseteq \mathbf {R}^{n}

is a smooth bounded domain. We obtain lower bounds for

λ
\lambda

in the case where

f
(
x
,

u
→

)

f(x,\vec u)

has linear growth, and relations between

λ
,
Ω

\lambda ,\Omega

, and ess sup

|

u
→

|

|\vec u|

when

f
(
x
,

u
→

)

f(x,\vec u)

has sub- or superlinear growth. The estimates are based on integration by parts and application of certain Sobolev inequalities. We briefly discuss extensions to higher order systems.
Jordan domains and the universal Teichmüller space
Authors: Flinn Barbara Brown 
Published: June 2010

Let

L
L

denote the lower half plane and let

B
(
L
)

B(L)

denote the Banach space of analytic functions

f
f

in

L
L

with

‖
f
‖

L

>
∞

{\left \| f \right \|_L} > \infty

, where

‖
f
‖

L

{\left \| f \right \|_L}

is the suprenum over

z
∈
L

z \in L

of the values

|

f
(
z
)

|

(
t
e
x
t

I
m

z

)
2

\left | {f(z)} \right |{(text{Im} z)^2}

. The universal Teichmüller space,

T
T

, is the subset of

B
(
L
)

B(L)

consisting of the Schwarzian derivatives of conformal mappings of

L
L

which have quasiconformal extensions to the extended plane. We denote by

J
J

the set
\[

{

S
f

:
f

is conformal in

L

and

f
(
L
)

is a Jordan domain

}

,

\left \{ {{S_f}:f{\text {is conformal in }}L{\text {and }}f(L){\text {is a Jordan domain}}} \right \},

\]
which is a subset of

B
(
L
)

B(L)

contained in the Schwarzian space

S
S

. In showing

S
−

T
¯

≠
∅

S - \bar T \ne \emptyset

, Gehring actually proves

S
−

J
¯

≠
∅

S - \bar J \ne \emptyset

. We give an example which demonstrates that

J
−

T
¯

≠
∅

J - \bar T \ne \emptyset

.
Completely unstable dynamical systems
Authors: Goel Sudhir K. , Neumann Dean A. 
Published: June 2010

We associate with the

C
r

(
r

≥

1
)

{C^r}\,(r\, \ge \,1)

dynamical system

ϕ
\phi

on an

m
m

-manifold

M
M

, the orbit space

M

/

ϕ

M/\phi

, defined to be the set of orbits of

ϕ
\phi

with the quotient topology. If

ϕ
\phi

is completely unstable,

M

/

ϕ

M/\phi

turns out to be a

C
r

(
m

−

1
)

{C^r}\,(m\, - \,1)

-nonseparated manifold. It is known that for a completely unstable flow

ϕ
\phi

on a contractible manifold

M
,

M

/

ϕ

M,\,M/\phi

is Hausdorff if and only if

ϕ
\phi

is parallelizable. In general, we place an order on the non-Hausdorff points of

M

/

ϕ

M/\phi

(essentially) by setting

p
¯

>

q
¯

\bar p > \bar q

if and only if

π

−
1

(

q
¯

)
⊆

J
+

(

π

−
1

(

p
¯

)
)

{\pi ^{ - 1}}(\bar q) \subseteq {J^ + }({\pi ^{ - 1}}(\bar p))

. Our result is that

(
M
,

ϕ
)

(M,\,\phi )

is topologically equivalent to

(

M
′

,

ϕ
′

)

(M’,\,\phi ’)

if and only if

M

/

ϕ

M/\phi

is order isomorphic to

M
′

/

ϕ
′

M’/\phi ’

.
(𝑛-1)-axial 𝑆𝑂(𝑛) and 𝑆𝑈(𝑛) actions on homotopy spheres
Authors: Ball R. D. 
Published: June 2010

Let

G
(
n
)
=
O
(
n
)

G(n) = O(n)

or

U
(
n
)

U(n)

and

S
G
(
n
)
=
S
O
(
n
)

SG(n) = SO(n)

or

S
U
(
n
)

SU(n)

. For each integer

m
⩾
1

m \geqslant 1

a family

{

S

γ
,
σ

:
γ
∈
H
,
σ
∈
K
}

\{ {S_{\gamma ,\sigma }}:\gamma \in H,\sigma \in K\}

of

(
n
−
1
)

(n - 1)

-axial

S
G
(
n
)

SG(n)

homotopy spheres

S

γ
,
σ

{S_{\gamma ,\sigma }}

is constructed. Each

S

γ
,
σ

{S_{\gamma ,\sigma }}

has fixed point set of dimension

(
m
−
1
)
⩾
0

(m - 1) \geqslant 0

and orbit space of dimension

r
=

1
2

n
(
n
−
1
)
+
(
m
−
1
)

r = \tfrac {1} {2}n(n - 1) + (m - 1)

(resp.

r
=

(
n
−
1

)
2

+
m
−
1

r = {(n - 1)^2} + m - 1

) if

S
G
(
n
)
=
S
O
(
n
)

SG(n) = SO(n)

(resp.

S
U
(
n
)

SU(n)

).

H
H

is

π

r
−
1

(
S
G
(
n
)

/

G
(
n
−
1
)
)

{\pi _{r - 1}}(SG(n)/G(n - 1))

.

K
K

is trivial if

S
G
(
n
)
=
S
O
(
n
)

SG(n) = SO(n)

and is a homotopy theoretically defined subgroup of sections of an

S
2

{S^2}

bundle depending only on

m
m

and

n
n

if

S
G
(
n
)
=
S
U
(
n
)

SG(n) = SU(n)

. Assume that

m
m

and

n
n

satisfy the mild restriction

§
5

\S 5

, (1). It is shown that the above family is universal for

(
n
−
1
)

(n - 1)

-axial

S
G
(
n
)

SG(n)

homotopy spheres and provides a classification analogous to the classification of fibre bundles: for each

(
n
−
1
)

(n - 1)

-axial

S
G
(
n
)

SG(n)

homotopy sphere

Σ
\Sigma

there is a

S

γ
,
σ

{S_{\gamma ,\sigma }}

and a unique equivariant stratified map

Σ
→

S

γ
,
σ

\Sigma \to {S_{\gamma ,\sigma }}

.

Σ
\Sigma

is equivariantly diffeomorphic to the pullback of

S

γ
,
σ

{S_{\gamma ,\sigma }}

via the map

B
(
Σ
)
→
B
(

S

γ
,
σ

)

B(\Sigma ) \to B({S_{\gamma ,\sigma }})

of orbit spaces. If

S
G
(
n
)
=
S
O
(
n
)

SG(n) = SO(n)

then

γ
\gamma

is unique (and

σ
=
1

\sigma = 1

). If

S
G
(
n
)
=
S
U
(
n
)

SG(n) = SU(n)

then

γ
\gamma

is unique modulo the image of
\[

π

r
−
1

S
(
U
(
n
−
2
)
×
U
(
2
)
)

/

U
(
k
−
1
)
×
U
(
1
)

in

H
.

{\pi _{r - 1}}S(U(n - 2) \times U(2))/U(k - 1) \times U(1)\quad {\text {in}}\;H.

\]
An example is given showing that the differentiable structure of the underlying smooth manifold of

S

γ
,
σ

{S_{\gamma ,\sigma }}

may be exotic.