bluuebunny/crossref_metadata_2025 huggingface dataset:
On nonisomorphic Room squares
Authors: Dinitz J. H. , Stinson D. R.
Published: June 2010

Let

NR

(
s
)

{\text {NR}}(s)

denote the number of nonisomorphic Room squares of side

s
s

. We prove that for

s
s

sufficiently large,

NR

(
s
)
⩾
exp
⁡
(
c

s
2

)

{\text {NR}}(s) \geqslant \exp (c{s^2})

for some absolute constant

c
c

. More precisely,

NR

(
s
)
⩾
.19
exp
⁡
(
.04

s
2

)

{\text {NR}}(s) \geqslant .19\exp (.04{s^2})

for

s
⩾
153

odd

s \geqslant 153\;{\operatorname {odd}}

; and

NR

(
s
)
⩾
.19
exp
⁡
(
.09

s
2

)

{\text {NR}}(s) \geqslant .19\exp (.09{s^2})

for

s
⩾
1001

 odd 

s \geqslant 1001{\text { odd }}

.
The uniqueness of multiplication in function algebras
Authors: Jarosz Krzysztof
Published: June 2010

Let

A
A

be a function algebra. We prove that the original multiplication of

A
A

is a unique multiplication on the underlying Banach space of

A
A

which produces a Banach algebra with the same unit as the original one.
A semibounded closed symmetric operator whose square has trivial domain
Authors: Chernoff Paul R.
Published: June 2010

The existence of closed symmetric operators

T
T

such that

D
(

T
2

)
=
(
0
)

D({T^2}) = (0)

was shown by Naimark. This paper gives a simple, explicit construction for such operators and shows that

T
T

can be semibounded.
A note on rank-one operators in reflexive algebras
Authors: Laurie Cecelia , Longstaff W. E.
Published: June 2010

It is shown that if the invariant subspace lattice of a reflexive algebra

A

\mathcal {A}

, acting on a separable Hilbert space, is both commutative and completely distributive, then the algebra generated by the rank-one operators of

A

\mathcal {A}

is dense in

A

\mathcal {A}

is any of the strong, weak, ultrastrong or ultraweak topologies. Some related density results are also obtained.
Note on the residues of the singularities of a Riemannian foliation
Authors: Mei Xiang Ming
Published: June 2010

We generalize the result of Lazarov and Pasternack [3] on the residues of the singularities of a Riemannian foliation and give an explicit formula of the residues of the connected components of the singular set of a Riemannian foliation without any restriction on the dimension of the connected components of its singular set. This formula is also the generalization of the residues formula of the zero set of the Killing vector field due to Baum and Cheeger [6].
A theorem on the cardinality of 𝜅-total spaces
Authors: Stephenson R. M.
Published: June 2010

Throughout this article,

κ
\kappa

denotes an arbitrary infinite cardinal number. In 1979, A. A. Gryzlov strengthened a well-known result of A. V. Arhangel’skii by proving that every compact

T
1

{T_1}

-space of pseudocharacter

κ
\kappa

has cardinality

⩽

2
κ

\leqslant {2^\kappa }

. Using techniques similar to Gryzlov’s, we prove that every

2
κ

{2^\kappa }

-total,

T
1

{T_1}

-space of pseudocharacter

⩽
κ

\leqslant \kappa

is compact and hence of cardinality

⩽

2
κ

\leqslant {2^\kappa }

. Some related results and examples are given.
A smooth scissors congruence problem
Authors: Greenberg Peter
Published: June 2010

Classifying space techniques are used to solve a smooth version of the classical scissors congruence problem.
Optimal linear extensions by interchanging chains
Authors: Rival Ivan
Published: June 2010

For a finite ordered set

P
P

how can a linear extension

L
=

C
1

⊕

C
2

L = {C_1} \oplus {C_2}

be constructed which minimizes the number

m
m

of chains

C
i

{C_i}

of

P
P

? While this question remains largely unanswered we show that a natural "greedy" algorithm is actually optimal for a far wider class of ordered sets than was hitherto suspected.
Equimultiplicity and hyperplanarity
Authors: Mulay S. B.
Published: June 2010

Let

R
R

be an excellent regular local domain containing a field. Let

F
F

be a nonzero principal ideal in

R
R

contained in

m
(
R
)

m(R)

. Then the

2
2

-codimensional equimultiple locus of

(
R
,
F
)

(R,F)

is hyperplanar.
The indices of torsion-free subgroups of Fuchsian groups
Authors: Burns R. G. , Solitar Donald
Published: June 2010

Elementary algebraic techniques are used to obtain the precise possible indices of torsion-free subgroups of finite index of finitely generated Fuchsian groups (and related groups).
A gloss on a theorem of Furstenberg
Authors: Dubins Lester E.
Published: June 2010

Certain refinements are offered for Furstenberg’s ergodic-theoretic version of Szemeredi’s theorem.
A note on countably normed nuclear spaces
Authors: Holmström Lasse
Published: June 2010

A modification of the Kōmura-Kōmura imbedding theorem is used to show that every countably normed nuclear space is isomorphic to a subspace of a nuclear Fréchet space with basis and a continuous norm. The space with basis can be chosen to be a quotient of

(
s
)

(s)

.
On the existence of nonsimple real eigenvalues for general Sturm-Liouville problems
Authors: Mingarelli A. B.
Published: June 2010

The Sturm-Liouville eigenvalue problem

−
y
+
q
(
t
)
y
=
λ
r
(
t
)
y

- y + q(t)y = \lambda r(t)y

,

t
∈
[
a
,
b
]

t \in [a, b]

, where

y
y

is required to satisfy a pair of homogeneous separated boundary conditions at

t
=
a

t = a

,

t
=
b

t = b

is considered when no sign restrictions are imposed upon the coefficients

q
q

,

r
r

. It will be shown that the general eigenvalue problem above can admit at most finitely many nonsimple real eigenvalues (in some cases none at all). Moreover, we will show by means of an example that nonsimple real eigenvalues may occur in the case when each of

q
q

and

r
r

changes sign in

(
a
,
b
)

(a, b)

and under Dirichlet boundary conditions.
Some tilings of the plane whose singular points form a perfect set
Authors: Breen Marilyn
Published: June 2010

Let

J

\mathcal {J}

be a tiling of the plane such that for every tile

T
T

of

J

\mathcal {J}

there correspond a tile

T
′

T’

of

J

\mathcal {J}

(not necessarily unique) and an integer

k
(
T
,

T
′

)

k(T,T’)

(depending on

T
T

and

T
′

T’

),

2
>
k

2 > k

, such that

T
T

meets

T
′

T’

in

k
(
T
,

T
′

)

k(T,T’)

connected components. Then the set of singular points of

J

\mathcal {J}

is a nowhere dense, perfect set.
A new omitting types theorem
Authors: Steinhorn Charles
Published: June 2010

An omitting types theorem for countable models of superstable theories containing an infinite set of indiscernibles is proved. Various corollaries and applications are given.
On partitions of plane sets into simple closed curves. II
Authors: Bankston Paul
Published: June 2010

We answer some questions raised in [1]. In particular, we prove: (i) Let

F
F

be a compact subset of the euclidean plane

E
2

{E^2}

such that no component of

F
F

separates

E
2

{E^2}

. Then

E
2

∖
F

{E^2}\backslash F

can be partitioned into simple closed curves iff

F
F

is nonempty and connected. (ii) Let

F
⊆

E
2

F \subseteq {E^2}

be any subset which is not dense in

E
2

{E^2}

, and let

S

\mathcal {S}

be a partition of

E
2

∖
F

{E^2}\backslash F

into simple closed curves. Then

S

\mathcal {S}

has the cardinality of the continuum. We also discuss an application of (i) above to the existence of flows in the plane.
A nonpolyhedral triangulated Möbius strip
Authors: Brehm Ulrich
Published: June 2010

We construct a triangulated Möbius strip with 9 vertices which is not embeddable into

R

3

{{\mathbf {R}}^3}

such that all edges are straight line segments. It even cannot be immersed polyhedrally into

R

3

{{\mathbf {R}}^3}

.
A note on the stable homotopy groups of 𝑀𝑆𝑝(𝑛)
Authors: Imaoka Mitsunori
Published: June 2010

The kernel of the epimorphism
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript 8 n plus 3 Superscript upper S Baseline left-parenthesis upper M upper S p left-parenthesis n right-parenthesis right-parenthesis right-arrow pi Subscript 4 n plus 3 Baseline left-parenthesis upper M upper S p right-parenthesis">

π

8
n
+
3

S

(
M
S
p
(
n
)
)
→

π

4
n
+
3

(
M
S
p
)

\pi _{8n + 3}^S(MSp(n)) \to {\pi _{4n + 3}}(MSp)

is a cyclic group of order

8
m
(
n
+
1
)

8m(n + 1)

for some integer

m
(
n
+
1
)

m(n + 1)

defined using the characteristic numbers of the symplectic cobordism classes, and this epimorphism splits for some

n
n

.
Shifted plane partitions of trapezoidal shape
Authors: Proctor Robert A.
Published: June 2010

The number of shifted plane partitions contained in the shifted shape

[
p
+
q
−
1
,
p
+
q
−
3
,
…
,
p
−
q
+
1
]

[p + q - 1,p + q - 3, \ldots ,p - q + 1]

with part size bounded by

m
m

is shown to be equal to the number of ordinary plane partitions contained in the shape

(
p
,
p
,
…
,
p
)

(p,p, \ldots ,p)

(
q

 rows

)

(q{\text { rows}})

with part size bounded by

m
m

. The proof uses known combinatorial descriptions of finite-dimensional representations of semisimple Lie algebras. A separate simpler argument shows that the number of chains of cardinality

k
k

in the poset underlying the shifted plane partitions is equal to the number of chains of cardinality

k
k

in the poset underlying the ordinary plane partitions. The first result can also be formulated as an equality of chain counts for a pair of posets. The pair of posets is obtained by taking order ideals in the other pair of posets.
Cohen-Macaulay algebraic monoids
Authors: Renner Lex E.
Published: June 2010

Let

E
E

be an irreducible, normal, algebraic monoid with group of units

G
=

G

l

2

(
K
)

G = {\text {G}}{{\text {l}}_2}(K)

,

S

l

2

(
K
)
×

K
∗

{\text {S}}{{\text {l}}_2}(K) \times {K^*}

or

PG

l

2

(
K
)
×

K
∗

{\text {PG}}{{\text {l}}_2}(K) \times {K^*}

. Then

E
E

is a Cohen-Macaulay algebraic variety.
Phragmén-Lindelöf theorem in a cohomological form
Authors: Lin Ching Her
Published: June 2010

The main result of this paper is as follows. Given functons

ϕ
1

(
ε
)
,
…
,

ϕ
ν

(
ε
)

{\phi _1}(\varepsilon ), \ldots ,{\phi _\nu }(\varepsilon )

which are holomorphic in sectors

S
1

,
…
,

S
p

{S_1}, \ldots ,{S_p}

, respectively, where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 1 union midline-horizontal-ellipsis union upper S Subscript nu Baseline equals left-brace epsilon colon StartAbsoluteValue arg epsilon EndAbsoluteValue greater-than pi slash 2 alpha comma 0 StartAbsoluteValue epsilon EndAbsoluteValue greater-than rho right-brace">

S
1

∪
⋯
∪

S
ν

=
{
ε
:

|

arg
⁡
ε

|

>
π

/

2
α
,
0

|

ε

|

>
ρ
}

{S_1} \cup \cdots \cup {S_\nu } = \{ \varepsilon :|\arg \varepsilon | > \pi /2\alpha ,0|\varepsilon | > \rho \}

for

α
>
1

\alpha > 1

,

ρ
>
0

\rho > 0

, set

ϕ

j
k

=

ϕ
j

−

ϕ
k

{\phi _{jk}} = {\phi _j} - {\phi _k}

if

S
j

∩

S
k

≠
∅

{S_j} \cap {S_k} \ne \emptyset

. Then

{

ϕ

j
k

}

\{ {\phi _{jk}}\}

satisfy cocycle conditions

ϕ

j
k

+

ϕ

k
l

=

ϕ

j
l

{\phi _{jk}} + {\phi _{kl}} = {\phi _{jl}}

whenever

S
j

∩

S
k

∩

S
l

≠
∅

{S_j} \cap {S_k} \cap {S_l} \ne \emptyset

. In addition to the conditions

|

ϕ
1

|

>

M
0

|{\phi _1}| > {M_0}

and

|

ϕ
ν

|

>

M
0

|{\phi _\nu }| > {M_0}

on the two rays of the boundary

(

i

.e

. 

arg
⁡
ε
=
π

/

2
α
)

({\text {i}}{\text {.e}}{\text {. }}\arg \varepsilon = \pi /2\alpha )

, and

|

ϕ
j

(
ε
)

|

⩽
A
exp
⁡
(
c

/

|

ε

|

)

|{\phi _j}(\varepsilon )| \leqslant A\exp (c/|\varepsilon |)

in

S
j

{S_j}

for some positive numbers

A
A

and

c
c

,

j
=
1
,
2
,
…
,
ν

j = 1,2, \ldots ,\nu

, if the

{

ϕ
j

}

\{ {\phi _j}\}

satisfy the conditions

{

ϕ

j
k

}
>

M
0

\{ {\phi _{jk}}\} > {M_0}

on

S
j

∩

S
k

(
≠
∅
)

{S_j} \cap {S_k}( \ne \emptyset )

, then we get

|

ϕ
j

|

>
M

|{\phi _j}| > M

on

S
S

,

j
=
1
,
2
,
…
,
ν

j = 1,2, \ldots ,\nu

. (From the cohomological point of view, we can get global results for

ϕ
j

{\phi _j}

,oce the local data on cocycles is known.)
Stability of isometries on Banach spaces
Authors: Gevirtz Julian
Published: June 2010

Let

X
X

and

Y
Y

be Banach spaces. A mapping

f
:
X
→
Y

f:X \to Y

is called an

ε
\varepsilon

-isometry if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue double-vertical-bar f left-parenthesis x 0 right-parenthesis minus f left-parenthesis x 1 right-parenthesis double-vertical-bar minus double-vertical-bar x 0 minus x 1 double-vertical-bar EndAbsoluteValue less-than-or-slanted-equals epsilon">

|

‖

f
(

x
0

)
−
f
(

x
1

)

‖

−

‖

x
0

−

x
1

‖

|

⩽
ε

|\left \| {f({x_0}) - f({x_1})} \right \| - \left \| {{x_0} - {x_1}} \right \|| \leqslant \varepsilon

for all

x
0

,

x
1

∈
X

{x_0},{x_1} \in X

. It is shown that there exist constants

A
A

and

B
B

such that if

f
:
X
→
Y

f:X \to Y

is a surjective

ε
\varepsilon

-isometry, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar f left-parenthesis left-parenthesis x 0 plus x 1 right-parenthesis slash 2 right-parenthesis minus left-parenthesis f left-parenthesis x 0 right-parenthesis plus f left-parenthesis x 1 right-parenthesis right-parenthesis slash 2 double-vertical-bar less-than-or-slanted-equals upper A left-parenthesis epsilon double-vertical-bar x 0 minus x 1 double-vertical-bar right-parenthesis Superscript 1 slash 2 Baseline plus upper B epsilon">

‖

f
(
(

x
0

+

x
1

)

/

2
)
−
(
f
(

x
0

)
+
f
(

x
1

)
)

/

2

‖

⩽
A

(
ε

‖

x
0

−

x
1

‖

)

1

/

2

+
B
ε

\left \| {f(({x_0} + {x_1})/2) - (f({x_0}) + f({x_1}))/2} \right \| \leqslant A{(\varepsilon \left \| {{x_0} - {x_1}} \right \|)^{1/2}} + B\varepsilon

for all

x
0

,

x
1

∈
X

{x_0},{x_1} \in X

. This, together with a result of Peter M. Gruber, is used to show that if

f
:
X
→
Y

f:X \to Y

is a surjective

ε
\varepsilon

-isometry, then there exists a surjective isometry

I
:
X
→
Y

I:X \to Y

for which

‖

f
(
x
)
−
I
(
x
)

‖

⩽
5
ε

\left \| {f(x) - I(x)} \right \| \leqslant 5\varepsilon

, thus answering a question of Hyers and Ulam about the stability of isometries on Banach spaces.
The Gauss map and nonholomorphic harmonic maps
Authors: Ishihara Tōru
Published: June 2010

J. Eells and J. C. Wood [2] gave the fundamental method of construction of certain harmonic maps from Riemann surfaces to complex projective spaces. In this paper, we will construct nonholomorphic harmonic maps of Kaehler manifolds to complex Grassmann manifolds as the Gauss maps for holomorphic immersions of Kaehler manifolds to complex projective spaces.
On homeomorphism spaces of Hilbert manifolds
Authors: Wong Raymond Y.
Published: June 2010

Let

M
M

be a Hilbert manifold modeled on the separable Hilbert space

l
2

{l^2}

. We prove the following Fibred Homeomorphism Extension Theorem: Let

M
×

B
n

→

B
n

M \times {B^n} \to {B^n}

be the product bundle over the

n
n

-ball

B
n

{B^n}

, then any (fibred) homeomorphism on

M
×

S

n
−
1

M \times {S^{n - 1}}

extends to a homeomorphism on all

M
×

B
n

M \times {B^n}

if and only if it extends to a mapping on all

M
×

B
n

M \times {B^n}

. Moreover, the size of the extension may be restricted by any open cover on

M
×

B
n

M \times {B^n}

. The result is then applied to study the space of homeomorphisms

H

(
M
)

\mathcal {H}(M)

under various topologies given on

H

(
M
)

\mathcal {H}(M)

. For instance, if

M
=

l
2

M = {l_2}

and

H

(

l
2

)

\mathcal {H}({l_2})

has the compact-open topology, the

H

(

l
2

)

\mathcal {H}({l_2})

is an absolute extensor for all metric spaces. A counterexample is provided to show that the statement above may not be generalized to arbitrary manifold

M
M

.
Reversing the Berry-Esseen inequality
Authors: Hall Peter , Barbour A. D.
Published: June 2010

We derive a lower bound to the rate of convergence in the central limit theorem. Our result is expressed in terms similar to those of the Berry-Esséen inequality, with the distance between two distributions on one side of the inequality and an easily calculated function of the summands on the other, related by a universal constant. The proof is based on Stein’s method.
A note on the ergodic theorem
Authors: Yoshimoto Takeshi
Published: June 2010

Let

M
M

be a positive regular shift-invariant method of summability and let

T
T

be a one-to-one transformation which maps

X
X

onto

X
X

and which is

B

\mathcal {B}

-bimeasurable, i.e.,

A
∈

B

A \in \mathcal {B}

if and only if

T
A
∈

B

TA \in \mathcal {B}

, where

(
X
,

B

)

(X,\mathcal {B})

is a measurable space. Then it is proved that if for a finite measure

μ
\mu

on

B

\mathcal {B}

the sequence

{

μ
(

T
m

A
)

}

\left \{ {\mu ({T^m}A)} \right \}

is

M
M

-summable for each

A
∈

B

A \in \mathcal {B}

, then for any real

r
⩾
1

r \geqslant 1

the sequence

{

f
∘

T
m

}

\left \{ {f \circ {T^m}} \right \}

is

(
C
,
r
)

(C,r)

-summable

μ
\mu

-almost everywhere for every bounded

B

\mathcal {B}

-measurable function

f
f

defined on

X
X

. The result includes the Blum-Hanson theorem.
Some weak equivalences for classifying spaces
Authors: Jekel Solomon M.
Published: June 2010

For topological categories

C

\mathcal {C}

with fractions we give a model for the loops on the classifying space

Ω
B

C

\Omega B\mathcal {C}

as a simplicial group which generalizes the fact that

Ω
B

C

=

C

\Omega B\mathcal {C} = \mathcal {C}

when

C

\mathcal {C}

is a topological group. This construction is applied to give some well-known, and some new, examples of weak equivalences between classifying spaces arising in the theory of foliations.
Immersions of highly connected manifolds
Authors: Wiegmann Reinhard
Published: June 2010

If

[

M
n

↬

R

2
n
−
k

]

[{M^n}\looparrowright {{\mathbf {R}}^{2n - k}}]

denotes the set of regular homotopy classes of immersions,

M
n

{M^n}

a

k
k

-connected compact manifold, we show by a direct geometric construction the correspondence
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper M Superscript n Baseline right-arrow with loop bold upper R Superscript 2 n minus k Baseline right-bracket right-arrow left-bracket upper S Superscript n Baseline right-arrow with loop bold upper R Superscript 2 n minus k Baseline right-bracket">

[

M
n

↬

R

2
n
−
k

]
→
[

S
n

↬

R

2
n
−
k

]

[{M^n}\looparrowright {{\bf {R}}^{2n - k}}] \to [{S^n}\looparrowright {{\bf {R}}^{2n - k}}]

for

0
≤
2
k
≤
n
−
2

0 \le 2k \le n - 2

.
Torsion-free abelian groups with prescribed finitely topologized endomorphism rings
Authors: Dugas Manfred , Göbel Rüdiger
Published: June 2010

We will show that any complete Hausdorff ring

R
R

which admits, as a basis of neighborhoods of 0, a family of right ideals

I
I

with

R

/

I

R/I

cotorsion-free can be realized as a topological endomorphism ring of some torsion-free abelian group with the finite topology. This theorem answers a question of A. L. S. Corner (1967) and can be used to provide examples in order to solve a problem (No. 72) in L. Fuchs’ book on abelian groups.
Abelian 𝑝-groups 𝐴 and 𝐵 such that 𝑇𝑜𝑟(𝐴,𝐺)≅𝑇𝑜𝑟(𝐵,𝐺), 𝐺 reduced
Authors: Cutler Doyle
Published: June 2010

Let

A
A

be an abelian

p
p

-group having all of its finite Ulm invariants nonzero. Let

C
C

be a countable direct sum of cyclic

p
p

-groups such that for each nonnegative integer

n
n

, the

n
n

th Ulm invariant of

C
C

is zero if the

n
n

th Ulm invariant of

A
A

is finite. Then for all reduced abelian groups

G
G

,

Tor

(
G
,
A
)
≅

Tor

(
G
,
A
⊕
C
)

{\operatorname {Tor}}(G,A) \cong {\text {Tor}}(G,A \oplus C)

.
A formula for Ramanujan’s tau function
Authors: Ewell John A.
Published: June 2010

A formula for Ramanujan’s tau function

τ
\tau

, defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript 1 Overscript normal infinity Endscripts tau left-parenthesis n right-parenthesis x Superscript n Baseline equals x product Underscript 1 Overscript normal infinity Endscripts left-parenthesis 1 minus x Superscript n Baseline right-parenthesis Superscript 24 Baseline left-parenthesis StartAbsoluteValue x EndAbsoluteValue greater-than 1 right-parenthesis">

∑
1
∞

τ
(
n
)

x
n

=

x

∏
1
∞

(
1
−

x
n

)

24

(

|
x
|

>
1
)

\sum \nolimits _1^\infty {\tau (n){x^n} = } x\prod _1^\infty {(1 - {x^n})^{24}}(\left | x \right | > 1)

, is presented. The author then observes that some of the known congruence properties of

τ
\tau

are immediate consequences of this formula representation.
Extreme points of a class of subordinate functions
Authors: Sheil-Small T.
Published: June 2010

It is shown that, if

F

(
z
)

F\left ( z \right )

is subordinate to

H

(

(

1
+
z

)

/

(

1
−
z

)

)

H\left ( {\left ( {1 + z} \right ) / \left ( {1 - z} \right )} \right )

in the unit disc, where

H

(
w
)

H\left ( w \right )

is a quadratic polynomial for which

H
′

(
w
)

≠
0

H’\left ( w \right ) \ne 0

in the right half-plane, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis z right-parenthesis equals integral Underscript upper T Endscripts upper H left-parenthesis StartFraction 1 plus z e Superscript minus i t Baseline Over 1 minus z e Superscript minus i t Baseline EndFraction right-parenthesis d mu">

F

(
z
)

=

∫
T

H

(

1
+
z

e

−
i
t

1
−
z

e

−
i
t

)

d
μ

F\left ( z \right ) = \int \limits _T {H\left ( {\frac {{1 + z{e^{ - it}}}}{{1 - z{e^{ - it}}}}} \right )d\mu }

\]
for a suitable probability measure

μ
\mu

on the unit circle

T
T

.
Boundary behaviour of level curves of harmonic functions
Authors: Walker W. J.
Published: June 2010

Sufficient conditions are obtained for a function harmonic in the disk and continuous at a boundary point to have a level curve emanating from the boundary point into the disk. The one-sided derivative of the level curve is studied at the boundary point.
Curvature and the backward shift operators
Authors: Misra G.
Published: June 2010

Let

φ
α

{\varphi _\alpha }

be a Möbius transformation of the unit disk

D

{\mathbf {D}}

,

|
α
|

>
1

\left | \alpha \right | > 1

. We characterize all the operators

T
T

in

B
1

(

D

)

{B_1}\left ( {\mathbf {D}} \right )

which are unitarily equivalent to

φ
α

(
T
)

{\varphi _\alpha }\left ( T \right )

for all

α
\alpha

with

|
α
|

>
1

\left | \alpha \right | > 1

, using curvature techniques.
Alexander modules
Authors: Sato Nobuyuki
Published: June 2010

The Alexander modules of a link are the homology groups of the universal abelian cover of the complement of the link. For a link of

n
n

-spheres in

S

n
+
2

{S^{n + 2}}

, we show that, if

n
⩾
2

n \geqslant 2

, the Alexander modules

A
2

,
…
,

A
n

{A_2}, \ldots ,{A_n}

and the torsion submodule of

A
1

{A_1}

are all of type

L
L

. This leads to a characterization, below the middle dimension, of the polynomial invariants of the link. These results were previously proven for the special case of boundary links.
On units of certain cubic fields and the Diophantine equation 𝑥³+𝑦³+𝑧³=3
Authors: Scarowsky Manny
Published: June 2010

The Diophantine equation

x
3

+

y
3

+

z
3

=
3

{x^3} + {y^3} + {z^3} = 3

is replaced by a sequence of parametrized Diophantine equations which can be factored in certain cubic fields. A unit in these fields is readily available. Some results about these fields and the parametrized equations are proved.
Mappings between Euclidean spaces that are one-to-one over the image of a dense subset
Authors: Bestvina Mladen , Walsh John J.
Published: June 2010

Examples are constructed that include, for

n
>
m
⩾
2

n > m \geqslant 2

, proper surjective maps

f
:

R

n

→

R

m

f:{{\mathbf {R}}^n} \to {{\mathbf {R}}^m}

that are one to one over the image of a dense subset

D
⊂

R

n

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

; i.e., the restriction of

f
f

yields a homeomorphism from

f

−
1

f
(
D
)

{f^{ - 1}}f(D)

onto

f
(
D
)

f(D)

. A routine Baire Category argument establishes that, for any

σ
\sigma

-compact subset

F
⊂

R

n

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

which

f
f

maps onto

R

m

{{\mathbf {R}}^m}

,

Int

(
F
)
≠
θ

{\operatorname {Int}}(F) \ne \theta

and, hence,

dim
⁡
F
=
n

\dim F = n

.
Realizing diagrams in the homotopy category by means of diagrams of simplicial sets
Authors: Dwyer W. G. , Kan D. M.
Published: June 2010

Given a small category

D

{\mathbf {D}}

, we show that a

D

{\mathbf {D}}

-diagram

X
¯

\bar X

in the homotopy category can be realized by a

D

{\mathbf {D}}

-diagram of simplicial sets iff a certain simplicial set

r

X
¯

r\bar X

is nonempty. Moreover, this simplicial set

r

X
¯

r\bar X

can be expressed as the homotopy inverse limit of simplicial sets whose homtopy types are quite well understood. There is also an associated obstruction theory. In the special case that

D

{\mathbf {D}}

is a group (i.e.

D

{\mathbf {D}}

has only one object and all its maps are invertible) these results reduce to the ones of G. Cooke.
On the dimension of limits of inverse systems
Authors: Yajima Yukinobu
Published: June 2010

We say that the limit of an inverse system

X
=

lim
←

{

X
λ

,

π
μ
λ

,
Λ

}

X = \underleftarrow {\lim }\left \{ {{X_\lambda },\pi _\mu ^\lambda ,\Lambda } \right \}

is cylindrical if each finite cozero cover of

X
X

has a

σ
\sigma

-locally finite refinement consisting of sets of the form

π
λ

−
1

(
U
)

\pi _\lambda ^{ - 1}(U)

, where

U
U

is a cozero-set in

X
λ

{X_\lambda }

and

π
λ

:
X
→

X
λ

{\pi _\lambda }:X \to {X_\lambda }

is the projection. We prove that if

X
X

is cylindrical, then

dim
⁡
X
=
sup

{

dim
⁡

X
λ

:
λ
∈
Λ

}

\dim X = \sup \left \{ {\dim {X_\lambda }:\lambda \in \Lambda } \right \}

.
Constructing prime-field planar configurations
Authors: Gordon Gary
Published: June 2010

An infinite class of planar configurations is constructed with distinct prime-field characteristic sets (i.e., configurations represented over a finite set of prime fields but over fields of no other characteristic). It is shown that if

p
p

is sufficiently large, then every subset of

k
k

primes between

p
p

and

f
(
p
,
k
)

f(p,k)

forms such a set (where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis p comma k right-parenthesis equals 2 Superscript left-bracket left-parenthesis StartRoot p EndRoot minus upper A k Super Superscript 3 slash 2 Superscript right-parenthesis slash upper B k Super Superscript 3 slash 2 Superscript right-bracket">

f
(
p
,
k
)
=

2

[
(

p

−
A

k

3

/

2

)

/

B

k

3

/

2

]

f(p,k) = {2^{[(\sqrt p - A{k^{3/2}})/B{k^{3/2}}]}}

for constants

A
A

and

B
B

). In particular, for every positive integer

k
k

, there exist infinitely many planar matroid configurations

C

i
,
k

{C_{i,k}}

with

|

χ

p
f

(

C

i
,
k

)

|

=
k

\left | {{\chi _{pf}}({C_{i,k}})} \right | = k

(where

χ

p
f

(
C
)

{\chi _{pf}}(C)

denotes the prime-field characteristic set of

C
C

). We also give a result concerning cofinite prime-field characteristic sets.
Coherent rings of dimension two
Authors: Ng Ho Kuen
Published: June 2010

We first continue the investigation of the finitely presented dimension, especially in the local case, then we propose a scheme for classifying commutative rings. Commutative coherent rings of finitely presented dimension two and global dimension two will be studied.
Localized group rings, the invariant basis property and Euler characteristics
Authors: Goodearl K. R.
Published: June 2010

The technique of embedding a complex group algebra into the von Neumann regular ring associated with the corresponding

W
∗

{W^ * }

group algebra is exploited to prove that certain localizations of a group ring

K
G

KG

possess the invariant basis property. From this it follows, using a method of S. Rosset, that certain

K
G

KG

-modules have zero Euler characteristic. The assumptions are that

K
K

is a commutative integral domain of characteristic zero, and that

G
G

has a nontrivial, torsion-free, abelian normal subgroup

A
A

. The main result of the paper is that the localization of

K
G

KG

obtained by inverting all elements of the form

α
−
a

\alpha - a

, where

α
\alpha

is a nonzero element of

K
K

and

a
a

is a nontrivial element of

A
A

, has the invariant basis property; more generally, this localization and all its matrix rings are directly finite. (M. Smith has extended the methods of this paper to cover the localization of

K
G

KG

obtained by inverting all nonzero elements of

K
A

KA

.) Given a

K
G

KG

-module

M
M

which has a finite free resolution, such that

M
M

is finitely generated over

K
K

, it is proved that the Euler characteristic of

M
M

is zero. This verifies an unpublished result of Rosset.
Ratios of dual generic degrees of a finite Coxeter group
Authors: Alvis Dean
Published: June 2010

A formula is obtained for the ratio of dual generic degrees of a finite Coxeter group.
Integral representation of linear functionals on spaces of sections
Authors: Seda Anthony Karel
Published: June 2010

An integral representation is given of functionals on the normed linear space of all sections, with compact support, of a Banach bundle

p
:
E
→
X

p:E \to X

. This representation is shown to be equivalent in a certain sense to the strong lifting property on

X
X

and to the disintegration of measures property on spaces

q
:
Y
→
X

q:Y \to X

over

X
X

.
Extending positive definite linear forms
Authors: Gil de Lamadrid Jesús
Published: June 2010

In the classical literature two properties, called here symmetry and bounded variation, of a positive definite linear form

a
′

a’

on an involutive Banach algebra

A

\mathcal {A}

are given. Together they form a necessary and sufficient condition that

a
′

a’

admit a positive definite linear extension to the involutive algebra obtained from

A

\mathcal {A}

by adjoining an identity. In this note we show that bounded variation alone suffices, in that it already implies symmetry.
Convergence of Fourier series expansion related to free groups
Authors: Kuhn G.
Published: June 2010

In

[
0
,
π
]

[0,\pi ]

we consider the complete orthogonal system

P
n

{P_n}

associated to the weight function
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi equals r left-parenthesis 2 r minus 1 right-parenthesis pi Superscript negative 1 Baseline normal s normal i normal n squared theta left-parenthesis r squared minus left-parenthesis 2 r minus 1 right-parenthesis normal c normal o normal s squared theta right-parenthesis Superscript negative 1">

ψ
=
r
(
2
r
−
1
)

π

−
1

s
i

n

2

θ

(

r
2

−
(
2
r
−
1
)

c
o

s

2

θ

)

−
1

\psi = r(2r - 1){\pi ^{ - 1}}{\rm {si}}{{\rm {n}}^2}\theta {({r^2} - (2r - 1){\rm {co}}{{\rm {s}}^2}\theta )^{ - 1}}

and we study mean and pointwise convergence of series expansions with respect to the system

P
n

{P_n}

in

L
p

(
[
0
,
π
]
,
d
ψ
)

{L^p}([0,\pi ],d\psi )

. This weight function, and the corresponding system

P
n

{P_n}

arise from the study of Gelfand transforms of radial functions on a finitely generated free group

F
r

{F_r}

and our results can be interpreted in terms of multipliers theory on

F
r

{F_r}

.
Characterizing the topology of infinite-dimensional 𝜎-compact manifolds
Authors: Mogilski Jerzy
Published: June 2010

A metric space

(
X
,
d
)

(X,d)

, which is a countable union of finite-dimensional compacta, is a manifold modelled on the space

l
2
f

=
{
(

x
i

)
∈

l
2

l_2^f = \{ ({x_i}) \in {l_2}

all but finitely many

x
i

=
0
}

{x_i} = 0\}

iff

X
X

is an ANR and the following condition holds: given

ε
>
0

\varepsilon > 0

, a pair of finite-dimensional compacta

(
A
,
B
)

(A,B)

and a map

f
:
A
→
X

f:A \to X

such that

f

|

B

f|B

is an embedding, there is an embedding

g
:
A
→
X

g:A \to X

such that

g

|

B
=
f

|

B

g\left | {B = f} \right |B

and

d
(
f
(
x
)
,
g
(
x
)
)
>
ε

d(f(x),g(x)) > \varepsilon

for all

x
∈
A

x \in A

. An analogous condition characterizes manifolds modelled on the space
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma equals StartSet left-parenthesis x Subscript i Baseline right-parenthesis element-of l 2 colon sigma-summation Underscript i equals 1 Overscript normal infinity Endscripts left-parenthesis i x Subscript i Baseline right-parenthesis squared greater-than normal infinity EndSet">

Σ
=
{
(

x
i

)
∈

l
2

:

∑

i
=
1

∞

(
i

x
i

)
2

>
∞
}

\Sigma = \{ ({x_i}) \in {l_2}:\sum _{i = 1}^\infty {(i{x_i})^2} > \infty \}

.
Applications of a set-theoretic lemma
Authors: Gruenhage Gary
Published: June 2010

A set-theoretic lemma is introduced and various applications are given, including: (1) a result of Erdös and Hajnal on the coloring number of a graph; (2) game characterizations of the coloring number of a graph; (3) K. Alster’s result that a point-countable collection of open, compact scattered spaces has a point-finite clopen refinement; (4) normal, locally compact, metacompact spaces which are scattered of finite height are paracompact.
Cyclic group actions on Riemann surfaces
Authors: Yahia Mahgoub
Published: June 2010

We study the actions of

Z

/

m

{\mathbf {Z}}/m

, the cyclic group of order

m
m

, on Riemann surfaces which generate the bordism group

U

2

Z

/

m

\mathcal {U}_2^{{\mathbf {Z}}/m}

. We analyse

U

2

Z

/

m

\mathcal {U}_2^{{\mathbf {Z}}/m}

by means of fixed point structure using the technique of "bordism with families of slice types" [4].
Krull versus global dimension in Noetherian P.I. rings
Authors: Goodearl K. R. , Small L. W.
Published: June 2010

The Krull dimension of any noetherian P.I. ring is bounded above by its global (homological) dimension (when finite).
Flat covers and flat cotorsion modules
Authors: Enochs Edgar
Published: June 2010

It is not known whether modules over an arbitrary ring have flat covers, however for certain modules over commutative noetherian rings they can be shown to exist. These covers, in turn, have an interesting connection with flat cotorsion modules. A complete description of flat cotorsion modules analogous to that given by Harrison for torsion free, cotorsion abelian groups will be given. In this article,

R
R

will denote a commutative noetherian ring.
Factorization of probability measures and absolutely measurable sets
Authors: Blackwell David , Maitra Ashok
Published: June 2010

We find necessary and sufficient conditions for a separable metric space

Y
Y

to possess the property that for any measurable space

(

X
,

A

)

\left ( {X,\mathcal {A}} \right )

and probability measure

P
P

on

X
×
Y

X \times Y

,

P
P

can be factored.
Dieudonné-Schwartz theorem in inductive limits of metrizable spaces
Authors: Qiu Jing Hui
Published: June 2010

The Dieudonné-Schwartz Theorem for bounded sets in strict inductive limits does not hold for general inductive limits

E
=

ind

lim

E

n

E = {\operatorname {ind}}\lim {{\text {E}}_{\text {n}}}

. It does if each

E
¯

n
E

⊂

E

m

(
n
)

\bar E_n^E \subset {E_{m\left ( n \right )}}

and all the

E
n

{E_n}

are Fréchet spaces. A counterexample shows that this condition is not necessary. When

E
E

is a strict inductive limit of metrizable spaces

E
n

{E_n}

, this condition is equivalent to the condition that each bounded set in

E
E

is contained in some

E
n

{E_n}

.
𝑚-accretive operators with 𝑚-dispersive resolvents
Authors: deLaubenfels Ralph
Published: June 2010

We characterize linear

m
m

-accretive operators with

m
m

-dispersive resolvents.

T
T

is linear and

m
m

-accretive, with

(
λ
+
T

)

−
1

{(\lambda + T)^{ - 1}}

m
m

-dispersive, if and only if the sequence

⟨

n
!
ϕ

(
λ
+
T
)

n
+
1

x

⟩

n
=
0

∞

\left \langle {n!\phi {{(\lambda + T)}^{n + 1}}x} \right \rangle _{n = 0}^\infty

equals the moments of a positive measure on the positive real line, for sufficiently many

ϕ
\phi

in

X
∗

{X^*}

,

x
x

in

X
X

.
A strong law of large numbers for martingales
Authors: Sheu Shey Shiung , Yao Yu Shan
Published: June 2010

We derive a moment inequality for the Skorohod representation theorem and apply it to obtain a strong law of large numbers for martingales.
On 2-knot groups with abelian commutator subgroups
Authors: Yoshikawa Katsuyuki
Published: June 2010

In this paper it is shown that if the commutator subgroup of a

2
2

-knot group is abelian, then it is isomorphic to

Z
⊕
Z
⊕
Z

Z \oplus Z \oplus Z

,

Z
α

{Z_\alpha }

,

Z
[
1

/

2
]

Z[1/2]

or

Z
[
1

/

2
]
⊕

Z
5

Z[1/2] \oplus {Z_5}

, where

α
\alpha

is an odd integer and

Z
[
1

/

2
]

Z[1/2]

is the additive group of the dyadic rationals.
A continuum 𝑋 which has no confluent Whitney map for 2^{𝑋}
Authors: Charatonik Włodzimierz J.
Published: June 2010

An example is shown of a continuum

X
X

which has no confluent Whitney map for

2
X

{2^X}

. This answers two problems asked by Nadler [N].
A short proof of the Cima-Wogen 𝐿(𝑓)=𝑐𝑖𝑟𝑐𝑙𝑒 theorem
Authors: Antonino José A. , Romaguera Salvador
Published: June 2010

In [1] Cima and Wogen showed that if

f
∈

f \in

ball

B
0

{B_0}

and

L
(
f
)

L(f)

contains a circle

γ
\gamma

, then

γ
=
L
(
f
)

\gamma = L(f)

. This note presents a new and straightforward proof of Cima and Wogen’s theorem.
On differentiability of Peano type functions. III
Authors: Cichoń Jacek , Morayne Michał
Published: June 2010

We show that for all positive natural numbers

m
m

,

n
n

the following two sentences are equivalent: (i)

2

ℵ
0

⩽

ℵ
n

{2^{{\aleph _0}}} \leqslant {\aleph _n}

; (ii) there exists an onto function

f
:

R
n

→

R

n
+
m

f:{R^n} \to {R^{n + m}}

(

R
R

the set of real numbers) such that at each point of

R
n

{R^n}

at least

n
n

coordinates of

f
f

are differentiable.
Extension of entire functions on nuclear locally convex spaces
Authors: Meise Reinhold , Vogt Dietmar
Published: June 2010

We prove that for a nuclear locally convex complex vector space every entire function is the pull-back of some entire function on an appropriate Banach space if and only if the entire functions on

E
E

have the following universal extension property: whenever

E
E

is a topological linear subspace of a locally convex space

F
F

with a fundamental system of seminorms induced by semi-inner products, then every entire function

f
f

on

E
E

can be extended to an entire function on

F
F

.
On the extension property of measurable spaces
Authors: Bzyl Włodzimierz , Mysior Adam
Published: June 2010

We prove that a metrizable measurable space has the extension property if and only if it is isomorphic to a Borel subset of the real line. It follows, in particular, that

(

R

,

P

(

R

)
)

({\mathbf {R}},\mathcal {P}({\mathbf {R}}))

does not have the extension property. Both results answer the questions raised by R. M. Shortt.
A combinatorial formula for the normal Euler class of a lattice 2-manifold in 4-space
Authors: Yusin Boris V.
Published: June 2010

For a lattice

2
2

-manifold immersed in

4
4

-space with all faces parallel to coordinate planes it is possible to assign a local index to each vertex type in a unique way so that the sum of vertex indices gives the normal Euler number of the immersion. Related results for simplicial immersions have been established by T. Banchoff.
Normal subgroups of 𝐷𝑖𝑓𝑓^{Ω}(𝑅³)
Authors: Mascaró Francisca
Published: June 2010

Let

Ω
\Omega

be a volume element on

R

3

{{\mathbf {R}}^3}

of infinite total

Ω
\Omega

-volume. We denote by

Dif

f

Ω

(

R

3

)

{\operatorname {Dif}}{{\text {f}}^\Omega }({{\mathbf {R}}^3})

the group of all

Ω
\Omega

-preserving diffeomorphisms of

R

3

{{\mathbf {R}}^3}

, by

Diff

c

Ω

(

R

3

)

{\operatorname {Diff}}_{\text {c}}^\Omega ({{\mathbf {R}}^3})

the subgroup of all elements with compact support and by

Diff

f

Ω

(

R

3

)

{\operatorname {Diff}}_{\text {f}}^{\Omega }(\mathbf {R}^3)

the subgroup of all elements whose support has finite

Ω
\Omega

-volume. We prove that there is no normal subgroup between

Diff

c

Ω

(

R

3

)

{\operatorname {Diff}}_{\text {c}}^\Omega ({{\mathbf {R}}^3})

and

Diff

f

Ω

(

R

3

)

{\operatorname {Diff}}_{\text {f}}^\Omega ({{\mathbf {R}}^3})

.
On the integrability of complex trigonometric series
Authors: Bray William O. , Stanojevic Vera B.
Published: June 2010

The classical criterion for the Fourier character of cosine series and sine series is generalized to complex trigonometric series. Our treatment unifies the ideas applied to cosine and sine series. A notion of complex convex sequences is developed generalizing a result of Young [1] concerning the integrability of cosine series.
Consecutive primitive roots in a finite field
Authors: Cohen Stephen D.
Published: June 2010

Every finite field of order

q
(
>
3
)

q( > 3)

such that

q
≢
7
(

mod

12
)

q \not \equiv 7(\mod 12)

and

q
≢
1
(

mod

60
)

q \not \equiv 1(\mod 60)

contains a pair of consecutive primitive roots.
Finite rank torsion-free abelian groups uniserial over their endomorphism rings
Authors: Hausen Jutta
Published: June 2010

An abelian group is

E
E

-uniserial if its lattice of fully invariant subgroups is totally ordered. Finite rank torsion-free reduced

E
E

-uniserial groups are characterized. Such a group is a free module over the center

C
C

of its endomorphism ring, and

C
C

is a strongly indecomposable discrete valuation ring. Properties similar to those of strongly homogeneous groups are derived.
Action of the automorphisms of a smooth domain in 𝐶ⁿ
Authors: Bedford Eric
Published: June 2010

A transformation rule relating the Bergman projection and an element of the Lie algebra of

Aut
⁡
(
Ω
)

\operatorname {Aut}(\Omega )

is given, and this is used to give a proof that the action of

Aut
⁡
(
Ω
)

\operatorname {Aut}(\Omega )

extends smoothly to

Ω
¯

\bar \Omega

.
Covering lemmas and the sharp function
Authors: Bagby Richard J. , Kurtz Douglas S.
Published: June 2010

We present some new measure-theoretic inequalities for families of cubes covering open sets in

R

n

{{\mathbf {R}}^n}

and use these inequalities to estimate the Hardy-Littlewood maximal function

M
f

Mf

in terms of the sharp function

f
#

{f^\# }

.
Positively curved Kaehler submanifolds
Authors: Ros Antonio
Published: June 2010

In this note we prove that if the holomorphic curvature of a compact Kaehler submanifold in the complex projective space is bigger than

1
2

\tfrac {1}{2}

, then it is totally geodesic.
Rational tilings by 𝑛-dimensional crosses. II
Authors: Szabó S.
Published: June 2010

The union of translates of a closed unit

n
n

-dimensional cube whose edges are parallel to the coordinate unit vectors

e

1

,
…
,

e

n

{{\mathbf {e}}_1}, \ldots ,{{\mathbf {e}}_n}

and whose centers are

i

e

j

,

|
i
|

≤
k
,
1
≤
j
≤
n

i{{\mathbf {e}}_j},\left | i \right | \leq k,1 \leq j \leq n

, is called a

(
k
,
n
)

(k,n)

-cross. A system of translates of a

(
k
,
n
)

(k,n)

-cross is called an integer (a rational) lattice tiling if its union is

n
n

-space and the interiors of its elements are disjoint, the translates form a lattice and each translation vector of the lattice has integer (rational) coordinates. In this paper we shall continue the examination of rational cross tilings begun in [2], constructing rational lattice tilings by crosses that have noninteger coordinates on several axes.
Exponential sums over primes in an arithmetic progression
Authors: Balog Antal , Perelli Alberto
Published: June 2010

In 1979 A. F. Lavrik obtained some estimates for exponential sums over primes in arithmetic progressions by an analytic method. In the present paper we give an estimate for the same sums, comparable with Lavrik’s estimate, by means of elementary methods like Vaughan’s identity.
Convex sets with the Lipschitz fixed point property are compact
Authors: Lin P. K. , Sternfeld Y.
Published: June 2010

Let

K
K

be a noncompact convex subset of a normed space

X
X

. It is shown that if

K
K

is not totally-bounded then there exists a Lipschitz self map

f
f

of

K
K

with

inf
⁡

{

‖

x
−
f

(
x
)

‖

:
x
∈
K

}

>
0

\operatorname {inf}\left \{ {\left \| {x - f\left ( x \right )} \right \|:x \in K} \right \} > 0

, while if

K
K

is totally-bounded then such a map does not exist, but still

K
K

lacks the fixed point property for Lipschitz mappings. It follows that a closed convex set in a normed space has the fixed point property for Lipschitz maps if and only if it is compact.
On fixed points of linear contractions
Authors: Radjavi Heydar , Rosenthal Peter
Published: June 2010

It is shown that a weakly closed convex semigroup of linear contractions on a separable Hilbert space has a common fixed point other than 0 if the operator 0 is not in the semigroup.
A holomorphic function with wild boundary behavior
Authors: Globevnik Josip
Published: June 2010

Let

B
B

be the open unit ball in

C

N

,
N
>
1

{{\mathbf {C}}^N},N > 1

. It is known that if

f
f

is a function holomorphic in

B
B

, then there are

x
∈
∂
B

x \in \partial B

and an arc

Λ
\Lambda

in

B
∪

{
x
}

B \cup \left \{ x \right \}

, with

x
x

as one endpoint along which

f
f

is constant. We prove Theorem. There exist an

r
>
0

r > 0

and a function

f
f

holomorphic in

B
B

with the property that, if

x
∈
∂
B

x \in \partial B

and

Λ
\Lambda

is a path with

x
x

as one endpoint, such that

Λ
−

{
x
}

\Lambda - \left \{ x \right \}

is contained in the open ball of radius

r
r

which is contained in

B
B

and tangent to

∂
B

\partial B

at

x
x

, then

lim

z
∈
Λ
,
z
→
x

f

(
z
)

\lim _{z \in \Lambda ,z \to x}f\left ( z \right )

does not exist.
Fibre tensor product bundles
Authors: Gelbaum Bernard R. , Kyriazis Athanasios
Published: June 2010

In analogy with fibre bundles, which are locally Cartesian products, fibre tensor product bundles are objects that are locally tensor products. These can be patched together via transition maps, etc., into an object very similar to the set of sections of a locally convex algebra bundle.
A Diophantine problem for Laurent polynomial rings
Authors: Pappas Peter
Published: June 2010

Let

R
R

be an integral domain of characteristic zero. We prove that the diophantine problem for the Laurent polynomial ring

R
[
T
,

T

−
1

]

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

with coefficients in

Z

[
T
]

{\mathbf {Z}}[T]

is unsolvable. Under suitable conditions on

R
R

we then show that either

Z

{\mathbf {Z}}

or

Z

[
i
]

{\mathbf {Z}}[i]

is diophantine over

R
[
T
,

T

−
1

]

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

.
A simple intuitive proof of a theorem in degree theory for gradient mappings
Authors: Scovel James C.
Published: June 2010

We give a simple, intuitive proof of a known theorem: the degree of the gradient of a coercive functional on a large ball in

R

n

{{\mathbf {R}}^n}

is one.
Set-theoretic complete intersections
Authors: Moh T. T.
Published: June 2010

In this article we establish that: (1) Every monomial curve in

P

k
n

{\mathbf {P}}_k^n

is a set-theoretic complete intersection, where

k
k

is a field of characteristic

p
p

(and thus generalize a result of R. Hartshorne [3]). (2) Let

k
k

be an algebraically closed field of characteristic

p
p

and

C
C

a curve of

P

k
n

{\mathbf {P}}_k^n

. If there is a linear projection

τ
:

P

k
n

→

P

k
2

\tau :{\mathbf {P}}_k^n \to {\mathbf {P}}_k^2

with center of

τ
\tau

disjoint of

C
C

,

τ
(
C
)

\tau (C)

is birational to

C
C

and

τ
(
C
)

\tau (C)

has only cusps as singularities, then

C
C

is a set-theoretic complete intersection (and thus generalize a result of D. Ferrand [2]).
On the multiplicities of the powers of a Banach space operator
Authors: Herrero Domingo A.
Published: June 2010

The multiplicities of the powers of a bounded linear operator

T
T

, acting on a complex separable infinite-dimensional Banach space

X

\mathfrak {X}

, satisfy the inequalities
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis asterisk asterisk right-parenthesis mu left-parenthesis upper T Superscript n Baseline right-parenthesis less-than-or-slanted-equals mu left-parenthesis upper T Superscript h n Baseline right-parenthesis less-than-or-slanted-equals h mu left-parenthesis upper T Superscript n Baseline right-parenthesis for all h comma n greater-than-or-slanted-equals 1 period">

(
∗
∗
)

μ
(

T
n

)
⩽
μ
(

T

h
n

)
⩽
h
μ
(

T
n

)

for

all

h
,
n
⩾
1.

( * * )\qquad \mu ({T^n}) \leqslant \mu ({T^{hn}}) \leqslant h\mu ({T^n})\quad {\text {for}}\;{\text {all}}\;h,n \geqslant 1.

\]
Nothing else can be said, in general, because simple examples show that for each sequence

{

μ
n

}

n
=
1

∞

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

, satisfying the inequalities

(
∗
∗
)

( * * )

, there exists

T
T

acting on

X

\mathfrak {X}

such that

μ
(

T
n

)
=

μ
n

\mu ({T^n}) = {\mu _n}

for all

n
⩾
1

n \geqslant 1

.