bluuebunny/crossref_metadata_2025 huggingface dataset:
Automorphisms of 𝐺𝐿_{𝑛}(𝑅)
Authors: Waterhouse William C.
Published: June 2010

Let R be a commutative ring in which 2 is invertible. Then all automorphisms of

G

L
n

(
R
)

G{L_n}(R)

for

n
⩾
3

n \geqslant 3

are of standard type.
A note on strongly 𝐸-reflexive inverse semigroups
Authors: O’Carroll L.
Published: June 2010

In contrast to the semilattice of groups case, an inverse semigroup S which is the union of strongly E-reflexive inverse subsemigroups need not be strongly E-reflexive. If, however, the union is saturated with respect to the Green’s relation

D

\mathcal {D}

, and in particular if the union is a disjoint one, then S is indeed strongly E-reflexive. This is established by showing that

D

\mathcal {D}

-saturated inverse subsemigroups have certain pleasant properties. Finally, in contrast to the E-unitary case, it is shown that the class of strongly E-reflexive inverse semigroups is not closed under free inverse products.
The computation of 𝑊_{∗}(𝜋,𝜔;𝑍) for 𝜋 an abelian 2-group
Authors: Gibbs David E.
Published: June 2010

Let

π
\pi

be a finite abelian 2-group and

ω
:
π
→

Z
2

=
{
+
1
,
−
1
}

\omega :\pi \to {Z_2} = \{ + 1, - 1\}

be a nontrivial homomorphism. Under these conditions, we compute the group

W
∗

(
π
,
ω
;
Z
)

{W_ \ast }(\pi ,\omega ;Z)

. We also show that

W
2

(
π
,
ω
;
Z
)

{W_2}(\pi ,\omega ;Z)

is isomorphic to

W
2

(

π
,
ω
;
Z

[

1
2

]

)

{W_2}\left ( {\pi ,\omega ;Z\left [ {\frac {1}{2}} \right ]} \right )

.
On the localisation sequence in 𝐾-theory
Authors: Snaith Victor
Published: June 2010

A description of the boundary map in Quillen’s localisation sequence is given in terms of classifying spaces of categories. Using this description the low dimensional part of the localisation sequence for a Dedekind domain is shown to coincide with the algebraically defined Bass-Tate sequence.
A curvature normal form for 4-dimensional Kähler manifolds
Authors: Johnson David L.
Published: June 2010

A curvature operator R is said to possess a normal form relative to some space of curvature operators

P

\mathcal {P}

if R is determined uniquely in

P

\mathcal {P}

by the critical points and critical values of the associated sectional curvature function. It is shown that any curvature operator of Kähler type in real dimension 4 with positive-definite Ricci curvature has a normal form relative to the space of all Kähler operators.
Almost disjoint refinement of families of subsets of 𝑁
Authors: Balcar Bohuslav , Vojtáš Peter
Published: June 2010

Without any set-theoretic assumptions, we prove that every uniform ultrafilter on the set N of all natural numbers has a Comfort system, that is, an almost disjoint refinement. Moreover, we describe one type of ideal such that the family of all subsets of N that are not contained in it has an almost disjoint refinement.
Normal closure of one-variable equations in free groups
Authors: Sibertin-Blanc C.
Published: June 2010

Let

w
(
x
)

w(x)

be a one-variable equation in a free group F of finite rank. Lyndon has proved that it is possible to associate effectively to

w
(
x
)

w(x)

the set of its solutions, whereas Appel and Lorenc have provided a simpler representation of the set inferred. In this paper, we invert the problem and demonstrate that if the elements of any set

S
⊂
F

S \subset F

are solutions of an equation

w
(
x
)

w(x)

, then

w
(
x
)

w(x)

belongs to the normal closure of finitely many short equations associated to S. A few consequences are given.
Taylor-Dirichlet series and algebraic differential-difference equations
Authors: Wadleigh Frank
Published: June 2010

It is proved that if a convergent Taylor-Dirichlet series
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript k equals 0 Overscript normal infinity Endscripts upper P Subscript k Baseline left-parenthesis s right-parenthesis e Superscript minus lamda Super Subscript k Superscript s Baseline comma s equals sigma plus i t comma lamda Subscript k Baseline element-of bold upper C comma upper P Subscript k Baseline left-parenthesis s right-parenthesis element-of bold upper C left-bracket s right-bracket comma script upper R left-parenthesis lamda Subscript k Baseline right-parenthesis up-arrow normal infinity comma">

∑

k
=
0

∞

P
k

(
s
)

e

−

λ
k

s

,

s
=
σ
+
i
t
,

λ
k

∈

C

,

P
k

(
s
)
∈

C

[
s
]
,

R

(

λ
k

)
↑
∞
,

\sum \limits _{k = 0}^\infty {{P_k}(s){e^{ - {\lambda _k}s}},\quad s = \sigma + it,{\lambda _k} \in {\mathbf {C}},{P_k}} (s) \in {\mathbf {C}}[s],\mathcal {R}({\lambda _k}) \uparrow \infty ,

\]
satisfies an algebraic differential-difference equation then the set of its exponents

{

λ
k

}

k
=
0

∞

\{ {\lambda _k}\} _{k = 0}^\infty

has a finite, linear, integral basis. This generalizes a theorem of A. Ostrowski. An application of the theorem to a problem of oscillation in neuro-muscular systems is indicated.
Cyclic vectors of Lambert’s weighted shifts
Authors: Yadav B. S. , Chatterjee S.
Published: June 2010

Let

B
(
H
)

B(H)

denote the Banach algebra of all bounded linear operators on an infinite-dimensional separable complex Hilbert space H, and let

l
2

{l^2}

be the Hilbert space of all square-summable complex sequences

x
=
{

x
0

,

x
1

,

x
2

,
…
}

x = \{ {x_0},{x_1},{x_2}, \ldots \}

. For an injective operator A in

B
(
H
)

B(H)

and a nonzero vector f in H, put
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w Subscript m Baseline equals double-vertical-bar upper A Superscript m Baseline f double-vertical-bar slash double-vertical-bar upper A Superscript m minus 1 Baseline f double-vertical-bar comma m equals 1 comma 2 comma ellipsis period">

w
m

=

‖

A
m

f

‖

/

‖

A

m
−
1

f

‖

,
m
=
1
,
2
,
…
.

{w_m} = \left \| {{A^m}f} \right \| / \left \| {{A^{m - 1}}f} \right \|,m = 1,2, \ldots .

The operator

T

A
,
f

{T_{A,f}}

on

l
2

{l^2}

, defined by

T

A
,
f

(
x
)
=
{

w
1

x
1

,

w
2

x
2

,
…
}

{T_{A,f}}(x) = \{ {w_1}{x_1},{w_2}{x_2}, \ldots \}

, is called a weighted (backward) shift with the weight sequence

{

w
m

}

m
=
1

∞

\{ {w_m}\} _{m = 1}^\infty

. This paper is concerned with the investigation of the existence of cyclic vectors of

T

A
,
f

{T_{A,f}}

. Also it is shown that if A satisfies certain nice conditions, then every transitive subalgebra of

B
(
H
)

B(H)

containing

T

A
,
f

{T_{A,f}}

coincides with

B
(
H
)

B(H)

.
Shape equivalence does not imply CE equivalence
Authors: Ferry Steve
Published: June 2010

We give an example of shape equivalent compacta X and Y such that there is no compactum Z with cell-like maps

Z
↠
X

Z \twoheadrightarrow X

and

Z
↠
Y

Z \twoheadrightarrow Y

.
Lifting semifree actions
Authors: Lashof R.
Published: June 2010

Necessary and sufficient conditions on the structure of a bundle are given, for lifting a semifree compact Lie group acting on the base to the total space.
Centralizer near-rings that are endomorphism rings
Authors: Maxson Carlton J. , Smith Kirby C.
Published: June 2010

For a finite ring R with identity and a finite unital R-module V the set

C
(
R
;
V
)
=
{
f
:
V
→
V

|

f
(
α
v
)
=
α
f
(
v
)

C(R;V) = \{ f:V \to V|f(\alpha v) = \alpha f(v)

for all

α
∈
R
,
v
∈
V
}

\alpha \in R,v \in V\}

is the centralizer near-ring determined by R and V. Those rings R such that

C
(
R
;
V
)

C(R;V)

is a ring for every R-module V are characterized. Conditions are given under which

C
(
R
;
V
)

C(R;V)

is a semisimple ring. It is shown that if

C
(
R
;
V
)

C(R;V)

is a semisimple ring then

C
(
R
;
V
)
=

End
R

(
V
)

C(R;V) = {\text {End}_R}(V)

.
Polynomial group laws. II
Authors: Nakao Zensho
Published: June 2010

Let V be a Zariski-open (i.e., cofinite) subset of an infinite field K. Call a map

m
:
V
×
V
→
V

m:V \times V \to V

separately polynomial if for each

x
∈
V

x \in V

the two partial maps

y
→
m
(
x
,
y
)
,
y
→
m
(
y
,
x
)

y \to m(x,y),y \to m(y,x)

are polynomial. If

m
:
V
×
V
→
V

m:V \times V \to V

is a separately polynomial group law, then either

V
=
K

V = K

and

m
(
x
,
y
)
=
x
+
y
+
k

m(x,y) = x + y + k

for some

k
∈
K

k \in K

or

V
=
K
−
{
k
}

V = K - \{ k\}

and

m
(
x
,
y
)
=
b
(
x
−
k
)
(
y
−
k
)
+
k

m(x,y) = b(x - k)(y - k) + k

for some

k
∈
K

k \in K

and

b
∈

K
∗

=
K
−
{
0
}

b \in {K^\ast } = K - \{ 0\}

.
On openness of 𝐻_{𝑛}-locus and semicontinuity of 𝑛th deviation
Authors: Ragusa Alfio
Published: June 2010

M. André has used the property

H
n

{H_n}

, namely the vanishing of certain homology groups, and the deviation

δ
n

{\delta _n}

to characterize some classes of rings. In the present paper the author establishes an inequality on the deviations and obtains a Nagata criterion for

H
n

{H_n}

-locus and its openness for quotients of complete intersection rings and excellent rings. The upper-semicontinuity for

δ
n

{\delta _n}

is also proved for the same classes of rings.
Variability regions for bounded analytic functions with applications to families defined by subordination
Authors: Abu-Muhanna Yusuf , MacGregor Thomas H.
Published: June 2010

We examine the set of points
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis phi left-parenthesis zeta right-parenthesis comma phi prime left-parenthesis zeta right-parenthesis comma ellipsis comma phi Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis zeta right-parenthesis right-parenthesis">

(
φ
(
ζ
)
,

φ
′

(
ζ
)
,
…
,

φ

(
n
)

(
ζ
)
)

(\varphi (\zeta ),\varphi ’(\zeta ), \ldots ,{\varphi ^{(n)}}(\zeta ))

where

|

ζ

|

>
1

|\zeta | > 1

and

φ
\varphi

varies over the class of functions analytic in the open unit disk and is either (1) uniformly bounded or (2) subordinate to a given univalent function. In each case boundary points of the set correspond to unique functions associated with finite Blaschke products. This yields information about the form of solutions to extremal problems over the classes, including the problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="max upper R e upper F left-parenthesis phi left-parenthesis zeta right-parenthesis comma phi prime left-parenthesis zeta right-parenthesis comma ellipsis comma phi Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis zeta right-parenthesis right-parenthesis">

max
Re
⁡
F
(
φ
(
ζ
)
,

φ
′

(
ζ
)
,
…
,

φ

(
n
)

(
ζ
)
)

\max \operatorname {Re} F(\varphi (\zeta ),\varphi ’(\zeta ), \ldots ,{\varphi ^{(n)}}(\zeta ))

\]
where

|

ζ

|

>
1

|\zeta | > 1

and F is analytic.
Some inequalities for entire functions
Authors: Saitoh Saburou
Published: June 2010

For any entire functions

φ
(
z
)

\varphi (z)

and

ψ
(
z
)

\psi (z)

on C with finite norm
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace StartFraction 1 Over pi EndFraction integral integral Underscript bold upper C Endscripts StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue squared exp left-parenthesis minus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis d x d y right-brace Superscript 1 slash 2 Baseline greater-than normal infinity comma">

{

1
π

∫

∫

C

|

f
(
z
)

|

2

exp
⁡
(
−

|

z

|

2

)
d
x

d
y

}

1

/

2

>
∞
,

{\left \{ {\frac {1}{\pi }\int {\int \limits _{\mathbf {C}} {|f(z){|^2}\exp ( - |z{|^2})dx\;dy} } } \right \}^{1/2}} > \infty ,

\]
we show that the inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row StartFraction 2 Over pi EndFraction integral integral Underscript bold upper C Endscripts StartAbsoluteValue phi left-parenthesis z right-parenthesis psi left-parenthesis z right-parenthesis EndAbsoluteValue squared exp left-parenthesis minus 2 StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis d x d y 2nd Row less-than-or-slanted-equals StartFraction 1 Over pi EndFraction integral integral Underscript bold upper C Endscripts StartAbsoluteValue phi left-parenthesis z right-parenthesis EndAbsoluteValue squared exp left-parenthesis minus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis d x d y StartFraction 1 Over pi EndFraction integral integral Underscript bold upper C Endscripts StartAbsoluteValue psi left-parenthesis z right-parenthesis EndAbsoluteValue squared exp left-parenthesis minus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis d x d y EndLayout">

2
π

∫

∫

C

|

φ
(
z
)
ψ
(
z
)

|

2

exp
⁡
(
−
2

|

z

|

2

)

d
x

d
y

⩽

1
π

∫

∫

C

|

φ
(
z
)

|

2

exp
⁡
(
−

|

z

|

2

)

d
x

d
y

1
π

∫

∫

C

|

ψ
(
z
)

|

2

exp
⁡
(
−

|

z

|

2

)

d
x

d
y

\begin {array}{*{20}{c}} {\frac {2}{\pi }\int {\int \limits _{\mathbf {C}} {|\varphi (z)\psi (z){|^2}\exp ( - 2|z{|^2})\;dx\;dy} } } \hfill \\ { \leqslant \frac {1}{\pi }\int {\int \limits _{\mathbf {C}} {|\varphi (z){|^2}\exp ( - |z{|^2})\;dx\;dy\frac {1}{\pi }\int {\int \limits _{\mathbf {C}} {|\psi (z){|^2}\exp ( - |z{|^2})\;dx\;dy} } } } } \hfill \\ \end {array}

\]
holds. This inequality is obtained as a special case of a general result. We also refer to some properties of a tensor product of spaces of entire functions.
Smoothness of certain degenerate elliptic equations
Authors: Lewis John L.
Published: June 2010

Given

p
>
1
,
p
≠
2

p > 1,p \ne 2

, let u be a solution to

div

(

|

grad

u

|

P
−
2

grad

u
)
=
0

{\text {div}}(|{\text {grad}}\;u{|^{P - 2}}{\text {grad}}\;u) = 0

on a domain D in Euclidean two space. We prove that if u is nonconstant and real analytic in D, then the gradient of u does not vanish in D. Some examples of Krol’ are used to show this result and a related result of Ural’tseva are nearly best possible.
A note on Hadamard products of univalent functions
Authors: Bshouty Daoud
Published: June 2010

An example is constructed to show that a modified Hadamard product of two normalized univalent functions with real coefficients may not be univalent.
A global existence theorem for Smoluchowski’s coagulation equations
Authors: White Warren H.
Published: June 2010

A countable system of nonlinear ordinary differential equations is shown to admit solutions valid over all positive time.
Best approximation of a normal operator in the Schatten 𝑝-norm
Authors: Bouldin Richard
Published: June 2010

Let A be a fixed normal operator and let

N

(
Λ
)

\mathfrak {N}(\Lambda )

denote the normal operators with spectrum contained in

Λ
\Lambda

. Provided there is some N in

N

(
Λ
)

\mathfrak {N}(\Lambda )

such that

A
−
N

A - N

belongs to the Schatten class

c
p

,
p
⩾
2

{c_p},p \geqslant 2

, the main result of this paper obtains a best approximation for A from

N

(
Λ
)

\mathfrak {N}(\Lambda )

with respect to the Schatten p-norm. A necessary and sufficient condition is given for A to have a unique best approximation in that case.
Ein Existenzsatz für gewöhnliche Differentialgleichungen in Banachräumen
Authors: Volkmann Peter
Published: June 2010

A positive answer will be given to the question in the books of Martin [4] and Deimling [2], whether initial value problems for ordinary differential equations in Banach spaces are locally solvable, provided the right-hand side of the equation is the sum of two operators of dissipative and compact type, respectively (see Theorem 2 below).
The homotopy Thom class of a spherical fibration
Authors: Marcum Howard J. , Randall Duane
Published: June 2010

We investigate the following problems. Given a spherical fibration, does the Whitehead square of its homotopy Thorn class vanish? If so, is the homotopy Thom class a cyclic homotopy class?
A topological characterization of a class of cardinals
Authors: Talamo Rodolfo
Published: June 2010

Let

m

\mathfrak {m}

be the first measurable cardinal. We say that a cardinal

α
\alpha

is Ulam-stable if, on the discrete space

D
(
α
)

D(\alpha )

of cardinal

α
\alpha

, every filter with

m

\mathfrak {m}

-intersection property can be extended to an ultrafilter with

m

\mathfrak {m}

-intersection property. The main result we prove is the following:

α
\alpha

is Ulam-stable if and only if its Hewitt-Nachbin realcompletion

υ
D
(
α
)

\upsilon D(\alpha )

is paracompact.
A hyperbolic GCM Lie algebra and the Fibonacci numbers
Authors: Feingold Alex Jay
Published: June 2010

The Fibonacci numbers are found to be involved with the Weyl-Macdonald-Kac denominator formula for a certain rank 2 Generalized Cartan Matrix Lie algebra.
A short proof of a generating function for Jacobi polynomials
Authors: Stanton Dennis
Published: June 2010

A short proof is given for Bailey’s bilinear generating function for Jacobi polynomials. It depends only upon the orthogonality relation for Jacobi polynomials and a quadratic transformation for a hypergeometric series. A q-analog is also stated.
An improved estimate for the Bloch norm of functions in Doob’s class
Authors: Hwang J. S. , Rung D. C.
Published: June 2010

For any fixed

0
>
ρ
>
2
π
,

D

(
ρ
)

0 > \rho > 2\pi ,\mathcal {D}(\rho )

is the family of all holomorphic functions in

Δ
\Delta

which satisfy (i)

f
(
0
)
=
0

f(0) = 0

, and (ii)

lim
_

z
→
τ

|

f
(
z
)

|

⩾
1

{\underline {\lim } _{z \to \tau }}|f(z)| \geqslant 1

, for all

τ
\tau

lying on some arc

Γ
f

⊆
∂
Δ

{\Gamma _f} \subseteq \partial \Delta

with arclength

|

Γ
f

|

⩾
ρ

|{\Gamma _f}| \geqslant \rho

. We showed that for each

f
∈

D

(
ρ
)

f \in \mathcal {D}(\rho )

there exists a point

z
f

∈
Δ

{z_f} \in \Delta

at which
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue f prime left-parenthesis z Subscript f Baseline right-parenthesis EndAbsoluteValue left-parenthesis 1 minus StartAbsoluteValue z Subscript f Baseline EndAbsoluteValue squared right-parenthesis greater-than-or-slanted-equals StartFraction 2 Over e EndFraction StartFraction sine left-parenthesis pi minus left-parenthesis rho slash 2 right-parenthesis right-parenthesis Over left-parenthesis pi minus left-parenthesis rho slash 2 right-parenthesis right-parenthesis EndFraction period">

|

f
′

(

z
f

)

|

(
1
−

|

z
f

|

2

)
⩾

2
e

sin
⁡
(
π
−
(
ρ

/

2
)
)

(
π
−
(
ρ

/

2
)
)

.

|f’({z_f})|(1 - |{z_f}{|^2}) \geqslant \frac {2}{e}\frac {{\sin (\pi - (\rho /2))}}{{(\pi - (\rho /2))}}.

\]
In this paper we improve this estimate by replacing the quantity

π
−
(
ρ

/

2
)

\pi - (\rho /2)

with a value

θ
(
ρ
)

\theta (\rho )

which lies between 0 and

π
−
(
ρ

/

2
)

\pi - (\rho /2)

and so improves the estimate. The value

θ
(
ρ
)

\theta (\rho )

is defined as the (unique) solution in this interval of the equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript rho Baseline left-parenthesis theta right-parenthesis equals log left-parenthesis cotangent left-parenthesis rho slash 4 right-parenthesis cotangent left-parenthesis theta slash 2 right-parenthesis right-parenthesis minus theta slash sine theta equals 0 period">

F
ρ

(
θ
)
=
log
⁡
(
cot
⁡
(
ρ

/

4
)
cot
⁡
(
θ

/

2
)
)
−
θ

/

sin
⁡
θ
=
0.

{F_\rho }(\theta ) = \log (\cot (\rho /4)\cot (\theta /2)) - \theta /\sin \theta = 0.

.
Hille-Wintner type comparison theorem for selfadjoint fourth order linear differential equations
Authors: Erbe L.
Published: June 2010

The well-known Hille-Wintner Theorem for second order linear differential equations is extended to fourth order selfadjoint equations.
𝐿¹-convergence of Fourier series with coefficients of bounded variation
Authors: Garrett J. W. , Rees C. S. , Stanojević Č. V.
Published: June 2010

It is shown that a Telyakovskii [2] theorem and its generalization are special cases of a corollary to a theorem of Garrett and Stanojević [4]. Another theorem of Garrett and Stanojević is generalized for the case of coefficients of bounded variation of order

m
⩾
1

m \geqslant 1

.
Banach spaces with the 4.3. intersection property
Authors: Lima Ȧsvald
Published: June 2010

We show that a finite-dimensional Banach space has the 4.3. intersection property if and only if it is isometric to an

l
∞

{l_\infty }

-sum of one- and two-dimensional spaces.
Inscribed and circumscribed circles to convex curves
Authors: Zamfirescu Tudor
Published: June 2010

A convex planar curve may have

2
,
3
,
…
,

c

2,3, \ldots ,\mathfrak {c}

contact points with its inscribed or circumscribed circle. One of these numbers appears in most cases: 3.
Homotopy nilpotency and associativity
Authors: Copeland Arthur H. , Shar Albert O.
Published: June 2010

If

(
Y
,
μ
)

(Y,\mu )

is an H-space then a measure of nonassociativity, called Homotopy Nuclear Nilpotency (HNN) is defined. We then calculate the HNN of thirteen H-spaces: the twelve classes of multiplications on

S
3

{S^3}

and the one class on

S

(

2
′

)

5

S_{(2’)}^5

, the 5-sphere localized away from 2.
On a question concerning countably generated 𝑧-ideals of 𝐶(𝑋)
Authors: Le Donne Attilio
Published: June 2010

In [D

1

_{1}

] the following question was asked: is every countably generated z-ideal of

C
(
X
)

C(X)

of the form

O
A

=

⋂

p
∈
A

O
p

{O^A} = \bigcap \nolimits _{p \in A} {{O^p}}

, for some zero-set A of

β
X

\beta X

? It is proved here that the answer is affirmative when X is normal and first countable; and an example is given, disproving the general conjecture. For terminology and notation see [GJ], [D

1

_{1}

].
On the lattice of 𝑓-proximities
Authors: Chattopadhyay K. C. , Vasudeva H. L.
Published: June 2010

Let (X, c) be a Čech closure space. By

M

f

(
X
,
c
)

{\mathfrak {M}_f}(X,c)

, we denote the family of all f-proximities on X which induce c. Under suitable restrictions on f, it is proved that

(

M

f

(
X
,
c
)
,
⊂
)

({\mathfrak {M}_f}(X,c), \subset )

is a complete distributive lattice.
Exponential sums related to binomial coefficient parity
Authors: Stein Alan H.
Published: June 2010

Let

α
(
n
)

\alpha (n)

be the number of 1’s in the binary expansion of n,

z
>
0

z > 0

and

ϕ
z

(
x
)
=

Σ

n
>
x

z

α
(
n
)

{\phi _z}(x) = {\Sigma _{n > x}}{z^{\alpha (n)}}

. Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta Subscript z Baseline equals left-parenthesis log left-parenthesis 1 plus z right-parenthesis right-parenthesis slash log 2 comma a left-parenthesis z right-parenthesis equals limit inf x Superscript minus theta Super Subscript z Superscript Baseline phi Subscript z Baseline left-parenthesis x right-parenthesis comma b left-parenthesis z right-parenthesis equals limit sup x Superscript minus theta Super Subscript z Superscript Baseline phi Subscript z Baseline left-parenthesis x right-parenthesis">

θ
z

=
(
log
⁡
(
1
+
z
)
)

/

log
⁡
2
,
a
(
z
)
=
lim
inf

x

−

θ
z

ϕ
z

(
x
)
,
b
(
z
)
=
lim
sup

x

−

θ
z

ϕ
z

(
x
)

{\theta _z} = (\log (1 + z))/\log 2,a(z) = \lim \inf {x^{ - {\theta _z}}}{\phi _z}(x),b(z) = \lim \sup {x^{ - {\theta _z}}}{\phi _z}(x)

. Then

0
>
a
(
z
)
⩽
1
⩽
b
(
z
)
>
2

0 > a(z) \leqslant 1 \leqslant b(z) > 2

. Furthermore, if

z
≠
1

z \ne 1

, then

a
(
z
)
>
b
(
z
)

a(z) > b(z)

.
A note on the existence of torsion-free modules with a prescribed type-set
Authors: De Munter-Kuyl Lucie
Published: June 2010

In [2], Ito has proved the existence of a completely anisotropic torsion-free abelian group of rank two with a prescribed type-set. We outline here an alternative construction in a more general setting.
The automorphism group of a near-ring
Authors: Scott S. D.
Published: June 2010

As for groups certain automorphisms of a near-ring N may be regarded as inner. The inner automorphisms of N form a normal subgroup of the automorphism group of N. Let V be a group,

Λ
\Lambda

the automorphism group of V, and N the near-ring of maps of V into V generated by

Λ
\Lambda

. It is shown that if

Λ
\Lambda

is a complete group (see [2, p. 94]), then it frequently happens that the automorphisms of N are all inner. Using this result the automorphism groups of certain near-rings may be found.
Beware the phony multiplication on Quillen’s 𝒜⁻¹𝒜
Authors: Thomason R. W.
Published: June 2010

This paper exposes a subtle but irreparable flaw in a frequently proposed construction of the graded ring structure on the algebraic K-groups of commutative rings.
On finitely generated and projective extensions of Banach algebras
Authors: Verdera Joan
Published: June 2010

We show that a finitely generated projective extension B of a commutative complex unitary Banach algebra A induces an open mapping

π
\pi

between the carrier spaces. We next prove that if

π
\pi

is a local homeomorphism then B contains an inertial subalgebra. Finally we present a necessary and sufficient condition for B to be uniform if A is.
Renorming the Banach space 𝑐₀
Authors: James Robert C.
Published: June 2010

Let

Φ
\Phi

be a subset of the unit sphere of

l
1

{l_1}

and let X be

c
0

{c_0}

, renormed by using

Φ
\Phi

and letting

x
=
y

x = y

if

|

|

|

x
−
y

|

|

|

=
0

|||x - y||| = 0

. Two conditions are given, which together imply X is “almost isometric” to a subspace of

c
0

{c_0}

. One condition is satisfied if

Φ
\Phi

is the unit sphere of a linear subset of

l
1

{l_1}

. Both conditions are satisfied if X is a quotient

c
0

/

W

{c_0}/W

and

Φ
\Phi

is the subset of the unit sphere whose members are zero on W.
Note on sequences of Mayer-Vietoris type
Authors: Dyer Eldon , Roitberg Joseph
Published: June 2010

In this largely expository note, we reexamine the construction of the homotopical Mayer-Vietoris sequence associated to a homotopy pullback. We show that in this situation, the Mayer-Vietoris sequence may be realized simply as the homotopy sequence of a suitable fibration. The usual approaches to constructing the Mayer-Vietoris sequence involve some auxiliary algebraic result, such as the Barratt-Whitehead lemma; the present approach avoids any such considerations. An additional beneficial feature of our approach is the attention paid to the bottom end of the Mayer-Vietoris sequence. Thus we are led to a cleaner proof of Proposition II.7.11 of [HMR]; moreover, we show that the converse of this latter result is also true. The homological Mayer-Vietoris sequence associated to a homotopy pushout may be established in a very similar manner, as we point out at the end of the paper.
Nonexistence theorems of higher order immersions of Dold manifolds
Authors: Kobayashi Teiichi
Published: June 2010

The purpose of this note is to improve partially the nonexistence theorems of J. J. Ucci and W.-L. Ting on immersions of Dold manifolds in Euclidean spaces.
On nowhere dense ccc 𝑃-sets
Authors: Dow Alan , van Mill Jan
Published: June 2010

We prove that no compact Hausdorff space can be covered by nowhere dense ccc P-sets. As an application it follows that if X is a compact Hausdorff space with a nonisolated P-point then

X
×
K

X \times K

is not homogeneous for any compact ccc space K.
When 𝑈(𝜅) can be mapped onto 𝑈(𝜔)
Authors: van Mill Jan
Published: June 2010

U
(
κ
)

U(\kappa )

can be mapped onto

U
(
ω
)

iff

cf

(
κ
)
=
ω

or

κ
⩾

2
ω

.

U(\omega ){\text {iff}}\;{\text {cf}}(\kappa ) = \omega {\text {or}}\kappa \geqslant {2^\omega }.

.
A Peano continuum homeomorphic to its own square but not to its countable infinite product
Authors: van Mill Jan
Published: June 2010

We give an example of a Peano continuum X with

X
2

≈
X

{X^2} \approx X

but

X
∞

≈
X

X^\infty \approx X

.
A note on sum-free sets of integers
Authors: Hu Mên Ch’ang
Published: June 2010

A set S of integers is said to be sum-free if

a
,
b
∈
S

a,b \in S

implies

a
+
b
∉
S

a + b \notin S

. Let

g
(
n
,
k
)

g(n,k)

denote the cardinality of a largest subset of

{
1
,
2
,
…
,
n
}

\{ 1,2, \ldots ,n\}

that can be partitioned into k sum-free sets. In this note we show that

g
(
n
,
2
)
=
n
−
[
n

/

5
]

g(n,2) = n - [n/5]

.
A characterization of minimal homogeneous Banach spaces
Authors: Feichtinger Hans G.
Published: June 2010

Let

G
G

be a locally compact group. It is shown that for a homogeneous Banach space

B
B

on

G
G

satisfying a slight additional condition there exists a minimal space

B

min

{B_{\min }}

in the family of all homogeneous Banach spaces which contain all elements of

B
B

with compact support. Two characterizations of

B

min

{B_{\min }}

are given, the first one in terms of "atomic" representations. The equivalence of these two characterizations is derived by means of certain (bounded) partitions of unity which are of interest for themselves.
Another 𝑞-extension of the beta function
Authors: Andrews George E. , Askey Richard
Published: June 2010

Another

q
q

-extension of the beta function is given. This one has a special case that is a symmetric extension of the symmetric beta distribution.
Some properties of 𝐻-spaces of rank 2
Authors: McGibbon C. A.
Published: June 2010

We study

H
H

-spaces with 3 cells. After one suspension the attaching map for the top cell is shown to have order at most 2. For such

H
H

-spaces localized away from 2, we obtain a new proof of a classification theorem due to Zabrodsky. We also show that under suitable restrictions in the 3 cell case, the vanishing of all Whitehead products implies the existence of a multiplication.
Spaces with dense conditionally compact subsets
Authors: Berner Andrew J.
Published: June 2010

A subset

S
S

of a topological space is said to be conditionally compact if every infinite subset of

S
S

has a limit point in the space. If a space has a dense conditionally compact subset, it follows that it is pseudocompact, but the converse is not true. Examples are given of spaces that are pseudocompact, do not have dense conditionally compact subsets, but do have compactifications that are products of first countable spaces. For locally compact spaces, though, with such compactifications, the continuum hypothesis implies that pseudocompactness is equivalent to having a dense conditionally compact subset. A locally compact pseudocompact space without a dense conditionally compact subset is described.
A variant of the chain rule for differential calculus
Authors: Graff Richard A.
Published: June 2010

A version of the chain rule is developed which can be applied to the construction of solutions to quasi-linear hyperbolic partial differential equations.
Representations of Baer ∗-semigroups
Authors: Gudder Stanley P. , Michel John R.
Published: June 2010

Various types of Hilbert space representations for a Baer *-semigroup

S
S

are defined. The representations are characterized in terms of *-positive functions on

S
S

which possess additivity and consistency properties.
The restriction of Whittaker modules to certain parabolic subalgebras
Authors: Wallach Nolan R.
Published: June 2010

Certain twisted cohomology spaces of modules for the standard maximal parabolic subalgebra of

s

l

(
n
,

C

)

\mathfrak {s}\mathfrak {l}(n,{\mathbf {C}})

are studied. These results are shown to imply new proofs of results of Kostant on Whittaker modules in the special case of real forms of products of

s

l

(
n
,

C

)

\mathfrak {s}\mathfrak {l}(n,{\mathbf {C}})

’s.
Remarks on bimeasures
Authors: Kluvánek Igor
Published: June 2010

Techniques of vector valued measures are used to obtain disintegrations of bimeasures.
On subharmonicity of the capacity of the spectrum
Authors: Słodkowski Zbigniew
Published: June 2010

It is shown that if

T
λ

{T_\lambda }

is an analytic operator valued function (or if

f
f

,

g
g

belong to a uniform algebra

A
A

) then

n
n

th diameters and logarithmic capacity of

σ
(

T
λ

)

\sigma ({T_\lambda })

(or of the set

g
(

f

−
1

(
λ
)
)

g({f^{ - 1}}(\lambda ))

) are subharmonic functions of

λ
\lambda

(on a suitable domain).
𝑀-ideals, the strong 2-ball property and some renorming theorems
Authors: Yost David
Published: June 2010

Examples are given of

M
M

-ideals in Banach spaces which do not possess the strong

2
2

-ball property. This solves a problem of Alfsen and Effros. A previous example is shown to be incorrect. The technique used to construct these examples is then employed to prove negative renorming theorems for Banach spaces. The following is representative: every separable Banach space has an equivalent norm which is strictly convex but not locally uniformly convex.
The states associated with approximately inner automorphisms
Authors: Choda Marie
Published: June 2010

Let

M
M

be a

I

I

1

{\text {I}}{{\text {I}}_1}

-factor acting standardly on a Hilbert space

H
H

. For an approximately inner automorphism

θ
\theta

of

M
M

, there exists a state

φ
\varphi

on

B
(
H
)

B(H)

associated with

θ
\theta

. If the symmetry

σ
\sigma

of

M
⊗
M

M \otimes M

is approximately inner on

M
⊗
M

M \otimes M

, then, by restricting the state associated with

σ
\sigma

to

B
(
H
)
⊗
I

B(H) \otimes I

, we have a hypertrace of

M
M

.
On a certain group structure for polynomials
Authors: Moh T. T.
Published: June 2010

A problem of C. C. Yang concerning level sets in complex analysis is generalized and algebraized to a problem about a group structure of polynomials. The group structure is established and several partial solutions of Yang’s problem follow at once.
Results on bi-univalent functions
Authors: Styer D. , Wright D. J.
Published: June 2010

When the class

σ
\sigma

of bi-univalent functions was first defined, it was known that functions of the form

ϕ
∘

ψ

−
1

∈
σ

\phi \circ {\psi ^{ - 1}} \in \sigma

when

ϕ
\phi

and

ψ
\psi

are univalent, map the unit disc

B

{\mathbf {B}}

onto a set containing

B

{\mathbf {B}}

, and satisfy

ϕ
(
0
)
=
ψ
(
0
)
=
0

\phi (0) = \psi (0) = 0

,

ϕ
′

(
0
)
=

ψ
′

(
0
)

\phi ’(0) = \psi ’(0)

. It is shown here that such functions form a proper subset of

σ
\sigma

, and that

σ
\sigma

is a proper subset of the set of functions of the form

ϕ
∘

ψ

−
1

\phi \circ {\psi ^{ - 1}}

, where

ϕ
\phi

and

ψ
\psi

are locally univalent, at most

2
2

-valent, each maps a subregion of

B

{\mathbf {B}}

univalently onto

B

{\mathbf {B}}

, and

ϕ
(
0
)
=
ψ
(
0
)
=
0

\phi (0) = \psi (0) = 0

,

ϕ
′

(
0
)
=

ψ
′

(
0
)

\phi ’(0) = \psi ’(0)

,

ψ

−
1

(
0
)
=
0

{\psi ^{ - 1}}(0) = 0

. It is also shown that there are

f
(
z
)
=
z
+

a
2

z
2

+
⋯

f(z) = z + {a_2}{z^2} + \cdots

in

σ
\sigma

with

|

a
2

|

>
4

/

3

\left | {{a_2}} \right | > 4/3

. However, doubt is cast that

|

a
2

|

\left | {{a_2}} \right |

can be as large as

3

/

2

3/2

.
Distal homomorphisms of nonmetric minimal flows
Authors: McMahon D. , Wu T. S.
Published: June 2010

In a recent paper, R. Ellis proved that the Furstenberg structure theorem for distal minimal flows with metric phase space held for flows with nonmetric phase space. In this paper the authors remove the metric constraint from the relativized Ellis-Furstenberg structure theorem for distal homomorphisms and for the Veech structure theorem for point-distal minimal flows.
AR-maps obtained from cell-like maps
Authors: Kozlowski George , van Mill Jan , Walsh John J.
Published: June 2010

The recent solution by J. van Mill of a problem of Borsuk involves using a convexification procedure in order to produce a map

f
f

from the Hilbert cube

Q
Q

to a non-

A
R

AR

X
X

so that each point-inverse

f

−
1

(
x
)

{f^{ - 1}}(x)

is a Hilbert cube. A different method of obtaining

A
R

AR

-maps from cell-like maps is described and is used to show that if there is a dimension raising cell-like map, then there is an integer

n
n

and a map

f
f

from

Q
Q

to a non-

A
R

AR

X
X

so that each point-inverse

f

−
1

(
x
)

{f^{ - 1}}(x)

is an

n
n

-cell or a point.
Boolean powers of groups
Authors: Lawrence John
Published: June 2010

A group is

B
B

-separating if a Boolean power of the group has a unique Boolean algebra. It is proved that a finite subdirectly irreducible group is

B
B

-separating if and only if it is non-Abelian.
Solvable subgroups in prime rings
Authors: Lanski Charles
Published: June 2010

Let

R
R

be a prime ring with center

Z
Z

and group of units

U
U

. The main theorem shows that any solvable normal subgroups of

U
U

must lie in

Z
Z

, provided that

R
R

is not a domain,

Z
Z

is large enough, and that the

Z
Z

-subalgebra generated by

U
U

contains a nonzero ideal of

R
R

. One consequence is the determination of the structure of

R
R

when

R
R

has an involution and the subgroup of

U
U

generated by the symmetric units is solvable.
Some inequalities related to Hölder’s inequality
Authors: Neugebauer C. J.
Published: June 2010

In this paper we generalize the Jodeit-Jones-Moser inequalities relating
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 less-than-or-slanted-equals x minus left-parenthesis integral Subscript 0 Superscript x Baseline f right-parenthesis Superscript p Baseline comma if double-vertical-bar f double-vertical-bar Subscript q Baseline less-than-or-slanted-equals 1">

0
⩽
x
−

(

∫
0
x

f

)

p

,

if

‖
f
‖

q

⩽
1

0 \leqslant x - {\left ( {\int _0^xf} \right )^p},{\text {if}}{\left \| f \right \|_q} \leqslant 1

.
Π-regular variation
Authors: Geluk J. L.
Published: June 2010

A function

U
:

R
+

→

R
+

U:{R^ + } \to {R^ + }

is said to be

Π
\Pi

-regularly varying with exponent

α
\alpha

if

U
(
x
)

x

−
α

U(x){x^{ - \alpha }}

is nondecreasing and there exists a positive function

L
L

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction upper U left-parenthesis lamda x right-parenthesis slash lamda Superscript alpha Baseline minus upper U left-parenthesis x right-parenthesis Over x Superscript alpha Baseline upper L left-parenthesis x right-parenthesis EndFraction right-arrow log lamda left-parenthesis x right-arrow normal infinity right-parenthesis for lamda greater-than 0 period">

U
(
λ
x
)

/

λ
α

−
U
(
x
)

x
α

L
(
x
)

→
log
⁡
λ

(
x
→
∞
)

for

λ

 > 0

.

\frac {{U(\lambda x)/{\lambda ^\alpha } - U(x)}} {{{x^\alpha }L(x)}} \to \log \lambda \quad (x \to \infty ){\text {for}}\lambda {\text { > 0}}{\text {.}}

\]
Suppose
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper U With caret left-parenthesis t right-parenthesis equals integral Subscript 0 Superscript normal infinity Baseline e Superscript minus t x Baseline d upper U left-parenthesis x right-parenthesis exists for t greater-than 0 period">

U
^

(
t
)
=

∫
0
∞

e

−
t
x

d
U
(
x
)

exists for 

t
>
0.

\hat {U}(t)=\int _0^\infty {e^{ - tx}}dU(x){\text {exists for }}t > 0.

\]
We prove that

U
U

is

Π
\Pi

-regularly varying iff

U
U

is

Π
\Pi

-regularly varying.
An Abel-Tauber theorem for partitions
Authors: Geluk J. L.
Published: June 2010

Suppose

Λ
=
{

λ
1

,

λ
2

,
…
}

\Lambda = \{ {\lambda _1},{\lambda _2}, \ldots \}

is a given set of real numbers such that

0
>

λ
1

>

λ
2

>
…
.

Let 

n
(
u
)
=

∑

λ
k

⩽
u

1

0 > {\lambda _1} > {\lambda _2} > \ldots .{\text {Let }}n(u) = {\sum _{{\lambda _k} \leqslant u}}1

and

P
(
u
)

P(u)

the number of solutions of

n
1

λ
1

+

n
2

λ
2

+
…
⩽
u

{n_1}{\lambda _1} + {n_2}{\lambda _2} + \ldots \leqslant u

in integers

n
i

⩾
0

{n_i} \geqslant 0

. An Abel-Tauber theorem concerning

n
(
u
)

n(u)

and log

P
(
u
)

P(u)

is proved for the case where

n
(
t
x
)

/

n
(
t
)
→
1
(
t
→
∞
)

n(tx)/n(t) \to 1(t \to \infty )

for

x
>
0

x > 0

.
A weakly infinite-dimensional compactum which is not countable-dimensional
Authors: Pol Roman
Published: June 2010

A compact metric space is constructed which is neither a countable union of zero-dimensional sets nor has an essential map onto the Hilbert cube.
Convergence in trace ideals
Authors: Simon B.
Published: June 2010

We give an elementary proof of a theorem of Arazy which presents necessary and sufficient conditions on a symmetric sequence so that the associated symmetrically normed trace ideal has the property that if

A
n

→
A

{A_n} \to A

in the weak operator topology and

‖

A
n

‖

→

‖
A
‖

\left \| {{A_n}} \right \| \to \left \| A \right \|

, then

‖

A
n

−
A

‖

→
0

\left \| {{A_n} - A} \right \| \to 0

.
Integral representation of multiplicative, involution preserving operators in 𝔏(ℭ₀(𝔖,𝔄),𝔅)
Authors: Choy Stephen T. L.
Published: June 2010

Bounded linear operators from the space of continuous vector-valued functions which preserve multiplication and involution are characterized in terms of their representing measures. A key role is played by the Arens product in the second dual of a Banach algebra.
On a theorem of Stanojević
Authors: Bray William O.
Published: June 2010

A new direct proof of a theorem of Stanojević is given. Consequently it is proven that the Fomin class

F

p

(
1
>
p
⩽
2
)

{\mathcal {F}_p}(1 > p \leqslant 2)

is a subclass of the class

B

V

∩

C

\mathcal {B}\mathcal {V} \cap \mathcal {C}

, where

C

\mathcal {C}

is the Garrett-Stanojević class and

B

V

\mathcal {B}\mathcal {V}

is the class of null sequences of bounded variation. This also provides a new direct proof of Fomin’s theorem.
On Mahler’s measure of a polynomial
Authors: Durand Alain
Published: June 2010

Let

P
P

be a polynomial with complex coefficients. We denote by

M
(
P
)

M(P)

the Mahler measure of

P
P

(resp. the maximum modulus of

P
P

on the disk

|
z
|

⩽
1
)

\left | z \right | \leqslant 1)

). We prove here that

M
(
P
)
=
inf

‖

P
Q

‖

M(P) = \inf \left \| {PQ} \right \|

where the infimum is taken over all polynomials

Q
Q

with complex coefficients satisfying

Q
(
0
)
=
1

Q(0) = 1

.
On the analytic cohomology of a domain in a Stein manifold
Authors: Eastwood Michael G.
Published: June 2010

Suppose

M
M

is an open subset of a Stein manifold without isolated points and that

V
V

is a holomorphic vector-bundle on

M
M

admitting a holomorphic connection. Then

H
q

(
M
,

O

(
V
)
)

{H^q}(M,\mathcal {O}(V))

is either zero or of infinite dimension.
A generalized Bessel inequality
Authors: Prestini Elena
Published: June 2010

To any pair of diadic intervals

[
I
,
ω
]

[I,\omega ]

with / Q

I
⊆
[
0
,
1
]

I \subseteq [0,1]

and

ω
=
[

N
ω

,

N
ω

+

|

I

|

−
1

)

\omega = [{N_\omega },{N_\omega } + |I{|^{ - 1}})

we associate the function
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript left-bracket upper I comma omega right-bracket Baseline left-parenthesis x right-parenthesis equals StartAbsoluteValue upper I EndAbsoluteValue Superscript negative 1 slash 2 Baseline e Superscript i upper N Super Subscript w Superscript x Baseline chi Subscript upper I Baseline left-parenthesis x right-parenthesis">

u

[
I
,
ω
]

(
x
)
=

|

I

|

−
1

/

2

e

i

N
w

x

χ
I

(
x
)

{u_{[I,\omega ]}}(x) = |I{|^{ - 1/2}}{e^{i{N_w}x}}{\chi _I}(x)

. In this paper we will give a condition under which a collection

B
B

of such pairs satisfies the inequality

B
B

for any

a
b

ab

in

f
f

.
Riemann 𝑅₁-summability of independent, identically distributed random variables
Authors: Cuzick Jack
Published: June 2010

Let

X
,

X
1

,

X
2

,
…

X,{X_1},{X_2}, \ldots

be i.i.d. random variables. It is shown that

E

|
X
|

log
+

log
+

|
X
|

>
∞

E\left | X \right |{\log ^ + }{\log ^ + }\left | X \right | > \infty

is a sufficient condition for Riemann

R
1

{R_1}

-summability of

{

X
n

}

\left \{ {{X_n}} \right \}

to

E
X

EX

. Counterexamples are provided which indicate that the strongest possible necessary condition of moment type is

E

|
X
|

>
∞

E\left | X \right | > \infty

. However under weak regularity conditions on the tails of the distribution of

X
X

the sufficient condition is also shown to be necessary.
Martingale convergence via the square function
Authors: Chen Louis H. Y.
Published: June 2010

By exploiting the natural setting of a convergence theorem of Burkholder, a direct and elementary proof of the theorem is given. This proof is also new for the martingale convergence theorem and the martingale transform convergence theorem which are corollaries to the above-mentioned convergence theorem.
Chains and discrete sets in zero-dimensional compact spaces
Authors: Bell Murray , Ginsburg John
Published: June 2010

Let

X
X

be a compact zero-dimensional space and let

B
(
X
)

B(X)

denote the Boolean algebra of all clopen subsets of

X
X

. Let

κ
\kappa

be an infinite cardinal. It is shown that if

B
(
X
)

B(X)

contains a chain of cardinality

κ
\kappa

then

X
×
X

X \times X

contains a discrete subset of cardinality

κ
\kappa

. This complements a recent result of J. Baumgartner and P. Komjath relating antichains in

B
(
X
)

B(X)

to the

π
\pi

-weight of

X
X

.
Cycle polynomials
Authors: Hwang F. K.
Published: June 2010

Let

G
G

be a graph consisting of

m
m

vertex-disjoint cycles with possibly different numbers of vertices on each cycle. We want to count the number of ways of selecting

k
k

vertices in

G
G

such that there are exactly

l
l

edges spanned by these

k
k

vertices. For

m
=
1

m = 1

, the problem is equivalent to the Whitworth bracelet problem with two colors and a closed-form solution is known. In this paper we show that the solution for the many-cycle case can be written as a sum of the solutions for single-cycle cases.
Über gewöhnliche Differentialungleichungen zweiter Ordnung
Authors: Lemmert Roland
Published: June 2010

A theorem about the separation of sub- and superfunctions

υ
\upsilon

and

w
w

by solutions of an ordinary differential equation of second order is proved, where

υ
⩾
w

\upsilon \geqslant w

throughout the given interval. Examples show that the condition imposed on the right side

f
f

of the equation is sharp, and that an analogous theorem is not true for Laplace’s equation, even in the case

f
≡
0
,
υ

f \equiv 0,\upsilon

sub- and

w
w

superharmonic.
A note on Noetherian P.I. rings
Authors: Braun Amiram
Published: June 2010

Let

R
=
F
{

x
1

,
…
,

x
k

}

R = F\{ {x_1}, \ldots ,{x_k}\}

be an affine semiprime p.i. ring, Krull-

dim
⁡
(
R
)
=
2

\dim (R) = 2

and

F
F

a central subfield. It is proved that

R
R

is noetherian iff

R
R

is a left (right) finite module over a commutative affine subring.
Boundedness of maximal functions and singular integrals in weighted 𝐿^{𝑝} spaces
Authors: Rubio de Francia José L.
Published: June 2010

Given a weight

w
(
x
)
>
0

w(x) > 0

in

R

n

{{\mathbf {R}}^n}

, necessary and sufficient conditions are found for the boundedness of the Hardy-Littlewood maximal function and singular integral operators from

L
p

(
w
)

{L^p}(w)

to some other weighted

L
p

{L^p}

space. The dual question is also considered and partially answered.
On the dimension of the zero or infinity tending sets for linear differential equations
Authors: Muldowney James S.
Published: June 2010

There are well-known conditions which guarantee that all solutions to a system of

n
n

differential equations

x
′

=
A
(
t
)
x

x’ = A(t)x

,

t
∈
[
0
,
ω
)

t \in [0,\omega )

, satisfy

lim

t
→
ω

|

x
(
t
)

|

=
0

{\lim _{t \to \omega }}|x(t)| = 0

. Under certain stability assumptions on the system, Hartman [2], Coppel [1] and Macki and Muldowney [4] give necessary and sufficient [sufficient] conditions that the system has at least one nontrivial solution satisfying

lim

t
→
ω

|

x
(
t
)

|

=
0
[
∞
]

\lim \limits _{t \to \omega } |x(t)| = 0[\infty ]

. These results are extended by studying a sequence of matrices

A

[
k
]

(
t
)

{A^{[k]}}(t)

,

k
=
1
,
…
,
n

k = 1, \ldots ,n

, related to

A
(
t
)

A(t)

such that, under the same stability assumptions as before, the given system has an

(
n
−
k
+
1
)

(n - k + 1)

-dimensional zero [infinity] tending solution set if and only if [if] all nontrivial solutions of the system

y
′

=

A

[
k
]

(
t
)
y

y’ = {A^{[k]}}(t)y

tend to zero [infinity].
Proof of a conjecture of Erdős about the longest polynomial
Authors: Bojanov B. D.
Published: June 2010

In 1939 P. Erdös conjectured that the Chebyshev polynomial

T
n

(
x
)

{T_n}(x)

has a maximal arc-length in

[
−
1
,
1
]

[ - 1,1]

among the polynomials of degree

n
n

which are bounded by 1 in

[
−
1
,
1
]

[ - 1,1]

. We prove this conjecture for every natural

n
n

.
Killing vector fields on complete Riemannian manifolds
Authors: Yorozu Shinsuke
Published: June 2010

We discuss Killing vector fields with finite global norms on complete Riemannian manifolds whose Ricci curvatures are nonpositive or negative.
A simple proof of the extension theorem of sequences of divided powers in characteristic 𝑝
Authors: Takeuchi Mitsuhiro
Published: June 2010

Using the idea of relative Hopf modules, a short proof of the extension theorem of sequences of divided powers in irreducible cocommutative Hopf algebras over a field of characteristic

p
>
0

p > 0

is presented.