bluuebunny/crossref_metadata_2025_split huggingface dataset:
Rediscovery of the Honduran Emerald Amazilia luciae in western Honduras: insights on the distribution, ecology, and conservation of a ’Critically Endangered’ hummingbird
Authors: ANDERSON DAVID L. , HOUSE PAUL , HYMAN ROBERT E. , STEINER RICARDO , HAWKINS H. ROSS , THORN SHERRY , REY MANUEL J. , ESPINAL MARIO R. , MARINEROS LEONEL E. 
Published: July 2010

SummaryThe Honduran Emerald Amazilia luciae is endemic to dry forests of Honduras and currently recognised as ‘Critically Endangered.’ Here we present the first modern assessment of its distribution, ecology, and conservation, based partly on our rediscovery of the species in western Honduras and on our observations in three Honduran departments. We found that dry forests inhabited by the emerald differed in structure and species composition between eastern and western Honduras, where we observed emeralds in open-canopied deciduous thorn forests and closed-canopied semi-deciduous woodlands, respectively. We interpret these differences in light of the geological and anthropogenic origins of dry forests in Honduras, and discuss the implications of such origins for the conservation of dry forests. Although our findings expand the known distribution and population size of the species, its status as ‘Critically Endangered’ is warranted due to its restricted distribution in dry forest fragments and increasing human pressures on this habitat.
The Influence of Illness Duration on Syndrome Clarity and Stability in Functional Psychosis: Does the Diagnosis Emerge and Stabilise with Time?
Authors: McGorry Patrick D. 
Published: July 2010

This review aims to examine the relationship between phase and duration of illness and the clarity and stability of syndromes in psychotic illness. Originating from clinical experience with first episode psychosis, where Neo-Kraepelinian templates and the underlying model have felt especially awkward and ill-fitting, a hypothesis that the clinical picture of functional psychosis might become clearer over time and stabilise in individual patients is proposed which involves the notion of differentiation. A broad range of psychopathological studies and historical papers which refer directly or indirectly to the evolution of the clinical picture in psychosis over time are reviewed. The findings are difficult to interpret, since few studies have focused directly on the issue; however there is evidence of evolution over time of the clinical picture, particularly within the schizophrenia spectrum, and instability beyond this. Further research is needed to clarify the longitudinal patterns of psychopathology in psychotic illness. The capacity for evolution and instability and consequent scope for reclassification of patients has important implications for research and psychoeducation in early psychosis.
Functions of finite 𝜆-type in several complex variables
Authors: Kujala Robert O. 
Published: July 2010

If

λ
:

R

+

→

R

+

\lambda :{{\bf {R}}^ + } \to {{\bf {R}}^ + }

is continuous and increasing then a meromorphic function f on

C
k

{C^k}

is said to be of finite

λ
\lambda

-type if there are positive constants s, A, B and R such that

T
f

(
r
,
s
)
≦
A
λ
(
B
r
)

{T_f}(r,s) \leqq A\lambda (Br)

for all

r
>
R

r > R

where

T
f

(
r
,
s
)

{T_f}(r,s)

is the characteristic of f. It is shown that if

λ
(
B
r
)

/

λ
(
r
)

\lambda (Br)/\lambda (r)

is bounded for r sufficiently large and

B
>
1

B > 1

, then every meromorphic function of finite

λ
\lambda

-type is the quotient of two entire functions of finite

λ
\lambda

-type. This theorem is the result of a careful and detailed analysis of the relation between the growth of a function and the growth of its divisors. The central fact developed in this connection is: A nonnegative divisor

ν
\nu

on

C
k

{C^k}

with

ν
(

0

)
=
0

\nu ({\bf {0}}) = 0

is the divisor of an entire function of finite

λ
\lambda

-type if and only if there are positive constants A, B and R such that
\[

N

ν

|

ξ

(
r
)
≦
A
λ
(
B
r
)
,

|

1
p

∑

s
>

|

z

|

≦
r

ν

|

ξ
(
z
)

z

−
p

|

≦
A
λ
(
B
r
)

r

−
p

+
A
λ
(
B
s
)

s

−
p

,

\begin {array}{*{20}{c}} {{N_{\nu |\xi }}(r) \leqq A\lambda (Br),} \\ {\left | {\frac {1}{p}\sum \limits _{s > |z| \leqq r} {\nu |\xi (z){z^{ - p}}} } \right | \leqq A\lambda (Br){r^{ - p}} + A\lambda (Bs){s^{ - p}},} \\ \end {array}

\]
for all

r
≧
s
>
R

r \geqq s > R

, all unit vectors

ξ
\xi

in

C
k

{C^k}

, and all natural numbers p. Here

ν

|

ξ

\nu |\xi

represents the lifting of the divisor

ν
\nu

to the plane via the map

z
↦
z
ξ

z \mapsto z\xi

and

N

ν

|

ξ

{N_{\nu |\xi }}

is the valence function of that divisor. Analogous facts for functions of zero

λ
\lambda

-type are also presented.
Representation of holomorphic functions by boundary integrals
Authors: Baernstein Albert 
Published: July 2010

Let

K
K

be a compact locally connected set in the plane and let

f
f

be a function holomorphic in the extended complement of

K
K

with

f
(
∞
)
=
0

f(\infty ) = 0

. We prove that there exists a sequence of measures

{

μ
n

}

\{ {\mu _n}\}

on

K
K

satisfying

lim

n
→
∞

|

|

μ
n

|

|

1

/

n

=
0

{\lim _{n \to \infty }}||{\mu _n}|{|^{1/n}} = 0

such that

f
(
z
)
=

∑

n
=
0

∞

∫
K

(
w
−
z
)

−
n
−
1

d

μ
n

(
w
)
(
z
∈
K
)

f(z) = \sum \nolimits _{n = 0}^\infty {\int _K {{{(w - z)}^{ - n - 1}}d{\mu _n}(w)(z \in K)} }

. It follows from the proof that two topologies for the space of functions holomorphic on

K
K

are the same. One of these is the inductive limit topology introduced by Köthe, and the other is defined by a family of seminorms which involve only the values of the functions and their derivatives on

K
K

. A key lemma is an open mapping theorem for certain locally convex spaces. The representation theorem and the identity of the two topologies is false when

K
K

is a compact subset of the unit circle which is not locally connected.
Regular functions 𝑓(𝑧) for which 𝑧𝑓’(𝑧) is 𝛼-spiral
Authors: Libera Richard J. , Ziegler Michael R. 
Published: July 2010

A function

f
(
z
)
=
z
+

Σ

n
=
2

∞

a
n

z
n

f(z) = z + \Sigma _{n = 2}^\infty {a_n}{z^n}

regular in the open unit disk

Δ
=
{
z
:

|

z

|

>
1
}

\Delta = \{ z:|z| > 1\}

is a (univalent)

α
\alpha

-spiral function for real

α
,

|

α

|

>
π

/

2

\alpha ,|\alpha | > \pi /2

, if

Re
⁡
{

e

i
α

z

f
′

(
z
)

/

f
(
z
)
}
>
0

\operatorname {Re} \{ {e^{i\alpha }}zf’(z)/f(z)\} > 0

for z in

Δ
\Delta

; in this case we write

f
(
z
)
∈

F

α

f(z) \in {\mathcal {F}_\alpha }

. A fundamental result of this paper shows that the transformation
\[

f
∗

(
z
)
=

a
z
f
(
(
z
+
a
)

/

(
1
+

a
¯

z
)
)

f
(
a
)
(
z
+
a
)

(
1
+

a
¯

z
)

e

−
2
i
α

{f_ \ast }(z) = \frac {{azf((z + a)/(1 + \bar az))}}{{f(a)(z + a){{(1 + \bar az)}^{{e^{ - 2i\alpha }}}}}}

\]
defines a function in

F

α

{\mathcal {F}_\alpha }

whenever

f
(
z
)

f(z)

is in

F

α

{\mathcal {F}_\alpha }

and a is in

Δ
\Delta

. If

g
(
z
)

g(z)

is regular in

Δ
,
g
(
0
)
=
0

\Delta ,g(0) = 0

and

g
′

(
0
)
=
1

g’(0) = 1

, then

g
(
z
)

g(z)

is in

G

α

{\mathcal {G}_\alpha }

if and only if

z

g
′

(
z
)

zg’(z)

is in

F

α

{\mathcal {F}_\alpha }

. The main result of the paper is the derivation of the sharp radius of close-to-convexity for each class

G

α

{\mathcal {G}_\alpha }

; it is given as the solution of an equation in r which is dependent only on

α
\alpha

. (Approximate solutions of this equation were made by computer and these suggest that the radius of close-to-convexity of the class

G

=

∪
α

G

α

\mathcal {G} = { \cup _\alpha }{\mathcal {G}_\alpha }

is approximately

.99097

+

.99097^{+}

.) Additional results are also obtained such as the radius of convexity of

G

α

{\mathcal {G}_\alpha }

, a range of

α
\alpha

for which

g
(
z
)

g(z)

in

G

α

{\mathcal {G}_\alpha }

is always univalent is given, etc. These conclusions all depend heavily on the transformation cited above and its analogue for

G

α

{\mathcal {G}_\alpha }

.
Locally uniform spaces
Authors: Williams James 
Published: July 2010

The axioms for a locally uniform space

(
X
,

U

)

(X,\mathcal {U})

may be obtained by localizing the last axiom for a uniform space to obtain

∀
x
∈
X
,
∀
U
∈

U

,
∃
V
∈

V

:
(
V
∘
V
)
[
x
]
⊆
U
[
x
]

\forall x \in X,\forall U \in \mathcal {U},\exists V \in \mathcal {V}:(V \circ V)[x] \subseteq U[x]

. With each locally uniform space one may associate a regular topology, just as one associates a completely regular topology with each uniform space. The topologies of locally uniform spaces with nested bases may be characterized using Boolean algebras of regular open sets. As a special case, one has that locally uniform spaces with countable bases have pseudo-metrizable topologies. Several types of Cauchy filters may be defined for locally uniform spaces, and a major portion of the paper is devoted to a study and comparison of their properties. For each given type of Cauchy filter, complete spaces are those in which every Cauchy filter converges; to complete a space is to embed it as a dense subspace in a complete space. In discussing these concepts, it is convenient to make the mild restriction of considering only those locally uniform spaces

(
X
,

V

)

(X,\mathcal {V})

in which each element of

V

\mathcal {V}

is a neighborhood of the diagonal in

X
×
X

X \times X

with respect to the relative topology; these spaces I have called NLU-spaces. With respect to the more general types of Cauchy filters, some NLU-spaces are not completable; this happens even though some completable NLU-spaces can still have topologies which are not completely regular. Examples illustrating these completeness situations and having various topological properties are obtained from a generalized construction. It is also shown that there is a largest class of Cauchy filters with respect to which each NLU-space has a completion that is also an NLU-space.
Successive remainders of the Newton series
Authors: Crofts G. W. , Shaw J. K. 
Published: July 2010

If

f
f

is analytic in the open unit disc

D
D

and

λ
\lambda

is a sequence of points in

D
D

converging to 0, then

f
f

admits the Newton series expansion

f
(
z
)
=
f
(

λ
1

)
+

∑

n
=
1

∞

Δ
λ
n

f
(

λ

n
+
1

)
(
z
−

λ
1

)
(
z
−

λ
2

)
⋯
(
z
−

λ
n

)

f(z) = f({\lambda _1}) + \sum \nolimits _{n = 1}^\infty {\Delta _\lambda ^nf({\lambda _{n + 1}})(z - {\lambda _1})(z - {\lambda _2}) \cdots (z - {\lambda _n})}

, where

Δ
λ
n

f
(
z
)

\Delta _\lambda ^nf(z)

is the

n
n

th divided difference of

f
f

with respect to the sequence

λ
\lambda

. The Newton series reduces to the Maclaurin series in case

λ
n

≡
0

{\lambda _n} \equiv 0

. The present paper investigates relationships between the behavior of zeros of the normalized remainders

Δ
λ
k

f
(
z
)
=

Δ
λ
k

f
(

λ

k
+
1

)
+

∑

n
=
k
+
1

∞

Δ
λ
n

f
(

λ

n
+
1

)
(
z
−

λ

k
+
1

)
⋯
(
z
−

λ
n

)

\Delta _\lambda ^kf(z) = \Delta _\lambda ^kf({\lambda _{k + 1}}) + \sum \nolimits _{n = k + 1}^\infty {\Delta _\lambda ^nf({\lambda _{n + 1}})(z - {\lambda _{k + 1}}) \cdots (z - {\lambda _n})}

of the Newton series and zeros of the normalized remainders

∑

n
=
k

∞

a
n

z

n
−
k

\sum \nolimits _{n = k}^\infty {{a_n}{z^{n - k}}}

of the Maclaurin series for

f
f

. Let

C
λ

{C_\lambda }

be the supremum of numbers

c
>
0

c > 0

such that if

f
f

is analytic in

D
D

and each of

Δ
λ
k

f
(
z
)
,

0
⩽
k
>
∞

\Delta _\lambda ^kf(z),\;0 \leqslant k > \infty

, has a zero in

|

z

|

⩽
c

|z| \leqslant c

, then

f
≡
0

f \equiv 0

. The corresponding constant for the Maclaurin series (

C
λ

{C_\lambda }

, where

λ
n

≡
0

{\lambda _n} \equiv 0

) is called the Whittaker constant for remainders and is denoted by

W
W

. We prove that

C
λ

⩾
W

{C_\lambda } \geqslant W

, for all

λ
\lambda

, and, moreover,

C
λ

=
W

{C_\lambda } = W

if

λ
∈

l
1

\lambda \in {l_1}

. In obtaining this result, we prove that functions

f
f

analytic in

D
D

have expansions of the form

f
(
z
)
=

∑

n
=
0

∞

Δ
λ
n

f
(

z
n

)

C
n

(
z
)

f(z) = \sum \nolimits _{n = 0}^\infty {\Delta _\lambda ^nf({z_n}){C_n}(z)}

, where

|

z
n

|

⩽
W

|{z_n}| \leqslant W

, for all

n
n

, and

C
n

(
z
)

{C_n}(z)

is a polynomial of degree

n
n

determined by the conditions

Δ
λ
j

C
k

(

z
j

)
=

δ

j
k

\Delta _\lambda ^j{C_k}({z_j}) = {\delta _{jk}}

.
Obstructions to embedding 𝑛-manifolds in (2𝑛-1)-manifolds
Authors: Maxwell J. W. 
Published: July 2010

Suppose

f
:
(

M
n

,
∂

M
n

)
→
(

Q

2
n
−
1

,
∂

Q

2
n
−
1

)

f:({M^n},\partial {M^n}) \to ({Q^{2n - 1}},\partial {Q^{2n - 1}})

is a proper PL map between PL manifolds

M
n

{M^n}

and

Q

2
n
−
1

{Q^{2n - 1}}

of dimension

n
n

and

2
n
−
1

2n - 1

respectively,

M
M

compact. J. F. P. Hudson has shown that associated with each such map

f
f

that is an embedding on

∂
M

\partial M

is an element

α
¯

(
f
)

\bar \alpha (f)

in

H
1

(
M
;

Z
2

)

{H_1}(M;{Z_2})

when

n
n

is odd and an element

β
¯

(
f
)

\bar \beta (f)

in

H
1

(
M
;
Z
)

{H_1}(M;Z)

when

n
n

is even. These elements are invariant under a homotopy relative to

∂
M

\partial M

. We show that, under slight additional assumptions on

M
,
Q

M,Q

and

f
,
f

f,f

is homotopic to an embedding if and only if

α
¯

(
f
)
=
0

\bar \alpha (f) = 0

for

n
n

odd and

β
¯

(
f
)
=
0

\bar \beta (f) = 0

for

n
n

even. This result is used to give a sufficient condition for extending an embedding

f
:
∂

M
n

→
∂

B

2
n
−
1

f:\partial {M^n} \to \partial {B^{2n - 1}}

(

B

2
n
−
1

{B^{2n - 1}}

denotes

(
2
n
−
1
)

(2n - 1)

-dimensional ball) to an embedding

F
:
(

M
n

,
∂

M
n

)
→
(

B

2
n
−
1

,
∂

B

2
n
−
1

)

F:({M^n},\partial {M^n}) \to ({B^{2n - 1}},\partial {B^{2n - 1}})

.
Inequalities associated with regular and singular problems in the calculus of variations
Authors: Bradley J. S. , Everitt W. N. 
Published: July 2010

An inequality of the form

∫
a
b

[
p

|

f
′

|

2

+
q

|

f

|

2

]
≥

μ
0

∫
a
b

|

f

|

2

(
f
∈
D
)

\smallint _a^b[p|f’{|^2} + q|f{|^2}] \geq {\mu _0}\smallint _a^b|f{|^2}\;(f \in D)

is established, where p and q are real-valued coefficient functions and f is a complex-valued function in a set D so chosen that both sides of the inequality are finite. The interval of integration is of the form

−
∞
>
a
>
b
≤
∞

- \infty > a > b \leq \infty

. The inequality is first established for functions in the domain of an operator in the Hilbert function space

L
2

(
a
,
b
)

{L^2}(a,b)

that is associated with the differential equation

−
(
p

y
′

)
′

+
q
y
=
λ
y

- (py’)’ + qy = \lambda y

, and the number

μ
0

{\mu _0}

in the inequality is the smallest number in the spectrum of this operator. An approximation theorem is given that allows the inequality to be established for the larger set of functions D. An extension of some classical results from the calculus of variations and some spectral theory is then used to give necessary and sufficient conditions for equality and to show that the constant

μ
0

{\mu _0}

is best possible. Certain consequences of these conclusions are also discussed.
Multipliers and linear functionals for the class 𝑁⁺
Authors: Yanagihara Niro 
Published: July 2010

Multipliers for the classes

H
p

{H^p}

are studied recently by several authors, see Duren’s book, Theory of

H
p

{H^p}

spaces, Academic Press, New York, 1970. Here we consider corresponding problems for the class

N
+

{N^ + }

of holomorphic functions in the unit disk such that
\[

lim

r
→
1

∫
0

2
π

log

+

|

f
(
r

e

i
θ

)

|

d
θ
=

∫
0

2
π

log

+

|

f
(

e

i
θ

)

|

d
θ
>
∞
.

\lim \limits _{r \to 1} \int _0^{2\pi } {{{\log }^ + }} |f(r{e^{i\theta }})|d\theta = \int _0^{2\pi } {{{\log }^ + }|f({e^{i\theta }})|} d\theta > \infty .

\]
Our results are: 1.

N
+

{N^ + }

is an

F
F

-space in the sense of Banach with the distance function
\[

ρ
(
f
,
g
)
=

1

2
π

∫
0

2
π

log
⁡
(
1
+

|

f
(

e

i
θ

)
−
g
(

e

i
θ

)

|

)

d
θ
.

\rho (f,g) = \frac {1}{{2\pi }}\int _0^{2\pi } {\log (1 + |f({e^{i\theta }}) - g({e^{i\theta }})|)} d\theta .

\]
2. A complex sequence

Λ
=
{

λ
n

}

\Lambda = \{ {\lambda _n}\}

is a multiplier for

N
+

{N^ + }

into

H
q

{H^q}

for a fixed

q
,
0
>
q
>
∞

q,0 > q > \infty

, if and only if

λ
n

=
O
(
exp
⁡
[
−
c

n

]
)

{\lambda _n} = O(\exp [ - c\sqrt n ])

for a positive constant

c
c

. 3. A continuous linear functional

ϕ
\phi

on the space

N
+

{N^ + }

is represented by a holomorphic function

g
(
z
)
=
Σ

b
n

z
n

g(z) = \Sigma {b_n}{z^n}

which satisfies

b
n

=
O
(
exp
⁡
[
−
c

n

]
)

{b_n} = O(\exp [ - c\sqrt n ])

for a positive constant

c
c

. Conversely, such a function

g
(
z
)
=
Σ

b
n

z
n

g(z) = \Sigma {b_n}{z^n}

defines a continuous linear functional on the space

N
+

{N^ + }

.
One-parameter semigroups holomorphic away from zero
Authors: Certain Melinda W. 
Published: July 2010

Suppose

T
T

is a one-parameter semigroup of bounded linear operators on a Banach space, strongly continuous on

[
0
,
∞
)

[0,\infty )

. It is known that

lim

sup

x
→
0

|

T
(
x
)
−
I

|

>
2

\lim {\sup _{x \to 0}}|T(x) - I| > 2

implies

T
T

is holomorphic on

(
0
,
∞
)

(0,\infty )

. Theorem I is a generalization of this as follows: Suppose

M
>
0
,
0
>
r
>
s

M > 0,0 > r > s

, and

ρ
\rho

is in (1,2). If

|

(
T
(
h
)
−
I

)
n

|

≤
M

ρ
n

|{(T(h) - I)^n}| \leq M{\rho ^n}

whenever

n
h

nh

is in

[
r
,
s
]
,
n
=
1
,
2
,
⋯
,
h
>
0

[r,s],n = 1,2, \cdots ,h > 0

, then there exists

b
>
0

b > 0

such that

T
T

is holomorphic on

[
b
,
∞
)

[b,\infty )

. Theorem II shows that, in some sense,

b
→
0

b \to 0

as

r
→
0

r \to 0

. Theorem I is an application of Theorem III: Suppose

M
>
0
,
0
>
r
>
s
,
ρ

M > 0,0 > r > s,\rho

is in (1,2), and

f
f

is continuous on

[
−
4
s
,
4
s
]

[ - 4s,4s]

. If

|

∑

q
=
0

n

(

n
q

)

(
−
1

)

n
−
q

f
(
t
+
q
h
)

|

≤
M

ρ
n

|\sum _{q = 0}^n \binom {n}{q}( - 1)^{n - q} f(t + qh)| \leq M \rho ^n

whenever

n
h

nh

is in

[
r
,
s
]

[r,s]

,

n
=
1

n = 1

,

2
2

, …,

h
>
0

h > 0

,

[
t
,
t
+
n
h
]
⊂
[
−
4
s
,
4
s
]

[t,t + nh] \subset [ - 4s,4s]

, then

f
f

has an analytic extension to an ellipse with center zero. Theorem III is a generalization of a theorem of Beurling in which the inequality on the differences is assumed for all

n
h

nh

. An example is given to show the hypothesis of Theorem I does not imply

T
T

holomorphic on

(
0
,
∞
)

(0,\infty )

.
The slimmest geometric lattices
Authors: Dowling Thomas A. , Wilson Richard M. 
Published: July 2010

The Whitney numbers of a finite geometric lattice L of rank r are the numbers

W
k

{W_k}

of elements of rank k and the coefficients

w
k

{w_k}

of the characteristic polynomial of L, for

0
≤
k
≤
r

0 \leq k \leq r

. We establish the following lower bounds for the

W
k

{W_k}

and the absolute values

w
k
+

=

(
−
1

)
k

w
k

w_k^ + = {( - 1)^k}{w_k}

and describe the lattices for which equality holds (nontrivially) in each case:
\[

W
k

≥

(

r

a
m
p
;
−

a
m
p
;
2

k

a
m
p
;
−

a
m
p
;
1

)

(
n
−
r
)
+

(

r

k

)

,

w
k
+

≥

(

r

a
m
p
;
−

a
m
p
;
1

k

a
m
p
;
−

a
m
p
;
1

)

(
n
−
r
)
+

(

r

k

)

,

{W_k} \geq \left ( {\begin {array}{*{20}{c}} r \hfill & - \hfill & 2 \hfill \\ k \hfill & - \hfill & 1 \hfill \\ \end {array} } \right )(n - r) + \left ( {\begin {array}{*{20}{c}} r \hfill \\ k \hfill \\ \end {array} } \right ),\quad w_k^ + \geq \left ( {\begin {array}{*{20}{c}} r \hfill & - \hfill & 1 \hfill \\ k \hfill & - \hfill & 1 \hfill \\ \end {array} } \right )(n - r) + \left ( {\begin {array}{*{20}{c}} r \hfill \\ k \hfill \\ \end {array} } \right ),

\]
where

n
=

W
1

n = {W_1}

is the number of points of L.
Rational points of commutator subgroups of solvable algebraic groups
Authors: Fauntleroy Amassa 
Published: July 2010

Let G be a connected algebraic group defined over a field k. Denote by

G
(
k
)

G(k)

the group of k-rational points of G. Suppose that A and B are closed subgroups of G defined over k. Then

[
A
,
B
]
(
k
)

[A,B](k)

is not equal to

[
A
(
k
)
,
B
(
k
)
]

[A(k),B(k)]

in general. Here [A,B] denotes the group generated by commutators

a
b

a

−
1

b

−
1

,
a
∈
A
,
b
∈
B

ab{a^{ - 1}}{b^{ - 1}},a \in A,b \in B

. We say that a field of k of characteristic p is p-closed if given any additive polynomial

f
(
x
)

f(x)

in

k
[
x
]

k[x]

and any element c in k, there exists an element

α
\alpha

in k such that

f
(
α
)
=
c

f(\alpha ) = c

. Theorem 1. Let G be a connected solvable algebraic group defined over the p-closed field k. Let A and B be closed connected subgroups of G, which are also defined over k, and suppose A normalizes B.

T
h
e
n

[
A
,
B
]

(
k
)
=
[
A
(
K
)
,
B
(
K
)
]

Then\;[A,B]\;(k) = [A(K),B(K)]

. 2. If G, A and B are as above and k is only assumed to be perfect then there exists a finite extension

k
0

{k_0}

of k such that if K is the maximal p-extension of

k
0

{k_0}

, then

[
A
,
B
]
(
K
)
=
[
A
(
K
)
,
B
(
K
)
]

[A,B](K) = [A(K),B(K)]

.
Perturbed semigroup limit theorems with applications to discontinuous random evolutions
Authors: Kertz Robert P. 
Published: July 2010

For

ε
>
0

\varepsilon > 0

small, let

U
ε

(
t
)

{U^\varepsilon }(t)

and

S
(
t
)

S(t)

be strongly continuous semigroups of linear contractions on a Banach space

L
L

with infinitesimal operators

A
(
ε
)

A(\varepsilon )

and

B
B

respectively, where

A
(
ε
)
=

A

(
1
)

+

ε

A

(
2
)

+
o
(
)

A(\varepsilon ) = {A^{(1)}} + {\varepsilon A^{(2)}} + o()

as

ε
→
0

\varepsilon \to 0

. Let

{
B
(
u
)
;
u
⩾
0
}

\{ B(u);u \geqslant 0\}

be a family of linear operators on

L
L

satisfying

B
(
ε
)
=
B
+

ε

Π

(
1
)

+

ε
2

Π

ε
(
2
)

+
o
(

ε
2

)

B(\varepsilon ) = B + {\varepsilon \Pi ^{(1)}} + {\varepsilon ^2}{\Pi ^{\varepsilon (2)}} + o({\varepsilon ^2})

as

ε
→
0

\varepsilon \to 0

. Assume that

A
(
ε
)
+

ε

−
1

B
(
)

A(\varepsilon ) + {\varepsilon ^{ - 1}}B()

is the infinitesimal operator of a strongly continuous contraction semigroup

T
ε

(
t
)

{T_\varepsilon }(t)

on

L
L

and that for each

f
∈
L
,

lim

λ
→
0

λ

∫
0
∞

e

−
λ
t

S
(
t
)
f
d
t
≡
P
f

f \in L,{\lim _{\lambda \to 0}}\lambda \int _0^\infty {{e^{ - \lambda t}}} S(t)fdt \equiv Pf

exists. We give conditions under which

T
ε

(
t
)

{T_\varepsilon }(t)

converges as

→
0

\to 0

to the semigroup generated by the closure of

P
(

A

(
1
)

+

Π

(
1
)

)

P({A^{(1)}} + {\Pi ^{(1)}})

on

R

(
P
)
∩

D

(

A

(
1
)

)
∩

D

(

Π

(
1
)

)

\mathcal {R}(P) \cap \mathcal {D}({A^{(1)}}) \cap \mathcal {D}({\Pi ^{(1)}})

. If

P
(

A

(
1
)

+

Π

(
1
)

)
f
=
0
,
B
h
=
−
(

A

(
1
)

+

Π

(
1
)

)
f

P({A^{(1)}} + {\Pi ^{(1)}})f = 0,Bh = - ({A^{(1)}} + {\Pi ^{(1)}})f

, and we let

V
^

f
=
P
(

A

(
1
)

+

Π

(
1
)

)
h

\hat Vf = P({A^{(1)}} + {\Pi ^{(1)}})h

, then we show that

T
ε

(
t

/

ε
)
f

{T_\varepsilon }(t/\varepsilon )f

converges as

ε
→
0

\varepsilon \to 0

to the strongly continuous contraction semigroup generated by the closure of

V

(
2
)

+

V
^

{V^{(2)}} + \hat V

. From these results we obtain new limit theorems for discontinuous random evolutions and new characterizations of the limiting infinitesimal operators of the discontinuous random evolutions. We apply these results in a model for the approximation of physical Brownian motion and in a model of the content of an infinite capacity dam.
𝐻^{𝑟,}^{∞}(𝑅)- and 𝑊^{𝑟,∞}(𝑅)-splines
Authors: Smith Philip W. 
Published: July 2010

Let E be a subset of R the real line and

f
:
E
→
R

f:E \to R

. Necessary and sufficient conditions are derived for

inf
(

‖

D
r

x
‖

L
∞

:
x

|

E

=
f
)

\inf (\left \|{D^r}x\right \|_{{L^\infty }}:x{|_E} = f)

to have a solution. When restricted to quasi-uniform partitions E, necessary and sufficient conditions are derived for the solution to be in

L
∞

{L^\infty }

. For finite partitions E it is shown that a solution to the

L
∞

{L^\infty }

infimum problem can be obtained by solving

inf
(

‖

D
r

x
‖

L
p

:
x

|

E

=
f
)

\inf (\left \|{D^r}x\right \|_{{L^p}}:x{|_E} = f)

and letting p go to infinity. In this way it was discovered that solutions to the

L
∞

{L^\infty }

problem could be chosen to be piecewise polynomial (of degree r or less). The solutions to the

L
p

{L^p}

problem are called

H

r
,
p

{H^{r,p}}

-splines and were studied extensively by Golomb in [3].