bluuebunny/crossref_metadata_2025 huggingface dataset:
Oversampling and preservation of tightness in affine frames
Authors: Catalán Raquel
Published: July 2002

The problem of how an oversampling of translations affects the bounds of an affine frame has been proposed by Chui and Shi. In particular, they proved that tightness is preserved if the oversampling factor is coprime with the dilation factor. In this paper we study, in the dyadic dilation case, oversampling of translation by factors which do not satisfy the above condition, and prove that tightness is preserved only in the case of affine frames generated by wavelets having frequency support with very particular properties.
Flat covers and cotorsion envelopes of sheaves
Authors: Enochs Edgar , Oyonarte Luis
Published: July 2002

In this paper we prove that any sheaf of modules over any topological space (in fact, any

O

\mathcal {O}

-module where

O

\mathcal {O}

is a sheaf of rings on the topological space) has a flat cover and a cotorsion envelope. This result is very useful, as we shall explain later in the introduction, in order to compute cohomology, due to the fact that the category of sheaves (

O

\mathcal {O}

-modules) does not have in general enough projectives.
Generic, almost primitive and test elements of free Lie algebras
Authors: Mikhalev Alexander , Umirbaev Ualbai , Yu Jie-Tai
Published: July 2002

We construct a series of generic elements of free Lie algebras. New almost primitive and test elements were found. We present an example of an almost primitive element which is not generic.
A uniqueness result concerning Schur ideals
Authors: Solazzo James
Published: July 2002

A subset of the set of all positive semi-definite matrices of a given size which is invariant under Schur (componentwise) multiplication by an arbitrary positive semi-definite matrix is said to be a Schur ideal. A subset of

k
k

-dimensional complex space is said to be

h
y
p
e
r
c
o
n
v
e
x

hyperconvex

if it arises as the set of possible values
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis w 1 comma ellipsis comma w Subscript k Baseline right-parenthesis equals left-parenthesis f left-parenthesis alpha 1 right-parenthesis comma ellipsis comma f left-parenthesis alpha Subscript k Baseline right-parenthesis right-parenthesis">

(

w

1

,
…
,

w

k

)
=
(
f
(

α

1

)
,
…
,
f
(

α

k

)
)

(w_{1}, \dots , w_{k}) = (f(\alpha _{1}), \dots , f(\alpha _{k}))

arising from restricting contractive elements

f
f

from some uniform algebra

A
A

to a finite set

{

α

1

,
…
,

α

k

}

\{ \alpha _{1}, \dots , \alpha _{k} \}

in the domain. When the uniform algebra is the disk algebra, the hyperconvex set is said to be a Pick body. Motivated by the classical Pick interpolation theorem, Paulsen has introduced a natural notion of duality between Schur ideals and hyperconvex sets. By using some recently developed results in operator algebras (matricial Schur ideals), we show that each Pick body has a unique affiliated Schur ideal.
Directive trees and games on posets
Authors: Ishiu Tetsuya , Yoshinobu Yasuo
Published: July 2002

We show that for any infinite cardinal

κ
\kappa

, every

(
κ
+
1
)

(\kappa +1)

-strategically closed poset is

κ

+

\kappa ^{+}

-strategically closed if and only if

◻

κ

\square _{\kappa }

holds. This extends previous results of Velleman, et.al.
The range of operators on von Neumann algebras
Authors: Bermúdez Teresa , Kalton N.
Published: July 2002

We prove that for every bounded linear operator

T
:
X
→
X

T:X\to X

, where

X
X

is a non-reflexive quotient of a von Neumann algebra, the point spectrum of

T
∗

T^*

is non-empty (i.e., for some

λ
∈

C

\lambda \in \mathbb C

the operator

λ
I
−
T

\lambda I-T

fails to have dense range). In particular, and as an application, we obtain that such a space cannot support a topologically transitive operator.
Regularity of viscosity solutions of a degenerate parabolic equation
Authors: Lu Yun-Guang , Qian Liwen
Published: July 2002

We study the Cauchy problem for the nonlinear degenerate parabolic equation of second order
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row u Subscript t Baseline equals u white up pointing triangle u minus gamma StartAbsoluteValue nabla u EndAbsoluteValue squared in normal upper Omega equals upper R Superscript upper N Baseline times upper R Superscript plus Baseline comma u left-parenthesis x comma 0 right-parenthesis equals u 0 left-parenthesis x right-parenthesis in upper R Superscript upper N Baseline comma EndLayout">

{

u
t

=
u
△
u
−
γ

|

∇
u

|

2

 
 
in

Ω
=

R
N

×

R

+

,
u
(
x
,
0
)
=

u

0

(
x
)
 
 
 
 
 
 
 
 
in

R

N

,

\left \{ \begin {array}{l} u_t=u\triangle u-\gamma |\nabla u|^2 \ \ \text {in} \hspace {0.3cm} \Omega =R^N \times R^{+}, u(x,0)=u_{0}(x)\ \ \ \ \ \ \ \ \text {in} \hspace {0.3cm} R^{N}, \end {array} \right .

\]
and present regularity results for the viscosity solutions.
The uncertainty principle on Riemannian symmetric spaces of the noncompact type
Authors: Sengupta J.
Published: July 2002

The uncertainty principle in

R

n

{\mathcal R}^{n}

says that it is impossible for a function and its Fourier transform to be simultaneously very rapidly decreasing. A quantitative assertion of this principle is Hardy’s theorem. In this article we prove various generalisations of Hardy’s theorem for Riemannian symmetric spaces of the noncompact type. In the case of the real line these results were obtained by Morgan and Cowling-Price.
Local boundary dilatation of quasiconformal maps in the disk
Authors: Cui Guizhen , Qi Yi
Published: July 2002

In this paper we partly give an affirmative answer to a problem proposed by F. Gardiner and N. Lakic by studying the gluing of quasiconformal maps.
On weak* convergence in 𝐻¹
Authors: Cima Joseph , Matheson Alec
Published: July 2002

A bounded sequence of functions in

H
1

H^1

which converges in measure on a set of positive measure of the unit circle converges weak

∗

{}^*

. An example is given to show that weak

∗

{}^*

convergence cannot be replaced by weak convergence.
Fixed points and powers of self-maps of 𝐻-spaces
Authors: Lupton Gregory , Oprea John
Published: July 2002

We give a new proof of the following result due to Duan: Let

f

:

X
→
X

f\,:\, X \to X

be any self-map of a finite connected

H
H

-space. Then

f

k

f^{k}

has a fixed point if

k
≥
2

k \geq 2

. Our proof is based on an approach due to Steve Halperin, whereby the Lefschetz number of a self-map is expressed in terms of the eigenvalues of the induced homomorphism of rational homotopy groups. This allows us to give a considerably shorter proof which avoids most of the technicalities of the original proof.
Self-similarity in inverse limit spaces of the tent family
Authors: Barge Marcy , Brucks Karen , Diamond Beverly
Published: July 2002

Taking inverse limits of the one-parameter family of tent maps of the interval generates a one-parameter family of inverse limit spaces. We prove that, for a dense set of parameters, these spaces are locally, at most points, the product of a Cantor set and an arc. On the other hand, we show that there is a dense

G
δ

G_\delta

set of parameters for which the corresponding space has the property that each neighborhood in the space contains homeomorphic copies of every inverse limit of a tent map.
Fibonacci numbers, Lucas numbers and integrals of certain Gaussian processes
Authors: Baringhaus Ludwig
Published: July 2002

We study the distributions of integrals of Gaussian processes arising as limiting distributions of test statistics proposed for treating a goodness of fit or symmetry problem. We show that the cumulants of the distributions can be expressed in terms of Fibonacci numbers and Lucas numbers.
A finiteness criterion for cohomology of Fréchet-Montel sheaves
Authors: Golovin V.
Published: July 2002

The paper deals with a finiteness criterion for the cohomology with compact supports of calculable Fréchet-Montel sheaves over locally compact topological spaces.
On maximal functions in Orlicz spaces
Authors: Kita Hiro-o
Published: July 2002

Let

Φ
(
t
)

\Phi (t)

and

Ψ
(
t
)

\Psi (t)

be the functions having the representations

Φ
(
t
)
=

∫

0

t

a
(
s
)
d
s

\Phi (t)=\int _{0}^{t} a(s)ds

and

Ψ
(
t
)
=

∫

0

t

b
(
s
)
d
s

\Psi (t)=\int _{0}^{t} b(s)ds

, where

a
(
s
)

a(s)

is a positive continuous function such that

∫

1

∞

a
(
s
)

s

d
s
=
+
∞

\int _{1}^{\infty }\frac {a(s)}{s}ds=+\infty

and

b
(
s
)

b(s)

is quasi-increasing. Then the maximal function

M
f

Mf

is a function in Orlicz space

L

Φ

L^{\Phi }

for all

f
∈

L

Ψ

f\in L^{\Psi }

if and only if there exists a positive constant

c

1

c_{1}

such that

∫

1

s

a
(
t
)

t

d
t
≤

c

1

b
(

c

1

s
)

\int _{1}^{s} \frac {a(t)}{t}dt\leq c_{1}b(c_{1}s)

for all

s
≥
1

s\geq 1

.
Some harmonic functions on Minkowski space
Authors: Glezen P. , Penner R.
Published: July 2002

This note presents elementary geometric descriptions of several simple families of harmonic functions on the upper sheet of the unit hyperboloid in Minkowski three-space. As is briefly discussed here, these calculations grew out of an earlier attempt to construct Poincaré series on punctured surfaces using Minkowski geometry.
The operator 𝑎(𝑥)\frac{𝑑}𝑑𝑥 on Banach space
Authors: Yao Fuyuan
Published: July 2002

The operator

a
(
x
)

d

d
x

a(x)\frac {d}{dx}

on

C
(
I
)

C(I)

, where

I
I

is an interval contained in the real line, is considered in many places. In this paper, we attempt to reconsider it in the subspace of

C
0

(
−
∞
,
∞
)

C_0(-\infty ,\infty )

containing all even functions, and show that it generates a strongly continuous semigroup. It is interesting that our main conditions seem contradictory to previous ones. It is due to the symmetry of the functions and the different domain of the operator than usual.
Coactions of Hopf algebras on Cuntz algebras and their fixed point algebras
Authors: Paolucci Anna
Published: July 2002

We study coactions of Hopf algebras coming from compact quantum groups on the Cuntz algebra. These coactions are the natural generalization to the coalgebra setting of the canonical representation of the unitary matrix group

U
(
d
)

U(d)

as automorphisms of the Cuntz algebra

O
d

O_d

.

In particular we study the fixed point subalgebra under the coaction of the quantum compact groups

U
q

(
d
)

U_q(d)

on the Cuntz algebra

O
d

O_d

by extending to any dimension

d
>
∞

d>\infty

a result of Konishi (1992).

Furthermore we give a description of the fixed point subalgebra under the coaction of

S

U
q

(
d
)

SU_q(d)

on

O
d

O_d

in terms of generators.
Hypercomplex structures on four-dimensional Lie groups
Authors: Barberis María
Published: July 2002

The purpose of this paper is to classify invariant hypercomplex structures on a

4
4

-dimensional real Lie group

G
G

. It is shown that the

4
4

-dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group

H

\mathbb H

of the quaternions, the multiplicative group

H

∗

{\mathbb H}^*

of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyperbolic spaces,

R

H
4

{\mathbb R}H^4

and

C

H
2

{\mathbb C}H^2

, respectively, and the semidirect product

C

⋊

C

{\mathbb C}\rtimes {\mathbb C}

. We show that the spaces

C

H
2

{\mathbb C}H^2

and

C

⋊

C

{\mathbb C}\rtimes {\mathbb C}\,

possess an

R

P
2

{\mathbb R}P^2

of (inequivalent) invariant hypercomplex structures while the remaining groups have only one, up to equivalence. Finally, the corresponding hyperhermitian

4
4

-manifolds are determined.
A new characterization of semisimple Lie algebras
Authors: Benayadi Said
Published: July 2002

Using Casimir elements, we characterize the semisimple Lie algebras among the quadratic Lie algebras. This characterization gives, in particular, a generalization of a consequence of Cartan’s second criterion.
Torsion-free duality is Warfield
Authors: Faticoni T. , Goeters H. , Vinsonhaler C. , Wickless W.
Published: July 2002

We show that, under certain natural conditions, a duality discovered by R. B. Warfield, Jr., is the only duality on categories of finite-rank torsion-free modules over Dedekind domains.
Ricci solitons on compact Kähler surfaces
Authors: Ivey Thomas
Published: July 2002

We classify the Kähler metrics on compact manifolds of complex dimension two that are solitons for the constant-volume Ricci flow, assuming that the curvature is slightly more positive than that of the single known example of a soliton in this dimension.
An embedding theorem of Sobolev type for an operator with singularity
Authors: Watanabe Shuji
Published: July 2002

We discuss spaces of Sobolev type which are defined by the operator with singularity:

D

=
d

/

d
x
−
(
c

/

x
)
R

\mathcal {D} = d/dx - (c/x)R

, where

R
u
(
x
)
=
u
(
−
x
)

Ru(x) = u(-x)

and

c
>
1

c > 1

. This operator appears in a one-dimensional harmonic oscillator governed by Wigner’s commutation relations. We study smoothness of

u
u

and continuity of

u

/

x

β

u / x^{\beta }

(

β
>
0

\beta > 0

) where

u
u

is in each space of Sobolev type, and obtain a generalization of the Sobolev embedding theorem. On the basis of a generalization of the Fourier transform, the proof is carried out. We apply the result to the Cauchy problems for partial differential equations with singular coefficients.
Continuous functions on compact groups
Authors: Blecher David
Published: July 2002

We show that every scalar valued continuous function

f
(
g
)

f(g)

on a compact group

G
G

may be written as

f
(
g
)
=
⟨
Φ
(
g
)
ξ
,
η
⟩

f(g) = \langle \Phi (g) \xi , \eta \rangle

for all

g
∈
G

g \in G

, where

ξ
,
η

\xi , \eta

are vectors in a separable Hilbert space

H

\mathcal {H}

, and

Φ
(
g
)

\Phi (g)

is a strongly continuous unitary valued function on

G
G

which is a product of unitary representations and antirepresentations of

G
G

on

H

\mathcal {H}

. This product is countable, but always converges uniformly on

G
G

. Moreover the supremum norm of

f
f

is matched by

‖
ξ
‖
‖
η
‖

\Vert \xi \Vert \Vert \eta \Vert

. This may be viewed as a ‘Fourier product representation’ for

f
f

, and complements a result of Eymard for the Fourier algebra. For ‘Fourier polynomials’ we show that the Hilbert space

\mathcal {}

may be taken to be finite dimensional, and the product finite, which is more or less obvious except in that we are able to match the correct norm. The main ingredients of the proof are the Peter-Weyl theory, Tannaka’s duality theorem, and a method developed with Paulsen using a characterization of operator algebras due to the author, Ruan and Sinclair. We also give the analogues of these formulae for compact quantum groups.
Functions operating from a complex Banach space to its real part
Authors: Briem Eggert
Published: July 2002

We consider functions operating from a complex Banach function space to its real part. We show among other things, that if

|

b

|

∈

Re

B

|b|\in \textrm {Re}B

for all

b
b

in an ultraseparating Banach function space

B
B

, then Re

B
=

C

R

(
X
)

B=C_ { \mathbf {R}}(X)

.
Rudin’s orthogonality problem and the Nevanlinna counting function
Authors: Bourdon Paul
Published: July 2002

Let

ϕ
\phi

be a holomorphic function taking the open unit disk

U
U

into itself. We show that the set of nonnegative powers of

ϕ
\phi

is orthogonal in

L
2

(
∂
U
)

L^2(\partial U)

if and only if the Nevanlinna counting function of

ϕ
\phi

,

N
ϕ

N_\phi

, is essentially radial. As a corollary, we obtain that the orthogonality of

{

ϕ
n

:
n
=
0
,
1
,
2
,
…
}

\{\phi ^n: n=0,1,2,\ldots \}

for a univalent

ϕ
\phi

implies

ϕ
(
z
)
=
α
z

\phi (z) = \alpha z

for some constant

α
\alpha

. We also show that if

{

ϕ
n

:
n
=
0
,
1
,
2
,
…
}

\{\phi ^n: n=0,1,2,\ldots \}

is orthogonal, then the closure of

ϕ
(
U
)

\phi (U)

must be a disk.
The Beurling-Malliavin density of a random sequence
Authors: Seip Kristian , Ulanovskii Alexander
Published: July 2002

A formula is given for the completeness radius of a random exponential system

{

t

l

e

i

ξ

n

t

}

l
=
0
,
n
∈

Z

p

n

−
1

\{t^{l}e^{i\xi _{n}t}\}_{l=0,n\in {\mathbb {Z}}}^{p_{n}-1}

in terms of the probability measures of

ξ

n

\xi _{n}

.
Beals-Cordes-type characterizations of pseudodifferential operators
Authors: Taylor Michael
Published: July 2002

We show that, if

U
U

is the representation of

S

O
e

(
n
+
1
,
1
)

SO_e(n+1,1)

on

L
2

(

S
n

)

L^2(S^n)

given by (2.11), and

P
P

is a bounded operator on

L
2

(

S
n

)

L^2(S^n)

, then

P
P

belongs to

O
P

S

1
,
0

0

(

S
n

)

OPS_{1,0}^0(S^n)

if and only if
\[

P
(
g
)
=
U
(
g
)
P
U
(
g

)

−
1

P(g)=U(g)PU(g)^{-1}

\]
is a

C
∞

C^\infty

function on

S

O
e

(
n
+
1
,
1
)

SO_e(n+1,1)

with values in the Banach space

L

(

L
2

(

S
n

)
)

\mathcal L(L^2(S^n))

.
Free quotients of 𝑆𝐿₂(𝑅[𝑥])
Authors: Krstic Sava , McCool James
Published: July 2002

It is shown that if

R
R

is an integral domain which is not a field, and

U
2

(
R
[
x
]
)

U_2(R[x])

is the subgroup of

S

L
2

(
R
[
x
]
)

SL_2(R[x])

generated by all unipotent elements, then the quotient group

S

L
2

(
R
[
x
]
)

/

U
2

(
R
[
x
]
)

SL_2(R[x])/U_2(R[x])

has a free quotient of infinite rank.
Gaussian polynomials and content ideals
Authors: Heinzer William , Huneke Craig
Published: July 2002

We prove that every regular Gaussian polynomial over a locally Noetherian ring has invertible content ideal. We do this by first proving that Gaussian polynomials over an approximately Gorenstein local ring have principal content ideal. We also show over locally Noetherian rings that a regular polynomial

f
f

of degree

n
n

is Gaussian if

c
(
f
g
)
=
c
(
f
)
c
(
g
)

c(fg) = c(f)c(g)

for polynomials

g
g

of degree at most

n
n

.
Estimating a skein module with 𝑆𝐿₂(ℂ) characters
Authors: Bullock Doug
Published: July 2002

We introduce a new technique for estimating the number of generators of the Kauffman bracket skein module of a three manifold; one which requires the construction of linear functionals on a simpler version of the module. Of particular interest is the use of representations of the fundamental group into

S

L
2

(

C

)

SL_2(\hspace {-.3pt}{\mathbb C}\hspace {.3pt})

to generate the functionals.
From countable compactness to absolute countable compactness
Authors: Rudin Mary , Stares Ian , Vaughan Jerry
Published: July 2002

We show that every countably compact space which is monotonically normal, almost 2-fully normal, radial

T
2

T_2

, or

T
3

T_3

with countable spread is absolutely countably compact. For the first two mentioned properties, we prove more general results not requiring countable compactness. We also prove that every monotonically normal, orthocompact space is finitely fully normal.
Optimal estimation of shell thickness in Cutland’s construction of Wiener measure
Authors: Li Bang-He , Li Ya-Qing
Published: July 2002

In Cutland’s construction of Wiener measure, he used the product of Gaussian measures on

∗

R
N

^*R^N

, where

N
N

is an infinite integer. It is mentioned by Cutland and Ng that for the product measure

γ
\gamma

,
\[

γ
(
{
x
:

R
1

≤
‖
x
‖
≤

R
2

}
)
≃
1
,

\gamma (\{x:R_1\le \|x\|\le R_2\})\simeq 1,

\]
where

R
1

=
1
−
(
log
⁡
N

)

1
2

N

−

1
2

R_1=1-(\log N)^{\frac 12} N^{-\frac 12}

and

R
2

=
1
+
M

N

−

1
2

R_2=1+MN^{-\frac 12}

with

M
M

any positive infinite number. We prove here that

R
1

R_1

may be replaced by

1
−
m

N

−

1
2

1-mN^{-\frac 12}

with

m
m

any positive infinite number. This is the optimal estimation for the shell thickness. It is also proved that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gamma left-parenthesis StartSet x colon double-vertical-bar x double-vertical-bar greater-than 1 EndSet right-parenthesis asymptotically-equals gamma left-parenthesis StartSet x colon double-vertical-bar x double-vertical-bar greater-than 1 EndSet right-parenthesis asymptotically-equals one half">

γ
(
{
x
:
‖
x
‖
>
1
}
)
≃
γ
(
{
x
:
‖
x
‖
>
1
}
)
≃

1
2

\gamma (\{x:\|x\|>1 \})\simeq \gamma (\{x:\|x\|>1\})\simeq \frac 12

. And for the *Lebesgue measure

μ
\mu

,

μ
(
{
x
:
‖
x
‖
≤
r
}
)

\mu (\{x:\|x\|\le r\})

is finite and not infinitesimal iff
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r equals left-parenthesis 2 pi e right-parenthesis Superscript negative one half Baseline upper N Superscript one half left-parenthesis 1 plus StartFraction 1 Over upper N EndFraction right-parenthesis Baseline e Superscript StartFraction a Over upper N EndFraction">

r
=
(
2
π
e

)

−

1
2

N

1
2

(
1
+

1
N

)

e

a
N

r=(2\pi e)^{-\frac 12}N^{\frac 12(1+\frac 1N)}e^{\frac aN}

with

a
a

finite, while for the *Lebesgue area of the sphere

S

N
−
1

(
r
)

S^{N-1}(r)

,

r
r

should be

(
2
π
e

)

−

1
2

N

1
2

e

a
N

(2\pi e)^{-\frac 12}N^{\frac 12} e^{\frac aN}

.
Continuous multiplicative mappings on 𝐶(𝑋)
Authors: Lešnjak Gorazd , Šemrl Peter
Published: July 2002

Let

X
X

and

Y
Y

be compact Hausdorff topological spaces, and let

C
(
X
)

C(X)

and

C
(
Y
)

C(Y)

be real Banach algebras of all real-valued continuous functions on

X
X

and

Y
Y

, respectively. The general form of continuous multiplicative mappings

Φ
:
C
(
X
)
→
C
(
Y
)

\Phi \colon C(X)\to C(Y)

is given.
Elementary abelian 2-group actions on flag manifolds and applications
Authors: Mukherjee Goutam , Sankaran Parameswaran
Published: July 2002

Let

N

∗

\mathcal N_\ast

denote the unoriented cobordism ring. Let

G
=
(

Z

/

2

)
n

G=(\mathbb Z/2)^n

and let

Z
∗

(
G
)

Z_\ast (G)

denote the equivariant cobordism ring of smooth manifolds with smooth

G
G

-actions having finite stationary points. In this paper we show that the unoriented cobordism class of the (real) flag manifold
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M equals upper O left-parenthesis m right-parenthesis slash left-parenthesis upper O left-parenthesis m 1 right-parenthesis times midline-horizontal-ellipsis times upper O left-parenthesis m Subscript s Baseline right-parenthesis right-parenthesis">

M
=
O
(
m
)

/

(
O
(

m
1

)
×
⋯
×
O
(

m
s

)
)

M=O(m)/(O(m_1)\times \dots \times O(m_s))

is in the subalgebra generated by

⨁

i
>

2
n

N

i

\bigoplus _{i>2^n}\mathcal N_i

, where

m
=
∑

m
j

m= \sum m_j

, and

2

n
−
1

>
m
≤

2
n

2^{n-1}>m\le 2^n

. We obtain sufficient conditions for indecomposability of an element in

Z
∗

(
G
)

Z_\ast (G)

. We also obtain a sufficient condition for algebraic independence of any set of elements in

Z
∗

(
G
)

Z_\ast (G)

. Using our criteria, we construct many indecomposable elements in the kernel of the forgetful map

Z
d

(
G
)
→

N

d

Z_d(G)\to \mathcal N_d

in dimensions

2
≤
d
>
n

2\le d>n

, for

n
>
2

n>2

, and show that they generate a polynomial subalgebra of

Z
∗

(
G
)

Z_\ast (G)

.
Incoherent negatively curved groups
Authors: Wise Daniel
Published: July 2002

In part 1, a construction of Rips is modified so that it produces a CAT

(
−
1
)

(-1)

group instead of a small-cancellation group. Thus, many of the applications of Rips’ construction to small-cancellation groups may be applied to CAT

(
−
1
)

(-1)

groups as well. Part 2 offers a simple way of producing incoherent groups.
Laplacians and Sobolev gradients
Authors: Neuberger J.
Published: July 2002

We describe a class of operators which fit the description of laplacians and which may be used to unify the construction of various Sobolev gradients.
Uniqueness of positive solutions for Sturm-Liouville boundary value problems
Authors: Wong Fu-Hsiang
Published: July 2002

Sufficient conditions for the uniqueness of positive solutions of singular Sturm-Liouville boundary value problems
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis BVP right-parenthesis EndLabel StartLayout Enlarged left-brace 1st Row left-parenthesis normal upper E right-parenthesis left-parenthesis StartAbsoluteValue u Superscript prime Baseline EndAbsoluteValue Superscript m minus 2 Baseline u Superscript prime Baseline right-parenthesis Superscript prime Baseline plus f left-parenthesis t comma u comma u Superscript prime Baseline right-parenthesis equals 0 comma in left-parenthesis theta 1 comma theta 2 right-parenthesis comma m greater-than-or-equal-to 2 comma left-parenthesis normal upper B normal upper C right-parenthesis StartLayout Enlarged left-brace 1st Row alpha 1 u left-parenthesis theta 1 right-parenthesis minus beta 1 u prime left-parenthesis theta 1 right-parenthesis equals 0 comma alpha 2 u left-parenthesis theta 2 right-parenthesis plus beta 2 u prime left-parenthesis theta 2 right-parenthesis equals 0 comma EndLayout EndLayout EndLayout">

(BVP)

{

(

E

)
(

|

u
′

|

m
−
2

u
′

)
′

+
f
(
t
,
u
,

u
′

)
=
0
,

in
(

θ
1

,

θ
2

)
,
m
≥
2
,
 
(

B
C

)

{

α
1

u
(

θ
1

)
−

β
1

u
′

(

θ
1

)
=
0
,
 

α
2

u
(

θ
2

)
+

β
2

u
′

(

θ
2

)
=
0
,

\begin{equation*} \begin {cases} (\mathrm E) (|u’|^{m-2}u’)’+f(t,u,u’)=0,\quad \text {in} (\theta _1,\theta _2),m\ge 2,\ (\mathrm {BC})\begin {cases} \alpha _1u(\theta _1)-\beta _1u’(\theta _1)=0,\ \alpha _2u(\theta _2)+\beta _2u’(\theta _2)=0, \end{cases} \end{cases} \tag {BVP} \end{equation*}

where

α
i

,

β
i

≥
0

\alpha _i,\beta _i\ge 0

and

α
i
2

+

β
i
2

≠
0

\alpha _i^2+\beta _i^2\not =0

(
i
=
1
,
2
)

(i=1,2)

, are established.
Differential forms on quotients by reductive group actions
Authors: Brion Michel
Published: July 2002

Let

X
X

be a smooth affine algebraic variety where a reductive algebraic group

G
G

acts with a smooth quotient space

Y
=
X

/

/

G

Y=X//G

. We show that the algebraic differential forms on

X
X

which are pull-backs of forms on

Y
Y

are exactly the

G
G

-invariant horizontal differential forms on

X
X

.
On 𝐽𝐵*-triples which are M-ideals in their biduals
Authors: Cabello Juan , Nieto Eduardo
Published: July 2002

The object of this paper is to investigateJB∗JB^*-triples which are M-ideals in their biduals.
An index theory for ℤ-actions
Authors: Kim In-Sook
Published: July 2002

This paper concerns an index theory for

Z

\Bbb Z

-actions induced by a homeomorphism of a compact space. We give a definition of a genus for uniform spaces and prove that the genus for compact spaces is an index. To this end we show a

Z

{\Bbb Z}

-version of the Borsuk-Ulam theorem and the existence of a continuous equivariant extension for these

Z

\Bbb Z

-actions.
A rigid space homeomorphic to Hilbert space
Authors: Nhu Nguyen , Sisson Paul
Published: July 2002

A rigid space is a topological vector space whose endomorphisms are all simply scalar multiples of the identity map. This is in sharp contrast to the behavior of operators on

ℓ

2

\ell _{2}

, and so rigid spaces are, from the viewpoint of functional analysis, fundamentally different from Hilbert space. Nevertheless, we show in this paper that a rigid space can be constructed which is topologically homeomorphic to Hilbert space. We do this by demonstrating that the first complete rigid space can be modified slightly to be an AR-space (absolute retract), and thus by a theorem of Dobrowolski and Torunczyk is homeomorphic to

ℓ

2

\ell _{2}

.
Ample and spanned vector bundles of top Chern number two on smooth projective varieties
Authors: Noma Atsushi
Published: July 2002

The purpose of this paper is to classify ample and spanned vector bundles of top Chern number two on smooth projective varieties of arbitrary dimension defined over an algebraically closed field of characteristic zero.
Additivity of quasi-measures
Authors: Grubb D. , LaBerge Tim
Published: July 2002

We prove that quasi-measures on compact Hausdorff spaces are countably additive. Contained in this result is a proof that every quasi-measure decomposes uniquely into a measure and a quasi-measure that has no smaller measure beneath it. We also show that it is consistent with the usual axioms of set-theory that quasi-measures on compact Hausdorff spaces are

ℵ
1

\aleph _1

-additive. Finally, we construct an example that places strong restrictions on other forms of additivity.
A note on a Jacobian identity
Authors: Ewell John
Published: July 2002

An identity involving eight-fold infinite products, first derived by Jacobi in his theory of theta functions, is the subject of this note. Three similar identities, including one that implies Jacobi’s identity, are presented.
On certain character sums over 𝔽_{𝕢}[𝕋]
Authors: Hsu Chih-Nung
Published: July 2002

LetFq{\mathbb F}_{q}be the finite field withqqelements and letA\mathbf {A}denote the ring of polynomials in one variable with coefficients inFq{\mathbb F}_{q}. LetPPbe a monic polynomial irreducible inA\mathbf {A}. We obtain a bound for the least degree of a monic polynomial irreducible inA\mathbf {A}(qqodd) which is a quadratic non-residue moduloPP. We also find a bound for the least degree of a monic polynomial irreducible inA\mathbf {A}which is a primitive root moduloPP.
Duality for full crossed products of 𝐶*-algebras by non-amenable groups
Authors: Nilsen May
Published: July 2002

Let

(
A
,
G
,
δ
)

(A, G, \delta )

be a cosystem and

(
A
,
G
,
α
)

(A, G,\alpha )

be a dynamical system. We examine the extent to which induction and restriction of ideals commute, generalizing some of the results of Gootman and Lazar (1989) to full crossed products by non-amenable groups. We obtain short, new proofs of Katayama and Imai-Takai duality, the faithfulness of the induced regular representation for full coactions and actions by amenable groups. We also give a short proof that the space of dual-invariant ideals in the crossed product is homeomorphic to the space of invariant ideals in the algebra, and give conditions under which the restriction mapping is open.
Regularization of semigroups that are strongly continuous for 𝑡>0
Authors: Kunstmann P.
Published: July 2002

Let

E
E

be a Banach space and

T
:
]
0
,
∞
[
→
L
(
E
)

T:]0,\infty [\to L(E)

a strongly continuous semigroup with

⋂

t
>
0

Kern
⁡

T
t

=
{
0
}

\bigcap _{t>0}\operatorname {Kern} T_t=\{0\}

. We show that the generator

A
A

of

(

T
t

)

(T_t)

generates a regularized semigroup. Our construction of a regularizing operator uses an existence result of J. Esterle.
A unified extension of two results of Ky Fan on the sum of matrices
Authors: Tam Tin-Yau
Published: July 2002

Let

A
A

be an

n
×
n

n\times n

Hermitian matrix with

λ
(
A
)
=
(

λ
1

(
A
)
,
…
,

λ
n

(
A
)
)

\lambda (A) = (\lambda _1(A), \dots , \lambda _n(A))

where

λ
1

(
A
)
≥
⋯
≥

λ
n

(
A
)

\lambda _1(A) \ge \cdots \ge \lambda _n(A)

are the ordered eigenvalues of

A
A

. A result of Ky Fan (1949) asserts that if

A
A

and

B
B

are

n
×
n

n\times n

Hermitian matrices, then

λ
(
A
+
B
)

\lambda (A+B)

is majorized by

λ
(
A
)
+
λ
(
B
)

\lambda (A) + \lambda (B)

. We extend the result in the framework of real semisimple Lie algebras in the following way. Let

g
\frak g

be a noncompact real semisimple Lie algebra with Cartan decomposition

g

=

t

+

p

{\frak g} = {\frak t} + {\frak p}

. We show that for any given

p
,
q
∈
p

p, q\in \frak p

,

a
+

(
p
+
q
)
≤

a
+

(
p
)
+

a
+

(
q
)

a_+(p+q)\le a_+(p) + a_+(q)

, where

a
+

(
x
)

a_+(x)

is the unique element corresponding to

x
∈
p

x\in \frak p

, in a fixed closed positive Weyl chamber

a

+

{\frak a}_+

of a maximal abelian subalgebra

a

{\frak a}

of

g

{\frak g}

in

p

{\frak p}

. Here the ordering

≤
\le

is induced by the dual cone

a

+
∗

{\frak a}_+^*

of

a

+

{\frak a}_+

. Fan’s result corresponds to the Lie algebra

s
l

(
n
,

C

)

{\frak {sl}}(n, {\Bbb C})

. The compact case is also discussed. As applications, two unexpected singular values inequalities concerning the sum of two real matrices and the sum of two real skew symmetric matrices are obtained.
Semigroup properties of factors in the polar decomposition or the operator De-Moivre formula
Authors: Catepillán Ximena , Szymański Wacław
Published: July 2002

We give necessary and sufficient conditions under which the factors in the polar decomposition of a semigroup homomorphism are themselves semigroup homomorphisms.
A generalization of 2-homogeneous continua being locally connected
Authors: Whittington Keith
Published: July 2002

An elementary proof is given that if each pair of points of a homogeneous metric continuum can be mapped by a homeomorphism into an arbitrarily small connected set, then the continuum is locally connected.
Splitting of the direct image of sheaves under the Frobenius
Authors: Bøgvad Rikard
Published: July 2002

A generalisation and a new proof are given of a recent result of J. F. Thomsen (1996), which says that for

L
L

a line bundle on a smooth toric variety

X
X

over a field of positive characteristic, the direct image

F
∗

L

F_*L

under the Frobenius morphism splits into a direct sum of line bundles. (The special case of projective space is due to Hartshorne.) Our method is to interpret the result in terms of Grothendieck differential operators
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D i f f Superscript left-parenthesis 1 right-parenthesis Baseline left-parenthesis upper L comma upper L right-parenthesis approximately-equals upper H o m Subscript upper O Sub Subscript upper X Sub Sub Superscript left-parenthesis 1 right-parenthesis Baseline left-parenthesis upper F Subscript asterisk Baseline upper L comma upper F Subscript asterisk Baseline upper L right-parenthesis">

Diff

(
1
)

⁡
(
L
,
L
)
≅

Hom

O

X

(
1
)

⁡
(

F
∗

L
,

F
∗

L
)

\operatorname {Diff}^{(1)} (L,L)\cong \operatorname {Hom}_{O_{X^{(1)}}}(F_*L,F_*L)

, and

T
T

-linearized sheaves.
Uniform factorization for compact sets of operators
Authors: Aron R. , Lindström M. , Ruess W. , Ryan R.
Published: July 2002

We prove a factorization result for relatively compact subsets of compact operators using the Bartle and Graves Selection Theorem, a characterization of relatively compact subsets of tensor products due to Grothendieck, and results of Figiel and Johnson on factorization of compact operators. A further proof, essentially based on the Banach-Dieudonné Theorem, is included. Our methods enable us to give an easier proof of a result of W.H. Graves and W.M. Ruess.
Gaussian estimates and regularized groups
Authors: Zheng Quan , Zhang Jizhou
Published: July 2002

We show that if a bounded analytic semigroup{T(z)}Re⁡z>0\{T(z)\}_{ \operatorname {Re}z>0}onL2(Ω)L^2({\boldsymbol {\Omega }} )(Ω⊂Rn)({\boldsymbol {\Omega }} \subset \mathbf {R} ^n)satisfies a Gaussian estimate of ordermmandApA_pis the generator of its consistent semigroup onLp(Ω)L^p({\boldsymbol {\Omega }} )(1≤p>∞)(1\le p>\infty ), theniApiA_pgenerates a(1−Ap)−α(1-A_p)^{-\alpha }-regularized group onLp(Ω)L^p({\boldsymbol {\Omega }} )whereα>2n|12−1p|\alpha >2n |\frac {1}{2}-\frac {1}{p}|. We obtain the estimate of(λ−Ap)−1(\lambda -A_p)^{-1}(|arg⁡λ|>π|\operatorname {arg}\lambda |>\pi) and thepp-independence ofσ(Ap)\sigma (A_p), and give applications to Schrödinger operators and elliptic operators of higher order.
Function algebras and the lattice of compactifications
Authors: Mendivil Franklin
Published: July 2002

We provide some conditions as to when

K
(
X
)
≅
K
(
Y
)

K(X) \cong K(Y)

for two locally compact spaces

X
X

and

Y
Y

(where

K
(
X
)

K(X)

is the lattice of all Hausdorff compactifications of

X
X

). More specifically, we prove that

K
(
X
)
≅
K
(
Y
)

K(X) \cong K(Y)

if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript asterisk Baseline left-parenthesis upper X right-parenthesis slash upper C 0 left-parenthesis upper X right-parenthesis approximately-equals upper C Superscript asterisk Baseline left-parenthesis upper Y right-parenthesis slash upper C 0 left-parenthesis upper Y right-parenthesis">

C
∗

(
X
)

/

C
0

(
X
)
≅

C
∗

(
Y
)

/

C
0

(
Y
)

C^*(X)/C_0(X) \cong C^*(Y)/C_0(Y)

. Using this result, we prove several extensions to the case where

K
(
X
)

K(X)

is embedded as a sub-lattice of

K
(
Y
)

K(Y)

and to where

X
X

and

Y
Y

are not locally compact. One major contribution is in the use of function algebra techniques. The use of these techniques makes the extensions simple and clean and brings new tools to the subject.
Ordinary differential equations with fractal noise
Authors: Klingenhöfer F. , Zähle M.
Published: July 2002

The differential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d x left-parenthesis t right-parenthesis equals a left-parenthesis x left-parenthesis t right-parenthesis comma t right-parenthesis d upper Z left-parenthesis t right-parenthesis plus b left-parenthesis x left-parenthesis t right-parenthesis comma t right-parenthesis d t">

d
x
(
t
)

=

a
(
x
(
t
)
,
t
)

d
Z
(
t
)

+

b
(
x
(
t
)
,
t
)

d
t

dx(t) \, = \, a(x(t),t) \,dZ(t) \:+\: b(x(t),t) \,dt

\]
for fractal-type functions

Z
(
t
)

Z(t)

is determined via fractional calculus. Under appropriate conditions we prove existence and uniqueness of a local solution by means of its representation

x
(
t
)

=

h
(
y
(
t
)
+
Z
(
t
)
,
t
)

x(t)\, =\, h(y(t)+Z(t),t)

for certain

C
1

C^1

-functions

h
h

and

y
y

. The method is also applied to Itô stochastic differential equations and leads to a general pathwise representation. Finally we discuss fractal sample path properties of the solutions.
Infinite-dimensional Lie algebras of generalized Block type
Authors: Osborn J. , Zhao Kaiming
Published: July 2002

This paper investigates a class of infinite-dimensional Lie algebras over a field of characteristic

0
0

which are called here Lie algebras of generalized Block type, and which genereralize a class of Lie algebras originally defined by Richard Block. Under certain natural restrictions, this class of Lie algebras is shown to break into five subclasses. One of these subclasses contains all generalized Cartan type

K
K

Lie algebras and some Lie algebras of generalized Cartan type

H
H

, and a second one is the class of Lie algebras of type

L
L

, which were previously defined and studied elsewhere by the authors. The other three types are hybrids of the first two types, and are new.
Arens regularity of weakly sequentially complete Banach algebras
Authors: Ülger A.
Published: July 2002

In this paper we prove the following result: Let

A
A

be a nonunital Banach algebra with a bounded approximate identity. Then

A
A

cannot be both Arens regular and weakly sequentially complete. The paper also contains some applications of this result.
𝐶¹ approximations of inertial manifolds via finite differences
Authors: Kobayasi Kazuo
Published: July 2002

We construct an inertial manifold for the evolution equation as a limit of the inertial manifolds for the difference approximations of the Trotter-Kato type and show that this limit is taken in a

C

1

C^{1}

topology.
Expanding a band
Authors: MacDonald Gordon
Published: July 2002

Any pure operator band can be expanded so that each component of the band is reflexive.
Free boundary value problems for analytic functions in the closed unit disk
Authors: Fournier Richard , Ruscheweyh Stephan
Published: July 2002

We shall prove (a slightly more general version of) the following theorem: let

Φ
\Phi

be analytic in the closed unit disk

D

¯

\overline {\mathbb {D}}

with

Φ
:
[
0
,
1
]
→
(
0
,
1
]

\Phi :[0,1]\rightarrow (0,1]

, and let

B
(
z
)

B(z)

be a finite Blaschke product. Then there exists a function

h
h

satisfying: i)

h
h

analytic in the closed unit disk

D

¯

\overline {\mathbb {D}}

, ii)

h
(
0
)
>
0

h(0)>0

, iii)

h
(
z
)
≠
0

h(z)\neq 0

in

D

¯

\overline {\mathbb {D}}

, such that
\[

F
(
z
)
:=

∫

0

z

h
(
t
)
B
(
t
)
d
t

F(z):=\int _{0}^{z}h(t)B(t)dt

\]
satisfies
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper F prime left-parenthesis z right-parenthesis EndAbsoluteValue equals normal upper Phi left-parenthesis StartAbsoluteValue upper F left-parenthesis z right-parenthesis EndAbsoluteValue squared right-parenthesis comma z element-of partial-differential double-struck upper D period">

|

F
′

(
z
)

|

=
Φ
(

|

F
(
z
)

|

2

)
,

z
∈
∂

D

.

|F’(z)|=\Phi (|F(z)|^2), \quad z\in \partial \mathbb {D}.

\]
This completes a recent result of Kühnau for

Φ
(
x
)
=
1
+
α
x

\Phi (x)=1+\alpha x

,

−
1
>
α
>
0

-1>\alpha >0

, where this boundary value problem has a geometrical interpretation, namely that

β
(
α
)
F
(
r
(
α
)
z
)

\beta (\alpha )F(r(\alpha )z)

preserves hyperbolic arc length on

∂

D

\partial \mathbb {D}

for suitable

β
(
α
)
,

\beta (\alpha ),

r
(
α
)

r(\alpha )

. For these important choices of

Φ
\Phi

we also prove that the corresponding functions

h
h

are uniquely determined by

B
B

, and that

z
h
(
z
)

zh(z)

is univalent in

D

\mathbb {D}

. Our work is related to Beurling’s and Avhadiev’s on conformal mappings solving free boundary value conditions in the unit disk.
𝐿^{𝑝} estimates for oscillatory integral operators
Authors: Cheng Leslie , Pan Yibiao
Published: July 2002

An endpoint boundedness result is established for a class of oscillatory integral operators.
Hilbert coefficients and the associated graded rings
Authors: Wang Hsin-Ju
Published: July 2002

Let

(
R
,

m

)

(R, \mathfrak {m})

be a

d
d

-dimensional Cohen-Macaulay local ring with infinite residue field. Let

I
I

be an

m

\mathfrak {m}

-primary ideal of

R
R

. In this paper, we prove that if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts lamda left-parenthesis upper I Superscript n Baseline slash upper I Superscript n minus 1 Baseline upper J right-parenthesis minus e 1 left-parenthesis upper I right-parenthesis equals 1">

∑

n
=
1

∞

λ
(

I
n

/

I

n
−
1

J
)
−

e
1

(
I
)
=
1

\sum _{n=1}^{\infty } \lambda (I^n/I^{n-1}J)-e_1(I)=1

for some minimal reduction

J
J

of

I
I

, then depth 

G
(
I
)
≥
d
−
2

G(I)\geq d-2

.
Factoring weakly compact operators and the inhomogeneous Cauchy problem
Authors: Bárcenas Diómedes
Published: July 2002

By using the technique of factoring weakly compact operators through reflexive Banach spaces we prove that a class of ordinary differential equations with Lipschitz continuous perturbations has a strong solution when the problem is governed by a closed linear operator generating a strongly continuous semigroup of compact operators.
On the coefficients of binary bent functions
Authors: Hou Xiang-dong
Published: July 2002

We prove a 2-adic inequality for the coefficients of binary bent functions in their polynomial representations. The 2-adic inequality implies a family of identities satisfied by the coefficients. The identities also lead to the discovery of some new affine invariants of Boolean functions on

Z

2
m

{\mathbf Z}_2^m

.
Group rings whose symmetric elements are Lie nilpotent
Authors: Lee Gregory
Published: July 2002

Let

F
G

FG

be the group ring of a group

G
G

over a field

F
F

, with characteristic different from

2
2

. Let

∗
\ast

denote the natural involution on

F
G

FG

sending each group element to its inverse. Denote by

(
F
G

)

+

(FG)^{+}

the set of symmetric elements with respect to this involution. A paper of Giambruno and Sehgal showed that provided

G
G

has no

2
2

-elements, if

(
F
G

)

+

(FG)^{+}

is Lie nilpotent, then so is

F
G

FG

. In this paper, we determine when

(
F
G

)

+

(FG)^{+}

is Lie nilpotent, if

G
G

does contain

2
2

-elements.
On the absence of invariant measures with locally maximal entropy for a class of ℤ^{𝕕} shifts of finite type
Authors: Robinson E. , Şahı̇n Ayşe
Published: July 2002

We prove that for a class of

Z

d

\mathbb Z^d

shifts of finite type,

d
>
1

d>1

, any invariant measure which is not a measure of maximal entropy can be perturbed a small amount in the weak* topology to an invariant measure of higher entropy. Namely, there are no invariant measures which are strictly local maxima for the entropy function.
Equivalence of a 𝐾-functional with the approximation behavior of some linear means for abstract Fourier series
Authors: Trebels Walter
Published: July 2002

Within the setting of abstract Cesàro-bounded Fourier series a

K
K

-functional is introduced and characterized by the convergence behavior of some linear means. Applications are given within the framework of Jacobi, Laguerre and Hermite expansions. In particular, Ditzian’s (1996) equivalence result in the setting of Legendre expansions is covered.
Further extension of the Heinz-Kato-Furuta inequality
Authors: Uchiyama Mitsuru
Published: July 2002

Let

T
T

be a bounded operator on a Hilbert space

H

,

\mathfrak {H},

and

A
,
B

A,B

positive definite operators. Kato has shown that if

‖
T
x
‖
≤
‖
A
x
‖

\Vert T x \Vert \leq \Vert A x \Vert

and

‖

T

∗

y
‖
≤
‖
B
y
‖

\Vert T^{*} y \Vert \leq \Vert B y \Vert

for all

x
,
y
∈

H

x, y \in \mathfrak {H}

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue left-parenthesis upper T x comma y right-parenthesis EndAbsoluteValue less-than-or-equal-to StartAbsoluteValue EndAbsoluteValue f left-parenthesis upper A right-parenthesis x StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue g left-parenthesis upper B right-parenthesis y StartAbsoluteValue EndAbsoluteValue comma">

|

(
T
x
,
y
)

|

≤

|

|

f
(
A
)
x

|

|

|

|

g
(
B
)
y

|

|

,

|(Tx , y)| \leq ||f(A)x|| \; ||g(B)y||,

where

f
(
t
)
,
g
(
t
)

f(t), g(t)

are operator monotone functions defined on

[
0
,
∞
)

[0, \infty )

such that

f
(
t
)
g
(
t
)
=
t

f(t) g(t)=t

. Furuta has shown that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue left-parenthesis upper T StartAbsoluteValue upper T EndAbsoluteValue Superscript alpha plus beta minus 1 Baseline x comma y right-parenthesis EndAbsoluteValue less-than-or-equal-to StartAbsoluteValue EndAbsoluteValue upper A Superscript alpha Baseline x StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue upper B Superscript beta Baseline y StartAbsoluteValue EndAbsoluteValue comma where 0 less-than-or-equal-to alpha comma beta less-than-or-equal-to 1 comma 1 less-than-or-equal-to alpha plus beta period">

|

(
T

|

T

|

α
+
β
−
1

x
,
y
)

|

≤

|

|

A

α

x

|

|

|

|

B

β

y

|

|

,
 where 
0
≤
α
,
β
≤
1
,
1
≤
α
+
β
.

|(T|T|^{\alpha +\beta -1}x,y)|\leq ||A^{\alpha }x|| ||B^{\beta }y||, \text { where } 0 \leq \alpha ,\beta \leq 1, 1\leq \alpha + \beta .

Let

f
(
t
)
,
g
(
t
)

f(t), g(t)

be any continuous operator monotone functions, and set

h
(
t
)
=
f
(
t
)
g
(
t
)

/

t

h(t) = f(t) g(t) /t

for

t
>
0.

t >0.

We will show that

T
h
(

|

T

|

)

Th(|T|)

is well defined and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue left-parenthesis upper T h left-parenthesis StartAbsoluteValue upper T EndAbsoluteValue right-parenthesis x comma y right-parenthesis EndAbsoluteValue less-than-or-equal-to StartAbsoluteValue EndAbsoluteValue f left-parenthesis upper A right-parenthesis x StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue g left-parenthesis upper B right-parenthesis y StartAbsoluteValue EndAbsoluteValue period">

|

(
T
h
(

|

T

|

)
x
,
y
)

|

≤

|

|

f
(
A
)
x

|

|

|

|

g
(
B
)
y

|

|

.

|(T h(|T|)x,y)| \leq ||f(A)x|| \; ||g(B)y||.

Moreover, we will extend this result for unbounded closed operators densely defined on

H

.

\mathfrak {H}.
Fundamental theorem of geometry without the 1-to-1 assumption
Authors: Chubarev Alexander , Pinelis Iosif
Published: July 2002

It is proved that any mapping of an

n
n

-dimensional affine space over a division ring

D

\mathbb {D}

onto itself which maps every line into a line is semi-affine, if

n
∈
{
2
,
3
,
…
}

n\in \{2,3,\dots \}

and

D

≠

Z

2

\mathbb {D}\ne \mathbb {Z}_{2}

. This result seems to be new even for the real affine spaces. Some further generalizations are also given. The paper is self-contained, modulo some basic terms and elementary facts concerning linear spaces and also – if the reader is interested in

D

\mathbb {D}

other than

R

\mathbb {R}

,

Z

p

\mathbb {Z}_{p}

, or

C

\mathbb {C}

– division rings.