bluuebunny/crossref_metadata_2025_split huggingface dataset:
On the ideal structure of 𝐶(𝑋)
Authors: Dietrich William E. 
Published: June 2010

The ideal structure of

C
(
X
)

C(X)

, the algebra of continuous functions from a completely regular Hausdorff space

X
X

to the scalars is analyzed by examining for fixed

A
⊂
β
X

A \subset \beta X

(the Stone-Čech compactification of

X
X

) the structure of the quotient

I
A

/

F
A

{I^A}/{F^A}

, where

I
A

[

F
A

]

{I^A}[{F^A}]

is the ideal of maps

f
∈
C
(
X
)

f \in C(X)

for which
\[

A
⊂

cl

β
X

Z
(
f
)

[
A
⊂

int

β
X

cl

β
X

Z
(
f
)
]
.

A \subset {\text {cl} _{\beta X}}Z(f)\quad [A \subset {\operatorname {int} _{\beta X}}{\text {cl} _{\beta X}}Z(f)].

\]
Unless it vanishes,

I
A

/

F
A

{I^A}/{F^A}

has no minimal or maximal ideals, and its Krull dimension is infinite. If

J
J

is an ideal of

C
(
X
)

C(X)

strictly between

F
A

{F^A}

and

I
A

{I^A}

, there are ideals

J
_

\underline {J}

and

J
¯

\bar J

of

C
(
X
)

C(X)

for which

F
A

⊂

J
_

⊂
J
⊂

J
¯

⊂

I
A

{F^A} \subset \underline {J} \subset J \subset \bar J \subset {I^A}

with all inclusions proper. For

K
⊂
C
(
X
)

K \subset C(X)

, let

Z
(
K
)
=
∩
{

cl

β
X

Z
(
f
)
:
f
∈
K
}

Z(K) = \cap \{ {\text {cl} _{\beta X}}Z(f):f \in K\}

. If

J
⫋
I

J \subsetneqq I

are ideals of

C
(
X
)

C(X)

with

Z
(
J
)
=
Z
(
I
)

Z(J) = Z(I)

and if

I
I

is semiprime, there is an ideal

K
K

strictly between

J
J

and

I
I

. If

I
I

and

J
J

are

Z
Z

-ideals,

K
K

can be chosen to be of the form

P
∩
I
,
P

P \cap I, P

a prime ideal of

C
(
X
)

C(X)

. The maximal ideals of a semiprime ideal

I
I

of

C
(
X
)

C(X)

are of the form

I
q

∩
I
,
q
∈
β
X
−
Z
(
I
)

{I^q} \cap I,q \in \beta X - Z(I)

. If

A
⊂
β
X

A \subset \beta X

is closed,

I
A

{I^A}

is a finitely generated ideal iff

A
A

is open.
Two extremal problems
Authors: Harrington Andrew , Ortel Marvin 
Published: June 2010

The class H consists of all functions f which are analytic in the disk D and satisfy
\[

∫
D

∫

|

f
(
z
)

|

d
x

d
y

>
∞
.

\int \limits _D {\int {|f(z)|dx\;dy} > \infty .}

\]
Linear extremal problems for functionals of the type

Λ
(
f
)
=
∫

∫
D

f
(
z
)
ϕ
(
z
)

d
x

d
y
,
ϕ
∈

L
∞

(
D
)
,
f
∈
H

\Lambda (f) = \smallint {\smallint _D}f(z)\phi (z)\;dx\;dy,\phi \in {L^\infty }(D),f \in H

, are studied.
Efficient generation of maximal ideals in polynomial rings
Authors: Davis E. D. , Geramita A. V. 
Published: June 2010

The cardinality of a minimal basis of an ideal I is denoted

ν
(
I
)

\nu (I)

. Let A be a polynomial ring in

n
>
0

n > 0

variables with coefficients in a noetherian (commutative with

1
≠
0

1 \ne 0

) ring R, and let M be a maximal ideal of A. In general

ν
(
M

A
M

)
+
1
⩾
ν
(
M
)
⩾
ν
(
M

A
M

)

\nu (M{A_M}) + 1 \geqslant \nu (M) \geqslant \nu (M{A_M})

. This paper is concerned with the attaining of equality with the lower bound. It is shown that equality is attained in each of the following cases: (1)

A
M

{A_M}

is not regular (valid even if A is not a polynomial ring), (2)

M
∩
R

M \cap R

is maximal in R and (3)

n
>
1

n > 1

. Equality may fail for

n
=
1

n = 1

, even for R of dimension 1 (but not regular), and it is an open question whether equality holds for R regular of dimension

>
1

> 1

. In case

n
=
1

n = 1

and

dim
⁡
(
R
)
=
2

\dim (R) = 2

the attaining of equality is related to questions in the K-theory of projective modules. Corollary to (1) and (2) is the confirmation, for the case of maximal ideals, of one of the Eisenbud-Evans conjectures; namely,

ν
(
M
)
⩽
max
{
ν
(
M

A
M

)
,
dim
⁡
(
A
)
}

\nu (M) \leqslant \max \{ \nu (M{A_M}),\dim (A)\}

. Corollary to (3) is that for R regular and

n
>
1

n > 1

, every maximal ideal of A is generated by a regular sequence—a result well known (for all

n
⩾
1

n \geqslant 1

) if R is a field (and somewhat less well known for R a Dedekind domain).
Ergodic equivalence relations, cohomology, and von Neumann algebras. I
Authors: Feldman Jacob , Moore Calvin C. 
Published: June 2010

Let

(
X
,

B

)

(X,\mathcal {B})

be a standard Borel space,

R
⊂
X
×
X

R \subset X \times X

an equivalence relation

∈

B

×

B

\in \mathcal {B} \times \mathcal {B}

. Assume each equivalence class is countable. Theorem 1:

∃
\exists

a countable group G of Borel isomorphisms of

(
X
,

B

)

(X,\mathcal {B})

so that

R
=
{
(
x
,
g
x
)
:
g
∈
G
}

R = \{ (x,gx):g \in G\}

. G is far from unique. However, notions like invariance and quasi-invariance and R-N derivatives of measures depend only on R, not the choice of G. We develop some of the ideas of Dye [1], [2] and Krieger [1]-[5] in a fashion explicitly avoiding any choice of G; we also show the connection with virtual groups. A notion of “module over R” is defined, and we axiomatize and develop a cohomology theory for R with coefficients in such a module. Surprising application (contained in Theorem 7): let

α
,
β

\alpha ,\beta

be rationally independent irrationals on the circle

T

\mathbb {T}

, and f Borel:

T

→

T

\mathbb {T} \to \mathbb {T}

. Then

∃
\exists

Borel

g
,
h
:

T

→

T

g,h:\mathbb {T} \to \mathbb {T}

with

f
(
x
)
=
(
g
(
a
x
)

/

g
(
x
)
)
(
h
(
β
x
)

/

h
(
x
)
)

f(x) = (g(ax)/g(x))(h(\beta x)/h(x))

a.e. The notion of “skew product action” is generalized to our context, and provides a setting for a generalization of the Krieger invariant for the R-N derivative of an ergodic transformation: we define, for a cocycle c on R with values in the group A, a subgroup of A depending only on the cohomology class of c, and in Theorem 8 identify this with another subgroup, the “normalized proper range” of c, defined in terms of the skew action. See also Schmidt [1].
𝐾-theory of hyperplanes
Authors: Dayton Barry H. , Weibel Charles A. 
Published: June 2010

Let R be the coordinate ring of a union of N hyperplanes in general position in

A

k

n
+
1

{\textbf {A}}_k^{n + 1}

. Then
\[

K
i

(
R
)

=

K
i

(
k
)

⊕

(

N

−

1

n

+

1

)

K

n
+
i

(
k
)
.

{K_i}(R)\, = \,{K_i}(k)\, \oplus \,\left ( {\begin {array}{*{20}{c}} {N\, - \,1} \\ {n\, + \,1} \\ \end {array} } \right )\,{K_{n + i}}(k).

\]
This formula holds for

K
0

,

K
1

,

K
i

(
i
>
0
)

{K_0},\,{K_1},\,{K_i}\,(i > 0)

, and for the Karoubi-Villamayor groups

K

V
i

(
i
∈

Z

)

K{V_i}\,(i \in \,{\mathbf {Z}})

. For

K
2

{K_2}

there is an extra summand

R
¯

/

R

\bar R/R

, where

R
¯

\bar R

is the normalization of R. For

K
3

{K_3}

the above is a quotient of

K
3

(
R
)

{K_3}(R)

. In §4 we show that

K
1

{K_1}

-regularity implies

K
0

{K_0}

-regularity, answering a question of Bass. We also show that

K
i

{K_i}

-regularity is equivalent to Laurent

K
i

{K_i}

-regularity for

i
⩽
1

i \leqslant 1

. The results of this section are independent of the rest of the paper.
On the existence of eigenvalues of differential operators dependent on a parameter
Authors: Strelitz Sh. , Abramovich S. 
Published: June 2010

In this paper we obtain results about the existence of eigenvalues for a system which depends polynomially on

λ
\lambda

,
\[

u
′

k

(
x
)

=

∑

j

=

1

n

b

k
j

(
x
,

λ
)

u
j

(
x
)
,

a
m
p
;

∑

i

=

0

p

∑

j

=

1

N

a

k
j

i

u
j

(

x
i

)

=

0
,

\begin {array}{*{20}{c}} {{{u’}_k}(x)\, = \,\sum \limits _{j\, = \,1}^n {{b_{kj}}(x,\,\lambda ){u_j}(x),} } & {\sum \limits _{i\, = \,0}^p {\sum \limits _{j\, = \,1}^N {a_{kj}^i{u_j}({x_i})\, = \,0,} } } \\ \end {array}

\]
,

k

=

1
,
.
.
.
,

N

k\, = \,1,...,\,N

. In order to get these results we prove that this system can be reduced to a standard system of the form
\[

y
′

k

(
x
)

=

∑

j
=
1

n

a

k
j

(
x
,

λ
)

y
j

(
x
)

,

a
m
p
;

y
k

(
0
)

=

a
k

(
λ
)
,

a
m
p
;

y
n

(
1
)

=

0
,

\begin {array}{*{20}{c}} {{{y’}_k}(x)\, = \,\sum \limits _{j = 1}^n {{a_{kj}}(x,\,\lambda )\,{y_j}(x)} ,} & {{y_k}(0)\, = \,{a_k}(\lambda ),} & {{y_n}(1)\, = \,0,} \\ \end {array}

\]

k

=

1
,
.
.
.
,

n

k\, = \,1,...,\,n

.
On the Hardy-Littlewood maximal function and some applications
Authors: Neugebauer C. J. 
Published: June 2010

With a monotone family

F

=

{

S
α

}
,

S
α

⊂

R

n

F\, = \,\{ {S_\alpha }\} ,\,{S_\alpha }\, \subset \,{{\textbf {R}}^n}

, we associate the Hardy-Littlewood maximal function

M
F

f
(
x
)

=

sup
α

(
1

/

|

S
α

|

)

∫

S
α

+

x

|
f
|

{M_F}f(x)\, = \,{\sup _\alpha }(1/\left | {{S_\alpha }} \right |)\int _{{S_\alpha }\, + \,x} {\left | f \right |}

. In general,

M
F

{M_F}

is not weak type (1.1). However, if we replace in the denominator

S
α

{S_\alpha }

by

S
F

∗

=

{
x

−

y
:

x
,

y

∈

S
α

}

S_F^ {\ast } \, = \,\{ x\, - \,y:\,x,\,y\, \in \,{S_\alpha }\}

, and denote the resulting maximal function by

M
F

∗

M_F^ {\ast }

, then

M
F

∗

M_F^ {\ast }

is weak type (1, 1) with weak type constant 1.
Subspaces of basically disconnected spaces or quotients of countably complete Boolean algebras
Authors: van Douwen Eric K. , van Mill Jan 
Published: June 2010

Under

MA

+

2

ω

=

ω
2

{\text {MA}}\,{\text { + }}\,{{\text {2}}^\omega }\, = \,{\omega _2}

there is a (compact) strongly zero-dimensional F-space of weight

2
ω

{2^\omega }

which cannot be embedded in any basically disconnected space. Dually, under

MA

+

2
ω

=

ω
2

{\text {MA}}\, + \,{2^\omega }\, = \,{\omega _2}

there is a weakly countably complete (or almost

σ
\sigma

-complete, or countable separation property) Boolean algebra of cardinality

2
ω

{2^\omega }

which is not a homomorphic image of any countably complete Boolean algebra. The key to our construction is the observation that if X is a subspace of a basically disconnected space and

β
ω

⊆

X

\beta \omega \, \subseteq \,X

then

β
ω

\beta \omega

is a retract of X. Dually, if B is a homomorphic image of a countably complete Boolean algebra, and if h is a homomorphism from B onto

P

(
ω
)

\mathcal {P}(\omega )

, the field of subsets of w, then there is an embedding

e
:

P

(
ω
)
→

B

e:\,\mathcal {P}(\omega ) \to \,B

such that

h

∘

e

=

i

d

P

(
ω
)

h\, \circ \,e\, = \,{\text {i}}{{\text {d}}_{\mathcal {P}(\omega )}}

.
Nonstandard extensions of transformations between Banach spaces
Authors: Tacon D. G. 
Published: June 2010

Let

X
X

and

Y
Y

be (infinite-dimensional) Banach spaces and denote their nonstandard hulls with respect to an

ℵ
1

\aleph _1

-saturated enlargement by

X
^

\hat X

and

Y
^

\hat Y

respectively. If

B

(
X
,
Y
)

\mathcal {B}(X,Y)

denotes the space of bounded linear transformations then a subset

S
S

of elements of

B

(
X
,
Y
)

\mathcal {B}(X,Y)

extends naturally to a subset

S
^

\hat S

of

B

(

X
^

,

Y
^

)

\mathcal {B}(\hat {X}, \hat {Y})

. This paper studies the behaviour of various kinds of transformations under this extension and introduces, in this context, the concepts of super weakly compact, super strictly singular and socially compact operators. It shows that

B

(
X
,
Y
)

^

⫋

B

(

X
^

,

Y
^

)

\widehat {\mathcal {B}(X,Y)} \subsetneqq \mathcal {B}(\hat {X}, \hat {Y})

provided

X
X

and

Y
Y

are infinite dimensional and contrasts this with the inclusion

K

(

H
^

)
⫋

K

(
H
)

^

\mathcal {K}(\hat H) \subsetneqq \widehat {\mathcal {K}(H)}

where

K

(
H
)

\mathcal {K}(H)

denotes the space of compact operators on a Hilbert space.
The embeddings of the discrete series in the principal series for semisimple Lie groups of real rank one
Authors: Baldoni Silva M. Welleda 
Published: June 2010

We consider the problem of finding all the “embeddings” of a discrete series representation in the principal series in the case of a simple real Lie group G of real rank one. More precisely, we solve the problem when G is

Spin
⁡
(
2
n
,

1
)
,

SU

(
n
,

1
)
,

SP

(
n
,

1
)

or

F
4

(
n

⩾

2
)

\operatorname {Spin} (2n,\,1),{\text {SU}}(n,\,1),\,{\text {SP}}(n,\,1)\,{\text {or}}\,{F_4}\,(n\, \geqslant \,2)

. The problem is reduced to considering only discrete series representations with trivial infinitesimal character, by means of tensoring with finite dimensional representations. Various other techniques are employed.
Boundary crossing probabilities for stationary Gaussian processes and Brownian motion
Authors: Cuzick Jack 
Published: June 2010

Let

X
(
t
)

X(t)

be a stationary Gaussian process,

f
(
t
)

f(t)

a continuous function, and

T
T

a finite or infinite interval. This paper develops asymptotic estimates for

P
(
X
(
t
)
⩾
f
(
t
)

P(X(t) \geqslant f(t)

, some

t
∈
T

t \in T

when this probability is small. After transformation to an Ornstein Uhlenbeck process the results are also applicable to Brownian motion. In that special case, if

W
(
t
)

W(t)

is Brownian motion,

f
f

is continuously differentiable, and

T
=
[
0
,
T
]

T = [0,T]

our estimate for

P
(
W
(
t
)
⩾
f
(
t
)

P(W(t) \geqslant f(t)

, some

t
∈
T
)

t \in T)

is
\[

Λ
=

∫
0
T

(
2
t
)

−
1

(
f
(
t
)

/

t

1

/

2

)
ϕ
(
f
(
t
)

/

t

1

/

2

)

d
t
+

I

{
(
f
(
t
)

/

t

1

/

2

)
′

|

t
=
T

>
0
}

Φ
∗

(
f
(
T
)

/

T

1

/

2

)

\Lambda = \int _0^T {{{(2t)}^{ - 1}}(f(t)/{t^{1/2}})\phi (f(t)/{t^{1/2}})} dt + {I_{\{ (f(t)/{t^{1/2}})’{|_{t = T}} > 0\} }}{\Phi ^ \ast }(f(T)/{T^{1/2}})

\]
provided

Λ
\Lambda

is small. Here

ϕ
\phi

is the standard normal density and

Φ
∗

{\Phi ^ \ast }

is its upper tail distribution. Our approach is to find an approximate first passage density and then compute crossing probabilities as a one-dimensional integral. In the case of boundaries without cusps, our results unify and extend separate results for crossings of constant levels developed by Pickands, and Qualls-Watanabe, and crossings of rapidly increasing barriers studied by Berman. Applications are also briefly explored.
Symmetry properties of the zero sets of nil-theta functions
Authors: Goodman Sharon 
Published: June 2010

Let

N
N

denote the three dimensional Heisenberg group, and let

Γ
\Gamma

be a discrete two-generator subgroup of

N
N

such that

N

/

Γ

N/\Gamma

is compact. Then we may decompose

L
2

(
N

/

Γ
)

{L^2}(N/\Gamma )

into primary summands with respect to the right regular representation

R
R

of

N
N

on

L
2

(
N

/

Γ
)

{L^2}(N/\Gamma )

as follows:

L
2

(
N

/

Γ
)
=
⊕

∑

m
∈

Z

H
m

(
Γ
)

{L^2}(N/\Gamma ) = \oplus \sum \nolimits _{m \in {\mathbf {Z}}} {{H_m}(\Gamma )}

. It can be shown that for

m
≠
0
,

H
m

(
Γ
)

m \ne 0,{H_m}(\Gamma )

is a multiplicity space for the representation

R
R

of multiplicity

|
m
|

\left | m \right |

. The distinguished subspace theory of

L

{\text {L}}

. Auslander and

J

{\text {J}}

. Brezin singles out a finite number of the decompositions of

H
m

(
Γ
)
,
m
≠
0

{H_m}(\Gamma ),m \ne 0

, which are in some ways nicer than the others. They define algebraically an integer valued function, called the index, on the set

Ω
m

{\Omega _m}

of irreducible closed

R
R

-invariant subspaces of

H
m

(
Γ
)

{H_m}(\Gamma )

such that the distinguished subspaces have index one. In this paper, we give an analytic-geometric interpretation of the index. Every space in

Ω
m

{\Omega _m}

contains a unique (up to constant multiple) special function, called a nil-theta function, that arises as a solution of a certain differential operator on

N

/

Γ

N/\Gamma

. These nil-theta functions have been shown to be closely related to the classical theta functions. Since the classical theta functions are determined (up to constant multiple) by their zero sets, it is natural to attempt to classify the spaces in

Ω
m

{\Omega _m}

using various properties of the zero sets of the nil-theta functions lying in these spaces. We define the index of a nil-theta function in

H
m

(
Γ
)

{H_m}(\Gamma )

using the symmetry properties of its zero set. Our main theorem asserts that the algebraic index of a space in

Ω
m

{\Omega _m}

equals the index of the unique nil-theta function lying in that space. We have thus an analytic-geometric characterization of the index. We then use these results to give a complete description of the zero sets of those nil-theta functions of a fixed index. We also investigate the behavior of the index under the multiplication of nil-theta functions; i.e. we discuss how the index of the nil-theta function

F
G

FG

relates to the indices of the nil-theta functions

F
F

and

G
G

.
Algebraic extensions of power series rings
Authors: Arnold Jimmy T. 
Published: June 2010

Let

D
D

and

J
J

be integral domains such that

D
⊂
J

D \subset J

and

J
[
[
X
]
]

J[[X]]

is not algebraic over

D
[
[
X
]
]

D[[X]]

. Is it necessarily the case that there exists an integral domain

R
R

such that

D
[
[
X
]
]
⊂
R
⊆
J
[
[
X
]
]

D[[X]] \subset R \subseteq J[[X]]

and

R
≅
D
[
[
X
]
]
[
[
{

Y
i

}

i
=
1

∞

]
]

R \cong D[[X]][[\{ {Y_i}\} _{i = 1}^\infty ]]

? While the general question remains open, the question is answered affirmatively in a number of cases. For example, if

D
D

satisfies any one of the conditions (1)

D
D

is Noetherian, (2)

D
D

is integrally closed, (3) the quotient field

K
K

of

D
D

is countably generated as a ring over

D
D

, or (4)

D
D

has Krull dimension one, then an affirmative answer is given. Further, in the Noetherian case it is shown that

J
[
[
X
]
]

J[[X]]

is algebraic over

D
[
[
X
]
]

D[[X]]

if and only if it is integral over

D
[
[
X
]
]

D[[X]]

and necessary and sufficient conditions are given on

D
D

and

J
J

in order that this occur. Finally if, for every positive integer

n
n

,

D
[
[

X
1

,
…
,

X
n

]
]
⊂
R
⊆
J
[
[

X
1

,
…
,

X
n

]
]

D[[{X_1}, \ldots ,{X_n}]] \subset R \subseteq J[[{X_1}, \ldots ,{X_n}]]

implies that

R
≆
D
[
[

X
1

,
…
,

X
n

]
]
[
[
{

Y
i

}

i
=
1

∞

]
]

R \ncong D[[{X_1}, \ldots ,{X_n}]][[\{ {Y_i}\} _{i = 1}^\infty ]]

, then it is shown that

J
[
[

X
1

,
…
,

X
n

]
]

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

is algebraic over

D
[
[

X
1

,
…
,

X
n

]
]

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

for every

n
n

.
The Hilbert transform and maximal function for approximately homogeneous curves
Authors: Weinberg David A. 
Published: June 2010

LetHγf(x)=p.v.∫−11f(x−γ(t))dt/t{\mathcal {H}_\gamma }f(x) = {\text {p}}{\text {.v}}{\text {.}}\int _{ - 1}^1 {f(x - \gamma (t))dt/t}andMγf(x)=sup1⩾h>0h−1∫0h|f(x−γ(t))|dt{\mathfrak {M}_\gamma }f(x) = {\sup _{1 \geqslant h > 0}}{h^{ - 1}}\int _0^h {|f(x - \gamma (t))|dt}. It is proved that forf∈S(Rn)f \in \mathcal {S}({{\mathbf {R}}^n}), the Schwartz class, and for an approximately homogeneous curveγ(t)∈Rn\gamma (t) \in {{\mathbf {R}}^n},‖Hγf‖2⩽C‖f‖2{\left \| {{\mathcal {H}_\gamma }f} \right \|_2} \leqslant C{\left \| f \right \|_2},‖Mγf‖2⩽C‖f‖2{\left \| {{\mathfrak {M}_\gamma }f} \right \|_2} \leqslant C{\left \| f \right \|_2}. A homogeneous curve is one which satisfies a differential equationγ1′(t)=(A/t)γ1(t){\gamma ’_1}(t) = (A/t){\gamma _1}(t),0>t>∞0 > t > \infty, whereAAis a nonsingular matrix all of whose eigenvalues have positive real part. An approximately homogeneous curveγ(t)\gamma (t)has the formγ1(t)+γ2(t){\gamma _1}(t) + {\gamma _2}(t), whereγ2(t){\gamma _2}(t)is a carefully specified "error", such thatγ2(j)\gamma _2^{(j)}is also restricted forj=2,…,n+1j = 2, \ldots ,n + 1. The approximately homogeneous curves generalize the curves of standard type treated by Stein and Wainger.
A hypersurface defect relation for a class of meromorphic maps
Authors: Biancofiore Aldo 
Published: June 2010

LetD1,…,Dq{D_1}, \ldots ,{D_q}be hypersurfaces of degreeppinPn{{\mathbf {P}}_n}with normal crossings. We prove for a certain class of meromorphic mapsf:Cm→Pnf:{{\mathbf {C}}^m} \to {{\mathbf {P}}_n}a defect relationδf(D1)+⋯+δf(Dq)⩽(n+1)/p{\delta _f}\left ( {{D_1}} \right ) + \cdots + {\delta _f}({D_q}) \leqslant (n + 1)/pconjectured by Ph. Griffiths and B. Shiffman.
Nilpotent inverse semigroups with central idempotents
Authors: Kowol G. , Mitsch H. 
Published: June 2010

An inverse semigroup

S
S

with central idempotents, i.e. a strong semilattice of groups, will be called nilpotent, if it is finite and if for each prime divisor

p
i

{p_i}

of the orders of the structure groups of

S
S

the sets

P
i

=
{
s
∈
S

|

o
(
s
)
=

p
i

k
s

,

k
s

⩾
0
}

{P_i} = \{ s \in S|o(s) = p_i^{{k_s}},\,{k_s} \geqslant 0\}

are subsemigroups of

S
S

. If

S
S

is a group, then

P
i

{P_i}

are exactly the Sylow

p
i

{p_i}

-subgroups of the group. A theory similar to that given by W. Burnside for finite nilpotent groups is developed introducing the concepts of ascending resp. descending central series in an inverse semigroup, and it is shown that almost all of the well-known properties of finite nilpotent groups do hold also for the class of finite inverse semigroups with central idempotents.
Minimum simplicial complexes with given abelian automorphism group
Authors: Miller Zevi 
Published: June 2010

Let

K
K

be a pure

n
n

-dimensional simplicial complex. Let

Γ
0

(
K
)

{\Gamma _0}(K)

be the automorphism group of

K
K

, and let

Γ
n

(
K
)

{\Gamma _n}(K)

be the group of permutations on

n
n

-cells of

K
K

induced by the elements of

Γ
0

(
K
)

{\Gamma _0}(K)

. Given an abelian group

A
A

we consider the problem of finding the minimum number of points

M
0

(
n
)

(
A
)

M_0^{(n)}(A)

in

K
K

such that

Γ
0

(
K
)
≅
A

{\Gamma _0}(K) \cong A

, and the minimum number of

n
n

-cells

M
1

(
n
)

(
A
)

M_1^{(n)}(A)

in

K
K

such that

Γ
n

(
K
)
≅
A

{\Gamma _n}(K) \cong A

. Write

A
=

∏

p
α

Z

p
α

e
(

p
α

)

A = {\prod _{{p^\alpha }}}{\mathbf {Z}}_{{p^\alpha }}^{e({p^\alpha })}

, where each factor

Z

p
α

{{\mathbf {Z}}_{{p^\alpha }}}

appears

e
(

p
α

)

e({p^\alpha })

times in the canonical factorization of

A
A

. For

A
A

containing no factors

Z

p
α

{{\mathbf {Z}}_{{p^\alpha }}}

satisfying

p
α

>
17

{p^\alpha } > 17

we find that

M
1

(
n
)

(
A
)
=

M
0

(
2
)

(
A
)
=

∑

p
α

p
α

e
(

p
α

)

M_1^{(n)}(A) = M_0^{(2)}(A) = {\sum _{{p^\alpha }}}{p^\alpha }e({p^\alpha })

when

n
⩾
4

n \geqslant 4

, and we derive upper bounds for

M
1

(
n
)

(
A
)

M_1^{(n)}(A)

and

M
0

(
n
)

(
A
)

M_0^{(n)}(A)

in the remaining possibilities for

A
A

and

n
n

.
Brownian motion with partial information
Authors: McConnell Terry R. 
Published: June 2010

We study the following problem concerning stopped

N
N

-dimensional Brownian motion: Compute the maximal function of the process, ignoring those times when it is in some fixed region

R
R

. Suppose this modified maximal function belongs to

L
q

{L^q}

. For what regions

R
R

can we conclude that the unrestricted maximal function belongs to

L
q

{L^q}

? A sufficient condition on

R
R

is that there exist

p
>
q

p > q

and a function

u
u

, harmonic in

R
R

, such that
\[

|

x

|

p

⩽
u
(
x
)
⩽
C

|

x

|

p

+
C
,

x
∈
R
,

|x{|^p} \leqslant u(x) \leqslant C|x{|^p} + C,\qquad x \in R,

\]
for some constant

C
C

. We give applications to analytic and harmonic functions, and to weak inequalities for exit times.
Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators
Authors: Lagarias J. C. 
Published: June 2010

This paper defines the notion of a best simultaneous Diophantine approximation to a vector

α
\alpha

in

R
n

R^n

with respect to a norm

‖

⋅

‖

\left \| \,\cdot \, \right \|

on

R
n

R^n

. Suppose

α
\alpha

is not rational and order the best approximations to

α
\alpha

with respect to

‖

⋅

‖

\left \|\, \cdot \, \right \|

by increasing denominators

1
=

q
1

>

q
2

>
⋯

1=q_1 > q_2 > \cdots

. It is shown that these denominators grow at least at the rate of a geometric series, in the sense that
\[

g

(

α
,

‖

⋅

‖

)

=

lim inf

k
→
∞

(

q
k

)

1

/

k

≥
1
+

1

2

n
+
1

g\left ( {\alpha ,\,\left \| {\,\cdot \,} \right \|} \right ) = \liminf \limits _{k \to \infty } {({q_k})^{1/k}} \geq 1 + \frac {1}{{{2^{n + 1}}}}

\]
. Let

g

(

‖

⋅

‖

)

g\left ( {\left \|\, \cdot \, \right \|} \right )

denote the infimum of

g

(

α
,

‖

⋅

‖

)

g\left ( {\alpha ,\,\left \| {\,\cdot \,} \right \|} \right )

over all

α
\alpha

in

R
n

R^n

with an irrational coordinate. For the sup norm

‖

⋅

‖

s

\left \|\, \cdot \,\right \|_s

on

R
2

R^2

it is shown that

g

(

‖

⋅

‖

s

)

≥
θ
=

1.270

+

g\left ( {\left \| \, \cdot \, \right \|}_s \right )\ge \theta =1.270^{+}

where

θ
4

=

θ

2

+
1

\theta ^4=\theta ^{2}+1

.