bluuebunny/crossref_metadata_2025 huggingface dataset:
Variations of cub filter on 𝒫_{𝜅}𝜆
Authors: Matsubara Yo
Published: June 2010

In his 1973 paper, Jech extended the notion of cub and stationary sets to such sets in

P

κ

λ

{\mathcal {P}_\kappa }\lambda

and showed that many of their properties are preserved. We study variations of cub filters in this paper. We make use of the partition property (a large cardinal hypothesis) to investigate the properties of these filters. In the last section we investigate the relation of our filter to supercompact filters on

P

ℵ
1

λ

{\mathcal {P}_{{\aleph _1}}}\lambda

under the Axiom of Determinacy. This motivates the formulation of a certain infinitary partition property, and this property implies the

λ
\lambda

-supercompactness of

ℵ
1

{\aleph _1}

.
The pseudocompact extension 𝛼𝑋
Authors: Aull C. E. , Sawyer J. O.
Published: June 2010

For any Tychonoff space we define

α
X
=
(
β
X
−
v
X
)
∪
X
=
β
X
−
(
v
X
−
X
)

\alpha X = (\beta X - vX) \cup X = \beta X - (vX - X)

. We show that

α
X

\alpha X

is the smallest pseudocompactification

Y
Y

of

X
X

contained is

β
X

\beta X

such that every free hyperreal

z
z

-ultrafilter on

X
X

converges in

Y
Y

and is the largest pseudocompactification

Y
Y

of

X
X

contained in

β
X

\beta X

such that every point in

Y
−
X

Y - X

is contained in a zero set of

Y
Y

which does not intersect

X
X

. A space

S
S

is defined to be

α
\alpha

-embedded in a space

X
X

if

α
S
⊂
β
X

\alpha S \subset \beta X

. Properties of

α
\alpha

-embeddings and its relation to

v
v

-embeddings of Blair

C
∗

{C^*}

-embeddings,

C
C

-embeddings, and well-embeddings are investigated. For instance, if

S
S

is

α
\alpha

-embedded and dense in

X
,
S

X,S

is fully well-embedded (for

P
,
R
⊂
X

P,R \subset X

, where

S
⊂
P
⊂
R
⊂
X

S \subset P \subset R \subset X

,

P
P

is well-embedded in

R
R

) in

X
X

iff

α
X
−
α
S
=
X
−
S

\alpha X - \alpha S = X - S

.
On the monodromy group of everywhere tangent lines to the octic surface in 𝑃³
Authors: D’Souza Harry
Published: June 2010

Let

S
0

{S_0}

be an octic surface in

P

3

,
G
=
G
(
1
,
3
)

{{\mathbf {P}}^3},G = G(1,3)

= Grassmannian of lines in

P

3

{{\mathbf {P}}^3}

, and

J

=
{
(
x
,
l
)

|

x
∈
l
∩

S
0

}
⊂

S
0

×
G

{\mathbf {J}} = \{ (x,l)|x \in l \cap {S_0}\} \subset {S_0} \times G

. Then

dim
⁡

J

=
5

\dim {\mathbf {J}} = 5

. Let

L

=

{
l

|

l

 is 

e
v
e
r
y
w
h
e
r
e

 tangent to 

S
0

}
−

⊂
G

{\mathbf {L}} = {\{ l|l{\text { is }}everywhere{\text { tangent to }}{S_0}\} ^ - } \subset G

. Let

π
2

:

S
0

×
G
→
G

{\pi _2}:{S_0} \times G \to G

be the projection onto the second factor. We denote its restriction to

J

{\mathbf {J}}

also by

π
2

{\pi _2}

. Then the locus of everywhere tangent lines is

π
2

(

L

)

{\pi _2}({\mathbf {L}})

. In this article we show that the monodromy group of these lines is the full symmetric group.
PFA implies all automorphisms are trivial
Authors: Shelah Saharon , Steprāns Juris
Published: June 2010

It is shown that PFA implies that all automorphisms of the Boolean algebra

P

(
ω
)

\mathcal {P}\left ( \omega \right )

/Finite are induced by almost-permutations of

ω
\omega

.
Fixed points for discontinuous quasi-monotone maps in 𝑅ⁿ
Authors: Hu Shou Chuan
Published: June 2010

Let

K
n

{K_n}

be the unit cube in

R
n

{R^n}

and

f
=
(

f
1

,

f
2

,
…
,

f
n

)
:

K
n

→

R
n

f = ({f_1},{f_2}, \ldots ,{f_n}):{K_n} \to {R^n}

. It is known that

f
f

has maximal and minimal fixed points in

K
n

{K^n}

if

f
:

K
n

→

K
n

f:{K_n} \to {K_n}

and

f
f

is monotone increasing. In this paper, a weaker condition, namely quasi-monotonicity, is considered and it is proved that the above mentioned conclusion is still true if

f
f

is either quasi-monotone and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit inf Underscript t right-arrow 0 Endscripts StartFraction left-bracket f Subscript i Baseline left-parenthesis x plus t e Subscript i Baseline right-parenthesis minus f Subscript i Baseline left-parenthesis x right-parenthesis right-bracket Over t EndFraction not-equals negative normal infinity comma">

lim

inf

t
→
0

[

f
i

(
x
+
t

e
i

)
−

f
i

(
x
)
]

t

≠
−
∞
,

\lim \inf \limits _{t \to 0} \frac {{[{f_i}(x + t{e_i}) - {f_i}(x)]}}{t} \ne - \infty ,

\]
or

−
f

- f

is quasi-monotone and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit sup Underscript t right-arrow 0 Endscripts StartFraction left-bracket f Subscript i Baseline left-parenthesis x plus t e Subscript i Baseline right-parenthesis minus f Subscript i Baseline left-parenthesis x right-parenthesis right-bracket Over t EndFraction not-equals plus normal infinity">

lim

sup

t
→
0

[

f
i

(
x
+
t

e
i

)
−

f
i

(
x
)
]

t

≠
+
∞

\lim \sup \limits _{t \to 0} \frac {{[{f_i}(x + t{e_i}) - {f_i}(x)]}}{t} \ne + \infty

\]
.
On stability of endomorphisms
Authors: Ikeda Hiroshi
Published: June 2010

In this note we prove a generalization of R. Mañé’s theorem. R. Mañé proved that

C
r

{C^r}

absolutely stable endomorphisms satisfy Axiom A. We prove that if an endomorphism

f
f

is both

C
r

{C^r}

structurally and infinitesimally stable, then

f
f

satisfies Axiom A.
GPIs having coefficients in Utumi quotient rings
Authors: Chuang Chen-Lian
Published: June 2010

Let

R
R

be a prime ring and let

U
U

be its Utumi quotient ring. We prove the following: (1) If

R
R

satisfies a GPI having all its coefficients in

U
U

, then

R
R

satisfies a GPI having all its coefficients in

R
R

. (2)

R
R

and

U
U

satisfy the same GPIs having their coefficients in

U
U

.
Idempotents in the reduced 𝐶*-algebra of a free group
Authors: Cohen Joel M. , Figà-Talamanca Alessandro
Published: June 2010

The purpose of this note is to give a simple proof of the theorem, originally due to M. Pimsner and D. Voiculescu, that the reduced

C
∗

{C^*}

-algebra of a free group has no nontrivial idempotents.
Note on subnormal weighted shifts
Authors: Fan Peng
Published: June 2010

The purpose of this note is two-fold: (1) To point out that Stampfli’s characterization of subnormal shifts can be reformulated in operator form which turns out to be the Spitkovskii’s characterization of subnormal operators; (2) To use this reformulation to give a new proof of the subnormality of a class of shifts.
Codimension two nonorientable submanifolds with nonnegative curvature
Authors: Baldin Yuriko Y. , Mercuri Francesco
Published: June 2010

We prove that a compact nonorientable

n
n

-dimensional submanifold of

R

n
+
2

{{\mathbf {R}}^{n + 2}}

with nonnegative curvature is a "generalized Klein bottle" if

n
≥
3

n \geq 3

.
A note on ₂𝑠𝑐(𝐸₁)
Authors: Barua Rana
Published: June 2010

We exhibit a set of reals recursive in

E

1

{{\mathbf {E}}_1}

which is not in

C

\mathcal {C}

, the smallest

σ
\sigma

-field containing analytic sets and closed under operation

A

\mathcal {A}

. As a consequence, a conjecture of Hinman is shown to be false.
Imbedding nondegenerate Jordan algebras in semiprimitive algebras
Authors: Martindale W. S. , McCrimmon K.
Published: June 2010

Zelmanov’s structure theory for prime Jordan algebras works directly with semiprimitive algebras, and the results are extended to nondegenerate algebras using properties of the free Jordan algebra. Here we show how Amitsur’s direct power trick of imbedding

J
J

in the algebra of all sequences from

J
J

can be used to imbed any nondegenerate algebra

J
J

in a semiprimitive

J
¯

\bar J

having exactly the same polynomial identities as

J
J

.
Elementary equivalence and profinite completions: a characterization of finitely generated abelian-by-finite groups
Authors: Oger Francis
Published: June 2010

In this paper, we show that any finitely generated abelian-by-finite group is an elementary submodel of its profinite completion. It follows that two finitely generated abelian-by-finite groups are elementarily equivalent if and only if they have the same finite images. We give an example of two finitely generated abelian-by-finite groups

G
,
H

G,H

which satisfy these properties while

G
×

Z

G \times {\bf {Z}}

and

G
×

Z

G \times {\bf {Z}}

are not isomorphic. We also prove that a finitely generated nilpotent-by-finite group is elementarily equivalent to its profinite completion if and only if it is abelian-by-finite.
Zeros of diagonal equations over finite fields
Authors: Wan Da Qing
Published: June 2010

Let

N
N

be the number of solutions

(

x
1

,
…
,

x
n

)

({x_1}, \ldots ,{x_n})

of the equation (1)
\[

(
1
)

c
1

x
1

d
1

+

c
2

x
2

d
2

+
⋯
+

c
n

x
n

d
n

=
c

(1)\quad {c_1}x_1^{{d_1}} + {c_2}x_2^{{d_2}} + \cdots + {c_n}x_n^{{d_n}} = c

\]
over the finite field

F
q

{F_q}

, where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d Subscript i Baseline vertical-bar left-parenthesis q minus 1 right-parenthesis comma c Subscript i Baseline element-of upper F Subscript q Superscript asterisk Baseline left-parenthesis i equals 1 comma ellipsis comma n right-parenthesis">

d
i

|

(
q
−
1
)
,

c
i

∈

F
q
∗

(
i
=
1
,
…
,
n
)

{d_i}|(q - 1),{c_i} \in F_q^*(i = 1, \ldots ,n)

, and

c
∈

F
q

c \in {F_q}

. If
\[

1

d
1

+

1

d
2

+
⋯
+

1

d
n

>
b
≥
1

\frac {1} {{{d_1}}} + \frac {1} {{{d_2}}} + \cdots + \frac {1} {{{d_n}}} > b \geq 1

\]
for some positive integer

b
b

, we prove that

q
b

|

N

{q^b}|N

. This result is an improvement of the theorem that

p

|

N

p|N

obtained by B. Morlaye [7] and also by J. R. Joly [3].
Convergence and divergence almost everywhere of spherical means for radial functions
Authors: Kanjin Yūichi
Published: June 2010

Let

d
>
1

d > 1

. It will be shown that the maximal operator

S
∗

{S^*}

of spherical means

S
R

,
R
>
0

{S_R},R > 0

, is bounded on

L
p

(

R

d

)

{L^p}({{\mathbf {R}}^d})

radial functions when

2
d

/

(
d
+
1
)
>
p
>
2
d

/

(
d
−
1
)

2d/(d + 1) > p > 2d/(d - 1)

, and it implies that, for every

L
p

(

R

d

)

{L^p}({{\mathbf {R}}^d})

radial function

f
(
t
)
,

S
R

f
(
t
)

f(t),{S_R}f(t)

converges to

f
(
t
)

f(t)

for a.e.

t
∈

R

d

t \in {{\mathbf {R}}^d}

when

2
d

/

(
d
+
1
)
>
p
≤
2

2d/(d + 1) > p \leq 2

. Also, it will be proved that there is an

L

2
d

/

(
d
+
1
)

(

R
d

)

{L^{2d/(d + 1)}}({R^d})

radial function

f
(
t
)

f(t)

with compact support such that

S
R

f
(
t
)

{S_R}f(t)

diverges for a.e.

t
∈

R
d

t \in {R^d}

.
Classification of one-dimensional hypergroups
Authors: Schwartz Alan L.
Published: June 2010

(
H
,
∗
)

(H, * )

is a hypergroup if

H
H

is a locally compact space and * is a binary operation with respect to which

M
(
H
)

M(H)

, the bounded Borel measures on

H
H

, becomes a Banach algebra with a number of additional properties so that * generalizes group convolution. The case when

H
H

is a one-dimensional set, that is, a circle or an interval (possibly unbounded), includes a large number of examples: the classical group algebras

M
(
R
)

M(R)

and

M
(
T
)

M(T)

(

T
T

is the unit circle), the subalgebra of

M
(
R
)

M(R)

consisting of even measures, the subalgebra of

M
(

R
n

)

M({R^n})

consisting of rotation invariant measures, and the subalgebra of

M
(

S
n

)

M({S^n})

(

S
n

{S^n}

is the unit sphere in

R

n
+
1

{R^{n + 1}}

) consisting of zonal measures. In addition to these, there are several continua of measure algebras unrelated to groups or geometry such as the measure algebras which arise in connection with Hankel transforms, ultraspherical and Jacobi polynomial series, and Sturm-Liouville expansions. The main result of this article is a classification of the one-dimensional hypergroups. It is shown that if a certain amount of "regularity" for * is assumed, then every one-dimensional hypergroup is commutative and, up to a change of variables, must be one of the following types: (i) the classical group algebra

M
(
R
)

M(R)

, (ii) the classical group algebra

M
(
T
)

M(T)

, (iii)

(
H

,
∗

)

(H{,^*})

, where

H
=
[
0
,
P
]

H = [0,P]

or

[
0
,
∞
)

[0,\infty )

,

δ
0

{\delta _0}

is an identity for *, and

0
∈
supp
⁡
(

δ
x

∗

δ
y

)

0 \in \operatorname {supp}({\delta _x}*{\delta _y})

if and only if

x
=
y

x = y

.
Analytic functions with rectifiable radial images
Authors: Ortel Marvin , Schneider Walter
Published: June 2010

We give a simple sufficient condition for an analytic function in the unit disk to have a radial image of finite length.
Characterizing shape preserving 𝐿₁-approximation
Authors: Zwick D.
Published: June 2010

With certain restrictions, a characterization of the best

L
1

{L_1}

-approximation to a continuous function from the set of

n
n

-convex functions is proved. Under these restrictions the best approximation is shown to be unique. The case

n
=
2

n = 2

(convex functions) is considered in more detail.
An upper bound for the projection constant
Authors: Lewis D. R.
Published: June 2010

There is a positive function

δ
(
n
)

\delta (n)

of exponential order such that, for any normed space

E
E

of dimension

n
≥
2

n \geq 2

, the projection constant of

E
E

satisfies

λ
(
E
)
≤

n

1

/

2

[
1
−
δ
(
n
)
]

\lambda (E) \leq {n^{1/2}}[1 - \delta (n)]

.
Occupation time and the Lebesgue measure of the range for a Lévy process
Authors: Port S. C.
Published: June 2010

We consider a Levy process on the line that is transient and with nonpolar one point sets. For

a
>
0

a > 0

let

N
(
a
)

N(a)

be the total occupation time of

[
0
,
a
]

[0,a]

and

R
(
a
)

R(a)

the Lebesgue measure of the range of the process intersected with

[
0
,
a
]

[0,a]

. Whenever

[
0
,
∞
]

[0,\infty ]

is a recurrent set we show

N
(
a
)

/

E
N
(
a
)
−
R
(
a
)

/

E
R
(
a
)

N(a)/EN(a) - R(a)/ER(a)

converges in the mean square to 0 as

a
→
∞

a \to \infty

. This in turn is used to derive limit laws for

R
(
a
)

/

E
R
(
a
)

R(a)/ER(a)

from those for

N
(
a
)

/

E
N
(
a
)

N(a)/EN(a)

.
An asymptotic expansion for the expected number of real zeros of a random polynomial
Authors: Wilkins J. Ernest
Published: June 2010

Let

ν
n

{\nu _n}

be the expected number of real zeros of a polynomial of degree

n
n

whose coefficients are independent random variables, normally distributed with mean 0 and variance 1. We find an asymptotic expansion for

ν
n

{\nu _n}

of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu Subscript n Baseline equals StartFraction 2 Over pi EndFraction log left-parenthesis n plus 1 right-parenthesis plus sigma-summation Underscript p equals 0 Overscript normal infinity Endscripts upper A Subscript p Baseline left-parenthesis n plus 1 right-parenthesis Superscript negative p">

ν
n

=

2
π

log
⁡
(
n
+
1
)
+

∑

p
=
0

∞

A
p

(
n
+
1
)

−
p

\nu _n = \frac {2}{\pi } \log (n + 1) + \sum \limits _{p = 0}^\infty {{A_p}{{(n + 1)}^{ - p}}}

\]
in which

A
0

=
0.625735818
,

A
1

=
0
,

A
2

=
−
0.24261274
,

A
3

=
0
,

A
4

=
−
0.08794067
,

A
5

=
0

{A_0} = 0.625735818,{A_1} = 0,{A_2} = - 0.24261274,{A_3} = 0,{A_4} = - 0.08794067,{A_5} = 0

. The numerical values of

ν
n

{\nu _n}

calculated from this expansion, using only the first four, or six, coefficients, agree with previously tabulated seven decimal place values

(
1
≤
n
≤
100
)

(1 \leq n \leq 100)

with an error of at most

10

−
7

{10^{ - 7}}

when

n
≥
30

n \geq 30

, or

n
≥
8

n \geq 8

.
ℒ-manifolds and cone-dual maps
Authors: Dragotti Sara , Magro Gaetano
Published: June 2010

Let

f
:
P
→
Q

f:P \to Q

be a simplicial map such that

D
(
α
,
f
)

D(\alpha ,f)

, the dual to

α
\alpha

with respect to

f
f

, is a cone, for each simplex

α
\alpha

of

Q
Q

. It is shown that if

P
P

is an

L
L

-manifold then

f
f

is approximable by PL-homeomorphisms, provided that

f
f

satisfies an extra condition on the boundary of

P
P

.
Simultaneous systems of representatives for families of finite sets
Authors: Nathanson Melvyn B.
Published: June 2010

Let

h
≥
2

h \geq 2

and

k
≥
1

k \geq 1

. Then there exists a real number

λ
=
λ

(

h
,
k

)

∈

(

0
,
1

)

\lambda = \lambda \left ( {h,k} \right ) \in \left ( {0,1} \right )

such that, if

S

=

{

S
i

}

i
=
1

s

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

and

T

=

{

T
j

}

j
=
1

t

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

are families of nonempty, pairwise disjoint sets with

|

S
i

|

≤
h

|{{S_i}}| \leq h

and

|

T
j

|

≤
k

|{{T_j}}| \leq k

and

S
i

⊈

T
j

{S_i} \nsubseteq {T_j}

for all

i
i

and

j
j

, then

N

(

S

,

T

)

≤

h
s

λ
t

N\left ( {\mathcal {S},\mathcal {T}} \right ) \leq {h^s}{\lambda ^t}

, where

N

(

S

,

T

)

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

is the number of 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}

. A conjecture about the best possible value of the constant

λ

(

h
,
k

)

\lambda \left ( {h,k} \right )

is proved in the case

h
>
k

h > k

. The necessity of the disjointness conditions for the families

S

\mathcal {S}

and

T

\mathcal {T}

is also demonstrated.
Weights of semi-invariants of the quotient division ring of an enveloping algebra
Authors: Nauwelaerts E. , Ooms A. I.
Published: June 2010

Let

L
L

be a finite dimensional Lie algebra over a field

k
k

of characteristic zero,

D
(
L
)

D(L)

the quotient division ring of

U
(
L
)

U(L)

. It is shown that the weights of the semi-invariants of

D
(
L
)

D(L)

form a finitely generated, free abelian group

Λ
D

(
L
)

{\Lambda _D}(L)

. It follows, among other things, that the semicenter of

D
(
L
)

D(L)

is isomorphic to the group algebra of

Λ
D

(
L
)

{\Lambda _D}(L)

over the center

Z
(
D
(
L
)
)

Z(D(L))

of

D
(
L
)

D(L)

.
The Hodge group of an abelian variety
Authors: Murty V. Kumar
Published: June 2010

Let

A
A

be a simple abelian variety of odd dimension, defined over

C

{\mathbf {C}}

. If the Hodge classes on

A
A

are intersections of divisors, then the semisimple part of the Hodge group of

A
A

is as large as it is allowed to be by endomorphisms and polarizations.
Fréchet differentiable points in Bochner function spaces 𝐿_{𝑝}(𝜇,𝑋)
Authors: Wang Jian Hua , Nan Chao Xun
Published: June 2010

In this paper, a characterization of Fréchet differentiable points of

L
p

(
μ
,
X
)
,
1
>
p
>
∞

{L_p}(\mu ,X),1 > p > \infty

, is given:

f
∈

L
p

(
μ
,
X
)
,
f
≠
0

f \in {L_p}(\mu ,X),f \ne 0

is a point of Fréchet differentiability of the norm if and only if the values

f
(
t
)

f(t)

are such almost everywhere in the support of

f
f

.
Subaveraging estimates for CR functions
Authors: Boggess A. , Dwilewicz R.
Published: June 2010

We give conditions on a CR submanifold

M
M

in

C

n

{{\mathbf {C}}^n}

and a compact submanifold

N
⊂
M

N \subset M

such that the average value on

N
N

of a CR function on

M
M

can be estimated uniformly by the

L
1

{L^1}

-norm of the CR function on a neighborhood of

N
N

in

M
M

. The conditions involve the Levi form of

M
M

and the transversality of

N
N

to the holomorphic tangent bundle of

M
M

.
The derivative of Bazilevič functions
Authors: London R. R. , Thomas D. K.
Published: June 2010

For

α

 > 

0

\alpha {\text { > }}0

, let

B
1

(
α
)

{B_1}(\alpha )

be the class of normalised analytic functions

f
f

defined in the open unit disc

D
D

such that

Re(

f
(
z
)

/

z

)

α
−
1

f
′

(
z
)

 > 

0

{\text {Re(}}f(z)/z{{\text {)}}^{\alpha - 1}}f’(z){\text { > }}0

for

z
∈
D

z \in D

. Sharp upper and lower bounds are obtained for

|

z

f
′

(
z
)

/

f
(
z
)

|

\left | {zf’(z)/f(z)} \right |

when

f
∈

B
1

(
α
)

f \in {B_1}(\alpha )

.
Ideals of multiplier algebras of simple AF 𝐶*-algebras
Authors: Lin Hua Xin
Published: June 2010

It is shown that the

C
∗

{C^ * }

-algebra

M
(
A
)

/

A

M(A)/A

, where

A
A

is a nonunital separable simple AF

C
∗

{C^ * }

-algebra and

M
(
A
)

M(A)

is the multiplier algebra of

A
A

, is simple if and only if

A
A

has a continuous scale or

A
A

is elementary. Some results concerning the ideal structure of

M
(
A
)

/

A

M(A)/A

are also obtained in the case that it is nontrivial.
Covering the boundary of a convex set by tiles
Authors: Lassak Marek
Published: June 2010

Let Euclidean space

E
n

{E^n}

be dissected by a finite family of hyper-planes. We estimate how many of the tiles obtained are sufficient to cover the boundary of a convex set

A
⊂

E
n

A \subset {E^n}

.
Existence of ad-nilpotent elements and simple Lie algebras with subalgebras of codimension one
Authors: Varea V. R.
Published: June 2010

For a perfect field

F
F

of arbitrary characteristic, the following statements are proved to be equivalent: (1) Any Lie algebra over

F
F

contains an ad-nilpotent element. (2) There are no simple Lie algebras over

F
F

having only abelian subalgebras. They are used to guarantee the existence of an ad-nilpotent element in every Lie algebra over a perfect field of type

(

C
1

)

({C_1})

of arbitrary characteristic (in particular, over any finite field). Furthermore, we give a sufficient condition to insure the existence of ad-nilpotent elements in a Lie algebra over any perfect field. As a consequence of this result we obtain an easy proof of the fact that the Zassenhaus algebras and

sl

(
2
,
F
)

{\text {sl}}(2,F)

are the only simple Lie algebras which have subalgebras of codimension 1, whenever the ground field

F
F

is perfect with

char

(
F
)
≠
2

{\text {char}}(F) \ne 2

. All Lie algebras considered are finite dimensional.
Modular group algebras of simply presented abelian groups
Authors: May Warren
Published: June 2010

Let

G
G

be a

p
p

-local Warfield group and

F
F

a perfect field of characteristic

p
p

. It is shown that if the group algebras FG and FH are isomorphic, then

G
G

and

H
H

are isomorphic.
Wiener-Hopf operators on a finite interval and Schatten-von Neumann classes
Authors: Peller Vladimir V.
Published: June 2010

Recently R. Rochberg has characterized Hankel and Toeplitz operators on the Paley-Wiener space which belong to the Schatten-von Neumann class

S
p

{S_p}

for

p
≥
1

p \geq 1

. These operators coincide up to Fourier transform with Wiener-Hopf operators on a finite interval. Using a different approach, we extend Rochberg’s result to all positive

p
p

.
𝐿² boundedness of highly oscillatory integrals on product domains
Authors: Prestini Elena
Published: June 2010

We prove

L
2

{L^2}

boundedness of the oscillatory singular integral
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T f left-parenthesis x comma y right-parenthesis equals double-integral Underscript upper D Subscript y Baseline Endscripts StartFraction exp left-parenthesis 2 pi i upper N left-parenthesis y right-parenthesis x prime right-parenthesis Over x prime y prime EndFraction f left-parenthesis x minus x Superscript prime Baseline comma y minus y Superscript prime Baseline right-parenthesis d x prime d y Superscript prime Baseline">

T
f
(
x
,
y
)
=

∬

D
y

exp
⁡
(
2
π
i
N
(
y
)

x
′

)

x
′

y
′

f
(
x
−

x
′

,
y
−

y
′

)
d

x
′

d

y
′

Tf(x,y) = \iint \limits _{{D_y}} {\frac {{\operatorname {exp} (2\pi iN(y)x’)}}{{x’y’}}}f(x - x’,y - y’)dx’dy’

\]
where

N
(
y
)

N(y)

is an arbitrary integer-valued

L
∞

{L^\infty }

function and where nothing is assumed on the dependency upon

y
y

of the domain of integration

D
y

{D_y}

. We also prove

L
2

{L^2}

boundedness of the corresponding maximal opertaor. Operators of this kind appear in a problem of a.e. convergence of double Fourier series.
𝐺_{𝜅} subspaces of hyadic spaces
Authors: Bell Murray G.
Published: June 2010

A hyadic space is a continuous image of a hyperspace of a compact space. For an infinite cardinal

κ
\kappa

, an intersection of at most

κ
\kappa

many open subsets of

X
X

is called a

G
κ

{G_\kappa }

subset of

X
X

. We construct, in ZFC, a compact separable space of uncountable

π
\pi

-weight and of cardinality continuum. This space is a

G
ω

{G_\omega }

subset of a hyadic space. We show that a compact space that does not contain any convergent sequences and which contains the Stone-Čech compactification of the countable discrete space cannot be imbedded as a

G
κ

{G_\kappa }

subset, where

κ
\kappa

is less than the continuum, of any hyadic space.
Characterizations of inductive posets with applications
Authors: Tasković Milan R.
Published: June 2010

This paper presents new characterizations of inductiveness of posets in terms of fixed apexes and fixed points. Applications in nonlinear functional analysis and fixed point theory are considered.
Constructive reflexivity of a uniformly convex Banach space
Authors: Ishihara Hajime
Published: June 2010

In this paper we consider a question about reflexivity of a Banach space within the framework of Bishop’s constructive mathematics and we give a partially affirmative answer to the question set by Bishop: "Is every uniformly convex Banach space reflexive?".
The Zalcman conjecture for close-to-convex functions
Authors: Ma Wan Cang
Published: June 2010

Let

S
S

be the class of functions

f
(
z
)
=
z
+
⋯

f(z) = z + \cdots

analytic and univalent in the unit disk

D
D

. For

f
(
z
)
=
z
+

a
2

z
2

+
⋯
∈
S

f(z) = z + {a_2}{z^2} + \cdots \in S

, Zalcman conjectured that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue a Subscript n Superscript 2 Baseline minus a Subscript 2 n minus 1 Baseline EndAbsoluteValue less-than-or-equal-to left-parenthesis n minus 1 right-parenthesis squared left-parenthesis n equals 2 comma 3 comma ellipsis right-parenthesis">

|

a
n
2

−

a

2
n
−
1

|

≤

(
n
−
1

)
2

(
n
=
2
,
3
,
…
)

|a_n^2 - {a_{2n - 1}}|\; \leq \;{(n - 1)^2}(n = 2,3, \ldots )

. This conjecture is verified for

n
≥
4

n \geq 4

and close-to-convex functions.
Unique solutions for a class of discontinuous differential equations
Authors: Bressan Alberto
Published: June 2010

This paper is concerned with the Cauchy Problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove x With dot left-parenthesis t right-parenthesis equals f left-parenthesis t comma x left-parenthesis t right-parenthesis right-parenthesis comma x left-parenthesis t 0 right-parenthesis equals x 0 element-of double-struck upper R Superscript n Baseline comma">

x
˙

(
t
)

=
f

(

t
,
x

(
t
)

)

,

x

(

t
0

)

=

x
0

∈

R

n

,

\dot x\left ( t \right ) = f\left ( {t,x\left ( t \right )} \right ),\quad x\left ( {{t_0}} \right ) = {x_0} \in {\mathbb {R}^n},

\]
where the vector field

f
f

may be discontinuous with respect to both variables

t
,
x

t,x

. If the total variation of

f
f

along certain directions is locally finite, we prove the existence of a unique solution, depending continuously on the initial data.
On the spectral picture of an irreducible subnormal operator
Authors: McGuire Paul
Published: June 2010

This paper extends the following result of R. F. Olin and J. E. Thomson: A compact subset

K
K

of the plane is the spectrum of an irreducible subnormal operator if and only if

R

(
K
)

\mathcal {R}(K)

has exactly one nontrivial Gleason part

G
G

such that

K
K

is the closure of

G
G

. The main result of this paper is that the only additional requirement needed for the pair

{

K
,

K
e

}

\left \{ {K,{K_e}} \right \}

to be the spectrum and essential spectrum, respectively, is that

K
e

{K_e}

be a compact subset of

K
K

which contains the boundary of

K
K

. Additionally, results are obtained on the question of which index values can be specified on the various components of the complement of

K
e

{K_e}

.
On a natural connection between the entropy spaces and Hardy space 𝑅𝑒𝐻¹
Authors: Dąbrowski Romuald
Published: June 2010

In 1983 B. Korenblum [7, 8] introduced a class of Banach function spaces associated with the notion of entropy (we will call these spaces and their norms entropy spaces and entropy norms, respectively). Entropy spaces were used in [8] as a tool for proving a new convergence test for Fourier series which includes classical tests of Dirichlet-Jordan and Dini-Lipschitz. In this paper we construct natural linear operators from the entropy spaces to Hardy space

Re

H
1

{\text {Re}}{H^1}

[5, 6]. In fact, these operators define multiplier type bounded embeddings of entropy spaces to

Re

H
1

{\text {Re}}{H^1}

. As a corollary we obtain a growth condition for Fourier coefficients of a continuous periodic function of bounded entropy norm (as announced in [4]). In particular, we show that if

f
f

is a continuous periodic function of bounded Shannon entropy norm (resp. of bounded Lipschitz entropy norm with exponent

q
,
0
>
q
>
1

q,0 > q > 1

), and

{

a
n

}

n
∈

Z

{\left \{ {{a_n}} \right \}_{n \in {\mathbf {Z}}}}

are the Fourier coefficients of

f
f

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n not-equals 0 Endscripts StartAbsoluteValue a Subscript n Baseline left-parenthesis log left-parenthesis vertical-bar n vertical-bar right-parenthesis slash n right-parenthesis EndAbsoluteValue greater-than normal infinity">

∑

n
≠
0

|

a
n

(

log(|n|)/n

)

|

>
∞

\sum \nolimits _{n \ne 0} {|{a_n}({\text {log(|n|)/n}})| > \infty }

(resp.

∑

n
≠
0

|

a
n

|

/

|n|

)
q

>
∞

\sum \nolimits _{n \ne 0} {|{a_n}|/{\text {|n|}}{)^q} > \infty }

). In §4 we study the relationship between the dual spaces of entropy spaces and space B.M.O. of functions of bounded mean oscillation. In §5 we conjecture that

Re
⁡

H
1

\operatorname {Re} {H^1}

is a direct limit of the entropy spaces in an appropriate sense.
On the spectral rigidity of 𝐶𝑃ⁿ
Authors: Perrone Domenico
Published: June 2010

Complex projective space

C

P
n

C{P^n}

with the Fubini-Study metric has recently been characterized by the spectrum of the Laplacian on

2
2

-forms. This important result was proved separately for

n
≠
2
,
8

n \ne 2,8

by B. Y. Chen and L. Vanhecke, and for

n
=
2

n = 2

and

n
=
8

n = 8

by S. I. Goldberg. In this paper, we give a new proof which does not distinguish the three cases. It makes strong use of a result of S. Kobayashi and T. Ochiai, and may be applied to the spectrum of the Laplacian on

1
1

-forms. Moreover, a characterization of

C

P
2

C{P^2}

by the spectrum of the Laplacian on

1
1

-forms is given.
Normal forms for skew-symmetric matrices and Hamiltonian systems with first integrals linear in momenta
Authors: Thompson G.
Published: June 2010

Using an appropriate notion of equivalence, those classical Hamiltonian systems which admit a first integral of motion polynomial of degree one in momentum are classified. The classification is effected by means of finding a normal form for a skew-symmetric matrix under the action of orthogonal symmetry.
Property C′′ and function spaces
Authors: Sakai Masami
Published: June 2010

For a Tychonoff space

X
X

, we show that each finite product of

X
X

has property

C

{\text {C}}

iff

C
p

(
X
)

{C_p}\left ( X \right )

has countable strong fan tightness, where

C
p

(
X
)

{C_p}\left ( X \right )

is the function space on

X
X

with the topology of pointwise convergence.
Compactifications of countable-dimensional and strongly countable-dimensional spaces
Authors: Engelking Ryszard , Pol Roman
Published: June 2010

Simple proofs of theorems on existence of compactifications of countable-dimensional and strongly countable-dimensional spaces are given.
An empty class of nonmetric spaces
Authors: Dow Alan
Published: June 2010

Let CSSM be the class of compact nonmetrizable spaces in which every subspace of cardinality at most

ω
1

{\omega _1}

is metrizable. We show that CSSM is empty.
Remarks on Besov spaces and best polynomial approximation
Authors: Ditzian Z. , Totik V.
Published: June 2010

Besov-type spaces that are generated by new moduli of smoothness are related to sequence spaces of

{

E
n

(
f
)

}

\left \{ {{E_n}\left ( f \right )} \right \}

where

E
n

(
f
)

{E_n}\left ( f \right )

is the appropriate best algebraic polynomial approximation.
A family of abelian varieties rationally isogenous to no Jacobian
Authors: Parish James L.
Published: June 2010

Let

E
d

{E_d}

, for any

d
∈

Q

(
i

)
∗

d \in {\mathbf {Q}}{(i)^*}

, be the curve

x
3

−
d
x

z
2

=

y
2

z

{x^3} - dx{z^2} = {y^2}z

, and let

g
g

be any positive integer. It is shown that, if

d
d

is not a square in

Q

(
i
)

{\mathbf {Q}}(i)

and

g
>
1

g > 1

, the abelian variety

E
d
g

E_d^g

is not isogenous over

Q

(
i
)

{\mathbf {Q}}(i)

to the Jacobian of any genus-

g
g

curve. The proof proceeds by showing that any curve whose Jacobian is isogenous to

E
d
g

E_d^g

over

Q

(
i
)

{\mathbf {Q}}(i)

must be hyperelliptic, and then showing that no hyperelliptic curve can have Jacobian isogenous to

E
d
g

E_d^g

over

Q

(
i
)

{\mathbf {Q}}(i)

.
Coefficients of symmetric functions of bounded boundary rotation
Authors: Koepf Wolfram
Published: June 2010

The well-known inclusion relation between functions with bounded boundary rotation and close-to-convex functions of some order is extended to

m
m

-fold symmetric functions. This leads solving the corresponding result for close-to-convex functions to the sharp coefficient bounds for

m
m

-fold symmetric functions of bounded boundary rotation at most

k
π

k\pi

when

k
≥
2
m

k \geq 2m

. Moreover it shows that an

m
m

-fold symmetric function of bounded boundary rotation at most

(
2
m
+
2
)
π

(2m + 2)\pi

is close-to-convex and thus univalent.
A counterexample to the conjecture of Woess on simple random walks on trees
Authors: Berman Kenneth A. , Konsowa Mokhtar
Published: June 2010

Let

T
T

be a locally finite tree with a countable number of vertices. The volume of

T
T

is the energy dissipation of the unit flow from the root of infinity that divides equally at every branching of the tree. It follows from Thomson’s Principle that if

T
T

contains an infinite leafless subtree whose volume is finite then

T
T

is transient. Woess [6] conjectured that the converse is also true. In this paper we give a counterexample to this conjecture by constructing a transient tree, such that every infinite leafless subtree has infinite volume.
Multipliers on weighted 𝐿_{𝑝}-spaces over locally compact Vilenkin groups
Authors: Onneweer C. W. , Quek T. S.
Published: June 2010

Let

G
G

be a locally compact Vilenkin group. We prove a multiplier theorem for power-weighted

L
p

(
G
)

{L_p}(G)

-spaces which extends an earlier result of this kind, due to Onneweer.
Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schrödinger operators with symmetric single-well potentials
Authors: Ashbaugh Mark S. , Benguria Rafael
Published: June 2010

We prove the optimal lower bound

λ
2

−

λ
1

≥
3

π
2

/

d
2

{\lambda _2} - {\lambda _1} \geq 3{\pi ^2}/{d^2}

for the difference of the first two eigenvalues of a one-dimensional Schrödinger operator

−

d
2

/

d

x
2

+
V
(
x
)

- {d^2}/d{x^2} + V(x)

with a symmetric single-well potential on an interval of length

d
d

and with Dirichlet boundary conditions. Equality holds if and only if the potential is constant. More generally, we prove the inequality

λ
2

[

V
1

]
−

λ
1

[

V
1

]
≥

λ
2

[

V
0

]
−

λ
1

[

V
0

]

{\lambda _2}[{V_1}] - {\lambda _1}[{V_1}] \geq {\lambda _2}[{V_0}] - {\lambda _1}[{V_0}]

in the case where

V
1

{V_1}

and

V
0

{V_0}

are symmetric and

V
1

−

V
0

{V_1} - {V_0}

is a single-well potential.
On a property of metric projections onto closed subsets of Hilbert spaces
Authors: Frerking J. , Westphal U.
Published: June 2010

Applying the theory of monotone operators to the metric projection

P
K

{P_K}

of a Hilbert space

H
H

onto a nonempty closed subset

K
K

of

H
H

we prove a kind of connectedness property of the set

{
x
∈
H
;

P
K

(
x
)

\{ x \in H;{P_K}(x)

is not a singleton or

P
K

{P_K}

is not upper semi-continuous at

x
}

x\}

which is a typical set for investigations in best approximation. A result of Balaganskii is extended.
On weighted norm inequalities for positive linear operators
Authors: Kerman R. , Sawyer E.
Published: June 2010

Let

T
T

be a positive linear operator defined for nonnegative functions on a

σ
\sigma

-finite measure space

(

X
,
m
,
μ

)

\left ( {X,m,\mu } \right )

. Given

1
>
p
>
∞

1 > p > \infty

and a nonnegative weight function

w
w

on

X
X

, it is shown that there exists a nonnegative weight function

v
v

, finite

μ
\mu

-almost everywhere on

X
X

, such that (1)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper X Endscripts left-parenthesis upper T f right-parenthesis Superscript p Baseline w d mu less-than-or-equal-to integral Underscript upper X Endscripts f Superscript p Baseline v d mu comma for all f less-than-or-equal-to 0">

∫
X

(

T
f

)

p

w
d
μ
≤

∫
X

f
p

v
d
μ

,

for all 

f
≤
0

\int _X {{{\left ( {Tf} \right )}^p}wd\mu \leq \int _X {{f^p}vd\mu } } ,\quad {\text {for all }}f\leq 0

\]
, if and only if there exists

ϕ
\phi

positive

μ
\mu

-almost everywhere on

X
X

with (2)
\[

∫
X

(

T
ϕ

)

p

w
d
μ
>
∞
.

\int \limits _X {{{\left ( {T\phi } \right )}^p}wd\mu > \infty .}

\]
In case (2) holds, we may take

v
=

ϕ

1
−
p

T
∗

[

(

T
ϕ

)

p
−
1

w

]

v = {\phi ^{1 - p}}{T^*}\left [ {{{\left ( {T\phi } \right )}^{p - 1}}w} \right ]

in (1). This partially answers a question of B. Muckenhoupt in [5]. Applications to some specific operators are also given.
On monoids of finite real character
Authors: Krause Ulrich
Published: June 2010

The property of Krull domains, to be determined already by its essential valuations, is extended to a class of monoids of finite real character containing Krull monoids. This is applied to construct a divisor theory for such monoids and also to characterize Krull domains purely multiplicatively as the integral domains for which the multiplicative monoid is a Krull monoid.
On the classification of homogeneous multipliers bounded on 𝐻¹(𝑅²)
Authors: Daly James E. , Phillips Keith
Published: June 2010

Necessary and sufficient conditions for Calderon-Zygmund singular integral operators to be bounded operators on

H
1

(

R

2

)

{H^1}({{\mathbf {R}}^2})

are investigated. Let

m
m

be a bounded measurable function on the circle, extended to

R

2

{{\mathbf {R}}^2}

by homogeneity

(
m
(
r
x
)
=
m
(
x
)
)

(m(rx) = m(x))

. If the Calderon-Zygmund singular integral operator

T
m

{T_m}

, defined by

T
m

f
=

F

−
1

(
m

F

(
f
)
)

{T_m}f = {\mathcal {F}^{ - 1}}(m\mathcal {F}(f))

, is bounded on

H
1

(

R

2

)

{H^1}({{\mathbf {R}}^2})

, then it is proved that

S
∗

m

{S^*}m

has bounded variation on the circle, where the Fourier transform of

S
S

on the circle is

S
^

(
n
)
=

(
−

isgn(

n
)

)

n
+
1

\widehat {S}(n) = {( - {\text {isgn(}}n))^{n + 1}}

. This implies that

m
m

must have an absolutely convergent Fourier series on the circle, and other relations on the Fourier series of

m
m

. Partial converses are also given. The problems are formulated in terms of distributions on the circle and on

R

2

{{\mathbf {R}}^2}

.
On the theorems of Šarkovskiĭ and Štefan on cycles
Authors: Gaweł Bolesław
Published: June 2010

New proofs of the well-known theorems of Šarkovskiĭ and Štefan on cycles of a continuous real mapping are given.
On the second derivatives of convex functions on Hilbert spaces
Authors: Kato Nobuyuki
Published: June 2010

Let

ϕ
\phi

be a proper l.s.c. convex function on a real Hilbert space

H
H

. We show that if

H
H

is separable, then

ϕ
\phi

is twice differentiable in some sense on a dense subset of the graph of

∂
ϕ

\partial \phi

.
On normal structure, fixed-point property and contractions of type (𝛾)
Authors: Khamsi M. A.
Published: June 2010

We prove that a Banach space

X
X

has normal structure provided it contains a finite codimensional subspace

Y
Y

such that all spreading models for

Y
Y

have normal structure. We show that a Banach space

X
X

is strictly convex if the set of fixed points of any nonexpansive map defined in any convex subset

C
⊂
X

C \subset X

is convex and give a sufficient condition for uniform convexity of a space in terms of nonexpansive map of type

(
γ
)

\left ( \gamma \right )

.
Uniqueness of bounded harmonic functions
Authors: Ortel Marvin , Schneider Walter
Published: June 2010

We prove the following theorem: A bounded harmonic function is identically zero if it tends to zero at a certain rate along a set of radii of positive measure. In particular, this uniqueness theorem does not require that the function in question have smooth boundary values or restricted derivatives.
Large groups of symmetries of handlebodies
Authors: Miller A. , Zimmermann B.
Published: June 2010

Let

V
g

{V_g}

be an orientable three-dimensional handlebody with genus

g
>
1

g > 1

. Let

N
(
g
)

N(g)

be the largest order among all finite groups which act effectively on

V
g

{V_g}

and preserve orientation. We show that

4
(
g
+
1
)
≤
N
(
g
)
≤
12
(
g
−
1
)

4(g + 1) \leq N(g) \leq 12(g - 1)

, and that

N
(
g
)

N(g)

equals either

8
(
q
−
1
)

8(q - 1)

or

12
(
g
−
1
)

12(g - 1)

when

g
g

is odd. Moreover each of the indicated upper and lower bounds are achieved for infinitely many genera

g
g

. The techniques which are used lead to more detailed results and also specialize to yield similar results for compact surfaces with nonempty boundary.
Terms in the Selberg trace formula for 𝑆𝐿(3,𝑍)\𝑆𝐿(3,𝑅)/𝑆𝑂(3,𝑅) associated to Eisenstein series coming from a maximal parabolic subgroup
Authors: Wallace D. I.
Published: June 2010

There are two types of Eisenstein series associated to

S
L

(
3
,

Z

)

\mathrm {SL}(3, \mathbf {Z})

. This paper deals with those which are built out of cuspidal Maass waveforms for

S
L

(
2
,

Z

)

\mathrm {SL}(2, \mathbf {Z})

. We compute the inner product of two of them over a truncated fundamental region and then compute the rate of divergence as the truncation parameter tends to infinity. The solution of this problem is of use in computing the details of the trace formula for

S
L

(
3
,

Z

)

\mathrm {SL}(3, \mathbf {Z})

.
A greedoid polynomial which distinguishes rooted arborescences
Authors: Gordon Gary , McMahon Elizabeth
Published: June 2010

We define a two-variable polynomial

f
G

(
t
,
z
)

{f_G}(t,z)

for a greedoid

G
G

which generalizes the standard one-variable greedoid polynomial

λ
G

(
t
)

{\lambda _G}(t)

. Several greedoid invariants (including the number of feasible sets, bases, and spanning sets) are easily shown to be evaluations of

f
G

(
t
,
z
)

{f_G}(t,z)

. We prove (Theorem 2.8) that when

G
G

is a rooted directed arborescence,

f
G

(
t
,
z
)

{f_G}(t,z)

completely determines the arborescence. We also show the polynomial is irreducible over

Z

[
t
,
z
]

{\mathbf {Z}}[t,z]

for arborescences with only one edge directed out of the distinguished vertex. When

G
G

is a matroid,

f
G

(
t
,
z
)

{f_G}(t,z)

coincides with the Tutte polynomial. We also give an example to show Theorem 2.8 fails for full greedoids. This example also shows

f
G

(
t
,
z
)

{f_G}(t,z)

does not distinguish rooted arborescences among the class of all greedoids.
On inner automorphisms of finite groups
Authors: Pettet Martin R.
Published: June 2010

It is shown that in certain classes of finite groups, inner automorphisms are characterized by an extension property and also by a dual lifting property. This is a consequence of the fact that for any finite group

G
G

and any prime

p
p

, there is a

p
p

-group

P
P

and a semidirect product

H
=
G
P

H = GP

such that

P
P

is characteristic in

H
H

and every automorphism of

H
H

induces an inner automorphism on

H

/

P

H/P

.
Mosco convergence and the Kadec property
Authors: Borwein Jonathan M. , Fitzpatrick Simon
Published: June 2010

We study the relationship between Wijsman convergence and Mosco convergence for sequences of convex sets. Our central result is that Mosco convergence and Wijsman convergence coincide for sequences of convex sets if and only if the underlying space is reflexive with the dual norm having the Kadec property.
Criteria for selfadjointness of the Dirac operator on pseudomanifolds
Authors: Chou Arthur W.
Published: June 2010

We study the Dirac operator on pseudomanifolds with piecewise constant curvature metric. The criteria for the self-adjointness of the Dirac operator are obtained and a vanishing theorem is proved. At the end we make some comments on the index theorem and the

A
^

\widehat A

-genus.
The asymptotics of the determinant function for a class of operators
Authors: Friedlander Leonid
Published: June 2010

Let

A
A

be an elliptic pseudodifferential operator on a closed manifold

M
M

and

ord

A
>
dim
⁡
M

{\text {ord}}A > \dim M

. We derive the asymptotics of

log
⁡
det
(
1
+
ε

A

−
1

)

\log \det (1 + \varepsilon {A^{ - 1}})

when

ε
→
∞

\varepsilon \to \infty

. The constant term of this asymptotics equals

−
log
⁡
det
A

- \log \det A

.
Hyperbolic surfaces and quadratic equations in groups
Authors: Gu Zhi-Bin
Published: June 2010

A group of a hyperbolic

2
2

-complex

K
K

is a group with its associated van Kampen diagrams satisfying a hyperbolic curvature condition and a link condition on the degree of the interior vertices. A solution of an equation

(

y
1

,
…
,

y
n

)
=
1

({y_1}, \ldots ,{y_n}) = 1

in

K
K

, where

W
W

is a path in a

2
2

-complex

B
B

, is a mapping

ζ
:
B
→
K

\zeta :B \to K

such that

ζ
W
=
W
(
ζ

y
1

,
…
,
ζ

y
n

)

\zeta W = W(\zeta {y_1}, \ldots ,\zeta {y_n})

is contractible in

K
K

. This solution

ζ
\zeta

is free if there is a mapping

h
:
B
→

K

(
1
)

h:B \to {K^{(1)}}

such that

W
(
h

y
1

,
…
,
h

y
n

)

W(h{y_1}, \ldots ,h{y_n})

is contractible in

K

(
1
)

{K^{(1)}}

and such that

ζ
=
π
h

\zeta = \pi h

, where

π
\pi

is the projection

π
:

K

(
1
)

→
K

\pi :{K^{(1)}} \to K

. Our main result is that each quadratic equation

W
=
1

W = 1

has only finitely many nonfree solutions in

K
K

. Our tool is essentially the cancellation diagrams on surfaces developed by the present author based on work of Schupp.
The torus lemma on calibrations, extended
Authors: Morgan Frank
Published: June 2010

The whole face

G

(
φ
)

G\left ( \varphi \right )

of

m
m

-planes calibrated by a torus

m
m

-form

φ
\varphi

is determined by the torus face

G
T

(
φ
)

{G_T}\left ( \varphi \right )

. Indeed,

G

(
φ
)

G\left ( \varphi \right )

results from applying a new closure operation to

G
T

(
φ
)

{G_T}\left ( \varphi \right )

.
Deformations of Dupin hypersurfaces
Authors: Pinkall Ulrich , Thorbergsson Gudlaugur
Published: June 2010

Examples are given of compact proper-Dupin hypersurfaces that are not Lie equivalent to an isoparametric hypersurface.
Group 𝐶*-algebras as algebras of “continuous functions” with noncommuting variables
Authors: Szymański Wojciech
Published: June 2010

It is shown that a system of commutation relations depending on the structure constants of a Lie algebra

g

\mathfrak {g}

leads to a

C
∗

{C^*}

-algebra which is isomorphic to the group

C
∗

{C^*}

-algebra of the simply connected Lie group associated with

g

\mathfrak {g}

.
On second-category sets
Authors: Komjáth P.
Published: June 2010

The existence of a measurable cardinal is equiconsistent to the existence of a second category set not decomposable into the union of uncountable many disjoint second category sets.
Cofinal families of compact subsets of an analytic set
Authors: Becker Howard
Published: June 2010

We consider a question of van Douwen regarding the minimum cardinality of cofinal families of compact sets in certain topological spaces. We show that the question cannot be answered in ZFC.
Unstable 𝑣₁-periodic homotopy groups of a Moore space
Authors: Thompson Robert D.
Published: June 2010

The

mod
⁡
p

v
1

\operatorname {mod} p\quad {v _1}

-periodic homotopy groups of a space

X
X

are defined by considering the homotopy classes of maps of a Moore space into

X
X

and then inverting the Adams self-map of a Moore space. In this paper the

mod
⁡
p

v
1

\operatorname {mod} p\quad {v _1}

-periodic homotopy groups of a Moore space are computed by using the Cohen-Moore-Neisendorfer splitting of the space of loops on a Moore space. The Adams map is shown to be compatible with this splitting and it is proved that the homomorphism of

v
1

{v _1}

-periodic homotopy groups induced by the Adams map is an isomorphism.