bluuebunny/crossref_metadata_2025 huggingface dataset:
Oscillation of solutions of a generalized Liénard equation
Authors: Benson Donald C.
Published: June 2010

The generalized Liénard equation,

x
¨

+
f
(
x
,

x
˙

)
+
h
(
x
)
=
0

\ddot x + f(x,\dot x) + h(x) = 0

, with

x
h
(
x
)
>
0

xh(x) > 0

and

y
f
(
x
,
y
)
>
0

yf(x,y) > 0

for nonzero x and y, is considered here, subject to the additional condition that

|

f
(
x
,
y
)

|

|f(x,y)|

is not greater than

k
(
x
)

|

y

|

α

k(x)|y{|^\alpha }

where

α
\alpha

is a positive number and

k
(
x
)

k(x)

is a continuous function which is positive for nonzero x. In case

α
≧
2

\alpha \geqq 2

, all solutions of this Liénard equation are oscillatory. In case

0
>
α
>
2

0 > \alpha > 2

, sufficient conditions are given which insure that all solutions are oscillatory.
Spaces with given homeomorphism groups
Authors: Thornton M. C.
Published: June 2010

A topological space X is constructed so that its group of homeomorphisms is isomorphic to a given finitely generated group G. If G has r generators and finite order n, X has

n
(
2
r
+
1
)

n(2r + 1)

points. The relation of X to certain covering spaces is considered.
Homotopy types of the deleted product of unions of two simplexes
Authors: Whitley W. T.
Published: June 2010

If X is a space, let

X
∗

=
X
×
X
−
D

{X^\ast } = X \times X - D

, where D is the diagonal. If f is a map on X to a space Y, let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Subscript f Superscript asterisk Baseline equals StartSet left-parenthesis x comma y right-parenthesis element-of upper X Superscript asterisk Baseline vertical-bar f left-parenthesis x right-parenthesis not-equals f left-parenthesis y right-parenthesis EndSet">

X
f
∗

=
{
(
x
,
y
)
∈

X
∗

|

f
(
x
)
≠
f
(
y
)
}

X_f^\ast = \{ (x,y) \in {X^\ast }|f(x) \ne f(y)\}

. In this paper we continue our investigation, begun in [6], of the homotopy types of

X
∗

{X^\ast }

and

X
f
∗

X_f^\ast

, and of a question due to Brahana [1, p. 236], as to when the homotopy types of

X
∗

{X^\ast }

and

X
f
∗

X_f^\ast

are the same. If X is the union of two nondisjoint simplexes, and if f is a simplicial map on X, we are able, using results and techniques developed in [6], to express the homotopy types of

X
∗

{X^\ast }

and

X
f
∗

X_f^\ast

in terms of spheres, and then to determine when the homotopy types of these spaces are the same.
On meromorphic functions commuting with elements of a function group
Authors: Smart John Roderick
Published: June 2010

The problem of whether there always exist meromorphic functions commuting with all substitutions of a function group is solved in the affirmative.
A property of 𝑦”’+𝑝(𝑥)𝑦^{’}+1\over2𝑝^{’}(𝑥)𝑦=0
Authors: Jones Gary D.
Published: June 2010

It is shown that if

y
′

+
p
(
x
)

y
′

+

1
2

p
′

(
x
)
y
=
0

y’ + p(x)y’ + \tfrac {1}{2}p’(x)y = 0

has an oscillatory solution, it has bases with 0, 1, 2, or 3 oscillatory elements.
A spherical surface measure inequality for convex sets
Authors: Fefferman Charles , Jodeit Max , Perlman Michael D.
Published: June 2010

Let the set C in the Euclidean space of n dimensions be closed, symmetric under reflection in the origin, and convex. The portion of the surface of the unit ball lying in C is shown to decrease in (the uniform) surface measure when C is replaced by AC, the image of C under any linear transformation A with norm no greater than one. Some cases of equality are discussed, and an application is given.
On a question of Erdős concerning cohesive basic sequences
Authors: Goldsmith D. L. , Gioia A. A.
Published: June 2010

For an arbitrary basic sequence

B

\mathcal {B}

, set

V
(

B

)
=
{
#

B
k

|

k
∈

Z
+

}

V(\mathcal {B}) = \{ \# {B_k}|k \in {Z^ + }\}

, where

#

B
k

\# {B_k}

is the number of pairs (a, b) in

B

\mathcal {B}

such that

a
b
=
k

ab = k

. It is proved that

V
(

B

)

V(\mathcal {B})

is unbounded if either

B

\mathcal {B}

is cohesive or

B

⊄

M

\mathcal {B} \not \subset \mathcal {M}

. The set

V
(

B

)

V(\mathcal {B})

is determined explicitly in these cases.
Mean convergence in 𝐿^{𝑝} spaces
Authors: Novinger W. P.
Published: June 2010

Let

(
X
,

B

,
μ
)

(X,\mathcal {B},\mu )

be a measure space and

{

f
n

}

\{ {f_n}\}

be a sequence in

L
p

(
X
,

B

,
μ
)
,
0
>
p
>
∞

{L^p}(X,\mathcal {B},\mu ),0 > p > \infty

. The author presents a very short proof of the familiar fact that if

f
n

→
f
μ

-a

.

e

.

{f_n} \to f\mu {\text {-a}}.{\text {e}}.

and

‖

f
n

‖

p

→

‖
f
‖

p

>
∞

{\left \| {{f_n}} \right \|_p} \to {\left \| f \right \|_p} > \infty

, then

‖

f
n

−
f

‖

p

→
0

{\left \| {{f_n} - f} \right \|_p} \to 0

.
On the behavior of solutions of sublinear second order differential equations
Authors: Teufel Hugo
Published: June 2010

The equations in question generalize
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x plus a left-parenthesis t right-parenthesis StartAbsoluteValue x EndAbsoluteValue Superscript gamma Baseline s g n x equals 0 comma 0 greater-than gamma greater-than 1 comma a left-parenthesis t right-parenthesis greater-than-over-equals 0 period">

x
+
a
(
t
)

|

x

|

γ

sgn
⁡
x
=
0
,

0
>
γ
>
1
,

a
(
t
)
≧
0.

x + a(t)|x{|^\gamma }\operatorname {sgn} x = 0,\quad 0 > \gamma > 1,\quad a(t) \geqq 0.

\]
. A comparison theorem and a uniqueness theorem for initial value problems are proved. Boundary value problems are studied. Oscillation is discussed via comparison.
On nonstrongly regular matrices
Authors: Bazinet J. , Siddiqi J. A.
Published: June 2010

Using the Rudin-Shapiro sequence, the existence of a regular but not strongly regular positive matrix that sums

{
exp
⁡
(
2
π
i
k
t
)
}

\{ \exp (2\pi ikt)\}

to 0 for all

t
∈
(
0
,
1
)

t \in (0,1)

is established. As a corollary it is shown that there exist matrices that sum all almost periodic sequences without possessing the Borel property and vice versa.
A normal form in the homeotopy group of a surface of genus 2, with applications to 3-manifolds
Authors: Birman Joan S.
Published: June 2010

It is shown that elements in the homeotopy group of a closed, compact, orientable 2-manifold of genus 2 can be put into a unique normal form which allows them to be enumerated systematically. As an application, the class of 3-manifolds which admit Heegaard splittings of genus 2 are shown to be denumerable, and a procedure is given for enumerating presentations for their fundamental groups.
Ideals generated by three elements
Authors: Kohn Peter
Published: June 2010

Let R be a commutative Noetherian ring. Any finitely generated module’s projective dimension is realized as the projective dimension of some cyclic module with three relations. This result was previously known for local rings.
Differentiable semigroups with a neighborhood of the identity homeomorphic to a Banach space
Authors: Neuberger J. W.
Published: June 2010

Suppose S is a topological semigroup with identity e, H is a Banach space and there is a homeomorphism g from an open subset of S containing e onto H. Theorem. Suppose that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V left-parenthesis x comma y right-parenthesis equals g left-parenthesis g Superscript negative 1 Baseline left-parenthesis x right-parenthesis g Superscript negative 1 Baseline left-parenthesis y right-parenthesis right-parenthesis">

V
(
x
,
y
)
=
g
(

g

−
1

(
x
)

g

−
1

(
y
)
)

V(x,y) = g({g^{ - 1}}(x){g^{ - 1}}(y))

for all x, y in H for which the second expression is defined. If V is

C

(
2
)

{C^{(2)}}

then there is an open subset D of S containing e so that if q is in D, then for some number t and some one-parameter subgroup f of S,

q
=
f
(
t
)

q = f(t)

.
Weak partition relations
Authors: Blass Andreas
Published: June 2010

The partition relation

ℵ
1

→
(

ℵ
1

)

3
,
2

2

{\aleph _1} \to ({\aleph _1})_{3,2}^2

, which was known to contradict the continuum hypothesis [1], is disproved without this hypothesis.
Continuous dependence on 𝐴 in the 𝐷₁𝐴𝐷₂ theorems
Authors: Sinkhorn Richard
Published: June 2010

It has been shown by Sinkhorn and Knopp and others that if A is a nonnegative square matrix such that there exists a doubly stochastic matrix B with the same zero pattern as A, then there exists a unique doubly stochastic matrix of the form

D
1

A

D
2

{D_1}A{D_2}

where

D
1

{D_1}

and

D
2

{D_2}

are diagonal matrices with positive main diagonals. Sinkhorn and Knopp have also shown that if A has at least one positive diagonal, then the sequence of matrices obtained by alternately normalizing the row and column sums of A will converge to a doubly stochastic limit. It is the intent of this paper to show that

D
1

A

D
2

{D_1}A{D_2}

and/or the limit of this iteration, when either exists, is continuously dependent upon the matrix A.
Gronwall’s inequality for systems of partial differential equations in two independent variables
Authors: Snow Donald R.
Published: June 2010

This paper presents a generalization for systems of partial differential equations of Gronwall’s classical integral inequality for ordinary differential equations. The proof is by reducing the vector integral inequality to a vector partial differential inequality and then using a vector generalization of Riemann’s method to obtain the final inequality. The final inequality involves a matrix function in the integrand which is a generalization of the scalar Riemann function. The proof includes a successive approximations argument to guarantee the existence and positivity property of this matrix function. The inequality is applied to prove a uniqueness theorem for a nonlinear vector hyperbolic partial differential equation, a comparison theorem for a linear hyperbolic vector partial differential equation, and a continuous dependence theorem for a nonlinear vector boundary value problem. The inequality also appears to have many applications in stability problems and in numerical solutions of partial differential equations. All of these results hold for the corresponding Volterra integral equations and the method of proof of the main result shows that the function on the right-hand side of the final inequality is the solution of the integral equation and hence is the maximal solution of the original inequality.
Integral as a certain type of a positive definite function
Authors: Saworotnow Parfeny P.
Published: June 2010

The integral with respect to an

H
∗

{H^ \ast }

-algebra valued measure on a compact Hausdorff space S is characterized as a certain type of positive definite function on the space

C
(
S
)

C(S)

of continuous functions on S.
Subsequential limit points of successive approximations
Authors: Guseman L. F. , Solomon J. L.
Published: June 2010

Results concerning the structure, cardinality, and invariance of the cluster set of the subsequential limit points of successive approximations are presented. These results along with some examples answer questions posed in a recent paper by F. T. Metcalf and T. D. Rogers.
Weak disjointness of transformation groups
Authors: Peleg Reuven
Published: June 2010

Two transformation groups (t.g.) are called weakly disjoint if their product is ergodic. We characterize this relation for a certain class of t.g. and then prove that for (X, T) and (Y, T) in a certain family of t.g. (X, T) and (Y, T) are disjoint iff they have no nontrivial common factor. Finally, we generalize some disjointness relations of [2] and [4].
The existence of oscillatory solutions for the equation 𝑑²𝑦/𝑑𝑡²+𝑞(𝑡)𝑦^{𝑟}=0,0<𝑟<1
Authors: Chiou Kuo Liang
Published: June 2010

This paper gives sufficient conditions for the existence of oscillatory solutions in the sublinear case of the second order differential equation

d
2

y

/

d

t
2

+
q
(
t
)

y
r

=
0

{d^2}y/d{t^2} + q(t){y^r} = 0

, where

q
(
t
)

q(t)

is non-negative and continuous and

0
>
r
>
1

0 > r > 1

. We use the technique of [3, Theorem 3.1] and obtain a result which extends [2, Corollary 1], [3, Theorem 3.1], and [3, Theorem 3.2].
Asymptotic behavior of solutions of perturbed linear systems
Authors: Bobisud L. E.
Published: June 2010

The existence of solutions of the system

y
′

+
A
y
=
f
(
t
,
y
)

y’ + Ay = f(t,y)

having the form

y
(
t
)
=
Z
(
t
)
a
(
t
)

y(t) = Z(t)a(t)

is proved, where

Z
(
t
)

Z(t)

satisfies

Z
′

+
A
Z
=
0

Z’ + AZ = 0

and the vector

a
(
t
)

a(t)

has limit

α
\alpha

as t increases. Estimates for the rate of convergence to zero of

a
(
t
)
−
α

a(t) - \alpha

and of

y
(
t
)
−
Z
(
t
)
α

y(t) - Z(t)\alpha

are obtained.
The Schur index and roots of unity
Authors: Janusz G. J.
Published: June 2010

A short proof is given for the main step in the proof of the theorem of Benard and Schacher which asserts that if the Schur index of a character

χ
\chi

of a finite group is m then the mth roots of unity lie in the field of values

Q
(
χ
)

Q(\chi )

.
Some unbounded functions of regular growth
Authors: DeKleine H. Arthur , Drobot Vladimir
Published: June 2010

The concept of regular growth for unbounded nondecreasing functions has its origin in the study of the asymptotic behavior of solutions for the second order equation

u
+
a
(
t
)
u
=
0

u + a(t)u = 0

. In this paper we give sufficient conditions for a continuous, differentiable function

a
(
t
)

a(t)

to possess the property that its logarithm increases regularly. We also show that the logarithm of a continuous unbounded concave or convex function increases regularly.
A random Trotter product formula
Authors: Kurtz Thomas G.
Published: June 2010

Let

X
(
t
)

X(t)

be a pure jump process with state space S and let

ξ
0

,

ξ
1

,

ξ
2

,
⋯

{\xi _0},{\xi _1},{\xi _2}, \cdots

be the succession of states visited by

X
(
t
)
,

Δ
0

Δ
1

⋯

X(t),{\Delta _0}{\Delta _1} \cdots

the sojourn times in each state,

N
(
t
)

N(t)

the number of transitions before t and

Δ
t

=
t
−

∑

k
=
0

N
(
t
)
−
1

Δ
k

{\Delta _t} = t - \sum \nolimits _{k = 0}^{N(t) - 1} {{\Delta _k}}

. For each

x
∈
S

x \in S

let

T
x

(
t
)

{T_x}(t)

be an operator semigroup on a Banach space L. Define
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript lamda Baseline left-parenthesis t comma w right-parenthesis equals upper T Subscript xi 0 Baseline left-parenthesis left-parenthesis 1 slash lamda right-parenthesis normal upper Delta 0 right-parenthesis upper T Subscript xi 1 Baseline left-parenthesis left-parenthesis 1 slash lamda right-parenthesis normal upper Delta 1 right-parenthesis midline-horizontal-ellipsis upper T Subscript xi Sub Subscript upper N left-parenthesis lamda t right-parenthesis Baseline left-parenthesis left-parenthesis 1 slash lamda right-parenthesis normal upper Delta Subscript lamda t Baseline right-parenthesis">

T
λ

(
t
,
w
)
=

T

ξ
0

(
(
1

/

λ
)

Δ
0

)

T

ξ
1

(
(
1

/

λ
)

Δ
1

)
⋯

T

ξ

N
(
λ
t
)

(
(
1

/

λ
)

Δ

λ
t

)

{T_\lambda }(t,w) = {T_{{\xi _0}}}((1/\lambda ){\Delta _0}){T_{{\xi _1}}}((1/\lambda ){\Delta _1}) \cdots T_{\xi _{N(\lambda t)}}((1/\lambda ){\Delta _{\lambda t}})

. Conditions are given under which

T
λ

(
t
,
w
)

{T_\lambda }(t,w)

will converge almost surely (or in probability) to a semigroup of operators as

λ
→
∞

\lambda \to \infty

. With

S
=
{
1
,
2
}

S = \{ 1,2\}

and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row upper X left-parenthesis t right-parenthesis equals 1 comma 2nd Row equals 2 comma EndLayout StartLayout 1st Row 2 n less-than-over-equals t greater-than 2 n plus 1 comma 2nd Row 2 n plus 1 less-than-over-equals t greater-than 2 n plus 2 comma EndLayout">

X
(
t
)
=
1
,

=
2
,

2
n
≦
t
>
2
n
+
1
,

2
n
+
1
≦
t
>
2
n
+
2
,

\begin {array}{*{20}{c}} \hfill {X(t) = 1,\quad } \\ \hfill { = 2,\quad } \\ \end {array} \begin {array}{*{20}{c}} {2n \leqq t > 2n + 1,} \hfill \\ {2n + 1 \leqq t > 2n + 2,} \hfill \\ \end {array}

\]

n
=
0
,
1
,
2
,
⋯

n = 0,1,2, \cdots

the result is just the “Trotter product formula".
Locally affine ring extensions of a Noetherian domain
Authors: Heinzer William
Published: June 2010

If

A
⊂
R

A \subset R

are integral domains with A noetherian, it is shown that if R is contained in an affine ring over A and if for each maximal ideal P of A with

S
=
A
∖
P
,

R
S

S = A\backslash P,{R_S}

is an affine ring over

A
P

{A_P}

, then R itself is affine over A.
A note on the 2/3 conjecture for starlike functions
Authors: McCarty Carl P. , Tepper David E.
Published: June 2010

Let

w
=
f
(
z
)
=
z
+

∑

n
=
2

∞

a
n

z
n

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

be regular and univalent for

|

z

|

>
1

|z| > 1

and map

|

z

|

>
1

|z| > 1

onto a region which is starlike with respect to

w
=
0

w = 0

. If

r
0

{r_0}

denotes the radius of convexity of

w
=
f
(
z
)
,

d
0

=
min

|

f
(
z
)

|

w = f(z),{d_0} = \min |f(z)|

for

|

z

|

=

r
0

|z| = {r_0}

, and

d
∗

=
inf

|

β

|

{d^ \ast } = \inf |\beta |

for

f
(
z
)
≠
β

f(z) \ne \beta

, then it has been conjectured that

d
0

/

d
∗

≧
2

/

3

{d_0}/{d^ \ast } \geqq 2/3

. It is shown here that

d
0

/

d
∗

≧
0.380
⋯

{d_0}/{d^\ast } \geqq 0.380 \cdots

which improves the old estimate

d
0

/

d
∗

≧
0.343
⋯

{d_0}/{d^\ast } \geqq 0.343 \cdots

. In addition an upper bound for

d
∗

{d^ \ast }

which depends on

|

a
2

|

|{a_2}|

is given.
A note on systems of linear integrodifferential equations
Authors: Wheeler Robert L.
Published: June 2010

Assume the existence and boundedness of a solution to a linear system of integrodifferential equations. Conditions are found which guarantee the solution is asymptotically almost periodic.
Uniqueness and norm convexity in the Cauchy problem for evolution equations with convolution operators
Authors: Strauss Monty J.
Published: June 2010

Uniqueness in the Cauchy problem is shown under suitable conditions for evolution equations of the form

u
t

(
x
,
t
)
−
B
(
t
,

D
x

)
u
(
x
,
t
)
=
0

{u_t}(x,t) - B(t,{D_x})u(x,t) = 0

, where B is a pseudo-differential operator of order

k
≧
0

k \geqq 0

in the x variables. This is proved as a corollary to a norm convexity relation. In the process of showing this, an extension to Hölder’s inequality is derived.
Quasi-cotripleable categories
Authors: Davis Robert C.
Published: June 2010

A category is quasi-cotripleable over the category of sets if it has all the properties of cotripleable categories except the right adjoint to the forgetful functor. Problems involving such categories are illustrated by categories of relational structures, and by categories of sets acted on by a monoid with open homomorphisms for maps. A characterization is given in terms of generalized operators and relations.
Applications of homotopy in sheaf theory
Authors: Parks James M.
Published: June 2010

Various mapping theorems for sheaf cohomology are obtained, where coefficients are in locally constant sheaves or limits of locally constant sheaves. Applications are made to the continuity of sheaf cohomology on homotopy-systems. A generalization of the Alexander-Čech cohomology continuity theorem is an immediate consequence.
A characterization of dual 𝐵*-algebras
Authors: McCharen Edith A.
Published: June 2010

Let A be a

B
∗

{B^\ast }

-algebra. The second conjugate space of A, denoted by

A

∗
∗

{A^{ \ast \ast }}

, is a

B
∗

{B^\ast }

-algebra under the Arens multiplication. A new proof is given that A is a dual algebra if and only if the natural image of A in

A

∗
∗

{A^{ \ast \ast }}

is an ideal in

A

∗
∗

{A^{\ast \ast }}

.
A model for closed orientable 3-manifolds of genus 1
Authors: Narvarte J. A.
Published: June 2010

In this paper we obtain a rectangular model for each closed orientable 3-manifold of genus 1 by identifications of the unit cube via unimodular plane involutions. The desirability of the model is indicated with regard to the fibring of such manifolds by circles, together with the development of a corresponding model for the connected sum of two such manifolds.
𝑃𝐷-minimal solutions of Δ𝑢=𝑃𝑢 on open Riemann surfaces
Authors: Ow Wellington H.
Published: June 2010

By means of the Royden compactification of an open Riemann surface R necessary and sufficient conditions are given for a Dirichlet-finite solution of

Δ
u
=
P
u

(
P
≧
0
,
P

≢

0
)

\Delta u = Pu\;(P \geqq 0, P\;{\nequiv }\;0)

to be PD-minimal on R. A relation between PD-minimal solutions and HD-minimal solutions is obtained. In addition it is shown that the dimension of the space of PD-solutions is the same as the number of P-energy nondensity points in the finite dimensional case.
The closed image of a metrizable space is 𝑀₁
Authors: Slaughter F. G.
Published: June 2010

J. Ceder introduced the notions of

M
1

{M_1}

space (a regular space with

σ
\sigma

-closure preserving base) and stratifiable space as natural generalizations of Nagata and Smirnov’s conditions for the metrizability of a regular space. Even though a topological space Y which is the image of a metrizable space under a closed, continuous mapping need not be metrizable, we show as our main result that Y will have a

σ
\sigma

-closure preserving base. It follows that one cannot obtain an example of a stratifiable space which is not

M
1

{M_1}

by constructing a quotient space from an upper semicontinuous decomposition of a metric space. In the course of establishing our major result, we obtain conditions under which the image of certain collections of sets under a closed, continuous mapping will be closure preserving.
Homotopic PL 𝑛-balls are isotopic
Authors: Dieffenbach Robert M.
Published: June 2010

The following theorem extends a result of Martin and Rolfsen [Proc. Amer. Math. Soc. 19 (1968), 1290-1292]. Theorem. Let

B
n

{B^n}

be a PL n-ball,

Q
q

{Q^q}

a

(
2
n
−
q
+
1
)

(2n - q + 1)

-connected PL q-manifold,

q
≧
n
+
2

q \geqq n + 2

. Suppose Q is either compact or open and that, for

i
=
0
,
1
,

H
i

:

B
n

→
Q
−

Q
˙

i = 0,1,{H_i}:{B^n} \to Q - \dot Q

is a locally unknotted PL embedding. If there exists a homotopy

H
:

B
n

×
I
→
Q

H:{B^n} \times I \to Q

between

H
0

{H_0}

and

H
1

{H_1}

such that

H
t

{H_t}

is fixed on

B
˙

n

{\dot B^n}

, then there exists a PL ambient isotopy

h
t

:
Q
→
Q

{h_t}:Q \to Q

, fixed on

H
0

(

B
˙

n

)
∪

Q
˙

{H_0}({\dot B^n}) \cup \dot Q

, such that

h
1

H
0

=

H
1

{h_1}{H_0} = {H_1}

. Locally unknotted is taken here to mean that there exists a triangulation (L, K) of

(
Q
,

H
i

(

B
n

)
)

(Q,{H_i}({B^n}))

with

K
˙

\dot K

full in K and

(

lk

(
ν
,
L
)

({\text {lk}}(\nu ,L)

,

lk

(
ν
,
K
)
)

{\text {lk}}(\nu ,K))

an unknotted sphere pair for all vertices

ν
∈
K
−

K
˙

\nu \in K - \dot K

.
𝐾-theory of commutative regular rings
Authors: Magid Andy R.
Published: June 2010

Pierce’s representation of a commutative regular ring as a sheaf of fields is used to compute the

K
K

-theory of the ring:

K
1

{K_1}

is units (Robert’s Theorem) and

K
2

{K_2}

is generated by symbols.
A class of partially ordered linear algebras
Authors: DeMarr Ralph
Published: June 2010

We consider a special type of partially ordered linear algebra which is like an algebra of real-valued functions. We show that various natural properties characterize this type of algebra. These natural properties relate the algebraic and order structures to each other.
Free derivation modules and a criterion for regularity
Authors: Moen Selmer
Published: June 2010

Let

k
k

be an algebraically closed field of characteristic zero,

R
R

an affine

k
k

-algebra. We prove that if the ideal of the variety of

R
R

can be generated by an

S
S

-sequence of forms in a polynomial ring

S
S

, and if the module of

k
k

-derivations of

R
R

into itself is a free

R
R

module, then

R
R

is regular.
Every 𝑛×𝑛 matrix 𝑍 with real spectrum satisfies ‖𝑍-𝑍*‖≤‖𝑍+𝑍*‖(log₂𝑛+0.038)
Authors: Kahan W.
Published: June 2010

The title’s inequality is proved for the operator bound-norm in a unitary space. An example is exhibited to show that the inequality cannot be improved by more than about 8
On matroids on edge sets of graphs with connected subgraphs as circuits
Authors: Sim oes-Pereira J. M. S.
Published: June 2010

It is proved that if

F

\mathcal {F}

is a finite family of connected, finite graphs, then a graph

G
G

exists such that the subgraphs of

G
G

isomorphic to a member of the family cannot be regarded as the circuits of a matroid on the edge set of

G
G

.
Peripherally compact tree-like spaces are continuum-wise connected
Authors: Bennett Ralph
Published: June 2010

The theorem stated in the title is proved, and examples are given to show the need for both peripheral compactness and tree-likeness.
Conditions for 𝐾𝑒𝑟(𝑅[𝑋]→𝑅[𝑐/𝑏]) to have a linear base
Authors: Ratliff L. J.
Published: June 2010

A number of necessary and/or sufficient conditions are given for the kernel of the homomorphism in the title to be generated by linear polynomials. Two applications are given.
A combinatorial analog of Lyapunov’s theorem for infinitesimally generated atomic vector measures
Authors: Loeb Peter A.
Published: June 2010

It is shown that the range of a measure obtained by the addition of infinitesimal vectors is convex up to infinitesimal errors.
Colength of derivation ideals
Authors: Kramer Kenneth
Published: June 2010

In this paper,

D
D

is a derivation acting on the formal power series ring

K
[
[

t
1

,
⋯
,

t
r

]
]

K[[{t_1}, \cdots ,{t_r}]]

over a field

K
K

of characteristic

p
≠
0

p \ne 0

. We conjecture that the colengths of the ideals (

(

D

p
n

t
1

,
⋯
,

D

p
n

t
r

)

({D^{{p^n}}}{t_1}, \cdots ,{D^{{p^n}}}{t_r})

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper D Superscript p Super Superscript n minus 1 Superscript Baseline t 1 comma midline-horizontal-ellipsis comma upper D Superscript p Super Superscript n minus 1 Superscript Baseline t Subscript r Baseline right-parenthesis">

(

D

p

n
−
1

t
1

,
⋯
,

D

p

n
−
1

t
r

)

({D^{{p^{n - 1}}}}{t_1}, \cdots ,{D^{{p^{n - 1}}}}{t_r})

are congruent modulo

p
n

{p^n}

, provided they are finite. We give a proof for the case

r
=
1

r = 1

and any

n
≧
1

n \geqq 1

, and for the case

n
=
1

n = 1

and any

r
≧
1

r \geqq 1

.
The dual and bidual of certain 𝐴*-algebras
Authors: Alexander Freda E.
Published: June 2010

It is well known that every

B
∗

{B^\ast }

-algebra is Arens’ regular and that its bidual is a

B
∗

{B^\ast }

-algebra. Wong has asked whether a dual

A
∗

{A^\ast }

-algebra of the first kind is Arens’ regular. It is shown that this is true in the topologically simple case; in the course of the proof it is shown that in this case the bidual is, modulo its radical, an

A
∗

{A^\ast }

-algebra of the first kind.
On the shape of torus-like continua and compact connected topological groups
Authors: Keesling James
Published: June 2010

In this paper it is shown that if

X
X

is a torus-like continuum, then

X
X

has the shape of a compact connected abelian topological group. Let

Π
\Pi

be a collection of compact connected Lie groups. In light of the above result it is natural to ask if a

Π
\Pi

-like continuum has the shape of a compact connected topological group. An example is given to show that this is not the case.
Solvable and nilpotent near-ring modules
Authors: Mason Gordon
Published: June 2010

The center of a unital near-ring module

R

M

{}_RM

is defined, leading to the construction of a lower central series and a definition of

R
R

-nilpotence. Likewise a suitable definition of commutators yields a derived series and

R
R

-solvability. When

(
R
,
+
)

(R, + )

is generated by elements which distribute over

M
M

the

R
R

-nilpotence (

R
R

-solvability) is shown to coincide with the nilpotence (solvability) of the underlying group. In this case, nilpotence has implications for

R
R

-normalizers and the Frattini submodule.
An extremal property of some capacitary measures in 𝐸_{𝑛}
Authors: Davis Burgess , Lewis John L.
Published: June 2010

The capacitary measure on an arc of the circle is known (via conformai mapping) to be that measure of a class of measures which has the largest potential at certain points of the plane. Here it is shown that the analogous result is true in

E
n

{E_n}

.
Disconjugacy and oscillation of third order differential equations with nonnegative coefficients
Authors: Etgen G. J. , Shih C. D.
Published: June 2010

The purpose of this paper is to establish conditions which imply that the third order linear differential equation with nonnegative coefficients defined on an infinite interval will fail to be disconjugate on any infinite subinterval. Assuming that the equation is not disconjugate on any infinite subinterval, conditions are presented which establish that the equation has oscillatory solutions. These results are in partial answer to questions raised by J. H. Barrett. The oscillation criteria obtained here are similar to the oscillation conditions established by A. C. Lazer.
Some order and topological properties of locally solid linear topological Riesz spaces
Authors: Aliprantis C. D.
Published: June 2010

A theorem of Luxemburg and Zaanen on normed Riesz spaces (Theorem 2.4 below) and one of Nakano (Theorem 2.3 below) have been extended by the author in [1] to metrizable locally solid linear topological Riesz spaces. This note gives an example which shows they cannot be further extended to nonmetrizable Hausdorff locally solid linear topological Riesz spaces.
An invariant of conformal mappings
Authors: Chen Bang-yen
Published: June 2010

A result of W. Blaschke on conformal invariants of a surface is generalized.
Pin and Pin′ cobordism
Authors: Giambalvo V.
Published: June 2010

The cobordism group

Ω
∗

Pi
⁡

n
′

\Omega _ \ast ^{\operatorname {Pi} n’}

of smooth manifolds with a Pin structure on the stable normal bundle is computed. The image

Ω
∗

Pi
⁡

n
′

→

N

∗

\Omega _ \ast ^{\operatorname {Pi} n’} \to {\mathfrak {N}_ \ast }

is determined, and some generators for

Ω
∗

Pin

\Omega _ \ast ^{\operatorname {Pin} }

and

Ω
∗

Pi
⁡

n
′

\Omega _ \ast ^{\operatorname {Pi} n’}

are given.
Further extending a complete convex metric
Authors: Dooley Robert A.
Published: June 2010

A metric

D
D

is convex if for every two points

x
,
z

x,z

there is a third point

y
y

such that

D
(
x
,
y
)
+
D
(
y
,
z
)
=
D
(
x
,
z
)

D(x,y) + D(y,z) = D(x,z)

. A generalized continuum is a connected, locally compact, metric space. Let

M
1

{M_1}

be a nonempty space with a complete convex metric

D
1

{D_1}

and let

M
2

{M_2}

be a nonempty locally connected generalized continuum. The following condition is shown to be necessary and sufficient for there to exist a complete convex metric for

M
1

∪

M
2

{M_1} \cup {M_2}

that extends

D
1

:

M
1

∩

M
2

{D_1}:{M_1} \cap {M_2}

is a nonempty subspace of both

M
1

{M_1}

and

M
2

{M_2}

which is closed in

M
2

{M_2}

and whose

M
2

{M_2}

boundary is closed in

M
1

{M_1}

.
Anosov flows and expansiveness
Authors: Norton Victor , O’Brien Thomas
Published: June 2010

We prove that an Anosov diffeomorphism of a compact manifold is expansive. We also show that a continuous flow on an infinite compact metric space cannot be expansive. We define a corresponding expansive concept for flows, that of an unstable flow (the word unstable is used here in a Lyapunov sense). We then prove that Anosov flows of compact manifolds are unstable.
The Baire order of the functions continuous almost everywhere
Authors: Mauldin R. D.
Published: June 2010

Let

Φ
\Phi

be the family of all real-valued functions defined on the unit interval

I
I

which are continuous except for a set of Lebesgue measure zero. Let

Φ
0

{\Phi _0}

be

Φ
\Phi

and for each ordinal

α
\alpha

, let

Φ
α

{\Phi _\alpha }

be the family of all pointwise limits of sequences taken from

⋃

γ
>
α

Φ
γ

\bigcup \nolimits _{\gamma > \alpha } {{\Phi _\gamma }}

Then

Φ

ω
1

{\Phi _{{\omega _1}}}

is the Baire family generated by

Φ
\Phi

. It is proven here that if

0
>
α
>

ω
1

0 > \alpha > {\omega _1}

, then

Φ
α

≠

Φ

ω
1

{\Phi _\alpha } \ne {\Phi _{{\omega _1}}}

. The proof is based upon the construction of a Borel measurable function

h
h

from

I
I

onto the Hilbert cube

Q
Q

such that if

x
x

is in

Q
Q

, then

h

−
1

(
x
)

{h^{ - 1}}(x)

is not a subset of an

F
σ

{F_\sigma }

set of Lebesgue measure zero.
The halting problem relativized to complements
Authors: Hay Louise
Published: June 2010

Let

H
A

=
{
e

|

domain

{
e
}
∩
A
≠
∅
}

{H^A} = \{ e|{\text {domain}}\{ e\} \cap A \ne \emptyset \}

. It is shown that there exists a set

A
A

of Turing degree

a
a

such that

H
A

{H^A}

is Turing-incomparable to

H

A
¯

{H^{\bar A}}

whenever

a
a

is an r.e. degree with

a
′

>

0
′

a’ > 0’

, or

a
≧
0

a \geqq 0

or

a
≧

0
′

a \geqq 0’

and

a
a

is r.e. in 0’. This contrasts with the fact that

H
A

{H^A}

is comparable to

H

A
¯

{H^{\bar A}}

for almost all

A
A

.
The Hoheisel phenomenon for generalized Dirichlet series
Authors: Moreno Carlos Julio
Published: June 2010

Hoheisel’s proof that the difference between two consecutive primes is of smaller order of magnitude than either prime depends on Littlewood’s estimate for the zero-free region of the Riemann zeta function and a density estimate for the number of zeros in certain rectangles in the critical strip. In this note we derive Hoheisel’s result without appealing to Littlewood’s theorem, thus enlarging the range of applicability of Hoheisel’s argument to a more general class of Dirichlet series. Applications of the results to number theory are given.
The equation of Ramanujan-Nagell and [𝑦²]
Authors: Mead D. G.
Published: June 2010

By arithmetizing Levi’s constructive test for membership in [

[

y
2

]

[{y^2}]

] we have translated the questions of whether a given power product is in [

[

y
2

]

[{y^2}]

] to determining whether a certain product of matrices is the zero matrix. This leads to number-theoretic problems, including the diophantine equations of the title

2
n

−
7
=

x
2

{2^n} - 7 = {x^2}

.
Finitely generated steady 𝔑-semigroups
Authors: Tamura Takayuki
Published: June 2010

In this paper the author proves that

S
S

is a finitely generated steady

N

\mathfrak {N}

-semigroup if and only if

S
S

is isomorphic to the direct product of a finite abelian group and the infinite cyclic semigroup; and also studies the homomorphisms of a finitely generated steady

N

\mathfrak {N}

-semigroup into another.
Bifurcation at eigenvalues of odd multiplicity
Authors: Westreich David
Published: June 2010

The most general form of the bifurcation problem is considered. Under reasonable hypotheses, “eigenvalues” of odd multiplicity are shown to be bifurcation points.
An example relating to Arhangel′skiĭ’s class 𝑀𝑂𝐵𝐼
Authors: Boone James R.
Published: June 2010

A space in Arhangel’skiĭ’s class MOBI is presented which fails to have two structural properties which are weaker than countable metacompactness. A space is countably metacompact if and only if it is countably

θ
\theta

-refinable and pointwise collectionwise normal for countable collections. This space does not have either of these properties.
An inverse function theorem for free groups
Authors: Birman Joan S.
Published: June 2010

Let

F
n

{F_n}

be a free group of rank

n
n

with free basis

x
1

,
⋯
,

x
n

{x_1}, \cdots ,{x_n}

. Let

{

y
1

,
⋯
,

y
k

}

\{ {y_1}, \cdots ,{y_k}\}

be a set of

k
≦
n

k \leqq n

elements of

F
n

{F_n}

, where each

y
i

{y_i}

is represented by a word

Y
i

(

x
1

,
⋯
,

x
n

)

{Y_i}({x_1}, \cdots ,{x_n})

in the generators

x
j

{x_j}

. Let

∂

y
i

/

∂

x
j

\partial {y_i}/\partial {x_j}

denote the free derivative of

y
i

{y_i}

with respect to

x
j

{x_j}

, and let

J

k
n

=

|

|

∂

y
i

/

∂

x
j

|

|

{J_{kn}} = ||\partial {y_i}/\partial {x_j}||

denote the

k
×
n

k \times n

Jacobian matrix. Theorem. If

k
=
n

k = n

, the set

{

y
1

,
⋯
,

y
n

}

\{ {y_1}, \cdots ,{y_n}\}

generates

F
n

{F_n}

if and only if

J

n
n

{J_{nn}}

has a right inverse. If

k
>
n

k > n

, the set

{

y
1

,
⋯
,

y
k

}

\{ {y_1}, \cdots ,{y_k}\}

may be extended to a set of elements which generate

F
n

{F_n}

only if

J

k
n

{J_{kn}}

has a right inverse. Several applications are given.
Locally compact groups without distinct isomorphic closed subgroups
Authors: Armacost D. L. , Bruner R. R.
Published: June 2010

In this note the structure of those locally compact topological groups which do not contain distinct isomorphic closed subgroups is determined.
For how many edges is a digraph almost certainly Hamiltonian?
Authors: Wright E. M.
Published: June 2010

K
n

(
r
)

{K_n}(r)

is the number of those Hamiltonian cycles in the complete digraph on

n
n

nodes, each of which has just

r
r

edges in common with a particular Hamiltonian cycle, and

B
(
n
,
r
)
=
n
!

/

{
r
!
(
n
−
r
)
!
}

B(n,r) = n!/\{ r!(n - r)!\}

. We show that

K
n

(
r
)
=
B
(
n
,
r
)

K

n
−
r

(
0
)

{K_n}(r) = B(n,r){K_{n - r}}(0)

and deduce that

K
n

(
0
)
∼
(
n
−
1
)
!

e

−
1

{K_n}(0)\sim (n - 1)!{e^{ - 1}}

for large

n
n

and that

K
n

(
r
)
∼
(
n
−
1
)
!

e

−
1

/

r
!

{K_n}(r)\sim (n - 1)!{e^{ - 1}}/r!

if

r
=
0
(
n
)

r = 0(n)

. An

(
n
,
q
)

(n,q)

digraph is one with

n
n

labelled nodes and

q
q

edges. From the result for

K
n

(
r
)

{K_n}(r)

, we deduce that, if

q

n

−
3

/

2

→
∞

q{n^{ - 3/2}} \to \infty

as

n
→
∞

n \to \infty

, then almost all

(
n
,
q
)

(n,q)

digraphs are Hamiltonian. If

q

n

−
3

/

2

→
c

q{n^{ - 3/2}} \to c

as

n
→
∞

n \to \infty

, then the proportion of

(
n
,
q
)

(n,q)

digraphs which are non-Hamiltonian is at most

1
−
exp
⁡
(
−

c

−
2

)

1 - \exp ( - {c^{ - 2}})

as

n
→
∞

n \to \infty

.
Perfect torsion theories
Authors: Bland Paul E.
Published: June 2010

The purpose of this paper is to introduce the study of perfect torsion theories on

Mod
⁡
R

\operatorname {Mod} R

and to dualize the concept of divisible module. A torsion theory on

Mod
⁡
R

\operatorname {Mod} R

is called perfect if every torsion module has a projective cover. It is shown for such a theory that the class of torsion modules is closed under projective covers if and only if the class of torsion free modules is closed under factor modules. In addition, it is shown that this condition on a perfect torsion theory is equivalent to its idempotent radical being an epiradical. Codivisible covers of modules are also introduced and we are able to show that any module which has a projective cover has a codivisible cover. Codivisible covers are then characterized in terms of the projective cover of the module and the torsion submodule of the kernel of the minimal epimorphism.
Metric inequalities and the zonoid problem
Authors: Witsenhausen H. S.
Published: June 2010

For normed spaces the hypermetric and quasihypermetric properties are equivalent and imply the quadrilateral property. The unit ball of a Minkowski space is a zonoid if and only if the dual space is hypermetric. The unit ball of

l
p
n

l_p^n

is not a zonoid for

n
=
3
,
p
>
log
⁡
3

/

log
⁡
2

n = 3,p > \log 3/\log 2

, and for

p
≦
2
−

(
2
n
log
⁡
2

)

−
1

+
o
(

n

−
1

)

p \leqq 2 - {(2n\log 2)^{ - 1}} + o({n^{ - 1}})

. The elliptic spaces

E

d

,
d
>
1

{\mathcal {E}^d},d > 1

, are not quasihypermetric.
Weighted representations of a primitive algebra
Authors: Goodaire E. G.
Published: June 2010

Let L be a diagonable subspace of an associative algebra A with identity over a field F; that is, L is spanned by a set of pairwise commuting elements, and the linear transformations ad

x
:
a
↦
a
x
−
x
a

x:a \mapsto ax - xa

for

x
∈
L

x \in L

are simultaneously diagonalizable. Denote the centralizer of L in A by

C

\mathcal {C}

. A module V over A or

C

\mathcal {C}

is L-weighted if for some nonzero

v
∈
V

v \in V

and map

λ
:
L
→
F
,
v

(
x
−
λ
(
x
)
1

)

n
(
x
)

=
0

\lambda :L \to F,v{(x - \lambda (x)1)^{n(x)}} = 0

for each

x
∈
L

x \in L

, and x-weighted if for some nonzero

v
∈
V
,
λ
∈
F

v \in V,\lambda \in F

and positive integer n,

v

(
x
−
λ
1

)
n

=
0

v{(x - \lambda 1)^n} = 0

. In this paper we give conditions under which the following statements are equivalent: 1. All irreducible modules over A and

C

\mathcal {C}

are L-weighted. 2. For each

x
∈
L

x \in L

, some irreducible A-module is x-weighted and x is algebraic over F.
Fixed points of certain self maps on an interval
Authors: Wong Chi Song
Published: June 2010

Let T be a self map on a bounded interval [a, b] with

a
,
b
∈
T
(
[
a
,
b
]
)

a,b \in T([a,b])

. Suppose that for any x, y in [a, b],
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper T left-parenthesis x right-parenthesis minus upper T left-parenthesis y right-parenthesis EndAbsoluteValue less-than-over-equals one half left-parenthesis StartAbsoluteValue x minus upper T left-parenthesis x right-parenthesis EndAbsoluteValue plus StartAbsoluteValue y minus upper T left-parenthesis y right-parenthesis EndAbsoluteValue right-parenthesis period">

|

T
(
x
)
−
T
(
y
)

|

≦

1
2

(

|

x
−
T
(
x
)

|

+

|

y
−
T
(
y
)

|

)
.

|T(x) - T(y)| \leqq \tfrac {1}{2}(|x - T(x)| + |y - T(y)|).

\]
It is shown without the continuity of T that the midpoint of [a, b] is a fixed point of T. A nontrivial example is given.
Operators with powers essentially similar to those of their adjoints
Authors: Patel S. M.
Published: June 2010

Let T be an operator on a Hilbert space H. In the present note the following result is obtained: If T is an operator such that for some integers

p
,
q
,
S

T

∗
p

=

T
q

S
+
K

p,q,S{T^{ \ast p}} = {T^q}S + K

, where 0 is not in the essential numerical range of S, and K is compact, then for any complex number

λ
\lambda

in the essential spectrum of T,

λ

∗
p

=

λ
q

{\lambda ^{ \ast p}} = {\lambda ^q}

.
The M.H.D. version of a theorem of H. Weyl
Authors: Conley Charles C. , Smoller Joel A.
Published: June 2010

In his discussion of shock waves in arbitrary fluids, H. Weyl proves a theorem concerning the behavior of the entropy function along the Hugoniot curve. The analogous result is proven for the M.H.D. case.
Topological algebras with a given dual
Authors: Chilana Ajit Kaur
Published: June 2010

Given an algebra E and a total subspace

E
′

E’

of its algebraic dual, we obtain necessary and sufficient conditions in terms of

E
′

E’

for the existence of an A-convex or a locally m-convex topology on E compatible with duality

(
E
,

E
′

)

(E,E’)

. It has also been proved that if E with the weak topology

w
(
E
,

E
′

)

w(E,E’)

is the closed linear hull of a bounded set and has hypocontinuous multiplication then it is locally m-convex.
The density character of function spaces
Authors: Noble N.
Published: June 2010

For topologies between the pointwise topology and the compact-open topology, the density character of

C
(
X
)

C(X)

(and, for certain spaces

Z
,
C
(
X
,
Z
)

Z,C(X,Z)

) is described in terms of a cardinal invariant of X. The Hewitt-Pondiczery theorem on the density character of product spaces follows as a corollary.
The antiautomorphism of the Steenrod algebra
Authors: Davis Donald M.
Published: June 2010

Combinatorial techniques are used to obtain some formulas for the canonical antiautomorphism of the Steenrod algebra.
Fully idempotent rings have regular centroids
Authors: Courter R. C.
Published: June 2010

We prove that the centroid of a ring all of whose ideals are idempotent is commutative and regular in the sense of von Neumann. The center of a fully idempotent ring is regular. Evidently every regular ring is fully idempotent. One nonregular example is Sasiada’s simple radical ring. A subring of the countably infinite row-finite matrices over Sasiada’s ring provides an example of a nonsimple, indecomposable, nonregular fully idempotent ring.
A note on groups with relatively compact conjugacy classes
Authors: Wilcox Theodore W.
Published: June 2010

In a more general form, the following structure theorem is proved. Let G be a locally compact group with small invariant neighborhoods. Then G has relatively compact conjugacy classes if and only if G is a direct product of a vector group V and a group L where L has a compact open normal subgroup K such that L/K has finite conjugacy classes.
Arcs defined by one-parameter semigroups of operators
Authors: Junghenn Hugo D. , Taam C. T.
Published: June 2010

Let

T
(
t
)
(
t
≧
0
)

T(t)(t \geqq 0)

be a one-parameter semigroup of continuous linear operators in a locally convex reflexive linear topological space

X
X

such that

T
(
c
)

T(c)

is an isomorphism (into) for some

c
>
0

c > 0

. It is proved that for any

x
∈
X
,
T
(
⋅
)
x

x \in X,T( \cdot )x

is of bounded variation on finite intervals if and only if

x
x

is in the domain of the infinitesimal generator of

T
(
t
)

T(t)

. The result is interpreted geometrically in terms of arc-length.
A counterexample concerning almost continuous functions
Authors: Herrington Larry L. , Hunsaker W. N. , Lindgren W. F. , Naimpally S. A.
Published: June 2010

An example is constructed of a function which is almost continuous in the sense of Singal and Singal but not in the sense of Stallings.
Open maps of chainable continua
Authors: Rosenholtz Ira
Published: June 2010

It is apparently “well known” that the image of the closed unit interval under an open map is homeomorphic to the closed unit interval (see [13], [11], and [15]). In this paper, we generalize this result to chainable continua. In particular, the fact that the open continuous image of a chainable continuum is also chainable is proved, answering a question of A. Lelek (see [10]). This fact, as well as its proof, implies that the open continuous image of the pseudo-arc is also a pseudo-arc. An additional corollary (of the proof) is that a local homeomorphism of a chainable continuum is actually a homeomorphism. The proofs are all very elementary.
A countably distributive complete Boolean algebra not uncountably representable
Authors: Gregory John
Published: June 2010

It is proved from the Continuum Hypothesis that there exists an

ω
\omega

-distributive complete Boolean algebra which is not

ω
1

{\omega _1}

-representable.
Class numbers and 𝜇-invariants of cyclotomic fields
Authors: Metsänkylä T.
Published: June 2010

We give a new upper bound for the

μ
\mu

-invariant of a cyclotomic field by estimating the first factor of the class number of the

p
p

th cyclotomic field (

p
p

an odd prime).
A property of transferable lattices
Authors: Grätzer G.
Published: June 2010

A lattice

K
K

is transferable if whenever

K
K

can be embedded into the ideal lattice of a lattice

L
L

, then

K
K

can be embedded in

L
L

. An element is called doubly reducible if it is both join- and meet-reducible. In this note it is proved that every lattice can be embedded into the ideal lattice of a lattice with no doubly reducible element. It follows from this result that a transferable lattice has no doubly reducible element.
The first derived equaliser of a pair of abelian group homomorphisms
Authors: Porter Tim
Published: June 2010

In this note we attack the problem of classifying a pair of abelian group homomorphisms by introducing a construction which we call “the derived equaliser". After developing fairly simple methods for the calculation of these objects, we work an example in a simple case.
Fixed points by a new iteration method
Authors: Ishikawa Shiro
Published: June 2010

The following result is shown. If

T
T

is a lipschitzian pseudo-contractive map of a compact convex subset

E
E

of a Hilbert space into itself and

x
1

{x_1}

is any point in

E
E

, then a certain mean value sequence defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Subscript n plus 1 Baseline equals alpha Subscript n Baseline upper T left-bracket beta Subscript n Baseline upper T x Subscript n Baseline plus left-parenthesis 1 minus beta Subscript n Baseline right-parenthesis x Subscript n Baseline right-bracket plus left-parenthesis 1 minus alpha Subscript n Baseline right-parenthesis x Subscript n">

x

n
+
1

=

α
n

T
[

β
n

T

x
n

+
(
1
−

β
n

)

x
n

]
+
(
1
−

α
n

)

x
n

{x_{n + 1}} = {\alpha _n}T[{\beta _n}T{x_n} + (1 - {\beta _n}){x_n}] + (1 - {\alpha _n}){x_n}

converges strongly to a fixed point of

T
T

, where

{

α
n

}

\{ {\alpha _n}\}

and

{

β
n

}

\{ {\beta _n}\}

are sequences of positive numbers that satisfy some conditions.