bluuebunny/crossref_metadata_2025 huggingface dataset:
Shifts in metabolic scaling, production, and efficiency across major evolutionary transitions of life
Authors: DeLong John P. , Okie Jordan G. , Moses Melanie E. , Sibly Richard M. , Brown James H.
Published: June 2010

The diversification of life involved enormous increases in size and complexity. The evolutionary transitions from prokaryotes to unicellular eukaryotes to metazoans were accompanied by major innovations in metabolic design. Here we show that the scalings of metabolic rate, population growth rate, and production efficiency with body size have changed across the evolutionary transitions. Metabolic rate scales with body mass superlinearly in prokaryotes, linearly in protists, and sublinearly in metazoans, so Kleiber’s 3/4 power scaling law does not apply universally across organisms. The scaling of maximum population growth rate shifts from positive in prokaryotes to negative in protists and metazoans, and the efficiency of production declines across these groups. Major changes in metabolic processes during the early evolution of life overcame existing constraints, exploited new opportunities, and imposed new constraints.
Expansive homeomorphisms on compact manifolds
Authors: O’Brien Thomas
Published: June 2010

In this paper theorems are proved which provide for lifting and projecting expansive homeomorphisms through pseudo-covering mappings so that the lift or projection is also an expansive homeomorphism. Using these techniques it is shown that the compact orientable surface of genus

2
2

admits an expansive homeomorphism.
The normal completions of certain partially ordered vector spaces
Authors: Waterman Alan G.
Published: June 2010

It is shown that the normal completions of certain partially ordered vector spaces are the same as certain other normal completions determined by Dilworth.
Strictly convex spaces via semi-inner-product space orthogonality
Authors: Torrance Ellen
Published: June 2010

Let

(
X
,

|

|

⋅

|

|

)

(X,|| \cdot ||)

be a normed space, and let

[
⋅
,
⋅
]

[ \cdot , \cdot ]

be any semi-inner-product on it. We show that

(
X
,

|

|

⋅

|

|

)

(X,|| \cdot ||)

is strictly convex if and only if

|

|

y
+
z

|

|

>

|

|

y

|

|

||y + z|| > ||y||

whenever

[
z
,
y
]
=
0

[z,y] = 0

and

z
≠
0

z \ne 0

, and if and only if

[
A
x
,
x
]
≠
0

[Ax,x] \ne 0

whenever

|

|

I
+
A

|

|

≦
1

||I + A|| \leqq 1

and

A
x
≠
0

Ax \ne 0

. The condition that

[
z
,
y
]
=
0

[z,y] = 0

can be replaced by a stronger or weaker condition.
Chains which are coset spaces of 𝑡𝑙-groups
Authors: Madell Robert L.
Published: June 2010

Let

G
G

be a lattice-ordered group with a topology with which

G
G

is a Hausdorff topological group and topological lattice. Let

N
N

be a closed prime convex

l
l

-subgroup of

G
G

and let

G

/

N

G/N

denote the topological chain of right cosets of

N
N

. It is shown that if

G
G

is locally compact, locally connected, or locally convex then

G

/

N

G/N

is either discrete or has precisely the interval topology.
Some conditions on an operator implying normality. II
Authors: Berberian S. K.
Published: June 2010

Conditions are given under which an operator with countable spectrum is normal, and applications are made to polynomially compact operators.
Reducing the rank of (𝐴-𝜆𝐵)
Authors: Thompson Gerald L. , Weil Roman L.
Published: June 2010

The rank of the

n
×
n

n \times n

matrix

(
A
−
λ
I
)

(A - \lambda I)

is

n
−
J
(
λ
)

n - J(\lambda )

when

λ
\lambda

is an eigenvalue occurring in

J
(
λ
)
≧
0

J(\lambda ) \geqq 0

Jordan blocks of the Jordan normal form of

A
A

. In our principal theorem we derive an analogous expression for the rank of

(
A
−
λ
B
)

(A - \lambda B)

for general,

m
×
n

m \times n

, matrices. When

J
(
λ
)
>
0
,
λ

J(\lambda ) > 0,\lambda

is a rank-reducing number of

(
A
−
λ
I
)

(A - \lambda I)

. We show how the rank-reducing properties of eigenvalues can be extended to

m
×
n

m \times n

matrix expressions

(
A
−
λ
B
)

(A - \lambda B)

. In particular we give a constructive way of deriving a polynomial

P
(
λ
,
A
,
B
)

P(\lambda ,A,B)

whose roots are the only rank-reducing numbers of

(
A
−
λ
B
)

(A - \lambda B)

. We name this polynomial the characteristic polynomial of

A
A

relative to

B
B

and justify that name.
On the cohomology Chern classes of the 𝐾-theory Chern classes
Authors: Smith Mi-soo Bae , Smith Larry
Published: June 2010

Let

ξ
\xi

be a vector bundle over a finite complex and

γ
i

ξ

{\gamma ^i}\xi

its

i
i

th-

K
K

theory Chern class. We first show that
\[

c
n

γ
i

ξ
=
(
i
−
1
)
!
S
(
n
,
i
)

c
n

ξ
+

decomposables

,

{c_n}{\gamma ^i}\xi = (i - 1)!S(n,i){c_n}\xi + {\text {decomposables}},

\]
where

S
(
n
,
i
)

S(n,i)

is a Stirling number of the second kind. We apply this result to show that certain multiples of the

e
e

-invariant of a map

S

2
m
−
1

→

S

2
n

{S^{2m - 1}} \to {S^{2n}}

must always be integral.
A 2-sphere of vetical order 5 bounds a 3-cell
Authors: Loveland L. D.
Published: June 2010

A subset

X
X

of

E
3

{E^3}

is said to have vertical order

n
n

if no vertical line contains more than

n
n

points of

X
X

. We prove that each

2
2

-sphere in

E
3

{E^3}

which has vertical order 5 bounds a

3
3

-cell.
On the operational calculus for groups of operators
Authors: Kantorovitz Shmuel
Published: June 2010

We study some compactness properties of the natural operational calculus for groups of operators.
Bounded in the mean solutions of \triangle𝑢=𝑃𝑢 on Riemannian manifolds
Authors: Chow Kwang-nan , Glasner Moses
Published: June 2010

Let

Φ
\Phi

be a convex positive increasing function and

d
=

lim

t
→
∞

Φ
(
t
)

/

t

d = {\lim _{t \to \infty }}\Phi (t)/t

. A harmonic function

u
u

on a Riemann surface

R
R

is called

Φ
\Phi

-bounded if

Φ
(

|

u

|

)

\Phi (|u|)

is majorized by a harmonic function on R. M. Parreau has shown that if

d
>
∞

d > \infty

(

d
=
∞

d = \infty

, resp.), then every positive (bounded, resp.) harmonic function on

R
R

reduces to a constant if and only if every

Φ
\Phi

-bounded harmonic function does. In this paper analogues of these results are given for the equation

Δ
u
=
P
u
(
P
≧
0
)

\Delta u = Pu(P \geqq 0)

on a Riemannian manifold.
A short proof of a theorem of Jennings
Authors: Formanek Edward
Published: June 2010

We give a short proof of the theorem of Jennings that the augmentation ideal of the group ring of a finitely generated torsion free nilpotent group is residually nilpotent.
Linear differential-difference operators and their adjoints
Authors: Hughes David K.
Published: June 2010

The formal adjoint for a first order matrix differential-difference operator is shown to be a true Hilbert space adjoint, and conditions under which such operators are selfadjoint (in a Hilbert space sense) are derived. Differential-difference operators whose domains are defined by a given initial function cannot be selfadjoint, whereas certain operators whose domains are defined only by conditions at the endpoints of an interval can be selfadjoint.
Ideal preserving automorphisms of postliminary 𝐶*-algebras
Authors: Elliott George A.
Published: June 2010

E. C. Lance has shown that an automorphism of a separable postliminary

C
∗

{C^ \ast }

-algebra which preserves closed two-sided ideals is universally weakly inner. This note extends Lance’s result to the nonseparable case.
Convergence of sequences of distributions
Authors: Dudley R. M.
Published: June 2010

It is shown that in L. Schwartz’s distribution space

D

′

\mathcal {D}’

, there exist sequentially open sets which are not open. The class of all sequentially open sets is not compatible with the vector space structure on

D

′

\mathcal {D}’

.
Explicit characterization of spherical curves
Authors: Breuer Shlomo , Gottlieb David
Published: June 2010

It is shown that the differential equation characterizing a spherical curve can be solved explicitly to express the radius of curvature of the curve in terms of its torsion.
Coefficients of meromorphic schlicht functions
Authors: Duren Peter L.
Published: June 2010

This paper presents an elementary proof of a known theorem on the coefficients of meromorphic schlicht functions: if

f
∈
Σ

f \in \Sigma

and

b
k

=
0

{b_k} = 0

for

1
≦
k
>
n

/

2

1 \leqq k > n/2

, then

|

b
n

|

≦
2

/

(
n
+
1
)

|{b_n}| \leqq 2/(n + 1)

.
Perturbation of complete orthonormal sets and eigenfunction expansions.
Authors: Kazdan Jerry L.
Published: June 2010

A technique is given for determining the asymptotic properties of vectors which are perturbations of a given basis—the eigenfunctions a selfadjoint operator. Its application is illustrated by a differential equation example, not using the Hilbert space norm. An estimate is also given for the codimension of the span of a perturbed set of vectors.
On an integral formula of Gauss-Bonnet-Grotemeyer
Authors: Chen Bang-yen
Published: June 2010

Let

e
(
p
)

e(p)

and

G
(
p
)

G(p)

be the unit outer normal and the Gauss-Kronecker curvature of an oriented closed even-dimensional hypersurface

M
M

of dimension

n
n

in

E

n
+
1

{E^{n + 1}}

. Then for a fixed unit vector

c
c

in

E

n
+
1

{E^{n + 1}}

, we have
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 right-parenthesis StartLayout 1st Row 1st Column integral Subscript upper M Superscript Baseline left-parenthesis c dot e right-parenthesis Superscript m Baseline upper G d upper V equals c Subscript n plus m Baseline chi left-parenthesis upper M right-parenthesis slash c Subscript m comma Baseline 2nd Column a m p semicolon for m equals 0 comma 2 comma 4 comma midline-horizontal-ellipsis comma 2nd Row 1st Column equals 0 comma 2nd Column a m p semicolon for m equals 1 comma 3 comma 5 comma midline-horizontal-ellipsis comma EndLayout">

(
1
)

∫
M

(

c

⋅

e)

m

G
d
V
=

c

n
+
m

χ
(
M
)

/

c

m
,

a
m
p
;

for

m
=
0
,
2
,
4
,
⋯
,

=
0
,

a
m
p
;

for

m
=
1
,
3
,
5
,
⋯
,

(1)\qquad \begin {array}{*{20}{c}} {\int _M^{} {{{({\text {c}}\cdot {\text {e)}}}^m}GdV = {c_{n + m}}\chi (M)/{c_{m,}}} } & {{\text {for}}\;m = 0,2,4, \cdots ,} \\ { = 0,} & {{\text {for}}\;m = 1,3,5, \cdots ,} \\ \end {array}

\]
where

c

⋅

e

{\text {c}} \cdot {\text {e}}

denotes the inner product of

c
c

and

e
e

the area of

m
m

-dimensional unit sphere, and

χ
(
M
)

\chi (M)

the Euler characteristic of

M
M

.
A homological characterization of Steinitz rings
Authors: Lenzing Helmut
Published: June 2010

A ring R (associative with an identity) is called a right Steinitz ring if any linearly independent subset of a free right R-module F can be extended to a basis of F. Steinitz rings have been investigated in a recent paper of Chwe and Neggers. In this note it is shown that the right Steinitz rings are exactly the right perfect, local rings.
A generalization of a theorem of Jacobson
Authors: Montgomery Susan
Published: June 2010

A well-known theorem of Jacobson asserts that a ring

R
R

in which

x

n
(
x
)

=
x

{x^{n(x)}} = x

for each

x
x

in

R
R

must be commutative. This paper gives a description of a ring with involution in which the above condition is imposed only on the symmetric elements. In particular, if

R
R

is primitive,

R
R

is either commutative or the

2
×
2

2 \times 2

matrices over a field, and, in general, any such

R
R

is locally finite and satisfies a polynomial identity of degree 8.
Conversion of the permanent into the determinant
Authors: Gibson P. M.
Published: June 2010

Let

A
A

be an

n
n

-square

(
0
,
1
)

(0,1)

-matrix with positive permanent. It is shown that if the permanent of

A
A

can be converted into a determinant by affixing

±
\pm

signs to the elements of

A
A

then

A
A

has at most

(

n
2

+
3
n
−
2
)

/

2

({n^2} + 3n - 2)/2

positive entries. Corollaries of this result are given.
Quaternion constituents of group algebras
Authors: Benard Mark
Published: June 2010

In this paper it is shown that each quaternion division algebra central over the rationals appears as a division ring constituent of some rational group algebra.
Sums of sets of continued fractions
Authors: Cusick T. W. , Lee R. A.
Published: June 2010

For each integer

k
≧
2

k \geqq 2

, let

S
(
k
)

S(k)

denote the set of real numbers

α
\alpha

such that

0
≦
α
≦

k

−
1

0 \leqq \alpha \leqq {k^{ - 1}}

and

α
\alpha

has a continued fraction containing no partial quotient less than k. It is proved that every number in the interval [0, 1] is representable as a sum of k elements of

S
(
k
)

S(k)

.
A model of Euclidean 2-space
Authors: Krishna Sastry M. S.
Published: June 2010

In this paper a model of Euclidean

2
2

-space, called the spin model, is introduced. To each complex-valued function

f
f

defined in an open subset of the complex plane is associated a function

f
~

\tilde f

mapping an open subset of the spin model space into the two-dimensional real vector space of two-rowed real column matrices. Cauchy’s theorem and Cauchy’s integral formula for an analytic function

f
f

are written as theorems involving the function

f
~

\tilde f

.
Valuation rings with zero divisors
Authors: Kelly Patrick H. , Larsen Max D.
Published: June 2010

Manis has developed a valuation theory for commutative rings which extends valuation theory for fields. However his results do not extend the characterization of valuation rings as domains of maximal partial homomorphisms. In this note we show that Manis’ theory also generalizes this aspect of valuation theory. Next we show that an overring W of a valuation ring V is not necessarily a valuation ring in any nice sense, but that W is contained in a valuation ring which is a large quotient ring of V.
On ℱ-abnormal maximal subgroups of a finite solvable group
Authors: Venzke Paul
Published: June 2010

Let

Δ
(
G
)

\Delta (G)

be the intersection of the nonnormal maximal subgroup of a finite group. W. Gaschütz has shown that

Δ
(
G
)

\Delta (G)

is nilpotent and that

Δ
(
G
)

/

Φ
(
G
)

\Delta (G)/\Phi (G)

is the center of

G

/

Φ
(
G
)

G/\Phi (G)

. This note, by considering the intersection of the

F

\mathfrak {F}

-abnormal maximal subgroups, generalizes these results for a saturated formation

F

\mathfrak {F}

.
Central measures on semisimple Lie groups have essentially compact support
Authors: Ragozin David L. , Rothschild Linda Preiss
Published: June 2010

In this paper it is shown that for a connected semisimple Lie group with no nontrivial compact quotient any finite central measure is a discrete measure concentrated on the center of the group. More generally, the largest possible support set for a central measure on any semisimple Lie group is determined. From these results it follows that the center of the algebra

L
1

(
H
)

{L_1}(H)

is trivial for any locally compact group H which has a noncompact connected simple Lie group as a homomorphic image.
A normal subgroup of a semisimple Lie group is closed
Authors: Ragozin David L.
Published: June 2010

The theorem of the title is proved.
Common fixed point theorems for almost weakly periodic nonexpansive mappings
Authors: Tan Kok Keong
Published: June 2010

The notions of normal structure, (convex) diminishing orbital diameters, regular orbital diameters (r.o.d.) have been generalized into a Hausdorff locally convex space

(
X
,
τ
)

(X,\tau )

whose topology

τ
\tau

is generated by a family

P

\mathcal {P}

of seminorms. Theorem 1. Let

K
⊆
X

K \subseteq X

be nonempty weakly compact convex with normal structure w.r.t.

P

\mathcal {P}

and

F

\mathcal {F}

be a (not necessarily finite nor commuting) family of almost weakly periodic nonexpansive mappings w.r.t.

P

\mathcal {P}

on K. Then

F

\mathcal {F}

has a common fixed point. Theorem 2. Let

K
⊆
X

K \subseteq X

be nonempty weakly compact convex and

F

\mathcal {F}

be a semigroup with identity of almost weakly periodic nonexpansive mappings w.r.t.

P

\mathcal {P}

on K. If

F

\mathcal {F}

has r.o.d. w.r.t.

P

\mathcal {P}

, then

F

\mathcal {F}

has a common fixed point. Corollary. If

K
⊆
X

K \subseteq X

is nonempty weakly compact convex and

F

=
{

f
1

,
⋯
,

f
n

}

\mathcal {F} = \{ {f_1}, \cdots ,{f_n}\}

is a finite commuting family of pointwise periodic nonexpansive mappings w.r.t.

P

\mathcal {P}

on K, then

F

\mathcal {F}

has a common fixed point.
Integrally closed and complete ordered quasigroups and loops
Authors: Hartman Phillip A.
Published: June 2010

We generalize the well-known results on embedding a partially ordered group in its Dedekind extension by showing that, with the appropriate definition of integral closure, any partially ordered quasigroup (loop) G can be embedded in a complete partially ordered quasigroup (loop) if and only if G is integrally closed. If G is directed as well, then its Dedekind extension is a complete lattice-ordered quasigroup (loop). Furthermore, any complete fully ordered quasigroup (loop) has, with one exception, the real numbers with their usual ordering as its underlying set. The quasigroup (loop) operation, however, need not be ordinary addition as it is in the group case. On the other hand, a complete, strongly power associative fully ordered loop is either the integers or the real numbers with ordinary addition.
Nonfactorization in subsets of the measure algebra
Authors: Burnham J. T.
Published: June 2010

In this note we unify and simplify some recent results showing the impossibility of factoring in certain convolution subalgebras of the group algebra of a nondiscrete LCAG. A new result is a direct proof of nonfactorization of the classical Hardy spaces, regarded as convolution algebras, on the circle. By considering the ideal of Hilbert-Schmidt operators in the algebra of compact operators on a Hilbert space we illustrate that nonfactorization is not peculiar to convolution.
On the range of a coercive maximal monotone operator in a nonreflexive Banach space
Authors: Gossez Jean-Pierre
Published: June 2010

It is shown that the range of a coercive everywhere defined maximal monotone operator from a (nonreflexive) Banach space into its dual is dense for the

w
e
a
k

∗

\mathrm {weak}^*

topology but not necessarily for the norm topology.
The convergence almost everywhere of Legendre series
Authors: Pollard Harry
Published: June 2010

It is proved that the Legendre series of an

L
p

{L^p}

function converges almost everywhere, provided

4

/

3
>
p
>
∞

4/3 > p > \infty

. The result fails if

1
≦
p
>
4

/

3

1 \leqq p > 4/3

.
Further comments on the continuity of distribution functions obtained by superposition
Authors: Huff Barthel W.
Published: June 2010

Let

{
X
(
t
)
}

\{ X(t)\}

be a differential process with discontinuous distributions and Y a nonnegative random variable independent of the process. The superposition

X
(
Y
)

X(Y)

has a continuous probability distribution if and only if the process has nonzero trend term and Y has continuous distribution. The nature of discontinuities of the probability distribution of the superposition is indicated.
On the number of solutions of Diophantine equations
Authors: Davis Martin
Published: June 2010

For any nontrivial set of cardinal numbers

≦

ℵ
0

\leqq {\aleph _0}

, it is shown that there is no algorithm for testing whether or not the number of positive integer solutions of a given polynomial Diophantine equation belongs to the set.
Existence of solutions of abstract measure differential equations
Authors: Sharma R. R.
Published: June 2010

In an earlier paper the author has introduced an abstract measure differential equation as a generalization of ordinary differential equations and measure differential equations, and proved a theorem for the existence and uniqueness of solutions of this equation. In the present paper the problem of the existence of solutions is further investigated.
Rings whose quasi-injective modules are injective
Authors: Byrd K. A.
Published: June 2010

A ring R is called a V-ring, respectively SSI-ring, respectively QII-ring if simple, respectively semisimple, respectively quasi-injective, right R-modules are injective. We show that R is SSI if and only if R is a right noetherian V-ring and that any SSI-ring is a finite ring direct sum of simple SSI-rings. We show that if R is left noetherian and SSI then R is QII provided R is hereditary and that in order for R to be hereditary it suffices that maximal right ideals of R be reflexive. An example of Cozzens is cited to show these rings need not be artinian.
Quasi-invariant Radon measures on groups
Authors: Gowrisankaran Chandra
Published: June 2010

Let G be a Hausdorff topological group which is a Baire space. It is proved that if there is a quasi-invariant Radon measure on G then G is locally compact. Examples of non-Baire groups with and without quasi-invariant measures are considered. In particular, it is shown that there is no

σ
\sigma

-finite measure on the Wiener space which preserves sets of measure zero under translation.
On the structure of idempotent semigroups
Authors: Warne R. J.
Published: June 2010

An idempotent semigroup (band) is a semigroup in which every element is an idempotent. We describe the structure of idempotent semigroups in terms of semilattices

Ω
\Omega

, partial chains

Ω
\Omega

of left zero semigroups, and partial chains

Ω
\Omega

of right zero semigroups. We also describe bands of maximal left zero semigroups in terms of partial chains

Ω
\Omega

of left zero semigroups and semilattices

Ω
\Omega

of right zero semigroups.
Triangular matrix algebras over Hensel rings
Authors: Wehlen Joseph A.
Published: June 2010

Let (R, m) be a local Hensel ring and A an algebra over R which is finitely generated and projective as an R-module. If A contains a complete set of mutually orthogonal primitive idempotents

e
1

,
⋯
,

e
n

{e_1}, \cdots ,{e_n}

indexed so that

e
i

N

e
j

⊆
m
A

{e_i}N{e_j} \subseteq mA

whenever

i
≧
j

i \geqq j

, we show that A is isomorphic to a generalized triangular matrix algebra and that A is the epimorphic image of a finitely generated, projective R-algebra B of Hochschild dimension less than or equal to one.
Some interdependencies of sum and product integrals
Authors: Helton Jon C.
Published: June 2010

In a recent paper Davis and Chatfield show if

∫
a
b

G
2

=
0

\smallint _a^b{G^2} = 0

, then

∫
a
b

G

\smallint _a^bG

exists if and only if

∏
a
b

(
1
+
G
)

\prod \nolimits _a^b {(1 + G)}

exists and is not zero. In this paper we extend that result and prove if

β
>
0
,

|

G

|

>
1
−
β

\beta > 0,|G| > 1 - \beta

on

[
a
,
b
]

[a,b]

and

∫
a
b

G
2

\smallint _a^b{G^2}

exists, then

∫
a
b

G

\smallint _a^bG

exists if and only if

∏
a
b

(
1
+
G
)

\prod \nolimits _a^b {(1 + G)}

exists and is not zero. Furthermore, if

∏
x
y

(
1
+
G
)

=
exp
⁡

∫
x
y

G

\prod \nolimits _x^y {(1 + G)} = \exp \smallint _x^yG

for

a
≦
x
>
y
≦
b

a \leqq x > y \leqq b

, then

∫
a
b

G
2

=
0

\smallint _a^b{G^2} = 0

.
Mean growth of inner functions
Authors: Herrero Domingo A.
Published: June 2010

Let A be a closed subset of the closed unit disc. It is shown that there exists a “universal growth function”

φ
(
r
,
A
)

\varphi (r,A)

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 minus integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue q left-parenthesis r e Superscript i x Baseline right-parenthesis EndAbsoluteValue squared d x slash 2 pi equals upper O left-parenthesis phi left-parenthesis r comma upper A right-parenthesis right-parenthesis">

1
−

∫
0

2
π

|

q
(
r

e

i
x

)

|

2

d
x

/

2
π
=
O
(
φ
(
r
,
A
)
)

1 - \smallint _0^{2\pi }|q(r{e^{ix}}){|^2}dx/2\pi = O(\varphi (r,A))

for all inner functions

q
(
z
)

q(z)

whose zeroes lie in

A
∩
{

|

z

|

>
1
}

A \cap \{ |z| > 1\}

and whose singularities in the unit circle lie on

A
∩
{

|

z

|

=
1
}

A \cap \{ |z| = 1\}

, if and only if the Lebesgue measure of

A
∩
{

|

z

|

=
1
}

A \cap \{ |z| = 1\}

is zero.
Adjoints of multipoint-integral boundary value problems
Authors: Brown R. C. , Krall Allan M.
Published: June 2010

The dual system to

L
y
=

y
′

+
P
y

Ly = y’ + Py

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript i equals 0 Overscript normal infinity Endscripts upper A Subscript i Baseline y left-parenthesis t Subscript i Baseline right-parenthesis plus integral Subscript 0 Superscript 1 Baseline upper K left-parenthesis t right-parenthesis y left-parenthesis t right-parenthesis d t equals 0">

∑

i
=
0

∞

A
i

y
(

t
i

)
+

∫
0
1

K
(
t
)
y
(
t
)

d
t
=
0

\sum \limits _{i = 0}^\infty {{A_i}y({t_i}) + \int _0^1 {K(t)y(t)\;dt = 0} }

\]
is found when the setting is

L
n
p

(
0
,
1
)
,
1
>
p
>
∞

L_n^p(0,1),1 > p > \infty

.
A new topology on 𝐵*-algebras arising from the Arens products
Authors: McCharen Edith A.
Published: June 2010

A locally convex topology

μ
\mu

is defined on a Banach algebra A. This topology arises naturally from considerations of the Arens products on the second conjugate space

A

∗
∗

{A^{ \ast \ast }}

of A. The main result states that if A is a

B
∗

{B^\ast }

-algebra on which the mapping

(
a
,
b
)
→
a
b

(a,b) \to ab

is

μ
\mu

-continuous for

‖
a
‖

≦
1

\left \| a \right \| \leqq 1

, then the completion of A with respect to the uniformity generated by

μ
\mu

is linearly isomorphic to

A

∗
∗

{A^{ \ast \ast }}

. An example is included which shows that this continuity condition does not hold in general as announced by P. C. Shields.
On a question of M. Hochster
Authors: Cowsik R. C.
Published: June 2010

It is shown that not all finite integral extensions of a Noetherian integrally closed domain A can be obtained by a process of adjoining integral elements whose monic irreducible equations over the quotient field have all the coefficients in the base ring and passing to A-subalgebras of the algebras so obtained.
On commutativity of endomorphism rings of ideals
Authors: Alamelu S.
Published: June 2010

Let R be a commutative noetherian ring with

Hom
R

(
I
,
I
)

{\operatorname {Hom}_R}(I,I)

commutative for all ideals I. Then the total quotient ring of R is quasi-Frobenius. This settles a conjecture of W. V. Vasconcelos [2].
A simple example of a universal Schwartz space
Authors: Randtke Daniel J.
Published: June 2010

A Schwartz space E is universal if every Schwartz space is topologically isomorphic to a linear subspace of some power

E
I

{E^I}

of E. In this paper concrete examples of universal Schwartz spaces are exhibited.
Countable torsion products of abelian 𝑝-groups
Authors: Lady E. L.
Published: June 2010

Let

A
1

,

A
2

,
⋯

{A_1},{A_2}, \cdots

be direct sums of cyclic p-groups, and

G
=
t

∏
1
∞

A
n

G = t\prod \nolimits _1^\infty {{A_n}}

be the torsion subgroup of the product. The product decomposition is used to define a topology on G in which each

G
[

p
r

]

G[{p^r}]

is complete and the restriction to

G
[

p
r

]

G[{p^r}]

of any endomorphism is continuous. Theorems are then derived dual to those proved by Irwin, Richman and Walker for countable direct sums of torsion complete groups. It is shown that every subgroup of

G
[
p
]

G[p]

supports a pure subgroup of G, and that if

G
=
H
⊕
K

G = H \oplus K

, then

H
≈
t

∏
1
∞

G
n

H \approx t\prod \nolimits _1^\infty {{G_n}}

, where the

G
n

{G_n}

are direct sums of cyclics. A weak isomorphic refinement theorem is proved for decompositions of G as a torsion product. Finally, in answer to a question of Irwin and O’Neill, an example is given of a direct decomposition of G that is not induced by a decomposition of

∏
1
∞

A
n

\prod \nolimits _1^\infty {{A_n}}

.
Nilpotent elements in Banach algebras
Authors: Behncke H.
Published: June 2010

Let

A

\mathfrak {A}

be an

A
∗

{A^\ast }

-algebra such that any maximal abelian

∗

^\ast

-subalgebra is regular and such that any quasinilpotent element x in

A

\mathfrak {A}

satisfies

x
N

=
0

{x^N} = 0

, with

N
>
∞

N > \infty

. Then any irreducible Hilbert space

∗

^\ast

-representation is at most N-dimensional. If

A

\mathfrak {A}

is a

C
∗

{C^\ast }

-algebra,

A

\mathfrak {A}

possesses transcendental quasinilpotent elements if there exists a

π
∈

A

^

\pi \in \hat {\mathfrak {A}}

with

dim
⁡
π
=
∞

\dim \pi = \infty

.
The product of commuting conical projections is a projection
Authors: Zarantonello Eduardo H.
Published: June 2010

The statement in the title is proved for two projections on closed convex cones in Hilbert space.
A note on Strassen’s version of the law of the iterated logarithm
Authors: Basu A. K.
Published: June 2010

Strassen’s law of the iterated logarithm is extended to stationary ergodic martingales and to a non-identically-distributed case.
Functions possessing restricted mean value properties
Authors: Heath David
Published: June 2010

A real-valued function

f
f

defined on an open subset of

R
N

{R^N}

is said to have the restricted mean value property with respect to balls (spheres) if, for each point

x
x

in the set, there exists a ball (sphere) with center

x
x

and radius

r
(
x
)

r(x)

such that the average value of

f
f

over the ball (sphere) is equal to

f
(
x
)

f(x)

. If

f
f

is harmonic then it has the restricted mean value property. Here new conditions for the converse implication are given.
Product integral approximations of solutions to linear operator equations
Authors: Martin Robert H.
Published: June 2010

In the paper we develop a class of iterative methods for approximating solutions of linear operator equations in a Banach space. The main techniques involve a product integral representation of solutions to linear Stieltjes integral equations, a variation of parameters formula, and the asymptotic convergence of solutions to the homogeneous integral equation.
Two radius of convexity problems
Authors: McCarty Carl P.
Published: June 2010

The sharp radius of convexity of functions with prescribed second coefficient is found for the two classes: functions starlike of order

α
\alpha

, and functions whose derivative has real part greater than

α
\alpha

.
Free-lattice-like sublattices of free products of lattices
Authors: Grätzer G. , Lakser H.
Published: June 2010

Sublattices of a free lattice are known to satisfy three particular conditions. It is shown that certain sublattices of a free product of lattices satisfy the same three conditions.
On integrated screens
Authors: Beidleman J. C.
Published: June 2010

Let

L

\mathcal {L}

be a screen with support

π
\pi

and let

F

\mathcal {F}

denote the saturated formation of finite solvable groups which is locally induced by

L

\mathcal {L}

. For each prime p, let

M

(
p
)
=

L

(
p
)
∩

F

\mathcal {M}(p) = \mathcal {L}(p) \cap \mathcal {F}

. Then

M

\mathcal {M}

is an integrated screen which locally induces

F

\mathcal {F}

and

M

⊆

L

\mathcal {M} \subseteq \mathcal {L}

. The purpose of this note is to prove the following theorems. Theorem 1. Assume that for each finite solvable group G the

L

\mathcal {L}

-izers of G satisfy the strict cover-avoidance property. Then

L

\mathcal {L}

is an integrated screen; that is

L

(
p
)
⊆

F

\mathcal {L}(p) \subseteq \mathcal {F}

for each prime p. Theorem 2. Assume that for each group G an

M

\mathcal {M}

-izer of an

L

\mathcal {L}

-izer of G is an

M

\mathcal {M}

-izer of G. Then

L

(
p
)
=

M

(
p
)

\mathcal {L}(p) = \mathcal {M}(p)

for each prime p.
Some remarks on Weierstrass points
Authors: Jenkins James A.
Published: June 2010

The author proves that, at a point

P
P

on a closed Riemann surface of genus

g
g

, if

h
h

is the first nongap at

P
P

and

k
k

is relatively prime to

h
h

, then

k
k

is a gap if

g
>

1
2

(
h
−
1
)
(
k
−
1
)

g > \tfrac {1}{2}(h - 1)(k - 1)

. A consequence is that at the Weierstrass points of a closed Riemann surface, if the first nongap is a prime, the situation mirrors that in the hyperelliptic case, at least in a limiting sense.
Units in arithmetic progression in an algebraic number field
Authors: Newman Morris
Published: June 2010

It is shown that a given algebraic number field of degree

n
≧
4

n \geqq 4

over the rationals can contain at most

n
n

units in arithmetic progression, and that this bound is sharp.
Automorphisms and tensor products of algebras
Authors: Bunce John W.
Published: June 2010

In this note we prove that if

A
A

is a complex Banach algebra with identity, then the automorphism on

A

⊗
^

A

A\hat \otimes A

determined by

θ
(
a
⊗
b
)
=
b
⊗
a

\theta (a \otimes b) = b \otimes a

is inner if and only if

A
=

M
n

(
C
)

A = {M_n}(C)

.
Countable interpolation sets and the Gleason metric
Authors: Oberlin D. M.
Published: June 2010

Countable interpolation sets of type 1 are characterized as those countable compact sets which intersect each part at most once. A metric condition is given which characterizes those countable compact sets which are interpolation sets for logmodular function algebras.
On smallest compactification for convergence spaces
Authors: Rao C. J. M.
Published: June 2010

In this note we obtain necessary and sufficient conditions for a convergence space to have a smallest Hausdorff compactification and to have a smallest regular compactification.
A coincidence theorem related to the Borsuk-Ulam theorem
Authors: Cohen Fred , Connett J. E.
Published: June 2010

A coincidence theorem generalizing the classical result of Borsuk on maps of

S
n

{S^n}

into

R
n

{R^n}

is proved, in which the antipodal map is replaced by a

Z
p

{Z_p}

-action on a space which is

(
n
−
1
)
(
p
−
1
)

(n - 1)(p - 1)

-connected.
𝑍₂-equivariant immersions and embeddings up to cobordism
Authors: Bix Michael C.
Published: June 2010

The manifolds in an additive basis for the cobordism ring of manifolds with involution are

Z
2

{Z_2}

-equivariantly immersed and embedded into representation spaces having the smallest possible number of nontrivial factors up to cobordism.
Parallel vector fields and periodic orbits
Authors: Schwartzman Sol
Published: June 2010

Let

V
V

be a parallel vector field on a compact Riemannian manifold without boundary. Suppose the Euler class over the reals of the normal bundle to

V
V

is different from zero. Then the flow defined by

V
V

has a periodic orbit.
Representations of decomposable forms
Authors: Waid Carter
Published: June 2010

Results connecting binary quadratic forms and their associated quadratic fields are extended to irreducible decomposable forms and their associated fields. A rational linear substitution that carries such a form into a nonzero rational multiple of itself is shown to correspond with a linear map which admits a unique decomposition as multiplication by a nonzero element of the field followed by an automorphism of the field. This correspondence is one-to-one whenever the form is nondegenerate.
Nonsplitting sequences of value groups
Authors: Mott Joe L.
Published: June 2010

If

K
K

is the quotient field of an integral domain

D
D

, then the value group

V
K

(
D
)

{V_K}(D)

of

D
D

in

K
K

is the group

K
∗

/

U
(
D
)

{K^ \ast }/U(D)

, partially ordered by

D
∗

/

U
(
D
)

{D^ \ast }/U(D)

, where

U
(
D
)

U(D)

denotes the group of units of

D
D

. This note shows that if the sequence
\[

(
1
)

{
1
}
→
G
→
H
→
J
→
{
1
}

(1)\quad \{ 1\} \to G \to H \to J \to \{ 1\}

\]
is lexicographically exact and if

H
H

is lattice ordered, then there is a Bezout domain

B
B

and a prime ideal

P
P

of

B
B

such that

V
K

(
B
)
=
H
,

V
K

(

B
P

)
=
J

{V_K}(B) = H,{V_K}({B_P}) = J

, and

V
k

(
B

/

P
)
=
G

{V_k}(B/P) = G

, where

k
k

denotes the residue field of

B
P

{B_P}

. Moreover,

B
B

is the direct sum of

B

/

P

B/P

and

P
P

, and

B
P

=
k
+
P

{B_P} = k + P

. In particular, the sequence (1) need not split, even with somewhat stringent restrictions on the integral domain

B
B

. This gives a negative answer to a question posed by R. Gilmer.
A generalization of an inequality of Coppel
Authors: Hewer G. A.
Published: June 2010

Upper and lower bounds for the solutions of a linear ordinary differential equation are determined from the solutions of upper and lower matrix comparison equations. The coefficients of the comparison equations are computed with the help of Lozinskiĭ’s logarithmic “norm”
\[

l
(
A
)
=

lim

h
→
+
0

[

|

I
+
h
A

|

−
1
]

/

h
,

l(A) = \lim \limits _{h \to + 0} [|I + hA| - 1]/h,

\]
and the concept of the “matricial norm” as a matrix of scalar norms. Using these estimates some new criteria for the stability of composite systems are obtained.
Graphs of measurable functions
Authors: Buckley J. J.
Published: June 2010

Necessary and sufficient conditions are given for the measurability of a function in terms of its graph.
Some uncomplemented uniform algebras
Authors: Etcheberry Alain
Published: June 2010

The uniform algebras specified by the Wermer embedding theorem are not linearly homeomorphic to a complemented subspace of any Banach space

C
(
Y
)

C(Y)

.
Holomorphic fiber bundles with hyperbolic fiber
Authors: Royden H. L.
Published: June 2010

The purpose of the present note is to show that holomorphic fiber bundles with hyperbolic fiber are almost trivial: They are trivial if the base is simply connected, and have a simple structure otherwise.
A note on semitopological classes
Authors: Crossley S. Gene
Published: June 2010

This paper shows that semitopological classes are subsemilattices of the lattice of topologies, and gives a new characterization for the finest topology in the semitopological class.
Planar and strongly uniform near-rings
Authors: Szeto George
Published: June 2010

The concept of a finite strongly uniform near-ring defined by G. Ferrero is generalized to the infinite case. A relation between planar near-rings and strongly uniform near-rings is studied. A structure theorem for an integral planar near-ring of M. Anshel and J. Clay is extended to a strongly uniform nearring.
Inequalities for a perturbation theorem of Paley and Wiener
Authors: Young Robert M.
Published: June 2010

A classical theorem of Paley and Wiener states that the set of functions

{

e

i

λ
n

t

}

n
=
−
∞

∞

\{ {e^{i{\lambda _n}t}}\} _{n = - \infty }^\infty

forms a basis for

L
2

(
−
π
,
π
)

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

whenever the following condition is satisfied:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis asterisk right-parenthesis StartAbsoluteValue EndAbsoluteValue sigma-summation c Subscript n Baseline left-parenthesis e Superscript i lamda Super Subscript n Superscript t Baseline minus e Superscript i n t Baseline right-parenthesis StartAbsoluteValue EndAbsoluteValue squared less-than-over-equals theta squared sigma-summation StartAbsoluteValue c Subscript n Baseline EndAbsoluteValue squared left-parenthesis 0 less-than-over-equals theta greater-than 1 right-parenthesis period">

(
∗
)

|

|

∑

c
n

(

e

i

λ
n

t

−

e

i
n
t

)

|

|

2

≦

θ
2

∑

|

c
n

|

2

(
0
≦
θ
>
1
)
.

( \ast )\quad ||\sum {{c_n}({e^{i{\lambda _n}t}} - {e^{int}})|{|^2} \leqq {\theta ^2}\,\sum {|{c_n}{|^2}} } \quad (0 \leqq \theta > 1).

\]
It is known that (

∗
\ast

) holds whenever

λ
n

{\lambda _n}

is real and

|

λ
n

−
n

|

≦
L
>

1
4

(
−
∞
>
n
>
∞
)

|{\lambda _n} - n| \leqq L > \frac {1}{4}( - \infty > n > \infty )

, and may fail to hold if

|

λ
n

−
n

|

=

1
4

|{\lambda _n} - n| = \frac {1}{4}

. In this note we show, more generally, that the condition

|

λ
n

−
n

|

>

1
4

|{\lambda _n} - n| > \frac {1}{4}

is also insufficient to ensure (

∗
\ast

).
A note on the sum of two closed lattice ideals
Authors: Lotz Heinrich P.
Published: June 2010

Suppose that

E
E

is a locally convex lattice. The main results established in this note are: (a) If

I
,
J

I,J

are

σ
(

E
′

,
E
)

\sigma (E’,E)

-closed lattice ideals in the dual

E
′

E’

of

E
E

, then

I
+
J

I + J

is

σ
(

E
′

,
E
)

\sigma (E’,E)

-closed. (b) If

E
E

is a Fréchet lattice (in particular, if

E
E

is a Banach lattice) and if

I
,
J

I,J

are closed lattice ideals in

E
E

, then

I
+
J

I + J

is closed.
Injective von Neumann algebras
Authors: Loebl Richard I.
Published: June 2010

Injective von Neumann algebras are defined, and a characterization of them as complemented subspaces of

L

(
H
)

\mathcal {L}(H)

is given. Several examples and applications are discussed.
A note on reductive operators
Authors: Gilfeather Frank
Published: June 2010

A bounded linear operator

A
A

on a Hilbert space is called reductive if every invariant subspace of

A
A

reduces it. This paper gives examples of operators which give an affirmative answer to the reductive question: If

A
A

is reductive, then is

A
A

normal?
The essential spectrum of some Toeplitz operators
Authors: Clancey Kevin F. , Morrel Bernard B.
Published: June 2010

The localization techniques of Douglas and Sarason are used to obtain the essential spectrum of the Toeplitz operator

T
φ

{T_\varphi }

for which

φ
\varphi

is the product of a continuous function and the characteristic function of a measurable subset of the unit circle. Examples are given of Toeplitz operators with one-dimensional self-commutator whose essential spectrum is the unit disk. Using an example of J. E. Brennan, the authors show the existence of a completely nonnormal, subnormal operator whose adjoint has no point spectrum.
Topological spaces in which Blumberg’s theorem holds
Authors: White H. E.
Published: June 2010

H. Blumberg proved that, if

f
f

is a real-valued function defined on the real line

R
R

, then there is a dense subset

D
D

of

R
R

such that

f

|

D

f|D

is continuous. J. C. Bradford and C. Goffman showed [3] that this theorem holds for a metric space

X
X

if and only if

X
X

is a Baire space. In the present paper, we show that Blumberg’s theorem holds for a topological space

X
X

having a

σ
\sigma

-disjoint pseudo-base if and only if

X
X

is a Baire space. Then we identify some classes of topological spaces which have

σ
\sigma

-disjoint pseudo-bases. Also, we show that a certain class of locally compact, Hausdorff spaces satisfies Blumberg’s theorem. Finally, we describe two Baire spaces for which Blumberg’s theorem does not hold. One is completely regular, Hausdorff, cocompact, strongly

α
\alpha

-favorable, and pseudo-complete; the other is regular and hereditarily Lindelöf.
𝜆 connectivity and mappings onto a chainable indecomposable continuum
Authors: Hagopian Charles L.
Published: June 2010

A continuum (i.e., a compact connected nondegenerate metric space)

M
M

is said to be

λ
\lambda

connected if any two of its points can be joined by a hereditarily decomposable continuum in

M
M

. Here we prove that a plane continuum is

λ
\lambda

connected if and only if it cannot be mapped continuously onto Knaster’s chainable indecomposable continuum with one endpoint. Recent results involving aposyndesis and decompositions to a simple closed curve are extended to

λ
\lambda

connected continua.
Refluent multifunctions on semitrees
Authors: Muenzenberger T. B. , Smithson R. E.
Published: June 2010

The concept of refluent on arcs is introduced for multifunctions on semitrees. This notion is then used to obtain a fixed point structure which has as corollaries some generalizations of known fixed point theorems in arcwise connected spaces. For example, we generalize: each continuous point closed multifunction on an arboroid has a fixed point. The final section of the paper develops the relationships between a number of classes of multifunctions on nested spaces.
Equivariant immersion and imbedding up to cobordism
Authors: Stong R. E.
Published: June 2010

This paper determines the smallest possible integer

r
r

so that a given manifold with involution is equivariantly bordant to an immersed or imbedded submanifold of some

R
s

×

R
r

{R^s} \times {R^r}

with involution

1
×
(
−
1
)

1 \times ( - 1)

.
The spectrum of the Laplacian for 1-forms
Authors: Tanno Shûkichi
Published: June 2010

Let

(
M
,
g
)

(M,g)

and

(

M
′

,

g
′

)

(M’,g’)

be compact orientable Riemannian manifolds with the same spectrum of the Laplacian for

1
1

-forms. We prove that, for

dim
⁡
M
=
2
,
3
,
16
,
17
,
⋯
,
93
,
(
M
,
g
)

\dim M = 2,3,16,17, \cdots ,93,(M,g)

is of constant curvature if and only if

(

M
′

,

g
′

)

(M’,g’)

is so.
Maxwell’s coefficients are conditional probabilities
Authors: Hersh Reuben
Published: June 2010

The capacitance coefficients of electrostatics are represented as conditional probabilities associated with Brownian motion. It follows, as an immediate consequence, that these coefficients depend monotonically on their domains.
On generating functions of classical polynomials
Authors: Verma Arun
Published: June 2010

An expansion of an arbitrary power series in terms of products of a power series and a polynomial

p
n

(
α
,
β
)

(
x
)

p_n^{(\alpha ,\beta )}(x)

(where

α
,
β

\alpha ,\beta

are not necessarily independent of

n
n

) is given.
Bounds for solutions of perturbed differential equations
Authors: Proctor T. G.
Published: June 2010

A modified form of the Alekseev variation of constants equation is used to relate the solutions of systems of the form

x
˙

=
f
(
t
,
x
,
λ
)
,
λ

\dot x = f(t,x,\lambda ),\lambda

in

R
m

{R^m}

and the perturbed system

y
˙

=
f
(
t
,
y
,
ψ
(
t
)
)
+
g
(
t
,
y
)

\dot y = f(t,y,\psi (t)) + g(t,y)

. Hypotheses are given on the

m
m

parameter family of differential equations

x
˙

=
f
(
t
,
x
,
λ
)

\dot x = f(t,x,\lambda )

so that if

ψ
˙

\dot \psi

and

g
g

are perturbation functions, bounds can be calculated for the solutions of the perturbed system.
On the depth of topological spaces
Authors: Gerlits J.
Published: June 2010

In this short note we prove that Kowalsky’s star-space of weight

ω
1

{\omega _1}

is a suitable counterexample to a conjecture of I. Juhàsz.
A note on the Dugundji extension theorem
Authors: Lutzer D. , Martin H.
Published: June 2010

We prove that if

A
A

is a closed, metrizable,

G
δ

{G_\delta }

-subspace of a collectionwise normal space

X
X

then there is a linear transformation

e
:
C
(
A
)
→
C
(
X
)

e:C(A) \to C(X)

such that for each

g
∈
C
(
A
)
,
e
(
g
)

g\in C(A),e(g)

extends

g
g

and the range of

e
(
g
)

e(g)

is contained in the closed convex hull of the range of

g
g

.
Zero-one laws for non-Gaussian measures
Authors: Zinn Joel
Published: June 2010

Some zero-one laws are proved for non-Gaussian measures on the space

R
∞

{R^\infty }

. Also included is a characterization of the generating Hilbert space of an abstract Wiener space in terms of the subgroups of positive measure.
Another exchange property for bases
Authors: Greene Curtis
Published: June 2010

Let

X
X

and

Y
Y

be bases of a combinatorial geometry of rank

n
n

. If

A
⊆
X

A \subseteq X

and

B
⊆
Y

B \subseteq Y

, with

|

A

|

+

|

B

|

≥
n
+
1

|A| + |B| \geq n + 1

, then there exist subsets

A
0

⊆
A

{A_0} \subseteq A

and

B
0

⊆
B

{B_0} \subseteq B

such that

(
X
−

A
0

)
∪

B
0

(X - {A_0}) \cup {B_0}

and

(
Y
−

B
0

)
∪

A
0

(Y - {B_0}) \cup {A_0}

are both bases.
Characterizations of compactness of the interval topology in semilattices
Authors: Muenzenberger T. B. , Smithson R. E.
Published: June 2010

In a semilattice two necessary and sufficient conditions for the interval topology to be compact are established. One is in terms of the fixed point property for increasing functions on the semilattice, and the other is in terms of completeness of the semilattice.
Fixed points of several classes of nonlinear mappings in Banach space
Authors: Kuhfittig Peter K. F.
Published: June 2010

In the first part of the paper conditions for the existence of ordinary and higher order fixed points of individual and commutative families of nonlinear operators are obtained. The second part deals with the existence of fixed points of an operator

T
:
C
→
X

T:C \to X

whose graph is closed in the Cartesian product topology induced by the strong topology in

C
C

and the weak topology in

X
X

. The convergence to fixed points of sequences of successive approximations is considered in both parts.
The heights of formal 𝐴-modules
Authors: Waterhouse William C.
Published: June 2010

Let

A
A

be a discrete valuation ring, finite over

Z

p

{{\mathbf {Z}}_p}

, acting on a commutative formal Lie group of height

h
h

. Then

h
h

is a multiple of

|

A
:

Z

p

|

|A:{{\mathbf {Z}}_p}|

; and if

A
A

acts on the tangent space by scalar multiplications, the dimension of the group is at most

h

/

|

A
:

Z

p

|

h/|A:{{\mathbf {Z}}_p}|

.