bluuebunny/crossref_metadata_2025_split huggingface dataset:
Composition of Blochs with bounded analytic functions
Authors: Kwon E. 
Published: July 2002

If

f
f

is a holomorphic self-map of the open unit disc and

1
≤
p
>
∞

1 \leq p > \infty

, then the following are equivalent.

(
1
)

h
∘
f
∈

H

2
p

(1)\,\,\,\, h\circ f \in H^{2p}

for all Bloch functions

h
h

.

(2)

s
u
p

r

∫

0

2
π

(
l
o
g

1

1
−
|
f
(
r

e

i
θ

)

|

2

)

p

d
θ

>
∞
.

\begin{equation*}\underset {{r} }{sup} \int _{0}^{2\pi } \left ( log \frac {1}{1 - \vert f(re^{i\theta })\vert ^{2}}\right )^{p} \,d\theta \,\, > \infty . \tag {2}\end{equation*}

(3)

∫

0

2
π

(

∫

0

1

(

f

#

)

2

(
r

e

i
θ

)

(
1
−
r
)
d
r
)

p

d
θ
>
∞
,

\begin{equation*}\int _{0}^{2\pi } \left ( \int _{0}^{1} (f^{\#})^{2}(re^{i\theta })\, (1-r) dr \right )^{p} d\theta > \infty , \tag {3}\end{equation*}

where

f

#

f^{\#}

is the hyperbolic derivative of

f
f

:

f

#

=
|

f
′

|

/

(
1
−
|
f

|

2

)

f^{\#} = \vert f’\vert / (1-\vert f\vert ^{2})

.
Banach spaces in which every 𝑝-weakly summable sequence lies in the range of a vector measure
Authors: Piñeiro C. 
Published: July 2002

Let

X
X

be a Banach space. For

1
>
p
>
+
∞

1>p>+\infty

we prove that the identity map

I
X

I_X

is

(
1
,
1
,
p
)

(1,1,p)

-summing if and only if the operator

x
∗

∈

X
∗

→
∑
⟨

x
n

,

x
∗

⟩

e
n

∈

l
q

x^*\in X^*\to \sum \langle x_n,x^*\rangle e_n\in l_q

is nuclear for every unconditionally summable sequence

(

x
n

)

(x_n)

in

X
X

, where

q
q

is the conjugate number for

p
p

. Using this result we find a characterization of Banach spaces

X
X

in which every

p
p

-weakly summable sequence lies inside the range of an

X

∗
∗

X^{**}

-valued measure (equivalently, every

p
p

-weakly summable sequence

(

x
n

)

(x_n)

in

X
X

, satisfying that the operator

(

α
n

)
∈

l
q

→
∑

α
n

x
n

∈
X

(\alpha _n)\in l_q\to \sum \alpha _nx_n\in X

is compact, lies in the range of an

X
X

-valued measure) with bounded variation. They are those Banach spaces such that the identity operator

I

X
∗

I_{X^*}

is

(
1
,
1
,
p
)

(1,1,p)

-summing.
On linear series on general 𝑘-gonal projective curves
Authors: Ballico E. , Keem C. 
Published: July 2002

Let

X
X

be a general

k
k

-gonal curve of genus

g
g

. Here we prove a strong upper bound for the dimension of linear series on

X
X

, i.e. we prove that

dim
⁡
(

W
d
r

(
X
)
)
≤
ρ
(
g
,
r
,
d
)
+
(
g
−
2
k
+
2
)
:=
g
−
(
r
+
1
)
(
r
+
g
−
d
)
+
(
g
−
2
k
+
2
)

\dim (W^r_d(X))\leq \rho (g,r,d)+(g-2k+2):=g-(r+1) (r+g-d)+(g-2k+2)

.
Almost linearity of 𝜀-bi-Lipschitz maps between real Banach spaces
Authors: Jun Kil-Woung , Park Dal-Won 
Published: July 2002

Let

X
X

and

Y
Y

be real Banach spaces. A map

f
f

between

X
X

and

Y
Y

is called an

ϵ
\epsilon

-bi-Lipschitz map if

(
1
−
ϵ
)
‖
x
−
y
‖
≤
‖
f
(
x
)
−
f
(
y
)
‖
≤
(
1
+
ϵ
)
‖
x
−
y
‖

(1-\epsilon )\|x-y\| \le \|f(x) -f(y)\| \le (1+\epsilon )\|x- y\|

for all

x
,
y
∈
X

x, y\in X

. In this note we show that if

f
f

is an

ϵ
\epsilon

-bi-Lipschitz map with

f
(
0
)
=
0

f(0)=0

from

X
X

onto

Y
Y

, then

f
f

is almost linear. We also show that if

f
:
X
⟶
Y

f:X\longrightarrow Y

is a surjective

ϵ
\epsilon

-bi-Lipschitz map with

f
(
0
)
=
0

f(0)=0

, then there exists a linear isomorphism

I
:
X
→
Y

I:X\to Y

such that
\[

‖
I
(
x
)
−
f
(
x
)
‖
≤
E
(
ϵ
,
α
)
(
‖
x

‖
α

+
‖
x

‖

2
−
α

)

\|I(x)-f(x)\| \le E(\epsilon , \alpha )(\|x\|^\alpha +\|x\|^{2-\alpha })

\]
where

E
(
ϵ
,
α
)
→
0

E(\epsilon ,\alpha )\to 0

as

ϵ
→
0

\epsilon \to 0

and

0
>
α
>
1

0>\alpha >1

.
Characterisation of compactness through convergence in Tychonoff spaces
Authors: Petalas C. , Vidalis T. 
Published: July 2002

In this article we first give a characterisation of compact spaces among

T

3.5

T_{3.5}

spaces by improving a theorem of J. Ewert. Then, with the aid of a new type of convergence, we give a characterisation of the pseudocompact and of the Lindelöf

T

3.5

T_{3.5}

spaces.
The local zeta function for the non-trivial characters associated with the singular Jordan algebras
Authors: Robinson Margaret 
Published: July 2002

This paper investigates the local integrals
\[

Z
m

(
t
,
χ
)
=

∫

H
m

(

O
C

)

χ
(
det
(
x
)
)

|

det
(
x
)

|

s

d
x

Z_m(t,\chi )=\int _{H_m(O_C)} \chi ( \det (x)) | \det (x) |^s dx

\]
where

O
C

O_C

represents the integers of a composition algebra over a non-archimedean local field

K
K

and

χ
\chi

is a non-trivial character on the units in the ring of integers of

K
K

extended to

K
∗

K^*

by setting

χ
(
π
)
=
1

\chi (\pi )=1

. The local zeta function for the trivial character is known for all composition algebras

C
C

. In this paper, we show in the quaternion case that

Z
(
t
,
χ
)
=
0

Z(t, \chi )=0

for all non-trivial characters and then compute the local zeta function in the ramified quadratic extension case for

χ
\chi

equal to the quadratic character. In this latter case,

Z
(
t
,
χ
)
=
0

Z(t, \chi )=0

for any character of order greater than

2
2

.
On pasting 𝐴_{𝑝}-weights
Authors: Schröder Bernd 
Published: July 2002

Theorem 3 gives a condition when two

A
p

A_p

-weights can be “pasted” together to yield another

A
p

A_p

-weight. It is subsequently used in Example 6 to give an example that shows that a necessary condition by Gohberg, Krupnik, and Spitkovsky is not sufficient.
Nonresonance problems for differential inclusions in separable Banach spaces
Authors: Ding Zouhua , Kartsatos Athanassios 
Published: July 2002

Let

X
X

be a real separable Banach space. The boundary value problem

(B)

a
m
p
;

x
′

∈
A
(
t
)
x
+
F
(
t
,
x
)
,

t
∈

R

+

,

a
m
p
;
U
x
=
a
,

\begin{equation*} \begin {split} &x’ \in A(t)x+F(t,x),~t\in \mathcal {R}_+,\\ &Ux = a, \end{split} \tag *{(B)} \end{equation*}

is studied on the infinite interval

R
+

=
[
0
,
∞
)
.

R_+=[0,\infty ).

Here, the closed and densely defined linear operator

A
(
t
)
:
X
⊃
D
(
A
)
→
X
,

t
∈

R

+

,

A(t):X\supset D(A)\to X,~t\in \mathcal {R}_+,

generates an evolution operator

W
(
t
,
s
)
.

W(t,s).

The function

F
:

R

+

×
X
→

2
X

F:\mathcal {R}_+\times X\to 2^X

is measurable in its first variable, upper semicontinuous in its second and has weakly compact and convex values. Either

F
F

is bounded and

W
(
t
,
s
)

W(t,s)

is compact for

t
>
s
,

t > s,

or

F
F

is compact and

W
(
t
,
s
)

W(t,s)

is equicontinuous. The mapping

U
:

C
b

(

R

+

,
X
)
→
X

U:C_b(\mathcal {R}_+,X)\to X

is a bounded linear operator and

a
∈
X

a\in X

is fixed. The nonresonance problem is solved by using Ma’s fixed point theorem along with a recent result of Przeradzki which characterizes the compact sets in

C
b

(

R

+

,
X
)
.

C_b(\mathcal {R}_+,X).
Non-commutative disc algebras and their representations
Authors: Popescu Gelu 
Published: July 2002

It is shown that the smallest closed subalgebra

A
l
g
(

I

K

,

V

1

,
…
,

V

n

)
⊂

B

(

K

)

(
n
=
2
,
3
,
…
,
∞
)

\begin{equation*}Alg(I_{ \mathcal {K}} ,V_{1},\dots ,V_{n})\subset \mathcal {B} (\mathcal {K}) \qquad (n=2,3,\dots ,\infty )\end{equation*}

generated by any sequence

V

1

,
…
,

V

n

V_{1},\dots , V_{n}

of isometries on a Hilbert space

K

\mathcal {K}

such that

V

1

V

1

∗

+
⋯
+

V

n

V

n

∗

≤

I

K

V_{1}V_{1}^{*}+\cdots +V_{n}V_{n}^{*}\le I_{\mathcal {K}}

is completely isometrically isomorphic to the non-commutative “disc” algebra

A

n

\mathcal {A} _{n}

introduced in Math. Scand. 68 (1991), 292–304. We also prove that for

n
≠
m

n\ne m

the Banach algebras

A

n

\mathcal {A} _{n}

and

A

m

\mathcal {A} _{m}

are not isomorphic. In particular, we give an example of two non-isomorphic Banach algebras which are completely isometrically embedded in each other. The completely bounded (contractive) representations of the “disc” algebras

A

n

(
n
=
2
,
3
,
…
,
∞
)

\mathcal {A} _{n} (n=2,3,\dots ,\infty )

on a Hilbert space are characterized. In particular, we prove that a sequence of operators

A

1

,

A

2

,
…

A_{1},A_{2},\dots

is simultaneously similar to a contractive sequence

T

1

,

T

2

,
…

T_{1},T_{2},\dots

(i.e.,

T

1

T

1

∗

+
⋯
+

T

n

T

n

∗

≤
I

T_{1}T_{1}^{*}+\cdots +T_{n}T_{n}^{*} \le I

) if and only if it is completely polynomially bounded. The first cohomology group of

A

n

\mathcal {A} _{n}

with coefficients in

C

\mathbb {C}

is calculated, showing, in particular, that the disc algebras are not amenable. Similar results are proved for the non-commutative Hardy algebras

F

n

∞

F_{n}^{\infty }

introduced in Math. Scand. 68 (1991), 292–304. The right joint spectrum of the left creation operators on the full Fock space is also determined.
Integral equations, implicit functions, and fixed points
Authors: Burton T. 
Published: July 2002

The problem is to show that (1)

V
(
t
,
x
)
=
S
(
t
,

∫
0
t

H
(
t
,
s
,
x
(
s
)
)

d
s
)

V(t,x) = S(t, \int _0^t H(t, s, x(s)) \, ds )

has a solution, where

V
V

defines a contraction,

V
~

\tilde V

, and

S
S

defines a compact map,

S
~

\tilde S

. A fixed point of

P
φ
=

S
~

φ
+
(
I
−

V
~

)
φ

P \varphi = \tilde S \varphi + (I - \tilde V) \varphi

would solve the problem. Such equations arise naturally in the search for a solution of

f
(
t
,
x
)
=
0

f(t, x) = 0

where

f
(
0
,
0
)
=
0

f(0,0) = 0

, but

∂
f
(
0
,
0
)

/

∂
x
=
0

\partial f(0,0) / \partial x = 0

so that the standard conditions of the implicit function theorem fail. Now

P
φ
=

S
~

φ
+
(
I
−

V
~

)
φ

P \varphi = \tilde S \varphi + ( I - \tilde V) \varphi

would be in the form for a classical fixed point theorem of Krasnoselskii if

I
−

V
~

I - \tilde V

were a contraction. But

I
−

V
~

I - \tilde V

fails to be a contraction for precisely the same reasons that the implicit function theorem fails. We verify that

I
−

V
~

I - \tilde V

has enough properties that an extension of Krasnoselskii’s theorem still holds and, hence, (1) has a solution. This substantially improves the classical implicit function theorem and proves that a general class of integral equations has a solution.
A sum packing problem of Erdös and the Conway-Guy sequence
Authors: Bohman Tom 
Published: July 2002

A set

S
S

of positive integers has distinct subset sums if the set

{

∑

x
∈
X

x
:
X
⊂
S
}

\left \{ \sum _{x \in X} x : X \subset S \right \}

has

2

|

S

|

2^{|S|}

distinct elements. Let
\[

f
(
n
)
=
min
{
max
S
:

|

S

|

=
n

and

S

has\hskip 2mm distinct\hskip 2mm subset \hskip 2mm sums
}
.

f(n) = \min \{ \max S: |S|=n \hskip 2mm \text {and} \hskip 2mm S \hskip 2mm \text {has\hskip 2mm distinct\hskip 2mm subset \hskip 2mm sums}\}.

\]
In 1931 Paul Erdős conjectured that

f
(
n
)
≥
c

2

n

f(n) \ge c2^{n}

for some constant

c
c

. In 1967 John Conway and Richard Guy constructed an interesting sequence of sets of integers. They conjectured that these sets have distinct subset sums and that they are close to the best possible (with respect to largest element). We prove that sets from this sequence have distinct subset sums. We also present some variations of this construction that give microscopic improvements in the best known upper bound on

f
(
n
)

f(n)

.
On an optimality property of Ramanujan sums
Authors: Bachman Gennady 
Published: July 2002

We evaluate

inf

b

n

∑

a
=
1

q

|

∑

n
=
1
(
n
,
q
)
=
1

q

b

n

e

2
π
i
a
n

/

q

|

\inf _{b_{n}}\sum _{a=1}^{q}|\sum _{\substack {n=1 (n,q)=1}}^{q} b_{n}e^{2\pi ian/q}|

, where the

inf
\inf

is taken over sequences

b

n

b_{n}

satisfying

b

n

≥
1

b_{n}\ge 1

. In particular we show that it is attained by taking

b

n

=
1

b_{n}=1

for all

n
n

, which reduces the summation over

n
n

to a Ramanujan sum

c

q

(
a
)
=

∑

n
=
1
(
n
,
q
)
=
1

q

e

2
π
i
a
n

/

q

c_{q}(a)=\sum _{\substack {n=1 (n,q)=1}}^{q}e^{2\pi ian/q}

.
Group homomorphisms inducing mod-𝑝 cohomology monomorphisms
Authors: Minh Pham 
Published: July 2002

Letf:G→Kf\colon G\to Kbe a homomorphism ofpp-groups such thatf(n):Hn(K,Z/p)→Hn(G,Z/p)f^{(n)}\colon H^n(K,\mathbf Z /p)\to H^n(G,\mathbf Z/p)is injective, for1≤n≤21\le n\le 2. We prove that the non-bijectivity offfimplies the existence of a quotientLLofGGcontainingKKas a proper direct factor. This gives a refined proof of a result of Evens, which asserts thatffis bijective iff(1)f^{(1)}is.
On the deleted product criterion for embeddability in ℝ^{𝕞}
Authors: Skopenkov A. 
Published: July 2002

For a space

K
K

let

K
~

=
{
(
x
,
y
)
∈
K
×
K

|

x
≠
y
}

\tilde K=\{(x,y)\in K\times K| x\not =y\}

. Let

Z

2

\mathbb {Z}_{2}

act on

K
~

\tilde K

and on

S

m
−
1

S^{m-1}

by exchanging factors and antipodes respectively. We present a new short proof of the following theorem by Weber: For an

n
n

-polyhedron

K
K

and

m
⩾

3
(
n
+
1
)

2

m\geqslant \frac {3(n+1)}{2}

, if there exists an equivariant map

F
:

K
~

→

S

m
−
1

F: \tilde K\rightarrow S^{m-1}

, then

K
K

is embeddable in

R

m

\mathbb {R}^{m}

. We also prove this theorem for a peanian continuum

K
K

and

m
=
2

m=2

. We prove that the theorem is not true for the 3-adic solenoid

K
K

and

m
=
2

m=2

.
A critical metric for the 𝐿²-norm of the curvature tensor on 𝑆³
Authors: Lamontagne François 
Published: July 2002

The

L
2

L^2

-norm of the curvature tensor
\[

R

(
g
)
=

1

(

V
o
l

(
M
)

)

n
−
4

n

∫

M

∣
R

∣

2

d
v
o

l

g

\mathcal {R}(g)=\frac {1}{(\mathrm {Vol}(M))^{\frac {n-4}{n}}}\int _{M}\mid R \mid ^{2}dvol_{g}

\]
defines a Riemannian functional on the space of metrics. This work exhibits a metric on

S
3

S^3

which is of Berger type but not of constant ricci curvature, and yet is critical for

R

\mathcal {R}

.
A generalization of Carleman’s uniqueness theorem and a discrete Phragmén-Lindelöf theorem
Authors: Korenblum B. , Mascuilli A. , Panariello J. 
Published: July 2002

Let

d
μ
≥
0

d\mu \geq 0

be a Borel measure on

[
0
,
∞
)

[0,\infty )

and

A

n

=

∫

0

∞

t

n

d
μ
(
t
)
>
∞

(
n
=
0
,
1
,
2
,
.
.
.
)

A_{n}=\int \limits _{0}^{\infty }t^{n}d\mu (t) > \infty ~~(n=0,1,2,...)

be its moments. T. Carleman found sharp conditions on the magnitude of

{

A

n

}

0

∞

\{A_{n}\}_{0}^{\infty }

for

d
μ

d\mu

to be uniquely determined by its moments. We show that the same conditions ensure a stronger property: if

A

n

′

=

∫

0

∞

t

n

d

μ

1

(
t
)

A_{n}’ =\int \limits _{0}^{\infty }t^{n} d\mu _{1} (t)

are the moments of another measure,

d

μ

1

≥
0
,

d\mu _{1} \geq 0,

with

lim sup

n
→
∞

|

A

n

−

A

n

′

|

1
n

=
ρ
>
∞
,

\limsup \limits _{n\to \infty } |A_{n}-A_{n}’|^{\frac {1}{n}}=\rho >\infty ,

then the measure

d
μ
−
d

μ

1

d\mu -d\mu _{1}

is supported on the interval

[
0
,
ρ
]
.

[0,\rho ].

This result generalizes both the Carleman theorem and a theorem of J. Mikusiński. We also present an application of this result by establishing a discrete version of a Phragmén-Lindelöf theorem.
Exotic cohomology for 𝐺𝐿_{𝑛}(ℤ[1/2])
Authors: Dwyer W. 
Published: July 2002

We show that for

n
=
32

n=32

the mod

2
2

group cohomology of

GL

n

⁡
(

Z

[
1

/

2
]
)

\operatorname {GL}_{n}(\mathbb {Z}[1/2])

is not detected on diagonal matrices.
Asymptotics for Sobolev orthogonal polynomials with coherent pairs: The Jacobi case, type 1
Authors: Pan K. 
Published: July 2002

Define

P
n

(
x
)

P_n(x)

and

Q
n

(
x
)

Q_n(x)

as the

n
n

th monic orthogonal polynomials with respect to

d
μ

d\mu

and

d
ν

d\nu

respectively. The pair

{
d
μ
,
d
ν
}

\{d\mu ,d\nu \}

is called a coherent pair if there exist non-zero constants

D
n

D_n

such that
\[

Q
n

(
x
)
=

P

n
+
1

′

(
x
)

n
+
1

+

D
n

P
n
′

(
x
)

n

,

n
≥
1.

Q_n(x)=\frac {P_{n+1}^\prime (x)}{n+1}+D_n\frac {P_n^\prime (x)}{n},\qquad n\ge 1.

\]
One can divide the coherent pairs into two cases: the Jacobi case and the Laguerre case. There are two types for each case: type 1 and 2. We investigate the asymptotic properties and zero distribution of orthogonal polynomials with respect to Sobolev inner product
\[

⟨
f
,
g
⟩
=

∫
a
b

f
(
x
)
g
(
x
)
d
μ
(
x
)
+
λ

∫
a
b

f
′

(
x
)

g
′

(
x
)
d
ν
(
x
)

\langle f,g\rangle =\int _a^b f(x)g(x)d\mu (x)+\lambda \int _a^b f’(x)g’(x)d\nu (x)

\]
for the coherent pair

{
d
μ
,
d
ν
}

\{d\mu ,d\nu \}

: the Jacobi case, type 1.
On scrambled sets and a theorem of Kuratowski on independent sets
Authors: Kato Hisao 
Published: July 2002

The measure of scrambled sets of interval self-maps

f
:
I
=
[
0
,
1
]
→
I

f:I=[0,1] \to I

was studied by many authors, including Smítal, Misiurewicz, Bruckner and Hu, and Xiong and Yang. In this note, first we introduce the notion of “

∗
\ast

-chaos" which is related to chaos in the sense of Li-Yorke, and we prove a general theorem which is an improvement of a theorem of Kuratowski on independent sets. Second, we apply the result to scrambled sets of higher dimensional cases. In particular, we show that if a map

f
:

I

k

→

I

k

(
k
≥
1
)

f: I^{k} \to I^{k}~(k\geq 1)

of the unit

k
k

-cube

I
k

I^k

is

∗
\ast

-chaotic on

I

k

I^{k}

, then for any

ϵ
>
0

\epsilon > 0

there is a map

g
:

I

k

→

I

k

g: I^{k} \to I^{k}

such that

f
f

and

g
g

are topologically conjugate,

d
(
f
,
g
)
>
ϵ

d(f,g) > \epsilon

and

g
g

has a scrambled set which has Lebesgue measure 1, and hence if

k
≥
2

k \geq 2

, then there is a homeomorphism

f
:

I

k

→

I

k

f: I^{k} \to I^{k}

with a scrambled set

S
S

satisfying that

S
S

is an

F

σ

F_{\sigma }

-set in

I
k

I^k

and

S
S

has Lebesgue measure 1.
Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in ℝ^{𝕕}
Authors: Xu Yuan 
Published: July 2002

We study the Fourier expansion of a function in orthogonal polynomial series with respect to the weight functions
\[

x

1

α

1

−
1

/

2

⋯

x

d

α

d

−
1

/

2

(
1
−

|

x

|

1

)

α

d
+
1

−
1

/

2

x_{1}^{\alpha _{1} -1/2} \cdots x_{d}^{\alpha _{d} -1/2}(1-|\mathbf {x}|_{1})^{\alpha _{d+1}-1/2}

\]
on the standard simplex

Σ

d

\Sigma ^{d}

in

R

d

\mathbb {R}^{d}

. It is proved that such an expansion is uniformly

(
C
,
δ
)

(C, \delta )

summable on the simplex for any continuous function if and only if

δ
>

|

α

|

1

+
(
d
−
1
)

/

2

\delta > |\alpha |_{1} + (d-1)/2

. Moreover, it is shown that

(
C
,

|

α

|

1

+
(
d
+
1
)

/

2
)

(C, |\alpha |_{1} + (d+1)/2)

means define a positive linear polynomial identity, and the index is sharp in the sense that

(
C
,
δ
)

(C,\delta )

means are not positive for

0
>
δ
>

|

α

|

1

+
(
d
+
1
)

/

2

0 >\delta >|\alpha |_{1} + (d+1)/2

.
Automorphic-differential identities and actions of pointed coalgebras on rings
Authors: Yanai Tadashi 
Published: July 2002

In this paper, we prove the following two results which generalize the theorem concerning automorphic-differential endomorphisms asserted by J. Bergen. Let

R
R

be a ring,

R

F

R _{\mathcal {F}}

its left Martindale quotient ring and

A

\mathfrak {A}

a right ideal of

R
R

having no nonzero left annihilator. (1) Let

C
C

be a pointed coalgebra which measures

R
R

such that the group-like elements of

C
C

act as automorphisms of

R
R

. If

R
R

is prime and

ξ
⋅

A

=
0

\xi \cdot \mathfrak {A}=0

for

ξ
∈
R
#
C

\xi \in R\#C

, then

ξ
⋅
R
=
0

\xi \cdot R=0

. Furthermore, if the action of

C
C

extends to

R

F

R _{\mathcal {F}}

and if

ξ
∈

R

F

#
C

\xi \in R _{\mathcal {F}}\#C

such that

ξ
⋅

A

=
0

\xi \cdot \mathfrak {A}=0

, then

ξ
⋅

R

F

=
0

\xi \cdot R _{\mathcal {F}}=0

. (2) Let

f
f

be an endomorphism of

R

F

R _{\mathcal {F}}

given as a sum of composition maps of left multiplications, right multiplications, automorphisms and skew-derivations. If

R
R

is semiprime and

f
(

A

)
=
0

f(\mathfrak {A})=0

, then

f
(
R
)
=
0

f(R)=0

.
A decreasing operator function associated with the Furuta inequality
Authors: Furuta Takayuki , Wang Derming 
Published: July 2002

Let

A
≥
B
≥
0

A\ge B\ge 0

with

A
>
0

A>0

and let

t
∈
[
0
,
1
]

t\in [0,1]

and

q
≥
0

q\ge 0

. As a generalization of a result due to Furuta, it is shown that the operator function
\[

G

p
,
q
,
t

(
A
,
B
,
r
,
s
)
=

A

−
r

/

2

{

A

r

/

2

(

A

−
t

/

2

B
p

A

−
t

/

2

)
s

A

r

/

2

}

(
q
−
t
+
r
)

/

[
(
p
−
t
)
s
+
r
]

A

−
r

/

2

G_{p,q,t}(A,B,r,s)=A^{-r/2}\{A^{r/2} (A^{-t/2} B^pA^{-t/2})^s A^{r/2}\}^{(q-t+r)/[(p-t)s+r]}A^{-r/2}

\]
is decreasing for

r
≥
t

r\ge t

and

s
≥
1

s\ge 1

if

p
≥
max
{
q
,
t
}

p\ge \max \{q,t\}

. Moreover, if

1
≥
p
>
t

1\ge p>t

and

q
≥
t

q\ge t

, then

G

p
,
q
,
t

(
A
,
B
,
r
,
s
)

G_{p,q,t}(A,B,r,s)

is decreasing for

r
≥
0

r\ge 0

and

s
≥

q
−
t

p
−
t

s\ge \frac {q-t}{p-t}

. The latter result is an extension of an earlier result of Furuta.
The qualitative analysis of a dynamical system of differential equations arising from the study of multilayer scales on pure metals II
Authors: Baker R. 
Published: July 2002

We provide a qualitative analysis of thenn-dimensional dynamical system:\[q˙i=−∑j=1naijqjk,qi(t)>0,i=1,…,n,\dot q_i=-\sum _{j=1}^n \frac {a_{ij}}{q_j^k},\quad q_i(t)>0,\qquad i=1,\dots , n,\]wherekkis an arbitrary positive integer. Under mild algebraic conditions on the constant matrixA=(aij)A=(a_{ij}), we show that every solutionq(t)\mathbf q(t),t∈[0,a)t\in [0,a), extends to a solution on[0,+∞)[0,+\infty ), such thatlimt→+∞qi(t)=+∞\lim _{t\to +\infty } q_i(t)=+\infty, fori=1,…,ni=1,\dots , n. Moreover, the difference between any two solutions approaches00ast→+∞t\to +\infty. We then use this result to give a complete qualitative analysis of a 3-dimensional dynamical system introduced by F. Gesmundo and F. Viani in modeling parabolic growth of three-oxide scales on pure metals.
Positive differentials, theta functions and 𝐻² Hardy kernels
Authors: Yamada Akira 
Published: July 2002

Let

R
R

be a planar regular region whose Schottky double

R
^

\hat {R}

has genus

g
g

and set

T
^

0

=
{
z
∈

C

g

|

−
1

z
∈

R

g

}

\hat {T}_0=\{z\in \mathbb {C}^g|\sqrt {-1}\,z\in \mathbb {R}^g \}

. For fixed

a
∈
R

a\in R

we determine the range of the function

F
(
e
)
=
θ
(
a
−

a
¯

+
e
)

/

θ
(
e
)
(
e
∈

T
^

0

)

F(e)=\theta (a-\bar {a}+e)/\theta (e) (e\in \hat {T}_0)

where

θ
(
z
)

\theta (z)

is the Riemann theta function on

R
^

\hat {R}

. Also we introduce two weighted Hardy spaces to study the problem when the matrix

(

∂
2

log
⁡
F

∂

z
i

∂

z
j

(
e
)
)

(\frac {\partial ^2\log F}{\partial z_i\partial z_j}(e))

is positive definite. The proof relies on new theta identities using Fay’s trisecants formula.
Random fluctuations of convex domains and lattice points
Authors: Iosevich Alex , Kinateder Kimberly 
Published: July 2002

In this paper, we examine a random version of the lattice point problem. LetH\mathcal Hdenote the class of all homogeneous functions inC2(Rn)C^2(\mathbb R^n)of degree one, positive away from the origin. LetΦ\Phibe a random element ofH\mathcal H, defined on probability space(Ω,F,P)(\Omega ,\mathcal F,P), and define\[FΦ(ω,⋅)(ξ)=∫{x:Φ(ω,x)≤1}e−i⟨x,ξ⟩dxF_{\Phi (\omega ,\cdot )}(\xi )=\int _{\{x\colon \Phi (\omega ,x)\le 1\}}e^{-i\langle x,\xi \rangle }dx\]forω∈Ω\omega \in \Omega. We prove that, ifE(|FΦ(ξ)|)≤C[ξ]n+12E(|F_\Phi (\xi )|)\le C[\xi ]^{\frac {n+1}{2}}, where[ξ]=1+|ξ|[\xi ]=1+|\xi |, then\[E(NΦ)(t)=Vtn+O(tn−2+2n+1)E(N_\Phi )(t)=Vt^n+O(t^{n-2+\frac {2}{n+1}})\]whereV=E(|{x:Φ(⋅,x)≤1}|)V=E(|\{x\colon \Phi (\cdot ,x)\le 1\}|), the expected volume. That is, on average,NΦ(t)=Vtn+O(tn−2+2n+1)N_\Phi (t)=Vt^n+O(t^{n-2+\frac {2}{n+1}}). We give explicit examples in which the Gaussian curvature of{x:Φ(ω,x)≤1}\{x\colon \Phi (\omega ,x)\le 1\}is small with high probability, and the estimateNΦ(t)=Vtn+O(tn−2+2n+1)N_\Phi (t)=Vt^n+O(t^{n-2+\frac {2}{n+1}})holds nevertheless.
Noetherian down-up algebras
Authors: Kirkman Ellen , Musson Ian , Passman D. 
Published: July 2002

Down-up algebras

A
=
A
(
α
,
β
,
γ
)

A= A(\alpha ,\beta ,\gamma )

were introduced by G. Benkart and T. Roby to better understand the structure of certain posets. In this paper, we prove that

β
≠
0

\beta \neq 0

is equivalent to

A
A

being right (or left) Noetherian, and also to

A
A

being a domain. Furthermore, when this occurs, we show that

A
A

is Auslander-regular and has global dimension 3.