bluuebunny/crossref_metadata_2025 huggingface dataset:
On differentiability of metric projections in 𝑅ⁿ. I. Boundary case
Authors: Shapiro Alexander
Published: June 2010

This paper is concerned with metric projections onto a closed subset

S
S

of a finite-dimensional normed space. Necessary and in a sense sufficient conditions for directional differentiability of a metric projection at a boundary point of

S
S

are given in terms of approximating cones. It is shown that if

S
S

is defined by a number of inequality constraints and a constraint qualification holds, then the approximating cone exists.
Exponentiation of reals: effects of base choice
Authors: Gurevič R.
Published: June 2010

Let

r
>
1

r > 1

and

X
r

{X_r}

be the minimal set of reals containing 1 and closed under

exp
r

:
x
↦

r
x

{\exp _r}:x \mapsto {r^x}

and addition. The behavior of

X
r

{X_r}

is studied. In particular among possible order types of

X
r

{X_r}

there are

ω
,

ω
ω

,
ω
+
q
,
ω
+
1
+
q

\omega ,{\omega ^\omega },\omega + q,\omega + 1 + q

, where

q
q

is the dense countable order without endpoints.
A note on the existence of 𝐺-maps between spheres
Authors: Waner Stefan
Published: June 2010

Let

G
G

be a finite group, and let

V
V

and

W
W

be finite-dimensional real orthogonal

G
G

-modules with

V
⊃
W

V \supset W

, and with unit spheres

S
(
V
)

S(V)

and

S
(
W
)

S(W)

respectively. The purpose of this note is to give necessary sufficient conditions for the existence of a

G
G

-map

J
:
S
(
V
)
→
S
(
W
)

J:S(V) \to S(W)

in terms of the Burnside ring of

G
G

and its relationship with

V
V

and

W
W

. Note that if

W
W

has a nonzero fixed point, such a

G
G

-map always exists, so for nontriviality, we assume this not the case.
On torsion-free abelian 𝑘-groups
Authors: Dugas Manfred , Rangaswamy K. M.
Published: June 2010

It is shown that a knice subgroup with cardinality

ℵ
1

{\aleph _1}

, of a torsion-free completely decomposable abelian group, is again completely decomposable. Any torsion-free abelian

k
k

-group of cardinality

ℵ
n

{\aleph _n}

has balanced projective dimension

≤
n

\leq n

.
Uniqueness theorems for subharmonic functions in unbounded domains
Authors: Gardiner S. J.
Published: June 2010

A theorem of Carlson says that a holomorphic function of exponential growth in the half-plane cannot approach zero exponentially along the boundary unless it vanishes identically. This paper presents a generalization of this result for subharmonic functions in a Greenian domain

Ω
\Omega

, using the Martin boundary, minimal fine topology and PWB solution to the

h
h

-Dirichlet problem. Applications of the general theorem to specific choices of

Ω
\Omega

, such as the half-space and strip, are presented in later sections.
Boundary value problems for first-order differential equations
Authors: Bobisud L. E. , O’Regan D.
Published: June 2010

Conditions sufficient to guarantee existence and uniqueness of solutions to multipoint boundary value problems for the first-order differential equation

y
′

=
h
(
t
,
y
)

y’ = h(t,y)

are given when

h
h

fails to be Lipschitz along a solution of

y
′

=
h
(
t
,
y
)

y’ = h(t,y)

and the initial-value problem thus has nonunique solutions.
A Steinhaus type theorem
Authors: Natarajan P. N.
Published: June 2010

The sequence space

Λ
r

,
r
≥
1

{\Lambda _r},r \geq 1

being a fixed integer, is defined as
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda Subscript r Baseline equals StartSet x equals left-brace x Subscript k Baseline right-brace element-of l Subscript normal infinity Baseline comma x Subscript k Baseline element-of upper K comma k equals 0 comma 1 comma 2 comma ellipsis comma StartAbsoluteValue x Subscript k plus r Baseline minus x Subscript k Baseline EndAbsoluteValue right-arrow 0 comma k right-arrow normal infinity EndSet comma">

Λ
r

=

{

x
=

{

x
k

}

∈

l
∞

,

x
k

∈
K
,
k
=
0
,
1
,
2
,
…
,

|

x

k
+
r

−

x
k

|

→
0
,
k
→
∞

}

,

{\Lambda _r} = \left \{ {x = \left \{ {{x_k}} \right \} \in {l_\infty },{x_k} \in K,k = 0,1,2, \ldots ,\left | {{x_{k + r}} - {x_k}} \right | \to 0,k \to \infty } \right \},

\]
where

K
K

is a complete, nontrivially valued field and

l
∞

{l_\infty }

is the space of bounded sequences with entries in

K
K

. In this paper, it is proved that given a regular matrix

A
=
(

a

n
k

)
,

a

n
k

∈
K
=

R

A = ({a_{nk}}),{a_{nk}} \in K = {\mathbf {R}}

or

C

{\mathbf {C}}

, there exists a sequence in

Λ
r

−

∪

i
=
1

r
−
1

Λ
i

{\Lambda _r} - \cup _{i = 1}^{r - 1}{\Lambda _i}

which is not

A
A

-summable. This is an improvement of the well-known Steinhaus theorem. It is, however, shown that this result fails to hold when

K
K

is a complete, nontrivially valued, nonarchimedean field, whereas it is known that the Steinhaus theorem continues to hold.
Similarity of parts to the whole for certain multiplication operators
Authors: Bourdon Paul S.
Published: June 2010

We show that the Bergman shift

B
B

, multiplication by

z
z

on the Bergman space

A
2

{A^2}

, is similar to its part

B

|

N

B\left | {_N} \right .

if and only if

N
=
φ

A
2

N = \varphi {A^2}

, where

φ
\varphi

is a finite product of interpolating Blaschke products. In addition, we show that

B
B

is not unitarily equivalent to any of its parts. For the analytic Toeplitz operator

T
f

{T_f}

on

H
2

{H^2}

, we obtain that

T
f

{T_f}

is similar to each of its parts if and only if

T
f

{T_f}

is unitarily equivalent to each of its parts if and only if

f
f

is a weak-star generator of

H
∞

{H^\infty }

.
A note on the Borsuk-Ulam theorem
Authors: Gauld David
Published: June 2010

Let

F

\mathcal {F}

denote the set of all maps from

S
n

{S^n}

to

R

n

{{\mathbf {R}}^n}

topologized by the usual metric, and

B

\mathcal {B}

the set of all nonempty closed subsets of

S
n

{S^n}

invariant with respect to the antipodal map. Let

β
:

F

→

B

\beta :\mathcal {F} \to \mathcal {B}

assign to each

f
∈

F

f \in \mathcal {F}

the set of all

x
x

for which

f

(
x
)

=
f

(

−
x

)

f\left ( x \right ) = f\left ( { - x} \right )

. The largest topology on

B

\mathcal {B}

for which

β
\beta

is continuous is identified: it is the upper semifinite topology.
Best constant for the ratio of the first two eigenvalues of one-dimensional Schrödinger operators with positive potentials
Authors: Ashbaugh Mark S. , Benguria Rafael
Published: June 2010

We prove the optimal upper bound

λ
2

/

λ
1

≤
4

{\lambda _2}/{\lambda _1} \leq 4

for the ratio of the first two eigenvalues of one-dimensional Schrödinger operators with nonnegative potentials. Equality holds if and only if the potential vanishes identically.
Linear operators preserving the numerical radius of matrices
Authors: Li Chi-Kwong
Published: June 2010

In this note we characterize all the linear operators on the linear space of

n
×
n

n \times n

complex matrices which preserve the numerical radius and all the linear operators on the real linear space of

n
×
n

n \times n

Hermitian matrices which preserve the numerical radius. From our results, we easily deduce V. J. Pellegrini’s characterization of all linear operators that preserve the numerical ranges of matrices and the result of Marcus and Moyls concerning the linear operators that preserve the eigenvalues of Hermitian matrices.
Group ring idempotents with minimal trace
Authors: Wallace D. A. R.
Published: June 2010

Let

G
G

be a group and let

K
K

be a field of characteristic 0. Let

e
≠
0

e \ne 0

be an idempotent of the group ring

K
(
G
)

K(G)

. It is shown that the trace of

e
e

attains its lower bound if and only if the support of

e
e

has a specific form.
On crossed products with property 𝑇
Authors: Watanabe Shōichi
Published: June 2010

Let

N
N

be a finite von Neumann algebra (with faithful normal normalized trace

τ
\tau

),

G
G

a countable discrete group, and

α
\alpha

a

τ
\tau

-preserving action of

G
G

on

N
N

such that

N

⋊
α

G

N{ \rtimes _\alpha }G

is a factor. It is proved that if

N

⋊
α

G

N{ \rtimes _\alpha }G

has Property

T

{\text {T}}

, then

G
G

has Kazhdan’s Property

T

{\text {T}}

.
Unconditional bases in 𝐿²(0,𝑎)
Authors: Hruščëv Sergeĭ V.
Published: June 2010

A method is given for producing unconditional bases in subspaces

K
θ

=

H
2

⊖
θ

H
2

{K_\theta } = {H^2} \ominus \theta {H^2}

of the Hardy space

H
2

,
θ

{H^2},\theta

being an inner function in the upper half-plane. For

θ
=
exp
⁡

(

i
a
z
)

\theta = \exp {\text {(}}iaz)

the space

K
θ

{K_\theta }

is the Fourier-Laplace transform of

L
2

(
0
,
a
)

{L^2}(0,a)

, which allows us to establish a necessary and sufficient condition for certain families of functions (including exponentials) to constitute unconditional bases in

L
2

(
0
,
a
)

{L^2}(0,a)

.
Littlewood-Paley operators on BMO
Authors: Kurtz Douglas S.
Published: June 2010

Two Littlewood-Paley operators, the area integral and the function

g
λ
∗

g_\lambda ^ *

, are considered as operators on functions of bounded mean oscillation. It is proved that the image of a BMO function under one of these operators is either equal to infinity almost everywhere or is in BMO.
On a converse result for Perron’s theorem for asymptotic stability for nonlinear differential equations
Authors: Seifert George
Published: June 2010

Two fairly simple proofs of a converse of Perron’s classical theorem for the exponential asymptotic stability of the trivial solution of a nonlinear system of ordinary differential equations are given.
An elementary proof of Słodkowski’s theorem
Authors: Casadio Tarabusi Enrico
Published: June 2010

Let

A

\mathfrak {A}

be a complex Banach algebra, and

σ
(
x
)

\sigma (x)

be the spectrum of

x
∈

A

x \in \mathfrak {A}

. We give a very short proof that if

f
:
G
→

A

f:G \to \mathfrak {A}

is holomorphic (

G
G

open in

C

{\mathbf {C}}

), then

σ
∘
f
:
G
→

2

C

\sigma \circ f:G \to {2^{\mathbf {C}}}

is Oka-analytic.
Completion of factor algebras of ideals
Authors: Balcar B. , Franek F.
Published: June 2010

Let

I

\mathfrak {I}

be a

κ
\kappa

-complete ideal over

κ
\kappa

. The structure of the completion of the Boolean algebra

℘
(
κ
)

/

I

\wp (\kappa )/\mathfrak {I}

is investigated with respect to properties of the ideal

I

\mathfrak {I}

and the cardinal

κ
\kappa

. It is shown that under certain conditions

Comp
⁡
(
℘
(
κ
)

/

I

)

\operatorname {Comp}(\wp (\kappa )/\mathfrak {I})

is isomorphic to a collapse algebra.
Some arithmetic properties of the minimal polynomials of Gauss sums
Authors: Wan Da Qing
Published: June 2010

For the minimal polynomial

f
(
x
)
=

x
k

+

b
1

x

k
−
1

+
⋯
+

b
k

f(x) = {x^k} + {b_1}{x^{k - 1}} + \cdots + {b_k}

of

∑

n
=
0

p
−
1

exp
⁡
(
2
π
i

n
k

/

p
)

\sum \nolimits _{n = 0}^{p - 1} {\exp (2\pi i{n^k}/p)}

over

Q
Q

, where

p
p

is a

prime
≡
1
(
mod
k
)

\operatorname {prime} \equiv 1(\bmod k)

, we evaluate

b
1

,

b
2

{b_1},{b_2}

and prove

p
|

b
i

(
i
=
1
,
…
,
k
)

\left . p \right |{b_i}(i = 1, \ldots ,k)

but

p
2

∤

b
j

(
j
=
2
,
k
)

{p^2}\nmid {b_j}(j = 2,k)

. Also, we raise the interesting conjecture that

p
2

∤

b
j

{p^2}\nmid {b_j}

for

k
≥
j
≥
2

k \geq j \geq 2

.
A non-Archimedean Stone-Banach theorem
Authors: Beckenstein Edward , Narici Lawrence
Published: June 2010

If the spaces

C
(
T
,
R
)

C(T,R)

and

C
(
S
,
R
)

C(S,R)

of continuous functions on

S
S

and

T
T

are linearly isometric, then

T
T

and

S
S

are homeomorphic. By the classical Stone-Banach theorem the only linear isometries of

C
(
T
,
R
)

C(T,R)

onto

C
(
S
,
R
)

C(S,R)

are of the form

x
→
a
(
x
∘
h
)

x \to a(x \circ h)

, where

h
h

is a homeomorphism of

S
S

onto

T
T

and

a
∈
C
(
S
,
F
)

a \in C(S,F)

is of magnitude 1 for all

s
s

in

S
S

. What happens if

R
R

is replaced by a field with a valuation? In brief, the result fails. We discuss "how" by way of developing a necessary and sufficient condition for the theorem to hold, along with some examples to illustrate the point.
A simple, direct proof of uniqueness for solutions of the Hamilton-Jacobi equations of eikonal type
Authors: Ishii Hitoshi
Published: June 2010

We present a new, direct proof of the uniqueness theorem for a class of Hamilton-Jacobi equations including the eikonal equation in geometric optics.
On 𝑛-dimensional Lorentz manifolds admitting an isometry group of dimension 𝑛(𝑛-1)/2+1
Authors: Matsuda Hiroo
Published: June 2010

We classify connected

n
n

-dimensional Lorentz manifolds admitting an isometry group of dimension

n
(
n
−
1
)

/

2
+
1

n(n - 1)/2 + 1

with compact isotropy subgroup

(
n
≥
5
)

(n \geq 5)

.
Infinite free set for small measure set mappings
Authors: Newelski Ludomir , Pawlikowski Janusz , Seredyński Witold
Published: June 2010

A set

A
⊂
X

A \subset X

is free for a function

F
:
X
→

P

(
X
)

F:X \to \mathcal {P}(X)

provided

x
∉
F
(
y
)

x \notin F(y)

for any distinct

x
,
y
∈
A

x,y \in A

. We show that, if

F
F

maps the reals into closed subsets of measure less than 1, then there is an infinite free set for

F
F

. This solves Problem 38(B) of Erdös and Hajnal [EH].
On the density of the distribution of 𝑝-stable seminorms, 0<𝑝<1
Authors: Lewandowski Maciej , Żak Tomasz
Published: June 2010

Let

μ
\mu

be a symmetric

p
p

-stable measure,

0
>
p
>
1

0 > p > 1

, on a locally convex separable linear metric space

E
E

and let

q
q

be a lower semicontinuous seminorm on

E
E

. It is known that

F
(
t
)
=
μ
{
x
:
q
(
x
)
>
t
}

F(t) = \mu \{ x:q(x) > t\}

is absolutely continuous with respect to the Lebesgue measure. We prove an explicit formula for the density

F
′

(
t
)

F’(t)

and give an asymptotic estimate of it at infinity.
Singular compactifications: the order structure
Authors: Chandler Richard E. , Faulkner Gary D.
Published: June 2010

This paper investigates the order structure of the collection of singular compactifications of a locally compact Hausdorff space. In particular we will prove several theorems showing that

β
X

\beta X

is the supremum of these simpler compactifications.
A note on order complete 𝑓-algebras
Authors: Lavrič Boris
Published: June 2010

Let

A
A

be an Archimedean uniformly complete

f
f

-algebra with unit element. Then the following conditions are equivalent (i)

A
A

is order complete. (ii) Every regular algebra ideal in

A
A

is an order ideal. (iii) Every finitely generated regular algebra ideal in

A
A

is a principal algebra ideal. The proof is based on the fact that the range of every injective orthomorphism in an order complete Riesz space is an order ideal.
The existence of minimal regular local overrings for an arbitrary domain
Authors: Johnston Bernard
Published: June 2010

It is shown that the set of regular local rings of dimension

n
n

containing an integral domain

D
D

, having the same quotient field as

D
D

, and ordered by containment satisfies the descending chain condition. The set of regular local rings of dimension

n
n

contained in a regular local ring of dimension

m
,
n
>
m

m,n > m

, need not satisfy the ascending chain condition, as is shown by example.
Group algebras and algebras of Golod-Shafarevich
Authors: Siderov Plamen N.
Published: June 2010

In [2], Golod, using results of Golod and Shafarevich [1], has constructed a finitely generated algebra

A
=
K

⟨

y
1

,
…
,

y
d

⟩

A = K\left \langle {{y_1}, \ldots ,{y_d}} \right \rangle

, over any field

K
K

, such that the ideal generated by

y
1

,
…
,

y
d

{y_1}, \ldots ,{y_d}

is nil, but

dim
K

A
=
∞

{\dim _K}A = \infty

. Moreover, when char

K
=
p
>
0

K = p > 0

, the subgroup

G
G

of the group of units of

A
A

, generated by

1
+

y
1

,
…
,
1
+

y
d

1 + {y_1}, \ldots ,1 + {y_d}

, is an infinite

p
p

-group. The main purpose of the present paper is to show that

K
[
G
]

K[G]

, the group algebra of

G
G

over

K
K

, is not isomorphic to

A
A

for "most" Golod-Shafarevich groups

G
G

.
Stability of solutions of linear delay differential equations
Authors: Kulenović M. R. S. , Ladas G. , Meimaridou A.
Published: June 2010

Consider the linear differential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 right-parenthesis ModifyingAbove x With dot left-parenthesis t right-parenthesis equals sigma-summation Underscript i equals 1 Overscript n Endscripts p Subscript i Baseline left-parenthesis t right-parenthesis x left-parenthesis t minus tau Subscript i Baseline right-parenthesis equals 0 comma t greater-than-or-slanted-equals t 0 comma">

(
1
)

x
˙

(
t
)
=

∑

i
=
1

n

p
i

(
t
)
x
(
t
−

τ
i

)

=
0
,

t
⩾

t
0

,

(1)\quad \dot x(t) = \sum \limits _{i = 1}^n {{p_i}(t)x(t - {\tau _i})} = 0,\quad t \geqslant {t_0},

\]
where

p
i

∈
C
(
[

t
0

,
∞
)
,

R

)

{p_i} \in C([{t_0},\infty ),{\mathbf {R}})

and

τ
i

⩾
0

{\tau _i} \geqslant 0

for

i
=
1
,
2
,
…
,
n

i = 1,2, \ldots ,n

. By investigating the asymptotic behavior first of the nonoscillatory solutions of (1) and then of the oscillatory solutions we are led to new sufficient conditions for the asymptotic stability of the trivial solution of (1). When the coefficients of (1) are all of the same sign, we obtain a comparison result which shows that the nonoscillatory solutions of (1) dominate the growth of the oscillatory solutions.
Cyclic operators, commutators, and absolutely continuous measures
Authors: Dombrowski J.
Published: June 2010

Commutator equations are used to study the relationship between the tridiagonal matrix structure of an unbounded cyclic selfadjoint operator and its spectrum. Sufficient conditions are given for absolute continuity. Results are related to the study of systems of orthogonal polyomials for which the measure of orthogonality is supported on an unbounded subset of the real line.
An undecidability result for power series rings of positive characteristic. II
Authors: Pheidas Thanases
Published: June 2010

We prove that the existential problem for a power series ring over an integral domain of positive characteristic with a predicate which represents the powers of the indeterminate is undecidable. We prove the same result for any ring which is contained in a power series ring and contains the corresponding ring of polynomials.
On the strong law of large numbers in Banach spaces
Authors: Godbole Anant P.
Published: June 2010

We study the relationship between the geometry of a real separable Banach space

B
B

(as manifested in its cotype, type, or logtype) and necessary or sufficient criteria for the validity of the Strong Law of Large Numbers (SLLN) for independent

B
B

-valued random variables, formulated in terms of the validity of a (verifiable) SLLN for real-valued random variables. Our results are the best possible of their kind and may be used in situations where the SLLN’s of Hoffman-Jørgensen and Pisier, and Kuelbs and Zinn are inconclusive.
Convergence and approximation results for measurable multifunctions
Authors: Lucchetti R. , Papageorgiou N. S. , Patrone F.
Published: June 2010

In this note we consider measurable multifunctions taking values in a separable Banach space. We show that if

F
(
ω
)
⊆

s

−
lim

F
n

(
ω
)

F(\omega ) \subseteq {\text {s}} - \lim {F_n}(\omega )

, then any Castaing representation of the

F
(
⋅
)

F( \cdot )

can be obtained as the strong limit of Castaing representations of the

F
n

{F_n}

. We also prove that any weakly measurable multifunction is the Kuratowski-Mosco limit of a sequence of countably simple multifunctions. Then we show that in reflexive Banach spaces this approximation property is equivalent to weak measurability. Finally we discuss the problem of measurability of the inferior and

w

{\text {w}}

-superior limits of a sequence of measurable multifunctions.
The 𝑙₁-completion of a metric combinatorial ∞-manifold
Authors: Sakai Katsuro
Published: June 2010

Let

K
K

be a simplicial complex. The realization

|
K
|

\left | K \right |

of

K
K

admits the metric
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d 1 left-parenthesis x comma y right-parenthesis equals sigma-summation Underscript upsilon element-of upper K Superscript 0 Baseline Endscripts StartAbsoluteValue x left-parenthesis upsilon right-parenthesis minus y left-parenthesis upsilon right-parenthesis EndAbsoluteValue comma">

d
1

(
x
,
y
)
=

∑

υ
∈

K
0

|

x
(
υ
)
−
y
(
υ
)

|

,

{d_1}(x,y) = \sum \limits _{\upsilon \in {K^0}} {\left | {x(\upsilon ) - y(\upsilon )} \right |,}

\]
where

x
(
υ
)

x(\upsilon )

and

y
(
υ
)
,
υ
∈

K
0

y(\upsilon ),\upsilon \in {K^0}

, are the barycentric coordinates of

x
x

and

y
y

respectively. The completion of the metric space

(

|
K
|

,

d
1

)

(\left | K \right |,{d_1})

is called the

l
1

{l_1}

-completion and is denoted by

|

K

|

¯

l
1

{\overline {|K|} ^{{l_1}}}

. In this paper, we prove that

|

K

|

¯

l
1

{\overline {|K|} ^{{l_1}}}

is an

l
2

{l_2}

-manifold if and only if

K
K

is a combinatorial

∞
\infty

-manifold.
Positive scalar curvature and 𝐾𝑂-characteristic numbers
Authors: Moriyoshi Hitoshi
Published: June 2010

Let

M
M

be an

(
8
k
+
2
)

(8k + 2)

-dimensional closed spin manifold and

N
N

an orientable hypersurface of

M
M

with the induced spin structure. If

M
M

admits a metric with positive scalar curvature and

N
N

represents a nonzero homology class of

H

8
k
+
1

(
M
;

Z

)

{H_{8k + 1}}(M;{\mathbf {Z}})

, then the KO-characteristic number

α
(
N
)

\alpha (N)

vanishes. This result relates to the conjecture by Gromov and Lawson on the vanishing of higher

A
^

\hat A

-genera.
Closed orbits of an Anosov flow and the fundamental group
Authors: Adachi Toshiaki
Published: June 2010

We show that closed orbits of a transitive Anosov flow generate the fundamental group of the base manifold.
Zeros of successive derivates of a class of real entire functions of exponential type
Authors: Shen Li-Chien
Published: June 2010

Using the method of steepest descent we prove that for a class of real entire functions of exponential type

τ
\tau

the spacings of the adjacent zeros of

f

(
n
)

{f^{(n)}}

converge to

π
τ

/

2

\pi \tau /2

.
A dense set of operators quasisimilar to normal + compact
Authors: Herrero Domingo A.
Published: June 2010

The algebra of all bounded linear operators acting on a complex separable infinite dimensional Hilbert space is the disjoint union of two dense subsets: Every operator in one of them is quasisimilar to an operator of the form "normal

+
+

compact," and every operator in the complement is not quasisimilar to an operator of that form.
Convexity and Banach envelope of the weak-𝐿_{𝑝} spaces
Authors: Ariño Miguel A.
Published: June 2010

The Banach envelope and a representation of the topological dual of the weak-

l
p

{l_p}

sequence spaces which involves the Lorentz sequence spaces are computed. The local convexity of weak-

L
p

{L_p}

spaces is studied also.
Symmetric Riemann surfaces, torsion subgroups and Schottky coverings
Authors: Heltai Blaise
Published: June 2010

We consider a torsion-free Fuchsian group

G
G

acting on

H
H

which admits an orientation reversing involution

j
j

. That is,

j
G
j
=
G

jGj = G

. Let

T
T

be the orientation preserving half of the torsion subgroup of the extended group

⟨

G
,
j

⟩

\left \langle {G,j} \right \rangle

. By considering invariant homology basis elements of the surface

H

/

G

H/G

, we show that the surface

H

/

T

H/T

is planar, and that the group

G

/

T

G/T

acts on

H

/

T

H/T

as a Schottky group.
On quasi-affine transforms of unilateral shifts
Authors: Takahashi Katsutoshi
Published: June 2010

It is shown that if a contraction

T
T

is a quasiaffine transform of a unilateral shift of finite multiplicity, then both the approximate point spectrum and the essential spectrum of

T
T

coincide with the unit circle.
Ground states for the prescribed mean curvature equation
Authors: Peletier L. A. , Serrin J.
Published: June 2010

We discuss the existence of radial ground states of problem

(

I

)

({\text {I}})

, under appropriate restrictions on the bifurcation parameter

λ
\lambda

and the exponent

q
q

.
A conservation result concerning bounded theories and the collection axiom
Authors: Buss Samuel R.
Published: June 2010

We present two proofs, one proof-theoretic and one model-theoretic, showing that adding the

B

Σ
1
0

B\Sigma _1^0

-collection axioms to any bounded first-order theory

R
R

of arithmetic yields an extension which is

∀

Σ
1
0

\forall \Sigma _1^0

-conservative over

R
R

.
A vanishing theorem for the conformal anomaly in dimension >2
Authors: Morava Jack , Tamanoi Hirotaka
Published: June 2010

We examine the conformal anomaly of quantum field theory in the light of Atiyah and Singer’s recent work on the chiral anomaly. We show that the bundle over the space of conformal structures defined on a given manifold, constructed as the index of the conformally invariant family of Laplace operators, is a torsion class in real

K
K

-theory, Indeed, its Chern forms are zero. These Chern forms transgress to define classes in certain local cohomology groups calculated recently by Bonora, Cotta-Ramusino, and Reina; a corollary is that, although these obstruction groups are nonzero, the obstruction to the existence of a gauge-covariant propagator for the Laplace operator is the trivial element of obstruction group.
A 𝑄-manifold local-compactification of a metric combinatorial ∞-manifold
Authors: Sakai Katsuro
Published: June 2010

Let

K
K

be a combinatorial

∞
\infty

-manifold, that is, a countable simplicial complex such that the star of each vertex is combinatorially equivalent to the countable-infinite full simplicial complex. Then the space

|
K
|

m

{\left | K \right |_m}

with the metric topology is a manifold modeled on the space

σ
\sigma

, where

σ
\sigma

is the subspace of the Hilbert cube

Q
=

I
ω

Q = {I^\omega }

which consists of all points having at most finitely many nonzero coordinates. In this paper, we give a local-compactification of

|
K
|

m

{\left | K \right |_m}

which is a

[
0
,
1
)

[0,1)

-stable

Q
Q

-manifold containing

|
K
|

m

{\left | K \right |_m}

as an f.d. cap set.
Denting points in tensor products of Banach spaces
Authors: Werner Dirk
Published: June 2010

Let dent

A
A

denote the set of denting points of a subset

A
A

of some Banach space. We prove
\[

dent cl co(

K
⊗
L
)
=

dent 

K
⊗

dent 

L

{\text {dent cl co(}}K \otimes L) = {\text {dent }}K \otimes {\text {dent }}L

\]
for closed, bounded, absolutely convex subsets

K
K

and

L
L

of Banach spaces

X
X

and

Y
Y

. Here the closure refers to the completion of

X
⊗
Y

X \otimes Y

w.r.t. some reasonable crossnorm.
Some remarks on category in topological spaces
Authors: Piotrowski Zbigniew , Szymański Andrzej
Published: June 2010

We prove some results concerning the covering, additivity, and the uniform numbers for general topological spaces.
Domain invariance in infinite-dimensional linear spaces
Authors: van Mill Jan
Published: June 2010

Let

X
X

be an infinite-dimensional locally convex linear space. It is known that

X
X

is homeomorphic to a subspace of

X
X

which is not open. We prove that every Banach space

B
B

contains a dense linear subspace

L
L

with the following property: If

U
⊆
L

U \subseteq L

is open and if

f
:
U
→
L

f:U \to L

is continuous and injective, then there exists a dense open

V
⊆
U

V \subseteq U

such that (a)

f
(
V
)

\begin {gathered}f(V) \end {gathered}

is open in

L

\begin {gathered} L \end {gathered}

, and (b)

f
:
V
→
f
(
V
)

f:V \to f(V)

is a homeomorphism. As a consequence,

L
L

and

L
×

R

L \times {\mathbf {R}}

are not homeomorphic.
Knots with finite weight commutator subgroups
Authors: Livingston Charles
Published: June 2010

An example of a knot in

S
3

{S^3}

is constructed which has a companion of winding number zero but for which the commutator subgroup of the fundamental group of the complement is of finite weight. This provides a counterexample to a conjecture made by Jonathan Simon.
Nonisomorphic algebraic structures on smooth manifolds
Authors: Bochnak J. , Kucharz W.
Published: June 2010

Given a compact smooth hypersurface

M
M

in

R

n
+
1

{{\mathbf {R}}^{n + 1}}

, we construct a family

{

X
k

}

,
k
=
1
,
2
,
…

\left \{ {{X_k}} \right \},k = 1,2, \ldots

, of nonsingular real algebraic subsets of

R

n
+
1

{{\mathbf {R}}^{n + 1}}

such that each

X
k

{X_k}

is isotopic to

M
M

but, for

k
≠
l
,

X
k

k \ne l,{X_k}

and

X
l

{X_l}

are not birationally equivalent.
A simple proof of Jacobi’s four-square theorem
Authors: Hirschhorn M. D.
Published: June 2010

Jacobi’s four-square theorem, which gives the number of representations of a positive integer as a sum of four squares, is shown to follow simply from the triple-product identity.
A necessary and sufficient condition for 𝑤*-bounded sets to be strongly bounded
Authors: Bosch Carlos , Kucera Jan
Published: June 2010

It is known that if a locally convex space

E
E

is quasi-complete then every

σ
(

E
′

,
E
)

\sigma (E’,E)

-bounded set in

E
′

E’

is

β
(

E
′

,
E
)

\beta (E’,E)

-bounded. This result cannot be reversed. Here we show that every

σ
(

E
′

,
E
)

\sigma (E’,E)

-bounded set is

β
(

E
′

,
E
)

\beta (E’,E)

-bounded iff

E
E

is fast complete.
Small perturbations and the eigenvalues of the Laplacian on large bounded domains
Authors: Kirsch Werner
Published: June 2010

Denote by

Δ
L
D

\Delta _L^D

the Laplacian on a hypercube in

R

d

{{\mathbf {R}}^d}

with side length

π
L

\pi L

. Also denote by

N

(

λ
,
A

)

N\left ( {\lambda ,A} \right )

the number of eigenvalues of the operator

A
A

below

λ
\lambda

. If

V
≥
0

V \geq 0

is a bounded function of compact support, (

V
>
0

V > 0

on a set of positive measure) then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N left-parenthesis minus normal upper Delta Subscript upper L Superscript upper D Baseline comma lamda right-parenthesis minus upper N left-parenthesis minus normal upper Delta Subscript upper L Superscript upper D Baseline plus upper V comma lamda right-parenthesis">

N

(

−

Δ
L
D

,
λ

)

−
N

(

−

Δ
L
D

+
V
,
λ

)

N\left ( { - \Delta _L^D,\lambda } \right ) - N\left ( { - \Delta _L^D + V,\lambda } \right )

is not bounded as

L
→
∞

L \to \infty

for dimension

d
>
1

d > 1

.
On the existence and convergence of pseudomoments for variables in the domain of normal attraction of an operator stable distribution
Authors: Weiner Daniel Charles
Published: June 2010

Integrals are constructed to replace absolute moments for variables in the domain of normal attraction of an operator stable law. These integrals, called pseudomoments, improve on the geometric information contained in absolute moments. Existence and convergence to appropriate values of these integrals are shown for the variables and their affine normalized sums.
Note on a paper of J. L. Palacios: “A correction note on: ‘Generalized Hewitt-Savage theorems for strictly stationary processes’ [Proc. Amer. Math. Soc. 63 (1977), no. 2, 313–316; MR0501304 (58 #18695)] by Isaac” [ibid. 88 (1983), no. 1, 138–140; MR0691294 (86a:60054)]
Authors: Isaac Richard
Published: June 2010

J. L. Palacios claimed [2] that the author’s paper [1] contained errors. This note refutes those claims by showing that Palacios misunderstood [1] and adopted assumptions different from those of [1].
Another view of metrizability
Authors: Hung H. H.
Published: June 2010

A fact long considered unsatisfactory about the classical metrization theorem of Alexandroff-Urysohn is that it expresses metrizability as a countable uniformity, uniformity itself being almost the former. In view of their unification, the classical theorems, with the exception of Arhangel’skiĭ’s regular open base theorem, are all really subject to the same criticism, to which our theorem here is an answer. We give a generalization here of Arhangel’skiĭ’s, of which Arhangel’skiĭ’s itself, the fundamental theorem of Alexandroff-Urysohn, A. H. Frink’s, and the Double Sequence Theorem of Nagata are all obvious special cases.
On 3-manifolds having surface bundles as branched coverings
Authors: Montesinos José María
Published: June 2010

We give a different proof of the result of Sakuma that every closed, oriented

3
3

-manifold

M
M

has a

2
2

-fold branched covering space

N
N

which is a surface bundle over

S
1

{S^1}

. We also give a new proof of the result of Brooks that

N
N

can be made hyperbolic. We give examples of irreducible

3
3

-manifolds which can be represented as

2
m

2m

-fold cyclic branched coverings of

S
3

{S^3}

for a number of different

m
m

’s as big as we like.
Products of a compact space and a metric space
Authors: Yajima Yukinobu
Published: June 2010

Let

X
×
Y

X \times Y

be the product of a compact space

X
X

and a metric space

Y
Y

. We consider a continuous closed image

Z
Z

of

X
×
Y

X \times Y

. Moreover, we consider a closed subspace

R
R

in

X
×
Y

X \times Y

which is a neighborhood retract of it. It is proved in this paper that

Z
Z

(respectively,

R
R

) is a Lašnev (metrizable) space iff all compact subspaces of

Z

(
R
)

Z\left ( R \right )

are metrizable.
Characters of 𝑝-groups
Authors: Ford Charles E.
Published: June 2010

The following theorem is proved for any prime

p
p

: Every irreducible rational represention of a finite

p
p

-group is the difference of two transitive permutation representations. Also given are two useful results about representations of

p
p

-groups, which are known to experts in the field.
Groups of units of modular group algebras
Authors: Bagiński Czesław
Published: June 2010

Let

U
(
G
)

U(G)

be the group of normalized units of the group algebra

F
[
G
]

F[G]

of a finite nonabelian

p
p

-group

G
G

over the field

F
=

G
F
(

p

)

F = {\operatorname {GF(}}p{\text {)}}

. The description is given of all

p
p

-groups,

p
>
2

p > 2

, for which

U
(
G
)

U(G)

does not contain subgroups isomorphic to the wreath product

Z
p

≀

Z
p

{Z_p}\wr {Z_p}

.
𝑘-to-1 functions on an arc
Authors: Katsuura Hidefumi , Kellum Kenneth R.
Published: June 2010

Recently Jo W. Heath [6] has shown that any

2
2

-to-

1
1

function from an arc onto a Hausdorff space must have infinitely many discontinuities. Here we investigate extending Heath’s result to

k
k

-to-

1
1

functions for

k
>
2

k > 2

. Examples show that in general Heath’s theorem cannot be extended even for functions from an arc into itself. However, if

f
f

is a

k
k

-to-

1
1

function

(
k
≥
2
)

(k \geq 2)

from an arc onto an arc, then we prove that

f
f

has infinitely many discontinuities.
Closed subspaces of finite codimension in some function algebras
Authors: Garimella Ramesh V. , Rao N. V.
Published: June 2010

We characterize all closed subspaces of finite codimension in some specific types of function algebras e.g. these include

C
(
X
)

C(X)

: algebra of continous functions on a compact Hausdorff space,

C
n

[
a
,
b
]

{C^n}[a,b]

: the algebra of

n
n

-times continuously differentiable functions on the closed interval

[
a
,
b
]

[a,b]

. Our work is a generalization of the well-known Gleason-Kahane-Želazko theorem [3, 6] for subspaces of codimension one in arbitrary unitary Banach algebras.
More quasireflexive subspaces
Authors: Bellenot Steven F.
Published: June 2010

It is shown that nonreflexive Banach spaces with a separable dual and the boundedly complete skipped blocking property have quasi-reflexive subspaces. In particular, Bourgain’s somewhat reflexive

L

∞

{\mathfrak {L}_\infty }

-spaces and Polish Banach spaces are somewhat quasi-reflexive.
Combinatorial properties for Blackwell sets
Authors: Shortt R. M.
Published: June 2010

Under the assumption of CH (continuum hypothesis) we produce a strongly Blackwell set whose product with a standard space and whose intersection with an analytic set axe not Blackwell. Previously, such examples were known to exist only under Martin’s Axiom (MA) and not-CH.
Finite covers of 3-manifolds containing essential surfaces of Euler characteristic =0
Authors: Kojima Sadayoshi
Published: June 2010

We give a short proof and a slight generalization of a theorem of John Luecke, that a compact connected orientable irreducible

3
3

-manifold containing an essential torus is finitely covered by a torus bundle or manifolds with unbounded first Betti numbers.
Necessary and sufficient conditions for the class number of a real quadratic field to be one, and a conjecture of S. Chowla
Authors: Mollin R. A.
Published: June 2010

Based on the fundamental unit of

Q

(

n

)

Q\left ( {\sqrt n } \right )

, an arbitrary real quadratic field, we provide a necessary condition for the class number

h

(
n
)

h\left ( n \right )

to be 1. For

n
=
4

m
2

+
1

n = 4{m^2} + 1

we prove the equivalence of three necessary and sufficient conditions for

h

(
n
)

h\left ( n \right )

to be 1. One of these conditions is that

−

x
2

+
x
+

m
2

- {x^2} + x + {m^2}

is prime for all integers

x
x

such that

1
>
x
>
m

1 > x > m

. This is the exact analogue of the complex quadratic field case. We discuss the connection with a conjecture of S. Chowla as well as with other related topics.
One parameter groups and the characterization of the sine function
Authors: Wallen Lawrence J.
Published: June 2010

A real

C
∞

{C^\infty }

function on

R

1

{{\mathbf {R}}^1}

, all of whose derivatives and (suitable) antiderivatives are uniformly bounded, is of the form

a
sin
⁡
(
x
−

x
0

)

a\sin (x - {x_0})

. We give an abstract version of this theorem.
On the ARG MIN multifunction for lower semicontinuous functions
Authors: Beer Gerald , Kenderov Petar
Published: June 2010

The epi-topology on the lower semicontinuous functions

L

(
X
)

L\left ( X \right )

on a Hausdorff space

X
X

is the restriction of the Fell topology on the closed subsets of

X
×
R

X \times R

to

L

(
X
)

L\left ( X \right )

, identifying lower semicontinuous functions with their epigraphs. For each

f
∈
L

(
X
)

f \in L\left ( X \right )

, let arg min

f
f

be the set of minimizers of

f
f

. With respect to the epi-topology, the graph of arg min is a closed subset of

L

(
X
)

×
X

L\left ( X \right ) \times X

if and only if

X
X

is locally compact. Moreover, if

X
X

is locally compact, then the epi-topology is the weakest topology on

L

(
X
)

L\left ( X \right )

for which the arg min multifunction has closed graph, and the operators

f
→
f
∨
g

f \to f \vee g

and

f
→
f
∧
g

f \to f \wedge g

are continuous for each continuous real function

g
g

on

X
X

.
The Gel′fand theorem and its converse for Kähler manifolds
Authors: Kowalski O. , Vanhecke L.
Published: June 2010

We characterize the locally Hermitian symmetric manifolds among the homogeneous Kähler manifolds

M
M

by each of the following properties: (i) all

A
0

(
M
)

{A_0}(M)

-invariant differential operators on

M
M

commute

(

A
0

(
M
)

({A_0}(M)

denotes the identity component of the group of all holomorphic isometries); (ii) all geodesies are orbits of one-parameter groups of holomorphic isometries.
On immersed compact submanifolds of Euclidean space
Authors: Rigoli Marco
Published: June 2010

Given an immersion

f
:
M
→

R

n

f:M \to {{\mathbf {R}}^n}

of a compact Riemannian manifold

M
M

we prove a simple criterion involving the tension field of

f
f

to determine whether or not

f
f

is an isometry.
On classification of quadratic harmonic maps of 𝑆³
Authors: Tóth Gábor
Published: June 2010

By the generalized Do Carmo-Wallach classification theorem polynomial harmonic maps between spheres can be parametrized by a finite-dimensional compact convex body. Here we describe the boundary of the parameter space in the first nonrigid range by exhibiting a large number of quadratic harmonic maps of

S
3

{S^3}

into spheres.
Bundles along the fiber in the PL category
Authors: Levitt Norman
Published: June 2010

We show that the notion of the bundle of tangents along the fiber over the total space of a fibering is not a good one when the fibering in question is PL. In particular it is shown that there are vector bundles over the same base space such that (1) the associated sphere bundles are PL-equivalent; (2) the consequent homeomorphism of total spaces is not covered by any PL-disk bundle equivalence of the respective bundles of tangents to the fiber manifolds.
Extremal problems for Lorentz classes of nonnegative polynomials in 𝐿² metric with Jacobi weight
Authors: Milovanović Gradimir V. , Petković Miodrag S.
Published: June 2010

Let

L
n

{L_n}

be the Lorentz class of nonnegative polynomials on

[
−
1
,
1
]

[-1,1]

. Extremal problems of Markov type, in

L
2

{L^2}

norm with Jacobi weight, on the set

L
n

{L_n}

or on its subset, are investigated.
Spectrum of the product of operators
Authors: Hladnik Milan , Omladič Matjaž
Published: June 2010

It is shown that the product of two operators on a Hilbert space has real spectrum if one of them is symmetric and the other is positive. Also, the product of two positive operators has positive spectrum.
On Hörmander’s ratio theorems
Authors: Jamjoom Fatma B. , Zaheer N.
Published: June 2010

In this paper we prove a theorem concerning the coincidence of ranges of the ratio of abstract homogeneous polynomials and the corresponding ratio of their polars. We employ the coincidence theorem on symmetric multilinear forms (proved recently by the authors [6 or 5]) to prove our theorem. The main result deduces two ratio theorems due to Hörmander [4], one being an improvement upon his version in the complex plane.
Analyticity of the interface of the porous media equation after the waiting time
Authors: Angenent Sigurd
Published: June 2010

We prove that the free boundary of a solution of the porous media equation, with compactly supported initial data, is real analytic after the waiting time. The proof uses the maximal regularity theory of Da Prato and Grisvard.
Regular variation in 𝑅^{𝑘}
Authors: Meerschaert Mark M.
Published: June 2010

Researchers investigating certain limit theorems in probability have discovered a multivariable analogue to Karamata’s theory of regularly varying functions. The method uses elements of real analysis and Lie groups to analyze the asymptotic behavior of functions and measures on

R

k

{{\mathbf {R}}^k}

. We present an account here which is independent of probabilistic considerations.
𝐿^{𝑝}-multipliers: a new proof of an old theorem
Authors: Schonbek Tomas P.
Published: June 2010

New proofs are given for the following results of Hirschman and Wainger: Let

ψ
∈

C
∞

(

R

n

)

\psi \in {C^\infty }({\mathbb {R}^n})

vanish in a neighborhood of the origin;

ψ
(
ξ
)
=
1

\psi (\xi ) = 1

for large

ξ
\xi

. Then
\[

|

ξ

|

−
β

ψ
(
ξ
)
exp
⁡
(
i

|

ξ

|

α

)

|\xi {|^{ - \beta }}\psi (\xi )\exp (i|\xi {|^\alpha })

\]
is a multiplier in

L
p

(

R

n

)

{L^p}({\mathbb {R}^n})

for

|

1

/

p
−
1

/

2

|

>
β

/

n
α

|1/p - 1/2| > \beta /n\alpha

; is not a multiplier in

L
p

(

R

n

)

{L^p}\left ( {{\mathbb {R}^n}} \right )

for

|

1

/

p
−
1

/

2

|

>
β

/

n
α

|1/p - 1/2| > \beta /n\alpha

.
On a conjecture of Graham
Authors: Sander J. W.
Published: June 2010

Let

a
1

>

a
2

>
⋯
>

a
n

{a_1} > {a_2} > \cdots > {a_n}

be a finite sequence of positive integers containing a prime power

p
d

{p^d}

with the property:

a
i

≠

p
k

a
j

{a_i} \ne {p^k}{a_j}

for all

i
,
j

i,j

and

k
>
0

k > 0

. Then

max

i
,
j

a
i

/

(

a
i

,

a
j

)

≥
n

{\max _{i,j}}{a_i}/\left ( {{a_i},{a_j}} \right ) \geq n

.
Le dual des fonctions holomorphes intégrables sur un domaine strictement pseudo-convexe
Authors: Coupet Bernard
Published: June 2010

We prove that on a strictly pseudoconvex domain with

C
4

{C^4}

boundary in

C
n

{C^n}

, the dual of the Bergman space

B
1

{B^1}

is Bloch space.
Extensions of the Berger-Shaw theorem
Authors: Hadwin Don , Nordgren Eric
Published: June 2010

We show how D. Voiculescu’s proof of the Berger-Shaw trace inequality for rationally cyclic nearly hyponormal operators can be presented using only elementary operator-theoretic concepts. In addition we show that if

T
T

is a hyponormal operator whose essential spectrum has zero area, then the question of whether

[

T
∗

,
T
]

[{T^ * },T]

is trace class depends only on the spectral picture of

T
T

. We also show how a special case of results of Helton-Howe can be derived from the BDF theory.
Summation methods and uniqueness in Vilenkin groups
Authors: Grubb D. J.
Published: June 2010

It is shown that the

σ
\sigma

-compact

U
U

-sets for two different summation methods on Vilenkin groups are the same.
The Hill-Penrose-Sparling CR manifolds
Authors: Eastwood Michael G.
Published: June 2010

This paper gives a simple formulation of the CR structures of Hill, Penrose, and Sparling. An elementary power series argument shows that they cannot be realized as hypersurfaces in an ambient complex manifold.
Simple proofs of Bernstein-type inequalities
Authors: Mohapatra R. N. , O’Hara P. J. , Rodriguez R. S.
Published: June 2010

A polynomial identity is established by the use of Lagrange interpolation. This identity is used to obtain simple proofs of Bernstein-type inequalities, one of which is an improvement of a recent result of Frappier, Rahman, and Ruscheweyh.
On an inequality of Osgood, Phillips and Sarnak
Authors: Widom Harold
Published: June 2010

In their work [4,5] on isospectral families for the Laplacian Osgood, Phillips and Sarnak needed and proved the following inequality:For any real-valued functionϕ\phibelonging to the Sobolev spaceH1/2(=W1/2,2){H_{1/2}}( = {W^{1/2,2}})of the unit circle and satisfying the side condition∫eϕeiθdθ=0\int {{e^\phi }{e^{i\theta }}d\theta = 0}one has\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="log StartFraction 1 Over 2 pi EndFraction integral e Superscript phi Baseline d theta minus StartFraction 1 Over 2 pi EndFraction integral phi d theta less-than-or-equal-to one half sigma-summation Underscript k equals 1 Overscript normal infinity Endscripts k vertical-bar ModifyingAbove phi With caret left-parenthesis k right-parenthesis vertical-bar Superscript 2 Baseline">log⁡12π∫eϕdθ−12π∫ϕdθ≤12∑k=1∞k|ϕ^(k)|2\log \frac {1} {{2\pi }}\int {{e^\phi }d\theta - \frac {1} {{2\pi }}\int \phi \,d\theta \leq \frac {1} {2}\sum \limits _{k = 1}^\infty {k|\hat \phi } (k){|^2}}\]whereϕ^(k)\hat \phi (k)is thekkth Fourier coefficient ofϕ\phi. Without the side condition the factor12\frac {1} {2}does not appear on the right side and the inequality is the first Lebedev-Milin inequality [1, §5.1]. The purpose of this note is to show that these are the first two of a series of inequalities which follow quickly from some theorems of G. Szegö on Toeplitz determinants.
The integrability of generalized Garrett-Stanojević sums
Authors: Singh Niranjan , Sharma K. M.
Published: June 2010

In this paper we have defined the generalized Garrett-Stanojević cosine sums
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript n Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript p equals 0 Overscript n Endscripts upper S Subscript p Superscript r minus 1 Baseline normal upper Delta Superscript r Baseline a Subscript p">

h
n

(
x
)

=

∑

p
=
0

n

S
p

r
−
1

Δ
r

a
p

{h_n}\left ( x \right ) = \sum \limits _{p = 0}^n {S_p^{r - 1}{\Delta ^r}{a_p}}

\]
and have proved that under suitable conditions

h
n

→
h

{h_n} \to h

in the

L
1

{L^1}

-norm, where

h

(
x
)

=

a
0

/

2
+

∑

n
=
1

∞

a
n

cos
⁡
n
x

h\left ( x \right ) = {a_0}/2 + \sum \nolimits _{n = 1}^\infty {{a_n}\cos nx}

. If

r
=
1

r = 1

, then

h
n

(
x
)

{h_n}\left ( x \right )

reduces to the modified cosine sums introduced by Rees and Stanojević.
A problem of Stallings on the direct square of a free group
Authors: Baumslag Gilbert
Published: June 2010

Let

F
F

be a free group of finite rank; let

Δ
(
F
)

\Delta (F)

denote the diagonal subgroup of

F
×
F

F \times F

, and let

A
A

and

B
B

be finitely presented subgroups of

F
×
F

F \times F

. It is shown that

A
∩
Δ
(
F
)

A \cap \Delta (F)

is finitely presented, that, if neither

A
A

nor

B
B

is free, then

A
∩
B

A \cap B

is finitely presented, and that there are examples where both

A
A

and

B
B

are free of finite rank such that

A
∩
B

A \cap B

is not finitely generated.