bluuebunny/crossref_metadata_2025 huggingface dataset:
Radii of immersed manifolds and nonexistence of immersions
Authors: Ishihara Tôru
Published: June 2010

Let M be a compact Riemannian manifold isometrically immersed in a complete Riemannian manifold N. By the radius of M in N, we mean the minimum of radii of closed geodesic balls in N which contain M. Using the concept of a radius, we will give a theorem about the nonexistence of isometric immersions, which is a generalization of J. D. Moore’s result.
On recurrence of a random walk in the plane
Authors: Chung Kai Lai , Lindvall Torgny
Published: June 2010

The purpose of this note is to establish a sufficient condition for recurrence of a random walk

(

S
n

)

({S_n})

in

R
2

{R^2}

. It follows from it that if

S
n

/

n

1

/

2

{S_n}/{n^{1/2}}

is asymptotically normal then we have recurrence.
Modules with Artinian prime factors
Authors: Armendariz Efraim P.
Published: June 2010

An R-module M has Artinian prime factors if

M

/

P
M

M/PM

is an Artinian module for each prime ideal P of R. For commutative rings R it is shown that Noetherian modules with Artinian prime factors are Artinian. If R is either commutative or a von Neumann regular V-ring then the endomorphism ring of a module with Artinian prime factors is a strongly

π
\pi

-regular ring.
Normal subgroups contained in Frattini subgroups are Frattini subgroups
Authors: Allenby R. B. J. T.
Published: June 2010

We prove that if N is a normal subgroup of the finite group G and if

N
⊆
Φ
(
G
)

N \subseteq \Phi (G)

, then there exists a finite group U such that

N
=
Φ
(
U
)

N = \Phi (U)

exactly. In particular, we see that the generalizations apparent in the conclusions of several recently stated theorems are illusory.
The ideal class groups of two cyclotomic fields
Authors: Gerth Frank
Published: June 2010

It is known that there are exactly two cyclotomic fields with class numbers equal to 8, namely the fields of 29th and 68th roots of unity. We show by elementary methods how to determine the structure of the ideal class groups of these fields.
Generalizations of a theorem of Mutylin
Authors: Warner Seth
Published: June 2010

We generalize Mutylin’s theorem that the only complete, locally bounded, additively generated topological fields are R and C by showing: (1) the only complete, locally bounded, additively generated topological division rings with left bounded commutator subgroup are R, C, and H; (2) a commutative, Hausdorff topological ring A with identity is a Banach algebra over R, equipped with the absolute value

|

.
.

|

p

|..{|^p}

for some

p
∈
(
0
,
1
]

p \in (0,1]

, if (and only if) A is complete, locally bounded, additively generated, and possesses an invertible topological nilpotent.
Linear convolution integral equations with asymptotically almost periodic solutions
Authors: Jordan G. S. , Madych W. R. , Wheeler R. L.
Published: June 2010

Let

μ
\mu

be a bounded Borel measure and f be asymptotically almost periodic. Conditions are found which ensure that certain bounded solutions of the linear convolution integral equation

g
∗
μ
=
f

g \ast \mu = f

are asymptotically almost periodic. This result is also extended to the case where the measure

μ
\mu

is replaced by a tempered distribution

τ
\tau

for which convolution with bounded functions makes sense.
A remark on the Bochner technique in differential geometry
Authors: Wu H.
Published: June 2010

It is observed that by pushing the standard arguments one step further, almost all the theorems in differential geometry proved with the help of Bochner’s technique can be sharpened.
Easy 𝑆 and 𝐿 groups
Authors: Roitman J.
Published: June 2010

We give a simple proof that the existence of strong S or L spaces implies the existence of strong S or L groups; in fact the algebraic structure can be varied quite a bit. We also construct, under CH, S and L groups whose squares are neither S nor L.
Differentiable decompositions of manifolds into totally 𝐶^{∞}-path disconnected subsets
Authors: Mielke M. V.
Published: June 2010

For

C
∞

{C^\infty }

-manifolds M, N, the set of all

C
s

{C^s}

-maps

M
→
N

M \to N

with totally

C
∞

{C^\infty }

-path disconnected fibers is shown to be dense in the set of all

C
s

{C^s}

-maps

M
→
N

M \to N

, if

dim
⁡
N
>
0

\dim N > 0

.
The length spectrum of a Riemann surface is always of unbounded multiplicity
Authors: Randol Burton
Published: June 2010

I show that the length spectrum of a Riemann surface is always of unbounded multiplicity, and indicate connections with recent work of Guillemin and Kazhdan.
An improved estimate for certain Diophantine inequalities
Authors: Liu Ming Chit , Ng Shu Ming , Tsang Kai Man
Published: June 2010

Let

λ
1

,
…
,

λ
8

{\lambda _1}, \ldots ,{\lambda _8}

be any nonzero real numbers such that not all

λ
j

{\lambda _j}

are of the same sign and not all ratios

λ
j

/

λ
k

{\lambda _j}/{\lambda _k}

are rational. If

η
,
α

\eta ,\alpha

are any real numbers with

0
>
α
>
3

/

70

0 > \alpha > 3/70

then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue eta plus normal upper Sigma Subscript j equals 1 Superscript 8 Baseline lamda Subscript j Baseline n Subscript j Superscript 3 Baseline EndAbsoluteValue greater-than left-parenthesis max n Subscript j Baseline right-parenthesis Superscript negative alpha">

|

η
+

Σ

j
=
1

8

λ
j

n
j
3

|

>

(
max

n
j

)

−
α

|\eta + \Sigma _{j = 1}^8{\lambda _j}n_j^3| > {(\max {n_j})^{ - \alpha }}

has infinitely many solutions in positive integers

n
j

{n_j}

.
A simple 𝐶*-algebra with no nontrivial projections
Authors: Blackadar Bruce E.
Published: June 2010

A

C
∗

{C^\ast }

-algebra is constructed which is separable, simple, nuclear, non-unital, and contains no nonzero projections. Some results on automorphisms of AF algebras are also obtained.
The Ceva property characterizes real, strictly convex Banach spaces
Authors: Valentine J. E. , Wayment S. G.
Published: June 2010

If a, b, c are distinct collinear points of a metric space, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis a comma b comma c right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column a b slash b c 2nd Column a m p semicolon if b is between a and c comma 2nd Row 1st Column minus left-parenthesis a b slash b c right-parenthesis 2nd Column a m p semicolon otherwise period EndLayout">

(
a
,
b
,
c
)
=

{

a
b

/

b
c

a
m
p
;

if

b

is

between

a

and

c
,

−
(
a
b

/

b
c
)

a
m
p
;

otherwise

.

(a,b,c) = \left \{ {\begin {array}{*{20}{c}} {ab/bc} \hfill & {{\text {if}}\;b\;{\text {is}}\;{\text {between}}\;a\;{\text {and}}\;c,} \hfill \\ { - (ab/bc)} \hfill & {{\text {otherwise}}.} \hfill \\ \end {array} } \right .

\]
A metric space satisfies the Ceva Property provided for each triple of noncollinear points p, q, r, if s,t,u are points distinct from p, q, r on the metric lines

L
(
p
,
q
)
,
L
(
q
,
r
)

L(p,q),L(q,r)

, and

L
(
r
,
p
)

L(r,p)

, respectively, then the metric lines

L
(
p
,
t
)
,
L
(
q
,
u
)

L(p,t),L(q,u)

, and

L
(
r
,
s
)

L(r,s)

have a common point if and only if

(
p
,
s
,
q
)
(
q
,
t
,
r
)
(
r
,
u
,
p
)
=
1

(p,s,q)(q,t,r)(r,u,p) = 1

, and

p
q

/

p
s
≠
p
u

/

p
r

pq/ps \ne pu/pr

. In the euclidean plane, the requirement that

p
q

/

p
s
≠
p
u

/

p
r

pq/ps \ne pu/pr

forces the lines

L
(
r
,
s
)

L(r,s)

and

L
(
q
,
u
)

L(q,u)

to have a common point. Thus the case of parallel lines is avoided and the Ceva Property is meaningful in an arbitrary metric space. The main result of the paper is that a complete, convex, externally convex, metric space is a strictly convex Banach space over the reals if and only if it satisfies the Ceva Property.
Regular operator equations: conditions for regularity
Authors: Karakostas George
Published: June 2010

Regular operator equations are causal equations admitting unique solutions and have the property that all of their limiting equations along solutions admit unique solutions. Sufficient conditions which guarantee that an operator equation

x
=
T
x

x = Tx

is regular are given in case

T
T

is a linear or a nonlinear operator.
Minimal entropy for endomorphisms of the circle
Authors: Ito Ryuichi
Published: June 2010

Let

f
f

be an endomorphism (continuous map) of the circle which has two periodic points of period

m
m

and

n
n

respectively such that

m
⩾
2
,
n
⩾
2

m \geqslant 2,n \geqslant 2

and

(
m
,
n
)
=
1

(m,n) = 1

, then topological entropy

h
(
f
)
⩾
log
⁡

μ

m
,
n

h(f) \geqslant \log {\mu _{m,n}}

where

μ

m
,
n

{\mu _{m,n}}

is the largest zero of the polynomial

x

m
+
n

−

x
m

−

x
n

−
1

{x^{m + n}} - {x^m} - {x^n} - 1

.
Geometric realization of 𝜋₀ℰ(ℳ)
Authors: Lee Kyung Bai
Published: June 2010

Let

M
M

be a closed flat Riemannian manifold,

ε
(
M
)

\varepsilon (M)

the group of self homotopy equivalences of

M
M

. Then there exists a subgroup

A
1

(
M
)

{A_1}(M)

of

Aff
⁡
(
M
)

\operatorname {Aff} (M)

such that the natural homomorphism of

A
1

(
M
)

{A_1}(M)

into

π
0

ε
(
M
)

{\pi _0}\varepsilon (M)

is a surjection with kernel a finite abelian group. Furthermore, this kernel can be identified with the structure group of the Calabi fibration.
Interpolation between 𝐻¹ and 𝐿^{∞}
Authors: MacCluer Barbara D.
Published: June 2010

The intermediate spaces in the Lions-Peetre method of interpolation between

H
1

{H^1}

and

L
∞

{L^\infty }

were identified by N. Rivière and Y. Sagher as Lorentz

L
(
p
,
q
)

L(p,q)

spaces. In this article we present a simplification of their proof of this result.
Remark on Lomonosov’s lemma
Authors: Longstaff W. E.
Published: June 2010

The famous lemma of V. I. Lomonosov states that if

A

\mathfrak {A}

is a transitive algebra of operators acting on a complex, infinite-dimensional Banach space

X
X

and

K
K

is a nonzero compact operator on

X
X

, then there is an

A
∈

A

A \in \mathfrak {A}

such that 1 is an eigenvalue of AK. Lomonosov’s proof uses Schauder’s fixed point theorem. A proof, using only elementary techniques, is given for the case where

K
K

has finite-rank.
Riemann’s function has an exponential bound
Authors: Beesack Paul R.
Published: June 2010

The Riemann function

υ
(
t
;
x
)

\upsilon (t;x)

of a hyperbolic characteristic initial value problem has been much used in recent years to provide upper bounds for functions which satisfy Gronwall-type integral inequalities. This note gives a direct proof of the fact that

υ
\upsilon

satisfies an inequality of the form

υ
(
t
;
x
)
⩽
exp
⁡
(

∫
t
x

b
(
s
)
d
s

\upsilon (t;x) \leqslant \exp (\int _t^x {b(s)ds}

.
The Kobayashi indicatrix at the center of a circular domain
Authors: Barth Theodore J.
Published: June 2010

The indicatrix of the Kobayashi infinitesimal metric at the center of a pseudoconvex complete circular domain coincides with this domain. It follows that a nonconvex complete circular domain cannot be biholomorphic to any convex domain. An example shows that a bounded pseudoconvex complete circular domain in

C

2

{{\mathbf {C}}^2}

need not be taut.
Existence of harmonic 𝐿¹ functions in complete Riemannian manifolds
Authors: Chung L. O.
Published: June 2010

We construct a complete Riemannian manifold which carries a nonconstant harmonic

L
1

L^1

function.
Chain of prime ideals in formal power series rings
Authors: de S. Doering Ada Maria , Lequain Yves
Published: June 2010

Let

R
R

be a Noetherian domain and

P
P

a prime ideal of

R
R

. Then

R
p

[
[

X
1

,
…
,

X
n

]
]

{R_p}[[{X_1}, \ldots ,{X_n}]]

has a maximal chain of prime ideals of length

r
r

if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R left-bracket left-bracket upper X 1 comma ellipsis comma upper X Subscript n Baseline right-bracket right-bracket Subscript left-parenthesis upper P comma upper X 1 comma ellipsis comma upper X Sub Subscript n Subscript right-parenthesis">

R

[
[

X
1

,
…
,

X
n

]

]

(
P
,

X
1

,
…
,

X
n

)

R{[[{X_1}, \ldots ,{X_n}]]_{(P,{X_1}, \ldots ,{X_n})}}

does, if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R left-bracket left-bracket upper X 1 comma ellipsis comma upper X Subscript n Baseline right-bracket right-bracket Subscript left-parenthesis upper P comma upper X 1 comma ellipsis comma upper X Sub Subscript n Subscript right-parenthesis">

R

[
[

X
1

,
…
,

X
n

]

]

(
P
,

X
1

,
…
,

X
n

)

R{[[{X_1}, \ldots ,{X_n}]]_{(P,{X_1}, \ldots ,{X_n})}}

does.
On the transpose map of matrix algebras
Authors: Tomiyama Jun
Published: June 2010

It is shown that for the transpose map

θ
(
n
)

\theta (n)

of the

n
×
n

n \times n

matrix algebra

M
n

{M_n}

, its

k
k

th multiplicity map

θ

(
n

)
k

\theta {(n)_k}

has exactly the norm

k
k

if

k
⩽
n

k \leqslant n

, hence the completely bounded norm of

θ
(
n
)

\theta (n)

written

‖

θ
(
n
)

‖

c
b

{\left \| {\theta (n)} \right \|_{cb}}

equals

n
n

. Some applications and related results are also proved.
On partitions of plane sets into simple closed curves
Authors: Bankston Paul
Published: June 2010

We investigate the conjecture that the complement in the euclidean plane

E
2

{E^2}

of a set

F
F

of cardinality less than the continuum

c
c

can be partitioned into simple closed curves iff

F
F

has a single point. The case in which

F
F

is finite was settled in [1] where it was used to prove that, among the compact connected two-manifolds, only the torus and the Klein bottle can be so partitioned. Here we prove the conjecture in the case where

F
F

either has finitely many isolated points or finitely many cluster points. Also we show there exists a self-dense totally disconnected set

F
F

of cardinality

c
c

and a partition of

E
2

∖
F

{E^2}\backslash F

into "rectangular" simple closed curves.
Diffeomorphisms without periodic points
Authors: Plante J. F.
Published: June 2010

It is proved that a compact smooth manifold admits a selfdiffeomorphism without periodic points if and only if its Euler characteristic is zero. When the manifold has dimension

≠
3

\ne 3

it is shown that such a diffeomorphism exists which is also volume preserving. The proof of this latter result uses a result of Gromov concerning the existence of nonsingular divergence-free vector fields, so an alternate proof of Gromov’s result is sketched.
Links of closed orbits of nonsingular Morse-Smale flows
Authors: Sasano Kazuhiro
Published: June 2010

The links which consist of closed orbits of certain nonsingular Morse-Smale flows on

S
3

{S^3}

are determined.
On the dimensions of projections of compact subsets of 𝑅^{𝑚}
Authors: Sternfeld Y.
Published: June 2010

In this note we continue the study of a problem considered earlier by G. Nöbling, S. Mardešić, C. Pixley and the author. Let

W
W

be an

n
n

-dimensional compact subset of

R

m

{{\mathbf {R}}^m}

, and let

{

i
1

,

i
2

,
…
,

i
k

}
⊂
{
1
,
2
,
…
,
m
}

\{ {i_1},{i_2}, \ldots ,{i_k}\} \subset \{ 1,2, \ldots ,m\}

. It is shown that if the projection

P

i
1
,
i
2
,
…
,
i
k

{P_{i1,i2, \ldots ,ik}}

satisfies a certain condition called normality on

W
W

, then there exist

i

k
+
1

{i_{k + 1}}

,

i

k
+
2

,
…
,

i
n

{i_{k + 2}}, \ldots ,{i_n}

so that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension upper P Subscript i Baseline 1 comma i Baseline 2 comma ellipsis comma i Sub Subscript k Subscript comma i Sub Subscript k plus 1 Subscript comma ellipsis comma i Sub Subscript n Subscript Baseline left-parenthesis upper W right-parenthesis equals n">

dim
⁡

P

i
1
,
i
2
,
…
,

i
k

,

i

k
+
1

,
…
,

i
n

(
W
)
=
n

\dim {P_{i1,i2, \ldots ,{i_k},{i_{k + 1}}, \ldots ,{i_n}}}(W) = n

. It is also shown that at least

(
k
n

)

(_k^n)

such normal projections do exist (for

1
⩽
k
⩽
n

1 \leqslant k \leqslant n

), and an example is constructed to show that the normality condition is not necessary.
Discrete sets of singular cardinality
Authors: Fleissner William G.
Published: June 2010

Let

κ
\kappa

be a singular cardinal. In Fleissner’s thesis, he showed that in normal spaces

X
X

, certain discrete sets

Y
Y

of cardinality

κ
\kappa

(called here sparse) which are

>
κ

> \kappa

-separated are, in fact, separated. In Watson’s thesis, he proves the same for countably paracompact spaces

X
X

. Here we improve these results by making no assumption on the space

X
X

. As a corollary, we get that assuming

V
=
L

V = L

,

ℵ
1

{\aleph _1}

,-paralindelöf

T
2

{T_2}

, spaces of character

⩽

ω
2

\leqslant {\omega _2}

, are collectionwise Hausdorff.
Continuous images of Lusin sets
Authors: Brown J. B. , Cox G. V.
Published: June 2010

It is the purpose of this note to further clarify the relationship between continuous images of Lusin sets and concentrated sets.
Additive mappings on rings of holomorphic functions
Authors: Burckel R. B. , Saeki Sadahiro
Published: June 2010

Additive maps

T
T

between rings of holomorphic functions, which satisfy the multiplier-like condition

T
(
z
f
)
=
ϕ
T
(
f
)

T(zf) = \phi T(f)

for some

ϕ
\phi

and all

f
f

, are characterized. No linearity or continuity assumptions are made. Earlier results by L. Bers on homomorphisms and by J. Becker on derivations are corollaries.
A note on new generating relations for functions of several variables
Authors: Shukla D. P.
Published: June 2010

In this paper we have derived generating relations for functions of several variables by using the operator

T
k

=
x
(
k
+
x
D
)

{T_k} = x(k + xD)

and the operational relations involving this operator. Some recent results of Srivastava and Panda [7] have been conveniently obtained by this method as well as some hitherto unknown results established.
On the hyperbolic Riesz means
Authors: El Kohen A.
Published: June 2010

We define the hyperbolic Riesz means in

R

2

{{\mathbf {R}}^2}

by

H
λ

f
=

(

m
λ

f
^

)
υ

{H_\lambda }f = {({m_\lambda }\hat f)^\upsilon }

where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m Subscript lamda Baseline left-parenthesis xi 1 comma xi 2 right-parenthesis equals left-parenthesis 1 minus left-parenthesis xi 1 xi 2 right-parenthesis squared right-parenthesis Subscript plus Superscript lamda Baseline comma lamda greater-than-or-slanted-equals 0">

m
λ

(

ξ
1

,

ξ
2

)
=
(
1
−

(

ξ
1

ξ
2

)
2

)
+
λ

,
λ
⩾
0

{m_\lambda }({\xi _1},{\xi _2}) = (1 - {({\xi _1}{\xi _2})^2})_ + ^\lambda ,\lambda \geqslant 0

, and show that

H
λ

{H_\lambda }

is bounded on

L
p

(

R

2

)

{L^p}({{\mathbf {R}}^2})

for

4
3

⩽
p
⩽
4

\tfrac {4}{3} \leqslant p \leqslant 4

and

λ
>

1
2

\lambda > \tfrac {1}{2}

or

1
>
p
>
∞

1 > p > \infty

and

λ
⩾
1

\lambda \geqslant 1

.
An improved estimate in the method of freezing
Authors: Vinograd Robert E.
Published: June 2010

Let

x
˙

=
A
(
t
)
x

\dot x = A(t)x

and

λ
k

(
t
)

{\lambda _k}(t)

be the eigenvalues of the matrix

A
(
t
)

A(t)

. The main result of the Method of Freezing [1] states that if

‖

A
(
t
)

‖

⩽
a

\left \| {A(t)} \right \| \leqslant a

. Re

λ
k

(
t
)
⩽

λ
0

{\lambda _k}(t) \leqslant {\lambda _0}

and

‖

A
(
t
)
−
A
(
s
)

‖

⩽
δ

|

t
−
s

|

\left \| {A(t) - A(s)} \right \| \leqslant \delta \left | {t - s} \right |

, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar x left-parenthesis t right-parenthesis double-vertical-bar less-than-or-slanted-equals double-vertical-bar x left-parenthesis t 0 right-parenthesis double-vertical-bar upper D Subscript delta Baseline exp left-parenthesis lamda 0 plus 2 a lamda Subscript delta Baseline right-parenthesis left-parenthesis t minus t 0 right-parenthesis left-parenthesis t greater-than-or-slanted-equals t 0 right-parenthesis">

‖

x
(
t
)

‖

⩽

‖

x
(

t
0

)

‖

D
δ

exp
⁡
(

λ
0

+
2
a

λ
δ

)
(
t
−

t
0

)

(
t
⩾

t
0

)

\left \| {x(t)} \right \| \leqslant \left \| {x({t_0})} \right \|{D_\delta }\exp ({\lambda _0} + 2a{\lambda _\delta })(t - {t_0})\quad (t \geqslant {t_0})

\]
for all solutions of the system, where
\[

λ
δ

=

(

C
n

⋅
δ

/

4

a
2

)

1

/

(
n
+
1
)

{\lambda _\delta } = {({C_n} \cdot \delta /4{a^2})^{1/(n + 1)}}

\]
. The previous best known value.

C
n

=
n
(
n
+
1
)

/

2

{C_n} = n(n + 1)/2

, is reduced to the substantially smaller value

2

n
n

e

−
n

/

(
n
−
1
)
!
>

2
n

/

π

2{n^n}{e^{ - n}}/(n - 1)! > \sqrt {2n/\pi }

.
Sums of three integer squares in complex quadratic fields
Authors: Estes Dennis R. , Hsia J. S.
Published: June 2010

We classify all complex quadratic number fields that have all their algebraic integers expressible as a sum of three integer squares. These fields are

F
=

Q

(

−
D

)

F = {\mathbf {Q}}(\sqrt { - D} )

,

D
D

a positive square-free integer congruent to

3
(

mod

8
)

3(\mod 8)

and such that

D
D

does not admit a positive proper factorization

D
≡

d
1

d
2

D \equiv {d_1}{d_2}

that satisfies simultaneously:

d
1

≡
5
,
7
(

mod

8
)

{d_1} \equiv 5,7(\mod 8)

and

(

d
2

/

d
1

)
=
1

({d_2}/{d_1}) = 1

.
Characterization of quasi-units in terms of equilibrium potentials
Authors: Arsove Maynard , Leutwiler Heinz
Published: June 2010

In the cone of nonnegative superharmonic functions on a bounded euclidean region

Ω
\Omega

, quasi-units were introduced as those elements invariant under the infinitesimal generator of the fundamental one-parameter semigroup of operators

S
λ

(
λ
⩾
0
)

{S_\lambda }(\lambda \geqslant 0)

. All harmonic measures and capacitary potentials are quasi-units, but the latter class has more extensive closure properties. Quasi-units arise naturally under various operations of classical potential theory and have important applications, for example in proving that the convex set of Green’s potentials

u
u

of positive mass distributions on

Ω
\Omega

with

u
⩽
1

u \leqslant 1

has as its extreme points precisely the capacitary potentials. Some new properties of quasi-units are developed here. In particular, it is shown that quasi-units can be characterized as limits of increasing sequences of continuous equilibrium potentials for which the equilibrium values tend to 1.
Realizability and nonrealizability of Dickson algebras as cohomology rings
Authors: Smith Larry , Switzer R. M.
Published: June 2010

Fix a prime

p
p

and let

V
V

be an

n
n

-dimensional vector space over

Z

/

p

{\mathbf {Z}}/p

. The general linear group

GL(

V

)

{\text {GL(}}V{\text {)}}

of

V
V

acts on the polynomial ring

P
(
V
)

P(V)

on

V
V

. The ring of invariants

P

(
V

)

GL

(
V
)

P{(V)^{{\text {GL}}(V)}}

has been computed by Dickson, and we denote it by

D
∗

(
n
)

{D^*}(n)

. If we grade

P
(
V
)

P(V)

by assigning the elements of

V
V

the degree 2, then

D
∗

(
n
)

{D^*}(n)

becomes a graded polynomial algebra on generators

Y
1

,
…
,

Y
n

{Y_1}, \ldots ,{Y_n}

of degrees

2

p
n

−
2

p

n
−
1

,
…
,
2

p
n

−
2

2{p^n} - 2{p^{n - 1}}, \ldots ,2{p^n} - 2

. The

mod

p

\mod p

Steenrod algebra acts on

P
(
V
)

P(V)

in a unique way compatible with the unstability condition and the Cartan formula. The

GL

(
V
)

{\text {GL}}(V)

action commutes with the Steenrod algebra action, and so

D
∗

(
n
)

{D^*}(n)

inherits the structure of an unstable polynomial algebra over the Steenrod algebra. In this note we determine explicit formulas for the action of the Steenrod algebra on the polynomial generators of

D
∗

(
n
)

{D^*}(n)

. As a consequence we are able to decide exactly which Dickson algebras can be

Z

/

p

{\mathbf {Z}}/p

cohomology rings.
A composition theorem for 𝛿-codes
Authors: Yang C. H.
Published: June 2010

If Golay complementary sequences (or equivalently a two-symbol

δ
\delta

-code) of length

n
n

and a Turyn

δ
\delta

-code of length

t
t

exist then four-symbol

δ
\delta

-codes of length

(
2
n
+
1
)

/

t

(2n + 1)/t

can be composed. Therefore new families of Hadamard matrices of orders

4
u
w

4uw

and

20
u
w

20uw

can be constructed, where

u
=
(

2

α
+
1

10
b

26
c

+
1
)
t

u = ({2^{\alpha + 1}}{10^b}{26^c} + 1)t

for odd

t
⩽
59

t \leqslant 59

or

t
=

2
d

10
e

26
f

+
1

t = {2^d}{10^e}{26^f} + 1

(all

a
a

,

b
b

,

c
c

,

d
d

,

e
e

, and

f
⩾
0

f \geqslant 0

), and

w
w

is the order of Williamson matrices.
Remarks on geodesics
Authors: Hartman Philip
Published: June 2010

For aC1{C^1}Riemann metric, we investigate questions concerning "curvature" and "C1{C^1}dependence of geodesics on initial conditions".
A characterization of uniform paracompactness
Authors: Fried J. , Frolík Z.
Published: June 2010

Main result: a uniform space

X
X

is uniformly paracompact [R] iff for some (and then any) compactification

K
K

of

X
X

and for any compact

C
⊂
K
∖
X

C \subset K\backslash X

closed disjoint sets

X
×
C

X \times C

and the diagonal

Δ
X

(
=
{

⟨

x
,
x

⟩

|

x
∈
X
}
)

{\Delta _X}( = \{ \left \langle {x,x} \right \rangle |x \in X\} )

can be separated by a uniformly continuous function on the semiuniform product

X
∗
K

X * K

.
Note on rotation set
Authors: Ito Ryuichi
Published: June 2010

Let

f
f

be an endomorphism of the circle of degree 1 and

f
¯

\bar f

be a lifting of

f
f

. We characterize the rotation set

ρ
(

f
¯

)

\rho (\bar f)

by the set of probability measures on the circle, and prove that if

ρ
+

(

f
¯

)

(

ρ
−

(

f
¯

)
)

{\rho _ + }(\bar f)\;({\rho _ - }(\bar f))

, the upper (lower) endpoint of

ρ
(

f
¯

)

\rho (\bar f)

, is irrational, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript plus Baseline left-parenthesis upper R Subscript theta Baseline f overbar right-parenthesis greater-than rho Subscript plus Baseline left-parenthesis f overbar right-parenthesis left-parenthesis rho Subscript minus Baseline left-parenthesis upper R Subscript theta Baseline f overbar right-parenthesis greater-than rho Subscript minus Baseline left-parenthesis f overbar right-parenthesis right-parenthesis">

ρ
+

(

R
θ

f
¯

)
>

ρ
+

(

f
¯

)

(

ρ
−

(

R
θ

f
¯

)
>

ρ
−

(

f
¯

)
)

{\rho _ + }({R_\theta }\bar f) > {\rho _ + }(\bar f)\;({\rho _ - }({R_\theta }\bar f) > {\rho _ - }(\bar f))

for any

θ
>
0

\theta > 0

, where

R
θ

(
x
)
=
x
+
θ

{R_\theta }(x) = x + \theta

. As a corollary, if

f
f

is structurally stable, then both

ρ
+

(

f
¯

)

{\rho _ + }(\bar f)

and

ρ
−

(

f
¯

)

{\rho _ - }(\bar f)

are rational.
A remark on the Hardy-Littlewood inequality
Authors: McMullen John R. , Sharp Bernice
Published: June 2010

We show that the inequality of the Littlewood conjecture holds precisely for compact abelian groups whose connected component has finite index.
Killing vector fields and holonomy algebras
Authors: Currás-Bosch Carlos
Published: June 2010

We prove that for each Killing vector field

X
X

on a complete Riemannian manifold, whose orthogonal distribution is involutive, the

(
1
,
1
)

(1,1)

skew-symmetric operator

A
X

{A_X}

associated to

X
X

by

A
X

=

L
X

−

∇
X

{A_X} = {L_X} - {\nabla _X}

lies in the holonomy algebra at each point. By using the same techniques, we also study when that operator lies in the infinitesimal and local holonomy algebras respectively.
A fixed point theorem for simplicial complexes whose fundamental groups are free products
Authors: Raymond Frank
Published: June 2010

A fixed point theorem for actions of toral and finite

p
p

-groups is obtained for finite simplicial complexes satisfying certain conditions on their fundamental groups and their universal covering spaces. Examples are constructed relating this result to other well-known results.
On a family of special linear systems on algebraic curves
Authors: Griffin Edmond E.
Published: June 2010

W

6
2

\mathcal {W}_6^2

is a scheme parametrizing pairs

L
→
C

L \to C

of smooth algebraic curves

C
C

of genus 10 together with line bundles

L
L

of degree 6 such that

H
0

(

C
,
L

)

⩾
3

{H^0}\left ( {C,L} \right ) \geqslant 3

. It is shown that one of the irreducible components of this scheme is nonreduced at every point.
Cyclic Stickelberger cohomology and descent of Kummer extensions
Authors: Childs Lindsay N.
Published: June 2010

Let

R
R

be a field,

S
=
R
[

ζ

]

S = R[{\rm {\zeta }}]

,

ζ

{\rm {\zeta }}

an

n
n

th root of unit,

Δ
=

G
a
l
(

S

/

R
)

\Delta = {\rm {Gal(}}S/R)

. The group of cyclic Kummer extensions of

S
S

on which

Δ
\Delta

acts, modulo those which descend to

R
R

, is isomorphic to a group of roots of unity and to a second group cohomology group of

Δ
\Delta

whose definition involves a "Stickelberger element".
On fixed rings of automorphisms
Authors: Puczyłowski Edmund
Published: June 2010

In this note we present simple proofs of some well-known results on fixed rings of finite automorphism groups of associative rings.
The Picard theorem for Riemann surfaces
Authors: Royden H. L.
Published: June 2010

Let

W
W

be a Riemann surface other than the sphere, plane, punctured plane or torus. Let

f
f

be a holomorphic map of the punctured disk

0
>

|
z
|

>
1

0 > \left | z \right | > 1

into

W
W

. Then

f
f

can be extended to a holomorphic map of the disk

|
z
|

>
1

\left | z \right | > 1

, possibly, into a Riemann surface

W
∗

{W^ * }

containing

W
W

. We give a new proof of this fact and explore some consequences of it.
A filter on [𝜆]^{𝜅}
Authors: Di Prisco C. A. , Marek W.
Published: June 2010

We define a filter on

[
λ
]

κ

{\left [ \lambda \right ]^\kappa }

with properties similar to those of the closed unbounded filter in

P
κ

(
λ
)

{P_\kappa }\left ( \lambda \right )

. This filter’s behaviour depends on set theoretical hypotheses.
Mackey functors and 𝐺-cohomology
Authors: Waner Stefan
Published: June 2010

The Bredon-Illman cohomology of universal

G
G

-spaces associated with a family of subgroups is related to derived functors in three fundamental categories of

G
G

-orbits. Analogous results for the

G
G

-cohomology of classifying spaces for

G
G

-covering spaces are also obtained.
Discrete ordered sets whose covering graphs are median
Authors: Bandelt Hans-J.
Published: June 2010

As is well known the covering graph (= Hasse diagram) of any median semilattice is a median graph, and every median graph is the covering graph of some median semilattice. The purpose of this note is to prove that an ordered set is a median semilattice whenever (i) no interval contains an infinite chain, (ii) each pair of elements is bounded below, and (iii) the covering graph is median.
A finite global Azumaya theorem in additive categories
Authors: Arnold David M.
Published: June 2010

Let

C

{\mathbf {C}}

be an additive category such that idempotent endomorphisms have kernels,

C
C

a class of objects of

C

{\mathbf {C}}

having Dedekind domains as endomorphism rings, and assume that if

X
X

and

Y
Y

are quasi-isomorphic objects of

C
C

then

Hom

(
X
,
Y
)

{\operatorname {Hom}}(X,Y)

is a torsion-free module over the endomorphism ring of

X
X

.

A
⊕
B
=

C
1

⊕
⋯
⊕

C
n

A \oplus B = {C_1} \oplus \cdots \oplus {C_n}

with each

C
i

{C_i}

in

C
C

, then

A
=

A
1

⊕
⋯
⊕

A
m

A = {A_1} \oplus \cdots \oplus {A_m}

, where each

A
j

{A_j}

is locally in

C
C

, and

End

(

A
j

)
≃

End

(

C
i

)

{\operatorname {End}}({A_j}) \simeq {\operatorname {End}}({C_i})

for some

i
i

. The proof includes a characterization of tiled orders. Moreover, there is a "local" uniqueness for finite direct sums of objects of

C
C

.
The spectral diameter in Banach algebras
Authors: Grabiner Sandy
Published: June 2010

The element

a
a

is in the center of the Banach algebra

A
A

modulo its radical if and only if there is an upper bound for the diameters of the spectra of

a
_

u
a

u

−
1

{a^\_}ua{u^{ - 1}}

for

u
u

invertible. Applications of this result are given to general Banach algebras and to the essential spectrum of operators on a Hilbert Space.
Bloch constants for meromorphic functions near an isolated singularity
Authors: Minda David
Published: June 2010

Suppose

f
f

is meromorphic in a punctured neighborhood of the origin and has an essential singularity at the origin. Given any

ε
>
0

\varepsilon > 0

we show that the Riemann surface of

f
f

contains an unramified disk of spherical radius

π

/

3
−
ε

\pi /3 - \varepsilon

. The number

π

/

3

\pi /3

can be replaced by

π

/

2

\pi /2

if

f
f

is locally schlicht and this value is best possible. If

f
f

is actually holomorphic, then the Riemann surface of

f
f

contains arbitrarily large unramified euclidean disks. These results generalize theorems of Valiron and Ahlfors dealing with holomorphic and meromorphic functions, respectively, on the complex plane which have an essential singularity at infinity.
Two problems of Dowker
Authors: Rudin Mary Ellen
Published: June 2010

An old question first raised by Hugh Dowker is asked again and two related problems are solved.
𝐿²𝑙𝑜𝑐-boundedness for a class of singular Fourier integral operators
Authors: El Kohen A.
Published: June 2010

We consider operators of the form∫−∞∞Ftϕ(t)dt\int _{-\infty }^\infty {{F_t}} \phi \left ( t \right )dtwhereFt{F_t}is a11-parameter family of Fourier integral operators andϕ(t)dt\phi \left ( t \right )dta tempered distribution on the real line. We extend the result given in [1].
A generalization of a theorem on naturally reductive homogeneous spaces
Authors: Kowalski Oldřich , Vanhecke Lieven
Published: June 2010

We prove that a homogeneous Riemannian manifold all of whose geodesics are orbits of one-parameter subgroups of isometries has volume-preserving local geodesic symmetries.
Stability theorems for conformal foliations
Authors: Blumenthal Robert A.
Published: June 2010

The global stability theorem of G. Reeb is no longer true if the codimension of the foliation is greater than one. However, in the presence of a complete transverse Riemannian structure, B. Reinhart obtained a global stability result. We prove global stability theorems for the much larger class of conformal foliations.
Versik processes and very weak Bernoulli processes with summable rates are independent
Authors: Dehling Herold , Denker Manfred , Philipp Walter
Published: June 2010

We show that very weak Bernoulli processes and Versik processes having dependence rates

o
(
1

/

n
)

o(1/n)

are independent or

2
2

-dependent respectively. The borderline case where the rate is

O
(
1

/

n
)

O(1/n)

is discussed in some detail.
𝐿^{𝑝}-computability in recursive analysis
Authors: Pour-El Marian Boykan , Richards Ian
Published: June 2010

L
p

{L^p}

-computability is defined in terms of effective approximation; e.g. a function

f
∈

L
p

[
0
,
1
]

f \in {L^p}[0,1]

is called

L
P

{L^P}

-computable if

f
f

is the effective limit in

L
p

{L^p}

-norm of a computable sequence of polynomials. Other families of functions can replace the polynomials; see below. In this paper we investigate conditions which are not based on approximation. For

p
>
1

p > 1

, we show that

f
f

is

L
p

{L^p}

-computable if and only if (a) the sequence of Fourier coefficients of

f
f

is computable, and (b) the

L
p

{L^p}

-norm of

f
f

is a computable real. We show that this fails for

p
=
1

p = 1

.
Mappings of the Sierpiński curve onto itself
Authors: Charatonik J. J.
Published: June 2010

Given two points

p
p

and

q
q

of the Sierpiński universal plane curve

S
S

, necessary and/or sufficient conditions are discussed in the paper under which there is a mapping

f
f

of

S
S

onto itself such that

f
(
p
)
=
q

f(p) = q

and

f
f

belongs to one of the following: homeomorphisms, local homeomorphisms, local homeomorphisms in the large sense, open, simple or monotone mappings.
On a result of S. Delsarte
Authors: Constantine Gregory , Kulkarni Ravi S.
Published: June 2010

For an isomorphism type of a finite abelian

p
p

-group

X
X

it is shown that the matrix

(

p

⟨

s
(
D
)
,
s
(
Y
)

⟩

)

({p^{\left \langle {s(D),s(Y)} \right \rangle }})

is nonsingular;

D
D

,

Y
∈

{

S

|

S
⩽
X

and

S
≠
X

}

Y \in \left \{ {S|S \leqslant X\;{\text {and}}\;S \ne X} \right \}

, the set of all proper isomorphism type of subgroups of

X
X

. Here

s
(
Y
)

s(Y)

denotes the signature of

Y
Y

. This completes the proof of a result of

S

{\text {S}}

. Delsarte which gives explicit formulas for the number of automorphisms of

X
X

, the number of subgroups of

X
X

isomorphic to

Y
Y

(and the number of homomorphisms from

Y
Y

into

X
X

) in terms of signatures.
On summability of Fourier series at a point
Authors: Dikshit G. D.
Published: June 2010

In this paper summability of Fourier series by a regular linear method of summation determined by a triangular matrix, has been studied and various results—some known and some new—on Cesàro and Nörlund summability have been deduced. A convergence criterion has also been obtained.
Homogeneity is not a Whitney property
Authors: Charatonik Włodzimierz J.
Published: June 2010

It is proved by showing a counterexample that homogeneity is not a Whitney property.
A resonance problem in which the nonlinearity may grow linearly
Authors: Ahmad Shair
Published: June 2010

The purpose of this paper is to study a semilinear two point boundary value problem of resonance type in which the nonlinear perturbation may grow linearly. A significant improvement of a recent result due to Cesari and Kannan is given.
A family of polynomials with concyclic zeros. II
Authors: Evans Ronald J. , Stolarsky Kenneth B.
Published: June 2010

Let

λ
1

,
…
,

λ
J

{\lambda _1}, \ldots ,{\lambda _J}

be nonzero real numbers. Expand
\[

E
(
z
)
=
∏

(
−
1
+
exp
⁡

λ
j

z
)

,

E(z) = \prod {( - 1 + \exp {\lambda _j}z)} ,

\]
rewrite products of exponentials as single exponentials, and replace every

exp
⁡
(
a
z
)

\exp (az)

by its approximation

(
1
+
a

n

−
1

z

)
n

{(1 + a{n^{ - 1}}z)^n}

, where

n
⩾
J

n \geqslant J

. The resulting polynomial has all zeros on the (possibly infinite) circle of radius

|
r
|

\left | r \right |

centered at

−
r

- r

, where

r
=
n

/

∑

λ
j

r = n/\sum {\lambda _j}

.
Error estimates for finite difference approximations to hyperbolic equations for large time
Authors: Layton William
Published: June 2010

Convergence results for bounded time intervals are well known for finite difference approximations to the Cauchy problem for hyperbolic equations. These results typically state that if the initial data is smooth and the approximation is stable in

L
2

{L^2}

and accurate of order

r
r

, then the error at time

t
t

is bounded by

C
(
t
,
f
)

h
r

C(t,f){h^r}

, where

f
f

is the initial data and

C
(
t
)
=
O
(
t
)

C(t) = O(t)

. This paper considers the error for long times. It is not possible for the error to be

O
(

h
r

)

O({h^r})

in

L
p

{L^p}

uniformly in

t
t

. However, it is shown here that if

Ω
\Omega

is a bounded domain the error in

L
p

(
Ω
)

{L^p}(\Omega )

is bounded by

C
(
Ω
,
f
)

h
r

C(\Omega ,f){h^r}

, where

C
C

is independent of

t
t

. Thus, the global error will grow as more timesteps are taken but the local error will remain uniformly bounded.
A remark on the maximal dilatation of a quasiconformal mapping
Authors: Vuorinen Matti
Published: June 2010

A lower bound for the maximal dilatation of a quasiconformal self-mapping of a uniform domain

G
G

in

R
n

{R^n}

with connected boundary is proved provided that the boundary is kept pointwise fixed by the mapping and that a prescribed point

a
a

is mapped on another point

b
b

in

G
G

. The lower bound is given in terms of the quasihyperbolic distance from

a
a

to

b
b

.
On a theorem of Ingham on nonharmonic Fourier series
Authors: Young Robert M.
Published: June 2010

A well-known result due to Ingham [3] shows that the system of complex exponentials

{

e

i

λ

n

t

}

\{ {e^{i\lambda _{n}t}}\}

is a basic sequence in

L
2

(
−
π
,
π
)

{L^2}( - \pi ,\pi )

whenever

λ

n
+
1

−

λ
n

⩾
γ
>
1

{\lambda _{n + 1}} - {\lambda _n} \geqslant \gamma > 1

. In this note, we show that the system need not be basic if

λ

n
+
1

−

λ
n

>
1

{\lambda _{n + 1}} - {\lambda _n} > 1

.
On inner product spaces over Dedekind domains of characteristic two
Authors: Wagner Richard C.
Published: June 2010

SupposeDDis a Dedekind domain of characteristic 2 and(M,φ)(M,\varphi )is an inner product space, i.e.MMis a finitely generated projectiveDDmodule supplied with a nonsingular symmetric bilinear formφ\varphi. It is shown that(M,φ)(M,\varphi )is determined up to isometry by the extension ofφ\varphitoF⊗DMF{ \otimes _D}M, whereFFis the quotient field ofDD, and the value moduleQ(M){\mathcal Q}(M)of allφ(m,m)\varphi (m,m)formminMM. In particular, a hyperbolic spaceH(M){\mathbf {H}}(M)is completely determined by the rank of the finitely generated projective moduleMM. As consequences, genera coincide with isometry classes, and ifN1{N_1}andN2{N_2}are isometric nonsingular submodules of(M,φ)(M,\varphi )such thatQ(N1⊥)=Q(N2⊥){\mathcal Q}({N_1}^ \bot ) = {\mathcal Q}({N_2}^ \bot ), thenN1⊥{N_1}^ \botandN2⊥{N_2}^ \botare isometric. Also, given anFFinner product space(V,φ)(V,\varphi )and aD(2){D^{(2)}}submodulePPofDD, a necessary and sufficient condition is given for the existence of aDDinner product space(M,Ψ)(M,\Psi )such that(FM,Ψ)≅(V,φ)(FM,\Psi ) \cong (V,\varphi )andQ(M)=P{\mathcal Q}(M) = P.
An elementary transformation of a special unimodular vector to its top coefficient vector
Authors: Rao Ravi A.
Published: June 2010

Let

R
R

be a commutative ring,

v

(
X
)

{\mathbf {v}}(X)

a unimodular

n
n

-vector

(
n
⩾
3
)

(n \geqslant 3)

over

R
[
X
]

R[X]

. Suppose the leading coefficients in

v

(
X
)

{\mathbf {v}}(X)

form a unimodular vector

L
(

v

)

L({\mathbf {v}})

over

R
R

. Then some element in

E
n

(
R
[
X
]
)

{E_n}(R[X])

will transform

v

(
X
)

{\mathbf {v}}(X)

to

L
(

v

)

L({\mathbf {v}})

.
Commutative ranges of analytic functions in Banach algebras
Authors: Fleming R. J. , Jamison J. E.
Published: June 2010

Let

A
A

denote a complex unital Banach algebra with Hermitian elements

(
A
)

(A)

. We show that if

F
F

is an analytic function from a connected open set

D
D

into

A
A

such that

F
(
z
)

F(z)

is normal

(
F
(
z
)
=
u
(
z
)
+
i
υ
(
z
)

(F(z) = u(z) + i\upsilon (z)

, where

u
(
z
)

u(z)

,

υ
(
z
)
∈
H
(
A
)

\upsilon (z) \in H(A)

and

u
(
z
)
υ
(
z
)
=
υ
(
z
)
u
(
z
)
)

u(z)\upsilon (z) = \upsilon (z)u(z))

for each

z
∈
D

z \in D

, then

F
(
z
)
F
(
w
)
=
F
(
w
)
F
(
z
)

F(z)F(w) = F(w)F(z)

for all

w
w

,

z
∈
D

z \in D

. This generalizes a theorem of Globevnik and Vidav concerning operator-valued analytic functions. As a corollary, it follows that an essentially normal-valued analytic function has an essentially commutative range.
Locally symmetric affine hypersurfaces
Authors: Verheyen P. , Verstraelen L.
Published: June 2010

The nondegenerate quadratic hypersurfaces and the improper affine hyperspheres are the only nondegenerate hypersurfaces of dimension greater than two of an affine space which are affine locally symmetric with respect to their induced connection.
Locally invariant topological groups and semidirect products
Authors: Bagley R. W. , Yang J. S.
Published: June 2010

We consider topological groups which have arbitrarily small invariant neighborhoods of the identity and those topological groups which admit continuous monomorphisms into such groups. We establish conditions under which the two corresponding classes of groups coincide. We apply these results to semidirect products. Since we do not assume local compactness in general, we use the symbol "[Sn]" rather than "[SIN]" for the class of groups with small invariant neighborhoods and the symbol "[In]" for those embeddable in Sn groups. We denote by "[N]" those groups

G
G

with the property: If

{

x
α

}

α
∈
D

{\{ {x_\alpha }\} _{\alpha \in D}}

is a net in

G
G

which converges to the identity and

{

y
α

}

α
∈
D

{\{ {y_\alpha }\} _{\alpha \in D}}

is any net such that

{

y
α

x
α

y
α

−
1

}

\{ {y_\alpha }{x_\alpha }y_\alpha ^{ - 1}\}

converges, then this net also converges to the identity. We also define a class of topological groups we term

S
(
U
)

S(U)

groups. The following are corollaries of our general results: (1) If

G
G

is locally compact,

G

/

G
0

G/{G_0}

is compact and

G
0

{G_0}

is an

N
N

groups, then

G
G

is an Sn group. (2) If

H
H

is a locally connected compact group,

G
G

is an Sn group, and if the semidirect product

H
Ⓢ
G

H\circledS G

is an

S
(
U
)

S(U)

group, then

H
Ⓢ
G

H\circledS G

is an Sn group. (3) If

H
Ⓢ
K

H\circledS K

is an Sn group for every compact group

K
K

, then every open subgroup of

H
H

is of finite index.
A generalized Jacobian criterion for regularity
Authors: Seibt Peter
Published: June 2010

For a commutative noetherian algebra

B
B

over a perfect field

A
A

regularity is equivalent to the flatness of

Ω

B

|

A

{\Omega _{B|A}}

plus

H
1

(
A
,
B
,
B
)
=
0

{H_1}(A,B,B) = 0

(simplicial homology). In characteristic 0 the homological condition is superfluous.
Nonabelian counterexamples to the Noether problem
Authors: Sonn Jack
Published: June 2010

Let

G
G

be a finite group having a cyclic quotient of order 8. Let

L
L

be the field given by adjoining indeterminates

{

x
g

|

g
∈
G
}

\{ {x_g}|g \in G\}

to

Q

{\mathbf {Q}}

. Then

L
G

{L^G}

is not a rational function field.
Abelian and Tauberian theorems for 0-regularly varying functions
Authors: Geluk J. L.
Published: June 2010

A general kernel Abelian and Tauberian theorem is proved for functions

f
f

satisfying

lim
¯

t
→
∞

f
(
t
x
)

/

f
(
t
)
>
∞

 for 

x
>
0

{\overline {\lim } _{t \to \infty }}f(tx)/f(t) > \infty {\text { for }}x > 0

.
Stability results for periodic second order linear differential equations
Authors: Erbe L. H.
Published: June 2010

Sufficient conditions for stability, asymptotic stability and instability of the trivial solution of a periodic second order linear differential equation are obtained.
A theorem on the tension field
Authors: Koufogiorgos Th. , Baikoussis Ch.
Published: June 2010

Suppose

M
M

and

N
N

are complete Riemannian manifolds;

M
M

with Ricci curvature bounded from below and

N
N

with sectional curvature bounded from above by a constant

K
0

{K_0}

. Let

f
:
M
→
N

f:M \to N

be a smooth map such that

f
(
M
)
⊂

B
R

f(M) \subset {B_R}

, where

B
R

{B_R}

is a normal ball in

N
N

and furthermore

R
>
π

/

2

K
0

R > \pi /2\sqrt {{K_0}}

if

K
0

>
0

{K_0} > 0

. If the energy density

e
(
f
)

e(f)

is bounded below by a positive constant, then there is a point

P
∈
M

P \in M

such that the tension field

τ
(
f
)

\tau (f)

at

P
P

is different from zero.
A hyperbolic 4-manifold
Authors: Davis Michael W.
Published: June 2010

There is a regular

4
4

-dimensional polyhedron with 120 dodecahedra as

3
3

-dimensional faces. (Coxeter calls it the "

120
120

-cell".) The group of symmetries of this polyhedron is the Coxeter group with diagram:
\[

[
u
n
k
]

[unk]

\]
For each pair of opposite

3
3

-dimensional faces of this polyhedron there is a unique reflection in its symmetry group which interchanges them. The result of identifying opposite faces by these reflections is a hyperbolic manifold

M
4

{M^4}

.
An area theorem for holomorphic functions
Authors: Yamashita Shinji
Published: June 2010

Let

D
⫋

C

D \subsetneqq \mathbf {C}

and

Δ
⫋

C

\Delta \subsetneqq \mathbf {C}

be open sets in the plane

C

\mathbf {C}

containing 0. Suppose that

D
D

is connected and

f
f

is holomorphic in

D
D

with

f
(
0
)
=
0

f(0) = 0

. Then, for

Q
=
f
(
D
)
∩
Δ

Q = f(D) \cap \Delta

and

P
=

f

−
1

(
Q
)

P = {f^{ - 1}}(Q)

, we have
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis upper P right-parenthesis plus upper A left-parenthesis upper Q right-parenthesis greater-than-or-slanted-equals pi left-bracket min left-brace right-brace comma of disdis left-parenthesis right-parenthesis comma of partial-differential upper D comma 0 comma of disdis left-parenthesis right-parenthesis comma of partial-differential upper Delta comma 0 right-bracket squared comma">

A
(
P
)
+
A
(
Q
)
⩾
π

[
min
{
dis
⁡
(
∂
D
,
0
)
,
dis
⁡
(
∂
Δ
,
0
)
}

]
2

,

A(P) + A(Q) \geqslant \pi {[\min \{ \operatorname {dis}(\partial D,0),\operatorname {dis}(\partial \Delta ,0)\} ]^2},

\]

A
(
⋅
)

A( \cdot )

denoting the area. The constant

π
\pi

is sharp and the equality holds if and only if

f
≡
0
,
dis
⁡
(
∂
D
,
0
)
⩽
dis
⁡
(
∂
Δ
,
0
)

f \equiv 0,\operatorname {dis}(\partial D,0) \leqslant \operatorname {dis}(\partial \Delta ,0)

, and

D
D

is a disk of center 0.