bluuebunny/crossref_metadata_2025 huggingface dataset:
On a property of rational functions. II
Authors: Rahman Q. I.
Published: June 2010

It is shown that if

r
n

(
z
)

{r_n}(z)

is a rational function of degree

n
n

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r Subscript n Baseline left-parenthesis 0 right-parenthesis equals 1 comma limit Underscript StartAbsoluteValue z EndAbsoluteValue right-arrow normal infinity Endscripts StartAbsoluteValue r Subscript n Baseline left-parenthesis z right-parenthesis EndAbsoluteValue equals 0">

r
n

(
0
)
=
1
,

lim

|

z

|

→
∞

|

r
n

(
z
)

|

=
0

{r_n}(0) = 1,{\lim _{|z| \to \infty }}|{r_n}(z)| = 0

and all its poles lie in

|

ζ
1

|

≦

|

z

|

≦
1

|{\zeta _1}| \leqq |z| \leqq 1

then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="max Underscript StartAbsoluteValue z EndAbsoluteValue equals rho greater-than StartAbsoluteValue zeta 1 EndAbsoluteValue Endscripts StartAbsoluteValue r Subscript n Baseline left-parenthesis z right-parenthesis EndAbsoluteValue greater-than-over-equals 1 slash left-parenthesis 1 minus rho Superscript n Baseline right-parenthesis">

max

|

z

|

=
ρ
>

|

ζ
1

|

|

r
n

(
z
)

|

≧
1

/

(
1
−

ρ
n

)

{\max _{|z| = \rho > |{\zeta _1}|}}|{r_n}(z)| \geqq 1/(1 - {\rho ^n})

.
Generalized Steenrod-Hopf invariants for stable homotopy theory
Authors: Krueger Warren M.
Published: June 2010

In his paper On the groups

J
(
X
)

J(X)

. IV, Adams suggested that one might try to continue his

d
d

and

e
e

invariants to a sequence of higher homotopy invariants, each defined upon the vanishing of its predecessors and each taking its value in a certain Ext group. Recently he pointed out the efficacy of relocating his

d
d

and

e
e

invariants in Ext groups formed over a certain abelian category of comodules. It is the purpose of this note to carry out the program suggested above in a setting of the sort just mentioned. More specifically, for each homology theory which is representable by a comutative ring spectrum and whose ring of cooperations is flat over the coefficient ring, a sequence of higher homotopy invariants is constructed whose first term is Adams’

e
e

invariant for this theory.
Lattice sums of homogeneous functions
Authors: Cahn Robert S.
Published: June 2010

An asymptotic estimate for a general lattice sum is derived using Fourier analysis techniques.
Absolute continuity of eigenvectors of time-varying operators
Authors: Acker Andrew F.
Published: June 2010

If

K
(
t
)

K(t)

is a compact, selfadjoint operator function of a real variable t with distinct eigenvalues at each t, we show that the eigenvalues and eigenvectors are absolutely continuous and that

{
K
(
t
)
}

\{ K(t)\}

is a commuting set provided that

K
(
t
)

K(t)

commutes with its time derivative

K
′

(
t
)

K’(t)

at each t. The distinct eigenvalue condition is shown to be necessary.
Combinatorial characterization of supercompact cardinals
Authors: Magidor M.
Published: June 2010

It is proved that supercompact cardinals can be characterized by combinatorial properties which are generalizations of ineffability.
Inequalities for Poisson kernels on symmetric spaces
Authors: Korányi Adam
Published: June 2010

Every symmetric space of noncompact type has a finite number of Furstenberg-Satake boundaries; to each of these there corresponds a Poisson kernel. Sharp Harnack-type inequalities are proved and it is shown that the Poisson kernel, in appropriate coordinates, is the square root of a rational function.
On modular lattices of order dimension two
Authors: Wille Rudolf
Published: June 2010

In this note, it is shown that a modular lattice has order dimension

≦
2

\leqq 2

if and only if it contains no subset isomorphic to one of five described partially ordered sets.
A simple proof of a theorem of Albert
Authors: Racine M. L.
Published: June 2010

A simple proof is given of the following theorem of Albert: An associative division algebra of degree 4 over its center is of order 4 in the Brauer group if and only if it cannot be written as a tensor product of quaternion algebras.
Functions with a concave modulus of continuity
Authors: White H. E.
Published: June 2010

In [1], C. Goffman proved that, if

σ
\sigma

is a modulus of continuity, then the set of all functions f in

C
[
0
,
1
]

C[0,1]

such that

m
(
{
x
:
f
(
x
)
=
g
(
x
)
}
)
=
0

m(\{ x:f(x) = g(x)\} ) = 0

(m denotes Lebesgue measure) for all g in

C
(
σ
)

C(\sigma )

, the set of all functions in

C
[
0
,
1
]

C[0,1]

having

σ
\sigma

as a modulus of continuity, is residual in

C
[
0
,
1
]

C[0,1]

. In the present article, we prove that, if

σ
\sigma

is a concave modulus of continuity and

0
>
K
>

24

−
1

0 > K > {24^{ - 1}}

, then the set of all functions f in

C
(
σ
)

C(\sigma )

such that

m
(
{
x
:
f
(
x
)
=
g
(
x
)
}
)
=
0

m(\{ x:f(x) = g(x)\} ) = 0

for all g in

C
(
K
σ
)

C(K\sigma )

is residual in

C
(
σ
)

C(\sigma )

. Using this result, we show that, if

0
>
α
>
1

0 > \alpha > 1

, then there are functions in

C
[
0
,
1
]

C[0,1]

which satisfy a Hölder condition of exponent

α
\alpha

such that

m
(
{
x
:
f
(
x
)
=
g
(
x
)
}
)
=
0

m(\{ x:f(x) = g(x)\} ) = 0

for all g in

C
[
0
,
1
]

C[0,1]

which satisfy a Hölder condition of exponent

>
α

> \alpha

.
Comparison theorems for special classes of nonselfadjoint elliptic equations
Authors: Kreith Kurt
Published: June 2010

Known comparisons for nonselfadjoint elliptic equations are strengthened in cases where the coefficients of these equations satisfy special conditions. These improved comparison theorems are also considered in the context of a related eigenvalue problem.
Some implications of the Euler-Poincaré characteristic for complete intersection manifolds
Authors: Chen Bang-yen , Ogiue Koichi
Published: June 2010

Studies on relations between Euler-Poincaré characteristic and codimension of a complete intersection manifold in a complex projective space.
On the weight of a topological space
Authors: Hodel R. E.
Published: June 2010

The main result in this paper states that the weight of a regular space is the product of three cardinal functions, namely the Lindelöf degree, the pluming degree, and the point separating weight.
An infinite complex and the spectral sequences for complex cobordism and 𝐾-theory
Authors: Johnson David Copeland
Published: June 2010

An infinite complex is given whose complex cobordism spectral sequence has differentials,

d
r

{d_r}

, nonzero for infinitely many positive integers,

r
r

, but whose complex

K
K

-theory spectral sequence has

d
r

{d_r}

nontrivial for only finitely many

r
r

.
Characterization of finite-dimensional 𝑍-sets
Authors: Kroonenberg Nelly
Published: June 2010

It is proved that closed finite-dimensional subsets of

Q
Q

and

l
2

{l_2}

are

Z
Z

-sets iff their complement is

1
1

-ULC. As a corollary, closed finite-dimensional sets of deficiency 1 are shown to be

Z
Z

-sets.
A note on the singular points of the function generated by the Bergman operator of the second kind
Authors: Rosenthal Paul
Published: June 2010

Let

ψ
=

P
2

(
f
)

\psi = {P_2}(f)

be Bergman’s operator of the second kind,

f
(
q
)

f(q)

is analytic at

q
=
0

q = 0

. In a previous paper [5] a theorem was obtained on the singularities of

ψ
\psi

when

ψ
\psi

was generated by a

f
(
q
)

f(q)

whose only singular points were poles. In this note we obtain a theorem on the singularities of

ψ
\psi

when

ψ
\psi

is generated by a

f
(
q
)

f(q)

whose singular points can be of more varied types.
On polynomials satisfying a Turán type inequality
Authors: Csordas George , Williamson Jack
Published: June 2010

For Legendre polynomials

P
n

(
x
)

{P_n}(x)

, P. Turán has established the inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta Subscript n Baseline left-parenthesis x right-parenthesis equals upper P Subscript n Superscript 2 Baseline left-parenthesis x right-parenthesis minus upper P Subscript n plus 1 Baseline left-parenthesis x right-parenthesis upper P Subscript n minus 1 Baseline left-parenthesis x right-parenthesis greater-than-over-equals 0 comma negative 1 less-than-over-equals x less-than-over-equals 1 comma n greater-than-over-equals 1 comma">

Δ
n

(
x
)
=

P
n
2

(
x
)
−

P

n
+
1

(
x
)

P

n
−
1

(
x
)
≧
0
,

−
1
≦
x
≦
1
,
n
≧
1
,

{\Delta _n}(x) = P_n^2(x) - {P_{n + 1}}(x){P_{n - 1}}(x) \geqq 0,\quad - 1 \leqq x \leqq 1,n \geqq 1,

\]
with equality only for

x
=
±
1

x = \pm 1

. This inequality has generated considerable interest, and analogous inequalities have been extended to various classes of polynomials: ultraspherical, Laguerre, Hermite, and a class of Jacobi polynomials. Our purpose here is to determine necessary and sufficient conditions for a general class of polynomials to satisfy a Turán type inequality and to characterize the generating functions of such a class.
Extensions of torsion-complete groups
Authors: Hill Paul , Megibben Charles
Published: June 2010

Problem 54 in L. Fuch’s Infinite abelian groups. Vol. II, is solved negatively. An abelian

p
p

-group without elements of infinite height can be an extension of one torsion-complete group by another without itself being torsion complete.
Proof of a polynomial conjecture
Authors: Kristiansen G. K.
Published: June 2010

Let a real polynomial have only real roots, all belonging to an interval

I
I

. An inequality is proved, relating the average value of the polynomial between two consecutive roots to its maximal absolute value in

I
I

.
The excess of sets of complex exponentials
Authors: Peterson David R.
Published: June 2010

Let

Λ
=
{

λ
n

}

\Lambda = \{ {\lambda _n}\}

be a complex sequence and denote its associated set of complex exponentials

{
exp
⁡
(
i

λ
n

x
)
}

\{ \exp (i{\lambda _n}x)\}

by

e
(
Λ
)

e(\Lambda )

. Redheffer and Alexander have shown that if

∑

|

λ
n

−

μ
n

|

>
∞

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

then

e
(
Λ
)

e(\Lambda )

and

e
(
μ
)

e(\mu )

have the same excess over their common completeness interval. This paper shows this result to be the best possible.
Cohesive sets: countable and uncountable
Authors: Shore Richard A.
Published: June 2010

We show that many uncountable admissible ordinals (including some cardinals) as well as all countable admissible ordinals have cohesive subsets. Exactly which cardinals have cohesive subsets, however, is shown to depend on set-theoretic assumptions such as

V
=
L

V = L

or a large cardinal axiom.
Derivations of 𝐴𝑊*-algebras
Authors: Deel James C.
Published: June 2010

It is proved that every derivation on an

A

W
∗

A{W^ \ast }

-algebra of type

I
I

1

\mathrm {II}_1

with central trace is inner. The proof employs a result on the algebraic decomposition of such algebras which is of interest even in the

W
∗

{W^ \ast }

case.
On function spectra
Authors: Hastings Harold M.
Published: June 2010

We construct an explicit internal mapping functor for the homotopy category of CW spectra; this, together with our earlier smash product, yields a symmetric monoidal closed category.
On generating free products of lattices
Authors: Grätzer G. , Sichler J.
Published: June 2010

For a lattice

K
K

let

g
(
K
)

g(K)

denote the cardinality of the smallest generating set of

K
K

. Let

L
L

be the free product of the lattices

A
A

and

B
B

. It is proved that

g
(
L
)
=
g
(
A
)
+
g
(
B
)

g(L) = g(A) + g(B)

. This is proved, in fact, for free products with respect to any given equational class of lattices. Some applications and generalizations are also given.
Exponential estimates for solutions of 𝑦^{”}-𝑞²𝑦=0
Authors: Read T. T.
Published: June 2010

It is shown for any nonnegative continuous function

q
q

on

[
0
,
∞
)

[0,\infty )

and any

c
>
1

c > 1

that any positive increasing solution

y
y

of

y
−

q
2

y
=
0

y - {q^2}y = 0

satisfies

y
(
x
)
≥
y
(
0
)
exp
⁡
(
c

∫
0
x

q
(
t
)
d
t
)

y(x) \geq y(0)\exp (c\int _0^x {q(t)dt)}

on the complement of a set of finite Lebesgue measure. It is also shown that if

lim
inf
(

∫
0
x

q
(
t
)
d
t

/

x
)
>
0

\lim \inf (\int _0^x {q(t)dt/x) > 0}

then the equation has an exponentially increasing solution and an exponentially decreasing solution.
Localizing equivariant bordism
Authors: Wheeler E. R.
Published: June 2010

The unitary bordism of a finite group is computed up to torsion and an equivariant Rohlin exact sequence is exhibited for groups of odd order.
A generalization of the 5-color theorem
Authors: Kainen Paul C.
Published: June 2010

We present a short topological proof of the

5
5

-color theorem using only the nonplanarity of

K
6

{K_6}

. As a bonus, we find that any graph which becomes planar upon the removal of 2 edges can be

5
5

-colored and that any graph which becomes planar when 5 edges are removed is

6
6

-colorable.
Localization properties of basic classes of 𝐶^{∞}-functions
Authors: Darst R. B.
Published: June 2010

Suppose that each of

M

=
C
{

M
n

}

\mathfrak {M} = C\{ {M_n}\}

and

K

=
C
{

K
n

}

\mathcal {K} = C\{ {K_n}\}

is a basic class of

C
∞

{C^\infty }

-functions defined on

I
=
[
0
,
2
π
]

I = [0,2\pi ]

and

M

\mathfrak {M}

is not a subset of

K

\mathcal {K}

. Then it is shown that

M

\mathfrak {M}

contains functions which are nowhere locally in

K

\mathcal {K}

. One of the corollaries asserts that there are quasi-analytic functions which are nowhere locally analytic. It is also shown that many non-quasi-analytic classes contain functions which are nowhere locally quasi-analytic.
𝑜-but not 𝑂-Tauberian theorems
Authors: Lorentz G. G. , Zeller K. L.
Published: June 2010

Let

A
A

be a regular matrix summability method. Relations between different Tauberian theorems for

A
A

are discussed. We show that most summability methods possess some

o
o

-Tauberian theorem with the property that the corresponding

O
O

-Tauberian theorem is not valid.
Adjoint functors induced by adjoint linear transformations
Authors: Palmquist Paul H.
Published: June 2010

Adjoint linear transformations between Hilbert spaces or, more generally, between dual systems of topological vector spaces induce contravariant functors which are adjoint on the right—essentially a Galois connection between the posets of subsets (or subspaces) of the spaces. Modulo scalars the passage from linear maps to functors is one-to-one; indeed, modulo scalars, two linear transformations are adjoint (hence both are weak continuous) if and only if the induced functors are adjoint.
Homotopy groups of compact Hausdorff spaces with trivial shape
Authors: Felt James E.
Published: June 2010

Given a collection

{

π
n

:
n
=
1
,
2
,
⋯
}

\{ {\pi _n}:n = 1,2, \cdots \}

of countable groups such that

π
i

{\pi _i}

is abelian and admits

π
1

{\pi _1}

as a group of operators for

i
≧
2

i \geqq 2

, we construct here an arcwise connected compact metric space of trivial shape whose

j
j

th homotopy group is isomorphic to

π
j

{\pi _j}

for

j
=
1
,
2
,
⋯

j = 1,2, \cdots

. The isomorphisms preserve the action of the first group on the higher groups. Thus, the homotopy groups of a compact metric space of trivial shape may be quite arbitrary.
Properties of nearly-compact spaces
Authors: Herrington Larry L.
Published: June 2010

A general product theorem for nearly-compact spaces and locally nearly-compact spaces is given along with other relating properties.
Analogue of Pontryagin character theory for topological semigroups
Authors: Bowman Thomas
Published: June 2010

The character semigroups of a wide class of abelian continuous inverse semigroups are determined. A duality theorem is given for which a special case gives necessary and sufficient conditions for duality to hold for locally compact semigroups.
Conjugate point properties for an even order linear differential equation
Authors: Johnson George W.
Published: June 2010

Consider an even order equation

D
n

y
=
0

{D_n}y = 0

, where

D
n

{D_n}

is defined recursively by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D 0 y equals y comma a Subscript k comma k plus 1 Baseline upper D Subscript k Baseline y equals left-brace left-parenthesis upper D Subscript k minus 1 Baseline y right-parenthesis prime minus normal upper Sigma Subscript i equals 1 Superscript k Baseline a Subscript k i Baseline upper D Subscript i minus 1 Baseline y right-brace">

D
0

y
=
y
,

a

k
,
k
+
1

D
k

y
=
{
(

D

k
−
1

y

)
′

−

Σ

i
=
1

k

a

k
i

D

i
−
1

y
}

{D_0}y = y,{a_{k,k + 1}}{D_k}y = \{ ({D_{k - 1}}y)’ - \Sigma _{i = 1}^k{a_{ki}}{D_{i - 1}}y\}

, where

a

i
j

{a_{ij}}

is continuous on
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket 0 comma normal infinity right-parenthesis comma a Subscript i comma i plus 1 Baseline greater-than 0 comma i equals 1 comma midline-horizontal-ellipsis comma n minus 1 comma a Subscript n comma n plus 1 Baseline identical-to 1">

[
0
,
∞
)
,

a

i
,
i
+
1

>
0
,
i
=
1
,
⋯
,
n
−
1
,

a

n
,
n
+
1

≡
1

[0,\infty ),{a_{i,i + 1}} > 0,i = 1, \cdots ,n - 1,{a_{n,n + 1}} \equiv 1

, and

a

i
j

≡
0

{a_{ij}} \equiv 0

for

j
>
i
+
1

j > i + 1

. The

k
k

th conjugate point function

η
k

(
a
)

{\eta _k}(a)

is defined and is shown to be a strictly increasing continuous function with domain

[
0
,
b
)

[0,b)

or

[
0
,
∞
)

[0,\infty )

. Extremal solutions are defined as nontrivial solutions with exactly

n
−
1
+
k

n - 1 + k

zeros on

[
a
,

η
k

(
a
)
]

[a,{\eta _k}(a)]

, and are shown to exist, to have odd order zeros at

a
a

and

η
k

(
a
)

{\eta _k}(a)

and exactly

k
−
1

k - 1

odd order zeros in

(
a
,

η
k

(
a
)
)

(a,{\eta _k}(a))

, thus establishing that

η
k

(
a
)
>

η

k
+
1

(
a
)

{\eta _k}(a) > {\eta _{k + 1}}(a)

if

η
k

(
a
)
>
∞

{\eta _k}(a) > \infty

.
Partitions with parts appearing a specified number of times
Authors: Moore Elmo
Published: June 2010

Restrictions of this type, which are known to produce partition identities when applied to all the parts, are shown to produce partition identities when only applied to certain parts.
A characterization of noncommutative quaternion rings
Authors: Kohls Carl W.
Published: June 2010

Noncommutative quaternion subrings of the ring of

4
×
4

4 \times 4

matrices over a formally real field are characterized in terms of the form of commutators and the multiplicity of characteristic roots.
Every direction a Julia direction
Authors: Cain Bryan E.
Published: June 2010

Let

f
(
z
)
=
exp
⁡
(
cosh
⁡
z
)

f(z) = \exp (\cosh z)

. If

N
N

is any

ϵ
\epsilon

-neighborhood of any ray through the origin with slope

m
≠
0
,
∞

m \ne 0,\infty

then

f

−
1

(
w
)
∩
N

{f^{ - 1}}(w) \cap N

is infinite if

w
≠
0

w \ne 0

.
On cyclic vectors of weighted shifts
Authors: Rabindranathan M.
Published: June 2010

Sufficient conditions on a sequence are given in order that the linear manifold spanned by its right translates is dense in certain Hilbert spaces of sequences.
The Silverman necessary condition for multiple integrals in the calculus of variations
Authors: Wilkins J. Ernest
Published: June 2010

A necessary condition for a weak relative minimum of a multiple integral in the calculus of variations, recently derived by Silverman, is shown to be a consequence of (and hence equivalent to) the well-known Legendre-Hadamard necessary condition in the special cases of a double, triple or quadruple integral.
Regularity of generalized stochastic processes and their derivatives
Authors: Pakula Lewis
Published: June 2010

If

X
X

is a generalized stochastic process which is regular in the prediction-theoretic sense then

P
(
d

/

d
x
)
X

P(d/dx)X

is regular for a differential operator

P
(
d

/

d
x
)

P(d/dx)

. This is used to study sufficient conditions for regularity of stationary processes. On the other hand, an example shows that the derivative of a (nonstationary) deterministic process may be regular.
Matrix group monotonicity
Authors: Berman Abraham , Plemmons Robert J.
Published: June 2010

Matrices for which the group inverse exists and is nonnegative are studied. Such matrices are characterized in terms of a generalization of monotonicity. In particular, nonnegative matrices with this property are characterized in terms of their nonnegative rank factorizations.
Free topological groups with no small subgroups
Authors: Morris Sidney A. , Thompson H. B.
Published: June 2010

The first author has shown that a quotient group of a topological group with no small subgroups can have small subgroups, thus answering a question of Kaplansky in the negative. The argument relied on showing that a free abelian topological group on any metric space has no small subgroups. Consequently any abelian metric group is a quotient of a group with no small subgroups. However metric groups with small subgroups exist in profusion! It is shown here that a necessary and sufficient condition for a free (free abelian) topological group on a topological space

X
X

to have no small subgroups is that

X
X

admits a continuous metric. Hence any topological group which admits a continuous metric is a quotient group of a group with no small subgroups.
Proximity spaces and topological functors
Authors: Hunsaker W. N. , Sharma P. L.
Published: June 2010

The purpose of this paper is to determine what natural functors

T
:

A

→

X

T:{\mathbf {A}} \to {\mathbf {X}}

are

(

E

,

M

)

(\mathcal {E},\mathfrak {M})

-topological, where

A

{\mathbf {A}}

is a subcategory of the category of proximity or uniform spaces and

X

{\mathbf {X}}

is an

(

E

,

M

)

(\mathcal {E},\mathfrak {M})

-category. We give necessary and sufficient conditions under which a point separating family of continuous functions can be nicely lifted to a proximally continuous family. Proximities having a finest compatible uniform structure are characterized.
Nonnegative idempotent matrices
Authors: DeMarr Ralph
Published: June 2010

Elementary ideas in the theory of partially ordered linear algebras are used to describe the structure of nonnegative idempotent matrices. In particular, we obtain a kind of “spectral decomposition” theorem for such matrices.
Borel structures and invariants for measurable transformations
Authors: Feldman Jacob
Published: June 2010

It is shown that, in a certain sense, there is no possibility of getting a complete set of conjugacy-invariants for measure-preserving transformations of a probability space.
Sets that remain maximal monotone under all monotone perturbations
Authors: Calvert Bruce
Published: June 2010

We characterize maximal monotone sets whose domain has nonempty interior and which remain maximal monotone under all perturbations.
Note on Boolean ultrapowers
Authors: Benda Miroslav
Published: June 2010

The paper gives a characterization of ultrafilters in Boolean algebras which make Boolean ultrapowers saturated. The description agrees with the definition of good ultrafilters if the Boolean algebra is atomic and is weaker if the Boolean algebra is atomless.
New distortion theorems for functions of positive real part and applications to the partial sums of univalent convex functions
Authors: Bernardi S. D.
Published: June 2010

New distortion theorems are obtained for the class of functions

p
(
z
)
=
1
+

c
n

z
n

+
⋯
(
n
≥
1
)

p(z) = 1 + {c_n}{z^n} + \cdots (n \geq 1)

which are analytic and

Re
p
(
z
)
>
α
(
0
≤
α
>
1
)

\text {Re} p(z) > \alpha (0 \leq \alpha > 1)

in the unit disk

|

z

|

>
1

|z| > 1

. These are used to obtain new results regarding the partial sums of univalent convex functions.
On (𝐿^{𝑝𝑜}(𝐴ₒ),𝐿^{𝑝₁}(𝐴₁))_{𝜃},_{𝑞}
Authors: Cwikel Michael
Published: June 2010

The Lions-Peetre formula for

(

L

p
0

(

A
0

)
,

L

p
1

(

A
1

)

)

θ
,
q

{({L^{{p_0}}}({A_0}),{L^{{p_1}}}({A_1}))_{\theta ,q}}

valid for

q
=
p
(
θ
)

q = p(\theta )

, where

1

/

p
(
θ
)
=
(
1
−
θ
)

/

p
0

+
θ

/

p
1

1/p(\theta ) = (1 - \theta )/{p_0} + \theta /{p_1}

, is shown to have no reasonable generalization for any

q
≠
p
(
θ
)

q \ne p(\theta )

.
Fixed point theorems in uniformly convex Banach spaces
Authors: Edelstein Michael
Published: June 2010

The notion of an asymptotic center is used to prove a number of results concerning the existence of fixed points under certain selfmappings of a closed and bounded convex subset of a uniformly convex Banach space.
A note on metric-fine spaces
Authors: Frolík Zdeněk
Published: June 2010

The coreflection into metric-fine spaces

X
X

is explicitly described, and it is shown that metric-fine proximally fine spaces are just the spaces

X
X

such that

f
:
X
→
Y

f:X \to Y

is uniformly continuous whenever the pre-images under

f
f

of zero sets are zero sets.
A note on the second smallest prime 𝑘th power nonresidue
Authors: Hudson Richard H.
Published: June 2010

Upper bounds for the second smallest prime

k
k

th power nonresidue, which we denote by

g
2

(
p
,
k
)

{g_2}(p,k)

, have been given by many authors. Theorem 1 represents an improvement of these bounds, at least for odd

k
k

. We also give specific estimates for

g
2

(
p
,
k
)

{g_2}(p,k)

, and an upper bound for the

n
n

th

(
n
≥
2
)

(n \geq 2)

smallest prime

k
k

th power nonresidue as a function of the first

n
−
1

n - 1

prime nonresidues. Upper bounds for

g
2

(
p
,
k
)

{g_2}(p,k)

should take on new interest since the author has shown elsewhere that the first two consecutive

k
k

th power nonresidues are bounded above by the product of the first two prime nonresidues.
The approximation of one-one measurable transformations by measure preserving homeomorphisms
Authors: White H. E.
Published: June 2010

This paper contains two results related to the material in [2]. Suppose

f
f

is a one-one transformation of the open unit interval

I
n

{I^n}

(where

n
≧
2

n \geqq 2

) onto

I
n

{I^n}

. 1. If

f
f

is absolutely measureable and

ε
>
0

\varepsilon > 0

, then there is an absolutely measurable homeomorphism

φ
ε

{\varphi _\varepsilon }

of

I
n

{I^n}

onto

I
n

{I^n}

such that

m
(
{
x
:
f
(
x
)
≠

φ
ε

(
x
)

m(\{ x:f(x) \ne {\varphi _\varepsilon }(x)

or

f

−
1

(
x
)
≠

φ
ε

−
1

(
x
)
}
)
>
ε

{f^{ - 1}}(x) \ne \varphi _\varepsilon ^{ - 1}(x)\} ) > \varepsilon

, where

m
m

denotes

n
n

-dimensional Lebesgue measure. 2. Suppose

μ
\mu

is either (1) a nonatomic, finite Borel measure on

I
n

{I^n}

such that

μ
(
G
)
>
0

\mu (G) > 0

for every nonempty open subset

G
G

of

I
n

{I^n}

, or (2) the completion of a measure of type (1). If

f
f

is

μ
\mu

-measure preserving and

ε
>
0

\varepsilon > 0

, then there is a

μ
\mu

-measure preserving homeomorphism

φ
ε

{\varphi _\varepsilon }

of

I
n

{I^n}

onto

I
n

{I^n}

such that

μ
(
{
x
:
f
(
x
)
≠

φ
ε

(
x
)
}
)
>
ε

\mu (\{ x:f(x) \ne {\varphi _\varepsilon }(x)\} ) > \varepsilon

.
On 𝐹𝐾-spaces which are boundedness domains
Authors: Meyers Glenn
Published: June 2010

An open question in summability theory is to characterize those FK-spaces,

E
E

, which are boundedness domains (i.e.,

E
=

m
A

E = {m_A}

for some infinite matrix

A
A

). As a partial solution to this problem we give necessary and sufficient conditions for an FK space

E
E

, which has the

T
T

-sectional boundedness property, to be equal to

m
A

{m_A}

for some row-finite

A
A

.
Quadratic Jordan algebras whose elements are all regular or nilpotent
Authors: McCrimmon Kevin
Published: June 2010

We prove that if

J
J

is a quadratic Jordan algebra whose elements are all either regular or nilpotent, and which satisfies a common multiple property (that whenever

z
z

is nilpotent and

υ
\upsilon

regular then

Im
⁡

U
υ

∩
Ker
⁡

U
z

≠
0

\operatorname {Im} {U_\upsilon } \cap \operatorname {Ker} {U_z} \ne 0

), then modulo the radical

R
R

the algebra

J

/

R

J/R

is either a domain or a form of a Jordan algebra determined by a traceless quadratic form in characteristic 2.
A maximum property for the characteristic initial value problem of the wave equation
Authors: Rhee H.
Published: June 2010

A maximum property is obtained for the solutions of the odd-dimensional wave equation in the interior of the characteristic cone.
Gleason parts and certain counterexamples in the big disc context
Authors: Milaszewicz J. P.
Published: June 2010

In the first part of this paper, Lemmas (1.1) and (1.4) provide a generalization of the fact that no Cauchy measures exist in the big disc. In the second part we show that this fact implies the existence of the H. Bohr and J. Favard counterexamples concerning harmonic and analytic almost periodic functions.
Variation of multiparameter Brownian motion
Authors: Yoder Lane
Published: June 2010

Lévy’s

N
N

-parameter Brownian motion in

d
d

-dimensional space is denoted by

W

(
N
,
d
)

{W^{(N,d)}}

. Using uniform partitions and a Vitali-type variation, Berman recently extended to

W

(
N
,
1
)

{W^{(N,1)}}

a classical result of Lévy concerning the relation between

W

(
1
,
1
)

{W^{(1,1)}}

and

2
2

-variation. With this variation

W

(
N
,
d
)

{W^{(N,d)}}

has variation dimension

2
N

2N

with probability one. An appropriate definition of weak variation is given using powers of the diameters of the images of sets which satisfy a parameter of regularity. A previous result concerning the Hausdorff dimensions of the graph and image is used to show the weak variation dimension of

W

(
N
,
d
)

{W^{(N,d)}}

is

2
N

2N

with probability one, extending the result for

W

(
1
,
1
)

{W^{(1,1)}}

of Goffman and Loughlin. If unrestricted partitions of the domain are used, the weak variation dimension of a function turns out to be the same as the Hausdorff dimension of the image.
Counting patterns with a given automorphism group
Authors: White Dennis E.
Published: June 2010

A formula, analogous to the classical Burnside lemma, is developed which counts orbit representatives from a set under a group action with a given stabilizer subgroup conjugate class. This formula is applied in a manner analogous to a proof of Pólya’s theorem to obtain an enumeration of patterns with a given automorphism group.
Free actions and complex cobordism
Authors: Lazarov Connor , Wasserman Arthur G.
Published: June 2010

Connor and Floyd have observed that a free action of a finite group

G
G

on a compact manifold

M
M

preserving a stable almost complex structure produces a stably almost complex quotient manifold

M

/

G

M/G

. Hence, the bordism group of such actions,

U
∗

G
,

free

U_ \ast ^{G,{\text {free}}}

, is just

U
∗

(
B
G
)

{U_ \ast }(BG)

. If

G
G

is not finite or abelian, but an arbitrary compact Lie group, the tangent bundle along the fibres gives trouble. Nevertheless, it is shown that if

H
∗

(
B
G
)

{H^ \ast }(BG)

is torsion free then

U
∗

G
,

free

≈

U
∗

(
B
G
)

U_ \ast ^{G,{\text {free}}} \approx {U_ \ast }(BG)

.
The order of the automorphism group of a central product
Authors: Hummel Kenneth G.
Published: June 2010

If a

p
p

-group

G
G

is the central product of nontrivial subgroups

H
H

and

A
A

, where

A
A

is abelian and

o
(
H
)

|

o
(
Aut
⁡
H
)

o(H)|o(\operatorname {Aut} H)

, then

o
(
G
)

|

o
(
Aut
⁡
G
)

o(G)|o(\operatorname {Aut} G)

.
Rearrangements of conditionally convergent real series with preassigned cycle type
Authors: Smith John Howard
Published: June 2010

For any conditionally convergent real series, any real number

r
r

, and any infinite cycle type, there is a permutation of the indices, of the given cycle type, which makes the series converge to

r
r

.
On the scalar curvature and sectional curvatures of a totally real submanifold
Authors: Houh Chorng Shi
Published: June 2010

For a totally real minimal submanifold of a complex space form, pinching for scalar curvature implies pinching for sectional curvatures.
Direct product decomposition of alternative rings
Authors: Myung Hyo Chul , Jimenez Luis R.
Published: June 2010

It is shown that any alternative ring

A
A

equipped with the relation

⩽
\leqslant

, defined by

x
⩽
y

x \leqslant y

if and only if

x
y
=

x
2

xy = {x^2}

, is isomorphic to a direct product of alternative division rings if and only if the relation

⩽
\leqslant

is a partial order on

A
A

such that

A
A

is hyperatomic and orthogonally complete.
Triples on reflective subcategories of functor categories
Authors: Newell David C.
Published: June 2010

We show that if

S

\mathcal {S}

is a cocontinuous triple on a full reflective subcategory of a functor category then the category of

S

\mathcal {S}

-algebras is again a full reflective subcategory of a functor category.
Hardy space expectation operators and reducing subspaces
Authors: Ball Joseph A.
Published: June 2010

In this paper we study the range of the isometry on

H
p

{H^p}

arising from an inner function which is zero at zero by composition. The range of such an isometry is characterized as a closed subspace

M

\mathfrak {M}

of

H
p

{H^p}

(weak-

∗

^ \ast

closed for

p
=
∞

p = \infty

) satisfying the following: (i) the constant function 1 is in

M

\mathfrak {M}

; (ii) if

f
∈

M

f \in \mathfrak {M}

and

g
∈

H
∞

∩

M

g \in {H^\infty } \cap \mathfrak {M}

, then

f
g
∈

M

fg \in \mathfrak {M}

; (iii) if

f
∈

M

f \in \mathfrak {M}

has inner-outer factorization

f
=
χ
⋅
F

f = \chi \cdot F

, then

χ
\chi

is in

M

\mathfrak {M}

; (iv) if

{

B
α

:
α
∈

A

}

\{ {B_\alpha }:\alpha \in \mathcal {A}\}

is a collection of inner functions in

M

\mathfrak {M}

, then the greatest common divisor of

{

B
α

:
α
∈

A

}

\{ {B_\alpha }:\alpha \in \mathcal {A}\}

is also in

M

\mathfrak {M}

; and (v) if

f
∈

M

,
B
∈

M

f \in \mathfrak {M},B \in \mathfrak {M}

, where

B
B

is inner and

B
¯

⋅
f
∈

H
p

\bar B \cdot f \in {H^p}

, then

B
¯

⋅
f
∈

M

\bar B \cdot f \in \mathfrak {M}

. The proof makes use of the fact that there exists a projection onto such a subspace satisfying the axioms of an expectation operator, which for

p
=
2

p = 2

, is simply the orthogonal projection. This characterization is applied to give an equivalent formulation of a conjecture of Nordgren concerning reducing subspaces of analytic Toeplitz operators.
On Kannan maps
Authors: Wong Chi Song
Published: June 2010

Let

K
K

be a (nonempty) weakly compact convex subset of a Banach space

B
B

. Let

T
T

be a self map on

K
K

such that for all

x
,
y

x,y

in
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K comma StartAbsoluteValue EndAbsoluteValue upper T left-parenthesis x right-parenthesis minus upper T left-parenthesis y right-parenthesis StartAbsoluteValue EndAbsoluteValue less-than-or-slanted-equals left-parenthesis StartAbsoluteValue EndAbsoluteValue x minus upper T left-parenthesis x right-parenthesis StartAbsoluteValue EndAbsoluteValue plus StartAbsoluteValue EndAbsoluteValue y minus upper T left-parenthesis y right-parenthesis StartAbsoluteValue EndAbsoluteValue right-parenthesis slash 2">

K
,

|

|

T
(
x
)
−
T
(
y
)

|

|

⩽
(

|

|

x
−
T
(
x
)

|

|

+

|

|

y
−
T
(
y
)

|

|

)

/

2

K,||T(x) - T(y)|| \leqslant (||x - T(x)|| + ||y - T(y)||)/2

. It is proved without the continuity of

T
T

and Zorn’s lemma that

T
T

has a fixed point if and only if

inf
{

|

|

x
−
T
(
x
)

|

|

:
x
∈
K
}
=
0

\inf \{ ||x - T(x)||:x \in K\} = 0

. A characterization of the existence of fixed points for such

T
T

is obtained in terms of close-to-normal structure. As consequences, the following results are obtained: (i)

T
T

has a unique fixed point if

B
B

is locally uniformly convex or more generally if

B
B

has the property

A

{\mathbf {A}}

: For any sequence

{

x
n

}

\{ {x_n}\}

in

B
,
{

x
n

}

B,\{ {x_n}\}

converges to a point

x
x

in

B
B

if it converges weakly to

x
x

and

{

|

|

x

|

|

}

\{ ||x||\}

converges to

|

|

x

|

|

||x||

; (ii)

T
T

has a unique fixed point if

B
B

is separable.
On the uniqueness of solutions of certain Diophantine equations
Authors: Muskat Joseph B. , Zee Yun Cheng
Published: June 2010

An arithmetic proof by L. E. Dickson of the uniqueness of the integral solutions of a certain quaternary quadratic form is generalized to include several similar forms which have appeared recently in cyclotomy.
An invariant for modules over a discrete valuation ring
Authors: Stanton R. O.
Published: June 2010

Warfield has recently defined a new class of invariants for mixed modules over a discrete valuation ring. These invariants, along with the Ulm invariants, enable Warfield to prove an analogue to Ulm’s theorem. Warfield’s definition contains two shortcomings. The invariants are defined for a limited class of modules. Moreover it is difficult to show that the invariants are well defined. This paper defines a new invariant which coincides with that of Warfield, and overcomes both difficulties.
The zeros of Jensen polynomials are simple
Authors: Csordas George , Williamson Jack
Published: June 2010

An entire function

f
(
z
)
=

Σ

k
=
0

∞

a
k

z

k
+
m

/

k
!

f(z) = \Sigma _{k = 0}^\infty {a_k}{z^{k + m}}/k!

is said to be in the class

L

−

P

\mathcal {L} - \mathcal {P}

(Laguerre-Pólya) if it can be represented in the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis equals c z Superscript m Baseline e Superscript minus alpha z squared plus beta z Baseline coproduct Underscript n Endscripts left-parenthesis 1 minus z slash z Subscript n Baseline right-parenthesis e Superscript z slash z Super Subscript n Superscript Baseline comma">

f
(
z
)
=
c

z
m

e

−
α

z
2

+
β
z

∐
n

(
1
−
z

/

z
n

)

e

z

/

z
n

 

,

f(z) = c{z^m}{e^{ - \alpha {z^2} + \beta z}}\coprod \limits _n {(1 - z/{z_n}){e^{z/{z_n}}}} {\text { }},

\]
where

α
≥
0
,
c
,
β

\alpha \geq 0,c,\beta

and

z
n

{z_n}

are real, and

Σ
n

z
n

−
2

>
∞

{\Sigma _n}z_n^{ - 2} > \infty

. A well-known result of Jensen asserts that the associated (Jensen) polynomials
\[

g
n

(
x
)
=

∑

k
=
0

n

(

n

k

)

a
k

x
k

{g_n}(x) = \sum \limits _{k = 0}^n {\left ( {\begin {array}{*{20}{c}} n \\ k \\ \end {array} } \right ){a_k}{x^k}}

\]
have only real zeros. Here we present an elementary proof of this fact; we also show that the zeros of

g
n

(
x
)

{g_n}(x)

are simple.
A tensor approach to interpolation phenomena in discrete abelian groups
Authors: Blei Ron C.
Published: June 2010

We show that a tensor algebra setting is an efficient tool that separates Sidonicity from other interpolation properties.
Stabilizing tensor products
Authors: Hastings Harold M.
Published: June 2010

Let

C
C

be a symmetric monoidal category with a suspension, and let SC be the resulting stable category. We shall give necessary and sufficient conditions for extending the symmetric monoidal structure to a monoidal structure on SC. These imply that the usual smash product on finite pointed CW complexes cannot be extended to a smash product (with

S
0

{S^0}

as unit) on finite spectra, hence not on Boardman spectra. This confirms a conjecture of Alex Heller.
Nonrealizability of some cyclic complex bordism modules
Authors: Zahler Raphael S.
Published: June 2010

It is shown that certain modules over the complex cobordism ring MU* cannot be realized in the sense that none of them can be isomorphic to the MU-cohomology module of a finite complex. Potential applications in stable homotopy theory are discussed.
Functions of Brownian motion
Authors: Walsh John B.
Published: June 2010

The class of continuous functions

h
h

for which

h
h

(Brownian motion) is a Markov process is determined.
Collisions are of first category
Authors: Saari Donald G.
Published: June 2010

It is shown that the set of initial conditions leading to collisions in the

n
n

-body problem of Newtonian mechanics is of (Baire) first category.
The number of proper minimal quasivarieties of groupoids
Authors: Shafaat A.
Published: June 2010

It is shown that if an algebra has more than one element, is freely generated in some variety by one element and has a cancellative endomorphism semigroup then it generates a minimal quasivariety. This is used to construct uncountably many minimal quasivarieties of groupoids that are not varieties.
A note on a theorem of Gehring and Lehto
Authors: Fadell A. G.
Published: June 2010

The concept of mesh approximate differential is defined as a modification of regular approximate differential. It is shown that for open continuous real-valued maps on open sets in

n
n

-space the concepts of mesh approximate differentiability and total differentiability are equivalent, and the Gehring-Lehto theorem is obtained as a corollary by a sharpening of a known theorem on regular approximate differentials.
Spherical harmonic expansion of the Poisson-Szegő kernel for the ball
Authors: Folland G. B.
Published: June 2010

By applying the theory of unitary spherical harmonics, we obtain an expansion of the Poisson-Szegö kernel for the unit ball in complex

n
n

-space in terms of Jacobi polynomials and hypergeometric functions.
On inverse limits of homotopy sets
Authors: Kahn Peter J.
Published: June 2010

An elementary proof is given that, under certain conditions on a space

F
F

, the homotopy set

[
X
,
F
]

[X,F]

maps bijectively onto the inverse limit of homotopy sets determined by the finite subcomplexes of

X
X

. The only other satisfactory proof known requires the Brown representability theorem.
Sums of stationary sequences cannot grow slower than linearly
Authors: Kesten Harry
Published: June 2010

It is shown that for a stationary sequence of random variables

X
1

,

X
2

,
⋯

{X_1},{X_2}, \cdots

one has
\[

lim
inf

n

−
1

∑

i
=
1

n

X
i

>
0

\lim \inf {n^{ - 1}}\sum \limits _{i = 1}^n {{X_i} > 0}

\]
a.e. on the set

{

Σ
1
n

X
i

→
∞
,
n
→
∞
}

\{ \Sigma _1^n{X_i} \to \infty ,n \to \infty \}

.
A reformulation of the Radon-Nikodým theorem
Authors: Lewin Jonathan , Lewin Mirit
Published: June 2010

The Radon-Nikodym theorems of Segal and Zaanen are principally concerned with the classification of those measures

μ
\mu

for which any

λ
≪
μ

\lambda \ll \mu

is given in the form
\[

(

i

)

λ
(
A
)
=

∫
A

g
d
μ

({\text {i}})\quad \lambda (A) = \int _A {gd\mu }

\]
for all sets

A
A

of finite

μ
\mu

measure. This paper is concerned with the characterization of those pairs

λ
,
μ

\lambda ,\mu

for which the equality (i) holds for every measurable set

A
A

, and introduces a notion of compatibility that essentially solves this problem. In addition, some applications are made to Radon-Nikodym theorems for regular Borel measures.
An example of nonexcisiveness in sheaf cohomology
Authors: Deo Satya
Published: June 2010

An example is constructed to show that the conditions of tautness, besides others, imposed on spaces in a result on

ϕ
\phi

-excisive couples obtained by Bredon [1] are needed.
Identities for conjugation in the Steenrod algebra
Authors: Straffin Philip D.
Published: June 2010

Let

χ
\chi

be the canonical conjugation in the Steenrod algebra

A

2

{\mathcal {A}_2}

. I prove the identity
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S q Superscript 2 Super Superscript n Baseline plus chi left-parenthesis upper S q Superscript 2 Super Superscript n Superscript Baseline right-parenthesis equals upper S q Superscript 2 Super Superscript n minus 1 Baseline chi left-parenthesis upper S q Superscript 2 Super Superscript n minus 1 Superscript Baseline right-parenthesis">

S

q

2
n

+
χ
(
S

q

2
n

)
=
S

q

2

n
−
1

χ
(
S

q

2

n
−
1

)

S{q^{{2^n}}} + \chi (S{q^{{2^n}}}) = S{q^{{2^{n - 1}}}}\chi (S{q^{{2^{n - 1}}}})

\]
and generalizations of this identity both in

A

2

{\mathcal {A}_2}

and in

A

p

{\mathcal {A}_p}

where

p
p

is an odd prime.
On infinite dimensional Ito’s formula
Authors: Lin T. F.
Published: June 2010

Ito’s formula for

Y
(
t
)
g
(
t
,
ξ
(
t
)
)

Y(t)g(t,\xi (t))

is given where

g
g

is a Hilbert space valued function,

ξ
(
t
)

\xi (t)

is a diffusion on Hilbert space and

Y
(
t
)

Y(t)

is an operator-valued stochastic integral w.r.t.

ξ
(
t
)

\xi (t)

. A stochastic integral representation for solution of a certain second order parabolic equation is also given.
An ℵ₀-categorical theory whose language is countably infinite
Authors: Parikh Rohit
Published: June 2010

In the following, we describe an

ℵ
0

{\aleph _0}

-categorical, complete, decidable theory

T
T

whose language is countably infinite.
On a uniqueness problem in the theory of linear integral equations
Authors: Stevens Robert R.
Published: June 2010

The primary purpose of this paper is to give sufficient conditions for a function

G
G

which ensure that if

∫
0
1

f
(
x
t
)
G
(
t
)
d
t
=
0

\int _0^1 {f(xt)G(t)dt = 0}

a.e. in

(
0
,
1
)

(0, 1)

then the function

f
f

is zero almost everywhere in

(
0
,
1
)

(0, 1)

. Several applications are given.
A new generalization of the Sturm comparison theorem to selfadjoint systems
Authors: Ahmad Shair , Lazer Alan C.
Published: June 2010

The Sturm Comparison Theorem is generalized to second order linear systems. It is based on a comparison of the elements of the matrices involved.
On Urysohn-closed and minimal Urysohn spaces
Authors: Joseph James E.
Published: June 2010

In this paper, several characterizations of Urysohn-closed and minimal Urysohn spaces are given along with some relating properties.
A subnormal semigroup without normal extension
Authors: Lubin Arthur
Published: June 2010

Theorem 1. There exists a subnormal semigroup with no commuting normal extension. Theorem 2. There exist two commuting quasinormal operators without commuting normal extension.
A converse of the Hilbert syzygy theorem
Authors: Cheng Charles Ching-an , Shapiro Jay
Published: June 2010

The following converse of the Hilbert Syzygy Theorem is proved. Suppose

K
K

is a noetherian commutative ring with identity that has finite global dimension, and suppose that

M
M

is a finitely generated abelian cancellative monoid. If

gl

dim
⁡
K
M
=
n
+

gl

dim
⁡
K

{\text {gl}}\dim KM = n + {\text {gl}}\dim K

then

M
M

is of the form

(

×

i
=
1

n

M
i

)
×
H

( \times _{i = 1}^n{M_i}) \times H

where

M
i

≅

Z

{M_i} \cong {\mathbf {Z}}

or

N

{\mathbf {N}}

and where

H
H

is a finite group with no

K
K

-torsion.
A note on characters of algebraic groups
Authors: Broughton S. A.
Published: June 2010

It is proven that a nowhere-vanishing polynomial function on a connected algebraic group, over an algebraically closed field, is a multiple of a (one-dimensional) character.
When is the semigroup ring perfect?
Authors: Okniński Jan
Published: June 2010

A characterization of perfect semigroup rings

A
[
G
]

A[G]

is given by means of the properties of the ring

A
A

and the semigroup

G
G

.
On the existence of Shilov boundaries
Authors: Wittmann Rainer
Published: June 2010

The existence of a carrier in the sense of potential theory is proved for rather general cones of functions. The existence of the Shilov boundary is an immediate consequence of this result.
On the pointwise convergence of a class of nonharmonic Fourier series
Authors: Young Robert M.
Published: June 2010

Extending a classical theorem of Levinson [1, Theorem XVIII], we show that when the numbers

{

λ
n

}

\left \{ {{\lambda _n}} \right \}

are given by

λ
n

=
n
+

1
4

(
n
>
0
)

{\lambda _n} = n + \tfrac {1}{4}(n > 0)

,

λ
0

=
0

{\lambda _0} = 0

, and

λ

−
n

=
−

λ
n

(
n
>
0
)

{\lambda _{ - n}} = - {\lambda _n}(n > 0)

, each function

f
f

in

L
2

(
−
π
,
π
)

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

has a unique nonharmonic Fourier expansion

f
(
t
)
∼

∑

−
∞

∞

c
n

e

i

λ
n

t

f(t) \sim \sum \nolimits _{ - \infty }^\infty {{c_n}{e^{i{\lambda _n}t}}}

, which is equiconvergent with its ordinary Fourier series, uniformly on each closed subinterval of

(
−
π
,
π
)

( - \pi ,\pi )

.
Relative 𝐶*-sums
Authors: Berberian S. K.
Published: June 2010

A "bounded choice" theorem is proved for

A

W

∗

{\text {A}}{{\text {W}}^ * }

-subalgebras of a

C

∗

{{\text {C}}^ * }

-sum.
On the product of 𝑓-proximities
Authors: Chattopadhyay K. C.
Published: June 2010

A unified theory of products of proximity spaces has been introduced. Under suitable conditions on grill operators

f
f

, it has been proved that an arbitrary product of

f
f

-proximities is an

f
f

-proximity.