bluuebunny/crossref_metadata_2025_split huggingface dataset:
Some observations on the melting behavior of isotactic polypropylene crystals
Authors: Mucha Maria 
Published: June 2010

AbstractThe results of DSC investigations carried out at different heating rates for isotactic polypropylene (IPP) samples with various degrees of morphology and crystallinity have been correlated with the results of SAXS experiments and optical microscopy observations. The SAXS experiments have been performed mainly to calculate the long‐period, thickness of crystalline layers and the degree of crystallinity. The multiple endotherms have been observed for samples crystallized at temperatures higher than 100°C. These endotherms are connected with different crystal types formed during crystallization of IPP. These crystals show superheating phenomena when the heating rate in DSC measurements increases. These effects lead to the conclusion that crystals of larger size and perfection grow in these samples rather than in those crystallized at lower temperatures. The latter ones show reorganization phenomena in DSC measurements. The DSC and SAXS investigations performed for the samples after their thermal oxidation in air at 130°C and in fuming nitric acid have shown the long‐period, thickness of crystalline layers and the degree of crystallinity changes correlated with the melting peak temperature variations. The polymer oxidation in air occurs mainly in the amorphous phase and causes the chain scissions. This effect results in the chain mobility increase and their recrystallization on the lamellae surfaces. At the same time (at sufficiently high temperature) the reorganization of the polymer crystals and the crystal form transformation occur and allow for elucidation of the observed changes in melting temperatures and DSC curves.
Preparation and properties of poly(silarylene siloxane)s
Authors: Koide N. , Lenz R. W. 
Published: June 2010

AbstractA series of exactly alternating silarylene‐siloxane homopolymers and copolymers was prepared by the step‐growth polymerization reaction of arylenedisilanols with bis(ureido)silanes. The polymers were obtained in high molecular weight and were found to have low glass transition temperatures and excellent thermal stabilities at high temperatures, particularly in an inert atmosphere. Copolymers containing vinylsiloxane units could be filled and crosslinked to form elastomers with useful mechanical properties and resistance to degradation in air at temperatures up to 250°C.
Farben im Metallleichtbau
Authors: Burkhardt Siegfried 
Published: June 2010

AbstractAuch im Industriebau spielt die Farbe eine wichtige Rolle zur architektonischen Gestaltung eines Gebäudes. Für die Außenhautdieser Bauwerke werden zum großen Teil Elemente aus dem Metallleichtbau verwendet. Bei einem großen Teil dieser Elemente handelt es sich um Sandwichelemente, Bauteile mit dünnen Metalldeckschichten und einem Dämmstoffkern zur Erzielung der Wärmedämmung. Die Deckschalen sind in aller Regel bandbeschichtet und können jede erdenkbare Farbe haben. Im Rahmen der Ursachenforschung von zwei zunächst nicht erklärbaren Schadensfällen wurde festgestellt, dass für die Helligkeit der eingesetzten Farbe der Kunststoffbeschichtungen infolge zweier Messverfahren unterschiedliche Ergebnissen gemessen wurden. Auch die allgemeinen bauaufsichtlichen Zulassungen gaben hierzu keine verlässlichen Angaben. Dies führte dazu, auf diese Messverfahren und vor allem auf die sich daraus ergebenden Schadensmöglichkeiten näher einzugehen.Colours in light metal construction. Also in the industrial construction the colour plays an important role in the architectural design of a building. For the lining of these buildings light metal elements are used largely. For the most part these elements are sandwich elements, components with thin metal top layers and a core of insulating material to achieve thermal insulation. Usually the top layer is coated and can have any conceivable colour. In the course of investigating two cases of damage, which could not be clarified initially, it was established that the brightness of the plastic‐coatings measured by two different measurement procedures gave different results. In addition the general technical approvals did not provide any reliable statements for this. This led to investigate these measurement methods and above all the possibilities of damage resulting.
Neuartige Cyclopentadienyl‐N‐silylphosphazen‐Komplexe der Seltenerdmetalle Yttrium und Lutetium
Authors: Hillesheim Nina S. , Elfferding Michael , Linder Thomas , Sundermeyer Jörg 
Published: June 2010

AbstractWe report the investigation of four new monoanionic cyclopentadienyl‐N‐silylphosphazene ligands [C5R4‐SiMe2‐NPR′3]– (CpSiNP–) 1–4. In this system the amido moiety of the classical dianionic chelate ligand (CpSiN2–) is displaced by an isoelectronic phosphazene donor function (CpSiNP–). The iminophosphorane‐functionalized tetramethylcyclopentadiene ligands have been prepared by the reaction of two equivalents of tris(tert‐butyl)iminophosphorane or tris(dimethylamino)iminophosphorane with tetramethyl or tert‐butyl‐substituted chlorodimethylsilyl‐1,3‐cyclopentadiene. The molecular structures of the protonated species of two representatives 1‐H and 2‐H were established by X‐ray diffraction analysis. Furthermore the metallation of the ligand [C5R4‐SiMe2‐NPtBu3]H (2‐H) was explored by the aryl elimination pathway using chelate stabilized arene complexes of rare‐earth metals (RE) [RE(dmba)3][RE = Y, Lu; (dmba–) = ortho‐metalated N,N‐dimethylbenzylamine]. The resulting cyclopentadienyl diaryl complexes of the type [(η5‐C5Me4‐SiMe2‐NPtBu3)RE(dmba)2] [RE = Y (5); Lu(6)] were completely characterized by NMR spectroscopy, elemental analysis and X‐ray diffraction analysis. The coordination number of the central atom in each case is nine. Five RE–C bonds to the Cp fragment, two RE–C bonds to the aryl‐carbon atom and two RE–N bonds to the chelating amino function are realized. The phosphazene unit with uncommonly large Si–N–P angles of 173° (5) and 166° (6) do not form a bond in terms of the Constrained‐Geometry ligand concept at least in these complexes. It seems that on the one hand the high coordination number and on the other hand the stable chelate structure of the coordinated amino‐aryl ligand is responsible for this result.
Factorising folds for faster functions
Authors: HUTTON GRAHAM , JASKELIOFF MAURO , GILL ANDY 
Published: June 2010

AbstractThe worker/wrapper transformation is a general technique for improving the performance of recursive programs by changing their types. The previous formalisation (A. Gill & G. Hutton, J. Funct. Program., vol. 19, 2009, pp. 227–251) was based upon a simple fixed-point semantics of recursion. In this paper, we develop a more structured approach, based upon initial-algebra semantics. In particular, we show how the worker/wrapper transformation can be applied to programs defined using the structured pattern of recursion captured by fold operators, and illustrate our new technique with a number of examples.
Stability of housekeeping genes and expression of locally produced growth factors and hormone receptors in goat preantral follicles
Authors: Frota Isana M. A. , Leitão Cintia C. F. , Costa José J. N. , Brito Ivina R. , van den Hurk Robert , Silva José R. V. 
Published: June 2010

SummaryThe aim of the present study was to investigate the stability of six housekeeping genes, and the relative expression of growth factors (EGF, GDF-9, BMP-15, VEGF, FGF-2, BMP-6, IGF-1 and KL) and hormone receptors (FSH, LH and GH) in goat preantral follicles. To evaluate to stability of housekeeping genes micro-dissected fresh follicles (150–200 μm) as well as follicles that have been in vitro cultured for 12 days were used. In addition, isolated fresh follicles were used to compare expression of various growth factors and hormone receptors before culture. Both fresh and cultured follicles were subjected to total RNA extraction and synthesis of cDNA. After amplification of cDNA by real-time PCR, the geNorm software program was used to evaluate the stability of glyceraldehyde-2-phosphate dehydrogenase (GAPDH), β-tubulin, β-actin, phosphoglycerokinase (PGK), 18S rRNA, ubiquitin (UBQ) and ribosomal protein 19 (RPL-19). In addition, follicular steady-state levels of mRNA from the various growth factors under study were compared. Results demonstrated that, in goat preantral follicles, UBQ and β-actin were the most suitable reference genes and thus could be used as parameters to normalize data from future in vitro studies. In contrast, 18S RNA appeared the least stable gene among the tested housekeeping genes. Analysis of mRNA for several hypophyseal hormone receptors in fresh preantral follicles showed significantly higher FSH-R mRNA levels than those of LH-R and GH-R, and no difference between GH-R and LH-R mRNA levels. In regard growth factor mRNA expression in goat preantral follicles, EGF mRNA levels appeared significantly lower than those of the other studied growth factors. Increasingly higher relative mRNA levels were observed for GDF-9, BMP-15, BMP-6, FGF-2, VEGF, Kl and IGF-1, successively. In conclusion, UBQ and β-actin are the most stable housekeeping genes in fresh and 12-days cultured caprine preantral follicles. Furthermore, in fresh follicles, high levels of FSH-R mRNA are detected while among eight growth factors, IGF-1 is the most highly expressed and EGF the weakest expressed compound.
Understanding Sample Surveys: Selective Learning about Social Science Research Methods
Authors: Currin-Percival Mary , Johnson Martin 
Published: June 2010

AbstractWe investigate differences in what students learn about survey methodology in a class on public opinion presented in two critically different ways: with the inclusion or exclusion of an original research project using a random-digit-dial telephone survey. Using a quasi-experimental design and data obtained from pretests and posttests in two public opinion courses, we test the hypothesis that students who participate in an original survey research project will have a stronger understanding of survey research methods than students who do not. To better assess the effect of the active learning element of the course, we estimate average treatment effects on the students who participated in the original survey project using nearest neighbor matching (Abadie et al. 2004) with student scores on a pretest. We find evidence of modest improvement in learning of survey methods in the course featuring the original survey research project; however, the major finding here is that a course featuring this kind of opportunity appeals to a different kind of student than a course that allows participants to stay closer to the classroom and library instead of the social science research laboratory. This discovery may have important implications for our understanding of the effects of active learning opportunities in other types of elective courses.
LATINO EMPLOYMENT AND RESIDENTIAL SEGREGATION IN METROPOLITAN LABOR MARKETS
Authors: Dickerson vonLockette Niki T. , Johnson Jacqueline 
Published: June 2010

AbstractThe spatial configuration of minorities relative to Whites in a metropolitan area, or residential segregation, has been identified as a significant barrier to access to employment opportunities for racial/ethnic minorities, including Latinos, in metropolitan labor markets. Dominating the research are tests of place stratification models that focus on segregated ethnic enclaves or the mismatch between minority communities and employment opportunities. Both approaches focus on predominantly Latino neighborhoods and communities, but overlook their structural location and isolation in the broader metropolitan labor market. This study examines whether and to what extent structural characteristics of metropolitan labor markets in which Mexicans, Puerto Ricans, and Cubans live and work shape their employment opportunities and whether or not these relationships vary across the three Latino native-origin groups. We utilize a unique dataset of the demographic, employment, educational, occupational, and industrial characteristics of the 95 largest US cities. The analyses feature both OLS regression to ascertain if varying levels of segregation across metropolitan areas in 2000 is associated with different levels of employment for Latinos, and a fixed-effects analysis to determine if changes in these structural factors between 1980 and 2000 within the same labor market affect the employment rates of Latinos in that metropolitan area. We find that segregation has a deleterious effect on Latino men's employment; in cities where segregation is worse, their employment rates are lower, and as the cities that they live in became more segregated over the 20 year period of study, their employment rates decreased.
Boundedness and dimension for weighted average functions
Authors: Stanford David P. 
Published: June 2010

The paper considers a weighted average property of the type

u
(

x
o

)
=
(

∫
B

u
w
d
x
)

/

(

∫
B

w
d
x
)

u({x_o}) = ({\smallint _B}uwdx)/({\smallint _B}wdx)

,

B
B

a ball in

E
n

{E^n}

with center

x
o

{x_o}

. A lemma constructing such functions is presented from which it follows that if

n
=
1

n = 1

and the weight function

w
w

is continuously differentiable but is not an eigenfunction of the

1
1

-dimensional Laplace operator, then

u
u

is constant. It is also shown that if

w
w

is integrable on

E
n

{E^n}

and

u
u

is bounded above or below,

u
u

is constant.
A characterization of 𝑁-compact spaces
Authors: Chew Kim-peu 
Published: June 2010

In this paper, we prove the following theorem: Theorem A. A

0
0

-dimensional space

X
X

is

N
N

-compact if and only if every clopen ultrafilter on

X
X

with the countable intersection property is fixed, where

N
N

is the space of all natural numbers. Two consequences of Theorem A are as follows: Theorem B. Suppose that

X
X

and

Y
Y

are

N
N

-compact spaces. A mapping

ϕ
\phi

from the Boolean ring

B

(
X
)

\mathfrak {B}(X)

of all clopen subsets of

X
X

onto the Boolean ring

B

(
X
)

\mathfrak {B}(X)

of all clopen subsets of

Y
Y

is an isomorphism with the property that

⋂

i
=
1

∞

A
i

=
∅
(

A
i

∈

B

(
X
)
)

\bigcap \nolimits _{i = 1}^\infty {{A_i} = \emptyset ({A_i} \in \mathfrak {B}(X))}

implies

⋂

i
=
1

∞

ϕ
(

A
i

)
=
∅

\bigcap \nolimits _{i = 1}^\infty \phi ({A_i}) = \emptyset

if and only if there exists a homeomorphism

h
h

from

X
X

onto

Y
Y

such that

ϕ
(
A
)
=
h
[
A
]

\phi (A) = h[A]

for each

A
A

in

B

(
X
)

\mathfrak {B}(X)

. Theorem C. A

0
0

-dimensional space

X
X

is

N
N

-compact if and only if the collection of all the countable clopen coverings of

X
X

is complete.
Real zeros of a random sum of orthogonal polynomials
Authors: Das Minaketan 
Published: June 2010

Let

c
0

,

c
1

,

c
2

,
⋯

{c_0},{c_1},{c_2}, \cdots

be a sequence of normally distributed independent random variables with mathematical expectation zero and variance unity. Let

P
k
∗

(
x
)
(
k
=
0
,
1
,
2
,
⋯
)

P_k^ \ast (x)(k = 0,1,2, \cdots )

be the normalised Legendre polynomials orthogonal with respect to the interval

(
−
1
,
1
)

( - 1,1)

. It is proved that the average number of the zeros of

c
0

P
0
∗

(
x
)
+

c
1

P
1
∗

(
x
)
+
⋯
+

c
n

P
n
∗

(
x
)

{c_0}P_0^ \ast (x) + {c_1}P_1^ \ast (x) + \cdots + {c_n}P_n^ \ast (x)

in the same interval is asymptotically equal to

(
3

)

−
1

/

2

n

{(3)^{ - 1/2}}n

when

n
n

is large.
Restrictions of 𝐿^{𝑝} transforms
Authors: Pigno Louis 
Published: June 2010

Let G be a locally compact abelian group with dual

Γ
\Gamma

, E a subset of

Γ
\Gamma

, and

ϕ
\phi

a complex-valued function defined on

Γ
\Gamma

. Assume

ϕ
\phi

has

σ
\sigma

-compact support. In this paper we prove that

ϕ
\phi

is a multiplier of type

(

L
1

,

L

p
1

∩

L

p
2

,
E
)

(
1
≦

p
1

≦
2
,
1
>

p
2

≦
∞
)

({L^1},{L^{{p_1}}} \cap {L^{{p_2}}},E)\;(1 \leqq {p_1} \leqq 2,1 > {p_2} \leqq \infty )

if and only if

ϕ
=

f
^

\phi = \hat f

a.e. on E for some

f
∈

L

p
1

(
G
)
∩

L

p
2

(
G
)

f \in {L^{{p_1}}}(G) \cap {L^{{p_2}}}(G)

. We give applications of this result to the problems of restrictions, uniqueness, inversion and characterization of

L
p

{L^p}

transforms.
A theorem on biquadratic reciprocity
Authors: Brown Ezra 
Published: June 2010

The following theorem on biquadratic reciprocity is proved: if

p
≡
q
≡
1

(
mod

4
)

p \equiv q \equiv 1 \pmod 4

are primes for which

(
p

|

q
)
=
1

(p|q) = 1

, and if

p
=

r
2

+
q

s
2

p = {r^2} + q{s^2}

for some integers r and s, then
\[

(
p

|

q
)

4

(
q

|

p
)

4

=
1
,

if

q
≡
1

(
mod

8
)
;

=

(
−
1
)

s

,

if

q
≡
5

(
mod

8
)
.

\begin {array}{*{20}{c}} \hfill {{{(p|q)}_4}{{(q|p)}_4} = 1,\qquad {\text {if}}\;q \equiv 1\;\pmod 8;} \\ \hfill { = {{( - 1)}^s},\quad {\text {if}}\;q \equiv 5 \pmod 8.} \\ \end {array}

\]
Simple expressions for the biquadratic character of some small primes are also obtained.
Solving integral equations by 𝐿 and 𝐿⁻¹ operators
Authors: Fox Charles 
Published: June 2010

(

1

)

g
(
u
)
=

∫
0
∞

k
(
u
x
)
f
(
x
)
d
x
=

1

2
π
i

∫
C

K
(
s
)
F
(
1
−
s
)

u

−
s

d
s
,

\begin{equation}\tag {$1$}g(u) = \int _0^\infty {k(ux)f(x)dx = \frac {1}{{2\pi i}}\int _C^{} {K(s)F(1 - s){u^{ - s}}ds,} } \end{equation}

where

g
(
u
)

g(u)

and

k
(
u
)

k(u)

are known and

f
(
x
)

f(x)

is to be found.

K
(
s
)

K(s)

is the Mellin transform of

k
(
x
)

k(x)

and

F
(
s
)

F(s)

of

f
(
x
)

f(x)

; hence the second equality. L and

L

−
1

{L^{ - 1}}

denote the Laplace transform and its inverse. If

(

2

)

K
(
s
)
=

∏

i
=
1

n

Γ
(

α
i

s
+

β
i

)

∏

j
=
1

m

Γ
(

α
j

s
+

β
j

)

\begin{equation}\tag {$2$}{{K(s) = \prod \limits _{i = 1}^n {\Gamma ({\alpha _i}s + {\beta _i})} } {\prod \limits _{j = 1}^m {\Gamma ({\alpha _j}s + {\beta _j})} }}\end{equation}

then I show that a suitable combination of L and

L

−
1

{L^{ - 1}}

operators, applied to (1), can eliminate

K
(
s
)

K(s)

from the second integrand. This leaves

F
(
1
−
s
)

F(1 - s)

standing free and the Mellin transform then obtains

f
(
x
)

f(x)

from

F
(
1
−
s
)

F(1 - s)

. This solution needs tables of Laplace transforms only. When (2) does not hold, an L and

L

−
1

{L^{ - 1}}

combination may turn (1) into an integral equation whose solution is already known.
On the coefficients of Bazilevič functions
Authors: Keogh F. R. , Miller Sanford S. 
Published: June 2010

Let

B
(
α
)

B(\alpha )

denote the class of normalized

(
f
(
0
)
=
0
,

f
′

(
0
)
=
1
)

(f(0) = 0,f’(0) = 1)

, Bazilevič functions of type

α
\alpha

defined in

Δ
:

|

z

|

>
1
,

i.e.

f
(
z
)
=

[
α

∫
0
z

P
(
ζ
)
g

(
ζ

)
α

ζ

−
1

d
ζ

]

1

/

α

\Delta :|z| > 1,{\text {i.e.}}f(z) = {[\alpha \smallint _0^zP(\zeta )g{(\zeta )^\alpha }{\zeta ^{ - 1}}d\zeta ]^{1/\alpha }}

where

g
(
ζ
)

g(\zeta )

is starlike in

Δ
,
P
(
ζ
)

\Delta ,P(\zeta )

is regular with

ℜ

P
(
ζ
)
>
0

\Re \,P(\zeta ) > 0

in

Δ
\Delta

and

α
>
0

\alpha > 0

. Let

B
m

(
α
)

{B_m}(\alpha )

denote the subclass of

B
(
α
)

B(\alpha )

which is m-fold symmetric

(
f
(

e

2
π
i

/

m

z
)
=

e

2
π
i

/

m

f
(
z
)
,
m
=
1
,
2
,
⋯
)

(f({e^{2\pi i/m}}z) = {e^{2\pi i/m}}f(z),m = 1,2, \cdots )

. Functions in

B
(
α
)

B(\alpha )

have been shown to be univalent. The authors obtain sharp coefficient inequalities for functions in

B
m

(
1

/

N
)

{B_m}(1/N)

where N is a positive integer. In addition an example of a Bazilevič function which is not close-to-convex is given.
Asymptotic estimates of the 𝑛-widths in Hilbert space
Authors: Jerome Joseph W. 
Published: June 2010

Let

Ω
⊂

R
m

\Omega \subset {R^m}

be a bounded open set satisfying the restricted cone property and let R be a nonnegative selfadjoint operator on

L
2

(
Ω
)

{L^2}(\Omega )

which is the realization of a uniformly elliptic operator A of order

v
v

with suitable coefficients and principal part

a
(
x
,
ξ
)

a(x,\xi )

. Let

R

\mathcal {R}

be the ellipsoid

{
f
:
(
R
f
,
f
)
≦
1
}

\{ f:(Rf,f) \leqq 1\}

. The

L
2

{L^2}

n-widths

d
n

(

R

)

{d_n}(\mathcal {R})

satisfy

d
n

(

R

)
∼

(
c

/

n

)

v

/

2
m

{d_n}(\mathcal {R}) \sim {(c/n)^{v/2m}}

where

c
=

∫
Ω

(

∫

0
>
a
(
x
,
ξ
)
>
1

d
ξ
)

d
x

c = \smallint _\Omega {(\smallint _{0 > a(x,\xi ) > 1} {d\xi )\;dx} }

. If

B
(
u
,
v
)

B(u,v)

is a nonnegative Hermitian coercive form over a subspace

V

\mathcal {V}

of the Sobolev space

W

k
,
2

(
Ω
)

{W^{k,2}}(\Omega )

, then the n-widths of

B

=
{
f
∈

V

:
B
(
f
,
f
)
≦
1
}

\mathcal {B} = \{ f \in \mathcal {V}:B(f,f) \leqq 1\}

satisfy,

0
>

(

c
′

)

k

/

m

≦
lim
inf

d
n

(

B

)

n

k

/

m

≦
lim
sup

d
n

(

B

)

n

k

/

m

≦

(
c

)

k

/

m

0 > {(c’)^{k/m}} \leqq \lim \inf {d_n}(\mathcal {B}){n^{k/m}} \leqq \lim \sup {d_n}(\mathcal {B}){n^{k/m}} \leqq {(c)^{k/m}}

. In some cases

c
′

=
c
=
c

c’ = c = c

where c is defined in terms of an elliptic operator of order 2k. The n-widths of

B

\mathcal {B}

in

W

j
,
2

(
Ω
)
,
0
≦
j
≦
k

{W^{j,2}}(\Omega ),0 \leqq j \leqq k

, are of order

O
(

n

−
(
k
−
j
)

/

m

)
,
n
→
∞

O({n^{ - (k - j)/m}}),n \to \infty

.
Distortion theorems for a special class of analytic functions
Authors: Shaffer Dorothy Browne 
Published: June 2010

Sharp bounds are derived for the derivative of analytic functions of class

P
α

{P_\alpha }

defined by the condition

|

p
(
z
)
−
1

/

2
α

|

≦
1

/

2
α
,
0
≦
α
≦
1
,
p
(
0
)
=
1

|p(z) - 1/2\alpha | \leqq 1/2\alpha ,0 \leqq \alpha \leqq 1,p(0) = 1

. These results are improved for the class of functions with missing terms. Application is made to the class of functions with derivative

∈

P
α

\in {P_\alpha }

and the radius of convexity is determined for this class.
Dirichlet problems for singular elliptic equations
Authors: Lo Chi Yeung 
Published: June 2010

Boundary value problems are formulated for the equation
\[

(
∗
)

L
[
u
]
=

∑

i
,
j
=
1

n

a

i
j

∂
2

u

∂

x
i

∂

x
j

+

∑

i
=
1

n
−
1

b
i

∂
u

∂

x
i

+

h

x
n

∂
u

∂

x
n

+
c
u
=
f

( \ast )\quad L[u] = \sum \limits _{i,j = 1}^n {{a_{ij}}\frac {{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}}} + \sum \limits _{i = 1}^{n - 1} {{b_i}\frac {{\partial u}}{{\partial {x_i}}}} + \frac {h}{{{x_n}}}\frac {{\partial u}}{{\partial {x_n}}} + cu = f

\]
in a bounded domain

G
G

in

E
n

{E_n}

with boundary

∂
G
=

S
1

∪

S
2

\partial G = {S_1} \cup {S_2}

where

S
1

{S_1}

is in

x
n

=
0

{x_n} = 0

and

S
2

{S_2}

is in

x
n

>
0

{x_n} > 0

. A uniqueness theorem is established for

(
∗
)

( \ast )

when boundary data is only given on

S
2

{S_2}

for
\[

h
(

x
1

,
⋯
,

x

n
−
1

,
0
)
≧
1
;

h({x_1}, \cdots ,{x_{n - 1}},0) \geqq 1;

\]
; whereas an existence and uniqueness theorem for the Dirichlet problem is proved for

h
(

x
1

,

x
2

,
⋯
,

x

n
−
1

,
0
)
>
1

h({x_1},{x_2}, \cdots ,{x_{n - 1}},0) > 1

.
Beurling generalized prime number systems in which the Chebyshev inequalities fail
Authors: Hall R. S. 
Published: June 2010

It is proved that there exist systems of generalized primes in which the asymptotic distribution of integers is

N
(
x
)
=
A
x
+
O
(
x
⋅

log

−
γ

x
)

N(x) = Ax + O(x \cdot {\log ^{ - \gamma }}x)

with

A
>
0

A > 0

and

γ
∈
[
0
,
1
)

\gamma \in [0,1)

and in which the Chebyshev inequalities
\[

lim

inf

x
→
∞

π
(
x
)
log
⁡
x

x

>
0
,

lim

sup

x
→
∞

π
(
x
)
log
⁡
x

x

>
∞

\lim \inf \limits _{x \to \infty } \frac {{\pi (x)\log x}}{x} > 0,\quad \lim \sup \limits _{x \to \infty } \frac {{\pi (x)\log x}}{x} > \infty

\]
do not hold.
A remark on 𝐶_{𝜎} spaces
Authors: Reich Simeon 
Published: June 2010

We give a simple new proof of the following result, conjectured by Effros and proved by Fakhoury: Let

E
E

be a

C
σ

{C_\sigma }

space and

Z
Z

the set of extreme points of the unit ball of

E
∗

{E^ \ast }

. Then

Z
∪
{
0
}
=
{
p
∈

E
∗

:

⟨

f
g
h
,
p

⟩

=

⟨

f
,
p

⟩

⟨

g
,
p

⟩

⟨

h
,
p

⟩

Z \cup \{ 0\} = \{ p \in {E^ \ast }:\left \langle {fgh,p} \right \rangle = \left \langle {f,p} \right \rangle \left \langle {g,p} \right \rangle \left \langle {h,p} \right \rangle

for all

f
,
g
,
h

f,g,h

in

E
}

E\}

.