bluuebunny/crossref_metadata_2025 huggingface dataset:
Sum of a double series
Authors: Sharma B. L.
Published: June 2010

In this paper we obtain the sum of a double series

F
(
1
,
1
)

F(1,1)

and, in a particular case, we get a new formula

4

F
3

(
1
)

_4{F_3}(1)

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 4 Baseline upper F 3 StartBinomialOrMatrix alpha comma beta minus alpha comma 1 slash 2 rho comma 1 slash 2 rho plus 1 slash 2 semicolon 1 Choose 1 slash 2 beta comma 1 slash 2 left-parenthesis 1 plus beta right-parenthesis comma 1 plus rho semicolon EndBinomialOrMatrix equals StartFraction normal upper Gamma left-parenthesis beta minus rho minus alpha right-parenthesis normal upper Gamma left-parenthesis beta right-parenthesis Over normal upper Gamma left-parenthesis beta minus rho right-parenthesis normal upper Gamma left-parenthesis beta minus alpha right-parenthesis EndFraction comma">

4

F
3

[

α
,
β
−
α
,
1

/

2
ρ
,
1

/

2
ρ
+
1

/

2
;
1

1

/

2
β
,
1

/

2
(
1
+
β
)
,
1
+
ρ
;

]

=

Γ
(
β
−
ρ
−
α
)
Γ
(
β
)

Γ
(
β
−
ρ
)
Γ
(
β
−
α
)

,

_4{F_3}\left [ {\begin {array}{*{20}{c}} {\alpha ,\beta - \alpha ,1/2\rho ,1/2\rho + 1/2;1} \\ {1/2\beta ,1/2(1 + \beta ),1 + \rho ;} \\ \end {array} } \right ] = \tfrac {{\Gamma (\beta - \rho - \alpha )\Gamma (\beta )}} {{\Gamma (\beta - \rho )\Gamma (\beta - \alpha )}},

\]
provided that

R
(
β
−
α
)
>
0

R(\beta - \alpha ) > 0

,

R
(
β
−
ρ
−
α
)
>
0

R(\beta - \rho - \alpha ) > 0

and

R
(
β
−
ρ
)
>
0

R(\beta - \rho ) > 0

. If

α
=
−
n

\alpha = - n

, the formula reduces to a known result due to Bailey [2].
On boundedness of solutions of second order differential equations in the limit circle case
Authors: Kwong Man Kam
Published: June 2010

A differential equation of the form

x
(
t
)
+
a
(
t
)
x
(
t
)
=
0
,
t
⩾
0

x(t) + a(t)x(t) = 0,t \geqslant 0

, is said to be in the limit circle case if all its solutions are square integrable on

[
0
,
∞
)

[0,\infty )

. It has been conjectured in [1] that all its solutions are bounded. J. Walter recently gave a counterexample. This paper gives a method of modifying any given equation in the limit circle case with bounded solutions to produce one with unbounded solutions.
Closure-preserving families and metacompactness
Authors: Potoczny Henry , Junnila Heikki
Published: June 2010

It is the purpose of this paper to show that if a space

X
X

admits a closure-preserving cover of compact, closed sets, then

X
X

is metacompact. This paper also provides a characterization of those closure-preserving covers of compact sets admitted by

σ
\sigma

-compact spaces.
Normal subgroups contained in the Frattini subgroup. II
Authors: Hill W. Mack
Published: June 2010

If

p
p

is an odd prime and

H
H

is a

p
p

-group with a characteristic subgroup

K
K

such that

|

K

|

>

|

K
∩
Z
(
H
)

|

=
p

|K| > |K \cap Z(H)| = p

, then

H
H

cannot be a normal subgroup contained in the Frattini subgroup of any finite group

G
G

.
On Λ(𝑝) sets with minimal constant in discrete noncommutative groups
Authors: Bożejko Marek
Published: June 2010

We compute the minimal constants for infinite

Λ
(
2
n
)

\Lambda (2n)

sets in discrete noncommutative groups and as a consequence we obtain an alternate proof of Leinert’s theorem on

Λ
(
∞
)

\Lambda (\infty )

sets.
Some nonprojective subgroups of free topological groups
Authors: Brown Ronald
Published: June 2010

For the free topological group on an interval

[
a
,
b
]

[a,b]

a family of closed, locally path-connected subgroups is given such that each group is not projective and so not free topological. Simplicial methods are used, and the test for nonprojectivity is nonfreeness of the group of path components. Similar results are given for the abelian case.
Uniformly convex totally ordered sets
Authors: Redfield R. H.
Published: June 2010

Since a uniformly convex totally ordered set is a uniform lattice, its uniform completion is totally ordered in a natural way. The assumption of uniform convexity is a small restriction because every nearly uniform ordered space is uniformly convex.
On a coradical of a finite-dimensional Hopf algebra
Authors: Radford David E.
Published: June 2010

Examples are constructed showing that the coradical of a finite-dimensional Hopf algebra over any algebraically closed field is not necessarily a subalgebra (hence the Jacobson radical is not a Hopf ideal in general). The square of the antipode may induce a permutation on the simple subcoalgebras of dimension

>
1

> 1

of arbitrarily high order.
Power-associativity of antiflexible rings
Authors: Çelik Hasan A. , Outcalt David L.
Published: June 2010

Conditions which force an antiflexible ring of characteristic

p
p

to be power-associative are determined.
On paracompact subsets of linear topological spaces
Authors: Graff Richard A.
Published: June 2010

It is shown that a connected open subset of a

σ
\sigma

-compact topological space is paracompact in the relative topology only if the subspace is

σ
\sigma

-compact. An application is made to demonstrate the existence of nonparacompact open subspaces, in the weak-star and bounded weak-star topologies, of the dual to a nonseparable Banach space. As a corollary, nonempty paracompact manifolds modeled on such a space always have open submanifolds which are not paracompact.
A change of rings theorem and the Artin-Rees property
Authors: Boratyński M.
Published: June 2010

A two-sided ideal

A

\mathfrak {A}

of the ring

R
R

is said to have the left

A
R

AR

property if for every left ideal

I
I

and every

k
k

there exists an

n
n

such that

A

n

∩
I
⊂

A

k

I

{\mathfrak {A}^n} \cap I \subset {\mathfrak {A}^k}I

. Let

R
R

be a left noetherian ring and

A

\mathfrak {A}

a two-sided ideal contained in its Jacobson radical. If

A

\mathfrak {A}

has the

AR
\operatorname {AR}

property then

l

gl

dim

R
⩽

p

dim

R

/

A

+

l

gl dim 

R

/

A

{\text {l}}\;{\text {gl}}\;{\text {dim}}\;R \leqslant {\text {p}}\;{\text {dim}}\;R/\mathfrak {A} + {\text {l}}\;{\text {gl dim }}R/\mathfrak {A}

, where

p

dim

R

/

A

{\text {p}}\;{\text {dim}}\;R/\mathfrak {A}

denotes the (left) projective dimension of the module

R

/

A

R/\mathfrak {A}

.
Boundary value problems for second order nonhomogeneous differential systems
Authors: Tefteller S. C.
Published: June 2010

This paper is a study of second order nonhomogeneous differential systems involving a parameter with boundary conditions specified at two points. By means of a polar coordinate transformation for this system, the existence of eigenvalues is established. The results of this study extend those of Max Mason in that selfadjointness of the problem is not necessary to insure a solution.
On infinitely divisible laws in 𝐶[0,1]
Authors: De Araujo Aloisio Pessoa
Published: June 2010

In Euclidean spaces, or in a separable Hilbert space, the characteristic function of an infinitely divisible distribution has the familiar form given by the Lévy-Khintchine formula. The Lévy measures

M
M

of this formula are characterized by the property that the integral of

min
[
1
,

|

|

x

|

|

2

]

\min [1,||x|{|^2}]

with respect to

M
M

is finite. This simple situation no longer holds in the Banach space

C
=
C
[
0
,
1
]

C = C[0,1]

where integrability of

min
[
1
,

|

|

x

|

|

]

\min [1,||x||]

is sufficient but integrability of

min
[
1
,

|

|

x

|

|

2

]

\min [1,||x|{|^2}]

is neither necessary nor sufficient. Certain other conditions which are sufficient to imply that

M
M

is the Lévy measure of a distribution on

C
C

can be obtained with the use of an integral formula of Garsia.
Some Baire spaces for which Blumberg’s theorem does not hold
Authors: White H. E.
Published: June 2010

First, in the second section, we describe a class of Baire spaces for which Blumberg’s theorem does not hold. Then, in the third section, we discuss Blumberg’s theorem for

P
P

-spaces.
A locally compact nonparacompact space for which the strict topology is Mackey
Authors: Wheeler Robert F.
Published: June 2010

A space as described in the title is used to construct counterexamples to several conjectures in the theory of locally convex spaces.
Singular integrals on nilpotent Lie groups
Authors: Strichartz Robert S.
Published: June 2010

Convolution operators

T
f
(
x
)
=
∫
f
(
x

y

−
1

)
K
(
y
)

d
y

Tf(x) = \smallint f(x{y^{ - 1}})K(y)\;dy

on a class of nilpotent Lie groups are shown to be bounded on

L
p

,

1
>
p
>
∞

{L^p},\;1 > p > \infty

, provided the Euclidean Fourier transform of

K
K

(with respect to a coordinate system in which the group multiplication is in a special form) satisfies familiar “multiplier” conditions.
Not every minimal Hausdorff space is 𝑒-compact
Authors: Stephenson R. M.
Published: June 2010

A topological space

X
X

is said to be

e
e

-compact with respect to a dense subset

D
D

provided either of the following equivalent conditions is satisfied: (i) every open cover of

X
X

has a finite subcollection which covers

D
D

; (ii) every ultrafilter on

D
D

converges to a point of

X
X

. If there exists a dense subset with respect to which a space

X
X

is

e
e

-compact, then

X
X

is called

e
e

-compact.

1

^{1}

Two problems recently raised by S. H. Hechler are the following. (a) Is every minimal Hausdorff space

e
e

-compact? (b) If there exists a Hausdorff space which is

e
e

-compact with respect to a space

D
D

, must

D
D

be completely regular? The main purpose of this paper is to provide a negative answer to (a) and to present some results which the author hopes will be of use in the solution to (b). These results can also be used to obtain a construction of

β
X

\beta X

for certain completely regular Hausdorff spaces

X
X

.
Rook theory. I. Rook equivalence of Ferrers boards
Authors: Goldman Jay R. , Joichi J. T. , White Dennis E.
Published: June 2010

We introduce a new tool, the factorial polynomials, to study rook equivalence of Ferrers boards. We provide a set of invariants for rook equivalence as well as a very simple algorithm for deciding rook equivalence of Ferrers boards. We then count the number of Ferrers boards rook equivalent to a given Ferrers board.
On Arnold’s formula for the dimension of a polynomial ring
Authors: Eakin Paul
Published: June 2010

If

R
R

is a commutative integral domain with quotient field

K
K

and

x
1

,
…
,

x
n

{x_1}, \ldots ,{x_n}

are indeterminates, then there exist

θ
1

,
…
,

θ
n

{\theta _1}, \ldots ,{\theta _n}

in

K
K

such that

dim
⁡
R
[

x
1

,
…
,

x
n

]
=
n
+
dim
⁡
R
[

θ
1

,
…
,

θ
n

]

\dim R[{x_1}, \ldots ,{x_n}] = n + \dim R[{\theta _1}, \ldots ,{\theta _n}]

.
On the plane term rank of a three dimensional matrix
Authors: Brualdi R. A. , Csima J.
Published: June 2010

A lower bound is obtained for the plane term rank of a three dimensional matrix in which each of the horizontal two dimensional matrices has constant row and column sums.
Certain multiple valued harmonic function
Authors: Caffarelli L. A.
Published: June 2010

The difference equation

v
(

x
+

)
−
v
(
x
)
=
u
(
x
)

v({x^ + }) - v(x) = u(x)

is solved for any harmonic

u
u

in the covering space of an unknotted curve.
Simultaneous spline approximation and interpolation preserving norms
Authors: Chui C. K. , Rozema E. R. , Smith P. W. , Ward J. D.
Published: June 2010

In this paper, it is proved that splines of order

k
(
k
⩾
2
)

k(k \geqslant 2)

have property SAIN. The proof of this result is based on the important properties of

B
B

-splines.
The devil and the angel of loops
Authors: Fook Leong
Published: June 2010

G
a

{G_a}

, the subloop generated by all the associators of a loop

G
G

, is singled out for study for the first time. If

G
G

is Moufang,

G
a

{G_a}

is found to be normal in

G
G

. The relation of

G
a

{G_a}

with the nucleus of

G
G

is also investigated.
Equivalence of certain discontinuous functions under closure
Authors: Miller Maurice Hugh
Published: June 2010

We show that no distinction can be made between the closure of a Darboux graph and the closure of a connected graph, and under certain conditions, we show the closure of a Darboux function is the closure of an almost continuous function. In showing this, a construction process is given to show how to turn certain non-almost continuous functions into almost continuous functions.
On endomorphisms of a solenoid
Authors: Wilson A. M.
Published: June 2010

Geometrically simple Bernoulli generators are constructed for certain ergodic endomorphisms of solenoids. An arbitrary ergodic solenoidal group automorphism is obtained as the limit of a sequence of such Bernoulli factors and hence, by a theorem of D. S. Ornstein, must be measure-theoretically isomorphic to a Bernoulli shift.
Some vector bundles on Jacobian varieties
Authors: Kempf George R.
Published: June 2010

Picard bundles on a Jacobian variety are invariant under some subgroups of its fundamental group.
Some remarks on uniqueness for a class of singular abstract Cauchy problems
Authors: Donaldson James A. , Goldstein Jerome A.
Published: June 2010

Of concern is the Cauchy problem for equations of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis t right-parenthesis plus alpha left-parenthesis t right-parenthesis u prime left-parenthesis t right-parenthesis plus upper S squared u left-parenthesis t right-parenthesis equals 0 left-parenthesis prime equals d slash d t right-parenthesis">

u
(
t
)
+
α
(
t
)

u
′

(
t
)
+

S
2

u
(
t
)
=
0

(
′

=
d

/

d
t
)

u(t) + \alpha (t)u’(t) + {S^2}u(t) = 0(’ = d/dt)

on a complex Hilbert space

X
X

.

S
S

is a selfadjoint operator on

X
X

while

α
\alpha

is a continuous function on

(
0
,
∞
)

(0,\infty )

which can be unbounded at

t
=
0

t = 0

. Uniqueness results are obtained for these equations by applying a uniqueness theorem for nonlinear equations. Furthermore, nonuniqueness examples for the linear abstract Euler-Poisson-Darboux equation, which is contained in this class, show that the uniqueness theorem is best possible.
Idempotent maximal ideals and independent sets
Authors: Graham Colin C.
Published: June 2010

Let

E
E

be a compact independent subset of a nondiscrete LCA group

G
G

. Let

G
p
E

GpE

be the subgroup of

G
G

generated algebraically by

E
E

. If

μ
\mu

is a continuous, regular, Borel measure on

G
p
E

GpE

with

μ
(
G
p
E
)
≠
0

\mu (GpE) \ne 0

, then there exists a maximal ideal

χ
\chi

of the algebra

M
(
G
)

M(G)

of regular Borel measures on

G
G

such that the restriction of

χ
\chi

to

L
1

(
μ
)
=
{
ν
∈
M
(
G
)
:
ν
≪
μ
}

{L^1}(\mu ) = \{ \nu \in M(G):\nu \ll \mu \}

is a nontrivial idempotent in

L
∞

(
μ
)

{L^\infty }(\mu )

. This result is used to give a new proof that

G
p
E

GpE

has zero Haar measure.
Generalized Morse sequences on 𝑛 symbols
Authors: Martin John C.
Published: June 2010

A class of bisequences on

n

{\text {n}}

symbols is constructed which includes the generalized Morse sequences introduced by Keane. The topological structure and endomorphisms of the resulting minimal symbolic flows are described.
On models ≡_{∞𝜔} to an uncountable model
Authors: Nadel Mark
Published: June 2010

It is shown that a model is

≡

∞
ω

{ \equiv _{\infty \omega }}

to an uncountable model provided there is an uncountable model of its complete theory with respect to some admissible fragment containing a copy of the given model.
Simple maximal quotient rings
Authors: Rubin Robert A.
Published: June 2010

In this paper we consider the question of when a ring

Λ
\Lambda

has a simple maximal left ring of quotients. In the first section we determine two necessary conditions; viz. that

Λ
\Lambda

be left nonsingular, and when

I
I

and

J
J

are nonzero ideals of

Λ
\Lambda

with

I
∩
J
=
0

I \cap J = 0

, then

I
+
J

I + J

is not left essential in

Λ
\Lambda

. In the second section we show that these conditions are also sufficient when

Λ
\Lambda

is of finite left Goldie dimension. In addition, for a left nonsingular ring of finite left Goldie dimension, we determine the ideal structure of the maximal left ring of quotients.
On rings whose left modules are direct sums of finitely generated modules
Authors: Fuller Kent R.
Published: June 2010

The relationship between rings of finite module type and rings whose left modules have decompositions that complement direct summands is examined by proving that the latter are precisely the rings of the title.
A new proof for an inequality of Jenkins
Authors: Leeman George B.
Published: June 2010

A new proof of Jenkins’ inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R e left-parenthesis e Superscript 2 i theta Baseline a 3 minus e Superscript 2 i theta Baseline a 2 squared minus tau e Superscript i theta Baseline a 2 right-parenthesis less-than-or-slanted-equals 1 plus three eighths tau squared minus one fourth tau squared log left-parenthesis tau slash 4 right-parenthesis comma 0 less-than-or-slanted-equals tau less-than-or-slanted-equals 4 comma">

Re
⁡
(

e

2
i
θ

a
3

−

e

2
i
θ

a
2
2

−
τ

e

i
θ

a
2

)
⩽
1
+

3
8

τ
2

−

1
4

τ
2

log
⁡
(
τ

/

4
)
,

0
⩽
τ
⩽
4
,

\operatorname {Re} ({e^{2i\theta }}{a_3} - {e^{2i\theta }}a_2^2 - \tau {e^{i\theta }}{a_2}) \leqslant 1 + \tfrac {3} {8}{\tau ^2} - \tfrac {1} {4}{\tau ^2}\log (\tau /4),\quad 0 \leqslant \tau \leqslant 4,

\]
for univalent functions

f
(
z
)
=
z
+

∑

n
=
2

∞

a
n

z
n

f(z) = z + \sum \nolimits _{n = 2}^\infty {{a_n}{z^n}}

is presented.
Isolated singularities of quadratic differentials arising from a module problem
Authors: Wiener Jeffrey Clayton
Published: June 2010

If

R
⊂
S

R \subset S

are Riemann surfaces, we will say that

z
0

∈
S
−
R

{z_0} \in S - R

is an isolated point boundary component of

R
R

if there exists a neighborhood

U
U

of

z
0

{z_0}

in

S
S

such that

U
−
{

z
0

}
⊂
R

U - \{ {z_0}\} \subset R

. We prove that the quadratic differential

Q

(
z
)

d

z
2

Q\left ( z \right )d{z^2}

obtained by solving the module problem

P
(

a
1

,
…
,

a
k

)

P({a_1}, \ldots ,{a_k})

applied to a free family of homotopy classes on

R
R

can be extended to

z
0

∈
S

{z_0} \in S

so that either

Q

(
z
)

Q\left ( z \right )

is regular at

z
0

{z_0}

or

Q
(
z
)

Q(z)

has a simple pole at

z
0

{z_0}

.
On Blaschke products diverging everywhere on the boundary of the unit disc
Authors: Linden C. N.
Published: June 2010

If the moduli of the zeros of a Blaschke product increase sufficiently slowly the arguments of the zeros may be so chosen that the product diverges everywhere on

{
z
:

|

z

|

=
1
}

\{ z:|z| = 1\}

.
Separation for differential operators and the 𝐿^{𝑝} spaces
Authors: Zettl Anton
Published: June 2010

The differential operator
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M y equals y Superscript left-parenthesis n right-parenthesis Baseline plus q Subscript n minus 1 Baseline y Superscript left-parenthesis n minus 1 right-parenthesis Baseline plus midline-horizontal-ellipsis plus q 1 y Superscript prime Baseline plus q 0 y">

M
y
=

y

(
n
)

+

q

n
−
1

y

(
n
−
1
)

+
⋯
+

q
1

y
′

+

q
0

y

My = {y^{(n)}} + {q_{n - 1}}{y^{(n - 1)}} + \cdots + {q_1}y’ + {q_0}y

\]
is said to be separated in

L
p

(
0
,
∞
)

{L^p}(0,\infty )

if

y
∈

L
p

(
0
,
∞
)
,

y

(
n
−
1
)

y \in {L^p}(0,\infty ),{y^{(n - 1)}}

absolutely continuous, and

M
y
∈

L
p

(
0
,
∞
)

My \in {L^p}(0,\infty )

imply that

q
i

y

(
i
)

∈

L
p

(
0
,
∞
)

{q_i}{y^{(i)}} \in {L^p}(0,\infty )

for all

i
=
0
,
1
,
…
,
n
−
1

i = 0,1, \ldots ,n - 1

. As a special case of our main result we obtain:

M
M

is separated in

L
p

(
0
,
∞
)

{L^p}(0,\infty )

if

q
i

=

b
i

+

s
i

{q_i} = {b_i} + {s_i}

where

b
i

{b_i}

is essentially bounded and

s
i

∈

L

p
i

(
0
,
∞
)

{s_i} \in {L^{pi}}(0,\infty )

for some

p
i

⩾
p
,
i
=
0
,
1
,
…
,
n
−
1

{p_i} \geqslant p,i = 0,1, \ldots ,n - 1

. The case

n
=
2
,
p
=
2
,

q
1

≡
0

n = 2,p = 2,{q_1} \equiv 0

is due to Everitt-Giertz-Weidmann [2]. In the same paper these authors show that this result is best possible in the sense that for any

p
p

satisfying

1
>
p
>
2

1 > p > 2

a function

q
q

exists in

L
p

(
0
,
∞
)

{L^p}(0,\infty )

such that

y
+
q
y

y + qy

is not separated in

L
2

(
0
,
∞
)

{L^2}(0,\infty )

.
Partial Mielnik spaces and characterization of uniformly convex spaces
Authors: Blass A. R. , Stanojevic C. V.
Published: June 2010

We characterize uniform convexity of normed linear spaces in terms of a functional inequality generalizing Clarkson’s inequality for

L
p

{L_p}

spaces. This inequality can be interpreted as saying that the unit sphere of the space carries a structure slightly weaker than a probability space in the sense of Mielnik. From this point of view, our result is analogous to an earlier characterization of inner product spaces. We also investigate briefly the abstract concept of partial probability space suggested by the main result.
Large basis dimension and metrizability
Authors: Gruenhage Gary
Published: June 2010

In this paper it is proved that if

X
X

is a regular Lindelöf space having finite large basis dimension, then

X
X

is metrizable if and only if it is a

Σ
\Sigma

-space or a

w
Δ

w\Delta

-space.
Bounds on positive integral solutions of linear Diophantine equations
Authors: Borosh I. , Treybig L. B.
Published: June 2010

Assuming the existence of a solution, we find bounds for small solutions

x
x

of the finite matrix equation

A
x
=
B

Ax = B

, where each entry of

A
,
B

A,B

is an integer, and

x
x

is a nontrivial column vector with nonnegative integer entries.
Curvature groups of a hypersurface
Authors: Goldberg Samuel I.
Published: June 2010

A cochain complex associated with the vector

1
1

-form determined by the first and second fundamental tensors of a hypersurface

M
M

in

E

n
+
1

{E^{n + 1}}

is introduced. Its cohomology groups

H
p

(
M
)

{H^p}(M)

, called curvature groups, are isomorphic with the cohomology groups of

M
M

with coefficients in a subsheaf

S

R

{\mathcal {S}_R}

of the sheaf

S

\mathcal {S}

of closed vector fields on

M
M

. If

M
M

is a minimal variety, the same conclusion is valid with

S

R

{\mathcal {S}_R}

replaced by a sheaf of harmonic vector fields. If the Ricci tensor is nondegenerate the

H
p

(
M
)

{H^p}(M)

vanish. If

S

R

≠
∅

{\mathcal {S}_R} \ne \emptyset

, and there are no parallel vector fields, locally, the

H
p

(
M
)

{H^p}(M)

are isomorphic with the corresponding de Rham groups.
Irreducibles in the Landweber Novikov algebra
Authors: Kozma Ilan
Published: June 2010

All the irreducible and reducible elements in the Landweber Novikov algebra are determined. A full set of relations mod reducibles is given.
The asymptotic expansion of the zeta-function of a compact semisimple Lie group
Authors: Cahn Robert S.
Published: June 2010

If

G
G

is a connected, simply connected, semisimple Lie group with metric given by the negative of the Killing form and zeta-function

Z
(
t
)

Z(t)

, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z left-parenthesis t right-parenthesis equals StartFraction upper V o l upper G Over left-parenthesis 4 pi t right-parenthesis Superscript dimension upper G slash 2 Baseline EndFraction exp StartAbsoluteValue delta EndAbsoluteValue squared t plus exponentially small error as t down-arrow 0 period">

Z
(
t
)
=

Vol
⁡
G

(
4
π
t
)

dim
⁡
G

/

2

exp
⁡

|

δ

|

2

t
+

exponentially small error as 

t
↓
0.

Z(t) = \frac {{\operatorname {Vol} G}} {{{{(4\pi t)}^{\dim G/2}}}}\exp |\delta {|^2}t + {\text {exponentially small error as }}t \downarrow 0.

\]
Every finite group is the automorphism group of some finite orthomodular lattice
Authors: Schrag Gerald
Published: June 2010

If

L
L

is a lattice, the automorphism group of

L
L

is denoted

Aut
⁡
(
L
)

\operatorname {Aut} (L)

. It is known that given a finite abstract group

H
H

, there exists a finite distributive lattice

D
D

such that

Aut
⁡
(
D
)
≅
H

\operatorname {Aut} (D) \cong H

. It is also known that one cannot expect to find a finite orthocomplemented distributive (Boolean) lattice

B
B

such that

Aut
⁡
(
B
)
≅
H

\operatorname {Aut} (B) \cong H

. In this paper it is shown that there does exist a finite orthomodular lattice

L
L

such that

Aut
⁡
(
L
)
≅
H

\operatorname {Aut} (L) \cong H

.
On large cyclic subgroups of finite groups
Authors: Bertram Edward A.
Published: June 2010

It is known that for each (composite)

n
n

every group of order

n
n

contains a proper subgroup of order greater than

n

1

/

3

{n^{1/3}}

. We prove that given

0
>
δ
>
1

0 > \delta > 1

, for almost all

n
⩽
x

n \leqslant x

, as

x
→
∞

x \to \infty

, every group

G
G

of order

n
n

contains a characteristic cyclic subgroup of square-free order

>

n

1
−
1

/

(
log
⁡
n
)

1
−
δ

> {n^{1 - 1/{{(\log n)}^{1 - \delta }}}}

, and provide an upper bound to the number of exceptional

n
n

. This leads immediately to a like density result for a lower bound to the number of conjugacy classes in

G
G

.
Lie and Jordan ideals in prime rings with derivations
Authors: Ahmad Mansoor
Published: June 2010

In this paper derivations on Lie and Jordan ideals of a prime ring

R
R

are studied. The following results are proved. (i) Let

R
R

be a prime ring of characteristic not

2
2

, and let

U
U

be a Lie or Jordan ideal of

R
R

. If

d
d

is a derivation defined on

U
U

, and if

a
a

is an element of the subring

T
(
U
)

T(U)

, generated by

U
U

, or

a
a

is an element of

R
R

, according as

U
U

is a Lie or Jordan ideal of

R
R

, such that

a
d
u
=
0

adu = 0

, for all

u
∈
U

u \in U

, then either

a
=
0

a = 0

or

d
u
=
0

du = 0

. (ii) Let

d
1

,

d
2

{d_1},{d_2}

be derivations defined for all

u
∈
U

u \in U

, and also for

u
2

{u^2}

and

u
3

{u^3}

if

U
U

is a Lie ideal of

R
R

, such that the iterate

d
1

d
2

{d_1}{d_2}

is also a derivation, satisfying the same conditions as

d
1

,

d
2

{d_1},{d_2}

. Let

d
1

(
u
)
∈
U

{d_1}(u) \in U

, whether

U
U

is a Lie or Jordan ideal of

R
R

. Then, at least, one of

d
1

(
u
)

{d_1}(u)

and

d
2

(
u
)

{d_2}(u)

is zero, for all

u
∈
U

u \in U

.
On groups of finite weight
Authors: Kutzko P.
Published: June 2010

A subset

S
S

of a group

G
G

is said to normally generate

G
G

if the smallest normal subgroup of

G
G

which contains

S
S

is

G
G

itself. If

α
\alpha

is minimal with the property that there exist a set of cardinality

α
\alpha

which normally generates

G
G

then

G
G

is said to have weight

α
\alpha

. It is shown that if

G
G

is a group of finite weight and if the lattice of those normal subgroups of

G
G

which are contained in the commutator subgroup

G
′

G’

of

G
G

satisfies the minimum condition then the weight of

G
G

is equal to the weight of

G

/

G
′

G/G’

.
Indecomposable Hilbert-Schmidt operators
Authors: Weiss Gary
Published: June 2010

In 1973, L. G. Brown, R. G. Douglas, and P. A. Fillmore characterized the set of all operators of the form

N
+
K

N + K

where

N
N

is a normal operator and

K
K

is a compact operator and they asked whether or not every Hilbert-Schmidt operator is the sum of a normal operator and a trace class operator. They later asked if, for every Hilbert-Schmidt operator

A
A

, there exists a normal operator

N
N

for which

A
⊕
N

A \oplus N

is the sum of a normal operator and a trace class operator. We produce a large class of Hilbert-Schmidt operators

A
A

none of which is the sum of a normal operator and a trace class operator, and furthermore, for each arbitrary operator

Q
,
A
⊕
Q

Q,A \oplus Q

is not the sum of a normal operator and a trace class operator. We then use this to show that their characterization of the operators

N
+
K

N + K

does not hold true if we replace the class of compact operators by the trace class or by any ideal

I
I

for which

I
≠

I

1

/

2

I \ne {I^{1/2}}

. In the case of the trace class, we show that even if the vanishing of the Helton and Howe trace invariant were added to the hypothesis of their characterization, it would not hold true.
Positively curved totally real minimal submanifolds immersed in a complex projective space
Authors: Ogiue Koichi
Published: June 2010

A sufficient condition for a complete totally real minimal submanifold immersed in a complex projective space to be totally geodesic is given in terms of sectional curvature.
Stone-Čech compactifications via adjunctions
Authors: Walker R. C.
Published: June 2010

The Stone-Čech compactification of a space

X
X

is described by adjoining to

X
X

continuous images of the Stone-Čech growths of a complementary pair of subspaces of

X
X

. The compactification of an example of Potoczny from [P] is described in detail.
Characterizing 𝐸₃ (the largest countable Π¹₃ set)
Authors: Guaspari David , Harrington Leo
Published: June 2010

Assume projective determinacy. For any real

α
\alpha

, let

λ
3
α

=
sup
{
ξ

|

ξ

\lambda _3^\alpha = \sup \{ \xi |\xi

is the type of a prewellordering of the reals which is

Δ
3
1

\Delta _3^1

in

α
}

\alpha \}

. Then,

C

3

{\mathcal {C}_3}

, the largest countable

Π
3
1

\Pi _3^1

set of reals, is equal to

{
α

|

∀
β
(

λ
3
α

⩽

λ
3
β

⇒
α

\{ \alpha |\forall \beta (\lambda _3^\alpha \leqslant \lambda _3^\beta \Rightarrow \alpha

is

Δ
3
1

\Delta _3^1

in

β
)
}

\beta )\}

. This result, which is true for all odd levels and generalizes a previously known characterization of

C

1

{\mathcal {C}_1}

, answers a question of Kechris.
Products of Baire spaces
Authors: Cohen Paul E.
Published: June 2010

Only the usual axioms of set theory are needed to prove the existence of a Baire space whose square is not a Baire space.
Polynomial Pell’s equations
Authors: Nathanson Melvyn B.
Published: June 2010

The polynomial Pell’s equation is

P
2

−
(

x
2

+
d
)

Q
2

=
1

{P^2} - ({x^2} + d){Q^2} = 1

, where

d
d

is an integer and the solutions

P
,
Q

P,Q

must be polynomials with integer coefficients. It is proved that this equation has nonconstant solutions if and only if

d
=
±
1
,
±
2

d = \pm 1, \pm 2

, and in these cases all solutions are determined.
The unity in rings with Gabriel and Krull dimension
Authors: Hansen F.
Published: June 2010

The main result is that all rings with right Krull dimension and divisible torsion free additive group have a right identity. Furthermore it will be proved that a simple ring with characteristic

0
0

, right Gabriel dimension

⩽
2

\leqslant 2

and finite right uniform dimension has a unity. This is false for higher Gabriel dimensions, as demonstrated by a counterexample. A similar construction gives an example for a ring with unity and Gabriel dimension, but without Krull dimension, all factor rings having finite uniform dimension.
A property of the periods of a Prym differential
Authors: Kempf George
Published: June 2010

The periods of Prym differentials can be used to prove the invariance of Picard bundles on Jacobian varieties.
Finite operators and amenable 𝐶*-algebras
Authors: Bunce John W.
Published: June 2010

In this paper we prove that the

C
∗

{C^\ast }

-algebra generated by the left regular representation of a discrete group is amenable if and only if the group is amenable. Theorems concerning finite operators and the relationship between finite operators and amenable

C
∗

{C^\ast }

-algebras are proved.
Strongly strictly cyclic weighted shifts
Authors: Hedlund James H.
Published: June 2010

An example is given of a unilateral weighted shift on complex Hilbert space which is strictly cyclic but not strongly strictly cyclic. Similar examples of weighted shifts on the sequence spaces

l
p

{l^p}

, for

1
>
p
>
∞

1 > p > \infty

, are indicated.
Some 𝑙-simple pathological lattice-ordered groups
Authors: Glass A. M. W. , McCleary Stephen H.
Published: June 2010

The purpose of this note is to explore two recently discovered lattice-ordered groups, and to establish that they are

l
l

-simple, i.e., have no proper

l
l

-ideals. These groups are “pathologically

o
−
2

o-2

-transitive” ordered permutation groups acting on the real line. The two groups exhibit even nastier properties than previous pathological groups, and serve as counterex-amples to several conjectures about

l
l

-groups.
A note on the equation 𝑥²=𝑦^{𝑞}+1
Authors: Chein E. Z.
Published: June 2010

It is proved here that the equation

x
2

=

y
q

+
1

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

has no solution in natural numbers

x
,
y

x,y

for which

q
q

is a prime

>
3

> 3

.
On asymptotic behavior for the Hawkins random sieve
Authors: Heyde C. C.
Published: June 2010

This paper is concerned with the Hawkins random sieve which is a probabilistic analogue of the sieve of Eratosthenes. Analogues of the prime number theorem and Mertens’ theorem have previously been obtained for this sieve by classical probabilistic methods. In the present paper, sharper results akin to the Riemann hypothesis are obtained by a more elegant martingale approach.
Groups with a finite number of Sylow 𝐿-subgroups
Authors: Schenkman Eugene
Published: June 2010

Two theorems on groups with a finite number of Sylow subgroups are proved.
On the density character of closed subgroups
Authors: Ginsburg John , Rajagopalan M. , Saks Victor
Published: June 2010

Answering a question posed by W. W. Comfort, G. L. Itzkowitz, and K. A. Ross, it is shown that for every infinite cardinal number

α
\alpha

there are a Hausdorff topological group

G
G

and a closed subgroup

H
H

of

G
G

such that

d
(
H
)
>
d
(
G
)
=
α

d(H) > d(G) = \alpha

(here

d
d

denotes “density character"). Specifically, for

G
G

we take the (appropriately topologized) free Abelian group

A
(
β
D
)

A(\beta D)

generated by the Stone-Čech compactification of the discrete space

D
D

with

|

D

|

=
α

|D| = \alpha

.
A compactness condition for solutions of ordinary differential equations
Authors: Jackson L. K.
Published: June 2010

It is proven that a sequence

{

y
k

(
x
)
}

\{ {y_k}(x)\}

of solutions of

y

(
n
)

=
f
(
x
,
y
,

y
′

,
…
,

y

(
n
−
1
)

)

{y^{(n)}} = f(x,y,y’, \ldots ,{y^{(n - 1)}})

with

{

y
k

(
x
)
}

\{ {y_k}(x)\}

uniformly bounded on a compact interval

[
c
,
d
]

[c,d]

has a bounded total variation sequence

{

V
c
d

(

y
k

)
}

\{ V_c^d({y_k})\}

provided solutions of the differential equation extend and

n
n

-point boundary value problems have at most one solution.
A characterization of minimal Hausdorff spaces
Authors: Herrington Larry L. , Long Paul E.
Published: June 2010

This paper gives a characterization of minimal Hausdorff spaces.
Small sets of homeomorphisms which control manifolds
Authors: Levitt Norman
Published: June 2010

Let

N
n

{N^n}

be a smooth connected paracompact manifold without boundary. A set

D
D

of self-homeomorphisms of

M
n

{M^n}

to itself is called controllable iff the semigroup generated by

D
D

acts transitively on

M
n

{M^n}

. Theorem A. There is a complete vector field

X
X

on

M
n

{M^n}

and a self-homeomorphism

H
H

so that the set

D
D

consisting of

H
,

H

−
1

H,{H^{ - 1}}

and

X
t

,
t
∈

R

{X_t},t \in {\mathbf {R}}

, is controllable. Theorem B. Let

n
≠
4

n \ne 4

and let

M
n

{M^n}

be compact and orientable. If

n
n

is even,

⩾
6

\geqslant 6

, let

M
n

{M^n}

be simply connected. If

n
≡
0
(
4
)

n \equiv 0(4)

, let signature

M
n

=
0

{M^n} = 0

. Then there is a vector field

X
X

on

M
n

{M^n}

and a self-homeomorphism

H
H

so that the set consisting of

H
,

H

−
1

H,{H^{ - 1}}

and

X
t

,
t
⩾
0

{X_t},t \geqslant 0

, is controllable.
The genus of an abstract intersection sequence
Authors: Percell Peter
Published: June 2010

An intersection sequence, denoted

I
S

IS

, is a combinatorial object associated with a normal immersion

f
:

S
1

→
M

f:{S^1} \to M

where

S
1

{S^1}

is an oriented circle and

M
M

is a closed, connected, oriented

2
2

-manifold. The genus of

I
S

IS

, denoted

γ
(
I
S
)

\gamma (IS)

, is defined to be the smallest number which is the genus of a manifold

M
M

admitting a realization

S
1

→
M

{S^1} \to M

of

I
S

IS

. A method is given for computing

γ
(
I
S
)

\gamma (IS)

from

I
S

IS

.
Quadratic zero-one laws for Gaussian measures and the distribution of quadratic forms
Authors: de Acosta Alejandro
Published: June 2010

Given a centered Gaussian measure on a vector space, the set of points where a sequence of quadratic forms converges to a constant has probability zero or one. The distribution of a quadratic form under a Gaussian measure is either point mass at zero or absolutely continuous.
On a formula in diffusion processes in population genetics
Authors: Holland Charles J.
Published: June 2010

A problem of recent interest has been the determination of the average values of certain quantities conditioned upon the fixation (or extinction) of the gene in the population. In this note we give another simple proof of formulas for the determination of these quantities.
Linear dynamical systems
Authors: Flytzanis E.
Published: June 2010

We consider a probability measure

m
m

on a Hilbert space

X
X

and a bounded linear transformation on

X
X

that preserves the measure. We characterize the linear dynamical systems

(
X
,
m
,
T
)

(X,m,T)

for the cases where either

X
X

is finite dimensional or

T
T

is unitary and we give an example where

T
T

is a weighted shift operator. We apply the results to the limit identification problem for a vector-valued ergodic theorem of A. Beck and J. T. Schwartz,

n

−
1

(

Σ
i
n

T
i

F
i

)
→

F
¯

{n^{ - 1}}(\Sigma _i^n{T^i}{F_i}) \to \overline F

a.s., where

F
i

{F_i}

is a stationary sequence of integrable

X
X

-valued random variables and

T
T

a unitary operator on

X
X

.
An ergodic theorem for noncommutative operators
Authors: Yoshimoto Takeshi
Published: June 2010

This note contains a proof of a weighted convergence theorem which includes the ordinary “noncommuting” ergodic theorem for noncommutative families of linear contraction operators.
Combinatorial functions and indecomposable cardinals
Authors: Ellentuck Erik
Published: June 2010

Combinatorial functions are used to replace indecomposable cardinals in certain types of set theoretic arguments. This allows us to extend decidability results from nonchoice set theories to set theories with a principle of linear ordering.
Simple proof that a 𝑝-adic Pascal’s triangle is 120deg rotatable
Authors: Hitotumatu Sin , Sato Daihachiro
Published: June 2010

An array of numbers which is obtained by the replacement of binomial or multinomial coefficients by their

p
p

-adic valuation is not only parallel translatable but also

120
∘

120^\circ

rotatable, without changing its configuration in the Pascal’s triangle.
On the uniqueness of conical vectors
Authors: Lepowsky J.
Published: June 2010

Certain conical vectors in induced modules for semisimple symmetric Lie algebras have been determined in a previous paper. Here the uniqueness aspects of those results are proved in a simpler way, and a refinement of one of the earlier results is given.
Continuous dependence of least squares solutions of linear boundary value problems
Authors: Kannan R. , Locker John
Published: June 2010

Let

u
λ

{u_\lambda }

be the unique least squares solution of minimal norm of the linear boundary value problem

L
u
−
λ
u
=
f

Lu - \lambda u = f

, where

L
L

is a selfadjoint differential operator in

L
2

[
a
,
b
]

{L^2}[a,b]

. Working in the Sobolev space

H
n

[
a
,
b
]

{H^n}[a,b]

, the alternative method is used to examine the continuous dependence of the

u
λ

{u_\lambda }

on the parameter

λ
\lambda

as

λ
→

λ
0

\lambda \to {\lambda _0}

, and the convergent and divergent cases are both characterized.
Asymptotic properties of linear fourth order differential equations
Authors: Ahmad Shair
Published: June 2010

Asymptotic behavior of solutions is studied for the general fourth order linear homogeneous differential equation with bounded coefficients. It is shown that roughly the same qualitative behavior as in the constant coefficient case holds.
A Korovkin type theorem on weak convergence
Authors: Meir A.
Published: June 2010

A weak convergence theorem is proved for a sequence of positive contractions on

L
1

(
0
,
1
)

{L_1}(0,1)

.
Cyclically monotone linear operators
Authors: Shiu Elias S. W.
Published: June 2010

A linear operator on a complex Hilbert space

H

\mathcal {H}

is called

n
n

-cyclically monotone if for each sequence

x
0

,

x
1

,
…
,

x

n
−
1

,

x
n

=

x
0

{x_0},{x_1}, \ldots ,{x_{n - 1}},{x_n} = {x_0}

of

n
n

elements in
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H comma normal upper Sigma Subscript j equals 0 Superscript n minus 1 Baseline upper R e left-parenthesis upper T x Subscript j Baseline minus x Subscript j plus 1 Baseline right-parenthesis greater-than-or-slanted-equals 0">

H

,

Σ

j
=
0

n
−
1

Re
⁡
(
T

x
j

−

x

j
+
1

)
⩾
0

\mathcal {H},\Sigma _{j = 0}^{n - 1}\operatorname {Re} (T{x_j} - {x_{j + 1}}) \geqslant 0

. We show that

T
T

is

n
n

-cyclically monotone if and only if

|

Arg
⁡
(
T
x
,
x
)

|

⩽
π

/

n
,
∀
x
∈

H

|\operatorname {Arg} (Tx,x)| \leqslant \pi /n,\forall x \in \mathcal {H}

. If

T
m

{T_m}

and

T
n

{T_n}

are

m
m

- and

n
n

-cyclically monotone operators, then the spectrum of the product

T
m

T
n

{T_m}{T_n}

lies in the sector

{
z
∈

C

:

|

Arg

z

|

⩽
π

/

m
+
π

/

n
}

\{ z \in {\mathbf {C}}:|\operatorname {Arg} \;z| \leqslant \pi /m + \pi /n\}

.
Products of Hermitian operators
Authors: Gray L. J.
Published: June 2010

Let

A
A

and

B
B

be selfadjoint operators on a Hilbert space. It is shown that

A
B

AB

and

B
A

BA

are not necessarily similar if their null spaces have equal dimension. If

A
A

and

B
B

are assumed to be Fredholm, then similarity can be established if additional conditions are satisfied.
Monotone decompositions of Hausdorff continua
Authors: Vought Eldon J.
Published: June 2010

A monotone, upper semicontinuous decomposition of a compact, Hausdorff continuum is admissible if the layers (tranches) of the irreducible subcontinua of

M
M

are contained in the elements of the decomposition. It is proved that the quotient space of an admissible decomposition is hereditarily arcwise connected and that every continuum

M
M

has a unique, minimal admissible decomposition

A

\mathcal {A}

. For hereditarily unicoherent continua

A

\mathcal {A}

is also the unique, minimal decomposition with respect to the property of having an arcwise connected quotient space. A second monotone, upper semicontinuous decomposition

G

\mathcal {G}

is constructed for hereditarily unicoherent continua that is the unique minimal decomposition with respect to having a semiaposyndetic quotient space. Then

G

\mathcal {G}

refines

G

\mathcal {G}

and

G

\mathcal {G}

refines the unique, minimal decomposition

L

\mathcal {L}

of FitzGerald and Swingle with respect to the property of having a locally connected quotient space (for hereditarily unicoherent continua).
Branchpoint covering theorems for confluent and weakly confluent maps
Authors: Eberhart C. A. , Fugate J. B. , Gordh G. R.
Published: June 2010

A branchpoint of a compactum

X
X

is a point which is the vertex of a simple triod in

X
X

. A surjective map

f
:
X
→
Y

f:X \to Y

is said to cover the branchpoints of

Y
Y

if each branchpoint in

Y
Y

is the image of some branchpoint in

X
X

. If every map in a class

F

\mathcal {F}

of maps on a class of compacta

C

\mathcal {C}

covers the branchpoints of its image, then it is said that the branchpoint covering property holds for

F

\mathcal {F}

on

C

\mathcal {C}

. According to Whyburn’s classical theorem on the lifting of dendrites, the branchpoint covering property holds for light open maps on arbitrary compacta. In this paper it is shown that the branchpoint covering property holds for (1) light confluent maps on arbitrary compacta, (2) confluent maps on hereditarily arcwise connected compacta, and (3) weakly confluent maps on hereditarily locally connected continua having closed sets of branchpoints. It follows that the weakly confluent image of a graph is a graph.
On the second homotopy module of two-dimensional CW complexes
Authors: Dyer M. N.
Published: June 2010

Let

X
X

be a connected

2
2

-dimensional

CW

{\text {CW}}

complex. This note reproves from a very simple point of view two classical theorems of H. Hopf relating the homology of the fundamental group

π
=

π
1

(
X
)

\pi = {\pi _1}(X)

of

X
X

and the Hurewicz map on

π
2

(
X
)

{\pi _2}(X)

. This point of view also allows the dual theorems to be proved. If

π
\pi

is a finite group, a new interpretation is given for

H
i

(
π
;
Z
)
(
i
=
2
,
3
)

{H_i}(\pi ;Z)(i = 2,3)

in terms of

π
2

(
X
)

{\pi _2}(X)

.
A Markov process which gives rise to a semigroup of expansive operators
Authors: Lin T. F.
Published: June 2010

Conditional expectation in a finite complex measure space

(
M
,

B

,
Q
)

(M,\mathcal {B},Q)

with

Q
(
M
)
=
1

Q(M) = 1

is studied. A Markov process which gives rise to a semigroup of expansive operators is constructed.
Automorphisms of the integral group ring of 𝑆_{𝑛}
Authors: Peterson Gary L.
Published: June 2010

In this paper it is shown that every normalized automorphism of the integral group ring of

S
n

{S_n}

can be written as a group automorphism followed by conjugation by a unit in the group algebra of

S
n

{S_n}

over the rationals.
Note on maximally almost periodic groups
Authors: Huang Ter Jenq
Published: June 2010

A simple proof is given that every locally compact maximally almost periodic group is unimodular.
Local coordinates around limit cycles of partial differential equations
Authors: Stokes Arnold
Published: June 2010

Given an isolated periodic solution (limit cycle)

p
=
p
(
x
,
t
)

p = p(x,t)

of a parabolic differential equation in a variable

u
=
u
(
x
,
t
)

u = u(x,t)

, local coordinates

(
s
=
s
(
t
)
,

w
=
w
(
x
,
t
)
)

(s = s(t),\;w = w(x,t))

are introduced so that

w
=
0
,

s
=
t
+

w = 0,\;s = t +

constant corresponds to

u
=
p

u = p

, and the equations for

s
,

w

s,\;w

are of the form

d
s

/

d
t
=
1
+

ds/dt = 1 +

higher-order terms, and

w
w

satisfies the variational equation for

p
p

on a subspace of codimension one. It is indicated how the method applies to ordinary differential equations, as motivation, and to hyperbolic equations, as an obvious extension.
On the hull of a linear differential operator
Authors: Humi Mayer
Published: June 2010

For a system of linear differential equations with almost periodic coefficients with respect to a group

G
G

we generalize previous results by Bochner and Sell. Furthermore we explore the relation between the solutions of the original system and those in its hull.
Limits of successive convolutions
Authors: Martin John C.
Published: June 2010

On an arbitrary compact, zero-dimensional, Abelian group, if

μ
0

,

μ
1

,
…

{\mu _0},{\mu _1}, \ldots

is a sequence of probability measures, a condition on these measures is given which is necessary and sufficient for each of the sequences

μ
t

,

μ
t

∗

μ

t
+
1

,

μ
t

∗

μ

t
+
1

∗

μ

t
+
2

,
…

{\mu _t},{\mu _t}{\ast }{\mu _{t + 1}},{\mu _t}{\ast }{\mu _{t + 1}}{\ast }{\mu _{t + 2}}, \ldots

of successive convolutions to converge to Haar measure in the weak-star topology. Some simple consequences of the theorem are noted.
On a characterization of barrelled spaces
Authors: Wilansky A.
Published: June 2010

We prove that a l.c. space

E
E

is barrelled if every closed graph linear map:

E
→
C
(
X
)

E \to C(X)

is continuous;

X
X

compact

T
2

{T_2}

. This generalizes Mahowald’s criterion, but the main contribution is the simplicity of the proof.
On the operator equation 𝑇𝑋-𝑋𝑉=𝐴
Authors: Apostol Constantin
Published: June 2010

A characterization of the operators

A
A

for which the equation

T
X
−
X
V
=
A

TX - XV = A

is solvable is given, where

T
T

is a fixed right invertible operator and

V
V

is a fixed unilateral shift.
Fixed points and iteration of a nonexpansive mapping in a Banach space
Authors: Ishikawa Shiro
Published: June 2010

The following result is shown. If

T
T

is a nonexpansive mapping from a closed convex subset

D
D

of a Banach space into a compact subset of

D
D

and

x
1

{x_1}

is any point in

D
D

, then the sequence

{

x
n

}

\{ {x_n}\}

defined by

x

n
+
1

=

2

−
1

(

x
n

+
T

x
n

)

{x_{n + 1}} = {2^{ - 1}}({x_n} + T{x_n})

converges to a fixed point of

T
T

. As a matter of fact, a theorem which includes this result is proved. Furthermore, a similar result is obtained under certain restrictions which do not imply the assumption on the compactness of

T
T

.