bluuebunny/crossref_metadata_2025 huggingface dataset:
Eilenberg-Mac Lane spectra
Authors: Margolis H. R.
Published: June 2010

Let

K
(

Z
p

)

K({Z_p})

be the Eilenberg-Mac Lane spectrum with homotopy

Z
p

{Z_p}

and let

A
=

H
∗

(
K
(

Z
p

)
;

Z
p

)

A = {H^ \ast }(K({Z_p});{Z_p})

—the

mod
p

\bmod p

Steenrod algebra. Let

X
X

be a locally finite spectrum. It is proven that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper K left-parenthesis upper Z Subscript p Baseline right-parenthesis comma upper X right-bracket right-arrow upper H o m Subscript upper A Baseline left-parenthesis upper H Superscript asterisk Baseline left-parenthesis upper X semicolon upper Z Subscript p Baseline right-parenthesis comma upper A right-parenthesis">

[
K
(

Z
p

)
,
X
]
→

Hom
A

(

H
∗

(
X
;

Z
p

)
,
A
)

[K({Z_p}),X] \to {\operatorname {Hom} _A}({H^ \ast }(X;{Z_p}),A)

\]
is an isomorphism. It is also proven that there is a unique decomposition

X
=
(
⊕
K
(

Z
p

)
)
⊕
Y

X = ( \oplus K({Z_p})) \oplus Y

where

H
∗

(
Y
;

Z
p

)

{H^ \ast }(Y;{Z_p})

as an

A
A

-module has no free summands.
Immersion in the metastable range and 2-localization
Authors: Glover Henry H. , Mislin Guido
Published: June 2010

Our purpose is to study immersion properties in the metastable range using the techniques of localization of homotopy types. The main theorem states that immersion of a manifold

M
M

in euclidean space in the metastable range depends only upon the homotopy type

M
2

{M_2}

, the localization of

M
M

at the prime 2.
Maximum antichains: a sufficient condition
Authors: Klass Michael J.
Published: June 2010

Given the finite partially ordered set

(
Q
,
≤
)

(Q, \leq )

, one might wish to know whether a maximal (nonextendible) antichain is a maximum antichain. Our result generalizes a theorem of Baker, which in turn constitutes a generalization of Sperner’s lemma.
Positive Cesàro means of numerical series
Authors: Askey Richard
Published: June 2010

If a series has positive Cesàro means of order

γ
\gamma

, then its Abel means have positive Cesàro means of order

α
\alpha

for

0
≤
r
≤
(
α
+
1
)

/

(
γ
+
1
)
,
−
1
>
α
>
γ

0 \leq r \leq (\alpha + 1)/(\gamma + 1), - 1 > \alpha > \gamma

.
Note on: “Positive Cesàro means of numerical series” (Proc. Amer. Math. Soc. 45 (1974), 63–68) by Richard Askey
Authors: Bustoz J.
Published: June 2010

We give a short proof of a result due to R. Askey.
Distortion properties of 𝑝-fold symmetric alpha-starlike functions
Authors: Coonce H. B. , Miller S. S.
Published: June 2010

Starlike functions

f
f

which are of Mocanu type

α
\alpha

and have power series of the form
\[

f
(
z
)
=
z
+

a

p
+
1

z

p
+
1

+

a

2
p
+
1

z

2
p
+
1

+
⋯
,

f(z) = z + {a_{p + 1}}{z^{p + 1}} + {a_{2p + 1}}{z^{2p + 1}} + \cdots ,

\]
where

p
=
1
,
2
,
3
,
⋯

p = 1,2,3, \cdots

, are shown to satisfy the relation

f
(
z
)
=

[
g
(

z
p

)

]

1

/

p

f(z) = {[g({z^p})]^{1/p}}

where

g
g

is of Mocanu type

p
α

p\alpha

with power series

g
(
z
)
=
z
+

b
2

z
2

+

b
3

z
3

+
⋯

g(z) = z + {b_2}{z^2} + {b_3}{z^3} + \cdots

. Distortion results dealing with the

1
4

\tfrac {1}{4}

-theorem and bounds on

|

f
(
z
)

|

|f(z)|

are obtained.
A note on Wallman extendible functions
Authors: Hajek Darrell W.
Published: June 2010

It is known that any continuous function into a

T
4

{T_4}

space has a unique continuous Wallman extension, and that any continuous Wallman extension of a continuous function with a

T
3

{T_3}

range must be unique. We show that for any

T
3

{T_3}

space

Y
Y

which is not

T
4

{T_4}

there exists a

T
3

{T_3}

space

X
X

and a continuous function

f
:
X
→
Y

f:X \to Y

which has no continuous Wallman extension.
Continuous surjections from Cantor sets to compact metric spaces
Authors: Schoenfeld Alan H.
Published: June 2010

We present here a new proof of the fact that any compact metric space is the continuous image of the Cantor discontinumm.
Functions which operate in the Fourier algebra of a discrete group
Authors: De-Michele Leonede , Soardi Paolo M.
Published: June 2010

In this paper we prove the following theorem: let

G
G

be a discrete amenable group with nontrivial almost-periodic compactification, and let

F
F

be a complex-valued function defined in

[
−
1
,
1
]

[ - 1,1]

; then

F
F

operates in

A
(
G
)

A(G)

if and only if

F
F

is real-analytic in a neighborhood of the origin and

F
(
0
)
=
0

F(0) = 0

.
Perturbations of semi-Fredholm operators by operators converging to zero compactly
Authors: Goldberg Seymour
Published: June 2010

Let

{

K
n

}

\{ {K_n}\}

be a sequence of bounded linear operators mapping a Banach space

X
X

into a Banach space such that

K
n

→
0

{K_n} \to 0

strongly and

{

K
n

x
n

}

\{ {K_n}{x_n}\}

is relatively compact for every bounded sequence

{

x
n

}
⊂
X

\{ {x_n}\} \subset X

; e.g.,

|

|

K
n

|

|

→
0

||{K_n}|| \to 0

. Given

T
T

a semi-Fredholm operator, it is shown that for all sufficiently large

n
,
T
+

K
n

n,T + {K_n}

has nullity and deficiency not exceeding that of

T
T

while the index of

T
+

K
n

T + {K_n}

equals that of

T
T

. Properties of the minimum modulus of

T
+

K
n

T + {K_n}

are also given.
Characterization of abstract composition operators
Authors: Ridge William C.
Published: June 2010

A composition operator on

L
p

(
X
,
μ
)

{L^p}(X,\mu )

is (roughly) an operator

T
T

induced by a point transformation

ϕ
\phi

on

X
X

by

T
f
=
f
⋅
ϕ

Tf = f \cdot \phi

. Characterizations are given of abstract Hilbert-space operators which can be represented (via unitary equivalence) as composition operators. Representation on

L
2

(
J
,
m
)

{L^2}(J,m)

(

J
J

an interval of the real line,

m
m

a Borel measure) and on

L
2

(
0
,
1
)

{L^2}(0,1)

(Lebesgue measure) are considered. Also, any bounded measure-algebra transformation which preserves disjoint unions is a sigma-homomorphism.
Approximating fixed points of nonexpansive mappings
Authors: Senter H. F. , Dotson W. G.
Published: June 2010

A condition is given for nonexpansive mappings which assures convergence of certain iterates to a fixed point of the mapping in a uniformly convex Banach space. A relationship between the given condition and the requirement of demicompactness is established.
Products of arcwise connected spaces
Authors: Lehman B.
Published: June 2010

It is proved that the arbitrary product of arcwise connected spaces is arcwise connected.
The Riesz summability of logarithmic type
Authors: Kwee B.
Published: June 2010

The series

Σ

n
=
1

∞

a
n

\Sigma _{n = 1}^\infty {a_n}

is said to be summable

(
L
)

(L)

to

s
s

if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis log left-parenthesis 1 minus x right-parenthesis right-parenthesis Superscript negative 1 Baseline normal upper Sigma Subscript n equals 1 Superscript normal infinity Baseline s Subscript n Baseline x Superscript n plus 1 slash n">

(
log
⁡
(
1
−
x
)

)

−
1

Σ

n
=
1

∞

s
n

x

n
+
1

/

n

{(\log (1 - x))^{ - 1}}\Sigma _{n = 1}^\infty {s_n}{x^{n + 1}}/n

, where

s
n

=

Σ

v
=
1

n

a
v

{s_n} = \Sigma _{v = 1}^n{a_v}

, converges for

0
≤
x
>
1

0 \leq x > 1

and tends to

s
s

when

x
→
1
−

x \to 1 -

. The aim of this paper is to discuss the relation between summability

(
L
)

(L)

and Riesz summability

(
R
,
log
⁡
n
,
κ
)

(R,\log n,\kappa )

. It is proved that

(
R
,
log
⁡
n
,
κ
)
⊆
(
L
)

(R,\log n,\kappa ) \subseteq (L)

holds for

0
≤
κ
≤
1

0 \leq \kappa \leq 1

and is false for

κ
>
1

\kappa > 1

. It is also proved that if

Σ

n
=
1

∞

a
n

=
s
(
L
)

\Sigma _{n = 1}^\infty {a_n} = s(L)

and bounded

(
R
,
log
⁡
n
,
κ
)

(R,\log n,\kappa )

for

κ
≥
0

\kappa \geq 0

then

Σ

n
=
1

∞

a
n

=
s
(
R
,
log
⁡
n
,
κ
+
δ
)

\Sigma _{n = 1}^\infty {a_n} = s(R,\log n,\kappa + \delta )

for every

δ
>
0

\delta > 0

.
On weakly continuous mappings
Authors: Noiri Takashi
Published: June 2010

P. E. Long and D. A. Carnahan studied several properties of almost continuous mappings in the sense of Singal (denoted by a.c.S.) in Proc. Amer. Math. Soc. 38 (1973), 413-418. In this note, it is shown that “a.c.S.” in some theorems of the above paper can be replaced by “weakly continuous".
Approximation by continued fractions
Authors: Nathanson Melvyn B.
Published: June 2010

Let

x
x

be a real irrational number whose continued fraction has infinitely many partial quotients not less than

k
k

. A simple proof is given that the inequality

|

x
−
p

/

q

|

>
(

k
2

+
4

)

−
1

/

2

q

−
2

|x - p/q| > (k^2 + 4)^{-1/2} q^{-2}

has infinitely many rational solutions

p

/

q

p/q

. The constant

(

k
2

+
4

)

−
1

/

2

(k^2 + 4)^{-1/2}

is best possible.
Modules over semihereditary Bezout rings
Authors: Shores Thomas S.
Published: June 2010

It is shown that every commutative semihereditary Bezout ring of Krull dimension at most one is an elementary divisor ring. A consequence is that the ring of polynomials in one indeterminate over a von Neumann regular ring is an elementary divisor ring.
A subdirect decomposition of semiprime rings and its application to maximal quotient rings
Authors: Rowen Louis Halle
Published: June 2010

Levy [2] has examined semiprime rings which are irredundant subdirect products of prime rings. In this note we look at the role of inessential prime ideals and see how every semiprime ring is a subdirect product of (i) a semiprime ring which is an irredundant subdirect product of prime rings, and (ii) a semiprime (nonprime) ring, all of whose prime ideals are essential. This leads to a direct sum decomposition of maximal left quotient rings of semiprime rings with left singular ideal zero.
On subdirect products of rings without symmetric divisors of zero
Authors: Yit Tao Cheng
Published: June 2010

A theorem of V. A. Andrunakievič and Ju. M. Rjabuhin asserts that a ring

R
R

is without nilpotent elements if and only if

R
R

is a subdirect product of skew-domains. In this paper we prove that a semiprime ring

R
R

with involution is a subdirect product of rings without symmetric divisors of zero if and only if

R
R

is compressible for its symmetric elements.
Lebesgue measure is a representing measure
Authors: Sidney S. J.
Published: June 2010

Lebesgue measure on the unit interval

I
I

is multiplicative on some maximal Dirichlet algebra on

I
I

. Related results are obtained.
Enumeration of posets generated by disjoint unions and ordinal sums
Authors: Stanley Richard P.
Published: June 2010

Let

f
n

{f_n}

be the number of

n
n

-element posets which can be built up from a given collection of finite posets using the operations of disjoint union and ordinal sum. A curious functional equation is obtained for the generating function

Σ

f
n

x
n

\Sigma {f_n}{x^n}

. Using a result of Bender, an asymptotic estimate can sometimes be given for

f
n

{f_n}

. The analogous problem for labeled posets is also considered.
An almost continuous function
Authors: Fleissner Richard J.
Published: June 2010

In this note, a function is constructed which is of the Cesàro type and is almost continuous in the sense of Stallings and in the sense of Husain.
Variation norm convergence of function sequences
Authors: Constantine Randolph
Published: June 2010

We prove that a pointwise convergent sequence of convex functions with a continuous limit converges with respect to the total variation norm. This yields a theorem on convexity-preserving operators which has as a corollary the result that a complex function

f
f

is absolutely continuous on

[
0
,
1
]

[0,1]

if and only if the sequence

B
.
(
f
)

B.(f)

of Bernstein polynomials of

f
f

converges to

f
f

with respect to the total variation norm.
On the serial completion of deleted Schauder bases by domain adjustment
Authors: Braun Ben-Ami
Published: June 2010

Given a Schauder basis in a Banach function space of a specified type, one can delete any finite number of elements of the basis and still preserve serial totality by making an arbitrarily small adjustment of the domain.
An extension of a theorem of Nagano on transitive Lie algebras
Authors: Sussmann Héctor J.
Published: June 2010

Let

M
M

be a real analytic manifold, and let

L
L

be a transitive Lie algebra of real analytic vector fields on

M
M

. A concept of completeness is introduced for such Lie algebras. Roughly speaking,

L
L

is said to be complete if the integral trajectories of vector fields in

L
L

are defined “as far as

L
L

permits". Examples of situations where this assumption is satisfied: (i)

L
L

= a transitive Lie algebra all of whose elements are complete vector fields, and (ii)

L
L

= the set

V
(
M
)

V(M)

of all real analytic vector fields on

M
M

. Our main result is: if

M
,

M
′

M,M’

are connected manifolds, then every Lie algebra isomorphism

F
:
L
→

L
′

F:L \to L’

between complete transitive Lie algebras of real analytic vector fields on

M
,

M
′

M,M’

which carries the isotropy subalgebra

L
m

{L_m}

of a point

m
m

of

M
M

to the isotropy subalgebra

L

m
′

{L_{m’}}

of

m
′

∈

M
′

m’\in M’

is induced by a (unique) real analytic diffeomorphism

f
:
M
→

M
′

f:M \to M’

such that

f
(
m
)
=

m
′

f(m) = m’

, provided that one of the following two conditions is satisfied: (l)

M
M

and

M
′

M’

are simply connected, or (2) the Lie algebras

L
L

and

L
′

L’

separate points. Nagano had proved this result for the case

L
=
V
(
M
)
,

L
′

=
V
(

M
′

)
,
M

L = V(M),L’ = V(M’),M

and

M
′

M’

compact.
Asymptotic distribution of normalized arithmetical functions
Authors: Erdős Paul , Galambos Janos
Published: June 2010

Let

f
(
n
)

f(n)

be an arbitrary arithmetical function and let

A
N

{A_N}

and

B
N

{B_N}

be sequences of real numbers with

0
>

B
N

→
+
∞

0 > {B_N} \to + \infty

with

N
N

. We give a sufficient condition for

(
f
(
n
)
−

A
N

)

/

B
N

(f(n) - {A_N})/{B_N}

to have a limiting distribution. The case when

f
(
n
)

f(n)

is defined by

f
(
n
)
=
Σ
g
(
d
)

f(n) = \Sigma g(d)

, where the summation is over all divisors

d
d

of

n
n

and

g
(
d
)

g(d)

is any given arithmetical function, is discussed in more detail. A concrete example is given as an application of our result, in which example

f
(
n
)

f(n)

is neither additive nor multiplicative. Our method of proof is to approximate

f
(
n
)

f(n)

by a suitably chosen additive function, as proposed in [4], and then to apply general theorems available for additive functions.
On the automorphism group of a Lie group
Authors: Wigner D.
Published: June 2010

It is proved that the automorphism group of a connected real or complex Lie group contains an open real or complex algebraic subgroup. It follows that the identity component of the group of complex automorphisms of a connected complex Lie group is a complex algebraic group.
Extreme points of classes of functions defined by subordination
Authors: Hallenbeck D. J.
Published: June 2010

The closed convex hull and extreme points of families of analytic functions, which are defined in terms of subordination, are determined. Integral representations are given for the hulls of these families in terms of probability measures on suitable sets. These results are used to solve extremal problems. The functions we consider are defined by subordination to certain classes of starlike and convex mappings.
A characterization of hypoelliptic differential operators with variable coefficients
Authors: White R. E.
Published: June 2010

Let

P
P

be a linear differential operator with coefficients in

C
∞

(
Ω
)

{C^\infty }(\Omega )

where

Ω
⊂

R

n

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

. We characterize the hypoelliptic operators in terms of the

∗
\ast

-hypoelliptic operators.

P
P

is defined to be

∗
\ast

-hypoelliptic on

Ω
\Omega

if and only if

u
∈

D

F
′

(
Ω
)

u \in {\mathcal {D}’_F}(\Omega )

and

P
u
∈

C
∞

(
Ω
)

Pu \in {C^\infty }(\Omega )

imply

u
∈

C
∞

(
Ω
)

u \in {C^\infty }(\Omega )

. We characterize the

∗
\ast

-hypoelliptic operators via a priori estimates. We prove

P
P

is hypoelliptic on

Ω
\Omega

if and only if for

u
∈

D

′

(
Ω
)

u \in \mathcal {D}’(\Omega )

and

P
u
∈

C
∞

(

Ω
′

)

Pu \in {C^\infty }(\Omega ’)

with

Ω
′

⊂
Ω

\Omega ’ \subset \Omega

, there exists for each

x
0

∈

Ω
′

{x_0} \in \Omega ’

a relatively compact open neighborhood

Ω

x
0

⊂

Ω
′

{\Omega _{{x_0}}} \subset \Omega ’

of

x
0

{x_0}

such that

P
P

is

∗
\ast

-hypoelliptic on

Ω

x
0

{\Omega _{{x_0}}}

.
A large algebraically closed field
Authors: Corzatt Clifton E.
Published: June 2010

A sequence is called an R.R.S. sequence if, roughly speaking, it is generated by some member of a set of recurrence formulas over the field

Q
(
i
)

Q(i)

which involves only rational operations. It is proved that the set of limits of all convergent R.R.S. sequences forms a countable algebraically closed field. Moreover, the field is shown to contain all numbers of the form

e
α

{e^\alpha }

, where

α
\alpha

is an algebraic number.
𝜖-continuity and shape
Authors: Felt James E.
Published: June 2010

In this paper a relationship is established between the concepts of

ε
\varepsilon

-continuity and shape for compact metric spaces, thus answering a question raised by Victor Klee.
Some problems on 𝐵-completeness
Authors: Mukherjee T. K.
Published: June 2010

In this paper, we give examples to show the following: 1. The product of two

B
B

-complete spaces is not necessarily

B
r

{B_r}

-complete. 2. The Mackey dual of a strict LF-space is not necessarily

B
r

{B_r}

-complete. 3. A separable, reflexive, and strict LF-space is not, in general,

B
r

{B_r}

-complete. The second point has reference to a problem of Dieudonné and Schwartz which asks essentially whether the Mackey dual of a strict LF-space is

B
B

-complete and which was answered in the negative by Grothendieck.
Proof of the Gelfand-Kirillov conjecture for solvable Lie algebras
Authors: Joseph A.
Published: June 2010

Let

g
g

be a solvable algebraic Lie algebra over the complex numbers

C

{\mathbf {C}}

. It is shown that the quotient field of the enveloping algebra of

g
g

is isomorphic to one of the standard fields

D

n
,
k

{D_{n,k}}

, being defined as the quotient field of the Weyl algebra of degree

n
n

over

C

{\mathbf {C}}

extended by

k
k

indeterminates. This proves the Gelfand-Kirillov conjecture for

g
g

solvable.
A characterization of the connectivity of a manifold in terms of large open cells
Authors: Summerhill R. Richard
Published: June 2010

If

k
k

and

n
n

are integers,

0
⩽
k
⩽
n
−
3

0 \leqslant k \leqslant n - 3

, and

M
n

{M^n}

is a topological

n
n

-manifold without boundary, it is shown that

M
M

is

k
k

-connected if and only if there is a “tame”

(
n
−
k
−
1
)

(n - k - 1)

-dimensional closed subset

X
X

in

M
M

such that

M
−
X

M - X

is homeomorphic to

E
n

{E^n}

.
Affine structures on three-step nilpotent Lie algebras
Authors: Scheuneman John
Published: June 2010

As part of the investigation of the parallel between solvable Lie theory and the theory of groups of affine motions, it is proved that every three-step nilpotent Lie algebra admits a faithful representation of a certain special kind. It follows immediately that every three-step nilpotent Lie group which is connected, simply connected, and of dimension

n
n

admits a representation as a simply transitive group of affine motions of

R
n

{R^n}

.
On the uniform ergodic theorem. II
Authors: Lin Michael
Published: June 2010

Let

{

T
t

}

\{ {T_t}\}

be a strongly continuous semigroup of bounded linear operators on a Banach space

X
X

, satisfying

lim

t
→
∞

|

|

T
t

|

|

/

t
=
0

{\lim _{t \to \infty }}||{T_t}||/t = 0

. We prove the equivalence of the following conditions: (1)

t

−
1

∫
0
t

T
r

d
r

{t^{ - 1}}\int _0^t {{T_r}dr}

converges uniformly as

t
→
∞

t \to \infty

. (2) The infinitesimal generator

A
A

has closed range. (3)

lim

λ
→

0
+

λ

R
λ

{\lim _{\lambda \to {0^ + }}}\lambda {R_\lambda }

exists in the uniform operator topology.
Generating functions for a special class of permutations
Authors: Carlitz L.
Published: June 2010

The paper contains a simple, direct derivation of the generating functions for a class of permutations generalizing the up-down permutations.
On the set of extreme points of a convex body
Authors: Collier James B.
Published: June 2010

We prove the following: Given a subset

X
X

of a compact

0
0

-dimensional metric space

Z
Z

and an integer

d
⩾
3

d \geqslant 3

, there is a homeomorphism of

Z
Z

into the boundary of a convex body

C
C

in

E
d

{E^d}

mapping

X
X

onto the set of extreme points of

C
C

if and only if

X
X

is a

G
δ

{G_\delta }

set with at least

d
+
1

d + 1

points.
Existence theorems for higher order boundary value problems
Authors: Schrader Keith , Umamaheswaram S.
Published: June 2010

In this paper the differential equation

y

(
n
)

=
f
(
x
,
y
)

{y^{(n)}} = f(x,y)

and associated boundary conditions

y

(
r
)

(

x
i

)
=

y

i
r

{y^{(r)}}({x_i}) = {y_{ir}}

for

i
=
1
,
2
,
⋯
,
k

i = 1,2, \cdots ,k

and

r
=
0
,
1
,
⋯
,
λ
(
i
)
−
1

r = 0,1, \cdots ,\lambda (i) - 1

where

λ
(
1
)
+
λ
(
2
)
+
⋯
+
λ
(
k
)
=
n

\lambda (1) + \lambda (2) + \cdots + \lambda (k) = n

are considered. Sufficient conditions on

f
f

are given to insure the existence of a solution to this

k
k

point boundary value problem. In the special cases when

k
=
2

k = 2

or

k
=
3

k = 3

sufficient conditions on

f
f

are given to insure both uniqueness and existence of solutions for certain of the boundary value problems.
A comparison theorem for certain functional differential equations
Authors: True Ernest D.
Published: June 2010

The oscillatory character of solutions to the functional differential equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis t right-parenthesis plus a left-parenthesis t right-parenthesis f left-parenthesis x left-parenthesis g left-parenthesis t right-parenthesis right-parenthesis right-parenthesis equals upper Q left-parenthesis t right-parenthesis">

x

(
n
)

(
t
)
+
a
(
t
)
f
(
x
(
g
(
t
)
)
)
=
Q
(
t
)

{x^{(n)}}(t) + a(t)f(x(g(t))) = Q(t)

is investigated, by comparison with the oscillatory character of solutions to

x

(
n
)

(
t
)
+
s
(
t
)
f
(
x
(
t
)
)
=
0

{x^{(n)}}(t) + s(t)f(x(t)) = 0

where

s
(
t
)
⩾
γ
a
(
t
)
,
0
>
γ
>
1

s(t) \geqslant \gamma a(t),0 > \gamma > 1

. Here,

Q
(
t
)

Q(t)

represents a bounded, oscillatory forcing function, and

g
(
t
)

g(t)

tends to

∞
\infty

as

t
→
∞

t \to \infty

or

g
(
t
)
⩾
t
−
c

g(t) \geqslant t - c

for large

t
t

but is otherwise arbitrary.
Omitting types: application to descriptive set theory
Authors: Mansfield Richard
Published: June 2010

The omitting types theorem of infinitary logic is used to prove that every small

Π
1
1

\Pi _1^1

set of analysis or any small

Σ
1

{\Sigma _1}

set of set theory is constructible.
Whitney number inequalities for geometric lattices
Authors: Dowling Thomas A. , Wilson Richard M.
Published: June 2010

Let

L
L

be a finite geometric lattice of rank

r
r

, and for

i
=
0
,
1
,
⋯
,
r

i = 0,1, \cdots ,r

, let

W
i

{W_i}

denote the number of elements of

L
L

with rank

i
i

. For

1
≤
k
≤
r
−
2

1 \leq k \leq r - 2

, we have
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W 1 plus upper W 2 plus midline-horizontal-ellipsis plus upper W Subscript k Baseline less-than-or-equal-to upper W Subscript r minus k Baseline plus midline-horizontal-ellipsis plus upper W Subscript r minus 2 Baseline plus upper W Subscript r minus 1">

W
1

+

W
2

+
⋯
+

W
k

≤

W

r
−
k

+
⋯
+

W

r
−
2

+

W

r
−
1

{W_1} + {W_2} + \cdots + {W_k} \leq {W_{r - k}} + \cdots + {W_{r - 2}} + {W_{r - 1}}

with equality if and only if the lattice

L
L

is modular. We give two further results concerning matchings of lattice elements of rank

≤
k

\leq k

into those of rank

≥
r
−
k

\geq r - k

, and observe that a middle term can be interpolated in the above inequality.
On complex strict and uniform convexity
Authors: Globevnik J.
Published: June 2010

Strict and uniform

c
c

-convexity of complex normed spaces are introduced as a natural generalization of strict and uniform convexity. It is proved that the complex space

L
1

(
S
,
σ
,
μ
)

{L_1}(S,\sigma ,\mu )

is uniformly

c
c

-convex. An application to analytic functions is given.
Property Γ and inner amenability
Authors: Effros Edward G.
Published: June 2010

If the regular group von Neumann algebra of a countable, infinite conjugacy class group satisfies Property

Γ
\Gamma

, then the group has a nontrivial mean which is invariant under inner automorphisms.
Structure diagrams for primitive Boolean algebras
Authors: Williams James
Published: June 2010

If

S
S

and

T
T

are structure diagrams for primitive Boolean algebras, call a homomorphism

f
f

from

S
S

onto

T
T

right-strong iff whenever

x
T
f
(
t
)

xTf(t)

, there is an

s
s

such that

f
(
s
)
=
x

f(s) = x

and

s
S
t

sSt

; let RSE denote the category of diagrams and onto right-strong homomorphisms. The relation “

S
S

structures

B

\mathfrak {B}

” between diagrams and Boolean algebras induces a 1-1 correspondence between the components of RSE and the isomorphism types of primitive Boolean algebras. Up to isomorphism, each component of RSE contains a unique minimal diagram and a unique maximal tree diagram. The minimal diagrams are like those given in a construction by William Hanf. The construction which is given for producing maximal tree diagrams is recursive; as a result, every diagram

S
S

structures a Boolean algebra recursive in

S
S

.
Note on the supplement to the law of cubic reciprocity
Authors: Williams Kenneth S.
Published: June 2010

A short proof is given of the supplement to the law of cubic reciprocity proved by Eisenstein in 1844.
Enumeration results in nilpotent algebras
Authors: Strade Helmut
Published: June 2010

We enumerate the number of subalgebras for some classes of finite nilpotent algebras. These classes are: alternative algebras over

G
F
(
q
)

GF(q)

, noncommutative Jordan algebras over

G
F
(
q
)

GF(q)

, algebras with only zero squares over some Dedekind domain.
An extension of Runge’s theorem
Authors: Rubel L. A.
Published: June 2010

Every uniformly continuously differentiable function on the compact set

X
X

in the complex plane is the uniform limit of rational functions with poles off

X
X

.
Nonselfadjoint representations of 𝐶*-algebras
Authors: Radjavi Heydar
Published: June 2010

The following strengthening of a result of B. A. Barnes is proved: If

ϕ
\phi

is a topologically irreducible representation of a

C
∗

{C^ \ast }

-algebra

A

\mathfrak {A}

on a Banach space such that

ϕ
(

A

)

\phi (\mathfrak {A})

contains a nonzero finite-rank operator, then

ϕ
\phi

is similar to an irreducible

∗

^ \ast

-representation of

A

\mathfrak {A}

(and is thus automatically continuous).
Dual spaces with the Krein-Milman property have the Radon-Nikodým property
Authors: Huff R. E. , Morris P. D.
Published: June 2010

The assertion in the title is proved.
Covering spaces of paracompact spaces
Authors: Brown David T.
Published: June 2010

This note proves that in the context of paracompact spaces, a definition used by the author implies the definition used by Saul Lubkin which does not involve the usual connectedness properties associated with covering spaces.
On the 𝑝-primary obstructions to finding a cross-section
Authors: Rigdon Robert
Published: June 2010

Let

q
:
T
→
B

q:T \to B

be a (weak) fibration, and let

p
p

be an odd prime. In this paper, we show that the existence of a fiber-preserving map

A
:
T
→
T

A:T \to T

having certain properties implies that the

p
p

-primary obstructions to a cross-section of

q
q

vanish. Assuming that

q
:
T
→
B

q:T \to B

has a cross-section, we prove a related theorem which bears on the problem of enumerating the homotopy classes of cross-sections of

q
q

.
Truncated polynomial rings over Poincaré algebras
Authors: Beem R. Paul
Published: June 2010

There are given, in certain cases, necessary and sufficient conditions for a truncated polynomial ring over a

Z
2

{Z_2}

-Poincaré algebra to again be a Poincaré algebra. Applications are a splitting theorem for Poincaré algebras and an algebraic bordism classification for real projective space bundles.
Orders in separable algebras
Authors: Tarsy Richard B.
Published: June 2010

The module

P
∗

/

m

P
∗

{P^ \ast }/m{P^ \ast }

, where

P
P

is an order in a separable algebra over the quotient field of an integrally closed, quasi-local domain, is studied. It is shown that if the domain is complete,

P
∗

/

m

P
∗

{P^ \ast }/m{P^ \ast }

contains one element from each isomorphism class of irreducible

P
P

modules. Also, in general, if the global dimension of

P
P

is finite, then it equals the homological dimension of

P
∗

/

m

P
∗

{P^ \ast }/m{P^ \ast }

.
The Hughes conjecture and groups with absolutely regular subgroups or 𝐸𝐶𝐹-subgroups
Authors: Gallian Joseph A.
Published: June 2010

Let

G
G

be a finite

p
p

-group and

H
p

(
G
)

{H_p}(G)

the subgroup generated by the elements of

G
G

of order different from

p
p

. Hughes conjectured that if

G
>

H
p

(
G
)
>
1

G > {H_p}(G) > 1

, then

|

G
:

H
p

(
G
)

|

=
p

|G:{H_p}(G)| = p

. Although the conjecture is not true for all

G
G

, it is shown here that if

G
G

has a subgroup

L
L

such that

|

L
:

L
p

|

≤

p

p
−
r

(
r
≥
1
)

|L:{L^p}| \leq {p^{p - r}}(r \geq 1)

and

p
r

≤

|

G
:
L

|

≤

p

r
+
p

{p^r} \leq |G:L| \leq {p^{r + p}}

or an ECF-subgroup

L
L

with

|

G
:

L
2

|

≤

p

p
+
2

|G:{L_2}| \leq {p^{p + 2}}

, then

G
G

satisfies the Hughes conjecture.
Some properties of certain relatively free groups
Authors: Hurley T. C.
Published: June 2010

It is shown that the groups

F

/

[

F
n

,

F
m

]

F/[{F_n},{F_m}]

, where

F
F

is a free group and

m
,
n

m,n

are positive integers such that

m
>
n
≤
2
m

m > n \leq 2m

, are residually “torsion-free and nilpotent", and the structure of their lower central factors is computed.
On reductive algebras containing compact operators
Authors: Rosenthal Peter
Published: June 2010

It is shown that a reductive operator algebra containing an injective compact operator is selfadjoint.
The Whitehead theorem for nilpotent spaces
Authors: Gersten S. M.
Published: June 2010

An easy argument is given for the theorem of the title.
𝒯 measure of Cartesian product sets
Authors: Ernst Lawrence R.
Published: June 2010

It is proven that there exists a subset

A
A

of Euclidean

2
2

-space such that the

2
2

-dimensional

T

\mathcal {T}

measure of the Cartesian product of an interval of unit length and

A
A

is greater than the

1
1

-dimensional

T

\mathcal {T}

measure of

A
A

. This shows that

T

\mathcal {T}

measure does not extend to Euclidean

3
3

-space the relation that area is the product of length by length. As corollaries, new proofs of some related but previously known results are obtained.
Complete domains with respect to the Carathéodory distance
Authors: Kim Dong S.
Published: June 2010

Concerning completeness with respect to the Carathéodory distance (

c
c

-completeness), the following theorems are shown. A bounded convex (in geometric sense) domain

D
D

in

C

n

(

R

2
n

)

{{\mathbf {C}}^n}({{\mathbf {R}}^{2n}})

is

c
c

-complete, so that it is boundedly holomorphic convex. To preserve

c
c

-completeness in complex spaces, it is sufficient to have a proper local biholomorphic mapping as follows: Let

α
\alpha

be a proper spread map of a

c
c

-hyperbolic complex space

(
X
,
A
)

(X,A)

onto a

c
c

-hyperbolic complex space

(

X
~

,

A
~

)

(\tilde X,\tilde A)

; then

X
X

is

c
c

-complete if and only if

X
~

\tilde X

is

c
c

-complete. We also show the following

D
D

to be domains of bounded holomorphy: let

(
X
,
A
;
α
)

(X,A;\alpha )

be a Riemann domain and

D
D

a domain in

X
X

with

α
(
D
)

\alpha (D)

bounded in

C

n

{{\mathbf {C}}^n}

. Let

B
(
D
)

B(D)

separate the points of

D
D

. Suppose there is a compact set

K
K

such that for any

x
∈
D

x \in D

there is an analytic automorphism

σ
∈
Aut
⁡
(
D
)

\sigma \in \operatorname {Aut} (D)

and a point

a
∈
K

a \in K

such that

σ
(
x
)
=
a

\sigma (x) = a

. Then

D
D

is a domain of bounded holomorphy.
Absolutely closed maps
Authors: Friedler Louis
Published: June 2010

An example is given of a continuous function

f
:
X
→
Y

f:X \to Y

which is closed, has point inverses

H
H

-closed, but which can be extended to a continuous function

F
:
Z
→
Y

F:Z \to Y

for some

Z
Z

which has

X
X

as a proper dense subset. A partial characterization of nonextendable functions is given in terms similar to Bourbaki’s theorem that perfect maps

f
:
X
→
Y

f:X \to Y

are those for which

f
×

i
Z

:
X
×
Z
→
Y
×
Z

f \times {i_Z}:X \times Z \to Y \times Z

is a closed map for all spaces

Z
Z

.
𝜙-Postnikov systems and extensions of 𝐻-spaces
Authors: Shar Albert O.
Published: June 2010

Let

f
:
X
→
Y

f:X \to Y

be a map of CW complexes and let

π
:

F
f

→
X

\pi :{F_f} \to X

be the fibration induced by

f
f

. The following theorems are proven: Theorem. Assume

F
(
=

F
f

)

F( = {F_f})

and

Ω
Y

\Omega Y

are simply connected and that (a)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Superscript asterisk Baseline colon upper H Superscript n Baseline left-parenthesis upper Y semicolon pi Subscript n Baseline left-parenthesis upper Y right-parenthesis right-parenthesis right-arrow upper H Superscript n Baseline left-parenthesis upper X semicolon pi Subscript n Baseline left-parenthesis upper Y right-parenthesis right-parenthesis">

f
∗

:

H
n

(
Y
;

π
n

(
Y
)
)
→

H
n

(
X
;

π
n

(
Y
)
)

{f^ \ast }:{H^n}(Y;{\pi _n}(Y)) \to {H^n}(X;{\pi _n}(Y))

is epic for all

n
n

, (b)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis i logical-and i right-parenthesis Superscript asterisk Baseline colon upper H Superscript n Baseline left-parenthesis upper F logical-and upper F semicolon c o kernel f Subscript n Sub Superscript asterisk Subscript Baseline right-parenthesis right-arrow upper H Superscript n Baseline left-parenthesis normal upper Omega upper Y logical-and normal upper Omega upper Y semicolon c o kernel f Subscript n Sub Superscript asterisk Subscript Baseline right-parenthesis">

(
i
∧
i

)
∗

:

H
n

(
F
∧
F
;
co
⁡
ker
⁡

f

n
∗

)
→

H
n

(
Ω
Y
∧
Ω
Y
;
co
⁡
ker
⁡

f

n
∗

)

{(i \wedge i)^ \ast }:{H^n}(F \wedge F;\operatorname {co} \ker {f_{{n^ \ast }}}) \to {H^n}(\Omega Y \wedge \Omega Y;\operatorname {co} \ker {f_{{n^ \ast }}})

is monic for all

n
n

(where

f

n
∗

:

π
n

(
X
)
→

π
n

(
Y
)
)

{f_{{n^ \ast }}}:{\pi _n}(X) \to {\pi _n}(Y))

). If

X
X

is an

H
H

-space then

F
F

is an

H
H

-space such that

π
:
F
→
X

\pi :F \to X

is an

H
H

-map. Theorem. Assume

Y
Y

is

(
p
−
1
)

(p - 1)

-connected,

F
F

is

(
q
−
1
)

(q - 1)

-connected

(
p
−
1
≥
2
,
q
−
1
≥
1
)

(p - 1 \geq 2,q - 1 \geq 1)

of dimension

>
min
(
2
p
−
1
,
p
+
q
−
1
)

> \min (2p - 1,p + q - 1)

, and

H
∗

(
Y
)

{H^ \ast }(Y)

is free. If

X
X

is an

H
H

-space and

f
∗

:

H
n

(
Y
)
→

H
n

(
X
)

{f^ \ast }:{H^n}(Y) \to {H^n}(X)

is onto for all

n
n

then

F
F

is an

H
H

-space and the map

π
:
F
→
X

\pi :F \to X

is an

H
H

-map. Analogous theorems are shown to hold for loop spaces.
Localizing prime idempotent kernel functors
Authors: Sim S. K.
Published: June 2010

In this note, we call a prime idempotent kernel functor a localizing prime if it has the so-called property (T) of Goldman. We generalize a theorem of Heinicke to characterize localizing prime idempotent kernel functors and present an example of a prime idempotent kernel functor on

M
o
d
-

⁡
R

\operatorname {Mod-}R

, the category of unitary right

R
R

-modules, which is not a localizing prime, even though

R
R

is a right artinian ring.
Oscillation criteria for second order nonlinear differential equations
Authors: Onose Hiroshi
Published: June 2010

Oscillation criteria are given for the second order nonlinear equation

y
(
t
)
+
a
(
t
)
f
(
y
(
t
)
)
=
0

y(t) + a(t)f(y(t)) = 0

, where the coefficient

a
(
t
)

a(t)

is not assumed to be nonnegative for all large values of

t
t

. These results are concerned with the interesting recent ones of Wong’s paper.
On the Cauchy problem of the differential operator 𝑆𝜇
Authors: Lee W. Y.
Published: June 2010

I. M. Gelfand and G. E. Shilov have obtained the uniqueness and correctness class of the Cauchy problem of the differential operator

i
(
∂

/

∂
x
)

i(\partial /\partial x)

. If

S
μ

{S_\mu }

is some particular differential operator, then the uniqueness class of the differential operator

S
μ

{S_\mu }

is given in this paper.
Strong regularity at nonpeak points
Authors: Wang James Li Ming
Published: June 2010

We construct a uniform algebra which is strongly regular at a nonpeak point.
The asymptotic behaviour of the reduced minimum modulus of a Fredholm operator
Authors: Förster K.-H. , Kaashoek M. A.
Published: June 2010

Let

γ
(
S
)

\gamma (S)

denote the reduced minimum modulus of a linear operator

S
S

acting in a complex Banach space

X
X

, and let

I
I

denote the identity on

X
X

. In this paper it is shown that for a (not necessarily bounded) Fredholm operator

T
T

acting in

X
X

, the limit

lim

n
→
∞

γ

(

T
n

)

1

/

n

{\lim _{n \to \infty }}\gamma {({T^n})^{1/n}}

exists and is equal to the supremum of all positive numbers

δ
\delta

such that the dimension of the null space and the codimension of the range of

T
−
λ
I

T - \lambda I

are constant on

0
>

|

λ

|

>
δ

0 > |\lambda | > \delta

.
Axiom of choice and complementation
Authors: Diaconescu Radu
Published: June 2010

It is shown that an intuitionistic model of set theory with the axiom of choice has to be a classical one.
Generalized algebraic operators
Authors: Ghatage P.
Published: June 2010

A class of polynomially bounded operators satisfying an

H
∞

{H^\infty }

function is introduced and some results relating to the

C
0

{C_0}

class of contractions introduced by Sz.-Nagy and Foias are generalized.
The second variation formula for harmonic mappings
Authors: Smith R. T.
Published: June 2010

The formula of the title is computed, and is used to calculate the index and nullity in several cases.
A characterisation of von Neumann regular Jordan triple systems
Authors: Meyberg Kurt
Published: June 2010

In this note we give a characterisation of the Jacobson radical of a Jordan triple system in terms of principal inner ideals. If

(

A

,
P
)

(\mathfrak {A},P)

is a Jordan triple system and

Rad
⁡

A

\operatorname {Rad} \mathfrak {A}

the Jacobson radical of

A

\mathfrak {A}

then

x
∈
Rad
⁡

A

x \in \operatorname {Rad} \mathfrak {A}

iff

P
(
x
)

A

=
P
(
x
+
P
(
x
)
y
)

A

P(x)\mathfrak {A} = P(x + P(x)y)\mathfrak {A}

for all

y
∈

A

y \in \mathfrak {A}

. We use this to give a new characterisation of von Neumann regular Jordan triple systems. In particular, this gives another most elementary and short proof that semi-simple Jordan triple systems with dcc on principal inner ideals are von Neumann regular.
Asymptotic behavior and nonoscillation of Volterra integral equations and functional differential equations
Authors: Izé A. F. , Freiria A. A.
Published: June 2010

It is proved that if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q Subscript i j Baseline left-parenthesis t comma s right-parenthesis rho Subscript j Baseline left-parenthesis s right-parenthesis left-bracket rho Subscript i Baseline left-parenthesis t right-parenthesis right-bracket Superscript negative 1">

q

i
j

(
t
,
s
)

ρ
j

(
s
)

[

ρ
i

(
t
)

]

−
1

{q_{ij}}(t,s){\rho _j}(s){[{\rho _i}(t)]^{ - 1}}

is bounded,

i
,
j
=
1
,
2
,
…
,
n

i,j = 1,2, \ldots ,n

, and

f
(
t
,
x
,
x
(
u
(
s
)
)
)

f(t,x,x(u(s)))

is “small",
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x left-parenthesis u left-parenthesis s right-parenthesis right-parenthesis equals left-parenthesis x 1 left-parenthesis u 1 left-parenthesis s right-parenthesis right-parenthesis comma x 2 left-parenthesis u 2 left-parenthesis s right-parenthesis right-parenthesis comma ellipsis comma x Subscript n Baseline left-parenthesis u Subscript n Baseline left-parenthesis s right-parenthesis right-parenthesis right-parenthesis">

x
(
u
(
s
)
)
=
(

x
1

(

u
1

(
s
)
)
,

x
2

(

u
2

(
s
)
)
,
…
,

x
n

(

u
n

(
s
)
)
)

x(u(s)) = ({x_1}({u_1}(s)),{x_2}({u_2}(s)), \ldots ,{x_n}({u_n}(s)))

\]
with

u
i

(
t
)
⩽
t

{u_i}(t) \leqslant t

and

lim

t
→
∞

u
i

(
t
)
=
∞

{\lim _{t \to \infty }}{u_i}(t) = \infty

, the solutions of the integral equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x left-parenthesis t right-parenthesis equals h left-parenthesis t right-parenthesis plus integral Subscript 0 Superscript t Baseline q left-parenthesis t comma s right-parenthesis f left-parenthesis s comma x left-parenthesis s right-parenthesis comma x left-parenthesis u left-parenthesis s right-parenthesis right-parenthesis right-parenthesis d s">

x

(
t
)

=
h
(
t
)
+

∫
0
t

q
(
t
,
s
)
f
(
s
,
x
(
s
)
,
x
(
u
(
s
)
)
)
d
s

x\left ( t \right ) = h(t) + \int _0^t {q(t,s)f(s,x(s),x(u(s)))ds}

\]
satisfy the conditions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x left-parenthesis t right-parenthesis equals h left-parenthesis t right-parenthesis plus rho left-parenthesis t right-parenthesis a left-parenthesis t right-parenthesis comma limit Underscript t right-arrow normal infinity Endscripts a left-parenthesis t right-parenthesis equals">

x
(
t
)
=
h
(
t
)
+
ρ
(
t
)
a
(
t
)
,

lim

t
→
∞

a
(
t
)
=

x(t) = h(t) + \rho (t)a(t),{\lim _{t \to \infty }}a(t) =

constant where

ρ
(
t
)

\rho (t)

is a nonsingular diagonal matrix chosen in such a way that

ρ

−
1

(
t
)
h
(
t
)

{\rho ^{ - 1}}(t)h(t)

is bounded. The results contain, in particular, some results on the asymptotic behavior, stability and existence of nonoscillatory solutions of functional differential equations.
Higher order estimates and smoothness of nonlinear wave equations
Authors: Reed Michael C.
Published: June 2010

A completely elementary proof is given for Segal’s nonlinear semigroup theorem.
An algorithm for partitions
Authors: Nathanson Melvyn B.
Published: June 2010

A class of algorithms is described to represent the positive integers as sums of elements in a prescribed sequence of integers, and results are obtained on the densities of the integers that can be represented by such partitions with a bounded number of summands.
Characterizing ℎ𝑜𝑙(Ω)
Authors: Carpenter Ronn
Published: June 2010

We find necessary and sufficient conditions for an algebra of complex valued functions defined on a

σ
\sigma

-compact

T
2

{T_2}

space to be algebraically and topologically equivalent to the algebra of analytic functions on a finitely connected domain in

C

{\text {C}}

.
Linear connections and almost complex structures
Authors: Terrier Jean-Marc
Published: June 2010

An almost complex structure is defined on

P
P

, the principal bundle of linear frames over an arbitrary even-dimensional smooth manifold

M
M

with a given linear connection. Complexifying connections are those which induce a complex structure on

P
P

. For two-dimensional manifolds, every linear connection is of this kind. In the special case where

M
M

itself is an almost complex manifold, a relationship between the two almost complex structures is found and provides a very simple proof of the fact that the existence of an almost complex connection without torsion implies the integrability of the given almost complex structure. As a second application, we give a geometrical interpretation of an identity between the torsion of an almost complex structure on

M
M

and the torsion of an almost complex connection over

M
M

.
Simple knots which are doubly-null-cobordant
Authors: Kearton C.
Published: June 2010

Using the Seifert matrix, a necessary and sufficient condition is given for a simple

(
2
q
−
1
)

(2q - 1)

-knot,

q
>
1

q > 1

, to be doubly-null-cobordant.
Linearization of homeomorphism groups
Authors: Chen Su Shing
Published: June 2010

A theorem of Beboutov and Kakutani is extended to a large class of homeomorphism groups.
On likelihood ratios of measures given by Markov chains
Authors: LePage R. , Mandrekar V.
Published: June 2010

In this note we study conditions for absolute continuity and singularity of two measures induced by finite Markov chains. These conditions are derived from a general result on singularity-continuity dichotomy.
Completely additive measure and integration
Authors: Shorb Alan McK.
Published: June 2010

This paper is an extension of the efforts to cast the theory of measure and integration into the framework of nonstandard analysis, begun by Robinson [7, particularly Theorem 3.5.2], and continued by Bernstein and Wattenberg, Loeb and Henson. The principal result, Theorem 3, is: There exists a completely additive measure function defined on all subsets of

R
R

which nearly agrees with Lebesgue measure and is nearly translation invariant on bounded sets. Its integral is defined for all sets and functions, and nearly agrees with the Lebesgue integral.
Primary ideals in rings of analytic functions
Authors: Alling Norman L.
Published: June 2010

Primary ideals in the ring of all analytic functions on a noncompact Riemann surface are analyzed with the aid of classical valuation theory.
A new generating function for a generalized function of two variables
Authors: Sharma B. L. , Abiodun R. F. A.
Published: June 2010

We discuss a new generating function for a generalized function of two variables and, in a particular case, obtain an interesting formula for a

G
G

-function,
Bounded multiplier convergence in measure of random vector series
Authors: Ryll-Nardzewski C. , Woyczyński W. A.
Published: June 2010

If the series

Σ

f
i

\Sigma {f_i}

of random vectors with values in a Banach space converges unconditionally in measure, then, for each

(

λ
i

)
ϵ

l
∞

({\lambda _i})\epsilon {l^\infty }

, the series

Σ

λ
i

f
i

\Sigma {\lambda _i}{f_i}

also converges unconditionally in measure.
Unique Hahn-Banach extensions and Korovkin’s theorem
Authors: Kurtz Lynn C.
Published: June 2010

This paper characterizes in terms of weak topologies those bounded linear functionals on a subspace which have unique Hahn-Banach extensions to the whole linear normed space. The relationship to the Choquet boundary is discussed, and a Korovkin type theorem is obtained.
Representation of a crinkled arc
Authors: Vitale Richard A.
Published: June 2010

Johnson [A crinkled arc, Proc. Amer. Math. Soc. 25 (1970), 375-376] has shown that under suitable normalizations all crinkled arcs are unitarily equivalent. Using this result, we find a general series expansion for a crinkled arc:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis t right-parenthesis equals StartRoot 2 EndRoot sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts x Subscript n Baseline StartFraction sine left-parenthesis n minus 1 slash 2 right-parenthesis pi t Over left-parenthesis n minus 1 slash 2 right-parenthesis pi EndFraction comma">

f
(
t
)
=

2

∑

n
=
1

∞

x
n

sin
⁡
(
n
−
1

/

2
)
π
t

(
n
−
1

/

2
)
π

,

f(t) = \sqrt {2} \sum _{n=1}^\infty x_n \tfrac {\sin (n - 1/2)\pi t}{(n - 1/2)\pi } ,

\]
where

{

x
n

}

\{ {x_n}\}

is an orthonormal set.
Identities in combinatorics. II. A 𝑞-analog of the Lagrange inversion theorem
Authors: Andrews George E.
Published: June 2010

A

q
q

-analog of Lagrange’s inversion theorem is obtained. It is applied to give a new proof of an expansion theorem due to Carlitz and to obtain formulae for certain combinatorial numbers studied by Carlitz.
On Lakshmikantham’s comparison for ordinary differential equations
Authors: Vidossich Giovanni
Published: June 2010

The paper describes the relation between Lakshmikantham’s comparison and other known facts.
On functional equations related to Mielnik’s probability spaces
Authors: Blakemore C. F. , Stanojevic C. V.
Published: June 2010

It is shown that the method used by C. V. Stanojevic to obtain a characterization of inner product spaces in terms of a Mielnik probability space of dimension

2
2

does not admit a generalization to dimension

n
>
2

n > 2

.