bluuebunny/crossref_metadata_2025 huggingface dataset:
The degree of copositive approximation by polynomials
Authors: Leviatan D.
Published: July 2010

Jackson type theorems are established for the approximation of a function

f
f

that changes sign finitely many times in

[
−
1
,
1
]

[ - 1,1]

by polynomials

p
n

{p_n}

which are copositive with it

f

p
n

⩾
0

 on 

[
−
1
,
1
]

f{p_n} \geqslant 0{\text { on }}[ - 1,1]

. The results yield the rate of nonconstrained approximation and are thus best possible in the same sense as in the nonconstrained case.
Logics with given centers and state spaces
Authors: Pták Pavel
Published: July 2010

Let

B
B

be a Boolean algebra and let

K
K

be a compact convex subset of a locally convex topological linear space. Then there exists a logic with the center Boolean isomorphic to

B
B

and with the state space affinely homeomorphic to

K
K

.
Regular Riemannian 𝑠-manifolds of noncompact type
Authors: Sánchez Cristián U.
Published: July 2010

In this note it is proven that a regular Riemannian

s
s

-manifold of noncompact type (see below) cannot be immersed isometrically and equivariantly in

R
n

{R^n}

.
Integral formulas and hyperspheres in a simply connected space form
Authors: Bivens Irl
Published: July 2010

Let

M
n

{M^n}

denote a connected compact hypersurface without boundary contained in Euclidean or hyperbolic

n
+
1

n + 1

space or in an open hemisphere of

S

n
+
1

{S^{n + 1}}

. We show that if two consecutive mean curvatures of

M
M

are constant then

M
M

is in fact a geodesic sphere. The proof uses the generalized Minkowski integral formulas for a hypersurface of a complete simply connected space form. These Minkowski formulas are derived from an integral formula for submanifolds in which the ambient Riemannian manifold

M
¯

\overline M

possesses a generalized position vector field; that is a vector field

Y
Y

whose covariant derivative is at each point a multiple of the identity. In addition we prove that if

M
¯

\overline M

is complete and connected with the covariant derivative of

Y
Y

exactly the identity at each point then

M
¯

\overline M

is isometric to Euclidean space.
Complete hypersurfaces with 𝑅𝑆=0 in 𝐸ⁿ⁺¹
Authors: Matsuyama Yoshio
Published: July 2010

A locally symmetric Riemannian manifold satisfies

R
R
=
0

RR = 0

and in particular

R
S
=
0

RS = 0

. The purpose of this paper is to show that the conditions

R
R
=
0

RR = 0

and

R
S
=
0

RS = 0

are equivalent for complete hypersurfaces in

E

n
+
1

{E^{n + 1}}

and to give by

R
S
=
0

RS = 0

some characterizations of locally symmetric hypersurfaces in

E

n
+
1

{E^{n + 1}}

.
An effective version of Hall’s theorem
Authors: Kierstead Henry A.
Published: July 2010

Manaster and Rosenstein [1972] constructed a recursively bipartite highly recursive graph that satisfies Hall’s condition for a bipartite graph to have a matching, but has no recursive matching. We discuss a natural extension of Hall’s condition which assures that every such graph has a recursive matching.
An algorithmically unsolvable problem in analysis
Authors: Lenard A. , Stillwell J.
Published: July 2010

The decision problem of distinguishing between the cases when the Laplace-Beltrami operator on the covering space of a compact manifold has 0 in its spectrum or is bounded away from 0 is algorithmically unsolvable in any class of manifolds that includes all

4
4

-dimensional ones. The proof depends on a result of Brooks connecting the spectrum with the amenability of the fundamental group.
Prophet inequalities and order selection in optimal stopping problems
Authors: Hill T. P.
Published: July 2010

A complete determination is made of the possible values for

E

(

sup

X
n

)

E\left ( {\sup {X_n}} \right )

and

sup

{

E

X
t

:
t

a

stop rule

}

\sup \left \{ {E{X_t}:t\;{\text {a}}\;{\text {stop rule}}} \right \}

for

X
1

,

X
2

,
…

{X_1},{X_2}, \ldots

independent uniformly bounded random variables; this yields results of Krengel, Sucheston, and Garling, and of Hill and Kertz as easy corollaries. In optimal stopping problems with independent random variables where the player is free to choose the order of observation of these variables it is shown that the player may do just as well with a prespecified fixed ordering as he can with order selections which depend sequentially on past outcomes. A player’s optimal expected gain if he is free to choose the order of observation is compared to that if he is not; for example, if the random variables are nonnegative and independent, he may never do better than double his optimal expected gain by rearranging the order of observation of a given sequence.
Nonincrease almost everywhere of certain measurable functions with applications to stochastic processes
Authors: Berman Simeon M.
Published: July 2010

Let

x
(
t
)

x(t)

,

0
⩽
t
⩽
1

0 \leqslant t \leqslant 1

, be a real valued measurable function having a local time

α

[
0
,
t
]

(
x
)

{\alpha _{[0,t]}}(x)

,

0
⩽
t
⩽
1

0 \leqslant t \leqslant 1

. If the latter is continuous in

t
t

for almost all

x
x

, then almost every

t
t

is not a point of increase of the function

x
(
⋅
)

x( \cdot )

.
Uniform convergence of distribution functions
Authors: Eisenberg Bennett , Gan Shi Xin
Published: July 2010

Necessary and sufficient conditions are given for uniform convergence of probability distribution functions.
The finite dimensionality of integral homology 3-manifolds
Authors: Walsh John J.
Published: July 2010

This note establishes that an integral homology

3
3

-manifold has covering dimension equal to three.
On the existence of equivariant embeddings of principal bundles into vector bundles
Authors: Hansen Vagn Lundsgaard , Møller Jesper Michael
Published: July 2010

Let

G
G

be a finite group and let

X
X

be, say, a connected CW-complex of dimension

k
⩾
1

k \geqslant 1

. Let

π
:
E
→
X

\pi :E \to X

be a principal

G
G

-bundle and

p
:
V
→
X

p:V \to X

an

m
m

-dimensional

G
G

-vector-bundle with trivial action of

G
G

on

X
X

. By an equivariant embedding of

π
\pi

into

p
p

we understand an equivariant embedding

h
:
E
→
V

h:E \to V

commuting with projections. We prove a general embedding theorem, a main special case of which is the following Theorem. If

k
>
m

k > m

and if the action of

G
G

on

V
V

is free outside the zero section for

p
p

, then any principal

G
G

-bundle

π
:
E
→
X

\pi :E \to X

can be embedded equivariantly into

p
:
V
→
X

p:V \to X

.
A space which contains no realcompact dense subspace
Authors: Terada Toshiji
Published: July 2010

A Tychonoff space which contains no realcompact space as a dense subspace is constructed. Let

D

R

\mathcal {D}\mathcal {R}

be the class of all spaces which contain some realcompact spaces as dense subspaces. Then, as a consequence of the above result, it follows that

D

R

\mathcal {D}\mathcal {R}

is not closed-hereditary.
A mapping theorem for Hilbert cube manifolds
Authors: Prasad V. S.
Published: July 2010

We show that every compact connected Hilbert cube manifold

M
M

can be obtained from the Hilbert cube

Q
Q

by making identifications on a face of

Q
Q

. Some applications of this result to measure preserving homeomorphisms on

M
M

are given: (1) The first is concerned with which measures on

M
M

are equivalent to each other by homeomorphisms. (2) The second application is about approximating invertible Borel measurable transformations of

M
M

by measure preserving homeomorphisms of

M
M

. (3) The final application is concerned with generic properties of measure preserving homeomorphisms of

M
M

.
On fibre inclusions and Kähler manifolds
Authors: Meier Willi
Published: July 2010

A map

i
:
X
→
Y

i:X \to Y

of

CW

{\text {CW}}

-complexes is said to be equivalent to a fibre inclusion if there exists a fibration (up to homotopy)

X
→
Y
→
B

X \to Y \to B

. Here some classes of maps of compact Kähler manifolds are presented which are not equivalent to a fibre inclusion.
A note on infinite-dimension under refinable maps
Authors: Kato Hisao
Published: July 2010

It is shown that refinable maps preserve weak infinite-dimension, but not strong infinite-dimension.
Rational automorphisms of Grassmann manifolds
Authors: Brewster Stephen , Homer William
Published: July 2010

The homotopy class of a self map of a complex projective space is well known to be classified by a degree detected in two dimensional cohomology. An analogous result is proved for the rationalization of the Grassmann manifold of complex

n
n

-planes in complex

N
N

-space, provided

N
≠
2
n

N \ne 2n

and the degree is not zero.
Circulants and difference sets
Authors: Newman Morris
Published: July 2010

Let

F
F

be any field,

f
(
x
)

f(x)

a polynomial over

F
F

of degree

⩽
υ
−
1

\leqslant \upsilon - 1

,

P
P

the

υ
×
υ

\upsilon \times \upsilon

full cycle, and

C
C

the

υ
×
υ

\upsilon \times \upsilon

circulant

f
(
P
)

f(P)

. Assume that if

F
F

is of finite characteristic

p
p

. then

(
p
,
υ
)
=
1

(p,\upsilon ) = 1

. It is shown that the rank of

C
C

over

F
F

is

υ
−
d

\upsilon - d

, where

d
d

is the degree of the greatest common divisor of

f
(
x
)

f(x)

and

x
υ

−
1

{x^\upsilon } - 1

. This result is used to determine the rank modulo a prime of the incidence matrix associated with a difference set. The notion of the degree of a difference set is introduced. Certain theorems connected with this notion are proved, and an open problem is stated. Some numerical results are appended.
Automorphism groups of ruled function fields and a problem of Zariski
Authors: Deveney James K.
Published: July 2010

Let

K
1

{K_1}

and

K
2

{K_2}

be finitely generated extensions of a field

K
K

and let

x
x

be transcendental over

K
1

{K_1}

and

K
2

{K_2}

, and assume

K
1

(
x
)
=

K
2

(
x
)

{K_1}(x) = {K_2}(x)

. The main results show that if

K
K

is infinite and the group of automorphisms of

K
2

{K_2}

over

K
K

is finite, or if

K
K

is finite and the group of automorphisms of

K
¯

K
2

\bar K{K_2}

over

K
¯

\bar K

(

K
¯

\bar K

the algebraic closure of

K
K

) is finite, then

K
1

{K_1}

equals

K
2

{K_2}

.
Commutative FPF rings arising as split-null extensions
Authors: Faith Carl
Published: July 2010

Let

R
=
(
B
,
E
)

R = (B,E)

be the split-null or trivial extension of a faithful module

E
E

over a commutative ring

B
B

.

R
R

is an FPF ring iff the partial quotient ring

B

S

−
1

B{S^{ - 1}}

with respect to the set

S
S

of elements of

B
B

with zero annihilator in

E
E

is canonically the endomorphism ring of

E
E

, that is

B

S

−
1

=

End
B

E

S

−
1

B{S^{ - 1}} = {\operatorname {End}_B}E{S^{ - 1}}

, every finitely generated ideal with zero annihilator in

E
E

is invertible in

B

S

−
1

B{S^{ - 1}}

, and

E
=
E

S

−
1

E = E{S^{ - 1}}

is an injective module over

B
B

. The proof uses the author’s characterization of commutative FPF rings [1] and also the characterization of self-injectivity of a split-null extension [3].
Morita equivalence and infinite matrix rings
Authors: Camillo Victor
Published: July 2010

This paper contains the proof of a theorem conjectured by William Stephenson in his thesis [3].
On the compactification of strongly pseudoconvex surfaces. II
Authors: Tan Vo Van
Published: July 2010

A complete answer to the following question is given: When is an algebraic surface

M
M

a compactification of some strongly pseudoconvex surface? In particular, we show this will not be the case if

M
M

is either

P

2

{{\mathbf {P}}_2}

, a quadric, an abelian surface, or a hyperelliptic surface. On the other hand, by constructing specific examples, we show that this will be the case for all other algebraic surfaces. Furthermore, we prove that any compactifiable strongly pseudoconvex surface is quasi-projective.
On the field of a 2-block. II
Authors: Basmaji B. G.
Published: July 2010

Every real

2
2

-block

B
B

of a finite metabelian group contains an irreducible character

θ
\theta

such that

Q
(
B
)
=
Q
(
θ
)

Q(B) = Q(\theta )

.
The Burau representation is unitary
Authors: Squier Craig C.
Published: July 2010

A slight modification of the Burau representation of the braid group is shown to be unitary relative to an explicitly defined Hermitian form. This gives a partial answer to the problem of identifying the image of the Burau representation and provides a tool for attacking the question of whether or not the Burau representation is faithful.
Nilpotency of derivatives on an ideal
Authors: Chung L. O. , Luh Jiang
Published: July 2010

Let

R
R

be a prime ring and

∂
\partial

a derivation of

R
R

. It is shown that if

∂
m

I
=
(
0
)

{\partial ^m}I = (0)

for some nonzero ideal

I
I

in

R
R

, then

∂
m

R
=
(
0
)

{\partial ^m}R = (0)

.
An algebraic periodicity theorem for spheres
Authors: Brennan Joseph P.
Published: July 2010

A periodicity theorem is given for spheres over a field of finite level generalizing results of Jouanolou. An extension of this result gives families of smooth affine varities with isomorphic

K
K

-groups.
Formules intégrales de Koppelman sur une variété de Stein
Authors: Laurent-Thiébaut Christine
Published: July 2010

En utilisant les noyaux définis par Henkin et Leiterer sur une variété de Stein, on démontre une formule intégrale globale du type "Koppelman" pour les

(
n
,
m
)

(n,m)

-formes

(
0
⩽
m
⩽
n
)

(0 \leqslant m \leqslant n)

où

n
n

est la dimension de la variété et on définit une solution fondamentale locale du

∂
¯

\bar \partial

.
The least area bounded by multiples of a curve
Authors: White Brian
Published: July 2010

For each positive integer

n
n

, we construct a smooth curve

Γ
\Gamma

in

R

4

{{\mathbf {R}}^4}

such that the least area of a surface (integral current) with boundary

n
Γ

n\Gamma

is less than

n

/

k

n/k

of the least area of a surface with boundary

k
Γ

(

1
⩽
k
>
n

)

k\Gamma \left ( {1 \leqslant k > n} \right )

.
Weak-star closed algebras and generalized Bergman kernels
Authors: Seddighi Karim
Published: July 2010

In this paper we will characterize the weak-star closed algebra

A

\mathcal {A}

generated by the canonical model associated with a generalized Bergman kernel defined on a domain

G
G

in the plane, whose spectrum is a spectral set. In fact,

A

\mathcal {A}

equals the space

H
∞

(

G
0

)

{H^\infty }\left ( {{G_0}} \right )

of all bounded analytic functions on an appropriate set

G
0

{G_0}

containing

G
G

.
Is there always a unique extremal Teichmüller mapping?
Authors: Strebel Kurt
Published: July 2010

A boundary correspondence of the circumference of the unit disk, which can be continued quasiconformally to the disk and which has the following property, is constructed: If there exists a Teichmüller mapping which is associated to a holomorphic quadratic differential and which is extremal for the boundary correspondence, then there is more than one such mapping.
Almost open mappings between normed spaces
Authors: Harte Robin
Published: July 2010

The almost open bounded linear operators between two normed spaces form an open set whose topological boundary is disjoint from the operators which are bounded below.
On the Borel 𝜎-field and the Baire property
Authors: Morgan John C.
Published: July 2010

There is no category base for which the

σ
\sigma

-field of sets with the Baire property coincides with the

σ
\sigma

-field of linear Borel sets.
A description of weights satisfying the 𝐴_{∞} condition of Muckenhoupt
Authors: Hruščev Sergei V.
Published: July 2010

A nonnegative weight

w
w

on

R
n

{R^n}

satisfies the

A
∞

{A_\infty }

condition iff
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript upper Q element-of script upper A Endscripts left-parenthesis StartAbsoluteValue upper Q EndAbsoluteValue Superscript negative 1 Baseline dot integral Underscript upper Q Endscripts w d x right-parenthesis dot exp left-brace StartFraction 1 Over StartAbsoluteValue upper Q EndAbsoluteValue EndFraction integral Underscript upper Q Endscripts log StartFraction 1 Over w EndFraction d x right-brace greater-than normal infinity period">

sup

Q
∈

A

(

|
Q
|

−
1

⋅

∫
Q

w
d
x

)

⋅
exp
⁡

{

1

|
Q
|

∫
Q

log
⁡

1
w

d
x

}

>
∞
.

\sup \limits _{Q \in \mathcal {A}} \left ( {{{\left | Q \right |}^{ - 1}} \cdot \int _Q {wdx} } \right ) \cdot \exp \left \{ {\frac {1}{{\left | Q \right |}}\int _Q {\log \frac {1}{w}dx} } \right \} > \infty .

\]
Here

A

\mathcal {A}

stands for a family of all cubes in

R
n

{R^n}

. Applications to BMO are considered.
On asymptotic behaviour of certain integro-differential equations
Authors: Yang En Hao
Published: July 2010

This paper deals with the asymptotic behaviour of some integro-differential equations which can be seen as perturbations of some nonlinear systems of ordinary differential equations. The main tools used here are two generalizations of the Pachpatte’s integral inequality and an Alekseev type nonlinear variation of constants formula.
Surjectivity of 𝜑-accretive operators
Authors: Park Jong An , Park Sehie
Published: July 2010

Let

X
X

and

Y
Y

be Banach spaces,

ϕ
:
X
→

Y
∗

\phi :X \to {Y^ * }

and

P
:
X
→
Y
:

P:X \to Y:

;

P
P

is said to be strongly

ϕ
\phi

-accretive if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper P x minus upper P y comma phi left-parenthesis x minus y right-parenthesis mathematical right-angle greater-than-or-slanted-equals c StartAbsoluteValue EndAbsoluteValue x minus y StartAbsoluteValue EndAbsoluteValue squared">

⟨
P
x
−
P
y
,

ϕ

(

x
−
y

)

⟩
⩾
c

|

|

x
−
y

|

|

2

\langle Px - Py,\;\phi \left ( {x - y} \right )\rangle \geqslant c{|| {x - y} ||^2}

for some

c
>
0

c > 0

and each

x
x

,

y
∈
X

y \in X

. These maps constitute a generalization simultaneously of monotone maps (when

Y
=

X
∗

Y = {X^ * }

) and accretive maps (when

Y
=
X

Y = X

). By applying the Caristi-Kirk fixed point theorem, W. O. Ray showed that a localized class of these maps must be surjective under appropriate geometric assumptions on

Y
∗

{Y^ * }

and continuity assumptions on the duality map. In this paper we show that such geometric assumptions can be removed without affecting the conclusion of Ray.
Analytic functions with large sets of Fatou points
Authors: Hwang J. S. , Lappan Peter
Published: July 2010

For a function

f
f

analytic in the unit disc

D
D

, and for each

λ
>
0

\lambda > 0

, let

L

(
λ
)

=

{

z
∈
D
:

|

f

(
z
)

|

=
λ

}

L\left ( \lambda \right ) = \left \{ {z \in D:\left | {f\left ( z \right )} \right | = \lambda } \right \}

denote a level set for

f
f

. We introduce a class

L

\mathcal {L}

, of functions characterized by geometric properties of a collection of sets

{

L

(

λ
n

)

}

\left \{ {L\left ( {{\lambda _n}} \right )} \right \}

, where

{

λ
n

}

\left \{ {{\lambda _n}} \right \}

is an unbounded sequence. We show that

L

1

{\mathcal {L}_1}

, is a proper subclass of the class

L

\mathcal {L}

of G. R. MacLane. Let

A
∞

{A_\infty }

denote the set of points

e

i
θ

{e^{i\theta }}

at which the function

f
f

has

∞
\infty

as an asymptotic value, and let

F

(
f
)

F\left ( f \right )

denote the set of Fatou points of

f
f

. We prove that for a function

f
f

in the class

L

1

{\mathcal {L}_1}

, if

Γ
\Gamma

is an arc of the unit circle such that

Γ
∩

A
∞

=
∅

\Gamma \cap {A_\infty } = \emptyset

, then almost every point of

Γ
\Gamma

belongs to

F

(
f
)

F\left ( f \right )

.
Removable singularities for 𝑛-harmonic functions and Hardy classes in polydiscs
Authors: Singman David
Published: July 2010

Let

ϕ
\phi

be any strongly convex function. For an open subset

G
G

of a polydisc

U
n

{U^n}

the Hardy class

H
ϕ

(
G
)

{H_\phi }\left ( G \right )

is the set of analytic functions

f
f

on

G
G

for which

ϕ
∘
log
⁡

|
f
|

\phi \circ \log \left | f \right |

has an

n
n

-harmonic majorant. It is shown that

H
ϕ

(

U
n

∖
E

)

=

H
ϕ

(

U
n

)

{H_\phi }\left ( {{U^n} \setminus E} \right ) = {H_\phi }\left ( {{U^n}} \right )

for any relatively closed

n
n

-negligible subset

E
E

of

U
n

{U^n}

.
Decomposability preserving curvature operators with an application to Einstein manifolds
Authors: Gabel Michael R. , Zoltek Stanley M.
Published: July 2010

In this paper we examine curvature operators that preserve decomposability. In particular, we prove that if at each point of an Einstein manifold

M
M

the sectional curvature operator is nonsingular and preserves decomposability, and the sectional curvature is either nonnegative or nonpositive, then

M
M

is a space of nonzero constant curvature.
A note on the topological degree at a critical point of mountainpass-type
Authors: Hofer Helmut
Published: July 2010

In this note we will show that the local degree at an isolated critical point given by the Ambrosetti-Rabinowitz mountainpass-theorem is

−
1

- 1

for a large class of functionals. This class includes for example those functionals arising in the study of second order elliptic equations.
Note on epimorphisms and monomorphisms in homotopy theory
Authors: Hilton Peter , Roitberg Joseph
Published: July 2010

We study epimorphisms

e
:
X
→
X

e:X \to X

and monomorphisms

m
:
X
→
X

m:X \to X

in the pointed homotopy category of path-connected CW-spaces. Our principal theorems allow us to infer that under suitable finiteness and fundamental group assumptions, such are in fact homotopy equivalences.
A “Riemann hypothesis” for triangulable manifolds
Authors: Sarkaria K. S.
Published: July 2010

Given a triangulable manifold we show how to find a triangulation whose characteristic polynomial has roots which are either real or on the line

Re
⁡
z
=
1

/

2

\operatorname {Re} z = 1/2

.
Closed maps and the character of spaces
Authors: Tanaka Yoshio
Published: July 2010

We give some necessary and sufficient conditions for the character of spaces to be preserved under closed maps.
A strong form of the Phragmén-Brouwer theorem
Authors: Dickman R. F.
Published: July 2010

In this paper we prove the following form of the Phragmen-Brouwer Theorem: a locally connected, connected normal

T
1

{T_1}

-space

X
X

is unicoherent if and only if for every pair of disjoint nonseparating continua

C
C

and

D
D

in

X
X

,

C
∪
D

C \cup D

does not separate

X
X

. Among the several corollaries is the proposition:

X
X

is multicoherent if and only if

X
X

is the union of a circular chain of continua

{

A
0

,

A
1

,

A
2

,

A
3

}

\left \{ {{A_0},{A_1},{A_2},{A_3}} \right \}

where no three of the

A
i

{A_i}

’s have a point in common.
A new 𝑞-Lagrange formula and some applications
Authors: Krattenthaler Christian
Published: July 2010

A new

q
q

-extension of the Lagrange-Bürmann expansion and related formulas are proved. Finally we give a method to find

q
q

-generalizations of Riordan’s inverse relations.
Monic polynomials and generating ideals efficiently
Authors: Nashier Budh
Published: July 2010

If

I
I

is an ideal containing a monic polynomial in

R
[
T
]

R[T]

where

R
R

is a semilocal ring, then

I
I

and

I

/

I
2

I/{I^2}

require the same minimal number of generators. An ideal containing a monic polynomial in a polynomial ring need not possess any minimal set of generators having a monic as a part of it.
Stationary points of plane forms
Authors: Colley Susan Jane
Published: July 2010

We apply formulas for multiple-points of general mappings to enumerate loci of stationary multiple-points of plane forms. This is accomplished by studying the normalization map of the plane form.
A remark on the structure of absolute Galois groups
Authors: Würfel Tilmann
Published: July 2010

Let the field

F
F

contain all

p
p

-power roots of unity for some prime

p
p

and suppose that the absolute Galois group

G
G

of

F
F

is a one-relator pro-

p
p

group. We use Merkurjev-Suslin’s theorem on the power norm residue symbol to show that

G
G

is an extension of a Demushkin group by a free pro-

p
p

group.
Some modular identities of Ramanujan useful in approximating 𝜋
Authors: Borwein Jon
Published: July 2010

We show how various modular identities due to Ramanujan may be used to produce simple high order approximations to

π
\pi

. Various specializations are considered and the Gaussian arithmetic geometric mean formula for

π
\pi

is rederived as a consequence.
Exponential sums and forms for varieties over finite fields
Authors: Sarkisian Richard G.
Published: July 2010

We prove that the roots of the

L
L

-function of an Artin-Schreier cover of an algebraic variety defined over a finite field differ from the roots of the zeta function of the cover by roots of unity.
Coincidence sets in the obstacle problem for the 𝑝-harmonic operator
Authors: Sakaguchi Shigeru
Published: July 2010

We consider the obstacle problem for the

p
p

-harmonic operator
\[

div

(

|

∇
⋅

|

p
−
2

∇
⋅

)

with

p
>
1
,

{\text {div}}\left ( {{{\left | {\nabla \cdot } \right |}^{p - 2}}\nabla \cdot } \right )\quad {\text {with}}\;p > 1,

\]
and show that the coincidence set is star shaped under certain conditions on the obstacle.
Weighted norm inequalities for certain integral operators. II
Authors: Heinig H. P.
Published: July 2010

Conditions on nonnegative weight functions

u
(
x
)

u(x)

and

υ
(
x
)

\upsilon (x)

are given which ensure that an inequality of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis integral StartAbsoluteValue upper T f left-parenthesis x right-parenthesis EndAbsoluteValue Superscript q Baseline u left-parenthesis x right-parenthesis d x right-parenthesis Superscript 1 slash q Baseline less-than-or-slanted-equals upper C left-parenthesis integral StartAbsoluteValue f left-parenthesis x right-parenthesis EndAbsoluteValue Superscript p Baseline upsilon left-parenthesis x right-parenthesis d x right-parenthesis Superscript 1 slash p">

(
∫

|

T
f
(
x
)

|

q

u
(
x
)

d
x

)

1

/

q

⩽
C

(
∫

|

f
(
x
)

|

p

υ
(
x
)

d
x

)

1

/

p

{(\smallint {\left | {Tf(x)} \right |^q}u(x)\;dx)^{1/q}} \leqslant C{(\smallint {\left | {f(x)} \right |^p}\upsilon (x)\;dx)^{1/p}}

holds for

1
⩽
q
>
p
>
∞

1 \leqslant q > p > \infty

, where

T
T

is an integral operator of the form

∫

−
∞

x

K
(
x
,
y
)
f
(
y
)
d
y

\int _{ - \infty }^x {K(x,y)f(y)dy}

or

∫
x
∞

K
(
y
,
x
)
f
(
y
)

d
y

\int _x^\infty {K(y,x)f(y)\;dy}

and

C
C

a constant independent of

f
f

. Specifically a number of inequalities for well-known classical operators are obtained. Inequalities of the above form for

1
⩽
p
⩽
q
>
∞

1 \leqslant p \leqslant q > \infty

were obtained in [1].
A small boundary for 𝐻^{∞} on a strictly pseudoconvex domain
Authors: Cupillari Antonella
Published: July 2010

Let

n
⩾
2

n \geqslant 2

and

D
⊂⊂

C

n

D \subset \subset {{\mathbf {C}}^n}

be a strictly pseudoconvex domain with

C
k

{C^k}

boundary for

k
>
2

k > 2

. There is a closed nowhere dense subset of the maximal ideal space of

L
∞

(

b

D
)

{L^\infty }({\text {b}}D)

which defines a closed boundary for

H
∞

(
D
)

{H^\infty }(D)

.
A note on minimal covers for sofic systems
Authors: Boyle Mike , Kitchens Bruce , Marcus Brian
Published: July 2010

We characterize the sofic systems which have minimal subshift-of-finite-type covers and derive some consequences.
Monotone 𝐿₁-approximation on the unit 𝑛-cube
Authors: Darst Richard B. , Huotari Robert
Published: July 2010

Let

Ω
\Omega

be the unit

n
n

-cube

[
0
,
1

]
n

{[0,1]^n}

, and let

M
M

be the set of all real-valued functions on

Ω
\Omega

, each of which is nondecreasing in each variable separately. If

f
:
Ω
→

R

f:\Omega \to \mathbb {R}

is continuous, we show that there exists an (essentially) unique, best

L
1

{L_1}

-approximation,

f
1

{f_1}

, to

f
f

by elements of

M
M

, and that

f
1

{f_1}

is continuous.
Weak spectral theory
Authors: Wong M. W.
Published: July 2010

We initiate the weak spectrum of a linear operator on

L
p

{L^p}

spaces,

1
⩽
p
>
∞

1 \leqslant p > \infty

. The weak spectrum of a pseudo-differential operator with symbol in

S

α
σ

m

S_{\alpha \sigma }^m

, where

−
∞
>
m
>
∞

- \infty > m > \infty

and

0
⩽
ρ
⩽
1

0 \leqslant \rho \leqslant 1

, is investigated.
Spherical quadrature and inversion of Radon transforms
Authors: Madych W. R. , Nelson S. A.
Published: July 2010

An equivalence is established between certain aspects of: (i) mechanical quadrature for integration on a sphere; (ii) a ridge function representation problem connected with inversion of Radon transform data.
Embedded minimal surfaces in 3-manifolds with positive scalar curvature
Authors: Rubinstein J. H.
Published: July 2010

Let

M
M

be a closed orientable Riemannian

3
3

-manifold with positive scalar curvature. We prove that any embedded closed minimal surface in

M
M

has a topological description as a generalized Heegaard surface. Also an existence theorem is proved which gives examples of such minimal surfaces.
Fold singularities in pseudo-Riemannian geodesic tubes
Authors: Kossowski Marek
Published: July 2010

For a general submanifold of a pseudo Riemannian manifold, the exponential mapping of the orthogonal bundle into the ambient manifold may fail to be a diffeomorphism on the zero section. Here we show that differential geometric information intrinsic to the submanifold determines when this map has a fold singularity.
Continuous functions on the space of probabilities
Authors: Bagchi S. C. , Rao B. V.
Published: July 2010

Weiss and Dubins discovered that any continuous function

g
(
P
)

g(P)

on the space of probabilities

P

\mathcal {P}

of a compact Hausdorff space

K
K

is of the form

∫
f

d

P
∞

\smallint f\;d{P^\infty }

for some continuous function

f
f

on

K
∞

{K^\infty }

. A short proof is given here.
Open and monotone fixed point free maps on uniquely arcwise connected continua
Authors: Mohler Lee , Oversteegen Lex G.
Published: July 2010

In this note we will construct uniquely arcwise connected continua admitting open and monotone fixed point free mappings, respectively. We will also show that each locally one-to-one map on a uniquely arcwise connected continuum has a fixed point.
A characterization of closed images of metric spaces
Authors: Foged L.
Published: July 2010

We prove that a regular topological space is the image of a metric space under a closed mapping if and only if it is a Fréchet space with a

σ
\sigma

-hereditarily closure-preserving

k
k

-network.
Inverse systems of absolute retracts and almost continuity
Authors: Akis Vladimir N.
Published: July 2010

Suppose that

Y
Y

is the inverse limit of a sequence of absolute retracts such that each bonding map is a retraction. We show that

Y
Y

is the almost continuous retract of the Hilbert cube. It follows that

Y
Y

, the cone over

Y
Y

, the suspension of

Y
Y

, and the product of

Y
Y

with any absolute retract must have the fixed point property.
A formally normal operator having no normal extension
Authors: Schmüdgen Konrad
Published: July 2010

We give an example of a formally normal operator

N
N

satisfying

dim
⁡

D

(

N
∗

)

/

D

(
N
)
=
1

\dim \mathcal {D}({N^ * })/\mathcal {D}(N) = 1

which has no normal extension in any larger Hilbert space.
The multiplicity of the Steinberg representation of 𝐺𝐿_{𝑛}𝐹_{𝑞} in the symmetric algebra
Authors: Kuhn N. J. , Mitchell S. A.
Published: July 2010

Let

S
(
V
)

S(V)

denote the symmetric algebra on the standard

n
n

-dimensional representation

V
V

of

G

L

n

F

q

{\text {G}}{{\text {L}}_n}{{\mathbf {F}}_q}

. The multiplicity series in

S
(
V
)

S(V)

for the Steinberg representation St of

G

L

n

F

q

{\text {G}}{{\text {L}}_n}{{\mathbf {F}}_q}

is determined. This series is defined by

F

S

t

(
t
)
=

∑

k
=
0

∞

a
k

t
k

{F_{{\text {S}}\text {t}}}(t) = \sum \nolimits _{k = 0}^\infty {{a_k}{t^k}}

where

a
k

a_k

is the multiplicity of St in the

k
k

th symmetric power

S
k

(
V
)

{S^k}(V)

. We show that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript upper S t Baseline left-parenthesis t right-parenthesis equals t Superscript r Baseline product Underscript i equals 1 Overscript n Endscripts left-parenthesis 1 minus t Superscript q Super Superscript i Superscript minus 1 Baseline right-parenthesis Superscript negative 1">

F

S

t

(
t
)
=

t
r

∏

i
=
1

n

(
1
−

t

q
i

−
1

)

−
1

{F_{{\text {S}}t}}(t) = {t^r}\prod \nolimits _{i = 1}^n {{{(1 - {t^{{q^i} - 1}})}^{ - 1}}}

, where

r
=

∑

i
=
1

n
−
1

(

q
i

−
1

)

r = \sum \nolimits _{i = 1}^{n - 1} {({q^i} - 1} )

. The proof involves a general property of Tits buildings and a computation of the invariants in

S
(
V
)

S(V)

of the parabolic subgroups of

G

L

n

F

q

{\text {G}}{{\text {L}}_n}{{\mathbf {F}}_q}

.
Deformations of exceptional Weierstrass points
Authors: Diaz Steven
Published: July 2010

Using first order deformation theory of pointed curves we show that the semigroup of a generic Weierstrass point whose semigroup has first nonzero element

k
k

consists only of multiples of

k
k

until after its greatest gap value, and that on the moduli space of curves two components of the divisor of points corresponding to curves possessing exceptional Weierstrass points intersect nontransversely.
Divisibility properties of additive bases
Authors: Nathanson Melvyn B.
Published: July 2010

Let

h
⩾
2

h \geqslant 2

. There exists an asymptotic basis

A
A

of order

h
h

such that

(

a
1

,
…
,

a
k

)

 > 1

({a_1}, \ldots ,{a_k}){\text { > 1}}

for all

a
1

…
,

a
k

∈
A

{a_1} \ldots ,{a_k} \in A

if and only if

k

 > 

h

k{\text { > }}h

. If

k
⩾
h

k \geqslant h

, the sumset

h
A

hA

contains only composite numbers. For

h
=
k

h = k

, there exists a set

A
A

of nonnegative integers with

(

a
1

,
…
,

a
h

)

 > 

1

({a_1}, \ldots ,{a_h}){\text { > }}1

for all

a
1

,
…
,

a
h

∈
A

{a_1}, \ldots ,{a_h} \in A

such that for every prime

p
p

the sumset

h
A

hA

contains all sufficiently large multiples of

p
p

.
Waring’s problem for finite intervals
Authors: Nathanson Melvyn B.
Published: July 2010

Let

f
(
n
,
k
,
s
)

f(n,k,s)

denote the cardinality of the smallest set

A
A

of nonnegative

k
k

-th powers such that every integer in

[
0
,
n
]

[0,n]

is a sum of

s
s

elements of

A
A

, and let

β
(
k
,
s
)
=

lim su

p

n
→
∞

log
⁡
f
(
n
,
k
,
s
)

/

log
⁡
n

\beta (k,s) = {\text {lim su}}{{\text {p}}_{n \to \infty }}\log f(n,k,s)/\log n

. Clearly,

β
(
k
,
s
)
⩾
1

/

s

\beta (k,s) \geqslant 1/s

. In this paper it is proved that

f
(
n
,
k
,
s
)

 > 

c

n

1

/

(
s
−
g
(
k
)
+
k
)

f(n,k,s){\text { > }}c{n^{1/(s - g(k) + k)}}

for all

n
⩾

n
1

(
k
,
s
)

n \geqslant {n_1}(k,s)

, where

g
(
k
)

g(k)

is defined as in Waring’s problem, and

β
(
k
,
s
)
∼
1

/

s

\beta (k,s) \sim 1/s

as

s
→
∞

s \to \infty

.
On the universality of words for the alternating groups
Authors: Droste Manfred
Published: July 2010

We prove the following theorem on the finite alternating groups

A
n

{A_n}

: For each pair

(
p
,
q
)

(p,q)

of nonzero integers there exists an integer

N
(
p
,
q
)

N(p,q)

such that, for each

n
⩾
N

n \geqslant N

, any even permutation

a
∈

A
n

a \in {A_n}

can be written in the form

a
=

b
p

⋅

c
q

a = {b^p} \cdot {c^q}

for some suitable elements

b
,
c
∈

A
n

b,c \in {A_n}

. A similar result is shown to be true for the finite symmetric groups

S
n

{S_n}

provided that

p
p

or

q
q

is odd.
Modules determined by their tops and socles
Authors: Bongartz K. , Smalø S. O.
Published: July 2010

We prove that a module in a preprojective component of the Auslander-Reiten quiver of an artin algebra is determined by its top and socle. Also other invariants determining such a module are given.
Some properties of FC-groups which occur as automorphism groups
Authors: Zimmerman Jay
Published: July 2010

We prove that if

G
G

is a group such that Aut

G
G

is a countably infinite torsion

F
C

FC

-group, then Aut

G
G

contains an infinite locally soluble, normal subgroup and hence a nontrivial abelian normal subgroup. It follows that a countably infinite subdirect product of nontrivial finite groups, of which only finitely many have nontrivial abelian normal subgroups, is not the automorphism group of any group.
The distance between the eigenvalues of Hermitian matrices
Authors: Bhatia Rajendra
Published: July 2010

It is shown that the minmax principle of Ky Fan leads to a quick simple derivation of a recent inequality of V. S. Sunder giving a lower bound for the spectral distance between two Hermitian matrices. This brings out a striking parallel between this result and an earlier known upper bound for the spectral distance due to L. Mirsky.
On semigroups generated by 𝑚-accretive operators in a strict sense
Authors: Watanabe Michiaki
Published: July 2010

Let

{

S
(
t
)
:
t
⩾
0

}

\left \{ {S(t):t \geqslant 0} \right \}

be a nonlinear semigroup generated by an

m
m

-accretive operator

A
A

in a real Banach space

(

X
,

|
⋅
|

)

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

. It is shown that (1) for any

x
∈
D
(
A
)

x \in D(A)

belongs to

L
p

(
0
,
T
;
V
)
(
T
>
0
)

{L^p}(0,T;V)(T > 0)

if

A
A

satisfies
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue u 1 minus u EndAbsoluteValue Superscript p Baseline plus upper C lamda double-vertical-bar u 1 minus u 2 double-vertical-bar Superscript p Baseline less-than-or-slanted-equals StartAbsoluteValue u 1 plus lamda upper A u 1 minus u 2 minus lamda upper A u 2 EndAbsoluteValue Superscript p">

|

u
1

−
u

|

p

+
C
λ

‖

u
1

−

u
2

‖

p

⩽

|

u
1

+
λ
A

u
1

−

u
2

−
λ
A

u
2

|

p

{\left | {{u_1} - u} \right |^p} + C\lambda {\left \| {{u_1} - {u_2}} \right \|^p} \leqslant {\left | {{u_1} + \lambda A{u_1} - {u_2} - \lambda A{u_2}} \right |^p}

\]

(
p

 > 1,

C

 > 

0
)

(p{\text { > 1,}}C{\text { > }}0)

for

λ

 > 

0

\lambda {\text { > }}0

and

u
i

∈
D
(
A
)
(
i
=
1
,
2
)

{u_i} \in D(A)(i = 1,2)

, where

(

V
,

‖
⋅
‖

)

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

is a Banach space including

D
(
A
)

D(A)

and included continuously in

X
X

; and that (2)

A
+
B

A + B

has similar properties to those of the above

A
A

if

B
B

is a Lipschitz continuous operator from

V
V

to

X
X

.
On the Dieudonné property for 𝐶(Ω,𝐸)
Authors: Kalton N. J. , Saab E. , Saab P.
Published: July 2010

In a recent paper, F. Bombal and P. Cembranos showed that if

E
E

is a Banach space such that

E
∗

{E^*}

is separable, then

C
(
Ω
,
E
)
C

C(\Omega ,E)C

, the Banach space of continuous functions from a compact Hausdorff space

Ω
\Omega

to

E
E

, has the Dieudonné property. They asked whether or not the result is still true if one only assumes that

E
E

does not contain a copy of

l
1

{l_1}

. In this paper we give a positive answer to their question. As a corollary we show that if

E
E

is a subspace of an order continuous Banach lattice, then

E
E

has the Dieudonné property if and only if

C
(
Ω
,
E
)

C(\Omega ,E)

has the same property.
A note on ideals of operators
Authors: Loebl Richard I.
Published: July 2010

An ideal

I
I

of

L

(
H
)

L\left ( H \right )

is said to be multiplicatively prime if

A
X
B
∈
I

AXB \in I

for all

X
∈
L

(
H
)

X \in L\left ( H \right )

implies

A
A

or

B
B

is in

I
I

. The only normable multiplicatively prime ideals are 0 and

K
K

, the compacts. Multiplicative primeness is related to other properties an ideal may possess.
Is the slit of a rational slit mapping in 𝑆 straight?
Authors: Srebro Uri
Published: July 2010

The question in the title is answered by showing that if

f
f

is a rational function in

C
^

{\mathbf {\hat C}}

and maps some disk injectively onto the complement of a set

E
E

of empty interior, then

degree
⁡
(
f
)
=
2

\operatorname {degree}(f) = 2

, and

E
E

is either a circular arc or a line segment in

C
^

=

C

∪
{
∞
}

{\mathbf {\hat C}} = {\mathbf {C}} \cup \{ \infty \}

.
Existence of solutions of 𝑥”+𝑥+𝑔(𝑥)=𝑝(𝑡),𝑥(0)=0=𝑥(𝜋)
Authors: Kannan R. , Ortega R.
Published: July 2010

We obtain criteria for the existence of solutions of

x
+
x
+
g
(
x
)
=
p
(
t
)
,
x
(
0
)
=
0
=
x
(
π
)

x + x + g(x) = p(t),x(0) = 0 = x(\pi )

, where

g
:
R
→
R

g:R \to R

is not necessarily bounded and does not necessarily have proper limits

g
(
∞
)

g(\infty )

and

g
(
−
∞
)

g( - \infty )

.
An error estimate for continued fractions
Authors: Gill John
Published: July 2010

New and improved truncation error bounds are derived for continued fractions

K
(

a
n

/

1
)

K({a_n}/1)

, where

a
n

→
0

{a_n} \to 0

. The geometrical approach is somewhat unusual in that it involves both isometric circles and fixed points of bilinear transformations.
Global stability of a biological model with time delay
Authors: Lenhart Suzanne M. , Travis Curtis C.
Published: July 2010

This paper gives necessary and sufficient conditions for global stability of certain logistic delay differential equations for all values of the delay.
Hopf formula and multitime Hamilton-Jacobi equations
Authors: Lions P.-L. , Rochet J.-C.
Published: July 2010

Problems arising in mathematical economics lead to the study of multi-time Hamilton-Jacobi equations. Using commutation properties of the semigroups for the standard equation, we propose a generalization of the Hopf formula that gives explicit solutions of these equations.
A finite but not stably finite 𝐶*-algebra
Authors: Clarke N. P.
Published: July 2010

In E. G. Effros’s paper [4] given at the 1980 Kingston Conference of the American Mathematical Society, he stated that "an example of a finite but not stably finite

C
∗

{C^*}

-algebra has yet to be found." This paper seeks to give an example of such an algebra by using a simple application of the duality between

K
K

and Ext-theory arising from the work of Brown, Douglas and Fillmore (see, for example, [2 and 3]).
Nonamenability and Borel paradoxical decompositions for locally compact groups
Authors: Paterson Alan L. T.
Published: July 2010

We show that a locally compact group

G
G

is not amenable if and only if it admits a Borel paradoxical decomposition.
Structure of Banach algebras 𝐴 satisfying 𝐴𝑥²=𝐴𝑥 for every 𝑥∈𝐴
Authors: Esterle J. , Oudadess M.
Published: July 2010

We give a complete characterization of Banach algebras which satisfy the condition

A

x
2

=
A
x

A{x^2} = Ax

for every

x
∈
A

x \in A

.
Some weakly compact operators between Banach lattices do not factor through reflexive Banach lattices
Authors: Talagrand Michel
Published: July 2010

We construct an example proving the claim of the title.
A new property equivalent to Lebesgue integrability
Authors: Schurle Arlo W.
Published: July 2010

Using the generalized Riemann approach to Lebesgue integration we define a new property which is equivalent to Lebesgue integrability for measurable functions. Roughly speaking, this property says that Riemann sums for sufficiently fine partitions of sufficiently small intervals can always be made arbitrarily small. We formulate this property in such a way that it applies to either Lebesgue integration or Perron integration, thus correcting a defect in earlier versions of this idea. The condition of measurability is used only in preliminary results to insure that the support of functions can always be assumed to be

G
δ

{G_\delta }

-sets.
On the continuity of Haar measure on topological groupoids
Authors: Seda Anthony Karel
Published: July 2010

It is shown that continuity of a family of invariant (Haar) measures on a topological groupoid

G
G

is equivalent to the continuity of the implied convolution product

f
∗
g

f * g

for all pairs of functions

f
f

and

g
g

. An example is given of a groupoid which admits no (continuous) Haar measure. It results, therefore, that the usual

C
∗

{C^ * }

-algebra associated with a Haar measure on

G
G

cannot, in general, be constructed. Some remarks are included concerning the construction of Haar measures on the holonomy groupoid of a foliated manifold.
Decomposition of charges and measures
Authors: Luschgy Harald
Published: July 2010

By means of a characterization of the band generated by a set of charges on an algebra of sets one can obtain unified proofs of some decomposition theorems for charges and measures.
Inner model operators and the continuum hypothesis
Authors: Becker Howard
Published: July 2010

Assuming AD, every inner model operator satisfies the continuum hypothesis.
Definable automorphisms of 𝒫(𝜔)/𝒻𝒾𝓃
Authors: Veličković Boban
Published: July 2010

We investigate definable automorphisms of

P

(
ω
)

/

fin

\mathcal {P}\left ( \omega \right )/{\text {fin}}

and show that e.g. every Borel automorphism is trivial. The existence of nontrivial projective automorphisms is consistent and independent from

ZFC + CH

{\text {ZFC + CH}}

.
A short computation of the norms of free convolution operators
Authors: Woess Wolfgang
Published: July 2010

Akemann and Ostrand in 1976 gave a formula for the norms of free convolution operators on the

L
2

{L^2}

-space of a discrete group. Using random walk techniques and generating functions, a short and elementary computation of this formula is given.
A characterization of semispan of continua
Authors: Duda Edwin
Published: July 2010

The main results of this paper are characterizations of semispan

ε
,
ε
⩾
0

\varepsilon ,\varepsilon \geqslant 0

, for those metric spaces which are atriodic hereditarily unicoherent continua. The results follow from Theorem 1 which gives conditions under which the union of two continua, each of semispan less than or equal to

ε
\varepsilon

, has semispan less than or equal to

ε
\varepsilon

.
The 3-dimensionality of certain codimension-3 decompositions
Authors: Daverman R. J.
Published: July 2010

It is proved that if

p
p

is a proper mapping of an

(
n
+
3
)

(n + 3)

-manifold

M
M

onto a metric space

B
B

such that each inverse set

p

−
1

b

{p^{ - 1}}b

has the shape of a closed, connected, orientable

n
n

-manifold, then

B
B

is

3
3

-dimensional.
A uniqueness theorem for monic Blaschke products
Authors: Horwitz Alan L. , Rubel Lee A.
Published: July 2010

If two monic Blaschke products of order

n
n

agree at

n
n

points of the open unit disc

D

{\mathbf {D}}

, then they must be identical.
On powers of characters and powers of conjugacy classes of a finite group
Authors: Blau Harvey I. , Chillag David
Published: July 2010

Two results are proved. The first gives necessary and sufficient conditions for a power of an irreducible character of a finite group to have exactly one irreducible constituent. The other presents necessary and sufficient conditions for a power of a conjugacy class of a finite group to be a single conjugacy class. Examples are given.
On the Diophantine equation 𝑥²ⁿ-𝒟𝓎²=1
Authors: Cao Zhen Fu
Published: July 2010

In this paper, it has been proved that if

n
>
2

n > 2

and Pell’s equation

u
2

−

D

v
2

=
−
1

{u^2} - \mathcal {D}{v^2} = - 1

has integer solution, then the equation

x

2
n

−

D

y
2

=
1

{x^{2n}} - \mathcal {D}{y^2} = 1

has only solution in positive integers

x
=
3

x = 3

,

y
=
22

y = 22

(when

n
=
5

n = 5

,

D

=
122

\mathcal {D} = 122

). That is proved by studying the equations

x
p

+
1
=
2

y
2

{x^p} + 1 = 2{y^2}

and

x
p

−
1
=
2

y
2

{x^p} - 1 = 2{y^2}

(

p
p

is an odd prime). In addition, some applications of the above result are given.
A note on GPIs and their coefficients
Authors: Lanski Charles
Published: July 2010

If

R
R

is a prime ring satisfying a GPI, then

R
R

satisfies a multilinear GPI having all its coefficients in

R
R

. Also, all

R
R

bimodules in the Martindale quotient ring of

R
R

satisfy the same multilinear GPIs.
Extending valuations to finite-dimensional division algebras
Authors: Wadsworth Adrian R.
Published: July 2010

LetDDbe a division algebra finite dimensional over its centerFF. It is shown that a (Krull) valuationvvonFFextends to a valuation onDDiffυ\upsilonextends uniquely to each commutative fieldKKwithF⊆K⊆DF \subseteq K \subseteq D.
On nearly absolutely isolated hypersurface singularities of dimension 2
Authors: Yang Jin Gen
Published: July 2010

A formula for the geometric genus of a nearly absolutely isolated hypersurface singularity of dimension 2 is found by using the canonical resolution. An upper bound for the fundamental cycle of such singularity is also given.
Some rings of differential operators which are Morita equivalent to the Weyl algebra 𝐴₁
Authors: Musson Ian M.
Published: July 2010

For a certain class of semigroup algebras

k
Λ

k\Lambda

, we show that the ring of all differential operators on

k
Λ

k\Lambda

is Morita equivalent to the first Weyl algebra

A
1

{A_1}

.