bluuebunny/crossref_metadata_2025 huggingface dataset:
Effect of nitrogen doping on TiOxNy thin film formation at reactive high-power pulsed magnetron sputtering
Authors: Stranak Vitezslav , Quaas Marion , Bogdanowicz Robert , Steffen Hartmut , Wulff Harm , Hubicka Zdenek , Tichy Milan , Hippler Rainer
Published: June 2010

The paper is focused on a study of formation of TiO
x
N
y
thin films prepared by pulsed magnetron sputtering of metallic Ti target. Oxygen and nitrogen were delivered into the discharge in the form of reactive gases O2 and N2. The films were deposited by high-power impulse magnetron sputtering working with discharge repetition frequency f = 250 Hz at low (p = 0.75 Pa) and high (p = 10 Pa) pressure. The substrates were on floating potential and thermally stabilized at room temperature during the deposition process. Post-deposition thermal annealing was not employed. The chemical composition from x-ray photoelectron spectroscopy diagnostic reveals formation of TiO
x
N
y
structure at low flow rate of oxygen in the discharge gas mixture. This result is confirmed by XRD investigation of N element's incorporation into the Ti–O lattice. Decrease in band-gap to values E
g ∼ 1.6 eV in TiO
x
N
y
thin film is attributed to formed Ti–N bonds. The discharge properties were investigated by time-resolved optical emission spectroscopy. Time evolution of particular spectral lines (Ar+, Ti+, Ti) is presented together with peak discharge current.
Nucleation and growth evolution of InP dots on InGaP/GaAs
Authors: Bortoleto J R R , Gazoto A , Brasil M J S P , Meneses E A , Cotta M A
Published: June 2010

We have investigated the evolution in the heteroepitaxy of InP on InGaP/GaAs (0 0 1) layers from the initial formation of a strained wetting layer (WL) up to the development of quantum dots. Atomic force microscopy and RHEED as well as continuous and time-resolved photoluminescence measurements provide evidence of four main stages for InP growth evolution. Our results indicate that the InP dot formation occurs according to the Stranski–Krastanov growth mode coupled to a WL erosion mechanism driven by the spatially variable strain field present in the WL. Moreover, the correlation of morphological and optical data indicates the stability of dot shapes and interfaces in this system.
Curvature-insensitive methodology for thermal-wave depth-profilometry in multi-layered curvilinear solids
Authors: Liu Liwang , Wang Chinhua , Yuan Xiao , Mandelis Andreas
Published: June 2010

A generalized similarity normalization (SN) methodology for characterizing depth profiles of continuously varying thermophysical properties in curvilinear (cylindrical and spherical) solids is presented. Specifically, the principle and the physical mechanism of the elimination of the surface curvature effect from the overall photothermal signal is introduced based on theoretical models of cylindrical, spherical and flat solids with multi-layer structures. The effects of the relative values of radii of curvature of the curvilinear solid, the thickness of the inhomogeneous surface layer and the measurement azimuthal angle on the validity of the technique are discussed in detail. Experimental reconstructions of thermophysical depth profiles of hardened cylindrical steel rods of various diameters are performed based on both curvilinear theory and the equivalent flat surface theory. The reconstructed results are compared and validated.
Long-range order and thermal stability of thin Co2FeSi films on GaAs(1 1 1)B
Authors: Jenichen B , Herfort J , Kumakura K , Trampert A
Published: June 2010

Co2FeSi/GaAs(1 1 1)B hybrid structures are grown by molecular-beam epitaxy and characterized by transmission electron microscopy (TEM) and x-ray diffraction. The lattice matched Co2FeSi films grow in a three-dimensional island growth mode at substrate temperatures T
S between T
S = 100 and 425 °C. The structures have a stable interface up to T
S = 275 °C. The films contain fully ordered L21 and partially ordered B2 phases. The spatial distribution of long-range order in Co2FeSi is characterized using a comparison of TEM images taken with superlattice reflections and the corresponding fundamental reflections. The spatial inhomogeneities of long-range order can be explained by local non-stoichiometry due to lateral segregation or stress relaxation without formation of extended defects.
On three basic results in the theory of stationary point processes
Authors: Leadbetter M. R.
Published: June 2010

A simple, unified treatment is given for three basic results in the theory of stationary streams of events (point processes). These results are the basic theorem of Khintchine concerning the existence of a stream intensity, Korolyuk’s theorem, and a fundamental lemma of Dobrushin giving sufficient conditions for a stationary stream to be orderly in the sense of Khintchine [2].
Boundary zeros of functions with derivative in 𝐻^{𝑝}
Authors: Caveny D. J. , Novinger W. P.
Published: June 2010

It is known that the set of boundary zeros of a function, analytic in the unit disc and with derivative in the Hardy class

H
p

{H^p}

, is a Carleson set provided

p
>
1

p > 1

. In this paper a proof is given which includes the case

p
=
1

p = 1

. Peak sets for such functions are investigated and sufficient conditions on subsets of the boundary are given, which guarantee that they are peak sets.
A note on isomorphisms of groups algebras
Authors: Wood Geoffrey V.
Published: June 2010

In this note, it is shown that, if

G
1

,

G
2

{G_1},\;{G_2}

are compact groups, and

C
(

G
1

)
,

C
(

G
2

)

C({G_1}),\;C({G_2})

are the (convolution) algebras of continuous, complex-valued functions on

G
1

{G_1}

and

G
2

{G_2}

respectively, then the existence of a norm-decreasing algebra-isomorphism of

C
(

G
1

)

C({G_1})

onto

C
(

G
2

)

C({G_2})

ensures that the groups are isomorphic. The corresponding theorem with

G
1

{G_1}

and

G
2

{G_2}

locally finite discrete groups is also proved.
A note on Palais’ axioms for section functors
Authors: Riddell R. C.
Published: June 2010

By a slight strengthening of one axiom, a technical slip is corrected in R. S. Palais’ proof of a basic lemma on functors from vector bundles over compact manifolds to Banach spaces of sections.
Nonconvex linear topologies with the Hahn Banach extension property.
Authors: Gregory D. A. , Shapiro J. H.
Published: June 2010

Let

⟨
E
,

E
′

⟩

\langle E,E’\rangle

be a dual pair of vector spaces. It is shown that whenever the weak and Mackey topologies on

E
E

are different there is a nonconvex linear topology between them. In particular this provides a large class of nonconvex linear topologies having the Hahn Banach Extension Property.
Norm convergent expansions for Gaussian processes in Banach spaces.
Authors: Jain Naresh C. , Kallianpur G.
Published: June 2010

Several authors have recently shown that Brownian motion with continuous paths on

[
0
,
1
]

[0,1]

can be expanded into a uniformly convergent (a.s.) orthogonal series in terms of a given complete orthonormal system (CONS) in its reproducing kernel Hilbert space (RKHS). In an earlier paper we generalized this result to Gaussian processes with continuous paths. Here we obtain such expansions for a Gaussian random variable taking values in an arbitrary separable Banach space. A related problem is also considered in which starting from a separable Hilbert space

H
H

with a measurable norm

|

|

⋅

|

|

1

|| \cdot |{|_1}

defined on it, it is shown that the corresponding abstract Wiener process has a

|

|

⋅

|

|

1

||\cdot |{|_1}

-convergent orthogonal expansion in terms of a CONS chosen from

H
H

.
A special basis for 𝐶([0,1])
Authors: Warren H. E.
Published: June 2010

This paper constructs a basis for

C
(
[
0
,
1
]
)

C([0,1])

which converges weakly to zero whose elements are nevertheless norm bounded away from zero.
Submanifolds in a euclidean hypersphere
Authors: Chen Bang-yen
Published: June 2010

Let

M
M

be an oriented closed

n
n

-dimensional submanifold of a euclidean (

(
n
+
N
)

(n + N)

-space

E

n
+
N

{E^{n + N}}

. Let

X
X

and

H
H

be the position vector field and the mean curvature vector field of

M
M

in

E

n
+
N

{E^{n + N}}

. Then

M
M

is contained in a hypersphere of

E

n
+
N

{E^{n + N}}

centered at

c
c

when and only when either

(
X
−
c
)
⋅
H
≧
−
1

(X - c) \cdot H \geqq - 1

or

(
X
−
c
)
⋅
H
≦
−
1

(X - c)\cdot H \leqq - 1

.
On the characteristic roots of real matrices
Authors: Schaefer H. H.
Published: June 2010

If

A
A

is a real

n
×
n

n \times n

-matrix whose absolute

|

A

|

|A|

has spectral radius 1, and if

ε
\varepsilon

is a unimodular characteristic value of

A
A

, then all odd (respectively, even) powers of

ε
\varepsilon

are characteristic values of

A
A

(respectively, of

|

A

|

|A|

). In particular, such

ε
\varepsilon

must be a

k
k

th root of unity for some

k
,
1
≦
k
≦
2
n

k, 1 \leqq k \leqq 2n

.
The orthomodular identity and metric completeness of the coordinatizing division ring
Authors: Morash Ronald P.
Published: June 2010

Let

F
F

be any division subring of the real quaternions

H
H

. Let

l
2

(
F
)

{l_2}(F)

denote the linear space of all square summable sequences from

F
F

and let

L
L

denote the lattice of all “

⊥
\bot

-closed” subspaces of

l
2

(
F
)

{l_2}(F)

, where “

⊥
\bot

” denotes the orthogonality relation derived from the

H
H

-valued form

(
a
,
b
)
=

∑

i
=
1

∞

a
i

b
¯

i

(a,b) = \sum \nolimits _{i = 1}^\infty {{a_i}{{\overline b }_i}}

where

a
,
b
∈

l
2

(
F
)
,
a
=
(

a
i

;
i
=
1
,
2
,
⋯
)

a,b \in {l_2}(F),a = ({a_i};i = 1,2, \cdots )

and

b
=
(

b
i

;
i
=
1
,
2
,
⋯
)

b = ({b_i};i = 1,2, \cdots )

. Then

L
L

is complete, orthocomplemented,

M
M

-symmetric, irreducible, atomistic, and separable, but

L
L

is orthomodular if and only if

F
F

is either the reals, the complex numbers, or the quaternions.
A comparison theorem
Authors: Leighton Walter , Ke William Oo Kian
Published: June 2010

In this paper the authors consider a pair of differential equations

y
1

+

p
1

(
x
)

y
1

=
0
,

y
2

+

p
2

(
x
)

y
2

=
0

{y_1} + {p_1}(x){y_1} = 0,{y_2} + {p_2}(x){y_2} = 0

, where

p
i

(
x
)

{p_i}(x)

are positive and continuous, and where solutions

y
1

(
x
)

{y_1}(x)

and

y
2

(
x
)

{y_2}(x)

have common consecutive zeros at

x
=
a

x = a

and

x
=
b

x = b

. They show that if the curves

y
=

p
1

(
x
)

y = {p_1}(x)

and

y
=

p
2

(
x
)

y = {p_2}(x)

have a single intersection (possibly a closed subinterval) and if

p
1

(
a
)
>

p
2

(
a
)
,

p
2

(
b
)
>

p
1

(
b
)

{p_1}(a) > {p_2}(a),{p_2}(b) > {p_1}(b)

, the first conjugate point of

a
+

ε

a + {\varepsilon }

(

ε

>
0

{\varepsilon } > 0

and small) for the second equation precedes that of the first.
Completion of norms for 𝐶(𝑋,𝑄)
Authors: Luchins Edith H.
Published: June 2010

Let

C
(
X
,
Q
)

C(X,Q)

denote the algebra of all continuous quaternion-valued functions vanishing at infinity on a locally compact Hausdorff space

X
X

. Under the natural norm (the sup norm) and under the spectral radius norm,

r
(
f
)

r(f)

, which is equivalent to the sup norm,

C
(
X
,
Q
)

C(X,Q)

is a Banach algebra. Let

δ
(
f
)

\delta (f)

be any multiplicative norm for

C
(
X
,
Q
)

C(X,Q)

; i.e., one under which it is a normed algebra. It is shown that

δ
(
f
)

\delta (f)

, whether or not it is complete, majorizes the natural norm and

r
(
f
)

r(f)

. Under certain conditions on the radical of the completion of

δ
(
f
)
,
δ
(
f
)

\delta (f),\delta (f)

is equivalent to the natural norm and

r
(
f
)

r(f)

.
Paracompactness and elastic spaces
Authors: Tamano Hisahiro , Vaughan J. E.
Published: June 2010

This paper gives a characterization of paracompactness, and introduces the notion of an elastic space which generalizes the concept of a stratifiable (in particular, metric) space.
Continuous and proper decompositions
Authors: Williams G. K.
Published: June 2010

If

X
X

is a locally connected, locally peripherally compact Hausdorff space and if

R
R

is an equivalence relation on

X
X

with fibers which are connected with compact boundaries, then it is shown that three definitions of continuity of

R
R

are equivalent. Some of the propositions used to obtain this result are then applied to get sufficient conditions for a decomposition of certain types of metric spaces to be proper.
Partition theorems for Euler pairs
Authors: Subbarao M. V.
Published: June 2010

This paper generalizes and extends the recent results of George Andrews on Euler pairs. If

S
1

{S_1}

and

S
2

{S_2}

are nonempty sets of natural numbers, we define

(

S
1

,

S
2

)

({S_1},{S_2})

to be an Euler pair of order

r
r

whenever

q
r

(

S
1

;
n
)
=
p
(

S
2

;
n
)

{q_r}({S_1};n) = p({S_2};n)

for all natural numbers

n
n

, where

q
r

(

S
1

;
n
)

{q_r}({S_1};n)

denotes the number of partitions of

n
n

into parts taken from

S
1

{S_1}

, no part repeated more than

r
−
1

r - 1

times

(
r
>
1
)

(r > 1)

, and

p
(

S
2

;
n
)

p({S_2};n)

the number of partitions of

n
n

into parts taken from

S
2

{S_2}

. Using a method different from Andrews’, we characterize all such pairs, and consider various applications as well as an extension to vector partitions.
Coefficients of functions with bounded boundary rotation
Authors: Noonan James W.
Published: June 2010

For

k
≧
2

k \geqq 2

denote by

V
k

{V_k}

the class of normalized functions analytic in the unit disc which have boundary rotation at most

k
π

k\pi

. For fixed

n
≦
(
k
+
6
)

/

4

n \leqq (k + 6)/4

we determine the maximum of the set of values of

|

a
n

|

\left | {{a_n}} \right |

, where

a
n

{a_n}

is the nth Taylor coefficient of a function in

V
k

{V_k}

.
An extremal property of stochastic integrals
Authors: Rosencrans Steven
Published: June 2010

In this paper we consider the stochastic integral

y
t

=

∫
0
t

e
(
s
,
b
)
d

b
s

{y_t} = \int _0^t {e(s,b)d{b_s}}

of a nonanticipating Brownian functional

e
e

that is essentially bounded with respect to both time and the Brownian paths. Let

f
f

be a convex function satisfying a certain mild growth condition. Then

E
f
(

y
t

)
≦
E
f
(

|

|

e

|

|

b
t

)

Ef({y_t}) \leqq Ef(||e||{b_t})

, where

b
t

{b_t}

is the position at time

t
t

of the Brownian path

b
b

. As a corollary, sharp bounds are obtained on the moments of

y
t

{y_t}

. The key point in the proof is the use of a transformation, derived from Itô’s lemma, that converts a hyperbolic partial differential equation into a parabolic one.
Semimodular posets and the Jordan-Dedekind chain condition
Authors: Haskins L. , Gudder S.
Published: June 2010

In this paper it is shown that an upper semimodular poset without infinite chains satisfies the Jordan-Dedekind chain condition. This corrects an error in Theorem 14, p. 40 of [1] and generalizes that theorem.
Loewy length of perfect rings
Authors: Osofsky B. L.
Published: June 2010

We construct a family of left and right perfect rings whose left and right Loewy lengths are arbitrary preassigned infinite ordinals.
On Lie rings satisfying the fourth Engel condition
Authors: Putcha Mohan S.
Published: June 2010

In this paper we prove that a Lie ring of characteristic prime to 2, 3 and 5, satisfying the fourth Engel condition, is nilpotent.
A functor to ringed spaces
Authors: Carns Gail L.
Published: June 2010

With the set of orders

O

\mathcal {O}

on a field and the Harrison topology induced from the set of all primes as a base space we define a ringed space

(

O

,

F

)

(\mathcal {O},\mathcal {F})

. For each field homomorphism we find an associated ringed space morphism producing a contravariant functor from the category of fields to the category of ringed spaces. An equivalence relation

∼
\sim

is defined on the set of orders and again a ringed space

(

O

/

∼
,

F

¯

)

(\mathcal {O}/\sim , \bar {\mathcal {F}})

and a contravariant functor from fields to ringed spaces is obtained along with a natural transformation from the first to the second functor. Finally, we obtain a ringed space morphism

(

O

/

∼
,

F

¯

)
→
(
Y
,

O

Y

)

(\mathcal {O}/\sim , \bar {\mathcal {F}}) \to (Y, \mathcal {O}_Y)

where Y is the spectrum of the ring of bounded elements and

O

Y

{\mathcal {O}_Y}

is the structure sheaf.
Direct sums of countably generated modules over complete discrete valuation rings
Authors: Bang Chang Mo
Published: June 2010

Throughout this paper,

R
R

will denote an arbitrary but fixed complete discrete valuation ring. We shall show that two reduced

R
R

-modules which are direct sums of countably generated

R
R

-modules are isomorphic if and only if they have the same Ulm invariants and the same basis type. This is a generalization of the celebrated Ulm and Kolettis theorem.
Logarithmic derivative of an entire function
Authors: Marden Morris
Published: June 2010

A representation for the logarithmic derivative

(

f
′

/

f
)

(f’/f)

of an entire function

f
f

of finite order, parametrically in terms of some zeros and critical points of

f
f

, is derived from the Hadamard representation and applied to Lucas’ type theorems and to growth estimates of

f
′

/

f

f’/f

.
Completeness of {𝑠𝑖𝑛𝑛𝑥+𝐾𝑖 𝑐𝑜𝑠𝑛𝑥}
Authors: Ginsberg Jonathan I.
Published: June 2010

Let

C

[

a
,
b

]

{\mathbf {C}}[ {a,b} ]

be the space of continuous functions on the interval

[

a
,
b

]

[ {a,b} ]

. It is shown that the set of functions

{

sin
⁡
n
x
+
K
i
cos
⁡
n
x

}

n
=
1

∞

,
K
≠
±
1

\{ {\sin nx + Ki\cos nx} \}_{n = 1}^\infty ,K \ne \pm 1

, is incomplete in

C

[

0
,
π
+
a

]
,
a
>
0

{\mathbf {C}}[ {0,\pi + a} ],a > 0

.
Functions which operate in the Fourier algebra of a compact group
Authors: Rider Daniel
Published: June 2010

Let

A
(
G
)

A(G)

be the Fourier algebra of a compact group

G
G

. It is shown that a function defined on a closed convex subset of the plane operates in

A
(
G
)

A(G)

if and only if it is real analytic. This was shown by Helson, Kahane, Katznelson and Rudin when

G
G

is locally compact and abelian and by Dunkl when

G
G

is compact and contains an infinite abelian subgroup. A direct proof is given of the following lemma which is all that is needed in order to apply the proof of Helson, Kahane, Katznelson and Rudin (

|

|

|

|

||\;||

is the Fourier algebra norm). Lemma. Let

r
>
0

r > 0

and

S
r

{S_r}

be the set of

f
∈
A
(
G
)

f \in A(G)

such that

f
f

is real and

|

|

f

|

|

=
r

||f|| = r

. Then
\[

sup

f
∈

S
r

|

|

e

i
f

|

|

=

e
r

.

\sup \limits _{f \in {S_r}} ||{e^{if}}|| = {e^r}.

\]
Counterexample to a question on commutators
Authors: McIntosh Alan
Published: June 2010

We show that it is possible for two selfadjoint operators A and B in a Hilbert space H with bounded commutator

A
B
−
B
A

AB - BA

to have the property that

|
A
|

B
−
B

|
A
|

\left | A \right |B - B\left | A \right |

is unbounded (where

|
A
|

\left | A \right |

denotes the positive square root of

A
2

{A^2}

). The proof reduces to showing that for all natural numbers n, there exist a bounded positive operator U and a bounded operator V satisfying

‖

U
V
−
V
U

‖

≧
n

‖

U
V
+
V
U

‖

\left \| {UV - VU} \right \| \geqq n\left \| {UV + VU} \right \|

.
On the existence of double singular integrals for kernels without smoothness
Authors: Walsh T.
Published: June 2010

Calderón and Zygmund have proved the pointwise convergence of singular integrals in

R
n

{R^n}

for locally integrable homogeneous kernels whose even part is locally in

L
L

log

L
L

by change to polar coordinates and use of the boundedness in

L
p

{L^p}

of the maximal operator of the one-dimensional Hilbert transformation. The present note shows how analogous results for double singular integrals can be derived from boundedness of the maximal operator of the double Hilbert transform.
A characterization of a space with countable infinity
Authors: Okuyama Akihiro
Published: June 2010

It is well known that, for a countable discrete space

N
N

,
\[

|

β
N
−
N

|

=

2

2

ℵ
0

.

|\beta N - N| = {2^{{2^{{\aleph _0}}}}}.

\]
So, any completely regular

T
1

{T_1}

space

X
X

with

β
X
−
X

|

≦

ℵ
0

\beta X - X| \leqq {\aleph _0}

does not contain any infinite discrete subspace. In this paper, we characterize those completely regular

T
1

{T_1}

spaces with countable infinity as follows: Such a space

X
X

is characterized by the two properties. (a)

X
X

is pseudocompact. (b) There exist a compact metric space

Y
Y

and a continuous map

f
f

from

X
X

onto

Y
Y

so that the subset

Y
0

=
{
y
:

f

−
1

(
y
)

{Y_0} = \{ y:{f^{ - 1}}(y)

of

Y
Y

is countable and

c

l

β
X

f

−
1

(
y
)
−

f

−
1

(
y
)

c{l_{\beta X}}{f^{ - 1}}(y) - {f^{ - 1}}(y)

is one point whenever

y
∈

Y
0

y \in {Y_0}

. (In particular, for any

y
y

in

Y
0

,
β
(

f

−
1

(
y
)
)
−

f

−
1

(
y
)

{Y_0},\beta ({f^{ - 1}}(y)) - {f^{ - 1}}(y)

is one point if

X
X

is normal.)
A general three-series theorem
Authors: Brown B. M.
Published: June 2010

Let

{
Ω
,

F

,
P
}

\{ \Omega ,\mathcal {F},P\}

be a probability space. The subset of

Ω
\Omega

on which an arbitrary sequence of random variables converges is shown to be equivalent to the intersection of three other sets, each specified by the almost sure convergence of a certain sequence of random variables. Kolmogorov’s three-series theorem, which gives necessary and sufficient conditions for the almost sure convergence of a sequence of sums of independent random variables, is obtainable as a particular case of the present result.
On some products involving primes
Authors: Uchiyama S.
Published: June 2010

Asymptotic formulae are given for the products

P
l

(
x
)
(
l
=
1
,
3
)

{P_l}(x)(l = 1,3)

defined below.
Note on a theorem of Pall
Authors: Williams Kenneth S.
Published: June 2010

A simple proof is given of Pall’s formula for the number of representations of a gaussian integer as the sum of two squares of gaussian integers.
Bounded approximation by polynomials whose zeros lie on a circle
Authors: Rubinstein Zalman , Saff E. B.
Published: June 2010

In a recent paper the first author gave an explicit construction of a sequence of polynomials having their zeros on the unit circumference which converge boundedly to a given bounded zero-free analytic function in the unit disk. In this paper we find the best possible uniform bound for such approximating polynomials and construct a sequence for which this bound is attained. The method is also applied to approximation of an analytic function in the unit disk by rational functions whose poles lie on the unit circumference. Some open problems are discussed.
On determination of the optimal factor of a nonnegative matrix-valued function
Authors: Salehi Habib
Published: June 2010

Let

F
=
[

f

i
j

]

F = [f_{ij}]

,

1
>
i

1 > i

,

j
≦
q

j \leqq q

, be a measurable, nonnegative definite

q
×
q

q \times q

matrix-valued function defined on the unit circle

C
C

. It is known that when

F

\mathbfit {F}

and

log
⁡
det

F

\log \det \textbf {F}

are in

L
1

(
C
)

L_1(C)

,

F

\mathbfit {F}

admits a factorization of the form

F
=

Φ

Φ

∗

F = \mathbf {\Phi } \mathbf {\Phi }^\ast

, where

Φ

\mathbf {\Phi }

is an optimal, full rank function in

L
2

0
+

(
C
)

L_2^{0+}(C)

. Under the additional assumption that

{
(

∏

i
=
1

q

f

i
i

)

/

det
F
}

\{ (\prod \nolimits _{i = 1}^q f_{ii})/\det F\}

is in

L
1

(
C
)

L_1(C)

, an iterative procedure which yields an infinite series for

Φ

\mathbf {\Phi }

in terms of

F

\mathbfit {F}

is given. The optimal function

Φ

\mathbf {\Phi }

plays a significant role in the multivariate prediction theory of stochastic processes. The present work generalizes the results of several authors concerning the determination of the optimal factor

Φ

\mathbf {\Phi }

.
On rings with a higher derivation
Authors: Satô Shizuka
Published: June 2010

Let

R
⊃

O

R \supset \mathcal {O}

be two rings with the unit 1. Then we set

R

(

O

,
R
)
=
{
x
∈
R
;

x
r

∈

O

\mathcal {R}(\mathcal {O},R) = \{ x \in R;{x^r} \in \mathcal {O}

for some integer

r
≧
1
}

r \geqq 1\}

. At first, it is shown that, under some assumptions,

d

O

⊂

O

d\mathcal {O} \subset \mathcal {O}

implies

d

R

(

O

,
R
)
⊂

R

(

O

,
R
)

d\mathcal {R}(\mathcal {O},R) \subset \mathcal {R}(\mathcal {O},R)

. Next, with the Lying-over Theorem on d-differential ideals, we show: Let (R, M) and

(

O

,
m
)

(\mathcal {O},m)

be two quasi-local rings and let d be a higher derivation of rank

∞
\infty

of the total quotient ring of R such that

d

O

⊂

O

d\mathcal {O} \subset \mathcal {O}

. Suppose that R is integral over

O

\mathcal {O}

and

O

\mathcal {O}

is dominated by R. Then

d
(
m
)
⊂
m

d(m) \subset m

implies

d
(
M
)
⊂
M

d(M) \subset M

.
Sigma-amenable locally compact groups
Authors: Jenkins J. W.
Published: June 2010

Let

G
G

denote a locally compact group and

σ
(
G
)

\sigma (G)

the semigroup of nonempty compact subsets of

G
G

. The combinatorial properties of the family of all groups

G
G

for which

σ
(
G
)

\sigma (G)

is amenable is studied. The relationship between amenability of

G
G

and amenability of

σ
(
G
)

\sigma (G)

is also investigated.
Remark on discrete subgroups
Authors: Wolf Joseph A.
Published: June 2010

One wants to know the extent to which a locally compact group G is determined by the isomorphism class of a discrete uniform subgroup

Γ
\Gamma

. Among other things, we show that if G has only finitely many components and K is a maximal compact subgroup then

Γ
\Gamma

determines the dimension of the space

G

/

K

G/K

. We then specialize our results to the case where

G

/

K

G/K

is a riemannian symmetric space.
A uniformization theorem for arbitrary Riemann surfaces with signature.
Authors: Wong C. K.
Published: June 2010

An arbitrary Riemann surface with signature can be represented as the quotient of a simply connected Riemann surface by a properly discontinuous group of conformal mappings with the natural projection satisfying some conditions. The representation is unique up to conjugation.
The iterates of a contraction and its adjoint
Authors: Holbrook John A. R.
Published: June 2010

We prove that when T is a contraction on Hilbert space the size of

|

(

(

T
∗

)
n

h
,
g
)

|

|({({T^\ast })^n}h,g)|

is controlled by that of

lim
sup

|

(

T
n

h
,
g
)

|

\lim \sup |({T^n}h,g)|

. We give an application to Fourier-Stieltjes coefficients. Important in the proof is a generalization of the technique of orthogonal projection.
Regular matrices with nowhere dense support
Authors: Rainwater John
Published: June 2010

A regular matrix has nowhere dense support in

β
N
∖
N

\beta N\backslash N

if and only if the largest entry in the mth row converges to 0.
Characterizations of the generalized convex kernel
Authors: Sparks Arthur G.
Published: June 2010

It is well known that the convex kernel

K
K

of a set

S
S

is the intersection of all maximal convex subsets of

S
S

. In this paper it is shown that the

n
n

th order kernel of a compact, simply-connected set

S
S

in the plane is an

L
n

{L_n}

set and is, in fact, the intersection of all maximal

L
n

{L_n}

subsets of

S
S

. Furthermore, it is shown that one does not have to intersect the family of all the maximal

L
n

{L_n}

subsets to obtain the

n
n

th order kernel, but that any subfamily thereof which covers the set is sufficient.
Countable paracompactness in product spaces
Authors: Zenor Phillip
Published: June 2010

The main purpose of this paper is to show that

X
ω

{X^\omega }

is normal if and only if (1)

X
n

{X^n}

is normal for each n, and (2)

X
ω

{X^\omega }

is countably paracompact. Furthermore,

X
ω

{X^\omega }

is perfectly normal if and only if

X
ω

{X^\omega }

is hereditarily countably paracompact. Also, the compact Hausdorff space X is metrizable if and only if

X
3

{X^3}

is hereditarily countably paracompact.
Concerning upper semicontinuous decompositions of irreducible continua.
Authors: Transue W. R. R. , Hinrichsen J. W. , Fitzpatrick B.
Published: June 2010

Let

K

\mathcal {K}

denote the class of all compact metric continua K such that there exists a monotone mapping from a compact metric irreducible continuum M onto an arc such that each point inverse is homeomorphic to K. It is shown that no connected 1-polyhedron other than an arc is an element of

K

\mathcal {K}

, but that

K

\mathcal {K}

contains certain locally connected continua.
On the length of a Hilbert ascending chain
Authors: Seidenberg A.
Published: June 2010

It is shown that if a bound

f
(
i
)

f(i)

is placed on the degrees of the elements in some basis of an ideal

A
i

{A_i}

in the polynomial ring

k
[

X
1

,
⋯
,

X
n

]

k[{X_1}, \cdots ,{X_n}]

over the field

k
,
i
=
0
,
1
,
2
,
⋯

k,i = 0,1,2, \cdots

, then a bound can be placed on the length of a strictly ascending chain

A
0

>

A
1

>
⋯

{A_0} > {A_1} > \cdots

. Moreover one could explicitly write down a formula for a bound

g
n

{g_n}

in terms of f and n.
Characterization of rings using quasiprojective modules. II
Authors: Golan Jonathan S.
Published: June 2010

Semiperfect rings, semihereditary rings, and hereditary rings, are characterized by properties of quasiprojective modules over their matrix rings.
Meromorphic solutions of a system of functional equations involving the modular group
Authors: Brady Michael M.
Published: June 2010

The purpose of this paper is to study the meromorphic solutions F of the system of functional equations (1)

F
[
T
(
z
)
]
=
T
[
F
(
z
)
]

F[T(z)] = T[F(z)]

, for all T in the modular group M and z in the upper-half plane. A formula is given which expresses all such solutions explicitly in terms of the functions automorphic relative to M. Then, an application of the functions that satisfy (1) is given, and some observations are made concerning solutions of the more general system of functional equations that results when M is replaced by an arbitrary Fuchsian group.
Wallman-type compactifications and products
Authors: Kost Frank
Published: June 2010

Y is a Wallman-type compactification (O. Frink, Amer. J. Math. 86 (1964), 602-607) of X in case there is a normal base Z for the closed sets of X such that the ultrafilter space from Z, denoted

ω
(
Z
)

\omega (Z)

, is topologically Y. It is not known if every compactification is Wallman-type. For

Z
α

{Z_\alpha }

a normal base for the closed sets of

X
α

{X_\alpha }

for each a belonging to an index set

Δ
\Delta

it is shown that the Tychonoff product space

∏

α
∈
Δ

ω
(

Z
α

)

{\prod _{\alpha \in \Delta }}\omega ({Z_\alpha })

is a Wallman compactification of

∏

α
∈
Δ

X
α

{\prod _{\alpha \in \Delta }}{X_\alpha }

. Also for

X
⊂
T
⊂
ω
(
Z
)

X \subset T \subset \omega (Z)

with Z a normal base for the closed sets of X, a proof that

ω
(
Z
)

\omega (Z)

is a Wallman-type compactification of T is indicated.
Compact group topologies for 𝑅
Authors: Hawley Douglas
Published: June 2010

There is a compact solenoidal group S that is group isomorphic to the additive real numbers R. The existence of S leads to a study of compact real groups; that is, compact groups which are group isomorphic to R. The compact real groups are characterized as products of S. Various conditions on a topological group G are given which are equivalent to G being compact real. It is shown that any compact real group is solenoidal, connected and separable. Many sets and functions on R which are measurable with respect to the usual topology are shown to be nonmeasurable with respect to any compact group topology.
A distortion theorem for analytic functions
Authors: Robertson M. S.
Published: June 2010

Let

f
(
z
)

f(z)

be a function analytic in the disk

E
{
z
:

|

z

|

>
1
}

E\{ z:|z| > 1\}

and for some real number

n
>
0

n > 0

let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue less-than-over-equals left-parenthesis 1 minus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis Superscript negative n Baseline comma z element-of upper E">

|

f
(
z
)

|

≦

(
1
−

|

z

|

2

)

−
n

,
z
∈
E

|f(z)| \leqq {(1 - |z{|^2})^{ - n}},z \in E

. In this paper it is shown that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue f prime left-parenthesis z right-parenthesis EndAbsoluteValue less-than-over-equals StartFraction left-parenthesis n plus 1 right-parenthesis Superscript n plus 1 Baseline Over n Superscript n Baseline EndFraction left-bracket 1 minus left-parenthesis StartFraction n Over n plus 1 EndFraction right-parenthesis Superscript 2 n Baseline left-parenthesis 1 minus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis Superscript 2 n Baseline StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue squared right-bracket division-sign left-parenthesis 1 minus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis Superscript n plus 1 Baseline comma">

|

f
′

(
z
)

|

≦

(
n
+
1
)

n
+
1

n
n

[

1
−

(

n

n
+
1

)

2
n

(
1
−

|

z

|

2

)

2
n

|

f
(
z
)

|

2

]

÷

(
1
−

|

z

|

2

)

n
+
1

,

|f’(z)| \leqq \frac {{{{(n + 1)}^{n + 1}}}}{{{n^n}}}\left [ {1 - {{\left ( {\frac {n}{{n + 1}}} \right )}^{2n}}{{(1 - |z{|^2})}^{2n}}|f(z){|^2}} \right ] \div {(1 - |z{|^2})^{n + 1}},

\]

z
∈
E

z \in E

. In the special case

n
=
1

n = 1

there is a constant

K
,
3
≦
K
≦
4

K,3 \leqq K \leqq 4

, so that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f prime left-parenthesis z right-parenthesis StartAbsoluteValue plus EndAbsoluteValue f left-parenthesis z right-parenthesis vertical-bar Superscript 2 Baseline less-than-over-equals upper K left-parenthesis 1 minus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis Superscript negative 2 Baseline period">

f
′

(
z
)

|

+

|

f
(
z
)

|

2

≦
K

(
1
−

|

z

|

2

)

−
2

.

f’(z)| + |f(z){|^2} \leqq K{(1 - |z{|^2})^{ - 2}}.

\]
This result has application in univalent function theory.
Oscillation properties of the 2-2 disconjugate fourth order selfadjoint differential equation
Authors: Schneider Leo J.
Published: June 2010

This paper contains a proof that either all, or none, of the nontrivial solutions of the fourth order linear selfadjoint differential equation have an infinite number of zeros on a half line, provided that no nontrivial solution has more than one double zero on that half line.
Unitary groups and differential operators
Authors: Kauffman Robert M.
Published: June 2010

Unitary groups generated by differential operators have special properties that can be used to study completeness of the set of eigenvectors of the infinitesimal generator. Unitary groups also occur in differential operator theory in another manner, associated with unitary equivalence of differential operators. We discuss what happens to

A
t

=

U
t

−
1

A

U
t

{A_t} = U_t^{ - 1}A{U_t}

as t approaches infinity provided that

A
t

{A_t}

is a differential operator for all t and A has certain properties.
On nonlinear equations of Hammerstein type in Banach spaces
Authors: Hess Peter
Published: June 2010

A new theorem on the existence and uniqueness of a solution of an equation of Hammerstein type

u
+
T
N
u
=
f

u + TNu = f

is given. Here N denotes a (nonlinear) monotone mapping of a real reflexive Banach space X into its conjugate space

X
∗

{X^ \ast }

and T a bounded monotone linear operator of

X
∗

{X^ \ast }

into X. It is not assumed that T or N is coercive.
On the topological completion of 𝑀-space products
Authors: Steiner A. K.
Published: June 2010

Using the continuum hypothesis, an example is given of two countably compact spaces X and Y such that

X
×
Y

X \times Y

is an M-space, but is not countably compact. If

μ
X

\mu X

denotes the completion of X with respect to its finest uniformity, the above example shows that

μ
(
X
×
Y
)

\mu (X \times Y)

is not necessarily equal to

μ
X
×
μ
Y

\mu X \times \mu Y

, even though

X
×
Y

X \times Y

is an M-space.
Continuity of systems of derivations on 𝐹-algebras
Authors: Carpenter R. L.
Published: June 2010

Let A be a commutative semisimple F-algebra with identity, and let

D
0

,

D
1

,
⋯

{D_0},{D_1}, \cdots

be a system of derivations from A into the algebra of all continuous functions on the spectrum of A. It is shown that the transformations

D
0

,

D
1

,
⋯

{D_0},{D_1}, \cdots

are necessarily continuous. This result is used to obtain a characterization of derivations on

Hol

(
Ω
)

{\text {Hol}}(\Omega )

where

Ω
\Omega

is an open polynomially convex subset of

C
n

{C^n}

.
On algebras of operators with totally ordered lattice of invariant subspaces
Authors: Conway John B.
Published: June 2010

For a Hilbert space

H

\mathcal {H}

, let

A

\mathcal {A}

be a weakly closed algebra of bounded operators on

H

\mathcal {H}

which contains the identity.

A

\mathcal {A}

is said to be transitive if no closed subspace of

H

\mathcal {H}

is invariant under

A

\mathcal {A}

. There are no known proper subalgebras of

B

(

H

)

\mathcal {B}(\mathcal {H})

which are transitive. In this paper it is shown that the only transitive algebra which satisfies a certain condition

β
\beta

is

B

(

H

)

\mathcal {B}(\mathcal {H})

. Furthermore, a generalization of condition

β
\beta

is given which characterizes those algebras with totally ordered lattice of invariant subspaces that are reflexive.
Cyclic atoms in orthomodular lattices
Authors: Catlin Donald E.
Published: June 2010

Let

P
(
H
)

P(H)

denote the projection lattice of a separable Hilbert space H. For each

x

∈
H

{\text {x}} \in H

, let

P

x

{P_{\text {x}}}

denote the projection onto the one dimensional subspace generated by x. If B is a Boolean sublattice of

P
(
H
)

P(H)

, then it is a theorem that whenever B is maximal in

P
(
H
)

P(H)

there exists a vector

x

0

∈
H

{{\text {x}}_0} \in H

, called a cyclic vector for B, such that the join in

P
(
H
)

P(H)

of all the

P

Q
(

x

0

)

{P_{Q({{\text {x}}_0})}}

as Q ranges through B is the identity operator I. In this paper we show that this theorem is an immediate corollary of a more general theorem in orthomodular lattice theory. In addition, a final theorem in the paper makes clear the necessity for the separability assumption on H.
Note on the Kloosterman sum
Authors: Williams Kenneth S.
Published: June 2010

The Kloosterman sum
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript x equals 0 semicolon left-parenthesis x comma p right-parenthesis equals 1 Overscript p Superscript alpha Baseline minus 1 Endscripts exp left-parenthesis 2 pi i n left-parenthesis x plus x overbar right-parenthesis slash p Superscript alpha Baseline right-parenthesis comma">

∑

x
=
0
;
(
x
,
p
)
=
1

p
α

−
1

exp
⁡
(
2
π
i
n
(
x
+

x
¯

)

/

p
α

)
,

\sum \limits _{x = 0;(x,p) = 1}^{{p^\alpha } - 1} {\exp (2\pi in(x + \bar x)/{p^\alpha }),}

\]
where p is an odd prime,

α
≧
2

\alpha \geqq 2

and

(
n
,
p
)
=
1

(n,p) = 1

, is evaluated in a very short direct way.
Two remarks on the group algebra of a finite group
Authors: Fields K. L.
Published: June 2010

If

K
⊆
Q
(

ζ
m

)

K \subseteq Q({\zeta _m})

, m least, we find the smallest n such that

M
n

(
K
)

{M_n}(K)

appears in

Q
G

QG

for some finite group G when m is either a prime power or not exactly divisible by a prime to the first power. We also show that every group of even order possesses a nontrivial real valued character of Schur index 1 over the rationals.
Can a 2-coherent Peano continuum separate 𝐸³?
Authors: Chewning W. C.
Published: June 2010

The fact that there are unicoherent continua which separate

E
2

{E^2}

is well known, e.g., a circle with a spiral converging onto it is such a continuum. In this paper we extend this pathology by describing a Peano continuum which separates

E
3

{E^3}

and has the property that however it is written as the union of two unicoherent Peano continua, their intersection is unicoherent.
Kronecker function rings and flat 𝐷[𝑋]-modules
Authors: Arnold J. T. , Brewer J. W.
Published: June 2010

Let

D
D

be an integral domain with identity. Gilmer has recently shown that in order that a

v
v

-domain

D
D

be a Prüfer

v
v

-multiplication ring, it is necessary and sufficient that

D
v

{D^v}

be a quotient ring of

D
[
X
]

D[X]

, where

D
v

{D^v}

is the Kronecker function ring of

D
D

with respect to the

v
v

-operation. In this paper the authors prove that in the above theorem it is possible to replace “a quotient ring of

D
[
X
]

D[X]

” with “a flat

D
[
X
]

D[X]

-module.” Moreover, it is shown that

D
v

{D^v}

is the only Kronecker function ring of

D
[
X
]

D[X]

which can ever be a flat

D
[
X
]

D[X]

-module.
Irreducible algebras of operators which contain a minimal idempotent.
Authors: Barnes Bruce A.
Published: June 2010

We prove that when A is a closed subalgebra of the bounded operators on a reflexive Banach space X, which acts irreducibly on X and contains a minimal idempotent, then every bounded operator with finite dimensional range on X is in A. We use this result to prove that every continuous irreducible representation of a GCR-algebra on a Hilbert space

H

\mathcal {H}

is similar to a

∗

^ \ast

-representation on

H

\mathcal {H}

.
Orthogonal systems of polynomials in finite fields
Authors: Niederreiter H.
Published: June 2010

The notion of an orthogonal system of polynomials in several variables in finite fields is introduced which generalizes a concept of orthogonality by Kurbatov and Starkov. Necessary and sufficient conditions for orthogonality in terms of character sums and permutation polynomials are given. Results of Carlitz on systems of equations in finite fields and earlier results of the author on permutation polynomials in several variables are generalized.
Topological algebras and Mackey topologies
Authors: Cochran Allan C.
Published: June 2010

Let E be a locally m-convex algebra with dual space

E
′

E’

. In a recent paper S. Warner asked if the finest locally m-convex topology on E compatible with

E
′

E’

was the mackey topology. It is shown that this is not the case. A similar result is given for this question in the A-convex algebra case. For any A-convex algebra, a construction is given of an associated locally m-convex algebra. It is shown that this associated locally m-convex topology is always the compact-open topology for the space

C
b

(
S
)

{C_b}(S)

with the strict topology.
The domain rank of open surfaces of infinite genus
Authors: Tondra Richard J.
Published: June 2010

In a recent paper it was shown that an open surface, i.e. a connected

2
2

-manifold without boundary, has finite domain rank if and only if it has finite genus. In the present paper, it is shown that the domain rank of any open surface of infinite genus is countably infinite.
Functions of direct integrals of operators
Authors: Chow T. R. , Gilfeather F.
Published: June 2010

This paper contains two results. The first one is that the unitary dilation of a direct integral of linear contraction operators is the direct integral of unitary dilations. For each linear contraction operator T on a Hilbert space, consider

f
(
T
)

f(T)

as a bounded linear operator. The second result states that if

T
=
∫
⊕
T
(
s
)
d
μ
(
s
)

T = \smallint \oplus T(s)d\mu (s)

is decomposable then so is

f
(
T
)

f(T)

and

f
(
T
)
=
∫
⊕
f
(
T
(
s
)
)
d
μ
(
s
)

f(T) = \smallint \oplus f(T(s))d\mu (s)

.
Embedding rational division algebras
Authors: Fein Burton
Published: June 2010

Necessary and sufficient conditions are given for two K-division rings, K an algebraic number field, to have precisely the same set of subfields. Using this, an example is presented of two K-division rings having precisely the same set of subfields such that only one of the division rings can be embedded in a Q-division ring.
On the group indices of the parasymplectic group of level 𝐹
Authors: Kodama Tetsuo
Published: June 2010

We obtain the group indices of a parasymplectic group to a principal congruence subgroup of level

q

\mathfrak {q}

and to an inhomogeneous congruence subgroup of level

q

\mathfrak {q}

.
Common fixed-points for equicontinuous semigroups of mappings
Authors: Mitchell Theodore
Published: June 2010

Let S be a semigroup of equicontinuous self maps of X, a compact Hausdorff space. It is shown that if S is left reversible (that is every pair of right ideals of S has nonempty intersection), then there is a compact group G of homeomorphisms of a retract Y of X with the property that S has a common fixed-point in X if and only if G has a common fixed-point in Y. As an application, it is proved that if F is a family of continuous commuting self maps of the closed unit interval I with the property that for each

f
∈
F

f \in F

, with one possible exception, the set of all iterates of f is equicontinuous, then I contains a common fixed-point of F.
Homomorphisms of rings of germs of analytic functions
Authors: Zame William R.
Published: June 2010

Let S and S’ be complex analytic manifolds with S Stein. Let

X
⊂
S

X \subset S

and

X
′

⊂

S
′

X’ \subset S’

be compact sets with X holomorphically convex. Denote by

O

(
X
)

\mathcal {O}(X)

(respectively

O

(

X
′

)

\mathcal {O}(X’)

) the ring of germs on X (respectively X’) of functions analytic near X (respectively X’). It is shown that each nonzero homomorphism of

O

(
X
)

\mathcal {O}(X)

into

O

(

X
′

)

\mathcal {O}(X’)

is given by composition with an analytic map defined in a neighborhood of X’ and taking values in S.
The closed ideals of some Dirichlet and hypo-Dirichlet algebras
Authors: Blumenthal Robert G.
Published: June 2010

We characterize the closed ideals of (i) the Dirichlet algebras discovered by A. Browder and J. Wermer and (ii) the hypo-Dirichlet algebras discovered recently by A. G. Brandstein. Our results show that the class of closed ideals of each of these algebras is surprisingly restricted.
Cauchy sequences in semimetric spaces
Authors: Burke Dennis K.
Published: June 2010

As the main result we prove that every semimetrizable space has a semimetric for which every convergent sequence has a Cauchy subsequence. This result is used to show that a

T
1

{T_1}

space X is semimetrizable if and only if it is a pseudo-open

π
\pi

-image of a metric space.
On polynomials which commute with a given polynomial
Authors: Boyce William M.
Published: June 2010

By extending a theorem of Jacobsthal, the following result is obtained: if g is a nonlinear polynomial, there is an integer

J
(
g
)
≧
1

J(g) \geqq 1

such that for each

m
>
0

m > 0

there are either

J
(
g
)

J(g)

or zero distinct polynomials of degree m which commute with g. A formula is given for computing

J
(
g
)

J(g)

from the coefficients of g.
Note on the projective limit on small categories
Authors: Laudal Olav Arnfinn
Published: June 2010

Let X be a small category and let

A

b

X

{\mathbf {A}}{{\mathbf {b}}^X}

be the category of covariant functors on X with values in Ab. Consider the projective limit functor proj

lim
X

:

A

b

X

→

A
b

{\lim _X}:{\mathbf {A}}{{\mathbf {b}}^X} \to {\mathbf {Ab}}

. The categories X for which proj

lim
X

{\lim _X}

is exact are characterized, proving a conjecture of Oberst.
On a converse of Young’s inequality
Authors: Hsu Ih Ching
Published: June 2010

A converse of Young’s inequality is proved through the formulation of a functional inequality.
Coefficients for the area theorem
Authors: Goodman A. W.
Published: June 2010

Let

f
(
z
)
=

∑

n
=
1

∞

a
n

z
n

f(z) = \sum \nolimits _{n = 1}^\infty {{a_n}{z^n}}

, and set
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis z right-parenthesis equals f left-parenthesis z Superscript negative p Baseline right-parenthesis Superscript minus slash 1 p Baseline equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts g Subscript n p minus 1 Baseline z Superscript 1 minus n p">

G
(
z
)
=
f

(

z

−
p

)

−

/

1
p

=

∑

n
=
0

∞

g

n
p
−
1

z

1
−
n
p

G(z) = f{({z^{ - p}})^{ - /1p}} = \sum \nolimits _{n = 0}^\infty {{g_{np - 1}}{z^{1 - np}}}

. This paper finds an explicit formula for

g

n
p
−
1

{g_{np - 1}}

in terms of the

a
n

{a_n}

. Such a formula (apparently previously unknown) may be very useful in the theory of univalent functions.