bluuebunny/crossref_metadata_2025 huggingface dataset:
𝐺𝐿(2,𝑍) action on a two torus
Authors: Park Kyewon Koh
Published: June 2010

We study the group

GL
⁡
(
2
,
Z
)

\operatorname {GL}(2,Z)

action on a two torus

T
2

{T^2}

with Lebesgue measure. We show that any measure-preserving transformation that commutes with the group action is the action of a matrix of the form

(

m

a
m
p
;
0

0

a
m
p
;
m

)

\left ( {\begin {array}{*{20}{c}} m & 0 \\ 0 & m \\ \end {array} } \right )

, where

m
m

is an integer. We also show that all factors come from these commuting transformations. Finally we show that the set of self-joinings consists of the product measure and the measures sitting on the graphs
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper K u comma upper M u right-parenthesis colon upper K equals Start 2 By 2 Matrix 1st Row 1st Column k 2nd Column a m p semicolon 0 2nd Row 1st Column 0 2nd Column a m p semicolon k EndMatrix comma upper M equals Start 2 By 2 Matrix 1st Row 1st Column m 2nd Column a m p semicolon 0 2nd Row 1st Column 0 2nd Column a m p semicolon m EndMatrix">

(
K
u
,
M
u
)
:
K
=

(

k

a
m
p
;
0

0

a
m
p
;
k

)

,
M
=

(

m

a
m
p
;
0

0

a
m
p
;
m

)

(Ku,Mu):K = \left ( {\begin {array}{*{20}{c}} k & 0 \\ 0 & k \\ \end {array} } \right ),M = \left ( {\begin {array}{*{20}{c}} m & 0 \\ 0 & m \\ \end {array} } \right )

, and

u
∈

T
2

}

u \in {T^2}\}

. We provide an example whose self-joinings consist only of the product measure and the diagonal measure.
Finite invariant measures on flows
Authors: Wong Robert S. Y.
Published: June 2010

We prove that an ergodic flow admits a finite invariant measure if and only if the Kreiger factor whose flow of weights is the flow contains a

III
1

{\text {III}_1}

subfactor which is the range of a faithful normal semifinite conditional expectation.
Special values of 𝐿-series
Authors: Hatcher Rhonda L.
Published: June 2010

The main result of this paper is an equation representing the value of the Dirichlet series

L
(
f
,
A
,
s
)

L(f,A,s)

at

s
=
k

s = k

in terms of the height pairings of special points on a vector bundle

V
V

associated with the quaternion algebra over

Q

{\mathbf {Q}}

ramified at

N
N

and

∞
\infty

.
On the strongly extreme points of convex bodies in separable Banach spaces
Authors: Godun B. V. , Lin Bor-Luh , Troyanski S. L.
Published: June 2010

Every separable Banach space admits an equivalent norm such that the unit ball with respect to this norm has at most countably many strongly extreme points. Every separable nonreflexive Banach space can be renormed so that its unit ball has at most countably many weakly strongly extreme points.
Closure sets of functions and a hierarchy of filters
Authors: Mignone Robert
Published: June 2010

This paper presents a transfinite extension of a filter generating closure property of functions. One consequence of this extension is a hierarchy of filters which coincide with filters generated by a directed set type closure property. At each level of this hierarchy a progressively stronger notion of normality is satisfied. Ideals with this stronger notion of normality each have a corresponding saturation characterization.
The minimal support for a continuous functional on a function space
Authors: Morishita Kazuhiko
Published: June 2010

Let

C
p

(
X
)

{C_p}(X)

be the function space with the pointwise convergent topology over a Tychonoff space

X
X

and

ξ
\xi

a continuous real-valued function on

C
p

(
X
)

{C_p}(X)

. A closed subset

S
S

of

X
X

is called a support for

ξ
\xi

if

ξ
(
f
)
=
ξ
(
g
)

\xi (f) = \xi (g)

holds for any pair

(
f
,
g
)

(f,g)

of elements of

C
p

(
X
)

{C_p}(X)

such that

f

|

S

=

g

|

S

{f_{|S}} = {g_{|S}}

. It is proven that the minimal support for any real-valued continuous function on the space

C
p

(
X
)

{C_p}(X)

exists.
On the Carleson measure characterization of BMOA functions on the unit ball
Authors: Jevtić Miroljub
Published: June 2010

An equivalent form of the well-known Carleson measure condition that characterizes BMOA functions on the unit ball is given. As an application, it is proved that the Toeplitz operator

T

φ
¯

,
φ
∈

H
∞

(
S
)

{T_{\bar \varphi }},\varphi \in {H^\infty }(S)

, is bounded on the Hardy-Sobolev space

H
α
1

,
0
>
α
>
∞

H_\alpha ^1,0 > \alpha > \infty

.
Fourier summation with kernels defined by Jacobi polynomials
Authors: Lasser R.
Published: June 2010

Trigonometric polynomial kernels defined by Jacobi polynomials are investigated. They generalize the classical Dirichlet kernel and the Fejér kernel. The asymptotic behavior of the corresponding Fourier summation obtained is leading to optimal kernels.
A characterization of Suzuki’s simple groups
Authors: Shi Wu Jie
Published: June 2010

In this short paper we have characterized Suzuki’s simple groups

S
z

(

2

2
m
+
1

)
,
m
⩾
1

{S_z}({2^{2m + 1}}),m \geqslant 1

using only the set

π
e

(
G
)

{\pi _e}(G)

of orders of elements in the group

G
G

. That is, we have Theorem 2. Let

G
G

be a finite group. Then

G
≃

S
z

(

2

2
m
+
1

)
,
m
⩾
1

G \simeq {S_z}({2^{2m + 1}}),m \geqslant 1

if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript e Baseline left-parenthesis upper G right-parenthesis equals left-brace 2 comma 4 comma a l l f a c t o r s o f left-parenthesis 2 Superscript 2 m plus 1 Baseline hyphen 1 right-parenthesis comma left-parenthesis 2 Superscript 2 m plus 1 Baseline minus 2 Superscript m plus 1 Baseline plus 1 right-parenthesis comma a n d left-parenthesis 2 Superscript 2 m plus 1 Baseline plus 2 Superscript m plus 1 Baseline plus 1 right-parenthesis right-brace">

π
e

(
G
)
=
{
2
,
4
,
a
l
l

f
a
c
t
o
r
s

o
f

(

2

2
m
+
1

 - 1),(

2

2
m
+
1

−

2

m
+
1

+
1
)
,

a
n
d

(

2

2
m
+
1

+

2

m
+
1

+
1
)
}

{\pi _e}(G) = \{ 2,4,all\;factors\;of\;{\text {(}}{{\text {2}}^{2m + 1}}{\text { - 1),(}}{{\text {2}}^{2m + 1}} - {2^{m + 1}} + 1),\;and\;({2^{2m + 1}} + {2^{m + 1}} + 1)\}

.
The Neumann Laplacian of a jelly roll
Authors: Simon Barry
Published: June 2010

We consider the Laplacian with Neumann boundary conditions of a bounded connected region obtained by removing a suitable infinite spiral from an annulus. We show that the spectrum has an absolutely continuous component.
On multiple Salié sums
Authors: Duke W.
Published: June 2010

This note shows that the Davenport-Hasse relation for Gauss sums is equivalent to the evaluation of some multidimensional exponential sums that generalize that of Salié.
On an extension of Minty’s theorem
Authors: Shimizu Makoto
Published: June 2010

The notion of monotone operator in a Hilbert space is extended, and a considerably broader class of possibly multivalued operators is introduced. It is shown that well-known Minty’s theorems on maximal monotone operators in Hilbert spaces can be extended to the cases of operators belonging to the class. Typical examples of partial differential operators are given to illustrate that the results can be applied to nonlinear equations, which involve nonmonotone differential operators.
Martingale transforms with unbounded multipliers
Authors: Chao J.-A. , Long R.-L.
Published: June 2010

The boundedness of martingale transforms with the "multiplier" sequence in various classes is studied.
Dimensions of topological groups containing the bouquet of two circles
Authors: Kimura Takashi
Published: June 2010

In this paper we prove the following: If a topological group

G
G

contains the bouquet

S
1

∨

S
1

{S^1} \vee {S^1}

, then

G
≥
2

G \geq 2

holds. This is a counterexample to a question of Bel’nov in the one-dimensional case.
Morava 𝐾-theory and the free loop space
Authors: McCleary John , McLaughlin Dennis A.
Published: June 2010

We generalize a result of Hopkins, Kuhn, and Ravenel relating the

n
n

th Morava

K
K

-theory of the free loop space of a classifying space of a finite group to the

(
n
+
1
)

(n + 1)

st Morava

K
K

-theory of the space. We show that the analogous result holds for any Eilenberg-Mac Lane space for a finite group. We also compute the Morava

K
K

-theory of the free loop space of a suspension, and comment on the general problem.
Small congruences and concreteness
Authors: Velebilová Magdalena
Published: June 2010

Let

K
_

\underline K

be a concrete category and

∼
\sim

a congruence on

K
_

\underline K

. Let

∼
\sim

be generated by a class

M
=

M
1

∪

M
2

M = {M_1} \cup {M_2}

of Pairs of

K
_

\underline K

-morphisms such that

{
dom
⁡
f
;
(
∃
g
)
(
(
f
,
g
)
∈

M
1

)
}

\{ \operatorname {dom} f;(\exists g)((f,g) \in {M_1})\}

and

{

rng
⁡
f
;
(
∃
g
)
(
(
f
,
g
)
∈

M
2

)

}

\left \{ {\operatorname {rng} f;(\exists g)((f,g) \in {M_2})} \right \}

are small sets. Then

K
_

/

∼

\underline K / \sim

is concrete. Consequently, if

∼
\sim

is generated by a small set of pairs of morphisms, then

K
_

/

∼

\underline K / \sim

is concrete.
Strong-principal bimodules of nest algebras
Authors: Gong Wei Bang , Zhu Jun
Published: June 2010

We shall show that

B
(
H
)

B(H)

can be represented by the strong closure of the linear span of the compounds of a fixed operator in

B
(
H
)

B(H)

and the rank one operators, composed only by the vectors of a certain orthonormal basis of

H
H

, in a nest algebra, even, under some assumption, in the radical of a nest algebra.
Invariant subspaces and perturbations
Authors: Noakes Lyle
Published: June 2010

We study the stability of proper closed invariant subspaces with respect to perturbations in norm of continuous operators on a Hubert space, using a nonlinear

C
∞

{C^\infty }

map of Banach spaces.
The 𝐶¹ closing lemma for endomorphisms with finitely many singularities
Authors: Wen Lan
Published: June 2010

The

C
1

{C^1}

closing lemma for endomorphisms with finitely many singularities is obtained by combining the

C
1

{C^1}

closing lemma for nonsingular endomorphisms together with a technique of L. S. Young.
On the Deskins index complex of a maximal subgroup of a finite group
Authors: Ballester-Bolinches A. , Ezquerro Luis M.
Published: June 2010

We analyze the influence of the maximal subgroups in the structure of a finite group

G
G

by means of the index complex introduced by Deskins in Proc. Sympos. Pure Math. vol. 1, Amer. Math. Soc. 1959, pp. 100-104.
Lifting primitivity of free nilpotent groups
Authors: Gupta C. K. , Gupta N. D.
Published: June 2010

Let

F

n
,
c

{F_{n,c}}

denote the free

n
n

-generator nilpotent group of class

c
c

. For

n
≥
c
+
m
−
1

n \geq c + m - 1

, every primitive system of

m
m

elements of

F

n
,
c

{F_{n,c}}

can be lifted to a primitive system of

m
m

elements of the free group

F
n

{F_n}

.
Stable measure of a small ball
Authors: Lewandowski M. , Ryznar M. , Żak T.
Published: June 2010

Let

μ
\mu

be a symmetric

p
p

-stable measure on a Banach space (

(
E
,

|

|

|

|

)

(E,||||)

. We prove that

μ
{

|

|

x

|

|

>
t
}
≤
K
t

\mu \{ ||x|| > t\} \leq Kt

, where the constant

K
K

is independent of all properties of

μ
\mu

except for the measure of the unit ball

μ
{

|

|

x

|

|

>
1
}

\mu \{ ||x|| > 1\}

.
On the existence of fixpoints of composite meromorphic functions
Authors: Bergweiler Walter
Published: June 2010

Let

f
f

be a meromorphic function with at least two poles, and let

g
g

be an entire transcendental function. It is proved that the composite function

f
∘
g

f \circ g

has infinitely many fixpoints.
A necessary and sufficient condition of nonresonance for a semilinear Neumann problem
Authors: Gossez J.-P. , Omari P.
Published: June 2010

We consider the Neumann problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row minus normal upper Delta u equals g left-parenthesis u right-parenthesis plus h left-parenthesis x right-parenthesis in normal upper Omega comma 2nd Row partial-differential u slash partial-differential nu equals 0 on b d r y normal upper Omega period EndLayout">

{

−
Δ
u
=
g
(
u
)
+
h
(
x
)

 in 

Ω
,

∂
u

/

∂
ν
=
0

on 

bdry
⁡
Ω
.

\left \{ {\begin {array}{*{20}{c}} { - \Delta u = g(u) + h(x){\text { in }}\Omega ,} \\ {\partial u/\partial \nu = 0\quad {\text {on }}\operatorname {bdry} \Omega .} \\ \end {array} } \right .

\]
Assuming some growth restriction on the nonlinearity

g
g

, we prove that a necessary and sufficient condition for the existence of a solution for every given

h
∈

L
∞

(
Ω
)

h \in {L^\infty }(\Omega )

is that

g
g

be unbounded from above and from below on

R

\mathbb {R}

.
Best approximation by subharmonic functions
Authors: Wilson J. M. , Zwick D.
Published: June 2010

Let

Ω
⊂

R

d

\Omega \subset {\mathbb {R}^d}

be a bounded domain. We prove existence of best subharmonic approximations in

L
∞

(
Ω
)

{L_\infty }(\Omega )

and, for functions continuous in

Ω
¯

\overline \Omega

, we characterize best continuous subharmonic approximations.
A note on the trace formulas for almost commuting operators
Authors: Xia Daoxing
Published: June 2010

A formula related to the cyclic cohomology for the trace of the product of commutators is established.
𝜅-topologies for right topological semigroups
Authors: Baker John , Hindman Neil , Pym John
Published: June 2010

Given a cardinal

κ
\kappa

and a right topological semigroup

S
S

with topology

τ
\tau

, we consider the new topology obtained by declaring any intersection of at most

κ
\kappa

members of

τ
\tau

to be open. Under appropriate hypotheses, we show that this process turns

S
S

into a topological semigroup. We also show that under these hypotheses the points of any subsemigroup

T
T

with card

T
≤
κ

T \leq \kappa

can be replaced by (new) open sets that algebraically behave like

T
T

. Examples are given to demonstrate the nontriviality of these results.
The flat strip theorem fails for surfaces with no conjugate points
Authors: Burns Keith
Published: June 2010

A compact

C
∞

{C^\infty }

surface with no conjugate points is constructed so that there are two homotopic closed geodesies that do not bound a flat annulus.
Nilpotent spaces: some inequalities on nilpotency degrees
Authors: Reynol Filho Augusto
Published: June 2010

Our purpose in this work is to compare the nilpotency degree of the action of

π
1

(
X
)

{\pi _1}(X)

on

π
n

(
X
)

(
2
≤
n
≤
7
)

{\pi _n}(X)\quad (2 \leq n \leq 7)

with the one of the action of

π
1

(
X
)

{\pi _1}(X)

on

H
n
(

X
~

)

Hn(\tilde X)

. We work in the category of the nilpotent spaces (here

X
~

\tilde X

means the universal cover of

X
X

). The main point in the proof of the main theorems, which yields such inequalities, is the reiterated use of the Serre spectral sequence.
Descending chain conditions for graded rings
Authors: Saorín M.
Published: June 2010

It is proved that if

R
R

is a perfect (resp. Artinian) strongly graded ring whose ground subring is, modulo its Jacobson radical, a finite direct product of finite-dimensional simple algebras over (nondenumerable) algebraically closed fields, then the grading group cannot contain an infinite abelian subgroup (resp. must be finite). These results extend those of A. Reid and D. S. Passman on twisted group algebras.
Hardy integral estimates for the Laplace transform
Authors: Bloom Steven
Published: June 2010

Weighted norm inequalities for the Laplace transform are obtained by reducing the Laplace operator to the Hardy operator. The conditions obtained include necessary and sufficient conditions when one of the weights is doubling, including, in particular, all the power weights.
Free subsemigroups of domains
Authors: Klein A. A.
Published: June 2010

It is proved that the multiplicative semigroup of the ring of polynomials in two commuting indeterminates over a noncommutative domain contains a noncommutative free subsemigroup.
A functional analysis proof of the existence of Haar measure on locally compact abelian groups
Authors: Izzo Alexander J.
Published: June 2010

A simple proof of the existence of Haar measure on locally compact abelian groups is given. The proof uses the Markov-Kakutani fixed-point theorem.
On changing fixed points and coincidences to roots
Authors: Brooks Robin , Wong Peter
Published: June 2010

The coincidence problem, finding solutions to

f
(
x
)
=
g
(
x
)

f(x) = g(x)

, can sometimes be converted to a root problem, finding solutions to

σ
(
x
)
=
a

\sigma (x) = a

. As an application, we show that for any two maps

f
,
g
:
M
→
M
,
N
(
f
,
g
)
=

|

L
(
f
,
g
)

|

f,g:M \to M,N(f,g) = |L(f,g)|

where

M
M

is a compact connected nilmanifold,

N
(
f
,
g
)

N(f,g)

and

L
(
f
,
g
)

L(f,g)

are the Nielsen and Lefschetz numbers, respectively, of

f
f

and

g
g

. This result in the case where

g
g

is the identity is due to D. Anosov.
A generalization of d’Alembert’s functional equation
Authors: Gudder Stanley
Published: June 2010

Recent investigations in the foundations of physics have shown that a quantum mechanical influence function satisfies a generalized d’Alembert functional equation. The present paper derives the solutions for this equation on a topological linear space.
The type of the maximal operators of a class of Walsh convolution operators
Authors: He Ze Lin , Mustard David
Published: June 2010

This paper discusses the properties of a class of

p
p

-adic Walsh convolution operators. The class consists of those

1
1

-parameter sets of operators with kernels that can be represented as the

p
p

-adic Walsh-Fourier integral of a uniformly quasi-convex function. The paper proves that the maximal operators associated with each

1
1

-parameter set are all of strong type

(
∞
,
∞
)

(\infty ,\infty )

and of weak type

(
1
,
1
)

(1,1)

.
Epimorphic adjunction of a weak order unit to an Archimedean lattice-ordered group
Authors: Ball Richard N. , Hager Anthony W. , Kizanis Ann
Published: June 2010

It is shown that an archimedean

l
l

-group

G
G

can be embedded into another,

H
H

, which has a weak unit, by an embedding that is epimorphic in archimedean

l
l

-groups if and only if there is countable

A
⊆
G

A \subseteq G

with

A
⊥

=
(
0
)

{A^ \bot } = (0)

. Then the extension

H
H

can always be chosen conditionally and laterally

σ
\sigma

-complete and the embedding essential, but can never be generated by

G
G

together with finitely many extra elements unless

G
G

already had a weak unit.
Measurable linear functionals and operators on Fréchet spaces
Authors: Wiśniewski Andrzej
Published: June 2010

The structure of measurable linear functionals and operators on Fréchet spaces with so-called stochastic bases is described.
The dual of a Gâteaux smooth Banach space is weak star fragmentable
Authors: Ribarska N. K.
Published: June 2010

The techniques of Preiss-Phelps-Namioka is used to prove that if a Banach space

E
E

admits a Gâteaux smooth norm then its dual

E
∗

{E^*}

, endowed the weak star topology, is fragmentable.
𝑘-hyponormality of weighted shifts
Authors: McCullough Scott , Paulsen Vern
Published: June 2010

An operator

T
T

is defined to be

k
k

-hyponormal if the operator matrix

(

[

T

∗
j

,

T
i

]

)

i
,
j
=
1

k

\left ( {\left [ {{T^{ * j}},{T^i}} \right ]} \right )_{i,j = 1}^k

is positive, where

[

A
,
B

]

=
A
B
−
B
A

\left [ {A,B} \right ] = AB - BA

. In A note on joint hyponormality, Proc. Amer. Math. Soc. 107 (1989), 187-195, we proved that

k
k

-hyponormality is equivalent to a Bram-type condition, namely, that the operator matrix

(

T

∗
j

T
i

)

i
,
j
=
0

k

\left ( {{T^{ * j}}{T^i}} \right )_{i,j = 0}^k

is positive. In this note we prove that for weighted shifts,

k
k

-hyponormality is equivalent to an Embry-type condition, namely, that the operator matrix

(

T

∗
i
+
j

T

i
+
j

)

i
,
j
=
0

k

\left ( {{T^{ * i + j}}{T^{i + j}}} \right )_{i,j = 0}^k

is positive. We give an example to show that this latter condition fails even for a rank one perturbation of a weighted shift. For weighted shifts this Embry condition reduces to the positivity of a sequence of

(

k
+
1

)

×

(

k
+
1

)

\left ( {k + 1} \right ) \times \left ( {k + 1} \right )

Hankel matrices and we use this reduction to give a new proof of one of the principal results of Curto.
A splitting theorem for the 𝑛-r.e. degrees
Authors: Cooper S. Barry
Published: June 2010

We prove a splitting theorem for the n-r.e. degrees, of which the Sacks Splitting Theorem [9] for the r.e. (= 1-r.e.) degrees is a special case. For background terminology and notation see [4] and [11].
Common fixed points for compatible maps on the unit interval
Authors: Jungck Gerald
Published: June 2010

Let

g
g

be a continuous self-map of the unit interval

I
I

. It is proved that

g
g

has a common fixed point with every continuous function

f
:
I
→
I

f:I \to I

that is nontrivially compatible with

g
g

iff every periodic point of

g
g

is a fixed point of

g
g

.
Invariant affine connections on symmetric spaces
Authors: Laquer H. Turner
Published: June 2010

The space of left invariant affine connections is determined for the compact irreducible Riemannian symmetric spaces.
Attractors in restricted cellular automata
Authors: Hurley Mike
Published: June 2010

The goal of this note is to extend previous results about the dynamics of cellular automata to "restricted cellular automata." Roughly speaking, a cellular automaton is a rule that updates a configuration of "states" that are arranged along the integer lattice in

R

\mathbb {R}

. In applications one often thinks of one of these states as "blank" or "quiescent," while the other "active" states evolve against a quiescent background. Often the physically relevant configurations are those with only a finite number of active states. If

X
0

{X_0}

is the set of all such states, and if a cellular automaton maps

X
0

{X_0}

to

X
0

{X_0}

, then its restriction to

X
0

{X_0}

is a restricted cellular automaton. The main results show that there are rather strong constraints on the collection of attractors for any restricted cellular automaton. These constraints parallel those described in [H1] for the unrestricted case.
Representations of a class of real 𝐵*-algebras as algebras of quaternion-valued functions
Authors: Kulkarni S. H.
Published: June 2010

For a compact Hausdorff space

X
X

, let

C
(
X
,

H

)

C(X,{\mathbf {H}})

denote the set of all quaternion-valued functions on

X
X

. It is proved that if a real

B
∗

{B^*}

-algebra

A
A

satisfies the following conditions: (i) the spectrum of every selfadjoint element is contained in the real line and (ii) every element in

A
A

is normal, then

A
A

is isometrically

∗
*

-isomorphic to a closed

∗
*

-subalgebra of

C
(
X
,

H

)

C(X,{\mathbf {H}})

for some compact Hausdorff

X
X

. In particular, a real

C
∗

{C^*}

-algebra in which every element is normal is isometrically

∗
*

-isomorphic to a closed

∗
*

-subalgebra of

C
(
X
,

H

)

C(X,{\mathbf {H}})

.
Operators on Banach spaces taking compact sets inside ranges of vector measures
Authors: Piñeiro Cándido
Published: June 2010

Let

X
X

and

Y
Y

be real Banach spaces. We prove that an operator

T
T

from

X
X

into

Y
Y

maps compact subsets of

X
X

into subsets of

Y
Y

that lie inside ranges of

Y
Y

-valued measures if and only if its dual operator

T
∗

{T^ * }

factors through a subspace of an

L
1

(
μ
)

{L^1}(\mu )

-space. In fact, we prove that every compact is taken into a subset of a compact range. We also prove that

T
∗

{T^ * }

is

1
1

-summing if and only if

T
T

maps compact subsets into subsets of

Y
Y

lying inside ranges of vector measures with bounded variation.
Nielsen numbers of periodic maps on solvmanifolds
Authors: Lee Kyung Bai
Published: June 2010

Let

f
:
M
→
M

f:M \to M

be a self-map of a solvmanifold

M
M

. Then the Lefschetz number

L
(
f
)

L(f)

and the Nielsen number

N
(
f
)

N(f)

of

f
f

satisfy

|

L
(
f
)

|

≤
N
(
f
)

|L(f)| \leq N(f)

. If

f
f

is homotopically periodic (i.e.,

f
k

≃

id

{f^k} \simeq {\text {id}}

, for some

k
≥
1

k \geq 1

), then

L
(
f
)
=
N
(
f
)

L(f) = N(f)

.
Essential chains and homotopy type of posets
Authors: Björner Anders
Published: June 2010

A short proof of a result of Billera, Kapranov, and Sturmfels (Cellular strings on polytopes, preprint, 1991), verifying a homotopy type conjecture of Baues (Geometry of loop spaces and the cobar construction, Mem. Amer. Math. Soc., no. 230, 1980), is given.
Classification theorem for Menger manifolds
Authors: Chigogidze A.
Published: June 2010

We introduce the notion of the

n
n

-homotopy kernel of a Menger manifold and prove the following theorem: Menger manifolds are

n
n

-homotopy equivalent if and only if the

n
n

-homotopy kernels are homeomorphic.
Minimal relative relation modules of finite 𝑝-groups
Authors: Yamin Mohammad
Published: June 2010

Consider

1
→
S
→
E
→
G
→
1

1 \to S \to E \to G \to 1

, where

G
G

is a finite

p
p

-group generated by

g
i

,

1
⩽
i
⩽
d

{g_i},\;1 \leqslant i \leqslant d

, and

E
E

a free product of cyclic groups

⟨

g
i

⟩
,
1
⩽
i
⩽
d

\langle {g_i}\rangle ,1 \leqslant i \leqslant d

. If

d
d

is the minimum number of generators for

G
G

, then we prove that the largest elementary abelian

p
p

-quotient

S

/

S
′

S
p

S/{S’}{S^p}

, regarded as an

F

p

G

{\mathbb {F}_p}G

-module via conjugation in

E
E

, is nonprojective and indecomposable.
An uncountable collection of mutually incomparable chainable continua
Authors: Awartani Marwan M.
Published: June 2010

We exhibit the existence of continuum many compactifications of the ray with the arc as remainder, no one of which maps onto any other. We also obtain continuum many pathwise connected circularly chainable continua no one of which maps onto any other. The involved constructions and proofs are combinatorial in nature.
A note on the cone multiplier
Authors: Mockenhaupt Gerd
Published: June 2010

In this paper we study the convolution operator given on the Fourier transform side by multiplication by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m Subscript alpha Baseline left-parenthesis x comma z right-parenthesis equals phi left-parenthesis z right-parenthesis left-parenthesis 1 minus StartAbsoluteValue x EndAbsoluteValue slash z right-parenthesis Subscript plus Superscript alpha Baseline comma left-parenthesis x comma z right-parenthesis element-of bold upper R squared times bold upper R comma alpha greater-than 0 comma">

m
α

(
x
,
z
)
=
ϕ
(
z
)
(
1
−

|

x

|

/

z

)
+
α

,

(
x
,
z
)
∈

R

2

×

R

,

α
>
0
,

{m_\alpha }(x,z) = \phi (z)(1 - |x|/z)_ + ^\alpha ,\qquad (x,z) \in {{\mathbf {R}}^2} \times {\mathbf {R}},\;\alpha > 0,

\]
where

ϕ
∈

C
0
∞

(
1
,
2
)

\phi \in C_0^\infty (1,2)

. We will prove that

m
α

{m_\alpha }

defines a bounded operator on

L
4

(

R

3

)

{L^4}({{\mathbf {R}}^3})

if

α
>

1
8

\alpha > \tfrac {1} {8}

. Furthermore, as a generalization of a result of C. Fefferman (Acta Math. 124 (1970), 9-36), we will show that an

(

L
2

,

L
p

)

({L^2},{L^p})

restriction theorem for compact

C
∞

{C^\infty }

submanifolds

M
⊂

R

n

M \subset {{\mathbf {R}}^n}

of arbitrary codimension imply results for multipliers having a singularity of the form

dist
⁡

(
x
,
M

)
α

\operatorname {dist} {(x,M)^\alpha }

near

M
M

.
Majorization and domination in the Bergman space
Authors: Korenblum Boris , Richards Kendall
Published: June 2010

Let

f
f

and

g
g

be functions analytic on the unit disk and let

|

|

⋅

|

|

|| \cdot ||

denote the Bergman norm. Conditions are identified under which there exists an absolute constant

c
c

, with

0
>
c
>
1

0 > c > 1

, such that the relationship
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue g left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-slanted-equals StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue left-parenthesis c less-than-or-slanted-equals StartAbsoluteValue z EndAbsoluteValue greater-than 1 right-parenthesis">

|

g
(
z
)

|

⩽

|

f
(
z
)

|

(
c
⩽

|

z

|

>
1
)

|g(z)| \leqslant |f(z)|(c \leqslant |z| > 1)

will imply

|

|

g

|

|

⩽

|

|

f

|

|

||g|| \leqslant ||f||

.
A strengthening of Leth and Malitz’s uniqueness condition for sequences
Authors: Khamsi M. A. , Nymann J. E.
Published: June 2010

A series

∑

a
n

\sum {{a_n}}

of nonnegative real numbers is determined up to a constant multiple by the comparisons of its subsums, provided that

a
i

⩽

∑

i
>
n

a
i

{a_i} \leqslant \sum \nolimits _{i > n} {{a_i}}

and

{

a
n

}

\{ {a_n}\}

decreases to

0
0

. This characterization is an improvement of Leth and Malitz’s results.
A note on commutativity of unbounded representations
Authors: Ōta Schōichi
Published: June 2010

Commutativity for unbounded representations is studied in terms of the Cayley transform.
Two weight Φ-inequalities for the Hardy operator, Hardy-Littlewood maximal operator, and fractional integrals
Authors: Lai Qinsheng
Published: June 2010

Suppose

Φ
\Phi

is an appropriate Young’s function and

w
(
x
)
,
v
(
x
)

w(x),v(x)

are nonnegative locally integrable functions. Let

T
T

denote one of three linear operators of special importance that map suitable functions on

R
n

{R^n}

into functions on

R
n

{R^n}

. For the Hardy operator

T
T

, we study the inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript normal infinity Baseline normal upper Phi left-parenthesis StartAbsoluteValue upper T f left-parenthesis x right-parenthesis EndAbsoluteValue right-parenthesis w left-parenthesis x right-parenthesis d x less-than-or-slanted-equals upper C integral Subscript 0 Superscript normal infinity Baseline normal upper Phi left-parenthesis StartAbsoluteValue f left-parenthesis x right-parenthesis EndAbsoluteValue right-parenthesis v left-parenthesis x right-parenthesis d x">

∫
0
∞

Φ
(

|

T
f
(
x
)

|

)
w
(
x
)

d
x
⩽
C

∫
0
∞

Φ
(

|

f
(
x
)

|

)
v
(
x
)

d
x

\int _0^\infty {\Phi (|Tf(x)|)w(x)\,dx \leqslant C\int _0^\infty {\Phi (|f(x)|)v(x)\,dx} }

\]
and for the Hardy-Littlewood maximal operator or fractional integrals

T
T

, we discuss the inequalities
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper R Superscript n Endscripts normal upper Phi left-parenthesis StartAbsoluteValue upper T left-parenthesis f v right-parenthesis left-parenthesis x right-parenthesis EndAbsoluteValue right-parenthesis w left-parenthesis x right-parenthesis d x less-than-or-slanted-equals upper C integral Underscript upper R Superscript n Baseline Endscripts normal upper Phi left-parenthesis StartAbsoluteValue f left-parenthesis x right-parenthesis EndAbsoluteValue right-parenthesis v left-parenthesis x right-parenthesis d x period">

∫

R
n

Φ
(

|

T
(
f
v
)
(
x
)

|

)
w
(
x
)

d
x
⩽
C

∫

R
n

Φ
(

|

f
(
x
)

|

)
v
(
x
)

d
x
.

\int _{{R^n}} {\Phi (|T(fv)(x)|)w(x)\,dx \leqslant C\int _{{R^n}} {\Phi (|f(x)|)v(x)\,dx.} }

\]
In all cases we obtain the necessary and sufficient conditions.
Strict radical classes of commutative rings
Authors: McConnell N. R.
Published: June 2010

We classify strongly hereditary strict radical classes for commutative rings and give some results for supernilpotent strict radical classes and strict radical classes contained in the nil radical.
Test elements in excellent rings with an application to the uniform Artin-Rees property
Authors: Aberbach Ian M.
Published: June 2010

Working in positive prime characteristic throughout, we show that excellent rings of dimension

2
2

or smaller have completely stable test elements and use this to show that excellent domains of dimension

3
3

have the uniform Artin-Rees property.
On closed subspaces of operator ranges
Authors: Harte Robin , Shannon Gerry
Published: June 2010

Necessary and sufficient for the closure of a linear subspace to lie in the range of a bounded linear operator is a certain "bounded preimage property" for the operator.
When is 𝐹[𝑥,𝑦] a unique factorization domain?
Authors: Beauregard Raymond A.
Published: June 2010

Although the commutative polynomial ring

F
[
x
,
y
]

F[x,y]

is a unique factorization domain (UFD) and the free associative algebra

F
⟨
x
,
y
⟩

F\langle x,y\rangle

is a similarity-UFD when

F
F

is a (commutative) field, it is shown that the polynomial ring

F
[
x
,
y
]

F[x,y]

in two commuting indeterminates is not a UFD in any reasonable sense when

F
F

is the skew field of rational quaternions.
Crinkled functions and intersections with polynomials
Authors: Fink A. M.
Published: June 2010

We prove that if

|

|

f

(
n
+
1
)

|

|

||{f^{(n + 1)}}||

is large and

|

|

f

(
n
)

|

|

∞

||{f^{(n)}}|{|_\infty }

is bounded, then there is a polynomial

p
p

of degree

n
n

such that

f
(
t
)
=
p
(
t
)

f(t) = p(t)

has many solutions.
Manifolds with pinched radial curvature
Authors: Machigashira Yoshiroh
Published: June 2010

We generalize Toponogov’s theorem to the context of radial curvature and obtain corresponding generalizations of classical sphere theorems.
The first cohomology groups of nest algebras on normed spaces
Authors: Han De Guang
Published: June 2010

In this short note we prove that all the first cohomology groups of nest algebras acting on a normed space are trivial when the coefficients lie in any weakly closed bimodule which contains the nest algebra.
Fractional differentiation in the self-affine case. III. The density of the Cantor set
Authors: Patzschke N. , Zähle M.
Published: June 2010

We compute the fractional density of the middle-third Cantor measure explicitly. Its numerical value is 0.62344....
A counterexample to the equivariant simple loop conjecture
Authors: Usadi Karin
Published: June 2010

In 1985 Gabai gave a complete proof of the Simple Loop Conjecture, which states that any map between closed surfaces, which does not induce an injection on the level of

π
1

{\pi _1}

, takes some noncontractible simple loop in the domain surface to a contractible loop in the target surface. In this paper we study an analogous result for the category of surfaces equipped with finite group actions and the maps which commute with the group structures (the "equivariant" maps). We find a counterexample to the equivariant analog of the Simple Loop Conjecture for the cyclic group of order

3
3

. The proof uses an equivariant analog of a theorem of Edmonds which gives a standard geometric representative for any homotopy class of surface maps of nonzero degree.
Taylor exactness and Kato’s jump
Authors: Harte Robin
Published: June 2010

The middle exactness condition of Joseph Taylor is related to the zero-jump condition of Tosio Kato, and some "commutative" Fredholm theory explored.
Schatten ideal behavior of a generalized Hardy operator
Authors: Nowak Krzysztof
Published: June 2010

We show the

S
p

{S^p}

-criteria and the cut-off result for an operator closely related to the classical Hardy inequality.
Linear transformations preserving potent matrices
Authors: Brešar Matej , Šemrl Peter
Published: June 2010

Linear transformations of

M
n

{M_n}

, the algebra of

n
×
n

n \times n

matrices over

C

\mathbb {C}

, which preserve the set of all potent matrices, are characterized.
Extendibility criterion for a projective module of rank one over 𝑅[𝑇] and 𝑅[𝑇,𝑇⁻¹]
Authors: Bhatwadekar S. M. , Varma P. L. N.
Published: June 2010

In this note we give a criterion for a finitely generated projective module

P

\mathcal {P}

of constant rank one over

R
[
T
]

R[T]

or

R
[
T
,

T

−
1

]

R[T,{T^{ - 1}}]

to be extended from

R
R

in terms of invertible ideals, when

R
R

is an integral domain. We show that if

I
I

is an invertible ideal of

R
[
T
]

R[T]

or

R
[
T
,

T

−
1

]

R[T,{T^{ - 1}}]

such that

I
∩
R
≠
0

I \cap R \ne 0

, then

I
I

is extended from

R
R

if and only if

I
∩
R

I \cap R

is an invertible ideal of

R
R

.
A proof of the existence of level 1 elliptic cohomology
Authors: Hovey Mark A.
Published: June 2010

Landweber provided two proofs of the existence of (level 2) elliptic cohomology (Lecture Notes in Math., vol. 1326, Springer-Verlag, New York, 1988, pp. 69-93). As Baker pointed out (J. Pure Appl. Algebra 63 (1990), 1-11), one of these proofs gives a level 1 elliptic cohomology theory as well. In this note we provide an alternative proof of the existence of level 1 elliptic cohomology. The idea here is to use Landweber’s direct proof of the existence of level 2 elliptic cohomology and an integrality argument to deduce the existence of level

1
1

elliptic cohomology from that.
The single-valued extension property and spectral manifolds
Authors: Sun Shan Li
Published: June 2010

We discuss the relation between the single-valued extension property (that is, Dunford’s property (A)) and spectral manifolds

X
T

(
F
)

{X_T}(F)

of a bounded linear operator. In particular, we prove that Dunford’s property (C) implies the property (A). We also prove that if

T
∈
B
(
X
)

T \in B(X)

has the property

(

β

∗

)

({\beta ^{\ast }})

introduced by Fong, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Subscript upper T Sub Superscript asterisk Superscript asterisk Baseline left-parenthesis upper F right-parenthesis equals upper X Subscript upper T Baseline left-parenthesis double-struck upper C minus upper F right-parenthesis Superscript up-tack">

X

T

∗

∗

(
F
)
=

X
T

(

C

∖
F

)
⊥

X_{{T^{\ast }}}^{\ast }(F) = {X_T}{(\mathbb {C}\backslash F)^ \bot }

for every closed set

F
F

in the complex plane

C

\mathbb {C}

.
Computational aspects of soap bubble deformations
Authors: Pinkall U. , Sterling I.
Published: June 2010

H
H

-surfaces are studied via soliton theory with the aid of a computer.
The Cantor tree, the 𝛾-property, and Baire function spaces
Authors: Ma Daniel K.
Published: June 2010

Let

X
⊆

2
ω

X \subseteq {2^\omega }

and

T
∪
X

T \cup X

be the Cantor tree over

X
X

. We show that

C
k

(
T
∪
X
)

{C_k}(T \cup X)

is a Baire space if and only if

X
X

is a

γ
\gamma

-set. We obtain from this result consistent examples of spaces

Y
Y

and

Z
Z

such that

C
k

(
Y
)

{C_k}(Y)

and

C
k

(
Z
)

{C_k}(Z)

are Baire spaces but

C
k

(
Y
)
×

C
k

(
Z
)

{C_k}(Y) \times {C_k}(Z)

is not a Baire space. It also follows that there are consistent examples of locally compact nonparacompact spaces

Y
Y

such that

C
k

(
Y
)

{C_k}(Y)

is Baire but not weakly

α
\alpha

-favorable.
Unbounded commuting operators and multivariate orthogonal polynomials
Authors: Xu Yuan
Published: June 2010

The multivariate orthogonal polynomials are related to a family of operators whose matrix representations are block Jacobi matrices. A sufficient condition is given so that these operators, in general unbounded, are commuting and selfadjoint. The spectral theorem for these operators is used to establish the existence of the measure of orthogonality in Favard’s theorem.
Lower bound on the error probability for families with bounded likelihood ratios
Authors: Rukhin Andrew L.
Published: June 2010

In the classification problem a new sharp lower bound for the error probability is derived. This bound depends only on the prior probabilities and on the support of pairwise likelihood ratios.
Some consequences of Harish-Chandra’s submersion principle
Authors: Rader Cary , Silberger Allan
Published: June 2010

Let

G
G

be a reductive

p

\mathfrak {p}

-adic group,

K
K

a good maximal compact subgroup,

K
1

⊂
K

{K_1} \subset K

any open subgroup, and

π
\pi

an admissible representation of

G
G

of finite type. In A submersion principle and its applications, Harish-Chandra proves the theorem that

∫
K

π
(
k
g

k

−
1

)

d
k

\int _K {\pi (kg{k^{ - 1}})\,dk}

is a finite-rank operator for

g
g

in the regular set

G
′

G’

in order to show that the character

Θ
π

(
g
)

{\Theta _\pi }(g)

is a locally constant class function on

G
′

G’

. From this, the authors derive the formula
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta left-parenthesis 1 right-parenthesis normal upper Theta Subscript pi Baseline left-parenthesis g right-parenthesis equals d left-parenthesis pi right-parenthesis integral Underscript upper G slash upper Z Endscripts integral Underscript upper K 1 Endscripts theta left-parenthesis x k g k Superscript negative 1 Baseline x Superscript negative 1 Baseline right-parenthesis d k d ModifyingAbove x With dot left-parenthesis g element-of upper G prime right-parenthesis">

θ
(
1
)

Θ
π

(
g
)
=
d
(
π
)

∫

G

/

Z

∫

K
1

θ
(
x
k
g

k

−
1

x

−
1

)

d
k

d

x
˙

(
g
∈

G
′

)

\theta (1){\Theta _\pi }(g) = d(\pi )\int _{G/Z} {\int _{{K_1}} {\theta (xkg{k^{ - 1}}{x^{ - 1}})\,dk\,d\dot x} \quad (g \in G’)}

for any

K
K

-finite matrix coefficient

θ
\theta

of a discrete series representation

π
\pi

with formal degree

d
(
π
)

d(\pi )

. They use another technical result of the paper to prove that invariant integrals of Schwartz space functions converge absolutely. None of these results depends upon a characteristic zero assumption.
Symmetric behavior in functions
Authors: Darji Udayan B.
Published: June 2010

S. Marcus raised the following problem: Find necessary and sufficient conditions for a set to be the set of points of symmetric continuity of some function

f
:
R
→
R

f:R \to R

. We show that there is no such characterization of topological nature. We prove that given a zero-dimensional set

M
⊆
R

M \subseteq R

, there exists a function

f
:
R
→
R

f:R \to R

whose set of points of symmetric continuity is topologically equivalent to

M
M

. Thus, there is no "upper bound" on the topological complexities of

M
M

. We also prove similar theorems about the set of points where a function may be symmetrically differentiable, symmetric, or smooth.
Characterizations of linear congruences on a free monoid
Authors: Cherubini Alessandra , Petrich Mario
Published: June 2010

Linear congruences on a free monoid

X

∗

{X^{\ast }}

coincide with the congruences on

X

∗

{X^{\ast }}

induced by nontrivial homomorphisms into the additive group of integers. For a finite

X
X

, we characterize abstractly several classes of linear congruences on

X

∗

{X^{\ast }}

, in particular,

π
\pi

-linear congruences, called

p
p

-linear and determined by Reis,

ξ
\xi

-linear congruences, introduced by Petrich and Thierrin, and general linear congruences, introduced by the authors. These characterizations include descriptions involving maximality as prefix congruences.
Minimal surfaces and 𝐻-surfaces in non-positively curved space forms
Authors: Palmer Bennett
Published: June 2010

We show that if the Gauss curvature of a surface of constant mean curvature in a nonpositively curved space form is sufficiently pinched, the surface is stable. In this case, we also give an upper bound for the inradius. We then show that the inradius of a stable minimal surface in Euclidean space, which is contained in a solid cylinder, is bounded above by a constant depending only on the radius of the cylinder.
Finite moments perturbations of 𝑦”=0 in Banach algebras
Authors: Spigler Renato , Vianello Marco
Published: June 2010

Rigorous asymptotics for a basis of

y
+
g
(
x
)
y
=
0
,

x
∈
[
1
,
+
∞
)

y + g(x)y = 0,\;x \in [1, + \infty )

, is derived in the framework of Banach algebras. The key assumption is

∫
1

+
∞

x
k

|

|

g
(
x
)

|

|

d
x
>
∞

\int _1^{ + \infty } {{x^k}} ||g(x)||dx > \infty

for

k
=
1

k = 1

or

k
=
2

k = 2

. Such results improve and generalize previous work on linear second-order matrix differential equations.
Univalent logharmonic ring mappings
Authors: Abdulhadi Zayid , Hengartner Walter , Szynal Jan
Published: June 2010

Univalent logharmonic ring mappings are characterized in terms of univalent starlike mappings. An existence and uniqueness theorem is also given.
On weak convergence in 𝐻¹(𝑅^{𝑑})
Authors: Jones Peter W. , Journé Jean-Lin
Published: June 2010

We prove that the a.e. convergence of a sequence of functions bounded in

H
1

(

R

d

)

{H^1}({{\mathbf {R}}^d})

to a function in

L
1

(

R

d

)

{L^1}({{\mathbf {R}}^d})

implies weak convergence.
Existence and weak-type inequalities for Cauchy integrals of general measures on rectifiable curves and sets
Authors: Mattila Pertti , Melnikov Mark S.
Published: June 2010

Ifμ\muis a finite complex Borel measure andΓ\Gammaa Lipschitz graph in the complex plane, then forλ>0\lambda > 0\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue StartSet z element-of normal upper Gamma colon sup Underscript epsilon greater-than 0 Endscripts StartAbsoluteValue integral Underscript StartAbsoluteValue zeta minus z EndAbsoluteValue greater-than-or-slanted-equals epsilon Endscripts left-parenthesis zeta minus z right-parenthesis Superscript negative 1 Baseline d mu zeta EndAbsoluteValue greater-than lamda EndSet EndAbsoluteValue less-than-or-slanted-equals c left-parenthesis normal upper Gamma right-parenthesis lamda Superscript negative 1 Baseline StartAbsoluteValue EndAbsoluteValue mu StartAbsoluteValue EndAbsoluteValue Subscript 1 Baseline period">|{z∈Γ:supε>0|∫|ζ−z|⩾ε(ζ−z)−1dμζ|>λ}|⩽c(Γ)λ−1||μ||1.\left | {\left \{ {z \in \Gamma :\sup \limits _{\varepsilon > 0} \left | {\int _{|\zeta - z| \geqslant \varepsilon } {{{(\zeta - z)}^{ - 1}}} d\mu \zeta } \right | > \lambda } \right \}} \right | \leqslant c(\Gamma ){\lambda ^{ - 1}}||\mu |{|_1}.\]It follows that for any finite Borel measureμ\muand any rectifiable curveΓ\Gammathe finite principal value\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript epsilon down-arrow 0 Endscripts integral Underscript StartAbsoluteValue zeta minus z EndAbsoluteValue greater-than-or-slanted-equals epsilon Endscripts left-parenthesis zeta minus z right-parenthesis Superscript negative 1 Baseline d mu zeta">limε↓0∫|ζ−z|⩾ε(ζ−z)−1dμζ\lim \limits _{\varepsilon \downarrow 0} \int _{|\zeta - z| \geqslant \varepsilon } {{{(\zeta - z)}^{ - 1}}d\mu \zeta }\]exists for almost all (with respect to length)z∈Γz \in \Gamma.
Polynomial approximation for a class of physical random variables
Authors: De Santis A. , Gandolfi A. , Germani A. , Tardelli P.
Published: June 2010

In white noise theory on Hilbert spaces, it is known that maps which are uniformly continuous around the origin in the S-topology constitute an important class of "physical" random variables. We prove that random variables having such a continuity property can be approximated in the gaussian measure by polynomial random variables. The proof relies on representing functions which are uniformly S-continuous around the origin as the composition of a continuous map with a Hilbert-Schmidt operator.
Connectedness of the space of minimal 2-spheres in 𝑆^{2𝑚}(1)
Authors: Kotani Motoko
Published: June 2010

Loo’s theorem asserts that the space of all branched minimal

2
2

-spheres of degree

d
d

in

S
4

(
1
)

{S^4}(1)

is connected. The main theorem in this paper is that the assertion is still true for

S

2
m

(
1
)

{S^{2m}}(1)

. It is shown that any branched minimal

2
2

-sphere in

S

2
m

(
1
)

{S^{2m}}(1)

can be deformed, preserving its degree, to a meromorphic function.
Spectral types of uniform distribution
Authors: Choe Geon H.
Published: June 2010

We investigate the spectral types of unitary operator

U
U

on

L
2

(

T

)

{L^2}(\mathbb {T})

defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper U f right-parenthesis left-parenthesis x right-parenthesis equals upper A left-parenthesis x right-parenthesis f left-parenthesis x plus theta right-parenthesis comma StartAbsoluteValue upper A left-parenthesis x right-parenthesis EndAbsoluteValue equals 1">

(
U
f
)
(
x
)
=
A
(
x
)
f
(
x
+
θ
)
,

|

A
(
x
)

|

=
1

(Uf)(x) = A(x)f(x + \theta ),|A(x)| = 1

a.e., where

T

\mathbb {T}

is the unit circle identified with the half open interval

[
0
,
1
)

[0,1)

and

θ
\theta

is irrational. It is shown that Veech’s result on the Kronecker-Weyl theorem modulo

2
2

is closely related to the spectral type of

U
U

.