bluuebunny/crossref_metadata_2025 huggingface dataset:
Stabilization in a gradient system with a conservation law
Authors: Pego Robert L.
Published: June 2010

Suppose

∑

μ
j

=
1

\sum {{\mu _j} = 1}

and

F
:

R

↦

R

F:{\mathbf {R}} \mapsto {\mathbf {R}}

is

C
1

{C^1}

with

F
′

F’

piecewise

C
1

{C^1}

. For the finite system of ordinary differential equations
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove u With dot Subscript i Baseline equals upper F prime left-parenthesis u Subscript i Baseline right-parenthesis minus sigma-summation Underscript j Endscripts mu Subscript j Baseline upper F prime left-parenthesis u Subscript j Baseline right-parenthesis equals 0 comma">

u
˙

i

=

F
′

(

u
i

)
−

∑
j

μ
j

F
′

(

u
j

)
=
0

,

{\dot u_i} = F’({u_i}) - \sum \limits _j {{\mu _j}F’({u_j}) = 0} ,

\]
I prove that every bounded solution stabilizes to some equilibrium as

t
→
∞

t \to \infty

. For this system,

∑

μ
j

u
j

\sum {{\mu _j}{u_j}}

is conserved and the quantity

∑

μ
j

F
(

u
j

)

\sum {{\mu _j}F({u_j})}

is nonincreasing and serves as a Lyapunov function, but the set of equilibria can be connected and degenerate. Essential use is made of a result related to one of Hale and Massat that an

ω
\omega

-limit set that lies in a

C
1

{C^1}

hyperbolic manifold of equilibria must be a singleton.
Bifurcation of limit cycles: geometric theory
Authors: Perko L. M.
Published: June 2010

Multiple limit cycles play a basic role in the theory of bifurcations. In this paper we distinguish between singular and nonsingular, multiple limit cycles of a system defined by a one-parameter family of planar vector fields. It is shown that the only possible bifurcation at a nonsingular, multiple limit cycle is a saddle-node bifurcation and that locally the resulting stable and unstable limit cycles expand and contract monotonically as the parameter varies in a certain sense. Furthermore, this same type of geometrical behavior occurs in any one-parameter family of limit cycles experiencing a saddle-node type bifurcation except possibly at a finite number of points on the multiple limit cycle.
A remark on the constants of the Littlewood-Paley inequality
Authors: Pichorides S. K.
Published: June 2010

If

f
∈

H
p

,
p
>
1

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

, and

γ
\gamma

denotes its Littlewood-Paley square function, then

|

|

γ

|

|

p

≤

B
p

|

|

f

|

|

p

||\gamma |{|_p} \leq {B_p}||f|{|_p}

with

B
p

=
0
(

(
p
−
1

)

−
1

)
,
p
→

1
+

{B_p} = 0({(p - 1)^{ - 1}}),p \to {1^ + }

.
The distribution of smooth numbers in arithmetic progressions
Authors: Balog Antal , Pomerance Carl
Published: June 2010

We estimate the number of integers

n
n

up to

x
x

in the arithmetic progression

a
(
mod
q
)

a(\bmod q)

with

n
n

free of prime factors exceeding

y
y

. For a wide range of the variables

x
,
y
,
q

x,y,q

, and

a
a

we show that this number is about

x

/

(
q

u
u

)

x/(q{u^u})

, where

u
=
log
⁡
x

/

log
⁡
y

u = \log x/\log y

.
A Liapunov function for three-dimensional feedback systems
Authors: Jiang Ji Fa
Published: June 2010

For a three-dimensional model of a positive feedback loop, Selgrade [11, 12] proved that every positive-time trajectory in the nonnegative orthant converges. Hirsch [6] gave another proof of this result under slightly different assumptions. This paper provides a new proof of Selgrade’s result that is much shorter and presents a generalization that can be applied to positive and negative feedback loops and other systems.
A note on a common fixed point theorem of Brodskiĭ and Mil′man and a lemma of Day
Authors: Leóng Tuck Sang
Published: June 2010

We should show that we may use a lemma of Day to prove, among others, a generalization of a common fixed point theorem of Brodskii and Milman when restricted to normed linear spaces that are uniformly convex in every direction.
Matrix summability of unbounded sequences
Authors: DeFranza J. , Zeller K.
Published: June 2010

A well-known result of Mazur and Orlicz states that a matrix method strictly stronger than convergence sums not only bounded sequences but unbounded sequences. We consider the question of whether a matrix method strictly stronger than convergence will also sum a sequence with series terms (differences) constituting an unbounded sequence. This is equivalent to the series to sequence convergence domain of the matrix containing an unbounded sequence. A simple criterion is given showing in many cases the answer is positive. Counterexamples of three types are considered; triangles that are not perfect, perfect row finite matrices, and perfect triangles.
Linear disjointness of polynomials
Authors: Abhyankar Shreeram S.
Published: June 2010

It is shown that any two bivariate polynomials can be made linearly disjoint by applying a linear transformation to one of the variables in one of the polynomials. From this it is deduced that the algebraic fundamental group of an affine line is closed relative to direct products.
𝑅-type summability methods, Cauchy criteria, 𝑃-sets and statistical convergence
Authors: Connor Jeff
Published: June 2010

A summability method

S
S

is called an

R
R

-type summability method if

S
S

is regular and

x
y

xy

is strongly

S
S

-summable to 0 whenever

x
x

is strongly

S
S

-summable to 0 and

y
y

is a bounded sequence. Associated with each

R
R

-type summability method

S
S

are the following two methods: convergence in

μ
\mu

-density and

μ
\mu

-statistical convergence where

μ
\mu

is a measure generated by

S
S

. In this note we extend the notion of statistically Cauchy to

μ
\mu

-Cauchy and show that a sequence is

μ
\mu

-Cauchy if and only if it is

μ
\mu

-statistically convergent. Let

W

(
A
)

=

A
¯

β

N

∩
β

N

∖

N

W\left ( A \right ) = {\overline A ^{\beta \mathbb {N}}} \cap \beta \mathbb {N}\backslash \mathbb {N}

for

A
⊂

N

A \subset \mathbb {N}

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper K equals intersection StartSet upper W left-parenthesis upper A right-parenthesis colon upper A subset-of-or-equal-to double-struck upper N comma chi Subscript upper A Baseline is strongly upper S minus summable to 1 EndSet">

K

 = 

∩

{

W

(
A
)

:
A
⊆

N

,

χ
A

is

strongly

S
−

summable

to

1

}

\mathcal {K}{\text { = }} \cap \left \{ {W\left ( A \right ):A \subseteq \mathbb {N}{\text {,}}{\chi _A}\;{\text {is}}\;{\text {strongly}}\;S - {\text {summable}}\;{\text {to}}\;1} \right \}

. Then

μ
\mu

-Cauchy is equivalent to convergence in

μ
\mu

-density if and only if every

G
δ

{G_\delta }

that contains

K

\mathcal {K}

in

β

N

∖

N

\beta \mathbb {N}\backslash \mathbb {N}

is a neighborhood of

K

\mathcal {K}

in

β

N

∖

N

\beta \mathbb {N}\backslash \mathbb {N}

. As an application, we show that the bounded strong summability field of a nonnegative regular matrix admits a Cauchy criterion.
On Archimedean ordered vector spaces and a characterization of simplices
Authors: Gierz Gerhard , Shekhtman Boris
Published: June 2010

We show that a convex subset

K
K

of a linear space is a simplex if and only if it is line compact and every nonempty intersection of two translates of

K
K

is a homothet of

K
K

. This answers a problem posed by Rosenthal. The proof uses a reformulation of this problem in terms of Archimedean ordered spaces
Non-Cohen-Macaulay symbolic blow-ups for space monomial curves
Authors: Morimoto Mayumi , Goto Shiro
Published: June 2010

Let

p

=

p

(

n
1

,

n
2

,

n
3

)

\mathfrak {p} = \mathfrak {p}({n_1},{n_2},{n_3})

denote the prime ideal in the formal power series ring

A
=
k
[
[
X
,
Y
,
Z
]
]

A = k[[X,Y,Z]]

over a field

k
k

defining the space monomial curve

X
=

T

n
1

,
Y
=

T

n
2

X = {T^{{n_1}}},Y = {T^{{n_2}}}

, and

Z
=

T

n
3

Z = {T^{{n_3}}}

with

GCD
⁡
(

n
1

,

n
2

,

n
3

)
=
1

\operatorname {GCD} ({n_1},{n_2},{n_3}) = 1

. Then the symbolic Rees algebra

R
s

(

p

)
=

⊕

n
≥
0

p

(
n
)

{R_s}(\mathfrak {p}) = { \oplus _{n \geq 0}}{\mathfrak {p}^{(n)}}

for

p

=

p

(

n
2

+
2
n
+
2
,

n
2

+
2
n
+
1
,

n
2

+
n
+
1
)

\mathfrak {p} = \mathfrak {p}({n^2} + 2n + 2,{n^2} + 2n + 1,{n^2} + n + 1)

is Noetherian but not Cohen-Macaulay if

ch

k
=
p
>
0

{\text {ch}}k = p > 0

and

n
=

p
e

n = {p^e}

with

e
≥
1

e \geq 1

. The same is true for

p

=

p

(

n
2

,

n
2

+
1
,

n
2

+
n
+
1
)

\mathfrak {p} = \mathfrak {p}({n^2},{n^2} + 1,{n^2} + n + 1)

if

ch

k
=
p
>
0

{\text {ch}}k = p > 0

and

n
=

p
e

≥
3

n = {p^e} \geq 3

.
Uniqueness of the uniform norm with an application to topological algebras
Authors: Bhatt S. J. , Karia D. J.
Published: June 2010

Any square-preserving linear seminorm on a unital commutative algebra is submultiplicative; and the uniform norm on a uniform Banach algebra is the only uniform

Q
Q

-algebra norm on it. This is proved and is used to show that (i) uniform norm on a regular uniform Banach algebra is unique among all uniform (not necessarily complete) norms and (ii) a complete uniform topological algebra that is a

Q
Q

-algebra is a uniform Banach algebra. Relevant examples, showing that the respective assumptions regarding regularity,

Q
Q

-algebra norm, and uniform property of topology cannot be omitted, have been discussed.
On some property of functions defined on 𝐑² that are ℐ-approximately continuous with respect to one variable
Authors: Carrese R. , Łazarow E.
Published: June 2010

Balcerzak, Lazarow, and Wilczyński proved that every separately

I

\mathcal {I}

-approximately continuous function is Baire 2. In this paper we shall prove that if

f
f

is a function

I

\mathcal {I}

-approximately continuous with respect to one of its variables and of the

α
\alpha

-class of Baire with respect to the other one, then

f
f

is of the

(
α
+
1
)

(\alpha + 1)

-class of Baire in

R
2

{R^2}

.
On the joint spectral radius. II
Authors: Chō Muneo , Huruya Tadasi , Wrobel Volker
Published: June 2010

In this paper we show that if

T

=
(

T
1

,
…
,

T
n

)

{\mathbf {T}} = ({T_1}, \ldots ,{T_n})

is a commuting

n
n

-tuple of operators on a Hilbert space such that

σ
(

T

)
=

Π

i
=
1

n

σ
(

T
i

)

\sigma ({\mathbf {T}}) = \Pi _{i = 1}^n\sigma ({T_i})

, then the algebraic joint spectral radius is equal to the geometric one.
Factorization of singular matrices
Authors: Sourour A. R. , Tang Kunikyo
Published: June 2010

We give a necessary and sufficient condition that a singular square matrix

A
A

over an arbitrary field can be written as a product of two matrices with prescribed eigenvalues. Except when

A
A

is a

2
×
2

2 \times 2

nonzero nilpotent, the condition is that the number of zeros among the eigenvalues of the factors is not less than the nullity of

A
A

. We use this result to prove results about products of hermitian and positive semidefinite matrices simplifying and strengthening some known results.
A converse of Lotz’s theorem on uniformly continuous semigroups
Authors: van Neerven J. M. A. M.
Published: June 2010

We prove the following partial converse to a theorem of Lotz: If every

C
0

{C_0}

-semigroup on a Banach lattice

E
E

with quasi-interior point is uniformly continuous, then

E
E

is isomorphic to a

C
(
K
)

C(K)

-space with the Grothendieck property.
Nontrivial solutions of semilinear elliptic equations with continuous or discontinuous nonlinearities
Authors: Mizoguchi Noriko
Published: June 2010

In this paper, we are concerned with the boundary value problem of the form

−
Δ
u
=
g
(
u
)

- \Delta u = g(u)

in

Ω
,
u

|

∂
Ω

=
0

\Omega ,u{|_{\partial \Omega }} = 0

, where

g
:

R

→

R

g:{\mathbf {R}} \to {\mathbf {R}}

is a continuous function, under assumptions of relations between

g
g

and the eigenvalues of

−
Δ

- \Delta

. If

g
g

is piecewise continuous on any bounded closed interval in

R

{\mathbf {R}}

, the above equation takes the form

−
Δ
u
∈
[

g
_

(
u
)
,

g
¯

(
u
)
]

- \Delta u \in [\underline {g}(u), \bar {g}(u)]

in

Ω
,
u

|

∂
Ω

=
0

\Omega ,u{|_{\partial \Omega }} = 0

. We obtain the existence of nontrivial solutions in both resonant and nonresonant cases at 0. Our theorems can be also applied when

g
g

is discontinuous (may be discontinuous at 0).
Spectral inclusion relations for 𝑇,𝑇|𝑌 and 𝑇/𝑌
Authors: Tanahashi Kôtarô , Yamagami Shigeru
Published: June 2010

Let

X
X

be a complex Banach space and

Y
Y

be an invariant subspace of a bounded linear operator

T
T

in

X
X

. Then it is easy to prove that (1)

σ
(
T
)
⊂
σ
(
T

|

Y
)
∪
σ
(
T

/

Y
)

\sigma (T) \subset \sigma (T|Y) \cup \sigma (T/Y)

, (2)

σ
(
T

|

Y
)
⊂
σ
(
T
)
∪
σ
(
T

/

Y
)

\sigma (T|Y) \subset \sigma (T) \cup \sigma (T/Y)

, and (3)

σ
(
T

|

Y
)
⊂
σ
(
T
)
∪
σ
(
T

/

Y
)

\sigma (T|Y) \subset \sigma (T) \cup \sigma (T/Y)

. In this paper, we first study these relations for unbounded linear operators: While (3) holds without any conditions, (1) and (2) do not hold in general. We shall make clear conditions on

T
T

that guarantee (1) and (2). Next we introduce the notion of extended spectrum for unbounded linear operators and prove similar results for the extended spectrum.
Second order ergodic theorems for ergodic transformations of infinite measure spaces
Authors: Aaronson Jon , Denker Manfred , Fisher Albert M.
Published: June 2010

For certain pointwise dual ergodic transformations

T
T

we prove almost sure convergence of the log-averages
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over log upper N EndFraction sigma-summation Underscript n equals 1 Overscript upper N Endscripts StartFraction 1 Over n a left-parenthesis n right-parenthesis EndFraction sigma-summation Underscript k equals 1 Overscript n Endscripts f ring upper T Superscript k Baseline left-parenthesis f element-of upper L 1 right-parenthesis">

1

log
⁡
N

∑

n
=
1

N

1

n
a

(
n
)

∑

k
=
1

n

f
∘

T
k

(

f
∈

L
1

)

\frac {1}{{\log N}}\sum \limits _{n = 1}^N {\frac {1}{{na\left ( n \right )}}\sum \limits _{k = 1}^n {f \circ {T^k}\left ( {f \in {L_1}} \right )} }

\]
and the Chung-Erdös averages
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over log a left-parenthesis upper N right-parenthesis EndFraction sigma-summation Underscript k equals 1 Overscript upper N Endscripts StartFraction 1 Over a left-parenthesis k right-parenthesis EndFraction f ring upper T Superscript k left-parenthesis f element-of upper L 1 Superscript plus Baseline right-parenthesis">

1

log
⁡
a

(
N
)

∑

k
=
1

N

1

a

(
k
)

f
∘

T
k

(

f
∈

L
1
+

)

\frac {1}{{\log a\left ( N \right )}}\sum \limits _{k = 1}^N {\frac {1}{{a\left ( k \right )}}f \circ {T^k}} \left ( {f \in L_1^ + } \right )

\]
towards

∫
f

\smallint f

, where

a

(
n
)

a\left ( n \right )

denotes the return sequence of

T
T

.
Crossed products of semisimple cocommutative Hopf algebras
Authors: Chin William
Published: June 2010

We provide a short proof of an analog of Nagata’s theorem for finite-dimensional Hopf algebras. The result, proved Hopf-algebraically by Sweedler and using group schemes by Demazure and Gabriel, says that a finite-dimensional cocommutative semisimple irreducible Hopf algebra is commutative. With mild base field assumptions such a Hopf algebra is just the dual of a

p
p

-group algebra. We give en route an easy proof of a version of Hochschild’s theorem on semisimple restricted enveloping algebras. Let

R

#
t

H

R{\# _t}H

denote a crossed product with an invertible cocycle

t
t

, where

H
H

is a semisimple cocommutative Hopf algebra

H
H

over a perfect field. The result above is applied to show that

R

#
t

H

R{\# _t}H

is semiprime if and only if

R
R

is

H
H

-semiprime. The approach relies on results on ideals of the crossed product that are stable under the action of the dual of

H
H

and the Fisher-Montgomery theorem for crossed products of finite groups.
A remark on singular integrals with complex homogeneity
Authors: Jones Roger L.
Published: June 2010

The Hilbert transform can be approximated by operators with Fourier multiplier given by

c
γ

sgn
⁡
(
ξ
)

|

ξ

|

i
γ

{c_\gamma }\operatorname {sgn} (\xi )|\xi {|^{i\gamma }}

. If we let

γ
→
0

\gamma \to 0

it is known that these operators converge to the Hilbert transform in

L
2

{L^2}

norm. It has also been shown that as

γ
→
0

\gamma \to 0

these operators may diverge even for functions in

L
2

{L^2}

. In this paper it is shown that we also will have divergence if we select any sequence

{

γ
n

}

n
=
1

∞

\{ {\gamma _n}\} _{n = 1}^\infty

such that

γ
n

→
0

{\gamma _n} \to 0

. The proof makes use of Bourgain’s entropy method.
On the almost split sequences for relatively projective modules over a finite group
Authors: Kleiner Mark
Published: June 2010

Let

G
G

be a finite group with a subgroup

H
H

. Given a field

k
k

of characteristic

p
p

dividing the order of

G
G

, denote by

mod
k
G

\bmod kG

the category of finite-dimensional over

k
k

left

G
G

-modules, and let

C

\mathcal {C}

be the full subcategory of

mod
k
G

\bmod kG

determined by the relatively projective modules. Let

0
→
L
→
M
→
N
→
0

0 \to L \to M \to N \to 0

be an exact sequence in

mod
k
G

\bmod kG

with

L
,
M
,
N
∈

C

L,M,N \in \mathcal {C}

. It is proved that the sequence is an almost split sequence in

C

\mathcal {C}

if and only if it is an almost split sequence in

mod
k
G

\bmod kG

. This implies, together with a recent result of Carlson and Happel, that

C

\mathcal {C}

has almost split sequences if and only if it is closed under extensions, i.e., if and only if

p
p

is coprime to either the order of

H
H

or the index of

H
H

in

G
G

.
On weighted 𝐿^{𝑝}-convergence of certain Lagrange interpolation
Authors: Min Guo Hua
Published: June 2010

The weighted

L
p

{L^p}

-convergence of Lagrange interpolation based on the zeros of

(
1
−

x
2

)

T
n

(
x
)

T
n
′

(
x
)

(1 - {x^2}){T_n}(x){T’_n}(x)

(where

T
n

(
x
)

{T_n}(x)

is the Tchebycheff polynomial of the first kind) is considered. The corresponding result of A. Bellen (J. Approx. Theory 33 (1981), 85-95) is improved and perfected.
Existence of solutions to 𝑢’(𝑡)=𝐴𝑢(𝑡) for 𝐴 weakly closed
Authors: Kanda Shigeo
Published: June 2010

We give an existence theorem of solutions to the evolution equation

(
d

/

d
t
)
u
(
t
)
=
A
u
(
t
)

(d/dt)u(t) = Au(t)

for a weakly closed operator

A
A

with some conditions on its resolvent in a reflexive Banach space.
Global exponential attractors for a class of almost-periodic parabolic equations in 𝑅^{𝑁}
Authors: Vuillermot Pierre-A.
Published: June 2010

In this article, we construct global exponential attractors for a class of almost-periodic semilinear reaction-diffusion equations with Neumann boundary conditions on bounded regions of

R

N

{\mathbb {R}^N}

. The class of problems that we analyze here contains, in particular, Fisher’s equations of population genetics.
Global invertibility of expanding maps
Authors: Hernández Jorge E. , Nashed M. Zuhair
Published: June 2010

We prove a global inversion theorem in reflexive Banach spaces utilizing a recent generalization of the interior mapping theorem. As a corollary, we provide, under a mild approximation property, a positive answer to an open problem that was stated by Nirenberg. We also establish global invertibility of an

α
\alpha

-expanding Fréchet differentiable map in Banach space under the assumption that the logarithmic norm of the derivative is negative.
Prescribing curvature with negative total curvature on open Riemann surfaces
Authors: Tang Jun Jie
Published: June 2010

On open Riemann surfaces, we obtain a necessary and sufficient condition for conformal metrics to have prescribed curvature and the desired singularities leading to negative total curvature.
Continuous independence and the Ilieff-Sendov conjecture
Authors: Miller Michael J.
Published: June 2010

A maximal polynomial is a complex polynomial that has all of its roots in the unit disk, one fixed root, and all of its critical points as far as possible from a fixed point. In this paper we determine a lower bound for the number of roots and critical points of a maximal polynomial that must lie on specified circles.
Essential maps and manifolds
Authors: Mertens Jean-François
Published: June 2010

Let

(
M
,
∂
M
)

(M,\partial M)

be a compact

n
n

-manifold with boundary, orientable over a field

K
K

with characteristic

q
q

. For

f
:
(
Y
,
∂
Y
)
→
(
M
,
∂
M
)

f:(Y,\partial Y) \to (M,\partial M)

, with

Y
Y

compact, and

(
X
,
∂
X
)

(X,\partial X)

a compact pair,

g
:
X
→
M

g:X \to M

, let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper P comma partial-differential upper P right-parenthesis equals StartSet left-parenthesis y comma x right-parenthesis element-of upper Y times left-parenthesis upper X comma partial-differential upper X right-parenthesis vertical-bar f left-parenthesis y right-parenthesis equals g left-parenthesis x right-parenthesis EndSet">

(
P
,
∂
P
)
=
{
(
y
,
x
)
∈
Y
×
(
X
,
∂
X
)

|

f
(
y
)
=
g
(
x
)
}

(P,\partial P) = \{ (y,x) \in Y \times (X,\partial X)|f(y) = g(x)\}

denote the fibered product, with

p
p

as the projection to

(
X
,
∂
X
)

(X,\partial X)

. In Čech-cohomology with coefficients

K
K

, we show that if

[
u
n
k

]
n

(
f
)

{[unk]^n}(f)

is injective then so is

[
u
n
k

]
∗

(
p
)

{[unk]^*}(p)

—and a number of strengthenings, which point to a concept of

q
q

-essential map from one compact space to another.
Multipliers on complemented Banach algebras
Authors: Tomiuk Bohdan J.
Published: June 2010

Let

A
A

be a semisimple right complemented Banach algebra,

L
A

{L_A}

the left regular representation of

A
A

, and

M
l

(
A
)

{M_l}\left ( A \right )

the left multiplier algebra of

A
A

. In this paper we are concerned with

L
A

{L_A}

and its relationship to

A
A

and

M
l

(
A
)

{M_l}\left ( A \right )

. We show that

L
A

{L_A}

is an annihilator algebra and that it is a closed ideal of

M
l

(
A
)

{M_l}\left ( A \right )

. Moreover,

L
A

{L_A}

and

M
l

(
A
)

{M_l}\left ( A \right )

have the same socle. We also show that the left multiplier algebra of a minimal closed ideal of

A
A

is topologically algebra isomorphic to

L

(
H
)

L\left ( H \right )

, the algebra of bounded linear operators on a Hilbert space

H
H

. Conditions are given under which

L
A

{L_A}

is right complemented.
On the Cartesian products of Lindelöf spaces with one factor hereditarily Lindelöf
Authors: Alster K.
Published: June 2010

E. Michael asked the following question: Is there a space

X
X

such that

Y
×
X

Y \times X

is Lindelöf for every hereditarily Lindelöf space

Y
Y

but

X
2

{X^2}

is not. The aim of this paper is to present a construction that provides such an example.
Riesz decomposition property implies asymptotic periodicity of positive and constrictive operators
Authors: Bartoszek Wojciech
Published: June 2010

Consider a linear and positive operator

T

{\mathbf {T}}

acting on an ordered,

F
F

-normed linear space

X

{\mathbf {X}}

. Assume that there exists an open neighborhood

U

∋

0

{\mathbf {U}} \ni {\mathbf {0}}

such that the trajectory

{

T

n

(

x

)

}

\left \{ {{{\mathbf {T}}^n}({\mathbf {x}})} \right \}

is attracted to a compact set

F

U

{{\mathbf {F}}_{\mathbf {U}}}

whenever

x

{\mathbf {x}}

is taken from

U

{\mathbf {U}}

and that the positive cone

X

+

{{\mathbf {X}}_ + }

is closed, proper, and reproducing. It is shown that if

(

X

,

X

+

)

({\mathbf {X}},{{\mathbf {X}}_ + })

has the Riesz Decomposition Property then

T

{\mathbf {T}}

has asymptotically periodic iterates.
On invertible hypercyclic operators
Authors: Herrero Domingo A. , Kitai Carol
Published: June 2010

Let

A
A

be an invertible (bounded linear) operator acting on a complex Banach space

X

\mathcal {X}

.

A
A

is called hypercyclic if there is a vector

y
y

in

X

\mathcal {X}

such that the orbit

Orb
⁡
(
A
;
y
)
:=
{
y
,
A
y
,

A
2

,
y
,
…
}

\operatorname {Orb}(A;y): = \{ y,Ay,{A^2},y, \ldots \}

is dense in

X

\mathcal {X}

. (

X

\mathcal {X}

is necessarily separable and infinite dimensional.) Theorem 1. The following are equivalent for an invertible operator A acting on

X

:
(

i

)
A

\mathcal {X}:({\text {i}})A

or

A

−
1

{A^{ - 1}}

is hypercyclic; (ii)

A
A

and

A

−
1

{A^{ - 1}}

are hypercyclic; (iii) there is a vector

z
z

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O r b left-parenthesis upper A semicolon z right-parenthesis Superscript minus Baseline equals upper O r b left-parenthesis upper A Superscript negative 1 Baseline semicolon z right-parenthesis Superscript minus Baseline equals script upper X">

Orb
⁡
(
A
;
z

)
−

=
Orb
⁡
(

A

−
1

;z

)
−

=

X

\operatorname {Orb}(A;z)^ - = \operatorname {Orb}({A^{ - 1}}{\text {;z}})^ - = \mathcal {X}

(the upper bar denotes norm-closure); (iv) there is a vector

y
y

in

X

\mathcal {X}

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper O r b left-parenthesis upper A semicolon y right-parenthesis union upper O r b left-parenthesis upper A Superscript negative 1 Baseline semicolon y right-parenthesis right-bracket Superscript minus Baseline equals script upper X period">

[
Orb
⁡
(
A
;
y
)
∪
Orb
⁡
(

A

−
1

;
y
)

]
−

=

X

.

[\operatorname {Orb}(A;y) \cup \operatorname {Orb}({A^{ - 1}};y)]^ - = \mathcal {X}.

\]
. Theorem 2. If

A
A

is not hypercyclic, then

A
A

and

A

−
1

{A^{ - 1}}

have a common nontrivial invariant closed subset.
Epimorphisms and monomorphisms in homotopy
Authors: Dydak Jerzy
Published: June 2010

The main result of this note is the following: Theorem A. If

f
:
X
→
Y

f:X \to Y

is an epimorphism of

H

C

W

∗

\mathcal {H}\mathcal {C}{\mathcal {W}^*}

, the homotopy category of pointed path-connected CW-spaces, and

π
1

(
f
)
:

π
1

(
X
)
→

π
1

(
Y
)

{\pi _1}(f):{\pi _1}(X) \to {\pi _1}(Y)

is a monomorphism, then

f
~

:

X
~

→

Y
~

\tilde f:\tilde X \to \tilde Y

is an epimorphism of

H

C

W

∗

\mathcal {H}\mathcal {C}{\mathcal {W}^*}

. As a straightforward consequence the following results of Dyer-Roitberg (Topology Appl. (to appear)) is derived: Theorem B. A map

f
:
X
→
Y

f:X \to Y

is an equivalence in

H

C

W

∗

\mathcal {H}\mathcal {C}{\mathcal {W}^*}

, the homotopy category of pointed path-connected CW-spaces, iff it is both an epimorphism and a monomorphism in

H

C

W

∗

\mathcal {H}\mathcal {C}{\mathcal {W}^*}

.
Characterization of separable metric 𝑅-trees
Authors: Mayer J. C. , Mohler L. K. , Oversteegen L. G. , Tymchatyn E. D.
Published: June 2010

An

R

{\mathbf {R}}

-tree

(
X
,
d
)

(X,d)

is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals.

R

{\mathbf {R}}

-trees arise naturally in the study of groups of isometries of hyperbolic space. Two of the authors had previously characterized

R

{\mathbf {R}}

-trees topologically among metric spaces. The purpose of this paper is to provide a simpler proof of this characterization for separable metric spaces. The main theorem is the following: Let

(
X
,
r
)

(X,r)

be a separable metric space. Then the following are equivalent: (1)

X
X

admits an equivalent metric

d

{\text {d}}

such that

(
X
,
d
)

(X,d)

is an

R

{\mathbf {R}}

-tree. (2)

X
X

is locally arcwise connected and uniquely arcwise connected. The method of proving that (2) implies (1) is to "improve" the metric

r
r

through a sequence of equivalent metrics of which the first is monotone on arcs, the second is strictly monotone on arcs, and the last is convex, as desired.
The symmetric genus of sporadic groups
Authors: Conder M. D. E. , Wilson R. A. , Woldar A. J.
Published: June 2010

Given a finite group

G
G

, the symmetric genus of

G
G

is defined to be the smallest integer

g
g

such that

G
G

acts faithfully on a closed orientable surface of genus

g
g

. Previous to this work, the task of determining the symmetric genus for the sporadic simple groups had been completed for all but nine groups:

J

3

{{\text {J}}_3}

,

McL
\operatorname {McL}

,

Suz
\operatorname {Suz}

,

O’N

{\text {O’N}}

,

Co
2

\operatorname {Co}_2

,

Fi

23

\operatorname {Fi}_{23}

,

Co
1

\operatorname {Co}_1

,

B

{\text {B}}

, and

M

{\text {M}}

. In the present paper the authors resolve the problem for six of these groups, viz.

J

3

{{\text {J}}_3}

,

McL
\operatorname {McL}

,

Suz
\operatorname {Suz}

,

O’N

{\text {O’N}}

,

Co
2

\operatorname {Co}_2

, and

Co
1

\operatorname {Co}_1

. Significant progress is also reported for the group

Fi

23

\operatorname {Fi}_{23}

.
Characterization of lower semicontinuous convex functions
Authors: Correa R. , Jofré A. , Thibault L.
Published: June 2010

We prove that a lower semicontinuous function defined on a reflexive Banach space is convex if and only if its Clarke subdifferential is monotone.
The framed braid group and 3-manifolds
Authors: Ko Ki Hyoung , Smolinsky Lawrence
Published: June 2010

The framed braid group on

n
n

strands is defined to be a semidirect product of the braid group

B
n

{B_n}

and

Z

n

{{\mathbf {Z}}^n}

. Framed braids represent

3
3

-manifolds in a manner analogous to the representation of links by braids. Consider two framed braids equivalent if they represent homeomorphic

3
3

-manifolds. The main result of this paper is a Markov type theorem giving moves that generate this equivalence relation.
A locally simply connected space and fundamental groups of one point unions of cones
Authors: Eda Katsuya
Published: June 2010

Let CX be the cone over a space

X
X

. Let a space

X
X

be first countable at

x
x

, then the following are equivalent: (1)

X
X

is locally simply connected at

x
x

; (2)

π
1

(

(

X
,
x

)

∨

(

X
,
x

)

,
x

)

{\pi _1}\left ( {\left ( {X,x} \right ) \vee \left ( {X,x} \right ),x} \right )

is naturally isomorphic to the free product

π
1

(

X
,
x

)

∗

π
1

(

X
,
x

)

{\pi _1}\left ( {X,x} \right ) * {\pi _1}\left ( {X,x} \right )

; (3)

π
1

(

(

C
X
,
x

)

∨

(

C
X
,
x

)

,
x

)

{\pi _1}\left ( {\left ( {CX,x} \right ) \vee \left ( {CX,x} \right ),x} \right )

is trivial. There exists a simply connected, locally simply connected Tychonoff space

X
X

with

x
∈
X

x \in X

, such that

(

X
,
x

)

∨

(

X
,
x

)

\left ( {X,x} \right ) \vee \left ( {X,x} \right )

is not simply connected.
The Hausdorff dimension of elliptic measure—a counterexample to the Oksendahl conjecture in 𝑅²
Authors: Sweezy Caroline
Published: June 2010

Two counterexamples to the Oksendahl conjecture in

R

2

{\mathbb {R}^2}

for elliptic measure are constructed. It is shown that there exists a strictly elliptic divergence form operator in a specially constructed quasi-disk such that the associated elliptic measure has as its support a set of Hausdorff dimension arbitrarily close to 2. The method is the construction of a quasi-conformal map from a quasi-disk whose boundary has high Hausdorff dimension to the unit disk. The

L
L

-operator is the pull-back of

Δ
\Delta

on the unit disk.
Proper holomorphic mappings and the Cowen-Douglas class
Authors: Krantz Steven G. , Salinas Norberto
Published: June 2010

We study the relationship between intersection theory for analytic varieties and membership of

n
n

-tuples of Toeplitz operators in the Cowen-Douglas class

B
m

(
Ω
)

{B_m}(\Omega )

. Connections with holomorphic mappings are discussed.
Modules and rings satisfying (accr)
Authors: Lu Chin-Pi
Published: June 2010

A module

M
M

over a ring

R
R

is said to satisfy (accr) if the ascending chain of residuals of the form

N
:
B
⊆
N
:

B
2

⊆
N
:

B
3

⊆
⋯

N: B \subseteq N:{B^2} \subseteq N:{B^3} \subseteq \cdots

terminates for every submodule

N
N

and every finitely generated ideal

B
B

of

R
R

. A ring satisfies (accr) if it does as a module over itself. This class of rings and modules satisfies various properties of Noetherian rings and modules. For each of the following rings, we investigate a necessary and sufficient condition for the ring to satisfy (accr): polynomial rings, power series rings, valuation rings, and Prüfer domains. We also prove that if

R
R

is a ring satisfying (accr), then every finitely generated

R
R

-module satisfies (accr).
Periodic orbits and the continuity of rotation numbers
Authors: Swanson Richard
Published: June 2010

The main result is that an annulus homeomorphism homotopic to the identity either has a well-defined and continuous assignment of rotation numbers on its chain recurrent set or there exists an interval of rotation numbers and periodic points corresponding to each reduced rational number in the interval. As a corollary, rotational discontinuities force the mapping to admit periodic points of all sufficiently large periods

n
n

. In a related result, we provide a criterion for the rotation set of an annulus homeomorphism to be nowhere dense.
𝐿^{𝑝} function decomposition on 𝐶^{∞} totally real submanifolds of 𝐶ⁿ
Authors: Calbeck William S.
Published: June 2010

For

1
>
p
>
∞

1 > p > \infty

we show that

L
p

{L^p}

functions defined on a

C
∞

{C^\infty }

totally real submanifold of

C

n

{\mathbb {C}^n}

can be locally decomposed into the sum of boundary values of holomorphic functions in wedges such that the boundary values are in

L
p

{L^p}

. The general case of a

C
∞

{C^\infty }

totally real submanifold is reduced to the flat case of

R

n

{\mathbb {R}^n}

in

C

n

{\mathbb {C}^n}

by an almost analytic change of variables.

L
p

{L^p}

results in the flat case are then obtained using Fourier multipliers. In transporting these results back to the manifold we lose analyticity, so it is necessary to solve a

∂
¯

\overline \partial

problem in an appropriate domain. This gives holomorphy in the wedges but produces a

C
∞

{C^\infty }

error on the edge. This

C
∞

{C^\infty }

function is then holomorphically decomposed using the FBI transform with a careful analysis to check that the functions are

C
∞

{C^\infty }

up to the edge and do not destroy the

L
p

{L^p}

behavior.
On induced representations of discrete groups
Authors: Binder Michael W.
Published: June 2010

The paper deals with induced representations

ind
H
G

⁡
σ

\operatorname {ind} _H^G\sigma

of a locally compact group

G
G

where

H
H

is an open subgroup. Using "elementary intertwining operators", we first describe the commutant

ind
H
G

⁡
σ
(
G

)
′

\operatorname {ind} _H^G\sigma (G)’

(also in the case of realizing the induced representations with positive definite measures). Then criteria for irreducibility and pairwise disjointness of induced representations are given. Finally, special attention is devoted to abelian subgroups

H
H

.
Analytic mappings on hyperfinite sets
Authors: Henson C. Ward , Ross David
Published: June 2010

Let

S
S

and

T
T

be hyperfinite sets in an

ℵ
1

{\aleph _1}

-saturated nonstandard universe. The following are equivalent: (i)

|

S

|

|

T

|

≈
1

\frac {{|S|}} {{|T|}} \approx 1

. (ii) There is a bijection from

|

S

|

|

T

|

≈
1.

\frac {{|S|}} {{|T|}} \approx 1.

onto

T
T

whose graph is Borel (over the internal subsets of

S
×
T

S \times T

). This follows from somewhat more general results about analytic partial functions on hyperfinite sets, the proofs of which use Choquet’s theorem on the capacitibility of analytic sets. This paper includes: proofs of the above results; an elementary direct construction for extensions of internal set functions to capacities; and a surprising corollary asserting the nonexistence of ergodic Borel transformations of a Loeb probability space.
The structure of some right Noetherian rings with Krull dimension one
Authors: Wangneo C. L.
Published: June 2010

In this note we prove a structure theorem for a right Noetherian ring

R
R

with krull dimension one having a right Artinian quotient ring and, moreover, that has a finitely generated, faithful, critical right module

U
U

with krull dimension

U
U

equal to one. We end this note with some examples that clarify certain features of this theorem.
A note on the Mackey dual of 𝐶(𝐾)
Authors: Wilkins Dave
Published: June 2010

Let

K
K

be a compact metric space, and let

τ
\tau

denote the Mackey topology on

M
(
K
)

M(K)

with respect to the

⟨
C
(
K
)
,
M
(
K
)
⟩

\langle C(K),M(K)\rangle

duality. That is,

τ
\tau

is the topology of uniform convergence on the weakly compact subsets of

C
(
K
)

C(K)

. Just as for the weak

∗

^{\ast }

topology, the dual space of

(
M
(
K
)
,
τ
)

(M(K),\tau )

is

C
(
K
)

C(K)

. However,

τ
\tau

is very different from weak

∗

^{\ast }

. Indeed, it is obvious that if

{

x
n

}

\{ {x_n}\}

is a sequence converging to

x
x

in

K
K

, then

δ
(

x
n

)

\delta ({x_n})

converges to

δ
(
x
)

\delta (x)

in the weak

∗

^{\ast }

topology, yet Kirk has shown (Pacific J. Math. 45 (1973), 543-554) that

{
δ
(
x
)

|

x
∈
K
}

\{ \delta (x)|x \in K\}

is closed and discrete in the Mackey topology. We obtain a further result along these lines: For each

A
⊂
K

A \subset K

set

Δ
A
=
{
δ
(
x
)
−
δ
(
y
)

|

x
≠
y
,
x
,
y
∈
A
}

\Delta A = \{ \delta (x) - \delta (y)|x \ne y,x,y \in A\}

. Let

D

\mathcal {D}

denote the totality of all subsets

A
A

of

K
K

with the property that

0
∈

Δ
A

¯

τ

0 \in {\overline {\Delta A} ^\tau }

. Then a closed set is in

D

\mathcal {D}

iff it is uncountable. Alternatively stated, a closed subset

A
A

of

K
K

is countable if and only if there is a weakly compact subset

L
L

of

C
(
K
)

C(K)

such that for every pair

x
,

y
∈
A
,

x
≠
y

x,\;y \in A,\;x \ne y

, there is an

h
∈
L

h \in L

with

|

h
(
x
)
−
h
(
y
)

|

⩾
1

|h(x) - h(y)| \geqslant 1

.
Refinements of the density and ℐ-density topologies
Authors: Ciesielski Krzysztof , Larson Lee
Published: June 2010

Given an arbitrary ideal

J

\mathcal {J}

on the real numbers, two topologies are defined that are both finer than the ordinary topology. There are nonmeasurable, non-Baire sets that are open in all of these topologies, independent of

J

\mathcal {J}

. This shows why the restriction to Baire sets is necessary in the usual definition of the

J

\mathcal {J}

-density topology. It appears to be difficult to find such restrictions in the case of an arbitrary ideal.
A martingale inequality related to exponential square integrability
Authors: Pipher Jill
Published: June 2010

We present an inequality for dyadic martingales (together with its continuous analog for functions on

R

n

{\mathbb {R}^n}

) which is shown to be equivalent to a result of Chang-Wilson-Wolff on exponential square integrability. The analog of this weighted inequality for double dyadic martingales is also proven. Finally, we discuss a possible connection between these inequalities and a theorem of Garnett.
Weak-invariant properties of the norm topology
Authors: Namioka I. , Pol R.
Published: June 2010

A property (P) relative to the norm topology of a Banach space is a weak-invariant if, whenever

A
A

and

B
B

are weakly homeomorphic subsets of (possibly different) Banach spaces and (

A
A

, norm) has property (P), then (

B
B

, norm) has property (P). We show that the property of being

σ
\sigma

-discrete and the property of being an absolute Souslin-

F

\mathcal {F}

space of weight

⩽

ℵ
1

\leqslant {\aleph _1}

, both relative to the norm topology, are weak-invariants. These conclusions are obtained from a result concerning maps of metrizable spaces into function spaces.
Links and nonshellable cell partitionings of 𝑆³
Authors: Armentrout Steve
Published: June 2010

A cell partitioning of a closed

3
3

-manifold

M
3

{M^3}

is a finite covering of

M
3

{M^3}

by

3
3

cells that fit together in a bricklike pattern. A cell partitioning

H
H

of

M
3

{M^3}

is shellable if

H
H

has a counting

⟨

h
1

,

h
2

,
…
,

h
n

⟩

\langle {h_1},{h_2}, \ldots ,{h_n}\rangle

such that if

1
⩽
i
>
n
,

h
1

∪

h
2

∪
⋯
∪

h
i

1 \leqslant i > n,\;{h_1} \cup {h_2} \cup \cdots \cup {h_i}

is a

3
3

-cell. The main result of this paper is a relationship between nonshellability of a cell partitioning

H
H

of

S
3

{S^3}

and the existence of links in

S
3

{S^3}

specially related to

H
H

. This result is used to construct a nonshellable cell partitioning of

S
3

{S^3}

.
A note on weighted Sobolev spaces, and regularity of commutators and layer potentials associated to the heat equation
Authors: Hofmann Steve
Published: June 2010

We give a simplified proof of recent regularity results of Lewis and Murray, namely, that certain commutators, and the boundary single layer potential for the heat equation in domains in

R

2

{\mathbb {R}^2}

with time dependent boundary, map

L
p

{L^p}

into an appropriate homogeneous Sobolev space. The simplification is achieved by treating directly only the case

p
=
2

p = 2

, but in a weighted setting.
Poincaré and Sobolev inequalities in product spaces
Authors: Shi Xian Liang , Torchinsky Alberto
Published: June 2010

Some local Poincaré and Sobolev inequalities involving weights in product spaces are established.
Cyclically presented groups embedded in one-relator products of cyclic groups
Authors: Campbell Colin M. , Heggie Patricia M. , Robertson Edmund F. , Thomas Richard M.
Published: June 2010

We consider the groups defined by the presentations
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle a comma b colon a squared equals b Superscript n Baseline equals a b Superscript negative 1 Baseline a b left-parenthesis a b a b Superscript negative 1 Baseline right-parenthesis Superscript alpha minus 1 Baseline a b squared a b Superscript negative 2 Baseline equals 1 mathematical right-angle">

⟨
a
,
b
:

a
2

=

b
n

=
a

b

−
1

a
b

(
a
b
a

b

−
1

)

α
−
1

a

b
2

a

b

−
2

=
1
⟩

\langle a,b:{a^2} = {b^n} = a{b^{ - 1}}ab{(aba{b^{ - 1}})^{\alpha - 1}}a{b^2}a{b^{ - 2}} = 1\rangle

\]
and investigate their structure for small values of

α
\alpha

. This forms part of a general investigation into the structure of groups defined by presentations of the form
\[

⟨
a
,
b
:

a
2

=

b
n

=
w
(
a
,
b
)
=
1
⟩
.

\langle a,b:{a^2} = {b^n} = w(a,b) = 1\rangle .

\]
Connections between these groups and the Fibonacci groups are also explored.
On the nonautonomous Volterra-Lotka competition equations
Authors: Ahmad Shair
Published: June 2010

A nonautonomous competitive Lotka-Volterra system of two equations is considered. It is shown that if the coefficients are continuous and satisfy certain inequalities, then any solution that is positive at some point has the property that one of its components vanishes while the other approaches a certain solution of the logistic equation.
Profinite orthomodular lattices
Authors: Choe Tae Ho , Greechie Richard J.
Published: June 2010

We prove that any compact topological orthomodular lattice

L
L

is zero dimensional. This leads one to show that

L
L

is profinite iff it is the product of finite orthomodular lattices with their discrete topologies. We construct a completion

L
¯

\overline L

of a residually finite orthomodular lattice

L
L

having the property that every element of

L
¯

\overline L

is the join of meets of elements of

L
L

. Necessary and sufficient conditions for

L
L

that

L
¯

\overline L

is the MacNeille completion are obtained.
The Hausdorff dimension of the nondifferentiability set of the Cantor function is [𝑙𝑛(2)/𝑙𝑛(3)]²
Authors: Darst Richard
Published: June 2010

The main purpose of this note is to verify that the Hausdorff dimension of the set of points

N

∗

{N^{\ast }}

at which the Cantor function is not differentiable is

[
ln
⁡
(
2
)

/

ln
⁡
(
3
)

]
2

{[\ln (2)/\ln (3)]^2}

. It is also shown that the image of

N

∗

{N^{\ast }}

under the Cantor function has Hausdorff dimension

ln
⁡
(
2
)

/

ln
⁡
(
3
)

\ln (2)/\ln (3)

. Similar results follow for a standard class of Cantor sets of positive measure and their corresponding Cantor functions.
Existence of a nontrivial solution to a strongly indefinite semilinear equation
Authors: Buffoni B. , Jeanjean L. , Stuart C. A.
Published: June 2010

Under general hypotheses, we prove the existence of a nontrivial solution for the equation

L
u
=
N
(
u
)

Lu = N(u)

, where

u
u

belongs to a Hilbert space

H
H

,

L
L

is an invertible continuous selfadjoint operator, and

N
N

is superlinear. We are particularly interested in the case where

L
L

is strongly indefinite and

N
N

is not compact. We apply the result to the Choquard-Pekar equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus normal upper Delta u left-parenthesis x right-parenthesis plus p left-parenthesis x right-parenthesis u left-parenthesis x right-parenthesis equals u left-parenthesis x right-parenthesis integral Underscript double-struck upper R cubed Endscripts StartFraction u squared left-parenthesis y right-parenthesis Over StartAbsoluteValue x minus y EndAbsoluteValue EndFraction d y comma u element-of upper H Superscript 1 Baseline left-parenthesis double-struck upper R cubed right-parenthesis comma u not-equals 0 comma">

−
Δ
u
(
x
)
+
p
(
x
)
u
(
x
)
=
u
(
x
)

∫

R

3

u
2

(
y
)

|

x
−
y

|

d
y
,

u
∈

H
1

(

R

3

)
,

u
≠
0
,

- \Delta u(x) + p(x)u(x) = u(x)\int _{{\mathbb {R}^3}} {\frac {{{u^2}(y)}} {{|x - y|}}dy,\qquad u \in {H^1}({\mathbb {R}^3}),\quad u \ne 0,}

\]
where

p
∈

L
∞

(

R

3

)

p \in {L^\infty }({\mathbb {R}^3})

is a periodic function.
Generalized spectrum and a problem of Apostol
Authors: Mbekhta Mostafa
Published: June 2010

We give a complete answer to a question of Apostol concerning the asymptotic behaviour of the reduced minimum modulus of a bounded operator on Hilbert space.
An application of set theory to the torsion product of abelian groups
Authors: Keef Patrick
Published: June 2010

The following problem of Fuchs is considered: relate the abelian groups

A
A

and

B
B

assuming

Tor
⁡
(
A
,
G
)
≅
Tor
⁡
(
B
,
G
)

\operatorname {Tor} (A,G) \cong \operatorname {Tor} (B,G)

for all reduced abelian groups

G
G

. A complete characterization is obtained in any set-theoretic universe in which

E
(
κ
)

E(\kappa )

is valid for a proper class of regular cardinals

κ
\kappa

.
Banach algebras which are not Wedderburnian
Authors: Yood Bertram
Published: June 2010

Let

A
A

be a Banach algebra with radical

R
R

. In 1951 Feldman exhibited an example in which it is impossible to find a closed subalgebra

K
K

of

A
A

such that

A
=
K
⊕
R

A = K \oplus R

. We provide other examples. Feldman’s algebra is commutative, but these examples are, in general, not commutative.
Foliations of 𝐸(5₂) and related knot complements
Authors: Cantwell John , Conlon Lawrence
Published: June 2010

The depth one foliations with a single compact leaf are classified for twist knots and pretzel knots

(
3
,
−
3
,
2
n
+
1
)

(3, - 3,2n + 1)

.
On sets of Vitali’s type
Authors: Cichoń J. , Kharazishvili A. , Węglorz B.
Published: June 2010

We consider the classical Vitali’s construction of nonmeasurable subsets of the real line

R

\mathbb {R}

and investigate its analogs for various uncountable subgroups of

R

\mathbb {R}

. Among other results we show that if

G
G

is an uncountable proper analytic subgroup of

R

\mathbb {R}

then there are Lebesgue measurable and Lebesgue nonmeasurable selectors for

R

/

G

\mathbb {R}/G

.
Measures of noncompactness and upper semi-Fredholm perturbation theorems
Authors: Galaz-Fontes Fernando
Published: June 2010

We introduce the concept of a perturbation function, which allows us to give a general approach to the question of obtaining perturbation theorems for upper semi-Fredholm operators. Also, we show that the usual measures of noncompactness of continuous linear operators, as well as other related quantities, are perturbation functions.
Tight extensions of normed spaces
Authors: Rao N. V.
Published: June 2010

In this note we show that bound extensions as defined by Kaufman (Acta Univ. (Szeged) 21 (1966), 163) and tight extensions as defined by Dress (Adv. in Math. 53 (1984), 322) are the same. Further we find that the property of being a bound extension is preserved under complexification.
On some cubic modular identities
Authors: Shen Li-Chien
Published: June 2010

Let

4
K

4K

and

2
i

K
′

2iK’

be the periods of

sn
⁡
z

\operatorname {sn} z

. By evaluating the functional equations

sn
2

z
+

cn
2

z
=
1

{\operatorname {sn} ^2}z + {\operatorname {cn} ^2}z = 1

and

k
2

sn
2

z
+

dn
2

z
=
1

{k^2}{\operatorname {sn} ^2}z + {\operatorname {dn} ^2}z = 1

at

z
=
i

K
′

/

3

z = iK’/3

, we deduce a set of cubic modular identities from which the familiar modular equation of degree

3
3

follows directly as a corollary.