bluuebunny/crossref_metadata_2025 huggingface dataset:
On perturbation of unstable second order linear differential equations
Authors: Hatvani L. , Pintér L.
Published: June 2010

In connection with a conjecture of J. M. Bownds, conditions will be given on the fundamental system of the solutions of the unstable differential equation

y
+
a
(
t
)
y
=
0

y + a(t)y = 0

which assure that the differential equation

x
+
a
(
t
)
x
=
g
(
t
,
x
,

x
′

)

x + a(t)x = g(t,x,x’)

has a solution with the property
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit sup left-parenthesis StartAbsoluteValue x left-parenthesis t right-parenthesis EndAbsoluteValue plus StartAbsoluteValue x prime left-parenthesis t right-parenthesis EndAbsoluteValue right-parenthesis equals normal infinity as t right-arrow normal infinity comma">

lim
sup
(

|

x
(
t
)

|

+

|

x
′

(
t
)

|

)
=
∞

as 

t
→
∞
,

\lim \sup (|x(t)| + |x’(t)|) = \infty \quad {\text {as }}t \to \infty ,

\]
provided that

g
(
t
,
x
,

x
′

)

g(t,x,x’)

is “sufficiently small".
A sheaf representation of distributive pseudocomplemented lattices
Authors: Cornish William H.
Published: June 2010

The main result of this paper shows that a distributive pseudocomplemented lattice

(
L
;
∨
,
∧

,
∗

,
0
,
1
)

(L; \vee , \wedge {,^ \ast },0,1)

, considered as an algebra of type

⟨
2
,
2
,
1
,
0
,
0
⟩

\langle 2,2,1,0,0\rangle

, can be represented as the algebra of all global sections in a certain sheaf. The stalks are the quotient algebras

L

/

Θ
(
O
(
P
)
)

L/\Theta (O(P))

, where

P
P

is a prime ideal in

L
L

. The base space is the set of prime ideals of

L
L

equipped with the topology whose basic open sets are of the form

P
:
P

P:P

prime in

L
,

x

∗
∗

∉
P

L,{x^{ \ast \ast }} \notin P

for some

x
∈
L

x \in L

.
Remarks on H. H. Schaefer’s conditions
Authors: Dai Taen Yu
Published: June 2010

We present two algebraic conditions, each characterizing a partially ordered linear algebra as an algebra of real-valued functions.
Nonimmersions of flag manifolds
Authors: Ilori S. A.
Published: June 2010

Certain useful quadratic identities in the cohomology of classifying spaces induce quadratic equations in the cohomology of a manifold

M
M

under the classifying map for the normal bundle of

M
M

. In low dimensional flag manifolds, one can show that the quadratic equation has no root, thus establishing a nonimmersion.
On topological methods in homological algebra
Authors: Edwards David A. , Hastings Harold M.
Published: June 2010

We give an appropriate extension of the concept of “tower of surjections” to arbitrary inverse systems. We introduce a natural closed model structure (in the sense of D. Quillen) on the category of pro-(Simplicial Abelian Groups) and interpret our condition as the definition of fibrant object.
Compactness of certain homogeneous spaces of locally compact groups
Authors: Sit Kwan-Yuk Claire
Published: June 2010

Let

H
H

be the fixed points of a family of automorphisms of a locally compact group

G
G

with

G

/

H

G/H

finite invariant measure. It is proved in this paper that when the

1
1

-component of

G
G

is open,

G

/

H

G/H

is compact.
Uniqueness theorems for harmonic functions of exponential growth
Authors: Zeilberger Doron
Published: June 2010

Two uniqueness theorems for harmonic functions of exponential growth are proved. The first is a generalization to

R
n

{R^n}

of a theorem proved by Boas [1] for

R
2

{R^2}

.
Extensions of minimal flows on manifolds
Authors: Perrizo William
Published: June 2010

Group extensions of vector field flows for which the orbit closure map forms a fiber bundle are constructed for the case of minimal flows on compact manifolds and compact Lie groups. Conditions for which minimal nontoral extensions exist are studied.
On mappings contractive in the sense of Kannan
Authors: Janos Ludvik
Published: June 2010

Let

f
:
X
→
X

f:X \to X

be a continuous compact mapping of a metric space

(
X
,
d
)

(X,d)

into itself with the property that

x
,
y
∈
X

x,y \in X

and

x
≠
y

x \ne y

implies
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d left-parenthesis f left-parenthesis x right-parenthesis comma f left-parenthesis y right-parenthesis right-parenthesis greater-than one half left-bracket d left-parenthesis x comma f left-parenthesis x right-parenthesis right-parenthesis plus d left-parenthesis y comma f left-parenthesis y right-parenthesis right-parenthesis right-bracket">

d
(
f
(
x
)
,
f
(
y
)
)
>

1
2

[
d
(
x
,
f
(
x
)
)
+
d
(
y
,
f
(
y
)
)
]

d(f(x),f(y)) > \tfrac {1} {2}[d(x,f(x)) + d(y,f(y))]

. It is shown that under these conditions

f
f

has a unique fixed point and, moreover,

f
f

is a Banach contraction relative to a suitable remetrization of the space

X
X

. A similar result concerning condensing mappings is also obtained.
A generalization of the arithmetic-geometric means inequality
Authors: Fink A. M. , Jodeit Max
Published: June 2010

It is shown that the arithmetic mean of

x
1

w
1

,
…
,

x
n

w
n

{x_1}{w_1}, \ldots ,{x_n}{w_n}

exceeds the geometric mean of

x
1

,
…
,

x
n

{x_1}, \ldots ,{x_n}

unless all the

x
x

’s are equal, where

w
1

,
…
,

w
n

{w_1}, \ldots ,{w_n}

depend on

x
1

,
…
,

x
n

{x_1}, \ldots ,{x_n}

and satisfy

0
⩽

w
i

>
1

0 \leqslant {w_i} > 1

unless

x
i

=
min

x
k

{x_i} = \min {x_k}

. This inequality is then applied to an integral inequality for functions

y
y

defined on

[
0
,

∞
)

[0,\;\infty )

with

y

(
k
)

{y^{(k)}}

convex and

0
0

at

0
0

for

0
⩽
k
>
n

0 \leqslant k > n

.
Meromorphic functions and smooth analytic functions
Authors: Kaufman Robert
Published: June 2010

Meromorphic functions with many zeroes can have logarithmic derivatives that are relatively smooth. We prove this, with a new construction of smooth analytic functions with many zeroes. Our examples belong to the theory of differential fields of functions.
A strong type of regularity for the 𝑃𝑊𝐵 solution of the Dirichlet problem
Authors: Armitage D. H.
Published: June 2010

Let

H
f

{H_f}

be the Perron-Wiener-Brelot solution of the Dirichlet problem for a resolutive function

f
f

on the boundary

∂
Ω

\partial \Omega

of a bounded domain

Ω
\Omega

in

E
n

{E^n}

. A point

y
y

of

∂
Ω

\partial \Omega

will be called strongly regular if

H
f

(
x
)
→
f
(
y
)
(
x
→
y
)

{H_f}(x) \to f(y)(x \to y)

whenever

f
f

is resolutive and continuous at

y
y

. Necessary and sufficient conditions for strong regularity are given.
An expression for the first conjugate point for an 𝑛th order nonlinear differential equation
Authors: Peterson Allan C.
Published: June 2010

It is proved that the first conjugate point for an

n
n

th order nonlinear differential equation is the infimum of the first conjugate points of the corresponding variational equations.
𝑆-limits and 𝐴-summability
Authors: Bell Howard T.
Published: June 2010

Certain classes of subspaces of

m
m

are examined that are the domains of uniqueness of certain Banach type limits. These subspaces are identified as the bounded convergence domains of

A

\mathcal {A}

-summability. As a corollary it is shown that the closure of

b
s

bs

in

m
m

is

f
0

{f_0}

. Also we characterize those matrices whose convergence domains include these spaces in a manner analogous to Lorentz’ characterization of strongly regular matrices.
Linear projections which implement balayage in Fourier transforms
Authors: Shapiro George S.
Published: June 2010

Let

Λ
\Lambda

be a closed and discrete or compact subset of a second countable

LCA

{\text {LCA}}

group

G
G

and

E
E

a subset of the dual group. Balayage is said to be possible for

(
Λ
,

E
)

(\Lambda ,\;E)

if for every finite measure

μ
\mu

on

G
G

there is some measure

ν
\nu

on

Λ
\Lambda

whose Fourier transform,

ν
^

\hat \nu

, agrees on

E
E

with

μ
^

\hat \mu

. If balayage is assumed possible just when

μ
\mu

is a point measure (with the norms of all the measures

ν
\nu

bounded by some constant), then there is a bounded linear projection,

B
Λ

{B_\Lambda }

, from the measures on

G
G

onto those on

Λ
\Lambda

with

(

B
Λ

μ

)
∧

=

μ
^

{({B_\Lambda }\mu )^ \wedge } = \hat \mu

on

E
E

. An application is made to balayage in product groups.
Contractive projections in square Banach spaces
Authors: Roy Nina M.
Published: June 2010

It is proved that if

X
X

is a square space and

P
P

is a contractive projection in

X
X

, then

P
X

PX

is square; and if

X
X

is regular, then

P
X

PX

is regular. It is also shown that a regular square space is isometric to the image, under a contractive projection, of a regular (square) Kakutani

M
M

-space. These results are analogous to those obtained for other classes of

L
1

{L_1}

-preduals by Lindenstrauss and Wulbert, and in this paper their diagram of

L
1

{L_1}

-preduals is enlarged so as to include the classes of square, regular square and regular

M
M

spaces.
On the spectral type of one-parameter groups on operator algebras
Authors: Evans David E.
Published: June 2010

We show that two unitary representations of

R

{\mathbf {R}}

of very different spectral type can give rise to the same one-parameter group of

∗

^{\ast }

-automorphisms of a

C

∗

{C^{\ast }}

-algebra.
On the commutation theorem for tensor products of von Neumann algebras
Authors: Rousseau R. , van Daele Alfons , Vanheeswijck L.
Published: June 2010

It is shown that the commutation theorem for tensor products of general von Neumann algebras follows trivially from the case of von Neumann algebras with a separating and cyclic vector.
More on the “zero-two” law
Authors: Foguel S. R.
Published: June 2010

The Ornstein-Sucheston “zero-two” law for Markov operators is extended, and its proof simplified.
On the structure of the fixed-point set of a nonexpansive mapping in a Banach space
Authors: Bruck Ronald E.
Published: June 2010

If

C
C

is a closed convex subset of a reflexive, strictly convex Banach space

E
E

, and

T
:
C
→
E

T:C \to E

is a nonexpansive mapping which has a fixed-point in the interior of

C
C

, then there exists a nonexpansive mapping

T

∗

:
E
→
E

{T^{\ast }}:E \to E

whose fixed-point set in

C
C

is the fixed-point set of

T
T

.
On Geroch’s counterexample to the algebraic Hopf conjecture
Authors: Klembeck Paul F.
Published: June 2010

In this paper we present a simplified version of Geroch’s counterexample to the algebraic Hopf conjecture which requires only a knowledge of the formal symmetries of Riemannian curvature tensors.
Elementary separation by elementary expansion
Authors: Marcus Leo
Published: June 2010

If

M
,

N

M,\;N

are countable, nonisomorphic models, such that

N
N

is not saturated and

M
M

is prime, then there is an expansion

M

∗

{M^{\ast }}

of

M
M

, having the same elementary submodels as

M
M

, such that there is no expansion of

N
N

elementarily equivalent to

M

∗

{M^{\ast }}

.
On a question of Rademacher concerning Dedekind sums
Authors: Pinzur Laurence
Published: June 2010

Rademacher raised the following question concerning the ordinary Dedekind sum

s
(
h
,
k
)

s(h,k)

. If

h
1

/

k
1

{h_1}/{k_1}

and

h
2

/

k
2

{h_2}/{k_2}

are adjacent Farey fractions such that

s
(

h
1

,

k
1

)
>
0

s({h_1},{k_1}) > 0

and

s

(

h
2

,

k
2

)

>
0

s\left ( {{h_2},{k_2}} \right ) > 0

, is it necessarily true that

s
(

h
1

+

h
2

,

k
1

+

k
2

)
⩾
0

s({h_1} + {h_2},{k_1} + {k_2}) \geqslant 0

? The answer to this question is found to be no. In fact, a characterization of all pairs of adjacent Farey fractions where the answer to Rademacher’s question is no is given.
Representations of compact groups on topological vector spaces: some remarks
Authors: Johnson Russell A.
Published: June 2010

A standard theorem concerning the decomposition of a representation of a compact group on a Hilbert space

E
E

is generalized to the case when

E
E

is locally convex and quasi-complete. As a corollary, it is shown that if

E
E

is topologically irreducible, then it is finite dimensional.
A note on integrable representations
Authors: Sund Terje
Published: June 2010

It is shown that if all the coordinate functions of an induced representation

π
=

Ind
K
G

⁡
(
γ
)

\pi = \operatorname {Ind} _K^G(\gamma )

are integrable, then the representation

γ
\gamma

has the same property.
A nonlinear generalization of Bihari’s inequality
Authors: Dhongade U. D. , Deo S. G.
Published: June 2010

The aim of the present note is to prove a nonlinear generalization of the well-known integral inequality due to Bihari. This generalization is useful in obtaining pointwise estimates of solutions of nonlinear Volterra integral equations.
An application of theorems of Schur and Albert
Authors: Markham Thomas L.
Published: June 2010

Suppose

Π
n

{\Pi _n}

is the cone of

n
×
n

n \times n

positive semidefinite matrices, and

int
⁡
(

Π
n

)

\operatorname {int} ({\Pi _n})

is the set of positive definite matrices. Theorems of Schur and Albert are applied to obtain some elements of

Π
n

{\Pi _n}

and

int
⁡
(

Π
n

)

\operatorname {int} ({\Pi _n})

. Then an analogue of Albert’s theorem is given for

M
M

-matrices, and finally a generalization is given for matrices of class

P
P

.
Projective moduli of certain quotient rings
Authors: Moore Nadine
Published: June 2010

The author considers some properties of extension rings

B
B

of a ring

A
A

that satisfy the condition that every maximal ideal of

B
B

is an extension of some ideal of

A
A

. Such extensions have been used by D. Lissner, K. Lønsted, N. Moore, and A. Simis to obtain rings for which the projective moduli are arbitrarily less than the dimension of the maximal spectra. It is shown that families of prime ideals of maximal type can be used to construct such extensions.
Weak containment and weak Frobenius reciprocity
Authors: Gootman Elliot C.
Published: June 2010

We study weak containment relations between unitary representations of a group

G
G

and a closed normal subgroup

K
K

by exploiting a property of

G
G

-ergodic quasi-invariant measures on the primitive ideal space of

K
K

. By this means, we prove that every irreducible representation of

G
G

is weakly contained in a representation induced from an irreducible representation of

K
K

if the quotient group

G

/

K

G/K

is amenable; and that the pair

(
G
,
K
)

(G,K)

satisfies a weak Frobenius reciprocity property if and only if

G

/

K

G/K

is amenable and

G
G

acts minimally on the primitive ideal space of

K
K

. If

G

/

K

G/K

is compact,

G
G

acts minimally if and only if the primitive ideal space of

K
K

is

T
1

{T_1}

.
Remarks on the Gauss-Lucas theorem in higher dimensional space
Authors: Goodman A. W.
Published: June 2010

A recent paper by J. B. Diaz and Dorothy Browne Shaffer extends the Gauss-Lucas Theorem to

n
n

-dimensional Euclidean space. The authors leave open certain natural questions concerning the existence of “zeros of the derivative". This paper answers three such questions, and suggests several other questions for further investigation.
The triangular representation of C. Apostol
Authors: Bouldin Richard
Published: June 2010

The existence of the triangular representation of C. Apostol is proved by arguments alternative to the original proof. The briefer development is, hopefully, more perspicuous than the original development.
Functional characterizations of the field of values and the convex hull of the spectrum
Authors: Johnson Charles R.
Published: June 2010

The only compact, convex set-valued homogeneous and translatable function of square complex matrices which is an indicator function for the matrices with positive semidefinite real part is the classical field of values. An analogous characterization of the convex hull of the spectrum is given.
A note on the spaces 𝐿_{𝑝} for 0<𝑝≤1
Authors: Kalton N. J.
Published: June 2010

It is shown that there is no Hausdorff vector topology

ρ
\rho

on the space

L
p

{L_p}

(where

0
>
p
⩽
1

0 > p \leqslant 1

) such that the unit ball of

L
p

{L_p}

is relatively compact for the topology

ρ
\rho

.
On maximality of Gorenstein sequences
Authors: Marinari Maria Grazia
Published: June 2010

It is well known that if

A
A

is a Gorenstein ring, then every ideal generated by a regular sequence

x

⊂
A

{\text {x}} \subset A

has irreducible (minimal) primary components. This feature led us to define a Gorenstein sequence of a ring

A
A

to be any ordered regular sequence

x

=
{

x
1

,
…
,

x
r

}
⊂
A

{\text {x}} = \{ {x_1}, \ldots ,{x_r}\} \subset A

such that for every

i
∈
{
1
,
…
,
r
}

i \in \{ 1, \ldots ,r\}

the ideal

(

x
1

,
…
,

x
i

)

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

has irreducible minimal primary components. We showed for Gorenstein sequences (

G

{\mathbf {G}}

-sequences for short) some parallels of well-known properties of regular sequences and moreover by means of

G

{\mathbf {G}}

-sequences we gave the following natural characterization of local Gorenstein rings: “A local ring

(
A
,

m

)

(A,\;\mathfrak {m})

is Gorenstein iff

m

\mathfrak {m}

contains a

G

 - sequence of length = 

K

 - 

dim
⁡
A

{\mathbf {G}}{\text { - sequence of length = }}K{\text { - }}\dim A

". In this note we are going to give some information about “maximality” of

G

{\mathbf {G}}

-sequences in a local ring

A
A

, producing sufficient conditions on

A
A

in order that the maximal

G

{\mathbf {G}}

-sequences of

A
A

all have the same length, i.e. in order to give a “good” definition of

G

{\mathbf {G}}

-depth

A
A

. Furthermore, we will state some results about the

G

{\mathbf {G}}

-depth behavior with respect to local flat ring homomorphisms.
An F. and M. Riesz type theorem for the unit ball in complex 𝑁-space
Authors: Kolaski Clinton
Published: June 2010

Let

M
B

{M_B}

denote Lebesgue measure on the open unit ball,

B
B

, in complex

N
N

-space and let

M
(
B
)

M(B)

denote the space of Borel measures on

B
B

. The volume Poisson kernel

χ
:

B
¯

×
B
→
(
0
,
∞
)

\chi :\overline B \times B \to (0,\infty )

is defined and then we prove Theorem. If

μ
∈
M
(
B
)

\mu \in M(B)

and if

μ
♯

(
w
)
=

∫
B

χ
(
z
,
w
)
d
μ
(
z
)

{\mu ^\sharp }(w) = {\smallint _B}\chi (z,w)d\mu (z)

is pluriharmonic in

B
B

, then

μ
♯

∈

L
′

(

M
B

)

{\mu ^\sharp } \in L’({M_B})

and

μ
=

μ
♯

⋅

M
B

\mu = {\mu ^\sharp } \cdot {M_B}

.
A Künneth formula for coproducts of simplicial groups
Authors: Katz Elyahu
Published: June 2010

Berstein [1] (and later Clark and Smith [9]) proved that over a field the homology of the coproduct of two simplicial groups is isomorphic to the coproduct of the augmented algebras obtained by taking the homology of these groups. It turns out that this result does not hold in general over the integers. However, we obtain a Kunneth like formula in which the homology of the coproduct of two groups is expressed in terms of the homology of the groups.
Determinateness and partitions
Authors: Prikry Karel
Published: June 2010

It is proved that the axiom of determinateness of Mycielski and Steinhaus for games in which players alternate in writing reals implies that

ω
→
(
ω

)
2
ω

\omega \to (\omega )_2^\omega

(i.e. for every partition of infinite sets of natural numbers into two classes there is an infinite set such that all its infinite subsets belong to the same class).
Multipliers on the rigid motions of the plane and their relations to multipliers on direct products
Authors: Rubin Richard L.
Published: June 2010

Hörmander multiplier theorems on

M
(
2
)

M(2)

and

R
2

×
T

{R^2} \times T

are developed. The relations between these theorems are studied. Applications to fractional Laplace operators and

R
3

{R^3}

Riesz transforms are given.
A class of functional equations and Mielnik probability spaces
Authors: Guccione S. J. , Stanojević Č. V.
Published: June 2010

Let

S
S

be the unit sphere of a normed real linear space

N
N

and let

(
S
,
p
)

(S,p)

be a Mielnik space of dimension two. For

p
(
x
,
y
)
=
f
(

|

|

x
+
y

|

|

)
,

x
,

y
∈
S

p(x,y) = f(||x + y||),\;x,\;y \in S

, where

f
f

is a continuous, strictly increasing function from

[
0
,
2
]

[0,2]

onto

[
0
,
1
]

[0,1]

, it has been shown that

(
S
,
p
)

(S,p)

being two dimensional is equivalent to

N
N

being an inner product space. In some polarization problems modeled on the unit sphere of an inner product space, the transition probability

p
(
x
,
y
)

p(x,y)

may not be as well behaved as

p
(
x
,
y
)
=
f
(

|

|

x
+
y

|

|

)

p(x,y) = f(||x + y||)

. In order to provide a more suitable setting, we have constructed wide classes of two-dimensional transitional probability spaces

(
S
,
p
)

(S,p)

, all having the same set of bases

B

\mathcal {B}

, with

p
=
ϕ
∘
f

p = \phi \circ f

where

ϕ
\phi

is a solution of a certain functional equation. In particular, for

p
(
x
,
y
)
=

|

|

x
+
y

|

|

2

/

4

p(x,y) = ||x + y||^2/4

, we answer a question due to B. Mielnik.
Unitaries and partial isometries in a real 𝑊*-algebra
Authors: Ekman Kenneth E.
Published: June 2010

The group of unitaries in a real

W

∗

{W^{\ast }}

-algebra without a finite type I direct summand is connected. This fact is used to characterize the components of the set of partial isometries in such algebras.
An existence theorem for boundary value problems of nonlinear ordinary differential equations
Authors: Klaasen Gene A.
Published: June 2010

Let

f
f

be continuous on

(
a
,
b
)
×

R
n

(a,b) \times {R^n}

and suppose solutions of initial value problems for

y

(
n
)

=
f
(
t
,
y
,
…
,

y

(
n
−
1
)

)

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

exist on

(
a
,
b
)

(a,b)

. Relaxing the assumption that solutions of initial value problems are unique, global existence of solutions of the boundary value problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Superscript left-parenthesis n right-parenthesis Baseline equals f left-parenthesis t comma y comma ellipsis comma y Superscript left-parenthesis n minus 1 right-parenthesis Baseline right-parenthesis comma y left-parenthesis t Subscript i Baseline right-parenthesis equals alpha Subscript i Baseline for 1 less-than-or-slanted-equals i less-than-or-slanted-equals n comma">

y

(
n
)

=
f
(
t
,
y
,
…
,

y

(
n
−
1
)

)
,
y
(

t
i

)
=

α
i

for 

1
⩽
i
⩽
n
,

{y^{(n)}} = f(t,y, \ldots ,{y^{(n - 1)}}),y({t_i}) = {\alpha _i}\quad {\text {for }}1 \leqslant i \leqslant n,

\]
is established assuming uniqueness of solutions of these problems and a compactness property of solutions of the differential equation.
The quadratic Putnam-Fuglede theorem
Authors: Daughtry John
Published: June 2010

We give a new sufficient condition for an invariant subspace for a normal operator to be reducing. This result constitutes a generalization of the Putnam-Fuglede Theorem to quadratic operator equations.
A mean ergodic theorem for families of contractions in Hilbert space
Authors: Blum J. R. , Reich J. I.
Published: June 2010

Let

G
G

be an LCA group and

h
h

a Hilbert space, and let

T
(
g
)

T(g)

be a function on

G
G

into the contractions on

H
H

. Let

{

σ
n

}

\{ {\sigma _n}\}

be a sequence of probability measures on

G
G

. Under suitable conditions on

T
(
g
)

T(g)

and the sequence

{

σ
n

}

\{ {\sigma _n}\}

we prove the strong convergence of the sequence

T
n

=
∫
T
(
g
)

σ
n

(
d
g
)

{T_n} = \smallint T(g){\sigma _n}(dg)

. In certain cases we identify the limiting operator.
Tauberian operators on Banach spaces
Authors: Kalton Nigel , Wilansky Albert
Published: June 2010

A Tauberian operator:

E
→
F

E \to F

(Banach spaces) is one which satisfies

T
g
∈
F
,
g
∈
E

Tg \in F,g \in E

imply

g
∈
E

g \in E

. The action of such operators and their pre-images on compact sets is studied in order to compare “Tauberian” with “weakly compact", an opposite property. Properties related to range closed are introduced which force operators with Tauberian-like properties to be Tauberian. Classes of spaces appear for which Tauberian is equivalent to semi-Fredholm. One example of this is the historical reason for the definition of these operators.
On the zeros of generalized axially symmetric potentials
Authors: McCoy Peter A.
Published: June 2010

Generalized axially symmetric potentials may be expanded as Fourier-Jacobi series in terms of the complete system

r
k

C
k

n

/

2
−
1

(
cos
⁡
θ
)

{r^k}C_k^{n/2 - 1}(\cos \theta )

on axisymmetric regions

Ω
⊂

E
n

(
n
⩾
3
)

\Omega \subset {E^n}(n \geqslant 3)

about the origin. The values of these potentials are characterized by the nonnegativity of sequences of determinants drawn from the Fourier coefficients in a manner analogous to the characterization of the values of analytic functions of one complex variable by the theorems of Carathéodory-Toeplitz and Schur.
A note on generalized resolvents for ordinary differential operators
Authors: Lee Sung J.
Published: June 2010

We give an explicit construction for the kernel of an arbitrary generalized resolvent for an ordinary symmetric differential operator. In particular, this avoids the use of approximation of selfadjoint operators on compact intervals. We also discuss integrability of functions which are adjoint to certain fundamental solutions.
Isoperimetric inequalities for a nonlinear eigenvalue problem
Authors: Bandle Catherine
Published: June 2010

An estimate for the spectrum of the two-dimensional eigenvalue problem

Δ
u
+
λ

e
u

=
0

\Delta u + \lambda {e^u} = 0

in

D
(
λ
>
0
)
,
u
=
0

D(\lambda > 0),u = 0

on

∂
D

\partial D

is derived, and upper and lower pointwise bounds for the solutions are constructed.
Integral closures of uncountable commutative regular rings
Authors: Lipshitz L.
Published: June 2010

Necessary and sufficient conditions are given for a commutative regular ring to have a prime integrally closed extension.
Semigroup of operators on dual Banach spaces
Authors: Lau Anthony To Ming
Published: June 2010

In this paper, we give a short and simple proof to a more general version of a recent result of Yeadon for semigroups of weak

∗

^{\ast }

-continuous operators on a dual Banach space. Our result has application to amenable groups and property

P

{\text {P}}

of a von Neumann algebra.
A homomorphism theorem for finite graphs
Authors: Aschbacher Michael
Published: June 2010

A homomorphism theorem for finite graphs is established and several group theoretical applications are derived. The proof is taken from a lemma of B. Fischer in his paper classifying groups generated by

3
3

-transpositions.
A short proof of the Fourier inversion formula
Authors: Robert Alain
Published: June 2010

We give an elementary proof of the Fourier inversion formula (on the real line) based on the Poisson summation formula.
The rate of growth of the denominators in the Oppenheim series
Authors: Galambos János
Published: June 2010

A Borel-Cantelli lemma is proved for a sequence of functions of the denominators in the Oppenheim expansion of real numbers. This is then applied to the study of the rate of growth of the denominators in the above series. The laws obtained are almost sure type, that is, valid for (Lebesgue) almost all

x
x

in the unit interval. The results are new even for the classical expansions of Engel, Sylvester and Cantor (product).
Pluriharmonic functions in balls
Authors: Rudin Walter
Published: June 2010

It is proved that a function is pluriharmonic in the open unit ball of

C

n

{{\mathbf {C}}^n}

if and only if it is harmonic with respect to both the ordinary Laplacian and the invariant Laplace-Beltrami operator.
On derivation algebras of Malcev algebras
Authors: Stitzinger Ernest L.
Published: June 2010

It is shown that if A is a Malcev algebra over a field of characteristic 0, then A is semisimple if and only if the derivation algebra

D

(
A
)

\mathfrak {D}(A)

is semisimple. It is then shown that A is semisimple if and only if

A
∗

=

L

(
A
)
+

D

(
A
)

{A^\ast } = \mathfrak {L}(A) + \mathfrak {D}(A)

is semisimple, where

L

(
A
)

\mathfrak {L}(A)

is the Lie multiplication algebra of A.
On a problem of Bruckner and Ceder concerning the sum of Darboux functions
Authors: Lipiński J. S.
Published: June 2010

The main purpose of this paper is to show that for some continuous function f and any preassigned, countable and dense set D of real numbers there exists a measurable function d which takes on every real value in every interval such that the range of

f
+
d

f + d

is D.
New subclasses of the class of close-to-convex functions
Authors: Chichra Pran Nath
Published: June 2010

In this paper we introduce new subclasses of the class of close-to-convex functions. We call a regular function

f
(
z
)

f(z)

an alpha-close-to-convex function if

(
f
(
z
)

f
′

(
z
)

/

z
)
≠
0

(f(z)f’(z)/z) \ne 0

for z in E and if for some nonnegative real number

α
\alpha

there exists a starlike function

ϕ
(
z
)
=
z
+
⋯

\phi (z) = z + \cdots

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R e left-bracket left-parenthesis 1 minus alpha right-parenthesis StartFraction z f prime left-parenthesis z right-parenthesis Over phi left-parenthesis z right-parenthesis EndFraction plus alpha StartFraction left-parenthesis z f prime left-parenthesis z right-parenthesis right-parenthesis prime Over phi prime left-parenthesis z right-parenthesis EndFraction right-bracket greater-than 0">

Re

[

(
1
−
α
)

z

f
′

(
z
)

ϕ
(
z
)

+
α

(
z

f
′

(
z
)

)
′

ϕ
′

(
z
)

]

>
0

\operatorname {Re} \;\left [ {(1 - \alpha )\frac {{zf’(z)}}{{\phi (z)}} + \alpha \frac {{(zf’(z))’}}{{\phi ’(z)}}} \right ] > 0

\]
for z in E. We have proved that all alpha-close-to-convex functions are close-to-convex and have obtained a few coefficient inequalities for

α
\alpha

-close-to-convex functions and an integral formula for constructing these functions. Let

F

α

{\mathfrak {F}_\alpha }

be the class of regular and normalised functions

f
(
z
)

f(z)

which satisfy

Re

(

f
′

(
z
)
+
α
z
f
(
z
)
)
>
0

\operatorname {Re} \;(f’(z) + \alpha zf(z)) > 0

for z in E.

f
(
z
)
∈

F

α

f(z) \in {\mathfrak {F}_\alpha }

gives

Re
⁡

f
′

(
z
)
>
0

\operatorname {Re} f’(z) > 0

for z in E provided

Re
⁡
α
⩾
0

\operatorname {Re} \alpha \geqslant 0

. A sharp radius of univalence of the class of functions

f
(
z
)

f(z)

for which

z

f
′

(
z
)
∈

F

α

zf’(z) \in {\mathfrak {F}_\alpha }

has also been obtained.
Linear groups as stabilizers of sets
Authors: Isaacs I. M.
Published: June 2010

A finite linear group over an infinite field is necessarily the full setwise stabilizer of some finite subset of the corresponding vector space. If the group is absolutely irreducible, the set can be taken to be a single orbit.
On some topological problems arising from maps of contractive type
Authors: Tan Kok Keong , Wong Chi Song
Published: June 2010

Characterizations of sequentially complete and of countably compact subsets of a uniform space are obtained. Applications to fixed point theory and coincidence theory are given.
Boundedness of value regions and convergence of continued fractions
Authors: Roach F. A.
Published: June 2010

If the elements of a continued fraction are restricted to lie within some region E of the complex plane, it is quite often possible to determine, with very little difficulty, where the approximants of the continued fraction lie. Generally, it is more difficult to determine whether every continued fraction with elements from this set E is convergent. In this paper, we give some results which, in certain cases, reduce the question of convergence to the question of whether the set of approximants is bounded.
Fibrations over a CWh-base
Authors: Schön Rolf
Published: June 2010

This note provides a short argument for the known fact that the total space of a fibration has the homotopy type of a CW-complex if base and fiber have.
A graph-theoretical generalization of a Cantor theorem
Authors: Fajtlowicz Siemion
Published: June 2010

In 1962 Gleason and Dilworth found a poset-theoretical generalization of the Cantor Theorem concerning the cardinality of power sets. In the present paper we prove a graph-theoretical generalization of both theorems.
Remark on measurable graph theorems
Authors: Lehn Jürgen
Published: June 2010

A theorem generalizing some known measurable graph theorems is proved.
A generalization of a theorem of S. N. Bernstein
Authors: Chandler J. D.
Published: June 2010

A rational approximation scheme is exhibited for a set

Δ
\Delta

which consists of a finite union of compact subintervals of the real line. This rational approximation scheme provides a characterization of the analytic functions on

Δ
\Delta

which generalizes S. N. Bernstein’s characterization of the analytic functions on

[
−
1
,
1
]

[ - 1,1]

.
Multipliers on dual 𝐴*-algebras
Authors: Tomiuk B. J.
Published: June 2010

Let A be an

A
∗

{A^\ast }

-algebra which is a dense

∗

^\ast

-ideal of a

B
∗

{B^\ast }

-algebra

A

\mathfrak {A}

. We use tensor products and the algebra

M
l

(
A
)

{M_l}(A)

of left multipliers on A to obtain a characterization of duality in A. We show, moreover, that if A is dual then

M
l

(
A
)

{M_l}(A)

is algebra isomorphic to the second conjugate space

A

∗
∗

{\mathfrak {A}^{\ast \ast }}

of

A

\mathfrak {A}

when

A

∗
∗

{\mathfrak {A}^{\ast \ast }}

is given Arens product.
A note on pseudocompact spaces and 𝑘_{𝑅}-spaces
Authors: Kato Akio
Published: June 2010

Utilizing the Stone-Čech compactification of an uncountable discrete space, we construct a pseudocompact space X which belongs to Frolík’s class

P

∗

{\mathfrak {P}^ \ast }

but

k
R

X

{k_R}X

is not pseudocompact.
A two-cardinal theorem and a combinatorial theorem
Authors: Shelah Saharon
Published: June 2010

We prove a new two-cardinal theorem, e.g.

(

ℵ
ω

,

ℵ
0

)
→
(

2

ℵ
0

,

ℵ
0

)

({\aleph _\omega },{\aleph _0}) \to ({2^{{\aleph _0}}},{\aleph _0})

. For this we prove a combinatorial theorem.
On a conjecture of Graham concerning greatest common divisors
Authors: Weinstein Gerald
Published: June 2010

Let

a
1

>

a
2

>
⋯
>

a
n

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

be a finite sequence of positive integers. R. L. Graham has conjectured that

max

i
,
j

{

a
i

/

(

a
i

,

a
j

)
}
⩾
n

{\max _{i,j}}\{ {a_i}/({a_i},{a_j})\} \geqslant n

. The following are proved: (1) If

a
l

=
p

{a_l} = p

, a prime, for some l and

p
≠
(

a
i

+

a
j

)

/

2
,
1
⩽
i
>
j
⩽
n

p \ne ({a_i} + {a_j})/2,1 \leqslant i > j \leqslant n

, then the conjecture holds. (2) Given a finite sequence of positive integers

k
1

>

k
2

>
⋯
>

k
m

{k_1} > {k_2} > \cdots > {k_m}

, where

k
I

=
p

{k_I} = p

, a prime, for some I and

k
m

>
n

{k_m} > n

, consider the set of all positive integral multiples of

k
1

,

k
2

,
…
,

k
m

{k_1},{k_2}, \ldots ,{k_m}

which are

>
n

> n

. Denote these multiples by

a
1

>

a
2

>
⋯
>

a
q

{a_1} > {a_2} > \cdots > {a_q}

. Define

P
n

{P_n}

to be a set consisting of the integers

a
1

>

a
2

>
⋯
>

a
q

>

a

q
+
1

>
⋯
>

a

q
+
r

{a_1} > {a_2} > \cdots > {a_q} > {a_{q + 1}} > \cdots > {a_{q + r}}

where r is maximal, such that

n
⩽

a

q
+
1

n \leqslant {a_{q + 1}}

and

max

i
,
j

{

a
i

/

(

a
i

,

a
j

)
}
>
n

{\max _{i,j}}\{ {a_i}/({a_i},{a_j})\} > n

. Thus
\[

P
n

=

P
n

(

k
1

,

k
2

,
…
,

k
m

)
.

{P_n} = {P_n}({k_1},{k_2}, \ldots ,{k_m}).

\]
Then (a)

|

P
n

|

⩾
n

|{P_n}| \geqslant n

for at most finitely many n. (b) If

|

P
n

|

>
n

|{P_n}| > n

for

n
>

l
.
c
.
m
.

{

k
1

,

k
2

,
…
,

k
m

}

n > \mathrm {l.c.m.}\{ k_1, k_2, \ldots , k_m \}

then

|

P
n

|

>
n

|{P_n}| > n

for all positive integers n.
The Markoff spectrum and minima of indefinite binary quadratic forms
Authors: Gbur Mary E.
Published: June 2010

Recently T.W. Cusick has shown that the Lagrange spectrum is the closure of the Markoff values of completely periodic doubly infinite sequences. In this note it is proved that if the Markoff value of a sequence M is attained at least three times, then M is a completely periodic sequence. We also show that if the minimum of an indefinite binary quadratic form is attained at least three times, then the form is equivalent to a rational form.
On the Schnirelmann density of the 𝑘-free integers
Authors: Diananda P. H. , Subbarao M. V.
Published: June 2010

Let

Q
k

(
n
)

{Q_k}(n)

be the number of k-free integers

⩽
n

\leqslant n

and

d
(

Q
k

)

d({Q_k})

the Schnirelmann density of the k-free integers. If

k
⩾
5

k \geqslant 5

, it is shown that

Q
k

(
n
)

/

n
=
d
(

Q
k

)

{Q_k}(n)/n = d({Q_k})

for some n satisfying

6
k

/

2
⩽
n
>

6
k

{6^k}/2 \leqslant n > {6^k}

and certain other properties, and that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d left-parenthesis upper Q Subscript k Baseline right-parenthesis greater-than-or-slanted-equals 1 minus 2 Superscript negative k Baseline minus 3 Superscript negative k Baseline minus 5 Superscript negative k Baseline plus left-parenthesis 3 Superscript negative k Baseline plus 2 dot 5 Superscript negative k Baseline right-parenthesis left-parenthesis 6 Superscript k Baseline minus 3 Superscript k Baseline plus 1 right-parenthesis Superscript negative 1 Baseline period">

d
(

Q
k

)
⩾
1
−

2

−
k

−

3

−
k

−

5

−
k

+
(

3

−
k

+
2
⋅

5

−
k

)

(

6
k

−

3
k

+
1

)

−
1

.

d({Q_k}) \geqslant 1 - {2^{ - k}} - {3^{ - k}} - {5^{ - k}} + ({3^{ - k}} + 2 \cdot {5^{ - k}}){({6^k} - {3^k} + 1)^{ - 1}}.

\]

d
(

Q
k

)

d({Q_k})

and the n for which

Q
k

(
n
)

/

n
=
d
(

Q
k

)

{Q_k}(n)/n = d({Q_k})

are found for

7
⩽
k
⩽
12

7 \leqslant k \leqslant 12

.
Two notes on nilpotency and standard algebras
Authors: Stitzinger Ernest L.
Published: June 2010

Two results on nilpotency which are known to hold for Jordan and alternative algebras are shown to hold for standard algebras as well. These results in the known cases arose when developing a Cartan theory for these classes.
On the heights of group characters
Authors: Haggarty R. J.
Published: June 2010

For a finite p-soluble group G we derive a bound on the heights of the irreducible complex characters of G lying in a p-block B. This bound depends on the prime p and the exponent d of a defect group of B. We show by examples that this bound is of the right order of magnitude.
Convergent sequences of 𝜏-smooth measures
Authors: Khurana Surjit Singh
Published: June 2010

It is proved that every

τ
\tau

-smooth, group-valued Borel measure on a regular Hausdorff space is regular; also it is proved that if a sequence of

τ
\tau

-smooth Borel measures on a regular Hausdorff space is convergent for regular open sets, then it is convergent for all Borel sets. For a completely regular Hausdorff space, it is proved that if a sequence of Borel

τ
\tau

-smooth measures is convergent for exactly open sets then it is convergent for all Borel sets.
A simple proof of a theorem of Ky Fan
Authors: Sehgal V. M.
Published: June 2010

A simple proof of a fixed point theorem of Ky Fan in locally convex spaces is constructed by means of his fixed point theorem for set valued functions.
A note on uniform paracompactness
Authors: Rice Michael D.
Published: June 2010

Three possible definitions for the paracompactness of a uniform space are presented and shown to be equivalent. A locally compact uniform space is shown to be uniformly paracompact if and only if it is uniformly locally compact. The set of points of a uniformly paracompact metric space that admit no compact neighborhood is shown to be compact; hence a metric topological group is uniformly paracompact if and only if it is locally compact.
An extension of Carlitz’s bipartition identity
Authors: Andrews George E.
Published: June 2010

Carlitz’s bipartition identity is extended to a multipartite partition identity by the introduction of the summatory maximum function:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s m a x left-parenthesis n 1 comma n 2 comma ellipsis comma n Subscript r Baseline right-parenthesis equals n 1 plus n 2 plus midline-horizontal-ellipsis plus n Subscript r Baseline minus left-parenthesis r minus 1 right-parenthesis min left-parenthesis n 1 comma n 2 comma ellipsis comma n Subscript r Baseline right-parenthesis period">

smax
⁡
(

n
1

,

n
2

,
…
,

n
r

)
=

n
1

+

n
2

+
⋯
+

n
r

−
(
r
−
1
)
min
(

n
1

,

n
2

,
…
,

n
r

)
.

\operatorname {smax}(n_1, n_2, \ldots , n_r) = n_1 + n_2 + \cdots + n_r - (r - 1)\min (n_1, n_2, \ldots , n_r).

\]
Let

π
0

(

n
1

,

n
2

,
…
,

n
r

)

\pi _0(n_1, n_2, \ldots , n_r)

denote the number of partitions of

(

n
1

,

n
2

,
…
,

n
r

)

({n_1},{n_2}, \ldots ,{n_r})

in which the minimum coordinate of each part is not less then the summatory maximum of the next part. Let

π
1

(

n
1

,

n
2

,
…
,

n
r

)

{\pi _1}({n_1},{n_2}, \ldots ,{n_r})

denote the number of partitions of

(

n
1

,

n
2

,
…
,

n
r

)

({n_1},{n_2}, \ldots ,{n_r})

in which each part has one of the

2
r
−
1

2r - 1

forms:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis a plus 1 comma a comma a comma ellipsis comma a right-parenthesis comma left-parenthesis a comma a plus 1 comma a comma ellipsis comma a right-parenthesis comma ellipsis comma left-parenthesis a comma a comma a comma ellipsis comma a plus 1 right-parenthesis comma left-parenthesis r a plus 2 comma r a plus 2 comma ellipsis comma r a plus 2 right-parenthesis comma left-parenthesis r a plus 3 comma r a plus 3 comma ellipsis comma r a plus 3 right-parenthesis comma ellipsis comma left-parenthesis r a plus r comma r a plus r comma ellipsis comma r a plus r right-parenthesis">

(
a
+
1
,
a
,
a
,
…
,
a
)
,
(
a
,
a
+
1
,
a
,
…
,
a
)
,
…
,
(
a
,
a
,
a
,
…
,
a
+
1
)
,
(
r
a
+
2
,
r
a
+
2
,
…
,
r
a
+
2
)
,
(
r
a
+
3
,
r
a
+
3
,
…
,
r
a
+
3
)
,
…
,
(
r
a
+
r
,
r
a
+
r
,
…
,
r
a
+
r
)

(a + 1,a,a, \ldots ,a),(a,a + 1,a, \ldots ,a), \ldots ,(a,a,a, \ldots ,a + 1),(ra + 2,ra + 2, \ldots ,ra + 2),(ra + 3,ra + 3, \ldots ,ra + 3), \ldots ,(ra + r,ra + r, \ldots ,ra + r)

. Theorem:

π
0

(

n
1

,
…
,

n
r

)
=

π
1

(

n
1

,
…
,

n
r

)

{\pi _0}({n_1}, \ldots ,{n_r}) = {\pi _1}({n_1}, \ldots ,{n_r})

.
Ranks of matrices over Ore domains
Authors: Bedoya H. , Lewin J.
Published: June 2010

Let R be a Noetherian Ore domain. Then rank M = inner rank M for every matrix M over R if and only if R is projective-free of global dimension at most 2.
On Ahlfors’ “second fundamental inequality”
Authors: Jenkins James A. , Oikawa Kôtaro
Published: June 2010

The authors have earlier given a generalization of Ahlfors’ “Second Fundamental Inequality” which reduced significantly the restrictions on the domains involved but retained the condition that certain determining functions for the domain be of bounded variation. In this paper it is shown that the condition of bounded variation can be replaced by that of finite

2

/

3

2/3

-variation on any closed interval and an appropriate new formal expression is given.
Comparison of two types of order convergence with topological convergence in an ordered topological vector space
Authors: May Roger W. , McArthur Charles W.
Published: June 2010

Birkhoff and Peressini proved that if

(
X
,

T

)

(X,\mathcal {T})

is a complete metrizable topological vector lattice, a sequence converges for the topology

T

\mathcal {T}

iff the sequence relatively uniformly star converges. The above assumption of lattice structure is unnecessary. A necessary and sufficient condition for the conclusion is that the positive cone be closed, normal, and generating. If, moreover, the space

(
X
,

T

)

(X,\mathcal {T})

is locally convex, Namioka [11, Theorem 5.4] has shown that

T

\mathcal {T}

coincides with the order bound topology

T

b

{\mathcal {T}_b}

and Gordon [4, Corollary, p. 423] (assuming lattice structure and local convexity) shows that metric convergence coincides with relative uniform star convergence. Omitting the assumptions of lattice structure and local convexity of

(
X
,

T

)

(X,\mathcal {T})

it is shown for the nonnecessarily local convex topology

T

ru

{\mathcal {T}_{{\text {ru}}}}

that

T

b

⊂

T

ru

=

T

{\mathcal {T}_{\text {b}}} \subset {\mathcal {T}_{{\text {ru}}}} = \mathcal {T}

and

T

b

=

T

ru

=

T

{\mathcal {T}_{\text {b}}} = {\mathcal {T}_{{\text {ru}}}} = \mathcal {T}

when

(
X
,

T

)

(X,\mathcal {T})

is locally convex.
Accessible points in the spectrum of a Fréchet algebra
Authors: Carpenter Ronn L.
Published: June 2010

For a point in the spectrum of a uniform Fréchet algebra, several types of accessibility are defined. The relationship between accessible points and peak points is studied as is the relationship of local accessibility to global accessibility.
The character of 𝜔₁ in first countable spaces
Authors: Fleissner William G.
Published: June 2010

We define a cardinal function

χ
(
P
,
Q
)

\chi (P,Q)

, where P and Q are properties of topological spaces. We show that it is consistent and independent that

χ
(

ω
1

,

first

countable

)
=

ω
1

\chi ({\omega _1},\;{\text {first}}\;{\text {countable}}) = {\omega _1}

.
More characterizations of 𝐻-closed spaces
Authors: Joseph James E.
Published: June 2010

Several characterizations of H-closed spaces are given along with some relating properties and examples.
The decidability of the theory of Boolean algebras with the quantifier “there exist infinitely many”
Authors: Weese Martin
Published: June 2010

By using the decidability of the weak second order theory of linear order we get the decidability of the theory of Boolean algebras with the additional quantifier

Q
0

{Q_0}

.
ℵ₀-categoricity of partially ordered sets of width 2.
Authors: Schmerl J. H.
Published: June 2010

A result of J. Rosenstein is that every

ℵ
0

{\aleph _0}

-categorical theory of linear order is finitely axiomatizable. We extend this to the case of partially ordered sets of width 2.
On certain weighted partitions and finite semisimple rings
Authors: Richmond L. B. , Subbarao M. V.
Published: June 2010

Let k be a fixed integer

⩾
1

\geqslant 1

and define

τ
k

(
n
)
=

Σ

d
k

/

n

1

{\tau _k}(n) = {\Sigma _{{d^k}/n}}1

. Thus

τ

1

(
n
)

{\tau _{1}}(n)

is the ordinary divisor function and

τ
k

(
n
)

{\tau _k}(n)

is the number of kth powers dividing n. We derive the asymptotic behaviour as

n
→
∞

n \to \infty

of

P
k

(
n
)

{P_k}(n)

defined by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts upper P Subscript k Baseline left-parenthesis n right-parenthesis x Superscript n Baseline equals product Underscript n equals 1 Overscript normal infinity Endscripts left-parenthesis 1 minus x Superscript n Baseline right-parenthesis Superscript minus tau Super Subscript k Superscript left-parenthesis n right-parenthesis Baseline period">

∑

n
=
0

∞

P
k

(
n
)

x
n

=

∏

n
=
1

∞

(
1
−

x
n

)

−

τ
k

(
n
)

.

\sum \limits _{n = 0}^\infty {{P_k}(n){x^n} = \prod \limits _{n = 1}^\infty {{{(1 - {x^n})}^{ - {\tau _k}(n)}}} .}

\]
Thus

P
k

(
n
)

{P_k}(n)

is the number of partitions of n where we recognize

τ
k

(
m
)

{\tau _k}(m)

different colours of the integer m when it occurs as a summand in a partition. The case

k
=
2

k = 2

is of special interest since the number

f
(
n
)

f(n)

of semisimple rings with n elements when

n
=

q
1

l
1

q
2

l
2

…

n = q_1^{{l_1}}q_2^{{l_2}} \ldots

is given by

f
(
n
)
=

P
2

(

l
1

)

P
2

(

l
2

)
…

f(n) = {P_2}({l_1}){P_2}({l_2}) \ldots

.
The sum of the reciprocals of a set of integers with no arithmetic progression of 𝑘 terms
Authors: Gerver Joseph L.
Published: June 2010

It is shown that for each integer

k
⩾
3

k \geqslant 3

, there exists a set

S
k

{S_k}

of positive integers containing no arithmetic progression of k terms, such that

Σ

n
∈

S
k

1

/

n
>
(
1
−
ε
)
k
log
⁡
k

{\Sigma _{n \in {S_k}}}1/n > (1 - \varepsilon )k\log k

, with a finite number of exceptional k for each real

ε
>
0

\varepsilon > 0

. This result is shown to be superior to that attainable with other sets in the literature, in particular Rankin’s sets

A

(
k
)

\mathcal {A}(k)

, which have the highest known asymptotic density for sets of positive integers containing no arithmetic progression of k terms.
Comparison of eigenvalues for systems of linear differential equations
Authors: Cheng Sui Sun
Published: June 2010

This paper is concerned with comparison theorems of an “integral type” concerning the smallest positive eigenvalues of systems of linear differential equations.
A new characterization of barrelled spaces
Authors: Pfister Helmut
Published: June 2010

A locally convex space E is barrelled if and only if

L
s

(
E
,
F
)

{L_s}(E,F)

is quasicomplete for every Banach space F.