bluuebunny/crossref_metadata_2025 huggingface dataset:
Some structure theorems for complete constant mean curvature surfaces with boundary a convex curve
Authors: Sa Earp Ricardo , Rosenberg Harold
Published: July 2010

Let

M
M

be a properly embedded, connected, complete surface in

R

3

{\mathbb {R}^3}

with non-zero constant mean curvature and with boundary a strictly convex plane curve

C
C

. It is shown that if

M
M

is contained in a vertical cylinder of

R

+
3

\mathbb {R}_ + ^3

, outside of some compact set of

R

3

{\mathbb {R}^3}

, and if

M
M

is contained in a half-space of

R

3

{\mathbb {R}^3}

determined by

C
C

, then

M
M

inherits the symmetries of

C
C

. In particular,

M
M

is a Delaunay surface if

C
C

is a circle. It is also shown that if

M
M

has a finite number of vertical annular ends and the area of the flat disc

D
D

bounded by

C
C

is not "too small," then

M
M

lies in a half-space.
Interconnections between the structure theory of set addition and rewritability in groups
Authors: Freiman Gregory A. , Schein Boris M.
Published: July 2010

An approach to groups and semigroups stemming from the structure theory of set addition turns out to have much in common with the so-called permutation or rewritable properties. We explain these connections and show how these properties take their place in a wider class of interesting and naturally arising problems. As an example, we characterize some classes of groups and group elements.
Split braids
Authors: Humphries Stephen P.
Published: July 2010

Let

B
n

{B_n}

be the group of braids on

n
n

strings with standard generators

σ
1

,
…
,

σ

n
−
1

{\sigma _1}, \ldots ,{\sigma _{n - 1}}

. For

i
∈
{
1
,
2
,
…
,
n
−
1
}

i \in \{ 1,2, \ldots ,n - 1\}

we let

B
n
i

B_n^i

be the subgroup of

B
n

{B_n}

generated by the elements

σ
1

,
…
,

σ

i
−
1

,

σ

i
+
1

,
…
,

σ

n
−
1

{\sigma _1}, \ldots ,{\sigma _{i - 1}},{\sigma _{i + 1}}, \ldots ,{\sigma _{n - 1}}

. In this paper we give an algorithm for deciding if, given

α
∈

B
n

\alpha \in {B_n}

there is

i
∈
{
1
,
2
,
…
,
n
−
1
}

i \in \{ 1,2, \ldots ,n - 1\}

such that

α
\alpha

is conjugate into

B
n
i

B_n^i

. We call such a braid a split braid. Such a split braid gives rise to a split link. This algorithm gives a partial solution to the problem of finding braids that represent reducible mapping classes. It also represents a contribution to the algebraic link problem and it gives a way of determining if a braid in

B
n

{B_n}

can be conjugated into the subgroup

B

n
−
1

{B_{n - 1}}

, which we identify with

B

n
−
1

n
−
1

B_{n - 1}^{n - 1}

.
On the integrability and 𝐿¹-convergence of complex trigonometric series
Authors: Móricz Ferenc
Published: July 2010

We prove that if a weakly even sequence

{

c
k

:
k
=
0
,
±
1
,
…
}

\{ {c_k}:k = 0, \pm 1, \ldots \}

of complex numbers is such that for some

p
>
1

p > 1

we have
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript m equals 1 Overscript normal infinity Endscripts 2 Superscript m slash q Baseline left-parenthesis sigma-summation Underscript k equals 2 Superscript m minus 1 Baseline Overscript 2 Superscript m Baseline minus 1 Endscripts StartAbsoluteValue normal upper Delta left-parenthesis c Subscript k Baseline plus c Subscript negative k Baseline right-parenthesis EndAbsoluteValue Superscript p Baseline right-parenthesis Superscript 1 slash p Baseline greater-than normal infinity comma StartFraction 1 Over p EndFraction plus StartFraction 1 Over q EndFraction equals 1 comma">

∑

m
=
1

∞

2

m

/

q

(

∑

k
=

2

m
−
1

2
m

−
1

|

Δ

(

c
k

+

c

−
k

)

|

p

)

1

/

p

>
∞
,

1
p

+

1
q

=
1
,

\sum \limits _{m = 1}^\infty {{2^{m/q}}} {\left ( {\sum \limits _{k = {2^{m - 1}}}^{{2^m} - 1} {{{\left | {\Delta \left ( {{c_k} + {c_{ - k}}} \right )} \right |}^p}} } \right )^{1/p}} > \infty ,\frac {1}{p} + \frac {1}{q} = 1,

\]
then the symmetric partial sums of the trigonometric series

(
∗
)

∑

k
=
−
∞

∞

c
k

e

i
k
x

( * )\sum \nolimits _{k = - \infty }^\infty {{c_k}{e^{ikx}}}

converge pointwise, except possibly at

x
=
0
(
mod
⁡
2
π
)

x = 0(\operatorname {mod} 2\pi )

, to a Lebesgue integrable function,

(
∗
)

( * )

is the Fourier series of its sum, and series

(
∗
)

( * )

converges in

L
1

(
−
π
,
π
)

{L^1}( - \pi ,\pi )

-norm if and only if

lim

|

k

|

→
∞

c
k

ln
⁡

|

k

|

=
0

{\lim _{|k| \to \infty }}{c_k}\ln |k| = 0

. In addition, we present new proofs of the theorems by J. Fournier and W. Self [6] and by ČC. V. Stanojević and V. B. Stanojević [10].
A few remarks on Riesz summability of orthogonal series
Authors: Szabłowski PawełJ.
Published: July 2010

We study convergence behavior of some sequences and series related to a given orthogonal series. Following the developed technique we define (in terms of fourth mixed moments only) a class of orthonormal functions

{

X
i

}

i
≥
1

{\left \{ {{X_i}} \right \}_{i \geq 1}}

such that the condition:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="there-exists k element-of double-struck upper N sigma-summation Underscript i greater-than-or-equal-to 1 Endscripts mu Subscript i Superscript 2 Baseline left-parenthesis ln Superscript left-parenthesis k right-parenthesis Baseline i right-parenthesis squared greater-than normal infinity">

∃
k
∈

N

∑

i
≥
1

μ
i
2

(

ln

(
k
)

i

)

2

>
∞

\exists k \in \mathbb {N}\sum \nolimits _{i \geq 1} {\mu _i^2} {\left ( {{{\ln }^{\left ( k \right )}}i} \right )^2} > \infty

implies almost everywhere convergence of the series

∑

i
≥
1

μ
i

X
i

\sum \nolimits _{i \geq 1} {{\mu _i}{X_i}}

, here for every

i
=
1
,
2
,
…
,
j
=
1
,
…
,
k

i = 1,2, \ldots ,j = 1, \ldots ,k

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ln Superscript left-parenthesis 1 right-parenthesis Baseline i equals ln Subscript 2 Baseline i comma ln Superscript left-parenthesis j right-parenthesis Baseline i equals ln Subscript 2 Baseline left-parenthesis max left-parenthesis 1 comma ln Superscript left-parenthesis j minus 1 right-parenthesis Baseline i right-parenthesis right-parenthesis period">

ln

(
1
)

i
=

ln
2

i
,

ln

(
j
)

i
=

ln
2

(
max
(
1
,

ln

(
j
−
1
)

i
)
)
.

{\ln ^{(1)}}i = {\ln _2}i,\quad {\ln ^{(j)}}i = {\ln _2}(\max (1,{\ln ^{(j - 1)}}i)).

\]
A full description of extreme points in 𝐶(Ω,𝐿^{𝜙}(𝜇))
Authors: Wisła Marek
Published: July 2010

Let

L
φ

(
μ
)

{L^\varphi }(\mu )

be an Orlicz space endowed with the Luxemburg norm. The main result of this paper reads as follows:

f
f

is an extreme point of the unit ball of the space of continuous functions from a compact Hausdorff space

Ω
\Omega

into

L
φ

(
μ
)

{L^\varphi }(\mu )

with the supremum norm if and only if the inverse image of the set of all extreme points of the unit ball of

L
φ

(
μ
)

{L^\varphi }(\mu )

under

f
f

is dense in

Ω
\Omega

.
Algebras realized by 𝑛 rational homotopy types
Authors: Lupton Gregory
Published: July 2010

I construct an example of a graded algebra that is realized as the rational cohomology algebra of exactly

n
n

rational homotopy types, for each natural number

n
n

. The algebras constructed have trivial multiplicative structure. Similar examples are given of graded Lie algebras realized as the rational homotopy Lie algebra of exactly

n
n

rational homotopy types.
Spectral radius formulae in quotient 𝐶*-algebras
Authors: Rakočević Vladimir
Published: July 2010

If

I
I

is a closed two-sided ideal in

C
∗

{C^ * }

-algebra

A
A

, we prove spectral radius formulae for the coset

x
+
I
(
x
∈
A
)

x + I(x \in A)

in the quotient algebra

A

/

I

A/I

. Then, as a corollary we get the main result of Mau-Hsiang Shin (Proc. Amer. Math. Soc. 100 (1987), 137-139).
Geodesics and bounded harmonic functions on infinite planar graphs
Authors: Northshield S.
Published: July 2010

It is shown there that an infinite connected planar graph with a uniform upper bound on vertex degree and rapidly decreasing Green’s function (relative to the simple random walk) has infinitely many pairwise finitely-intersecting geodesic rays starting at each vertex. We then demonstrate the existence of nonconstant bounded harmonic functions on the graph.
Geodesics in Euclidean space with analytic obstacle
Authors: Albrecht Felix , Berg I. D.
Published: July 2010

In this note we are concerned with the behavior of geodesies in Euclidean

n
n

-space with a smooth obstacle. Our principal result is that if the obstacle is locally analytic, that is, locally of the form

x
n

=
f
(

x
1

,
…
,

x

n
−
1

)

{x_n} = f({x_1}, \ldots ,{x_{n - 1}})

for a real analytic function

f
f

, then a geodesic can have, in any segment of finite arc length, only a finite number of distinct switch points, points on the boundary that bound a segment not touching the boundary. This result is certainly false that for a

C
∞

{C^\infty }

boundary. Indeed, even in

E
2

{E^2}

, where our result is obvious for analytic boundaries, we can construct a

C
∞

{C^\infty }

boundary so that the closure of the set of switch points is of positive measure.
A note on the homology of free abelianized extensions
Authors: Hartley Brian , Stöhr Ralph
Published: July 2010

Let

G
G

be a group given by a free presentation

G
=
F

/

N

G = F/N

, and

N
′

N’

the commutator subgroup of

N
N

. The quotient

F

/

N
′

F/N’

is called a free abelianized extension of

G
G

. We study the integral homology of

F

/

N
′

F/N’

. In particular, if

G
G

has no elements of order

p
p

(

p
p

an odd prime), we determine the

p
p

-torsion in dimension

p
2

{p^2}

in terms of the modulo

p
p

homology of

G
G

. This extends results of Kuz’min [5, 6] describing the

p
p

-torsion in smaller dimensions. Our approach is based on examining the homology of

G
G

with coefficients in symmetric powers of the augmentation ideal, which we relate to the integral homology of

F

/

N
′

F/N’

.
2^{2^{𝜔}} nonisomorphic short ordered commutative domains whose quotient fields are long
Authors: Ciesielski Krzysztof
Published: July 2010

A linearly ordered set is short if it does not contain any monotonic sequence of length

ω
1

{\omega _1}

, and it is long if it contains a monotonic sequence of length

α
\alpha

for every ordinal

α
>

(

2
ω

)
+

\alpha > {({2^\omega })^ + }

. We prove that there exists a family

F

{\mathbf {F}}

of power

2

2
ω

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

of long ordered fields of size

2
ω

{2^\omega }

that are pairwise nonisomorphic (as fields) and such that every field

F
∈

F

F \in {\mathbf {F}}

has

2

2
ω

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

nonisomorphic short subdomains whose field of quotients is

F
F

. The generalization of this result for higher cardinals is also discussed. This generalizes the author’s result of [Ci].
On the Sendov conjecture for sixth degree polynomials
Authors: Brown Johnny E.
Published: July 2010

The Sendov conjecture asserts that if

p
(
z
)
=

∏

k
=
1

n

(
z
−

z
k

)

p(z) = \prod _{k = 1}^n(z - {z_k})

is a polynomial with zeros

|

z
k

|

≤
1

\left | {{z_k}} \right | \leq 1

, then each disk

|

z
−

z
k

|

≤
1
,
(
1
≤
k
≤
n
)

\left | {z - {z_k}} \right | \leq 1,(1 \leq k \leq n)

contains a zero of

p
′

(
z
)

p’(z)

. This conjecture has been verified in general only for polynomials of degree

n
=
2
,
3
,
4
,
5

n = 2,3,4,5

. If

p
(
z
)

p(z)

is an extremal polynomial for this conjecture when

n
=
6

n = 6

, it is known that if a zero

|

z
j

|

≤

λ
6

=
0.626997
…

\left | {{z_j}} \right | \leq {\lambda _6} = 0.626997 \ldots

then

|

z
−

z
j

|

≤
1

\left | {z - {z_j}} \right | \leq 1

contains a zero of

p
′

(
z
)

p’(z)

. (The conjecture for

n
=
6

n = 6

would be proved if

λ
6

=
1

{\lambda _6} = 1

.) It is shown that

λ
6

{\lambda _6}

can be improved to

λ
6

=
63

/

64
=
0.984375

{\lambda _6} = 63/64 = 0.984375

.
Entropy of semipatterns or how to connect the dots to minimize entropy
Authors: Ashley Jonathan , Coven Ethan M. , Geller William
Published: July 2010

In this paper we solve the following problem: Given a map

φ
\varphi

from a finite subset of the reals into the reals, how do you connect the dots in the graph of

φ
\varphi

in order to minimize the topological entropy of the resulting map of the interval?
A global pinching theorem for compact minimal surfaces in 𝑆³
Authors: Hsu Yi-Jung
Published: July 2010

Let

M
M

be a compact minimally immersed surface in the unit sphere

S
3

{S^3}

, and let

S
S

denote the square of the length of the second fundamental form of

M
M

. We prove that if

‖
S
‖

2

≤
2

2

π

{\left \| S \right \|_2} \leq 2\sqrt 2 \pi

, then

M
M

is either the equatorial sphere or the Clifford torus.
On the commutativity of ultraproducts with direct products
Authors: Hébert Michel
Published: July 2010

We show that an ultraproduct of direct products of structures is elementarily equivalent to a direct product (naturally defined over an ultraproduct of sets!) of ultraproducts of these structures.
On isomorphisms of inductive limit 𝐶*-algebras
Authors: Thomsen Klaus
Published: July 2010

We prove that for a large class of inductive limit

C
∗

{C^*}

-algebras, including inductive limits of finite direct sums of interval and circle algebras, any

∗
*

-isomorphism is induced from an approximate intertwining, in the sense of Elliott, between the inductive systems defining the algebras.
The loxodromic term of the Selberg trace formula for 𝑆𝐿(3,𝑍)\𝑆𝐿(3,𝑅)/𝑆𝑂(3,𝑅)
Authors: Wallace D. I.
Published: July 2010

In this paper we calculate the contribution to the trace formula of those orbital integrals coming from matrices in

SL
⁡
(
3
,

Z

)

\operatorname {SL} (3,\mathbb {Z})

with two complex eigenvalues and one real one, none of which are equal to zero. These correspond to mixed cubic number fields and will be seen to occur with multiplicity equal to the class number of a certain order in the number field.
Two new open mapping theorems for analytic multivalued functions
Authors: Baribeau Line , Harbottle Sarah
Published: July 2010

It is conjectured that the image

U
U

of a domain

D
D

, under an analytic multivalued function

K
K

, satisfies

U
∩
∂
U
⊂

∩

λ
∈
D

K
(
λ
)

U \cap \partial U \subset { \cap _{\lambda \in D}}K(\lambda )

if the values of

K
K

are polynomially convex sets. We prove that this conjecture is true in the two special cases when

K
K

is convex-valued and polar-valued.
An order-theoretic characterization of Peano continua
Authors: Holte Sarah E.
Published: July 2010

A characterization of Peano continua as images of dendrites is obtained which allows us to characterize Peano continua order-theoretically.
Commutator approximants
Authors: Maher P. J.
Published: July 2010

This paper deals with minimizing

|

|

B
−
(
A
X
−
X
A
)

|

|

p

||B - (AX - XA)|{|_p}

, where

A
A

and

B
B

are fixed,

B
∈

C

p

B \in {\mathcal {C}_p}

, and

X
X

varies such that

A
X
−
X
A
∈

C

p

AX - XA \in {\mathcal {C}_p}

. (Here,

C

p

{\mathcal {C}_p}

denotes the von Neumann-Schatten class and

‖
‖

p

{\left \| \right \|_p}

denotes its norm.) The main result (Theorem 3.2) says that if

A
A

is normal and

A
B
=
B
A

AB = BA

then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper B minus left-parenthesis upper A upper X minus upper X upper A right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript p Baseline comma 1 less-than-or-equal-to p greater-than normal infinity">

|

|

B
−
(
A
X
−
X
A
)

|

|

p

,
1
≤
p
>
∞

||B - (AX - XA)|{|_p},1 \leq p > \infty

, is minimized if and for

1
>
p
>
∞

1 > p > \infty

only if,

A
X
−
X
A
=
0

AX - XA = 0

; and that the map
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X right-arrow StartAbsoluteValue EndAbsoluteValue upper B minus left-parenthesis upper A upper X minus upper X upper A right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript p Superscript p Baseline comma 1 greater-than p greater-than normal infinity">

X
→

|

|

B
−
(
A
X
−
X
A
)

|

|

p
p

,
1
>
p
>
∞

X \to ||B - (AX - XA)||_p^p,1 > p > \infty

, has a critical point at

X
=
V

X = V

if and only if

A
V
−
V
A
=
0

AV - VA = 0

.
Groups with many rewritable products
Authors: Curzio Mario , Longobardi Patrizia , Maj Mercede , Rhemtulla Akbar
Published: July 2010

For any integer

n
≥
2

n \geq 2

, denote by

R
n

{R_n}

the class of groups

G
G

in which every infinite subset

X
X

contains

n
n

elements

x
1

,
…
,

x
n

{x_1}, \ldots ,{x_n}

such that the product

x
1

…

x
n

=

x

σ
(
1
)

⋯

x

σ
(
n
)

{x_1} \ldots {x_n} = {x_{\sigma (1)}} \cdots {x_{\sigma (n)}}

for some permutation

σ
≠
1

\sigma \ne 1

. The case

n
=
2

n = 2

was studied by B. H. Neumann who proved that

R
2

{R_2}

is precisely the class of centre-by-finite groups. Here we show that

G
∈

R
n

G \in {R_n}

for some

n
n

if and only if

G
G

contains an FC-subgroup

F
F

of finite index such that the exponent of

F

/

Z
(
F
)

F/Z(F)

is finite, where

Z
(
F
)

Z(F)

denotes the centre of

F
F

.
On the Rosenthal compacta and analytic sets
Authors: Krawczyk Adam
Published: July 2010

We consider pointwise convergence in a separable Rosenthal compactum. The main result is that if

X
⊂

R

ω
ω

X \subset {{\mathbf {R}}^{{\omega ^\omega }}}

is a Rosenthal compactum,

Y
⊂
X

Y \subset X

is countable dense and

x
∈
X

x \in X

, then the following are equivalent: (i)

Y
Y

has a countable base at

x
x

. (ii)

{
(

Y
i

)
∈

Y
ω

:

lim

i
→
∞

y
i

=
x
}

\{ ({Y_i}) \in {Y^\omega }:{\lim _{i \to \infty }}{y_i} = x\}

is analytic, when

Y
Y

has the discrete topology.
Free summands in maximal Cohen-Macaulay approximations and Eisenbud operators over hypersurface rings
Authors: Martsinkovsky Alex
Published: July 2010

The Eisenbud operator of a module over a complete hypersurface ring completely determines the delta invariants of this module.
A note on coherent rings of dimension two
Authors: Zhao Yi Cai
Published: July 2010

It is shown that a commutative coherent domain cannot have Ng dimension 2 and a commutative coherent regular ring of Ng dimension 2 cannot have finite indecomposable decomposition.
Compact and weakly compact homomorphisms between algebras of differentiable functions
Authors: González Manuel , Gutiérrez Joaquín M.
Published: July 2010

It is proved that the weakly compact continuous homomorphisms between algebras of

k
k

-times continuously differentiable functions on Banach spaces are induced by constant mappings.
Reflexivity of commutative subspace lattices
Authors: Haydon Richard
Published: July 2010

A short proof is given of Arveson’s reflexivity theorem for strongly closed commutative subspace lattices.
Finite groups in which the degrees of the nonlinear irreducible characters are distinct
Authors: Berkovich Yakov , Chillag David , Herzog Marcel
Published: July 2010

Finite groups in which all the nonlinear irreducible characters have equal degrees were described by Isaacs, Passman, and others. The purpose of this article is to consider the other extreme, namely, to characterize all finite groups in which all the nonlinear irreducible characters have distinct degrees.
The automorphism group of a function field
Authors: Madan Manohar , Rosen Michael
Published: July 2010

Let

k
k

be an algebraically closed field,

K
K

a function field in one variable over

k
k

, and

G
G

a nontrivial finite group. It is proven that there exist infinitely many Galois extensions

L

/

K

L/K

such that

Gal
⁡
(
L

/

K
)

\operatorname {Gal} (L/K)

is isomorphic to

G
G

, and

Gal
⁡
(
L

/

K
)
=

Aut
k

(
L
)

\operatorname {Gal} (L/K) = {\operatorname {Aut} _k}(L)

. This extends to arbitrary characteristic, a result first proven in the case

k
=

C

k = \mathbb {C}

by Greenberg in 1974.
Trigonometric polynomials and lattice points
Authors: Cilleruelo J. , Córdoba A.
Published: July 2010

In this paper we study the distribution of lattice points on arcs of circles centered at the origin. We show that on such a circle of radius

R
R

, an arc whose length is smaller than

2

R

1

/

2
−
1
(
4
[
m

/

2
]
+
2
)

\sqrt 2 {R^{1/2 - 1(4[m/2] + 2)}}

contains, at most,

m
m

lattice points. We use the same method to obtain sharp

L
4

{L^4}

-estimates for uncompleted, Gaussian sums
Imbedding of any vector field in a scalar semilinear parabolic equation
Authors: Poláčik P.
Published: July 2010

The scalar semilinear parabolic equation
\[

u
t

=
Δ
u
+
f
(
x
,
u
,
∇
u
)
,

x
∈
Ω
,

t
>
0
,

{u_t} = \Delta u + f(x,u,\nabla u),\quad x \in \Omega ,\quad t > 0,

\]
on a smooth bounded convex domain

Ω
⊂

R

N

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

under Neumann boundary condition (2)
\[

∂
u

∂
[
u
n
k
]

=
0

on 

∂
Ω

\quad \frac {{\partial u}}{{\partial [unk]}} = 0\quad {\text {on }}\partial \Omega

\]
is considered. For any prescribed vector field

H
H

on

R

N

{\mathbb {R}^N}

, a function

f
f

is found such that the flow of (1), (2) has an invariant

N
N

-dimensional subspace and the vector field generating the flow of (1), (2) on this invariant subspace coincides, in appropriate coordinates, with

H
H

.
A note on Krylov-Tso’s parabolic inequality
Authors: Escauriaza Luis
Published: July 2010

We show that if

u
u

is a solution to
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript i comma j equals 1 Overscript n Endscripts a Subscript i j Baseline left-parenthesis x comma t right-parenthesis upper D Subscript i j Baseline u left-parenthesis x comma t right-parenthesis minus upper D Subscript t Baseline u left-parenthesis x comma t right-parenthesis equals phi left-parenthesis x right-parenthesis">

∑

i
,
j
=
1

n

a

i
j

(
x
,
t
)

D

i
j

u
(
x
,
t
)
−

D
t

u
(
x
,
t
)
=
ϕ
(
x
)

\sum \nolimits _{i,j = 1}^n {{a_{ij}}(x,t){D_{ij}}u(x,t) - {D_t}u(x,t) = \phi (x)}

on a cylinder

Ω
T

=
Ω
×
(
0
,
T
)

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

, where

Ω
\Omega

is a bounded open set in

R

n

,
T
>
0

{{\mathbf {R}}^n},T > 0

, and

u
u

vanishes continuously on the parabolic boundary of

Ω
T

{\Omega _T}

. Then the maximum of

u
u

on the cylinder is bounded by a constant

C
C

depending on the ellipticity of the coefficient matrix

(

a

i
j

(
x
,
t
)
)

({a_{ij}}(x,t))

, the diameter of

Ω
\Omega

, and the dimension

n
n

times the

L
n

{L^n}

norm of

ϕ
\phi

in

Ω
\Omega

.
On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries
Authors: Escauriaza L. , Fabes E. B. , Verchota G.
Published: July 2010

We show that if

u
u

is a weak solution to

div
⁡
(
A
∇
u
)
=
0

\operatorname {div} (A\nabla u) = 0

on an open set

Ω
\Omega

containing a Lipschitz domain

D
D

, where

A
=
k
I

χ
D

+
I

χ

Ω

/

D

(
k
>
0
,
k
≠
1
)

A = kI{\chi _D} + I{\chi _{\Omega /D}}(k > 0,k \ne 1)

. Then, the nontangential maximal function of the gradient of

u
u

lies in

L
2

(
∂
D
)

{L^2}(\partial D)

.
Pseudocontraction and homotopy of the sin(1/𝑥) curve
Authors: Katsuura Hidefumi
Published: July 2010

We will prove that the space

sin
⁡
1

/

x

\sin 1/x

curve is not pseudocontractible using itself as the parameter space and that it has finitely many different homotopy equivalent classes of maps.
The first interval of stability of a periodic equation of Duffing type
Authors: Ortega Rafael
Published: July 2010

Consider the differential equation of a nonlinear oscillator with linear friction and a

T
T

-periodic external force. We find optimal bounds on the derivative of the restoring force in order to obtain a unique

T
T

-periodic solution that is asymptotically stable.
A characterization of weighted voting
Authors: Taylor Alan , Zwicker William
Published: July 2010

A simple game is a structure

G
=
(
N
,
W
)

G = (N,W)

where

N
=
{
1
,
…
,
n
}

N = \{ 1, \ldots ,n\}

and

W
W

is an arbitrary collection of subsets of

N
N

. Sets in

W
W

are called winning coalitions and sets not in

W
W

are called losing coalitions.

G
G

is said to be a weighted voting system if there is a function

w
:
N
→

R

w:N \to \mathbb {R}

and a "quota"

q
∈

R

q \in \mathbb {R}

so that

X
∈
W

X \in W

iff

∑

{
w
(
x
)
:
x
∈
X
}
≥
q

\sum {\{ w(x):x \in X\} \geq q}

. Weighted voting systems are the hypergraph analogue of threshold graphs. We show here that a simple game is a weighted voting system iff it never turns out that a series of trades among (fewer than

2

2
n

{2^{{2^n}}}

not necessarily distinct) winning coalitions can simultaneously render all of them losing. The proof is a self-contained combinatorial argument that makes no appeal to the separating of convex sets in

R

n

{\mathbb {R}^n}

or its algebraic analogue known as the Theorem of the Alternative.
Generalized first boundary value problem for Schrödinger equation
Authors: Ren Yan Xia
Published: July 2010

In this paper, we have obtained two main results by using probabilistic methods: (i) For a domain, we obtained a representation formula of the bounded solution to the first boundary value problem for Schrödinger equation; (ii) For

α
∈

R
1

\alpha \in {R^1}

, under certain conditions, we proved that the bounded solution having limit

α
\alpha

at infinity to the generalized first boundary value problem for Schrödinger equation exists and is unique, and it is represented in explicit formula. The results of this paper are generalizations of Chung and Rao.
Gaussian periods and units in certain cyclic fields
Authors: Lazarus Andrew J.
Published: July 2010

We analyze the property of period-unit integer translation (there exists a Gaussian period

η
\eta

and rational integer

c
c

such that

η
+
c

\eta + c

is a unit) in simplest quadratic, cubic, and quartic fields of arbitrary conductor. This is an extension of work of E. Lehmer, R. Schoof, and L. C. Washington for prime conductor. We also determine the Gaussian period polynomial for arbitrary conductor.
Zero-dimensionality in commutative rings
Authors: Gilmer Robert , Heinzer William
Published: July 2010

If

{

R
α

}

α
∈
A

{\left \{ {{R_\alpha }} \right \}_{\alpha \in A}}

is a family of zero-dimensional subrings of a commutative ring

T
T

, we show that

∩

α
∈
A

R
α

{ \cap _{\alpha \in A}}{R_\alpha }

is also zero-dimensional. Thus, if

R
R

is a subring of a zero-dimensional subring

[
u
n
k
]

T

[unk]\;T

(a condition that is satisfied if and only if a power of

r
T

rT

is idempotent for each

r
∈
R

r \in R

, then there exists a unique minimal zero-dimensional subring

R
0

{R^0}

of

T
T

containing

R
R

. We investigate properties of

R
0

{R^0}

as an

R
R

-algebra, and we show that

R
0

{R^0}

is unique, up to

R
R

-isomorphism, only if

R
R

itself is zero-dimensional.
Noncompact chain recurrence and attraction
Authors: Hurley Mike
Published: July 2010

Both this paper and Chain recurrence and attraction in noncompact spaces, [Ergodic Theory Dynamical Systems (to appear)] are concerned with the question of extending certain results obtained by C. Conley for dynamical systems on compact spaces to systems on arbitrary metric spaces. The basic result is the analogue of Conley’s theorem that characterizes the chain recurrent set of

f
f

in terms of the attractors of

f
f

and their basins of attraction. The point of view taken in the above-mentioned paper was that the given metric was of primary importance rather than the topology that it generated. The purpose of this note is to give results that depend on the topology induced by a metric rather than on the particular choice of the metric.
Global attractivity in nonlinear delay difference equations
Authors: Kocić V. Lj. , Ladas G.
Published: July 2010

We obtain a set of sufficient conditions under which all positive solutions of the nonlinear delay difference equation

x

n
+
1

=

x
n

f
(

x

n
−
k

)
,
n
=
0
,
1
,
2
,
…

{x_{n + 1}} = {x_n}f({x_{n - k}}),n = 0,1,2, \ldots

, are attracted to the positive equilibrium of the equation. Our result applies, for example, to the delay logistic model

N

t
+
1

=
α

N
t

/

(
1
+
β

N

t
−
k

)

{N_{t + 1}} = \alpha {N_t}/(1 + \beta {N_{t - k}})

and to the delay difference equation

x

n
+
1

=

x
n

e

r
(
1
−

x

n
−
k

)

{x_{n + 1}} = {x_n}{e^{r(1 - {x_{n - k}})}}

.
The Fourier-Bessel series representation of the pseudo-differential operator (-𝑥⁻¹𝐷)^{𝜈}
Authors: Singh O. P. , Pandey J. N.
Published: July 2010

For a certain Fréchet spaceFFconsisting of complex-valuedC∞{C^\infty }functions defined onI=(0,∞)I = (0,\infty )and characterized by their asymptotic behaviour near the boundaries, we show that: (I) The pseudo-differential operator(−x−1D)ν,ν∈R,D=d/dx{( - {x^{ - 1}}D)^\nu },\nu \in \mathbb {R},D = d/dx, is an automorphism (in the topological sense) onFF; (II)(−x−1D)ν{( - {x^{ - 1}}D)^\nu }is almost an inverse of the Hankel transformhν{h_\nu }in the sense that\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript nu Baseline ring left-parenthesis x Superscript negative 1 Baseline upper D right-parenthesis Superscript nu Baseline left-parenthesis phi right-parenthesis equals h 0 left-parenthesis phi right-parenthesis comma for-all phi element-of upper F comma for-all nu element-of double-struck upper R semicolon">hν∘(x−1D)ν(φ)=h0(φ),∀φ∈F,∀ν∈R;{h_\nu } \circ {({x^{ - 1}}D)^\nu }(\varphi ) = {h_0}(\varphi ),\quad \forall \varphi \in F,\quad \forall \nu \in \mathbb {R};\](III)(−x−1D)ν{( - {x^{ - 1}}D)^\nu }has a Fourier-Bessel series representation on a subspaceFb⊂F{F_b} \subset Fand also on its dualFb′{F’_b}.
On the Kostant convexity theorem
Authors: Ziegler François
Published: July 2010

A quick proof that the coadjoint orbits of a compact connected Lie group project onto convex polytopes in the dual of a Cartan subalgebra.
The iterated total squaring operation
Authors: Lomonaco Luciano
Published: July 2010

In this paper we prove a formula that expresses the iterated total squaring operation in terms of modular invariant theory and provide an alternative proof of a classical result of Múi’s.
Polydiscs and nontangential limits
Authors: GowriSankaran Kohur
Published: July 2010

A well-known result states that for all bounded

n
n

-harmonic functions on the polydisc

D

n

{\mathbb {D}^n}

the nontangential limits exist for (Lebesgue) almost every element of the

n
n

-torus. In this paper it is shown that a similar result is not in general valid for bounded quotients of two positive

n
n

-harmonic functions. Necessary and sufficient conditions on a

n
n

-harmonic function

u
>
0

u > 0

are given to ensure the existence "almost everywhere" of the nontangential limits of the quotients

w

/

u

w/u

in the case (i) for all

n
n

-harmonic functions

w
w

such that

w

/

u

w/u

is bounded and in the case (ii) for all

n
n

-harmonic functions

w
w

that are

u
u

-quasi-bounded.’
Locally complete intersection multiple structures on smooth algebraic curves
Authors: Boratyński M.
Published: July 2010

The aim of this paper is to characterize the class of the locally complete intersection multiple structures on smooth curves contained in a smooth three-dimensional variety.
Some remarks on the Daugavet equation
Authors: Wojtaszczyk P.
Published: July 2010

We show that direct sums in

l
1

{l_1}

and

l
∞

{l_\infty }

sense of Banach spaces that satisfy the Daugavet Equation (1) also satisfy it. We also show that most uniform algebras satisfy (1) for weakly compact operators.
A characterization of the sphere in terms of single-layer potentials
Authors: Shahgholian Henrik
Published: July 2010

Let

Ω
\Omega

be a bounded smooth domain in

R

n

{\mathbb {R}^n}

, and suppose the single-layer potential of

∂
Ω

\partial \Omega

coincides for

y
∉

Ω
¯

y \notin \overline \Omega

with the function

c

|

y

|

−
1

(
c
>
0
)

c|y{|^{ - 1}}(c > 0)

. Then

∂
Ω

\partial \Omega

is a sphere centered at the origin.
Nonlinear 𝑝-Laplacian problems on unbounded domains
Authors: Yu Lao Sen
Published: July 2010

We consider the

p
p

-Laplacian problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus d i v left-parenthesis a left-parenthesis x right-parenthesis StartAbsoluteValue nabla u EndAbsoluteValue Superscript p minus 2 Baseline nabla u right-parenthesis plus b left-parenthesis x right-parenthesis vertical-bar u StartAbsoluteValue u equals f left-parenthesis x comma u right-parenthesis comma x element-of normal upper Omega comma u EndAbsoluteValue Superscript p minus 2 Baseline Subscript partial-differential normal upper Omega Baseline equals 0 comma limit Underscript StartAbsoluteValue x EndAbsoluteValue right-arrow normal infinity Endscripts u equals 0 comma">

−
d
i
v
(
a
(
x
)

|

∇
u

|

p
−
2

∇
u
)
+
b
(
x
)

|

u

|

p
−
2

u
=
f
(
x
,
u
)
,

x
∈
Ω
,

u

|

∂
Ω

=
0
,

lim

|

x

|

→
∞

u
=
0
,

- div(a(x)|\nabla u{|^{p - 2}}\nabla u) + b(x)|u{|^{p - 2}}u = f(x,u),\quad x \in \Omega ,\quad u{|_{\partial \Omega }} = 0,\quad \lim \limits _{|x| \to \infty } u = 0,

\]
where

1
>
p
>
n
,
Ω
(
⊂

R
n

)

1 > p > n,\Omega ( \subset {R^n})

is an exterior domain. Under certain conditions, we show the existence of solutions for this problem via critical point theory.
Linear differential equations with exceptional fundamental sets. II
Authors: Steinmetz Norbert
Published: July 2010

We prove a sharp order estimate for entire functions of completely regular growth, whose zeros are distributed near finitely many raysarg⁡z=ωj\arg z = {\omega _j}in terms of the anglesωj{\omega _j}. This result then leads immediately to a proof of a conjecture of Hellerstein and Rossi concerning the distribution of zeros of the solutions of linear differential equations with polynomials coefficients.
Buzano’s inequality and bounds for roots of algebraic equations
Authors: Fujii Masatoshi , Kubo Fumio
Published: July 2010

A new bound for roots of algebraic equations will be given as a consequence of an inequality due to Buzano.
A min-max principle with a relaxed boundary condition
Authors: Ghoussoub N.
Published: July 2010

A standard Min-Max procedure to find critical points for a

C
1

{C^1}

functional

φ
\varphi

verifying a compactness condition of Palais-Smale type on a smooth Banach manifold

X
X

consists of finding an appropriate class

F

\mathcal {F}

of compact subsets of

X
X

, all containing a fixed boundary

B
B

, and then showing that the value

c
=

inf

A
∈

F

sup

x
∈
A

φ
(
x
)

c = {\inf _{A \in \mathcal {F}}}{\sup _{x \in A}}\varphi (x)

is a critical level, provided it satisfies

sup

φ
(
B
)
>
c

\sup \,\varphi (B) > c

. In this paper, we refine this procedure by relaxing the boundary condition.
Strong type endpoint bounds for analytic families of fractional integrals
Authors: Grafakos Loukas
Published: July 2010

In

R

2

{\mathbb {R}^2}

we consider an analytic family of fractional integrals, whose convolution kernel is obtained by taking some transverse derivatives of arclength measure on the parabola

(
t
,

t
2

)

(t,{t^2})

multiplied by

|

t

|

γ

|t{|^\gamma }

and doing so in a homogeneous way. We determine the exact range of

p
,

q

p,\;q

for which the analytic family maps

L
p

{L^p}

to

L
q

{L^q}

. We also resolve a similar issue on the Heisenberg group.
Numerical dynamics of ordinary differential equations with singularity
Authors: Hockett Kevin
Published: July 2010

We investigate the global dynamics of Euler’s method and second order Runge-Kutta when applied to certain nonlinear ordinary differential equations that possess a singularity. We show that the numerics admit spurious ‘chaotic’ dynamics in the form of subshifts of finite type with positive topological entropy independent of the choice of stepsize. We show that using a higher order method can, in fact, increase the topological entropy of the numerical dynamics. Techniques from complex analytic dynamics give some insight into this phenomenon.
A Whitney stratification and equisingular family of quasi-ordinary singularities
Authors: Ban Chunsheng
Published: July 2010

Let

(
V
,
0
)
⊂
(

C

d
+
1

,
0
)

(V,0) \subset ({{\mathbf {C}}^{d + 1}},0)

be a quasi-ordinary singularity and

π
:
(
V
,
0
)
→
(

C

d

,
0
)

\pi :(V,0) \to ({{\mathbf {C}}^d},0)

a quasi-ordinary projection.

C

d

{{\mathbf {C}}^d}

has a natural Whitney stratification given by the multiplicities of the discriminant locus of

π
\pi

. It is proved that the pullback of this stratification gives a Whitney stratification of

(
V
,
0
)

(V,0)

. Then using this result, an equisingular family of quasi-ordinary singularities is studied.
Structural monoids associated to equational varieties
Authors: Dehornoy Patrick
Published: July 2010

Monoids of elementary transformations are associated to certain equational varieties in order to describe the corresponding free objects; in some regular cases, these monoids reduce to groups, and the word problem for the free objects of the variety is connected with the existence of normal forms for the members of the associated group.
A note on solutions of the Schrödinger equation
Authors: Hansen Wolfhard
Published: July 2010

Using a suitable geometric series a simple proof for the continuity of

L
1

{L^1}

-solutions of the Schrödinger equation

L
u
:=
Δ
u
−
V
u

Lu: = \Delta u - Vu

(

V
V

being a Kato function) is obtained. It works as well for uniformly elliptic operators having Hölder continuous coefficients and Kato measures instead of Kato functions. Moreover, it is shown that results on singularities are immediate consequences of the local existence of continuous Green functions for

L
L

.
A note on Riesz potentials and the first eigenvalue
Authors: Chen Jie Cheng
Published: July 2010

In this paper, we consider the boundedness of Riesz potentials on positively curved manifolds. As an application, we get the greatest lower bound of the essential spectrum of a positively curved manifold.
More smoothly real compact spaces
Authors: Kriegl Andreas , Michor Peter W.
Published: July 2010

A topological space

X
X

is called

A

\mathcal {A}

-real compact if every algebra homomorphism from

A

\mathcal {A}

to the reals is an evaluation at some point of

X
X

, where

A

\mathcal {A}

is an algebra of continuous functions. Our main interest lies on algebras of smooth functions. Arias-de-Reyna has shown that any separable Banach space is smoothly real compact. Here we generalize this result to a huge class of locally convex spaces including arbitrary products of separable Fréchet spaces.
The approximation order of box spline spaces
Authors: Ron A. , Sivakumar N.
Published: July 2010

Let

M
M

be a box spline associated with an arbitrary set of directions and suppose that

S
(
M
)

S(M)

is the space spanned by the integer translates of

M
M

. In this note, the subspace of all polynomials in

S
(
M
)

S(M)

is shown to be the joint kernal of a certain collection of homogeneous differential operators with constant coefficients. The approximation order from the dilates of

S
(
M
)

S(M)

to smooth functions is thereby characterized. This extends a well-known result of de Boor and Höllig (

B
B

-splines from parallelepipeds, J. Analyse Math. 42 (1982/83), 99-115), on box splines with integral direction sets. The argument used is based on a new relation, valid for any compactly supported distribution

ϕ
\phi

, between the semidiscrete convolution

ϕ

∗
′

\phi \ast ’

and the distributional convolution

ϕ
∗

\phi \ast

.
Unconditional convergence and weakly compact maps
Authors: Díaz Santiago
Published: July 2010

We characterize the barrelled spaces

E
E

such that every linear continuous map from

E
E

to

l
1

{l_1}

carries bounded sets into relatively compact sets. These characterizations involve unconditional convergence of series, Edgar’s ordering for Banach spaces, perfect sequence spaces, copies of

c
0

{c_0}

, and Orlicz-Pettis topologies.
On products of polynomials
Authors: Masser D. W. , Wolbert J.
Published: July 2010

For a complex polynomial

P
P

in a single variable, let

H
(
P
)

H(P)

be the maximum of the absolute values of its coefficients. Given nonnegative integers

n
1

{n_1}

and

n
2

{n_2}

, it is well known that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu left-parenthesis n 1 comma n 2 right-parenthesis equals inf upper H left-parenthesis upper P 1 upper P 2 right-parenthesis slash left-parenthesis upper H left-parenthesis upper P 1 right-parenthesis upper H left-parenthesis upper P 2 right-parenthesis right-parenthesis greater-than 0">

μ
(

n
1

,

n
2

)
=
inf

H
(

P
1

P
2

)

/

(
H
(

P
1

)
H
(

P
2

)
)
>
0

\mu ({n_1},{n_2}) = \inf \,H({P_1}{P_2})/(H({P_1})H({P_2})) > 0

, where the infimum is taken over all such polynomials

P
1

{P_1}

and

P
2

{P_2}

of degrees

n
1

{n_1}

and

n
2

{n_2}

respectively. We determine here the exact values of

μ
(
1
,
n
)

\mu (1,n)

for every

n
n

.
Integrable mean periodic functions on locally compact abelian groups
Authors: Rana Inder K. , Navada K. Gowri
Published: July 2010

Let

G
G

be a locally compact abelian group with a Haar measure

λ
G

{\lambda _G}

. A function

f
f

on

G
G

is said to be mean-periodic if there exists a nonzero finite regular measure

μ
\mu

of compact support on

G
G

such that

f

∗

μ
=
0

f{\ast }\mu = 0

. It is known that there exist no nontrivial integrable mean periodic functions on

R

n

{{\mathbf {R}}^n}

. We show that there exist nontrivial integrable mean periodic functions on

G
G

provided

G
G

has nontrivial proper compact subgroups. Let

f
∈

L
1

(
G
)

f \in {L_1}(G)

be mean periodic with respect to a nonzero finite measure

μ
\mu

of compact support. If

μ
(
G
)
≠
0

\mu (G) \ne 0

and

λ
G

(
supp
⁡
(
μ
)
)
>
0

{\lambda _G}(\operatorname {supp} (\mu )) > 0

, then there exists a compact subgroup

K
K

of

G
G

such that

f

∗

λ
K

=
0

f{\ast }{\lambda _K} = 0

, i.e.,

f
f

is mean periodic with respect to

λ
K

{\lambda _K}

, where

λ
K

{\lambda _K}

denotes the normalized Haar measure of

K
K

. When

G
G

is compact, abelian and meterizable, we show that there exists continuous (hence integrable and almost periodic) functions on

G
G

that are not mean periodic.
Shapes of stars
Authors: Kojima Sadayoshi , Yamashita Yasushi
Published: July 2010

In this paper we construct a natural geometric structure for the space of shapes of a star-shaped polygon. Roughly speaking we find: The set of similarity classes of marked stars forms a hyperbolic right angle pentagon bundle over the space of external angle sets of inscribed pentagons. The assignment of the shape of its fiber to each angle set forms a hyperbolic plane bundle over the Teichmüller space of hyperbolic right angle pentagons. Any automorphism induced by renumbering is compatible with these geometric structures.
The bounded and thin Whitehead theorems
Authors: Anderson Douglas R. , Munkholm Hans Jørgen
Published: July 2010

This paper deals with finite-dimensional CW complexes equipped with reference maps to a fixed metric space and maps between such complexes that respect the reference maps up to a bounded distortion. We prove two Whitehead Theorems for such maps

f
f

. The Bounded Whitehead Theorem allows one to decide whether

f
f

is a bounded homotopy equivalence. The Thin Whitehead Theorem allows one to decide when a map of bound zero admits homotopy inverses of arbitrarily small bound (also on the homotopies). Both theorems come in two versions: One that deals with homotopy in all dimensions; one where homotopy in dimensions at least two is replaced by homology of "universal covers".
All meager filters may be null
Authors: Bartoszyński Tomek , Goldstern Martin , Judah Haim , Shelah Saharon
Published: July 2010

We show that in the Cohen model the sum of two nonmeasurable sets is always nonmeager. As a consequence we show that it is consistent with ZFC that all filters which have the Baire property are Lebesgue measurable. We also show that the existence of a Sierpinski set implies that there exists a nonmeasurable filter which has the Baire property.
Interspersions and dispersions
Authors: Kimberling Clark
Published: July 2010

An array

A
=
(

a

i
j

)

A = ({a_{ij}})

of all the positive integers is an interspersion if the terms of any two rows, from some point on, alternate in size, and a dispersion if, for a suitable sequence

(

s
n

)

({s_n})

, the recurrence

a
j

=

s

a

j
−
1

{a_j} = {s_{{a_{j - 1}}}}

holds for each entry

a
j

{a_j}

of each row of

A
A

, for

j
⩾
2

j \geqslant 2

. An array is proved here to be an interspersion if and only if it is a dispersion. Such arrays whose rows satisfy certain recurrences are considered.
New proofs of weighted inequalities for the one-sided Hardy-Littlewood maximal functions
Authors: Martín-Reyes F. J.
Published: July 2010

In this note we give a simple proof of the characterization of the weights for which the one-sided Hardy-Littlewood maximal functions apply

L
p

(
V
)

{L^p}(V)

into weak-

L
p

(
U
)

{L^p}(U)

and a direct proof of the characterization of the weights for which the one-sided Hardy-Littlewood maximal functions apply

L
p

(
W
)

{L^p}(W)

into

L
p

(
W
)

{L^p}(W)

.
On isolated points of the spectrum of a bounded linear operator
Authors: Schmoeger Christoph
Published: July 2010

For a bounded linear operator

A
A

on a Banach space we characterize the isolated points in the spectrum of

A
A

, the Riesz points of

A
A

, and the poles of the resolvent of

A
A

.
A characterization of the uniquely ergodic endomorphisms of the circle
Authors: Alsedà Lluís , Mañosas Francesc , Szlenk Wiesław
Published: July 2010

We characterize the uniquely ergodic endomorphisms of the circle in terms of their periodic orbits.
Multipliers on subsemigroups of the real line
Authors: Dinh Hung T.
Published: July 2010

We show that multipliers on subsemigroups of the real line can be extended to multipliers on groups.
Examples of Buchsbaum quasi-Gorenstein rings
Authors: Herrmann Manfred , Trung Ngô Viet
Published: July 2010

The paper shows the existence of Buchsbaum quasi-Gorenstein rings of any admissible depth.
The reduction number of an ideal and the local cohomology of the associated graded ring
Authors: Marley Thomas
Published: July 2010

Let

(
R
,
m
)

(R,m)

be a local ring and

I
I

an

m
m

-primary ideal. A result of Trung shows that if the local cohomology of

g

r
I

(
R
)

g{r_I}(R)

satisfies certain conditions, then the reduction number of

I
I

is independent of the minimal reduction chosen. These conditions consist of

t
=
dim
⁡
R
−
grade

g

r
I

(
R

)
+

t = \dim R - \operatorname {grade} \;g{r_I}{(R)^ + }

inequalities. We show that if

R
R

is Cohen-Macaulay, then one of these inequalities is always satisied, while another can often be easily checked. Applications are then given in two-dimensional Cohen-Macaulay rings. For instance, we show that if the Hilbert function of

I
I

equals the Hilbert polynomial of

I
I

for all integers greater than

1
1

, then the reduction number is independent of the choice of minimal reduction.
An example of a harmonic map between Euclidean balls
Authors: Melas Antonios D.
Published: July 2010

Here we construct a homeomorphism of the unit sphere in three-dimensional Euclidean space to itself whose harmonic extension in the unit ball is not a diffeomorphism.
On purely infinite 𝐶*-algebras
Authors: Jeong Ja A , Lee Sa Ge
Published: July 2010

We find conditions under which a quotient

C

∗

{C^{\ast }}

-algebra

A

/

I

A/I

of a purely infinite

C

∗

{C^{\ast }}

-algebra

A
A

becomes purely infinite. S. Zhang proved that if

A
A

is a

σ
\sigma

-unital, simple, purely infinite

C

∗

{C^{\ast }}

-algebra, then a hereditary

C

∗

{C^{\ast }}

-subalgebra

A
x

{A_x}

is a stable or unital for each

x
x

of

A
+

{A_ + }

. We prove the converse for a completely

σ
\sigma

-unital infinite simple

C

∗

{C^{\ast }}

-algebra.
Notes on 𝜋-quasi-normal subgroups in finite groups
Authors: Ren Yong Cai
Published: July 2010

Let

G
G

be a finite group and let

π
\pi

be a set of primes. A subgroup

H
H

of

G
G

is called

π
\pi

-quasi-normal in

G
G

if

H
H

permutes with every Sylow

p
p

-subgroup of

G
G

for every

p
p

in

π
\pi

. In this paper, we investigate how

π
\pi

-quasi-normality conditions on some subgroups of

G
G

affect the structure of

G
G

.