bluuebunny/crossref_metadata_2025_split huggingface dataset:
Synthesis of polyurethanes and investigation of their hydrolytic stability
Authors: Masař B. , Čefelín P. , Lipatova T. E. , Bakalo L. A. , Lugovskaya G. G. 
Published: June 2010

AbstractPolyurethane films were prepared with poly(tetramethylene oxide) prepolymer, terminated with hexamethylene diisocyanate, and a polyfunctional diester of oxalic acid and glycerol, di‐(2,3‐dihydroxypropyl) oxalate (DHPO), or its mixtures with 3,6‐dioxaoctane‐1,8‐diol (DOD) as chain extender. Some mechanical characteristics of these films were ascertained. These samples are more readily hydrolyzed than those prepared with DOD alone; the hydrolyzability can be regulated by changing the DHPO/DOD ratio. The change in structure during the hydrolysis was investigated by differential thermal analysis (DTA).
Surface characterization of poly(hydroxyethyl methacrylate) and related polymers. I. Contact angle methods in water
Authors: Andrade J. D. , King R. N. , Gregonis D. E. , Coleman D. L. 
Published: June 2010

AbstractThe characterization of the gel‐water interface is considered, particularly with respect to obtaining the interfacial free energy between the gel and water, γ utilizing contact angle data. The question of contact‐angle‐induced deformation of the three‐phase region is examined and visual documentation of such deformation is presented. Air in water and octane in water contact‐angle data are used to estimate γdsv γpsv γsv, and γsw for the gel‐water interface as a function of bulk water fraction in the gel. It is shown that γ °0 ± 5 erg/cm2 when the bulk water fraction is greater than 0.2. All interfacial parameters determined appear to reach a constant value at a bulk water fraction of about 0.4. These results are for poly(hydroxyethyl methacrylate), poly(methoxyethyl methacrylate), poly(dihydroxypropyl methacrylate), and selected copolymers at 25°C in distilled water.
Polyarylacetylenes: Structure and properties
Authors: Simionescu C. I. , Percec V. 
Published: June 2010

AbstractConsiderable confusion exists in the literature concerning the structure and properties of polyarylacetylenes. Taking into account the significant structural aspects from the literature data with our recent results on the polypentadeuterophenylacetylene isomers, we present our findings concerning this class of polymer.
Organofunctional phosphazenes and organofunctional phosphazene polymers
Authors: Allen Christopher W. 
Published: June 2010

AbstractRecent work has shown that the range of synthesis of new and useful phosphazene derivatives can be dramatically expanded by incorporation of an organofunctional substituent on the phosphazene ring. This substituent can serve as the site for various synthetic transformations leading to new monomeric and polymeric phosphazene moieties. Initial work on propenylphosphazenes and their copolymerization with organic monomers is briefly reviewed. The synthesis of new improved phosphazene monomers which lead to flame retardant copolymers is discussed. In addition to electron‐rich olefins such as alkoxylvinylphosphazenes, novel alkynlphosphazene/styrene copolymers were prepared. The vinyl alcohol derivative N3P3Cl5OCH=CH2 and related compounds were prepared and characterized. Both homopolymerization and a broad range of copolymerization reactions of N3P3Cl5OCH=CH2 are presented. The properties of these systems can be modified by reactions of the cyclophosphazene units in the polymer.
Diffusion‐limited gelation
Authors: Witten T. A. 
Published: June 2010

AbstractDiffusion‐limited aggregation is a means of forming continuous random networks of low density. Recent work on diffusion‐limited aggregation is reviewed. These aggregates have universal power‐law density correlations. This implies scaling laws relating an aggregate's size to its mass when it is grown from a point, line, or surface. The mass of the growing interface scales as a power of the total mass, but not in the way suggested by simple estimates. A mean field model of the process yields a density that falls inversely as the distance from the origin. It is speculated that the mean field solution may be stable above some spatial dimension dc.
Homodinuclear f–f and Heterodinuclear f–p Lanthanide Helicates
Authors: Albrecht Markus 
Published: June 2010

AbstractLanthanide helicates are of interest due to their ability to emit either in the visible or in the NIR region. In this context it is important to prepare heterodinuclear helicates and to use them for the investigation of intramolecular communication processes between different complex units. In here we describe the ligand synthesis and self‐assembly of homo‐ and heterodinuclear helicates based on β‐diketonate, catechol‐acyl hydrazone and 2‐amido‐8‐hydroxyquinoline ligands. Strongly emitting homodinuclear europium(III) complexes are obtained in addition to heterodinuclear coordination compounds. Some of the latter exhibit intramolecular energy transfer behaviour with internal quenching of an aluminium trisquinolinate and enhancement of the NIR emission at an ytterbium(III), neodymium(III) or erbium(III) complex unit.
The Impact of Obesity on Gynecologic Cancer Screening: An Integrative Literature Review
Authors: Aldrich Tess , Hackley Barbara 
Published: June 2010

Introduction: Evidence indicates lower rates of breast and cervical cancer screening among obese compared to nonobese women. This integrative review examines the association between gynecologic cancer screening and body weight, as well as potential barriers to screening.Methods: A literature search of standard computerized databases was conducted for peer‐reviewed articles published between 1950 and January 2009.Results: Twenty‐three studies met the criteria for review. Of the 17 studies that evaluated rates of cervical cancer screening, 13 found obese women significantly less likely than their nonobese counterparts to have had a recent Papanicolaou test, a trend that was stronger in white women when compared to African American women. Eight of the 15 studies examining routine mammography found an inverse association between increasing body weight and recent screening, although findings generally pertained only to women who were white and/or severely obese. Possible barriers to care included embarrassment and perceived weight stigma in the clinical setting, lack of appropriately sized examination equipment, and poor patient–provider communication.Discussion: Further research is needed to clarify the challenges that obese women face in accessing care and to evaluate strategies such as ensuring the availability of appropriate equipment and supplies, the use of alternative screening methodologies, and more culturally sensitive counseling approaches that may improve screening rates in obese women.J Midwifery Womens Health 2010;55:344–356 © 2010 by the American College of Nurse‐Midwives.
Training the Next Generation of Teaching Professors: A Comparative Study of Ph.D. Programs in Political Science
Authors: Ishiyama John , Miles Tom , Balarezo Christine 
Published: June 2010

AbstractIn this article, we investigate the graduate curricula of political science programs and 122 Ph.D.-granting political science programs in the United States and how they seek to prepare political science teachers. We first investigate whether the department offers a dedicated political science course at the graduate level on college teaching, and whether the presence of this class correlates with the size of the department, the size of the university, the ranking of the department, and so on. We find that whether a program offers a graduate course on teaching is inversely related to the research productivity of a department, and that departments at public institutions are more likely to offer such courses than are departments at private institutions. Second, we conduct content analysis of a sample of syllabi from departments that offer such courses to ascertain the kinds of topics that are covered. Finally, we briefly describe some model programs that seek to prepare graduate students for teaching careers that integrate graduate student teacher training throughout the Ph.D. program.
Getting the Most from Classroom Simulations: Strategies for Maximizing Learning Outcomes
Authors: Wedig Timothy 
Published: June 2010

AbstractClassroom simulations can make a significant contribution to learning outcomes in political science courses, provided that they are firmly linked to course content and learning objectives. This article offers a step-by-step decision framework for instructors seeking to use simulations as a core component of their courses, including selection of an exercise, pre-simulation preparation, instructor role during a simulation, and techniques for debriefing after the exercise. Options such as online and face-to-face, synchronous and asynchronous, distributed and single classroom, and individual and team formats are compared, with a focus on their associated learning outcomes.
FORECASTING AND PREDICTING THE ELECTION OF AN AFRICAN AMERICAN PRESIDENT
Authors: Walton Hanes , Allen Josephine A. V. , Puckett Sherman C. , Deskins Donald R. , Starks Robert T. 
Published: June 2010

AbstractBeginning with the 1972 presidential election and for each election thereafter Harvard University's Institute of Politics in the John F. Kennedy School of Government has held a post-election symposium where all of the campaign managers, pollsters, political consultants and media advisors for all of the primary and general election candidates come together with leading journalists, electronic and print, and political commentators and pundits to discuss and dialogue about what occurred during the election among the candidates, nominees, and the winner and losers. The symposia have allowed campaign managers to describe what happened and forecast for the forthcoming presidential election. After the multi-day symposium a book length transcript is published. In 1992 the Annenberg School of Communication at the University of Pennsylvania launched its own symposia where the campaign managers of the Democratic and Republican nominees in the general election are invited to discuss what happened and forecast for the next presidential election. The ten books in the Harvard series and the three books in the Pennsylvania series are used as data sources in this article to determine if any of the campaign managers forecasted and/or predicted an African American presidential candidate, even when such candidates had appeared in previous years. While our findings uncover some of the interests and concerns of presidential campaign managers and advisors since 1972, the overriding focus of the symposia has been on the nature, scope, and significance of the African American electorate.
On an integral formula of Gauss-Bonnet-Grotemeyer
Authors: Chen Bang-yen 
Published: June 2010

Let

e
(
p
)

e(p)

and

G
(
p
)

G(p)

be the unit outer normal and the Gauss-Kronecker curvature of an oriented closed even-dimensional hypersurface

M
M

of dimension

n
n

in

E

n
+
1

{E^{n + 1}}

. Then for a fixed unit vector

c
c

in

E

n
+
1

{E^{n + 1}}

, we have
\[

(
1
)

∫
M

(

c

⋅

e)

m

G
d
V
=

c

n
+
m

χ
(
M
)

/

c

m
,

a
m
p
;

for

m
=
0
,
2
,
4
,
⋯
,

=
0
,

a
m
p
;

for

m
=
1
,
3
,
5
,
⋯
,

(1)\qquad \begin {array}{*{20}{c}} {\int _M^{} {{{({\text {c}}\cdot {\text {e)}}}^m}GdV = {c_{n + m}}\chi (M)/{c_{m,}}} } & {{\text {for}}\;m = 0,2,4, \cdots ,} \\ { = 0,} & {{\text {for}}\;m = 1,3,5, \cdots ,} \\ \end {array}

\]
where

c

⋅

e

{\text {c}} \cdot {\text {e}}

denotes the inner product of

c
c

and

e
e

the area of

m
m

-dimensional unit sphere, and

χ
(
M
)

\chi (M)

the Euler characteristic of

M
M

.
Mean growth of inner functions
Authors: Herrero Domingo A. 
Published: June 2010

Let A be a closed subset of the closed unit disc. It is shown that there exists a “universal growth function”

φ
(
r
,
A
)

\varphi (r,A)

such that

1
−

∫
0

2
π

|

q
(
r

e

i
x

)

|

2

d
x

/

2
π
=
O
(
φ
(
r
,
A
)
)

1 - \smallint _0^{2\pi }|q(r{e^{ix}}){|^2}dx/2\pi = O(\varphi (r,A))

for all inner functions

q
(
z
)

q(z)

whose zeroes lie in

A
∩
{

|

z

|

>
1
}

A \cap \{ |z| > 1\}

and whose singularities in the unit circle lie on

A
∩
{

|

z

|

=
1
}

A \cap \{ |z| = 1\}

, if and only if the Lebesgue measure of

A
∩
{

|

z

|

=
1
}

A \cap \{ |z| = 1\}

is zero.
Adjoints of multipoint-integral boundary value problems
Authors: Brown R. C. , Krall Allan M. 
Published: June 2010

The dual system to

L
y
=

y
′

+
P
y

Ly = y’ + Py

,
\[

∑

i
=
0

∞

A
i

y
(

t
i

)
+

∫
0
1

K
(
t
)
y
(
t
)

d
t
=
0

\sum \limits _{i = 0}^\infty {{A_i}y({t_i}) + \int _0^1 {K(t)y(t)\;dt = 0} }

\]
is found when the setting is

L
n
p

(
0
,
1
)
,
1
>
p
>
∞

L_n^p(0,1),1 > p > \infty

.
Inequalities for a perturbation theorem of Paley and Wiener
Authors: Young Robert M. 
Published: June 2010

A classical theorem of Paley and Wiener states that the set of functions

{

e

i

λ
n

t

}

n
=
−
∞

∞

\{ {e^{i{\lambda _n}t}}\} _{n = - \infty }^\infty

forms a basis for

L
2

(
−
π
,
π
)

{L^2}( - \pi ,\pi )

whenever the following condition is satisfied:
\[

(
∗
)

|

|

∑

c
n

(

e

i

λ
n

t

−

e

i
n
t

)

|

|

2

≦

θ
2

∑

|

c
n

|

2

(
0
≦
θ
>
1
)
.

( \ast )\quad ||\sum {{c_n}({e^{i{\lambda _n}t}} - {e^{int}})|{|^2} \leqq {\theta ^2}\,\sum {|{c_n}{|^2}} } \quad (0 \leqq \theta > 1).

\]
It is known that (

∗
\ast

) holds whenever

λ
n

{\lambda _n}

is real and

|

λ
n

−
n

|

≦
L
>

1
4

(
−
∞
>
n
>
∞
)

|{\lambda _n} - n| \leqq L > \frac {1}{4}( - \infty > n > \infty )

, and may fail to hold if

|

λ
n

−
n

|

=

1
4

|{\lambda _n} - n| = \frac {1}{4}

. In this note we show, more generally, that the condition

|

λ
n

−
n

|

>

1
4

|{\lambda _n} - n| > \frac {1}{4}

is also insufficient to ensure (

∗
\ast

).
On the weak type (1,1) inequality for conjugate functions
Authors: Davis Burgess 
Published: June 2010

A theorem of Kolmogorov states that there is a positive constant

K
K

such that if

f
~

\tilde f

is the conjugate function of an integrable real valued function

f
f

on the unit circle then

m
{

|

f
~

|

≧
λ
}
≦
K

|

|

f

|

|

1

/

λ
,
λ
>
0

m\{ |\tilde f| \geqq \lambda \} \leqq K||f|{|_1}/\lambda ,\lambda > 0

. It is shown that the smallest possible value for

K
K

in this theorem, the so called weak type (1, 1) norm of the conjugate function operator, is

(
1
+

3

−
2

+

5

−
2

+
⋯
)

/

(
1
−

3

−
2

−

5

−
2

−
⋯
)
≈
1.347

(1 + {3^{ - 2}} + {5^{ - 2}} + \cdots )/(1 - {3^{ - 2}} - {5^{ - 2}} - \cdots ) \approx 1.347

. This number is also shown to be the weak type (1, 1) norm of the Hilbert transform operator on functions defined on the real line. The proof uses P. Levy’s result that Brownian motion in the plane is conformally invariant.
On generalizing alternative rings
Authors: Rodabaugh D. J. 
Published: June 2010

Consider a ring

R
R

that satisfies the identity

(
x
,
x
,
x
)
=
0

(x,x,x) = 0

and any two of the three identities:

(
w
x
,
y
,
z
)
+
(
w
,
x
,
[
y
,
z
]
)
−
w
(
x
,
y
,
z
)
−
(
w
,
y
,
z
)
x
=
0
;
(
[
w
,
x
]
,
y
,
z
)
+
(
w
,
x
,
y
z
)
−
y
(
w
,
x
,
z
)
−
(
w
,
x
,
y
)
z
=
0
;
(
w
,
x
⋅
y
,
z
)
−
x
⋅
(
w
,
y
,
z
)
−
y
⋅
(
w
,
x
,
z
)
=
0

(wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x = 0;([w,x],y,z) + (w,x,yz) - y(w,x,z) - (w,x,y)z = 0;(w,x \cdot y,z) - x \cdot (w,y,z) - y \cdot (w,x,z) = 0

. In this paper, we prove that if

R
R

has characteristic prime to 6 then

R
R

semiprime with idempotent

e
e

implies

R
R

has a Peirce decomposition in which the modules multiply as they do in an alternative ring. If in addition

R
R

is prime with idempotent

e
≠
0
,
1

e \ne 0,1

then

R
R

is alternative.
The Hardy class of a function with slowly-growing area
Authors: Hansen Lowell J. 
Published: June 2010

In this paper we show that if

f
f

is analytic on the open unit disk and if the area of

[
{

|

z

|

≤
R
}
∩

image of

f
]

[\{ |z| \leq R\} \cap {\text {image of }}f]

grows sufficiently slowly as a function of

R
R

, then

f
f

belongs to the Hardy class

H
p

{H^p}

for all

p
p

satisfying

0
>
p
>
+
∞

0 > p > + \infty

.
The effect of retarded actions on nonlinear oscillations
Authors: Sficas Y. G. , Staïkos V. A. 
Published: June 2010

We study the oscillatory and asymptotic behavior of nonlinear retarded differential equations of the form

x

(
n
)

(
t
)
+

(
−
1

)

n
+
1

p
(
t
)
⋅
ϕ
(
x
[
g
(
t
)
]
)
=
0

{x^{(n)}}(t) + {( - 1)^{n + 1}}p(t) \cdot \phi (x[g(t)]) = 0

under certain conditions which do not hold in the particular case

g
(
t
)
≡
t

g(t) \equiv t

of ordinary differential equations.
A generalization of Banach’s contraction principle
Authors: Ćirić Lj. B. 
Published: June 2010

Let

T
:
M
→
M

T:M \to M

be a mapping of a metric space

(
M
,
d
)

(M,d)

into itself. A mapping

T
T

will be called a quasi-contraction iff

d
(
T
x
,
T
y
)
⩽
q
max
{
d
(
x
,
y
)
;
d
(
x
,
T
x
)
;
d
(
y
,
T
y
)
;
d
(
x
,
T
y
)
;
d
(
y
,
T
x
)
}

d(Tx,Ty) \leqslant q\max \{ d(x,y);d(x,Tx);d(y,Ty);d(x,Ty);d(y,Tx)\}

for some

q
>
1

q > 1

and all

x
,
y
∈
M

x,y \in M

. In the present paper the mappings of this kind are investigated. The results presented here show that the condition of quasi-contractivity implies all conclusions of Banach’s contraction principle. Multi-valued quasi-contractions are also discussed.
Inequalities for polynomials satisfying 𝑝(𝑧)≡𝑧ⁿ𝑝(1/𝑧)
Authors: Govil N. K. , Jain V. K. , Labelle G. 
Published: June 2010

If

p
(
z
)
=

Σ

v
=
0

n

a
v

z
v

p(z) = \Sigma _{v = 0}^n {{a_v}} {z^v}

is a polynomial of degree

n
n

, then it is known that

Max

|

z

|

=
1

|

p
′

(
z
)

|

≤
n

Max

|

z

|

=
1

|

p
(
z
)

|

{\operatorname {Max} _{|z| = 1}}|p’(z)| \leq n{\operatorname {Max} _{|z| = 1}}|p(z)|

. In this paper we obtain the analogous inequality for a subclass of polynomials satisfying

p
(
z
)
≡

z
n

p
(
1

/

z
)

p(z) \equiv {z^n}p(1/z)

. Some other inequalities have also been obtained.
Oscillation of a forced second order nonlinear differential equation
Authors: Rankin Samuel M. 
Published: June 2010

Sufficient conditions are given which insure that every solution of

(
a
(
t
)

y
′

)
′

+
p
(
t
)
f
(
y
)
g
(

y
′

)
=
r
(
t
)

(a(t)y’)’ + p(t)f(y)g(y’) = r(t)

has arbitrarily large zeros. We seem to have a partial answer to a question posed by Kartsatos [4]. An example is given illustrating the result.
Similarity of certain operators in 𝑙^{𝑝}
Authors: Kantorovitz Shmuel 
Published: June 2010

Let M be the multiplication operator in

l
p

,
1
⩽
p
⩽
∞

{l^p},1 \leqslant p \leqslant \infty

, i.e.,

M
:
x
=
{

x
k

}
→
{
k

x
k

}

M:x = \{ {x_k}\} \to \{ k{x_k}\}

. Let

w
=
{

w
j

}

j
=
0

∞

w = \{ {w_j}\} _{j = 0}^\infty

be a weight, i.e., a positive sequence such that

w
1

>

w
0

=
1

{w_1} > {w_0} = 1

and

w

n
+
m

⩽

w
n

w
m

{w_{n + m}} \leqslant {w_n}{w_m}

. For

ζ
∈
C

\zeta \in C

, define

N
w
ζ

N_w^\zeta

on

l
p

{l^p}

by
\[

(

N
w
ζ

x

)
k

=

∑

j
=
1

k

(

w
k

w
j

)

(

ζ
−
1
+
k
−
j

k
−
j

)

x
j

(
k
=
1
,
2
,
…
)
.

{(N_w^\zeta x)_k} = \sum \limits _{j = 1}^k {\left ( {\frac {{{w_k}}}{{{w_j}}}} \right )} \left ( {\begin {array}{*{20}{c}} {\zeta - 1 + k - j} \\ {k - j} \\ \end {array} } \right ){x_j}\quad (k = 1,2, \ldots ).

\]
Then

{

N
w
ζ

;
ζ
∈
C
}

\{ N_w^\zeta ;\zeta \in C\}

is a holomorphic group of operators, and for any function g holomorphic on the spectrum of

N
w
ζ

,
M
+
g
(

N
w
ζ

)

N_w^\zeta ,M + g(N_w^\zeta )

is similar to

M
+
g
(
1
)
I

M + g(1)I

.
A generalization of a theorem of Janos and Edelstein
Authors: Park Sehie 
Published: June 2010

A generalization of Edelstein’s version of a theorem of Janos and its converse are obtained: Theorem. Let X be a compact metrizable topological space, and f, g be continuous self-maps of X such that

g
f
=
f
g

gf = fg

and f is bijective. Then g is injective and

⋂
1
∞

g
n

X
=
{

x
0

}

\bigcap \nolimits _1^\infty {{g^n}X = \{ {x_0}\} }

, where

x
0

∈
X

{x_0} \in X

, iff, given

λ
,
0
>
λ
>
1

\lambda ,0 > \lambda > 1

, a homeomorphism h of X into

l
2

{l_2}

exists such that
\[

‖

h
(
g
x
)
−
h
(
g
y
)

‖

=
λ

‖

h
(
f
x
)
−
h
(
f
y
)

‖

\left \| {h(gx) - h(gy)} \right \| = \lambda \left \| {h(fx) - h(fy)} \right \|

\]
for all

x
,
y
∈
X

x,y \in X

.
Extremal properties of real axially symmetric harmonic functions in 𝐸³
Authors: McCoy Peter A. 
Published: June 2010

The set

H

\mathcal {H}

consists of all real harmonic functions defined by

U
(
r
,
θ
)
=

Σ

k
=
0

∞

a
k

r
k

P
k

(
cos
⁡
θ
)

U(r,\theta ) = \Sigma _{k = 0}^\infty {a_k}{r^k}{P_k}(\cos \theta )

which are regular in

Σ
\Sigma

, the open unit sphere about the origin

E
3

{E^3}

. Two problems arise concerning

H

\mathcal {H}

and a subset

H

∗

{\mathcal {H}_\ast }

whose members U have the first

n
+
1

n + 1

coefficients

a
0

,
…
,

a
n

{a_0}, \ldots ,{a_n}

specified. (1) For

U
∈

H

U \in \mathcal {H}

, determine

I
(
U
)
=
inf
{
U
(
r
,
θ
)

|

(
r
,
θ
)
∈
Σ
}

I(U) = \inf \{ U(r,\theta )|(r,\theta ) \in \Sigma \}

as the limit of a monotone sequence of constants

{

λ
k

(

a
0

,
…
,

a
k

)

}

k
=
0

∞

\{ {\lambda _k}({a_0}, \ldots ,{a_k})\} _{k = 0}^\infty

which can be computed algebraically. (2) Find

U
0

∈

H

∗

{U_0} \in {\mathcal {H}_\ast }

and the constant
\[

λ
n

(

a
0

,
…
,

a
n

)
=

Sup

{

I
(
U
)

|

U
∈

H

∗

}

=
I
(

U
0

)
.

{\lambda _n}({a_0}, \ldots ,{a_n}) = {\operatorname {Sup}}\left \{ {I(U)|U \in {\mathcal {H}_\ast }} \right \} = I({U_0}).

\]
These are answered by means of the Bergman Integral Operator Method and applications of the Methods of Ascent and Descent to the classical Carathéodory-Fejér problem regarding extremal properties of harmonic functions in

E
2

{E^2}

.
Vector valued functions equivalent to measurable functions
Authors: Uhl J. J. 
Published: June 2010

Let X be a Banach space with dual

X
∗

{X^\ast }

and let

(
Ω
,
Σ
,
μ
)

(\Omega ,\Sigma ,\mu )

be a finite measure space. Suppose

f
:
Ω
→
X

f:\Omega \to X

is weakly measurable. There exists a (norm) measurable

g
:
Ω
→
X

g:\Omega \to X

such that

⟨

x
∗

,
f
⟩
=
⟨

x
∗

,
g
⟩

\langle {x^\ast },f\rangle = \langle {x^\ast },g\rangle

a.e. for each

x
∗

∈

X
∗

{x^\ast } \in {X^\ast }

if and only if each set A of positive

μ
\mu

-measure has a subset B of positive

μ
\mu

-measure such that there is a weakly compact convex subset K of X with the property that
\[

⟨

x
∗

,
f
⟩
⩽

sup

x
∈
K

⟨

x
∗

,
x
⟩

\langle {x^\ast },f\rangle \leqslant \sup \limits _{x \in K} \;\langle {x^\ast },x\rangle

\]

μ
\mu

-almost everywhere on B for each

x
∗

∈

X
∗

{x^\ast } \in {X^\ast }

.
Another extremal property of perfect splines
Authors: Goodman T. N. T. , Lee S. L. 
Published: June 2010

Let

t

=
{

t
i

}
,
i
=
1
,
2
,
…
,
n
+
k

{\mathbf {t}} = \{ {t_i}\} ,i = 1,2, \ldots ,n + k

, be a given sequence in [a, b],

f
0

∈

W
∞
k

[
a
,
b
]
,
A
>

‖

f
0

(
k
)

‖

∞

{f_0} \in W_\infty ^k[a,b],A > {\left \| {f_0^{(k)}} \right \|_\infty }

, and

F
=
{
f
∈

W
∞
k

[
a
,
b
]
:
f

|

t

=

f
0

|

t

F = \{ f \in W_\infty ^k[a,b]:f|{\mathbf {t}} = {f_0}|{\mathbf {t}}

and

‖

f

(
k
)

‖

∞

⩽
A
}

{\left \| {{f^{(k)}}} \right \|_\infty } \leqslant A\}

. We show that F contains precisely two perfect splines g, h of degree k with

|

g

(
k
)

|

=

|

h

(
k
)

|

=
A

|{g^{(k)}}| = |{h^{(k)}}| = A

and n interior nodes, and for all

f
∈
F

f \in F

,

min
(
g
(
x
)
,
h
(
x
)
)
⩽
f
(
x
)
⩽
max
(
g
(
x
)
,
h
(
x
)
)
;
∀
x
∈
[
a
,
b
]

\min (g(x),h(x)) \leqslant f(x) \leqslant \max (g(x),h(x));\forall x \in [a,b]

.