bluuebunny/crossref_metadata_2025 huggingface dataset:
Perturbations preserving asymptotics of spectrum with remainder
Authors: Ramm A. G.
Published: June 2010

Let

A
A

be a compact linear operator on a Hilbert space

H
H

,

s
n

(
A
)
=

λ
n

1

/

2

(

A
∗

A
)

{s_n}(A) = \lambda _n^{1/2}({A^* }A)

,

Q
Q

be a linear operator,

|

Q
f

|

⩽
c

|

A
f

|

a

|

f

|

1
−
a

|Qf| \leqslant c|Af{|^a}|f{|^{1 - a}}

,

a
>
0

a > 0

,

c
>
0

c > 0

,

∀
f
∈
H

\forall f \in H

. Let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s Subscript n Baseline left-parenthesis upper A right-parenthesis equals c n Superscript negative r Baseline left-brace 1 plus upper O left-parenthesis n Superscript negative q Baseline right-parenthesis right-brace comma r comma q greater-than 0 comma upper B equals upper A left-parenthesis upper I plus upper Q right-parenthesis period">

s
n

(
A
)
=
c

n

−
r

{
1
+
O
(

n

−
q

)
}
,

r
,
q
>
0
,
B
=
A
(
I
+
Q
)
.

{s_n}(A) = c{n^{ - r}}\{ 1 + O({n^{ - q}})\} ,\quad r,q > 0,B = A(I + Q).

\]
Then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s Subscript n Baseline left-parenthesis upper B right-parenthesis equals s Subscript n Baseline left-parenthesis upper A right-parenthesis left-brace 1 plus upper O left-parenthesis n Superscript negative gamma Baseline right-parenthesis right-brace comma gamma equals min left-parenthesis q comma StartFraction r a Over 1 plus r a EndFraction right-parenthesis period">

s
n

(
B
)
=

s
n

(
A
)
{
1
+
O
(

n

−
γ

)
}
,

γ
=
min

(

q
,

r
a

1
+
r
a

)

.

{s_n}(B) = {s_n}(A)\{ 1 + O({n^{ - \gamma }})\} ,\quad \gamma = \min \left ( {q,\frac {{ra}} {{1 + ra}}} \right ).

\]
Some applications of this result to the spectral theory of elliptic operators are given.
Lipschitz spaces and mixed Lebesgue spaces
Authors: Madych W. R.
Published: June 2010

It is shown that translation invariant linear operators which improve Lipschitz classes behave almost as well as the corresponding fractional Riesz transforms when applied to the mixed Lebesgue spaces. These results partially generalize some of the theorems concerning Riesz transforms and mixed Lebesgue classes due to Adams and Bagby, Lizorkin, and others.
A uniqueness result in conformal mapping. II
Authors: Jenkins James A.
Published: June 2010

This paper gives an elementary proof of the result that for a function

f
f

in the family

Σ
\Sigma

the diameter of the complement of the image of

|
z
|

>
1

\left | z \right | > 1

by

w
=
f
(
z
)

w = f(z)

attains its minimal value 2 only for

f
(
z
)
=
z
+
c

f(z) = z + c

,

c
c

constant.
A difference between minimal and ordinary fine topology in function theory
Authors: Hwang J. S.
Published: June 2010

After Choquet (see Brelot and Doob [6, p. 404]), we recently have presented an alternative answer of Doob’s problem [8] by showing that there is a Blaschke product

B
B

having the minimal fine cluster value 0 and the angular limit 1 at

∞
\infty

. In our construction, the zeros of

B
B

lie on both the first and fourth quadrant. Naturally, we may ask if a product having the same property, but having zeros which lie on only one quadrant, can be constructed. We show that the answer to this question is no for the minimal fine topology, but yes for the ordinary one. There is a significant difference between these two topologies in function theory.
Hilbertian interpretations of manuals
Authors: Gudder Stanley P.
Published: June 2010

We characterize manuals which admit an interpretation in a manual on a Hilbert space. This characterization is given in terms of a certain set of states that the manual supports.
A note on linear systems on 𝐾-3 surfaces
Authors: Tannenbaum Allen
Published: June 2010

A simple necessary and sufficient condition is given for a general member of the complete linear system

|

Y

|

|Y|

to be irreducible and nonsingular where

Y
Y

is a reduced, connected curve on a

K
−
3

K-3

surface.
An inequality for invariant factors
Authors: Thompson Robert C.
Published: June 2010

A divisibility relation is proved connecting the invariant factors of integral matrices

A
A

,

B
B

,

C
C

when

C
=
A
B

C = AB

.
Asymptotic evaluation of Gauss polynomials
Authors: Gallagher Patrick X. , Ramsey L. Thomas , Wells Benjamin B.
Published: June 2010

An asymptotic evaluation is given for the polynomials

G
(
α
)
=

∑

j
=
1

n

e
(

j
2

/

n
)
e
(
j
α
)

G(\alpha ) = \sum \nolimits _{j = 1}^n {e({j^2}/n)e(j\alpha )}

for

4

|

n

4|n

.
On the Hahn-Banach extension property
Authors: Borwein Jonathan M.
Published: June 2010

A self-contained and brief proof is given of the equivalence of the Hahn-Banach extension property (HB) and the conditional order completeness of the range space (LUB). Various other equivalences are discussed.
The left regular *-representation of an inverse semigroup
Authors: Wordingham J. R.
Published: June 2010

The left regular

∗

^ *

-representation of the semigroup algebra of an inverse semigroup is faithful. Clifford semigroups with a particular type of semilattice are shown to have the weak containment property if and only if each subgroup is amenable.
Certain Schur-Hadamard multipliers in the spaces 𝐶_{𝑝}
Authors: Arazy Jonathan
Published: June 2010

Let

f
f

be a continuously differentiable function on

[
−
1
,
1
]

[ - 1,1]

satisfying

|

f
′

(
t
)

|

⩽
C

|
t
|

α

\left | {f’(t)} \right | \leqslant C{\left | t \right |^\alpha }

for some

0
>
C

0 > C

,

α
>
∞

\alpha > \infty

and all

−
1
⩽
t
⩽
1

- 1 \leqslant t \leqslant 1

, and let

λ
=
(

λ
i

)
∈

l
r

\lambda = ({\lambda _i}) \in {l_r}

satisfy

−
1
⩽

λ
i

⩽
1

- 1 \leqslant {\lambda _i} \leqslant 1

for all

i
i

. Then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Subscript f comma lamda Baseline equals left-parenthesis StartFraction f left-parenthesis lamda Subscript i Baseline right-parenthesis minus f left-parenthesis lamda Subscript j Baseline right-parenthesis Over lamda Subscript i Baseline minus lamda Subscript j Baseline EndFraction right-parenthesis">

a

f
,
λ

=

(

f
(

λ
i

)
−
f
(

λ
j

)

λ
i

−

λ
j

)

{a_{f,\lambda }} = \left ( {\frac {{f({\lambda _i}) - f({\lambda _j})}} {{{\lambda _i} - {\lambda _j}}}} \right )

\]
is a Schur-Hadamard multiplier from

C

p
1

{C_{{p_1}}}

into

C

p
2

{C_{{p_2}}}

for all

p
1

{p_1}

,

p
2

{p_2}

satisfying

1
⩽

p
2

⩽
2
⩽

p
1

⩽
∞

1 \leqslant {p_2} \leqslant 2 \leqslant {p_1} \leqslant \infty

and

p
2

−
1

⩽

p
1

−
1

+
α

/

r

p_2^{ - 1} \leqslant p_1^{ - 1} + \alpha /r

.
Measure-theoretic quantifiers and Haar measure
Authors: Lyons Russell
Published: June 2010

Measure-theoretic quantifiers are introduced as convenient notation and to facilitate certain applications of Fubini’s theorem. They are used to prove the uniqueness of Haar measure and to give some conditions involving translation which imply absolute continuity of another measure.
On zero sets for the ball algebra
Authors: Alexander H.
Published: June 2010

Rudin has constructed a function in the ball algebra in

C

2

{{\mathbf {C}}^2}

whose zero set has infinite volume and has asked if such functions exist in

C

n

{{\mathbf {C}}^n}

. Using some recent work of Ryll and Wojtaszczyk on homogeneous polynomials we shall extend Rudin’s proof to

C

n

{{\mathbf {C}}^n}

for arbitrary

n
n

.
On Dini’s theorem and a metric on 𝐶(𝑋) topologically equivalent to the uniform metric
Authors: Beer Gerald
Published: June 2010

Let

X
X

be a compact metric space and let

U
C
(
X
)

UC(X)

denote the u.s.c. real valued functions on

X
X

. Let

τ
\tau

be a topology on

U
C
(
X
)

UC(X)

.

Ω
⊂
U
C
(
X
)

\Omega \subset UC(X)

is called a Dini class of functions induced by

τ
\tau

if (1)

Ω
\Omega

is

τ
\tau

-closed, (2)

C
(
X
)
⊂
Ω

C(X) \subset \Omega

, (3) for each

h
∈
Ω

h \in \Omega

whenever

{

h
n

}

\{ {h_n}\}

is a decreasing sequence of u.s.c. functions convergent pointwise to

h
h

, then

{

h
n

}
τ

\{ {h_n}\} \tau

-converges to

h
h

. By Dini’s theorem the topology of uniform convergence on

U
C
(
X
)

UC(X)

induces

C
(
X
)

C(X)

as its Dini class of functions. As a main result, when

X
X

is locally connected we show that the hyperspace topology on

U
C
(
X
)

UC(X)

obtained by identifying each u.s.c. function with the closure of its graph induces a larger Dini class of functions than

C
(
X
)

C(X)

, even though the restriction of this topology to

C
(
X
)

C(X)

agrees with the topology of uniform convergence.
Attractors with positively Lyapunov stable trajectories
Authors: Nadzieja Tadeusz
Published: June 2010

We show that a finite dimensional attractor in locally connected metric space consisting of positively Lyapunov stable trajectories is a torus.
Riemannian metrics induced by two immersions
Authors: do Carmo M. , Dajczer M.
Published: June 2010

We consider the situation where a Riemannian manifold

M
n

{M^n}

can be isometrically immersed into spaces

N

n
+
1

(
c
)

{N^{n + 1}}(c)

and

N

n
+
q

(

c
~

)

{N^{n + q}}(\tilde c)

with constant curvatures

c
>

c
~

c > \tilde c

,

q
⩽
n
−
3

q \leqslant n - 3

, and show that this implies the existence, at each point

p
∈
M

p \in M

, of an umbilic subspace

U
p

⊂

T
p

M

{U_p} \subset {T_p}M

, for both immersions, with

U
p

⩾
n
−
q

{U_p} \geqslant n - q

. In particular, if

M
n

{M^n}

can be isometrically immersed as a hypersurface into two spaces of distinct constant curvatures,

M
n

{M^n}

is conformally flat.
A characteristic property of the sphere
Authors: Saban Giacomo
Published: June 2010

Let

S
S

be any connected piece of surface in Euclidean three-space, of class

C
3

{C^3}

and

g

i
j

{g_{ij}}

,

l

i
j

{l_{ij}}

be the coefficients of the first and second fundamental forms of

S
S

. If these coefficients satisfy the system of differential equations obtained by interchanging the

g

i
j

{g_{ij}}

and

l

i
j

{l_{ij}}

having same indices in the Mainardi-Codazzi equations,

S
S

is part of a sphere. Furthermore, if two metrics on

S
S

satisfy a similar condition, they are proportional.
Weakly homogeneous models
Authors: Pillay Anand
Published: June 2010

I consider some notions of weak homogeneity, which generalise

ω
\omega

-homogeneity. I first analyse a specific such notion, called almost homogeneity, in the context of

ω
\omega

-stable theories. (Almost homogeneity is just like

ω
\omega

-homogeneity, but using strong types in place of types.) Then in a more general context, I prove for weakly homogeneous countable models some classification results which are known for

ω
\omega

-homogeneous countable models, in particular the result that the isomorphism type of such a model is determined by the types which it realises.
Quasicontinuous functions associated with a Hunt process
Authors: Le Jan Yves
Published: June 2010

A stochastic characterization is given for quasi-continuous functions associated with Hunt processes.
A singular stochastic integral equation
Authors: Nualart David , Sanz Marta
Published: June 2010

This note is devoted to the discussion of the stochastic differential equation

X
d
X
+
Y
d
Y
=
0

XdX + YdY = 0

,

X
X

and

Y
Y

being continuous local martingales. A method to construct solutions of this equation is given.
When is a Kirillov orbit a linear variety?
Authors: Felix Rainer
Published: June 2010

It is well known that a Kirillov orbit is a linear variety if and only if the corresponding irreducible representation is square integrable modulo its kernel ([1], Theorem 1.1). Now we give a new representation-theoretic criterion for a Kirillov orbit being a linear variety in terms of weak containment and tensor products of group representations.
Uniformly contractive fixed points in compact metric spaces
Authors: Leader Solomon
Published: June 2010

Let

f
f

be a continuous self-map on a compact metric space. Equivalent conditions are given for the existence of a uniformly contractive fixed point

w
(

f
n

x
→
w

w({f^n}x \to w

uniformly for all

x
x

). Corollaries give sufficient conditions for uniformly contractive fixed points subsuming some known results.
A pathological area preserving 𝐶^{∞} diffeomorphism of the plane
Authors: Handel Michael
Published: June 2010

The pseudocircle

P
P

is an hereditarily indecomposable planar continuum. In particular, it is connected but nowhere locally connected. We construct a

C
∞

{C^\infty }

area preserving diffeomorphism of the plane with

P
P

as a minimal set. The diffeomorphism

f
f

is constructed as an explicit limit of diffeomorphisms conjugate to rotations about the origin. There is a well-defined irrational rotation number for

f

|

P

f|P

even though

f

|

P

f|P

is not even semi-conjugate to a rotation of

S
1

{S^1}

. If we remove the requirement that our diffeomorphisms be area preserving, we may alter the example so that

P
P

is an attracting minimal set.
Integral polynomial generators for the homology of 𝐵𝑆𝑈
Authors: Kochman Stanley O.
Published: June 2010

Explicit formulas are given for polynomial generators of

H
∗

B
S
U

{H_ * }BSU

as specific polynomials in the canonical polynomial generators of

H
∗

B
U

{H_ * }BU

. The method is also applied to

H
∗

(
B
S
U
;
R
)

{H_ * }(BSU;R)

for any coefficient ring

R
R

and to

H
∗

(
B
S
O
;

Z
2

)

{H_ * }(BSO;{Z_2})

.
On ideal class groups of 2-power exponent
Authors: Earnest A. G. , Körner O. H.
Published: June 2010

It is shown that for a fixed totally real algebraic number field

k
k

and a fixed positive integer

t
t

, there exist only finitely many totally imaginary quadratic extensions

K
K

of

k
k

having ideal class groups of exponent

2
t

{2^t}

.
On double centralizer subgroups of some finite 𝑝-groups
Authors: Cheng Ying
Published: June 2010

Let

A
A

be a maximal abelian normal subgroup of a finite

p
p

-group

G
(
p
>
2
)

G(p > 2)

such that

[
G
,
A
]

[G,A]

is cyclic. Then (i)

C
G

(

C
G

(
D
)
)
=
D

{C_G}({C_G}(D)) = D

and

[
G
:

C
G

(
D
)
]
=
[
D
:
Z
(
G
)
]

[G:{C_G}(D)] = [D:Z(G)]

for every

Z
(
G
)
⩽
D
⩽
G

Z(G) \leqslant D \leqslant G

; (ii)

[
G
:
Z
(
G
)
]
=

[
G
,
A

]
2

[G:Z(G)] = {[G,A]^2}

and every faithful absolutely irreducible representation of

G
G

is of degree

[
G
:
A
]

[G:A]

. The case

p
=
2

p = 2

will also be mentioned.
Ideals and centralizing mappings in prime rings
Authors: Mayne Joseph H.
Published: June 2010

Let

R
R

be a prime ring and

U
U

be a nonzero ideal of

R
R

. If

T
T

is a nontrivial automorphism or derivation of

R
R

such that

u

u
T

−

u
T

u

u{u^T} - {u^T}u

is in the center of

R
R

and

u
T

{u^T}

is in

U
U

for every

u
u

in

U
U

, then

R
R

is commutative. If

R
R

does not have characteristic equal to two, then

U
U

need only be a nonzero Jordan ideal.
Holomorphic mappings of domains with generic corners
Authors: Webster S. M.
Published: June 2010

The boundary behavior of a biholomorphic mapping

f
f

between two domains with real analytic, generic, nondegenerate corners in

C

n

{{\mathbf {C}}^n}

is considered. Under certain minimal regularity assumptions on

f
f

it is shown that

f
f

continues holomorphically past the boundary.
Distance estimates for von Neumann algebras
Authors: Rosenoer Shlomo
Published: June 2010

It is shown that for certain von Neumann algebras

A

\mathcal {A}

, there is a constant

C
C

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dist left-parenthesis upper T comma script upper A right-parenthesis less-than-or-slanted-equals upper C sup Underscript upper P element-of lat script upper A Endscripts double-vertical-bar upper P Superscript up-tack Baseline upper T upper P double-vertical-bar for all upper T in script upper B left-parenthesis script upper H right-parenthesis period">

dist

(
T
,

A

)
⩽
C

sup

P
∈

lat

A

‖

P
⊥

T
P

‖

for

all

T

in

B

(

H

)

.

{\text {dist}}(T, \mathcal {A}) \leqslant C\sup \limits _{P \in {\text {lat}}\mathcal {A}} \left \| {{P^ \bot }TP} \right \|\quad {\text {for}}\;{\text {all}}\;T\;{\text {in}}\;\mathcal {B}{\text {(}}\mathcal {H}{\text {)}}{\text {.}}

\]
On factorizations of selfadjoint ordinary differential operators
Authors: Granata Antonio
Published: June 2010

Consider an ordinary linear differential operator

L
L

, of order

n
⩾
1

n \geqslant 1

, represented by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L u identical-to a Subscript n Baseline left-parenthesis t right-parenthesis u Superscript left-parenthesis n right-parenthesis Baseline plus midline-horizontal-ellipsis plus a 0 left-parenthesis t right-parenthesis u for-all u element-of upper C Superscript n Baseline left-parenthesis a comma b right-parenthesis">

L
u
≡

a
n

(
t
)

u

(
n
)

+
⋯
+

a
0

(
t
)
u

∀
u
∈

C
n

(
a
,
b
)

Lu \equiv {a_n}(t){u^{(n)}} + \cdots + {a_0}(t)u\;\forall u \in {C^n}(a,b)

, with real-valued coefficients

a
k

∈

C
k

(
a
,
b
)

{a_k} \in {C^k}(a,b)

,

0
⩽
k
⩽
n

0 \leqslant k \leqslant n

,

a
n

≠
0

{a_n} \ne 0

on

(
a
,
b
)

(a,b)

. According to a classical result, if

L
L

is formally selfadjoint on

(
a
,
b
)

(a,b)

then it has a factorization of the type
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L u identical-to p Subscript n Baseline left-bracket p Subscript n minus 1 Baseline left-parenthesis midline-horizontal-ellipsis left-parenthesis p 1 left-parenthesis p 0 u right-parenthesis prime right-parenthesis prime midline-horizontal-ellipsis right-parenthesis prime right-bracket prime for-all u element-of upper C Superscript n Baseline left-parenthesis a comma b right-parenthesis">

L
u
≡

p
n

[

p

n
−
1

(
⋯
(

p
1

(

p
0

u

)
′

)
′

⋯

)
′

]
′

∀
u
∈

C
n

(
a
,
b
)

Lu \equiv {p_n}[{p_{n - 1}}( \cdots ({p_1}({p_0}u)’)’ \cdots )’]’\forall u \in {C^n}(a,b)

, where the

p
k

{p_k}

’s are sufficiently-smooth and everywhere nonzero functions on

(
a
,
b
)

(a,b)

such that

p
k

=

p

n
−
k

{p_k} = {p_{n - k}}

(
k
=
0
,
…
,
n
)

(k = 0, \ldots ,n)

. In this note we shall examine this result critically and show by means of counterexamples that the different classical proofs are either merely local or purely heuristic. A proof, which is both rigorous and global, is inferred from recent results on canonical factorizations of disconjugate operators. In addition, information is obtained on the behavior of the

p
k

{p_k}

’s at the endpoints of

(
a
,
b
)

(a,b)

which may prove useful in applications.
Nuclear faces of state spaces of 𝐶*-algebras
Authors: Batty C. J. K.
Published: June 2010

Let

B
1

{B_1}

and

B
2

{B_2}

be maximal abelian subalgebras of

C
∗

{C^* }

-algebras

A
1

{A_1}

and

A
2

{A_2}

, and suppose that for each pure state

ψ
1

{\psi _1}

of

B
1

{B_1}

, the von Neumann algebra

p

ψ
1

A
1

∗
∗

p

ψ
1

{p_{{\psi _1}}}A_1^{**}{p_{{\psi _1}}}

is injective, where

p

ψ
1

{p_{{\psi _1}}}

is the common support in

A
1

∗
∗

A_1^{**}

of all the states of

A
1

{A_1}

which extend

ψ
1

{\psi _1}

. Then

B
1

⊗

B
2

{B_1} \otimes {B_2}

is a maximal abelian subalgebra of any

C
∗

{C^*}

-tensor product

A
1

⊗
β

A
2

{A_1}{ \otimes _\beta }{A_2}

.
The Radon transform on a family of curves in the plane. II
Authors: Cormack A. M.
Published: June 2010

A further discussion of Radon’s problem for curves in the plane given, for fixed

(
p
,
ϕ
)

(p,\phi )

, by

r
α

cos
⁡
{
α
(
θ
−
ϕ
)
}
=

p
α

{r^\alpha }\cos \{ \alpha (\theta - \phi )\} = {p^\alpha }

.

α
\alpha

real,

α
≠
0

\alpha \ne 0

. Functions yielding null transforms, and zeros of the Fourier components of the transforms are given for general

α
\alpha

. and several orthogonal expansions are given for

α
=
±
1

/

m

\alpha = \pm 1/m

.

m
=
1
,
2
,
3
…
.

m = 1,2,3 \ldots .
The minimal normal filter on 𝑃_{𝜅}𝜆
Authors: Carr Donna M.
Published: June 2010

Let

κ
\kappa

be an uncountable regular cardinal, let

C

F
κ

C{F_\kappa }

be the cub filter on

κ
\kappa

and let

F
S

F
κ

FS{F_\kappa }

be the filter generated by

{
{
β
>
κ
:
β
>
α
}
:
α
>
κ
}

\{ \{ \beta > \kappa :\beta > \alpha \} :\alpha > \kappa \}

. It is well known that

C

F
k

C{F_k}

is normal, that

C

F
κ

=
Δ
F
S

F
κ

C{F_\kappa } = \Delta FS{F_\kappa }

and hence that every normal filter on

κ
\kappa

extends

C

F
κ

C{F_\kappa }

. Jech extended some of these results to the context of

P
κ

λ

{P_\kappa }\lambda

. Let

λ
\lambda

be a cardinal

⩾
κ

\geqslant \kappa

and let

C

F

κ
λ

C{F_{\kappa \lambda }}

denote the cub filter on

P

κ
λ

{P_{\kappa \lambda }}

as defined by Jech; he showed that

C

F

κ
λ

C{F_{\kappa \lambda }}

is normal and that every normal ultrafilter on

P

κ
λ

{P_{\kappa \lambda }}

extends

C

F

κ
λ

C{F_{\kappa \lambda }}

. In this paper we extend these results further. In particular, we show that

F

κ
λ

=
Δ
Δ
F
S

F

κ
λ

{F_{\kappa \lambda }} = \Delta \Delta FS{F_{\kappa \lambda }}

where

F
S

F

κ
λ

FS{F_{\kappa \lambda }}

is the filter generated by

{
{
y
∈

P
κ

λ
:
x
⊂
y
}
:
x
∈

P
κ

λ
}

\{ \{ y \in {P_\kappa }\lambda :x \subset y\} :x \in {P_\kappa }\lambda \}

, and that every normal filter on

P
κ

λ

{P_\kappa }\lambda

extends

C
F

P

κ
λ

CF{P_{\kappa \lambda }}

. Finally, we show that for any

λ
⩾
κ

\lambda \geqslant \kappa

and any ideal

I
I

on

P
κ

λ
,
∇
∇
∇
I
=
∇
∇
I

{P_\kappa }\lambda ,\nabla \nabla \nabla I = \nabla \nabla I
Counting fixed points
Authors: Rosenholtz Ira
Published: June 2010

In this paper, we develop a machine which enables us to predict, in many cases, the exact number of fixed points of a local diffeomorphism. Though much more general, our technique applies in particular to locally expansive maps on compact, connected, orientable differentiable manifolds.
On the irreducibility of an induced representation. II
Authors: Quigg John C.
Published: June 2010

A unitary representation induced from a normal subgroup of a separable locally compact group is irreducible if and only if the inducing representation is irreducible and the restriction of the induced representation to the normal subgroup is multiplicity-free.
Maximal intersecting families of finite sets and 𝑛-uniform Hjelmslev planes
Authors: Drake David A. , Sane Sharad S.
Published: June 2010

The following theorem is proved. The collection of lines of an

n
n

-uniform projective Hjelmslev plane is maximal when considered as a collectiion of mutually intersecting sets of equal cardinality.
Representing triples of a symplectic manifold
Authors: Karrer Guido
Published: June 2010

For a symplectic manifold

M
M

and its associated real Lie algebra

P
(
M
,
ω
)

P(M,\omega )

(its Poisson algebra) a definition of first-order representations and a structure theorem for the representation ring is given.
Discrete generalized Cesàro operators
Authors: Rhaly H. C.
Published: June 2010

For

|
λ
|

⩽
1

\left | \lambda \right | \leqslant 1

,

A
λ
∗

A_\lambda ^*

is the operator defined formally on the Hardy space

H
2

{H^2}

by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A Subscript lamda Superscript asterisk Baseline f right-parenthesis left-parenthesis z right-parenthesis equals minus left-parenthesis lamda minus z right-parenthesis Superscript negative 1 Baseline integral Subscript lamda Superscript z Baseline f left-parenthesis s right-parenthesis d s comma StartAbsoluteValue z EndAbsoluteValue greater-than 1 period">

(

A
λ
∗

f
)
(
z
)
=
−

(
λ
−
z

)

−
1

∫
λ
z

f
(
s
)

d
s
,

|
z
|

>
1.

(A_\lambda ^*f)(z) = - {(\lambda - z)^{ - 1}}\int _\lambda ^z {f(s)\,} ds,\quad \left | z \right | > 1.

\]
If

λ
=
1

\lambda = 1

, then the usual identification of

H
2

{H^2}

with

l
2

{l^2}

takes

A
1

{A_1}

onto the discrete Cesàro operator. Here we answer questions about boundedness, spectra, unitary equivalence, compactness, and subnormality for the operators

A
λ

{A_\lambda }

.
The translation invariant uniform approximation property for compact groups
Authors: Travaglini Giancarlo
Published: June 2010

We show that, unlike the abelian case, the translation invariant uniform approximation property fails in a strong way for

L
1

{L^1}

of a compact connected semisimple Lie group.
On unicity of complex polynomial 𝐿₁-approximation along curves
Authors: Kroó A.
Published: June 2010

We study the unicity of best polynomial

L
1

{L_1}

-approximation of complex continuous functions along curves

γ
\gamma

in the complex plane. A sufficient condition on

γ
\gamma

is given which implies unicity. In particular our result includes the known cases of circle and line segment.
On a gap Tauberian theorem of Lorentz and Zeller
Authors: Keagy T. A.
Published: June 2010

G. G. Lorentz and K. L. Zeller have stated an

O
O

-Tauberian theorem which places a restriction on the rate of absolute convergence of the row sums of a regular summability method. In this note, we prove a theorem that has as a corollary an extension of the above result in which this restriction is deleted.
Approximate topology on 𝑅𝑒𝑝(𝐴)
Authors: Lee Sa Ge
Published: June 2010

Let

A
A

be a

C
∗

{C^*}

-algebra and Rep

(
A
)

(A)

the set of all nondegenerate representations of

A
A

. We define a new topology on Rep

(
A
)

(A)

and study the relations with the point weak topology of Rep

(
A
)

(A)

.
On the relation between left thickness and topological left thickness in semigroups
Authors: Wong James C. S.
Published: June 2010

In this paper, we establish an interesting relation between left thickness and topological left thickness in semigroups by showing that a Borel subset

T
T

of a locally compact semigroup

S
S

is topological left thick in

S
S

iff a certain subset

M
T

{M_T}

associated with

T
T

is left thick in a semigroup

S
1

{S_1}

containing

S
S

, or equivalent, iff

M
T

{M_T}

contains a left ideal of

S
1

{S_1}

. Our results contain a topological analogue of a result of H. Junghenn in [Amenability of function spaces on thick subsemigroups, Proc. Amer. Math. Soc. 75 (1979), 37-41]. However, even in the case of discrete semigroups, our results are more general and in a way more natural than those of Junghenn’s. Furthermore, the fact that

M
T

{M_T}

is left thick iff it contains a left ideal in

S
1

{S_1}

is quite surprising, since in general, a left thick subset need not contain a left ideal although the converse is always true.
Generic existence of a solution for a differential equation in a scale of Banach spaces
Authors: Benavides Tomás Domínguez
Published: June 2010

Let

{

X
s

:
α
⩽
s
⩽
β

}

\left \{ {{X_s}:\alpha \leqslant s \leqslant \beta } \right \}

be a scale of Banach spaces,

J
J

a real interval,

U
U

an open subset of

J
×

X
s

J \times {X_s}

for some

s
s

. In this paper we prove that the existence of solutions for
\[

x
′

=
A
(
t
)
x
+
f
(
t
,
x
)
,

x
(

t
0

)
=

x
0

,

x’ = A(t)x + f(t,x),\quad x({t_0}) = {x_0},

\]
is a generic property, when

A
(
t
)

A(t)

is an operator satisfying
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper A left-parenthesis t right-parenthesis EndAbsoluteValue Subscript upper L left-parenthesis upper X Sub Subscript s Sub Sub Superscript prime Sub Subscript Subscript semicolon upper X Sub Subscript s Subscript right-parenthesis Baseline less-than-or-slanted-equals upper M left-parenthesis s prime minus s right-parenthesis Superscript negative 1 Baseline left-parenthesis upper M greater-than 0 independent of s comma s prime comma t right-parenthesis">

|

A
(
t
)

|

L
(

X

s
′

;

X
s

)

⩽
M

(

s
′

−
s

)

−
1

(
M
>
0

independent

of

s
,

s
′

,
t
)

{\left | {A(t)} \right |_{L({X_{s’}};{X_s})}} \leqslant M{(s’ - s)^{ - 1}}\quad (M > 0\;{\text {independent}}\;{\text {of}}\;s,s’,t)

\]
in the scale

{

X
s

}

\left \{ {{X_s}} \right \}

and

f
:
J
×
U
→

X
β

f:J \times U \to {X_\beta }

is continuous.
A necessary and sufficient condition for uniform approximation by certain rational modules
Authors: Carmona Doménech J. J.
Published: June 2010

Let

X
X

be a compact subset of

C

{\mathbf {C}}

with empty interior and let

g
g

be a complex function of class

C
2

{C^2}

in a neighborhood of

X
X

. For

Z
=

{

z
∈
X

|

∂
g
(
z
)

/

∂

z
¯

=
0

}

Z = \left \{ {z \in X|\partial g(z)/\partial \bar z = 0} \right \}

, we prove that

R
(
X
)
+
g
R
(
X
)

R(X) + gR(X)

is uniformly dense in

C
(
X
)

C(X)

if and only if

R
(
Z
)
=
C
(
Z
)

R(Z) = C(Z)

.
A characterization of the least cardinal for which the Baire category theorem fails
Authors: Miller Arnold W.
Published: June 2010

Let

κ
\kappa

be the least cardinal such that the real line can be covered by

κ
\kappa

many nowhere dense sets. We show that

κ
\kappa

can be characterized as the least cardinal such that "infinitely equal" reals fail to exist for families of cardinality

κ
\kappa

.
Invariant measures for affine foliations
Authors: Goldman William M. , Hirsch Morris W. , Levitt Gilbert
Published: June 2010

A (transversely) affine foliation is a foliation with an atlas whose coordinate changes are locally affine. Such foliations arise naturally in the study of affine structures on manifolds. In this paper we prove that an affine foliation with nilpotent affine holonomy group always admits a nontrivial transverse measure. Two proofs are given: one for noncompact manifolds, and another, valid for compact manifolds with

(
G
,
X
)

(G,X)

-foliations (not necessarily affine) having nilpotent holonomy. These results are applied to prove that a certain cohomology class on a compact affine manifold with nilpotent holonomy is nonzero. Examples are discussed.
Sheaf theoretic cohomological dimension and finitistic spaces
Authors: Deo Satya
Published: June 2010

For a topological

n
n

-manifold

X
X

, we proved earlier [7] that

Di

m

Z

(
X
)
=
n
+
1

{\text {Di}}{{\text {m}}_{\mathbf {Z}}}(X) = n + 1

, if

n
>
0

n > 0

; and, for a zero-dimensional manifold (discrete space) we observed that

Di

m

Z

(
X
)
=
0

{\text {Di}}{{\text {m}}_{\mathbf {Z}}}(X) = 0

. The question was later raised as to what are those paracompact spaces, besides discrete one, for which

Di

m

Z

(
X
)
=
0

{\text {Di}}{{\text {m}}_{\mathbf {Z}}}(X) = 0

. In this paper we prove that there is none, i.e., if

X
X

is not discrete then

Di

m

Z

(
X
)
>
0

{\text {Di}}{{\text {m}}_{\mathbf {Z}}}(X) > 0

. Another question which cropped up only recently in the cohomological theory of topological transformation groups is whether or not there exists a finitistic space which is not of finite (sheaf theoretic) integral cohomological dimension. We show that this question is related to a famous unsolved problem of cohomological dimension theory.
On simple loops on a solid torus of general genus
Authors: Kaneto Takeshi
Published: June 2010

Let

l
l

be a simple loop on the boundary of a solid torus

T
T

of genus

g
g

. Let

m

{\mathbf {m}}

be a complete system of oriented meridian disks of

T
T

. Let

W
(
l
,

m

)

W(l,{\mathbf {m}})

be a word obtained by reading the intersections

l
∩

m

l \cap {\mathbf {m}}

along

l
l

. We shall give a natural method for realizing the cyclic reductions of

W
(
l
,

m

)

W(l,{\mathbf {m}})

geometrically. This yields a simple proof of Whitehead-Zieschang’s theorem related to the minimality of the intersections of two systems

{

l
1

,
…
,

l
n

}

\{ {l_1}, \ldots ,{l_n}\}

and

m

{\mathbf {m}}

.
A generalization of a theorem of Ayoub and Chowla
Authors: Redmond Don
Published: June 2010

Let

X

1

\mathcal {X}1

and

X

2

\mathcal {X}2

be characters modulo

q
1

{q_1}

and

q
2

{q_2}

, respectively, where

q
1

{q_1}

and

q
2

{q_2}

are positive integers. Let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis n right-parenthesis equals sigma-summation Underscript d vertical-bar n Endscripts script upper X Baseline 1 left-parenthesis d right-parenthesis script upper X Baseline 2 left-parenthesis n slash d right-parenthesis period">

f
(
n
)
=

∑

d

|

n

X

1
(
d
)

X

2
(
n

/

d
)
.

f(n) = \sum \limits _{d|n} \mathcal {X}1 (d)\mathcal {X}2(n/d).

\]
In this paper we shall give an estimate for the sum
\[

∑

n
⩽
x

f
(
n
)

log
⁡
(
x

/

n
)
.

\sum \limits _{n \leqslant x} {f(n)} \log (x/n).

\]
Quantitative behaviour of the norms of an analytic measure
Authors: Pigno Louis , Smith Brent
Published: June 2010

A Littlewood-Paley type inequality for the quotient norms of an analytic measure is obtained; one consequence of this inequality is the classical theorem of F. and M. Riesz.
Factorization in codimension one ideals of group algebras
Authors: Willis George
Published: June 2010

It is shown that if

G
G

is a locally compact group and

I
I

is a closed, two-sided ideal with codimension one in

L
1

(
G
)

{L^1}(G)

, then

I
2

=
I

{I^2} = I

.
Finite type functions as limits of exponential sums
Authors: Newman D. J.
Published: June 2010

The functions

f
(
z
)

f(z)

of type 1 which are bounded on the real axis are usually thought of as spanned by the exponentials

e

i
λ
z

{e^{i\lambda z}}

,

λ
∈
[
−
1
,
1
]

\lambda \in [ - 1,1]

. We find the correct topology in which this "spanning" occurs.
A counterexample to a conjecture in linear second-order differential equations
Authors: Szabó Kálmán
Published: June 2010

The aim of the present paper is to give a simple counterexample to a conjecture [3] in linear second-order differential equations.
Inequalities of Gauss-Bonnet type for a convex domain
Authors: Dekster B. V.
Published: June 2010

Let

N
N

be a compact convex

n
n

-dimensional Riemannian manifold with a boundary

∂
N

\partial N

having normal curvatures

⩾
κ
>
0

\geqslant \kappa > 0

. Suppose the sectional curvature

>
−

κ
2

> - {\kappa ^2}

in

N
N

. Let

H
H

be the integral mean curvature of

∂
N

\partial N

,

V
V

be the volume of

N
N

,

k

s
c

{k_{sc}}

be the scalar curvature and

k
¯

R

(
p
)

{\bar k_R}(p)

,

p
∈
N

p \in N

, be the maximum Ricci curvature at

p
p

. Then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H greater-than-or-slanted-equals StartFraction n minus 2 Over 2 EndFraction kappa squared upper V minus StartFraction 1 Over 2 left-parenthesis n minus 1 right-parenthesis EndFraction integral Underscript upper N Endscripts k Subscript s c Baseline d upper V comma upper H greater-than-or-slanted-equals left-parenthesis n minus 2 right-parenthesis kappa squared upper V minus StartFraction 1 Over n minus 1 EndFraction integral Underscript upper N Endscripts k overbar Subscript upper R Baseline d upper V period">

H
⩾

n
−
2

2

κ
2

V
−

1

2
(
n
−
1
)

∫
N

k

s
c

d
V

,

H
⩾
(
n
−
2
)

κ
2

V
−

1

n
−
1

∫
N

k
¯

R

d
V
.

H \geqslant \frac {{n - 2}} {2}{\kappa ^2}V - \frac {1} {{2(n - 1)}}\int _N {{k_{sc}}\;dV} ,\quad H \geqslant (n - 2){\kappa ^2}V - \frac {1} {{n - 1}}\int _N {{{\bar k}_R}\;dV.}

\]
Let

N
−

{N_ - }

denote

N
N

with nonpositive sectional curvature. Let

G
G

be the integral Gauss curvature of

∂

N
−

\partial {N_ - }

. Then

G
⩾
−

κ

n
−
2

∫

N
−

k
¯

R

d
V

G \geqslant - {\kappa ^{n - 2}}\int _{N - } {{{\bar k}_R}\;dV}

. These three estimates are sharp. For a ball in

3
3

-dimensional hyperbolic space, the ratio of the right-hand part of each estimate to its left-hand part (i.e.

V
(

κ
2

+
3
)

/

2
H

V({\kappa ^2} + 3)/2H

,

V
(

κ
2

+
1
)

/

H

V({\kappa ^2} + 1)/H

and

2
κ
V

/

G

2\kappa V/G

respectively) approaches 1 as the

radius

→
∞

{\operatorname {radius}} \to \infty

. The same ratios for the estimates
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H greater-than-or-slanted-equals minus StartFraction 1 Over 2 left-parenthesis n minus 1 right-parenthesis EndFraction integral Underscript upper N Endscripts k Subscript s c Baseline d upper V and upper H greater-than-or-slanted-equals minus StartFraction 1 Over n minus 1 EndFraction integral Underscript upper N Endscripts k overbar Subscript upper R Baseline d upper V">

H
⩾
−

1

2
(
n
−
1
)

∫
N

k

s
c

d
V

and

H
⩾
−

1

n
−
1

∫
N

k
¯

R

d
V

H \geqslant - \frac {1} {{2(n - 1)}}\int _N {{k_{sc}}\;} dV\quad {\text {and}}\quad H \geqslant - \frac {1} {{n - 1}}\int _N {{{\bar k}_R}\;dV}

\]
(rougher ones but without

κ
\kappa

) approach

3
4

\tfrac {3} {4}

and

1
2

\tfrac {1} {2}

respectively.
Orthogonal expansions of vectors in a Hilbert space for non-Gaussian measures
Authors: Okazaki Yoshiaki
Published: June 2010

Let

H

\mathcal {H}

be a separable Hilbert space and

μ
\mu

a probability Radon measure on

H

\mathcal {H}

of second order. Then there exist

(

a
n

)
∈

l
2

({a_n}) \in {l^2}

, an O.N.S.

(

x
n

)
⊂

H

({x_n}) \subset \mathcal {H}

and an O.N.S.

(

ξ
n

)
⊂
H

({\xi _n}) \subset H

such that the orthogonal series

∑

n
=
1

∞

a
n

ξ
n

(
x
)

x
n

\sum \nolimits _{n = 1}^\infty {{a_n}{\xi _n}(x){x_n}}

converges in

H

\mathcal {H}

μ
\mu

-almost everywhere and it holds that

x
=

∑

n
=
1

∞

a
n

ξ
n

(
x
)

x
n

x = \sum \nolimits _{n = 1}^\infty {{a_n}{\xi _n}(x){x_n}}

,

μ
\mu

-almost everywhere, where

H
H

is the generating Hubert space of

μ
\mu

. In the case where

μ
\mu

is a Gaussian measure, a similar result was proved by Kuelbs [2] in general Banach spaces.
Geometric taming of compacta in 𝐸ⁿ
Authors: Wright David G.
Published: June 2010

We investigate

k
k

-dimensional compacta in

E
n

(
k
⩽
n
−
3
)

{E^n}(k \leqslant n - 3)

that satisfy geometric properties. We prove that such a compactum

X
X

in

E
n

{E^n}

is tamely embedded if each point of

X
X

can be touched by the tip of a cone from the complement of

X
X

. Furthermore, we show that a

k
k

-dimensional compactum

Y
Y

in

E
n

(
k
⩽
n
−
3
)

{E^n}(k \leqslant n - 3)

is tame if

Y
Y

has vertical order

n
−
k
−
2

n - k - 2

.
A short proof that compact quasidevelopable spaces are metrizable
Authors: Bennett H. R.
Published: June 2010

A new characterization of quasidevelopable spaces is given that allows an easier proof that compact quasidevelopable spaces are metrizable.
Spaces for which the generalized Cantor space 2^{𝐽} is a remainder
Authors: Ünlü Yusuf
Published: June 2010

Let

X
X

be a locally compact noncompact space,

m
m

be an infinite cardinal and

|

J

|

=
m

| J | = m

. Let

F
(
X
)

F(X)

be the algebra of continuous functions from

X
X

into

R

{\mathbf {R}}

which have finite range outside of an open set with compact closure and let

I
(
X
)
=
{
g
∈
F
(
X
)
:
g

I(X) = \{ g \in F(X):g

vanishes outside of an open set with compact closure}. Conditions on

R
(
X
)
=
F
(
X
)

/

I
(
X
)

R(X) = F(X)/I(X)

and internal conditions are obtained which characterize when

X
X

has

2
J

{2^J}

as a remainder.
The nonexistence of “liftings” for arithmetic density
Authors: Weizsäcker H.
Published: June 2010

P. Erdös, answering a question of D. Maharam, proved that arithmetic density as a set function does not admit an analogue of measure theoretic liftings [1, Theorem 4]. Here we give a quick proof of this which works in a slightly more general situation.
An analytic characterization of groups with no finite conjugacy classes
Authors: Cowie E. R.
Published: June 2010

Let

A
A

be a unital Banach algebra and

G

\mathcal {G}

the group of isometries in

A
A

. The norm in

A
A

is uniquely maximal if

G

\mathcal {G}

is not contained in any larger bounded group in

A
A

and there is no equivalent norm on

A
A

with the same group of isometries. We use a group theory result of B. H. Neumann to prove that the discrete measure algebra

l
1

(
G
)

{l^1}(G)

is uniquely maximal if and only if

G
G

has no finite conjugacy classes.
Finite jumps in Milnor number imply vanishing folds
Authors: O’Shea Donal B.
Published: June 2010

Let

{

X
t

}

\left \{ {{X_t}} \right \}

be a family of isolated hypersurface singularities in which the Milnor number is not constant. It is proved that there must be a vanishing fold centered at any

t
=

t
0

t = {t_0}

at which the Milnor number of the

X
t

{X_t}

changes discontinuously. This is much stronger than the condition that the Whitney conditions fail.
The norm map on Jacobians
Authors: Rosen Michael
Published: June 2010

Let

π
:
Γ
→

Γ
0

\pi :\Gamma \to {\Gamma _0}

be an unramified normal cover of smooth projective curves. Let

π
∗

:
J
→

J
0

{\pi _*}:J \to {J_0}

be the induced map on Jacobians. Let

H
H

be the kernel of

π
∗

{\pi _*}

and

H
0

{H^0}

the connected component of

H
H

. We prove that

H

/

H
0

H/{H^0}

is isomorphic to

G

/

[
G
,
G
]

G/[G,G]

where

G
G

is the covering group of

Γ

/

Γ
0

\Gamma /{\Gamma _0}

.
Cohomology of Heisenberg Lie algebras
Authors: Santharoubane L. J.
Published: June 2010

The cohomology of Heisenberg Lie algebras is studied and we obtain the description of cocycles, coboundaries and cohomological spaces.
Reductive weak decomposable operators are spectral
Authors: Tanahashi Kôtarô
Published: June 2010

We show that if a bounded linear operator

T
T

on a complex Hilbert space is reductive and weak decomposable, then

T
T

is a spectral operator with a normal scalar part. This is a generalization of a result due to Jafarian [3].
Polynômes de Jacobi, interprétation combinatoire et fonction génératrice
Authors: Foata Dominique , Leroux Pierre
Published: June 2010

This classical generating function for the Jacobi polynomials is derived by purely combinatorial methods.
Tracial positive linear maps of 𝐶*-algebras
Authors: Choi Man Duen , Tsui Sze Kai
Published: June 2010

A positive linear map

Φ
:

A

→

B

\Phi :\mathfrak {A} \to \mathfrak {B}

between two

C
∗

{C^ * }

-algebras is said to be tracial if

Φ
(

A
1

A
2

)
=
Φ
(

A
2

A
1

)

\Phi ({A_1}{A_2}) = \Phi ({A_2}{A_1})

for all

A
i

∈

A

{A_i} \in \mathfrak {A}

. A tracial positive linear map

A

→

B

(

H

)

\mathfrak {A} \to \mathcal {B}\left (\mathcal {H} \right )

is analyzed as the composition of a tracial positive linear map

A

→
C
(
X
)

\mathfrak {A} \to C(X)

followed by a positive linear map

C
(
X
)
→

B

(

H

)

C(X) \to \mathcal {B}\left ( \mathcal {H} \right )

.
Stokes’ theorem and parabolicity of Riemannian manifolds
Authors: Glasner Moses
Published: June 2010

A noncompact Riemannian

n
n

-manifold is parabolic if and only if Stokes’ theorem is valid for every square integrable

(
n
−
1
)

(n - 1)

-form with integrable derivative.
On Boolean algebras of projections and scalar-type spectral operators
Authors: Ricker W.
Published: June 2010

It is shown that the weakly closed operator algebra generated by an equicontinuous

σ
\sigma

-complete Boolean algebra of projections on a quasi-complete locally convex space consists entirely of scalar-type operators. This extends W. Badé’s well-known theorem that the same assertion is valid for Banach spaces; however, the technique of proof here differs from his method, which extends only to metrizable spaces.
On a generalized moment problem
Authors: Hwang J. S.
Published: June 2010

The well-known Müntz-Szász theorem asserts that the sequence of powers

x

n
p

{x^{{n_p}}}

is complete on

[
a
,
b
]

[a,b]

, where

a
⩾
0

a \geqslant 0

, if and only if (1)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 right-parenthesis sigma-summation Underscript p equals 1 Overscript normal infinity Endscripts StartFraction 1 Over n Subscript p Baseline EndFraction equals normal infinity comma where 0 greater-than n 1 greater-than n 2 greater-than midline-horizontal-ellipsis period">

(
1
)

∑

p
=
1

∞

1

n
p

=
∞
,

where

0
>

n
1

>

n
2

>
⋯
.

(1)\quad \sum \limits _{p = 1}^\infty {\frac {1} {{{n_p}}} = \infty ,\quad {\text {where}}\;0 > {n_1} > {n_2} > \cdots .}

\]
Let

f
(
x
)

f(x)

be absolutely continuous,

|

f
′

(
x
)

|

⩾
k
>
0

\left | {f’(x)} \right | \geqslant k > 0

, and

f
(
a
)
f
(
b
)
⩾
0

f(a)f(b) \geqslant 0

. We prove that under the assumption (1) the sequence

{

f

(
x
)

n
p

}

\left \{ {f{{(x)}^{{n_p}}}} \right \}

is complete on

[
a
,
b
]

[a,b]

if and only if

f
(
x
)

f(x)

is monotone on

[
a
,
b
]

[a,b]

.
On the operator ranges of analytic functions
Authors: Hwang J. S.
Published: June 2010

Following Doob, we say that a function

f
(
z
)

f(z)

analytic in the unit disk

U
U

has the property

K
(
ρ
)

K(\rho )

if

f
(
0
)
=
0

f(0) = 0

and for some arc

γ
\gamma

on the unit circle whose measure

|
γ
|

⩾
2
ρ
>
0

\left | \gamma \right | \geqslant 2\rho > 0

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit inf Underscript j right-arrow normal infinity Endscripts StartAbsoluteValue f left-parenthesis z Subscript j Baseline right-parenthesis EndAbsoluteValue greater-than-or-slanted-equals 1 where z Subscript j Baseline right-arrow z element-of gamma and z Subscript j Baseline element-of upper U period">

lim

inf

j
→
∞

|

f
(

z
j

)

|

⩾
1

where

z
j

→
z
∈
γ

and

z
j

∈
U
.

\lim \inf \limits _{j \to \infty } \left | {f({z_j})} \right | \geqslant 1\quad {\text {where}}\;{z_j} \to z \in \gamma \;{\text {and}}\;{z_j} \in U.

\]
Let

H
H

be a Hilbert space over the complex field,

A
A

an operator whose spectrum is included in

U
U

,

|

|

A

|

|

|| A ||

the operator norm of

A
A

, and

f
(
A
)

f(A)

the usual Riesz-Dunford operator. We prove that there is no function with the property

K
(
ρ
)

K(\rho )

satisfying
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 minus StartAbsoluteValue EndAbsoluteValue upper A StartAbsoluteValue EndAbsoluteValue right-parenthesis StartAbsoluteValue EndAbsoluteValue f prime left-parenthesis upper A right-parenthesis StartAbsoluteValue EndAbsoluteValue less-than-or-slanted-equals 1 slash n for all StartAbsoluteValue EndAbsoluteValue upper A StartAbsoluteValue EndAbsoluteValue greater-than 1 comma">

(
1
−

|

|

A

|

|

)

|

|

f
′

(
A
)

|

|

⩽
1

/

n

for

all

|

|

A

|

|

>
1
,

(1 - || A ||)|| {f’(A)} || \leqslant 1/n\quad {\text {for}}\;{\text {all}}\;|| A || > 1,

\]
where

n
>
N
(
ρ
)
=
log
⁡
(
1

/

(
1
−
cos
⁡
ρ
)
)

n > N(\rho ) = \log (1/(1 - \cos \rho ))

. We also show that if

f
f

has the property

K
(
ρ
)

K(\rho )

then the operator range of

f
(
A
)

f(A)

covers a ball of radius

k
(
ρ
)
=

3

/

(
4
N
(
ρ
)
)

k(\rho ) = \sqrt 3 /(4N(\rho ))

. These two results generalize our previous solutions of two long open problems of Doob [1]. Finally, we prove that the operator range of any

4
4

-fold univalent function is not convex. This extends our solution to Ky Fan’s Problem [4].
Closable operators and semigroups
Authors: Falkner Neil
Published: June 2010

We show that a linear operator is closable iff it is possible to weaken the topology on its range in a certain nice way so as to render the operator continuous. We apply this result to show that if

E
E

is a sequentially complete locally convex Hausdorff space and

(
L
(
t
)

)

0
⩽
t
>
∞

{(L(t))_{0 \leqslant t > \infty }}

is a locally equicontinuous semigroup of class

(

C
0

)

({C_0})

in

E
E

with generator

S
S

and if

x
∈
E

x \in E

(not necessarily belonging to the domain of

S
S

) then the function

u
(
t
)
=
L
(
t
)
x

u(t) = L(t)x

is a solution, in a generalized sense, of the initial value problem

u
′

(
t
)
=
S
u
(
t
)

u’(t) = Su(t)

,

u
(
0
)
=
x

u(0) = x

, and that such a generalized solution is unique. It should be noted here that

u
(
t
)

u(t)

may fail to belong to the domain of

S
S

so we must assign a suitable meaning to the expression

S
u
(
t
)

Su(t)

.
Entropy for smooth abelian actions
Authors: Fried David
Published: June 2010

We generalize the usual notions of metric and topological entropy for flows to actions of compactly generated abelian Lie groups. Unlike previous generalizations, ours is nontrivial for smooth actions of

R

n

{{\mathbf {R}}^n}

,

n
⩾
2

n \geqslant 2

. We prove some elementary properties of our definitions and we relate them to characteristic exponents and the entropy conjecture.
A note on neighbourhoods of univalent functions
Authors: Fournier Richard
Published: June 2010

Using a notion of neighbourhood of analytic functions due to Stephan Ruscheweyh we examine conditions under which neighbourhoods of a certain class of convex functions are included in a class of starlike functions.
Green’s functions coupled to self-dual Maxwell fields
Authors: Ginsberg Matthew L.
Published: June 2010

A twistor construction is given which gives an explicit form for the spin

n

/

2

n/2

propagators for massless fields coupled to a self-dual Maxwell field.
Hereditary homotopy equivalences
Authors: Sieradski Allan J.
Published: June 2010

This paper introduces the notion of hereditary homotopy equivalence which provides a homotopy-theoretic reformulation of the existence of a Cohen-Lyndon basis for a group presentation.