bluuebunny/crossref_metadata_2025 huggingface dataset:
𝜆-equivariant bordism of maps
Authors: Rinne Robert L.
Published: June 2010

It is shown that

λ
\lambda

-equivariant maps may be classified up to bordism where the acting groups are abelian of odd order.
Equivalent types of invariant means on locally compact groups
Authors: Renaud P. F.
Published: June 2010

For

G
G

a locally compact amenable group, we establish the equivalence of left invariant means and topologically left invariant means on

L
∞

(
G
)

{L^\infty }(G)

.
Multiplier operators on 𝐵*-algebras
Authors: Malviya B. D. , Tomiuk B. J.
Published: June 2010

The purpose of this paper is to give a characterization of the dual

B
∗

{B^\ast }

-algebra and the algebra of bounded linear operators on Hilbert space in terms of their multipliers.
A note on the asymptotic behavior of an integral equation
Authors: Shilepsky Carol
Published: June 2010

Assume the existence and boundedness of a solution to an integral equation. Conditions are found which ensure the solution has a limit at infinity.
Extremal problems for certain classes of analytic functions
Authors: Brickman Louis
Published: June 2010

An extremal problem for the set of probability measures on the unit circle is formulated and solved. The result contains as special cases many of the published results concerning extremal problems for analytic functions with positive real part, starlike functions, and other classes of functions associated with integral formulas.
On the first boundary value problem for [ℎ(𝑥,𝑥^{’},𝑡)]^{’}= 𝑓(𝑥,𝑥^{’},𝑡)
Authors: Ford Wayne T.
Published: June 2010

The boundary value problem for

[
h
(
x
,

x
′

,
t
)

]
′

=
f
(
x
,

x
′

,
t
)

[h(x,x’,t)]’ = f(x,x’,t)

is studied with

x
(
0
)
=
x
(
1
)
=
0

x(0) = x(1) = 0

. It is assumed that substitution of functions u and v in

L
2

(
0
,
1
)

{L_2}(0,1)

into h and f produces the functions

h
[
u
(
⋅
)
,
v
(
⋅
)
,
⋅
]

h[u( \cdot ),v( \cdot ), \cdot ]

and

f
[
u
(
⋅
)
,
v
(
⋅
)
,
⋅
]

f[u(\cdot ),v(\cdot ),\cdot ]

in

L
2

(
0
,
1
)

{L_2}(0,1)

such that this map from

L
2

(
0
,
1
)
×

L
2

(
0
,
1
)

{L_2}(0,1) \times {L_2}(0,1)

into

L
2

(
0
,
1
)
×

L
2

(
0
,
1
)

{L_2}(0,1) \times {L_2}(0,1)

is hemicontinuous. Existence and uniqueness are shown in

H
0
1

(
0
,
1
)

H_0^1(0,1)

under the assumption that constants

λ
\lambda

and

η
\eta

exist such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket left-parenthesis upper V minus v right-parenthesis left-bracket h left-parenthesis upper U comma upper V comma t right-parenthesis minus h left-parenthesis u comma v comma t right-parenthesis right-bracket plus left-parenthesis upper U minus u right-parenthesis left-bracket f left-parenthesis upper U comma upper V comma t right-parenthesis minus f left-parenthesis u comma v comma t right-parenthesis right-bracket right-bracket greater-than-over-equals lamda left-parenthesis upper V minus v right-parenthesis squared minus eta left-parenthesis upper U minus u right-parenthesis squared">

[
(
V
−
v
)
[
h
(
U
,
V
,
t
)
−
h
(
u
,
v
,
t
)
]
+
(
U
−
u
)
[
f
(
U
,
V
,
t
)
−
f
(
u
,
v
,
t
)
]
]
≧
λ

(
V
−
v

)
2

−
η

(
U
−
u

)
2

[(V - v)[h(U,V,t) - h(u,v,t)] + (U - u)[f(U,V,t) - f(u,v,t)]] \geqq \lambda {(V - v)^2} - \eta {(U - u)^2}

\]
whenever t lies between zero and one while u, v, U and V are arbitrary. Also, it is assumed that

λ
\lambda

and

λ

π
2

−
η

\lambda {\pi ^2} - \eta

are positive.
On semibounded differential operators
Authors: Hochstadt Harry
Published: June 2010

It is shown that regular ordinary differential operators have a semibounded spectrum. The proof requires fewer prerequisites than other proofs found in the literature and also yields estimates on the lower bound of the spectrum.
On one-relator groups having elements of finite order
Authors: Fischer J. , Karrass A. , Solitar D.
Published: June 2010

Let G be a one-relator group having torsion. It is easy to show that there exists a normal subgroup N which is of finite index and torsion-free. We prove that N is free iff G is the free product of a free group and a finite cyclic group; N is a proper free product iff G is a proper free product of a free group and a one-relator group. In the proof, use is made of the following result: the elements of finite order in G generate a group which is the free product of conjugates of a finite cyclic group.
Multivalued operations and universal coalgebra
Authors: Davis Robert C.
Published: June 2010

We define a type of representation of a semigroup by relations on a set, more general than the representation by transformations, which leads to a category cotripleable over the category of sets. This result motivates a generalization to higherorder operations and a concept of cotheory resembling that of theory in universal algebra.
Representation of a binary quadratic form as a sum of two squares
Authors: Williams Kenneth S.
Published: June 2010

Let

ϕ
(
x
,
y
)

\phi (x,y)

be an integral binary quadratic form. A short proof is given of Pall’s formula for the number of representations of

ϕ
(
x
,
y
)

\phi (x,y)

as the sum of squares of two integral linear forms.
Linear processes generated by independent random variables
Authors: Woyczyński W. A.
Published: June 2010

In a recent paper R. Dudley gave a characterization of those sequences of independent and identically distributed random variables which are

l
p

{l_p}

-compatible for

p
≧
1

p \geqq 1

. In the present note we extend his result into

p
∈
(
0
,
1
]

p \in (0,1]

and provide some conditions (necessary or sufficient) for

l
φ

{l_\varphi }

-compatibility of a sequence of independent random variables not necessarily identically distributed.
On the order of some error functions related to 𝑘-free integers
Authors: Joshi V. S.
Published: June 2010

Let

Δ
k

(
x
)

{\Delta _k}(x)

and

Δ

k
′

(
x
)

\Delta {’_k}(x)

be the error functions in the asymptotic formulae for the number and the sum of k-free integers not exceeding x. We prove that on the assumption of Riemann hypothesis, we have
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta prime Subscript k Baseline left-parenthesis x right-parenthesis minus x normal upper Delta Subscript k Baseline left-parenthesis x right-parenthesis equals upper O left-parenthesis x Superscript 1 plus 1 slash 2 k plus epsilon Baseline right-parenthesis">

Δ

k
′

(
x
)
−
x

Δ
k

(
x
)
=
O
(

x

1
+
1

/

2
k
+
ε

)

\Delta {’_k}(x) - x{\Delta _k}(x) = O({x^{1 + 1/2k + \varepsilon }})

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over x EndFraction integral Subscript 1 Superscript x Baseline normal upper Delta Subscript k Baseline left-parenthesis t right-parenthesis d t equals upper O left-parenthesis x Superscript 1 slash 2 k plus epsilon Baseline right-parenthesis comma">

1
x

∫
1
x

Δ
k

(
t
)
d
t
=
O
(

x

1

/

2
k
+
ε

)
,

\frac {1}{x}\int _1^x {{\Delta _k}(t)dt = O({x^{1/2k + \varepsilon }}),}

\]
for arbitrary

ε
>
0

\varepsilon > 0

.
On restricted uniqueness for systems of ordinary differential equations
Authors: Bownds J. M. , Díaz J. B.
Published: June 2010

A uniqueness theorem is proved, for not necessarily Lipschitzian systems of ordinary differential equations,

y
′

=
f

y’ = f

. This theorem compares with one of Okamura and Murakami, in that, here, at the expense of assuming a certain additional smoothness for f on open sets, no assumption is made regarding the existence of an auxiliary positive definite (Lyapunov) function. An example compares the relative applicability of the two theorems.
Compact Riemann surfaces with conformal involutions
Authors: Gilman Jane
Published: June 2010

In this paper surfaces which have conformal involutions are characterized by their period matrices.
A hyperspace for convergence spaces
Authors: Gazik R. J.
Published: June 2010

The purpose of this note is to introduce a convergence structure

h
(
t
)

h(t)

on the collection

C
(
E
)

C(E)

of nonempty, compact subsets of a Hausdorff convergence space (E, t). It is shown that if (E, t) is topological, then

h
(
t
)

h(t)

agrees with the Vietoris topology on

C
(
E
)

C(E)

. It is proved that

(
C
(
E
)
,
h
(
t
)
)

(C(E),h(t))

is Hausdorff, that it inherits regularity from (E, t) and that it is compact whenever (E, t) is compact and regular.
A property of a class of nonlinear difference equations
Authors: Howard F. T.
Published: June 2010

Let

g
(
n
)

g(n)

be a rational function of

n
n

whose denominator is divisible by the same power of 2 for each

n
n

and let

a
1

,

a
2

,
⋯

{a_1},{a_2}, \cdots

be any sequence of rational numbers such that for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Subscript n Baseline equals g left-parenthesis n right-parenthesis left-parenthesis a 1 a Subscript n minus 1 Baseline plus a 2 a Subscript n minus 2 Baseline plus midline-horizontal-ellipsis plus a Subscript n minus 1 Baseline a 1 right-parenthesis">

a
n

=
g
(
n
)
(

a
1

a

n
−
1

+

a
2

a

n
−
2

+
⋯
+

a

n
−
1

a
1

)

{a_n} = g(n)({a_1}{a_{n - 1}} + {a_2}{a_{n - 2}} + \cdots + {a_{n - 1}}{a_1})

. In this paper we determine the exact power of 2 dividing the denominator of

a
n

{a_n}

for each

n
n

and prove congruences

(
mod

4
)

\pmod 4

and

(
mod

8
)

\pmod 8

.
Global dimension of triangular orders over a discrete valuation ring
Authors: Jategaonkar Vasanti A.
Published: June 2010

We characterize triangular

R
R

-orders of finite global dimension in

n
×
n

n \times n

matrix rings over the quotient field of DVR

R
R

and obtain a precise upper bound for their global dimension, viz.

n
−
1

n - 1

. We also characterize triangular

R
R

-orders of highest global dimension.
Coverable semigroups
Authors: Lau A. Y. W.
Published: June 2010

The concept of a semilattice having small semilattices has been studied and some equivalences of this property have been investigated. In the process of investigating semilattices, the author found a class of semigroups called coverable semigroups, and the interesting fact about it is that a necessary and sufficient condition for a compact semilattice to have small semilattices is to be coverable. Also by virtue of the concept of coverable semigroups, one can show that the usual interval is costable instead of the more intricate method of proving it by producing an arc.
Order summability and almost convergence
Authors: Bell Howard T.
Published: June 2010

The object of this paper is to compare the summability methods of almost convergence and order summability. This is done by examining a method of summability which includes both methods. The main result is that every monotone Fourier effective matrix is strongly regular.
A criterion for the positivity of sine polynomials
Authors: Steinig John
Published: June 2010

A sufficient condition is derived for a sine polynomial with real coefficients to be nonnegative on

(
0
,
π
)

(0,\pi )

.
Mackey compactness in Banach spaces
Authors: Howard Joe
Published: June 2010

If

A
′

A’

, a subset of a conjugate Banach space

X
′

X’

, is sequentially compact in the Mackey topology

(
τ
(

X
′

,
X
)
)

(\tau (X’,X))

, then

A
′

A’

is conditionally compact in the Mackey topology. The converse is not true.
Injective dimension of quaternion orders
Authors: Ramras Mark
Published: June 2010

Tarsy has shown that if

R
R

is a discrete valuation ring with quotient field

K
K

, and

Σ
\Sigma

is a quaternion

K
K

-algebra, then the finitistic global dimension of any

R
R

-order in

Σ
\Sigma

is 1. In this paper we allow

R
R

to be any regular local ring of dimension

n
n

and study the

R
R

-free orders

Λ
\Lambda

in

Σ
\Sigma

. First we show that the finitistic global dimension of

Λ
\Lambda

is

n
n

. Our main result concerns the injective dimension of

Λ
\Lambda

(considered as either a left or a right

Λ
\Lambda

-module). Let

M

\mathfrak {M}

denote the maximal ideal of

R
R

. Then the injective dimension of

Λ
\Lambda

is

n
n

, unless

Λ

/

M

Λ

\Lambda /\mathfrak {M}\Lambda

is a commutative local ring whose socle is not principal. In this case, the injective dimension of

Λ
\Lambda

is

∞
\infty

.
Prime generalized alternative rings 𝐼 with nontrivial idempotent
Authors: Smith Harry F.
Published: June 2010

A generalized alternative ring

I
I

is a nonassociative ring

R
R

in which the identities
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis w x comma y comma z right-parenthesis plus left-parenthesis w comma x comma left-bracket y comma z right-bracket right-parenthesis minus w left-parenthesis x comma y comma z right-parenthesis minus left-parenthesis w comma y comma z right-parenthesis x semicolon left-parenthesis left-bracket w comma x right-bracket comma y comma z right-parenthesis plus left-parenthesis w comma x comma y z right-parenthesis minus y left-parenthesis w comma x comma z right-parenthesis minus left-parenthesis w comma x comma y right-parenthesis z">

(
w
x
,
y
,
z
)
+
(
w
,
x
,
[
y
,
z
]
)
−
w
(
x
,
y
,
z
)
−
(
w
,
y
,
z
)
x
;
(
[
w
,
x
]
,
y
,
z
)
+
(
w
,
x
,
y
z
)
−
y
(
w
,
x
,
z
)
−
(
w
,
x
,
y
)
z

(wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x;([w,x],y,z) + (w,x,yz) - y(w,x,z) - (w,x,y)z

; and

(
x
,
x
,
x
)

(x,x,x)

are identically zero. It is demonstrated here that if

R
R

is a ring of this type with characteristic different from two and three, then

R
R

semiprime with idempotent

e
e

implies that

R
R

has a Peirce decomposition relative to

e
e

. Furthermore, if

R
R

is prime and

e
≠
0
,
1

e \ne 0,1

; then

R
R

must be alternative.
Continuous dependence of fixed point sets
Authors: Markin J. T.
Published: June 2010

The stability of the fixed point sets of a uniformly convergent sequence of set valued contractions is proved under the assumption that the maps are defined on a closed bounded subset

B
B

of Hilbert space and take values in the family of nonempty closed convex subsets of

B
B

.
The amalgamation property for 𝐺-metric spaces
Authors: Hung H. H.
Published: June 2010

Let G be a (totally) ordered (abelian) group. A G-metric space

(
X
,
ρ
)

(X,\rho )

consists of a nonempty set X and a G-metric

ρ
:
X
×
X
→
G

\rho :X \times X \to G

(satisfying the usual axioms of a metric, with G replacing the ordered group of real numbers). That the amalgamation property holds for the class of all metric spaces is attributed, by Morley and Vaught, to Sierpiński. The following theorem is proved. Theorem. The class of all G-metric spaces has the amalgamation property if, and only if, G is either the ordered group of the integers or the ordered group of the reals.
On periodic solutions of autonomous Hamiltonian systems of ordinary differential equations
Authors: Clark David C.
Published: June 2010

For the system

x
(
t
)
+
grad
⁡
U
(
x
(
t
)
)
=
0

x(t) + \operatorname {grad} U(x(t)) = 0

lower bounds are obtained for the number of pairs

±
x
(
t
)

\pm x(t)

of odd, periodic solutions, with the period prescribed. These bounds are in terms of the behavior of

U
(
x
)

U(x)

near the origin and far away from the origin. It is assumed that

U
(
x
)

U(x)

is even, and two different types of behavior of

U
(
x
)

U(x)

far away from the origin are considered.
A new proof of a regularity theorem for elliptic systems
Authors: Uhlenbeck K.
Published: June 2010

We give a proof, which makes use of the Riesz-Thorin theorem, for a smoothness theorem for solutions of elliptic systems in divergence form with bounded measurable coefficients. The results imply an important theorem in two dimensions due to Morrey [3]. Meyers has used a similar technique to get these results for elliptic equations [4].
An isolated bounded point derivation
Authors: O’Farrell Anthony G.
Published: June 2010

For a compact subset

X
X

of the plane,

R
(
X
)

R(X)

denotes the class of uniform limits on

X
X

of rational functions with poles off

X
X

.

R
(
X
)

R(X)

is a function algebra on

X
X

. An example

X
X

is constructed such that

R
(
X
)

R(X)

admits a bounded point derivation at exactly one point of

X
X

.
Regular closed maps
Authors: Dickman R. F.
Published: June 2010

A subset

A
A

of

X
X

is far from the remainder if whenever

U

\mathcal {U}

is a free open ultrafilter in

X
X

there exists

U
∈

U

U \in \mathcal {U}

such that

A
∩

c

l

X

U
=
∅

A \cap {\text {c}}{{\text {l}}_X}U = \emptyset

. A map is regular closed provided that the image of every regular closed set is closed. In this note we use some recent results of G. Viglino to show that every map can be extended to a regular closed map with far from the remainder point inverses. We also relate these maps to several other interesting classes of maps.
A new proof of the construction theorem for Stone algebras
Authors: Katriňák Tibor
Published: June 2010

A simple proof is given of Chen’s and Grätzer’s theorem, which gives a method to construct a Stone algebra from a Boolean algebra and a distributive lattice with 1 by certain connective conditions between the two given lattices.
A case in which irreducibility of an analytic germ implies irreducibility of the tangent cone
Authors: Draper Richard
Published: June 2010

There are simple examples in which a variety is irreducible at a point but has a reducible tangent cone. The following theorem is proved. If

X
p

{X_p}

is an irreducible analytic germ and if the Jacobian ideal beomes principal on the normalization then the tangent cone of

X
X

at

p
p

is irreducible. If, moreover, the singular set of

X
X

is a manifold at

p
p

then

X
X

is Whitney

a
,
b

a,b

-regular along the singular set at

p
p

.
Projective compact distributive topological lattices
Authors: Choe Tae Ho
Published: June 2010

In the category of all compact distributive topological lattices and their continuous lattice-homomorphisms, it is shown that every projective object is either zero-dimensional or not

I
I

-compact.
Groups in which every maximal partial order is exhaustive
Authors: Sherman Gary J.
Published: June 2010

Let

G
G

be a group. A partial order

P
P

on

G
G

is exhaustive if each nongeneralized periodic element of

G
G

is either positive or negative with respect to

P
P

. The class of groups on which every maximal partial order is exhaustive is characterized and shown to be closed under homomorphisms. An example is given to show that the class of groups on which every maximal right partial order is exhaustive is not closed under direct products.
On valuation rings that contain zero divisors
Authors: Huckaba James A.
Published: June 2010

Let

R
R

be a commutative ring with identity. A new proof is given of the theorem due to Samuel and Griffin which states that

R
R

is integrally closed in its total quotient ring if and only if

R
R

is the intersection of

B
B

-valuation rings. We then prove the main result of the paper: If

K
K

is a

π
\pi

-regular ring, then

K
K

admits only Prüfer rings as valuation rings.
Determinantal varieties, monomial semigroups, and algebras associated with ideals
Authors: Barshay Jacob
Published: June 2010

This paper is concerned with the Rees and symmetric algebras of powers of ideals generated by

A
A

-sequences. These algebras are represented as quotients of polynomial rings over

A
A

by ideals defined by minors of matrices. Their Krull dimensions are computed when

A
A

is a finite domain over a field. When

A
A

is a polynomial ring over a Cohen-Macaulay ring

A
0

{A_0}

and the

A
A

-sequence consists of indeterminates, the Rees algebra is shown to be Cohen-Macaulay. If furthermore

A
0

{A_0}

is a finite domain over a field, the symmetric algebra is shown to be Cohen-Macaulay only for

A
A

-sequences and squares of

A
A

-sequences of length two. Connections with algebras generated by monomials and Veronese varieties are pointed out.
Prime rings with involution whose symmetric zero-divisors are nilpotent
Authors: Cohn P. M.
Published: June 2010

Let

k
k

be a field and

R
R

the

k
k

-algebra generated by

x
x

and

y
y

with the single defining relation

x
2

=
0

{x^2} = 0

. Using free ring techniques we prove that the set of left zero-divisors of

R
R

is

R
x

Rx

. There is a unique involution fixing

x
,
y

x,y

and this makes

R
R

into a prime ring with involution whose symmetric zero-divisors are nilpotent (answering a question by W. S. Martindale). This example also provides us with a subfunctor of the identity whose value is a onesided ideal (answering a question by R. Baer).
The compactness of the set of arc cluster sets of an arbitrary function
Authors: Gresser John T.
Published: June 2010

It is known that if f is a continuous complex-valued function defined in the open unit disk D, then the set

C

f

(
ζ
)

(
ζ
∈
∂
D
)

{\mathfrak {C}_f}(\zeta )\;(\zeta \in \partial D)

of all arc cluster sets of f at

ζ
\zeta

is compact in a natural topology for all but at most a countable number of points

ζ
∈
∂
D

\zeta \in \partial D

. We show that if f is an arbitrary complex-valued function defined on an arbitrary subset Z of the plane, then

C

f

(
p
)

{\mathfrak {C}_f}(p)

is compact for all but at most a countable number of points

p
∈
Z
∪
∂
Z

p \in Z \cup \partial Z

.
Equations which characterize inner product spaces
Authors: Senechalle David A.
Published: June 2010

It is shown that if

N
N

is a normed linear space and there is a point

y
y

of norm 1 such that an inequality of the type
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a squared StartAbsoluteValue EndAbsoluteValue x StartAbsoluteValue EndAbsoluteValue squared less-than-over-equals limit Underscript u right-arrow 0 Endscripts upper G left-parenthesis left-brace StartAbsoluteValue EndAbsoluteValue b Subscript i Baseline u x plus c Subscript i Baseline y StartAbsoluteValue EndAbsoluteValue right-brace Subscript i equals 1 Superscript n Baseline right-parenthesis less-than-over-equals b squared StartAbsoluteValue EndAbsoluteValue x StartAbsoluteValue EndAbsoluteValue squared">

a
2

|

|

x

|

|

2

≦

lim

u
→
0

G
(
{

|

|

b
i

u
x
+

c
i

y

|

|

}

i
=
1

n

)
≦

b
2

|

|

x

|

|

2

{a^2}||x|{|^2} \leqq {\lim _{u \to 0}}G(\{ ||{b_i}ux + {c_i}y||\} _{i = 1}^n) \leqq {b^2}||x|{|^2}

holds for all

x
x

in

N
N

(where

0
>
a
≦
b

0 > a \leqq b

, the

c
i

{c_i}

’s are nonzero and

G
G

and

|

|

⋅

|

|

|| \cdot ||

satisfy a certain twice-differentiability condition), then

N
N

is isomorphic to an inner product space and

inf

|

|

T

|

|

⋅

|

|

T

−
1

|

|

≦
b

/

a

\inf ||T|| \cdot ||{T^{ - 1}}|| \leqq b/a

, where the infimum is taken over all linear homeomorphisms

T
T

between

N
N

and an inner product space. In the event that

a
=
b
=
1

a = b = 1

, the inequality reduces to an equation which characterizes inner product spaces. An example shows that these results do not follow without the twice-differentiability condition on

G
G

.
Primitive ideals in von Neumann algebras
Authors: Halpern Herbert
Published: June 2010

For a von Neumann algebra it is shown that the set of primitive ideals containing a fixed maximal ideal of the center is sequentially closed in the order topology defined on the set of all ideals containing the maximal ideal. As a corollary, it is shown that every ideal generated by a sequence of elements of a von Neumann algebra and a maximal ideal of the center is either primitive or simple modulo a primitive ideal.
A correction to: “On Fricke moduli”
Authors: Keen Linda
Published: June 2010

In this paper, the “twist parameter” for combining Fuchsian groups is redefined correctly in terms of traces of the elements of the groups.
A note on multivalence of functions of bounded index
Authors: Fricke Gerd H.
Published: June 2010

This paper examines the relationship between the concept of bounded index and the radius of univalence, respectively

p
p

-valence, of entire functions and their derivatives at arbitrary points in the plane.
𝑆-algebras on sets in 𝐶ⁿ
Authors: Chalice Donald R.
Published: June 2010

We give conditions which are necessary and sufficient for polynomial approximation of any continuous function on a compact subset of

C
n

{C^n}

.
Isomorphism theorems for infinite order differential operators
Authors: Buckholtz J. D.
Published: June 2010

The operators studied are infinite order linear differential operators with constant coefficients. We require that the characteristic function of the operator be analytic in a neighborhood of 0 and have a radius of convergence which is less than that of its reciprocal. It is shown that such operators may be regarded as isomorphisms between Banach spaces of entire functions. These Banach spaces, in every case, are isomorphic to the sequence space

c
0

{c_0}

. Further, translation in the domain plane defines an automorphism on both the domain and range space of the operator.
Transitive actions on highly connected spaces
Authors: Schneider Victor
Published: June 2010

Let

G
G

be a compact, connected Lie group and

H
H

a closed subgroup of

G
G

. It is shown that if

G

/

H

G/H

is highly connected relative to

Rk
⁡
(
G
)
−
Rk
⁡
(
H
)
,
G

/

H

\operatorname {Rk} (G) - \operatorname {Rk} (H),G/H

splits as a product of homogeneous spaces of simple Lie groups. This is used to show that the only transitive, effective actions on a large class of products of spheres are products of the known actions on the individual spheres.
Extending a Jordan ring homomorphism
Authors: Lewand Robert
Published: June 2010

In this paper a homomorphism from an ideal

B

\mathcal {B}

of a quadratic Jordan algebra

J

\mathcal {J}

without

2
2

-torsion over a ring

Φ
\Phi

onto a unital quadratic Jordan algebra

J

′

\mathcal {J}’

without

2
2

-torsion is extended to a homomorphism from

J

\mathcal {J}

to

J

′

\mathcal {J}’

. We then show if

D
D

is any class of quadratic Jordan algebras without

2
2

-torsion, then the upper radical property determined by

D
D

is hereditary.
Decomposable and spectral operators on a Hilbert space
Authors: Wadhwa Bhushan L.
Published: June 2010

A necessary and sufficient condition for a decomposable operator, on a Hilbert space, to be equal to a normal plus a commuting quasi-nilpotent operator is found.
A class of wild closed curves that span orientable surfaces
Authors: Harrold O. G.
Published: June 2010

A classical result in the topology of manifolds asserts that every polygonal closed curve in three-space bounds an orientable surface. In this note we relax the condition that the curve be locally tame and obtain a partial generalization.
Rund forms over real algebraic function fields in one variable
Authors: Elman Richard
Published: June 2010

The isometry types of rund quadratic forms over an arbitrary real algebraic function field in one variable are completely determined.
The noncharacteristic Cauchy problem for parabolic equations in two space variables
Authors: Colton David
Published: June 2010

An integral representation is obtained for the solution of the noncharacteristic Cauchy problem for second order parabolic equations in two space variables with entire, time independent coefficients. This is accomplished through the use of contour integration techniques and the calculus of residues in the space of several complex variables.
Equational theories of algebras with distributive congruences
Authors: Padmanabhan R. , Quackenbush R. W.
Published: June 2010

If an equational class of algebras has the distributive or permutable congruence property then it is well known that it satisfies certain conditions, known as Mal’cev-type conditions. In this note such Mal’cev-type conditions are used to find minimal bases for certain equational theories of algebras. A typical result states that every finitely based equational theory of algebras with distributive and permutable congruences is one-based.
On the derivative of a polynomial
Authors: Govil N. K.
Published: June 2010

If

p
(
z
)
=

Σ

v
=
0

n

a
v

z
v

p(z) = \Sigma _{v = 0}^n {{a_v}{z^v}}

is a polynomial of degree

n
n

having all its zeros in

|

z

|

≦
K
≦
1

|z| \leqq K \leqq 1

, then it is known that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="max Underscript StartAbsoluteValue z EndAbsoluteValue equals 1 Endscripts StartAbsoluteValue p prime left-parenthesis z right-parenthesis EndAbsoluteValue greater-than-over-equals left-parenthesis n slash left-parenthesis 1 plus upper K right-parenthesis right-parenthesis max Underscript StartAbsoluteValue z EndAbsoluteValue equals 1 Endscripts StartAbsoluteValue p left-parenthesis z right-parenthesis EndAbsoluteValue">

max

|

z

|

=
1

|

p
′

(
z
)

|

≧
(
n

/

(
1
+
K
)
)

max

|

z

|

=
1

|

p
(
z
)

|

{\max _{|z| = 1}}|p’(z)| \geqq (n/(1 + K)){\max _{|z| = 1}}|p(z)|

. In this paper we consider the case when

K
>
1

K > 1

and obtain a sharp result.
The local range set of a meromorphic function
Authors: Brown Leon , Gauthier P. M.
Published: June 2010

Let

f
f

be a function meromorphic in a domain

G
G

of the Riemann sphere. The global range set of

f
f

is the set of values assumed infinitely often by

f
f

, and similarly the local range set of

f
f

at a boundary point

p
p

is the set of values assumed infinitely often in every neighborhood of

p
p

. Obviously any range set is a

G
δ

{G_\delta }

set. In this paper we show that every

G
δ

{G_\delta }

set is the local range set of some meromorphic function. This contrasts with the situation for the global range set. Our methods rely on prime end theory and Arakélian’s approximation theorems.
Multipliers of closed ideals in group algebras
Authors: Friedberg Stephen H.
Published: June 2010

The purpose of this paper is to show that the multipliers of a special class of closed ideals in group algebras are “trivial", and that a result of Y. Meyer for groups of the form

R
n

×

T
m

{R^n} \times {T^m}

concerning the existence of nontrivial multipliers cannot be extended to any disconnected group.
A prime-divisor function
Authors: Knopfmacher J.
Published: June 2010

This note studies the asymptotic mean values over arithmetical progressions, the general distribution of values, and the maximum order of magnitude, of a certain natural prime-divisor function of positive integers.
Boundedly holomorphic convex Riemann domain
Authors: Kim Dong S.
Published: June 2010

A boundedly holomorphic convex Riemann domain with a bounded spread map is a Stein manifold of bounded type.
An order-theoretic description of Marinescu spaces
Authors: Feldman W. A. , Porter J. F.
Published: June 2010

It is shown that any inductive limit

E
E

in the category of convergence spaces of real locally convex topological vector spaces (i.e., any Marinescu space) can be embedded in a partially ordered vector space so that convergence in

E
E

can be characterized as an order-theoretic convergence. The order-theoretic convergence in question is a modification of classical order convergence.
Formal 3-deformations of 2-polyhedra
Authors: Wright Perrin
Published: June 2010

A formal deformation of one polyhedron to another is a finite sequence of expansions and collapses, beginning with one polyhedron and ending with the other. If a formal deformation exists between two 2-dimensional polyhedra, it is possible to choose a deformation through polyhedra of dimension at most four. It is desired to reduce this number to three. We give a partial result in that direction.
Representation of 𝐴-convex algebras
Authors: Cochran Allan C.
Published: June 2010

Algebraic properties of

A
A

-convex algebras are developed via a functor to locally

m
m

-convex algebras. The Gel’fand-Mazur theorem holds for

A
A

-convex algebras, and this fact allows a Gel’fand-type representation theorem for a subclass of uniformly

A
A

-convex algebras. Connections to existing functional representation theory are also obtained.
Two types of hyperinvariant subspaces
Authors: Kauffman Robert M.
Published: June 2010

Let

A
A

be a bounded operator in a Banach space

B
B

. Suppose that

A
A

has the single valued extension property. Given a closed set

F
F

in the complexes, define

σ
A

(
F
)

{\sigma _A}(F)

to be the set of all

x
x

in

B
B

such that there is an analytic function

x
(
λ
)

x(\lambda )

from the complement of

F
F

to

B
B

with

(
A
−
λ
I
)
x
(
λ
)
=
x

(A - \lambda I)x(\lambda ) = x

.

A
A

is said to have property

Q
Q

if

σ
A

(
F
)

{\sigma _A}(F)

is a closed subset of

B
B

for every

F
F

. Let

A
A

be, again, a bounded operator in a Banach space

B
B

. Given a real number

b
b

, define

S
A

(
b
)

{S_A}(b)

to be the set of all

x
x

in

B
B

such that

exp
⁡
(
−
c
t
)
exp
⁡
(
A
t
)
x

\exp ( - ct)\exp (At)x

is a bounded function from the nonnegative reals to

B
B

for all

c
>
b

c > b

.

A
A

is said to have property

P
\operatorname {P}

if

S
A

(
b
)

{S_A}(b)

is a closed subspace of

B
B

for all

b
b

. These two properties are discussed in this paper.
On Akcoglu and Sucheston’s operator convergence theorem in Lebesgue space
Authors: Satō Ryōtarō
Published: June 2010

Let

T
T

be a bounded linear operator on an

L
1

{L_1}

-space and

τ
\tau

its linear modulus. It is proved that if the adjoint of

τ
\tau

has a strictly positive subinvariant function then the following two conditions are equivalent: (i)

T
n

{T^n}

converges weakly; (ii)

(
1

/

n
)

Σ

i
=
1

n

T

k
i

(1/n)\Sigma _{i = 1}^n{T^{{k_i}}}

converges strongly for any strictly increasing sequence

k
1

,

k
2

,
⋯

{k_1},{k_2}, \cdots

of nonnegative integers.
Operators with a strictly cyclic vector
Authors: Barnes Bruce
Published: June 2010

In this paper, operators that have a strictly cyclic vector in a Banach space

X
X

are studied. It is shown that the spectrum of such operators has some special properties. When

X
X

is a Hilbert space, the connection between having a strictly cyclic vector and irreducibility is explored. If

X
X

is an infinite dimensional Hilbert space, it is shown that no hyponormal operator on

X
X

has a strictly cyclic vector.
The Schnirelmann density of the (𝑘,2)-integers
Authors: Diananda P. H.
Published: June 2010

The Schnirelmann density of the

(
k
,
2
)

(k,2)

-integers is found for each

k
≧
4

k \geqq 4

.
On the representation of nonseparable analytic sets
Authors: Hansell R. W.
Published: June 2010

Recently, the author considered extending the concepts of a Borel and analytic set for nonseparable metric spaces by allowing

σ
\sigma

-discrete families to replace countable families in the classical definitions. The resulting class of “extended Borel sets” was shown to lead to a new class of sets, intermediate to the Borel and analytic sets. In the present paper we show that the corresponding class of “extended analytic sets” does not lead to a new class of sets but actually coincides with the standard class of analytic sets. Thus their importance lies in the fact that they provide a useful “representation” for the analytic sets in arbitrary spaces. Several such representations are shown to lead to the same class of analytic sets.
A non-Hausdorff Ascoli theorem for 𝑘₃-spaces
Authors: Fox Geoffrey , Morales Pedro
Published: June 2010

The paper establishes an Ascoli theorem in the space of continuous functions on a

k
3

{k_3}

-space to a regular space. The theorem, in terms of even continuity and the compact open topology

τ
c

{\tau _c}

, properly contains the Ascoli theorems of Myers, Gale, Kelley-Morse, Bagley-Yang, and the

k
3

{k_3}

-space theorem of Noble.
On the absolute summability of the allied integral by a functional Nörlund method
Authors: Lal Shiva Narain
Published: June 2010

In this paper a theorem on the absolute summability of an integral associated with the allied integral of a Fourier integral by a functional Nörlund method has been established. Our theorem generalises an earlier result due to Boĭcun on the absolute Nörlund summability of the allied integral.
A note on periodic solutions for delay-differential systems
Authors: Gustafson G. B. , Schmitt K.
Published: June 2010

Let

f
(
t
,
x
,
y
)
:
[
0
,
∞
)
×

R
n

×

R
n

→

R
n

f(t,x,y):[0,\infty ) \times {R^n} \times {R^n} \to {R^n}

be continuous and 1-periodic in t,

τ
(
t
)
:
[
0
,
∞
)
→
[
0
,
h
]
(
0
>
h
≦
1
)

\tau (t):[0,\infty ) \to [0,h](0 > h \leqq 1)

continuous and 1-periodic. A simple geometric condition (Theorem 1) is given for the existence of 1-periodic solutions

x
(
t
)

x(t)

of the nonlinear delay-differential system

x
′

(
t
)
=
f
(
t
,
x
(
t
)
,
x
(
t
−
τ
(
t
)
)
)

x’(t) = f(t,x(t),x(t - \tau (t)))

, with

x
(
t
)

x(t)

in a given bounded convex open set G in

R
n

{R^n}

. The addition of a Lipschitz condition in x and monotonicity in y allows one to calculate

x
(
t
)

x(t)

by a monotone sequence of successive approximations (Theorem 2). Extensions to a more general functional differential equation

x
′

(
t
)
=
g
(
t
,
x
(
t
)
,

x
t

)

x’(t) = g(t,x(t),{x_t})

are given.
The problem of eigenvalues in some singular homogeneous Volterra integral equations
Authors: Chambers Ll. G.
Published: June 2010

It is shown that when the kernel of a homogeneous Volterra integral equation is singular, it is possible for there to be a continuous spectrum of eigenvalues.
Prime ideals in uniform algebras
Authors: Dietrich William E.
Published: June 2010

A uniform algebra on a compact metric space has infinite Krull dimension and exactly

2
c

{2^c}

nonmaximal prime ideals.
Cohomology, maximal ideals, and point evaluations
Authors: Nagel Alexander
Published: June 2010

We consider algebras A of continuous complex valued functions, which are given as the set of global sections of a sheaf

S

\mathcal {S}

on a topological space X. Under the hypothesis that all the higher cohomology groups of the sheaf are zero, we investigate the relationship between ideals in A, kernels of algebra homomorphisms of A into the complex numbers C, and sets of functions vanishing at a point of X. As applications, we obtain some simple proofs of theorems about ideals in certain algebras of holomorphic functions.
Holomorphic solutions of functional differential systems near singular points
Authors: Grimm L. J. , Hall L. M.
Published: June 2010

Functional analysis techniques are used to prove a theorem, analogous to the Harris-Sibuya-Weinberg theorem for ordinary differential equations, which yields as corollaries a number of existence theorems for holomorphic solutions of linear functional differential systems of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z Superscript upper D Baseline y prime left-parenthesis z right-parenthesis equals upper A left-parenthesis z right-parenthesis y left-parenthesis z right-parenthesis plus upper B left-parenthesis z right-parenthesis y left-parenthesis alpha z right-parenthesis plus upper C left-parenthesis z right-parenthesis y prime left-parenthesis alpha z right-parenthesis">

z
D

y
′

(
z
)
=
A
(
z
)
y
(
z
)
+
B
(
z
)
y
(
α
z
)
+
C
(
z
)

y
′

(
α
z
)

{z^D}y’(z) = A(z)y(z) + B(z)y(\alpha z) + C(z)y’(\alpha z)

in the neighborhood of the singularity at

z
=
0

z = 0

.
A note on zero-dimensional spaces with the star-finite property
Authors: Reichel Hans-Christian
Published: June 2010

The paper provides necessary and sufficient conditions for a weakly zero-dimensional metrizable space to be strongly paracompact, i.e., to have the star-finite property. The characterizations use special basis properties of uniformities which induce the topology of X, and yield further characteristics of the class of all metric spaces with ind

X
=
0

X = 0

and Ind

X
>
0

X > 0

.
Recovering a function from its gram determinants
Authors: Grünbaum F. Alberto
Published: June 2010

We prove that any real valued symmetric function defined on

A
×
A
,
A

A \times A,A

an arbitrary set, can be determined up to a phase factor from the knowledge of its Gram determinants.
A simple alternative problem for finding periodic solutions of second order ordinary differential systems
Authors: Bebernes J. W.
Published: June 2010

Existence of solutions for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x equals f left-parenthesis t comma x comma x Superscript prime Baseline right-parenthesis comma x left-parenthesis 0 right-parenthesis equals x left-parenthesis 1 right-parenthesis comma x prime left-parenthesis 0 right-parenthesis equals x prime left-parenthesis 1 right-parenthesis">

x
=
f
(
t
,
x
,

x
′

)
,
x
(
0
)
=
x
(
1
)
,

x
′

(
0
)
=

x
′

(
1
)

x = f(t,x,x’),x(0) = x(1),x’(0) = x’(1)

are proven by considering a simple alternative problem to which Leray-Schauder degree arguments can be directly applied.
Monotonicity of quadrature approximations
Authors: Newman D. J.
Published: June 2010

For the general quadrature rules of Newton-Cotes we show that the convergence is monotone for the appropriate class of functions. Included are the well-known trapezoidal and Simpson rules.
Remarks on “Operators with inverses similar to their adjoints”
Authors: DePrima C. R.
Published: June 2010

A result in the cited paper [7] which purports to furnish a sufficient condition for an operator

T
T

on a Hilbert space

H

\mathfrak {H}

for which

T

−
1

S
=
S

T
∗

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

with

0
∉
cl
⁡
(
W
(
S
)
)

0 \notin \operatorname {cl} (W(S))

to be unitary is shown to be false. Various conditions under which such a result does hold are explored.
Starlike functions
Authors: McCarty Carl P.
Published: June 2010

Let

S

∗

[
α
]

{\mathcal {S}^\ast }[\alpha ]

denote the class of functions

f
(
z
)
=
z
+

∑

n
=
2

∞

a
n

z
n

f(z) = z + \sum \nolimits _{n = 2}^\infty {{a_n}{z^n}}

analytic in

|

z

|

>
1

|z| > 1

and for which

|

z

f
′

(
z
)

/

f
(
z
)
−
1

|

>
1
−
α

|zf’(z)/f(z) - 1| > 1 - \alpha

for

|

z

|

>
1

|z| > 1

and

α
∈
[
0
,
1
)

\alpha \in [0,1)

. Sharp results concerning coefficients, distortion, and the radius of convexity are obtained. Furthermore, it is shown that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n equals 2 Overscript normal infinity Endscripts left-bracket left-parenthesis n minus alpha right-parenthesis slash left-parenthesis 1 minus alpha right-parenthesis right-bracket StartAbsoluteValue a Subscript n Baseline EndAbsoluteValue greater-than 1">

∑

n
=
2

∞

[
(
n
−
α
)

/

(
1
−
α
)
]

|

a
n

|

>
1

\sum \nolimits _{n = 2}^\infty {[(n - \alpha )/(1 - \alpha )]|{a_n}| > 1}

is a sufficient condition for

f
(
z
)
∈

S

∗

[
α
]

f(z) \in {\mathcal {S}^\ast }[\alpha ]

.
On the dominated ergodic theorem in 𝐿₂ space
Authors: Akcoglu M. A. , Sucheston L.
Published: June 2010

Let

T
T

be a contraction on

L
2

{L_2}

of a

σ
\sigma

-finite measure space,

A
n

(
T
)

{A_n}(T)

the operator
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 slash n right-parenthesis left-parenthesis upper T Superscript 0 Baseline plus midline-horizontal-ellipsis plus upper T Superscript n Baseline right-parenthesis comma upper S left-parenthesis upper T right-parenthesis f">

(
1

/

n
)
(

T
0

+
⋯
+

T
n

)
,
S
(
T
)
f

(1/n)({T^0} + \cdots + {T^n}),S(T)f

the function

sup
n

|

A
n

(
T
)
f

|

{\sup _n}|{A_n}(T)f|

. Theorem 1. Assume that, whatever be the measure space,

S
(
U
)
f
∈

L
2

S(U)f \in {L_2}

for each unitary operator

U
U

on

L
2

{L_2}

and each function

f
∈

L
2

f \in {L_2}

. Then there exists a universal constant

K
K

such that

|

|

S
(
T
)
f

|

|

≦
K

|

|

f

|

|

||S(T)f|| \leqq K||f||

for each contraction

T
T

on

L
2

{L_2}

and each

f
∈

L
2

f \in {L_2}

. Theorem 2. Let

T
T

be a contraction on

L
2

{L_2}

and let

U
U

be a unitary dilation of

T
T

acting on a Hilbert space

H
H

containing

L
2

{L_2}

. If all expressions of the form

Σ

n
=
1

∞

P
n

A
n

(
U
)

\Sigma _{n = 1}^\infty {{P_n}{A_n}(U)}

, where

P
n

{P_n}

are mutually orthogonal projections, are bounded operators on

H
H

, then for each

f
∈

L
2

,
S
(
T
)
f
∈

L
2

f \in {L_2}, S(T)f \in {L_2}

and

A
n

(
T
)
f

{A_n}(T)f

converges a.e.
Functions with a closed graph
Authors: Baggs Ivan
Published: June 2010

Let

X
X

be a

T
2

{T_2}

Baire space. A set

F
⊂
X

F \subset X

is closed and nowhere dense in

X
X

if

F
F

is the set of points of discontinuity of a function with a closed graph from

X
X

into

R
n

{R^n}

. Although the converse does not hold in general, it does hold when

X
X

is the real line.
Higher derivations on finitely generated integral domains
Authors: Brown W. C.
Published: June 2010

In this paper, we prove the following theorem: Let

A
=
k
[

x
1

,
⋯
,

x
t

]

A = k[{x_1}, \cdots ,{x_t}]

be a finitely generated integral domain over a field k of characteristic zero. Then A regular, i.e. the local ring

A
q

{A_q}

is regular for all primes

q
⊆
A

q \subseteq A

, is equivalent to the following two conditions: (1) No nonminimal prime of A is differential, and (2)

der
n

⁡
(
A

/

k
)
=

D
e
r

n

(
A

/

k
)

\operatorname {der}^n (A/k) = \mathrm {Der}^n (A/k)

for all n. Here

Der
n

⁡
(
A

/

k
)

\operatorname {Der}^n (A/k)

denotes the A-module of all nth order derivations of A into A which are zero or k, and

der
n

⁡
(
A

/

k
)

\operatorname {der}^n(A/k)

denotes the A-submodule of

Der
n

⁡
(
A

/

k
)

\operatorname {Der}^n(A/k)

generated by composites

δ
1

∘
⋯
∘

δ
j

(
1
≦
j
≦
n
)

{\delta _1} \circ \cdots \circ {\delta _j}(1 \leqq j \leqq n)

of first order derivations

δ
i

{\delta _i}

.
Convex matrix functions
Authors: Watkins William
Published: June 2010

The purpose of this paper is to prove convexity properties for the tensor product, determinant, and permanent of hermitian matrices.
Proof of an inequality for trigonometric polynomials
Authors: Kristiansen G. K.
Published: June 2010

Let a trigonometric polynomial have only real roots. An inequality is proved, relating the average value of the trigonometric polynomial between two consecutive roots to its maximal absolute value.
Convergence rate for a large deviation probability
Authors: Kostka David G.
Published: June 2010

It is known that a condition more stringent than a finite variance is needed to show, by a direct application of large deviation estimates, the convergence of the series used in classical proofs of the law of the iterated logarithm. However, the series still converges if the variables have only a finite variance.
Valuations and rings of quotients
Authors: Brown David E.
Published: June 2010

Valuations on a commutative ring, as defined by Manis, are considered in the special case where the domain of the valuation mapping is a ring of quotients of a given ring

R
R

. We consider relations between valuation mappings on various rings of quotients of a given ring. It is also shown that if

K
K

is any von Neumann regular ring of quotients of

R
R

, then any pair of nonassociates of

R
R

can be separated by valuations on

K
K

if and only if these elements are nonassociates in the integral closure of

R
R

in

K
K

.