bluuebunny/crossref_metadata_2025 huggingface dataset:
The undecidability of theories of groupoids with an extra predicate
Authors: Garfunkel Solomon , Schmerl James H.
Published: June 2010

Let T be any theory in the language of groupoids, and let

T
′

T’

be the same theory considered now in the language with an extra unary predicate. If some model of T has a substructure which is an infinite cancellative groupoid, then

T
′

T’

is hereditarily undecidable.
The 𝐻_{𝑝}-problem for groups with certain central factors cyclic
Authors: Gallian Joseph A.
Published: June 2010

Let G be a group and

H
p
(
G
)

Hp(G)

the subgroup generated by the elements of G of order different from p. Hughes conjectured that if

G
>
H
p
(
G
)
>
1

G > Hp(G) > 1

, then

|

G
:
H
p
(
G
)

|

=
p

|G:Hp(G)| = p

. In this paper it is shown that if G is a finite p-group and certain central factors of G are cyclic or if the normal subgroups of G of a certain order are two generated, then the Hughes conjecture is true for G.
Area of Bernstein-type polynomials
Authors: Price Martin E.
Published: June 2010

Bernstein polynomials in one variable are known to be total-variation diminishing when compared to the approximated function f. Here we consider the two variable case and give a counterexample to show they are not area-diminishing. Sufficient conditions are then given on a continuous function f to insure convergence in area. A similar theorem is proved for Kantorovitch polynomials in the case f is summable.
On separability in linear topological spaces
Authors: Lohman Robert H. , Stiles Wilbur J.
Published: June 2010

This paper contains (1) an example of a separable linear topological space with a closed nonseparable linear subspace, and (2) a proof of the fact that every metrizable subspace of a separable linear topological space is separable.
Continuity of certain connected functions and multifunctions
Authors: Hagan Melvin R.
Published: June 2010

In this paper it is proved that if X is a 1st countable, locally connected,

T
1

{T_1}

-space and Y is a

σ
\sigma

-coherent, sequentially compact

T
1

{T_1}

-space, then any nonmingled connectedness preserving multifunction from X onto Y with closed point values and connected inverse point values is upper semicontinuous. It follows that any monotone, connected, single-valued function from X onto Y is continuous. Let X be as above and let Y be a sequentially compact

T
1

{T_1}

-space with the property that if a descending sequence of connected sets has a nondegenerate intersection, then this intersection must contain at least three points. If f is a monotone connected single-valued function from X onto Y, then f is continuous. An example of a noncontinuous monotone connected function from a locally connected metric continuum onto an hereditarily locally connected metric continuum is given.
An example of a local flow on a manifold
Authors: Blackmore Denis L.
Published: June 2010

Let p be a point of a smooth n-dimensional manifold. If n is even it is easy to construct a local flow about p such that p is an isolated critical point and no orbit except the stationary one at p has p as a limit point. We call such a flow a nonnull flow about p (NN-flow). Mendelson has conjectured that NN-flows do not exist on odd dimensional manifolds. We show that Mendelson’s conjecture is false by constructing an NN-flow on any smooth manifold whose dimension is an odd integer exceeding one.
A note on subadditive functions
Authors: Wong James S. W.
Published: June 2010

A short proof is given for the following result: Let

ϕ
\phi

be subadditive on

(
0
,
b
)

(0,b)

and

ϕ
(
t
)
>
t
,
t
∈
(
0
,
b
)

\phi (t) > t,t \in (0,b)

. Then

sup

a
≦
t
≦
b

ϕ
(
t
)

/

t
>
1

{\sup _{a \leqq t \leqq b}}\phi (t)/t > 1

for

0
>
a
>
b

0 > a > b

.
Regular neighborhoods and topological manifolds
Authors: Weller Glenn P.
Published: June 2010

This paper defines a notion of topological regular neighborhood for compact triangulable subspaces of topological manifolds, proves a uniqueness theorem for such neighborhoods, and shows that this theorem is in a sense the best possible. These results have obvious applications to regular neighborhoods in manifolds having more than one piecewise linear structure.
Pseudo-matchings of a bipartite graph
Authors: Brace Alan , Daykin D. E.
Published: June 2010

Let G be a graph whose edges (x, y) have

x
∈
X
,
y
∈
Y
,

|

X

|

=

|

Y

|

>
∞

x \in X,y \in Y,|X| = |Y| > \infty

. A (t, u) cover of G is a set of t edges which cover

≧
u

\geqq u

vertices in both X and Y. We give conditions on the valency (minimum local degree) and the number of edges which ensure a (t, u) cover or that a Hamiltonian circuit exists.
Unique factorization in graded power series rings
Authors: Landweber Peter S.
Published: June 2010

It is shown that the graded ring

R
[

x
1

,

x
2

,
⋯
]
[
[
t
]
]

R[{x_1},{x_2}, \cdots ][[t]]

of homogeneous power series is a graded UFD if R is a regular UFD, the degrees of the

x
i

{x_i}

are positive and tend to

∞
\infty

, and t has degree

−
1

- 1

. In particular this applies to

M

U
∗

(
C

P
∞

)

M{U^ \ast }(C{P^\infty })

and

B

P
∗

(
C

P
∞

)

B{P^ \ast }(C{P^\infty })

.
On (𝑛,𝑛)-zeros of solutions of linear differential equations of order 2𝑛
Authors: Ridenhour Jerry R.
Published: June 2010

Sufficient conditions on the coefficients

p

2
n

,

p

2
n
−
1

,
⋯
,

p
0

{p_{2n}},{p_{2n - 1}}, \cdots ,{p_0}

are given which guarantee that no nontrivial solution of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript 2 n Baseline y Superscript left-parenthesis 2 n right-parenthesis Baseline plus p Subscript 2 n minus 1 Baseline y Superscript left-parenthesis 2 n minus 1 right-parenthesis Baseline plus midline-horizontal-ellipsis plus p 0 y equals 0">

p

2
n

y

(
2
n
)

+

p

2
n
−
1

y

(
2
n
−
1
)

+
⋯
+

p
0

y
=
0

{p_{2n}}{y^{(2n)}} + {p_{2n - 1}}{y^{(2n - 1)}} + \cdots + {p_0}y = 0

has two distinct zeros each of order at least

n
n

. These conditions are in the form of

n
n

inequalities which are satisfied by linear combinations of the coefficients and their derivatives.
Convolutions of continuous measures and sums of an independent set
Authors: Rago James Michael
Published: June 2010

Let

E
E

be a compact independent subset of an l.c.a. group

G
;

μ
1

,
⋯
,

μ

n
+
1

G;{\mu _1}, \cdots ,{\mu _{n + 1}}

continuous regular bounded Borel measures on

G
G

; and

k
1

,
⋯
,

k
n

{k_1}, \cdots ,{k_n}

integers. Let

k
i

×
E
=
{

k
i

x

|

x
∈
E
}

{k_i} \times E = \{ {k_i}x|x \in E\}

. We prove (1)

μ
1

∗
⋯
∗

μ

n
+
1

(

k
1

×
E
+
⋯
+

k
n

×
E
)
=
0

{\mu _1} \ast \cdots \ast {\mu _{n + 1}}({k_1} \times E + \cdots + {k_n} \times E) = 0

(the proof is a combinatorial argument). As a corollary of (1) we obtain (2) if

H
H

is any closed nondiscrete subgroup of

G
G

, then the intersection of

H
H

with the group generated by

E
E

has zero

H
H

-Haar measure.
The volume of a region defined by polynomial inequalities
Authors: Rothaus O. S.
Published: June 2010

Let

P
(
x
)

P(x)

be a polynomial on

R
n

{R^n}

with nonnegative coefficients. We develop a simple necessary and sufficient condition that the set

S
=
{
x
∈

R
n

|

x
i

≧
0
,
P
(
x
)
≦
1
}

S = \{ x \in {R^n}|{x_i} \geqq 0,P(x) \leqq 1\}

shall have finite volume. A corresponding result where

P
(
x
)

P(x)

is replaced by a collection of polynomials is an easy corollary. Finally, the necessary and sufficient conditions for the special case that P is a product of linear forms is also given.
On Δ(𝑥,𝑛)=𝜙(𝑥,𝑛) -𝑥𝜙(𝑛)/𝑛
Authors: Suryanarayana D.
Published: June 2010

Let

Δ
(
x
,
n
)
=
φ
(
x
,
n
)
−
x
φ
(
n
)

/

n

\Delta (x,n) = \varphi (x,n) - x\varphi (n)/n

, where

φ
(
x
,
n
)

\varphi (x,n)

denotes the number of positive integers

≦
x

\leqq x

and prime to

n
,
φ
(
n
)
=
φ
(
n
,
n
)

n,\varphi (n) = \varphi (n,n)

. In this paper, lower and upper bounds for

Δ
(
x
,
n
)

\Delta (x,n)

, which hold for all values of

x
≧
1

x \geqq 1

and

n
≧
2

n \geqq 2

, are established.
Sums of quotients of additive functions
Authors: De Koninck Jean-Marie
Published: June 2010

Denote by

ω
(
n
)

\omega (n)

and

Ω
(
n
)

\Omega (n)

the number of distinct prime factors of

n
n

and the total number of prime factors of

n
n

, respectively. Given any positive integer

α
\alpha

, we prove that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript 2 less-than-over-equals n less-than-over-equals x Endscripts normal upper Omega left-parenthesis n right-parenthesis slash omega left-parenthesis n right-parenthesis equals x plus x sigma-summation Underscript i equals 1 Overscript alpha Endscripts a Subscript i Baseline slash left-parenthesis log log x right-parenthesis Superscript i Baseline plus upper O left-parenthesis x slash log log x right-parenthesis Superscript alpha plus 1 Baseline right-parenthesis comma">

∑

2
≦
n
≦
x

Ω
(
n
)

/

ω

(
n
)
=
x
+
x

∑

i
=
1

α

a
i

/

(
log
⁡
log
⁡
x
)

i

+
O

(
x

/

log
⁡
log
⁡
x
)

α
+
1

)
,

\sum \limits _{2 \leqq n \leqq x} {\Omega (n)/\omega } (n) = x + x\sum \limits _{i = 1}^\alpha {{a_i}/{{(\log \log x)}^i} + O{{(x/\log \log x)}^{\alpha + 1}}),}

\]
where

a
1

=

∑
p

1

/

p
(
p
−
1
)

{a_1} = \sum \nolimits _p {1/p(p - 1)}

and all the other

a
i

{a_i}

’s are computable constants. This improves a previous result of R. L. Duncan.
Some criteria for the nonexistence of certain finite linear groups
Authors: Blau Harvey I.
Published: June 2010

Let

p
p

be a prime and

G
G

a finite group, not of type

L
2

(
p
)

{L_2}(p)

, with a cyclic Sylow

p
p

-subgroup

P
P

. Assume that

G
=

G
′

G = G’

. The purpose of this note is to put some rather stringent lower bounds on the degree

d
d

of a faithful indecomposable representation of

G
G

over a field of characteristic

p
p

given certain conditions on the normalizer

N
N

and the centralizer

C
C

of

P
P

in

G
G

. In particular, if the center of

G
G

has order 2 and

|

N
:
C

|

=
p
−
1

|N:C| = p - 1

, then

d
≧
p
−
1

d \geqq p - 1

.
On periodicity of entire functions
Authors: Yang Chung Chun
Published: June 2010

A sequence

S
=
{

s
n

}

S = \{ {s_n}\}

is said to be a periodic set of period

τ
(
≠
0
)

\tau ( \ne 0)

if and only if

S
∗

=
{

s
n

+
τ
}

{S^\ast } = \{ {s_n} + \tau \}

can be rearranged to be a sequence to coincide with

S
S

. Let

F
F

be the class of all entire functions

f
f

satisfying the growth condition:
\[

lim

r
→
∞

sup
log
⁡
log
⁡
log
⁡
M
(
r
,
f
)

/

log
⁡
r
>
1.

\lim \limits _{r \to \infty } \sup \log \log \log M(r,f)/\log r > 1.

\]
In this paper it is shown that if

f
∈
F

f \in F

and the zero sets of

f
f

and

f
′

f’

both are periodic sets with the same period

τ
\tau

, then

f
f

can be expressed as

f
(
z
)
=

e

c
z

g
(
z
)

f(z) = {e^{cz}}g(z)

, where

c
c

is a constant and

g
(
z
)

g(z)

is a periodic entire function with period

τ
\tau

. A counterexample is exhibited to show that the above condition is a necessary one.
The ring of holomorphic functions on a Stein compact set as a unique factorization domain
Authors: Dales H. G.
Published: June 2010

Let

Γ
\Gamma

be the ring of germs of analytic functions on a Stein compact subset

K
K

of a complex-analytic space. Necessary and sufficient conditions on

K
K

for

Γ
\Gamma

to be a unique factorization domain are given.
Estimates for the imaginary parts of the zeros of a polynomial
Authors: Giroux André
Published: June 2010

Estimates for the imaginary parts of the zeros of a polynomial are obtained as a function of its

L
2

{L^2}

norm on an interval with respect to an arbitrary weight. The resulting inequality is sharp. Orthogonal polynomials with respect to the given weight are used as the main tool. Simple consequences are deduced and

L
p

{L^p}

norms are also considered.
Relative size of the Shilov boundary of a function algebra
Authors: Zame William R.
Published: June 2010

A notion of size for subsets of the spectrum of a function algebra is described, relative to which each open subset of the Shilov boundary has the same size as the spectrum.
On a multiplication decomposition theorem in a Dedekind 𝜎-complete partially ordered linear algebra
Authors: Dai Taen Yu
Published: June 2010

Suppose a Dedekind

σ
\sigma

-complete partially ordered linear algebra (dsc-pola) satisfies a certain multiplication decomposition property (see definition below), then we show that this partially ordered linear algebra actually has the same structure of a special class of real matrix algebras, consisting of elements that can be decomposed as diagonal part plus nilpotent part

w
w

, such that

w
2

=
0

{w^2} = 0

.
On the join of subnormal subgroups
Authors: Van Werkhooven A. J.
Published: June 2010

Let

G

\mathfrak {G}

be the class of finitely generated groups. If the join of finitely many subnormal

X

=
s
n

X

\mathfrak {X} = sn\mathfrak {X}

subgroups is always an

X

\mathfrak {X}

-group and

N

=
{
s
n
,
q
,

n
0

}

N

⊆

G

\mathfrak {N} = \{ sn,q,{n_0}\} \mathfrak {N} \subseteq \mathfrak {G}

, then the join of finitely many subnormal

X

N

\mathfrak {X}\mathfrak {N}

-subgroups is an

X

N

\mathfrak {X}\mathfrak {N}

-group. If the subnormal coalition class

X

\mathfrak {X}

and the class

N

=
{
s
n
,
q
,

n
0

}

N

\mathfrak {N} = \{ sn,q,{n_0}\} \mathfrak {N}

are such that whenever

A
∈

X

N

A \in \mathfrak {X}\mathfrak {N}

, A has a maximum subnormal

X

\mathfrak {X}

-subgroup, then

X

(

N

∧

G

)

\mathfrak {X}(\mathfrak {N} \wedge \mathfrak {G})

is a subnormal coalition class

(

N

∧

G

(\mathfrak {N} \wedge \mathfrak {G}

is the class of finitely generated

N

\mathfrak {N}

-groups).
Problem 26 of L. Fuchs
Authors: Hsü Chin Shui
Published: June 2010

This solves the following problem: Which Abelian groups are the inverse limits of Abelian groups, each of which is a finite direct sum of quasi-cyclic and bounded Abelian groups? (Here quasi-cyclic means isomorphic to some

Z
(

p
∞

)

Z({p^\infty })

.) A necessary and sufficient condition for an Abelian group to be such is that it takes the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript r Baseline circled-plus normal upper Pi Subscript p Baseline upper H o m Subscript z Baseline left-parenthesis upper A Subscript p Baseline comma upper Z left-parenthesis p Superscript normal infinity Baseline right-parenthesis right-parenthesis">

A
r

⊕

Π
p

Hom
z

⁡
(

A
p

,
Z
(

p
∞

)
)

{A_r} \oplus \Pi _p\operatorname {Hom}_z({A_p},Z({p^\infty }))

where

A
r

{A_r}

is complete and reduced, the

A
p

{A_p}

are torsion-free and the direct product is taken over the set of prime numbers.
The extension of regular holomorphic maps
Authors: Royden H. L.
Published: June 2010

Let

f
f

be a holomorphic map of a

k
k

-dimensional polydisk in

C
n

{C^n}

into an

n
n

-dimensional complex manifold

M
M

which is regular at the origin. It is shown that

f
f

can be extended to a neighborhood in

C
n

{C^n}

of a slightly smaller polydisk so that the extension is also regular. If

f
f

is an embedding or immersion, then the extension may also be taken to be an embedding or immersion.
Metric and symmetric spaces
Authors: Harley Peter W.
Published: June 2010

In this paper we give an alternative proof, without reference to Urysohn’s lemma, of the metrization theorem of Bing [2], Nagata [6], and Smirnov [8] via the theory of symmetric spaces as developed by H. Martin in [5].
Lusin and Suslin topologies on a set
Authors: Gowrisankaran Kohur
Published: June 2010

It is proved that the Borel subsets of two Suslin topologies on a set

X
X

are identical if and only if there is a Suslin topology on

X
X

finer than both the given topologies. Properties of the supremum topology are given under suitable conditions on the subdivisions of the spaces.
A problem of Martin concerning strongly convex metrics on 𝐸³
Authors: Tymchatyn E. D. , Friberg B. O.
Published: June 2010

If

d
d

is a strongly convex metric on

E
3

{E^3}

and

C
C

is a simple closed curve in

E
3

{E^3}

such that

C
C

is the union of three line segments then

C
C

is unknotted.
Conformally invariant metrics and prime ends
Authors: Minda Carl David
Published: June 2010

Let

R
R

be a Riemann surface such that the group of conformal self-mappings of

R
R

acts transitively on

R
R

. If

d
d

is a metric on

R
R

which is invariant under all conformal automorphisms of

R
R

and which induces the given topology on

R
R

, then it is shown that the metric space

⟨

R
,
d

⟩

\left \langle {R,d} \right \rangle

is complete. This result is used to show that the prime end compactification of a simply connected Riemann surface

R
R

cannot be obtained by completion of a metric space

⟨

R
,
d

⟩

\left \langle {R,d} \right \rangle

, where

d
d

defines the given topology on

R
R

and is conformally invariant.
On a weakly closed subset of the space of 𝜏-smooth measures
Authors: Grömig Wolfgang
Published: June 2010

It is known that a lot of topological properties devolve from a basic space

X
X

to the family

M
τ

(
X
)

{M_\tau }(X)

of all

τ
\tau

-smooth Borel measures endowed with the weak topology (or to certain subspaces of

M
τ

(
X
)

{M_\tau }(X)

). The aim of this paper is to show that among these topological properties there cannot be properties which are hereditary on closed subsets but not on countable products of

X
X

, e.g. normality, paracompactness, the Lindelöf property, local compactness and

σ
\sigma

-compactness. For this purpose it is proved that the countable product space

X
N

{X^N}

is homeomorphic to a closed subset of

M
τ

(
X
)

{M_\tau }(X)

. A further consequence of this result is for example that, for the family

M
τ
1

(
X
)

M_\tau ^1(X)

of probability measures in

M
τ

(
X
)

{M_\tau }(X)

, compactness, local compactness and

σ
\sigma

-compactness are equivalent properties.
A relation in 𝐻*(𝑀𝑂⟨8⟩,𝑍²)
Authors: Giambalvo V.
Published: June 2010

It is shown that

H
∗

(
M
O
⟨
8
⟩
,

Z
2

)

{H^ \ast }(MO\langle 8\rangle ,{Z_2})

does not split as a module over the Steenrod algebra into a direct sum of modules, each having a single generator.
Representation of universal algebras by sheaves
Authors: Swamy U. Maddana
Published: June 2010

It is proved that every (universal) algebra

A
A

with distributive and permutable structure lattice is isomorphic with the algebra of all global sections with compact supports of a sheaf of homomorphic images of

A
A

over a topological space. This completely generalises the corresponding result of Klaus Keimel for

l
l

-rings.
On polynomial density in 𝐴_{𝑞}(𝐷)
Authors: Metzger Thomas A.
Published: June 2010

Let

D
D

be a bounded Jordan domain. Define

A
q

(
D
)

{A_q}(D)

, the Bers space, to be the Banach space of holomorphic functions on

D
D

, such that

∬
D

|

f

|

λ
D

2
−
q

d
x
d
y

\iint _D {|f|\lambda _D^{2 - q}dxdy}

is finite, where

λ
D

(
z
)

{\lambda _D}(z)

is the Poincaré metric for

D
D

. It is well known that the polynomials are dense in

A
q

(
D
)

{A_q}(D)

for

2
≦
q
>
∞

2 \leqq q > \infty

and we shall prove they are dense in

A
q

(
D
)

{A_q}(D)

for

1
>
q
>
2

1 > q > 2

if the boundary of

D
D

is rectifiable. Also some remarks are made in case the boundary of

D
D

is not rectifiable.
Approximation of multipliers
Authors: Gaudry Garth I. , Inglis Ian R.
Published: June 2010

We note some necessary and sufficient conditions concerning norm approximation of Fourier multipliers, and give an example to show that

M
q

(
Z
)

{M_q}(Z)

, the space of Fourier multipliers of type

(
q
,
q
)

(q,q)

, is not norm dense in

M
p

(
Z
)

{M_p}(Z)

when

1
≦
q
>
p
≦
2

1 \leqq q > p \leqq 2

. An extension of this example to more general groups is indicated.
A Riesz product proof of the Wiener-Pitt theorem
Authors: Graham Colin C.
Published: June 2010

A Riesz product proof of the Wiener-Pitt theorem is given.
Extreme measurable selections
Authors: Johnson Jerry A.
Published: June 2010

The extreme points of the set of measurable selections for a set-valued mapping are characterized. As a corollary, the extreme points of the unit ball of the space of “vector-valued

L
p

{L^p}

functions” are characterized, thus generalizing results of Sundaresan.
Arens multiplication and a characterization of 𝑤*-algebras
Authors: Palmer T. W.
Published: June 2010

Let

A

\mathfrak {A}

be a Banach algebra which is the dual of a normed linear space

X

\mathfrak {X}

. Suppose the multiplication in

A

\mathfrak {A}

is a continuous function of each factor separately in the weak* topology. We show that the natural projection of

A

∗
∗

=

X

∗
∗
∗

{\mathfrak {A}^{ \ast \ast }} = {\mathfrak {X}^{ \ast \ast \ast }}

onto

A

=

X

∗

\mathfrak {A} = {\mathfrak {X}^ \ast }

is a homomorphism with respect to either Arens’ multiplication. From this we derive a simple proof of a variant form of Sakai’s characterization of

W
∗

{W^ \ast }

-algebras.
On restricted weak type (1,1)
Authors: Moon K. H.
Published: June 2010

Let

{

S
k

}

k
≧
1

{\{ {S_k}\} _{k \geqq 1}}

be a sequence of linear operators defined on

L
1

(

R
n

)

{L^1}({R^n})

such that for every

f
∈

L
1

(

R
n

)
,

S
k

f
=
f
∗

g
k

f \in {L^1}({R^n}),{S_k}f = f \ast {g_k}

for some

g
k

∈

L
1

(

R
n

)
,
k
=
1
,
2
,
⋯

{g_k} \in {L^1}({R^n}),k = 1,2, \cdots

, and

T
f
(
x
)
=

sup

k
≧
1

|

S
k

f
(
x
)

|

Tf(x) = {\sup _{k \geqq 1}}|{S_k}f(x)|

. Then the inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m left-brace x element-of upper R Superscript n Baseline semicolon upper T f left-parenthesis x right-parenthesis greater-than y right-brace less-than-over-equals upper C y Superscript negative 1 Baseline integral Underscript upper R Superscript n Baseline Endscripts StartAbsoluteValue f left-parenthesis t right-parenthesis EndAbsoluteValue d t">

m
{
x
∈

R
n

;
T
f
(
x
)
>
y
}
≦
C

y

−
1

∫

R
n

|

f
(
t
)

|

d
t

m\{ x \in {R^n};Tf(x) > y\} \leqq C{y^{ - 1}}\smallint _{{R^n}} {|f(t)|dt}

holds for characteristic functions f (T is of restricted weak type (1, 1)) if and only if it holds for all functions

f
∈

L
1

(

R
n

)

f \in {L^1}({R^n})

(T is of weak type (1, 1)). In particular, if

S
k

f

{S_k}f

is the kth partial sum of Fourier series of f, this theorem implies that the maximal operator T related to

S
k

{S_k}

is not of restricted weak type (1, 1).
A short proof and generalization of a measure theoretic disjointization lemma
Authors: Kupka Joseph
Published: June 2010

This paper presents general conditions under which a subfamily may be selected from an infinite family of nonnegative, finitely additive measures such that this subfamily has the same cardinality as the original family, and such that the members of this subfamily are, in a certain sense, dis jointly supported. The generalized continuum hypothesis is required for the general result, but not for a special case of this result which had previously been obtained by Rosenthal, and for which the present techniques yield a much shorter proof.
On the boundedness of 𝑝-integrable automorphic forms
Authors: Rajeswara Rao K. V.
Published: June 2010

For a Fuchsian group, a criterion is obtained in order that every

p
p

-integrable automorphic form be bounded. This encompasses the known results for

p
=
1

p = 1

. The condition implies an interesting inequality between the Bergman kernel and the Poincaré line element of a Riemann surface.
Nonlinear perturbation of 𝑚-accretive operators
Authors: Fitzgibbon W. E.
Published: June 2010

Let

X
X

be a reflexive Banach space. Conditions sufficient to guarantee that the sum

A
+
B

A + B

, of two

m
m

-accretive operators

A
A

and

B
B

is

m
m

-accretive are provided. The basic requirements are that the operator

B
B

be bounded in some sense relative to

A
A

and that

A
A

and

B
B

be weakly closed.
Nonimmersions of low dimensional flag manifolds
Authors: Connell Fred J.
Published: June 2010

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

M
M

under the classifying map for the normal bundle of

M
M

. In low dimensional flag manifolds, one can show that the quadratic equation has no root, thus establishing a nonimmersion.
On a subclass of Bazilevič functions
Authors: Eenigenburg P. J. , Miller S. S. , Mocanu P. T. , Reade M. O.
Published: June 2010

The authors show that certain Bazilevič functions are spiral-like. Then the authors study the growth and Hardy classes of those special functions.
Generating function for generalized function of two variables
Authors: Sharma B. L. , Abiodun R. F. A.
Published: June 2010

In this paper we obtain a generating function for a generalized function of two variables. The result is very general in character and includes as particular cases some of the results recently given by Meijer [4], Carlitz [7] and Srivastava [5].
The Radon-Nikodym property and dentable sets in Banach spaces
Authors: Davis W. J. , Phelps R. R.
Published: June 2010

In order to prove a Radon-Nikodym theorem for the Bochner integral, Rieffel [5] introduced the class of “dentable” subsets of Banach spaces. Maynard [3] later introduced the strictly larger class of “

s
s

-dentable” sets, and extended Rieffel’s result to show that a Banach space has the Radon-Nikodym property if and only if every bounded nonempty subset of

E
E

is

s
s

-dentable. He left open, however, the question as to whether, in a space with the Radon-Nikodym property, every bounded nonempty set is dentable. In the present note we give an elementary construction which shows this question has an affirmative answer.
A category analogue of the Hewitt-Savage zero-one law
Authors: Rao M. Bhaskara , Rao K. P. S. Bhaskara
Published: June 2010

A category analogue of the Hewitt-Savage zero-one law is obtained in this paper.
A universal space for 𝐺-actions in which a normal subgroup acts freely
Authors: Rinne Robert L.
Published: June 2010

A universal space is constructed and it is used to show that

N

∗
G

(
X
,
A
,
ψ
)

\mathfrak {N}_ \ast ^G(X,A,\psi )

is computable when

G
G

is a finite super-solvable group.
Nilpotency in endomorphism rings
Authors: Gordon Robert
Published: June 2010

Nil subrings of the endomorphism ring of a module with finite Krull dimension sequence are nilpotent. This includes the case of a module with finite Krull dimension as well as noetherian modules. The method used is to embed the endomorphism ring, modulo a nilpotent ideal, in the endomorphism ring of an artinian object of a Grothendieck category.
On pure subgroups of 𝐿𝐶𝐴 groups
Authors: Armacost D. L.
Published: June 2010

In this note we determine the structure of those locally compact abelian (LCA) groups in which no nontrivial closed subgroup is pure and the structure of those LCA groups in which every closed subgroup is pure.
The compact 3-manifolds covered by 𝑆²×𝑅¹
Authors: Tollefson Jeffrey L.
Published: June 2010

The classification of all free actions by a finite group on

S
2

×

S
1

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

follows from the observation that there exist only four compact

3
3

-manifolds which have

S
2

×

R
1

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

for a universal covering space.
An application of the maximum principle to the geometry of plane curves
Authors: Johnson Harold H.
Published: June 2010

The maximum principle of control theory is used to find necessary and sufficient conditions for a plane curve, which has bounded piecewise continuous curvature and prescribed initial and terminal points and directions, to have minimal length. This result is used to prove that such a closed curve having length

L
L

and curvature

k
k

satisfying

|

k

|

≦
K

|k| \leqq K

can be contained in a circle of radius

R
R

, where

R
≦
L

/

4
−
(
π
−
2
)

/

2
K

R \leqq L/4 - (\pi - 2)/2K

.
On the existence of point countable bases in Moore spaces
Authors: Reed G. M.
Published: June 2010

In this paper, the author answers in the negative two questions raised by E. E. Grace and R. W. Heath concerning the existence of point countable bases in Moore spaces. These answers are obtained by a general construction technique developed by the author which associates to each first countable

T
2

{T_2}

-space a Moore space.
A class of manifolds covered by Euclidean space
Authors: Maxwell J. W.
Published: June 2010

The following is the main result: Theorem 1. Suppose

W
n

{W^n}

is a PL manifold which has homotopy type

K
(
Π
,
1
)
,
W

K(\Pi ,1),W

has one end

∞
,

π
1

\infty ,{\pi _1}

is essentially constant at

∞
\infty

, and the induced homomorphism

π
1

(
∞
)
→

π
1

(
W
)

{\pi _1}(\infty ) \to {\pi _1}(W)

is an isomorphism. Then the universal cover of

W
W

is PL homomorphic to

R
n

{R^n}

provided

n
≥
5

n \geq 5

.
Necessary and sufficient conditions for factorability of nonnegative operator-valued functions on Banach space
Authors: Miamee A. G. , Salehi H.
Published: June 2010

Several necessary and sufficient conditions for factorability of positive operator-valued functions on a Banach space are given. These characterizations involve the analysis of quasi square root and their corresponding invariant subspaces. The work extends to the Banach space a recent theorem of Yu. A. Rozanov and a certain earlier result of R. Payen. It also reveals the connection between their works on the factorization problem.
Matrices with prescribed entries and characteristic polynomial
Authors: Dias da Silva J. A.
Published: June 2010

It is proved that there exists an

n
×
n

n \times n

matrix over an arbitrary field with

n
−
1

n - 1

prescribed entries and prescribed characteristic polynomial.
Splitting groups by integers
Authors: Hamaker W. , Stein S.
Published: June 2010

A question concerning tiling Euclidean space by crosses raised this algebraic question: Let

G
G

be a finite abelian group and

S
S

a set of integers. When do there exist elements

g
1

,

g
2

,
⋯
,

g
n

{g_1},{g_2}, \cdots ,{g_n}

in

G
G

such that each nonzero element of

G
G

is uniquely expressible in the form

s

g
i

s{g_i}

for some

s
s

in

S
S

and some

g
i

{g_i}

? The question is answered for a broad (but far from complete) range of

S
S

and

G
G

.
On dilution and Cesàro summation
Authors: Isbell John R.
Published: June 2010

The problem whether a real sequence

(

s
i

)

({s_i})

has a dilution which is

(
C
,
1
)

(C,1)

summable to a number

s
s

is transformed by means of two sequences measuring the oscillation of

(

s
i

)

({s_i})

about

s
s

. (If it does not oscillate, the condition, known, is that

s
s

is a limit point of

(

s
i

)

({s_i})

.) For the

j
j

th consecutive block of

s
i

{s_i}

on one side of

s
,

α
j

s,{\alpha _j}

is the minimum of their distances from

s
,

β
j

s,{\beta _j}

the sum of distances. Then there must exist positive numbers

p
j

{p_j}

such that

β
j

+

p
j

α
j

=
o
(

p
1

+
⋯
+

p

j
−
1

)

{\beta _j} + {p_j}{\alpha _j} = o({p_1} + \cdots + {p_{j - 1}})

. The necessary condition and the sufficient condition coincide for very smooth sequences at

α
i

log
⁡

β
i

=
o
(
i
)

{\alpha _i}\log {\beta _i} = o(i)

.
Another proof of Szegő’s theorem for a singular measure
Authors: Holland Finbarr
Published: June 2010

It is shown that the set

{

e

int

:
n
⩾
1
}

\{ {e^{\operatorname {int} }}:n \geqslant 1\}

spans

L

2

(
σ
)

{\mathfrak {L}^2}(\sigma )

if

σ
\sigma

is a singular measure on the unit circle. The proof makes no appeal either to the F. and M. Riesz theorem on measures or to Hilbert space methods.
Riesz seminorms with Fatou properties
Authors: Aliprantis C. D.
Published: June 2010

A seminormed Riesz space

L
ρ

{L_\rho }

satisfies the

σ
\sigma

-Fatou property (resp. the Fatou property) if

θ
≤

u
n

↑
u

\theta \leq {u_n} \uparrow u

in

L
L

(resp.

θ
≤

u
α

↑
u

in

L

\theta \leq {u_\alpha } \uparrow u\;{\text {in}}\;L

) implies

ρ
(

u
n

)
↑
ρ
(
u
)

\rho ({u_n}) \uparrow \rho (u)

(resp.

ρ
(

u
α

)
↑
ρ
(
u
)

\rho ({u_\alpha }) \uparrow \rho (u)

). The following results are proved: (i) A normed Riesz space

L
ρ

{L_\rho }

satisfies the

σ
\sigma

-Fatou property if, and only if, its norm completion does and

L
ρ

{L_\rho }

has

(

A

,
0
)

({\mathbf {A}},0)

. (ii) The quotient space

L
ρ

/

I
ρ

{L_\rho }/{I_\rho }

has the Fatou property if

L
ρ

{L_\rho }

is Archimedean with the Fatou property.

(

I
ρ

=
{
u
ε
L
:
ρ
(
u
)
=
0
}
.
)

({I_\rho } = \{ u\varepsilon L:\rho (u) = 0\} .)

(iii) If

L
ρ

{L_\rho }

is almost

σ
\sigma

-Dedekind complete with the

σ
\sigma

-Fatou property, then

L
ρ

/

I
ρ

{L_\rho }/{I_\rho }

has the

σ
\sigma

-Fatou property. A counterexample shows that (iii) may be false for Archimedean Riesz spaces.
Bounded solutions of the equation Δ𝑢=𝑝𝑢 on a Riemannian manifold
Authors: Kwon Young K.
Published: June 2010

Given a nonnegative

C
1

{C^1}

-function

p
(
x
)

p(x)

on a Riemannian manifold

R
R

, denote by

B
p

(
R
)

{B_p}(R)

the Banach space of all bounded

C
2

{C^2}

-solutions of

Δ
u
=
p
u

\Delta u = pu

with the sup-norm. The purpose of this paper is to give a unified treatment of

B
p

(
R
)

{B_p}(R)

on the Wiener compactification for all densities

p
(
x
)

p(x)

. This approach not only generalizes classical results in the harmonic case

(
p
≡
0
)

(p \equiv 0)

, but it also enables one, for example, to easily compare the Banach space structure of the spaces

B
p

(
R
)

{B_p}(R)

for various densities

p
(
x
)

p(x)

. Typically, let

β
(
p
)

\beta (p)

be the set of all

p
p

-potential nondensity points in the Wiener harmonic boundary

Δ
\Delta

, and

C
p

(
Δ
)

{C_p}(\Delta )

the space of bounded continuous functions

f
f

on

Δ
\Delta

with

f

|

Δ
−
β
(
p
)
≡
0

f|\Delta - \beta (p) \equiv 0

. Theorem. The spaces

B
p

(
R
)

{B_p}(R)

and

C
p

(
Δ
)

{C_p}(\Delta )

are isometrically isomorphic with respect to the sup-norm.
Some inequalities for the multiplicator of a finite group. II
Authors: Jones Michael R.
Published: June 2010

In a previous article of the same title we gave a series of upper and lower bounds for the multiplicator of a finite group. §2 of this article gives an improvement of one of these results. In §3, we give a result that connects the multiplicator of a finite group with that of a normal subgroup whose factor-group is cyclic.
An inequality for analytic functions
Authors: Kamowitz Herbert
Published: June 2010

If

F
F

denotes the boundary value of a function

f
∈

H
p

,
1
≤
p
≤
∞

f \in {H^p}, 1 \leq p \leq \infty

, the infimum of the measure of

{
θ

|

|

F
(
θ
)

|

>
A
}

\{ \theta |\;|F(\theta )| > A\}

for given
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A comma 0 greater-than upper A greater-than StartAbsoluteValue f left-parenthesis 0 right-parenthesis EndAbsoluteValue comma StartAbsoluteValue EndAbsoluteValue f StartAbsoluteValue EndAbsoluteValue Subscript upper H Sub Superscript p Subscript Baseline equals 1">

A
,
0
>
A
>

|

f
(
0
)

|

,

|

|

f

|

|

H
p

=
1

A, 0 > A > |f(0)|, ||f|{|_{{H^p}}} = 1

, is determined.
Hereditary completeness and quasi-reflexivity
Authors: Mukherjee T. K.
Published: June 2010

In this note, we prove that the countable direct sum of quasi-reflexive Banach spaces is hereditarily complete and hence the separated countable inductive limit of quasi-reflexive Banach spaces is complete.
A characterization of parallel ovaloids
Authors: Koutroufiotis Dimitri
Published: June 2010

Two ovaloids

S
,

S
¯

S,\bar S

can be mapped diffeomorphically onto each other by equal inner normals. If, under this mapping, principal directions are preserved and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket left-parenthesis p minus p overbar right-parenthesis minus left-parenthesis c k 1 Superscript negative 1 Baseline minus k overbar Subscript 1 Superscript negative 1 Baseline right-parenthesis right-bracket left-bracket left-parenthesis p minus p overbar right-parenthesis minus left-parenthesis c k 2 Superscript negative 1 Baseline minus k overbar Subscript 2 Superscript negative 1 Baseline right-parenthesis right-bracket less-than-or-equal-to 0">

[
(
p
−

p
¯

)
−
(
c

k
1

−
1

−

k
¯

1

−
1

)
]
[
(
p
−

p
¯

)
−
(
c

k
2

−
1

−

k
¯

2

−
1

)
]
≤
0

[(p - \bar p) - (ck_1^{ - 1} - \bar k_1^{ - 1})][(p - \bar p) - (ck_2^{ - 1} - \bar k_2^{ - 1})] \leq 0

\]
everywhere on the unit sphere for a certain constant

c
c

, then

c
=
1

c = 1

and

p
−

p
¯

p - \bar p

= constant. Here

p
,

p
¯

p,\bar p

are the support functions,

k
1

{k_1}

, and

k
2

{k_2}

the principal curvatures of

S
,

k
¯

1

S,{\bar k_1}

and

k
¯

2

{\bar k_2}

the corresponding principal curvatures of

S
¯

\bar S

. Various characterizations of the sphere are obtained as corollaries.
On the classification of knots
Authors: Perko Kenneth A.
Published: June 2010

Linking numbers between branch curves of irregular covering spaces of knots are used to extend the classification of knots through ten crossings and to show that the only amphicheirals in Reidemeister’s table are the seven identified by Tait in 1884. Diagrams of the 165 prime

10
10

-crossing knot types are appended. (Murasugi and the author have proven them prime; Conway claims proof that the tables are complete.) Including the trivial type, there are precisely 250 prime knots with ten or fewer crossings.
The inverse limit and first derived functor, a short exact sequence
Authors: Lieberman Gerald
Published: June 2010

This paper provides a new topological proof that a certain sequence involving the inverse limit and first derived functor is short exact.
Completely outer groups of automorphisms acting on 𝑅/𝐽(𝑅)
Authors: Osterburg J.
Published: June 2010

Let

R
R

be a ring with unit,

J
(
R
)

J(R)

its Jacobson radical, and assume

R

/

J
(
R
)

R/J(R)

Artinian. Let

G
G

be a finite group of automorphisms of

R
R

that induces a completely outer group on

R

/

J
(
R
)

R/J(R)

. Then

R
R

is

G
G

-Galois over the fixed ring,

S
S

, if

R
R

is projective over the usual crossed product,

Δ
\Delta

, or, if the order of

G
G

is invertible in

R
R

, or if

R
R

is Artinian.
Summability methods fail for the 2ⁿ𝑡ℎ partial sums of Fourier series
Authors: Newman D. J.
Published: June 2010

Although the Fourier series of a continuous function need not converge everywhere, it was an important discovery of Fejér that this series must be Cesàro summable. Indeed, it is a frequent occurrence that convergence may be restored to an expansion by use of an appropriate summability method. What we show in this note is that the very opposite phenomenon can occur. Namely, that if one considers only the

2
n

{2^n}

th partial sums of the Fourier series, there is no summability method whatever which produces convergence for all continuous functions.
Trace class perturbations of isometries and unitary dilations
Authors: Carey Richard W.
Published: June 2010

Let

U
U

and

V
V

be isometries acting on a Hilbert space. If

U
−
V

U - V

belongs to the trace class we show that the absolutely continuous parts of the corresponding minimal unitary dilations are unitarily equivalent.
On the topology of the even-dimensional complex quadrics
Authors: Lai Hon Fei
Published: June 2010

Neat proofs are given of the explicit structures of the homotopy groups and cohomology rings of the even-dimensional complex quadrics.
Two nonequivalent conditions for weight functions
Authors: Fefferman Charles , Muckenhoupt Benjamin
Published: June 2010

A nonnegative function on the real line satisfies the condition

A

∞

{{\mathbf {A}}_\infty }

if, given

ε
>
0

\varepsilon > 0

, there exists a

δ
>
0

\delta > 0

such that if

I
I

is an interval,

E
⊂
I

E \subset I

, and

|

E

|

>
δ

|

I

|

|E| > \delta |I|

, then

∫
E

W
≤
ε

∫
I

W

\int _E {W \leq \varepsilon \int _I W }

. A nonnegative function on the real line satisfies the condition

A

{\mathbf {A}}

if for every interval

I
,

∫

2
I

W
≤
C

∫
I

W

I,\int _{2I} {W \leq C} \int _I W

, where

2
I

2I

is the interval with the same center as

I
I

and twice as long, and

C
C

is independent of

I
I

. An example is given of a function that satisfies

A

{\mathbf {A}}

but not

A

∞

{{\mathbf {A}}_\infty }

.
How separable is a space? That depends on your set theory!
Authors: Tall Franklin D.
Published: June 2010

A. Wilansky has raised the question of the productive behaviour of the property of having a countable set, such that each point is a sequential limit point of the set. The set-theoretic consistency and independence of the proposition that this property is preserved by products of

ℵ
1

{\aleph _1}

factors is established.
The image of unitary bordism in unoriented bordism—the equivariant case
Authors: Beem R. Paul , Wheeler E. R.
Published: June 2010

It is shown that the image of unitary

G
G

-bordism in unoriented

G
G

-bordism is the set of squares in unoriented

G
G

-bordism if

G
G

is

Z
2

{Z_2}

and properly contains the squares if

G
G

is an odd order finite group.
Steinhaus type theorems for summability matrices
Authors: Maddox I. J.
Published: June 2010

Necessary and sufficient conditions are given for an infinite matrix to sum all bounded strongly summable sequences. It is shown that the Borel matrix does not sum all such sequences. A corollary is that the bounded summability field of the Borel method is strictly contained in that of the

(
C
,
1
)

(C,1)

method. Also, it is proved that no coregular matrix can almost sum all bounded sequences—a generalization of Steinhaus’ theorem.
Alternating forms and one-relator groups
Authors: Sonn Jack
Published: June 2010

This paper considers epimorphisms of finitely generated one-relator groups

G
=
F

/

R
,
F

G = F/R,F

free,

R
⊆

F
p

[
F
,
F
]

R \subseteq {F^p}[F,F]

for some rational prime

p
p

. The main result is a lower bound for the difference rank

G
G

-rank

G
′

G’

when

G
′

G’

is a one-relator group homomorphic to

G
G

. This generalizes a known result for the case

G
′

G’

a free group.
A note on spaces with normal product with some compact space
Authors: Thomason R. W.
Published: June 2010

For compact

X
X

with

log
⁡

|

X

|

≧
ℵ
,
X
×
Z

\log |X| \geqq \aleph ,X \times Z

is normal only if

Z
Z

is

ℵ
\aleph

-collectionwise normal. If

Z
Z

is also semimetric or

ℵ
\aleph

-metacompact, it is then

ℵ
\aleph

-paracompact.
Triple points and singularities of projections of smoothly immersed surfaces
Authors: Banchoff Thomas
Published: June 2010

For a transversal smooth immersion of a closed

2
2

-dimensional surface into Euclidean

3
3

-space, the number of triple points is congruent modulo 2 to the Euler characteristic. The approach of this paper includes an introduction to normal Euler classes of smoothly immersed manifolds by means of singularities of projections.
The cohomology rings of certain finite permutation representations
Authors: Blowers James V.
Published: June 2010

In this paper the concept of join of two permutation representations is defined and the cohomology of this join is computed and shown to have trivial cup-products. This computation is then used to compute the cohomology groups of the

p
p

-Sylow subgroup of a symmetric group of order

n
n

acting on the set of

n
n

elements, and it is shown that the ring structure on these groups is not finitely generated, although it is transitive.
Separable Menger-regular Hausdorff curves are metrizable
Authors: Cornette J. L.
Published: June 2010

It is shown that every connected, separable, locally compact and locally peripherally finite Hausdorff space is metrizable.
On semiprime Jordan rings 𝐻(𝑅) with ACC
Authors: Britten Daniel J.
Published: June 2010

Let

R
R

be a

2
2

-torsion free semiprime associative ring with involution*. Conditions are put on the Jordan ring

H
(
R
)

H(R)

of symmetric elements which imply the existence of a ring of quotients which is a direct sum of involution simple Artinian rings. Using this result we obtain a Jordan ring of quotients for

H
(
R
)

H(R)

.
Subrings of Noetherian rings
Authors: Formanek Edward , Jategaonkar Arun Vinayak
Published: June 2010

Let

S
S

be a subring of a ring

R
R

such that

R
R

is a finitely generated right

S
S

-module. Clearly, if

S
S

is a right Noetherian ring then so is

R
R

. Generalizing a result of P. M. Eakin, we show that if

R
R

is right Noetherian and

S
S

is commutative then

S
S

is Noetherian. We also show that if

R
S

{R_S}

has a finite generating set

{

u
1

,
⋯
,

u
m

}

\{ {u_1}, \cdots ,{u_m}\}

such that

u
i

S
=
S

u
i

{u_i}S = S{u_i}

for

1
≤
i
≤
m

1 \leq i \leq m

, then a right

R
R

-module is Noetherian, Artinian or semisimple iff it is respectively so as a right

S
S

-module. This yields a result of Clifford on group algebras.
On the upper bound of the number of real roots of a random algebraic equation with infinite variance. II
Authors: Samal G. , Mishra M. N.
Published: June 2010

Let

N
n

{N_n}

be the number of real roots of

∑

v
=
0

n

a
v

ξ
v

x
v

=
0

\sum \nolimits _{v = 0}^n {{a_v}{\xi _v}{x^v}} = 0

where

ξ
v

{\xi _v}

’s are independent random variables identically distributed with a common characteristic function

exp
⁡
(
−
C

|

t

|

α

)
;
C

\exp ( - C|t{|^\alpha });C

is a positive constant,

a
0

,

a
1

,
⋯
,

a
n

{a_0},{a_1}, \cdots ,{a_n}

are nonzero real numbers such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k Subscript n Baseline equals max Underscript 0 less-than-over-equals v less-than-over-equals n Endscripts StartAbsoluteValue a Subscript v Baseline EndAbsoluteValue equals upper O left-parenthesis n Superscript beta Baseline slash log n right-parenthesis">

k
n

=

max

0
≦
v
≦
n

|

a
v

|

=
O
(

n
β

/

log
⁡
n
)

{k_n} = {\max _{0 \leqq v \leqq n}}|{a_v}| = O({n^\beta }/\log n)

. Then (i)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="probability left-brace upper S u p Subscript n greater-than n 0 Baseline upper N Subscript n Baseline slash left-parenthesis log n right-parenthesis squared greater-than mu right-brace greater-than mu prime slash n 0 Superscript 3 alpha minus 2 minus beta Baseline comma 1 less-than-over-equals alpha less-than-over-equals 2 comma 0 greater-than beta greater-than 1">

Pr
{

Sup

n
>

n
0

N
n

/

(
log
⁡
n

)
2

>
μ
}
>

μ
′

/

n
0

3
α
−
2
−
β

,
1
≦
α
≦
2
,
0
>
β
>
1

\Pr \{ {\operatorname {Sup} _{n > {n_0}}}{N_n}/{(\log n)^2} > \mu \} > \mu ’/n_0^{3\alpha - 2 - \beta },1 \leqq \alpha \leqq 2,0 > \beta > 1

; (ii)

Pr
{

N
n

/

(
log
⁡
n

)
2

>
μ
}
>

μ
′

/

n
,
α
≧
1

\Pr \{ {N_n}/{(\log n)^2} > \mu \} > \mu ’/n,\alpha \geqq 1

; (iii)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="probability left-brace upper N Subscript n Baseline slash left-parenthesis log n right-parenthesis squared greater-than mu right-brace greater-than mu prime slash n Superscript 3 alpha minus 1 minus beta Baseline comma 1 less-than-over-equals alpha less-than-over-equals 2">

Pr
{

N
n

/

(
log
⁡
n

)
2

>
μ
}
>

μ
′

/

n

3
α
−
1
−
β

,
1
≦
α
≦
2

\Pr \{ {N_n}/{(\log n)^2} > \mu \} > \mu ’/{n^{3\alpha - 1 - \beta }},1 \leqq \alpha \leqq 2

.
Galois theory and the existence of maximal unramified subalgebras
Authors: Kreimer H. F.
Published: June 2010

Let

B
B

be a commutative ring with 1, let

G
G

be a finite group of automorphisms of

B
B

, and let

A
A

be the subring of

G
G

-invariant elements of

B
B

. There exists a

G
G

-stable, unramified

A
A

-subalgebra of

B
B

which contains every unramified

A
A

-subalgebra of

B
B

.
Almost linear operators and functionals on 𝒞([0,1])
Authors: Baxter J. R. , Chacon R. V.
Published: June 2010

Let

C

(
M
)

\mathcal {C}(M)

be the bounded continuous functions on a topological space

M
M

. “Almost linear” operators (and functionals) on

C

(
M
)

\mathcal {C}(M)

are defined. Almost linearity does not imply linearity in general. However, it is shown that if

M
=
[
0
,
1
]

M = [0,1]

then any almost linear operator (or functional) must be linear. Specifically, if (a)

|

|

f

|

|

→
0

||f|| \to 0

implies

T
(
f
)
→
0

T(f) \to 0

, (b)

T
(
f
+
g
)
=
T
(
f
)
+
T
(
g
)

T(f + g) = T(f) + T(g)

whenever

f
g
=
0

fg = 0

, (c)

T
(
f
+
g
)
=
T
(
f
)
+
T
(
g
)

T(f + g) = T(f) + T(g)

whenever

g
g

is constant, and

M
=
[
0
,
1
]

M = [0,1]

, then

T
T

is linear. An application is given to convergence of measures.