bluuebunny/crossref_metadata_2025 huggingface dataset:
Distortions properties of alpha-starlike functions
Authors: Miller Sanford S.
Published: June 2010

Let

α
\alpha

be real and suppose that

f
(
z
)
=
z
+

Σ
2
∞

a
n

z
n

f(z) = z + \Sigma _2^\infty {a_n}{z^n}

is regular in the unit disc

D
D

with

f
(
z
)

f
′

(
z
)
≠
0

f(z)f’(z) \ne 0

in

0
>

|

z

|

>
1

0 > |z| > 1

. If
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R e left-bracket left-parenthesis 1 minus alpha right-parenthesis z f prime left-parenthesis z right-parenthesis slash f left-parenthesis z right-parenthesis plus alpha left-parenthesis left-parenthesis z f left-parenthesis z right-parenthesis slash f prime left-parenthesis z right-parenthesis right-parenthesis plus 1 right-parenthesis right-bracket greater-than 0">

Re
⁡
[
(
1
−
α
)
z

f
′

(
z
)

/

f
(
z
)
+
α
(
(
z
f
(
z
)

/

f
′

(
z
)
)
+
1
)
]
>
0

\operatorname {Re} [(1 - \alpha )zf’(z)/f(z) + \alpha ((zf(z)/f’(z)) + 1)] > 0

for

z
∈
D

z \in D

, then

f
(
z
)

f(z)

is said to be an alpha-starlike function. These functions are univalent and they very naturally unify the classes of starlike

(
α
=
0
)

(\alpha = 0)

and convex

(
α
=
1
)

(\alpha = 1)

functions. The author obtains the

1
4

\tfrac {1}{4}

-theorem, sharp bounds on

|

f
(
z
)

|

|f(z)|

and

|

f
′

(
z
)

|

|f’(z)|

, and growth conditions on

M
(
r
)

M(r)

.
A note on Hall’s lemma
Authors: Gaier Dieter
Published: June 2010

Let H be the half plane

{
z
:
Re
⁡
z
>
0
}

\{ z:\operatorname {Re} z > 0\}

. Let y be a Jordan arc joining

z
=
0

z = 0

and

z
=

e

i
α

(
0
≦
α
>
π

/

2
)

z = {e^{i\alpha }}\;(0 \leqq \alpha > \pi /2)

in

H
∩
{

|

z

|

≦
1
}

H \cap \{ |z| \leqq 1\}

. Let

γ
∗

{\gamma ^\ast }

be the segment

z
=
i
y

(
0
≦
y
≦
1
)

z = iy\;(0 \leqq y \leqq 1)

of the imaginary axis. If

ω
(
z
,
γ
)

\omega (z,\gamma )

is the harmonic measure of

γ
\gamma

with respect to

H
∖
γ

H\backslash \gamma

and

ω
(
z
,

γ
∗

)

\omega (z,{\gamma ^\ast })

the harmonic measure of

γ
∗

{\gamma ^\ast }

with respect to H, then

ω
(
x
+
i
y
,
γ
)
>
ω
(
x
−
i

|

y

|

,

γ
∗

)

\omega (x + iy,\gamma ) > \omega (x - i|y|,{\gamma ^\ast })

.
Operators with inverses similar to their adjoints
Authors: Singh U. N. , Mangla Kanta
Published: June 2010

If

T
T

is an invertible operator on a Hilbert space such that

S

−
1

T

−
1

S
=

T
∗

{S^{ - 1}}{T^{ - 1}}S = {T^ \ast }

and

0
∉

Cl

(
W
(
S
)
)

0 \notin {\text {Cl}}(W(S))

for some invertible operator

S
S

, where

Cl

(
W
(
S
)
)

{\text {Cl}}(W(S))

denotes the closure of the numerical range of

S
S

and

T
∗

{T^ \ast }

is the adjoint of

T
T

, then it is shown that

T
T

is similar to a unitary operator. In fact, this has been proved as a corollary to a more general result, which also includes the corresponding result of J. P. Williams for selfadjoint operators.
The Gauss map in spaces of constant curvature
Authors: Weiner Joel L.
Published: June 2010

Let

N
N

be a complete simply connected Riemannian manifold of constant sectional curvature

≠
0

\ne 0

. Let

M
M

be an immersed Riemannian hypersurface of

N
N

. The Gauss map on

M
M

based at a point

p
p

in

N
N

is defined. Suppose a Gauss map on

M
M

has constant rank less than the dimension of

M
M

; then

M
M

is generated by Riemannian submanifolds with constant sectional curvature. The sectional curvature of each of these generating submanifolds of

M
M

has the same sign as the sectional curvature of

N
N

.
Comparing almost continuous functions
Authors: Long Paul E. , Carnahan Donald A.
Published: June 2010

Three definitions as to when a function from one topological space to another is called almost continuous are cited from the literature. Similarities and dissimilarities of two of these definitions are discussed.
The homotopy groups of spaces whose cohomology is a 𝑍_{𝑝} truncated polynomial algebra
Authors: Shar Albert
Published: June 2010

On such a space one can define a “Hopf” invariant homomorphism

h
:

π

q
n
−
1

(
K
)
→
Z

h:{\pi _{qn - 1}}(K) \to Z

in two ways. We prove both definitions are equivalent and show that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript p Baseline pi Subscript i Baseline left-parenthesis upper K right-parenthesis asymptotically-equals Subscript p Baseline pi Subscript i minus 1 Baseline left-parenthesis upper S Superscript n minus 1 Baseline right-parenthesis circled-plus Subscript p Baseline pi Subscript i Baseline left-parenthesis upper S Superscript q n minus 1 Baseline right-parenthesis">

p

π
i

(
K
)
≃

p

π

i
−
1

(

S

n
−
1

)
⊕

p

π
i

(

S

q
n
−
1

)

{}_p{\pi _i}(K) \simeq {}_p{\pi _{i - 1}}({S^{n - 1}}) \oplus {}_p{\pi _i}({S^{qn - 1}})

if and only if there is an

α
∈

π

q
n
−
1

(
K
)

\alpha \in {\pi _{qn - 1}}(K)

such that

(
h
(
α
)
,
p
)
=
1

(h(\alpha ),p) = 1

. As immediate corollaries of this we get a result of Toda on the homotopy groups of the reduced product spaces of spheres and a well-known result of Serre on the odd primary parts of the homotopy groups of spheres.
On weakly flat 1-spheres
Authors: Daverman Robert J.
Published: June 2010

With his cellularity criterion, McMillan has provided a characterization of those complementary domains of an

(
n
−
1
)

(n - 1)

-sphere in the

n
n

-sphere

S
n

{S^n}

that are open

n
n

-cells

(
n
≠
4
)

(n \ne 4)

. Paul F. Duvall has used the cellularity criterion to characterize those

k
k

-spheres

Σ
\Sigma

in

S
n

(
2
≦
k
≦
n
−
3
)

{S^n}(2 \leqq k \leqq n - 3)

such that

S
n

−
Σ

{S^n} - \Sigma

is homeomorphic to the complement of the standard

k
k

-sphere in

S
n

{S^n}

. The purpose of this note is to indicate how McMillan’s methods can be reapplied to handle the case

k
=
1
(
n
≠
4
)

k = 1(n \ne 4)

as well.
A metric formula for the Godbillon-Vay invariant for foliations
Authors: Reinhart Bruce L. , Wood John W.
Published: June 2010

The Godbillon-Vey invariant for a foliation of codimension 1 is acohomology class defined by a

3
3

-form. On a Riemannian manifold, this form can be expressed in terms of the curvature and torsion of the normal curve family and the second fundamental form of the leaves.
Continued fractions with sequences of partial quotients
Authors: Hirst K. E.
Published: June 2010

Three results are proved concerning the Hausdorff fractional dimension of sets of continued fractions whose partial quotients belong to given sequences.
On Kneser’s addition theorem in groups
Authors: Diderrich George T.
Published: June 2010

The following theorem is proved. Theorem A. Let

G
G

be a group written additively with finite nonempty subsets

A
,
B

A,B

. Assume that

B
B

is commutative, i.e.

b
1

+

b
2

=

b
2

+

b
1

{b_1} + {b_2} = {b_2} + {b_1}

, for

b
1

,

b
2

∈
B

{b_1},{b_2} \in B

. Then there exists an Abelian subgroup

H
H

of

G
G

such that

A
+
B
+
H
=
A
+
H
+
B
=
A
+
B

A + B + H = A + H + B = A + B

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper A plus upper B EndAbsoluteValue greater-than-over-equals StartAbsoluteValue upper A plus upper H EndAbsoluteValue plus StartAbsoluteValue upper B plus upper H EndAbsoluteValue minus StartAbsoluteValue upper H EndAbsoluteValue">

|

A
+
B

|

≧

|

A
+
H

|

+

|

B
+
H

|

−

|

H

|

|A + B| \geqq |A + H| + |B + H| - |H|

. This is Kneser’s theorem, if

G
G

is Abelian. Also, as an application of the above theorem, the following is proved. Theorem B. Let

G
G

be a finite group of order

υ
(
υ
>
1
)

\upsilon (\upsilon > 1)

and let

a
1

,
⋯
,

a
υ

{a_1}, \cdots ,{a_\upsilon }

be a sequence (repeats are allowed) of nonzero elements of

G
G

. The set

S
S

of sums

a

i
1

+
⋯
+

a

i
t

{a_{{i_1}}} + \cdots + {a_{{i_t}}}

where

1
≦

i
1

>
⋯
>

i
t

≦
υ

1 \leqq {i_1} > \cdots > {i_t} \leqq \upsilon

and

1
≦
t
≦
υ

1 \leqq t \leqq \upsilon

must contain a nontrivial subgroup

H
H

of

G
G

. Finally, the Kemperman

d
d

-transform, a transform similar to the Dyson

e
e

-transform, is introduced and evidence is given to support the conjecture that Theorem A remains true, if the commutativity of

B
B

is dropped.
On composition series in finite groups
Authors: Bauman Steven
Published: June 2010

Theorem. Let

G
G

be a finite group and

H
H

a solvable subgroup of

G
G

. Suppose that the Schreier conjecture holds. Then

G
G

is solvable iff

G
G

has an

H
H

-composition series.
Equivalence of integrals
Authors: Chatfield J. A.
Published: June 2010

Suppose

R
R

is the set of real numbers and

N
N

is the set of nonnegative real numbers, each of

G
G

and

F
F

is a function from

R
×
R

R \times R

to

N
N

. All integrals considered are of the subdivision-refinement type. This paper gives necessary and sufficient conditions for

∫
a
b

F
=

∫
a
b

G

\int _a^b {F = } \int _a^b G

. A necessary and sufficient condition for

∫
a
b

G
2

=
0

\int _a^b {{G^2} = 0}

is also given.
On Bazilevič functions
Authors: Singh Ram
Published: June 2010

In this paper we study two subclasses of the class

B
B

—the so-called class of Bazilevič functions. It is known that

B
B

is a subclass of

S
S

, the class of univalent functions in

E
=
{
z

|

|

z

|

>
1
}

E = \{ z | |z| > 1 \}

. Some well-known results pertaining to various subclasses of

S
S

will follow as corollaries to the theorems which are obtained here.
Lower bounds for solutions of hyperbolic inequalities in unbounded regions
Authors: Murray Amy C.
Published: June 2010

This paper considers

C
2

{C^2}

solutions

u
=
u
(
t
,
x
)

u = u(t,x)

of the differential inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper L u EndAbsoluteValue less-than-over-equals k 1 left-parenthesis t comma x right-parenthesis StartAbsoluteValue u EndAbsoluteValue plus k 2 left-parenthesis t comma x right-parenthesis StartAbsoluteValue EndAbsoluteValue nabla u StartAbsoluteValue EndAbsoluteValue">

|

L
u

|

≦

k
1

(
t
,
x
)

|

u

|

+

k
2

(
t
,
x
)

|

|

∇
u

|

|

|Lu| \leqq {k_1}(t,x)|u| + {k_2}(t,x)||\nabla u||

. The coefficients of the hyperbolic operator

L
L

depend on both

t
t

and

x
x

. Explicit lower bounds are given for the energy of

u
u

in a region of

x
x

-space expanding at least as fast as wave-fronts for

L
L

. These bounds depend on the asymptotic behavior of

k
1

,

k
2

{k_1},{k_2}

, and the coefficients of

L
L

. They do not require boundary conditions on

u
u

.
Measurable solutions of functional equations related to information theory
Authors: Kannappan Pl. , Ng C. T.
Published: June 2010

Measurable solutions of functional equations connected with Shannon’s measure of entropy, directed divergence or information gain and inaccuracy are found.
Ideals of square summable power series in several variables
Authors: Radlow James
Published: June 2010

Let

C

(
z
)

\mathcal {C}(z)

be the Hilbert space of formal power series in

z
1

,
⋯
,

z
r

(
r
≧
1
)

{z_1}, \cdots ,{z_r}(r \geqq 1)

. An ideal of

C

(
z
)

\mathcal {C}(z)

is a vector subspace

M

\mathcal {M}

of

C

(
z
)

\mathcal {C}(z)

which contains

z
1

f
(
z
)
,
⋯
,

z
r

f
(
z
)

{z_1}f(z), \cdots ,{z_r}f(z)

whenever it contains

f
(
z
)

f(z)

. If

B
(
z
)

B(z)

is a formal power series such that

B
(
z
)
f
(
z
)

B(z)f(z)

belongs to

C

(
z
)

\mathcal {C}(z)

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper B left-parenthesis z right-parenthesis f left-parenthesis z right-parenthesis StartAbsoluteValue EndAbsoluteValue equals StartAbsoluteValue EndAbsoluteValue f left-parenthesis z right-parenthesis StartAbsoluteValue EndAbsoluteValue">

|

|

B
(
z
)
f
(
z
)

|

|

=

|

|

f
(
z
)

|

|

||B(z)f(z)|| = ||f(z)||

, then the set

M

(
B
)

\mathcal {M}(B)

of all products

B
(
z
)
f
(
z
)

B(z)f(z)

is a closed ideal of

C

(
z
)

\mathcal {C}(z)

. In the case

r
=
1

r = 1

, Beurling showed that every closed ideal is of this form for some such

B
(
z
)

B(z)

. Here we give conditions under which a closed ideal is of the form

M

(
B
)

\mathcal {M}(B)

for

r
≧
2

r \geqq 2

.
On normal derivations
Authors: Anderson Joel
Published: June 2010

Let

Δ
T

{\Delta _T}

be the derivation on

B

(

H

)

\mathfrak {B}(\mathcal {H})

defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta Subscript upper T Baseline left-parenthesis upper X right-parenthesis equals upper T upper X minus upper X upper T left-parenthesis upper T comma upper X element-of German upper B left-parenthesis script upper H right-parenthesis right-parenthesis">

Δ
T

(
X
)
=
T
X
−
X
T
(
T
,
X
∈

B

(

H

)
)

{\Delta _T}(X) = TX - XT(T,X \in \mathfrak {B}(\mathcal {H}))

. We prove that if

T
T

is an isometry or a normal operator, then the range of

Δ
T

{\Delta _T}

is orthogonal to the null space of

Δ
T

{\Delta _T}

. Also, we prove that if

T
T

is normal with an infinite number of points in its spectrum then the closed linear span of the range and the null space of

Δ
T

{\Delta _T}

is not all of

B

(

H

)

\mathfrak {B}(\mathcal {H})

.
On the homotopy type of irregular sets
Authors: Duvall P. F. , Husch L. S.
Published: June 2010

If

M
M

is an open connected manifold and

h
h

is a homeomorphism of

M
M

onto itself such that

h
h

is positively regular on

M
M

and the set of irregular points,

Irr
⁡
(
h
)

\operatorname {Irr} (h)

, is a nonseparating compactum, then it is shown that

Irr
⁡
(
h
)

\operatorname {Irr} (h)

is a strong deformation retract of

M
M

.
On splitting cotorsion radicals
Authors: Ramamurthi V. S.
Published: June 2010

For a category of modules, the notion, dual to that of a torsion radical, has been called a cotorsion radical. In this paper, the following two properties are examined for a cotorsion radical

ρ
\rho

: (1) If

N
N

is a submodule of

M
M

and

ρ
(
M
)
=
M

\rho (M) = M

, then

ρ
(
N
)
=
N

\rho (N) = N

. (2) The exact sequence

0
→
ρ
(
M
)
→
M
→
M

/

ρ
(
M
)
→
0

0 \to \rho (M) \to M \to M/\rho (M) \to 0

splits for each module

M
M

.
Bianalytic functions with exceptional values
Authors: Krajkiewicz P.
Published: June 2010

Let

f
(
z
,

z
¯

)

f(z,\bar z)

be a bianalytic function which omits the value zero in some deleted neighborhood

A
A

of an isolated singularity

z
0

{z_0}

. It is shown that there is a function

g
(
z
)

g(z)

analytic on

A
A

and a function

h
(
z
,

z
¯

)

h(z,\bar z)

bianalytic on

A
A

with a nonessential singularity at

z
0

{z_0}

such that

f
(
z
,

z
¯

)
=
g
(
z
)
h
(
z
,

z
¯

)

f(z,\bar z) = g(z)h(z,\bar z)

on

A
A

.
Bounded, conservative, linear operators and the maximal group. II
Authors: Kelly E. P. , Hogan D. A.
Published: June 2010

Let

V
V

denote an infinite dimensional Banach space over the complex field,

B
[
V
]

B[V]

the bounded linear operators on

V
V

and

F
F

a closed subspace of

V
V

. An element of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper T Subscript upper F Baseline equals left-brace upper T vertical-bar upper T element-of upper B left-bracket upper V right-bracket comma upper T left-parenthesis upper F right-parenthesis subset-of-or-equal-to upper F right-brace">

T

F

=
{
T

|

T
∈
B
[
V
]
,
T
(
F
)
⊆
F
}

{\mathcal {T}_F} = \{ T|T \in B[V],T(F) \subseteq F\}

is called a conservative operator. Some sufficient conditions for

T
∈

T

F

T \in {\mathcal {T}_F}

to be in the boundary,

B

\mathcal {B}

, of the maximal group,

M

\mathcal {M}

, of invertible elements are determined. For example, if

T
∈

T

F

T \in {\mathcal {T}_F}

, is such that (i)

V
V

is the topological direct sum of

R

(
T
)

\mathcal {R}(T)

and

N
(
T
)
≠
{
θ
}

N(T) \ne \{ \theta \}

, (ii)

T
T

is an automorphism on

R

(
T
)
∩
F

\mathcal {R}(T) \cap F

, then

T
∈

B

T \in \mathcal {B}

. Also, the complement of the closure of

M

\mathcal {M}

is discussed. This is an extension of another paper by the same authors [6].
The absolute continuity of a class of Toeplitz operators
Authors: Rosenblum Marvin
Published: June 2010

It is shown that certain generalized Toeplitz operators are the real parts of subnormal operators that have no normal part, and hence are absolutely continuous. It follows that the Toeplitz operators associated with the real parts of nonconstant bounded holomorphic functions in the polydisc are absolutely continuous.
On cohomology groups of certain subcomplexes of Dolbeault complexes
Authors: Mong Shaw
Published: June 2010

The paper first shows that for a path connected topological group acting analytically on a complex manifold

M
M

, the induced action on the cohomology groups of antiholomorphic forms on

M
M

is trivial. By the same method it then shows that the cohomology groups of complex

{
ker
⁡

∂

p
,
∗

}

\{ \ker {\partial ^{p,\ast }}\}

can be injected into the Dolbeault cohomology groups.
Realizability of representations in cyclotomic fields
Authors: Fein Burton
Published: June 2010

In this paper sufficient conditions are given for a real valued complex irreducible character

χ
\chi

to be written in

Q
(
χ
)
(

ε
q

)
,

ε
q

Q(\chi )({\varepsilon _q}),{\varepsilon _q}

a primitive

q
q

th root of unity,

q
q

a prime dividing the order of the group.
A note on the Radon-Nikodym theorem
Authors: Cho Tae Geun , Tong A.
Published: June 2010

This paper gives a necessary and sufficient condition in order that a bounded linear mapping from

L
1

(
μ
)

{L^1}(\mu )

into a Banach space be compact. It is applied to provide a slightly improved form of the Radon-Nikoydm theorem for vector valued measures and to give a sufficient condition in order that the range of a vector valued measure of bounded variation be compact.
An integral equation arising in potential theory
Authors: Jones H. T. , Specht E. J.
Published: June 2010

This paper gives an integral equation, the solution of which is a solution of a classical problem in potential theory: Given a region with boundary

B

\mathcal {B}

, what distribution of charge on

B

\mathcal {B}

will produce a potential having specified values on

B

\mathcal {B}

? The paper also indicates briefly how the integral equation is useful in simplifying certain proofs and extending certain theorems in potential theory.
Distance spheres in complex projective spaces
Authors: Weinstein Alan
Published: June 2010

Distance spheres in complex projective spaces are counterexamples to the odd-dimensional extension of a lemma of Klingenberg.
Schauder bases and norm ideals of compact operators
Authors: Holub J. R.
Published: June 2010

This paper studies the relationship between unitarily invariant crossnorms on the tensor product of Hilbert spaces and the corresponding symmetric sequence spaces. It is shown that problems concerning the behavior of norm ideals of compact operators on Hilbert space can often be treated successfully by translating the problem to one concerned with basis theory (and hence, perhaps, to one more transparent). For example, Schatten has asked for necessary and sufficient conditions that the conjugate space of a minimal norm ideal again be a minimal norm ideal. Using the ideas developed here we give such a characterization.
Sufficient conditions for periodicity of a Killing vector field
Authors: Lynge Walter C.
Published: June 2010

Let

X
X

be a complete Killing vector field on an

n
n

-dimensional connected Riemannian manifold. Our main purpose is to show that if

X
X

has as few as

n
n

closed orbits which are located properly with respect to each other, then

X
X

must have periodic flow. Together with a known result, this implies that periodicity of the flow characterizes those complete vector fields having all orbits closed which can be Killing with respect to some Riemannian metric on a connected manifold

M
M

. We give a generalization of this characterization which applies to arbitrary complete vector fields on

M
M

.
Spheres which are loop spaces 𝑚𝑜𝑑 𝑝
Authors: Wilkerson Clarence
Published: June 2010

If

S

(
p
)

2
n
−
1

S_{(p)}^{2n - 1}

has a loop space structure, than

n

|

p
−
1

n|p - 1

.
Pontryagin classes of vector bundles over 𝐵𝑆𝑝(𝑛)
Authors: O’Neill Duane
Published: June 2010

Let

X
X

be a finite skeleton of the classifying space of

BSp
⁡
(
n
)
,

γ
0

→
BSp
⁡
(
n
)

\operatorname {BSp} (n),{\gamma _0} \to \operatorname {BSp} (n)

, the classifying bundle for

Sp
⁡
(
n
)

\operatorname {Sp} (n)

vector bundles and

γ
→
X

\gamma \to X

the restriction of

γ
0

{\gamma _0}

over

X
X

. If

ξ
→
X

\xi \to X

is another

Sp
⁡
(
n
)

\operatorname {Sp} (n)

vector bundle, the Pontryagin classes

p
q

(
ξ
)

{p_q}(\xi )

must be congruent to

d
1
q

p
q

(
γ
)

d_1^q{p_q}(\gamma )

modulo certain odd primes. Equality obtains if

ξ
\xi

is the restriction over

X
X

of a

ξ
0

→

BSp

(
n
)

{\xi _0} \to {\operatorname {BSp}}(n)

. In particular,

Sp

(
m
)

{\operatorname {Sp}}(m)

vector bundles

θ
\theta

over

BSp

(
n
)

{\operatorname {BSp}}(n)

have

p
(
θ
)
=
1

p(\theta ) = 1

if

m
>
n

m > n

.
Imbedding classes and 𝑛-minimal complexes
Authors: Ummel Brian R.
Published: June 2010

Algebraic and geometrical techniques are used to study examples (new and previously conjectured) of

n
n

-dimensional simplicial complexes which cannot be topologically imbedded in Euclidean

2
n

2n

-space, but each proper subcomplex of any of them can be imbedded in Euclidean

2
n

2n

-space.
A two-color theorem for analytic maps in 𝑅ⁿ
Authors: Cain Bryan E.
Published: June 2010

f
1

,
⋯
,

f
k

{f_1}, \cdots ,{f_k}

are real analytic functions on

R
n

{R^n}

th then the connected components of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n Baseline minus left-bracket f 1 Superscript negative 1 Baseline left-parenthesis 0 right-parenthesis union midline-horizontal-ellipsis union f Subscript k Superscript negative 1 Baseline left-parenthesis 0 right-parenthesis right-bracket">

R
n

∖
[

f
1

−
1

(
0
)
∪
⋯
∪

f
k

−
1

(
0
)
]

{R^n}\backslash [f_1^{ - 1}(0) \cup \cdots \cup f_k^{ - 1}(0)]

can be “colored” with two colors so that two components will have different colors whenever their common boundary contains a topological

(
n
−
1
)

(n - 1)

-manifold.
A lifting formula for the Hilbert symbol
Authors: Bender Edward A.
Published: June 2010

The Hilbert symbol in an extension field is expressed in terms of norms, traces and Hilbert symbols in the ground field.
A condition for the existence of a strongly embedded subgroup
Authors: Aschbacher Michael
Published: June 2010

Let

D
D

be a normal set of involutions in a finite group. Form the involutory graph with vertex set

D
D

by joining distinct commuting elements of

D
D

. Assume the product of any two such elements is in

D
D

, and the graph is disconnected. Then the group generated by

D
D

contains a strongly embedded subgroup. Two corollaries are proved.
Characterization of certain groupoids and loops
Authors: Kannappan Pl.
Published: June 2010

Groupoids (loops) possessing the left inverse property (lip) (also the right inverse property) and crossed-inverse (CI) loops are characterized as subvarieties of the variety of groupoids satisfying a single identity.
Homogeneous solution of a nonlinear differential equation
Authors: Reid James L.
Published: June 2010

The nonlinear second order differential equation satisfied by the homogeneous function

y
=

[
a

u
m

+
m
b

u
j

v
n

+
c

v
m

]

k

/

m

,
m
=
j
+
n

y = {[a{u^m} + mb{u^j}{v^n} + c{v^m}]^{k/m}},m = j + n

, is obtained. Functions

u
u

and

v
v

satisfy independently the linear equation

y
¨

+
r
(
t
)

y
˙

+
q
(
t
)
y
=
0

\ddot y + r(t)\dot y + q(t)y = 0

. The nonlinear equation derived contains previous results as special cases of

r
(
t
)

r(t)

, of the constants

a
,
b

a,b

, and

c
c

, and of the numbers

k
k

and

m
m

.
A second order nonlinear oscillation theorem
Authors: Wong James S. W.
Published: June 2010

An oscillation criterion is given for the second order nonlinear equation

X
+
a
(
t
)

|

x

|

γ

sgn
⁡
x
=
0
,
γ
>
1

X + a(t)|x{|^\gamma }\operatorname {sgn} x = 0,\gamma > 1

, where the coefficient

a
(
t
)

a(t)

is not assumed to be nonnegative for all large values of

t
t

.
Axiomatic cohomology for Banach modules
Authors: Craw I. G.
Published: June 2010

A characterisation is given of the cohomology functor for Banach modules over a fixed Banach algebra. The result is used to simplify the proofs of two known isomorphism theorems.
Real-linear operators on quaternionic Hilbert space
Authors: Powers N. C.
Published: June 2010

The main result is that any continuous real-linear operator

A
A

on a quaternionic Hilbert space has a unique decomposition

A
=

A
0

+

i
1

A
1

+

i
2

A
2

+

i
3

A
3

A = {A_0} + {i_1}{A_1} + {i_2}{A_2} + {i_3}{A_3}

, where the

A
ν

{A_\nu }

are continuous linear operators and

(

i

1
,

,

i
2

,

i
3

)

({i_{1,}},{i_2},{i_3})

is any right-handed orthonormal triad of vector quaternions. Other results concern the place of the colinear and complex-linear operators in this characterisation and the effect on the

A
ν

{A_\nu }

of a rotation of the triad of vector quaternions. A new result concerning symplectic images of a quaternionic Hilbert space is also presented.
On the variational method for the existence of solutions of nonlinear equations of Hammerstein type
Authors: de Figueiredo Djairo G. , Gupta Chaitan P.
Published: June 2010

Let

X
X

be a real Banach space and

X
∗

{X^ \ast }

its conjugate Banach space. Let

A
A

be an unbounded monotone linear mapping from

X
X

to

X
∗

{X^ \ast }

and

N
N

a potential mapping from

X
∗

{X^ \ast }

to

X
X

. In this paper we establish the existence of a solution of the equation

u
+
A
N
u
=
v

u + ANu = v

for a given

v
v

in

X
∗

{X^ \ast }

using variational method. Our method consists in using a splitting of

A
A

via an auxiliary Hilbert space and solving an equivalent equation in this auxiliary Hilbert space. In §2, we prove the same result in the case when

X
X

is a Hilbert space using the natural splitting of

A
A

in terms of its square root. We do this to compare and contrast the proofs in the two cases.
Intersecting unions of maximal convex sets
Authors: Breen Marilyn
Published: June 2010

Hare and Kenelly have characterized the intersection of the maximal starshaped subsets of a set

S
S

, where

S
S

is compact, simply connected and planar, and Sparks has solved the general problem for maximal

L
n

{L_n}

sets. In this paper, a similar question is examined for unions of maximal convex sets: Let

S
S

be a subset of

R
2

,

C

{R^2},\mathcal {C}

the collection of all maximal convex subsets of

S
S

, and

N

=
{
A
∪
B
:
A
,
B

\mathcal {N} = \{ A \cup B:A,B

distinct members of

C

}

\mathcal {C}\}

. Then

∩

N

{ \cap ^\mathcal {N}}

is expressible as a union of three or fewer convex sets.
On products of powers in groups
Authors: Lyndon Roger , McDonough Thomas , Newman Morris
Published: June 2010

In this note we show that a product of

N
N

h powers in a group cannot in general be expressed as a product of fewer

N
N

th powers. This extends a result of Lyndon and Newman [1].
On the boundedness and unboundedness of certain convolution operators on nilpotent Lie groups
Authors: Goodman Roe
Published: June 2010

One method of proving irreducibility of the “principal series” representations of semisimple Lie groups involves showing that a certain nonintegrable function on a nilpotent subgroup

X
X

cannot be regularized to give a bounded convolution operator on

L
2

(
X
)

{L_2}(X)

. This note gives an elementary proof of this unboundedness property for the groups

X
X

which occur in real-rank one semisimple groups.
Reducing decompositions for strictly cyclic operators
Authors: Bouldin Richard
Published: June 2010

If

T
T

is a strictly cyclic operator on

H
H

then

H
H

has a direct sum decomposition

H
1

⊕

H
2

{H_1} \oplus {H_2}

where

H
1

{H_1}

and

H
2

{H_2}

are invariant under

T
T

if and only if the spectrum of

T
T

is not connected. If

λ
\lambda

is a reducing eigenvalue for the strictly cyclic operator

T
T

then the multiplicity of

λ
\lambda

is one and

λ
\lambda

is an isolated point of the spectrum of

T
T

.
On special linear characters of free groups of rank 𝑛≥4
Authors: Whittemore Alice
Published: June 2010

Let

F
n

{F_n}

be a free group of rank

n
n

. In a recent paper R. Horowitz has shown that for

n
≦
3

n \leqq 3

the ideal

I
n

{I_n}

in the ring of special linear characters of

F
n

{F_n}

consisting of those polynomials in the characters which vanish for all representations of

F
n

{F_n}

by subgroups of

S
L
(
2
,
C
)

SL(2,C)

is principal. In this paper the case

n
=
4

n = 4

is investigated; it is shown that for

n
>
3
,

I
n

n > 3,{I_n}

is not principal.
Normal subgroups of Fuchsian groups with fixed parabolic class number
Authors: Berger Marc
Published: June 2010

Let

F
F

be a finitely generated Fuchsian group of the first kind. We provided necessary and sufficient conditions such that, for any

t
≧
1
,
F

t \geqq 1,F

has only a finite number of normal subgroups with

t
t

parabolic classes.
A note on topological dynamics and limiting equations
Authors: Imdadi S. M. Shamim , Rama Mohana Rao M.
Published: June 2010

A theorem on uniform asymptotic stability of the null solution of a system of differential equations is proved while assuming that the null solution of a limiting equation is uniformly asymptotically stable. This generalizes some of the results of L. Markus.
On a problem of F. Riesz concerning proximity structures
Authors: Thron W. J.
Published: June 2010

It is shown that every separated Lodato proximity is induced by the elementary proximity on a

T
1

{T_1}

bicompactification of the original space.
On the reduction of complex bordism to unoriented bordism
Authors: Johnson David Copeland
Published: June 2010

The image of the natural transformation from the complex bordism of a CW complex

X
,
M

U
∗

(
X
)

X,M{U_ \ast }(X)

, to its unoriented bordism,

N
∗

(
X
)

{N_ \ast }(X)

, is contained in a subgroup identified with

H
∗

(
X
;
Z
)
⊗

(

N
∗

)
2

{H_ \ast }(X;Z) \otimes {({N_ \ast })^2}

. A characterization is given for the CW complexes for which the image and the subgroup coincide.
An extension of Haynsworth’s determinant inequality
Authors: Hartfiel D. J.
Published: June 2010

Let

A
A

and

B
B

be positive definite hermitian matrices of order

n
n

. This paper improves a lower bound given for

|

A
+
B

|

|A + B|

by E. V. Haynsworth.
Asymptotic inversion of Laplace transforms: a class of counterexamples
Authors: Lew John S.
Published: June 2010

Let

f
f

be a complex-valued locally integrable function on

[
0
,
+
∞
)

[0, + \infty )

, and let

L
f

Lf

be its Laplace transform, whenever and wherever it exists. We review some known methods, exact and approximate, for recovering

f
f

from

L
f

Lf

. Since numerical algorithms need auxiliary information about

f
f

near

+
∞

+ \infty

, we note that the behavior of

f
f

near

+
∞

+ \infty

depends on the behavior of

L
f

Lf

near 0 +, hence that our ability to retrieve

f
f

is limited by the class of momentless functions, namely, all functions

f
f

such that

L
f
(
s
)

Lf(s)

converges absolutely for

Re
⁡
(
s
)
>
0

\operatorname {Re} (s) > 0

and satisfies
\[

L
f
(
s
)
=
o
(

s
n

)

 near 

0
+

for

n
=
0
,
1
,
2
,
⋯
.

Lf(s) = o({s^n}){\text { near }}0 + \quad {\text {for}}\;n = 0,1,2, \cdots .

\]
We discuss the space

Z
Z

of momentless functions and complex distributions, then construct a family of elements in this space which defy various plausible conjectures.
Local automorphisms are differential operators on some Banach spaces
Authors: Wells John C. , DePrima Charles R.
Published: June 2010

If

E
E

belongs to a certain category of Banach spaces (the

B
∞

{B^\infty }

-smooth spaces) which include Hilbert spaces and if

F
F

is any normed space, it is proved that any local linear automorphism of

C
∞

(
E
,
F
)

{C^\infty }(E,F)

is a differential operator. This generalizes a result of J. Peetre when

E
=

R
n

E = {R^n}

and

F
=
R

F = R

.
A study of monotonically normal spaces
Authors: Borges Carlos R.
Published: June 2010

We obtain various characterizations of monotonically normal spaces which not only answer various questions of Zenor but also allow an elementary proof of a result of Heath and Lutzer. We also prove that elastic spaces are monotonically normal.
Abstract stationary processes
Authors: Saworotnow Parfeny P.
Published: June 2010

Concept of the multivariate stationary process is generalized to an infinite-dimensional case. Representation theorems are derived from the theory of Hilbert modules.
Duality in 𝐵*-algebras
Authors: McKilligan Sheila A.
Published: June 2010

Let

X
X

be a locally compact Hausdorff space and let

C
0

(
X
)

{C_0}(X)

be the algebra of continuous functions on

X
X

vanishing at infinity. Then

C
0

(
X
)

{C_0}(X)

is a dual algebra if and only if the operator

μ
→
f
d
μ

\mu \to fd\mu

is weakly completely continuous on

C
0

(
X

)
∗

{C_0}{(X)^ \ast }

for all

f
∈

C
0

(
X
)

f \in {C_0}(X)

. This improves a recent result of P. K. Wong and provides a description of dual

B
∗

{B^ \ast }

-algebras.
A characterization of four-dimensional Einstein manifolds
Authors: Yen Yi Ching
Published: June 2010

It is shown that for any

P
P

on a four-dimensional Einstein manifold

M
M

, there exists an orthonormal basis

{

Y
i

}

\{ {Y_i}\}

in

T
p

(
M
)

{T_p}(M)

such that all the sectional curvatures

K
(

Y
i

,

Y
j

)

K({Y_i},{Y_j})

are equal.
Fields of constants of infinite higher derivations
Authors: Deveney James K.
Published: June 2010

Let

K
K

be a field of characteristic

p
≠
0

p \ne 0

, and let

P
P

be its maximal perfect subfield. Let

h
h

be a subfield of

K
K

containing

P
P

such that

K
K

is separable over

h
h

. We prove: Every regular subfield of

K
K

containing

h
h

is the field of constants of a set of higher derivations on

K
K

if and only if (1) the transcendence degree of

K
K

over

h
h

is finite, and (2)

K
K

has a separating transcendency basis over

h
h

. This result leads to a generalization of the Galois theory developed in [4].
Identities in implicative semilattices
Authors: Margaris Angelo
Published: June 2010

An effective procedure is given for deciding whether or not an equation in the theory of implicative semilattices is an identity.
The structure of a lattice-ordered group as determined by its prime subgroups
Authors: Pierce Keith R.
Published: June 2010

We characterize by structure theorems the classes of all lattice-ordered groups in which (a) every prime subgroup is principal, (b) every proper prime subgroup is principal, and (c) every minimal prime subgroup is principal. These classes are also characterized by the structure of the root system of regular subgroups.
On invariant linear manifolds
Authors: Fillmore P. A.
Published: June 2010

For a linear transformation

A
A

on a Banach space, let

L

(
A
)

\mathcal {L}(A)

be the lattice of (not necessarily closed) invariant subspaces of

A
A

. For

A
A

bounded it is shown that if

L

(
A
⊕
A
)
⊂

L

(
T
⊕
T
)

\mathcal {L}(A \oplus A) \subset \mathcal {L}(T \oplus T)

, or if

L

(
A
)
⊂

L

(
T
)

\mathcal {L}(A) \subset \mathcal {L}(T)

and

T
T

commutes with

A
A

, then

T
T

is a polynomial in

A
A

. In the case of a Hilbert space, if

L

(
A
)
⊂

L

(

A
∗

)

\mathcal {L}(A) \subset \mathcal {L}({A^ \ast })

then

A
∗

{A^ \ast }

is a polynomial in

A
A

.
The fundamental ideal and 𝜋₂ of higher dimensional knots
Authors: Lomonaco S. J.
Published: June 2010

Let

(

S
4

,
k
(

S
2

)
)

({S^4},k({S^2}))

be a knot formed by spinning a polyhedral arc

α
\alpha

about the standard

2
2

-sphere

S
2

{S^2}

in the

3
3

-sphere

S
3

{S^3}

. Then the second homotopy group of

S
4

−
k
(

S
2

)

{S^4} - k({S^2})

as a

Z

π
1

Z{\pi _1}

-module is isomorphic to each of the following: (1) The fundamental ideal modulo the left ideal generated by

a
−
1

a - 1

, where

a
a

is the image in

π
1

(

S
4

−
k
(

S
2

)
)

{\pi _1}({S^4} - k({S^2}))

of a generator of

π
1

(

S
2

−
α
)

{\pi _1}({S^2} - \alpha )

. (2) The first homology group of the kernel of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi 1 left-parenthesis upper S cubed minus k left-parenthesis upper S squared right-parenthesis right-parenthesis right-arrow pi 1 left-parenthesis upper S Superscript 4 Baseline minus k left-parenthesis upper S squared right-parenthesis right-parenthesis">

π
1

(

S
3

−
k
(

S
2

)
)
→

π
1

(

S
4

−
k
(

S
2

)
)

{\pi _1}({S^3} - k({S^2})) \to {\pi _1}({S^4} - k({S^2}))
Criteria for compactness and for discreteness of locally compact amenable groups
Authors: Granirer Edmond
Published: June 2010

Let

G
G

be a locally compact group
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P left-parenthesis upper G right-parenthesis equals left-brace 0 less-than-over-equals phi element-of upper L 1 left-parenthesis upper G right-parenthesis semicolon integral phi left-parenthesis x right-parenthesis d x equals 1 right-brace">

P
(
G
)
=
{
0
≦
ϕ
∈

L
1

(
G
)
;
∫

ϕ
(
x
)
d
x
=
1
}

P(G) = \{ 0 \leqq \phi \in {L_1}(G);\int {\phi (x)dx = 1\} }

and

(

l
a

f
)
(
x
)
=

a

f
(
x
)
=
f
(
a
x
)

({l_a}f)(x) = {}_af(x) = f(ax)

for all

a
,
x
∈
G

a,x \in G

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper L Superscript normal infinity Baseline left-parenthesis upper G right-parenthesis .0 less-than-over-equals normal upper Psi element-of upper L Superscript normal infinity Baseline left-parenthesis upper G right-parenthesis Superscript asterisk Baseline comma normal upper Psi left-parenthesis 1 right-parenthesis equals 1">

f
∈

L
∞

(
G
)
.0
≦
Ψ
∈

L
∞

(
G

)
∗

,
Ψ
(
1
)
=
1

f \in {L^\infty }(G).0 \leqq \Psi \in {L^\infty }{(G)^ \ast },\Psi (1) = 1

is said to be a [topological] left invariant mean ([TLIM] LIM) if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Psi left-parenthesis Subscript a Baseline f right-parenthesis equals normal upper Psi left-parenthesis f right-parenthesis left-bracket normal upper Psi left-parenthesis phi asterisk f right-parenthesis equals normal upper Psi left-parenthesis f right-parenthesis">

Ψ

(

a

f
)
=
Ψ
(
f
)
[
Ψ
(
ϕ
∗
f
)
=
Ψ
(
f
)

\Psi {{\text {(}}_a}f) = \Psi (f)[\Psi (\phi \ast f) = \Psi (f)

] for all

a
∈
G
,
ϕ
∈
P
(
G
)
,
f
∈

L
∞

(
G
)

a \in G,\phi \in P(G),f \in {L^\infty }(G)

. The main result of this paper is the Theorem. Let

G
G

be a locally compact group, amenable as a discrete group. If

G
G

contains an open

σ
\sigma

-compact normal subgroup, then LIM = TLIM if and only if

G
G

is discrete. In particular if

G
G

is an infinite compact amenable as discrete group then there exists some

Ψ
∈
L
I
M

\Psi \in LIM

which is different from normalized Haar measure. A harmonic analysis type interpretation of this and related results are given at the end of this paper.

2

^{2}
Noninvertible knots of codimension 2
Authors: Kearton C.
Published: June 2010

A proof of the noninvertibility of the pretzel knot

(
25
,
−
3
,
13
)

(25, - 3,13)

is given which applies to the knots obtained by repeatedly spinning the pretzel knot.
Analytic functions, ideals, and derivation ranges
Authors: Weber R. E.
Published: June 2010

When

A
A

is in the Banach algebra

B

(

H

)

\mathcal {B}(\mathcal {H})

of all bounded linear operators on a Hilbert space

H

\mathcal {H}

, the derivation generated by

A
A

is the bounded operator

Δ
A

{\Delta _A}

on

B

(

H

)

\mathcal {B}(\mathcal {H})

defined by

Δ
A

(
X
)
=
A
X
−
X
A

{\Delta _A}(X) = AX - XA

. It is shown that (i) if

B
B

is an analytic function of

A
A

, then the range of

Δ
B

{\Delta _B}

is contained in the range of

Δ
A

{\Delta _A}

; (ii) if

U
U

is a nonunitary isometry, then the range of

Δ
U

{\Delta _U}

, contains nonzero left ideals; (iii) if

U
U

and

V
V

are isometries with orthogonally complemented ranges, then the span of the ranges of the corresponding derivations is all of

B

(

H

)

\mathcal {B}(\mathcal {H})

.
Oscillation in first order nonlinear retarded argument differential equations
Authors: Shreve Warren E.
Published: June 2010

A result of Ladas, Lakshmikantham, and Papadakis [1] concerning oscillation caused by lag in linear first order retarded argument differential equations is generalized to the sublinear case. Examples showing that such generalization to the superlinear case is impossible are given.
An inequality for the distribution of the Brownian gradient function
Authors: Davis Burgess
Published: June 2010

The Brownian gradient function of a harmonic function u in the unit disc is shown to be distributionally about as large as the classical area function of u. This distribution function inequality strengthens some integral inequalities of Burkholder and Gundy.
More on tight isometric immersions of codimension two
Authors: Chen Chang Shing
Published: June 2010

We continue our investigation on tight isometric immersion of a nonnegatively curved compact manifold

M
n

{M^n}

into

R

n
+
2

{R^{n + 2}}

. Under some minor restrictions, we prove that the immersion is a product embedding of convex hypersurfaces. For surfaces in

R
4

{R^4}

, the restrictions are unnecessary.
Relations in stable homotopy modules
Authors: Kahn Donald W.
Published: June 2010

The “generating hypothesis” of Freyd implies various fascinating statements in stable homotopy, particularly about the stable homotopy ring of spheres and the modules over this ring. The present paper extends some of these consequences. It also investigates the question of whether a possible counterexample to the hypothesis might be detected by its action on homology groups; for this, we find a connection with work of Joel Cohen on coherent graded rings.
The model companion of the theory of commutative rings without nilpotent elements
Authors: Lipshitz L. , Saracino D.
Published: June 2010

We show that the theory of commutative rings without nilpotent elements has a model companion. The model companion is decidable and is the model completion of the theory of commutative regular rings.
𝑛-Gorenstein rings
Authors: Foxby Hans Bjørn
Published: June 2010

The object of this note is to study commutative noetherian n-Gorenstein rings. The first result is: if each module satisfying Samuel’s conditions

(

a

i

)

({{\text {a}}_i})

for some

i
≦
n

i \leqq n

is an ith syzygy, then the ring is n-Gorenstein. This is the converse to a theorem of Ischebeck. The next result characterizes n-Gorenstein rings in terms of commutativity of certain rings of endomorphisms. This answers a question of Vasconcelos. Finally the last result deals with embedding of finitely generated modules into finitely generated modules of finite projective dimension.
A note on Dedekind rings
Authors: Matsuda Ryûki
Published: June 2010

We prove the following theorem: let

R
R

be a commutative ring without zero-divisors. If every ideal in

R
R

is a product of prime ideals

≠
R
,
R

\ne R,R

has the identity.
Uniform convergence for a hyperspace
Authors: Gazik R. J.
Published: June 2010

In this note a uniform convergence in the collection

C
(
E
)

C(E)

of nonempty, compact subsets of a separated uniform convergence space E is defined. This convergence is compared with the hyperspace convergence on

C
(
E
)

C(E)

and it is shown that the two convergences agree on Richardson’s class

Γ
\Gamma

. In the case of a regular

T
1

{T_1}

topological space (E, t) this means that there is a uniform convergence structure on E, which induces t, such that uniform convergence in

C
(
E
)

C(E)

is convergences in the Vietoris topology on

C
(
E
)

C(E)

.
The ergodic decomposition of conservative Baire measures
Authors: Sturgeon Michael L.
Published: June 2010

Certain topological conditions on a Markov transition function are shown sufficient for an integral representation of conservative invariant Baire measures. The analysis incorporates the Choquet-Bishop-de Leeuw extension of the Krein-Milman theorem.
Clifford translations of symmetric spaces
Authors: Ozols V.
Published: June 2010

A direct proof, not using the classification of symmetric spaces, is given for the following characterization of Clifford translations in a symmetric space

M
M

: An isometry

g
g

is a Clifford translation of

M
M

if and only if the centralizer

Z
(
g
)

Z(g)

of

g
g

in the isometry group of

M
M

is transitive on

M
M

. The proof uses a geodesic characterization of Clifford translations, and the subgroups

T

(
h
)

{T^{(h)}}

of J. de Siebenthal.
On Dirichlet’s theorem and infinite primes
Authors: Waid Carter
Published: June 2010

It is shown that Dirichlet’s theorem on primes in an arithmetic progression is equivalent to the statement that every unit of a certain quotient ring

Z
¯

\bar Z

of the nonstandard integers is the image of an infinite prime. The ring

Z
¯

\bar Z

is the completion of

Z
Z

relative to the “natural” topology on

Z
Z

.
A comparison of metrics on Teichmüller space
Authors: Linch Michele
Published: June 2010

The length in the Weil-Petersson metric of the Teichmüller geodesic between two points is computed, yielding the result that the Weil-Petersson metric is dominated by a constant multiple of the Teichmüller metric.
Fourier transforms and measure-preserving transformations
Authors: McGehee O. Carruth
Published: June 2010

There exists a continuous function

f
f

on the real line, vanishing at infinity, such that, for every measure-preserving transformation

h
h

, the composition

f
∘
h

f \circ h

fails to be a Fourier transform. This fact is a consequence of a theorem about measurable functions which is obtained from the theory of idempotents.
A generalization of the Rudin-Carleson theorem
Authors: Hag Per
Published: June 2010

The purpose of this paper is to prove a common generalization of a theorem due to T. W. Gamelin [3] and a theorem due to Z. Semadeni [5]. Both these results are generalizations of E. Bishop’s abstract version of the well-known RudinCarleson theorem [2].
Locally modular lattices and locally distributive lattices
Authors: Maeda Shûichirô
Published: June 2010

A locally modular (resp. locally distributive) lattice is a lattice with a congruence relation and each of whose equivalence class has sufficiently many elements and is a modular (resp. distributive) sublattice. Both the lattice of all closed subspaces of a locally convex space and the lattice of projections of a locally finite von Neumann algebra are locally modular. The lattice of all

T
1

{T_1}

-topologies of an infinite set is locally distributive.
𝑛-normal lattices
Authors: Cornish William H.
Published: June 2010

An

n
n

-normal lattice is a distributive lattice with 0 such that each prime ideal contains at most

n
n

minimal prime ideals. A relatively

n
n

-normal lattice is a distributive lattice such that each bounded closed interval is an

n
n

-normal lattice. The main results of this paper are: (1) a distributive lattice

L
L

with 0 is

n
n

-normal if and only if for any

x
0

,

x
1

,
⋯
,

x
n

ε
L

{x_0},{x_1}, \cdots ,{x_n}\varepsilon L

such that

x
i

∧

x
j

=
0

{x_i} \wedge {x_j} = 0

for any
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i not-equals j comma i comma j equals 0 comma midline-horizontal-ellipsis comma n comma left-parenthesis x 0 right-bracket Superscript asterisk Baseline logical-or left-parenthesis x 1 right-bracket Superscript asterisk Baseline logical-or midline-horizontal-ellipsis logical-or left-parenthesis x Subscript n Baseline right-bracket Superscript asterisk Baseline equals upper L">

i
≠
j
,
i
,
j
=
0
,
⋯
,
n
,

(

x
0

]
∗

∨

(

x
1

]
∗

∨
⋯
∨

(

x
n

]
∗

=
L

i \ne j,i,j = 0, \cdots ,n,{({x_0}]^ \ast } \vee {({x_1}]^ \ast } \vee \cdots \vee {({x_n}]^ \ast } = L

, (2) a distributive lattice

L
L

is relatively

n
n

-normal if and only if for any

n
+
1

n + 1

incomparable prime ideals

P
0

,

P
1

,
…
,

P
n

,

P
0

∨

P
1

∨
…
∨

P
n

=
L

{P_0},{P_1}, \ldots ,{P_n},{P_0} \vee {P_1} \vee \ldots \vee {P_n} = L

.