bluuebunny/crossref_metadata_2025 huggingface dataset:
On the hyperbolicity and the Schottky property of complex spaces
Authors: Thai Do Duc
Published: July 2010

The purpose of this paper is to study the hyperbolicity and the tautness of spaces that have the Schottky property. Moreover, the hyperbolicity of compact complex spaces is characterized by the classical theorem of Bloch.
A correction: “A monotone principle of fixed points” [Proc. Amer. Math. Soc. 94 (1985), no. 3, 427–432; MR0787887 (87c:54065)] by M. R. Tasković
Authors: Turinici Mihai
Published: July 2010

Some technical aspects of the main result in Taskovic’s A monotone principle of fixed points [Proc. Amer. Math. Soc. 94 (1985), 427-432] are investigated.
Differential equations with nonlinear boundary conditions
Authors: Fečkan Michal
Published: July 2010

This paper shows the existence of multiple unbounded branches of solutions for certain equations via the Nielsen fixed-point theory.
A factorization constant for 𝑙ⁿ_{𝑝}, 0<𝑝<1
Authors: Peck N. T.
Published: July 2010

We prove that if PT is a factorization of the identity operator on

ℓ
p
n

\ell _p^n

through

ℓ
∞
k

,
0
>
p

\ell _\infty ^k,0 > p

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper P double-vertical-bar double-vertical-bar upper T double-vertical-bar greater-than-or-equal-to upper C n Superscript 1 slash p minus 1 slash 2 Baseline left-parenthesis log n right-parenthesis Superscript negative 1 slash 2">

‖
P
‖

‖
T
‖

≥
C

n

1

/

p
−
1

/

2

(
log
⁡
n

)

−
1

/

2

\left \| P \right \| \left \| T \right \| \geq C{n^{1/p - 1/2}}{(\log n)^{ - 1/2}}

. This is a corollary of a more general result on factoring the identity operator on a quasi-normed space X through

ℓ
∞
k

\ell _\infty ^k

.
A construction of modular representations of classical Lie algebras
Authors: Peters Karl M. , Shi Zhiyong
Published: July 2010

In this paper, we construct and analyze new classes of modular representations of classical Lie algebras of type C and type A. These representations include a class of pointed torsion free representations, a class of irreducible nonrestricted representations, and a class of indecomposable representations of arbitrary high dimension. The construction is based on the realization of these Lie algebras in the modular Weyl algebras acting on truncated polynomial algebras. We also classify all the irreducible representations of the modular Weyl algebra.
Primes in arithmetic progression and uniform distribution
Authors: Wodzak M. A.
Published: July 2010

Let f be an entire, non-polynomial function with real coefficients. Let

p
~

\tilde p

run through the primes such that

p
~

≡
h
mod
k
,
(
h
,
k
)
=
1

\tilde p \equiv h \bmod k,(h,k) = 1

, and let

M
>
1

M > 1

. If there exists

α
∈
(
1
,
4

/

3
)

\alpha \in (1,4/3)

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue less-than-or-equal-to exp left-parenthesis left-parenthesis log StartAbsoluteValue z EndAbsoluteValue right-parenthesis Superscript alpha Baseline right-parenthesis">

|

f
(
z
)

|

≤
exp
⁡
(

(
log
⁡

|

z

|

)
α

)

|f(z)| \leq \exp ({(\log |z|)^\alpha })

for all

|

z

|

>
M

|z| > M

, then the sequence

{
f
(

p
~

)
}

\{ f(\tilde p)\}

is uniformly distributed modulo 1.
Some more Mathieu group coverings in characteristic two
Authors: Abhyankar Shreeram S. , Yie Ikkwon
Published: July 2010

Explicit equations are given for unramified coverings of the affine line in characteristic two with Mathieu groups of degrees 23 and 24 as Galois groups.
A note on the 𝑀₂₃- and 𝐹𝑖₂₃-minimal parabolic geometries
Authors: Rowley Peter , Walker Louise
Published: July 2010

Characterizations of the minimal parabolic geometries for the sporadic simple groups

M

23

{{\text {M}}_{23}}

and

Fi

23

{\text {Fi}}_{23}

are presented.
Profinite groups with restricted centralizers
Authors: Shalev Aner
Published: July 2010

Let G be a profinite group in which every centralizer

C
G

(
x
)

(
x
∈
G
)

{C_G}(x)\;(x \in G)

is either finite or of finite index. It is shown that G is finite-by-abelian-by-finite. Moreover, if, in addition, G is a just-infinite pro-p group, then it has the structure of a p-adic space group whose point group is cyclic or generalized quaternion.
Generators for all principal congruence subgroups of 𝑆𝐿(𝑛,𝑍) with 𝑛≥3
Authors: Sury B. , Venkataramana T. N.
Published: July 2010

We show that there is a uniform bound for the numbers of generators for all principal congruence subgroups of

SL

(
n
,
Z
)

{\text {SL}}(n,Z)

for

n
≥
3

n \geq 3

. On the other hand, we show that the numbers are unbounded if we work with all arithmetic subgroups of

SL

(
n
,
Z
)

{\text {SL}}(n,Z)

.
Observing a preferred distribution
Authors: Dinwoodie I. H.
Published: July 2010

We give asymptotic results for the number of empirical laws that one must observe in order for one of them to be close to a certain distribution.
A curve for which Coleman’s effective Chabauty bound is sharp
Authors: Grant David
Published: July 2010

We show that Coleman’s effective Chabauty bound is sharp for the curve

C
:

y
2

=
x
(
x
−
1
)
(
x
−
2
)
(
x
−
5
)
(
x
−
6
)

C:{y^2} = x(x - 1)(x - 2)(x - 5)(x - 6)

defined over

Q

\mathbb {Q}

, by considering its reduction

mod

7

\bmod \;7

. We also show that the Jacobian of C is absolutely simple.
Minimal tori in 𝑆²×𝑆¹
Authors: Hsieh Chun Chung , Wang Ai Nung
Published: July 2010

Choi and Schoen (1985) have shown that for a positively Ricci curved three-fold M, the family of embedded minimal surfaces of a fixed genus is compact. It is of interest to know how much the positivity condition can be relaxed.
A complement theorem in the universal Menger compactum
Authors: Sher R. B.
Published: July 2010

A. Chigogidze has shown that two Z-sets in the universal Menger compactum of dimension

k
+
1

k + 1

have the same k-shape if and only if their complements are homeomorphic. We show that this result holds for weak Z-sets. The class of weak Z-sets, defined herein and analogous to the weak Z-sets in Q, contains but is larger than the class of Z-sets. We give some examples of weak Z-sets in the universal Menger compactum and in Q that are not Z-sets.
The representation of homeomorphisms on the interval as finite compositions of involutions
Authors: Young Sam W.
Published: July 2010

It is known that every homeomorphism on the interval is the composition of at most four involutions and that every decreasing homeomorphism is the composition of at most three involutions. We will characterize those homeomorphisms that are the composition of two involutions. It is then demonstrated that there exist three involutions that together generate a dense subgroup of the topological group of all homeomorphisms on the interval.
A construction related to the cosine problem
Authors: Kolountzakis Mihail N.
Published: July 2010

We give a constructive proof of the fact that for any sequence of positive integers

n
1

,

n
2

,
…
,

n
N

{n_1},{n_2}, \ldots ,{n_N}

there is a subsequence

m
1

,
…
,

m
r

{m_1}, \ldots ,{m_r}

for which
\[

−

min
x

∑
1
r

cos
⁡

m
j

x
≥
C
N
,

- \min \limits _x \sum \limits _1^r {\cos {m_j}x \geq CN,}

\]
where C is a positive constant. Uchiyama previously proved the above inequality with the right-hand side replaced by

C

N

C\sqrt N

. We give a polynomial time algorithm for the selection of the subsequence

m
j

{m_j}

.
An explicit formula for the Picard group of the cyclic group of order 𝑝²
Authors: Stolin Alexander
Published: July 2010

We give a formula for the Picard group of the integer group ring of the cyclic group of order

p
2

{p^2}

for any odd prime p. As a corollary one gets a formula for properly irregular prime p in terms of Bernoulli numbers.
On the absolutes of compact spaces with a minimally acting group
Authors: Bandlow Ingo
Published: July 2010

If an

ω
\omega

-bounded group G acts continuously on a compact Hausdorff space X and the orbit of every point is dense in X, then X is coabsolute to a Cantor cube.
Special points in compact spaces
Authors: Bell Murray
Published: July 2010

Given a collection

C

\mathcal {C}

, of cardinality

κ
\kappa

, of subsets of a compact space X, we prove the existence of a point x such that whenever

C
∈

C

C \in \mathcal {C}

and

X
∈

C
¯

X \in \bar C

, there exists a

G
λ

{G_\lambda }

-set Z with

λ
>
κ

\lambda > \kappa

and

x
∈
Z
⊂

C
¯

x \in Z \subset \bar C

. We investigate the case when

C

\mathcal {C}

is the collection of all cozerosets of X and also when X is a dyadic space. We apply this result to homogeneous compact spaces. Another application is a characterization of

2

ω
1

{2^{{\omega _1}}}

among dyadic spaces.
Branched surfaces and Thurston’s norm on homology
Authors: Tollefson Jeffrey L. , Wang Ningyi
Published: July 2010

Given a closed, irreducible, orientable 3-manifold M, let x denote the Thurston norm on

H
2

(
M
;
R
)

{H_2}(M;R)

. Suppose g, h, and f are three homology classes of

H
2

(
M
;
Z
)

{H_2}(M;Z)

carried by a single face of the x-unit sphere in

H
2

(
M
;
R
)

{H_2}(M;R)

. In this paper it is shown that there exists a taut, oriented branched surface carrying representatives of g and h and a semi-taut oriented branched surface carrying representatives of all three homology classes.
Elliptic genera of level 𝑁 and Jacobi polynomials
Authors: von Oehsen J. Barr
Published: July 2010

In this work, we study Hirzebruch’s level N elliptic genera and show that the image of the complex projective spaces under the level 3 genus can be realized very compactly in terms of Jacobi polynomials. To obtain these results we examine a differential equation which the level 3 logarithm satisfies.
Detecting negative curvature in groups via local conditions
Authors: Shawcroft Paul
Published: July 2010

We characterize negative curvature of groups in terms of properties which can be checked locally in the Cayley graph. We prove the equivalence of these properties with the more familiar thin triangles property. We also present an algorithm which, when given a finite presentation of a group G, will recognize these properties in finite time if G is negatively curved.
A fixed-point theorem for central elements in quasisimple groups
Authors: Blau Harvey I.
Published: July 2010

It is shown that with a few exceptions (which are listed), any central element in a quasisimple finite group fixes some conjugacy class of the group under multiplication.
The Lyapounov central limit theorem for factorizable arrays
Authors: Wesołowski Jacek
Published: July 2010

A sort of the Lyapounov central limit theorem for row-wise factorizable triangular arrays is obtained. Also a new version of the classical Lyapounov theorem for independent random variables, being a tool in the proof of the main result, seems to be of independent interest.
A quasi-commutative ring that is not neo-commutative
Authors: Kaplansky Irving
Published: July 2010

An example of a ring that is quasi-commutative but not neo-commutative is given.
The maximal ideal space of 𝑙𝑖𝑝_{𝐴}(𝑋,𝛼)
Authors: Mahyar H.
Published: July 2010

Let X be a compact subset of the complex plane

C

\mathbb {C}

, and let

0
>
α
>
1

0 > \alpha > 1

. We show that the maximal ideal space of

lip
A

⁡
(
X
,
α
)

\operatorname {lip}_A(X,\alpha )

is X.
The infimal value functional and the uniformization of hit-and-miss hyperspace topologies
Authors: Beer Gerald , Tamaki Robert
Published: July 2010

We give necessary and sufficient conditions for the uniformizability of hit-and-miss and proximal hit-and-miss hyperspace topologies defined on the nonempty closed subsets

CL

(
X
)

{\text {CL}}(X)

of a Hausdorff uniform space

⟨
X
,

U

⟩

\langle X,\mathcal {U}\rangle

. In the case of uniformizability, one can always find a family

F

\mathcal {F}

of continuous functions on X into [0, 1] so that the hyperspace topology is the weak topology induced by

{

m
f

:
f
∈

F

}

\{ {m_f}:f \in \mathcal {F}\}

, where for each f,

m
f

:

CL

(
X
)
→
[
0
,
1
]

{m_f}:{\text {CL}}(X) \to [0,1]

is the infimal value functional defined by

m
f

(
A
)
=
inf
{
f
(
x
)
:
x
∈
A
}

{m_f}(A) = \inf \{ f(x):x \in A\}

.
All autohomeomorphisms of connected Menger manifolds are stable
Authors: Sakai Katsuro
Published: July 2010

Generalizing the result of Chigogidze (1991), we prove that all autohomeomorphisms of connected Menger manifolds are stable (in the sense of Brown and Gluck).
Towers on trees
Authors: Goldstern Martin , Johnson Mark J. , Spinas Otmar
Published: July 2010

We show that (under MA) for any

c

\mathfrak {c}

many dense sets in Laver forcing

L

\mathbb {L}

there exists a

σ
\sigma

-centered

Q
⊆

L

Q \subseteq \mathbb {L}

such that all the given dense sets are dense in Q. In particular, MA implies that

L

\mathbb {L}

satisfies MA and does not collapse the continuum and the additivity of the Laver ideal is the continuum. The same is true for Miller forcing and for Mathias forcing. In the case of Miller forcing this involves the correction of the wrong proof of Judah, Miller, and Shelah, Sacks, Laver forcing, and Martin’s Axiom, Arch. Math. Logic 31 (1992), Theorem 4.1, p. 157.
Fixed points arising only in the growth of first countable spaces
Authors: Watson Stephen
Published: July 2010

We construct a Tychonoff first countable space X and an autohomeomorphism f with no fixed points (either a translation or a reflection) such that

β
f

\beta f

does have a fixed point answering a question of Krawczyk and Steprāns. We do this by replacing each point of Mrowka’s construction of a first countable space whose growth has size one with a copy of the integers which can be translated.
Zero entropy and directional Bernoullicity of a Gaussian 𝑍²-action
Authors: Ferenczi S. , Kamiński B.
Published: July 2010

We give an example of a Gaussian

Z

2

{\mathbb {Z}^2}

-action

Φ
\Phi

with zero entropy which is weakly mixing, rigid and such that every non-trivial measure-preserving transformation

Φ
g

{\Phi ^g}

defined by

Φ
,
g
∈

Z

2

\Phi , g \in {\mathbb {Z}^2}

, is a Bernoulli shift with infinite entropy.
Exceptional invariants in the parabolic invariant theory of conformal geometry
Authors: Bailey Toby N. , Gover A. Rod
Published: July 2010

We provide a construction for the exceptional invariants of certain modules for a parabolic subgroup of a pseudo-orthogonal group. The invariant theory of these modules has applications in conformal geometry.
Local derivations of nest algebras
Authors: Han De Guang , Wei Shu Yun
Published: July 2010

Let X be an arbitrary reflexive Banach space, and let

N

\mathcal {N}

be a nest on X. Denote by

D

(

N

)

\mathcal {D}(\mathcal {N})

the set of all derivations from

Alg
⁡

N

\operatorname {Alg}\mathcal {N}

into

Alg
⁡

N

\operatorname {Alg}\mathcal {N}

. For

N
⊂

N

N \subset \mathcal {N}

, we set

N
−

=
∨
{
M
∈

N

:
M
⊂
N
}

{N_ - } = \vee \{ M \in \mathcal {N}:M \subset N\}

. We also write

0
−

=
0

{0_ - } = 0

. Finally, for

E
,
F
∈

N

E, F \in \mathcal {N}

define

(
E
,
F
]
=
{
K
∈

N

:
E
⊂
K
⊆
F
}

(E,F] = \{ K \in \mathcal {N}:E \subset K \subseteq F\}

. In this paper we prove that a sufficient condition for

D

(

N

)

\mathcal {D}(\mathcal {N})

to be (topologically) algebraically reflexive is that for all

0
≠
E
∈

N

0 \ne E \in \mathcal {N}

and for all

X
≠
F
∈

N

X \ne F \in \mathcal {N}

, there exist

M
∈
(
0
,
E
]

M \in (0,E]

and

N
∈
(
F
,
X
]

N \in (F,X]

, such that

M
−

⊂
M

{M_ - } \subset M

and

N
−

⊂
N

{N_ - } \subset N

. In particular, we prove that this condition automatically holds for nests acting on finite-dimensional Banach spaces.
Admissibility, the locally convex approximation property, and the 𝐴𝑅-property in linear metric spaces
Authors: Nhu Nguyen To
Published: July 2010

We introduce the notion of the locally convex approximation property (the LCAP) for convex sets in linear metric spaces. The LCAP is an extension of the notion of admissibility of Klee. We prove that any convex set with the LCAP is an AR.
Filters and games
Authors: Bartoszyński Tomek , Scheepers Marion
Published: July 2010

We obtain game-theoretic characterizations for meagerness and rareness of filters on

ω
\omega

.
On disguised inverted Wishart distribution
Authors: Gupta A. K. , Ofori-Nyarko S.
Published: July 2010

Let

A
∼

W
p

(
n
,
Σ
)

A \sim {W_p}(n,\Sigma )

and

A
=
Z

Z
′

A = ZZ’

where Z is a lower triangular matrix with positive diagonal elements. Further, let

B
=

A

−
1

=

W
′

W

B = {A^{ - 1}} = W’W

have inverted Wishart distribution so that

W
=

Z

−
1

W = {Z^{ - 1}}

. In this paper we derive the distribution of

M
=
W
Σ

W
′

M = W\Sigma W’

. It is also shown that

n
−
p
+
1

n
p

T
′

M
T
∼

F

p
,
n
−
p
+
1

\frac {{n - p + 1}}{{np}}T’MT \sim {F_{p,n - p + 1}}

where

T
∼

N
p

(
0
,

I
p

)

T \sim {N_p}(0,{I_p})

is independent of M.
Nonextendability of the Bers isomorphism
Authors: Zhang Chaohui
Published: July 2010

Let G be a torsion free finitely generated Fuchsian group of the first kind of type (p, n). The purpose of this paper is to show that the Bers isomorphism of the Bers fiber space

F
(
G
)

F(G)

onto the Bers embedding of

T
(

G
˙

)

T(\dot G)

has no continuous extension to the boundary, provided that

dim
⁡
T
(
G
)
≥
1

\dim T(G) \geq 1

, where Ġ is another torsion free finitely generated Fuchsian group of the first kind of type

(
p
,
n
+
1
)

(p,n + 1)

.
The noncommutativity of Hecke algebras associated to Weyl groups
Authors: Anderson Michael R.
Published: July 2010

We prove that the Hecke algebra

H

(
W
,

W
J

)

\mathcal {H}(W,{W_J})

, where W is a Weyl group of spherical type and

W
J

{W_J}

is a standard parabolic subgroup of W of corank

≥
2

\geq 2

, is noncommutative.
Criterion for the resolvent set of nonsymmetric tridiagonal operators
Authors: Aptekarev A. I. , Kaliaguine V. , Van Assche W.
Published: July 2010

We study nonsymmetric tridiagonal operators acting in the Hilbert space

ℓ
2

{\ell ^2}

and describe the spectrum and the resolvent set of such operators in terms of a continued fraction related to the resolvent. In this way we establish a connection between Padé approximants and spectral properties of nonsymmetric tridiagonal operators.
On submanifolds with harmonic mean curvature
Authors: Barros Manuel , Garay Oscar J.
Published: July 2010

The classification of curves in

E
m

{E^m}

with harmonic mean curvature vector field in the normal bundle is obtained and then it is used to obtain some applications.
Bounds for the Betti numbers of generalized Cohen-Macaulay ideals
Authors: Hoa Lê Tuan , Miró-Roig Rosa M.
Published: July 2010

Upper bounds for the Betti numbers of generalized Cohen-Macaulay ideals are given. In particular, for the case of non-degenerate, reduced and irreducible projective curves we get an upper bound which only depends on their degree.
A residue map and its applications to some one-dimensional rings
Authors: Huang I-Chiau
Published: July 2010

A residue map is used to study canonical modules of the ring

k
[
[

X

t
1

,
…
,

X

t
n

]
]

k[[{X^{{t_1}}}, \ldots ,{X^{{t_n}}}]]

. A simple proof of a well-known numerical criterion for

k
[
[

X

t
1

,
…
,

X

t
n

]
]

k[[{X^{{t_1}}}, \ldots ,{X^{{t_n}}}]]

to be Gorenstein is given.
A random Banach-Steinhaus theorem
Authors: Velasco M. V. , Villena A. R.
Published: July 2010

In an earlier paper, we began a study of linear random operators which have a certain probability of behaving as continuous operators. In this paper we study the pointwise limit in probability of a sequence of such operators, extending the Banach-Steinhaus theorem in a stochastical sense.
Rings with flat socles
Authors: Xiao Yu Fei
Published: July 2010

We show that the class of rings with flat socles is closed under the formation of polynomial extensions, direct products, and excellent extensions. We also point out that if a ring R can be embedded into a semisimple Artinian ring, then all finitely generated flat R-modules are projective.
A note on the generalized dumbbell problem
Authors: Anné Colette
Published: July 2010

This note is devoted to the calculation of the asymptotics of the small eigenvalues and corresponding eigenfunctions for the Laplace operator with Neumann boundary conditions on a domain obtained by adding several thin channels between given bounded domains.
Younger mates and the Jacobian conjecture
Authors: Cheng Charles Ching-an , McKay James H. , Wang Stuart Sui Sheng
Published: July 2010

Let

F
,
G
∈

C

[
x
,
y
]

F,G \in \mathbb {C}[x,y]

. If the Jacobian determinant of F and G is 1, then G is said to be a Jacobian mate of F. If, in addition, G has degree less than that of F, then G is said to be a younger mate of F. In this paper, a necessary and sufficient condition is given for a polynomial to have a younger mate. This also gives rise to a formula for the younger mate if it exists. Furthermore, a conjecture concerning the existence of a younger mate is shown to be equivalent to the Jacobian conjecture.
Variation of the discrete eigenvalues of normal operators
Authors: Elsner L. , Friedland S.
Published: July 2010

The Hoffman-Wielandt inequality, which gives a bound for the distance between the spectra of two normal matrices, is generalized to normal operators A, B on a separable Hilbert space, such that

A
−
B

A - B

is Hilbert-Schmidt.
Eigenvalues of Laplacians on a closed Riemannian manifold and its nets
Authors: Fujiwara Koji
Published: July 2010

We show that the eigenvalues of the Laplacian of a closed manifold M is approximated in a certain sense by the eigenvalues of the Laplacian of the graph of a

1
n

\frac {1}{n}

-net in M as

n
→
∞

n \to \infty

. Our approximation needs no assumption on M except for dimension.
Infinite differentiability in polynomially bounded o-minimal structures
Authors: Miller Chris
Published: July 2010

Infinitely differentiable functions definable in a polynomially bounded o-minimal expansion

ℜ
\Re

of the ordered field of real numbers are shown to have some of the nice properties of real analytic functions. In particular, if a definable function

f
:

R

n

→

R

f:{\mathbb {R}^n} \to \mathbb {R}

is

C
N

{C^N}

at

a
∈

R

n

a \in {\mathbb {R}^n}

for all

N
∈

N

N \in \mathbb {N}

and all partial derivatives of f vanish at a, then f vanishes identically on some open neighborhood of a. Combining this with the Abhyankar-Moh theorem on convergence of power series, it is shown that if

ℜ
\Re

is a polynomially bounded o-minimal expansion of the field of real numbers with restricted analytic functions, then all

C
∞

{C^\infty }

functions definable in

ℜ
\Re

are real analytic, provided that this is true for all definable functions of one variable.
Indecomposable coalgebras, simple comodules, and pointed Hopf algebras
Authors: Montgomery Susan
Published: July 2010

We prove that every coalgebra C is a direct sum of coalgebras in such a way that the summands correspond to the connected components of the Ext quiver of the simple comodules of C. This result is used to prove that every pointed Hopf algebra is a crossed product of a group over the indecomposable component of the identity element.
Completeness of metrizable pre-images of van Douwen-complete spaces
Authors: Tsuda Kōichi
Published: July 2010

We shall show the recurrence of complete metrizability of irreducible closed pre-images of van Douwen-complete spaces. As its corollary we shall show that every van Douwen-complete space is

G
δ

{G_\delta }

in any Lašnev space.
Noetherian domains with many more elements than height-one primes
Authors: Anderson D. D.
Published: July 2010

Extending previous results of L. Claborn and H.W. Lenstra, Jr., we show that if D is a Krull domain with a set of height-one primes

X

(
1
)

{X^{(1)}}

that satisfies either (1) D contains a subset k with

|

k

|

>

|

X

(
1
)

|

|k| > |{X^{(1)}}|

and for

μ
≠
λ
∈
k
,
μ
−
λ

\mu \ne \lambda \in k,\mu - \lambda

is a unit, or (2)

|

D

|

>

|

X

(
1
)

|

ℵ
0

|D| > |{X^{(1)}}{|^{{\aleph _0}}}

, then D is a Euclidean domain. We also show that any Noetherian ring satisfying (1) or Noetherian domain satisfying (2) has Krull dimension at most one.
On the 𝐶^{∞} wave-front set of solutions of first-order nonlinear PDEs
Authors: Asano Claudio Hirofume
Published: July 2010

Let

Ω
⊂

R

m
+
1

\Omega \subset {{\mathbf {R}}^{m + 1}}

be a neighborhood of the origin and assume

u
∈

C
2

(
Ω
)

u \in {C^2}(\Omega )

is a solution of the nonlinear PDE
\[

u
t

=
f
(
x
,
t
,
u
,

u
x

)
,

{u_t} = f(x,t,u,{u_x}),

\]
where

f
(
x
,
t
,

ζ
0

,
ζ
)

f(x,t,{\zeta _0},\zeta )

is

C
∞

{C^\infty }

in the variables

(
x
,
t
)
∈

R

m

×

R

(x,t) \in {{\mathbf {R}}^m} \times {\mathbf {R}}

and holomorphic in the variables

(

ζ
0

,
ζ
)
∈

C

×

C

m

({\zeta _0},\zeta ) \in {\mathbf {C}} \times {{\mathbf {C}}^m}

. We present a proof that
\[

W
F
(
u
)
⊂
char
⁡
(

L
u

)
,

WF(u) \subset \operatorname {char}({L^u}),

\]
where

W
F
(
u
)

WF(u)

denotes the

C
∞

{C^\infty }

wave-front set of u and

char
⁡
(

L
u

)

\operatorname {char}({L^u})

is the characteristic set of the linearized operator
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript u Baseline equals StartFraction partial-differential Over partial-differential t EndFraction minus sigma-summation Underscript j equals 1 Overscript m Endscripts left-parenthesis StartFraction partial-differential f Over partial-differential zeta Subscript j Baseline EndFraction right-parenthesis left-parenthesis x comma t comma u comma u Subscript x Baseline right-parenthesis StartFraction partial-differential Over partial-differential x Subscript j Baseline EndFraction period">

L
u

=

∂

∂
t

−

∑

j
=
1

m

(

∂
f

∂

ζ
j

)

(
x
,
t
,
u
,

u
x

)

∂

∂

x
j

.

{L^u} = \frac {\partial }{{\partial t}} - \sum \limits _{j = 1}^m {\left ( {\frac {{\partial f}}{{\partial {\zeta _j}}}} \right )} (x,t,u,{u_x})\frac {\partial }{{\partial {x_j}}}.

\]
On the inequality |𝑝(𝑧)|≤𝑝(|𝑧|) for polynomials
Authors: De Angelis Valerio
Published: July 2010

We derive a simple test to check if the inequality

|

p
(
z
)

|

≤
p
(

|

z

|

)

|p(z)| \leq p(|z|)

holds for all sufficiently small or sufficiently large values of

|

z

|

|z|

, where p is a Laurent polynomial in one variable with real coefficients and

z
∈

C

z \in \mathbb {C}

. Some examples and applications are also given.
Statistics for special 𝑞,𝑡-Kostka polynomials
Authors: Fishel Susanna
Published: July 2010

Kirillov and Reshetikhin introduced rigged configurations as a new way to calculate the entries

K

λ
μ

(
t
)

{K_{\lambda \mu }}(t)

of the Kostka matrix. Macdonald defined the two-parameter Kostka matrix whose entries

K

λ
μ

(
q
,
t
)

{K_{\lambda \mu }}(q,t)

generalize

K

λ
μ

(
t
)

{K_{\lambda \mu }}(t)

. We use rigged configurations and a formula of Stembridge to provide a combinatorial interpretation of

K

λ
μ

(
q
,
t
)

{K_{\lambda \mu }}(q,t)

in the case where

μ
\mu

is a partition with no more than two columns. In particular, we show that in this case,

K

λ
μ

(
q
,
t
)

{K_{\lambda \mu }}(q,t)

has nonnegative coefficients.
An operator still not satisfying Lomonosov’s hypothesis
Authors: Hadwin Don
Published: July 2010

It was shown that quasianalytic unilateral shifts do not satisfy the hypothesis of a famous invariant subspace theorem of V. Lomonosov. We prove that these same shifts do not satisfy the hypothesis of Lomonosov’s recent improvement of his theorem.
Generalized Swan’s theorem and its application
Authors: Manoharan P.
Published: July 2010

Swan’s theorem verifies the equivalence between finitely generated projective modules over function algebras and smooth vector bundles. We define

A

(
r
)

{A^{(r)}}

-maps that correspond to usual non-linear differential operators of degree r under the equivalence of Swan’s theorem and thus generalize Swan’s theorem to include non-linear differential operators as morphisms. An

A

(
r
)

{A^{(r)}}

-manifold structure is introduced on the space of sections of a fiber bundle through charts with

A

(
r
)

{A^{(r)}}

-maps as transition homeomorphisms. A characterization for all the smooth maps between the spaces of sections of vector bundles, whose kth derivatives are linear differential operators of degree r in each variable, is given in terms of

A

(
r
)

{A^{(r)}}

-maps.
The dimension subalgebra problem for enveloping algebras of Lie superalgebras
Authors: Riley David M.
Published: July 2010

Let L be an arbitrary Lie superalgebra over a field of characteristic different from 2. Denote by

ω
u
(
L
)

\omega u(L)

the ideal generated by L in its universal enveloping algebra

U
(
L
)

U(L)

. It is shown that

L
∩
ω
u

(
L

)
n

=

γ
n

(
L
)

L \cap \omega u{(L)^n} = {\gamma _n}(L)

for each

n
≥
1

n \geq 1

, where

γ
n

(
L
)

{\gamma _n}(L)

is the nth term of the lower central series of L. We also prove that

ω
u
(
L
)

\omega u(L)

is a residually nilpotent ideal if and only if L is residually nilpotent. Both these results remain true in characteristic 2 provided we take L to be an ordinary Lie algebra.
On monocomposition algebras
Authors: Walcher Sebastian
Published: July 2010

Monocomposition algebras satisfy a weaker version of composition with a certain quadratic form. The main result of this paper is that the construction of unital nondegenerate monocomposition algebras requires a family of skew-symmetric linear maps with certain additional conditions. Some classes of examples are discussed.
On the numbers of 2-weights, unipotent conjugacy classes, and irreducible Brauer 2-characters of finite classical groups
Authors: An Jian Bei , Conder Marston
Published: July 2010

A closed expression for the number of unipotent conjugacy classes of a classical group G is given and Alperin’s weight conjecture is confirmed for G globally and for a symplectic or odd-dimensional orthogonal group block by block.
Transference of maximal multipliers on Hardy spaces
Authors: Fan Da Shan , Wu Zhi Jian
Published: July 2010

Based on the atomic decomposition of the Hardy space, we give a simple proof for a theorem of Liu and Lu (Studia Math. 105 (1993), 121-134), which discusses the relation between the maximal operators on

R

n

{\mathbb {R}^n}

and on

T

n

{\mathbb {T}^n}

. More significantly, our proof shows that condition (1) in Liu and Lu’s Theorem 1 is superfluous.
Fonctions totalement croissantes: une correction
Authors: Grenier Jean-Pierre
Published: July 2010

On montre, à l’aide d’un contre exemple qu’un des énoncés classique de la théorie des fonctions totalement croissantes (A. Revuz) est faux, puis on en donne une version corrigée. Using a counterexample we show that a classical result about the totally increasing functions (A. Revuz) is false, then we give a right version of it.
Purely infinite simple 𝐶*-crossed products
Authors: Jeong Ja A
Published: July 2010

Let

α
\alpha

be an outer action by a finite group G on a simple

C
∗

{C^ \ast }

-algebra A with

R
R
(
A
)
=
0

RR(A) = 0

. If A is purely infinite, then the

C
∗

{C^ \ast }

-crossed product

A

×
α

G

A{ \times _\alpha }G

is purely infinite. The converse is also true if G is a finite abelian group.
The Borel classes of Mahler’s 𝐴, 𝑆, 𝑇, and 𝑈 numbers
Authors: Ki Haseo
Published: July 2010

In this article we examine theA, S, T, andUsets of Mahler’s classification from a descriptive set theoretic point of view. We calculate the possible locations of these sets in the Borel hierarchy.Aturns out to beΣ20\Sigma _2^0-complete, whileUprovides a rare example of a naturalΣ30\Sigma _3^0-complete set. We produce an upperbound ofΣ40\Sigma _4^0forSand show thatTisΠ40\Pi _4^0but notΣ30\Sigma _3^0. Our main result is based on a deep theorem of Schmidt that allows us to guarantee the existence of theTnumbers.
A counterexample to a compact embedding theorem for functions with values in a Hilbert space
Authors: Migórski Stanisław
Published: July 2010

A counterexample to a compactness embedding result of Nagy is provided.
Finding discontinuities from tomographic data
Authors: Ramm A. G.
Published: July 2010

A method for derivation of inversion formulas for the Radon transform of

f
(
x
)

f(x)

is given. This method allows one to construct functions

ψ
(
x
)

\psi (x)

which are easy to compute and which have the same wave fronts as

f
(
x
)

f(x)

. This, in turn, allows one to calculate the singular support S of

f
(
x
)

f(x)

by computing

ψ
(
x
)

\psi (x)

given tomographic data. A simple geometrical relation between the singular supports of f and

f
^

\hat f

, its Radon transform, is formulated (the duality law).
A remark on the Hausdorff-Young inequality
Authors: Sjölin Per
Published: July 2010

We shall prove a sharp Hausdorff-Young inequality of Beckner type for functions on

T

\mathbb {T}

with small support.
Banach spaces of polynomials without copies of 𝑙¹
Authors: Valdivia Manuel
Published: July 2010

Let X be a Banach space. For a positive integer m, let

P

w
∗

(
m

X
∗

)

{\mathcal {P}_{{w^ \ast }}}{(^m}{X^ \ast })

denote the Banach space formed by all m-homogeneous polynomials defined on

X
∗

{X^ \ast }

whose restrictions to the closed unit ball

B
(

X
∗

)

B({X^ \ast })

of

X
∗

{X^ \ast }

are continuous for the weak-star topology. For each one of such polynomials, its norm will be the supremum of the absolute value in

B
(

X
∗

)

B({X^ \ast })

. In this paper the bidual of

P

w
∗

(
m

X
∗

)

{\mathcal {P}_{{w^ \ast }}}{(^m}{X^ \ast })

is constructed when this space does not contain a copy of

l
1

{l^1}

. It is also shown that, whenever X is an Asplund space,

P

w
∗

(
m

X
∗

)

{\mathcal {P}_{{w^ \ast }}}{(^m}{X^ \ast })

is also Asplund.
A note on the boundary representation of a continuous spatial semigroup of *-endomorphisms of ℬ(ℋ)
Authors: Alevras Alexis
Published: July 2010

We prove that the equivalence class of the boundary representation

π
α

,

U

{\pi _\alpha },\mathcal {U}

of an

E
0

{E_0}

-semigroup

α
\alpha

is independent of the intertwining semigroup of isometries

U

\mathcal {U}

.
On solutions of elliptic equations that decay rapidly on paths
Authors: Armitage D. H.
Published: July 2010

Let

P
(
D
)

P(D)

be an elliptic differential operator on

R

n

{\mathbb {R}^n}

with constant coefficients. It is known that if u is a solution of

P
(
D
)
u
=
0

P(D)u = 0

on an unbounded domain and if u decays uniformly and sufficiently rapidly, then

u
=
0

u = 0

. In this note it is shown that the same conclusion holds if u decays rapidly, but not a priori uniformly, on a sufficiently large set of unbounded paths.
LCM-stability of power series extensions characterizes Dedekind domains
Authors: Condo John T.
Published: July 2010

In this paper we prove the following main result. A (commutative integral) domain R is a Dedekind domain if and only if

R
[
[
X
]
]
⊂
T
[
[
X
]
]

R[[X]] \subset T[[X]]

is LCM-stable for each domain T containing R as a subring.
A characterization of Clifford minimal hypersurfaces in 𝑆⁴
Authors: Li Hai Zhong
Published: July 2010

In this note we give a characterization of Clifford minimal hypersurfaces in

S
4

{S^4}

by the Ricci curvature condition.
Boundary value problems for elliptic pseudo-differential operators
Authors: Taira Kazuaki
Published: July 2010

The purpose of this paper is to study boundary value problems for elliptic pseudo-differential operators which originate from the problem of existence of Markov processes in probability theory. We prove existence and uniqueness theorems for these boundary value problems in the framework of Sobolev spaces. Our approach permits us to interpret sufficient conditions for the unique solvability of boundary value problems in terms of Markovian motion.
The Haar measure on a compact quantum group
Authors: Van Daele A.
Published: July 2010

Let A be a

C

∗

{{\text {C}}^ \ast }

-algebra with an identity. Consider the completed tensor product

A

⊗
¯

A

A\bar \otimes A

of A with itself with respect to the minimal or the maximal

C

∗

{{\text {C}}^ \ast }

-tensor product norm. Assume that

Δ
:
A
→
A

⊗
¯

A

\Delta :A \to A\bar \otimes A

is a non-zero

∗

^ \ast

-homomorphism such that

(
Δ
⊗
ι
)
Δ
=
(
ι
⊗
Δ
)
Δ

(\Delta \otimes \iota )\Delta = (\iota \otimes \Delta )\Delta

where

ι
\iota

is the identity map. Then

Δ
\Delta

is called a comultiplication on A. The pair

(
A
,
Δ
)

(A,\Delta )

can be thought of as a ’compact quantum semi-group’. A left invariant Haar measure on the pair

(
A
,
Δ
)

(A,\Delta )

is a state

φ
\varphi

on A such that

(
ι
⊗
φ
)
Δ
(
a
)
=
φ
(
a
)
1

(\iota \otimes \varphi )\Delta (a) = \varphi (a)1

for all

a
∈
A

a \in A

. We show in this paper that a left invariant Haar measure exists if the set

Δ
(
A
)
(
A
⊗
1
)

\Delta (A) (A \otimes 1)

is dense in

A

⊗
¯

A

A\bar \otimes A

. It is not hard to see that, if also

Δ
(
A
)
(
1
⊗
A
)

\Delta (A) (1 \otimes A)

is dense, this Haar measure is unique and also right invariant in the sense that

(
φ
⊗
ι
)
Δ
(
a
)
=
φ
(
a
)
1

(\varphi \otimes \iota )\Delta (a) = \varphi (a)1

. The existence of a Haar measure when these two sets are dense was first proved by Woronowicz under the extra assumption that A has a faithful state (in particular when A is separable).
A reduction theorem on purely singular splittings of cyclic groups
Authors: Woldar Andrew J.
Published: July 2010

A set M of nonzero integers is said to split a finite abelian group G if there is a subset S of G for which

M
⋅
S
=
G
∖
{
0
}

M \cdot S = G\backslash \{ 0\}

. If, moreover, each prime divisor of

|

G

|

|G|

divides an element of M, we call the splitting purely singular. It is conjectured that the only finite abelian groups which can be split by

{
1
,
…
,
k
}

\{ 1, \ldots ,k\}

in a purely singular manner are the cyclic groups of order

1
,
k
+
1

1,k + 1

and

2
k
+
1

2k + 1

. We show that a proof of this conjecture can be reduced to a verification of the case

gcd
(

|

G

|

,
6
)
=
1

\gcd (|G|,6) = 1

.
Solutions of several problems in the theory of compact positive operators
Authors: Abramovich Y. A. , Wickstead A. W.
Published: July 2010

We construct a compactly dominated compact operator S on a Dedekind complete Banach lattice whose modulus

|

S

|

|S|

is not compact. We also construct a compactly dominated compact operator without modulus.
On representations of elementary subgroups of Chevalley groups over algebras
Authors: Chen Yu
Published: July 2010

It is shown that every nontrivial linear or projective representation of the elementary subgroup of a Chevalley group over an algebra containing an infinite field must have degree greater than or equal to the square root of the dimension of the corresponding Chevalley-Demazure group scheme adding 1 and the equality emerges only if the Chevalley group is of type

A
n

{A_n}

for

n
≥
1

n \geq 1

.
On the holomorphy conjecture for Igusa’s local zeta function
Authors: Denef Jan , Veys Willem
Published: July 2010

To a polynomial f over a p-adic field K and a character

χ
\chi

of the group of units of the valuation ring of K one associates Igusa’s local zeta function

Z
(
s
,
f
,
χ
)

Z(s,f,\chi )

, which is a meromorphic function on

C

\mathbb {C}

. Several theorems and conjectures relate the poles of

Z
(
s
,
f
,
χ
)

Z(s,f,\chi )

to the monodromy of f; the so-called holomorphy conjecture states roughly that if the order of

χ
\chi

does not divide the order of any eigenvalue of monodromy of f, then

Z
(
s
,
f
,
χ
)

Z(s,f,\chi )

is holomorphic on

C

\mathbb {C}

. We prove mainly that if the holomorphy conjecture is true for

f
(

x
1

,
…
,

x

n
−
1

)

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

, then it is true for

f
(

x
1

,
…
,

x

n
−
1

)
+

x
n
k

f({x_1}, \ldots ,{x_{n - 1}}) + x_n^k

with

k
≥
3

k \geq 3

, and we give some applications.
Measures satisfying a refined doubling condition and absolute continuity
Authors: Jonsson Alf
Published: July 2010

It is shown that for certain subsets

F
⊂

R

n

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

, two measures with support F satisfying a refined doubling condition are necessarily mutually absolutely continuous. This is contrary to the situation with measures satisfying the usual doubling condition, in which case no such result is available.
Best constants in Kahane-Khintchine inequalities for complex Steinhaus functions
Authors: Peškir Goran
Published: July 2010

Let

{

φ
k

}

k
≥
1

{\{ {\varphi _k}\} _{k \geq 1}}

be a sequence of independent random variables uniformly distributed on

[
0
,
2
π
[

[0,2\pi [

, and let

‖
∙
‖

ψ

{\left \| \bullet \right \|_\psi }

denote the Orlicz norm induced by the function

ψ
(
x
)
=
exp
⁡
(

|

x

|

2

)
−
1

\psi (x) = \exp (|x{|^2}) - 1

. Then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar sigma-summation Underscript k equals 1 Overscript n Endscripts z Subscript k Baseline e Superscript i phi Super Subscript k Superscript Baseline double-vertical-bar Subscript psi Baseline less-than-or-equal-to StartRoot 2 EndRoot left-parenthesis sigma-summation Underscript k equals 1 Overscript n Endscripts StartAbsoluteValue z Subscript k Baseline EndAbsoluteValue squared right-parenthesis Superscript 1 slash 2">

‖

∑

k
=
1

n

z
k

e

i

φ
k

‖

ψ

≤

2

(

∑

k
=
1

n

|

z
k

|

2

)

1

/

2

{\left \| {\sum \limits _{k = 1}^n {{z_k}{e^{i{\varphi _k}}}} } \right \|_\psi } \leq \sqrt 2 {\left ( {\sum \limits _{k = 1}^n {|{z_k}{|^2}} } \right )^{1/2}}

\]
for all

z
1

,
…
,

z
n

∈

C

{z_1}, \ldots ,{z_n} \in {\mathbf {C}}

and all

n
≥
1

n \geq 1

. The constant

2

\sqrt 2

is shown to be the best possible. The method of proof relies upon a combinatorial argument, Taylor expansion, and the central limit theorem. The result is additionally strengthened by showing that the underlying functions are Schur-concave. The proof of this fact uses a result on the multinomial distribution of Rinott, and Schur’s proposition on the sum of convex functions. The estimates obtained throughout are shown to be the best possible. The result extends and generalizes to provide similar inequalities and estimates for other Orlicz norms.
Oscillation results on linear systems of difference equations
Authors: Kong Qingkai
Published: July 2010

We consider the difference systeman+1−an+∑i=1mQian−ki=0{a_{n + 1}} - {a_n} + \sum \nolimits _{i = 1}^m {{Q_i}{a_{n - {k_i}}} = 0}whereki∈Z,Qi∈Rr×r,i=1,…,m{k_i} \in \mathbb {Z},{Q_i} \in {\mathbb {R}^{r \times r}},i = 1, \ldots ,m, and systems of more general forms. We establish explicit conditions for the systems to be oscillatory. Those conditions are necessary and sufficient for special cases.
Symmetric orthogonal polynomials and the associated orthogonal 𝐿-polynomials
Authors: Sri Ranga A.
Published: July 2010

We show how symmetric orthogonal polynomials can be linked to polynomials associated with certain orthogonal L-polynomials. We provide some examples to illustrate the results obtained. Finally as an application, we derive information regarding the orthogonal polynomials associated with the weight function

(
1
+
k

x
2

)

(
1
−

x
2

)

−
1

/

2

,
k
>
0

(1 + k{x^2}) {(1 - {x^2})^{ - 1/2}},k > 0

.
Indices of holomorphic vector fields relative to invariant curves on surfaces
Authors: Suwa Tatsuo
Published: July 2010

The local index of a holomorphic vector field relative to a (possibly singular) invariant complex analytic curve on a complex surface is defined and it is shown that, for a compact curve invariant by a one-dimensional singular foliation on a surface, the sum of the indices is equal to its self-intersection number. An interpretation of the indices in terms of holonomy is also given.
On the determinant of elliptic boundary value problems on a line segment
Authors: Burghelea D. , Friedlander L. , Kappeler T.
Published: July 2010

In this paper we present a formula for the determinant of a matrix-valued elliptic differential operator of even order on a line segment [0, T] with boundary conditions.
Explicit formulas for the Szegő kernel on certain weakly pseudoconvex domains
Authors: Francsics Gábor , Hanges Nicholas
Published: July 2010

The objective of this paper is to determine the Szegő kernel of the domain
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D equals left-brace left-parenthesis z comma zeta comma w right-parenthesis element-of double-struck upper C Superscript n plus m plus 1 Baseline semicolon normal fraktur upper I m w greater-than double-vertical-bar z double-vertical-bar squared plus double-vertical-bar zeta double-vertical-bar Superscript 2 p Baseline right-brace">

D

=
{
(
z
,
ζ
,
w
)
∈

C

n
+
m
+
1

;
ℑ

m

w
>

‖
z
‖

2

+

‖
ζ
‖

2
p

}

\mathcal {D} = \{ (z,\zeta ,w) \in {\mathbb {C}^{n + m + 1}};\Im {\text {m}}w > {\left \| z \right \|^2} + {\left \| \zeta \right \|^{2p}}\}

explicitly in closed form.
An inverse problem in bifurcation theory. III
Authors: Kamimura Yutaka
Published: July 2010

This note is concerned with a nonlinear boundary value problem for a simple ordinary differential equation. A sufficient condition for the second bifurcating curve to determine a nonlinear term uniquely is obtained.
A counterexample on the semicontinuity of minima
Authors: Luque-Vásquez Fernando , Hernández-Lerma Onésimo
Published: July 2010

Let X and Y be metric spaces,

Φ
\Phi

a multifunction from X to Y, and v a real-valued function on

X
×
Y

X \times Y

. We give an example in which

Φ
\Phi

is continuous, and v is continuous, inf-compact and bounded below, but the minimum function

v
∗

(
x
)
:=

inf

y
∈
Φ
(
x
)

v
(
x
,
y
)

{v^ \ast }(x): = {\inf _{y \in \Phi (x)}}v(x,y)

on X is not lower semicontinuous.
A compact Hausdorff space without 𝑃-points in which 𝐺_{𝛿}-sets have interior
Authors: Watson Stephen
Published: July 2010

We construct a compact Hausdorff space which has no P-points and yet in which every nonempty

G
δ

{G_\delta }

set has nonempty interior.
Compactifications of the ray with the arc as remainder admit no 𝑛-mean
Authors: Awartani M. M. , Henderson David W.
Published: July 2010

An n-mean on X is a function

F
:

X
n

→
X

F:{X^n} \to X

which is idempotent and symmetric. In 1970 P. Bacon proved that the

sin
⁡
(
1

/

x
)

\sin (1/x)

continuum admits no 2-mean. In this paper, it is proved that if X is any metric space which contains an open line one of whose boundary components in X is an arc, then X admits no n-mean,

n
≥
2

n \geq 2

.
Computing the Euler characteristic of a manifold with boundary
Authors: Levine Harold
Published: July 2010

Given a stable map, f, from an even-dimensional, compact manifold with boundary, M, into

R

2

{\mathbb {R}^2}

, the Euler characteristic of

M
,
χ
(
M
)

M,\chi (M)

, is expressed in terms of projectivized rotation numbers of f restricted to

S
(
f
)

S(f)

, the singular curves of f, and of f restricted to

S
(
f

|

∂
M
)

S(f|\partial M)

, the singular curves of f restricted to the boundary of M.
Ultrametrics and geometric measures
Authors: Movahedi-Lankarani H. , Wells R.
Published: July 2010

LetZbe a locally connected, locally compact, and separable metric space equipped with a geometric measurev. It is shown that if a subsetYofZis bi-Lipschitz isomorphic to an ultrametric space, thenν(Y)=0\nu (Y) = 0.
Duality on noncompact manifolds and complements of topological knots
Authors: Venema Gerard A.
Published: July 2010

Let

Σ
\Sigma

be the image of a topological embedding of

S

n
−
2

{S^{n - 2}}

into

S
n

{S^n}

. In this paper the homotopy groups of the complement

S
n

−
Σ

{S^n} - \Sigma

are studied. In contrast with the situation in the smooth and piecewise-linear categories, it is shown that the first nonstandard homotopy group of the complement of such a topological knot can occur in any dimension in the range 1 through

n
−
2

n - 2

. If the first nonstandard homotopy group of the complement occurs above the middle dimension, then the end of

S
n

−
Σ

{S^n} - \Sigma

must have a nontrivial homotopy group in the dual dimension. The complement has the proper homotopy type of

S
1

×

R

n
−
1

{S^1} \times {\mathbb {R}^{n - 1}}

if both the complement and the end of the complement have standard homotopy groups in every dimension below the middle dimension. A new form of duality for noncompact manifolds is developed. The duality theorem is the main technical tool used in the paper.