bluuebunny/crossref_metadata_2025 huggingface dataset:
Near-rings of invariants. II
Authors: Maxson C. J.
Published: June 2010

Let

E
E

be the equivalence relation determined by the orbits of a group

G
G

acting on a set

X
X

and let

K
K

be a group. Using a "sandwich" function
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi colon upper X slash upper E right-arrow upper K comma upper M left-parenthesis upper X comma upper G comma upper K comma phi right-parenthesis equals left-brace f colon upper X right-arrow upper K vertical-bar f is constant on each upper E hyphen class right-brace">

φ
:
X

/

E
→
K
,

M
(
X
,
G
,
K
,
φ
)
=
{
f
:
X
→
K

|

f

is constant on each

E

-class

}

\varphi :X/E \to K,\;M(X,G,K,\varphi ) = \{ f:X \to K|f\;{\text {is constant on each}}\;E{\text {-class}}\}

is a near-ring, the near-ring of invariants determined by

(
X
,
G
,
K
,
φ
)

(X,G,K,\varphi )

. In this paper we continue the study of the transfer of information between the structure of the near-ring of invariants and properties of the group action

(
G
,
X
)

(G,X)

, the group

K
K

, and the function

φ
:
K
→
X

/

E

\varphi :K \to X/E

.
Horocycles on Riemann surfaces
Authors: Seppälä Mika , Sorvali Tuomas
Published: June 2010

By the Collar Theorem, every puncture on a hyperbolic Riemann surface with punctures has a horocyclic neighborhood of area

2
2

. Furthermore two such neighborhoods associated to different punctures are disjoint. This result can be improved if we omit the condition that horocyclic neighborhoods of different punctures must be disjoint. Using arguments of the second author we show, in this paper, that each puncture of a hyperbolic Riemann surface has a horocyclic neighborhood of area

4
4

.
An elementary simultaneous approximation theorem
Authors: Kilgore Theodore
Published: June 2010

We will give an elementary and direct proof that for

f
∈

C
q

[
−
1
,
1
]

f \in {C^q}[ - 1,1]

there exists a sequence of polynomials

P
n

{P_n}

of degree at most

n

(
n
>
2
q
)

n\;(n > 2q)

such that for

k
=
0
,
…
,
q

k = 0, \ldots ,q

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue f Superscript left-parenthesis k right-parenthesis Baseline left-parenthesis x right-parenthesis minus upper P Subscript n Superscript left-parenthesis k right-parenthesis Baseline left-parenthesis x right-parenthesis EndAbsoluteValue less-than-or-slanted-equals upper M Subscript q comma k Baseline left-parenthesis StartFraction StartRoot 1 minus x squared EndRoot Over n EndFraction right-parenthesis Superscript q minus k Baseline upper E Subscript n minus q Baseline left-parenthesis f Superscript left-parenthesis q right-parenthesis Baseline right-parenthesis comma">

|

f

(
k
)

(
x
)
−

P
n

(
k
)

(
x
)

|

⩽

M

q
,
k

(

1
−

x
2

n

)

q
−
k

E

n
−
q

(

f

(
q
)

)
,

|{f^{(k)}}(x) - P_n^{(k)}(x)| \leqslant {M_{q,k}}{\left ( {\frac {{\sqrt {1 - {x^2}} }} {n}} \right )^{q - k}}{E_{n - q}}({f^{(q)}}),

\]
with

M

q
,
k

{M_{q,k}}

depending only upon

q
q

and

k
k

. Moreover
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Superscript left-parenthesis q right-parenthesis Baseline left-parenthesis plus-or-minus 1 right-parenthesis equals upper P Subscript n Superscript left-parenthesis q right-parenthesis Baseline left-parenthesis plus-or-minus 1 right-parenthesis">

f

(
q
)

(
±
1
)
=

P
n

(
q
)

(
±
1
)

{f^{(q)}}( \pm 1) = P_n^{(q)}( \pm 1)

.
Cardinalities of 𝜂₁-ordered fields
Authors: Perović Žikica
Published: June 2010

We show, using GCH, that a cardinal

κ
\kappa

is the cardinality of an

η
1

{\eta _1}

-ordered field iff

κ
ω

=
κ

{\kappa ^\omega } = \kappa

. We also show, without using GCH, that a cardinal

κ
\kappa

is the cardinality of a maximally valued

η
1

{\eta _1}

-ordered field only if

κ
ω

=
κ

{\kappa ^\omega } = \kappa

.
On maximal 𝑘-ideals of semirings
Authors: Sen M. K. , Adhikari M. R.
Published: June 2010

For a semiring

S
S

with commutative addition, conditions are considered such that

S
S

has nontrivial

k
k

-ideals or maximal

k
k

-ideals, among others, by the help of the congruence class semiring

S

/

A

S/A

defined by an ideal

A
A

of

S
S

. Moreover, all maximal

k
k

-ideals of the semiring of nonnegative integers are described.
On the number of 𝐷𝑇𝑟-orbits containing directing modules
Authors: Peng Lian Gang , Xiao Jie
Published: June 2010

In this note we prove that all directing modules over an artin algebra are distributed to finitely many

DTr
\operatorname {DTr}

-orbits.
𝐶*-extreme points of some compact 𝐶*-convex sets
Authors: Farenick D. R. , Morenz Phillip B.
Published: June 2010

In the

C

∗

{C^{\ast }}

-algebra

M
n

{M_n}

of complex

n
×
n

n \times n

matrices, we consider the notion of noncommutative convexity called

C

∗

{C^{\ast }}

-convexity and the corresponding notion of a

C

∗

{C^{\ast }}

-extreme point. We prove that each irreducible element of

M
n

{M_n}

is a

C

∗

{C^{\ast }}

-extreme point of the

C

∗

{C^{\ast }}

-convex set it generates, and we classify the

C

∗

{C^{\ast }}

-extreme points of any

C

∗

{C^{\ast }}

-convex set generated by a compact set of normal matrices.
A simple formula for cyclic duality
Authors: Elmendorf A. D.
Published: June 2010

We give a simple formula for duality in an easily described covering category of the cyclic category and show that the formula given descends to the cyclic category.
Nonresonance conditions on the potential for a second-order periodic boundary value problem
Authors: Omari Pierpaolo , Zanolin Fabio
Published: June 2010

We consider the periodic problem\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row negative u equals f left-parenthesis u right-parenthesis plus h left-parenthesis t right-parenthesis comma 2nd Row u left-parenthesis 0 right-parenthesis equals u left-parenthesis 2 pi right-parenthesis comma u prime left-parenthesis 0 right-parenthesis equals u prime left-parenthesis 2 pi right-parenthesis comma EndLayout">−u=f(u)+h(t),u(0)=u(2π),u′(0)=u′(2π),\begin {array}{*{20}{c}} { - u = f(u) + h(t),} \\ {u(0) = u(2\pi ),\qquad u’(0) = u’(2\pi ),} \\ \end {array}\]and prove its solvability for any givenhh, under new assumptions on the asymptotic behaviour of the potential of the nonlinearityff, with respect to two consecutive eigenvalues of the associated linear problem.
A sharp bound for the regularity index of fat points in general position
Authors: Catalisano M. V. , Trung N. V. , Valla G.
Published: June 2010

A bound is given for the regularity index of the coordinate ring of a set of fat points in general position in

P

k
n

{\mathbf {P}}_k^n

. The bound is attained by points on a rational normal curve.
Almost isometric copies of 𝑙_{∞} in some Banach spaces
Authors: Hudzik H. , Mastyło M.
Published: June 2010

It is shown that any

σ
\sigma

-complete Banach lattice, with an order semicontinuous norm containing an isomorphic copy of

l
∞

{l_\infty }

, contains an almost isometric copy of

l
∞

{l_\infty }

. It is also proved that any Fenchel-Orlicz space (resp. the subspace of finite elements of any Fenchel-Orlicz space) generated by an Orlicz function not satisfying the suitable

Δ
2

{\Delta _2}

-condition contains an almost isometric copy of

l
∞

{l_\infty }

(resp.

c
0

{c_0}

).
Differentiation of Zygmund functions
Authors: Ullrich David C.
Published: June 2010

The "little-

o
o

Zygmund class"

λ

∗

{\lambda ^{\ast }}

contains a nowhere-differentiable function.
A numerical characterization of hypersurface singularities
Authors: Elías Juan
Published: June 2010

In this note we give a numerical characterization of hypersurface singularities in terms of the normalized Hilbert-Samuel coefficients, and we interpret this result from the point of view of rigid polynomials.
On Teichmüller contraction
Authors: Gardiner Frederick P.
Published: June 2010

Universal Teichmüller space is the space of quasi-symmetric homeomorphisms

Q
S

QS

of a circle factored by those Möbius transformations that preserve the circle. Another Teichmüller space, which also has universal properties, is

Q
S

QS

factored by the closed subgroup

S
S

of symmetric homeomorphisms. Teichmüller’s metric for

Q
S

mod
S

QS\,\bmod S

is the boundary dilatation metric. Sullivan’s coiling property for Beltrami lines and the Hamilton-Reich-Strebel necessary and sufficient condition for extremality are proved for

Q
S
mod
S

QS\bmod S

. The coiling property implies a contraction principle for certain types of self-mappings of Teichmüller space. It is also shown that the boundary dilatation metric has an infinitesimal form and that this metric is the integral of its infinitesimal form.
On mapping cones of suspension elements of finite order in the homotopy groups of a wedge of spheres
Authors: Bokor Imre
Published: June 2010

The genus of the mapping cone

C
f

{C_f}

of a map

f
:

S

m
−
1

→
⋁

S
n

(
m
>
n
>
1
)

f:{S^{m - 1}} \to \bigvee {S^n}(m > n > 1)

representing a suspension element of finite order in

π

m
−
1

(
⋁

S
n

)

{\pi _{m - 1}}(\bigvee {S^n})

is classified by a subgroup

G
f

{G_f}

of

π

m
−
1

(

S
n

)

{\pi _{m - 1}}({S^n})

depending only on the homotopy type of

C
f

{C_f}

. The group

G
f

{G_f}

finds application in proving that the genus of

C
f

{C_f}

is trivial whenever

C
f

{C_f}

has sufficiently many

n
n

-cells, the number being limited by the torsion subgroup of

π

m
−
1

(

S
n

)

{\pi _{m - 1}}({S^n})

.
Embeddings and immersions of a 2-sphere in 4-manifolds
Authors: Gan Dan Yan , Guo Jian Han
Published: June 2010

Let

M
M

be

C

P
2

#
(
−
C

P
2

)
#

P
1

#
⋯
#

P

m
+
k

C{P^2}\# ( - C{P^2})\# {P_1}\# \cdots \# {P_{m + k}}

, where

P
1

,
…
,

P

m
+
k

{P_1}, \ldots ,{P_{m + k}}

are copies of

−
C

P
2

- C{P^2}

. Let

h
,
g
,

g
1

,
…
,

g

m
+
k

h,g,{g_1}, \ldots ,{g_{m + k}}

be the images of the standard generators of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 2 left-parenthesis upper C upper P squared semicolon upper Z right-parenthesis comma upper H 2 left-parenthesis minus upper C upper P squared semicolon upper Z right-parenthesis comma upper H 2 left-parenthesis upper P 1 semicolon upper Z right-parenthesis comma ellipsis comma upper H 2 left-parenthesis upper P Subscript m plus k Baseline semicolon upper Z right-parenthesis">

H
2

(
C

P
2

;
Z
)
,

H
2

(
−
C

P
2

;
Z
)
,

H
2

(

P
1

;
Z
)
,
…
,

H
2

(

P

m
+
k

;
Z
)

{H_2}(C{P^2};Z),\;{H_2}( - C{P^2};Z),\;{H_2}({P_1};Z), \ldots ,{H_2}({P_{m + k}};Z)

in

H
2

(
M
;
Z
)

{H_2}(M;Z)

respectively. Let

ξ
=
p
h
+
q
g
+

∑

i
=
1

m

r
i

g
i

\xi = ph + qg + \sum \nolimits _{i = 1}^m {{r_i}{g_i}}

be an element of

H
2

(
M
;
Z
)

{H_2}(M;Z)

. Suppose

ξ
2

=
l
>
0
,

p
2

−

q
2

⩾
8
,

|

p

|

−

|

q

|

⩾
2

{\xi ^2} = l > 0,\,{p^2} - {q^2} \geqslant 8,\,|p| - |q| \geqslant 2

, and

r
i

≠
0

{r_i} \ne 0

. If

2
(
m
+
l
−
2
)
⩾

p
2

−

q
2

2(m + l - 2) \geqslant {p^2} - {q^2}

, then

ξ
\xi

cannot be represented by a smoothly embedded

2
2

-sphere. If

2
(
m
+
r
+
[
(
l
−
r
−
1
)

/

4
]
−
1
)
⩾

p
2

−

q
2

2(m + r + [(l - r - 1)/4] - 1) \geqslant {p^2} - {q^2}

for some

r
r

with

0
⩽
r
⩽
l
−
1

0 \leqslant r \leqslant l - 1

, then for a normal immersion

f
f

of a

2
2

-sphere representing

ξ
\xi

the number of points of positive self-intersection

d
f

⩾
[
(
l
−
r
−
1
)

/

4
]
+
1

{d_f} \geqslant [(l - r - 1)/4] + 1

.
An inequality for some nonnormal operators—extension to normal approximate eigenvalues
Authors: Fujii Masatoshi , Furuta Takayuki , Seo Yūki
Published: June 2010

An inequality of use in testing convergence of eigenvector calculations is extended. It is an improvement of Bernstein and Furuta’s results for selfajoint operators and dominant operators respectively.
A transitivity theorem for algebras of elementary operators
Authors: Magajna Bojan
Published: June 2010

Let

A

\mathcal {A}

be a

C

∗

{C^{\ast }}

-algebra and

E

\mathcal {E}

the algebra of all elementary operators on

A

\mathcal {A}

, and let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove a With right-arrow equals left-parenthesis a 1 comma ellipsis comma a Subscript n Baseline right-parenthesis comma ModifyingAbove b With right-arrow equals left-parenthesis b 1 comma ellipsis comma b Subscript n Baseline right-parenthesis element-of script upper A Superscript n">

a
→

=
(

a
1

,
…
,

a
n

)
,

b
→

=
(

b
1

,
…
,

b
n

)
∈

A

n

\vec a = ({a_1}, \ldots ,{a_n}),\;\vec b = ({b_1}, \ldots ,{b_n}) \in {\mathcal {A}^n}

. It is proved that

b
→

\vec b

is contained in the closure of the set

{
(
E

a
1

,
…
,
E

a
n

)
:
E
∈

E

}

\{ (E{a_1}, \ldots ,E{a_n}):E \in \mathcal {E}\}

if and only if each complex linear combination

∑

j
=
1

n

λ
j

b
j

\sum \nolimits _{j = 1}^n {{\lambda _j}} {b_j}

is contained in the closed two-sided ideal generated by

∑

j
=
1

n

λ
j

a
j

\sum \nolimits _{j = 1}^n {{\lambda _j}} {a_j}

. In particular, a bounded linear operator on

A

\mathcal {A}

preserves all closed two-sided ideals if and only if it is in the strong closure of

E

\mathcal {E}

.
Typical intersections of continuous functions with monotone functions
Authors: Hejný M.
Published: June 2010

For each parameter

Φ
\Phi

a typical continuous function intersects every monotone function in a

(
Φ
)

(\Phi )

-uniformly symmetrically porous set.
On a combinatorial problem associated with the odd order theorem
Authors: Glauberman George , Norton Simon P.
Published: June 2010

We prove a conjecture about finite fields that arose in Péterfalvi’s study of the Feit-Thompson Theorem.
Singer groups, an approach from a group of multipliers of even order
Authors: Ho Chat Yin
Published: June 2010

The order of a Sylow

2
2

-subgroup of a multiplier group of an abelian Singer group is at most half the order of the order of a Sylow

2
2

-subgroup of the automorphism group of the Singer group. We determine the situation when equality occurs. We also study the effect of a nontrivial Sylow

2
2

-subgroup of a multiplier group on the structure of the Singer group.
Double points on hyperbolic surfaces
Authors: Jørgensen Troels , Sandler Hanna
Published: June 2010

It is shown that every proper intersection point of two closed geodesics on an orientable hyperbolic surface is either a simultaneous double point of two closed geodesics of equal length or a quadruple point, and this phenomenon persists under deformations of the surface.
Every attractor of a flow on a manifold has the shape of a finite polyhedron
Authors: Günther Bernd , Segal Jack
Published: June 2010

It is shown that the class of compacta which can occur as attractors of continuous flows on topological manifolds coincides with the class of finite dimensional compacta having the shape of a finite polyhedron.
On the rationalization of the circle
Authors: Casacuberta Carles
Published: June 2010

We give an example showing that, for a nilpotent group

G
G

and a set of primes

P
P

, the

P
P

-localization homomorphism

l
:
G
→

G
P

l:G \to {G_P}

need not induce an isomorphism in cohomology with arbitrary (twisted)

Z

P

{{\mathbf {Z}}_P}

-module coefficients. From this fact we infer that, in the pointed homotopy category of connected CW-complexes, the inclusion of the subcategory of spaces whose higher homotopy groups are

Z

P

{{\mathbf {Z}}_P}

-modules and whose fundamental group is uniquely

P
′

{P’}

-radicable does not admit a left adjoint.
On strongly discrete subsets of 𝜔*
Authors: Rabus Mariusz
Published: June 2010

We prove that it is consistent with Martin’s Axiom and

¬

C
H

\neg \mathrm {CH}

that there is a strongly discrete subspace

A
⊆

ω

∗

A \subseteq {\omega ^{\ast }}

of cardinality

ℵ
1

{\aleph _1}

such that the closure of

A
A

is not homeomorphic with

β

ω
1

\beta {\omega _1}

. We also prove that MA and

¬

C
H

\neg \mathrm {CH}

imply that there is no convergent strongly discrete subset of

ω

∗

{\omega ^{\ast }}

.
Boundedness of the Riesz potential on a complete manifold with nonnegative Ricci curvature
Authors: Li Jiayu
Published: June 2010

In this paper we obtain a necessary and sufficient condition for the boundedness of the Riesz potential on a complete manifold with nonnegative Ricci curvature.
On the shape of the unit sphere in 𝑄(Δ)
Authors: Goldberg Lisa R.
Published: June 2010

We show that the unit sphere in the Banach space of

L
1

{L^1}

holomorphic quadratic differentials on the disk is weakly uniformly convex with exponent

1

/

2

1/2

at certain points.
Hilbert’s tenth problem for rational function fields in characteristic 2
Authors: Videla Carlos R.
Published: June 2010

In Hilbert’s Tenth problem for fields of rational functions over finite fields (Invent. Math. 103 (1991)) Pheidas showed that Hilbert’s Tenth problem over a field of rational functions with constant field a finite field of characteristic other than

2
2

is undecidable. We show that the same holds for characteristic

2
2

.
Countable sections for locally compact group actions. II
Authors: Kechris Alexander S.
Published: June 2010

In this paper we study the structure of the orbit equivalence relation induced by a Borel action of a second countable locally compact group on a standard Borel space.
Tychonoff’s theorem
Authors: Wright David G.
Published: June 2010

A simple proof of Tychonoff’s Theorem is given. The proof is, in spirit, much like Tychonoff’s original proof, which is also given.
Determinant type generalizations of Heinz-Kato theorem via Furuta inequality
Authors: Furuta Takayuki
Published: June 2010

A capital letter means a bounded linear operator on a complex Hilbert space

H
H

. By a nice application of the Furuta inequality, we give two kinds of determinant type generalizations (Theorems 1 and 2 in

§
1

\S 1

) of the famous and well-known Heinz-Kato theorem containing the terms

T
,

|

T

|

T,\;|T|

, and

|

T

∗

|

|{T^{\ast }}|

.
Weights in cohomology groups arising from hyperplane arrangements
Authors: Kim Minhyong
Published: June 2010

The formalism of weights allows very simple analysis of the cohomology of hyperplane complements in a uniform fashion for different cohomology theories. An

l
l

-adic analogue of Arnold’s conjecture on the torsion-freeness of these cohomology groups is one of the consequences.
A functional equation from probability theory
Authors: Baker John A.
Published: June 2010

The functional equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis left-parenthesis 1 right-parenthesis right-parenthesis EndLabel f left-parenthesis x right-parenthesis equals product Underscript j equals 1 Overscript upper N Endscripts left-bracket f left-parenthesis beta Subscript j Baseline x right-parenthesis right-bracket Superscript gamma Super Subscript j EndLayout">

(

(
1
)

)

f
(
x
)
=

∏

j
=
1

N

[
f
(

β
j

x
)
]

γ
j

\begin{equation}\tag {$(1)$} f(x) = \prod \limits _{j = 1}^N {{{[f({\beta _j}x)]}^{{\gamma _j}}}}\end{equation}

has been used by Laha and Lukacs (Aequationes Math. 16 (1977), 259-274) to characterize normal distributions. The aim of the present paper is to study (1) under somewhat different assumptions than those assumed by Laha and Lukacs by using techniques which, in the author’s opinion, are simpler than those employed by the afore-mentioned authors. We will prove, for example, that if

0
>

β
j

>
1

0 > {\beta _j} > 1

and

γ
j

>
0

{\gamma _j} > 0

for

1
≤
j
≤
N
,
∑

j
=
1

N

β
j
k

γ
j

=
1

1 \leq j \leq N, \sum {_{j = 1}^N\beta _j^k{\gamma _j} = 1}

, where k is a natural number,

f
:

R

→
[
0
,
+
∞
)

f:\mathbb {R} \to [0, + \infty )

, (1) holds for

x
∈

R

x \in \mathbb {R}

and

f

(
k
)

(
0
)

{f^{(k)}}(0)

exists then either

f
≡
0

f \equiv 0

or there exists a real constant c such that

f
(
x
)
=
exp
⁡
(
c

x
k

)

f(x) = \exp (c{x^k})

for all

x
∈

R

x \in \mathbb {R}

.
A norm on the fundamental group of non-Haken 3-manifolds
Authors: Jones Kerry N.
Published: June 2010

A canonical (presentation-independent) conjugacy-invariant norm is constructed on the fundamental group of any

3
3

-manifold which is orientable, irreducible, has infinite fundamental group, and contains no incompressible surface. More generally, this norm exists on any torsion-free group whose commutator quotient is finite. This norm is then computed explicitly in an example which shows that the induced metric on the group is not quasi-isometric to any word metric.
On the left ideal in the universal enveloping algebra of a Lie group generated by a complex Lie subalgebra
Authors: Tirao Juan
Published: June 2010

Let

G
0

{G_0}

be a connected Lie group with Lie algebra

g
0

{g_0}

and let h be a Lie subalgebra of the complexification g of

g
0

{g_0}

. Let

C
∞

(

G
0

)
h

{C^\infty }{({G_0})^h}

be the annihilator of h in

C
∞

(

G
0

)

{C^\infty }({G_0})

and let

A

=

A

(

C
∞

(

G
0

)
h

)

\mathcal {A} = \mathcal {A}({C^\infty }{({G_0})^h})

be the annihilator of

C
∞

(

G
0

)
h

{C^\infty }{({G_0})^h}

in the universal enveloping algebra

U

(
g
)

\mathcal {U}(g)

of g. If h is the complexification of the Lie algebra

h
0

{h_0}

of a Lie subgroup

H
0

{H_0}

of

G
0

{G_0}

then

A

=

U

(
g
)
h

\mathcal {A} = \mathcal {U}(g)h

whenever

H
0

{H_0}

is closed, is a known result, and the point of this paper is to prove the converse assertion. The paper has two distinct parts, one for

C
∞

{C^\infty }

, the other for holomorphic functions. In the first part the Lie algebra

h
¯

0

{\bar h_0}

of the closure of

H
0

{H_0}

is characterized as the annihilator in

g
0

{g_0}

of

C
∞

(

G
0

)
h

{C^\infty }{({G_0})^h}

, and it is proved that

h
0

{h_0}

is an ideal in

h
¯

0

{\bar h_0}

and that

h
¯

0

=

h
0

⊕
v

{\bar h_0} = {h_0} \oplus v

where v is an abelian subalgebra of

h
¯

0

{\bar h_0}

. In the second part we consider a complexification G of

G
0

{G_0}

and assume that h is the Lie algebra of a closed connected subgroup H of G. Then we establish that

A

(

O

(
G

)
h

)
=

U

(
g
)
h

\mathcal {A}(\mathcal {O}{(G)^h}) = \mathcal {U}(g)h

if and only if

G

/

H

G/H

has many holomorphic functions. This is the case if

G

/

H

G/H

is a quasi-affine variety. From this we get that if H is a unipotent subgroup of G or if G and H are reductive groups then

A

(

C
∞

(

G
0

)
h

)
=

U

(
g
)
h

\mathcal {A}({C^\infty }{({G_0})^h}) = \mathcal {U}(g)h

.
Hypersurfaces satisfying the equation Δ𝑥=𝑅𝑥+𝑏
Authors: Park Joonsang
Published: June 2010

We prove that a hypersurface in a space form or in Lorentzian space whose immersion

x
x

satisfies

Δ
x
=
R
x
+
b

\Delta x = Rx + b

is minimal or isoparametric. In particular, we locally classify such hypersurfaces which are not minimal.
On the translates of a set which meet it in a set of positive measure
Authors: Foran James
Published: June 2010

Given a singular Borel regular measure

m
a

{m_a}

on

R
n

{R^n}

and a Borel subset E of

R
n

{R^n}

, it is shown that the set of vectors x for which

m
a

(
(
E
+
x
)
∩
E
)
>
0

{m_a}((E + x) \cap E) > 0

is of Lebesgue measure 0. This fact is then used to show that subsets of finite, nonzero, Hausdorff s-measure are nonmeasurable sets with respect to any approximating measure

s
−

m
δ

s - {m_\delta }

.
Generic properties of the Nicoletti and Floquet boundary value problems
Authors: Bielawski Dariusz
Published: June 2010

The existence of solutions and the existence and uniqueness of solutions are considered as generic properties of an abstract boundary value problem. Applications to the Nicoletti and the Floquet boundary value problems are given.
High-dimensional knots with 𝜋₁≅𝑍 are determined by their complements in one more dimension than Farber’s range
Authors: Richter William
Published: June 2010

The surgery theory of Browder, Lashof, and Shaneson reduces the study of high-dimensional smooth knots

Σ
n

→

S

n
+
2

{\Sigma ^n} \to {S^{n + 2}}

with

π
1

≅

Z

{\pi _1} \cong \mathbb {Z}

to homotopy theory. We apply Williams’s Poincaré embedding theorem to a highly connected Seifert surface. Then such knots are determined by their complements if the

Z

\mathbb {Z}

-cover of the complement is

[
(
n
+
2
)

/

3
]

[(n + 2)/3]

-connected; we improve Farber’s work by one dimension.
The Takaga operator, Bernoulli sequences, smoothness conditions and fractal curves
Authors: Deliu Anca , Wingren Peter
Published: June 2010

We map Lipschitz spaces and functions of bounded variation by the operator
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma colon phi left-parenthesis x right-parenthesis right-arrow sigma-summation Underscript 0 Overscript normal infinity Endscripts 2 Superscript negative n Baseline phi left-parenthesis 2 Superscript n Baseline x right-parenthesis comma x element-of left-bracket 0 comma 1 right-bracket">

σ
:
φ
(
x
)
→

∑
0
∞

2

−
n

φ
(

2
n

x
)
,
x
∈
[
0
,
1
]

\sigma :\varphi (x) \to \sum \nolimits _0^\infty {{2^{ - n}}} \varphi ({2^n}x),x \in [0,1]

, and we estimate the Hausdorff measure of

σ
(
φ
)

\sigma (\varphi )

. We furthermore introduce a class of continuous and nowhere differentiable functions on [0,1] which we call

T

\mathcal {T}

. We make a refined analysis of the fractal and smoothness properties of the functions in

T

\mathcal {T}

and study the relationship between the two. We show that all the functions in

T

\mathcal {T}

have box dimension equal to

1
2

\frac {1}{2}

, with respect to the dimension family

{
t

/

(
log
⁡

1
t

)
s

:
s
∈

R

+

}

\{ t/{(\log \frac {1}{t})^s}:s \in {\mathbb {R}^ + }\}

, but that their order of smoothness covers a wide range.
On induced characters
Authors: Berkovich Yakov
Published: June 2010

Suppose that H is a normal subgroup of a finite group G,

φ
∈

Irr

(
H
)

\varphi \in {\text {Irr}}(H)

, and

Irr

(

φ
G

)

{\text {Irr}}({\varphi ^G})

is the set of all irreducible constituents of the induced character

φ
G

{\varphi ^G}

. If

|

Irr

(

φ
G

)

|

>

|

G
:
H

|

/

4

|{\text {Irr}}({\varphi ^G})| > |G:H|/4

then

G

/

H

G/H

is solvable.
Near-rings associated with matched pairs on ring modules
Authors: Maxson C. J. , van der Walt A. P. J.
Published: June 2010

Let G be a module over a ring R, let

C

=
{

C
i

}
,
i
∈
I

\mathcal {C} = \{ {C_i}\}, i \in I

, be a family of submodules of G, and let

H

=
{

H
i

}
,
i
∈
I

\mathcal {H} = \{ {H_i}\}, i \in I

, where

H
i

{H_i}

is a subgroup of

Hom
R

⁡
(

C
i

,
G
)

\operatorname {Hom}_R({C_i},G)

with certain properties. To each such pair

(

C

,

H

)

(\mathcal {C},\mathcal {H})

, a near-ring

M
(

C

,

H

)

M(\mathcal {C},\mathcal {H})

is associated, which is a generalization of the near-ring of homogeneous functions determined by (G, R). The transfer of information from module properties of

G
R

{G_R}

reflected in

(

C

,

H

)

(\mathcal {C},\mathcal {H})

to structural properties of

M
(

C

,

H

)

M(\mathcal {C},\mathcal {H})

is investigated.
Comparison of lognormal population means
Authors: Chen Hanfeng
Published: June 2010

Comparisons of two lognormal population means are investigated. For large samples, the conventional test for significance of the population means is the ordinary Student t-test with normal critical value. By Chen and Loh’s result (Ann. Statist. 20 (1992)), however, the transformed t-test based on log-data is asymptotically more powerful than the ordinary t-test. In this paper, a new power transformation (hence a new transformed t-test) is proposed. The new transformed t-test is proven to be asymptotically more powerful than the one based on log-data. Both small sample and large sample properties of the proposed estimate for the power-transformation parameter are studied. A simulation study shows that the advantages of the new test over the t-test based on log-data are overwhelming and evident for sizes of the two samples as small as 20 and 30, or even 10 and 15. The simulation results also show that the new test has greater asymptotic power than Rao’s efficient score test.
A characterization of 𝜎-symmetrically porous symmetric Cantor sets
Authors: Evans Michael J. , Humke Paul D. , Saxe Karen
Published: June 2010

The purpose of this paper is to characterize those symmetric Cantor sets which are

σ
\sigma

-symmetrically porous in terms of a defining sequence of deleted proportions. In contrast to other notions of porosity, a symmetric Cantor set can be

σ
\sigma

-symmetrically porous without being symmetrically porous.
Linked pairs of contractible polyhedra in 𝑆ⁿ
Authors: Guilbault Craig R.
Published: June 2010

B. Mazur has described a geometrically linked pair of compact contractible polyhedra in

S
4

{S^4}

. In this note we exhibit an even more extreme type of linking between compact contractible polyhedra in

S
n

,
n
≥
5

{S^n},n \geq 5

.
Sur un théorème de Mitiagin, Rolewicz et Żelazko
Authors: Allouche M. , Couture R. , Tsai C. T.
Published: June 2010

We establish the analogue of a theorem of Mitiagin, Rolewicz, and Zelazko for a family of nonmetrizable topological algebras. These algebras are composed of infinitely differentiable functions and are, as topological vector spaces, Silva spaces. The condition of m-convexity is replaced by one on the collection of bounded subsets of the algebra. Nous démontrons l’analogue du théorème de Mitiagin, Rolewicz et Zelazko pour une famille d’algèbres topologiques non-métrisables. Ces algèbres sont formées de fonctions indéfiniment dérivables et sont, en tant qu’espaces vectoriels topologiques, des espaces de Silva. La condition de m-convexité est remplacée par une autre portant sur l’ensemble des parties bornées de l’algèbre.
Hölder-continuity of the solutions for operators which are a sum of squares of vector fields plus a potential
Authors: Citti G. , Di Fazio G.
Published: June 2010

In this paper we study the local Hölder-regularity of weak solutions to

L

u
+
V
u
=
0

\mathcal {L}u + Vu = 0

where

L

\mathcal {L}

is a Hörmander hypoelliptic operator and the potential V belongs to a new class of functions which is the natural extension of Morrey spaces to this situation. We improve a recent result of Citti, Garofalo, and Lanconelli.
On a complementarity problem in Banach space
Authors: Nanda Sudarsan
Published: June 2010

The paper gives a result on the existence and uniqueness of solutions for the complementarity problems associated with hemicontinuous monotone mappings of convex cones. We correct the assumptions and proof given in an earlier paper of A. T. Dash and S. Nanda [A complementarity problem in mathematical programming in Banach space, J. Math. Anal. Appl. 98 (1984), 328-331].
Borsuk’s antipodal and fixed-point theorems for set-valued maps
Authors: Chang Shiow-Yu
Published: June 2010

The purpose of this paper is to obtain the extensions of Borsuk’s antipodal and fixed-point theorems for set-valued maps.
Existence of smooth invariant measures for geodesic flows of foliations of Riemannian manifolds
Authors: Walczak PawełG.
Published: June 2010

We construct a nontrivial smooth finite measure invariant under the geodesic flow of a foliation

F

\mathcal {F}

of a compact Riemannian manifold

M
M

assuming that the transverse mean curvature of

F

\mathcal {F}

is distributed "nicely" along some leaf geodesics.
A counterexample in Jacobson radicals
Authors: Schultz Phillip
Published: June 2010

A plausible conjecture states that an element of the Jacobson radical of the endomorphism ring of an abelian p-group increases the height of any non-zero element of the socle. I construct a counter-example.
A gap theorem for ends of complete manifolds
Authors: Cai Mingliang , Colding Tobias Holck , Yang DaGang
Published: June 2010

Let

(

M
n

,
o
)

({M^n},o)

be a pointed open complete manifold with Ricci curvature bounded from below by

−
(
n
−
1
)

Λ
2

- (n - 1){\Lambda ^2}

(for

Λ
≥
0

\Lambda \geq 0

) and nonnegative outside the ball

B
(
o
,
a
)

B(o,a)

. It has recently been shown that there is an upper bound for the number of ends of such a manifold which depends only on

Λ
a

\Lambda a

and the dimension n of the manifold

M
n

{M^n}

. We will give a gap theorem in this paper which shows that there exists an

ε
=
ε
(
n
)
>
0

\varepsilon = \varepsilon (n) > 0

such that

M
n

{M^n}

has at most two ends if

Λ
a
≤
ε
(
n
)

\Lambda a \leq \varepsilon (n)

. We also give examples to show that, in dimension

n
≥
4

n \geq 4

, such manifolds in general do not carry any complete metric with nonnegative Ricci Curvature for any

Λ
a
>
0

\Lambda a > 0

.
Triangular UHF algebras over arbitrary fields
Authors: Baker R. L.
Published: June 2010

Let K be an arbitrary field. Let

(

q
n

)

({q_n})

be a sequence of positive integers, and let there be given a family

{

Ψ

n
m

|

n
≥
m
}

\{ {\Psi _{nm}}|n \geq m\}

of unital K-monomorphisms
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Psi Subscript n m Baseline colon upper T Subscript q Sub Subscript m Subscript Baseline left-parenthesis upper K right-parenthesis right-arrow upper T Subscript q Sub Subscript n Subscript Baseline left-parenthesis upper K right-parenthesis">

Ψ

n
m

:

T

q
m

(
K
)
→

T

q
n

(
K
)

{\Psi _{nm}}:{T_{{q_m}}}(K) \to {T_{{q_n}}}(K)

such that

Ψ

n
p

Ψ

p
m

=

Ψ

n
m

{\Psi _{np}}{\Psi _{pm}} = {\Psi _{nm}}

whenever

m
≤
n

m \leq n

, where

T

q
n

(
K
)

{T_{{q_n}}}(K)

is the K-algebra of all

q
n

×

q
n

{q_n} \times {q_n}

upper triangular matrices over K. A triangular UHF (TUHF) K-algebra is any K-algebra that is K-isomorphic to an algebraic inductive limit of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper T equals limit Underscript right-arrow Endscripts left-parenthesis upper T Subscript q Sub Subscript n Subscript Baseline left-parenthesis upper K right-parenthesis semicolon normal upper Psi Subscript n m Baseline right-parenthesis">

T

=

lim
→

(

T

q
n

(
K
)
;

Ψ

n
m

)

\mathcal {T} = \lim \limits _ \to ({T_{{q_n}}}(K);{\Psi _{nm}})

. The first result of the paper is that if the embeddings

Ψ

n
m

{\Psi _{nm}}

satisfy certain natural dimensionality conditions and if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S equals limit Underscript right-arrow Endscripts left-parenthesis upper T Subscript p Sub Subscript n Subscript Baseline left-parenthesis upper K right-parenthesis semicolon normal upper Phi Subscript n m Baseline right-parenthesis">

S

=

lim
→

(

T

p
n

(
K
)
;

Φ

n
m

)

\mathcal {S} = \lim \limits _ \to ({T_{{p_n}}}(K);{\Phi _{nm}})

is an arbitrary TUHF K-algebra, then

S

\mathcal {S}

is K-isomorphic to

T

\mathcal {T}

, only if the supernatural number,

N
[
(

p
n

)
]

N[({p_n})]

, of

(

q
n

)

({q_n})

is less than or equal to the supernatural number,

N
[
(

p
n

)
]

N[({p_n})]

, of

(

p
n

)

({p_n})

. Thus, if the embeddings

Φ

n
m

{\Phi _{nm}}

also satisfy the above dimensionality conditions, then

S

\mathcal {S}

is K-isomorphic to

T

\mathcal {T}

, only if

N
[
(

p
n

)
]
=
N
[
(

q
n

)
]

N[({p_n})] = N[({q_n})]

. The second result of the paper is a nontrivial "triangular" version of the fact that if p, q are positive integers, then there exists a unital K-monomorphism

Φ
:

M
q

(
K
)
→

M
p

(
K
)

\Phi :{M_q}(K) \to {M_p}(K)

, only if

q

|

p

q|p

. The first result of the paper depends directly on the second result.
Perfect images of Čech-analytic spaces
Authors: Hansell R. W. , Pan Shiho
Published: June 2010

A completely regular Hausdorff space X is Čech-analytic if X is the result of the Souslin operation applied to the locally compact sets in some (equivalently, any) compactification. We prove that Čech-analytic spaces are preserved under general perfect maps, thus settling a question raised by the late Z. Frolík.
Local cohomology of Rees algebras and Hilbert functions
Authors: Johnston Bernard , Verma Jugal
Published: June 2010

Let I be an ideal primary to the maximal ideal in a local ring. We utilize two well-known theorems due to J.-P. Serre to prove that the difference between the Hilbert function and the Hilbert polynomial of I is the alternating sum of the graded pieces of the graded local cohomology (with respect to its positively-graded ideal) of the Rees ring of I. This gives new insight into the higher Hilbert coefficients of I. The result is inspired by one due to J. D. Sally in dimension two and is implicit in a paper by D. Kirby and H. A. Mehran, where very different methods are used.
Continuity of random derivations
Authors: Velasco M. V. , Villena A. R.
Published: June 2010

In this paper we extend the well-known Johnson-Sinclair theorem in a stochastic sense showing that stochastically derivative linear random operators on semisimple Banach algebras are stochastically continuous. Besides, we prove that probably derivative linear random operators on semisimple Banach algebras are probably continuous.
The arc length of the lemniscate {|𝑝(𝑧)|=1}
Authors: Borwein Peter
Published: June 2010

We show that the length of the set
\[

{

z
∈

C

:

|

∏

i
=
1

n

(
z
−

α
i

)

|

=
1

}

\left \{ {z \in \mathbb {C}:|\prod \limits _{i = 1}^n {(z - {\alpha _i})| = 1} } \right \}

\]
is at most

8
π
e
n

8\pi en

. This gives the correct rate of growth in a long-standing open problem of Erdös, Herzog, and Piranian and improves the previous bound of

74

n
2

74{n^2}

due to Pommerenke.
Traces and subadditive measures on projections in 𝐽𝐵𝑊-algebras and von Neumann algebras
Authors: Bunce L. J. , Hamhalter J.
Published: June 2010

Let

P
(
M
)

P(M)

be the projection lattice of an arbitrary JBW-algebra or von Neumann algebra M. It is shown that the tracial states of M correspond by extension precisely to the subadditive probability measures on

P
(
M
)

P(M)

. The analogous result for normal semifinite traces is also proved.
On crossed products of Hopf algebras
Authors: Lorenz Maria E. , Lorenz Martin
Published: June 2010

Let

B
=
A

#
σ

H

B = A{\# _\sigma }H

denote a crossed product of the associative algebra A with the Hopf algebra H. We investigate the weak dimension and the global dimension of B and show that

wdim

B
≤

wdim

H
+

wdim

A

{\text {wdim}}\;B \leq {\text {wdim}}\;H + {\text {wdim}}\;A

and

l.gldim

B
≤

r.gldim

H
+

l.gldim

A

{\text {l.gldim}} \; B \leq {\text {r.gldim}} \; H + {\text {l.gldim}} \; A

.
On the rank of an element of a free Lie algebra
Authors: Shpilrain Vladimir
Published: June 2010

Let L be a free Lie algebra over an arbitrary field K, and let

{

x
1

,
…
,

x
n

,
…
}
,
n
≥
2

\{ {x_1}, \ldots ,{x_n}, \ldots \} ,n \geq 2

, be a free basis of L. We define the rank of an element u of L as the least number of free generators on which the image of u under an arbitrary automorphism of L can depend. We prove that for a homogeneous element u of degree

m
≥
2

m \geq 2

, to have rank

n
≥
2

n \geq 2

is equivalent to another property which in the most interesting and important case when the algebra

L
=

L
n

L = {L_n}

has a finite rank

n
≥
2

n \geq 2

looks as follows: an arbitrary endomorphism

ϕ
\phi

of

L
n

{L_n}

is an automorphism if and only if u belongs to

(
ϕ
(

L
n

)

)
m

{(\phi ({L_n}))^m}

. This yields in particular a simple algorithm for finding the rank of a homogeneous element and also for finding a particular automorphic image of this element realizing the rank.
Local derivation of nest algebras
Authors: Zhu Jun
Published: June 2010

We show that every strongly continuous local derivation on a nest algebra is a derivation.
Gaussian estimates and holomorphy of semigroups
Authors: Ouhabaz El-Maati
Published: June 2010

We show that if a selfadjoint semigroup T on

L
2

(
Ω
)

{L^2}(\Omega )

satisfies a Gaussian estimate
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper T left-parenthesis t right-parenthesis f EndAbsoluteValue less-than-or-equal-to upper M upper G left-parenthesis b t right-parenthesis StartAbsoluteValue f EndAbsoluteValue comma 0 less-than-or-equal-to t less-than-or-equal-to 1 comma f element-of upper L squared left-parenthesis normal upper Omega right-parenthesis">

|

T
(
t
)
f

|

≤
M
G
(
b
t
)

|

f

|

,
0
≤
t
≤
1
,
f
∈

L
2

(
Ω
)

|T(t)f| \leq MG(bt)|f|,0 \leq t \leq 1,f \in {L^2}(\Omega )

(where

G
=
G

(
t

)

t
≥
0

G = G{(t)_{t \geq 0}}

is the Gaussian semigroup on

L
2

(

R
N

)

{L^2}({R^N})

and

Ω
\Omega

is an open set of

R
N

{R^N}

), then T defines a holomorphic semigroup of angle

π
2

\frac {\pi }{2}

on

L
p

(
Ω
)

{L^p}(\Omega )

. We obtain by duality the same result on

C
0

(
Ω
)

{C_0}(\Omega )

. Applications to uniformly elliptic operators and Schrödinger operators are given.
A remark on distribution of zeros of solutions of linear differential equations
Authors: Zheng Jian Hua
Published: June 2010

The main purpose of this paper is to prove a sharp estimate of the order

ρ
(
w
)

\rho (w)

of a transcendental solution w in the complex plane of an n th-order linear differential equation with polynomial coefficients in terms of the distribution of its Stokes rays, under the assumption that zero is not a Nevanlinna deficient value of w. If, in addition, there are only two Stokes rays and if all the solutions of the equation have order at most

ρ
(
w
)

\rho (w)

, then we can conclude that the coefficients of the equation are all constants.
A generalization of outer parallel sets of a convex set
Authors: Sangwine-Yager J. R.
Published: June 2010

An outer parallel set is formed by adding outer normals of fixed length to the boundary points of a convex set. In this paper the length of the outer normals varies as a function of the direction. This construction yields a geometric interpretation of the dual quermassintegrals of Lutwak and bounds on certain mixed volumes. An inequality of Firey will follow from one of the mixed volume bounds.
Vassiliev invariants and similarity of knots
Authors: Ohyama Yoshiyuki
Published: June 2010

We show that for any knot K and any natural number n, we can construct infinitely many knots, all of whose finite type invariants of order at most n coincide with those of K.
A note on Morita equivalence of twisted 𝐶*-dynamical systems
Authors: Kaliszewski S.
Published: June 2010

We present an elementary proof that every twisted

C
∗

{C^ \ast }

-dynamical system is Morita equivalent to an ordinary system. As a corollary we prove the equivalence

C
0

(
G

/

H
,
A
)

×

α
~

,

u
~

G
∼
A

×

α
,
u

H

{C_0}(G/H,A){ \times _{\tilde \alpha ,\tilde u}}G \sim A{ \times _{\alpha ,u}}H

for Busby-Smith twisted dynamical systems, generalizing an important result of Green.
Portraits of frames
Authors: Aldroubi Akram
Published: June 2010

We introduce two methods for generating frames of a Hilbert space

H

\mathcal {H}

. The first method uses bounded operators on

H

\mathcal {H}

. The other method uses bounded linear operators on

l
2

{l_2}

to generate frames of

H

\mathcal {H}

. We characterize all the mappings that transform frames into other frames. We also show how to construct all frames of a given Hilbert space

H

\mathcal {H}

, starting from any given one. We illustrate the results by giving some examples from multiresolution and wavelet theory.
The theorems of Carathéodory and Gluskin for 0<𝑝<1
Authors: Bastero Jesús , Bernués Julio , Peña Ana
Published: June 2010

In this note we prove the p-convex analogue of both Caratheodory’s convexity theorem and Gluskin’s theorem concerning the diameter of Minkowski compactum.
Oscillation and nonoscillation criteria for delay differential equations
Authors: Elbert Á. , Stavroulakis I. P.
Published: June 2010

Oscillation and nonoscillation criteria for the first-order delay differential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x prime left-parenthesis t right-parenthesis plus p left-parenthesis t right-parenthesis x left-parenthesis tau left-parenthesis t right-parenthesis right-parenthesis equals 0 comma t greater-than-or-equal-to t 0 comma tau left-parenthesis t right-parenthesis greater-than t comma">

x
′

(
t
)
+
p
(
t
)
x
(
τ
(
t
)
)
=
0
,

t
≥

t
0

,
τ
(
t
)
>
t
,

x’(t) + p(t)x(\tau (t)) = 0,\quad t \geq {t_0},\tau (t) > t,

\]
are established in the case where
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript tau left-parenthesis t right-parenthesis Superscript t Baseline p left-parenthesis s right-parenthesis d s greater-than-or-equal-to StartFraction 1 Over e EndFraction and limit Underscript t right-arrow normal infinity Endscripts integral Subscript tau left-parenthesis t right-parenthesis Superscript t Baseline p left-parenthesis s right-parenthesis d s equals StartFraction 1 Over e EndFraction period">

∫

τ
(
t
)

t

p
(
s
)
d
s
≥

1
e

and

lim

t
→
∞

∫

τ
(
t
)

t

p
(
s
)
d
s
=

1
e

.

\int _{\tau (t)}^t {p(s)ds \geq \frac {1}{e}\quad {\text {and}}\quad \lim \limits _{t \to \infty } \int _{\tau (t)}^t {p(s)ds = \frac {1}{e}.} }

\]
The existence of flat covers over Noetherian rings of finite Krull dimension
Authors: Xu Jin Zhong
Published: June 2010

Bass characterized the rings R with the property that every left R-module has a projective cover. These are the left perfect rings. A ring is left perfect if and only if the class of projective R-modules coincides with the class of flat R-modules, so the projective covers over these rings are flat covers. This prompts the conjecture that over any ring R, every left R-module has a flat cover. Known classes of rings for which the conjecture holds include Von Neumann regular rings (trivially), the left perfect rings (Bass), Prufer domains (Enochs), and then more generally, all right coherent rings of finite weak global dimension (Belshoff, Enochs, Xu). In this paper we show that the conjecture holds for all commutative Noetherian rings of finite Krull dimension and so for all local rings and all coordinate rings of affine algebraic varieties.
The module of derivations of a Stanley-Reisner ring
Authors: Brumatti Paulo , Simis Aron
Published: June 2010

An explicit description is given of the module

Der
⁡
(
k
[

X
_

]

/

I
,
k
[

X
_

]

/

I
)

\operatorname {Der}(k[\underline X ]/I,k[\underline X ]/I)

of the derivations of the residue ring

k
[

X
_

]

/

I

k[\underline X]/I

, where I is an ideal generated by monomials whose exponents are prime to the characteristic of the field k (this includes the case of square free monomials in any characteristic and the case of arbitrary monomials in characteristic zero). In the case where I is generated by square free monomials, this description is interpreted in terms of the corresponding abstract simplicial complex

Δ
\Delta

. Sharp bounds for the depth of this module are obtained in terms of the depths of the face rings of certain subcomplexes

Δ
i

{\Delta _i}

related to the stars of the vertices

v
i

{v_i}

of

Δ
\Delta

. The case of a Cohen-Macaulay simplicial complex

Δ
\Delta

is discussed in some detail: it is shown that

Der
⁡
(
k
[
Δ
]
,
k
[
Δ
]
)

\operatorname {Der}(k[\Delta ],k[\Delta ])

is a Cohen-Macaulay module if and only if

depth

Δ
i

≥
dim
⁡
Δ
−
1

{\text {depth}}{\Delta _i} \geq \dim \Delta - 1

for every vertex

v
i

{v_i}

. A measure of triviality of the complexes

Δ
i

{\Delta _i}

is introduced in terms of certain star corners of

v
i

{v_i}

. A curious corollary of the main structural result is an affirmative answer in the present context to the conjecture of Herzog-Vasconcelos on the finite projective dimension of the

k
[

X
_

]

/

I

k[\underline X ]/I

-module

Der
⁡
(
k
[

X
_

]

/

I
,
k
[

X
_

]

/

I
)

\operatorname {Der}(k[\underline X ]/I,k[\underline X ]/I)

.