bluuebunny/crossref_metadata_2025 huggingface dataset:
A nonembedding theorem for finite groups
Authors: Stitzinger Ernest L.
Published: June 2010

Let

N
N

be the class of nilpotent groups with the following properties: (1) The center of

N
,

Z
⊥

(
N
)

N,{Z_ \bot }(N)

is of prime order. (2) There exists an abelian characteristic subgroup

A
A

of

N
N

such that

Z
1

(
N
)
⊂
A
⊆

Z
2

(
N
)

{Z_1}(N) \subset A \subseteq {Z_2}(N)

where

Z
2

(
N
)

{Z_2}(N)

is the second term in the upper central series of

N
N

. The main result shown is the following:

N
∈

X

N \in \mathfrak {X}

, then

N
N

cannot be an invariant subgroup contained in the Frattini subgroup of a finite group.
Note on metric dimension
Authors: Prosser Reese T.
Published: June 2010

The metric dimension of a compact metric space is defined here as the order of growth of the exponential metric entropy of the space. The metric dimension depends on the metric, but is always bounded below by the topological dimension. Moreover, there is always an equivalent metric in which the metric and topological dimensions agree. This result may be used to define the topological dimension in terms of the metric entropy as the infimum of the metric dimension taken over all metrics.
Representation of linear sets as critical sets
Authors: Kaufman Robert
Published: June 2010

A class of linear sets investigated by Besicovitch and Taylor is related to the critical set of differentiable mappings of a specified degree of smoothness. An example is constructed to show that certain results on Hausdorff measure are nearly best-possible.
An inequality for the Riemann-Stieltjes integral
Authors: Darst Richard , Pollard Harry
Published: June 2010

Let

g
g

and

h
h

be real valued and continuous on the interval

[
a
,
b
]

[a,b]

, and suppose that the variation,

V
[
h
]

V[h]

, of

h
h

on

[
a
,
b
]

[a,b]

is finite. By completely elementary methods, it is shown that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V left-bracket h right-bracket dot sup Underscript Subscript a less-than-over-equals alpha greater-than beta less-than-over-equals b Endscripts left-parenthesis g left-parenthesis beta right-parenthesis minus g left-parenthesis alpha right-parenthesis right-parenthesis">

V
[
h
]
⋅

sup

a
≦
α
>
β
≦
b

(
g
(
β
)
−
g
(
α
)
)

V[h] \cdot {\sup _{_{a \leqq \alpha > \beta \leqq b}}}(g(\beta ) - g(\alpha ))

is an upper bound for

∫
a
b

(
h
−
inf
h
)
d
g

\int _a^b {(h - \inf h)dg}

.
Homeomorphic measures in metric spaces
Authors: Oxtoby John C.
Published: June 2010

For any nonatomic, normalized Borel measure

μ
\mu

in a complete separable metric space

X
X

there exists a homeomorphism

h
:

N

→
X

h:\mathfrak {N} \to X

such that

μ
=
λ

h

−
1

\mu = \lambda {h^{ - 1}}

on the domain of

μ
\mu

, where

N

\mathfrak {N}

is the set of irrational numbers in

(
0
,
1
)

(0,1)

and

λ
\lambda

denotes Lebesgue-Borel measure in

N

\mathfrak {N}

. A Borel measure in

N

\mathfrak {N}

is topologically equivalent to

λ
\lambda

if and only if it is nonatomic, normalized, and positive for relatively open subsets.
A growth property of the Nevanlinna characteristic
Authors: Weitsman Allen
Published: June 2010

We prove here a relation between the rate of growth of the Nevanlinna characteristic of a meromorphic function

f
(
z
)

f(z)

, the size of a deficient value

τ
\tau

, and the sets where

f
(
z
)

f(z)

is close to

τ
\tau

.
Stability of solutions of linear systems with retarded arguments
Authors: Abramowich John
Published: June 2010

A criterion is given for the stability of the trivial solution of a linear system of differential equations with variable coefficients and variable delays under the assumptions of continuity and boundedness of the coefficients and continuous differentiability of the delays. The criterion is a condition on the roots of a certain “quasi-polynomial,” i.e. a polynomial in a variable and exponentials of that variable. The derivation is based on Lyapunov’s direct method.
Conullity of operators on some 𝐹𝐾-spaces
Authors: Brown H. I. , Stratton H. H.
Published: June 2010

The notion of conullity for a subclass of the algebra of matrix operators on the space of convergent sequences is well known in summability theory. In this paper the space of convergent sequences is replaced by a general (locally convex)

F
K

FK

-space and the following question is studied: Given a subalgebra of the algebra of all continuous linear operators on this

F
K

FK

-space, is there a class of operators in this subalgebra whose behavior is “conull-like"? The question is answered in the case when the

F
K

FK

-space has a suitable (Schauder) basis and also in some other special cases.
Lattice-invariant properties of topological spaces
Authors: Wong Yim-ming
Published: June 2010

W. J. Thron proved in 1962 that regularity and normality are lattice-invariant properties but not

T
0

{T_0}

iexcl and

T
1

{T_1}

. In the present paper it is proved that complete regularity, compactness, local compactness, Lindelöf, second countability and connectedness are lattice-invariant properties. It is also proved that Hausdorff, complete normality, separability, and first countability are not lattice-invariant properties.
Extensions of a ring by a ring with a bimodule structure
Authors: Kohls Carl W.
Published: June 2010

A type of ring extension is considered that was introduced by J. Szendrei and generalizes many familiar examples, including the complex extension of the real field. We give a method for constructing a large class of examples of this type of extension, and show that for some rings all possible examples are obtained by this method. An abstract characterization of the extension is also given, among rings defined on the set product of two given rings.
The annihilator of radical powers in the modular group ring of a 𝑝-group
Authors: Hill E. T.
Published: June 2010

We show that if

N
N

is the radical of the group ring and

L
L

is the exponent of

N
N

, then the annihilator of

N
w

{N^w}

is

N

L
−
w
+
1

{N^{L - w + 1}}

. As corollaries we show that the group ring has exactly one ideal of dimension one and if the group is cyclic, then the group ring has exactly one ideal of each dimension.
Oscillating solutions of third order differential equations
Authors: Utz W. R.
Published: June 2010

Third order ordinary linear differential equations are investigated with respect to the property of having a basis for solutions that consist of three oscillating solutions yet some nontrivial solution of the equation is nonoscillatory.
Weak 𝐴-convex algebras
Authors: Cochran Allan C.
Published: June 2010

Necessary and sufficient conditions are given in terms of

E
′

E’

that a weak topology

w
(
E
,

E
′

)

w(E,E’)

on an algebra

E
E

be

A
A

-convex. The main condition is that each element

g
g

of

E
′

E’

contain a weakly closed subspace

L
L

of finite codimension such that

g
g

is bounded on all multiplicative translates of

L
L

. For weak topologies,

A
A

-convexity (which assumes only separate continuity of multiplication) is equivalent to joint continuity of multiplication.
Positively curved deformations of invariant Riemannian metrics
Authors: Weinstein Alan
Published: June 2010

Let

K
γ

{K_\gamma }

denote the sectional curvature function of the Riemannian metric

γ
\gamma

on a manifold

M
M

. Suppose

M
M

admits no metric

γ
\gamma

invariant under the action of a compact group

G
G

and having

K
γ

>
0

{K_\gamma } > 0

. It is shown that a

G
G

-invariant metric

γ
(
0
)

\gamma (0)

with

K

γ
(
0
)

≧
0

{K_{\gamma (0)}} \geqq 0

cannot be embedded in a

1
1

-parameter family

γ
(
t
)

\gamma (t)

for which

[
d

K

γ
(
t
)

/

d
t

]

t
=
0

{[d{K_{\gamma (t)}}/dt]_{t = 0}}

is positive wherever

K

γ
(
0
)

{K_{\gamma (0)}}

is zero.
An upper asymptotic estimate of Brownian path variation
Authors: Stackelberg Olaf P.
Published: June 2010

Let

X
(
t
,
ω
)

X(t,\omega )

be standard Brownian motion, and denote by

B

n

{\mathfrak {B}_n}

the family of all partitions of

[
0
,
1
]

[0, 1]

with minimum distance between successive partition points

≧
1

/

n

\geqq 1/n

. Then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P left-bracket limit sup Underscript n right-arrow normal infinity Endscripts sup Underscript upper P element-of German upper B Subscript 2 n Baseline StartFraction log log n Over log squared n EndFraction sigma-summation Underscript t Subscript i Baseline element-of upper P Endscripts left-parenthesis upper X left-parenthesis t Subscript i Baseline right-parenthesis minus upper X left-parenthesis t Subscript i minus 1 Baseline right-parenthesis right-parenthesis squared less-than-over-equals upper K Endscripts right-bracket equals 1">

P
[
lim

sup

n
→
∞

sup

P
∈

B

2
n

log
⁡
log
⁡
n

log
2

⁡
n

∑

t
i

∈
P

(
X
(

t
i

)
−
X
(

t

i
−
1

)

)
2

≦
K

]
=
1

P[ \lim \sup _{n \to \infty } \sup _{P \in \mathfrak {B}_{2n} \frac {\log \log n}{\log ^2 n} \sum \limits _{t_i \in P} (X(t_i) - X(t_{i - 1}))^2 \leqq K} ] = 1

\]
where

K
K

is an appropriate constant.
The degrees of the factors of certain polynomials over finite fields.
Authors: Mills W. H.
Published: June 2010

Neal Zierler has discovered that the polynomial

x

585

+
x
+
1

{x^{585}} + x + 1

over

GF
⁡
(
2
)

\operatorname {GF} (2)

is the product of

13
13

irreducible factors of degree

45
45

and that the polynomial

x

16513

+
x
+
1

{x^{16513}} + x + 1

over

GF
⁡
(
2
)

\operatorname {GF} (2)

is the product of

337
337

irreducible factors of degree

49
49

. We prove a general theorem that includes these results, as well as some other well known results, as special cases.
The big Picard theorem for polyanalytic functions
Authors: Bosch W. , Krajkiewicz P.
Published: June 2010

Let

f
,
g

f,g

, and

h
h

be polyanalytic in an annular neighborhood

A
A

of a complex number

z
0

{z_0}

, finite or infinite, such that

g
g

and

h
h

do not have an essential singularity at

z
0

{z_0}

and

g
−
h

g-h

is not identically zero on

A
A

. It is shown that if

f
−
g

f-g

and

f
−
h

f-h

never vanish on

A
A

, then

z
0

{z_0}

is not an essential singularity of

f
f

.
A global existence theorem for autonomous differential equations in a Banach space
Authors: Martin R. H.
Published: June 2010

Let

E
E

be a Banach space and let

A
A

be a continuous function from

E
E

into

E
E

. Sufficient conditions are given to insure that the differential equation

u
′

(
t
)
=
A
u
(
t
)

u’(t) = Au(t)

has a unique solution on

[
0
,
∞
)

[0,\infty )

for each initial value in

E
E

. One consequence of this result is that if

−
A

-A

is monotonic, then

−
A

-A

is

m
m

-monotonic and

A
A

is the generator of a nonexpansive semigroup of operators.
Remarks on pseudo-contractive mappings
Authors: Kirk W. A.
Published: June 2010

Let

X
X

be a Banach space,

D
⊂
X

D \subset X

. A mapping

U
:
D
→
X

U:D \to X

is said to be pseudo-contractive if for all

u
,
v
∈
D

u,v \in D

and all

r
>
0

r > 0

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue u minus v StartAbsoluteValue EndAbsoluteValue less-than-over-equals StartAbsoluteValue EndAbsoluteValue left-parenthesis 1 plus r right-parenthesis left-parenthesis u minus v right-parenthesis minus r left-parenthesis upper U left-parenthesis u right-parenthesis minus upper U left-parenthesis v right-parenthesis right-parenthesis StartAbsoluteValue EndAbsoluteValue">

|

|

u
−
v

|

|

≦

|

|

(
1
+
r
)
(
u
−
v
)
−
r
(
U
(
u
)
−
U
(
v
)
)

|

|

||u - v|| \leqq ||(1 + r)(u - v) - r(U(u) - U(v))||

. This concept is due to F. E. Browder, who showed that

U
:
X
→
X

U:X \to X

is pseudo-contractive if and only if

I
−
U

I - U

is accretive. In this paper it is shown that if

X
X

is a uniformly convex Banach,

B
B

a closed ball in

X
X

, and

U
U

a Lipschitzian pseudo-contractive mapping of

B
B

into

X
X

which maps the boundary of

B
B

into

B
B

, then

U
U

has a fixed point in

B
B

. This result is closely related to a recent theorem of Browder.
On noninvertible links with invertible proper sublinks
Authors: Whitten W. C.
Published: June 2010

The title supports the primary objective of this paper. Specifically, for each

μ
≧
3

\mu \geqq 3

, a noninvertible, oriented, ordered link of

μ
\mu

components tamely imbedded in

S
3

{S^3}

is exhibited with the property that each proper sublink is invertible. The case

μ
=
2

\mu = 2

has been covered in a previous paper.
On the boundedness of an iterative procedure for solving a system of linear inequalities
Authors: Block H. D. , Levin S. A.
Published: June 2010

In this paper it is proved that the perceptron error-correction procedure stays bounded, even when no solution to the system of linear inequalities exists. This supplements earlier papers by B. Efron and by M. Minsky and S. Papert.
On elementary groups
Authors: Stitzinger Ernest L.
Published: June 2010

Bechtell has defined a group

G
G

to be elementary if the Frattini subgroup of each subgroup of

G
G

is the identity. In this note we prove the following: If the derived group of

G
G

is nilpotent, then necessary and sufficient conditions that

G
G

be elementary are that the Frattini subgroup of

G
G

be the identity and that the Frattini subgroup of some Carter subgroup

K
K

of

G
G

be equal to the derived group of

K
K

.
A lower Jackson bound on (-∞,∞)
Authors: Byrnes J. S. , Newman D. J.
Published: June 2010

We produce a lower bound for the degree of uniform polynomial approximation to continuous functions on the whole real line using the weight function

exp
⁡
(
−

|

x

|

α

)
,
α
≧
2

\exp ( - |x{|^\alpha }),\alpha \geqq 2

. This lower bound has the same order of magnitude as the upper bound produced previously by Džrbašyan.
No infinite dimensional 𝑃 space admits a Markuschevich basis
Authors: Johnson William B.
Published: June 2010

Theorem. Let

X
X

be a Banach space. If

X
X

is a Grothendieck space and

X
X

admits a Markuschevich basis then

X
X

is reflexive. This theorem is used to prove the conjecture of J. A. Dyer [1] stated in the title.
Seminormal operators with compact self-commutators
Authors: Clancey Kevin
Published: June 2010

Putnam’s inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi StartAbsoluteValue EndAbsoluteValue upper T Superscript asterisk Baseline upper T minus upper T upper T Superscript asterisk Baseline StartAbsoluteValue EndAbsoluteValue less-than-over-equals m e a s 2 left-parenthesis sp left-parenthesis upper T right-parenthesis right-parenthesis">

π

|

|

T
∗

T
−
T

T
∗

|

|

≦

meas
2

⁡
(

sp

(
T
)
)

\pi ||{T^ \ast }T - T{T^ \ast }|| \leqq \operatorname {meas}_2({\text {sp}}(T))

is verified for seminormal operators

T
T

when

T
∗

T
−
T

T
∗

{T^ \ast }T - T{T^ \ast }

is compact. The spectrum of the seminormal singular integral operator defined for

f
∈

L
2

(
0
,
1
)

f \in {L^2}(0,1)

by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T f left-parenthesis s right-parenthesis equals s f left-parenthesis s right-parenthesis plus i left-parenthesis StartFraction phi left-parenthesis s right-parenthesis Over pi i EndFraction integral Subscript 0 Superscript 1 Baseline StartFraction f left-parenthesis t right-parenthesis ModifyingAbove phi With bar left-parenthesis t right-parenthesis Over s minus t EndFraction d t right-parenthesis comma s element-of left-bracket 0 comma 1 right-bracket comma">

T
f
(
s
)
=
s
f
(
s
)
+
i

(

ϕ
(
s
)

π
i

∫
0
1

f
(
t
)

ϕ
¯

(
t
)

s
−
t

d
t

)

,

s
∈
[
0
,
1
]
,

Tf(s) = sf(s) + i\left ( {\frac {{\phi (s)}}{{\pi i}}\int _0^1 {\frac {{f(t)\bar \phi (t)}}{{s - t}}dt} } \right ),\quad s \in [0,1],

\]
where

ϕ
\phi

is any fixed essentially bounded measurable function on

[
0
,
1
]

[0,1]

, is computed.
Isotopy and homeomorphism
Authors: Kibler Dennis
Published: June 2010

If

X
X

and

Y
Y

are isotopically equivalent topological spaces, they are not necessarily homeomorphic. If

X
X

and

Y
Y

are compact without boundary manifolds or

n
n

-pure simplicial complexes, then isotopy equivalence implies topological equivalence. An example of compact manifolds with boundaries which are not homeomorphic but are isotopically equivalent is given.
Surfaces of finite domain rank
Authors: Tondra Richard J.
Published: June 2010

In this paper it is shown that a surface has finite domain rank if and only if it is homeomorphic to a domain of a compact or closed surface.
Cardinals 𝑚 such that 2𝑚=𝑚
Authors: Halpern J. D. , Howard Paul E.
Published: June 2010

In this paper we compare characterizations of cardinals

m
m

satisfying

2
⋅
m
=
m

2 \cdot m = m

with certain characterizations of Dedekind infinite cadinals. It is also shown that a strengthening of

∀
m

\forall m

(

m
m

an infinite cardinal

⇒
2
⋅
m
=
m

\Rightarrow 2 \cdot m = m

) implies the axiom of choice.
Index-dependent parameters of Laguerre and related polynomial sets.
Authors: Brown J. W. , Goldberg J. L.
Published: June 2010

It is known that linearity of the function

γ
(
n
)

\gamma (n)

is sufficient for the set

{

L
n

(
γ
(
n
)
)

(
x
)
}

\{ L_n^{(\gamma (n))}(x)\}

of generalized Laguerre polynomials to be of type zero as defined by I. M. Sheffer. We prove here that linearity is also necessary. This result is exhibited as a special case in the broader context of generalized Appell representations introduced by R. P. Boas, Jr. and R. C. Buck.
Determining the cellularity of a 𝑖-complex by properties of its arcs.
Authors: Sher R. B.
Published: June 2010

We show that a

1
1

-complex

K
K

topologically embedded in the interior of a topological

n
n

-manifold

M
,
n
≥
3

M,n \geq 3

, satisfies the cellularity criterion if for each arc

A
A

in

K
,
M
−
A

K,M - A

is

1
1

-LC at an endpoint of

A
A

. This condition is satisfied if each arc in

K
K

is LPU at an endpoint. An example is given to show that it is not sufficient to suppose that each arc in

K
K

satisfies the cellularity criterion.
A note on stable homotopy modules
Authors: Kahn Donald W.
Published: June 2010

We establish under very general conditions that a space may be changed by adding one cell, so that the stable homotopy module becomes infinitely generated over the stable homotopy ring. We give an application to the study of ideals in the stable homotopy ring.
An odd theorem
Authors: Eaves B. Curtis
Published: June 2010

Let

C
C

be a bounded convex polyhedral set and let

f
:
C
→
C

f:C \to C

be continuous and piecewise linear. Using notions from complementary pivot theory, it is shown that if each fixed point of

f
f

lies interior to some piece of linearity, then

f
f

has an odd number of fixed points. In addition, an algorithm is given for computing a fixed point of

f
f

.
Tree-like continua and cellularity
Authors: Summerhill R. Richard
Published: June 2010

In this paper the equivalence of tree-like and cellular is proved for

1
1

-dimensional continua in

E
n

{E^n}

. More precisely, if

X
X

is a tree-like continuum, then the collection of all embeddings

h
:
X
→

E
n

,
n
≧
3

h:X \to {E^n},n \geqq 3

, such that

h
[
X
]

h[X]

is cellular in

E
n

{E^n}

is a dense

G
δ

{G_\delta }

-subset of the collection of all maps from

X
X

into

E
n

{E^n}

. Conversely, if

X
X

is a

1
1

-dimensional cellular subset of

E
n

{E^n}

, then

X
X

is a tree-like continuum.
The Garden-of-Eden theorem for finite configurations
Authors: Amoroso S. , Cooper G.
Published: June 2010

In [1] Moore showed that the existence of mutually erasable configurations in a two-dimensional tessellation space is sufficient for the existence of Garden-of-Eden configurations. In [2] Myhill showed that the existence of mutually indistinguishable configurations is both necessary and sufficient for the existence of Garden-of-Eden configurations. After redefining the basic concepts with some minor changes in terminology, and after restating the main results from [l] and [2], we shall establish the equivalence between the existence of mutually erasable configurations and the existence of mutually indistinguishable configurations. This implies that the converse of Moore’s result is true as well. We then show that by limiting the universe to the set of all finite configurations of the tessellation array, both of the above conditions remain sufficient, but neither is then necessary. Finally, we establish a necessary and sufficient condition for the existence of Garden-of-Eden configurations when only finite configurations are considered.
A sufficient condition for nonabelianness of fundamental groups of differentiable manifolds
Authors: Chen Kuo-tsai
Published: June 2010

In this paper we prove that, if

H
γ

(
X
)

{H^\gamma }(X)

denotes the

r
r

th deRham cohomology group of a connected manifold

X
X

and if the cup product
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 1 Baseline left-parenthesis upper X right-parenthesis logical-and Subscript upper R Baseline upper H Superscript 1 Baseline left-parenthesis upper X right-parenthesis right-arrow upper H squared left-parenthesis upper X right-parenthesis">

H
1

(
X
)

∧
R

H
1

(
X
)
→

H
2

(
X
)

{H^1}(X){ \wedge _R}{H^1}(X) \to {H^2}(X)

is not injective, then

π
1

(
X
)

{\pi _1}(X)

is not abelian. As a corollary, if

b
r

{b_r}

is the

r
r

th Betti number, then

1
2

b
1

(

b
1

−
1
)
>

b
2

\frac {1}{2}{b_1}({b_1} - 1) > {b_2}

implies

π
1

(
X
)

{\pi _1}(X)

being nonabelian.
On a generalisation of Hermite polynomials
Authors: Lahiri Maya
Published: June 2010

In this paper, the author introduces a generalisation of the Hermite polynomials. Hypergeometric representations, a new generating relation and

n
n

th order differential formulae for the generalised polynomials have also been derived therein.
Semiadjoint functors and quintuples
Authors: Davis Robert
Published: June 2010

We obtain a generalization of adjoint functors and triples by axiomatizing the behavior of Albanese varieties, obtain a few basic properties, and show that abelian varieties form a generalized type of triplable category over complete abstract varieties.
Addendum to “On the Frattini subgroup”
Authors: Cossey John , Whittemore Alice
Published: June 2010

Let

F
F

be a free group,

R
R

a normal subgroup of

F
F

and

V
V

a fully invariant subgroup of

R
R

. In a recent paper the authors calculated the Frattini subgroup of

F

/

V

F/V

under suitable conditions on

R
R

and

V
V

. This paper presents information on the Frattini subgroup of subgroups of

F

/

V

F/V

under the same conditions.
Counterexamples on the rank of a manifold
Authors: Bredon Glen E.
Published: June 2010

For any given integer

r
r

a closed manifold is constructed which has a smooth free action of the

r
r

-torus, and hence has rank at least

r
r

, but for which -1 is not a multiple root of the Poincaré polynomial.
Overrings of principal ideal domains
Authors: Brungs H.-H.
Published: June 2010

All rings between a (right and left) principal ideal domain

R
R

and its skewfield

Q
(
R
)

Q(R)

of quotients are quotient rings of

R
R

with respect to Ore-systems in

R
R

.
Multiplicative linear functionals on convolution algebras
Authors: Aizley Paul
Published: June 2010

It is shown that semicharacters on the semigroup

S
S

lead in a natural way to multiplicative linear functionals on

l
(
S
)

l(S)

, the convolution algebra of all complex valued functions on

S
S

. A theorem of D. H. Lehmer and a theorem of M. Tainiter follow as special cases.
The Jacobi sums of order twenty-two
Authors: Zee Yun-cheng
Published: June 2010

This paper completes the analysis of the Jacobi sums of order 22 outlined by L. E. Dickson. Furthermore, it is shown that a prime

p
p

of the form

22
f
+
1

22f + 1

has the binary quadratic decomposition

4
p
=

u
2

+
11

v
2

4p = {u^2} + 11{v^2}

and that certain Jacobi sums can be evaluated in terms of

u
u

and

v
v

, where

u
u

satisfies

u
≡
9

(
mod

1
)
1

u \equiv 9\pmod 11

.
Two remarks about hereditary orders
Authors: Jacobinski H.
Published: June 2010

In the first remark it is shown that, over a Dedekind ring, hereditary orders in a separable algebra are precisely the “maximal” orders under a relation stronger than inclusion (Theorem 1). At the same time simple proofs for known structure theorems of hereditary orders are obtained. In the second remark a complete classification is given of lattices over a hereditary order, provided the underlying Dedekind ring is contained in an algebraic number field and the lattices satisfy the Eichler condition (Theorem 2).
Topological invariant means on locally compact groups and fixed points.
Authors: Wong James C. S.
Published: June 2010

A locally compact group

G
G

is said to have the fixed point property if whenever

G
G

acts affinely on a compact convex subset

S
S

of a separated locally convex space

E
E

with the map

G
×
S
→
S

G \times S \to S

jointly continuous, there is a fixed point for the action. N. Rickert has proved that

G
G

has this fixed point property if

G
G

is amenable. In this paper, we study the fixed point property for actions of the algebras

L
1

(
G
)

{L_1}(G)

and

M
(
G
)

M(G)

and prove some fixed point theorems for locally compact groups.
Zero sets of functions from non-quasi-analytic classes
Authors: Hughes Robert B.
Published: June 2010

It is well known that any closed subset of the line is the zero set of a

C
∞

{C^\infty }

-function. One can also specify the orders of the zeros at the isolated points. The present paper improves this result by replacing the class of

C
∞

{C^\infty }

-functions by any non-quasi-analytic class of

C
∞

{C^\infty }

-functions.
Products of 𝔪-compact spaces
Authors: Saks Victor , Stephenson R. M.
Published: June 2010

Some results are given on the closure under suitably restricted products of a class of spaces similar to one considered by Z. Frolík and, more recently, by N. Noble. An answer is given to the following question of Gulden, Fleischman, and Weston: Does there exist

M

>

ℵ
0

\mathfrak {M} > {\aleph _0}

and an

M

\mathfrak {M}

-compact space

X
X

such that some subset

A
A

of

X
X

of cardinality

≦

M

\leqq \mathfrak {M}

is contained in no compact subset of

X
X

? It is shown that for every

M

≧

ℵ
0

\mathfrak {M} \geqq {\aleph _0}

there is a topological group which has this property.
A sufficient condition for an arc to be nearly polyhedral
Authors: McPherson James M.
Published: June 2010

We give a simple geometric condition for an arc to be wild at an endpoint, with penetration index at least three.
A fixed point theorem for manifolds
Authors: Jaworowski Jan W.
Published: June 2010

A Lefschetz type fixed point theorem is proved extending a recent theorem by Robert F. Brown. It deals with compact maps of the form

f
:
(
M
−
U
,
X
)
→
(
M
,
M
−
U
)

f:(M - U,X) \to (M,M - U)

, where

M
M

is an

n
n

-manifold,

X
X

is an

(
n
−
2
)

(n - 2)

-connected ANR which is closed in

M
M

and

U
U

is an unbounded component of

M
−
U

M - U

. The map

f
f

defines maps

u
:
M
−
U
→
M
−
U

u:M - U \to M - U

and

v
:
M
→
M

v:M \to M

; the Lefschetz numbers of

u
u

and

v
v

are defined and are shown to be equal; and if this number is nonzero then

f
f

has a fixed point.
Fields with few extensions
Authors: Knopfmacher J. , Sinclair A. M.
Published: June 2010

We show that a valued field

Λ
\Lambda

with only a finite number of nonisomorphic valued extensions is equal to the complex field

C

\mathbfit {C}

or is real closed with

C
=
Λ
(
√
(
−
1
)
)

C = \Lambda (\surd ( - 1))

.
The maximal ideals in quaternion orders
Authors: Riley John A.
Published: June 2010

Let

R
R

be a Noetherian, integrally closed local domain, and

Λ
\Lambda

an

R
R

-order in a generalized quaternion algebra over the quotient field of

R
R

. In this note, it is proved that: (a)

Λ
\Lambda

has at most two maximal ideals; and (b) in case

Λ
\Lambda

does have exactly two maximal ideals, the corresponding residue class rings are commutative fields.
Decompositions of Abelian 𝑝-groups
Authors: Stringall R. W.
Published: June 2010

Using some elementary properties of endomorphism rings and their radical ideals, an equivalence between the category of

p
p

-rings and the category of Boolean rings and some examples introduced by the author, it is shown that for every countable atomic Boolean algebra there is a

p
p

-group without elements of infinite height, standard basic subgroup and no proper isomorphic subgroups which contains a maximal lattice of summands isomorphic to the given Boolean algebra. Moreover, it is established that this lattice is representative in the sense that it determines, up to isomorphism, all the summands of the group.
On roots and subsemigroups of nilpotent groups
Authors: Kuczkowski Joseph E.
Published: June 2010

E
ω

,

U
ω

{E_\omega },{U_\omega }

and

D
ω

{D_\omega }

semigroups are defined by extrapolating the definitions of their group counterparts; and a class

n
n

semigroup is defined to be a subsemigroup of a class

n
n

group. The purpose of this paper is to show that a class

n

E
ω

n{E_\omega }

semigroup generates an

E
ω

{E_\omega }

group and that a class

n
n

semigroup is

U
ω

{U_\omega }

if and only if it generates a

U
ω

{U_\omega }

group.
On Burnside’s lemma
Authors: Herzog Marcel
Published: June 2010

Burnside’s lemma on characters of finite groups is generalized, leading to the following theorem: if

G
G

is a simple group of order divisible by exactly 3 primes, and if one of the Sylow subgroups of

G
G

is cyclic, then for each Sylow subgroup

P
P

of

G
G

we have

C
G

(
P
)
=
Z
(
P
)

{C_G}(P) = Z(P)

.
An inequality for complex linear groups of small degree
Authors: Blau Harvey I.
Published: June 2010

Let

G
G

be a finite complex irreducible linear group of degree less than

p
−
1

p-1

for some fixed prime

p
p

, whose order is divisible by

p
p

to the first power only, and which has no normal Sylow

p
p

-subgroup. An inequality of Brauer, which bounds

p
p

by a function of the number of conjugate classes of

p
p

-elements, is improved.
Identities of group algebras
Authors: Milnes Paul
Published: June 2010

In this note the converse to the following well-known proposition about locally compact topological groups G is proved: if G is discrete, then the group

C
∗

{C^ \ast }

-algebra

C
∗

(
G
)

{C^\ast }(G)

has an identity.
Abelian differentials with double zeros
Authors: Farkas Hershel M.
Published: June 2010

In this note we give a simple proof of the following theorem: The locus of points in the Torelli space of compact Riemann surfaces of genus

g
≧
2

g \geqq 2

whose underlying surfaces do not permit a basis for the abelian differentials of first kind each of whose elements is a differential with double zeros, has positive codimension in the Torelli space.
On an asymptotic property of a Volterra integral equation
Authors: Izé A. F.
Published: June 2010

It is proved that if

q
(
t
−
s
)

q(t - s)

is bounded and

f
(
t
,
x
)

f(t,x)

is “small,” the solutions of the integral equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x left-parenthesis t right-parenthesis equals a left-parenthesis t right-parenthesis plus integral Subscript 0 Superscript t Baseline q left-parenthesis t minus s right-parenthesis f left-parenthesis s comma x left-parenthesis s right-parenthesis right-parenthesis d s">

x
(
t
)
=
a
(
t
)
+

∫
0
t

q
(
t
−
s
)
f
(
s
,
x
(
s
)
)
d
s

x(t) = a(t) + \int _0^t {q(t - s)f(s,x(s))ds}

satisfies the conditions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x left-parenthesis t right-parenthesis equals h left-parenthesis t right-parenthesis plus rho left-parenthesis t right-parenthesis a left-parenthesis t right-parenthesis comma limit Underscript t right-arrow normal infinity Endscripts a left-parenthesis t right-parenthesis equals a">

x
(
t
)
=
h
(
t
)
+
ρ
(
t
)
a
(
t
)
,

lim

t
→
∞

a
(
t
)
=
a

x(t) = h(t) + \rho (t)a(t),{\lim _{t \to \infty }}a(t) = a

constant where

ρ
(
t
)

\rho (t)

is a nonsingular diagonal matrix chosen in such a way that

ρ
(
t
)
h
(
t
)

\rho (t)h(t)

is bounded. The results are extended to the more general integral equation

x
(
t
)
=
h
(
t
)
+

∫
0
t

F
(
t
,
s
,
x
(
s
)
)
d
s

x(t) = h(t) + \int _0^t {F(t,s,x(s))ds}

and contain, in particular, some results on the boundedness, asymptotic behavior and existence of nonoscillatory solution of differential equations.
Short proof of Sobczyk’s theorem
Authors: Veech William A.
Published: June 2010

A new proof is given of Sobczyk’s Theorem, which asserts that

c
0

{c_0}

is complemented in any separable Banach space which contains it (isomorphically) as a closed subspace.
Extending free circle actions on spheres to 𝑆³ actions
Authors: Conrad Bruce
Published: June 2010

Let

X
X

be a PL homotopy

C

P

2
k
+
1

C{P^{2k + 1}}

corresponding by Sullivan’s classification to the element

(

N
1

,

α
2

,

N
2

,
⋯
,

α
k

,

N
k

)

({N_1},{\alpha _2},{N_2}, \cdots ,{\alpha _k},{N_k})

of

Z
⊕

Z
2

⊕
Z
⊕
⋯
⊕

Z
2

⊕
Z

Z \oplus {Z_2} \oplus Z \oplus \cdots \oplus {Z_2} \oplus Z

. Theorem 1. The topological circle action on

S

4
k
+
3

{S^{4k + 3}}

with orbit space

X
X

is the restriction of an

S
3

{S^3}

action with a triangulable orbit space iff

α
i

=
0
,
i
=
2
,
⋯
,
k

{\alpha _i} = 0,i = 2, \cdots ,k

; and

N
1

≡
0
mod
2

{N_1} \equiv 0\bmod 2

; and

∑

(
−
1

)
i

N
i

=
0

\sum {( - 1)^i}{N_i} = 0

. If

X
X

admits a smooth structure and satisfies the hypotheses of Theorem 1, a certain smoothing obstruction arising from the integrality theorems vanishes for the corresponding

S
3

{S^3}

action.
Separation of roots and oscillation in ordinary linear differential equations of second order
Authors: Breuer Shlomo , Gottlieb David
Published: June 2010

New criteria for the oscillatory or nonoscillatory behavior of ordinary, linear differential equations of second order and the location of their roots are obtained with the aid of new types of Sturmian majorant comparison equations. Several known results are obtained as special cases under less stringent conditions.
The selfadjoint operators of a von Neumann algebra form a conditionally complete lattice
Authors: Olson Milton Philip
Published: June 2010

The bounded resolutions of the identity in a von Neumann algebra can be ordered by

{

E
S

(
u
)
}
≼
{

E
T

(
u
)
}

\{ {E_S}(u)\} \preccurlyeq \{ {E_T}(u)\}

if

E
S

(
u
)
≧

E
T

(
u
)
,
u
∈
R

{E_S}(u) \geqq {E_T}(u),u \in R

. The selfadjoint operators in the algebra are partially ordered by this relation and are shown to form a conditionally complete lattice. The lattice operations are (essentially) defined by

E

V

S
α

(
u
)
=
∧

E

S
α

(
u
)

{E_{V{S_\alpha }}}(u) = \wedge {E_{{S_\alpha }}}(u)

for all

u
u

contained in

R
R

. This order is called spectral order and agrees with the usual order on commutative subalgebras. For positive operators,

S
S

is greater than or equal to

T
T

in spectral order if and only if

S
n

{S^n}

is greater than or equal to

T
n

{T^n}

in the usual order for all

n
≧
1

n \geqq 1

. Kadison’s well-known counterexample is shown to fail. The operator lattice defined by spectral order differs from a vector lattice in the fact that

S
≽
T

S \succcurlyeq T

does not imply that

S
+
C
≽
T
+
C

S + C \succcurlyeq T + C

.
Commuting operator solutions of algebraic equations
Authors: Riddell R. C. , Insley R. B.
Published: June 2010

Let

G
(
w
,
z
)

G(w,z)

be a complex polynomial, and

S
S

a bounded operator of scalar type on a complex Banach space, whose spectrum avoids the points

λ
\lambda

for which

G
(
λ
,
z
)
=
0

G(\lambda ,z) = 0

has multiple roots

z
z

. The form of a bounded operator

T
T

which commutes with

S
S

and satisfies

G
(
S
,
T
)
=
0

G(S,T) = 0

is established.
Unique factorization monoids and domains
Authors: Johnson R. E.
Published: June 2010

It is the purpose of this paper to construct unique factorization (uf) monoids and domains. The principal results are: (1) The free product of a well-ordered set of monoids is a uf-monoid iff every monoid in the set is a uf-monoid. (2) If

M
M

is an ordered monoid and

F
F

is a field, the ring

F
[
[
M
]
]

F[[M]]

of all formal power series with well-ordered support is a uf-domain iff

M
M

is naturally ordered (i.e., whenever

b
>
a

b > a

and

a
M

⋂
b

M
≠
∅

aM{ \bigcap ^b}M \ne \emptyset

, then

a
M
⊂
b
M
)

aM \subset bM)

.
Quotients in Noetherian lattice modules
Authors: Johnson J. A.
Published: June 2010

In this paper we obtain a generalization of the fact that if

M
M

is a maximal (proper) ideal of a Noetherian ring

R
R

, then the ring

M

/

M
A

M/MA

is a vector space over

R

/

M

R/M

for all ideals

A
A

of the ring

R
R

.
Functions which are Fourier-Stieltjes transforms
Authors: Friedberg Stephen H.
Published: June 2010

Let

G
G

be a locally compact abelian group,

G
^

\hat G

the dual group,

M
(
G
)

M(G)

the algebra of regular bounded Borel measures on

G
G

, and

M

(
G

)
∧

M{(G)^\wedge }

the algebra of Fourier-Stieltjes transforms. The purpose of this paper is to characterize those continuous functions on

G
^

\hat G

which belongs to

M
(
X

)
∧

M(X)^\wedge

, where

X
X

is a closed subset of

G
G

and

M
(
X
)
=
{
μ
∈
M
(
G
)

M(X) = \{ \mu \in M(G)

: the support of

μ
\mu

is contained in

X
}

X\}

.
𝑀₀(𝐺) is not a prime 𝐿-ideal of measures
Authors: Graham Colin C.
Published: June 2010

A technique of Hewitt and Zuckerman is used to show that if

G
G

is any locally compact abelian group with dual

Γ
\Gamma

, then there exist nonzero positive regular Borel measures

μ
,
v

\mu ,v

on

G
G

, each one of which is mutually singular with each measure

ω
\omega

whose Fourier-Stieltjes transform vanishes at infinity on

Γ
\Gamma

and such that the Fourier-Stieltjes transform of the convolution

μ
∗
v

\mu \ast v

vanishes at infinity on

Γ
\Gamma

.
On a problem of S. Ulam
Authors: Fournier Gilles
Published: June 2010

Giving a negative answer to a problem of S. Ulam, the author provides an example of two noncomplete subspaces of R which are not isometric but whose squares are isometric.
A zero-one law for Gaussian processes
Authors: Jain Naresh C.
Published: June 2010

Let

P
0

{P_0}

be a Gaussian probability measure on the measurable space

(
X
,
B
(
X
)
)

(X,B(X))

, where X is a linear space of realvalued functions over a complete separable metric space T, and

B
(
X
)

B(X)

is the

σ
\sigma

-algebra generated by sets of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet x element-of upper X colon left-parenthesis x left-parenthesis t 1 right-parenthesis comma midline-horizontal-ellipsis comma x left-parenthesis t Subscript n Baseline right-parenthesis right-parenthesis element-of upper B Superscript n Baseline EndSet semicolon upper B Superscript n">

{
x
∈
X
:
(
x
(

t
1

)
,
⋯
,
x
(

t
n

)
)
∈

B
n

}
;

B
n

\{ x \in X:(x({t_1}), \cdots ,x({t_n})) \in {B^n}\} ;{B^n}

being the Borel sets of

R
n

,
n
≧
1

{R^n},n \geqq 1

. The covariance

R
(
s
,
t
)

R(s,t)

is assumed continuous on

T
×
T

T \times T

. If G is a subgroup of X and belongs to the

σ
\sigma

-algebra

B
0

(
X
)

{B_0}(X)

(the completion of

B
(
X
)

B(X)

with respect to

P
0

{P_0}

), then it is shown that

P
0

(
G
)
=
0

{P_0}(G) = 0

or 1.
Some consequences of 𝑑𝑖𝑚𝑝𝑟𝑜𝑗Ω(𝐴)<∞
Authors: Knighten Carol M.
Published: June 2010

Let

X
X

be an affine variety over a field

k
k

and

x
x

a point on

X
X

. We are interested in relating the properties of

Ω

(
X

)
x

\Omega {(X)_x}

, the Kähler module of differentials of

x
x

, with geometric properties of

X
X

at

x
x

. Lipman has given necessary and sufficient conditions for

Ω

(
X

)
x

\Omega {(X)_x}

to be respectively torsion free and reflexive in the case where

X
X

is locally a complete intersection at

x
x

. We give a generalization of these results for the case where the projective dimension (dim proj) of

Ω

(
X

)
x

\Omega {(X)_x}

is finite.
A free rank 4 Hopf algebra with antipode of order 4
Authors: Radford David E.
Published: June 2010

Let k be an arbitrary commutative ring. In this paper we construct a free rank 4 Hopf algebra H over k with antipode of order 4 and also discuss its symmetry. H is not semisimple.
The word and conjugacy problems for the knot group of any tame, prime, alternating knot
Authors: Weinbaum C. M.
Published: June 2010

The decision problems in the title are solved using the solutions of the word and conjugacy problems obtained by Lyndon and Schupp, respectively, for certain classes of groups.
Perfect matrix methods
Authors: Fleming D. J. , Jessup P. G.
Published: June 2010

Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Subscript i Baseline equals left-parenthesis delta Subscript i j Baseline right-parenthesis Subscript j equals 1 Superscript normal infinity Baseline comma normal upper Delta equals left-parenthesis e Subscript i Baseline right-parenthesis Subscript i equals 1 Superscript normal infinity">

e
i

=
(

δ

i
j

)

j
=
1

∞

,
Δ
=
(

e
i

)

i
=
1

∞

{e_i} = ({\delta _{ij}})_{j = 1}^\infty ,\Delta = ({e_i})_{i = 1}^\infty

and let A be an infinite matrix which maps E into E where E is an FK-space with Schauder basis

Δ
\Delta

. Necessary and sufficient conditions in terms of the matrix A are obtained for E to be dense in the summability space

E
A

=
{
x

|

A
x
∈
E
}

{E_A} = \{ x | {Ax \in E\} }

and conditions are obtained to guarantee that

E
A

{E_A}

has Schauder basis

Δ
\Delta

. Finally it is shown that if weak and strong sequential convergence coincide in E then in

E
A

{E_A}

the series

∑

k

x
k

e
k

\sum {_k{x_k}{e_k}}

converges to x strongly if and only if it converges to x weakly.
Central functions in group algebras
Authors: Mosak Richard D.
Published: June 2010

In this paper we show that for a locally compact group G, the group algebra

L
1

(
G
)

{L_1}(G)

has nontrivial center if and only if G possesses a compact neighborhood of 1, invariant under inner automorphisms. Moreover, G has a basis of such neighborhoods at 1 if and only if

L
1

(
G
)

{L_1}(G)

has an approximate identity consisting of functions in the center of

L
1

{L_1}

. This constitutes part of a program of finding conditions on the group algebra which characterize groups satisfying various compactness conditions (see e.g., [3]).
Energy-finite solutions of Δ𝑢=𝑃𝑢 and Dirichlet mappings
Authors: Glasner Moses
Published: June 2010

Let R, S be noncompact Riemannian m-manifolds and let

T
:
R
→
S

T:R \to S

be a Dirichlet mapping. Consider a nonnegative locally bounded measurable m-form P on R and set

Q
=

T
∗

−
1

P

Q = T_\ast ^{ - 1}P

, the pull-back of P under

T

−
1

{T^{ - 1}}

. Denote by

P
E
(
R
)

(
Q
E
(
S
)

PE(R)\;(QE(S)

resp.) the space of energy-finite solutions of

Δ
u
=
P
u

\Delta u = Pu

on R (

Δ
u
=
Q
u

\Delta u = Qu

on S resp.). The spaces

P
E
(
R
)

PE(R)

and

Q
E
(
S
)

QE(S)

are isomorphic, the isomorphism being bicontinuous with respect to the energy norms and preserves the sup norm of bounded solutions.
On a sequence of almost deterministic pairwise independent random variables
Authors: Joffe A.
Published: June 2010

We show the existence of infinite sequences of pairwise independent random variables (which are not constant almost everywhere) such that the whole sequence is determined with probability one by the knowledge of any pair of members of that sequence.
Products of normal supersolvable subgroups
Authors: Friesen D. K.
Published: June 2010

It is shown in this paper that if G is a finite group which is the product of two normal supersolvable subgroups of relatively prime index, then G is supersolvable.
Construction of quasigroups using the singular direct product
Authors: Lindner Charles C.
Published: June 2010

The idea of a discrete

w
(
x
,
y
)
=
v
(
x
,
y
)

w(x,y) = v(x,y)

quasigroup is given along with a generalization of A. Sade’s singular direct product. These notions are then used to construct certain types of quasigroups. In particular an algebraic generalization of E. H. Moore’s construction of Steiner triple systems is obtained.
A note on some operator theory in certain semi-inner-product spaces.
Authors: Koehler D. O.
Published: June 2010

Let X be a smooth uniformly convex Banach space and let

[
⋅
,
⋅
]

[\cdot ,\cdot ]

be the unique semi-inner-product generating the norm of X. If A is a bounded linear operator on X,

A
†

{A^\dagger }

mapping X to X is called the generalized adjoint of A if and only if

[
A
(
x
)
,
y
]
=
[
x
,

A
†

(
y
)
]

[A(x),y] = [x,{A^\dagger }(y)]

for all x and y in X. In this setting adjoint abelian (iso abelian) operators [5] are characterized as those operators A for which

A
†

=
A
(

A
†

=

A

−
1

{A^\dagger } = A({A^\dagger } = {A^{ - 1}}

, i.e. the invertible isometries). It is also shown that the compression spectrum of an operator is contained in its numerical range.
Concerning diagonal similarity of irreducible matrices
Authors: Hartfiel D. J.
Published: June 2010

If

A
=
(

a

i
j

)

A = ({a_{ij}})

is an

n
×
n

n \times n

irreducible matrix, then there are positive numbers

d
1

,

d
2

,
⋯

{d_1},{d_2}, \cdots

,

d
n

{d_n}

so that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript k Endscripts d Subscript i Baseline a Subscript i k Baseline d Subscript k Superscript negative 1 Baseline equals sigma-summation Underscript k Endscripts d Subscript k Baseline a Subscript k i Baseline d Subscript i Superscript negative 1">

∑
k

d
i

a

i
k

d
k

−
1

=

∑
k

d
k

a

k
i

d
i

−
1

\sum \nolimits _k {{d_i}{a_{ik}}d_k^{ - 1} = } \sum \nolimits _k {{d_k}{a_{ki}}d_i^{ - 1}}

for each

i
∈
{
1
,
2
,
⋯
,
n
}

i \in \{ 1,2, \cdots ,n\}

. Further, the numbers

d
1

,

d
2

,
⋯

{d_1},{d_2}, \cdots

,

d
n

{d_n}

are unique up to scalar multiples.
On the automorphism group of a finite 𝑝-group with the central quotient metacyclic
Authors: Davitt Richard M. , Otto Albert D.
Published: June 2010

In this paper it is shown that if G is a finite p-group

(
p
>
2
)

(p > 2)

such that the central quotient group of G is metacyclic and nontrivial, then the order of G divides the order of the automorphism group of G.
Polynomial extremal problems in 𝐿^{𝑝}
Authors: Beller E.
Published: June 2010

For

p
>
2

p > 2

, let

m

p
,
n

{m_{p,n}}

be the minimum of the

L
p

{L^p}

norm all nth degree polynomials

∑
n

a
k

e

i
k
t

{\sum ^n} {a_k} {e^{ikt}}

which satisfy

|

a
k

|

=
1
,
k
=
0
,
1
,
⋯
,
n

|{a_k}| = 1, k = 0, 1, \cdots , n

. We exhibit certain polynomials

P
n

{P_n}

whose

L
p

{L^p}

norm

(
2
>
p
>
∞
)

( 2 > p > \infty )

is asymptotic to

√

n

\surd {n}

, thereby proving that

m

p
,
n

{m_{p,n}}

is itself asymptotic to

√

n

\surd {n}

. We also show that the sup norm of (essentially) the same polynomials is asymptotic to

(
1.1716
…
)
×
√

n

(1.1716 \ldots ) \times \surd {n}

.
On vector Lyapunov functions
Authors: Deo S. G.
Published: June 2010

It has been proved that the use of a vector Lyapunov function is more advantageous in certain situations rather than a scalar function. Moreover, each function needs to satisfy less rigid requirements. In this paper a new situation has been considered where vector Lyapunov functions play a further useful role. For this purpose, a new type of stability, namely, strict partial stability has been defined. The principal tool employed is the second method of Lyapunov and a comparision theorem of a more general type.
Galois extensions and the ramification sequence of some wildly ramified 𝜋-adic fields
Authors: Davis R. D. , Wishart E. F.
Published: June 2010

In this paper the authors have found necessary and sufficient conditions for a pth degree Eisenstein extension of an arbitrary

π
\pi

-adic field to be normal. In addition we have found where the Galois automorphisms must appear in the ramification sequence of the extended ring.
Torsion-free subgroups of triangle groups
Authors: Feuer R. D.
Published: June 2010

Certain torsion-free subgroups of various triangle groups are considered, the proof of their existence, and in some cases their calculation outlined. There is one section which treats certain specific triangle groups, and one which treats the general case.
Idempotents in group rings
Authors: Passman D. S.
Published: June 2010

In this note we offer an elementary entirely self-contained proof of a theorem of Kaplansky on idempotents in complex group rings.