bluuebunny/crossref_metadata_2025_split huggingface dataset:
Findings From the Analysis of the American College of Nurse‐Midwives' Membership Surveys: 2006–2008
Authors: Schuiling Kerri Durnell , Sipe Theresa Ann , Fullerton Judith 
Published: June 2010

Introduction: The American College of Nurse‐Midwives (ACNM) Core Data Survey is an annual membership survey that collects demographic and selected workforce data about certified nurse‐midwives (CNMs), certified midwives (CMs), and students enrolled in midwifery education programs accredited by the Accreditation Commission for Midwifery Education, who are members of the organization. This article presents findings from the analysis of membership data for the years 2006 to 2008.Methods: An e‐mail invitation to participate in the online survey was sent to all ACNM members who provided ACNM with an e‐mail address. A paper copy of the survey was available upon request. The survey instrument for the years 2006 to 2008 focused on five categories: demographics, certification, education, employment, and licensure.Results: ACNM member respondents continue to remain predominantly white and female. The average age of CNMs/CMs for 2008 is 51 years, and the majority holds a master's degree as their highest degree.Discussion: Very few advances have been made in the effort to increase the diversity of ACNM membership. The number of CNMs earning doctoral degrees (including the doctor of nursing practice degree) is increasing. A majority of CNMs/CMs continue to identify a broad domain of clinical midwifery practice as their primary responsibility in their primary employment, and hospitals and physician practices remain the largest employers of midwives. Salaries for midwifery‐related work are rising, but it is unclear if midwives are earning more because salaries are higher or because the higher salaries reflect market wage adjustments that occur over time.
An analytic model of plasma-neutral coupling in the heliosphere plasma
Authors: SHAIKH DASTGEER , DASGUPTA B. 
Published: June 2010

AbstractWe have developed an analytic model to describe coupling of plasma and neutral fluids in the partially ionized heliosphere plasma medium. The sources employed in our analytic model are based on a κ-distribution as opposed to the Maxwellian distribution function. Our model uses the κ-distribution to analytically model the energetic neutral atoms that result in the heliosphere partially ionized plasma from charge exchange with the protons and subsequently produce a long tail, which is otherwise not describable by the Maxwellian distribution. We present our analytic formulation and describe major differences in the sources emerging from these two distinct distributions.
Does an EMILY's List Endorsement Predict Electoral Success, or Does EMILY Pick the Winners?
Authors: Hannagan Rebecca J. , Pimlott Jamie P. , Littvay Levente 
Published: June 2010

AbstractWomen's political action committees (PACs)—those committees founded by women to raise money for women candidates—have been and will likely continue to be an important part of American electoral politics. In this article, we investigate the impact of EMILY's List, because it is the standard bearer of women's PACs and is commonly cited as crucial to women's electoral success. Empirical studies of EMILY's List impact to date have largely assumed causal inference by using traditional linear models. We use a propensity score–matching model to leverage on causality and find that an EMILY endorsement helps some candidates and hurts others. Our findings set the stage for further and more nuanced investigations of when, where, and how EMILY's List can enhance the likelihood of electoral success for women.
Assessing Student Learning Outcomes and Documenting Success through a Capstone Course
Authors: Sum Paul E. , Light Steven Andrew 
Published: June 2010

AbstractColleges and universities are increasingly intentional about meeting well-articulated and consistent general education goals and documenting substantive learning outcomes. Institutional imperatives to document the successful teaching of essential knowledge and skill sets frequently fall to faculty and departments, posing new challenges in an environment of time and resource constraints. A capstone course is an increasingly common method to measure student learning and assess programmatic and institutional success. We provide concrete suggestions to design a capstone course and assess student learning outcomes. After describing the structure of the course and four innovative assignments, we present the results of assessment conducted through the capstone. We further the conversation on the development of best practices and how political science departments can align institutional and programmatic goals and lead the way in university assessment.
A study to evaluate the provision of psychosocial supervision within an Early Intervention team
Authors: Neil Sandra T. , Nothard Sarah , Glentworth David , Stewart Elaine 
Published: June 2010

AbstractPsychosocial Interventions (PSIs) and PSI supervision underpin the delivery of early interventions for people experiencing psychosis. Early Intervention (EI) teams are relatively new in the NHS and there is currently a lack of empirical research into PSI supervision in this area. This study aimed to elicit staff views of PSI supervision and to identify any unmet supervision needs within a newly developed EI team in the UK. Semi-structured interviews were conducted with 16 multidisciplinary team members. Descriptive statistics and a thematic analysis were used to analyse the responses. The different types of supervision available to team members, gaps in the provision of PSI supervision and aspects that supervisees found helpful and unhelpful about PSI supervision are discussed as are ideas for improving the provision of PSI supervision in EI teams. The limitations of the study and ideas for further research are also outlined.
A real analogue of the Gel′fand-Neumark theorem
Authors: Ono Tamio 
Published: June 2010

Let

A
A

be a real Banach

∗

^{\ast }

-algebra enjoying the following three conditions:

|

|

x

∗

x

|

|

=

|

|

x

∗

|

|

|

|

x

|

|

,

S
p

x

∗

x
≧
0

||{x^{\ast }}x|| = ||{x^{\ast }}||\;||x||,\;Sp{x^{\ast }}x \geqq 0

, and

|

|

x

∗

|

|

=

|

|

x

|

|

(
x
∈
A
)

||{x^{\ast }}|| = ||x||\;(x \in A)

. It is shown, after Ingelstam, Palmer, and Behncke, as a real analogue of the Gelfand-Neumark theorem, that

A
A

is isometrically

∗

^{\ast }

-isomorphic onto a real

C

∗

{C^{\ast }}

-algebra acting on a suitable real (or complex) Hilbert space. The converse is obvious.
Operators from Banach spaces to complex interpolation spaces
Authors: Williams Vernon 
Published: June 2010

Given a closed linear operator

A
A

with dense domain in a Banach space

X
X

, M. Schechter [4] utilized the Lebesgue integral to construct a family of bounded linear operators from

X
X

to the Calderón complex interpolation space

(
X
,
D
(
A
)

)
8

{(X,D(A))_8}

[2], where

D
(
A
)

D(A)

, the domain of

A
A

in

X
X

, is a Banach space under the norm
\[

|

|

x

|

|

D
(
A
)

=

|

|

x

|

|

+

|

|

A
x

|

|

.

||x|{|_{D(A)}} = ||x|| + ||Ax||.

\]
In this paper we utilize the complex functional calculus, which provides a more natural setting, to construct a similar family of operators. At the same time we achieve a strengthening of the Schechter result, for in the proof of our theorem we make no use of the adjoint

A
∗

{A^ \ast }

of

A
A

and consequently do not require the domain of

A
A

to be dense in

X
X

. A completely analogous procedure would permit the removal from the Schechter theorem, referred to above, of the hypothesis that the domain of

A
A

is dense in

X
X

.
On the saturation class for spline functions
Authors: Richards Franklin 
Published: June 2010

Let

S

k

(

Δ
n

)

{\mathcal {S}_k}({\Delta _n})

be the space of piecewise polynomials of degree at most k on [0, 1] possessing jumps at

1

/

n
,
2

/

n
,
⋯
,
n
−
1

/

n

1/n,2/n, \cdots ,n - 1/n

. Recently Gaier has shown that under the supremum norm

‖

f
−

S

k

(

Δ
n

)

‖

+

‖

f
−

S

k

(

Δ

n
+
1

)

‖

≧
c

n

−
k
−
1

\left \| {f - {\mathcal {S}_k}({\Delta _n})} \right \| + \left \| {f - {\mathcal {S}_k}({\Delta _{n + 1}})} \right \| \geqq c{n^{ - k - 1}}

unless f is a polynomial of degree at most k. Here we show if

0
>
α
≦
1

0 > \alpha \leqq 1

, then

‖

f
−

S

k

(

Δ
n

)

‖

=
O
(

n

−
k
−
α

)

\left \| {f - {\mathcal {S}_k}({\Delta _n})} \right \| = O({n^{ - k - \alpha }})

if and oniy if

f
∈

C
k

[
0
,
1
]

f \in {C^k}[0,1]

and

f

(
k
)

{f^{(k)}}

satisfies a Lipschitz condition of order

α
\alpha

. In addition, a result similar to Gaier’s is given.
Extensions of left uniformly continuous functions on a topological semigroup
Authors: Wiley Samuel J. 
Published: June 2010

For any topological semigroup S with separately continuous operation, let

C
(
S
)

C(S)

denote the set of all bounded continuous real valued functions on S with the supremum norm and let

LUC

(
S
)

{\text {LUC}}(S)

denote the set of all f in

C
(
S
)

C(S)

such that whenever

{
s
(
γ
)
}

\{ s(\gamma )\}

is a net in S which converges to some s in S, then

sup
{

|

f
(
s
(
γ
)
t
)
−
f
(
s
t
)

|

:
t
∈
S
}

\sup \{ |f(s(\gamma )t) - f(st)|:t \in S\}

converges to 0. In this paper we prove that if S is an abelian subsemigroup of a compact topological group and

f
∈

LUC

(
S
)

f \in {\text {LUC}}(S)

, then there is an

F
∈

LUC

(
G
)

F \in {\text {LUC}}(G)

where

F
(
s
)
=
f
(
s
)

F(s) = f(s)

for all

s
∈
S

s \in S

. We also show whenever there is an extension of the type indicated above, there is a norm preserving extension.
A Carleson measure theorem for Bergman spaces
Authors: Hastings William W. 
Published: June 2010

Let

μ
\mu

be a finite, positive measure on

U
n

{U^n}

, the unit polydisc in

C

n

{{\mathbf {C}}^n}

, and let

σ
n

{\sigma _n}

be

2
n

2n

-dimensional Lebesgue volume measure on

U
n

{U^n}

. For

1
⩽
p
⩽
q
>
∞

1 \leqslant p \leqslant q > \infty

a necessary and sufficient condition on

μ
\mu

is given in order that

{
∫

U
n

f
q

(
z
)
d
μ
(
z
)

}

1

/

q

⩽

C
{
∫

U
n

f
p

(
z
)
d

σ
n

(
z
)

}

1

/

p

\{ \int {_{{U^n}}{f^q}(z)d\mu (z){\} ^{1/q}} \leqslant } C\{ \int {_{{U^n}}{f^p}(z)d{\sigma _n}(z){\} ^{1/p}}}

for every positive

n
n

-subharmonic function

f
f

on

U
n

{U^n}

.
An elementary method for estimating error terms in additive number theory
Authors: Hayashi Elmer K. 
Published: June 2010

Let

R
k

(
n
)

{R_k}(n)

denote the number of ways of representing the integers not exceeding

n
n

as the sum of

k
k

members of a given sequence of nonnegative integers. Using only elementary methods, we prove a general theorem from which we deduce that, for every

ϵ
>
0

\epsilon > 0

,
\[

R
k

(
n
)
−
c

n
β

≠
o
(

n

β
(
1
−
β
)
(
1
−
1

/

k
)

/

(
1
−
β
+
β

/

k
)
−
ϵ

)

{R_k}(n) - c{n^\beta } \ne o({n^{\beta (1 - \beta )(1 - 1/k)/(1 - \beta + \beta /k) - \epsilon }})

\]
where

c
c

is a positive constant and

0
>
β
>
1

0 > \beta > 1

.
The derivative of a bounded holomorphic function in the disk
Authors: Yamashita Shinji 
Published: June 2010

Let a nonconstant function

f
f

be holomorphic and bounded,

|

f

|

>
1

|f| > 1

in

D
:

|

z

|

>
1

D:|z| > 1

. We shall estimate

f

∗

(
z
)
=
(
1
−

|

z

|

2

)

|

f
′

(
z
)

|

/

(
1
−

|

f
(
z
)

|

2

)

{f^{\ast }}(z) = (1 - |z{|^2})|f’(z)|/(1 - |f(z){|^2})

at each point

z
ϵ
D

z\epsilon D

((1) in Theorem 1). The function

d
d

appearing in the estimate concerns the sizes of the schlicht disks on the Riemannian image

F

\mathcal {F}

of

D
D

by

f
f

. Boundary properties of

f
f

and

f

∗

{f^{\ast }}

will be stated in Theorems 2 and 3; use is made of the cluster sets of

d
d

.
A lattice theoretic characterization of an integral operator
Authors: Lessner Lawrence 
Published: June 2010

We are concerned here with obtaining necessary and sufficient conditions for a linear operator,

K
:

L

(

X

1

,

A

1

,

μ
1

)
→
M
(

X

2

,

A

2

,

μ
2

)

K:\mathcal {L}({{\text {X}}_1},\;{\mathcal {A}_1},\;{\mu _1}) \to M({{\text {X}}_2},\;{\mathcal {A}_2},\;{\mu _2})

, to be represented by an integral,

K
(
f
)
=
∫
k
(
x
,

y
)
f
(
y
)

d
y

K(f) = \smallint k(x,\;y)f(y)\;dy

, with an

A

2

×

A

1

{\mathcal {A}_2} \times {\mathcal {A}_1}

measurable kernel

k
(
x
,

y
)

k(x,\;y)

. That such conditions are developed in a lattice theoretic context will be shown to be quite natural. Our direction will be to characterize an integral operator by its action pointwise: i.e.,

K
(
)
(
x
)

K()(x)

is a linear functional on a subspace of the essentially bounded functions. Such a development leads one to define the kernel,

k
(
x
,

y
)

k(x,\;y)

, in a pointwise fashion also, and as a result we are confronted with the question of the

A

2

×

A

1

{\mathcal {A}_2} \times {\mathcal {A}_1}

measurability of

k
(
x
,

y
)

k(x,\;y)

.
Covariances of generalized processes with orthogonal values
Authors: Pakula Lewis 
Published: June 2010

A general form for the covariance of a generalized process with orthogonal values is found in the case where the covariance

B
B

depends on test functions and their first derivatives. Specifically, if

B
(
ϕ
,
ϕ
)
=
∫

ϕ
2

d

μ
0

+
∫

ϕ

ϕ
′

d

μ
1

+
∫

ϕ
2

d

μ
2

⩾
0

for

ϕ
ϵ

D

(

R

)

B(\phi ,\phi ) = \int {{\phi ^2}d{\mu _0} + \int {\phi \phi ’d{\mu _1} + \int {{\phi ^2}d{\mu _2} \geqslant 0} } } \;{\text {for }}\phi \epsilon \mathcal {D}({\mathbf {R}})

and Radon measures

μ
0

,

μ
1

,

μ
2

{\mu _0},{\mu _1},{\mu _2}

, then there exist Radon measures

ν
0

,

ν
1

,

ν
2

{\nu _0},{\nu _1},{\nu _2}

such that

B
(
ϕ
,
ϕ
)
=
∫

ϕ
2

d

ν
0

+
∫

ϕ

ϕ
′

d

ν
1

+
∫

ϕ

′
2

d

ν
2

B(\phi ,\phi ) = \int {{\phi ^2}d{\nu _0} + \int {\phi \phi ’d{\nu _1} + \int {\phi {’^2}d{\nu _2}} } }

and, moreover,

∫

f
2

d

ν
0

+
∫

f
g
d

ν
1

+
∫

g
2

d

ν
2

⩾
0

\int {{f^2}d{\nu _0} + \int {fgd{\nu _1} + \int {{g^2}d{\nu _2} \geqslant 0} } }

for all

f
,
g
ϵ

D

(

R

)

f,g\epsilon \mathcal {D}({\mathbf {R}})

.
On decomposability of compact perturbations of operators
Authors: Radjabalipour M. 
Published: June 2010

Let

A
A

be a Hilbert-space operator satisfying the growth condition

|

|

(
z
−
A

)

−
1

|

|

⩽
exp
⁡
{
K

[
dist
⁡
(
z
,

J
)

]

−
S

}
,

z
∉
J

||{(z - A)^{ - 1}}|| \leqslant \exp \{ K{[\operatorname {dist} (z,\;J)]^{ - S}}\} ,\;z \notin J

, where

J
J

is a

C
2

{C^2}

Jordan curve, and

K
>
0
,

s
ϵ
(
0
,

1
)

K > 0,\;s\epsilon (0,\;1)

are two constants. Let

T
=
A
+
B

T = A + B

for some

B
ϵ

C
p

,

1
⩽
p
>
∞

B\epsilon {C_p},\;1 \leqslant p > \infty

. It is shown that

T
T

is strongly decomposable if and only if

σ
(
T
)

\sigma (T)

does not fill the “interior” of

J
J

.
Operator radii of commuting products
Authors: Okubo K. , Ando T. 
Published: June 2010

Operator radii

w
ρ

(
T
)

{w_\rho }(T)

for a bounded linear operator

T
T

on a Hilbert space were introduced in connection with unitary

ρ
\rho

-dilations. We shall be concerned with universal estimates for the ratios
\[

w
ρ

(
S
T
)

/

(

w
σ

(
S
)
⋅

w
ρ

(
T
)
)

{w_\rho }(ST)/({w_\sigma }(S) \cdot {w_\rho }(T))

\]
for commuting operators

S
,
T

S,T

and

σ
,
ρ
>
0

\sigma ,\rho > 0

.
The final value problem for Sobolev equations
Authors: Lagnese John 
Published: June 2010

Let

A
A

and

B
B

be

m
m

-accretive linear operators in a complex Hilbert space

H
H

with

D
(
A
)
⊂
D
(
B
)

D(A) \subset D(B)

. The method of quasi-reversibility is used to obtain a solution to the Sobolev equation

(
d

/

d
t
)
[
(
I
+
B
)
u
(
t
)
]
+
A
u
(
t
)
=
0
,
0
>
t
>
1

(d/dt)[(I + B)u(t)] + Au(t) = 0,0 > t > 1

, which approximates a specified final value

u
(
1
)
=
f

u(1) = f

. In general, when

D
(
A
)
⊂
D
(
B
)

D(A) \subset D(B)

, it is not possible to find a solution which achieves exactly the final value

u
(
1
)
=
f

u(1) = f

.
Polynomial density in Bers spaces
Authors: Burbea Jacob 
Published: June 2010

Let D be a bounded Jordan domain such that

∫

∫

D

λ
D

2
−
q

d
x

d
y

>
∞

\smallint \;\smallint {\;_D}\lambda _D^{2 - q}\;dx\;dy\; > \infty

for

q
>
1

q > 1

. Here

λ
D

(
z
)

{\lambda _D}(z)

is the Poincaré metric for D. Define

A
q
p

(
D
)

A_q^p(D)

, the Bers space, to be the Fréchet space of holomorphic functions f on D, such that

‖
f
‖

q
,
p

p

=
∫

∫

D

λ
D

2
−
q
p

|

f

|

p

d
x

d
y

\left \| f \right \|_{q,p}^p = \smallint \;\smallint {\;_D}\lambda _D^{2 - qp}|f{|^p}\;dx\;dy

is finite,

0
>
p
>
∞
,
q
p
>
1

0 > p > \infty ,qp > 1

. It is well known that the polynomials are dense in

A
q
p

(
D
)

A_q^p(D)

for

q
p
⩾
2

qp \geqslant 2

. We show that they are dense in

A
q
p

(
D
)

A_q^p(D)

for

q
p
>
1

qp > 1

irrespective whether the boundary of D is rectifiable or not.
Regularity of solutions to an abstract inhomogeneous linear differential equation
Authors: Webb G. F. 
Published: June 2010

Let

T
(
t
)
,
t
⩾
0

T(t),t \geqslant 0

, be a strongly continuous semigroup of linear operators on a Banach space X with infinitesimal generator A satisfying

T
(
t
)
X
⊂
D
(
A
)

T(t)X \subset D(A)

for all

t
>
0

t > 0

. Let f be a function from

[
0
,
∞
)

[0,\infty )

to X of strong bounded variation. It is proved that

u
(
t
)

=

def

T
(
t
)
x
+

∫

t
0

T
(
t
−
s
)
f
(
s
)
d
s
,
x
∈
X

u(t){ = ^{{\text {def}}}}T(t)x + {\smallint ^{t0}}T(t - s)f(s)ds,x \in X

, is strongly differentiable and satisfies

d
u
(
t
)

/

d
t
=
A
u
(
t
)
+
f
(
t
)

du(t)/dt = Au(t) + f(t)

for all but a countable number of

t
>
0

t > 0

.
On the Radon-Nikodým theorem and locally convex spaces with the Radon-Nikodým property
Authors: Chi G. Y. H. 
Published: June 2010

Let F be a quasi-complete locally convex space,

(
Ω
,
Σ
,
μ
)

(\Omega ,\Sigma ,\mu )

a complete probability space, and

L
1

(
μ
;
F
)

{L^1}(\mu ;F)

the space of all strongly integrable functions

f
:
Ω
→
F

f:\Omega \to F

with the Egoroff property. If F is a Banach space, then the Radon-Nikodým theorem was proved by Rieffel. This result extends to Fréchet spaces. If F is dual nuclear, then the Lebesgue-Nikodým theorem for the strong integral has been established. However, for nonmetrizable, or nondual nuclear spaces, the Radon-Nikodým theorem is not available in general. It is shown in this article that the Radon-Nikodým theorem for the strong integral can be established for quasi-complete locally convex spaces F having the following property: (CM) For every bounded subset

B
⊂

l
N
1

{
F
}

B \subset l_N^1\{ F\}

, the space of absolutely summable sequences, there exists an absolutely convex compact metrizable subset

M
⊂
F

M \subset F

such that

Σ

i
=
1

∞

p
M

(

x
i

)
>
1
,
∀
(

x
i

)
∈
B

\Sigma _{i = 1}^\infty {p_M}({x_i}) > 1,\forall ({x_i}) \in B

. In fact, these spaces have the Radon-Nikodým property, and they include the Montel (DF)-spaces, the strong duals of metrizable Montel spaces, the strong duals of metrizable Schwartz spaces, and the precompact duals of separable metrizable spaces. When F is dual nuclear, the Radon-Nikodým theorem reduces to the Lebesgue-Nikodým theorem. An application to probability theory is considered.
A problem on Noetherian local rings of characteristic 𝑝
Authors: Goto Shiro 
Published: June 2010

Let (A, m, k) be a one-dimensional Noetherian local ring of characteristic p (

p
>
0

p > 0

, a prime number) and assume that the Frobenius endomorphism F of A is finite. Further assume that the field k is algebraically closed and that it is contained in A. Let B denote A when it is regarded as an A-algebra by F. Then, if

Hom
A

⁡
(
B
,
A
)
≅
B

\operatorname {Hom}_A(B,A) \cong B

as B-modules, A is a Macaulay local ring and

r
(
A
)
≡

dim
k

⁡

Ext
A
1

⁡
(
k
,
A
)
⩽
max
{
♯

Ass

A
^

−
1
,
1
}

r(A) \equiv \dim _k\operatorname {Ext}_A^1(k,A) \leqslant \max \{ \sharp {\text {Ass}}\hat A - 1,1\}

where Â denotes the m-adic completion of A. Thus, in case

♯

Ass

A
^

⩽
2
,
A

\sharp {\text {Ass}}{\mkern 1mu} \hat A \leqslant 2,A

is a Gorenstein local ring if and only if

Hom
A

⁡
(
B
,
A
)
≅
B

\operatorname {Hom}_A(B,A) \cong B

as B-modules. If

♯

Ass

A
^

⩾
3

\sharp {\text {Ass}}\hat A \geqslant 3

this assertion is not true and the counterexamples are given.
Behavior of maximally defined solutions of a nonlinear Volterra equation
Authors: Herdman Terry L. 
Published: June 2010

This paper is concerned with the behavior of solutions of an n-dimensional nonlinear Volterra integral equation
\[

x
(
t
)
=
f
(
t
)
+

∫
0
t

g
(
t
,
s
,
x
(
s
)
)
d
s
,

t
⩾
0.

x(t) = f(t) + \int _0^t {g(t,s,x(s))ds,\quad t \geqslant 0.}

\]
In particular, sufficient conditions for a solution

x
(
t
)

x(t)

on its maximal interval of existence

[
0
,
T
)

[0,T)

to possess the property that

|

x
(
t
)

|

→
+
∞

|x(t)| \to + \infty

as

t
→

T
−

t \to {T^ - }

are obtained. Thus these additional conditions give a positive answer to the problem posed by Miller [3, p. 145]. One can construct examples, satisfying the hypotheses given in [3], which provide a negative answer to this problem, see Artstein [1, Appendix A].
Some abstract Cauchy problems in exceptional cases
Authors: Bragg Louis R. 
Published: June 2010

Let X be a Banach space and let

A
=

B
2

A = {B^2}

in which B is the infinitesimal generator of a strongly continuous group in X with dense domain

D

(
A
)

\mathcal {D}(A)

. This paper develops solutions of the abstract Euler-Poisson-Darboux problem
\[

u

t
t

(
t
)
+

1
−
2
m

t

u
t

(
t
)
=
A
u
(
t
)
,

a
m
p
;

t
>
0
,

m
=
1
,
2
,
3
,
…
,

‖

u
(
t
)
−
ϕ

‖

→
0

as

t
→
0
,

a
m
p
;

ϕ
∈

D

(

A
r

)
,

r

>
m
,

\begin {array}{*{20}{c}} {{u_{tt}}(t) + \frac {{1 - 2m}}{t}{u_t}(t) = Au(t),} & {t > 0,\quad m = 1,2,3, \ldots ,} \hfill \\ {\left \| {u(t) - \phi } \right \|\to 0\;{\text {as}}\;t \to 0,} & {\phi \in \mathcal {D}({A^r}),\quad r\; > m,} \hfill \\ \end {array}

\]
and associated Cauchy problem in terms of solutions of related abstract wave problems. Connections between solutions of certain abstract hypergeometric equations play an important role in these developments. J. B. Diaz and H. Weinberger and E. K. Blum have obtained solutions of the standard Euler-Poisson-Darboux problem (i.e.

A
=

Δ
n

A = {\Delta _n}

, the Laplacian) in the exceptional cases.
An abstract semilinear Volterra integrodifferential equation
Authors: Webb G. F. 
Published: June 2010

The abstract semilinear Volterra integrodifferential equation
\[

u
′

(
t
)
=
A
u
(
t
)
+

∫
0
t

g
(
t
−
s
,
u
(
s
)
)
d
s
+
f
(
t
)
,

t
⩾
0
,
u
(
0
)
=
x
∈
X
,

u’(t) = Au(t) + \int _0^t {g(t - s,u(s))ds + f(t),\quad t \geqslant 0,u(0) = x \in X,}

\]
is investigated, where A is the infinitesimal generator of a semigroup of linear operators in a Banach space X and g is nonlinear and unbounded in its second place. Some results are proved concerning local existence, global existence, continuous dependence upon initial values, and asymptotic stability. The method used treats the equation in the domain of A with the graph norm employing results from linear semigroup theory concerning abstract inhomogeneous linear differential equations.
Uniform asymptotic stability in functional differential equations
Authors: Burton T. A. 
Published: June 2010

The classical uniform asymptotic stability result for a system of functional differential equations

(

1

)

x
′

=
F
(
t
,

x
t

)

\begin{equation}\tag {$1$} x’ = F(t,{x_t})\end{equation}

calls for a Liapunov functional

V
(
t
,
ϕ
)

V(t,\phi )

satisfying

W
(

|

ϕ
(
0
)

|

)
⩽
V
(
t
,
ϕ
)
⩽

W
1

(

|

ϕ
(
0
)

|

)
+

W
2

(

|

|

|

ϕ

|

|

|

)
,

V

(
1
)

′

⩽
−

W
3

(

|

ϕ
(
0
)

|

)

W(|\phi (0)|) \leqslant V(t,\phi ) \leqslant {W_1}(|\phi (0)|) + {W_2}(|||\phi |||),{V’_{(1)}} \leqslant - {W_3}(|\phi (0)|)

, and

|

f
(
t
,

x
t

)

|

|f(t,{x_t})|

bounded for

|

|

|

x
t

|

|

|

|||{x_t}|||

bounded. We show that it is not necessary to require

|

f
(
t
,

x
t

)

|

|f(t,{x_t})|

bounded. Here,

|

|

|

⋅

|

|

|

||| \cdot |||

is the

L
2

{L^2}

-norm.