bluuebunny/crossref_metadata_2025 huggingface dataset:
A short proof of Macdonald’s conjecture for the root systems of type 𝐴
Authors: Stembridge John R.
Published: June 2010

We give a new proof of I. G. Macdonald’s conjecture for the root systems of type

A
A

(or equivalently, the equal parameter

q
q

-Dyson Theorem) that is short, elementary and direct. We also give a short proof of the equal parameter case of a constant term identity due to D. Bressoud and I. Goulden.
A conjecture of S. Chowla via the generalized Riemann hypothesis
Authors: Mollin R. A. , Williams H. C.
Published: June 2010

S. Chowla conjectured that if

p
=

m
2

+
1

p = {m^2} + 1

is prime and

m
>
26

m > 26

, then

h
K

{h_K}

, the class number of

K
=
Q
(

p

)

K = Q(\sqrt p )

, is greater than 1. We prove this conjecture under the assumption of the Riemann hypothesis for

ς
K

\varsigma K

, the zeta function of

K
K

, i.e. the generalized Riemann hypothesis (GRH).
Smoothing curves by reflexive sheaves
Authors: Bolondi Giorgio
Published: June 2010

We give smoothing theorems for singular projective curves, by using curvilinear sheaves. This allows us to smooth reducible curves obtained by liaison addition.
A new proof of McKenna’s theorem
Authors: Zeng Guang Xin
Published: June 2010

Using a new and simpler method, the following result is shown: Let

(
K
,
C
)

(K,C)

be a formally real field with core

C
C

which has only a finite number of orderings. Then

(
K
,
C
)

(K,C)

has the Weak Hilbert Property if and only if

K
K

is dense in every real closure of

(
K
,
C
)

(K,C)

. This result contains the main theorem of McKenna in [1]
Fourier-Laplace transforms and the Bergman spaces
Authors: Saitoh Saburou
Published: June 2010

The Fourier-Laplace transforms on

R
n

(
n
≥
2
)

{R^n}(n \geq 2)

whose images belong to the Bergman spaces are investigated from the point of view of a general theory of integral transforms. The central problems are to give the expressions of the Bergman kernels in terms of the Fourier-Laplace transforms, and to investigate the relationship between the domains and the ranges in the expressions.
Large roots yield large coefficients. An addendum to: “The roots of a polynomial vary continuously as a function of the coefficients” [Proc. Amer. Math. Soc. 100 (1987), no. 2, 390–392; MR0884486 (88h:30006)]
Authors: Harris Gary , Martin Clyde
Published: June 2010

We provide the details to the argument that the map

σ
^

\hat \sigma

, defined in the above named paper, is open. We do this by including two trivial arguments that a polynomial with a big root must also have a big coefficient.
A generalization of the Poincaré-Birkhoff theorem
Authors: Winkelnkemper H. E.
Published: June 2010

We substitute Poincaré’s twist hypothesis by the weakest possible topological one: that the homeomorphism in question not be conjugate to a translation.
The gap between 𝑐𝑚𝑝𝑋 and 𝑑𝑒𝑓𝑋 can be arbitrarily large
Authors: Kimura Takashi
Published: June 2010

We give an example of a separable metrizable space

X
X

with def

X
−

cmp

X
=
n

X - {\text {cmp}}\;X = n

for every

n
∈

N

n \in \mathbb {N}

.
An elementary section of a bundle
Authors: Rigas A.
Published: June 2010

We use the Cayley algebra and triality to provide an explicit section of a principal

G
2

{G_2}

-bundleover

S
7

{S^7}

. This section is the basic ingredient for a direct, elementary, proof that

π
6

G
2

≅

Z

3

,

π
6

S
U
(
3
)
≅

Z

6

{\pi _6}{G_2} \cong {{\mathbf {Z}}_3},{\pi _6}SU(3) \cong {{\mathbf {Z}}_6}

and

π
6

S
3

≅

Z

12

{\pi _6}{S^3} \cong {{\mathbf {Z}}_{12}}
Weakly normal filters and the closed unbounded filter on 𝑃_{𝜅}𝜆
Authors: Abe Yoshihiro
Published: June 2010

Assuming that

κ
\kappa

is supercompact and

λ
\lambda

is inaccessible, we present two isomorphic fine measures on

P
κ

λ

{P_\kappa }\lambda

containing the closed unbounded filter. Some remarks on the (strongly) closed unbounded filter and weakly normal filters are added.
On the Young-Fenchel transform for convex functions
Authors: Beer Gerald
Published: June 2010

Let

Γ
(
X
)

\Gamma (X)

be the proper lower semicontinuous convex functions on a reflexive Banach space

X
X

. We exhibit a simple Vietoris-type topology on

Γ
(
X
)

\Gamma (X)

, compatible with Mosco convergence of sequences of functions, with respect to which the Young-Fenchel transform (conjugate operator) from

Γ
(
X
)

\Gamma (X)

to

Γ
(

X
∗

)

\Gamma ({X^*})

is a homeomorphism. Our entirely geometric proof of the bicontinuity of the transform halves the length of Mosco’s proof of sequential bicontinuity, and produces a stronger result for nonseparable spaces.
Moment problems for compact sets
Authors: Chandler J. D.
Published: June 2010

The solvability of the Hausdorff moment problem for an arbitrary compact subset of Euclidean

n
n

-space is shown to be equivalent to the nonnegativity of a family of quadratic forms derived from the given moment sequence and the given compact set. A variant theorem for the one-dimensional case and an analogous theorem for the trigonometric moment problem are also given. The one-dimensional theorems are similar to theorems of Kreĭn and Nudel’man [11], but the proofs, unlike those in [11], do not depend on the existence of a standard form for polynomials which are nonnegative on a given compact set.
A new lattice for the Hall-Janko group
Authors: Lindsey J. H.
Published: June 2010

Six by six matrix generators with entries in the rational numbers adjoined by a fifth root of unity, and most of these entries 0, are given for the proper central extension of the cyclic group of order 2 by the Hall-Janko group. They are found to preserve a

24
24

-dimensional lattice giving a packing of unit spheres with density about one fourth that of the Leach lattice.
Yet more on the differentiability of convex functions
Authors: Rainwater John
Published: June 2010

Generic differentiability theorems are obtained for convex functions which are defined and locally Lipschitzian on the convex subset

N
(
C
)

N(C)

of nonsupport points of a closed convex subset

C
C

of a Banach space

E
E

, which is assumed to be either an Asplund space (for Fréchet differentiability) or to be weakly compactly generated (for Gateaux differentiability).
The holomorphic extension of CR functions near a point of higher type
Authors: Boggess A.
Published: June 2010

Suppose

M
M

is a submanifold of

C
n

{C^n}

with real codimension at least one. A geometric description is given of the local hull of holomorphy of an open subset of

M
M

which contains a point of higher type in which all Hormander numbers are the same. This result is proved as a consequence of examining the relationship between the hypoanalytic wave front sets of CR functions on

M
M

and CR extension to a manifold of one higher dimension than

M
M

.
Closed sets without measurable matching
Authors: Laczkovich M.
Published: June 2010

We construct a rectangle in the unit square such that its perimeter contains a matching (i.e. the graph of a bijection of the unit interval onto itself) but does not contain a Borel matching or a matching measurable with respect to the linear measure.
Topological invariance of weights for weighted homogeneous isolated singularities in 𝐶³
Authors: Saeki Osamu
Published: June 2010

We show that the weights of a weighted homogeneous polynomial

f
f

in

C

3

{{\mathbf {C}}^3}

with an isolated singularity are local topological invariants of

(

C

3

,

f

−
1

(
0
)

)

\left ( {{{\mathbf {C}}^3},{f^{ - 1}}\left ( 0 \right )} \right )

at the origin.
A Kähler deformation of 𝐶𝑃ⁿ
Authors: Mori H. , Watanabe Y.
Published: June 2010

We shall construct a one-parameter family of complete Kähler metrics on the

n
n

-dimensional complex projective space

C

P
n

C{P^n}

, including the Fubini-study metric. We shall get, as a corollary, many noncanonical

S

C
m

S{C^m}

Kähler structures on

C

P
n

C{P^n}

.
Borel and analytic one-one parametrizations of the countable sets of reals
Authors: Becker Howard
Published: June 2010

We consider whether or not the parametrizations of the title exist. In the Borel case, they do not exist. In the analytic case, this is independent of ZFC.
The virtual 𝑍-representability of certain 3-manifold groups
Authors: Baker Mark D.
Published: June 2010

We use results on the cohomology of principal congruence subgroups of

PS

L

2

(

Z

[
ω
]
)
,

ω
2

+
ω
+
1
=
0

{\text {PS}}{{\text {L}}_2}({\mathbf {Z}}[\omega ]),{\omega ^2} + \omega + 1 = 0

, to prove the existence of a large class of closed, orientable

3
3

-manifolds with virtually

Z

{\mathbf {Z}}

-representable fundamental groups. In particular, these manifolds have finite covers with positive first Betti number.
Equivalence of cocycles under an irrational rotation
Authors: Baggett Larry , Merrill Kathy
Published: June 2010

This paper describes a method for studying the equivalence relation among cocycles for an irrational rotation. A parameterized family of cocycles is presented, which meets the equivalence class of each piecewise absolutely continuous function whose derivative is

L
2

{L^2}

. The difficulties in describing the equivalence among the elements of this family is shown to reduce to the analogous problem for describing equivalence among step functions, thereby relating this paper to the earlier work of Veech, Petersen, Merrill, and others.
Random approximations and random fixed point theorems for non-self-maps
Authors: Lin Tzu-Chu
Published: June 2010

Recently, Sehgal and Singh [18] and Papageorgiou [16] considered different random versions of a very interesting theorem of Fan [4]. Instead of compact convex domain, here we consider a continuous condensing or non-expansive random map defined on a closed ball (or closed convex set with bounded range). We prove it is true for certain spaces. As applications of our theorems, some random fixed point theorems of non-self-maps are derived.
A note on extreme points of subordination classes
Authors: Hallenbeck D. J.
Published: June 2010

Let

s
(
F
)

s(F)

denote the set of functions subordinate to a univalent function

F
F

in

Δ
\Delta

in the unit disc. Let

B
B

denote the set of functions

ϕ
(
z
)

\phi (z)

analytic in

Δ
\Delta

satisfying

|

ϕ
(
x
)

|

>
1

|{\phi (x)}| > 1

and

ϕ
(
0
)
=
0

\phi (0) = 0

. Let

D
=
F
(
Δ
)

D = F(\Delta )

and

λ
(
w
,
∂
D
)

\lambda (w,\partial D)

denote the distance between

w
w

and

∂
D

\partial D

(boundary of

D
D

). We prove that if

ϕ
\phi

is an extreme point of

B
B

then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript 2 pi Baseline log lamda left-parenthesis upper F left-parenthesis phi left-parenthesis e Superscript i t Baseline right-parenthesis right-parenthesis comma partial-differential upper D right-parenthesis d t equals negative normal infinity">

∫
0

2
π

log
⁡
λ
(
F
(
ϕ
(

e

i
t

)
)
,
∂
D
)
d
t
=
−
∞

\int _0^{2\pi } {\log \lambda (F(\phi ({e^{it}})),\partial D)dt = - \infty }

. As a corollary we prove that if

F
∘
ϕ

F \circ \phi

is an extreme point of

s
(
F
)

s(F)

then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript 2 pi Baseline log lamda left-parenthesis upper F left-parenthesis phi left-parenthesis e Superscript i t Baseline right-parenthesis right-parenthesis comma partial-differential upper D right-parenthesis d t equals negative normal infinity">

∫
0

2
π

log
⁡
λ
(
F
(
ϕ
(

e

i
t

)
)
,
∂
D
)
d
t
=
−
∞

\int _0^{2\pi } {\log \lambda (F(\phi ({e^{it}})),\partial D)dt = - \infty }

.
Singular integrals in product domains and the method of rotations
Authors: Krug Donald
Published: June 2010

Singular integrals with kernels of the form

K
(
x
,
y
)

K(x,y)

where

K
K

satisfies conditions to be a bounded singular integral operator in each of its variables have been much studied lately. In this paper we use the classical method of rotations to give a proof that kernels of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis x comma y right-parenthesis equals normal upper Omega left-parenthesis x comma y right-parenthesis slash StartAbsoluteValue x EndAbsoluteValue Superscript n Baseline StartAbsoluteValue y EndAbsoluteValue Superscript m">

K
(
x
,
y
)
=
Ω
(
x
,
y
)

/

|

x

|

n

|

y

|

m

K(x,y) = \Omega (x,y)/|x{|^n}|y{|^m}

correspond to bounded singular integral operators.
Simultaneous systems of representatives for finite families of finite sets
Authors: Jia Xing De
Published: June 2010

Let

h
≥
2

h \geq 2

and

k
≥
1

k \geq 1

. It is proved that if

S

=
{

S
i

}

i
=
1

s

\mathcal {S} = \{ {S_i}\} _{i = 1}^s

and

T

=
{

T
j

}

j
=
1

t

\mathcal {T} = \{ {T_j}\} _{j = 1}^t

are two families of nonempty, pairwise disjoint sets such that

|

S
i

|

≤
h
,

|

T
j

|

≤
k

|{S_i}| \leq h,|{T_j}| \leq k

and

S
i

⊈

T
j

{S_i} \nsubseteq {T_j}

for all

i
i

and

j
j

, then the number

N
(

S

,

T

)

N(\mathcal {S},\mathcal {T})

of the sets

X
X

such that

X
X

is a minimal system of representatives for

S

\mathcal {S}

and

X
X

is simultaneously a system of representatives for

T

\mathcal {T}

that satisfies
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N left-parenthesis script upper S comma script upper T right-parenthesis less-than-or-equal-to h Superscript s Baseline left-parenthesis 1 minus left-parenthesis h minus r right-parenthesis slash h Superscript q plus 1 Baseline right-parenthesis Superscript t">

N
(

S

,

T

)
≤

h
s

(
1
−
(
h
−
r
)

/

h

q
+
1

)
t

N(\mathcal {S},\mathcal {T}) \leq {h^s}{(1 - (h - r)/{h^{q + 1}})^t}

, where

k
=
q
(
h
−
1
)
+
r

k = q(h - 1) + r

with

0
≤
r
≤
h
−
2

0 \leq r \leq h - 2

. This was conjectured by M. B. Nathanson [3] in 1985.
The isomorphism question for modular group algebras of metacyclic 𝑝-groups
Authors: Bagiński Czesław
Published: June 2010

Let

F
[
G
]

F[G]

be a group algebra of a finite

p
p

-group

G
G

over the field

F
=
G
F
(
p
)

F = GF(p)

. If

G
≃
H

G \simeq H

, then clearly

F
[
G
]
≃
F
[
H
]

F[G] \simeq F[H]

. However, it is not known whether the converse is true. The answer for metacyclic

p
p

-groups,

p
>
3

p > 3

, is given.
Order relation in quadratic Jordan rings and a structure theorem
Authors: González Santos , Martínez Consuelo
Published: June 2010

It is shown that the relation defined by

x
≤
y

x \leq y

if and only if

V
x

x
=

V
x

y

{V_x}x = {V_x}y

and

U
x

x
=

U
x

y
=

U
y

x

{U_x}x = {U_x}y = {U_y}x

is an order relation for quadratic Jordan algebras without nilpotent elements, which extends our previous one for linear Jordan algebras, and reduces to the usual Abian order for associative algebras. We prove that a quadratic Jordan algebra is isomorphic to a direct product of division algebras if and only if the algebra has no nilpotent elements and is hyperatomic and orthogonally complete.
Uniformly persistent semidynamical systems
Authors: Fonda Alessandro
Published: June 2010

A necessary and sufficient condition is given for a semidynamical system to be uniformly persistent. As a consequence some known results are improved.
Hölder continuity of quasiconformal mappings of the unit ball
Authors: Anderson G. D. , Vamanamurthy M. K.
Published: June 2010

The authors obtain a modified form of Gehring’s Hölder-continuity theorem for the family of

K
K

-quasiconformal self-mappings of the unit ball in

n
n

-space, such that the distortion coefficient is independent of

n
n

and depends only on

K
K

.
Norms of free operators
Authors: Picardello Massimo A. , Pytlik Tadeusz
Published: June 2010

We give a short and elementary proof of the formula for the norm of a free convolution operator on

L
2

{L^2}

of a discrete group. The formula was obtained in 1976 by C. Akemann and Ph. Ostrand, and by several other authors afterwards.
Representation of Borel isomorphism by a probability measure
Authors: Shortt R. M.
Published: June 2010

A theorem of Chuaqui on the representation of equivalence relations by probability measures is applied to Borel isomorphism. Using a result of Cenzer and Mauldin, we show that under

V
=
L

V = L

, the isomorphism types of analytic sets cannot be linearly ordered.
A characterization of preordered fields with the weak Hilbert property
Authors: Zeng Guangxin
Published: June 2010

By introducing a new notion "local density", we show the following result: Let

(
K
,
S
)

(K,S)

be a preordered field, then

(
K
,
S
)

(K,S)

has the weak Hilbert property if and only if

(
K
,
S
)

(K,S)

is locally dense. From this, a theorem of McKenna in reference [1] and one of Prestel in reference [2] will be easily proved.
The maximal quotient ring of regular group rings. II
Authors: Cedó Ferran
Published: June 2010

Let

K
K

be a commutative field. Let

G
G

be a locally finite group without elements of order

p
p

in case char

K
=
p
>
0

K = p > 0

. In this paper it is proved that for some large classes of groups

G
G

(

Δ
\Delta

-hypercentral, residually finite and groups satisfying Min-

q
q

for all primes

q
q

) the Type

I

∞

{{\text {I}}_\infty }

part of the maximal quotient ring of the group ring

K
[
G
]

K[G]

is zero.
A proof of the uniformization theorem for arbitrary plane domains
Authors: Fisher Yuval , Hubbard John H. , Wittner Ben S.
Published: June 2010

We present a simple constructive proof of the Uniformization Theorem which works for plane domains. The proof is a combination of covering space theory and Koebe’s constructive proof of the Riemann mapping theorem, and the resulting algorithm can be used to estimate the Poincaré metric for the domain.
Inverses of generators
Authors: deLaubenfels Ralph
Published: June 2010

Let

A
A

be a (possibly unbounded) linear operator on a Banach space

X
X

that generates a bounded holomorphic semigroup of angle

θ
(
0
>
θ
≤
π

/

2
)

\theta (0 > \theta \leq \pi /2)

. We show that, if the range of

A
A

is dense, then

A
A

is one-to-one, and

A

−
1

{A^{ - 1}}

(defined on the range of

A
A

) generates a bounded holomorphic semigroup of angle

θ
\theta

, given by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Superscript z upper A Super Superscript negative 1 Baseline equals integral e Superscript negative w Baseline left-parenthesis w upper A plus z right-parenthesis Superscript negative 1 Baseline StartFraction d w Over 2 pi i EndFraction comma">

e

z

A

−
1

=
∫

e

−
w

(
w
A
+
z
)

−
1

d
w

2
π
i

,

{e^{z{A^{ - 1}}}} = \int {{e^{ - w}}{{(wA + z)}^{ - 1}}\frac {{dw}}{{2\pi i}},}

\]
over an appropriate curve. When

X
X

is reflexive, it is sufficient that

A
A

be one-to-one.
Commensurate sequences of characters
Authors: Pełczyński A.
Published: June 2010

If

(

a
j

)

({a_j})

and

(

b
j

)

({b_j})

are sequences of characters on compact abelian groups

S
S

and

T
T

respectively such that for every sequence of scalars
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis alpha Subscript j Baseline right-parenthesis StartAbsoluteValue EndAbsoluteValue sigma-summation alpha Subscript j Baseline a Subscript j Baseline StartAbsoluteValue EndAbsoluteValue Subscript normal infinity Baseline equivalent-to StartAbsoluteValue EndAbsoluteValue sigma-summation alpha Subscript j Baseline b Subscript j Baseline StartAbsoluteValue EndAbsoluteValue Subscript normal infinity">

(

α
j

)

|

|

∑

α
j

a
j

|

|

∞

≍

|

|

∑

α
j

b
j

|

|

∞

({\alpha _j})||\sum {\alpha _j}{a_j}|{|_\infty } \asymp ||\sum {\alpha _j}{b_j}|{|_\infty }

tnen for every

1
≤
p
>
∞

1 \leq p > \infty

and every sequence

(

x
j

)

({x_j})

of elements of an arbitrary Banach space

X
X

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper S Endscripts double-vertical-bar sigma-summation x Subscript j Baseline a Subscript j Baseline double-vertical-bar Superscript p Baseline d s equivalent-to integral Underscript upper T Endscripts double-vertical-bar sigma-summation x Subscript j Baseline b double-vertical-bar Superscript p Baseline d t period">

∫
S

‖

∑

x
j

a
j

‖

p

d
s
≍

∫
T

‖

∑

x
j

b

‖

p

d
t
.

{\int _S {\left \| {\sum {{x_j}{a_j}} } \right \|} ^p}ds \asymp {\int _T {\left \| {\sum {{x_j}b} } \right \|} ^p}dt.

\]
This result generalizes a result of Pisier [Pi 1] for Sidon sets. For topological Sidon sets on

R

{\mathbf {R}}

a slightly stronger result holds.
Hypercontractivity estimates for nonselfadjoint diffusion semigroups
Authors: Pinsky Ross
Published: June 2010

We use Gross’ results on logarithmic Sobolev inequalities and hypercontractivity to show that the hypercontractivity of a positive recurrent diffusion semigroup depends essentially only on its invariant probability measure and a given diffusion matrix and its diffusion matrix. Thus all diffusion semigroups (including the unique selfadjoint one) with a given invariant probability measure and a given diffusion matrix are essentially equally hypercontractive.
A proof of the isosystolic inequality for the Klein bottle
Authors: Sakai Takashi
Published: June 2010

We give a simple proof of the isosystolic inequality for the Klein bottle, which was first obtained by C. Bavard.
Stable maps into the Hilbert cube
Authors: Garity Dennis J. , Rohm Dale M.
Published: June 2010

A map into the Hilbert cube is stable if each composition with projection onto a finite number of factors is stable. We prove that a map from a compact metric space into the Hilbert cube is stable if and only if it is universal. As a consequence, the composition of a stable map with any self homeomorphism of the Hilbert cube is also stable.
Existence of symmetric skew balanced starters for odd prime powers
Authors: Du Ding Zhu , Hwang F. K.
Published: June 2010

Strong starters and skew starters have been widely used in various combinatorial designs. In particular skew balanced starters and symmetric skew balanced starters are crucially used in the construction of completely balanced Howell rotations. Let

n
=

2
m

k
+
1

n = {2^m}k + 1

be an odd prime power where

m
≥
2

m \geq 2

and

k
k

is an odd number. The existence of symmetric skew balanced starters for

G
F
(
n
)

GF(n)

has been proved for

m
≥
2

m \geq 2

and

k
≠
1
,
3
,
9

k \ne 1,3,9

. In this paper, we present a new approach which gives a uniform proof of the existence of symmetric skew balanced starters for all

m
≥
2

m \geq 2

and

k
≥
3

k \geq 3

.
Some consequences of the standard polynomial
Authors: Chang Qing
Published: June 2010

The standard polynomial of degree

m
m

is the polynomial
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation left-brace sign left-parenthesis rho right-parenthesis x Subscript rho left-parenthesis 1 right-parenthesis Baseline x Subscript rho left-parenthesis 2 right-parenthesis Baseline midline-horizontal-ellipsis x Subscript rho left-parenthesis m right-parenthesis Baseline vertical-bar rho element-of upper S Subscript m Baseline right-brace">

∑

{

sign(

ρ

)

x

ρ
(
1
)

x

ρ
(
2
)

⋯

x

ρ
(
m
)

|

ρ
∈

S
m

}

\sum {\{ {\text {sign(}}\rho {\text {)}}{x_{\rho (1)}}{x_{\rho (2)}} \cdots {x_{\rho (m)}}|\rho \in {S_m}} \}

, where

S
m

{S_m}

is the symmetric group on

m
m

letters. We show that the polynomial
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation left-brace sign left-parenthesis rho sigma right-parenthesis x Subscript rho left-parenthesis 1 right-parenthesis Baseline y Subscript sigma left-parenthesis 1 right-parenthesis Baseline x Subscript rho left-parenthesis 2 right-parenthesis Baseline y Subscript sigma left-parenthesis 2 right-parenthesis Baseline midline-horizontal-ellipsis x Subscript rho left-parenthesis m right-parenthesis Baseline y Subscript sigma left-parenthesis m right-parenthesis Baseline vertical-bar rho comma sigma element-of upper S Subscript m Baseline right-brace">

∑

{

sign(

ρ
σ

)

x

ρ
(
1
)

y

σ
(
1
)

x

ρ
(
2
)

y

σ
(
2
)

⋯

x

ρ
(
m
)

y

σ
(
m
)

|

ρ
,
σ
∈

S
m

}

\sum {\{ {\text {sign(}}\rho \sigma {\text {)}}{x_{\rho (1)}}{y_{\sigma (1)}}{x_{\rho (2)}}{y_{\sigma (2)}} \cdots {x_{\rho (m)}}{y_{\sigma (m)}}|\rho ,\sigma \in {S_m}\} }

\]
is a consequence of the standard polynomial of degree

m
m

. We also show that certain polynomials of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation StartSet sign left-parenthesis rho right-parenthesis x Subscript rho left-parenthesis 1 right-parenthesis Baseline x Subscript rho left-parenthesis 2 right-parenthesis Baseline midline-horizontal-ellipsis x Subscript rho left-parenthesis n right-parenthesis Baseline vertical-bar rho element-of upper Q EndSet">

∑
{

sign(

ρ

)

x

ρ
(
1
)

x

ρ
(
2
)

⋯

x

ρ
(
n
)

|

ρ
∈
Q
}

\sum \{ {\text {sign(}}\rho {\text {)}}{x_{\rho (1)}}{x_{\rho (2)}} \cdots {x_{\rho (n)}}|\rho \in Q\}

, where

n
≥
m

n \geq m

and

Q
Q

is a suitable subset of

S
n

{S_n}

, are consequences of the standard polynomial of degree

m
m

.
A Korovkin type approximation theorem for set-valued functions
Authors: Keimel Klaus , Roth Walter
Published: June 2010

This paper is a contribution to the problem of approximating continuous functions

F
F

defined on a compact Hausdorff space

X
X

, where the value

F
(
x
)

F(x)

is a compact convex set in

R

n

{{\mathbf {R}}^n}

for every

x
x

in

X
X

. More specifically we show how to transfer Korovkin type approximation theorems for real-valued continuous functions to this set-valued situation.
Isothermic surfaces and the Gauss map
Authors: Palmer Bennett
Published: June 2010

We give a necessary and sufficient condition for the Gauss map of an immersed surface

M
M

in

n
n

-space to arise simultaneously as the Gauss map of an anti-conformal immersion of

M
M

into the same space. The condition requires that the lines of curvature of each normal section lie on the zero set of a harmonic function. The result is applied to a class of surfaces studied by S. S. Chern which admit an isometric deformation preserving the principal curvatures.
Volumes of flows
Authors: Johnson David L.
Published: June 2010

If

F
F

is an oriented nonsingular flow on a Riemannian manifold

M
M

, the volume of

F
F

is defined as the

n
n

-dimensional measure of the unit vector field tangent to

F
F

, as a section of

T
∗

(
M
)

{T_ * }\left ( M \right )

with the induced metric. It is shown that, for any metric of the two-dimensional torus, and for any homotopy class of flows on the torus, there is a unique smooth flow of minimal volume within the homotopy class. It has been shown that the Hopf foliation on the round threesphere absolutely minimizes the volume of flows on

S
3

{S^3}

. In higher dimensions this is not the case; the Hopf fibrations are not even local minima of the volume functional for flows on the round five-sphere. It is not known whether a volume-minimizing flow on

S
5

{S^5}

exists.
On 𝒜-submodules for reflexive operator algebras
Authors: Han De Guang
Published: June 2010

In [2] the authors described all weakly closed

A

\mathcal {A}

-submodules of

L

(
H
)

L\left ( H \right )

for a nest algebra

A

\mathcal {A}

in terms of order homomorphisms of Lat

A

\mathcal {A}

. In this paper we prove that for any reflexive algebra

A

\mathcal {A}

which is

σ
\sigma

-weakly generated by rank-one operators in

A

\mathcal {A}

, every

σ
\sigma

-weakly closed

A

\mathcal {A}

-submodule can be characterized by an order homomorphism of Lat

A

\mathcal {A}

. In the case when

A

\mathcal {A}

is a reflexive algebra with a completely distributive subspace lattice and

M

\mathcal {M}

is a

σ
\sigma

-weakly closed ideal of

A

\mathcal {A}

, we obtain necessary and sufficient conditions for the commutant of

A

\mathcal {A}

modulo

M

\mathcal {M}

to be equal to AlgLat

M

\mathcal {M}

.
An inequality for some nonnormal operators
Authors: Furuta Takayuki
Published: June 2010

An inequality of use in testing convergence of eigenvector calculations is improved. If

e
λ

{e_\lambda }

is a unit eigenvector corresponding to an eigenvalue

λ
\lambda

of a dominant operator

A
A

on a Hilbert space

H
H

, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue left-parenthesis g comma e Subscript lamda Baseline right-parenthesis EndAbsoluteValue squared less-than-or-equal-to StartFraction StartAbsoluteValue EndAbsoluteValue g StartAbsoluteValue EndAbsoluteValue squared StartAbsoluteValue EndAbsoluteValue upper A g StartAbsoluteValue EndAbsoluteValue squared minus StartAbsoluteValue left-parenthesis g comma upper A g right-parenthesis EndAbsoluteValue squared Over StartAbsoluteValue EndAbsoluteValue left-parenthesis upper A minus lamda upper I right-parenthesis g StartAbsoluteValue EndAbsoluteValue squared EndFraction">

|

(
g
,

e
λ

)

|

2

≤

|

|

g

|

|

2

|

|

A
g

|

|

2

−

|

(
g
,
A
g
)

|

2

|

|

(
A
−
λ
I
)
g

|

|

2

|(g,{e_\lambda }){|^2} \leq \frac {{||g|{|^2}||Ag|{|^2} - |(g,Ag){|^2}}}{{||(A - \lambda I)g|{|^2}}}

\]
for all

g
g

in

H
H

for which

A
g
≠
λ
g

Ag \ne \lambda g

. The equality holds if and only if the component of

g
g

orthogonal to

e
λ

{e_\lambda }

is also an eigenvector of

A
A

. This result is an improvement of Bernstein’s result for selfadjoint operators.
A new characterization of trees
Authors: Ward L. E.
Published: June 2010

It is proved that a continuum is a tree if and only if for each pair of nondegenerate subcontinua

K
K

and

L
L

with

K
⊂
L

K \subset L

, it follows that

K
K

contains a cutpoint of

L
L

.
Automorphisms of Grassmannians
Authors: Cowen Michael J.
Published: June 2010

For a complex vector space

V

\mathcal {V}

of dimension

n
n

, the group of holomorphic automorphisms of the Grassmannian

Gr
⁡
(
p
,

V

)

\operatorname {Gr}(p,\mathcal {V})

can be identified with the subgroup of

P

G
1
(

∧
p

V

)

{\mathbf {P}}G1({\wedge ^p}\mathcal {V})

preserving the Grassmannian. Using this, Chow showed

Aut
⁡

(Gr

(
p
,

V

)
)
=

P

Gl
⁡
(

V

)

\operatorname {Aut} {\text {(Gr}}(p,\mathcal {V})) = {\mathbf {P}}\operatorname {Gl} (\mathcal {V})

for

n
≠
2
p

n \ne 2p

, and

P

G
1
(

V

)

{\mathbf {P}}G1(\mathcal {V})

is a normal subgroup of index 2 in

A
u
t
(
G
r

⁡
(
p
,

V

)
)

\operatorname {Aut(Gr}(p,\mathcal {V}))

for

n
=
2
p

n = 2p

. We prove a version of Chow’s result for a separable Hilbert space

H

\mathcal {H}

. Theorem.

P

Gl
⁡
(

H

)

{\mathbf {P}}\operatorname {Gl} (\mathcal {H})

is the subgroup of

P

Gl
⁡
(

∧
p

H

)

{\mathbf {P}}\operatorname {Gl} ({\wedge ^p}\mathcal {H})

which preserves

Gr
⁡
(
p
,

H

)

\operatorname {Gr}(p,\mathcal {H})

. That is, if

R
R

is an invertible linear operator on

∧
p

H

{\wedge ^p}\mathcal {H}

which preserves decomposable

p
p

-vectors, then there exists

S
S

, an invertible linear operator on

H

\mathcal {H}

, such that

R
=

∧
p

S

R = {\wedge ^p}S

.
Sturm-Liouville equations with Besicovitch almost-periodicity
Authors: Dzurnak A. , Mingarelli A. B.
Published: June 2010

In this article we extend a former result [Proc. Amer. Math. Soc. 97, (1986), 269-272] dealing with the oscillation of (Bohr) almost-periodic Sturm-Liouville operators to the generalization of such as considered by Besicovitch. This includes all the classical extensions of almost periodic functions as considered by Stepanoff and Weyl.
A note on Roe’s characterization of the sine function
Authors: Howard Ralph
Published: June 2010

Let

f

(
n
)

n
=
0
,
±
1
,
±
2
,
…

{f^{(n)}}n = 0, \pm 1, \pm 2, \ldots

be a sequence of complex valued functions on the real line with

(
d

/

d
x
)

f

(
n
)

=

f

(
n
+
1
)

(d/dx){f^{(n)}} = {f^{(n + 1)}}

and satisfying inequalities
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue f Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis x right-parenthesis EndAbsoluteValue less-than-or-equal-to upper M Subscript n Baseline left-parenthesis 1 plus StartAbsoluteValue x EndAbsoluteValue right-parenthesis Superscript k">

|

f

(
n
)

(
x
)

|

≤

M
n

(
1
+

|

x

|

)
k

|{f^{(n)}}(x)| \leq {M_n}{(1 + |x|)^k}

where as

n
→
∞

n \to \infty

the growth conditions

lim
_

M
n

(
1
+
ε

)

−
n

=
0

\underline {\lim } {M_n}{(1 + \varepsilon )^{ - n}} = 0

and

lim
_

M

−
n

(
1
+
ε

)

−
n

=
0

\underline \lim {M_{ - n}}{(1 + \varepsilon )^{ - n}} = 0

hold for all

ε
>
0

\varepsilon > 0

. Then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Superscript left-parenthesis 0 right-parenthesis Baseline left-parenthesis x right-parenthesis equals p left-parenthesis x right-parenthesis e Superscript i x Baseline plus q left-parenthesis x right-parenthesis e Superscript minus i x">

f

(
0
)

(
x
)
=
p
(
x
)

e

i
x

+
q
(
x
)

e

−
i
x

{f^{(0)}}(x) = p(x){e^{ix}} + q(x){e^{ - ix}}

where

p
p

and

q
q

are polynomials of degree at most

k
k

.
Finiteness of index and total scalar curvature for minimal hypersurfaces
Authors: Tysk Johan
Published: June 2010

Let

M
n

,
n
≥
3

{M^n},n \geq 3

, be an oriented minimally immersed complete hypersurface in Euclidean space. We show that for

n
=
3
,
4
,
5
,

 or 

6

n = 3,4,5,{\text { or }}6

, the index of

M
n

{M^n}

is finite if and only if the total scalar curvature of

M
n

{M^n}

is finite, provided that the volume growth of

M
n

{M^n}

is bounded by a constant times

r
n

{r^n}

, where

r
r

is the Euclidean distance function. We also note that this result does not hold for

n
≥
8

n \geq 8

. Moreover, we show that the index of

M
n

{M^n}

is bounded by a constant multiple of the total scalar curvature for all

n
≥
3

n \geq 3

, without any assumptions on the volume growth of

M
n

{M^n}

.
A generalization of the Wedderburn-Artin theorem
Authors: Jain S. K. , López-Permouth S. R.
Published: June 2010

The structure of rings such that each of its homomorphic images has the property that each cyclic right module over it is essentially embeddable in a direct summand is determined. Such rings are precisely (i) right uniserial rings, (ii)

n
×
n

n \times n

matrix rings over two-sided uniserial rings with

n
>
1

n > 1

, or (iii) sums of rings of the types (i) and (ii).
The simple-zero theorem for support points in Σ
Authors: Leung Y. J. , Schober G.
Published: June 2010

The purpose of this article is to confirm a simple-zero conjecture raised by the second author for support points in

Σ
\Sigma

. The conjecture implies that the omitted arcs of an extremal function in

Σ
\Sigma

for a linear problem can branch out in, at most, three equiangular directions at a time.
Définissabilité avec paramètres extérieurs dans 𝑄_{𝑝} et 𝑅
Authors: Delon Françoise
Published: June 2010

Let

N
N

be an elementary extension of the field

Q

p

{{\mathbf {Q}}_p}

and

A
A

a definable subset of

N
n

{N^n}

. Then

A
∩

Q

p
n

A \cap {\mathbf {Q}}_p^n

is definable in

Q

p

{{\mathbf {Q}}_p}

.
On the spans and width of simple triods
Authors: West Thelma
Published: June 2010

In 1964 the concept of the span of a metric space was introduced by A. Lelek. Since that time, some modified versions of the span have been considered. To date, the metric spaces for which the various spans have been explictly calculated have mainly been objects for which the span is zero. In this paper, we calculate and estimate the spans of simple triods. Questions have been raised concerning the relationships between these various versions of spans. For instance, it was conjectured that the surjective span is always at least a half of the span. We find that these spans are equal for some types of simple triods. Another problem concerning simple triods asks if the surjective semispan and the surjective span of them are always equal. This problem is solved positively for some classes of simple triods. In 1961 the width of a treelike continuum was introduced by C. E. Burgess. The relationship of the width and the span is only vaguely known. We involve the widths of simple triods in our estimations of the span.
Hyperplane sections of weakly normal varieties in positive characteristic
Authors: Cumino Caterina , Greco Silvio , Manaresi Mirella
Published: June 2010

It is known that the general hyperplane sections of a weakly normal projective algebraic variety are weakly normal if the ground field has characteristic zero. This is shown to be false in positive characteristic; counterexamples can be obtained by intersecting "bad" weakly normal varieties with suitable linear spaces.
Linear extremal problems for harmonic mappings of the disk
Authors: Duren Peter , Schober Glenn
Published: June 2010

Sharp bounds for Fourier coefficients and distortion are established for harmonic mappings of the unit disk onto itself.
Hypo-analytic pseudodifferential operators
Authors: Berhanu S.
Published: June 2010

Let

Ω
\Omega

be a hypo-analytic manifold of dimension

m
m

equipped with a hypo-analytic structure whose structure bundle

T
′

T’

has dimension

m
m

. This paper introduces hypo-analytic pseudodifferential operators and it is shown that such operators preserve the hypo-analyticity of a distribution.
Generalization of a classical theorem of Pólya and Szegő
Authors: Mera Ruben N.
Published: June 2010

An entire function of exponential type bounded on the real axis is bounded along every horizontal line. A generalization of this theorem to the class of entire functions of finite order is given.
On the number of good rational approximations to algebraic numbers
Authors: Mueller Julia , Schmidt W. M.
Published: June 2010

We study rational approximations

x

/

y

x/y

to algebraic and, more generally, to real numbers

ξ
\xi

. Given

δ
>
0

\delta > 0

, and writing

L
=
log
⁡
(
1
+
δ
)

L = \log (1 + \delta )

, the number of approximations with

|

ξ
−
(
x

/

y
)

|

>

y

−
2
−
δ

|\xi - (x/y)| > {y^{ - 2 - \delta }}

is

≤

L

−
1

log
⁡
log
⁡
H
+

c
1

(
δ
,
r
)

\leq {L^{ - 1}}\log \log H + {c_1}(\delta ,r)

if

ξ
\xi

is algebraic of degree

≤
r

\leq r

and of height

H
H

, and is

≤

L

−
1

log
⁡
log
⁡
B
+

c
2

(
δ
)

\leq {L^{ - 1}}\log \log B + {c_2}(\delta )

if

ξ
\xi

is real and we restrict to approximations with

y
≤
B

y \leq B

. It turns out that the dependency on

H
H

resp.

B
B

in these estimates is the best possible, i.e., that the summands

L

−
1

log
⁡
log
⁡
H

{L^{ - 1}}\log \log H

resp.

L

−
1

log
⁡
log
⁡
B

{L^{ - 1}}\log \log B

are optimal.
A generalization of nonharmonic Fourier series
Authors: Benzinger Harold E.
Published: June 2010

It is shown that the theory of nonharmonic Fourier series is a special case of a general method for perturbing bases in Banach spaces, and that even in the classical case, there are many ways of perturbing ordinary Fourier series while still preserving the norm and pointwise convergence properties.
Can the Weyl algebra be a fixed ring?
Authors: Smith S. P.
Published: June 2010

If a finite soluble group acts as automorphisms of a domain, then the invariant subring is not isomorphic to the first Weyl algebra

C

[
t
,
d

/

d
t
]

{\mathbf {C}}[t,d/dt]

.
On extending actions
Authors: Vaught Robert
Published: June 2010

Consider a Polish topological group

G
G

acting via

J
J

on a substandard (= countably generated) Borel space. Theorem 1. Any such "Borel action" can be extended to a Borel action

J
′

:
G
×

X
′

→

X
′

J’:G \times X’ \to X’

where

X
′

X’

is coanalytic. (Theorem 3 gives an analogue for continuous actions.) Corollary 2. The result "in any Borel action, orbits are Borel" implies the (well-known) result "all such orbits are absolutely Borel".
Tangles and the Gromov invariant
Authors: Adams Colin C.
Published: June 2010

Previous results about mutant knots are extended to show that the Gromov invariant of a knot or link is preserved when a chain of tangles which make up the knot or link is permuted.
When are Rees congruences principal?
Authors: Reis C. M.
Published: June 2010

Let

ρ
I

{\rho _I}

be the Rees congruence modulo the ideal

I
I

of the free monoid

X
∗

{X^*}

. In this short note we give necessary and sufficient conditions, in terms of the partial order induced by division on the complement of

I
I

, for

ρ
I

{\rho _I}

to be principal. In particular, we prove that if

I
I

is principal, so is

ρ
I

{\rho _I}

.
Weighted inequalities for Riemann-Liouville fractional integrals of order one and greater
Authors: Martín-Reyes F. J. , Sawyer E.
Published: June 2010

A simple characterization is given for two-weight norm inequalities for generalized Hardy operators
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript phi Baseline f left-parenthesis x right-parenthesis equals integral Subscript 0 Superscript x Baseline phi left-parenthesis StartFraction t Over x EndFraction right-parenthesis f left-parenthesis t right-parenthesis d t">

T
φ

f
(
x
)
=

∫
0
x

φ
(

t
x

)
f
(
t
)
d
t

{T_\varphi }f(x) = \smallint _0^x\varphi (\tfrac {t}{x})f(t)dt

, where

φ
:
(
0
,
1
)
→
(
0
,
∞
)

\varphi :(0,1) \to (0,\infty )

is nonincreasing and satisfies

φ
(
a
b
)
≤
D
[
φ
(
a
)
+
φ
(
b
)
]

\varphi (ab) \leq D[\varphi (a) + \varphi (b)]

for

0
>
a
,
b
>
1

0 > a,b > 1

. Included in particular are the Riemann-Liouville fractional integrals.
Prime length of crossed products
Authors: Welsh Charles C.
Published: June 2010

In this paper we show that the prime length of a crossed product

R
∗
G

R*G

, where

R
R

is a right Noetherian ring and

G
G

is a polycyclic-by-finite group, is bounded by the plinth length of

G
G

and the prime length of

R
R

. We begin by considering prime ideals in group rings of finitely generated Abelian groups, and generalize a theorem of J. E. Roseblade. We then use the description of prime ideals in crossed products given by D. S. Passman to achieve the result.
Weakly constricted operators and Jamison’s convergence theorem
Authors: Sine Robert
Published: June 2010

If

T
T

is a linear contraction on a complex

C
(
X
)

C(X)

space which is irreducible and if

{

T
n

f
}

\{ {T^n}f\}

converges weakly to zero then the convergence is actually in norm. If

T
T

is weakly constricted and irreducible then

T
T

is actually strongly constricted. If

T
T

is weakly almost periodic and irreducible and either

T
T

or

T
∗

{T^*}

has a unimodular point spectral value then

T
T

is strongly almost periodic.
Isometries of spaces of weak * continuous functions
Authors: Cambern Michael , Jarosz Krzysztof
Published: June 2010

If

X
X

is a compact Hausdorff space and

E
∗

{E^*}

a Banach dual, we denote by

C
(
X
,
(

E
∗

,

σ
∗

)
)

C(X,({E^*},{\sigma ^*}))

the Banach space of continuous functions from

X
X

to

E
∗

{E^*}

when the latter space is given its weak * topology. It is shown that if

E
1
∗

,

E
2
∗

E_1^*,E_2^*

have trivial centralizers and satisfy a topological condition introduced by Namioka and Phelps, then, given an isometry

T
T

mapping

C
(

X
1
∗

,
(

E
1

,

σ
∗

)
)

C(X_1^*,({E_1},{\sigma ^*}))

onto

C
(

X
2

,
(

E
2
∗

,

σ
∗

)
)

C({X_2},(E_2^*,{\sigma ^*}))

and given

F
∈
C
(

X
1

,
(

E
1
∗

,

σ
∗

)
)

F \in C({X_1},(E_1^*,{\sigma ^*}))

, there exists a dense

G
δ

{G_\delta }

in

X
2

{X_2}

on which
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis asterisk right-parenthesis left-parenthesis upper T upper F right-parenthesis left-parenthesis x right-parenthesis equals upper U left-parenthesis x right-parenthesis upper F ring normal upper Phi left-parenthesis x right-parenthesis comma">

(
∗
)

(
T
F
)
(
x
)
=
U
(
x
)
F
∘
Φ
(
x
)
,

(*)\quad (TF)(x) = U(x)F \circ \Phi (x),

\]
where

Φ
\Phi

is a homeomorphism of

X
2

{X_2}

onto

X
1

{X_1}

and

U
(
⋅
)

U( \cdot )

an operator-valued function independent of

F
F

. If the

E
i
∗

E_i^*

are separable then the

U
(
x
)

U(x)

are surjective isometries and

x
→
U
(
x
)

x \to U(x)

is continuous. When the

X
i

{X_i}

are metric the representation (*) holds on all of

X
2

{X_2}

.
Maximal abelian subalgebras with simple normalizer
Authors: Longo Roberto
Published: June 2010

All infinite factors with separable predual contain a maximal Abelian * subalgebra whose normalizer generates a simple subfactor
The Cauchy transform on bounded domains
Authors: Anderson J. M. , Hinkkanen A.
Published: June 2010

Suppose that

f
f

is in

L
2

(
Δ
)

{L^2}(\Delta )

where

Δ
\Delta

is the unit disk, and that

f
=
0

f = 0

outside

Δ
\Delta

. We show that then the Cauchy transform

C

f

\mathcal {C}\,f

of

f
f

, when restricted to

Δ
\Delta

, satisfies
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue script upper C f StartAbsoluteValue EndAbsoluteValue Subscript 2 Baseline less-than-or-equal-to left-parenthesis 2 slash alpha right-parenthesis StartAbsoluteValue EndAbsoluteValue f StartAbsoluteValue EndAbsoluteValue Subscript 2">

|

|

C

f

|

|

2

≤
(
2

/

α
)

|

|

f

|

|

2

||\mathcal {C}\,f|{|_2} \leq (2/\alpha )||f|{|_2}

, where

α
≈
2.4048

\alpha \approx 2.4048

is the smallest positive zero of the Bessel function

J
0

{J_0}

. This inequality is sharp.
On the global dimension of fixed rings
Authors: Lorenz Martin
Published: June 2010

Let

G
G

be a finite group acting on a

k
k

-algebra

R
R

, and let

S
=

R
G

S = {R^G}

denote the fixed subring of

R
R

. Our main interest is in the case where

|
G
|

\left | G \right |

is not invertible in

R
R

. Instead, we assume that

R
R

is flat over

S
S

and that the trivial

k
G

kG

-module

k
k

has a periodic projective resolution. (For a field

k
k

of characteristic

p
p

, the latter condition holds precisely if the Sylow

p
p

-subgroups of

G
G

are cyclic or generalized quaternion.) We use a periodicity result for Extgroups, established here in a more general setting that is independent of group actions, to estimate the global dimension of

S
S

in this case.
Two definability results in the equational context
Authors: Hébert M. , McKenzie R. N. , Weaver G. E.
Published: June 2010

Let

τ
\tau

be a type bounded by an infinite regular cardinal

α
\alpha

,

V

{\mathbf {V}}

be a variety in

τ
,

τ
′

⊆
τ

\tau ,\tau ’ \subseteq \tau

and

V
′

{\mathbf {V’}}

the class of all

τ
′

\tau ’

-reducts of the algebras in

V

{\mathbf {V}}

. We show that the operations in

τ
∖

τ
′

\tau \backslash \tau ’

are explicitely definable in

V

{\mathbf {V}}

by pure formulas (i.e. existential-positive without disjunction) if and only if they are implicitely definable and

V
′

{\mathbf {V’}}

is closed under unions of

α
\alpha

-chains (if and only if every

τ
′

\tau ’

-homomorphisms between algebras in

V

{\mathbf {V}}

are

τ
\tau

-homomorphisms, as J. Isbell has shown). It follows that the operations in

τ
∖

τ
′

\tau \backslash \tau ’

are equivalent (in

V

{\mathbf {V}}

) to

τ
′

\tau ’

-terms if and only if every algebra in the

(

τ
′

−
)

(\tau ’ - )

variety generated by

V
′

{\mathbf {V’}}

has a unique

τ
\tau

-expansion in

V

{\mathbf {V}}

.
One criterion for univalency
Authors: Nunokawa Mamoru , Obradović Milutin , Owa Shigeyoshi
Published: June 2010

In this note we give one criterion for the functions

f

(
z
)

=
z
+

a
2

z
2

+
⋯

f\left ( z \right ) = z + {a_2}{z^2} + \cdots

to be univalent in

|
z
|

>
1

\left | z \right | > 1

.
Monomial space curves in 𝑃³_{𝑘} as binomial set theoretic complete intersections
Authors: Thoma Apostolos
Published: June 2010

We give a necessary and sufficient condition for monomial curves in

P
k
3

P_k^3

to be set theoretic complete intersections on a binomial surface. Using this condition, we prove that the twisted cubic curve is the only smooth monomial curve which is a set theoretic complete intersection on a binomial surface, in characteristic zero.
Second order elliptic equations with degenerate weight
Authors: Allegretto W.
Published: June 2010

We consider the eigenvalue problem:

−
Δ
u
−
q
u
=
λ
ω
u
,
u
∈

H
˙

1
,
2

(
Ω
)

- \Delta u - qu = \lambda \omega u,u \in \dot {H}^{1,2}(\Omega )

, in a smooth bounded domain

Ω
⊂

R

n

\Omega \subset {{\mathbf {R}}^n}

. We allow

−
Δ
−
q

- \Delta - q

to have negative spectrum and assume

ω
≥
0

\omega \geq 0

in

Ω
,
ω
≡
0

\Omega ,\omega \equiv 0

in a subdomain of

Ω
\Omega

. Under suitable regularity conditions, we establish several results for the spectrum of this problem. In particular, we give: a min.max. formula for

λ
\lambda

; a precise estimate on the number of negative

λ
\lambda

; an estimate for the location of negative

λ
\lambda

. An example concludes the paper.