bluuebunny/crossref_metadata_2025 huggingface dataset:
A note on identifications of metric spaces
Authors: Siwiec Frank
Published: June 2010

A space

X
X

is said to be

σ
M
K

\sigma MK

provided that

X
X

has a countable closed cover

C

\mathcal {C}

of metrizable subspaces such that if

K
K

is a compact subset of

X
X

, there is a

C
∈

C

C \in \mathcal {C}

for which

K
⊂
C

K \subset C

. A Hausdorff space is

σ
M
K

\sigma MK

and Fréchet if and only if it is representable as a closed image of a metric space obtained by identifying a discrete collection of closed sets with hemicompact boundaries to points.
A transfer principle for simple properties of theories
Authors: Nadel Mark E.
Published: June 2010

A notion of simple property of theories is introduced and it is shown that if

P

{\text {P}}

is a simple property of theories,

A
A

countable admissible, and

M
M

a structure in

A
A

, then

Th
A

(
M
)

{\operatorname {Th} _A}(M)

has property

P

{\text {P}}

iff

Th

∞
ω

(
M
)

{\operatorname {Th} _{\infty \omega }}(M)

has property

P

{\text {P}}

.
Nonnegative matrices each of whose positive diagonals has the same sum
Authors: Hedrick Mark Blondeau
Published: June 2010

The author shows that if

A
A

is a fully indecomposable nonnegative matrix each of whose positive diagonals has sum

M
M

and when

a

i
j

=
0

{a_{ij}} = 0

, the sum of each positive diagonal in the submatrix of

A
A

obtained by deleting the

i

th

i{\text {th}}

row and

j

th

j{\text {th}}

column is less than

M
M

, then there is a unique positive matrix

B
B

such that its rank is at most two, each of its diagonals has sum

M
M

, and

a

i
j

=

b

i
j

{a_{ij}} = {b_{ij}}

when

a

i
j

>
0

{a_{ij}} > 0

. The author then compares his results to those obtained by Sinkhorn and Knopp who carried out a similar analysis for positive diagonal products.
On Lebesgue summability for double series
Authors: Kohn M. J.
Published: June 2010

In [2] a two dimensional analogue of Lebesgue’s theorem on differentiation of formally integrated trigonometric series was established. Here we show that a stronger analogue holds.
A theorem on continuous decompositions of the plane into nonseparating continua
Authors: Smith Michel
Published: June 2010

E. Dyer [2] proved that there is no continuous decomposition of a compact irreducible continuum into decomposable continua which is an arc with respect to its elements. The author extends Dyer’s result to the plane. Consider a continuous decomposition of the plane into nonseparating compact continua. R. L. Moore [6] has shown that the decomposition space is homeomorphic to the plane. Using Moore’s result it is shown that the union of the elements of each arc in the decomposition space is an irreducible continuum. It follows then, from Dyer’s result, that there is no continuous decomposition of the plane into nonseparating compact decomposable continua.
On the topological completion
Authors: Curzer Howard , Hager Anthony W.
Published: June 2010

Let

X
X

be a Tychonoff space. As is well known, the points of the Stone-Čech compactification

β
X

\beta X

“are” the zero-set ultrafilters of

X
X

, and the points of the Hewitt real-compactification

υ
X

\upsilon X

are the zero-set ultrafilters which are closed under countable intersection. It is shown here that a zero-set ultrafilter is a point of the Dieudonné topological completion

δ
X

\delta X

iff the family of complementary cozero sets is

σ
\sigma

-discretely, or locally finitely, additive. From this follows a characterization of those dense embeddings

X
⊂
Y

X \subset Y

such that each continuous metric space-valued function on

X
X

extends over

Y
Y

, and a somewhat novel proof of the Katětov-Shirota Theorem.
Uniform distribution on locally compact groups
Authors: Rindler Harald
Published: June 2010

In this paper we give a short proof of the existence of uniformly distributed sequences in separable locally compact groups, a result first stated by Benzinger [1]. The existence of u.d. sequences in separable compact groups was proved by Veech [7] whose results also imply the above result. We also prove the existence of well-distributed sequences.
A uniqueness result for topological groups
Authors: Kallman Robert R.
Published: June 2010

We give a rapid proof of a general result which has an easy corollary that the

p
p

-adic integers have a unique topology in which they are a complete separable metric group.
Strictly convex normed linear spaces
Authors: Gudder S. , Strawther D.
Published: June 2010

A new characterization of strict convexity for complex normed linear spaces in terms of duality maps is given. It is then shown that many of the latest characterizations of strict convexity follow as simple corollaries.
Composition operators induced by rational functions
Authors: Singh R. K.
Published: June 2010

A necessary and sufficient condition for a rational function to define a composition operator on

L
p

(
μ
)

{L^p}(\mu )

is given in this paper, where

μ
\mu

is the Lebesgue measure on the Borel subsets of the real line. In particular, all polynomials inducing composition operators are completely determined.
A factorization of the direct limit of Hilbert cubes
Authors: Heisey Richard E.
Published: June 2010

We show that the countable direct limit of Hilbert cubes

Q
∞

{Q^\infty }

is homeomorphic to the product of the Hilbert cube

Q
Q

and the countable direct limit of lines

R
∞

{R^\infty }

. As a consequence, two open subsets of

R
∞

{R^\infty }

have the same homotopy type if and only if their products with

Q
Q

are homeomorphic. Combined with a theorem of D. W. Henderson our result implies that

X
×
Q
×

R
∞

≅
Q
×

R
∞

X \times Q \times {R^\infty } \cong Q \times {R^\infty }

, where

X
X

is any locally compact, separable AR (metric).
Homology of complex projective hypersurfaces with isolated singularities
Authors: Miller John L.
Published: June 2010

We use information concerning the homology of links and Milnor fibers at each singularity of a hypersurface

V
~

\tilde V

, of degree

d
d

in

C

P

n
+
1

C{P^{n + 1}}

with only isolated singularities, to determine

V
~

\tilde V

’s homology except for the question of torsion in

H
n

(

V
~

;
Z
)

{H_n}(\tilde V;Z)

.
A proof of the compact leaf conjecture for foliated bundles
Authors: Uomini R.
Published: June 2010

Given an oriented fiber bundle

M
M

whose fiber is a connected,

m
m

-dimensional manifold, and a codimension

n
n

foliation of

M
M

which is transverse to the fibers of

M
M

and all of whose leaves are compact, we will show that there is an upper bound on the orders of the holonomy groups of the leaves.
The finite intersection property on annihilator right ideals
Authors: Zelmanowitz J. M.
Published: June 2010

In this article the finite intersection property on annihilator right ideals will be shown to be an adequate substitute for more stringent chain conditions on such ideals. One application of the investigation will produce a new characterization of orders in semisimple artinian rings, another will generate new classes of absolutely torsion-free rings.
Projective modules
Authors: Jøndrup S.
Published: June 2010

In this note we prove that if

R
R

is a ring satisfying a polynomial identity and

P
P

is a projective left

R
R

-module such that

P
P

is finitely generated modulo the Jacobson radical, then

P
P

is finitely generated. As a corollary we get that if

R
R

is a ring still satisfying a polynomial identity and

M
M

is a finitely generated flat

R
R

-module such that

M

/

J
M

M/JM

is

R

/

J

R/J

-projective, then

M
M

is

R
R

-projective,

J
J

denotes the Jacobson radical.
Sur les fonctions de deux variables dont les coupes sont des dérivées
Authors: Grande Zbigniew
Published: June 2010

This paper concerns relationships between the measurability of a function

f
:

I
2

→
R

f:{I^2} \to R

(when

I
=
⟨
0
,
1
⟩

I = \langle 0,1\rangle

and

R
R

is the set of all real numbers) and its cross sections

f

x
0

(
y
)
=
f
(

x
0

,
y
)

{f_{{x_0}}}(y) = f({x_0},y)

and

f

y
0

(
x
)
=
f
(
x
,

y
0

)

{f^{{y_0}}}(x) = f(x,{y_0})

. A function

g
:
I
→
R

g:I \to R

is said to have property

(

K

)

({\mathbf {K}})

if for each measurable set

A
⊂
I

A \subset I

of positive measure the function

g
g

is ponctuellement-discontinue (i.e., the set of continuities is dense) on the closure of the set of all density points of

A
A

. The main result is: If a function

f
:

I
2

→
R

f:{I^2} \to R

is bounded and each

f
x

{f_x}

has property

(

K

)

({\mathbf {K}})

and each

f
y

{f^y}

is a derivative, then

f
f

is Lebesgue measurable.
A multiplier theorem for 𝑆𝑈(𝑛)
Authors: Weiss Norman J.
Published: June 2010

Let

G
=

SU

(
n
)

G = {\text {SU}}(n)

, let

g

\mathfrak {g}

be its Lie algebra and let

m
m

be a function on

g

\mathfrak {g}

, invariant under the adjoint action of

G
G

, which is continuous at the points of

G
^

\hat G

(which can be imbedded in

g

\mathfrak {g}

). If

1
⩽
p
>
2
[
1
−

(
n
+
2

)

−
1

]

1 \leqslant p > 2[1 - {(n + 2)^{ - 1}}]

and

m
m

is a multiplier for the

Ad
G

{\operatorname {Ad} _G}

-invariant

L
p

{L^p}

functions on

g

\mathfrak {g}

, then the restriction of a translate of

m
m

to

G
^

\hat G

is a multiplier for the central

L
p

{L^p}

functions on

G
G

.
On the Stone-Weierstrass theorem for the strict and superstrict topologies
Authors: Haydon R. G.
Published: June 2010

Sentilles has introduced topologies

β
0

,

β

{\beta _0},\;\beta

and

β
1

{\beta _1}

on the space

C
b

(
S
)

{C_b}(S)

of all bounded, continuous, real-valued functions on the completely regular space

S
S

, which yield as dual spaces the three important spaces of measures,

M
t

(
S
)
,

M
τ

(
S
)

{M_t}(S),\;{M_\tau }(S)

and

M
σ

(
S
)

{M_\sigma }(S)

, respectively. A number of authors have proved a Stone-Weierstrass theorem for

β
0

{\beta _0}

, the coarsest of the three topologies. In this paper, it is shown that the superstrict topology

β
1

{\beta _1}

does not obey the Stone-Weierstrass theorem, except perhaps when

β
1

=
β

{\beta _1} = \beta

. Examples are then given to show that the situation for

β
\beta

itself is rather complicated.
Existence of a fixed point for nonexpansive mappings with closed values
Authors: Markin J. T.
Published: June 2010

Fixed point existence and fixed point stability results are presented for nonexpansive mappings of a Banach space

B
B

into the family of nonempty closed bounded convex subsets of

B
B

, where

B
B

is assumed separable, strictly convex, and reflexive with a weakly continuous duality mapping.
A note on Borel’s density theorem
Authors: Furstenberg Harry
Published: June 2010

We prove the following theorem of Borel: If

G
G

is a semisimple Lie group,

H
H

a closed subgroup such that the quotient space

G

/

H

G/H

carries finite measure, then for any finite-dimensional representation of

G
G

, each

H
H

-invariant subspace is

G
G

-invariant. The proof depends on a consideration of measures on projective spaces.
Almost-complex substructures on the sphere
Authors: Dibag I.
Published: June 2010

The paper solves completely the existence problem of almost-complex substructures on spheres.
Extension of hyperfunction solutions of linear differential equations with constant coefficients
Authors: Kawai Takahiro
Published: June 2010

We prove the local extensibility across a

k
k

-convex hypersurface of solutions of a system of linear differential equations with constant coefficients which satisfies some algebraic conditions determined by

k
k

. A global extensibility is also proved in the case of completely

k
k

-convex set.
A limit theorem for measurable random processes and its applications
Authors: Grinblat L. Š.
Published: June 2010

Let the measurable random processes

ξ
1

(
t
)
,
…
,

ξ
n

(
t
)
,
…

{\xi _1}(t), \ldots ,{\xi _n}(t), \ldots

and

ξ
(
t
)

\xi (t)

be defined on

[
0
,

1
]

[0,\;1]

. There exists

C
C

such that for all

n
n

and

t
t

we have

E

|

ξ
n

(
t
)

|

p

⩽
C
,

p
⩾
1

E|{\xi _n}(t){|^p} \leqslant C,\;p \geqslant 1

. The following assertion is valid: if for any finite set of points

t
1

,
…
,

t
k

⊂
[
0
,

1
]

{t_1}, \ldots ,{t_k} \subset [0,\;1]

the joint distribution of

ξ
n

(

t
1

)
,
…
,

ξ
n

(

t
k

)

{\xi _n}({t_1}), \ldots ,{\xi _n}({t_k})

converges to the joint distribution of

ξ
(

t
1

)
,
…
,
ξ
(

t
k

)

\xi ({t_1}), \ldots ,\xi ({t_k})

, and if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E StartAbsoluteValue xi Subscript n Baseline left-parenthesis t right-parenthesis EndAbsoluteValue Superscript p Baseline right-arrow upper E StartAbsoluteValue xi left-parenthesis t right-parenthesis EndAbsoluteValue Superscript p">

E

|

ξ
n

(
t
)

|

p

→
E

|

ξ
(
t
)

|

p

E|{\xi _n}(t){|^p} \to E|\xi (t){|^p}

for all

t
∈
[
0
,

1
]

t \in [0,\;1]

, then for any continuous functional

f
f

on

L
p

[
0
,

1
]

{L_p}[0,\;1]

the distribution of

f
(

ξ
n

(
t
)
)

f({\xi _n}(t))

converges to the distribution of

f
(
ξ
(
t
)
)

f(\xi (t))

. This statement immediately implies the convergence of distributions in some limit theorems for the sums of independent random variables (for example, in one of the theorems of P. Erdös and M. Kac) and in some statistical criteria (for example, in the

ω
2

{\omega ^2}

-criterion of Cramér and von Mises).
Truth and infinity
Authors: Rucker R. v. B.
Published: June 2010

This paper formalizes Gödel’s 1946 conjecture that every set-theoretic sentence is decidable from the present axioms plus some true axioms of infinity; and we prove a weak variant of this conjecture to be true for every

ZF

{\text {ZF}}

universe. We then make precise the extent to which unbound quantifiers can be taken to range only over ordinals in

ZF

{\text {ZF}}

, obtaining a sort of normal-form theorem. The last section relates these results to the problem of how wide the class of all sets is.
Applications of the Stone-Čech compactification to free topological groups
Authors: Hardy J. P. L. , Morris Sidney A. , Thompson H. B.
Published: June 2010

In this note the Stone-Čech compactification is used to produce short proofs of two theorems on the structure of free topological groups. The first is: The free topological group on any Tychonoff space

X
X

contains, as a closed subspace, a homeomorphic copy of the product space

X
n

{X^n}

. This is a generalization of a result of B. V. S. Thomas. The second theorem proved is C. Joiner’s, Fundamental Lemma.
On analytic functions into 𝑙^{𝑝}-spaces
Authors: Globevnik J.
Published: June 2010

Let

Δ
\Delta

be the open unit disc in the complex plane and let

1
⩽
p
>
∞

1 \leqslant p > \infty

. We construct an example of a discontinuous function from

Δ
\Delta

into

l
p

{l^p}

which is analytic as a function from

Δ
\Delta

into

l

p
′

{l^{p’}}

for any

p
′

>
p

p’ > p

and whose derivatives (into

l

p
′

{l^{p’}}

) of all orders are functions into

l
p

{l^p}

.
A singular integral inequality on a bounded interval
Authors: Bradley J. S. , Everitt W. N.
Published: June 2010

An inequality of the form (1.1) is established, where

p
,
q

p,q

are real-valued functions on an interval

[
a
,
b
)

[a,b)

of the real line, with

−
∞
>
a
>
b
>
∞
,
p
(
x
)
>
0

- \infty > a > b > \infty ,p(x) > 0

on

[
a
,
b
)
,

μ
0

[a,b),{\mu _0}

is a real number and

f
f

is a complex-valued function in a linear manifold so chosen that all three integrals in (1.1) are absolutely convergent. The problem is singular in that while

p

−
1

∈
L
(
a
,
b
)

{p^{ - 1}} \in L(a,b)

we require

q
q

to have a behavior at

b
b

of such a form that

q
∉
L
(
a
,
b
)

q \notin L(a,b)

.
Some stability theorems for nonharmonic Fourier series
Authors: Young Robert M.
Published: June 2010

The theory of nonharmonic Fourier series in

L
2

(
−
π
,
π
)

{L^2}( - \pi ,\pi )

is concerned with the completeness and expansion properties of sets of complex exponentials

{

e

i

λ
n

t

}

\{ {e^{i{\lambda _n}t}}\}

. It is well known, for example, that the completeness of the set

{

e

i

λ
n

t

}

\{ {e^{i{\lambda _n}t}}\}

ensures that of

{

e

i

μ
n

t

}

\{ {e^{i{\mu _n}t}}\}

whenever

∑

|

λ
n

−

μ
n

|

>
∞

\sum {|{\lambda _n} - {\mu _n}| > \infty }

. In this note we establish two results which guarantees that if

{

e

i

λ
n

t

}

\{ {e^{i{\lambda _n}t}}\}

is a Schauder basis for

L
2

(
−
π
,
π
)

{L^2}( - \pi ,\pi )

, then

{

e

i

μ
n

t

}

\{ {e^{i{\mu _n}t}}\}

is also a Schauder basis whenever

{

μ
n

}

\{ {\mu _n}\}

is “sufficiently close” to

{

λ
n

}

\{ {\lambda _n}\}

.
Projections from 𝐿^{𝑝}(𝐺) onto central functions
Authors: Wells Benjamin B.
Published: June 2010

Let

G
G

denote a locally compact

[
F
C

]
B
−

[FC]_B^ -

-group. Then for every

1
⩽
p
⩽
∞

1 \leqslant p \leqslant \infty

there is a projection from

L
p

(
G
)

{L^p}(G)

onto the

B
B

-central functions in

L
p

(
G
)

{L^p}(G)

.
A noninvertible unweighted bilateral shift
Authors: Gellar Ralph , Silber Robert
Published: June 2010

A noninvertible unweighted bilateral shift is constructed in a Nakano

l

p
i

{l^{{p_i}}}

space.
On the facial structure of a convex body
Authors: Collier J. B.
Published: June 2010

The family formed by taking the relative interior of each face of a

d
d

-dimensional convex body

C
C

is a partition of

C
C

. It is shown here that the subfamily consisting of all the

(
d
−
2
)

(d - 2)

-dimensional sets has a quotient topology which is paracompact and this is used to prove a property of the set of extreme points when

d
=
3

d = 3

.
Scalar curvatures on 𝑂(𝑀),𝐺₂(𝑀)
Authors: Rigas Alcibiades
Published: June 2010

We show that every

C
∞

f
:

G
2

(
M
)
→

R

,

M
n

{C^\infty }f:{G_2}(M) \to {\mathbf {R}},{M^n}

a compact connected riemannian manifold

n
⩾
3

n \geqslant 3

, is the scalar curvature function of some complete riemannian metric on

G
2

(
M
)

{G_2}(M)

, the grassmann bundle of

2
2

planes over

M
M

, except possibly when

K
=

constant 

⩾
0

K = {\text {constant }} \geqslant 0

. A similar result holds for

O
(
M
)

O(M)

bundle of orthonormal frames on

M
M

.
𝛽𝑅 is a topological Galois space
Authors: Sharma P. L.
Published: June 2010

The result given in the title of this note is proved. An example of two nonhomeomorphic Hausdorff Galois spaces having isomorphic homeomorphism groups follows.
Acyclic models and excision
Authors: Schön Rolf
Published: June 2010

This note gives a proof of the excision theorem in singular homology using the method of acyclic models and avoiding the commonly used barycentric subdivision operators.
Open subsets of 𝑅^{∞} are stable
Authors: Heisey Richard E.
Published: June 2010

Let

U
U

be an open subset of

R
∞

=
dir
⁡
lim

R
n

{R^\infty } = \operatorname {dir} \lim {R^n}

, where

R
R

denotes the reals. We show that

U
×

R
∞

U \times {R^\infty }

is homeomorphic to

U
U

. Combined with previous work of the author we obtain the corollary that two open subsets of

R
∞

{R^\infty }

are homeomorphic if and only if they have the same homotopy type.
Balanced valuations and flows in multigraphs
Authors: Jaeger F.
Published: June 2010

A balanced valuation of a multigraph

H
H

is a mapping

w
w

of its vertex-set

V
(
H
)

V(H)

into

R
R

such that

∀
S
⊆
V
(
H
)

\forall S \subseteq V(H)

the number of edges of

H
H

with exactly one vertex in

S
S

is greater than or equal to

|

∑

v
∈
S

w
(
v
)

|

|{\sum _{v \in S}}w(v)|

; we apply the theory of flows in networks to obtain known and new results on balanced valuations such as: A cubic multigraph has chromatic index

3
3

if and only if it has a balanced valuation with values in

{
−
2
,
+
2
}

\{ - 2, + 2\}

(Bondy [5]). Every planar cubic

2
2

-edge connected multigraph has a balanced valuation with values in

{
−
5

/

3
,
+
5

/

3
}

\{ - 5/3, + 5/3\}

. Every planar

5
5

-regular

4
4

-edge connected multigraph has a balanced valuation with values in

{
−
3
,
+
3
}

\{ - 3, + 3\}

.
The preparation theorem and the freeness of 𝐴[[𝑋]]/𝐼
Authors: Cox S. H.
Published: June 2010

Let

I
I

be a nonzero ideal of

A
[
[
X
]
]

A[[X]]

, the ring of formal power series over a commutative Noetherian ring

A
A

. These are equivalent: (i)

I
I

is generated by a single series

f
=

a
0

+

a
1

X
+
…

f = {a_0} + {a_1}X + \ldots

such that for some

s
,

a
s

s,\;{a_s}

is a unit, the first

s
s

coefficients

a
0

,
…
,

a

s
−
1

{a_0}, \ldots ,{a_{s - 1}}

of

f
f

lie in the Jacobson radical of

A
A

and

A
A

is complete in the adic topology defined by the ideal they generate. (ii)

A
[
[
X
]
]

/

I

A[[X]]/I

is a free

A
A

-module.
Monic and monic free ideals in a polynomial semiring in several variables
Authors: Dale Louis
Published: June 2010

The study of monic and monic free ideals in a polynomial semiring

S
[
x
]

S[x]

, where

S
S

is a commutative semiring with an identity, is extended to

S
[

x
1

,

x
2

,
…
,

x
n

]

S[{x_1},{x_2}, \ldots ,{x_n}]

. Structure theorems are given for monic, monic free, and monic free

k
k

-ideals in

S
[

x
1

,

x
2

,
…
,

x
n

]

S[{x_1},{x_2}, \ldots ,{x_n}]

. It is shown that each monic free

k
k

-ideal in

S
[

x
1

,
…
,

x
n

]
,

S

S[{x_1}, \ldots ,{x_n}],\;S

a strict semiring, is the sum of a finite number of ideals

B
i

{B_i}

such that each

B
i

{B_i}

is the union of a proper infinite ascending chain of ideals.
Structure theorem for 𝐴-compact operators
Authors: Lakhani G. D.
Published: June 2010

A contraction

T
T

defined on a complex Hilbert space is called

A
A

-compact if there exists a nonzero function

f
f

analytic in the open unit disc and continuous on the closed disc such that

f
(
T
)

f(T)

is a compact operator. In this paper, the factorization of

f
f

is used to obtain a structure theorem which describes the spectrum of

T
T

.
The orbit space of a sphere by an action of 𝑍_{𝑝^{𝑠}}
Authors: Willson Stephen J.
Published: June 2010

Let

X
X

be a finite

CW

{\text {CW}}

complex with the

Z

p
r

{Z_{{p^r}}}

homology of an

n
n

-sphere. Suppose

Z

p
s

{Z_{{p^s}}}

acts cellularly on

X
X

. The homology of the orbit space

X

/

Z

p
s

X/{Z_{{p^s}}}

with coefficients

Z

p
r

{Z_{{p^r}}}

is computed.
Deformations of generalized complete intersections
Authors: Mandelbaum R. , Schaps M.
Published: June 2010

The concept of a generalized complete intersection (GCI) of affine schemes is introduced. The proofs of the following theorems are then sketched. Theorem 1. Suppose

X
X

is a

G
C
I

GCI

of affine Cohen-Macaulay schemes of codimension

⩽
2

\leqslant 2

. Then if the versal deformation space of

X
X

exists, it is smooth. Theorem 2. Suppose

X
X

is a

G
C
I

GCI

of determinantal schemes. Then if

dim
⁡
X

\dim X

is sufficiently low,

X
X

is smoothable. In particular, if

X
X

is as in Theorem 1 and

dim
⁡
X
>
4

\dim X > 4

then

X
X

is smoothable.
Rational Chebyshev approximation to certain entire functions of zero order on the positive real axis. II
Authors: Reddy A. R.
Published: June 2010

A sufficient condition is given on the growth of certain entire functions which guarantee a certain rate of convergence of the error in approximating reciprocals of a class of entire functions by reciprocals of polynomials under the uniform norm on the positive real axis.
A note on regular methods of summability and the Banach-Saks property
Authors: Erdős P. , Magidor M.
Published: June 2010

Using the Galvin-Prikry partition theorem from set theory it is proved that every bounded sequence in a Banach space has a subsequence such that either every subsequence of which is summable or no subsequence of which is summable.
A sufficient condition for hyperinvariance
Authors: Longstaff W. E.
Published: June 2010

A linear transformation on a finite-dimensional complex linear space has the property that all of its invariant subspaces are hyperinvariant if and only if its lattice of invariant subspaces is distributive [1]. It is shown that an operator on a complex Hilbert space has this property if its lattice of invariant subspaces satisfies a certain distributivity condition.
A geometric equivalent of the invariant subspace problem
Authors: Nordgren Eric A. , Radjavi Heydar , Rosenthal Peter
Published: June 2010

It is shown that every operator has an invariant subspace if and only if every pair of idempotents has a common invariant subspace.
Minimal surfaces in tori by Weyl groups
Authors: Nagano T. , Smyth B.
Published: June 2010

We report a method of constructing compact minimal surfaces immersed in flat Riemannian tori of arbitrary dimension

n
⩾
3

n \geqslant 3

by the use of the Weyl groups of compact simple Lie groups, inspired by H. A. Schwarz [5].
On positively turning immersions
Authors: Quine J. R.
Published: June 2010

Let

γ
:

S
1

→

C

\gamma :{S^1} \to {\mathbf {C}}

be a

C
2

{C^2}

immersion of the circle. Let

k
k

be the number of zeros of

γ
\gamma

and suppose

d
arg
⁡
γ
(

e

i
θ

)

/

d
θ
>
0

d\arg \gamma ({e^{i\theta }})/d\theta > 0

for

γ
(

e

i
θ

)
≠
0

\gamma ({e^{i\theta }}) \ne 0

; then

twn
⁡
γ
=
k

/

2
+

(
2
π

)

−
1

∫
A

d
arg
⁡
γ

\operatorname {twn} \gamma = k/2 + {(2\pi )^{ - 1}}{\smallint _A}d\arg \gamma

where

γ
\gamma

is the tangent winding number, and

A
=

S
1

−

γ

−
1

(
0
)

A = {S^1} - {\gamma ^{ - 1}}(0)

. This generalizes the theorem of Cohn that if

p
p

is a self-inversive polynomial, the number of zeros of

p
′

p’

in

|

z

|

>
1

|z| > 1

is the same as the number of zeros of

p
p

in

|

z

|

>
1

|z| > 1

. For

k
=
0

k = 0

, this is a topological generalization of Lucas’ theorem. We show how

(
2
π

)

−
1

∫
A

d
arg
⁡
γ

{(2\pi )^{ - 1}}{\smallint _A}d\arg \gamma

represents a generalization of the notion of the winding number of

γ
\gamma

about

0
0

.
Periodic points of continuous maps and Lindemann’s independence theorem for exponentials
Authors: Neumann Dean A.
Published: June 2010

We give a simple proof of the sufficiency of known conditions for the existence of periodic points of a continuous map, using a classical theorem of Lindemann on transcendental numbers.
Rotundity and smoothness in conjugate spaces
Authors: Smith M. A.
Published: June 2010

It is shown that the James space

J
J

admits an equivalent norm such that its third conjugate space is rotund.
Approximation by invertibles
Authors: Gartenberg Michael
Published: June 2010

The uniform closure of the set of invertibles of certain

C

∗

{C^{\ast }}

-algebras is characterized. It is shown that indices on

G

(
C
(
A
)
)

\mathcal {G}(C(A))

, where

A
A

is a closed annulus in the plane and on

M
2

S
2

×

S
1

M_2^{{S^2} \times {S^1}}

have continuous extensions to elements which are not invertible.
Fixed points of Anosov maps of certain manifolds
Authors: Sondow Jonathan D.
Published: June 2010

Lemma. If

H
H

is a graded exterior algebra on odd generators with augmentation ideal

J
J

and

h
:
H
→
H

h:H \to H

is an algebra homomorphism inducing

J

/

J
2

→
J

/

J
2

J/{J^2} \to J/{J^2}

with eigenvalues

{

λ
i

}

\{ {\lambda _i}\}

, then the Lefschetz number

L
(
h
)
=
Π
(
1
−

λ
i

)

L(h) = \Pi (1 - {\lambda _i})

. The lemma is applied to an Anosov map or diffeomorphism of a compact manifold with real cohomology

H
H

to give sufficient conditions that none of the eigenvalues

λ
i

{\lambda _i}

be a root of unity and that there exist a fixed point. In particular, every Anosov diffeomorphism of a compact connected Lie group has a fixed point.
Countably generated families
Authors: Mauldin R. Daniel
Published: June 2010

This paper discusses some interrelationships between various statements involving sets generated by rectangles, universal spaces, and real-valued measures on the continuum. Borel sets on ordinal spaces are also discussed.
Summability integrals
Authors: Brauer George
Published: June 2010

An integral on the set of natural numbers

N
N

is defined. If

E
E

is a subset of

N
N

such that the characteristic function

χ
E

{\chi _E}

of

E
E

is evaluated to some number

σ
\sigma

by a regular nonnegative summation matrix

A
A

, then

μ
A

(
E
)

{\mu _A}(E)

is defined to be

σ
;

μ
A

\sigma ;{\mu _A}

defines a finitely additive measure on

N
N

. If

s
s

is a sequence which can be written as a linear combination of characteristic functions

Σ

i
=
1

n

a
i

χ

E
i

\Sigma _{i = 1}^n{a_i}{\chi _{{E_i}}}

, where each sequence

χ

E
i

{\chi _{{E_i}}}

is evaluated by

A
A

, then

∫
N

s
d

μ
A

{\smallint _N}sd{\mu _A}

is defined to be

Σ

a
i

μ
A

(

E
i

)

\Sigma {a_i}{\mu _A}({E_i})

. Finally the definition of the integral is naturally extended to

L
(
A
)

L(A)

, the class of sequences which can be approximated by linear combinations of characteristic functions [2, pp. 85-88]. It is shown that if

A
A

and

B
B

are two nonnegative regular matrices such that the convergence field of

A
A

includes that of

B
B

, then

L
(
A
)

L(A)

includes

L
(
B
)

L(B)

provided

B
B

is normal. Finally for a nonnegative regular matrix

A
=
(

a

n
k

)

A = ({a_{nk}})

, the spaces

L
p

(
A
)

{L^p}(A)

spanned by sequences such that

{

Σ

k
=
0

∞

a

n
k

|

s
k

|

p

}

\{ \Sigma _{k = 0}^\infty {a_{nk}}|{s_k}{|^p}\}

is bounded and

lim

Σ

k
=
0

∞

a

n
k

s
k
p

\lim \Sigma _{k = 0}^\infty {a_{nk}}s_k^p

exists are studied. It is shown that if

p
p

is greater than one, then the sequences in

L

p
′

(
A
)

{L^{p’}}(A)

give rise to a set of bounded linear functionals on

L
p

(
A
)

{L^p}(A)

which are weak star dense in the dual of

L
p

(
A
)

{L^p}(A)

.
On the 𝐿²-boundedness of pseudo-differential operators
Authors: Childs A. G.
Published: June 2010

A. P. Calderón and R. Vaillancourt have established

L
2

{L^2}

-boundedness in case the symbol has bounded partial derivatives to order

3
3

in any variable. H. O. Cordes has proved

L
2

{L^2}

-boundedness if this order is

1
1

. It is shown here that it suffices for the symbol to satisfy a uniform Hölder continuity condition of order

1
2

+
δ
,

δ
>
0

\tfrac {1} {2} + \delta ,\;\delta > 0

.
Multipliers of Segal algebras
Authors: Tewari U. B.
Published: June 2010

We show that there exists a noncompact locally compact abelian group

G
G

and a Segal algebra

S
(
G
)

S(G)

on

G
G

whose multiplier algebra properly contains the measure algebra.
A note on unconditionally converging series in 𝐿_{𝑝}
Authors: Ørno Peter
Published: June 2010

Theorem. A series

∑

f
i

\sum {{f_i}}

in

L
p

[
0
,

1
]
(
1
⩽
p
⩽
2
)

{L_p}[0,\,1](1 \leqslant p \leqslant 2)

is unconditionally convergent if and only if for each

i
i

and for all
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t element-of left-bracket 0 comma 1 right-bracket comma f Subscript i Baseline left-parenthesis t right-parenthesis equals alpha Subscript i Baseline g left-parenthesis t right-parenthesis w Subscript i Baseline left-parenthesis t right-parenthesis">

t
∈
[
0
,

1
]
,

f
i

(
t
)
=

α
i

g
(
t
)

w
i

(
t
)

t \in [0,\,1],\;{f_i}(t) = {\alpha _i}g(t){w_i}(t)

where

(

α
i

)
∈

l
2

,

g
∈

L
2

[
0
,

1
]

({\alpha _i}) \in {l_2},\;g \in {L_2}[0,\,1]

and

(

w
i

)

({w_i})

is an orthonormal sequence in

L
2

[
0
,

2
]

{L_2}[0,\,2]

. This characterization allows the generalization (to u.c. series in

L
p

[
0
,

1
]

{L_p}[0,\,1]

) of several classical theorems concerning almost everywhere convergence of orthogonal series in

L
2

{L_2}

.
The space lip 𝛼 and certain other spaces have duals with Cesàro bases
Authors: Buntinas Martin
Published: June 2010

Banach sequence spaces whose duals are Banach sequence spaces with Toeplitz bases are characterized. For example, the duals of the lip

α
\alpha

spaces, for

0
>
α
>
1

0 > \alpha > 1

, are shown to have Cesàro bases. Also reflexive spaces with a Toeplitz basis are characterized and an equivalent form of the well-known theorem of F. and M. Riesz on the absolute continuity of measures is given.
The 𝑄-topology, Whyburn type filters and the cluster set map
Authors: Herrmann Robert A.
Published: June 2010

We use nonstandard topology and the

Q
Q

-topology to characterize normal, almost-normal, regular, almost-regular, semiregular spaces. The cluster [resp.

θ
\theta

-cluster] set relation is used to characterize regular, almost-regular [resp. strongly-regular] spaces. The Whyburn [resp. Dickman] filter bases are characterized and it is shown that the cluster [resp.

θ
\theta

-cluster] set relation restricted to the domain of the Whyburn [resp. Dickman] filter bases is an essentially continuous [resp. strongly

θ
\theta

-continuous] map iff the space is Hausdorff [resp. Urysohn].
A note on the Denjoy integrability of abstractly-valued functions
Authors: Morrison T. J.
Published: June 2010

We present necessary and sufficient conditions on a Banach space

X
X

that the classes of strongly measurable

X
X

-valued Denjoy-Pettis integrable and Denjoy-Gelfand integrable functions coincide.
The solution of 𝑦²+²ⁿ=𝑥³
Authors: Rabinowitz Stanley
Published: June 2010

All solutions to the diophantine equation

(

∗

)

y
2

+
γ

2
n

=

x
3

;

γ
=
±
1
,

\begin{equation}\tag {$\ast $}{y^2} + \gamma {2^n} = {x^3};\quad \gamma = \pm 1,\end{equation}

are found.
A curiosity concerning the degrees of the characters of a finite group
Authors: Fields K. L.
Published: June 2010

Let G be a finite group with irreducible characters

{
…
,
χ
,
…
}

\{ \ldots ,\chi , \ldots \}

and

K
=

Q

(
…
,
χ
,
…
)

K = {\mathbf {Q}}( \ldots ,\chi , \ldots )

the field generated over the rationals by their values. We will prove: If

K
=

Q

K = \mathbf {Q}

(or if

[
K
:

Q

]

[K:{\mathbf {Q}}]

is odd) then

∏

χ
(
1
)

o
d
d

χ
(
1
)

\prod \limits _{\chi (1)\;odd} {\chi (1)}

is a perfect square.
Local completeness of operator algebras
Authors: Behncke H. , Cuntz J.
Published: June 2010

A normed

∗
\ast

-algebra

A

\mathcal {A}

is called a local

C
∗

{C^\ast }

-algebra, if all its maximal commutative

∗
\ast

-subalgebras are

C
∗

{C^\ast }

-algebras. It is shown that any local

C
∗

{C^\ast }

-algebra dense in

K

(

H

)

\mathcal {K}(\mathcal {H})

, the algebra of compact operators on the Hilbert space

H

\mathcal {H}

equals

K

(

H

)

\mathcal {K}(\mathcal {H})

. The same result holds also for local

C
∗

{C^\ast }

-algebras dense in

A

W
∗

A{W^\ast }

-algebras without a

II
1

{\text {II}_1}

summand.
Number of odd binomial coefficients
Authors: Harborth Heiko
Published: June 2010

Let

F
(
n
)

F(n)

denote the number of odd numbers in the first n rows of Pascal’s triangle, and

θ
=
(
log
⁡
3
)

/

log
⁡
2
)

\theta = (\log 3)/\log 2)

. Then

α
=
lim
sup
F
(
n
)

/

n
θ

=
1

\alpha = \lim \sup F(n)/{n^\theta } = 1

, and

β
=
lim
inf
F
(
n
)

/

n
θ

=
0.812

556

…
.

\beta = \lim \inf F(n)/{n^\theta } = 0.812\;556\; \ldots .
A series and its associated continued fraction
Authors: Davison J. L.
Published: June 2010

Let

α
=
(
1
+
√
5
)

/

2

\alpha = (1 + \surd 5)/2

. In this paper it is proved that
\[

[
u
n
k
]

[unk]

\]
where

t
n

=

2

f

n
−
2

{t_n} = {2^{{f_{n - 2}}}}

and

(

f
n

)

({f_n})

is the Fibonacci sequence. It is also shown that

T
(
α
)

T(\alpha )

is transcendental.
Generators and relations for the special linear group over a division ring
Authors: Green Sherry M.
Published: June 2010

Let

R

\mathcal {R}

be a division ring,

n
⩾
3

n \geqslant 3

an integer, and let

SL

(
n
,

R

)

{\text {SL}}(n,\mathcal {R})

be the special linear group over

R

\mathcal {R}

. In this paper a presentation in terms of generators and relations is given for

SL

(
n
,

R

)

{\text {SL}}(n,\mathcal {R})

.
An example of a weight with type III centralizer
Authors: Haagerup Uffe
Published: June 2010

We give an example of a weight

ϕ
\phi

on the hyperfinite type

II
∞

{\text {II}_\infty }

factor R, such that the centralizer

M
ϕ

{M_\phi }

of

ϕ
\phi

is a type III von Neumann algebra.
On majorization and normality of operators
Authors: Radjabalipour Mehdi
Published: June 2010

Recent results of C. R. Putnam are used to find some conditions for normality of operators. The emphasis is on the classes of spectral operators (defined by N. Dunford) and M-hyponormal operators (defined by J. G. Stampfli).
Sets which can be extended to 𝑚-convex sets
Authors: Breen Marilyn
Published: June 2010

Let S be a compact set in

R
d

,

T
0

⊆
S

{R^d},{T_0} \subseteq S

. Then

T
0

{T_0}

lies in an m-convex subset of S if and only if every finite subset of

T
0

{T_0}

lies in an m-convex subset of S. For S a closed set in

R
d

{R^d}

and

T
0

⊆
S

{T_0} \subseteq S

, let

T
1

=
{
P
:
P

{T_1} = \{ P:P

a polytope in S having vertex set in

T
0

,
dim
⁡
P
⩽
d
−
1
}

{T_0},\dim P \leqslant d - 1\}

. If for every three members of

T
1

{T_1}

, at least one of the corresponding convex hulls
\[

conv

{

P
i

∪

P
j

}
,

1
⩽
i
>
j
⩽
3.

{\text {conv}}\{ {P_i} \cup {P_j}\} ,\quad 1 \leqslant i > j \leqslant 3.

\]
lies in S, then

T
0

{T_0}

lies in a 3-convex subset of S. An analogous result holds for m-convex sets provided ker

S
≠
∅

S \ne \emptyset

.
A solution of a problem of Steenrod for cyclic groups of prime order
Authors: Arnold James E.
Published: June 2010

Given a

Z
[
G
]

Z[G]

module A, we will say a simply connected CW complex X is of type (A, n) if X admits a cellular G action, and

H
~

i

(
X
)
=
0
,
i
≠
n
,

H
n

(
X
)
≃
A

{\tilde H_i}(X) = 0,i \ne n,{H_n}(X) \simeq A

as

Z
[
G
]

Z[G]

modules. In [5], R. Swan considers the problem posed by Steenrod of whether or not there are finite complexes of type (A, n) for all finitely generated A and finite G. Using an invariant defined in terms of

G
0

(
Z
[
G
]
)

{G_0}(Z[G])

, solutions were obtained for

A
=

Z
p

A = {Z_p}

(p-prime) and

G
⊆
Aut

(

Z
p

)

G \subseteq \operatorname {Aut}\;({Z_p})

. The question of infinite complexes of type (A, n) was left open. In this paper we obtain the following complete solution for

Z
[

Z
p

]

Z[{Z_p}]

modules: There are complexes of type

(
A
,
n
)

(
n
⩾
3
)

(A,n)\;(n \geqslant 3)

, and there are finite complexes of type (A, n) if and only if the invariant which corresponds to Swan’s invariant for these modules vanishes.
Taming free circle actions
Authors: Edmonds Allan L.
Published: June 2010

It is shown that an arbitrary free action of the circle group on a closed manifold of dimension at least six is concordant to a “tame” action (so that the orbit space is a manifold). A consequence is that the concordance classification of arbitrary free actions of the circle on a simply connected manifold is the same as the equivariant homeomorphism classification of free tame actions.
Applications of group actions on finite complexes to Hilbert cube manifolds
Authors: Newman Leon Stagg
Published: June 2010

For any compact Hilbert cube manifold M such that

H
~

∗

(
M
,

Z
p

)

{\tilde H_\ast }(M,{Z_p})

, there exists an embedding g of M into the Hilbert cube Q such that

g
(
M
)

g(M)

is the fixed point set of a semifree periodic homeomorphism of Q with period p. A counterexample is given to the conjecture that any two proper homotopic period p homeomorphisms of a Hilbert cube manifold such that the homeomorphisms revolve trivially about a unique fixed point are equivalent. A counterexample is also given for the case where the fixed point set is empty.
An application of a theorem of R. E. Zink
Authors: White H. E.
Published: June 2010

In §1 we discuss a measure theoretic analogue of Blumberg’s theorem; in §2 we discuss a topological analogue of the Saks-Sierpinski theorem.
On zero-one laws
Authors: Morgan John C.
Published: June 2010

An extension of a topological zero-one law due to M. and K.P.S. Bhaskara Rao and of the Hewitt-Savage zero-one law is presented.
On odd dimensional surgery with finite fundamental group
Authors: Hausmann Jean-Claude
Published: June 2010

One proves that, for any finite group G and homomorphism

ω
:
G
→

Z

/

2

Z

\omega :G \to {\mathbf {Z}}/2{\mathbf {Z}}

, the natural homomorphism
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript 2 k plus 1 Superscript h Baseline left-parenthesis bold upper Z upper G comma omega right-parenthesis right-arrow upper L Subscript 2 k plus 1 Superscript h Baseline left-parenthesis bold upper Q upper G comma omega right-parenthesis">

L

2
k
+
1

h

(

Z

G
,
ω
)
→

L

2
k
+
1

h

(

Q

G
,
ω
)

L_{2k + 1}^h({\mathbf {Z}}G,\omega ) \to L_{2k + 1}^h({\mathbf {Q}}G,\omega )

between Wall surgery groups is identically zero. Some results concerning the exponent of

L

2
k
+
1

h

(

Z

G
;
ω
)

L_{2k + 1}^h({\mathbf {Z}}G;\omega )

are deduced.
Splittings of cyclic 𝑝-algebras
Authors: Saltman David J.
Published: June 2010

In this note we investigate the finite p-groups appearing as Galois groups of maximal subfields of cyclic p-algebras and show that for a fixed cyclic p-algebra, all possible such groups appear.
Dense subgroups of Lie groups
Authors: Zerling David
Published: June 2010

Let G be a dense analytic subgroup with compact center of an analytic group L. Then there exist closed vector subgroups W and U of G and a (CA) closed normal analytic subgroup M of G, which contains the center of G, such that

G
=
M
W
U
,
M
W
∩
U
=
M
∩
W
=
{
e
}

G = MWU,MW \cap U = M \cap W = \{e\}

, and WU is a closed vector subgroup of G. Moreover,

L
=
M
W

U
¯

L = MW\bar U

, where MW is a closed normal analytic subgroup of L and

U
¯

\bar U

is a toral group, such that

M
W
∩

U
¯

MW \cap \bar U

is finite.
Derivations, homomorphisms, and operator ideals
Authors: Hoover T. B.
Published: June 2010

Let

A

\mathfrak {A}

be a

C
∗

{C^\ast }

-algebra of operators on a Hilbert space, and let

C
p

{C_p}

be the Schatten p-ideal. It is shown that every derivation from

A

\mathfrak {A}

to

C
p

{C_p}

is inner. A similar argument shows that two

C
∗

{C^\ast }

-homomorphisms which agree modulo

C
p

{C_p}

are equivalent.
Hyperinvariant subspaces of reductive operators
Authors: Moore Robert L.
Published: June 2010

T. B. Hoover has shown that if A is a reductive operator, then

A
=

A
1

⊕

A
2

A = {A_1} \oplus {A_2}

, where

A
1

{A_1}

is normal and all the invariant subspaces of

A
2

{A_2}

are hyperinvariant. A new proof is presented of this result, and several corollaries are derived. Among these is the fact that if A is hyperinvariant and T is polynomially compact and

A
T
=
T
A

AT = TA

, then

A
∗

T
=
T

A
∗

{A^ \ast }T = T{A^ \ast }

. It is also shown that every reductive operator is quasitriangular.
Perturbation theory for generalized Fredholm operators. II
Authors: Caradus S. R.
Published: June 2010

If T is a bounded linear operator with generalized inverse S, i.e.

T
S
T
=
T

TST = T

, we obtain conditions so that

T
−
U

T - U

has a generalized inverse. When T is a Fredholm operator, the conditions become simply the requirement that

I
−
U
S

I - US

(or equivalently

I
−
S
U

I - SU

) is a Fredholm operator. This result includes the classical perturbation theorems where U is required to have a small norm or to be compact.
Uniformities induced by cozero and Baire sets
Authors: Hager Anthony W.
Published: June 2010

This paper treats the cozero- and Baire-fine uniform spaces, those X such that each cozero (resp., Baire) function on X is uniformly continuous. The emphasis is on the general method, with the results about coz and Ba as corollaries. Some of these, stated just for coz: The coz functor out of Unif has no left adjoint, but its restrictions to precompact, and to separable, spaces do. A space is coz-fine iff it is proximally fine and each finite coz-cover is uniform. A cozero field

A

\mathcal {A}

has a compatible coz-fine uniform space iff the meet of two completely additive

A

\mathcal {A}

-covers is another.
An example of an infinite Lie group
Authors: Pisanelli Domingos
Published: June 2010

We study the complex l.c.s. X of germs of holomorphic mappings around the origin of

C
n

{C^n}

, with values in

C
n

{C^n}

, vanishing at the origin. We show that X is isomorphic to

M
(
n
,
C
)
×

H
2

M(n,C) \times {H_2}

, where

M
(
n
,
C
)

M(n,C)

is the set of complex matrices

n
×
n

n \times n

and

H
2

{H_2}

is the vector topological subspace of X of germs with vanishing jacobian matrix at the origin. We study the subset

Ω
\Omega

of invertible germs of X. We show that

Ω
\Omega

is open, connected and that

π
1

(
Ω
)
=

Z

{\pi _1}(\Omega ) = {\mathbf {Z}}

. We define in

Ω
\Omega

a topological and a Lie group structure. We determine its infinitesimal transformation, the differential equation of its law of composition and a fundamental bound of its right side. This work is a part of a larger research on infinite Lie groups, which started with a summary of results in [P

1

_{1}

]. In a subsequent paper we shall study the covering group of

Ω
\Omega

.
Congruence function fields of genus 𝑔 and class number 𝑔+1
Authors: Leitzel James R. C.
Published: June 2010

Congruence function fields of genus g and class number

g
+
1

g + 1

are fully classified. As an application we determine explicitly the real quadratic function fields with this property and of odd characteristic for which the ring of integers is a unique factorization domain.
General trinomials having symmetric Galois group
Authors: Smith John Howard
Published: June 2010

Let a and b be independent transcendentals over a field F and let

n
>
k

n > k

be integers

>
1

> 1

with

n
,
k
,
n
−
k

n,k,n - k

and the characteristic of F pairwise relatively prime. Then the Galois group of

x
n

+
a

x
k

+
b

{x^n} + a{x^k} + b

over

F
(
a
,
b
)

F(a,b)

is

S
n

{S_n}

.
Functional equations for polynomials
Authors: McKennon Kelly , Dearden Bruce
Published: June 2010

The set of all continuous symmetric multilinear forms of degree m on a real topological vector space V are shown to be in one-to-one correspondence with the family of continuous scalar-valued functions on V satisfying a certain functional equation. If V is n-dimensional, these functions are precisely those which can be represented by m-homogeneous polynomials of degree n (with respect to some basis of V). The connection between this family of generalized polynomials and the mth derivatives of a scalar-valued function is discussed.
Separable criteria for 𝐺-diagrams over commutative rings
Authors: Roark Charles Winfred
Published: June 2010

Let S be a commutative, separable algebra over the commutative ring R and finitely generated and projective as an R-module. Suppose G is a group of ring automorphisms of S stabilizing R setwise. It is shown that for the ring of invariants

S
G

{S^G}

to be a strongly separable extension of

R
G

{R^G}

it is necessary that

R
⋅

S
G

R \cdot {S^G}

be R-separable; and it is shown that this condition is sufficient when R and S are finitely generated algebras over an algebraically closed field and G is a linearly reductive algebraic group acting rationally on S.
Measurability of partial derivatives
Authors: Marcus Moshe , Mizel Victor J.
Published: June 2010

Let f be a real function defined in

R
n

{R_n}

. In this note we give a sufficient condition in order that the set of points where the partial derivative

∂
f

/

∂

x
i

\partial f/\partial {x_i}

exists is Lebesgue measurable and

∂
f

/

∂

x
i

\partial f/\partial {x_i}

is a measurable function on this set. This result unifies and extends a number of previous results.
A characterisation of Riesz proximities
Authors: Chattopadhyay K. C. , Vasudeva H. L.
Published: June 2010

The purpose of this note is to characterise separated Riesz proximities generated by clusters.
An 𝐿_{𝜔₁𝜔} complete and consistent theory without models
Authors: Makkai M. , Mycielski J.
Published: June 2010

We simplify the example of Ryll-Nardzewski of an

L

ω
1

ω

{L_{{\omega _1}\;\omega }}

theory with the above properties. We include other facts on the relationship of logic to 01-laws and on the

L

ω
1

ω

{L_{{\omega _1}\;\omega }}

definability of some sets which are meager but of full measure.
A combinatorial problem of Shields and Pearcy
Authors: Schanuel Stephen H.
Published: June 2010

Pearcy and Shields asked the following question. If

x
1

,
…
,

x
n

{x_1}, \ldots ,{x_n}

are positive real numbers, can one always delete a subset D (possibly empty) such that the following two conditions are satisfied: (1)

∑

1

/

x
i

⩽
n

\sum \;1/{x_i} \leqslant n

(sum over all deleted terms), (2)

∑

x
i

>
1

\sum \;{x_i} > 1

(sum over any interval of consecutive terms disjoint from D)? In this note we show that this is always possible.