bluuebunny/crossref_metadata_2025 huggingface dataset:
Positive solutions of positive linear equations
Authors: Nelson Paul
Published: June 2010

Let

B
B

be a real vector lattice and a Banach space under a semimonotonic norm. Suppose

T
T

is a linear operator on

B
B

which is positive and eventually compact,

y
y

is a positive vector, and

λ
\lambda

is a positive real. It is shown that

(
λ
I
−
T

)

−
1

y

{(\lambda I - T)^{ - 1}}y

is positive if, and only if,

y
y

is annihilated by the absolute value of any generalized eigenvector of

T
∗

{T^\ast }

associated with a strictly positive eigenvalue not less than

λ
\lambda

. A strictly positive eigenvalue is a positive eigenvalue having an associated positive eigenvector. For the case of

B
=

L
p

B = {L^p}

this yields the result that

(
λ
I
−
T

)

−
1

y
≧
0

{(\lambda I - T)^{ - 1}}y \geqq 0

if, and only if,

y
y

is almost everywhere zero on a certain set which depends on

λ
\lambda

but is otherwise fixed.
An algebraic proof that [Ω^{𝑈}]₂=𝔑²
Authors: Porter Don
Published: June 2010

It is often desirable to find the image of one cobordism theory in another. Milnor settled the first problem of this sort, the image of complex cobordism in unoriented cobordism, by construction of concrete generating manifolds. When such constructions are too difficult, it still may be possible to solve the problem using more algebraic methods. This note offers a proof of Milnor’s result which depends on the Adams spectral sequence and requires no ad hoc construction of manifolds.
The class number of 𝑄(√-𝑝), for 𝑃≡1(𝑚𝑜𝑑8) a prime
Authors: Brown Ezra
Published: June 2010

Let

h
(
−
p
)

h( - p)

be the class number of the quadratic field

Q
(
√
−
p
)

Q(\surd - p)

, where

p
≡
1

(
mod

8
)

p \equiv 1\pmod 8

is a prime. Write

p
=

a
2

+

b
2

=
2

e
2

−

d
2

p = {a^2} + {b^2} = 2{e^2} - {d^2}

, where

a
≡
e
≡
d
≡
b
+
1
≡
1

(
mod

2
)

a \equiv e \equiv d \equiv b + 1 \equiv 1\pmod 2

and

e
>
0

e > 0

. We prove that

h
(
−
p
)
≡
0

h( - p) \equiv 0

or

4

(
mod

8
)

4\pmod 8

according as

(
e

|

p
)
=
1

(e|p) = 1

or

−
1

- 1

; using this, we prove that

h
(
−
p
)
≡
(
p
−
1
)

/

2
+
b

(
mod

8
)

h( - p) \equiv (p - 1)/2 + b\pmod 8

. The proofs are elementary, relying on the theory of composition of binary quadratic forms.
A note on finitely generated ideals which are locally principal
Authors: Brewer James W. , Rutter Edgar A.
Published: June 2010

Let

R
R

be a commutative ring with identity

1
≠
0

1 \ne 0

and let

A
A

be a nonzero ideal of

R
R

. A problem of current interest is to relate the notions of “projective ideal", “flat ideal” and “multiplication ideal". In this note we prove two results which show that the maximal ideals containing the annihilator of

A
A

can play an important role in determining the relationship between these concepts. As a consequence we are able to prove that a finitely generated multiplication ideal in a semi-quasi-local ring is principal, that a finitely generated flat ideal having only a finite number of minimal prime divisors is projective and that for Noetherian rings or semihereditary rings, finitely generated multiplication ideals with zero annihilator are invertible.
Separation of nonassociates by valuations
Authors: Brown David E. , Larsen Max D.
Published: June 2010

In many classical integral domains, given two nonassociates it is possible to find a valuation on the quotient field of the domain which is nonnegative on the domain and for which the nonassociates have different values. Recent work by Griffin, Harrison, and Manis has extended valuation theory to commutative rings with identity which contain zero divisors. In this paper we investigate the separation of nonassociates by valuations for the extended valuation theory. Our main result states that if

R
R

is a ring with a von Neumann regular total quotient ring, then nonassociates can be separated by valuations if and only if there is no unit in the integral closure of

R
R

which is not a unit in

R
R

.
The hyperspace of a pseudoarc is a Cantor manifold
Authors: Nishiura Togo , Rhee Choon Jai
Published: June 2010

The following theorem which was conjectured by C. Eberhart and S. B. Nadler, Jr., in [EN] is proved. Theorem. The hyperspace of nonvoid subcontinua of a pseudoarc is a two-dimensional Cantor manifold.
Cohomology operations on 𝑝-fold sums
Authors: Thomas Emery
Published: June 2010

A formula is given for evaluating higher order cohomology operations on integral classes that are p-fold multiples, p a prime.
Strict essential minima
Authors: O’Malley R. J.
Published: June 2010

A simple proof is given of the fact that the set of strict essential minima of a real function of n variables is of measure zero. The proof uses only that a continuous function on a compact set has a maximum and the elementary fact, which seems to be new, that each set of positive measure contains a compact set which has positive upper density at each of its points.
An abstract measure differential equation
Authors: Sharma R. R.
Published: June 2010

An abstract measure differential equation is introduced as a generalization of ordinary differential equations and measure differential equations. The existence and extension of solutions of this equation are considered.
The spectra of unbounded hyponormal operators
Authors: Putnam C. R.
Published: June 2010

A bounded operator

T
T

on a Hilbert space is said to be completely hyponormal if

T
∗

T
−
T

T
∗

≧
0

{T^\ast }T - T{T^\ast } \geqq 0

and if

T
T

has no nontrivial reducing space on which it is normal. If

0
0

is in the spectrum of such an operator

T
T

and if the spectrum of

T
T

near

0
0

is not “too dense,” then the unbounded operator

T

−
1

{T^{ - 1}}

acts as though it were bounded. In particular, under certain conditions,

T

−
1

{T^{ - 1}}

has a rectangular representation with absolutely continuous real and imaginary parts whose spectra are the closures of the projections of the spectrum of

T

−
1

{T^{ - 1}}

onto the coordinate axes.
Quotient groups of topological groups with no small subgroups
Authors: Morris Sidney A.
Published: June 2010

It is shown here that a quotient group of a topological group with no small subgroups can have small subgroups.
On the saturation class for spline functions
Authors: Richards Franklin
Published: June 2010

Let

S

k

(

Δ
n

)

{\mathcal {S}_k}({\Delta _n})

be the space of piecewise polynomials of degree at most k on [0, 1] possessing jumps at

1

/

n
,
2

/

n
,
⋯
,
n
−
1

/

n

1/n,2/n, \cdots ,n - 1/n

. Recently Gaier has shown that under the supremum norm
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar f minus script upper S Subscript k Baseline left-parenthesis normal upper Delta Subscript n Baseline right-parenthesis double-vertical-bar plus double-vertical-bar f minus script upper S Subscript k Baseline left-parenthesis normal upper Delta Subscript n plus 1 Baseline right-parenthesis double-vertical-bar greater-than-over-equals c n Superscript negative k minus 1">

‖

f
−

S

k

(

Δ
n

)

‖

+

‖

f
−

S

k

(

Δ

n
+
1

)

‖

≧
c

n

−
k
−
1

\left \| {f - {\mathcal {S}_k}({\Delta _n})} \right \| + \left \| {f - {\mathcal {S}_k}({\Delta _{n + 1}})} \right \| \geqq c{n^{ - k - 1}}

unless f is a polynomial of degree at most k. Here we show if

0
>
α
≦
1

0 > \alpha \leqq 1

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar f minus script upper S Subscript k Baseline left-parenthesis normal upper Delta Subscript n Baseline right-parenthesis double-vertical-bar equals upper O left-parenthesis n Superscript negative k minus alpha Baseline right-parenthesis">

‖

f
−

S

k

(

Δ
n

)

‖

=
O
(

n

−
k
−
α

)

\left \| {f - {\mathcal {S}_k}({\Delta _n})} \right \| = O({n^{ - k - \alpha }})

if and oniy if

f
∈

C
k

[
0
,
1
]

f \in {C^k}[0,1]

and

f

(
k
)

{f^{(k)}}

satisfies a Lipschitz condition of order

α
\alpha

. In addition, a result similar to Gaier’s is given.
Constructions of disjoint Steiner triple systems
Authors: Doyen Jean
Published: June 2010

Let

D
∗

(
v
)

{D^ \ast }(v)

denote the maximum number of pairwise disjoint and isomorphic Steiner triple systems of order v. The main result of this paper is a lower bound for

D
∗

(
v
)

{D^\ast }(v)

, namely

D
∗

(
6
t
+
3
)
≧
4
t
−
1

{D^\ast }(6t + 3) \geqq 4t - 1

or

4
t
+
1

4t + 1

according as

2
t
+
1

2t + 1

is or is not divisible by 3, and

D
∗

(
6
t
+
1
)
≧
t

/

2

{D^ \ast }(6t + 1) \geqq t/2

or t according as t is even or odd. Some other related problems are studied or proposed for study.
Extensions of left uniformly continuous functions on a topological semigroup
Authors: Wiley Samuel J.
Published: June 2010

For any topological semigroup S with separately continuous operation, let

C
(
S
)

C(S)

denote the set of all bounded continuous real valued functions on S with the supremum norm and let

LUC

(
S
)

{\text {LUC}}(S)

denote the set of all f in

C
(
S
)

C(S)

such that whenever

{
s
(
γ
)
}

\{ s(\gamma )\}

is a net in S which converges to some s in S, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup left-brace right-brace colon vertical-bar vertical-bar minus minus of ff left-parenthesis right-parenthesis times times of ss left-parenthesis right-parenthesis gamma t of ff left-parenthesis right-parenthesis times times st colon element-of element-of tS">

sup
{

|

f
(
s
(
γ
)
t
)
−
f
(
s
t
)

|

:
t
∈
S
}

\sup \{ |f(s(\gamma )t) - f(st)|:t \in S\}

converges to 0. In this paper we prove that if S is an abelian subsemigroup of a compact topological group and

f
∈

LUC

(
S
)

f \in {\text {LUC}}(S)

, then there is an

F
∈

LUC

(
G
)

F \in {\text {LUC}}(G)

where

F
(
s
)
=
f
(
s
)

F(s) = f(s)

for all

s
∈
S

s \in S

. We also show whenever there is an extension of the type indicated above, there is a norm preserving extension.
Linear isotropy group of an affine symmetric space
Authors: Nagasawa Jun
Published: June 2010

Let K be a subgroup of the general linear group

GL

(
n
)

{\text {GL}}(n)

. The author found a necessary and sufficient condition that there exist an n-dimensional simply connected affine symmetric space M such that K coincides with the linear isotropy group of all affine automorphisms of M at some point in M.
Equivariant bordism and cyclic groups
Authors: Landweber Peter S.
Published: June 2010

For a finite cyclic group

G
G

the equivariant complex bordism module

Ω
∗
U

(
G
)

\Omega _\ast ^U(G)

is shown to be a free module over

Ω
∗
U

\Omega _\ast ^U

.
Continuum embedded eigenvalues in a spatially cutoff 𝑃(𝜙)₂ field theory
Authors: Simon Barry
Published: June 2010

We prove the existence of nontrivial two-dimensional spatially cutoff self-coupled Boson field theory Hamiltonians with bound states embedded in the continuum.
Stone’s theorem for a group of unitary operators over a Hilbert space
Authors: Salehi Habib
Published: June 2010

The spectral representation for a group of unitary operators acting on a Hilbert space where the parameter set is a separable real Hilbert space is obtained. The usual spectral representation of such a group of unitary operators is when the parameter set is a locally compact abelian group (Stone’s theorem). The main result used in the proof is the Bochner theorem on the representation of positive definite functions on a real Hilbert space.
Total stability in nondifferentiable systems
Authors: McCann Roger C.
Published: June 2010

A definition of total stability in nondifferentiable dynamical systems is given. A prolongation is defined which characterizes the total stability of compact sets. A compact set which is the intersection of compact asymptotically stable sets is shown to be totally stable.
The signature of fiber bundles
Authors: Schafer James A.
Published: June 2010

Let

Y
↪
Z

π
→

X

Y\hookrightarrow Z\underrightarrow \pi X

be a locally trivial fiber bundle in the category of oriented topological manifolds. It is shown that if the identity component of the structure group

G
G

has finite index, then (signature of

Z
Z

)=(signature of

X
X

)

◼
\blacksquare

(signature of

Y
Y

).
On averaging Lefschetz numbers
Authors: Scholz U. Kurt
Published: June 2010

Let (E, p, X) be a regular covering space where E is a connected metric ANR (absolute neighborhood retract) and let

f
:
X
→
X

f:X \to X

be a map. This paper investigates the relationship between the Lefschetz number of f and those of its lifts, i.e. maps

f
′

:
E
→
E

f’:E \to E

so that

p

f
′

=
f
p

pf’ = fp

. In particular, it is shown that to a lift

f
′

:
E
→
E

f’:E \to E

one may associate a class of lifts

L

(

f
′

)

\mathfrak {L}(f’)

with the property that the Lefschetz number of f is equal to the average of the Lefschetz numbers of maps in

L

(

f
′

)

\mathfrak {L}(f’)

.
Nodal algebras defined by skew-symmetric bilinear forms
Authors: Goldman Jerry
Published: June 2010

This paper studies nodal algebras defined by skew-symmetric bilinear forms, a subclass of the class of Kokoris algebras. The ideals of such algebras are classified and a characterization of the automorphisms of these algebras is given.
Criteria for unique metric lines in Banach spaces
Authors: Andalafte E. Z. , Valentine J. E.
Published: June 2010

A metric space

M
M

has the monotone property if for each point

p
p

and line

L
L

of

M
M

the distance

p
x

px

between

p
p

and a point

x
x

of

L
L

is monotone increasing as

x
x

recedes along either half-line of

L
L

determined by the foot of

p
p

on

L
L

. It is shown that a Banach space (over the reals) has the monotone property if and only if it has unique metric lines. Using previously known results, additional equivalents of the monotone property are obtained and new proofs of some older criteria for unique metric lines result.
The ring of polynomial over a von Neumann regular ring
Authors: McCarthy P. J.
Published: June 2010

It is shown that the ring of polynomials in one indeterminate over a commutative von Neumann regular ring with identity element is semihereditary.
Finiteness in prime ideals in rings of global dimension two
Authors: Sheng Hu
Published: June 2010

Let

A
A

be a commutative ring with identity. The main result states conditions that ensure the finiteness of prime ideals in a coherent ring

A
A

of global dimension two. Precisely, any ideal containing two noncomparable prime ideals is finitely generated. As a corollary it follows that a Krull domain of global dimension two is noetherian. Another corollary is that if

A
A

is not semihereditary it contains a finitely generated maximal ideal.
Halving the Milnor manifolds and some conjectures of Ray
Authors: Segal David M.
Published: June 2010

The peculiar properties of the

2
j

−
2

{2^j} - 2

dimensional generators of unitary bordism (the

2
2

-primary Milnor generators) are related to the

2
j

−
3

{2^j} - 3

dimensional indecomposable torsion classes of Alexander. This result is then used to confirm a conjecture of Ray concerning the generalised homology spectral sequence for

M
S

p
∗

(
M
U
)

MS{p_\ast }(MU)

. Finally it is noted that Ray’s conjecture to the effect that all classes in

M
S

p
∗

MS{p_\ast }

are detectable by KO-characteristic numbers must fail.
The poset of skew fields generated by a free algebra
Authors: Fisher James L.
Published: June 2010

This paper shows that the poset of skew fields generated by a free algebra contains a subposet isomorphic to the lattice of all subsets of an infinite set.
Finite groups whose subnormal subgroups permute with all Sylow subgroups
Authors: Agrawal Ram K.
Published: June 2010

As a generalization of

(
t
)

(t)

-groups and of

(
q
)

(q)

-groups, a group

G
G

is called a

(
π
−
q
)

(\pi - q)

-group if every subnormal subgroup of

G
G

permutes with all Sylow subgroups of

G
G

. It is shown that if

G
G

is a finite solvable

(
π
−
q
)

(\pi - q)

-group, then its hypercommutator subgroup

D
(
G
)

D(G)

is a Hall subgroup of odd order and every subgroup of

D
(
G
)

D(G)

is normal in

G
G

; conversely, if a group

G
G

has a normal Hall subgroup

N
N

such that

G

/

N

G/N

is a

(
π
−
q
)

(\pi - q)

-group and every subnormal subgroup of

N
N

is normal in

G
G

, then

G
G

is a

(
π
−
q
)

(\pi - q)

-group.
A note on hyponormal operators
Authors: Whitley Robert
Published: June 2010

A short, elementary proof that a hyponormal operator with a compact imaginary part is normal.
On a game theoretic notion of complexity for compact convex sets
Authors: Kalai Ehud , Smorodinsky Meir
Published: June 2010

The notion of complexity for compact convex sets introduced by Billera and Bixby is considered. It is shown that for

n
≥
3

n \geq 3

there are sets in

R
n

{R^n}

of complexity

n
n

. Also for

n
=
3

n = 3

the maximal complexity is 3.
Unitary harmonic numbers
Authors: Hagis Peter , Lord Graham
Published: June 2010

If

d
∗

(
n
)

{d^ \ast }(n)

and

σ
∗

(
n
)

{\sigma ^ \ast }(n)

denote the number and sum, respectively, of the unitary divisors of the natural number

n
n

then the harmonic mean of the unitary divisors of

n
n

is given by

H
∗

(
n
)
=
n

d
∗

(
n
)

/

σ
∗

(
n
)

{H^ \ast }(n) = n{d^ \ast }(n)/{\sigma ^ \ast }(n)

. Here we investigate the properties of

H
∗

(
n
)

{H^ \ast }(n)

, and, in particular, study those numbers

n
n

for which

H
∗

(
n
)

{H^ \ast }(n)

is an integer.
A note on proper maps
Authors: Ho Chung Wu
Published: June 2010

The author establishes some necessary and sufficient conditions on a Hausdorff space

Y
Y

which force every open proper map into

Y
Y

to be surjective. Using this result, the author then shows that a local homeomorphism from a path connected space into a simply connected space is a global homeomorphism onto if and only if the map is proper.
Point partition numbers and girth
Authors: Cook R. J.
Published: June 2010

In recent papers D. R. Lick and A. T. White have introduced point partition numbers as generalizations of the chromatic number and the point-arboricity of a graph. In particular they proved that an analogue of Heawood’s theorem holds for the point partition numbers. In the present paper it is shown that the bounds provided by their result may be improved for graphs of large girth. Finally, using a method of Erdös, it is shown that there exist graphs with large girth and large point-partition number.
On *-rings satisfying the square root axiom
Authors: Maeda Shûichirô
Published: June 2010

It was mentioned by Kaplansky that the parallelogram law is a key property for developing the dimension theory on the lattice of projections of a Baer

∗

^{\ast }

-ring, and he proved that this law follows from a pair of axioms: the EP and SR axioms. In this paper, it is shown that this law follows from only the SR axiom.
The boundary of the numerical range
Authors: Lancaster John S.
Published: June 2010

This work includes two results which indicate a set theoretic relationship between the boundary of the numerical range and the essential numerical range. Several applications are derived.
Codimension of compact 𝑀-semilattices
Authors: Lea J. W. , Lau A. Y. W.
Published: June 2010

This paper is a generalization of [5] and gives a partial answer to Question 31 in [1], i.e., if

S
S

is a compact

M
M

-semilattice of finite codimension and

x
≠
y

x \ne y

, then there exists a closed subsemilattice

A
A

of

S
S

such that

A
A

separates

x
x

and

y
y

in

S
S

and

cd
⁡
A
>
cd
⁡
S

\operatorname {cd} A > \operatorname {cd} S

.
Quadratic forms in harmonic analysis and the Bochner-Eberlein theorem
Authors: Horn Roger A.
Published: June 2010

New characterizations are given of the classes of functions on a locally compact Abelian group which are Fourier-Stieltjes transforms of: bounded measures, nonnegative bounded measures, and integrable functions. These characterizations are all of the same form, viz., that a positive definite Hermitian form dominate a symmetric quadratic form.
A Carleson measure theorem for Bergman spaces
Authors: Hastings William W.
Published: June 2010

Let

μ
\mu

be a finite, positive measure on

U
n

{U^n}

, the unit polydisc in

C

n

{{\mathbf {C}}^n}

, and let

σ
n

{\sigma _n}

be

2
n

2n

-dimensional Lebesgue volume measure on

U
n

{U^n}

. For

1
⩽
p
⩽
q
>
∞

1 \leqslant p \leqslant q > \infty

a necessary and sufficient condition on

μ
\mu

is given in order that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace integral Subscript upper U Sub Superscript n Subscript Baseline f Superscript q Baseline left-parenthesis z right-parenthesis d mu left-parenthesis z right-parenthesis right-brace Superscript 1 slash q Baseline less-than-or-slanted-equals upper C left-brace integral Subscript upper U Sub Superscript n Subscript Baseline f Superscript p Baseline left-parenthesis z right-parenthesis d sigma Subscript n Baseline left-parenthesis z right-parenthesis right-brace Superscript 1 slash p Baseline">

{
∫

U
n

f
q

(
z
)
d
μ
(
z
)

}

1

/

q

⩽

C
{
∫

U
n

f
p

(
z
)
d

σ
n

(
z
)

}

1

/

p

\{ \int {_{{U^n}}{f^q}(z)d\mu (z){\} ^{1/q}} \leqslant } C\{ \int {_{{U^n}}{f^p}(z)d{\sigma _n}(z){\} ^{1/p}}}

for every positive

n
n

-subharmonic function

f
f

on

U
n

{U^n}

.
An elementary method for estimating error terms in additive number theory
Authors: Hayashi Elmer K.
Published: June 2010

Let

R
k

(
n
)

{R_k}(n)

denote the number of ways of representing the integers not exceeding

n
n

as the sum of

k
k

members of a given sequence of nonnegative integers. Using only elementary methods, we prove a general theorem from which we deduce that, for every

ϵ
>
0

\epsilon > 0

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript k Baseline left-parenthesis n right-parenthesis minus c n Superscript beta Baseline not-equals o left-parenthesis n Superscript beta left-parenthesis 1 minus beta right-parenthesis left-parenthesis 1 minus 1 slash k right-parenthesis slash left-parenthesis 1 minus beta plus beta slash k right-parenthesis minus epsilon Baseline right-parenthesis">

R
k

(
n
)
−
c

n
β

≠
o
(

n

β
(
1
−
β
)
(
1
−
1

/

k
)

/

(
1
−
β
+
β

/

k
)
−
ϵ

)

{R_k}(n) - c{n^\beta } \ne o({n^{\beta (1 - \beta )(1 - 1/k)/(1 - \beta + \beta /k) - \epsilon }})

\]
where

c
c

is a positive constant and

0
>
β
>
1

0 > \beta > 1

.
A note on the geometric criteria for the factoriality of an affine ring
Authors: Hornell James
Published: June 2010

Let

R
R

be a normal integral domain finitely generated over a field

k
k

, let

U
U

be an affine algebraic variety with coordinate ring

R
R

, and let

V
V

be a projective completion of

U
U

over

k
k

.

R
R

is factorial if and only if the class group of

V
V

over

k
k

is generated by the

k
k

-rational cycles at infinity with respect to

U
U

. If

k
k

is algebraically closed,

R
R

is shown to be factorial if and only if the Picard group of

V
V

is zero and the Néron-Severi group of

V
V

is generated by the

k
k

-rational cycles at infinity. If

k
k

is finitely generated over its prime field, some well-known arithmetic results are applied to show the existence of affine localizations of

V
V

which have a factorial coordinate ring over

k
k

. The relationship between the existence of an affine localization of

V
V

with a factorial coordinate ring, and the birationality of

V
V

is also discussed.
The derivative of a bounded holomorphic function in the disk
Authors: Yamashita Shinji
Published: June 2010

Let a nonconstant function

f
f

be holomorphic and bounded,

|

f

|

>
1

|f| > 1

in

D
:

|

z

|

>
1

D:|z| > 1

. We shall estimate
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Superscript asterisk Baseline left-parenthesis z right-parenthesis equals left-parenthesis 1 minus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis StartAbsoluteValue f prime left-parenthesis z right-parenthesis EndAbsoluteValue slash left-parenthesis 1 minus StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue squared right-parenthesis">

f

∗

(
z
)
=
(
1
−

|

z

|

2

)

|

f
′

(
z
)

|

/

(
1
−

|

f
(
z
)

|

2

)

{f^{\ast }}(z) = (1 - |z{|^2})|f’(z)|/(1 - |f(z){|^2})

at each point

z
ϵ
D

z\epsilon D

((1) in Theorem 1). The function

d
d

appearing in the estimate concerns the sizes of the schlicht disks on the Riemannian image

F

\mathcal {F}

of

D
D

by

f
f

. Boundary properties of

f
f

and

f

∗

{f^{\ast }}

will be stated in Theorems 2 and 3; use is made of the cluster sets of

d
d

.
A loop condition for embedded compacts
Authors: Coram D. , Daverman R. J. , Duvall P. F.
Published: June 2010

In this paper we consider a version

(

SLC

)

({\text {SLC}})

of the cellularity criterion which applies to embeddings of arbitrary compacta. It is shown that if the continuum

X
⊂

E
n

,

dim
⁡
X
⩽
n
−
3

X \subset {E^n},\;\dim X \leqslant n - 3

, has the shape of a finite

k
k

-complex

K
,

2
k
+
1
⩽
n

K,\;2k + 1 \leqslant n

, and satisfies the

SLC

{\text {SLC}}

, then

X
X

has arbitrarily small neighborhoods which are regular neighborhoods of a copy of

K
K

.
The Schnirelmann density of the sums of three squares
Authors: Wagstaff Samuel S.
Published: June 2010

The number in the title is

5

/

6

5/6

, which is the same as the asymptotic density of the set. The best possible upper and lower bounds (of the form

5
x

/

6
+
A
+
B
log
⁡
x

5x/6 + A + B\log x

) are obtained for the number of positive integers less than

x
x

which are a sum of three squares.
Smooth interpolating curves of prescribed length and minimum curvature
Authors: Jerome Joseph W.
Published: June 2010

It is shown that, among all smooth curves of length not exceeding a prescribed upper bound which interpolate a finite set of planar points, there is at least one which minimizes the curvature in the

L
2

{L^2}

sense. Thus, we show to be sufficient for the solution of the problem of minimum curvature a condition, viz., prescribed length, which has been known to be necessary for at least a decade. The proof extends immediately to curves in

R

n

,
n
>
2

{{\mathbf {R}}^n},n > 2

.
Unimaximal orders
Authors: Fossum T. V.
Published: June 2010

Let

R
R

be a Dedekind domain with quotient field

K
K

, and let

A
A

be a separable

K
K

-algebra. An

R
R

-order

Λ
\Lambda

in

A
A

is said to be unimaximal if

Λ
\Lambda

is contained in a unique maximal

R
R

-order in

A
A

. Unimaximal orders are given characterizations which are applied to determine those finite groups

G
G

of order

n
n

for which

R
G

RG

is unimaximal, where

K
K

is an algebraic number field containing a primitive

n
n

th root of unity.
On the applications of coalgebras to group algebras
Authors: Rosenberg Alan
Published: June 2010

This paper examines the coalgebra structure on

k

[
G

]
∘

k{[G]^ \circ }

and relates it to the group theoretic properties of

G
G

. In particular it is shown that there is an intimate relation between

k
[
G
]

k[G]

being proper and

G
G

being residually finite. We use this to derive a series of conditions on the group to guarantee it being residually finite.
A common fixed-point theorem for compact convex semigroups of nonexpansive mappings
Authors: Bruck Ronald E.
Published: June 2010

Let

C
C

be a bounded closed convex subset of a strictly convex Banach space and let

S
S

be a semigroup of nonexpansive self-mappings of

C
C

which is convex and compact in the topology of weak point-wise convergence. If

S
S

has the property that

co

¯

R

(

s
1

)
∩

co

¯

R

(

s
2

)
≠
∅

\overline {\operatorname {co} \,} \mathcal {R}({s_1}) \cap \overline {\operatorname {co} \,} \mathcal {R}({s_2}\;) \ne \emptyset

whenever

s
1

,

s
2

ϵ
S

{s_1},\;{s_2}\epsilon S

, then

S
S

has a common fixed point and

F
(
S
)

F(S)

is a nonexpansive retract of

C
C

.
A finite version of Schur’s theorem
Authors: Kulkarni R. S.
Published: June 2010

A necessary and sufficient condition for constancy of curvature in terms of umbilicity of small metric spheres is given.
Integration is differentiable
Authors: Gottlieb Allan
Published: June 2010

The integration map taking vector fields to flows is well known to be continuous. In this paper an elementary proof is given that the map is actually smooth if the domain and range are given appropriate topologies.
On the sum of Darboux functions
Authors: Bruckner A. M. , Ceder J.
Published: June 2010

The main purpose is to indicate how badly the sum of a continuous function with a Darboux function can fail to be a Darboux function.
A class of strong differentiability spaces
Authors: Collier James B.
Published: June 2010

It is shown that if the dual of a Banach space

X
X

is weakly compactly generated, then each convex function on

X
X

is Fréchet differentiable on a dense

G
δ

{G_\delta }

subset of its domain of continuity.
Characterizing a circle with the double midset property
Authors: Loveland L. D. , Valentine J. E.
Published: June 2010

A short and elementary proof is given to show that a space

X
X

is a circle with the natural geodesic metric if

X
X

is a nondegenerate, complete, convex metric space with the double midset property.
A general proof of Bing’s shrinkability criterion
Authors: Marin A. , Visetti Y. M.
Published: June 2010

This paper gives a proof of the general Bing shrinkability criterion, including a proof of the fundamental theorem that a shrinkable compact upper semicontinuous decomposition of a complete metric space is realized by a pseudo-isotopy of the space.
Strongly homogeneous spaces
Authors: Kitai Carol
Published: June 2010

Spaces satisfying various conditions have previously been called strongly homogeneous spaces and many results about the group of homeomorphisms of such spaces have been proved. However spaces may satisfy some “strongly homogeneous” condition without being homogeneous. In this paper we give a definition of strong homogeneity which implies homogeneity and includes most of the natural examples previously studied.
Compatible group topologies
Authors: Sharpe Kevin J.
Published: June 2010

Two topologies defined on some space are compatible if they contain in common a Hausdorff topology. The following result is proved for two compatible group topologies

A

1

{\mathcal {A}_1}

and

A

2

{\mathcal {A}_{_2}}

. Suppose

A

1

{\mathcal {A}_1}

is locally compact and

A

2

{\mathcal {A}_2}

is locally countably compact, and there is a non-void

A

2

{\mathcal {A}_2}

-open set contained in some

A

1

{\mathcal {A}_1}

-Lindelöf set. Then

A

1

⊆

A

2

{\mathcal {A}_1} \subseteq {\mathcal {A}_2}

. This result is a stronger version of a theorem by Kasuga, in which two group topologies are shown to be equal if both of them are locally compact and

σ
\sigma

-compact, and they are compatible.
A lattice theoretic characterization of an integral operator
Authors: Lessner Lawrence
Published: June 2010

We are concerned here with obtaining necessary and sufficient conditions for a linear operator,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K colon script upper L left-parenthesis upper X Subscript 1 Baseline comma script upper A 1 comma mu 1 right-parenthesis right-arrow upper M left-parenthesis upper X Subscript 2 Baseline comma script upper A 2 comma mu 2 right-parenthesis">

K
:

L

(

X

1

,

A

1

,

μ
1

)
→
M
(

X

2

,

A

2

,

μ
2

)

K:\mathcal {L}({{\text {X}}_1},\;{\mathcal {A}_1},\;{\mu _1}) \to M({{\text {X}}_2},\;{\mathcal {A}_2},\;{\mu _2})

, to be represented by an integral,

K
(
f
)
=
∫
k
(
x
,

y
)
f
(
y
)

d
y

K(f) = \smallint k(x,\;y)f(y)\;dy

, with an

A

2

×

A

1

{\mathcal {A}_2} \times {\mathcal {A}_1}

measurable kernel

k
(
x
,

y
)

k(x,\;y)

. That such conditions are developed in a lattice theoretic context will be shown to be quite natural. Our direction will be to characterize an integral operator by its action pointwise: i.e.,

K
(
)
(
x
)

K()(x)

is a linear functional on a subspace of the essentially bounded functions. Such a development leads one to define the kernel,

k
(
x
,

y
)

k(x,\;y)

, in a pointwise fashion also, and as a result we are confronted with the question of the

A

2

×

A

1

{\mathcal {A}_2} \times {\mathcal {A}_1}

measurability of

k
(
x
,

y
)

k(x,\;y)

.
Tauberian conclusions
Authors: Jukes K. A. , Maddox I. J.
Published: June 2010

Littlewood’s celebrated Tauberian theorem states that

∑

a
n

=
s

\sum {a_n} = s

(Abel) and

n

a
n

=
O
(
1
)

n{a_n} = O(1)

imply

s
n

=

∑

k
=
1

n

a
k

{s_n} = \sum _{k = 1}^n{a_k}

converges to

s
s

, the Tauberian condition

n

a
n

=
O
(
1
)

n{a_n} = O(1)

being best possible. We investigate ’best possibility’ of the conclusion

s
n

−
s
=
o
(
1
)

{s_n} - s = o(1)

, replacing the usual Tauberian condition by

(

q
n

a
n

)
ϵ
E

({q_n}{a_n})\epsilon E

where

(

q
n

)

({q_n})

is a positive sequence and

E
E

a given sequence space.
Covariances of generalized processes with orthogonal values
Authors: Pakula Lewis
Published: June 2010

A general form for the covariance of a generalized process with orthogonal values is found in the case where the covariance

B
B

depends on test functions and their first derivatives. Specifically, if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B left-parenthesis phi comma phi right-parenthesis equals integral phi squared d mu 0 plus integral phi phi prime d mu 1 plus integral phi squared d mu 2 greater-than-or-slanted-equals 0 for phi epsilon script upper D left-parenthesis bold upper R right-parenthesis">

B
(
ϕ
,
ϕ
)
=
∫

ϕ
2

d

μ
0

+
∫

ϕ

ϕ
′

d

μ
1

+
∫

ϕ
2

d

μ
2

⩾
0

for 

ϕ
ϵ

D

(

R

)

B(\phi ,\phi ) = \int {{\phi ^2}d{\mu _0} + \int {\phi \phi ’d{\mu _1} + \int {{\phi ^2}d{\mu _2} \geqslant 0} } } \;{\text {for }}\phi \epsilon \mathcal {D}({\mathbf {R}})

and Radon measures

μ
0

,

μ
1

,

μ
2

{\mu _0},{\mu _1},{\mu _2}

, then there exist Radon measures

ν
0

,

ν
1

,

ν
2

{\nu _0},{\nu _1},{\nu _2}

such that

B
(
ϕ
,
ϕ
)
=
∫

ϕ
2

d

ν
0

+
∫

ϕ

ϕ
′

d

ν
1

+
∫

ϕ

′
2

d

ν
2

B(\phi ,\phi ) = \int {{\phi ^2}d{\nu _0} + \int {\phi \phi ’d{\nu _1} + \int {\phi {’^2}d{\nu _2}} } }

and, moreover,

∫

f
2

d

ν
0

+
∫

f
g
d

ν
1

+
∫

g
2

d

ν
2

⩾
0

\int {{f^2}d{\nu _0} + \int {fgd{\nu _1} + \int {{g^2}d{\nu _2} \geqslant 0} } }

for all

f
,
g
ϵ

D

(

R

)

f,g\epsilon \mathcal {D}({\mathbf {R}})

.
On decomposability of compact perturbations of operators
Authors: Radjabalipour M.
Published: June 2010

Let

A
A

be a Hilbert-space operator satisfying the growth condition
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue left-parenthesis z minus upper A right-parenthesis Superscript negative 1 Baseline StartAbsoluteValue EndAbsoluteValue less-than-or-slanted-equals exp left-brace upper K left-bracket d i s t left-parenthesis z comma upper J right-parenthesis right-bracket Superscript negative upper S Baseline right-brace comma z not-an-element-of upper J">

|

|

(
z
−
A

)

−
1

|

|

⩽
exp
⁡
{
K

[
dist
⁡
(
z
,

J
)

]

−
S

}
,

z
∉
J

||{(z - A)^{ - 1}}|| \leqslant \exp \{ K{[\operatorname {dist} (z,\;J)]^{ - S}}\} ,\;z \notin J

, where

J
J

is a

C
2

{C^2}

Jordan curve, and

K
>
0
,

s
ϵ
(
0
,

1
)

K > 0,\;s\epsilon (0,\;1)

are two constants. Let

T
=
A
+
B

T = A + B

for some

B
ϵ

C
p

,

1
⩽
p
>
∞

B\epsilon {C_p},\;1 \leqslant p > \infty

. It is shown that

T
T

is strongly decomposable if and only if

σ
(
T
)

\sigma (T)

does not fill the “interior” of

J
J

.
A note on Tutte’s unimodular representation theorem
Authors: Brylawski Tom
Published: June 2010

A short, direct, and constructive proof is given of a result of W. T. Tutte (Lectures on matroids, J. Res. Nat. Bur. Standards Sect. B 69B (1965), 1-47). Theorem. Let

G
G

be an abstract finite combinatorial geometry whose dependencies can be represented by vectors over the field with two elements as well as by vectors over another field of characteristic other than two. Then

G
G

may be represented simultaneously over every field by the column vectors of a totally unimodular matrix.
Modules satisfying both chain conditions with respect to a torsion theory
Authors: Golan Jonathan S.
Published: June 2010

Goldman [3] has introduced the notion of the length of a module with respect to a torsion theory and has studied finitely-generated modules over left noetherian rings which have finite length. In this note we simplify the proofs of some of Goldman’s results and generalize them by removing both the finite-generation and noetherianness conditions.
Some finitely presented non-3-manifold groups
Authors: Heil Wolfgang H.
Published: June 2010

For every pair of integers

(
m
,
n
)

(m,n)

, G. Baumslag and D. Solitar exhibited two-generator one-relator groups

G

m
,
n

{G_{m,n}}

that are not residually finite or Hopfian. It is shown that these groups do not provide counterexamples to the conjecture that

3
3

-manifold groups are residually finite (or at least Hopfian) by showing that they have subgroups which are not

3
3

-manifold groups.
Uniform absolute continuity in spaces of set functions
Authors: Stein James D.
Published: June 2010

Let

X
X

be a regular topological space,

K
K

a collection of bounded regular measures defined on the Borel sets of

X
X

. The following conditions are equivalent. (1) Let

M
(
X
)

M(X)

denote the Borel measures,

M

(
X

)
+

M{(X)^ + }

the nonnegative members of

M
(
X
)

M(X)

. There is a

λ
∈
M

(
X

)
+

\lambda \in M{(X)^ + }

such that

K
K

is uniformly

λ
\lambda

-continuous. (2) If

{

U
n

|

n
=
1
,
2
,
…
}

\{ {U_n}|n = 1,2, \ldots \}

is a disjoint sequence of open sets, then

lim

n

→
∞

μ
(

U
n

)
=
0

{\lim _{{n^{ \to \infty }}}}\mu ({U_n}) = 0

uniformly for

μ
∈
K

\mu \in K

. (3) If

E
E

is a Borel subset of

X
X

and

ϵ
>
0

\epsilon > 0

, there is a compact set

F
⊆
E

F \subseteq E

such that

|

μ

|

(
E
∼
F
)
>
ϵ

|\mu |(E \sim F) > \epsilon

for

μ
∈
K

\mu \in K

. (4) If

{

E
n

|

n
=
1
,
2
,
…
}

\{ {E_n}|n = 1,2, \ldots \}

is a disjoint sequence of Borel sets, then

lim

n
→
∞

μ
(

E
n

)
=
0

{\lim _{n \to \infty }}\mu ({E_n}) = 0

uniformly for

μ
∈
K

\mu \in K

.
Solutions of linear differential equations in function fields of one variable
Authors: Singer Michael F.
Published: June 2010

Formal power series techniques are used to investigate the algebraic relationships between a function satisfying a linear differential equation and its derivatives. We are able to derive some conclusions, among them that an elliptic function satisfies no linear differential equation over a liouvillian extension of the complex numbers.
Completely decomposable flat modules over locally factorial domains
Authors: Lady E. L.
Published: June 2010

Rank

1
1

flat modules are classified for a locally factorial noetherian domain by extending the concept of divisor. A direct sum of rank one flat modules in called completely decomposable. A summand of a completely decomposable module is a direct sum of homogeneous components but need not be completely decomposable.
On upper gauge density
Authors: Pu H. W. , Pu H. H.
Published: June 2010

Let

g
g

be a diametric gauge over a metric space

(
X
,
ρ
)

(X,\rho )

. It is proved, in this paper, that the upper gauge density

D
(
A
,
x
)
=
0

D(A,x) = 0

for almost all points of the complement of

A
A

provided that

A
A

is in a certain family which contains all Borel sets of finite measure. Also, a relation between conditions for a diametric gauge and certain regularity conditions is given.
Inverse theorems for functions in 𝐿_{𝑝} and other spaces
Authors: Ditzian Z.
Published: June 2010

Inverse theorem proved by R. De Vore for

C
(
−
∞
,
∞
)

C( - \infty ,\infty )

is proved to be true also for

L
p

{L_p}

. This is an answer to a problem posed by De Vore in his book, The approximation of continuous functions by positive linear operators.
On a theorem of Brickman-Fillmore
Authors: Hwang Antonio
Published: June 2010

Let

V
V

be a finite dimensional vector space over an arbitrary field. We show that if

dim
⁡
V
⩽
3

\dim V \leqslant 3

and if

A
,
B

A,B

and

C
C

are pairwise commuting linear transformations on

V
V

such that every subspace invariant for both

A
A

and

B
B

is also invariant for

C
C

, then

C
C

is a polynomial in

A
A

and

B
B

. (Brickman and Fillmore proved that if

B
=
0

B = 0

then this statement is true for any finite dimensional vector space

V
V

.) An example shows that this is not true for

dim
⁡
V
>
3

\dim V > 3

.
Extensions of monotone operator functions
Authors: Chandler J. D.
Published: June 2010

It is shown that a monotone operator function

f
f

defined on an open subset

Δ
\Delta

of the real numbers may be extended to a monotone operator function on the convex hull of

Δ
\Delta

.
On Neumer’s theorem
Authors: Juhász I.
Published: June 2010

In this note we propose to show that Neumer’s theorem on regressive functions is actually a topological fact, by formulating and proving an entirely topological statement of which Neumer’s theorem is an immediate corollary.
On the order and degree of solutions to pure equations
Authors: Risman Lawrence J.
Published: June 2010

Let

K
K

be a field. Let

θ
\theta

be an element of a field extension of

K
K

. The order of

θ
\theta

over

K
K

is the smallest positive integer

m
m

such that

θ
m

{\theta ^m}

lies in

K
K

, or

∞
\infty

. We compare the order

m
m

of

θ
\theta

to the degree

h
h

of

θ
\theta

over

K
K

. Clearly

h
⩽
m

h \leqslant m

. Theorem. Let

K
K

be a field. Let

θ
\theta

be an element of degree

h
h

and order

m
m

over

K
K

. Let

p
p

be a prime. Let

p
e

{p^e}

be the maximum power of

p
p

dividing

h
h

, and suppose

p
s

{p^s}

divides

m
m

. (1) If the characteristic of

K
K

is

p
p

, then

s
⩽
e

s \leqslant e

. (2) If

s
>
e

s > e

and

p
p

is odd, then

K
(
θ
)

K(\theta )

contains a primitive

p
p

th root of unity

u
u

not in

K
K

. Moreover

K
(
u
)

K(u)

contains a primitive

p

s
−
e

{p^{s - e}}

root of unity. (3) If

s
>
e

s > e

and

p
=
2

p = 2

, then

−
1

- 1

is not a square in

K
K

and

K
(
θ
)

K(\theta )

contains

i
=

−
1

i = \sqrt { - 1}

. Moreover

−
1

- 1

is a

2

s
−
e

{2^{s - e}}

power in

K
(
i
)

K(i)

.
On the rate of growth of the Walsh antidifferentiation operator
Authors: Penney R.
Published: June 2010

In [1] Butzer and Wagner introduced a concept of differentiation and antidifferentiation of Walsh-Fourier series. Antidifferentiation is accomplished by convolving (in the sense of the Walsh group) against a function

Ω
\Omega

. In this paper we study growth and the continuity properties of

Ω
\Omega

showing that

Ω
\Omega

is bounded from below by

−
1

- 1

, is continuous in

(
0
,
1
)

(0,1)

and grows at most like

log
⁡
1

/

x

\log 1/x

as

x
→
0

x \to 0

. We use this information to study continuity properties of differentiable functions.
Numerical range of a weighted shift with periodic weights
Authors: Ridge William C.
Published: June 2010

Calculation of the numerical range of a weighted shift is reduced to the solution of a polynomial equation when the weights form a periodic sequence, or approach a periodic sequence from below.
An almost continuous function 𝑓: 𝑆ⁿ→𝑆^{𝑚} which commutes with the antipodal map
Authors: Kellum Kenneth R.
Published: June 2010

It is shown that if

n
,
m
⩾
1

n,m \geqslant 1

are integers, then there exists an almost continuous function from the

n
n

-sphere

S
n

{S^n}

onto

S
m

{S^m}

which commutes with the antipodal map.
Embeddings of compacta with shape dimension in the trivial range
Authors: Venema Gerard A.
Published: June 2010

In this paper a loop condition is defined which generalizes the cellularity criterion and applies to compacta with nontrivial shape. It is shown that if

X
,
Y
⊂

E
n

,
n
⩾
5

X,Y \subset {E^n},n \geqslant 5

, are compacta which satisfy this loop condition and whose shape classes include a space having dimension in the trivial range with respect to

n
n

, then

Sh
⁡
(
X
)
=
Sh
⁡
(
Y
)

\operatorname {Sh} (X) = \operatorname {Sh} (Y)

is equivalent to

E
n

−
X
≈

E
n

−
Y

{E^n} - X \approx {E^n} - Y

. An application is given to compacta with the shape of a compact connected abelian topological group.
A note on countably generated complete Boolean algebras
Authors: Jech Thomas J.
Published: June 2010

In this note, we construct (assuming the G.C.H.) a countably generated cardinal preserving complete Boolean algebra of cardinality

ℵ

ω
+
2

{\aleph _{\omega + 2}}

.
Operator radii of commuting products
Authors: Okubo K. , Ando T.
Published: June 2010

Operator radii

w
ρ

(
T
)

{w_\rho }(T)

for a bounded linear operator

T
T

on a Hilbert space were introduced in connection with unitary

ρ
\rho

-dilations. We shall be concerned with universal estimates for the ratios
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w Subscript rho Baseline left-parenthesis upper S upper T right-parenthesis slash left-parenthesis w Subscript sigma Baseline left-parenthesis upper S right-parenthesis dot w Subscript rho Baseline left-parenthesis upper T right-parenthesis right-parenthesis">

w
ρ

(
S
T
)

/

(

w
σ

(
S
)
⋅

w
ρ

(
T
)
)

{w_\rho }(ST)/({w_\sigma }(S) \cdot {w_\rho }(T))

\]
for commuting operators

S
,
T

S,T

and

σ
,
ρ
>
0

\sigma ,\rho > 0

.
On nonexpansive mappings
Authors: Karlovitz L A.
Published: June 2010

A generalized Hilbert space property is used to analyze nonexpansive mappings in certain settings. In particular it is shown that in

l
1

{l_1}

and in the important, recently defined, space

J
0

{J_0}

, a nonexpansive self-mapping of a bounded weak

∗

^{\ast }

closed convex subset has a fixed point.
On compactness and sequential compactness
Authors: Levine Norman
Published: June 2010

Let

X
X

be a compact space of cardinality less than or equal to

ℵ
1

{\aleph _1}

. It is shown that

X
X

is sequentially compact.
Recursive Euler and Hamilton paths
Authors: Bean Dwight R.
Published: June 2010

We employ recursion theoretic arguments to show that Hamilton paths for locally finite graphs are more difficult to find, in general, than Euler paths. A locally finite graph is recursive if we can effectively decide whether or not any two given vertices are adjacent, and highly recursive if we can effectively find all vertices adjacent to any given vertex. We find that there are recursive planar graphs with Euler or Hamilton paths but no such recursive paths. There are even particularly simple classes of connected, planar, highly recursive graphs for which we can show there is no effective way to decide about the existence of Euler or Hamilton paths. However, we obtain the following contrast: If a highly recursive graph has an Euler path we can effectively find a recursive Euler path; whereas, there is a planar, highly recursive graph with Hamilton paths but no recursive Hamilton path.
The final value problem for Sobolev equations
Authors: Lagnese John
Published: June 2010

Let

A
A

and

B
B

be

m
m

-accretive linear operators in a complex Hilbert space

H
H

with

D
(
A
)
⊂
D
(
B
)

D(A) \subset D(B)

. The method of quasi-reversibility is used to obtain a solution to the Sobolev equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis d slash d t right-parenthesis left-bracket left-parenthesis upper I plus upper B right-parenthesis u left-parenthesis t right-parenthesis right-bracket plus upper A u left-parenthesis t right-parenthesis equals 0 comma 0 greater-than t greater-than 1">

(
d

/

d
t
)
[
(
I
+
B
)
u
(
t
)
]
+
A
u
(
t
)
=
0
,
0
>
t
>
1

(d/dt)[(I + B)u(t)] + Au(t) = 0,0 > t > 1

, which approximates a specified final value

u
(
1
)
=
f

u(1) = f

. In general, when

D
(
A
)
⊂
D
(
B
)

D(A) \subset D(B)

, it is not possible to find a solution which achieves exactly the final value

u
(
1
)
=
f

u(1) = f

.
Inertial coefficient rings and the idempotent lifting property
Authors: Kirkman Ellen E.
Published: June 2010

A commutative ring

R
R

with identity is called an inertial coefficient ring if every finitely generated

R
R

-algebra

A
A

with

A

/

N

A/N

separable over

R
R

contains a separable

R
R

-subalgebra

S
S

of

A
A

such that

A
=
S
+
N

A = S + N

, where

N
N

is the Jacobson radical of

A
A

. We say

A
A

has the idempotent lifting property if every idempotent in

A

/

N

A/N

is the image of an idempotent in

A
A

. Our main theorem is that any finitely generated algebra over an inertial coefficient ring has the idempotent lifting property.