bluuebunny/crossref_metadata_2025 huggingface dataset:
Multiplicative perturbations of linear Volterra equations
Authors: Rhandi Abdelaziz
Published: July 2010

We prove that the following problems are essentially equivalent:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column left-bracket upper V upper O right-bracket Subscript upper C upper T 2nd Column a m p semicolon u left-parenthesis t right-parenthesis equals x plus integral Subscript 0 Superscript t Baseline a left-parenthesis t minus s right-parenthesis upper C upper T u left-parenthesis s right-parenthesis d s comma 2nd Row 1st Column left-bracket upper V upper O right-bracket Subscript upper T upper C 2nd Column a m p semicolon v left-parenthesis t right-parenthesis equals y plus integral Subscript 0 Superscript t Baseline a left-parenthesis t minus s right-parenthesis upper T upper C v left-parenthesis s right-parenthesis d s comma EndLayout">

[
VO
]

C
T

a
m
p
;

u
(
t
)
=
x
+

∫
0
t

a
(
t
−
s
)
C
T
u
(
s
)

d
s
,

[
VO
]

T
C

a
m
p
;

v
(
t
)
=
y
+

∫
0
t

a
(
t
−
s
)
T
C
v
(
s
)

d
s
,

\begin {array}{*{20}{c}} {{{[\operatorname {VO} ]}_{CT}}} & {\qquad \qquad u(t) = x + \int _0^t {a(t - s)CTu(s)\,ds,} } \\ {{{[\operatorname {VO} ]}_{TC}}} & {\qquad \qquad v(t) = y + \int _0^t {a(t - s)TCv(s)\,ds,} } \\ \end {array}

\]
where

T
T

is an unbounded closed linear operator in a Banach space

X
X

with dense domain

D
(
T
)
,

C

D(T),\;C

is a bounded linear operator on

X
X

, and

a
∈

L

loc

1

(
[
0
,
∞
)
,

R

)

a \in L_{\operatorname {loc} }^1([0,\infty ),\mathbb {R})

, which is exponentially bounded. We give some applications of our abstract theorem to second-order differential operators on the line.
Oscillation criteria for Hamiltonian matrix difference systems
Authors: Erbe L. H. , Yan Peng Xiang
Published: July 2010

We obtain some oscillation criteria for the Hamiltonian difference system
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row normal upper Delta upper Y left-parenthesis t right-parenthesis equals upper B left-parenthesis t right-parenthesis upper Y left-parenthesis t plus 1 right-parenthesis plus upper C left-parenthesis t right-parenthesis upper Z left-parenthesis t right-parenthesis comma 2nd Row normal upper Delta upper Z left-parenthesis t right-parenthesis equals minus upper A left-parenthesis t right-parenthesis upper Y left-parenthesis t plus 1 right-parenthesis minus upper B Superscript asterisk Baseline left-parenthesis t right-parenthesis upper Z left-parenthesis t right-parenthesis comma EndLayout">

{

Δ
Y
(
t
)
=
B
(
t
)
Y
(
t
+
1
)
+
C
(
t
)
Z
(
t
)
,

Δ
Z
(
t
)
=
−
A
(
t
)
Y
(
t
+
1
)
−

B

∗

(
t
)
Z
(
t
)
,

\left \{ \begin {gathered} \Delta Y(t) = B(t)Y(t + 1) + C(t)Z(t), \hfill \\ \Delta Z(t) = - A(t)Y(t + 1) - {B^{\ast }}(t)Z(t), \hfill \\ \end {gathered} \right .

\]
where

A
,
B
,
C
,
Y
,
Z

A,B,C,Y,Z

are

d
×
d

d \times d

matrix functions. As a corollary, we establish the validity of an earlier conjecture for a second-order matrix difference system.
Boundary behavior of holomorphic functions of 𝐴^{𝑝}_{𝑞,𝑠}
Authors: Hu Zhang Jian
Published: July 2010

In this paper we prove that the Sobolov spaces

A

q
,
s

p

(
D
)

A_{q,s}^p(D)

on bounded strongly pseudoconvex domains

D
D

are continuously contained in

BMOA
⁡
(
∂
D
)

\operatorname {BMOA} (\partial D)

for

0
>
p
>
∞
,
q
⩾
0

0 > p > \infty ,q \geqslant 0

, and

s
=
(
n
+
q
)

/

p

s = (n + q)/p

.
Every needle point space contains a compact convex AR-set with no extreme points
Authors: Nhu Nguyen To , Tri Le Hoang
Published: July 2010

Every needle point contains a compact convex AR-set without any extreme points. In particular the following spaces contain such compact convex sets: (i) the spaces

L
p

,

0
⩽
p
>
1

{L_p},\;0 \leqslant p > 1

; (ii) the linear metric space constructed by Roberts (Studia Math. 60 (1977), 255-266).
A generalization of the isoperimetric inequality on 𝑆² and flat tori in 𝑆³
Authors: Enomoto Kazuyuki
Published: July 2010

A geometric inequality is proved for closed curves on

S
2

{S^2}

which are regularly homotopic to simple closed curves. This generalizes the classical isoperimetric inequality for simple closed curves on

S
2

{S^2}

. The proof is based on the study of flat tori in

S
3

{S^3}

and their images under the Gauss map.
Isoperimetric inequalities for immersed closed spherical curves
Authors: Weiner Joel L.
Published: July 2010

Let

α
:

S
1

→

S
2

\alpha :{S^1} \to {S^2}

be a

C
2

{C^2}

immersion with length

L
L

and total curvature

K
K

. If

α
\alpha

is regularly homotopic to a circle traversed once then

L
2

+

K
2

⩾
4

π
2

{L^2} + {K^2} \geqslant 4{\pi ^2}

with equality if and only if

α
\alpha

is a circle traversed once. If

α
\alpha

has nonnegative geodesic curvature and multiple points then

L
+
K
⩾
4
π

L + K \geqslant 4\pi

with equality if and only if

α
\alpha

is a great circle traversed twice.
Strong moment problems for rapidly decreasing smooth functions
Authors: Durán Ana Lía , Estrada Ricardo
Published: July 2010

It is shown that the existence of rapidly decreasing smooth solutions of various moment problems follows from the theorem of Ritt on the existence of analytic functions with a prescribed asymptotic power series at the vertex of a given sector.
Approximation of fixed points of strongly pseudocontractive mappings
Authors: Chidume C. E.
Published: July 2010

Let

E
E

be a real Banach space with a uniformly convex dual, and let

K
K

be a nonempty closed convex and bounded subset of

E
E

. Let

T
:
K
→
K

T:K \to K

be a continuous strongly pseudocontractive mapping of

K
K

into itself. Let

{

c
n

}

n
=
1

∞

\{ {c_n}\} _{n = 1}^\infty

be a real sequence satisfying: (i)

0
>

c
n

>
1

0 > {c_n} > 1

for all

n
⩾
1

n \geqslant 1

; (ii)

∑

n
=
1

∞

c
n

=
∞

\sum \nolimits _{n = 1}^\infty {{c_n} = \infty }

; and (iii)

∑

n
=
1

∞

c
n

b
(

c
n

)
>
∞

\sum \nolimits _{n = 1}^\infty {{c_n}b({c_n}) > \infty }

, where

b
:
[
0
,
∞
)
→
[
0
,
∞
)

b:[0,\infty ) \to [0,\infty )

is some continuous nondecreasing function satisfying

b
(
0
)
=
0
,

b
(
c
t
)
⩽
c
b
(
t
)

b(0) = 0,\,b(ct) \leqslant cb(t)

for all

c
⩾
1

c \geqslant 1

. Then the sequence

{

x
n

}

n
=
1

∞

\{ {x_n}\} _{n = 1}^\infty

generated by

x
1

∈
K

{x_1} \in K

,
\[

x

n
+
1

=
(
1
−

c
n

)

x
n

+

c
n

T

x
n

,

n
⩾
1
,

{x_{n + 1}} = (1 - {c_n}){x_n} + {c_n}T{x_n},\qquad n \geqslant 1,

\]
converges strongly to the unique fixed point of

T
T

. A related result deals with the Ishikawa iteration scheme when

T
T

is Lipschitzian and strongly pseudocontractive.
A note on liftings of linear continuous functionals
Authors: Osswald Horst
Published: July 2010

We show that for each bounded Loeb space

(
Λ
,

L
ν

(

A

)
,

ν
^

)

(\Lambda ,{L_\nu }(\mathfrak {A}),\hat \nu )

a functional

φ
∈

L
∞

(
Λ

)
′

\varphi \in {L_\infty }{(\Lambda )’}

has a lifting if and only if

φ
∈

L
1

(
Λ
)

\varphi \in {L_1}(\Lambda )

. If

p
∈
[
1
,
∞
[

p \in [1,\infty [

, then every

φ
∈

L
p

(
Λ

)
′

\varphi \in {L_p}{(\Lambda )’}

has a lifting.
A note on a type of approximate identity in the Fourier algebra
Authors: Forrest Brian , Skantharajah Mahatheva
Published: July 2010

Let

A
(
G
)

A(G)

denote the Fourier algebra of the locally compact group

G
G

. We show that for a large class of groups an ideal

I
I

in

A
(
G
)

A(G)

has a

Δ
\Delta

-weak bounded approximate identity if and only if it has a bounded approximate identity.
Compact subgroups of Lie groups and locally compact groups
Authors: Hofmann Karl H. , Terp Christian
Published: July 2010

We show that the set of compact subgroups in a connected Lie group is inductive. In fact, a locally compact group

G
G

has the inductivity property for compact subgroups if and only if the factor group

G

/

G
0

G/{G_0}

modulo the identity component has it.
On open three manifolds of quasi-positive Ricci curvature
Authors: Zhu Shun-Hui
Published: July 2010

It is proved that an open three-manifold of Ricci curvature nonnegative and positive at one point is diffeomorphic to the three-dimensional Euclidean space.
𝐴_{∞}-condition for the Jacobian of a quasiconformal mapping
Authors: Heinonen Juha , Koskela Pekka
Published: July 2010

We show that the Jacobian

J
f

{J_f}

of a quasi-conformal mapping

f
:

B

n

→
D

f:{{\mathbf {B}}^n} \to D

is an

A
∞

{A_\infty }

-weight in

B

n

{{\mathbf {B}}^n}

if and only if

D
D

is a John domain. A similar question concerning

J

f
−
1

{J_{f - 1}}

is also studied.
A weighted inequality for the Kakeya maximal operator
Authors: Vargas A. M.
Published: July 2010

Let

K

δ

{\mathcal {K}_\delta }

be the Kakeya Maximal Operator defined as the supremum of averages over parallelepipeds of eccentricity

δ
\delta

. We show that

K

δ

{\mathcal {K}_\delta }

satisfies
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue script upper K Subscript delta Baseline f StartAbsoluteValue EndAbsoluteValue Subscript upper L Sub Superscript p Subscript left-parenthesis omega right-parenthesis Baseline less-than-or-slanted-equals upper C Subscript n comma p Baseline left-parenthesis 1 slash delta right-parenthesis Superscript n slash p minus 1 Baseline left-parenthesis log left-parenthesis 1 slash delta right-parenthesis right-parenthesis Superscript alpha Super Subscript n Baseline StartAbsoluteValue EndAbsoluteValue f StartAbsoluteValue EndAbsoluteValue Subscript upper L Sub Superscript p Subscript left-parenthesis script upper K Sub Subscript delta Subscript omega right-parenthesis">

|

|

K

δ

f

|

|

L
p

(
ω
)

⩽

C

n
,
p

(
1

/

δ

)

n

/

p
−
1

(
log
⁡
(
1

/

δ
)

)

α
n

|

|

f

|

|

L
p

(

K

δ

ω
)

||{\mathcal {K}_\delta }f|{|_{{L^p}(\omega )}} \leqslant {C_{n,p}}{(1/\delta )^{n/p - 1}}{(\log (1/\delta ))^{{\alpha _n}}}||f|{|_{{L^p}({\mathcal {K}_\delta }\omega )}}

for all

p
⩽
(
n
+
1
)

/

2

p \leqslant (n + 1)/2

with some constants

C

n
,
p

,

α
n

{C_{n,p}},\;{\alpha _n}

, independent of

f
f

and the weight

ω
\omega

.
𝑄-universal quasivarieties of algebras
Authors: Adams M. E. , Dziobiak W.
Published: July 2010

A quasivariety of algebras of finite type is

Q
Q

-universal if its lattice of subquasivarieties has, as a homomorphic image of a sublattice, the lattice of subquasivarieties of any quasivariety of algebras of finite type. A sufficient condition for a quasivariety to be

Q
Q

-universal is given, thereby adding, amongst others, the quasivarieties of de Morgan algebras, Kleene algebras, distributive

p
p

-algebras, distributive double

p
p

-algebras, Heyting algebras, double Heyting algebras, lattices containing the modular lattice

M

3
,
3

,

M
V

{M_{3,3}},\,MV

-algebras, and commutative rings with unity to the known

Q
Q

-universal quasivarieties.
Hypersurfaces with constant mean curvature in spheres
Authors: Alencar Hilário , do Carmo Manfredo
Published: July 2010

Let

M
n

{M^n}

be a compact hypersurface of a sphere with constant mean curvature

H
H

. We introduce a tensor

ϕ
\phi

, related to

H
H

and to the second fundamental form, and show that if

|
ϕ
|

2

⩽

B
H

{\left | \phi \right |^2} \leqslant {B_H}

, where

B
H

≠
0

{B_H} \ne 0

is a number depending only on

H
H

and

n
n

, then either

|
ϕ
|

2

≡
0

{\left | \phi \right |^2} \equiv 0

or

|
ϕ
|

2

≡

B
H

{\left | \phi \right |^2} \equiv {B_H}

. We also characterize all

M
n

{M^n}

with

|
ϕ
|

2

≡

B
H

{\left | \phi \right |^2} \equiv {B_H}

.
Entire propagator
Authors: Autret L. , Emamirad H.
Published: July 2010

We construct an entire solution for an evolutionary Cauchy problem when the initial data is an entire vector of an appropriate operator in a Banach space. We give a sufficient condition for an entire domination. Our results are applied to some partial differential equations including the Dirac system.
A cohomological class of vector bundles
Authors: Miró-Roig Rosa M.
Published: July 2010

The goal of this paper is to give a cohomological characterization of

F

n
,
t

{F_{n,t}}

, where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript n comma t Baseline colon equals upper K e r left-parenthesis left-parenthesis n plus t semicolon n right-parenthesis script upper O Subscript bold upper P Sub Superscript n Subscript Baseline left-parenthesis negative t right-parenthesis right-arrow script upper O Subscript bold upper P Sub Superscript n Subscript Baseline right-parenthesis">

F

n
,
t

:=
Ker
⁡
(
(
n
+
t
;
n
)

O

P

n

(
−
t
)
→

O

P

n

)

{F_{n,t}}: = \operatorname {Ker} ((n + t;n){\mathcal {O}_{{{\mathbf {P}}^n}}}( - t) \to {\mathcal {O}_{{{\mathbf {P}}^n}}})

.
Analytic continuation of Riemann’s zeta function and values at negative integers via Euler’s transformation of series
Authors: Sondow Jonathan
Published: July 2010

We prove that a series derived using Euler’s transformation provides the analytic continuation of

ζ
(
s
)

\zeta (s)

for all complex

s
≠
1

s \ne 1

. At negative integers the series becomes a finite sum whose value is given by an explicit formula for Bernoulli numbers.
An infinite loop space machine for theories with noncontractible multiplication
Authors: Kříž Igor
Published: July 2010

In this paper, we develop a new kind of infinite loop space machine starting with theories with possibly noncontractible multiplication. As an application, we give an example of an infinite loop space machine with ’coherent’ homotopy inverses.
A characterization of function rings with Boolean domain
Authors: Carson Andrew B.
Published: July 2010

In §1 we characterize (effectively in terms of omitted logical types) those countable rings that can be represented as certain specified functions from their Boolean spectra to some member of a universal class of indecomposable rings that has the amalgamation property. In §2 we show that this characterization fails for uncountable rings and give an alternate (although less interesting) one that does hold for all cardinalities.
Borel liftings of the measure algebra and the failure of the continuum hypothesis
Authors: Carlson T. , Frankiewicz R. , Zbierski P.
Published: July 2010

It is proved that the failure of the continuum hypothesis is consistent with the existence of a Borel lifting for the Lebesgue measure algebra and an embedding of the Lebesgue measure algebra into

℘
(
ω
)

/

finite

\wp (\omega )/{\text {finite}}

.
A characterization of 𝑀_{𝑝}-groups
Authors: Hanaki Akihide
Published: July 2010

We characterize finite groups, all irreducible characters of which over an algebraically closed field of characteristic p are monomial.
A homogeneous, globally solvable differential operator on a nilpotent Lie group which has no tempered fundamental solution
Authors: Müller Detlef
Published: July 2010

We present an example of a homogeneous, left-invariant differential operator on the Heisenberg group

H
3

{H_3}

which admits fundamental solutions but no tempered ones. This answers a question raised by Corwin in the negative.
Characterization of classical type orthogonal polynomials
Authors: Kwon K. H. , Littlejohn L. L. , Lee J. K. , Yoo B. H.
Published: July 2010

We characterize the classical type orthogonal polynomials

{

P
n

(
x
)

}
0
∞

\{ {P_n}(x)\} _0^\infty

satisfying a fourth-order differential equation of type
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript i equals 0 Overscript 4 Endscripts script l Subscript i Baseline left-parenthesis x right-parenthesis y Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis x right-parenthesis equals lamda Subscript n Baseline y left-parenthesis x right-parenthesis">

∑

i
=
0

4

ℓ
i

(
x
)

y

(
i
)

(
x
)
=

λ
n

y
(
x
)

\sum \limits _{i = 0}^4 {{\ell _i}(x){y^{(i)}}(x) = {\lambda _n}y(x)}

\]
where

ℓ
i

(
x
)

{\ell _i}(x)

are polynomials of degree

⩽
i

\leqslant i

and

λ
n

{\lambda _n}

is a constant. They are only the orthogonal polynomials satisfying an orthogonality of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle tau 2 comma upper P Subscript m Superscript Baseline upper P Subscript n Superscript Baseline mathematical right-angle plus mathematical left-angle tau 1 comma upper P prime Subscript m Baseline upper P prime Subscript n mathematical right-angle plus mathematical left-angle tau 0 comma upper P Subscript m Baseline upper P Subscript n Baseline mathematical right-angle equals 0 for m not-equals n">

⟨

τ
2

,

P
m

P
n

⟩
+
⟨

τ
1

,

P
m
′

P
n
′

⟩
+
⟨

τ
0

,

P
m

P
n

⟩
=
0

for

m
≠
n

\langle {\tau _2},P_m^{}P_n^{}\rangle + \langle {\tau _1},P_m’P_n’\rangle + \langle {\tau _0},{P_m}{P_n}\rangle = 0\quad {\text {for}}\;m \ne n

\]
where

τ
0

,

τ
1

{\tau _0},{\tau _1}

, and

τ
2

{\tau _2}

are moment functionals.
The associated graded ring and the index of a Gorenstein local ring
Authors: Ding Songqing
Published: July 2010

Let

(
R
,

m

,
k
)

(R,\mathfrak {m},k)

be a Gorenstein local ring. It is shown that if the associated graded ring

G
(
R
)

G(R)

of

R
R

is Cohen-Macaulay, then the index of

R
R

is equal to the generalized Loewy length of

R
R

.
Goldie’s theorem for alternative rings
Authors: Essannouni H. , Kaidi A.
Published: July 2010

It is known that the socle of a semiprime Goldie ring is generated by a central idempotent and that a prime Goldie ring with a nonzero socle is a simple artinian ring. We prove the extension of these results to alternative rings. We also give an analogue of Goldie’s theorem for alternative rings. A Goldie-like theorem was obtained earlier by the authors for noetherian alternative rings by a quite different method.
The conjugation operator on 𝐴_{𝑞}(𝐺)
Authors: Gupta Sanjiv Kumar , Madan Shobha , Tewari U. B.
Published: July 2010

Let G be a compact abelian group and

Γ
\Gamma

its dual. For

1
≤
q
>
∞

1 \leq q > \infty

, the space

A
q

(
G
)

{A_q}(G)

is defined as
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript q Baseline left-parenthesis upper G right-parenthesis equals left-brace f vertical-bar f element-of upper L Superscript 1 Baseline left-parenthesis upper G right-parenthesis comma ModifyingAbove f With caret element-of l Subscript q Baseline left-parenthesis normal upper Gamma right-parenthesis right-brace">

A
q

(
G
)
=
{
f

|

f
∈

L
1

(
G
)
,

f
^

∈

l
q

(
Γ
)
}

{A_q}(G) = \{ f|f \in {L^1}(G),\quad \hat f \in {l_q}(\Gamma )\}

\]
with the norm
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar f double-vertical-bar Subscript upper A Sub Subscript q Baseline equals double-vertical-bar f double-vertical-bar Subscript upper L Sub Superscript 1 Baseline plus double-vertical-bar ModifyingAbove f With caret double-vertical-bar Subscript l Sub Subscript q">

‖
f
‖

A
q

=

‖
f
‖

L
1

+

‖

f
^

‖

l
q

{\left \| f \right \|_{{A_q}}} = {\left \| f \right \|_{{L^1}}} + {\left \| {\hat f} \right \|_{{l_q}}}

. We prove: Let G be a compact, connected abelian group and P any fixed order on

Γ
\Gamma

. If

q
>
2

q > 2

and

ϕ
\phi

is a Young’s function, then the conjugation operator H does not extend to a bounded operator from

A
q

(
G
)

{A_q}(G)

to the Orlicz space

L
ϕ

(
G
)

{L^\phi }(G)

.
A geometric approach to the multivariate Müntz problem
Authors: Kroó András
Published: July 2010

For a countable set

Ω
⊂

R

n

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

denote by

P
(
Ω
)

P(\Omega )

the space of polynomials spanned by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript omega Baseline comma omega element-of normal upper Omega left-parenthesis x equals left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis element-of double-struck upper R Superscript n Baseline comma omega equals left-parenthesis omega 1 comma ellipsis comma omega Subscript n Baseline right-parenthesis element-of normal upper Omega comma x Superscript omega Baseline equals product Underscript i equals 1 Overscript n Endscripts x Subscript i Superscript omega Super Subscript i Superscript Baseline right-parenthesis">

x
ω

,
ω
∈
Ω
(
x
=
(

x
1

,
…
,

x
n

)
∈

R

n

,
ω
=
(

ω
1

,
…
,

ω
n

)
∈
Ω
,

x
ω

=

∏

i
=
1

n

x
i

ω
i

)

{x^\omega }, \omega \in \Omega (x = ({x_1}, \ldots ,{x_n}) \in {\mathbb {R}^n}, \omega = ({\omega _1}, \ldots ,{\omega _n}) \in \Omega , {x^\omega } = \prod _{i = 1}^nx_i^{{\omega _i}})

. In this paper we investigate the question of the density of

P
(
Ω
)

P(\Omega )

in

C
(
K
)

C(K)

, the space of real valued continuous functions endowed with the supremum norm on compact set

K
⊂

R

n

K \subset {\mathbb {R}^n}

. In case

n
=
1

n = 1

the classical theorem of Müntz gives an elegant necessary and sufficient condition for density. This problem (closely related to the distribution of zeros of Fourier transforms) is much more complex in the multivariate setting. We shall present an extension of Müntz’ condition to the case

n
>
1

n > 1

which will suffice for density. This, in particular, will enable us to construct "optimally sparse" lattice point sets

Ω
\Omega

for which density holds.
𝑛× oversampling preserves any tight affine frame for odd 𝑛
Authors: Chui Charles K. , Shi Xian Liang
Published: July 2010

If

ψ
\psi

generates an affine frame

ψ

j
,
k

(
x
)
=

2

j

/

2

ψ
(

2
j

x
−
k
)
,
j
,
k
∈

Z

{\psi _{j,k}}(x) = {2^{j/2}}\psi ({2^j}x - k),j,k \in \mathbb {Z}

, of

L
2

(

R

)

{L^2}(\mathbb {R})

, we prove that

{

n

−
1

/

2

ψ

j
,
k

/

n

}

\{ {n^{ - 1/2}}{\psi _{j,k/n}}\}

is also an affine frame of

L
2

(

R

)

{L^2}(\mathbb {R})

with the same frame bounds for any positive odd integer n. This establishes the result stated as the title of this paper. A counterexample of this statement for

n
=
2

n = 2

is also given.
A matricial identity involving the self-commutator of a commuting 𝑛-tuple
Authors: Curto Raúl E. , Jian Ren Yi
Published: July 2010

For a commuting n-tuple

a
=
(

a
1

,
…
,

a
n

)

a = ({a_1}, \ldots ,{a_n})

of elements of a unital

C
∗

{C^ \ast }

-algebra

A

\mathcal {A}

, we establish a matricial identity linking the self-commutator of a to the

2

n
−
1

×

2

n
−
1

{2^{n - 1}} \times {2^{n - 1}}

matrix

a
^

\hat a

that detects the Taylor invertibility of a. As a consequence, we obtain a simple proof of a result of D. Xia (Oper. Theory: Adv. Appl. 48 (1990), 423-448), which states that for commuting t-hyponormal n-tuples,

σ
T

(
a
)
=

σ
r

(
a
)

{\sigma _T}(a) = {\sigma _r}(a)

.
The existence of nonmeasurable sets for invariant measures
Authors: Penconek Marcin , Zakrzewski Piotr
Published: July 2010

We prove that if G is a locally compact Polish group acting in a reasonable way on a set X, then for every countably additive,

σ
\sigma

-finite, G-invariant measure on X there exist nonmeasurable sets. In particular, the latter is true when X is a compact, metric, metrically homogeneous space, and G is the group of its isometries.
Some particular entries of the two-parameter Kostka matrix
Authors: Stembridge John R.
Published: July 2010

Macdonald has defined a two-parameter refinement of the Kostka matrix, denoted

K

λ
,
μ

(
q
,
t
)

{K_{\lambda ,\mu }}(q,t)

. The entries are rational functions of q and t, but he has conjectured that they are in fact polynomials with nonnegative integer coefficients. We prove two results that support this conjecture. First, we prove that if

μ
\mu

is a partition with at most two columns (or at most two rows), then

K

λ
,
μ

(
q
,
t
)

{K_{\lambda ,\mu }}(q,t)

is indeed a polynomial. Second, we provide a combinatorial interpretation of

K

λ
,
μ

(
q
,
t
)

{K_{\lambda ,\mu }}(q,t)

for the case in which

μ
\mu

is a hook. This interpretation proves in this case that not only are the entries polynomials, but also that their coefficients are nonnegative integers.
Krull dimension of modules over involution rings
Authors: Beĭdar K. I. , Puczyłowski E. R. , Smith P. F.
Published: July 2010

The following question of Lanski is answered positively in the case when a ring R with involution

∗

^ \ast

is Noetherian with respect to two-sided

∗
\ast

-ideals. Let R be a ring with

∗

^ \ast

and invertible 2, and let

S
¯

\bar S

be the subring of R generated by the symmetric elements in R. Does any left R-module have the same Krull dimension when considered as an R-module and

S
¯

\bar S

-module?
Degeneration of pseudo-Laplace operators for hyperbolic Riemann surfaces
Authors: Ji Lizhen
Published: July 2010

For finite volume, noncompact Riemann surfaces with their canonical hyperbolic metrics, there is a notion of pseudo-Laplace operators which include all embedded eigenvalues

(
>

1
4

)

(> \frac {1}{4})

of the Laplacian as a part of their eigenvalues. Similarly, we define pseudo-Laplace operators for compact hyperbolic Riemann surfaces with short geodesics. Then, for any degenerating family of hyperbolic Riemann surfaces

S
l

(
l
≥
0
)

{S_l} (l \geq 0)

, we show that normalized pseudoeigenfunctions and pseudoeigenvalues of

S
l

{S_l}

converge to normalized pseudoeigenfunctions and pseudoeigenvalues of

S
0

{S_0}

as

l
→
0

l \to 0

. In particular, normalized embedded eigenfunctions and their embedded eigenvalues of

S
0

{S_0}

can be approximated by normalized pseudoeigenfunctions and pseudoeigenvalues of

S
l

{S_l}

and

l
→
0

l \to 0

.
The conjugacy problem for HNN extensions with infinite cyclic associated groups
Authors: Horadam K. J. , Farr G. E.
Published: July 2010

Under suitable recursive conditions, the conjugacy problem for HNN extensions of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper G comma t Subscript i Baseline comma i element-of upper I colon x Superscript k Super Subscript i Superscript Baseline equals t Subscript i Superscript negative 1 Baseline x Superscript l Super Subscript i Superscript Baseline t Subscript i Baseline comma k Subscript i Baseline comma l Subscript i Baseline greater-than-or-slanted-equals 1 comma i element-of upper I mathematical right-angle">

⟨
G
,

t
i

,
i
∈
I
:

x

k
i

=

t
i

−
1

x

l
i

t
i

,

k
i

,

l
i

⩾
1
,

i
∈
I
⟩

\langle G,{t_i},i \in I:{x^{{k_i}}} = t_i^{ - 1}{x^{{l_i}}}{t_i},\;{k_i},\;{l_i} \geqslant 1,\;i \in I\rangle

is solvable if and only if the inverse subsemigroup generated by

{
(

k
i

,

l
i

)
,

i
∈
I
}

\{ ({k_i},{l_i}),\;i \in I\}

has solvable extended word problem in the semigroup
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S equals left-parenthesis double-struck upper N times double-struck upper N comma left-parenthesis a comma b right-parenthesis left-parenthesis c comma d right-parenthesis equals left-parenthesis a c slash gcd left-parenthesis b comma c right-parenthesis comma b d slash gcd left-parenthesis b comma c right-parenthesis right-parenthesis right-parenthesis period">

S
=
(

N

×

N

,

(
a
,
b
)
(
c
,
d
)
=
(
a
c

/

gcd
(
b
,
c
)
,

b
d

/

gcd
(
b
,
c
)
)
)
.

S = (\mathbb {N} \times \mathbb {N},\;(a,b)(c,d) = (ac/\gcd (b,c),\;bd/\gcd (b,c))).

\]
Furthermore,

S
S

is isomorphic to the direct sum of countably many copies of the bicyclic semigroup, which has a central place in the theory of inverse semigroups. This new approach to the conjugacy problem is used to determine several classes of HNN extensions with infinitely many stable letters and solvable conjugacy problem.
A discrete fractal in 𝑍¹₊
Authors: Khoshnevisan Davar
Published: July 2010

In this paper, we show that the level sets of mean zero finite variance random walks in

R

1

{\mathbb {R}^1}

form a discrete fractal in the sense of Barlow and Taylor. Analogously to the Brownian motion result, the Hausdorff dimension of the level sets is almost surely equal to

1
2

\tfrac {1} {2}

.
Positive solutions of nonlinear second order ordinary differential equations
Authors: Zhao Z.
Published: July 2010

We prove an existence theorem for positive solutions of nonlinear second order ordinary differential equations by using the perturbation method and the Schauder fixed point theorem.
Auréole of a quasi-ordinary singularity
Authors: Ban Chunsheng
Published: July 2010

The auréole of an analytic germ

(
X
,
x
)
⊂
(

C

n

,
0
)

(X,x) \subset ({\mathbb {C}^n},0)

is a finite family of subcones of the reduced tangent cone

|

C

X
,
x

|

|{C_{X,x}}|

such that the set

D

X
,
x

{D_{X,x}}

of the limits of tangent hyperplanes to

X
X

at

x
x

is equal to

∪

(
Proj

C
α

)
∨

\cup {(\operatorname {Proj} \,{C_\alpha })^ \vee }

. The auréole for a case of quasi-ordinary singularity is computed.
Representation of a real 𝐵*-algebra on a quaternionic Hilbert space
Authors: Kulkarni S. H.
Published: July 2010

Let A be a real

B
∗

{B^ \ast }

-algebra containing a

∗
\ast

-subalgebra that is

∗
\ast

-isomorphic to the real quaternion algebra

H

\mathbb {H}

. Suppose the spectrum of every self-adjoint element in A is contained in the real line. Then it is proved that there exists a quaternionic Hilbert space X and an isometric

∗
\ast

-isomorphism

π
\pi

of A onto a closed

∗
\ast

-subalgebra of

B
L
(
X
)

BL(X)

, the algebra of all bounded linear operators on X. If, in addition to the above hypotheses, every element in A is normal, then A is also proved to be isometrically

∗
\ast

-isomorphic to

C
(
Y
,

H

)

C(Y,\mathbb {H})

, the algebra of all continuous

H

\mathbb {H}

-valued functions on a compact Hausdorff space Y.
Density of infimum-stable convex cones
Authors: Prolla João B.
Published: July 2010

Let X be a compact Hausdorff space and let A be a linear subspace of

C
(
X
;

R

)

C(X;\mathbb {R})

containing the constant functions, and separating points from probability measures. Then the inf-lattice generated by A is uniformly dense in

C
(
X
;

R

)

C(X;\mathbb {R})

. We show that this is a corollary of the Choquet-Deny Theorem, thus simplifying the proof and extending to the nonmetric case a result of McAfee and Reny.
An uncertainty principle on hyperbolic space
Authors: Sun Li Min
Published: July 2010

In this paper, we establish an uncertainty principle on hyperbolic space

H
n

=
S

O
e

(
n
,
1
)

/

S
O
(
n
)

{H^n} = S{O_e}(n,1)/SO(n)

, which prohibits f from being confined to small neighborhoods around any point in

H
n

{H^n}

under certain assumptions on the Fourier transform

f
~

\tilde f

, where f is a normalized

L
2

{L^2}

function on

H
n

{H^n}

.
Integration of vector-valued pseudo-almost periodic functions
Authors: Zhang Chuan Yi
Published: July 2010

A necessary and sufficient condition is given to show that the indefinite integral of a vector-valued pseudo-almost periodic function is again pseudo-almost periodic. Then we use this result to answer a question about weakly almost periodic functions.
A noniterative 2-adic statement of the 3𝑁+1 conjecture
Authors: Bernstein Daniel J.
Published: July 2010

Associated with the

3
N
+
1

3N + 1

problem is a permutation

Φ
\Phi

of the 2-adic integers. The

3
N
+
1

3N + 1

conjecture is equivalent to the conjecture that 3Q is an integer if

Φ
(
Q
)

\Phi (Q)

is a positive integer. We state a new definition of

Φ
\Phi

. To wit: Q and

N
=
Φ
(
Q
)

N = \Phi (Q)

are linked by the equations

Q
=

2

d
0

+

2

d
1

+
⋯

Q = {2^{{d_0}}} + {2^{{d_1}}} + \cdots

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N equals left-parenthesis negative 1 slash 3 right-parenthesis 2 Superscript d 0 Baseline plus left-parenthesis negative 1 slash 9 right-parenthesis 2 Superscript d 1 Baseline plus left-parenthesis negative 1 slash 27 right-parenthesis 2 Superscript d 2 Baseline plus midline-horizontal-ellipsis">

N
=
(
−
1

/

3
)

2

d
0

+
(
−
1

/

9
)

2

d
1

+
(
−
1

/

27
)

2

d
2

+
⋯

N = ( - 1/3){2^{{d_0}}} + ( - 1/9){2^{{d_1}}} + ( - 1/27){2^{{d_2}}} + \cdots

with

0
≤

d
0

>

d
1

>
⋯

0 \leq {d_0} > {d_1} > \cdots

. We list four applications of this definition.
On a definite integral of a hypergeometric function
Authors: Shen Li-Chien
Published: July 2010

The connections between the elliptic functions and the hypergeometric series

2

F
1

(

1
2

,

1
2

;
1
;
x
)

_2{F_1}(\tfrac {1} {2},\tfrac {1} {2};1;x)

is well known and classical. In this note, we investigate its relation with

2

F
1

(

1
4

,

3
4

;
1
;
x
)

_2{F_1}(\tfrac {1} {4},\tfrac {3} {4};1;x)

. We find that it is less ideal than the classical case and discuss the flaws.
Combinatorics and topology of line arrangements in the complex projective plane
Authors: Artal-Bartolo Enrique
Published: July 2010

We use some results about Betti numbers of coverings of complements of plane projective curves to discuss the problem of how combinatorics determine the topology of line arrangement, finding a counterexample to a conjecture of Orlik.
The set of all 𝑚×𝑛 rectangular real matrices of rank 𝑟 is connected by analytic regular arcs
Authors: Evard J.-Cl. , Jafari F.
Published: July 2010

It is well known that the set of all square invertible real matrices has two connected components. The set of all

m
×
n

m \times n

rectangular real matrices of rank

r
r

has only one connected component when

m
≠
n

m \ne n

or

r
>
m
=
n

r > m = n

. We show that all these connected components are connected by analytic regular arcs. We apply this result to establish the existence of

p
p

-times differentiable bases of the kernel and the image of a rectangular real matrix function of several real variables.
Interpolation with meromorphic matrix functions
Authors: Ball Joseph A. , Clancey Kevin F.
Published: July 2010

A complete solution is given to a first-order pole-zero meromorphic matrix function interpolation problem on a closed Riemann surface. The solution to the interpolation problem is constructed from the solution to a natural linear homogeneous system.
On universal primitive functions
Authors: Gan Xiao-Xiong , Stromberg Karl R.
Published: July 2010

This paper generalizes Marcinkiewicz’s universal primitive on pointwise a.e. convergence directly to higher-dimensional spaces. It is also proved that the set of all universal primitive functions with respect to some given nonzero null sequence is residual and, hence, dense in the Banach space

C
(

I
n

,

R

m

)
∀
n
,
m
∈

N

C({I^n},{\mathbb {R}^m})\forall n,m \in \mathbb {N}

.
On Gödel’s second incompleteness theorem
Authors: Jech Thomas
Published: July 2010

In this note we give a short proof of Gödel’s Second Incompleteness Theorem.
An orthomodular lattice admitting no group-valued measure
Authors: Navara Mirko
Published: July 2010

We construct a finite orthomodular lattice L such that, for each commutative group G, there is no nontrivial G-valued measure on L. This result extends a result of R. J. Greechie (Orthogonal lattices admitting no states, J. Combin. Theory Ser. A 10 (1971), 119-132), and also sheds light on recent investigations in the noncommutative measure theory.
Common subspaces of 𝐿_{𝑝}-spaces
Authors: Koldobsky Alexander
Published: July 2010

For

n
≥
2
,
p
>
2

n \geq 2,p > 2

, and

q
>
2

q > 2

does there exist an n-dimensional Banach space different from Hilbert spaces which is isometric to subspaces of both

L
p

{L_p}

and

L
q

{L_q}

? Generalizing the construction from the paper Zonoids whose polars are zonoids by R. Schneider (Proc. Amer. Math. Soc. 50 (1975), 365-368) we give examples of such spaces. Moreover, for any compact subset Q of

(
0
,
∞
)
∖
{
2
k
,
k
∈
N
}

(0,\infty )\backslash \{ 2k,k \in N\}

we can construct a space isometric to subspaces of

L
q

{L_q}

for all

q
∈
Q

q \in Q

simultaneously.
Minimal surfaces with the Ricci condition in 4-dimensional space forms
Authors: Sakaki Makoto
Published: July 2010

Let

X
N

(
c
)

{X^N}(c)

denote the N-dimensional simply connected space form of constant curvature c. We consider a problem to classify those minimal surfaces in

X
N

(
c
)

{X^N}(c)

which are locally isometric to minimal surfaces in

X
3

(
c
)

{X^3}(c)

. In this paper we solve this problem in the case where

N
=
4

N = 4

, and give a result also in higher codimensional cases.
On lifting the hyperelliptic involution
Authors: Accola Robert D. M.
Published: July 2010

Let

W
p

{W_p}

stand for a compact Riemann surface of genus p. (1) Let

W
q

{W_q}

be hyperelliptic, and let n be a positive integer. Then there exists an unramified covering of n sheets,

W
p

→

W
q

{W_p} \to {W_q}

, where

W
p

{W_p}

is hyperelliptic. (2) Let

W

2
n
+
1

→

W
2

{W_{2n + 1}} \to {W_2}

be an unramified Galois covering with a dihedral group as Galois group, and let n be odd. Then

W

2
n
+
1

{W_{2n + 1}}

is elliptic hyperelliptic (bi-elliptic). (3) Let

W
4

→

W
2

{W_4} \to {W_2}

be an unramified non-Galois covering of three sheets. Then

W
4

{W_4}

is hyperelliptic.
An inverse spectral problem for Sturm-Liouville operators with discontinuous coefficients
Authors: Carlson Robert
Published: July 2010

For generic Sturm-Liouville problems with piecewise constant leading coefficients, the leading coefficient can be determined up to a finite ambiguity from the eigenvalues of the problem.
Approximation by semi-Fredholm operators
Authors: Galaz-Fontes Fernando
Published: July 2010

We present a simple approach for calculating some known distances from a bounded operator defined on a separable Hilbert space to certain sets related to semi-Fredholm operators.
Support points and double poles
Authors: Goh Say Song
Published: July 2010

This paper gives some sufficient conditions for support points of the class S of univalent functions to be rotations of the Koebe function

k
(
z
)
=
z

(
1
−
z

)

−
2

k(z) = z{(1 - z)^{ - 2}}

. If f is a support point associated with a continuous linear functional L and if the function

Φ
(
w
)
=
L
(

f
2

/

(
f
−
w
)
)

\Phi (w) = L({f^2}/(f - w))

does not have a double pole, then under some mild additional assumptions, a rational support point f must be a rotation of the Koebe function. The situation is more complicated when

Φ
\Phi

has a double pole. However, we are able to prove the two-functional conjecture for derivative functionals, where

Φ
\Phi

has a double pole.
On the hypoellipticity of convolution equations in the ultradistribution spaces of 𝐿^{𝑞} growth
Authors: Kovačević Dušanka
Published: July 2010

We consider convolution equations in the ultradistribution spaces

D

L
q

′

(

M
p

)

\mathcal {D}_{{L^q}}^{’({M_p})}

and

D

L
q

′

{

M
p

}

,

q
∈
[
1
,
∞
]

\mathcal {D}_{{L^q}}^{’\{ {M_p}\} },\;q \in [1,\infty ]

, of Beurling and Roumieu type of

L
q

{L^q}

growth. Our main aim is to find conditions for convolution operators to be hypoelliptic in

D

L
∞

′

(

M
p

)

\mathcal {D}_{{L^\infty }}^{’({M_p})}

and

D

L
∞

′

{

M
p

}

\mathcal {D}_{{L^\infty }}^{’\{ {M_p}\} }

respectively, in terms of their Fourier transforms.
The 𝐻^{𝑝}-boundedness of the Riesz means for the eigenfunction expansions for elliptic operators with constant coefficients
Authors: Liu He Ping , Lu Shan Zhen
Published: July 2010

In this paper we study the bounded property of the Riesz means for the eigenfunction expansions for elliptic operators on the Hardy spaces. Our result generalizes the classical result due to Sjölin and Stein-Taibleson-Weiss on the Bochner-Riesz means.
Arithmetic groups of higher 𝑄-rank cannot act on 1-manifolds
Authors: Witte Dave
Published: July 2010

Let

Γ
\Gamma

be a subgroup of finite index in

SL
n

(

Z

)

{\text {SL}_n}(\mathbb {Z})

with

n
≥
3

n \geq 3

. We show that every continuous action of

Γ
\Gamma

on the circle

S
1

{S^1}

or on the real line

R

\mathbb {R}

factors through an action of a finite quotient of

Γ
\Gamma

. This follows from the algebraic fact that central extensions of

Γ
\Gamma

are not right orderable. (In particular,

Γ
\Gamma

is not right orderable.) More generally, the same results hold if

Γ
\Gamma

is any arithmetic subgroup of any simple algebraic group G over

Q

\mathbb {Q}

, with

Q

-

rank

(
G
)
≥
2

\mathbb {Q} \text {-} {\text {rank}}(G) \geq 2

.
Functors of sub-descent type and dominion theory
Authors: Johnson P. B.
Published: July 2010

Necessary and sufficient conditions are given for the Eilenberg-Moore comparison functor

Φ
\Phi

arising from a functor U (having a left adjoint) to be a Galois connection in the sense of J. R. Isbell, in which case the functor U is said to be of subdescent type. These conditions, when applied to a contravariant hom-functor

U
=

C

(
−
,
B
)
:

C

op

→

S
e
t

U = {\mathbf {C}}( - ,B):{{\mathbf {C}}^{{\text {op}}}} \to {\mathbf {Set}}

, read like a kind of functional completeness axiom for the object B. In order to appreciate this result, it is useful to consider the full subcategory

d
o
m

B

⊂

C

{\mathbf {dom}_B} \subset {\mathbf {C}}

of so-called B-dominions, consisting of certain canonically arising regular subobjects of powers of the object B. The functor

U
=

C

(
−
,
B
)

U = {\mathbf {C}}( - ,B)

is of subdescent type if and only if the object B is a regular cogenerator for the category

d
o
m

B

{\mathbf {dom}_B}

, in which case

d
o
m

B

{\mathbf {dom}_B}

is the reflective hull of B in C and, moreover, the category

d
o
m

B

{\mathbf {dom}_B}

admits a Stone-like representation as (being contravariantly equivalent, via the comparison functor

Φ
\Phi

, to) a full, reflective subcategory of the category of algebras for the triple in Set induced by the functor U.
Coalgebras over the higher rank symplectic groups
Authors: Paolucci Anna Maria
Published: July 2010

In these notes we generalize the construction obtained for the deformation of the symplectic group

Sp
⁡
(
2
)

\operatorname {Sp}(2)

to the case of any N even:

N
=
2
k

N = 2k

. We characterize the bialgebras

A
q

(

Sp

(
k
)
)

{A_q}({\operatorname {Sp}}(k))

by generators and relations. We consider the deformation of the algebra of polynomials on the group

Sp

(
k
)
:

Pol

(

Sp
q

(
k
)
)

{\operatorname {Sp}}(k):{\operatorname {Pol}}({\operatorname {Sp}_q}(k))

is a

Hopf
∗

{\operatorname {Hopf}^ \ast }

-algebra and we build

∗

^ \ast

-representations of it by means of a Verma module construction.
The structure of measurable mappings on metric spaces
Authors: Wiśniewski Andrzej
Published: July 2010

The purpose of this paper is to investigate the conditions under which every measurable mapping on a metric space X with the measure

μ
\mu

is a limit of a sequence of continuous mappings, with respect to the convergence

μ
\mu

-almost everywhere.
Global approximation in harmonic spaces
Authors: Gardiner Stephen J. , Goldstein Myron , GowriSankaran Kohur
Published: July 2010

This paper characterizes, in terms of thinness, compact sets K in a suitable harmonic space

Ω
\Omega

which have the following property: functions which are harmonic (resp. continuous and superharmonic) on a neighbourhood of K can be uniformly approximated on K by functions which are harmonic (resp. continuous and superharmonic) on

Ω
\Omega

. The corresponding problems of approximating functions which are continuous on K and harmonic (resp. superharmonic) on the interior

K
˚

\mathring {K}

are also solved.
Rates of growth of p.i. algebras
Authors: Berele Allan
Published: July 2010

Let

A
A

be any p.i. algebra in characteristic zero. Then the

GK

{\text {GK}}

-dimension of finitely generated subalgebras is linearly bounded in the number of generators.
Strong unboundedness of unbounded analytic functions
Authors: Gordon A. Ya.
Published: July 2010

It is proved that if f is an unbounded analytic function in the open unit disc

D

\mathbb {D}

, then there must exist a sequence

(

z
n

)

({z_n})

in

D

\mathbb {D}

such that for every

j
=
0
,
1
,
2
,
…
,

f

(
j
)

(

z
n

)
→
∞

j = 0,1,2, \ldots ,{f^{(j)}}({z_n}) \to \infty

as

n
→
∞

n \to \infty

. This answers affirmatively a question asked by J. Langley and L. Rubel in 1984.
On 𝐻^{𝑝}(𝐑ⁿ)-multipliers of mixed-norm type
Authors: Onneweer C. W. , Quek T. S.
Published: July 2010

For a function m in

L
∞

(

R

n

)

{L^\infty }({{\mathbf {R}}^n})

, an appropriately chosen function

η
\eta

in

C
∞

(

R

n

)

{C^\infty }({{\mathbf {R}}^n})

and

δ
>
0

\delta > 0

we define

m
δ

{m_\delta }

by

m
δ

(
ξ
)
=
m
(
δ
ξ
)
η
(
ξ
)

{m_\delta }(\xi ) = m(\delta \xi )\eta (\xi )

. We show that if

0
>
p
≤
1

0 > p \leq 1

and if the sequence

(
(

m

2
n

)

∅
^

)

((m_{2^n})\hat \emptyset )

belongs to a certain mixed-norm space, depending on p, then m is a Fourier multiplier for the corresponding Hardy space

H
p

(

R

n

)

{H^p}({{\mathbf {R}}^n})

. Moreover, we prove the sharpness of our multiplier theorem. Comparable results had been proved earlier for multipliers for Hardy spaces defined on a locally compact Vilenkin group.
A note on generalized solutions of singular Hamiltonian systems
Authors: Tanaka Kazunaga
Published: July 2010

We consider T-periodic solutions of singular Hamiltonian systems with weak force
\[

q
¨

+
∇
V
(
q
,
t
)
=
0
,

\ddot q + \nabla V(q,t) = 0,

\]
where

V
(
q
,
t
)
∼
−
1

/

|

q

|

α

V(q,t) \sim - 1/|q{|^\alpha }

near

q
=
0

q = 0

with

α
∈
(
0
,
2
)

\alpha \in (0,2)

. In particular, we study some properties of generalized T-periodic solutions, which were introduced by Bahri and Rabinowitz.
On the Müntz rational approximation rate
Authors: Zhou S. P.
Published: July 2010

The present paper constructs a counterexample to show that a conjecture of Newman concerning rational approximation rate of arbitrary Markov system is generally not true.
On the failure of the Urysohn-Menger sum formula for cohomological dimension
Authors: Dranišnikov A. N. , Repovš D. , Ščepin E. V.
Published: July 2010

We prove that the classical Urysohn-Menger sum formula,

dim
⁡
(
A
∪
B
)
⩽
dim
⁡
A
+
dim
⁡
B
+
1

\dim (A \cup B) \leqslant \dim A + \dim B + 1

, which is also known to be true for cohomological dimension over the integers (and some other abelian groups), does not hold for cohomological dimension over an arbitrary abelian group of coefficients. In particular, we prove that there exist subsets

A
,

B
⊂

R

4

A,\;B \subset {\mathbb {R}^4}

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4 equals dimension Subscript double-struck upper Q slash double-struck upper Z Baseline left-parenthesis upper A union upper B right-parenthesis greater-than dimension Subscript double-struck upper Q slash double-struck upper Z Baseline upper A plus dimension Subscript double-struck upper Q slash double-struck upper Z Baseline upper B plus 1 equals 3">

4
=

dim

Q

/

Z

(
A
∪
B
)
>

dim

Q

/

Z

A
+

dim

Q

/

Z

B
+
1
=
3

4 = {\dim _{\mathbb {Q}/\mathbb {Z}}}(A \cup B) > {\dim _{\mathbb {Q}/\mathbb {Z}}}A + {\dim _{\mathbb {Q}/\mathbb {Z}}}B + 1 = 3

.
Union theorem for cohomological dimension: a simple counterexample
Authors: Dydak Jerzy
Published: July 2010

An elementary counterexample to the Union Theorem for cohomological dimension with coefficients in

Z

/

2
∞

{\mathbf {Z}}/{2^\infty }

is presented.
On location and scale maximum likelihood estimators
Authors: Gupta A. K. , Székely G. J.
Published: July 2010

The maximum likelihood estimators of location and scale parameters have the following undesirable properties: (i) they are not always consistent, and (ii) the asymptotic correlation of the MLE’s of the "unrelated" location and scale parameters can be arbitrarily close to one. In this paper methods to overcome these undesirable properties have been studied.
New examples of complete Ricci solitons
Authors: Ivey Thomas
Published: July 2010

The Ricci soliton condition reduces to a set of ODEs when one assumes that the metric is a doubly-warped product of a ray with a sphere and an Einstein manifold. If the Einstein manifold has positive Ricci curvature, we show there is a one-parameter family of solutions which give complete noncompact Ricci solitons.
Polynomial modules over the Steenrod algebra and conjugation in the Milnor basis
Authors: Monks Kenneth G.
Published: July 2010

Let

P
s

=

F

2

[

x
1

,
…
,

x
s

]

{P_s} = {\mathbb {F}_2}[{x_1}, \ldots ,{x_s}]

be the

mod

2

\bmod \;2

cohomology of the s-fold product of

R

P

∞

\mathbb {R}{{\text {P}}^\infty }

with the usual structure as a module over the Steenrod algebra. A monomial in

P
s

{P_s}

is said to be hit if it is in the image of the action

A
¯

⊗

P
s

→

P
s

\bar A \otimes {P_s} \to {P_s}

where

A
¯

\bar A

is the augmentation ideal of A. We extend a result of Wood to determine a new family of hit monomials in

P
s

{P_s}

. We then use similar methods to obtain a generalization of antiautomorphism formulas of Davis and Gallant.