bluuebunny/crossref_metadata_2025 huggingface dataset:
On the blow up of 𝑢_{𝑡} at quenching
Authors: Deng Keng , Levine Howard A.
Published: June 2010

Let

Ω
\Omega

be a bounded convex domain in

R

n

{{\mathbf {R}}^n}

with smooth boundary. We consider the problems

(
C
)

:

u
t

=
Δ
u
+
φ

(
u
)

\left ( C \right ):{u_t} = \Delta u + \varphi \left ( u \right )

in

Ω
×

(

0
,
T

)

\Omega \times \left ( {0,T} \right )

, while

u
=
0

u = 0

on

∂
Ω
×

(

0
,
T

)

\partial \Omega \times \left ( {0,T} \right )

and

u

(

x
,
0

)

=

u
0

(
x
)

u\left ( {x,0} \right ) = {u_0}\left ( x \right )

. Here
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis u right-parenthesis colon left-parenthesis negative normal infinity comma upper A right-parenthesis right-arrow left-parenthesis 0 comma normal infinity right-parenthesis left-parenthesis upper A greater-than 0 right-parenthesis">

φ

(
u
)

:

(

−
∞
,
A

)

→

(

0
,
∞

)

(

A
>
0

)

\varphi \left ( u \right ):\left ( { - \infty ,A} \right ) \to \left ( {0,\infty } \right )\left ( {A > 0} \right )

satisfies

φ
′

(
u
)

≥
0
,
φ

(
u
)

≥
0

\varphi ’\left ( u \right ) \geq 0,\varphi \left ( u \right ) \geq 0

, and

lim

u
→

A
−

φ

(
u
)

=
+
∞

{\lim _{u \to {A^ - }}}\varphi \left ( u \right ) = + \infty

, while

u
0

{u_0}

satisfies

Δ

u
0

(
x
)

+
φ

(

u
0

(
x
)

)

≥
0

\Delta {u_0}\left ( x \right ) + \varphi \left ( {{u_0}\left ( x \right )} \right ) \geq 0

. We show that if

u
u

quenches (reaches

A
A

in finite time), then the quenching points are in a compact subset of

Ω
\Omega

and

u
t

{u_t}

blows up. We also extend the result to the third boundary value problem.
On two-generator discrete groups of Möbius transformations
Authors: Tan De Lin
Published: June 2010

Assume that Möbius transformations

f
f

and

g
g

generate a discrete group. We obtain the following generalizations of Jørgensen’s inequalities. If

tr

(
f
g

f

−
1

g

−
1

)
≠
1

{\text {tr}}(fg{f^{ - 1}}{g^{ - 1}}) \ne 1

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue t r squared left-parenthesis f right-parenthesis minus 2 EndAbsoluteValue plus StartAbsoluteValue trace left-parenthesis f g f Superscript negative 1 Baseline g Superscript negative 1 Baseline right-parenthesis minus 1 EndAbsoluteValue greater-than-or-equal-to 1">

|

t

r

2

(
f
)
−
2

|

+

|

tr

(
f
g

f

−
1

g

−
1

)
−
1

|

≥
1

\left | {{\text {t}}{{\text {r}}^2}(f) - 2} \right | + \left | {{\text {tr}}(fg{f^{ - 1}}{g^{ - 1}}) - 1} \right | \geq 1

. If

tr

(
f
g

f

−
1

g

−
1

)
=
1

{\text {tr}}(fg{f^{ - 1}}{g^{ - 1}}) = 1

, then either

t

r

2

(
f
)
=
2

ort

r

2

(
f
)
=
1

or

|

t

r

2

(
f
)
−
2

|

1
2

{\text {t}}{{\text {r}}^2}(f) = 2{\text {ort}}{{\text {r}}^2}(f) = 1{\text {or}}\left | {{\text {t}}{{\text {r}}^2}(f) - 2} \right |\frac {1}{2}

and

|

t

r

2

(
f
)
−
1

|

>

1
2

\left | {{\text {t}}{{\text {r}}^2}(f) - 1} \right | > \frac {1}{2}

. If

t

r

2

(
f
)
≠
1

{\text {t}}{{\text {r}}^2}(f) \ne 1

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue t r squared left-parenthesis f right-parenthesis minus 1 EndAbsoluteValue plus StartAbsoluteValue trace left-parenthesis f g f Superscript negative 1 Baseline g Superscript negative 1 Baseline right-parenthesis EndAbsoluteValue greater-than-or-equal-to 1">

|

t

r

2

(
f
)
−
1

|

+

|

tr

(
f
g

f

−
1

g

−
1

)

|

≥
1

\left | {{\text {t}}{{\text {r}}^2}(f) - 1} \right | + \left | {{\text {tr}}(fg{f^{ - 1}}{g^{ - 1}})} \right | \geq 1

. If

t

r

2

(
f
)
=
1

{\text {t}}{{\text {r}}^2}(f) = 1

then either
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="trace left-parenthesis f g f Superscript negative 1 Baseline g Superscript negative 1 Baseline right-parenthesis equals 0 ortr left-parenthesis f g f Superscript negative 1 Baseline g Superscript negative 1 Baseline right-parenthesis equals 1">

tr

(
f
g

f

−
1

g

−
1

)
=
0

ortr

(
f
g

f

−
1

g

−
1

)
=
1

{\text {tr}}(fg{f^{ - 1}}{g^{ - 1}}) = 0{\text {ortr}}(fg{f^{ - 1}}{g^{ - 1}}) = 1

and

|

tr

(
f
g

f

−
1

g

−
1

)
−
1

|

1
2

\left | {{\text {tr}}(fg{f^{ - 1}}{g^{ - 1}}) - 1} \right |\frac {1}{2}

.
On the Popov-Pommerening conjecture for groups of type 𝐴_{𝑛}
Authors: Tan Lin
Published: June 2010

The present paper gives an affirmative answer to the Popov-Pommerening conjecture in the case where the reductive group

G
G

in the conjecture is of type

A
n

{A_n}

with

n
≤
4

n \leq 4

, and provides a subgroup

H
H

of

G

L
5

(
k
)

G{L_5}(k)

such that the algebra

A
H

{A^H}

is finitely generated, but is not spanned by the invariant standard bitableaux.
On metric spaces with uniform normal structure
Authors: Khamsi M. A.
Published: June 2010

In this work, we prove that metric spaces with uniform normal structure have a kind of intersection property, which is equivalent to reflexivity in Banach spaces.
Inner gradings and Galois extensions with normal basis
Authors: Beattie Margaret
Published: June 2010

We prove that for

G
G

a finite group, a

G
G

-graded Azumaya algebra over a commutative ring has inner grading if and only if an associated Galois extension has normal basis.
Arveson nests and operator factorization along commutative subspace lattices
Authors: Daughtry John , Johns Ronald
Published: June 2010

Similar commutative subspace lattices (CSL’s) are shown to be unitarily equivalent if certain sublattices (which may be taken to be nests!) are unitarily equivalent and a technical condition is satisfied. This result provides a connection between existing results for arbitrary similarities of countable CSL’s and similarities of general CSL’s by operators near the identity. One consequence is the generalizaton to CSL’s of a theorem of David Pitts on the relationship between similarity and unitary equivalence of nests he calls "injective."
New combinatorial interpretations of two analytic identities
Authors: Agarwal A. K.
Published: June 2010

Two generalized partition theorems involving partitions with "

n
+
1

n + 1

copies of

n
n

" and "

n
+
2

n + 2

copies of

n
n

", respectively, are proved. These theorems have potential of yielding infinite Rogers-Ramanujan type identities on MacMahon’s lines. Five particular cases are also discussed. Among them three are known and two provide new combinatorial interpretations of two known

q
q

-identities.
The isometries of 𝐻¹_{}ℋ
Authors: Cambern Michael , Jarosz Krzysztof
Published: June 2010

Let

H

\mathcal {H}

be a finite-dimensional complex Hilbert space. In this article we characterize the linear isometries of the Banach space

H

H

1

H_\mathcal {H}^1

onto itself. We show that

T
T

is such an isometry iff

T
T

is of the form

T
F
(
z
)
=
U
F
(
ψ
(
z
)
)

ψ
′

(
z
)

TF(z) = UF(\psi (z))\psi ’(z)

, for

F
∈

H

H

1

F \in H_\mathcal {H}^1

and

z
z

in the unit disc, where

ψ
\psi

is a conformal map of the disc onto itself, and

U
U

is a unitary operator on

H

\mathcal {H}

.
Hypertranscendence of the functional equation 𝑔(𝑥²)=[𝑔(𝑥)]²+𝑐𝑥
Authors: Borwein Peter
Published: June 2010

The functional equation

g
(

x
2

)
=

[
g
(
x
)

]
2

+
c
x

g({x^2}) = {[g(x)]^2} + cx

has a unique nontrivial solution that is analytic at zero. We show, for

c
>
0

c > 0

, that the solution of this equation satisfies no algebraic differential equation.
Convergence to ends for random walks on the automorphism group of a tree
Authors: Cartwright Donald I. , Soardi P. M.
Published: June 2010

Let

μ
\mu

be a probability on a free group

Γ
\Gamma

of rank

r
≥
2

r \geq 2

. Assume that

Supp
⁡

(
μ
)

\operatorname {Supp} \left ( \mu \right )

is not contained in a cyclic subgroup of

Γ
\Gamma

. We show that if

(

X
n

)

n
≥
0

{\left ( {{X_n}} \right )_{n \geq 0}}

is the right random walk on

Γ
\Gamma

determined by

μ
\mu

, then with probability 1,

X
n

{X_n}

converges (in the natural sense) to an infinite reduced word. The space

Ω
\Omega

of infinite reduced words carries a unique probability

ν
\nu

such that

(

Ω
,
ν

)

\left ( {\Omega ,\nu } \right )

is a frontier of

(

Γ
,
μ

)

\left ( {\Gamma ,\mu } \right )

in the sense of Furstenberg [10]. This result extends to the right random walk

(

X
n

)

\left ( {{X_n}} \right )

determined by a probability

μ
\mu

on the group

G
G

of automorphisms of an arbitrary infinite locally finite tree

T
T

. Assuming that

Supp
⁡

(
μ
)

\operatorname {Supp} \left ( \mu \right )

is not contained in any amenable closed subgroup of

G
G

, then with probability 1 there is an end

ω
\omega

of

T
T

such that

X
n

υ

{X_n}\upsilon

converges to

ω
\omega

for each

υ
∈
T

\upsilon \in T

. Our methods are principally drawn from [9] and [10].
A simple proof of Livingston’s inequality for Carathéodory functions
Authors: Delsarte Philippe , Genin Yves
Published: June 2010

The Livingston determinant inequality involving the Maclaurin coefficients of a Carathéodory function are derived in a straightforward manner by use of the Riesz-Herglotz representation and the Schwarz inequality. The result is extended to the case of matrix-valued functions.
Complete Möbius strips minimally immersed in 𝑅³
Authors: Ishihara Tōru
Published: June 2010

In the present paper, we will determine all complete minimal immersions of a Möbius strip into

R

3

{{\mathbf {R}}^3}

with finite total curvature.
Partitions with equal products. II
Authors: Kelly John B.
Published: June 2010

The following theorem is proved: Let

k
≥
3

k \geq 3

and

r
r

be positive integers. There exist infinitely many integers having

r
r

partitions into

k
k

parts such that the products of the integers in each partition are equal. Moreover, these partitions are mutually disjoint, i.e., no integer occurs in more than one of them. Of some additional interest is a lemma stating that a certain class of elliptic curves has positive rank over

Q

{\mathbf {Q}}

.
An example on null sets of parabolic measures
Authors: Wu Jang-Mei
Published: June 2010

A set

E
E

on the real line of Hausdorff dimension 1 is constructed, such that the graph of

E
E

on the boundary of any

C
2

{C^2}

domain

{

t
>
τ
(
x
)

}

\left \{ {t > \tau (x)} \right \}

, or on the boundary of any

Lip

1
2

{\text {Lip}}\tfrac {1}{2}

domain

{

x
>
χ
(
t
)

}

\left \{ {x > \chi (t)} \right \}

, is null with respect to the parabolic measure associated with any parabolic operator

L
=
a
(
x
,
t
)

∂
2

/

∂

x
2

−
∂

/

∂
t

L = a(x,t){\partial ^2}/\partial {x^2} - \partial /\partial t

on

R

2

;
a

{{\mathbf {R}}^2};a

is Hölder continuous and

0
>

Λ
1

≤
a
≤

Λ
2

>
∞

0 > {\Lambda _1} \leq a \leq {\Lambda _2} > \infty

for some constants

Λ
1

{\Lambda _1}

and

Λ
2

{\Lambda _2}

.
The oblique derivative problem for the heat equation in Lipschitz cylinders
Authors: Brown Russell M.
Published: June 2010

We consider a class of initial-boundary value problems for the heat equation on

(
0.
T
)
×
Ω

(0.T) \times \Omega

with

Ω
\Omega

a bounded Lipschitz domain in

R

n

{{\mathbf {R}}^n}

. On the lateral boundary,

(
0
,
T
)
×
∂
Ω
=

Σ
T

(0,T) \times \partial \Omega = {\Sigma _T}

, we specify

⟨

α
,
∇
u

⟩

\left \langle {\alpha ,\nabla u} \right \rangle

where

∇
u

\nabla u

denotes the spatial gradient of the solution and

α
:

Σ
T

→
{
x
:

|

x

|

=
1
}

\alpha :{\Sigma _T} \to \{ x:|x| = 1\}

is a continuous vector field satisfying

⟨

α
,
ν

⟩

≥
μ
>
0

\left \langle {\alpha ,\nu } \right \rangle \geq \mu > 0

with

ν
\nu

the unit normal to

∂
Ω

\partial \Omega

. On the initial surface,

{
0
}
×
Ω

\{ 0\} \times \Omega

, we require that the solution vanish. The lateral data is taken from

L
p

(

Σ
T

)

{L^p}({\Sigma _T})

. For

p
∈
(
2
−
,
∞
)

p \in (2 - ,\infty )

, we show existence and uniqueness of solutions to this problem with estimates for the parabolic maximal function of the spatial gradient of the solution.
Smooth polynomial paths with nonanalytic tangents
Authors: McLeod Robert M. , Meisters Gary H.
Published: June 2010

We prove that there exist

C
∞

{C^\infty }

functions

φ
:

R

t

×

R

x

→

R

\varphi :{{\mathbf {R}}_t} \times {{\mathbf {R}}_x} \to {\mathbf {R}}

such that although

φ

(

t
,
x

)

\varphi \left ( {t,x} \right )

is a polynomial in

x
x

for each

t
t

in
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper R comma ModifyingAbove phi With dot left-parenthesis 0 comma x right-parenthesis identical-to left-parenthesis partial-differential phi slash partial-differential t right-parenthesis left-parenthesis 0 comma x right-parenthesis">

R

,

φ
˙

(

0
,
x

)

≡

(

∂
φ

/

∂
t

)

(

0
,
x

)

{\mathbf {R}},\dot \varphi \left ( {0,x} \right ) \equiv \left ( {\partial \varphi /\partial t} \right )\left ( {0,x} \right )

need not even be analytic in

x
x

let alone polynomial. It was shown earlier by one of the authors [Meisters] that this cannot happen if

φ
\varphi

satisfies the group-property (even locally) of flows, namely if

φ

(

s
,
φ

(

t
,
x

)

)

=
φ

(

s
+
t
,
x

)

\varphi \left ( {s,\varphi \left ( {t,x} \right )} \right ) = \varphi \left ( {s + t,x} \right )

.
A class of simple Lie algebras of characteristic three
Authors: Brown Gordon
Published: June 2010

We show the existence of a certain infinite class of simple Lie algebras of characteristic three. These algebras, although of neither classical nor Cartan type, resemble algebras of Cartan type in their relationship to each other.
The image of 𝐻_{∗}(𝐵𝑆𝑈;𝑍_{𝑝}) in 𝐻_{∗}(𝐵𝑈;𝑍_{𝑝})
Authors: Papastavridis Stavros
Published: June 2010

In this note we construct explicit polynomial generators for the image of

H
∗

(

B
S
U
;

Z
p

)

{H_ * }\left ( {BSU;{Z_p}} \right )

inside

H
∗

(

B
U
;

Z
p

)

{H_ * }\left ( {BU;{Z_p}} \right )

.
Short proofs of two hypergeometric summation formulas of Karlsson
Authors: Ekhad Shalosh B.
Published: June 2010

Karlsson [2] gave elegant proofs of two hypergeometric summation formulas conjectured by Gosper, that were mentioned in [1]. Here I give new proofs that are much shorter, but less elegant.
Pairwise incompatible generic families
Authors: Guzicki Wojciech
Published: June 2010

Let

M
M

be a countable model of

Z

F

−

{\mathbf {Z}}{{\mathbf {F}}^ - }

. There exists a family

F

\mathcal {F}

of

2

2
ω

{2^{{2^\omega }}}

models of

Z

F

−

{\mathbf {Z}}{{\mathbf {F}}^ - }

each obtained from

M
M

by adjoining an

M
M

-generic family of

2
ω

{2^\omega }

Cohen reals, such that no two distinct models in

F

\mathcal {F}

have a common extension to a model of

Z

F

−

{\mathbf {Z}}{{\mathbf {F}}^ - }

with the same ordinals.
Brangesian spaces in the polydisk
Authors: Singh Dinesh
Published: June 2010

In this paper we extend to the polydisk

D
2

{D^2}

a theorem of L. de Branges which characterizes the class of all Hilbert spaces that are contractively contained in the classical Hardy space

H
2

{H^2}

of the disk and which are invariant under the shift

S
S

acting as an isometry. Our theorem characterizes Hilbert spaces which are vector subspaces of

H
2

(

D
2

)

{H^2}({D^2})

and which are invariant under the operators of multiplication by the coordinate functions whose actions are isometric and which doubly commute. We do not use contractivity.
On a problem of S. Banach from The Scottish book
Authors: Kazarian K. S.
Published: June 2010

Denote byB(Φ)B\left ( \Phi \right )the closure inL2{L^2}of linear combinations of functions of the orthonormal systemΦ={Φn}n=1∞\Phi = \left \{ {{\Phi _n}} \right \}_{n = 1}^\infty. Denote byB(Φ)⊥B{(\Phi )^ \bot }the orthogonal complement ofB(Φ)B(\Phi )inL2{L^2}. We prove the following theorem: Letφ\varphibe a nonnegative measurable function defined on[0,+∞)\left [ {0, + \infty } \right )such thatlimx→+∞x2/φ(x)=0{\text {li}}{{\text {m}}_{x \to + \infty }}{x^2}/\varphi \left ( x \right ) = 0andNNis a positive integer orN=+∞N = + \infty. Then there is a uniformly bounded orthonormal systemΦ(N)={Φn}n=1∞{\Phi ^{\left ( N \right )}} = \left \{ {{\Phi _n}} \right \}_{n = 1}^\inftysuch thatdim⁡B(Φ(N))⊥=N\operatorname {dim} B{({\Phi ^{(N)}})^ \bot } = Nand, for every nontrivial functionfffrom<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B left-parenthesis normal upper Phi Superscript left-parenthesis upper N right-parenthesis Baseline right-parenthesis Superscript up-tack Baseline comma integral phi left-parenthesis StartAbsoluteValue f left-parenthesis t right-parenthesis EndAbsoluteValue right-parenthesis d t equals plus normal infinity">B(Φ(N))⊥,∫φ(|f(t)|)dt=+∞B{({\Phi ^{(N)}})^ \bot },\int {\varphi \left ( {\left | {f\left ( t \right )} \right |} \right )} dt = + \infty.
On the impossibility of ruler-only constructions
Authors: Baston V. J. , Bostock F. A.
Published: June 2010

The paper points out that there are several interpretations in the literature of what is meant by a geometric construction. It is shown that these differences in interpretation are important, since certain classic results (including the Mohr-Mascheroni Theorem) are true under one but false under another.
Arc-smoothness and contractibility in Whitney levels
Authors: Illanes Alejandro
Published: June 2010

Let

X
X

be a continuum. Let

2
x

{2^x}

(resp.,

C
(
X
)

C(X)

) be the space of all nonempty closed subsets (resp., subcontinua) of

X
X

. In this paper we prove that if

X
X

is an arc-smooth continuum, then there exists an admissible Whitney map

μ
:

2
x

→

R

\mu :{2^x} \to {\mathbf {R}}

such that

μ

|

C
(
X
)
:
C
(
X
)
→

R

\mu |C(X):C(X) \to {\mathbf {R}}

is admissible and for every

t
∈
(
0
,
μ
(
X
)
)
,

μ

−
1

(
t
)

t \in (0,\mu (X)),{\mu ^{ - 1}}(t)

and

(
μ

|

C
(
X
)

)

−
1

(
t
)

{(\mu |C(X))^{ - 1}}(t)

are arc-smooth. This answers a question by J. T. Goodykoontz, Jr. Also we give an example of a contractible continuum

X
X

such that, for every Whitney map

υ
:
C
(
X
)
→

R

\upsilon :C(X) \to {\mathbf {R}}

there exists

t
∈
(
0
,
υ
(
X
)
)

t \in (0,\upsilon (X))

such that

υ

−
1

(
t
)

{\upsilon ^{ - 1}}(t)

is not contractible.
Proper hereditary shape equivalences preserve small weak infinite-dimensionality
Authors: Millspaugh Richard P.
Published: June 2010

A space is said to be small weakly infinite dimensional if it has a basis

B
B

such that the collection of finite unions of elements of

B
B

is inessential. A characterization of small weak infinite dimensionality is given for locally compact spaces. This characterization is then used to prove that if

f
:
X
→
Y

f:X \to Y

is a proper hereditary shape equivalence from a metric space

X
X

which is small weakly infinite dimensional onto a locally compact metric space

Y
Y

, then

Y
Y

is small weakly infinite dimensional.
Finite coverings by cosets of normal subgroups
Authors: Parmenter M. M.
Published: June 2010

In this brief note, we characterize those groupsGGwhich can be covered by finitely many cosetsaiMi{a_i}{M_i}of maximal normal subgroupsMi{M_i}, where the covering is irredundant and not allMi{M_i}are equal. This refines an earlier result of Brodie, Chamberlain, and Kappe, who characterized those groups which can be covered by finitely many proper normal subgroups.
A characterization of the approximation order of translation invariant spaces of functions
Authors: Jia Rong Qing
Published: June 2010

Let

S
S

be a space of functions on

R

\mathbb {R}

with the following properties: (i)

S
S

is translation invariant, i.e.,

f
∈
S

f \in S

implies

f
(
⋅
±
1
)
∈
S

f( \cdot \pm 1) \in S

; (ii)

dim
⁡
S

|

[
0
,
1
]

>
∞

\operatorname {dim} S{|_{[0,1]}} > \infty

; (iii)

S
S

is closed under uniform convergence on compact sets. In this paper we characterize the approximation order of

S
S

by proving the following: Theorem.

S
S

provides approximation of order

k
k

if and only if

S
S

contains a compactly supported function

ψ
\psi

such that the Fourier transform

ψ
^

\hat \psi

of

ψ
\psi

satisfies

ψ
^

(
0
)
=
1

\hat \psi (0) = 1

and

D
α

ψ
^

(
2
π
j
)
=
0

{D^\alpha }\hat \psi (2\pi j) = 0

for

0
≤
α
>
k

0 \leq \alpha > k

and

j
∈

Z

∖
{
0
}

j \in \mathbb {Z}\backslash \{ 0\}

. This result extends a corresponding result of de Boor and DeVore, who proved the above theorem for the case

k
=
1

k = 1

.
Forced oscillations with rapidly vanishing nonlinearities
Authors: Kannan R. , Nagle Kent
Published: June 2010

We obtain sufficient conditions for the existence of periodic solutions of nonlinear problems where the nonlinearity vanishes infinitely often.
Enumeration of Hamiltonian cycles and paths in a graph
Authors: Liu C. J.
Published: June 2010

First, we show that the determinant of a given matrix can be expanded by its principal minors together with a set of arbitrary parameters. The enumeration of Hamiltonian cycles and paths in a graph is then carried out by an algebraic method. Three types of nonalgebraic representation are formulated. The first type is given in terms of the determinant and permanent of a parametrized adjacent matrix. The second type is presented by a determinantal function of multivariables, each variable having domain 0, 1. Formulas of the third type are expressed by spanning trees of subgraphs. When applying the formulas to a complete multipartite graph, one can easily find the results.
A precomposition analysis of linear operators on 𝑙^{𝑝}
Authors: Hudson Steve M.
Published: June 2010

Given a function

g
g

, the operator that sends the function

f
(
x
)

f(x)

to the function

f
(
g
(
x
)
)

f(g(x))

is called a precomposition operator. If

g
g

preserves measure on its domain, at least approximately, then this operator is bounded on all the

L
p

{L^p}

spaces. We ask which operators can be written as an average of precomposition operators. We give sufficient, almost necessary conditions for such a representation when the domain is a finite set. The class of operators studied approximate many commonly used positive operators defined on

L
p

{L^p}

of the real line, such as maximal operators. A major tool is the combinatorial theorem of distinct representatives, commonly called the marriage theorem. A strong connection between this theorem and operators of weak-type 1 is demonstrated.
Zeros of solutions of a second order nonlinear differential inequality
Authors: Wong Fu Hsiang
Published: June 2010

Under suitable assumptions on

r
,
g

r,g

, and

F
F

, we show that every zero of a solution of the nonlinear differential inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis r left-parenthesis t right-parenthesis y prime left-parenthesis t right-parenthesis right-parenthesis prime plus g left-parenthesis t right-parenthesis upper F left-parenthesis y left-parenthesis t right-parenthesis right-parenthesis less-than-or-equal-to 0 left-parenthesis greater-than-or-equal-to 0 right-parenthesis">

(
r
(
t
)

y
′

(
t
)

)
′

+
g
(
t
)
F
(
y
(
t
)
)
≤
0
(
≥
0
)

(r(t)y’(t))’ + g(t)F(y(t)) \leq 0( \geq 0)

\]
is simple.
The Rochlin invariant of surgered, sewn link exteriors
Authors: Woodard Mark R.
Published: June 2010

One method of producing

3
3

-manifolds is the sewing-up construction of W. R. Brakes. This involves identifying the boundary components of the exterior of a

2
2

-component link. Under certain assumptions, this process yields homology handles which can be surgered to obtain homology spheres. The main result is a formula for the Rochlin invariant of homology spheres obtained by this construction when the link is proper.
The gap between the first two eigenvalues of a one-dimensional Schrödinger operator with symmetric potential
Authors: Abramovich S.
Published: June 2010

We prove the inequality

λ
2

[

V
1

]
−

λ
1

[

V
1

]
≥

λ
2

[

V
0

]
−

λ
1

[

V
0

]

{\lambda _2}[{V_1}] - {\lambda _1}[{V_1}] \geq {\lambda _2}[{V_0}] - {\lambda _1}[{V_0}]

for the difference of the first two eigenvalues of one-dimensional Schrödinger operators

−

d
2

d

x
2

+

V
i

(
x
)
,
i
=
0
,
1

- \frac {{{d^2}}}{{d{x^2}}} + {V_i}(x),i = 0,1

, where

V
1

{V_1}

and

V
0

{V_0}

are symmetric potentials on

(
a
,
b
)

(a,b)

and on

(
a
,
(
a
+
b
)

/

2
)

(a,(a + b)/2)

, and

V
0

−

V
1

{V_0} - {V_1}

is decreasing on

(
a
,
(
3
a
+
b
)

/

4
)

(a,(3a + b)/4)

.
The hyperspaces of subcontinua of the pseudo-arc and of solenoids of pseudo-arcs are Cantor manifolds
Authors: Krupski Paweł
Published: June 2010

New proofs of the above facts are based on specific homogeneity properties of the pseudo-arc and of solenoids of pseudo-arcs.
Approximation of fixed points of asymptotically nonexpansive mappings
Authors: Schu Jürgen
Published: June 2010

Let

T
T

be an asymptotically nonexpansive self-mapping of a non-empty closed, bounded, and starshaped (with respect to zero) subset of a smooth reflexive Banach space possessing a duality mapping that is weakly sequentially continuous at zero. Then, if id-

T
T

is demiclosed and

T
T

satisfies a strengthened regularity condition, the iteration process

z

n
+
1

:=

μ

n
+
1

T
n

(

z
n

)

{z_{n + 1}}: = {\mu _{n + 1}}{T^n}({z_n})

converges strongly to some fixed point of

T
T

, provided

(

μ
n

)

({\mu _n})

has certain properties.
The free lattice-ordered group over a nilpotent group
Authors: Darnel Michael R.
Published: June 2010

We show that the free lattice-ordered group over a finitely generated torsionfree nilpotent group is

l
l

-solvable of some finite rank.
On intersections of compacta in Euclidean space
Authors: Dranishnikov A. N.
Published: June 2010

Let

X

{\mathbf {X}}

be a codimension-three tame compactum in Euclidean space

E

n

{{\mathbf {E}}^n}

. If

dim
⁡

X

×

Y

>
n

\operatorname {dim} {\mathbf {X}} \times {\mathbf {Y}} > n

, then every map

f
:

Y

→

E

n

f:{\mathbf {Y}} \to {{\mathbf {E}}^n}

can be approximated by map

g
g

with

X

∩
Im
⁡
g
=
∅

{\mathbf {X}} \cap \operatorname {Im} g = \emptyset

.
Systems of equations in the predual of a von Neumann algebra
Authors: Marsalli Michael
Published: June 2010

A von Neumann algebra

A

\mathcal {A}

on a separable, complex Hilbert space

H

\mathcal {H}

has property

A

n

{{\mathbf {A}}_n}

if for every

n
×
n

n \times n

array

{

f

i
,
j

}

\{ {f_{i,j}}\}

of elements in the predual there exists sequences

{

x
i

}
,
{

y
j

}

\{ {x_i}\} ,\{ {y_j}\}

in

H

\mathcal {H}

such that

f

i
,
j

(
A
)
=
(
A

x
i

,

y
j

)

{f_{i,j}}(A) = (A{x_i},{y_j})

for all

A
A

in

A

\mathcal {A}

and

0
≤
i
,
j
>
n

0 \leq i,j > n

. We show that the von Neumann algebras with property

A

ℵ
0

{{\mathbf {A}}_{{\aleph _0}}}

are the von Neumann algebras with properly infinite commutant. We describe how these properties are transformed by the tensor product. We characterize the abelian von Neumann algebras with property

A

n

{{\mathbf {A}}_n}

.
Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition
Authors: del Pino Manuel A. , Manásevich Raúl F.
Published: June 2010

We show existence for a nonlinear fourth-order boundary value problem under a nonresonance condition involving a two-parameter linear eigenvalue problem. We also state extensions of this result to certain higher-order P.D.E. cases.
A symbolic calculus for layer potentials on 𝐶¹ curves and 𝐶¹ curvilinear polygons
Authors: Lewis Jeff E.
Published: June 2010

A symbolic calculus for some algebras of Mellin operators on the finite interval

J
≡

[

0
,
1

]

J \equiv \left [ {0,1} \right ]

is developed. The algebras are ample enough to include singular integral operators and analytic double layer potentials and their adjoints on

C
1

{C^1}

curves and piecewise

C
1

{C^1}

curves with corners. Fredholmness and the index of the operators on

L
p

(
J
)

{L^p}\left ( J \right )

are completely determined by the principal symbol on

L
p

(
J
)

,

Smb

l

1

/

p

{L^p}\left ( J \right ),{\text {Smb}}{{\text {l}}^{1/p}}

.
Multiplicative decomposition of probability measures
Authors: Luschgy Harald
Published: June 2010

We investigate decompositions of probability measures having the feature that the component of interest in the decomposition of a product probability measure is the product of the corresponding components in the decomposition of the factors.
A remark on incomparable ultrafilters in the Rudin-Keisler order
Authors: Butkovičová Eva
Published: June 2010

If

2

>

c

>

c

{2^{ > {\mathbf {c}}}} > {\mathbf {c}}

and

p
p

is an ultrafilter on

ω
\omega

of character

c

{\mathbf {c}}

then there exist many ultrafilters that are incomparable with

p
p

in the Rudin-Keisler order.
A growth condition for Fourier coefficients of functions of bounded entropy norm
Authors: Lang W. Christopher
Published: June 2010

A simple growth condition is found for the Fourier coefficients of continuous

1
1

-periodic functions of bounded

k
k

-entropy norm. This growth condition is equivalent to the condition given by Dabrowski. The entropy norms of certain random Fourier series are considered.
Construction of characteristic functions for classes of infinitely differentiable functions
Authors: Allouche M. , Couture R. , Tsai C. T.
Published: June 2010

Characteristic functions are constructed for classes of infinitely differentiable functions defined on a half-line and for Paley-Wiener classes. A corresponding result is given for normal operators defined in Hilbert space.
The space (𝑙_{∞}/𝑐₀,𝑤𝑒𝑎𝑘) is not a Radon space
Authors: de María José L. , Rodríguez-Salinas Baltasar
Published: June 2010

Talagrand [10] gives an example of a Banach space with weak topology which is not a Radon space, independently of their weight. This result gives an answer to a question formulated by Schwartz [9]. In this paper, following the papers of Drewnowski and Roberts [1] and Talagrand [10], we prove that the classical space (

l
∞

/

c
0

{l_\infty }/{c_0}

, weak) is not a Radon space.
Interpretations into 𝑝-algebras
Authors: Cartagena O. , González C. , Lewin R. , Rubilar O.
Published: June 2010

We prove that there is only one interpretation from the variety of

p
p

-algebras into itself.
On the relationship between density and weak density in Boolean algebras
Authors: Bozeman Kyle
Published: June 2010

Given a homogeneous, complete Boolean algebra

B
B

, it is shown that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d left-parenthesis upper B right-parenthesis less-than-or-equal-to min left-parenthesis 2 Superscript greater-than wd left-parenthesis upper B right-parenthesis Baseline comma sup left-brace right-brace colon lamda of cc left-parenthesis right-parenthesis upper B colon separator separator greater-than greater-than lamda wd left-parenthesis right-parenthesis upper B right-parenthesis">

d

(
B
)
≤
min
(

2

>

wd

(
B
)

,
sup
{

λ

c
(
B
)

:
λ
>

wd

(
B
)
}
)

{\text {d}}(B) \leq \min ({2^{ > {\text {wd}}(B)}},\sup \{ {\lambda ^{c(B)}}:\lambda > {\text {wd}}(B)\} )

in ZFC, where

d
(
B
)

d(B)

is the density,

wd

(
B
)

{\text {wd}}(B)

is the weak density, and

c
(
B
)

c(B)

is the cellularity of

B
B

. A corollary to this result is that

d
(
B
)
=

wd

(
B
)

d(B) = {\text {wd}}(B)

in ZFC+GCH.
Creation and annihilation operators for anharmonic oscillators
Authors: Friedlander Leonid
Published: June 2010

Positive elliptic operators from the Helffer-Robert classes of pseudodifferential operators on the line can be approximately factored as products of creation and annihilation operators. Operators of anharmonic oscillators belong to these classes.
The stability of certain functional equations
Authors: Baker John A.
Published: June 2010

The aim of this paper is to prove the stability (in the sense of Ulam) of the functional equation:
\[

f
(
t
)
=
α
(
t
)
+
β
(
t
)
f
(
ϕ
(
t
)
)
,

f(t) = \alpha (t) + \beta (t)f(\phi (t)),

\]
where

α
\alpha

and

β
\beta

are given complex valued functions defined on a nonempty set

S
S

such that

sup
{

|

β
(
t
)

|

:
t
∈
S
}
>
1

\sup \{ |\beta (t)|:t \in S\} > 1

and

ϕ
\phi

is a given mapping of

S
S

into itself.
The spectral extension property and extension of multiplicative linear functionals
Authors: Meyer Michael J.
Published: June 2010

Let

A

\mathcal {A}

be a commutative Banach algebra. Denote the spectral radius of an element

a
a

in

A

\mathcal {A}

by

ρ

A

(
a
)

{\rho _\mathcal {A}}(a)

. An extension of

A

\mathcal {A}

is a Banach algebra

B

\mathcal {B}

such that

A

\mathcal {A}

is algebraically, but not necessarily continuously, embedded in

B

\mathcal {B}

. We view

A

\mathcal {A}

as a subalgebra of

B

\mathcal {B}

. If

B

\mathcal {B}

is an extension of

A

\mathcal {A}

then

S

p

B

(
a
)
∪
{
0
}
⊆
S

p

A

(
a
)
∪
{
0
}

S{p_\mathcal {B}}(a) \cup \{ 0\} \subseteq S{p_\mathcal {A}}(a) \cup \{ 0\}

and thus

ρ

B

(
a
)
≤

ρ

A

(
a
)
,
∀
a
∈

A

{\rho _\mathcal {B}}(a) \leq {\rho _\mathcal {A}}(a),\forall a \in \mathcal {A}

. Let us say that

A

\mathcal {A}

has the spectral extension property if

ρ

B

(
a
)
=

ρ

A

(
a
)

{\rho _\mathcal {B}}(a) = {\rho _\mathcal {A}}(a)

for all

a
∈

A

a \in \mathcal {A}

and all extensions

B

\mathcal {B}

of

A

\mathcal {A}

, that

A

\mathcal {A}

has the strong spectral extension property if

S

p

B

(
a
)
∪
{
0
}
=
S

p

A

(
a
)
∪
{
0
}

S{p_\mathcal {B}}(a) \cup \{ 0\} = S{p_\mathcal {A}}(a) \cup \{ 0\}

for all

a
∈

A

a \in \mathcal {A}

and all extensions

B

\mathcal {B}

of

A

\mathcal {A}

, and that

A

\mathcal {A}

has the multiplicative Hahn-Banach property if every multiplicative linear functional

χ
\chi

on

A

\mathcal {A}

has a multiplicative linear extension to every commutative extension

B

\mathcal {B}

of

A

\mathcal {A}

. We give characterizations of these properties for semisimple commutative Banach algebras.
Uniform 𝐿²-weighted Sobolev inequalities
Authors: Chiarenza Filippo , Ruiz Alberto
Published: June 2010

We prove the inequality (1) for weights

w
w

in a class which contains the class

J
p

,
p
>
(
n
−
1
)

/

2

{J_p},p > (n - 1)/2

, introduced by C. Fefferman and D. H. Phong in studying eigenvalues of Schrödinger operators. In our case,

C
C

is independent of the lower order terms of

P
P

. As a consequence we prove unique continuation theorem for solutions of

Δ
+
V
,
V

\Delta + V,V

in the same class.
A commutative Banach algebra with factorization of elements but not of pairs
Authors: Ouzomgi S. I.
Published: June 2010

We find a one-point Gleason part

ϕ
\phi

off the Šilov boundary of

H
∞

(
Δ
)

{H^\infty }(\Delta )

such that the maximal ideal

M
ϕ

{M_\phi }

factors but such that pairs do not factor in

M
ϕ

{M_\phi }

.
Projectively normal adjunction surfaces
Authors: Andreatta Marco , Ballico Edoardo
Published: June 2010

On a projective surface

S
S

polarized by a very ample line bundle

L
L

one can consider the adjoint bundles

(

K
S

⊗
L

)

⊗
n

=

L

⊗
n

{({K_S} \otimes L)^{ \otimes n}} = {L^{ \otimes n}}

and the adjunction mappings associated to them. Suppose these mappings are embeddings (it is well known when this is the case: see [So-VdV]). We prove that these embeddings are projectively normal for

n
≥
2

n \geq 2

and we describe some counterexamples for

n
=
1

n = 1

. For

n
≥
2

n \geq 2

we show that the ideals of the image of

S
S

are generated by quadrics.
On 𝑟-separated sets in normed spaces
Authors: Arias-de-Reyna Juan
Published: June 2010

The separation of a bounded set

A
A

in a metric space

δ
(
A
)

\delta (A)

is defined as the supremum of the numbers

r
>
0

r > 0

such that there exists a sequence

(

x
n

)

({x_n})

in

A
A

such that

d
(

x
n

,

x
m

)
>
r

d({x_n},{x_m}) > r

for every

n
≠
m

n \ne m

. We prove for every bounded set

A
A

in a Banach space that

δ
(
A
)
=
δ
(

co

(
A
)
)

\delta (A) = \delta ({\text {co}}(A))

where

co

(
A
)

{\text {co}}(A)

denotes the convex hull of

A
A

. This yields a generalization of Darbo’s fixed point theorem.
A local classification of 2-type surfaces in 𝑆³
Authors: Hasanis Th. , Vlachos Th.
Published: June 2010

The only spherical surfaces in

E
4

{E^4}

that are either of

1
1

-type or of

2
2

-type are portions of ordinary spheres, minimal surfaces in

S
3

{S^3}

, and Riemannian products of two plane circles of different radii.
The rate of spatial decay of nonnegative solutions of nonlinear parabolic equations and inequalities
Authors: Lair Alan V.
Published: June 2010

Let

L
L

be a uniformly parabolic linear partial differential operator. We show that nonnegative solutions of the differential inequality

L
u
≤
c
(
u
+

|

∇
u

|

)

Lu \leq c(u + |\nabla u|)

on

R

n

×
(
0
,
T
)

{{\mathbf {R}}^n} \times (0,T)

for which

u
(
x
,
T
)
=

0

(
exp
⁡

(

−
δ

|

x

|

2

)
)

u(x,T) = {\mathbf {0}}(\exp {\text {(}} - \delta |x{|^2}))

must be identically zero if the constant

δ
\delta

is sufficiently large. An analogous result is given for nonlinear systems.
On noncontinuous chaotic functions
Authors: Ceder Jack
Published: June 2010

A function

f
:
[
0
,
1
]
→
[
0
,
1
]

f:[0,1] \to [0,1]

is constructed such that for each two distinct points

x
x

and

y
y

in

[
0
,
1
]

[0,1]

the sequence
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace StartAbsoluteValue f Superscript n Baseline left-parenthesis x right-parenthesis minus f Superscript n Baseline left-parenthesis y right-parenthesis EndAbsoluteValue right-brace Subscript n equals 0 Superscript normal infinity">

{

|

f
n

(
x
)
−

f
n

(
y
)

|

}

n
=
0

∞

\left \{ {\left | {{f^n}(x) - {f^n}(y)} \right |} \right \}_{n = 0}^\infty

is dense in

[
0
,
1
]

[0,1]

. Here

f
n

{f^n}

is the

n
n

th iterate of

f
f

. Moreover a Baire 2 function can be constructed so that the above condition is valid for all distinct

x
x

and

y
y

in a dense open subset of

[
0
,
1
]

[0,1]

.
Good approximations and continued fractions
Authors: Kraaikamp Cor , Liardet Pierre
Published: June 2010

Let

(

q
n

)
n

{({q_n})_n}

be the sequence of best approximation denominators of an irrational number

α
\alpha

. The set of real numbers

x
x

for which

q
n

x
→
0

{q_n}x \to 0

(
mod
1
)

(\bmod 1)

is studied. It is shown that a number

x
x

belongs to

α

Z

(
mod

1)

\alpha \mathbb {Z}(\bmod {\text {1)}}

if and only if a simple condition on the speed of the convergence related to an arithmetic property of

α
\alpha

is satisfied. This set is uncountable whenever

α
\alpha

has unbounded partial quotients.
On measurable local homomorphisms
Authors: Sasvári Zoltán
Published: June 2010

We prove that every measurable local homomorphism between locally compact groups is continuous.
Integral means, bounded mean oscillation, and Gel′fer functions
Authors: Girela Daniel
Published: June 2010

A Gelfer function

f
f

is a holomorphic function in the unit disc

D
=
{
z
:

|

z

|

>
1
}

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

such that

f
(
0
)
=
1

f(0) = 1

and

f
(
z
)
+
f
(
w
)
≠
0

f(z) + f(w) \ne 0

for all

z
,
w

z,w

in

D
D

. The family

G
G

of Gelfer functions contains the family

P
P

of holomorphic functions

f
f

in

D
D

with

f
(
0
)
=
1

f(0) = 1

and Re

f
>
0

f > 0

in

D
D

. Yamashita has recently proved that if

f
f

is a Gelfer function then

f
∈

H
p

,
0
>
p
>
1

f \in {H^p},0 > p > 1

while

log
⁡
f
∈
BMOA

\log f \in \operatorname {BMOA}

and

‖

log
⁡
f

‖

BMO
⁡

A

2

≤
π

/

2

{\left \| {\log f} \right \|_{\operatorname {BMO}{{\text {A}}_2}}} \leq \pi /\sqrt 2

. In this paper we prove that the function

λ
(
z
)
=
(
1
+
z
)

/

(
1
−
z
)

\lambda (z) = (1 + z)/(1 - z)

is extremal for a very large class of problems about integral means in the class

G
G

. This result in particular implies that

G
⊂

H
p

,
0
>
p
>
1

G \subset {H^p},0 > p > 1

, and we use it also to obtain a new proof of a generalization of Yamashita’s estimation of the BMOA norm of

log
⁡
f
,
f
∈
G

\log f,f \in G

.
Finite order vanishing of boundary values of holomorphic mappings
Authors: Lakner Mitja
Published: June 2010

Suppose that

f
f

is a holomorphic function on a half-disc in the complex plane that extends continuously to the diameter

E
E

, such that the extension maps

E
E

to a double cone with the vertex at the origin. If the extension

f

|

E

f{|_E}

has an isolated zero at the origin, then

f
f

vanishes to finite order at 0.
von Neumann algebras which are second dual spaces
Authors: Chu Cho-Ho
Published: June 2010

A

σ
\sigma

-finite von Neumann algebra is a second dual if and only if it is atomic.
Concerning continuous images of rim-metrizable continua
Authors: Tuncali H. Murat
Published: June 2010

Mardesic (1962) proved that if

X
X

is a continuous, Hausdorff, infinite image of a compact ordered space

K
K

under a light mapping in the sense of ordering, then

ω
(
X
)
=
ω
(
K
)

\omega (X) = \omega (K)

. He also proved (1967) that a continuous, Hausdorff image of a compact ordered space is rim-metrizable. Treybig (1964) proved that the product of two infinite nonmetrizable compact Hausdorff spaces cannot be a continuous image of a compact ordered space. We prove some analogues of these results for continuous Hausdorff images of rim-metrizable spaces.
On a conjecture of Tarski on products of cardinals
Authors: Jech Thomas , Shelah Saharon
Published: June 2010

We look at an old conjecture of A. Tarski on cardinal arithmetic and show that if a counterexample exists, then there exists one of length

ω
1

+
ω

{\omega _1} + \omega

.
Relating spans of some continua homeomorphic to 𝑆ⁿ
Authors: West Thelma
Published: June 2010

It has been conjectured that

σ
∗

(
X
)
≥

1
2

σ
0
∗

(
X
)

{\sigma ^ * }(X) \geq \tfrac {1}{2}\sigma _0^ * (X)

for each nonempty connected metric space

X
X

. In this paper we show that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Superscript asterisk Baseline left-parenthesis upper X right-parenthesis greater-than-or-equal-to left-parenthesis StartRoot 3 EndRoot slash 2 right-parenthesis sigma 0 Superscript asterisk Baseline left-parenthesis upper X right-parenthesis">

σ
∗

(
X
)
≥
(

3

/

2
)

σ
0
∗

(
X
)

{\sigma ^ * }(X) \geq (\sqrt 3 /2)\sigma _0^ * (X)

when

X
⊂

R
n

X \subset {R^n}

is homeomorphic to

S

n
−
1

{S^{n - 1}}

for

n
=
2
,
3
,
…

n = 2,3, \ldots

and

A
A

is convex where

A
A

is the bounded component of

R
n

−
X

{R^n} - X

. We also show that under certain conditions a lower bound for the ratio

σ
∗

(
X
)

/

σ
0
∗

(
X
)

{\sigma ^ * }(X)/\sigma _0^ * (X)

is larger than

3

/

2

\sqrt 3 /2

. It has also been conjectured that

σ
∗

(
X
)
≥
σ
(
X
)

/

2

{\sigma ^ * }(X) \geq \sigma (X)/2

and that

σ
0
∗

(
X
)
≥

σ
0

(
X
)

/

2

\sigma _0^ * (X) \geq {\sigma _0}(X)/2

for each nonempty connected metric space

X
X

. We show that these two inequalities hold when

X
⊂

R
n

X \subset {R^n}

is homeomorphic to

S

n
−
1

{S^{n - 1}}

for

n
=
3
,
4
,
…

n = 3,4, \ldots

and

A
A

is convex where

A
A

is the bounded component of

R
n

−
X

{R^n} - X

.
Cofinality in normal almost compact spaces
Authors: Fleissner W. , Kulesza J. , Levy R.
Published: June 2010

A regular space is said to be a NAC space if, given any pair of disjoint closed subsets, one of them is compact. The standard example of a noncompact NAC space is an ordinal space of uncountable cofinality. The cofinality of an arbitrary noncompact NAC space is defined, and the extent to which cofinality in NAC spaces behaves like cofinality of ordinal spaces is discussed.
A note on continuous mappings and the property of J. L. Kelley
Authors: Kato Hisao
Published: June 2010

In this paper, it is proved that if

X
X

is a continuum and

ω
\omega

is any Whitney map for

C
(
X
)

C(X)

, then the following are equivalent: (1)

X
X

has property [K]. (2) There exists a (continuous) mapping

F
:
X
×
I
×
[
0
,
ω
(
X
)
]
→
C
(
X
)

F:X \times I \times [0,\omega (X)] \to C(X)

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis StartSet x EndSet times upper I times StartSet t EndSet right-parenthesis equals StartSet upper A element-of omega Superscript negative 1 Baseline left-parenthesis t right-parenthesis vertical-bar x element-of upper A EndSet">

F
(
{
x
}
×
I
×
{
t
}
)
=
{
A
∈

ω

−
1

(
t
)

|

x
∈
A
}

F(\{ x\} \times I \times \{ t\} ) = \{ A \in {\omega ^{ - 1}}(t)|x \in A\}

for each

x
∈
X

x \in X

and

t
∈
[
0
,
ω
(
X
)
]

t \in [0,\omega (X)]

, where

I
=
[
0
,
1
]

I = [0,1]

. (3) For each

t
∈
[
0
,
ω
(
X
)
]

t \in [0,\omega (X)]

, there is an onto map

f
:
X
×
I
→

ω

−
1

(
t
)

f:X \times I \to {\omega ^{ - 1}}(t)

such that

f
(
{
x
}
×
I
)
=
{
A
∈

ω

−
1

(
t
)

|

x
∈
A
}

f(\{ x\} \times I) = \{ A \in {\omega ^{ - 1}}(t)|x \in A\}

for each

x
∈
X

x \in X

. Some corollaries are obtained also.
Extensions of isomorphisms between affine algebraic subvarieties of 𝑘ⁿ to automorphisms of 𝑘ⁿ
Authors: Kaliman Shulim
Published: June 2010

We derive a criterion, when an isomorphism between two closed affine algebraic subvarieties in an affine space can be extended to an automorphism of the space.
Unknotting number one knots are prime: a new proof
Authors: Zhang Xingru
Published: June 2010

An alternative proof for unknotting number one knots being prime is given.
Simple 𝐶*-algebras with continuous scales and simple corona algebras
Authors: Lin Hua Xin
Published: June 2010

It is shown that the corona algebra

M

(
A
)

/

A

M\left ( A \right )/A

of a separable simple

C
∗

{C^ * }

-algebra

A
A

is simple if and only if

A
A

has a continuous scale or

A
A

is elementary. It is also shown that simple

C
∗

{C^ * }

-algebras with continuous scales are algebraically simple.
The derivative of the exponential map
Authors: Herz Carl
Published: June 2010

We give a quick analytic derivation of the formula for the derivative of the exponential of a vector field on a manifold.
An integral inequality
Authors: Wilkins J. Ernest
Published: June 2010

We furnish conditions on the functions

p
(
t
)
,
f
(
t
)

p(t),f(t)

, and

g
(
t
)

g(t)

that are sufficient for the validity of the inequality,

α
2

δ
≥

γ
2

β

{\alpha ^2}\delta \geq {\gamma ^2}\beta

, in which

α
,
β
,
γ

\alpha ,\beta ,\gamma

, and

δ
\delta

respectively, are the integrals over a measurable set

E
E

of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis t right-parenthesis g left-parenthesis t right-parenthesis comma p left-parenthesis t right-parenthesis g squared left-parenthesis t right-parenthesis comma p left-parenthesis t right-parenthesis f left-parenthesis t right-parenthesis">

p
(
t
)
g
(
t
)
,
p
(
t
)

g
2

(
t
)
,
p
(
t
)
f
(
t
)

p(t)g(t),p(t){g^2}(t),p(t)f(t)

, and

p
(
t
)

f
2

(
t
)

p(t){f^2}(t)

.
Finitely additive Radon-Nikodým theorem and concentration function of a probability with respect to a probability
Authors: Berti Patrizia , Regazzini Eugenio , Rigo Pietro
Published: June 2010

An "exact" Radon-Nikodým theorem is obtained for a pair

(
m
,
μ
)

(m,\mu )

of finitely-additive probabilities, using a notion of concentration function of

μ
\mu

with respect to

m
m

. In addition, some direct consequences of that theorem are examined.
Regular states and countable additivity on quantum logics
Authors: Dvurečenskij Anatolij , Neubrunn Tibor , Pulmannová Sylvia
Published: June 2010

We give a counterexample of the result of Béaver and Cook concerning a generalization of the Alexandroff theorem for regular, finitely-additive states on quantum logics using states on the system of all splitting subspaces of an incomplete inner-product space. Moreover, we introduce another type of state regularity which entails countable additivity of states on logics.
Chain conditions on essential submodules
Authors: Osofsky Barbara L.
Published: June 2010

For

ℵ
\aleph

an infinite cardinal and

M
M

a unital right module over a ring

R
R

with 1 or an object in an

A

B

5

\mathcal {A}\mathcal {B}5

category, we show that every well ordered ascending (respectively descending) chain of essential submodules of

M
M

has cardinality less than

ℵ
\aleph

if and only if every well ordered ascending (respectively descending) chain of submodules of

M

/

socle
⁡

(
M
)

M/\operatorname {socle}\left ( M \right )

has cardinality less than

ℵ
\aleph

. We use this to show that a CS module with an

ℵ
\aleph

-chain condition on essential submodules is a direct sum of a module with that same chain condition on all submodules plus a semisimple module. Thus a CS module with fewer than

ℵ
\aleph

generators has an

ℵ
\aleph

-chain condition on essential submodules if and only if it has that same

ℵ
\aleph

-chain condition on all submodules. As an application in the case of an

ℵ
0

{\aleph _0}

-chain condition, we describe the endomorphism ring of a continuous module with acc on essential submodules.