bluuebunny/crossref_metadata_2025 huggingface dataset:
A characterization of the nondifferentiability set of the Cantor function
Authors: Eidswick J. A.
Published: June 2010

The points of nondifferentiability of the Cantor ternary function are characterized in terms of the spacing of 0’s and 2’s in ternary expansions. One consequence is an illustration of Morse’s intermediate-value theorem for Dini derivates.
On the location of the singularities of the function generated by the Bergman operator of the second kind
Authors: Rosenthal Paul
Published: June 2010

Let

g
(
x
,
y
)
=

P
2

(
f
)

g(x,y) = {P_2}(f)

be Bergman’s operator of the second kind,

f
(
q
)

f(q)

analytic at

q
=
0

q = 0

. The purpose of this paper is to generalize a previous result of the author on the location of the singularities of

g
(
x
,
y
)

g(x,y)

when

f
(
q
)

f(q)

had only a simple pole.

f
(
q
)

f(q)

now is assumed to be a rational function whose poles are distributed along the arc of a circle. An order relation is also obtained for

g
(
x
,
y
)

g(x,y)

for certain fixed

x
x

and

y
y

sufficiently large and positive.
The orthogonal invariants of a curve in Hilbert space
Authors: Grünbaum F. Alberto
Published: June 2010

Let

x
(
t
)
,
y
(
t
)
(
t
∈
R
)

x(t),y(t)(t \in R)

be a pair of continuous curves in H, not passing through the origin. We consider the problem of deciding when one curve is obtained by rotating the other one.
Categories of 𝐻-spaces
Authors: Malraison Pierre J.
Published: June 2010

Fuchs’ [1] category of H-spaces and homotopy classes of shm-maps is shown equivalent to a simpler category of fractions.
Homotopy groups of the isotropy groups of annulus
Authors: Lee Jong P.
Published: June 2010

We compute the isotopy groups of various subspaces of the isotropy group at an interior point of an annulus. We also prove that if

a
a

and

x
x

are interior points of a disk

D
D

, then

π
0

[
H
(
D
−
a
,
x
)
]
=

Z
2

{\pi _0}[H(D - a,x)] = {Z_2}

and

π
n

[
H
(
D
−
a
,
x
)
]
=
0

{\pi _n}[H(D - a,x)] = 0

for

n
≧
1

n \geqq 1

where

H
(
D
−
a
,
x
)

H(D - a,x)

is the isotropy group at

x
x

.
On the invariance of certain classes of functions under composition
Authors: Chaika Milton , Waterman Daniel
Published: June 2010

Certain classes of functions are mapped into themselves by any change of variable. For some classes of this type which are of interest in the study of Fourier series, it is shown that the necessary and sufficient condition that

g
∘
f

g \circ f

be in the class for each

f
f

of that class is that

g
∈
Lip
⁡
1

g \in \operatorname {Lip} 1

.
Local holonomy groups of induced connections
Authors: Liu Mu Chou
Published: June 2010

There are two naturally induced connections on the tangent bundle, the so called Jacobi connection and the Sasaki connection. By using the elementary theory of systems of linear differential equations, we completely determine the local holonomy group of these two induced connections, and find some relation to the local holonomy group of the manifold itself. There is an induced connection on the vector bundle of linear maps of the fibers. We also investigate the properties of the holonomy group of this bundle.
A theorem of Hurwitz and Radon and orthogonal projective modules
Authors: Geramita A. V. , Pullman N. J.
Published: June 2010

We find the maximum number of orthogonal skew-symmetric anticommuting integer matrices of order n for each natural number n and relate this to finding free direct summands of certain generic projective modules.
Extending continuous linear functionals in convergence inductive limit spaces
Authors: Kranzler S. K. , McDermott T. S.
Published: June 2010

Let

E
n

{E_n}

be an increasing sequence of locally convex linear topological spaces such that the dual

E
n
′

{E’_n}

of each has a Fréchet topology (not necessarily compatible with the dual system

(

E
n
′

,

E
n

)
)

({E’_n},{E_n}))

weaker than the Mackey topology. Let

E
=

⋃

n
=
1

∞

E
n

,
F

E = \bigcup \nolimits _{n = 1}^\infty {{E_n},F}

be a subspace of

E
E

and

τ
\tau

the inductive limit convergence structure on

E
E

. Conditions are given which insure that every

τ
\tau

-continuous linear functional on

F
F

has a

τ
\tau

-continuous linear extension to

E
E

. This result generalizes a theorem of C. Foias and G. Marinescu.
On scattered spaces
Authors: Kannan V. , Rajagopalan M.
Published: June 2010

We show that each

0
0

-dimensional Hausdorff space which is scattered can be mapped continuously in a one-to-one way onto a scattered

0
0

-dimensional Hausdorff space of the same weight as its cardinality. This gives an easier and a new proof of the fact that a countable regular space admits a coarser compact Hausdorff topology if and only if it is scattered. We also show that a

0
0

-dimensional, Lindelöf, scattered first-countable Hausdorff space admits a scattered compactification. In particular we give a more direct proof than that of Knaster, Urbanik and Belnov of the fact that a countable scattered metric space is a subspace of

[
1
,
Ω
)

[1,\Omega )

, and deduce a result of W. H. Young as a corollary.
A two-dimensional non-Noetherian factorial ring
Authors: Gilmer Robert
Published: June 2010

Let

R
R

be a commutative ring with identity and let

G
G

be an abelian group of torsion-free rank

α
\alpha

. If

{

X
λ

}

\{ {X_\lambda }\}

is a set of indeterminates over

R
R

of cardinality

α
\alpha

, then the group ring of

G
G

over

R
R

and the polynomial ring

R
[
{

X
λ

}
]

R[\{ {X_\lambda }\} ]

have the same (Krull) dimension. The preceding result and a theorem due to T. Parker and the author imply that for each integer

k
≧
2

k \geqq 2

, there is a

k
k

-dimensional non-Noetherian unique factorization domain of arbitrary characteristic.
Information flow in one-dimensional Markov systems
Authors: Dawson D. A.
Published: June 2010

The information flow in discrete Markov systems provides a method for determining that such a system has a unique invariant measure. Estimates are obtained for the information flow and conditions under which there is a unique invariant measure for a one-dimensional Markov system are obtained.
Prime quadratics associated with complex quadratic fields of class number two
Authors: Hendy M. D.
Published: June 2010

We establish a necessary and sufficient relation between those quadratic fields of class number two, and some quadratic polynomials

f
(
x
)

f(x)

) which take only prime values for small positive integers.
On extensions of an inequality among means
Authors: Chan F. , Goldberg D. , Gonek S.
Published: June 2010

An inequality of Fan relates the arithmetic and geometric means of x and

1
−
x

1 - x

. An extension to generalized means is conjectured. This conjecture is proven for several special cases. In addition, some counterexamples are given.
On isomorphic groups and homeomorphic spaces
Authors: Yang J. S.
Published: June 2010

Let

C
(
X
,
G
)

C(X,G)

denote the group of continuous functions from a topological space

X
X

into a topological group

G
G

with the pointwise multiplication. Some classes of SQ-pairs and properties of the corresponding topological group

C
(
X
,
G
)

C(X,G)

with the compact-open topology are investigated. We also show that the existence of a group isomorphism between groups

C
(
X
,
G
)

C(X,G)

and

C
(
Y
,
G
)

C(Y,G)

implies the existence of a homeomorphism between

X
X

and

Y
Y

, if

(
X
,
G
)

(X,G)

and

(
Y
,
G
)

(Y,G)

are SQ-pairs.
On Carmichael’s conjecture
Authors: Pomerance Carl
Published: June 2010

A sufficient condition is given for a natural number

x
x

in order that the equation

φ
(
x
)
=
φ
(
y
)

\varphi (x) = \varphi (y)

has only the solution

y
=
x

y = x

. It is conjectured that no natural numbers satisfy this sufficient condition.
Closed images of the Wallman compactification
Authors: Harris Douglas
Published: June 2010

The class of closed canonical images of the Wallman compactification of a space is given an internal characterization, and the class of locally compact-small spaces is introduced and shown to play for such compactifications the same role that locally compact spaces play for Hausdorff compactifications.
Asymptotic stability for some critical autonomous differential equations
Authors: Winston Elliot
Published: June 2010

Liapunov functions are constructed and used to prove stability theorems for critical autonomous systems in which the linear part of the right-hand side has a zero eigenvalue.
On two theorems of Paley
Authors: Rivière N. M. , Sagher Y.
Published: June 2010

A strengthening of Paley’s theorem for the Fourier coefficients of an

L
p

{L^p}

function is presented. The result is then applied to prove strong versions of recent results of P. L. Duren, and of J. H. Hedlund on

(

L
p

,

L
q

)

({L^p},{L^q})

multipliers.
0-divisors in groups rings
Authors: Sinha I.
Published: June 2010

If

G
G

is any group with two finite subgroups

H
,
K
,
K
≦
G
,
(

|

H

|

,

|

K

|

)
=
1

H,K,K \leqq G,(|H|,|K|) = 1

, then RG has

∩
α

G

α

≠
0

{ \cap _\alpha }{\mathfrak {G}_\alpha } \ne 0

, where

G

\mathfrak {G}

is the augmentation ideal.
Polynomials over finite fields which commute with translations
Authors: Wells Charles
Published: June 2010

The polynomials over a finite field which commute with translation by an element of the field are characterized. A generalization of a long-known theorem about centralizers of permutations is used in obtaining the characterization.
Rings whose modules are projective over endomorphism rings
Authors: Snider Robert L.
Published: June 2010

Let

R
R

be either an Artin ring or a commutative ring. We show that every

R
R

-module is projective over its endomorphism ring if and only if

R
R

is uniserial. We study rings, all of whose modules are projective over their endomorphism rings. We show that for Artin or commutative rings this is equivalent to being uniserial. We remark that we know of no examples other than uniserial rings and conjecture that all such rings are uniserial. Our proof uses the complete ring of quotients as in [4]. Our work is partially motivated by the work of Sally and Vasconcelos on commutative rings whose ideals are projective over their endomorphism rings [7]. We use

J
J

throughout to denote the Jacobson radical and

Soc
⁡
(
R
)

\operatorname {Soc} (R)

to denote the socle of

R
R

.
𝑃-sets in 𝐹^{’}-spaces
Authors: Atalla Robert E.
Published: June 2010

A

P
P

-set is one which is interior to any zero set which contains it. An

F
′

F’

-space may be characterized as one in which the closure of a cozero set is a

P
P

-set. We study applications of

P
P

-sets to the topology of

F
′

F’

-spaces, and certain set-theoretical operations under which the class of

P
P

-sets is stable. A. I. Veksler has shown that in a basically disconnected space the closure of an arbitrary union of

P
P

-sets is a

P
P

-set, while in

F
′

F’

-spaces we are only able to prove this for countable unions. Our main result is an example of a set in the compact

F
F

-space

β
N
∖
N

\beta N\backslash N

which is not a

P
P

-set, but which is the closure of a union of

P
P

-sets. The set is related to the almost-convergent functions of G. G. Lorentz.
On a sublattice of the lattice of normal Fitting classes
Authors: Makan A. R.
Published: June 2010

Let

L

{\mathbf {L}}

be the set of all Fitting classes

F

\mathfrak {F}

with the following two properties: (i)

F

⊇

N

\mathfrak {F} \supseteq \mathfrak {N}

, the class of all finite nilpotent groups, and (ii) every

F

\mathfrak {F}

-avoided, complemented chief factor of any finite soluble group

G
G

is partially

F

\mathfrak {F}

-complemented in

G
G

. It is shown that

L

{\mathbf {L}}

is a complete sublattice of the complete lattice

N

{\mathbf {N}}

of all nontrivial normal Fitting classes, and, moreover, it is lattice isomorphic to the subgroup lattice of the Frattini factor group of a certain abelian torsion group due to H. Lausch.
A generalization of a theorem by D. K. Faddeev
Authors: Behnen Konrad
Published: June 2010

In this paper we give a simple proof of the statement
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow normal infinity Endscripts integral upper K Subscript n Baseline left-parenthesis x comma y right-parenthesis f left-parenthesis y right-parenthesis d mu left-parenthesis y right-parenthesis equals f left-parenthesis x right-parenthesis">

lim

n
→
∞

∫

K
n

(
x
,
y
)
f
(
y
)
d
μ
(
y
)
=
f
(
x
)

{\lim _{n \to \infty }}\int {{K_n}(x,y)f(y)d\mu (y) = f(x)}

for

μ
\mu

-almost all

x
x

under weaker and more general assumptions than those of the usual Faddeev theorems.
On compact* spaces and compactifications
Authors: Salbany Sergio
Published: June 2010

The space

β
X

\beta X

of

Z
Z

-ultrafilters on

X
X

with the standard filter space topology is shown to be compact*. Without considering the reflection associated with compact* spaces, we also prove that products of compact* spaces are compact*, in response to a request for a direct proof.
On the failure of the first principle of separation for coanalytic sets
Authors: Maitra Ashok
Published: June 2010

In this note we present a new example of a pair of disjoint coanalytic sets which are not Borel separable, i.e., coanalytic sets

D
D

and

H
H

such that

D
∩
H
=
ϕ

D \cap H = \phi

and such that there is no Borel set

E
E

for which

D
⊆
E

D \subseteq E

and

E
∩
H
=
ϕ

E \cap H = \phi

.
A short proof of the classical edge of the wedge theorem
Authors: Bedford Eric
Published: June 2010

By solving the

∂
¯

\bar \partial

-equation and using Bochner’s theorem on tube domains, one may derive an easy proof of the edge of the wedge theorem in 2 variables.
On direct sums of reductive operators
Authors: Wiggen Thomas P.
Published: June 2010

An example is given to show that the direct sum of two (distinct) reductive operators need not be reductive. The conjecture that

A
⊕
A

A \oplus A

is reductive if

A
A

is reductive is shown to be equivalent to the reductive operator conjecture (every reductive operator is normal).
Normal Moore spaces in the constructible universe
Authors: Fleissner William
Published: June 2010

Assuming the axiom of constructibility, points in closed discrete subspaces of certain normal spaces can be simultaneously separated. This is a partial result towards the normal Moore space conjecture.
Some formulas in the Steenrod algebra
Authors: Peterson F. P.
Published: June 2010

In this paper we prove that certain sums of Steenrod operations are zero. We also show how certain Milnor basis elements can be expanded in the Adem basis.
An annular function which is the sum of two normal functions
Authors: Lappan Peter
Published: June 2010

It is known that a nonconstant normal function cannot have a Koebe value. An example is presented of an annular function which is the sum of two normal holomorphic functions. This example shows that a sum of two normal functions can result in a nonconstant function which has the Koebe value

∞
\infty

.
Simple knots without unique minimal surfaces
Authors: Lyon Herbert C.
Published: June 2010

We construct infinitely many simple knots and infinitely many nonsimple knots, all of genus one, and all having both knotted and unknotted minimal spanning surfaces. The complements of the simple knots all contain closed incompressible surfaces of genus two.
Some phenomena in homotopical algebra
Authors: Varadarajan K.
Published: June 2010

In [6] D. G. Quillen developed homotopy theory in categories satisfying certain axioms. He showed that many results in classical homotopy theory (of topological spaces) go through in his axiomatic set-up. The duality observed by Eckmann-Hilton in classical homotopy theory is reflected in the axioms of a model category. In [7] we developed the theory of numerical invariants like the Lusternik-Schnirelmann category and cocategory etc. for such model categories and in [8] we dealt with applications of this theory to injective and projective homotopy theory of modules as developed by Hilton [2], [3, Chapter 13]. Contrary to the general expectations there are many aspects of classical homotopy theory which cannot be carried over to Quillen’s axiomatic set-up. This paper deals with some of these phenomena.
Global dimension of differential operator rings
Authors: Goodearl K. R.
Published: June 2010

This paper is concerned with finding the global homological dimension of the ring of differential operators

R
[
θ
]

R[\theta ]

over a differential ring

R
R

with a single derivation. Examples are constructed to show that

R
[
θ
]

R[\theta ]

may have finite dimension even when

R
R

has infinite dimension. For a commutative noetherian differential algebra

R
R

over the rationals, with finite global dimension

n
n

, it is shown that the global dimension of

R
[
θ
]

R[\theta ]

is the supremum of

n
n

and one plus the projective dimensions of the modules

R

/

P

R/P

, where

P
P

ranges over all prime differential ideals of

R
R

. One application derives the global dimension of the Weyl algebra over a commutative noetherian ring

S
S

of finite global dimension, where

S
S

either is an algebra over the rationals or else has positive characteristic.
Convexity of vector-valued functions
Authors: Hsu Ih Ching , Kuller Robert G.
Published: June 2010

Let

(

B

,
≪
)

(\mathcal {B}, \ll )

be a Banach lattice, and

(
a
,
b
)

(a,b)

be an open interval on the real line. A function

F
:
(
a
,
b
)
→

B

F:(a,b) \to \mathcal {B}

is defined to be weakly convex if there exists a nonnegative nondecreasing continuous function

G
:
(
a
,
b
)
→

B

G:(a,b) \to \mathcal {B}

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-bracket upper F left-parenthesis upper S right-parenthesis right-bracket plus t p left-bracket upper G left-parenthesis s right-parenthesis right-bracket less-than-or-equal-to p left-bracket upper F left-parenthesis s plus t right-parenthesis right-bracket">

p
[
F
(
S
)
]
+
t
p
[
G
(
s
)
]
≤
p
[
F
(
s
+
t
)
]

p[F(S)] + tp[G(s)] \leq p[F(s + t)]

, whenever

s
s

and

s
+
t

s + t

are in

(
a
,
b
)

(a,b)

for each positive linear functional

p
p

on

B

\mathcal {B}

. A representation theorem is proved as follows: If

F
F

is weakly convex on

(
a
,
b
)

(a,b)

and is bounded on an interval contained in

(
a
,
b
)

(a,b)

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper B right-parenthesis integral Subscript a plus epsilon Superscript x Baseline upper G left-parenthesis s right-parenthesis d m equals upper F left-parenthesis x right-parenthesis minus upper F left-parenthesis a plus epsilon right-parenthesis">

(
B
)

∫

a
+
ϵ

x

G
(
s
)
d
m
=
F
(
x
)
−
F
(
a
+
ϵ
)

(B)\int _{a + \epsilon }^x {G(s)dm = F(x) - F(a + \epsilon )}

, where

(
B
)

∫

a
+
ϵ

x

G
(
s
)
d
m

(B)\int _{a + \epsilon }^x {G(s)dm}

is the Bochner integral of

G
G

on

[
a
,
ϵ
,
x
]

[a, \epsilon ,x]

with

0
>
ϵ

0 > \epsilon

and

a
>
ϵ
>
x
>
b

a > \epsilon > x > b

.
Pointwise bounds on eigenfunctions and wave packets in 𝑁-body quantum systems. II
Authors: Simon Barry
Published: June 2010

We provide a simple proof (and mild improvement) of Schnol’s result that

L
2

{L^2}

eigenfunctions of

−
Δ
+
V

- \Delta + V

are

O
(
exp
⁡
(
−
a
r
)
)

O(\exp ( - ar))

for any

a
>
0

a > 0

whenever

V
→
∞

V \to \infty

as

r
→
∞

r \to \infty

.
Maximal cancellative subsemigroups and cancellative congruences
Authors: Putcha Mohan S.
Published: June 2010

A subsemigroup

T
T

of a commutative semigroup

S
S

is called a mild ideal if for any

a
∈
S
,
a
T
∩
T
≠
ϕ

a \in S,aT \cap T \ne \phi

. It is shown here that any maximal cancellative subsemigroup

T
T

of a commutative, idempotent-free, archimedean semigroup

S
S

must be a mild ideal of

S
S

. Maximal cancellative subsemigroups exist in abundance due to Zorn’s lemma. It is also shown that if

T
T

is mild ideal of a commutative semigroup

S
S

, then every cancellative congruence of

T
T

has a unique extension to a cancellative congruence of

S
S

.
The F. and M. Riesz theorem
Authors: Forelli Frank
Published: June 2010

We weaken the hypothesis of the F. and M. Riesz theorem.
A characterization of strictly convex Banach spaces
Authors: Strawther D. , Gudder S.
Published: June 2010

A very short proof is given for Petryshyn’s characterization of strictly convex Banach spaces [2].
Derivations of 𝐶*-algebras which are not determined by multipliers in any quotient algebra
Authors: Tomiyama Jun
Published: June 2010

An example is given to show that, in a

C
∗

{C^ \ast }

-algebra without unit, a derivation is not necessarily determined by a multiplier in some quotient algebra.
Characters and generosity of permutation groups
Authors: Saxl Jan
Published: June 2010

A necessary and sufficient character condition is obtained for a group

G
G

to be generously

k
k

-fold transitive. This is similar to an old theorem of Frobenius on multiply transitive groups.
On the coefficients of meromorphic univalent functions
Authors: Thomas D. K.
Published: June 2010

Let

f
∈
Σ

f \in \Sigma

, the class of all analytic univalent functions defined in

γ
=
{
z
:

|

z

|

>
1
}

\gamma = \{ z:|z| > 1\}

. For

f
,
g
∈
Σ

f,g \in \Sigma

define

h
h

in

γ
\gamma

by

h
(
z
)
=
f

(
z

)

1
−
α

g

(
z

)
α

,
0
>
α
>
1

h(z) = f{(z)^{1 - \alpha }}g{(z)^\alpha },0 > \alpha > 1

. If

h
(
z
)
=
z
+

Σ

n
=
0

∞

c
n

z

−
n

h(z) = z + \Sigma _{n = 0}^\infty {c_n}{z^{ - n}}

, it is shown that

Σ

n
=
1

∞

n

|

c
n

|

2

>
∞

\Sigma _{n = 1}^\infty n|{c_n}{|^2} > \infty

. This result is used to show that if

B
α

{B_\alpha }

denotes the class of all meromorphic Bazilevič functions of type

α
\alpha

and

f
∈

B
α

f \in {B_\alpha }

with

f
(
z
)
=
z
+

Σ

n
=
0

∞

a
n

z

−
n

f(z) = z + \Sigma _{n = 0}^\infty {a_n}{z^{ - n}}

, then

n

a
n

=
O
(
1
)

n{a_n} = O(1)

as

n
→
∞

n \to \infty

, the result being best possible.
Regular functions 𝑓(𝑧) for which 𝑧𝑓^{’}(𝑧) is 𝛼-spiral-like
Authors: Chichra Pran Nath
Published: June 2010

Let

F

α
λ

\mathfrak {F}_\alpha ^\lambda

be the class of functions

f
(
z
)
=
z
+

a
2

z
2

+
⋯

f(z) = z + {a_2}{z^2} + \cdots

which are regular in

E
=
{
z

/

|

z

|

>
1
}

E = \{ z/|z| > 1\}

and satisfy
\[

Re
⁡
{

e

i
α

(
1
+
z
f
(
z
)

/

f
′

(
z
)
)
}
>
λ
cos
⁡
α

\operatorname {Re} \{ {e^{i\alpha }}(1 + zf(z)/f’(z))\} > \lambda \cos \alpha

\]
for some

α
,

|

α

|

>
π

/

2

\alpha ,|\alpha | > \pi /2

, and for some

λ
,
0
≤
λ
>
1

\lambda ,0 \leq \lambda > 1

. The author finds a range on

α
\alpha

for which

f
(
z
)

f(z)

in

F

α
λ

\mathfrak {F}_\alpha ^\lambda

is univalent in

E
E

. In particular, the author improves upon the range on a for which

f
(
z
)
∈

F

α
0

f(z) \in \mathfrak {F}_\alpha ^0

is known to be univalent in

E
E

. Also a corresponding result is obtained for those functions

f
(
z
)

f(z)

in

F

α
λ

\mathfrak {F}_\alpha ^\lambda

for which

f
(
0
)
=
0

f(0) = 0

.
Weakly compact groups of operators
Authors: Gillespie T. A. , West T. T.
Published: June 2010

It is shown that the weakly closed algebra generated by a weakly compact group of operators on a Banach space is reflexive and equals its second commutant. Also, an example is given to show that the generator of a monothetic weakly compact group of operators need not have a logarithm in the algebra of all bounded linear operators on the underlying space.
An exchange theorem for independence structures
Authors: McDiarmid C. J. H.
Published: June 2010

I give a simple proof of the following result, which extends a result of C. Greene [1]. Let

X
X

and

Y
Y

be sets in an independence structure

E

\mathcal {E}

and let

X
=

X
1

∪

X
2

X = {X_1} \cup {X_2}

be a partition of

X
X

. Then there exists a partition

Y
=

Y
1

∪

Y
2

Y = {Y_1} \cup {Y_2}

of

Y
Y

such that for

i
=
1
,
2
,

X
i

∩

Y
i

=
ϕ

i = 1,2,{X_i} \cap {Y_i} = \phi

and

X
i

∪

Y
i

{X_i} \cup {Y_i}

is in

E

\mathcal {E}

.
Some linear subordination results for classes of univalent functions
Authors: Byers Robert B.
Published: June 2010

In this note we determine necessary and sufficient conditions on complex numbers

λ
\lambda

and

μ
\mu

such that

λ
z

/

(
1
−
μ

a
2

z
)

\lambda z/(1 - \mu {a_2}z)

is subordinate to

f
(
z
)
=
z
+

a
2

z
2

+
⋯

f(z) = z + {a_2}{z^2} + \cdots

for all functions

f
f

in certain classes of univalent functions.
Linear differential equations where nonoscillation is equivalent to eventual disconjugacy
Authors: Ridenhour Jerry R.
Published: June 2010

Conditions on

n
n

th order linear differential equations are given which imply that nonoscillation is equivalent to eventual disconjugacy. These conditions are in the form of assumptions that certain boundary-value functions are infinite for all values of the argument.
Nearly comonotone approximation
Authors: Roulier John A.
Published: June 2010

This paper obtains estimates on the degree of nearly comonotone approximation which extend and improve the estimate obtained by Newman, Passow, and Raymon. In particular, the restriction that

f
∈

Lip
M

1

f \in {\operatorname {Lip} _M}1

is removed, and estimates for the degree of nearly comonotone approximation are obtained for all proper piecewise monotone functions. It is also shown that if

f
′

f’

exists and is continuous on the interval, then the ordinary polynomials of best approximation form a nearly comonotone sequence.
Inequalities concerning the characters of a finite group
Authors: Fields K. L.
Published: June 2010

Given a finite group we provide explicit bounds (in terms of the group order and numbers of conjugacy classes and involutions) for (a) the number of real valued characters of type

R

{\mathbf {R}}

; (b) the sum of the degrees of the irreducible characters; (c) the sum of the entries of the character table; (d) the sums (b), (c) restricted to real valued characters. We also provide a bound on the number of elements of order

2
n

2n

in terms of the number of elements of order

n
n

.
On the Padé table of 𝑐𝑜𝑠𝑧
Authors: Magnus Arne , Wynn Jan
Published: June 2010

The Padé table for

cos
⁡
z

\cos z

and related functions is written in a form that involves the binomial coefficients, the Euler numbers and Bernoulli numbers. Identities involving the corresponding persymmetric determinants and their minors are developed.
Schur’s theorem for nearly Kähler manifolds
Authors: Naveira A. M. , Hervella L. M.
Published: June 2010

The classical theorem of Schur on Kähler manifolds is generalized to nearly Kähler manifolds, thus solving a conjecture of A. Gray [3, p. 289].
Nonlinear Volterra integral equations with positive definite kernels
Authors: Staffans Olof J.
Published: June 2010

Conditions are given for the asymptotic stability of the solution of a nonlinear Volterra integro-differential equation with convolution kernel. These conditions are less restrictive than most of the known ones for this equation.
Convergence, closed projections and compactness
Authors: Vaughan J. E.
Published: June 2010

In this paper several results of N. Noble concerning closed projections are extended using generalizations of first countable, Fréchet, and sequential spaces. We also consider compactness conditions defined by requiring that certain nets have cluster points.
Hereditarily closure-preserving collections and metrization
Authors: Burke D. , Engelking R. , Lutzer D.
Published: June 2010

In this paper we present a generalization of the Nagata-Smirnov metrization theorem. We prove that a regular

T
1

{T_1}

-space is metrizable if and only if it has a base of open sets which is the union of countably many hereditarily closure-preserving subcollections. In addition, we investigate intersections of hereditarily closure-preserving collections of open sets.
Test modules
Authors: Cheatham T. , Cumbie R.
Published: June 2010

The results of this paper arose from an investigation of the class of

Σ
\Sigma

-modules, i.e. those modules

M
M

for which

Hom
R

(
M
,
−
)

{\operatorname {Hom} _R}(M, - )

commutes with direct sums. A module

T
T

is called a test module if

Hom
R

(
M
,
−
)

{\operatorname {Hom} _R}(M, - )

commutes with direct sums of copies of

T
T

only when

M
M

is a

Σ
\Sigma

-module. Test modules are characterized and their relation to cogenerators is investigated.
Oscillation and comparison for second order differential equations
Authors: Schrader Keith
Published: June 2010

Consider the equations
\[

(
1
)

x
=
f
(
t
,
x
,

x
′

)

(1)\quad x = f(t,x,x’)

\]
and
\[

(
2
)

x
=
g
(
t
,
x
,

x
′

)

(2)\quad x = g(t,x,x’)

\]
where

f
,
g
:
[
a
,
+
∞
)
×

R
2

→
R

f,g:[a, + \infty ) \times {R^2} \to R

are continuous. Assume that solutions of initial value problems for (1) and for (2) are unique and extend to

[
a
,
+
∞
)

[a, + \infty )

. Let

f
(
t
,
0
,
0
)
=
0
=
g
(
t
,
0
,
0
)

f(t,0,0) = 0 = g(t,0,0)

for

t
ϵ
[
a
,
+
∞
)

t \epsilon [a, + \infty )

and

f
(
t
,
x
,

x
′

)

/

x
≤
g
(
t
,
x
,

x
′

)

/

x

f(t,x,x’)/x \leq g(t,x,x’)/x

for

|

x

|

>
0

|x| > 0

and

(
t
,
x
,

x
′

)

(t,x,x’)

in the domain of

f
f

and

g
g

. Under these hypotheses it can be shown that if every solution of (2) has a zero on an interval

I
⊂
[
a
,
+
∞
)

I \subset [a, + \infty )

then it follows that every solution of (1) has a zero on

I
I

. In particular this shows that under these hypotheses (2) is oscillatory (every solution has a zero on

[
a
+
n
,
+
∞
)

[a + n, + \infty )

for each positive integer

n
n

) implies (1) is oscillatory.
Counterexample to a theorem on differential equations in Hilbert space
Authors: Knight William J.
Published: June 2010

A counterexample is given to a theorem by Felix Browder concerning solutions of differential equations with values in Hilbert space. Several valid conclusions (as near the original conclusions as possible) are pointed out.
A characterization of the kernel of a closed set
Authors: Breen Marilyn
Published: June 2010

Let

S
S

be a closed subset of some linear topological space such that int ker

S
≠
ϕ

S \ne \phi

and ker

S
≠
S

S \ne S

Let

C

\mathcal {C}

denote the collection of all maximal convex subsets of

S
S

and, for any fixed

k
≥
1

k \geq 1

, let

M

=
{

A
1

∪
⋯
∪

A
k

:

A
1

,
…
,

A
k

\mathfrak {M} = \{ {A_1} \cup \cdots \cup {A_k}:{A_1}, \ldots ,{A_k}

distinct members of

C

}

\mathcal {C}\}

. Then

M

≠
ϕ

\mathfrak {M} \ne \phi

and

∩

M

=
ker
⁡
S

\cap \mathfrak {M} = \ker S

.
Fixed points of antitone mappings
Authors: Shmuely Zahava
Published: June 2010

We present a family of antitone mappings defined on complete atomic lattices which have the fixed point property. Two commuting mappings of the family have a common fixed point. An example is given of three commuting mappings of the family which do not have a common fixed point.
A note on Dilworth’s embedding theorem
Authors: Trotter William T.
Published: June 2010

The dimension of a poset

X
X

is the smallest positive integer

t
t

for which there exists an embedding of

X
X

in the cartesian product of

t
t

chains. R. P. Dilworth proved that the dimension of a distributive lattice

L
=

2
_

X

L = {\underline 2 ^X}

is the width of

X
X

. In this paper we derive an analogous result for embedding distributive lattices in the cartesian product of chains of bounded length. We prove that for each

k
⩾
2

k \geqslant 2

, the smallest positive integer

t
t

for which the distributive lattice

L
=

2
_

X

L = {\underline 2 ^X}

can be embedded in the cartesian product of

t
t

chains each of length

k
k

equals the smallest positive integer

t
t

for which there exists a partition

X
=

C
1

∪

C
2

∪
⋯
∪

C
t

X = {C_1} \cup {C_2} \cup \cdots \cup {C_t}

where each

C
i

{C_i}

is a i a chain of at most

k
−
1

k - 1

points.
Wildly ramified 𝑍/2 actions in dimension two
Authors: Artin M.
Published: June 2010

The rings of power series which are invariant under an automorphism of order 2 are described by equations having a standard form.
Annular functions in probability
Authors: Howell Russell W.
Published: June 2010

A function

f
f

holomorphic in the open unit disk

U
U

is said to be strongly annular if there exists a sequence

{

C
n

}

\{ {C_n}\}

of concentric circles converging outward to the boundary of

U
U

such that the minimum of

|

f

|

|f|

on

C
n

{C_n}

tends to infinity as

n
n

increases. We show here that such functions with Maclaurin coefficients

±
1

\pm 1

form a residual set in the space of functions with coefficients

±
1

\pm 1

. We also show that the set of

t
t

in

[
0
,
1
]

[0,1]

for which

∑

r
n

(
t
)

z
n

\sum {{r_n}(t){z^n}}

is strongly annular (

r
n

{r_n}

is the

n
n

th Rademacher function) is residual, and measurable with measure either

0
0

or

1
1

.
Quasi-nilpotent sets in semigroups
Authors: Chow H. L.
Published: June 2010

In a compact semigroup

S
S

with zero

0
0

, a subset

A
A

of

S
S

is called quasi-nilpotent if the closed semigroup generated by

A
A

contains

0
0

. A probability measure

μ
\mu

on

S
S

is called nilpotent if the sequence

(

μ
n

)

({\mu ^n})

converges to the Dirac measure at

0
0

. It is shown that a probability measure is nilpotent if and only if its support is quasi-nilpotent. Consequently, the set of all nilpotent measures on

S
S

is convex and everywhere dense in the set of all probability measures on

S
S

and the union of their supports is

S
S

.
On iterates of convolutions
Authors: Foguel Shaul R.
Published: June 2010

Let

μ
\mu

be a signed measure, of total variation one, on a locally compact Abelian group. We study in this note the ideal

I
=
{
τ
:
τ
≪
m

a
n
d

|

|

μ
n

∗
τ

|

|

→
0
}

I = \{ \tau :\tau \ll m\;and\; ||{\mu ^n} \ast \tau || \to 0\}

where

m
m

is the Haar measure.
Inequalities for the powers of nonnegative Hermitian operators
Authors: Kwong Man Kam
Published: June 2010

In the set of bounded Hermitian operators from a Hilbert space

H
H

into itself, we define three types of ordering by means of the cones of nonnegative, positive definite and positive invertible operators respectively. Our main theorem shows that for all three types of ordering, if

A
A

is “greater” than

B
B

, then

A
r

{A^r}

is “greater” than

B
r

{B^r}

for all real numbers

r
≤
1

r \leq 1

. This generalizes the results of Heinz [3] and Kato [4].
A four-vertices theorem for ruled surfaces
Authors: Jha P.
Published: June 2010

The skewness of distribution of an orientable closed ruled surface satisfies a four-vertices theorem.
Minimal 𝑁-compact spaces
Authors: Rao K. P. S. Bhaskara , Raha Asit Baran
Published: June 2010

In this note it is established that minimal

N
N

-compact spaces are compact.
A maximal realcompactification with 0-dimensional outgrowth
Authors: Rayburn Marlon C.
Published: June 2010

If

X
X

is a rimhard space, then it has a realcompactification with properties similar to those of the Freudenthal compactification of a rimcompact space.
Disk-like products of 𝜆 connected continua. I, II
Authors: Hagopian Charles L.
Published: June 2010

A continuum

X
X

is

λ
\lambda

connected if each two of its points can be joined by a heteditarily decomposable subcontinuum of

X
X

. We prove that continua

X
X

and

Y
Y

are atriodic and hereditarily unicoherent when the topological product

X
×
Y

X \times Y

is disk-like. From this result and a theorem of R. H. Bing’s it follows that

λ
\lambda

connected continua

X
X

and

Y
Y

are arc-like if and only if

X
×
Y

X \times Y

is disk-like.
Homomorphic images of 𝜎-complete Boolean algebras
Authors: Koppelberg Sabine
Published: June 2010

It is a well-known theorem of R. S. Pierce that, for every infinite cardinal

α
,

α

ℵ
0

=
α

\alpha ,{\alpha ^{{\aleph _0}}} = \alpha

if and only if there is a complete Boolean algebra

B
B

s.t. card

B
=
α

B = \alpha

(see [3, Theorem 25.4]). Recently, Comfort and Hager proved [1] that, for every infinite

σ
\sigma

-complete Boolean algebra

B
,

(
card
⁡
B

)

ℵ
0

=
card
⁡
B

B,{(\operatorname {card} B)^{{\aleph _0}}} = \operatorname {card} B

. We extend this result to the class of homomorphic images of

σ
\sigma

-complete algebras, following closely Comfort’s and Hager’s proof. As a corollary, an improvement of Shelah’s theorem on the cardinality of ultraproducts of finite sets [2] is derived (Theorem 2).

1

^{1}
Powers of a matrix with coefficients in a Boolean ring
Authors: Almkvist Gert
Published: June 2010

The best possible integer

u
n

{u_n}

such that

f

u
n

=
1

{f^{{u_n}}} = 1

, if

f
f

is an invertible

n
×
n

n \times n

matrix with coefficients in a Boolean ring, is determined. The period of linear recursive sequences in a Boolean ring (e.g. the trace sequence

{
Tr
⁡

f
k

}
1
∞

\{ \operatorname {Tr} {f^k}\} _1^\infty

) is computed.
Certain overrings of right hereditary, right Noetherian rings are 𝑉-rings
Authors: Hansen Friedhelm
Published: June 2010

It will be shown that a particular quotient ring of a ring

R
R

is a nontrivial

V
V

-ring if

R
R

is a special kind of a right hereditary, right Noetherian ring. Another result states that all overrings of a right and left Goldie

V
V

-ring, which is semiartinian modulo every essential right ideal, are

V
V

-rings.
Duality between 𝐻^{𝑝} and 𝐻^{𝑞} and associated projections
Authors: Pranger Walter
Published: June 2010

If

U

/

G

U/G

represents a Riemann surface as the disk

U
U

modulo a discontinuous group

G
G

and if

L
p

/

G

{L^p}/G

denotes the

L
p

{L^p}

functions on the circle which are

G
G

invariant, then it is shown that

L
p

/

G
=

N
p

⊕

K
p

{L^p}/G = {N_p} \oplus {K_p}

if and only if

H
p

/

G

{H^p}/G

and

H
¯

q

/

G

{\bar H^q}/G

are naturally dual. Here

K
p

{K_p}

is the subset of

L
p

/

G

{L^p}/G

consisting of those functions which are invariant and whose conjugates are invariant;

N
p

{N_p}

is

E
(

H
p

)
∩
E
(

H
¯

0
p

)

E({H^p}) \cap E(\bar H_0^p)

where

E
E

is the conditional expectation operator.

H
p

{H^p}

is the space of boundary values of holomorphic functions and

1
>
p
>
∞

1 > p > \infty

.
Spectra of compact composition operators
Authors: Caughran James G. , Schwartz Howard J.
Published: June 2010

Let

ϕ
\phi

be holomorphic and map the open unit disk into itself, and let

C
ϕ

:
f
→
f
∘
ϕ

{C_\phi }:f \to f \circ \phi

be the composition operator on

H
2

{H^2}

generated by

ϕ
\phi

. If

C
ϕ

{C_\phi }

is a compact operator then

(
1
)
ϕ
(

z
0

)
=

z
0

(1)\phi ({z_0}) = {z_0}

for some

z
0

ϵ
D

{z_0} \epsilon D

;
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 2 right-parenthesis sigma left-parenthesis upper C Subscript phi Baseline right-parenthesis equals left-brace phi prime left-parenthesis z 0 right-parenthesis Superscript n Baseline colon n equals 0 comma 1 comma 2 comma ellipsis right-brace union StartSet 0 EndSet">

(
2
)
σ
(

C
ϕ

)
=
{

ϕ
′

(

z
0

)
n

:
n
=
0
,
1
,
2
,
…
}
∪
{
0
}

(2)\sigma ({C_\phi }) = \{ \phi ’{({z_0})^n}:n = 0,1,2, \ldots \} \cup \{ 0\}

.
A generalization of Lusin’s theorem
Authors: Wage Michael L.
Published: June 2010

In this note we characterize

σ
\sigma

-finite Riesz measures that allow one to approximate measurable functions by continuous functions in the sense of Lusin’s theorem. We call such measures Lusin measures and show that not all

σ
\sigma

-finite measures are Lusin measures. It is shown that if a topological space

X
X

is either normal or countably paracompact, then every measure on

X
X

is a Lusin measure. A counterexample is given to show that these sufficient conditions are not necessary.
A note on a basis problem
Authors: Anderson J. M.
Published: June 2010

It is shown that the functions

{
exp
−

λ
ν

x

}

ν
=
1

∞

\{ \exp - {\lambda _\nu }x\} _{\nu = 1}^\infty

form a basis for the subspace of

L

2

(
0
,
∞
)

{\mathcal {L}_2}(0,\infty )

which they span if and only if
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="inf Underscript mu Endscripts product Underscript nu equals 1 semicolon nu not-equals mu Overscript normal infinity Endscripts StartAbsoluteValue StartFraction lamda Subscript nu Baseline minus lamda Subscript mu Baseline Over lamda overbar Subscript nu Baseline plus lamda Subscript mu Baseline EndFraction EndAbsoluteValue equals delta greater-than 0 period">

inf
μ

∏

ν
=
1
;
ν
≠
μ

∞

|

λ
ν

−

λ
μ

λ
¯

ν

+

λ
μ

|

=
δ
>
0.

\inf \limits _\mu \prod \limits _{\nu = 1;\nu \ne \mu }^\infty {|\frac {{{\lambda _\nu } - {\lambda _\mu }}}{{{{\bar \lambda }_\nu } + {\lambda _\mu }}}| = \delta > 0.}

\]
The proof uses certain estimates concerning interpolation in

H
2

{H_2}

due to Shapiro and Shields. The proof makes explicit a construction embedded in a paper of Binmore [1, Theorems 9-12.].
A totally real surface in 𝐶𝑃² that is not totally geodesic
Authors: Ludden Gerald D. , Okumura Masafumi , Yano Kentaro
Published: June 2010

An example of a totally real surface immersed in complex projective space is given. This surface is not totally geodesic. The relation of this example to previous theorems on totally real submanifolds is given.
A generalization of absolute retracts
Authors: Martin John R.
Published: June 2010

In this paper the concept of an absolute retract is generalized to a new concept which we call an absolute approximate retract. It is shown that for the class of compact metric spaces, the class of absolute approximate retracts contains the class of absolute retracts and lies in the class of contractible Peano continua.
Equivalence of 5-dimensional 𝑠-cobordisms
Authors: Scharlemann Martin
Published: June 2010

The classification of

5
5

-dimensional

h
h

-cobordisms given by Cappell, Lashof, and Shaneson is here strengthened and extended to

s
s

-cobordisms when the ends of the

s
s

-cobordism are smooth.
The Picard sequences of a fibration
Authors: Magid Andy R.
Published: June 2010

A fibration, in the Zariski topology, of algebraic varieties leads to an exact sequence of cohomology of the relative units functor. Under suitable hypotheses, the exact sequence can be interpreted as a sequence including the Picard groups of the base variety, the total space, and the fibre.
Cyclotomic splitting fields for group characters
Authors: Benard Mark
Published: June 2010

This paper is concerned with cyclotomic splitting fields for a real-valued irreducible character of a finite group. The fields considered are of the form

Q
(

ϵ
m

)

Q({\epsilon _m})

, where

m
m

is either an odd prime or a power of

2
2

.
On a problem of J. L. Taylor
Authors: Izuchi Keiji
Published: June 2010

Let

S
S

be the structure semigroup of a measure algebra

M
(
G
)

M(G)

and

K
K

be the union of all maximal groups of

S
S

. Taylor proposed the following problem: Are there L. C. A. groups

G
G

with nontrivial measures concentrated on

K
K

? The purpose of this paper is to give a positive solution to this problem.
A curvature obstruction to covering deformations of connections by metric deformations
Authors: Ehrlich Paul E.
Published: June 2010

A curvature obstruction to covering a deformation of connections of a Levi-Civita connection of a Riemannian metric by a metric deformation is derived and used to study deformations of connections related to the deformation of connections arising from a conformal deformation of a Riemannian metric.
Semilattices on Peano continua
Authors: Williams W. Wiley
Published: June 2010

A continuum is cell-cyclic if every cyclic element is an

n
n

-cell for some integer

n
n

. It is shown that every cell-cyclic Peano continuum admits a topological semilattice.
The Lévy-Baxter theorem for Gaussian random fields: a sufficient condition
Authors: Kawada Takayuki
Published: June 2010

For a function

F
F

of some class and for a Gaussian random field

{
X
(

t

)
;

t

ϵ

[
0
,

1

]
N

}

\{ X({\text {t}});\;{\text {t}}\;\epsilon {[0,\;1]^N}\}

,

F
F

-variation

V

F
,
n

(
X
)

{V_{F,n}}(X)

is defined as

V

F
,
n

(
X
)
=

∑

i

F
(

X

i

,
n

)

{V_{F,n}}(X) = {\sum _{\text {i}}}F({X_{{\text {i}},n}})

, where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Subscript i comma n Baseline equals normal upper Delta Subscript upper N Superscript s Baseline midline-horizontal-ellipsis normal upper Delta 1 Superscript s Baseline upper X left-parenthesis s i right-parenthesis comma s equals 2 Superscript negative n">

X

i

,
n

=

Δ
N
s

⋯

Δ
1
s

X
(
s

i

)
,

s
=

2

−
n

{X_{{\text {i}},n}} = \Delta _N^s \cdots \Delta _1^sX(s{\text {i}}),\;s = {2^{ - n}}

, and

Δ
k
s

\Delta _k^s

is the difference operator in the

k
th

k\operatorname {th}

component of

i

=
(

i
1

,
⋯
,

i
N

)

{\text {i}} = ({i_1}, \cdots ,{i_N})

,

(
1
⩽

i
k

⩽

2
n

;
k
=
1
,
…
,
N
)

(1 \leqslant {i_k} \leqslant {2^n};k = 1, \ldots ,N)

. Here is presented a sufficient condition for the existence with probability

1
1

of the limit of the normalization of

V

F
,
n

(
X
)

{V_{F,n}}(X)

as

n
→
∞

n \to \infty

.
Contracted ideals and purity for ring extensions
Authors: Brewer J. W. , Costa D. L.
Published: June 2010

In this paper an example is given of a pair of commutative noetherian rings

R
⊆
S

R \subseteq S

with

S
S

a finite

R
R

-module and

I
S
∩
R
=
I

IS \cap R = I

for each ideal

I
I

of

R
R

, but having the property that

0
→
R
→
S

0 \to R \to S

is not a pure sequence of

R
R

-modules. Purity of the sequence

0
→
R
→
S

0 \to R \to S

is equivalent to

R
[
X
]

R[X]

being “ideally closed” in

S
[
X
]
,

X

S[X],\;X

an indeterminate. Therefore, the example renders appealing the proposition that for

R
R

noetherian and

S
S

a noetherian torsion-free

R
R

-algebra containing

R
R

, if

α
S
∩
R
=
α
R

\alpha S \cap R = \alpha R

for each non-zero-divisor

α
ϵ
R

\alpha \epsilon R

, then the extension

R
[
X
]
⊆
S
[
X
]

R[X] \subseteq S[X]

has the same properties. Finally, it is also shown that for

R
R

noetherian and

0
→
R
→
S

0 \to R \to S

pure, with

S
S

an

R
R

-algebra, then

R
[
[

X
1

,
…
,

X
n

]
]

R[[{X_1}, \ldots ,{X_n}]]

is pure in

S
[
[

X
1

,
…
,

X
n

]
]

S[[{X_1}, \ldots ,{X_n}]]

for each positive integer

n
n

.
Measurability of norm proved by Haar functions
Authors: Finlayson H. C.
Published: June 2010

A proof is given of the measurability of the supremum norm on Wiener space.
Operators satisfying certain growth conditions
Authors: Patel S. M. , Gupta B. C.
Published: June 2010

Let

T
T

be an operator on a complex Hilbert space

H
H

. Some growth conditions on operator radius of the resolvent of

T
T

are studied. Moreover, it is shown that the conjecture, due to V. Istrătescu, that for operators

T
T

satisfying growth condition

(

G

1

)

({{\text {G}}_1})

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript StartAbsoluteValue EndAbsoluteValue x StartAbsoluteValue EndAbsoluteValue equals 1 Endscripts left-brace StartAbsoluteValue EndAbsoluteValue upper T x StartAbsoluteValue EndAbsoluteValue squared minus StartAbsoluteValue left-parenthesis upper T x comma x right-parenthesis EndAbsoluteValue squared right-brace equals upper R Subscript upper T Superscript 2 Baseline comma">

sup

|

|

x

|

|

=
1

{

|

|

T
x

|

|

2

−

|

(
T
x
,

x
)

|

2

}
=

R
T
2

,

\sup \limits _{||x|| = 1} \;\{ ||Tx|{|^2} - |(Tx,\;x){|^2}\} = R_T^2,

\]
where

R
T

{R_T}

is the radius of the smallest circular disk containing the spectrum

σ
(
T
)

\sigma (T)

, turns out to be false.
Rational approximation to solutions of linear differential equations with algebraic coefficients
Authors: Schmidt Wolfgang M.
Published: June 2010

Let

K
K

be the field of formal series

α
=

a
k

t
k

+
⋯
+

a
0

+

a

−
1

t

−
1

+
⋯

\alpha = {a_k}{t^k} + \cdots + {a_0} + {a_{ - 1}}{t^{ - 1}} + \cdots

and let

|

|

|\;\;|

be the valuation with

|

α

|

=

2
k

|\alpha | = {2^k}

if

a
k

≠
0

{a_k} \ne 0

. Suppose

α
ϵ
K

\alpha \epsilon K

satisfies an

m

th

m{\text {th}}

order linear differential equation whose coefficients are algebraic functions of

t
t

. Then for

ϵ
>
0

\epsilon > 0

there are only finitely many rational functions

p
(
t
)

/

q
(
t
)

p(t)/q(t)

with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue alpha minus p left-parenthesis t right-parenthesis slash q left-parenthesis t right-parenthesis EndAbsoluteValue greater-than StartAbsoluteValue q left-parenthesis t right-parenthesis EndAbsoluteValue Superscript negative 2 minus 4 m minus epsilon">

|

α
−
p
(
t
)

/

q
(
t
)

|

>

|

q
(
t
)

|

−
2
−
4
m
−
ϵ

|\alpha - p(t)/q(t)| > |q(t){|^{ - 2 - 4m - \epsilon }}

.