bluuebunny/crossref_metadata_2025 huggingface dataset:
Noncommutative 𝐻² spaces
Authors: Marsalli Michael
Published: July 2002

Let

M

\mathcal {M}

be a von Neumann algebra with a faithful, finite, normal tracial state

τ
\tau

, and let

A

\mathcal {A}

be a finite, maximal subdiagonal algebra of

M

\mathcal {M}

. Let

H
2

H^2

be the closure of

A

\mathcal {A}

in the noncommutative Lebesgue space

L
2

(

M

,
τ
)

L^2(\mathcal {M},\tau )

. Then

H
2

H^2

possesses several of the properties of the classical Hardy space on the circle, including a commutant lifting theorem, some results on Toeplitz operators, an

H
1

H^1

factorization theorem, Nehari’s Theorem, and harmonic conjugates which are

L
2

L^2

bounded.
Intersection of sets with 𝑛-connected unions
Authors: Horvath Charles , Lassonde Marc
Published: July 2002

We show that if

n
n

sets in a topological space are given so that all the sets are closed or all are open, and for each

k
≤
n

k\le n

every

k
k

of the sets have a

(
k
−
2
)

(k-2)

-connected union, then the

n
n

sets have a point in common. As a consequence, we obtain the following starshaped version of Helly’s theorem: If every

n
+
1

n+1

or fewer members of a finite family of closed sets in

R

n

\mathbb {R}^n

have a starshaped union, then all the members of the family have a point in common. The proof relies on a topological KKM-type intersection theorem.
On an optimality property of Ramanujan sums
Authors: Bachman Gennady
Published: July 2002

We evaluate
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="inf Underscript b Subscript n Endscripts sigma-summation Underscript a equals 1 Overscript q Endscripts StartAbsoluteValue sigma-summation Underscript StartLayout 1st Row n equals 1 left-parenthesis n comma q right-parenthesis equals 1 EndLayout Overscript q Endscripts b Subscript n Baseline e Superscript 2 pi i a n slash q Baseline EndAbsoluteValue">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c Subscript q Baseline left-parenthesis a right-parenthesis equals sigma-summation Underscript StartLayout 1st Row n equals 1 left-parenthesis n comma q right-parenthesis equals 1 EndLayout Overscript q Endscripts e Superscript 2 pi i a n slash q">

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}

.
Best approach regions for potential spaces
Authors: Raposo José , Soria Javier
Published: July 2002

We characterize the approach regions so that the non-tangential maximal function is of weak-type on potential spaces, for which we use a simple argument involving Carleson measure estimates.
Fixed point theory for compact upper semi-continuous or lower semi-continuous set valued maps
Authors: O’Regan Donal
Published: July 2002

Fixed point theory is presented for compact u.s.c. and l.s.c. set valued maps.
Rings with finite essential socle
Authors: Pardo José , Asensio Pedro
Published: July 2002

Let

R
R

be a ring such that every direct summand of the injective envelope

E
=
E
(

R
R

)

E=E(R_R)

has an essential finitely generated projective submodule. We show that, if the cardinal of the set of isomorphism classes of simple right

R
R

-modules is no larger than that of the isomorphism classes of minimal right ideals, then

R
R

R_R

cogenerates the simple right

R
R

-modules and has finite essential socle. This extends Osofsky’s theorem which asserts that a right injective cogenerator ring has finite essential right socle. It follows from our result that if

R
R

R_R

is a CS cogenerator, then

R
R

R_R

is already an injective cogenerator and, more generally, that if

R
R

R_R

is CS and cogenerates the simple right

R
R

-modules, then it has finite essential socle. We show with an example that in the latter case

R
R

R_R

need not be an injective cogenerator.
Finite dimensional representations of 𝑈_{𝑞}(𝑠𝑙(2,1))
Authors: Zou Yi
Published: July 2002

Structures of the finite dimensional simple weight

U

q

(
s
l
(
2
,
1
)
)

U_{q}(sl(2,1))

-modules are studied in detail for both the generic and the roots of 1 cases.
Topological entropy for geodesic flows under a Ricci curvature condition
Authors: Paeng Seong-Hun
Published: July 2002

It is known that the topological entropy for the geodesic flow on a Riemannian manifold

M
M

is bounded if the absolute value of sectional curvature

|

K

M

|

|K_{M}|

is bounded. We replace this condition by the condition of Ricci curvature and injectivity radius.
Dual local completeness
Authors: Saxon Stephen , Ruiz L.
Published: July 2002

The 1971 articles in which Saxon-Levin and Valdivia independently proved their Theorem feature two conditions equivalent to dual local completeness. One became Ruess’ property (LC). The other is among new characterizations previously known only as necessary conditions.
A new fixed point theorem and its applications
Authors: Wu Xian
Published: July 2002

In this paper, we first give a new fixed theorem of lower semicontinuous multivalued mappings, and then, as its applications we obtain some new equilibrium theorems for abstract economies and qualitative games.
Conjugacy classes of symmetries in orthogonal groups
Authors: James Donald
Published: July 2002

The number of conjugacy classes of symmetries in the integral orthogonal group of an indefinite

Z

\Bbb Z

-lattice is determined. The results are applied to the extended Bianchi groups.
Products with linear and countable type factors
Authors: Purisch S. , Rudin M.
Published: July 2002

The basic theorem presented shows that the product of a linearly ordered space and a countable (regular) space is normal. We prove that the countable space can be replaced by any of a rather large class of countably tight spaces. Examples are given to prove that monotone normality cannot replace linearly ordered in the base theorem. However, it is shown that the product of a monotonically normal space and a monotonically normal countable space is normal.
A non-metrizable space whose countable power is 𝜎-metrizable
Authors: Gruenhage Gary
Published: July 2002

We answer a question of A.V. Arhangel’skii by finding a non-metrizable space

X
X

such that

X

ω

X^{\omega }

is the countable union of metrizable spaces.
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 that<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Superscript left-parenthesis n right-parenthesis Baseline colon upper H Superscript n Baseline left-parenthesis upper K comma bold upper Z slash p right-parenthesis right-arrow upper H Superscript n Baseline left-parenthesis upper G comma bold upper Z slash p right-parenthesis">f(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.
Spectral properties of continuous refinement operators
Authors: Jia R. , Lee S. , Sharma A.
Published: July 2002

This paper studies the spectrum of continuous refinement operators and relates their spectral properties with the solutions of the corresponding continuous refinement equations.
Extendibility of homogeneous polynomials on Banach spaces
Authors: Kirwan Pádraig , Ryan Raymond
Published: July 2002

We study the

n
n

-homogeneous polynomials on a Banach space

X
X

that can be extended to any space containing

X
X

. We show that there is an upper bound on the norm of the extension. We construct a predual for the space of all extendible

n
n

-homogeneous polynomials on

X
X

and we characterize the extendible 2-homogeneous polynomials on

X
X

when

X
X

is a Hilbert space, an

L

1

\mathcal L_1

-space or an

L

∞

\mathcal L_\infty

-space.
A generalization of Banchoff’s triple point theorem
Authors: Akhmetiev P. , Rimányi R. , Szűcs A.
Published: July 2002

Consider an immersion of a surface into

S

3

S^{3}

. Banchoff’s theorem states that the parity of the number of triple points and the parity of the Euler characteristic of the surface coincide. Here we generalize this theorem to codimension 1 immersions of arbitrary even dimensional manifolds in spheres. The proof is an analogue of a proof of Banchoff’s theorem circulated in preprint form due to R. Fenn and P. Taylor in 1977.
Closure ordering and the Kostant-Sekiguchi correspondence
Authors: Barbasch Dan , Sepanski Mark
Published: July 2002

Let

S
S

be a real semisimple Lie group with Lie algebra

s

=

k

+

p

\mathfrak {s}=\mathfrak {k} +\mathfrak {p}

. The Kostant–Sekiguchi correspondence is a bijection between nilpotent

S
S

orbits on

s

\mathfrak {s}

and nilpotent

K

C

K_{\mathbb {C}}

orbits on

p

C

\mathfrak {p}_{\mathbb {C}}

. In this note we prove that the closure relations among nilpotent orbits are preserved under the Kostant–Sekiguchi correspondence. The techniques rely on work of M. Vergne and P. Kronheimer.
Some harmonic 𝑛-slit mappings
Authors: Dorff Michael
Published: July 2002

The class

S
H

S_H

consists of univalent, harmonic, and sense-preserving functions

f
f

in the unit disk,

Δ
\Delta

, such that

f
=
h
+

g
¯

f=h+\overline {g}

where

h
(
z
)
=
z
+

∑
2
∞

a
k

z
k

h(z)=z+\sum _2^\infty a_kz^k

,

g
(
z
)
=

∑
1
∞

b
k

z
k

g(z)=\sum _1^\infty b_kz^k

.

S
H
O

S_H^O

will denote the subclass with

b
1

=
0

b_1=0

. We present a collection of

n
n

-slit mappings

(
n
≥
2
)

(n \geq 2)

and prove that the

2
2

-slit mappings are in

S
H

S_H

while for

n
≥
3

n \geq 3

the mappings are in

S
H
O

S_H^O

. Finally we show that these mappings establish the sharpness of a previous theorem by Clunie and Sheil-Small while disproving a conjecture about the inner mapping radius.
The cohomology of the Morava stabilizer group 𝕊₂ at the prime 3
Authors: Gorbounov Vassily , Siegel Stephen , Symonds Peter
Published: July 2002

We compute the cohomology of the Morava stabilizer group

S

2

\mathbb {S}_2

at the prime

3
3

by resolving it by a free product

Z

/

3
∗

Z

/

3

\mathbb {Z}/3*\mathbb {Z}/3

and analyzing the “relation module.”
A generalization of Lyapunov’s convexity theorem with applications in optimal stopping
Authors: Kühn Zuzana , Rösler Uwe
Published: July 2002

Lyapunov proved that the range of

n
n

finite measures defined on the same

σ
\sigma

-algebra is compact, and if each measure

μ

i

\mu _{i}

also is atomless, then the range is convex. Although both conclusions may fail for measures on different

σ
\sigma

-algebras of the same set, they do hold if the

σ
\sigma

-algebras are nested, which is exactly the setting of classical optimal stopping theory.
Nil subsets of graded algebras
Authors: Montgomery S. , Small L.
Published: July 2002

We prove that if

A
A

is a Noetherian

Z

\mathbf Z

-graded algebra, then the Jacobson radical of

A
A

is nilpotent under mild hypotheses on

A
0

A_0

. We also consider affine PI-algebras graded by torsion groups. Finally we prove a Nullstellensatz-type theorem for enveloping algebras of Lie color algebras.
Fusion in the character table
Authors: Navarro Gabriel
Published: July 2002

Suppose that

P
P

is a Sylow

p
p

-subgroup of a finite

p
p

-solvable group

G
G

. If

g
∈
P

g \in P

, then the number of

G
G

-conjugates of

g
g

in

P
P

can be read off from the character table of

G
G

.
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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R left-parenthesis g right-parenthesis equals StartFraction 1 Over left-parenthesis normal upper V normal o normal l left-parenthesis upper M right-parenthesis right-parenthesis Superscript StartFraction n minus 4 Over n EndFraction Baseline EndFraction integral Underscript upper M Endscripts bar upper R bar d v o l Subscript g">

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}

.
Normalizers of nest algebras
Authors: Coates Keith
Published: July 2002

For a nest

N

\mathcal {N}

with associated nest algebra

A

N

\mathcal {A}_{\mathcal {N}}

, we define

S

N

\mathcal {S}_{\mathcal {N}}

, the normalizer of

A

N

\mathcal {A}_{\mathcal {N}}

. We develop a characterization of elements of

S

N

\mathcal {S}_{\mathcal {N}}

based on certain order homomorphisms of

N

\mathcal {N}

into itself. This characterization enables us to prove several structure theorems.
Radiation conditions and uniqueness for stationary oscillations in elastic plates
Authors: Constanda Christian
Published: July 2002

Sommerfeld-type radiation conditions are indicated for the solutions of the system governing the small stationary oscillations in plates with transverse shear deformation, and a uniqueness theorem is proved in the case of the corresponding exterior Dirichlet and Neumann problems.
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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript n Baseline equals integral Subscript 0 Superscript normal infinity Baseline t Superscript n Baseline d mu left-parenthesis t right-parenthesis greater-than normal infinity left-parenthesis n equals 0 comma 1 comma 2 comma period period period right-parenthesis">

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.
The stability radius of a quasi-Fredholm operator
Authors: Poon Pak
Published: July 2002

We extend the technique used by Kordula and Müller to show that the stability radius of a quasi-Fredholm operator

T
T

is the limit of

γ
(

T
n

)

1

/

n

\gamma (T^n)^{1/n}

as

n
→
∞

n\rightarrow \infty

. If

0
0

is an isolated point of the Apostol spectrum

σ
γ

(
T
)

\sigma _\gamma (T)

, then the above limit is non-zero if and only if

T
T

is quasi-Fredholm.
The regular element property
Authors: Richman Fred
Published: July 2002

The property that an ideal whose annihilator is zero contains a regular element is examined from the point of view of constructive mathematics. It is shown that this property holds for finitely presented algebras over discrete fields, and for coherent, Noetherian, strongly discrete rings that contain an infinite field.
Conjugate Hardy’s inequalities with decreasing weights
Authors: Cerdà Joan , Martín Joaquim
Published: July 2002

We prove that for a decreasing weight

ω
\omega

on

R

+

\mathbf {R}^+

, the conjugate Hardy transform is bounded on

L
p

(
ω
)

L_p(\omega )

(

1
≤
p
>
∞

1\leq p>\infty

) if and only if it is bounded on the cone of all decreasing functions of

L
p

(
ω
)

L_p(\omega )

. This property does not depend on

p
p

.
Some extremal problems in 𝐿^{𝑝}(𝑤)
Authors: Cheng R. , Miamee A. , Pourahmadi M.
Published: July 2002

Fix a positive integer

n
n

and

1
>
p
>
∞

1>p>\infty

. We provide expressions for the weighted

L

p

L^{p}

distance
\[

inf

f

∫

0

2
π

|

1
−
f

|

p

w

d
λ
,

\inf _{f} \int ^{2 \pi }_{0} | 1 - f |^p w\,d\lambda ,

\]
where

d
λ

d\lambda

is normalized Lebesgue measure on the unit circle,

w
w

is a nonnegative integrable function, and

f
f

ranges over the trigonometric polynomials with frequencies in
\[

S
1

=
{
…
,
−
3
,
−
2
,
−
1
}
∪
{
1
,
2
,
3
,
…
,
n
}
,

S_1 = \{ \ldots , -3, -2, -1\}\cup \{ 1, 2, 3,\ldots , n\},

\]

\[

S
2

=
{
…
,
−
3
,
−
2
,
−
1
}
∖
{
−
n
}
,

S_2=\{ \ldots , -3, -2,-1\}\setminus \{-n\},

\]
or
\[

S
3

=
{
…
,
−
3
,
−
2
,
−
1
}
∪
{
n
}
.

S_3 =\{ \ldots , -3, -2, - 1\}\cup \{n\}.

\]
These distances are related to other extremal problems, and are shown to be positive if and only if

log
⁡
w

\log w

is integrable. In some cases they are expressed in terms of the series coefficients of the outer functions associated with

w
w

.
Some remarks on a boundedness assumption for monotone dynamical systems
Authors: Dancer E.
Published: July 2002

We study the consequences of the assumption that all forward orbits are bounded for monotone dynamical systems. In particular, it turns out that this assumption has more implications than is immediately apparent.
Monotonic dynamical systems under spatial discretization
Authors: Diamond P. , Kloeden P. , Kozyakin V. , Pokrovskii A.
Published: July 2002

We estimate the probability of replicating the asymptotic behaviour of a dynamical system generated by a monotonic mapping for randomly centered roundoff lattices.
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.
𝐶*-algebras that are only weakly semiprojective
Authors: Loring Terry
Published: July 2002

We show that the

C

∗

C^{*}

-algebra of continuous functions on the Cantor set is a weakly semiprojective

C

∗

C^{*}

-algebra that is not semiprojective.
Some properties of ordinary sense slice 1-links: Some answers to problem (26) of Fox
Authors: Ogasa Eiji
Published: July 2002

We prove that, for any ordinary sense slice 1-link

L
L

, we can define the Arf invariant, and Arf(

L
L

)=0. We prove that, for any

m
m

-component 1-link

L

1

L_{1}

, there exists a

3
m

3m

-component ordinary sense slice 1-link

L

2

L_{2}

of which

L

1

L_{1}

is a sublink.
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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q Subscript n Baseline left-parenthesis x right-parenthesis equals StartFraction upper P prime Subscript n plus 1 Baseline left-parenthesis x right-parenthesis Over n plus 1 EndFraction plus upper D Subscript n Baseline StartFraction upper P prime Subscript n Baseline left-parenthesis x right-parenthesis Over n EndFraction comma n greater-than-or-equal-to 1 period">

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle f comma g mathematical right-angle equals integral Subscript a Superscript b Baseline f left-parenthesis x right-parenthesis g left-parenthesis x right-parenthesis d mu left-parenthesis x right-parenthesis plus lamda integral Subscript a Superscript b Baseline f prime left-parenthesis x right-parenthesis g prime left-parenthesis x right-parenthesis d nu left-parenthesis x right-parenthesis">

⟨
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.
The primality of subfactors of finite index in the interpolated free group factors
Authors: Ştefan Marius
Published: July 2002

In this paper we prove that any II

1

_{1}

-subfactor of finite index in the interpolated free group factor

L
(

F

t

)

L(F_{t})

is prime for any

1
>
t
≤
∞

1>t\leq \infty

i.e., it is not isomorphic to tensor products of II

1

_{1}

-factors.
A splitting theorem for equifocal submanifolds in simply connected compact symmetric spaces
Authors: Ewert Heiko
Published: July 2002

A submanifold in a symmetric space is called equifocal if it has a globally flat abelian normal bundle and its focal data is invariant under normal parallel transportation. This is a generalization of the notion of isoparametric submanifolds in Euclidean spaces. To each equifocal submanifold, we can associate a Coxeter group, which is determined by the focal data at one point. In this paper we prove that an equifocal submanifold in a simply connected compact symmetric space is a non-trivial product of two such submanifolds if and only if its associated Coxeter group is decomposable. As a consequence, we get a similar splitting result for hyperpolar group actions on compact symmetric spaces. These results are an application of a splitting theorem for isoparametric submanifolds in Hilbert spaces by Heintze and Liu.
Mergelyan pairs for harmonic functions
Authors: Gardiner Stephen
Published: July 2002

Let

Ω
⊆

R

n

\Omega \subseteq \mathbb R^n

be open and

E
⊆
Ω

E\subseteq \Omega

be a bounded set which is closed relative to

Ω
\Omega

. We characterize those pairs

(
Ω
,
E
)

(\Omega ,E)

such that, for each harmonic function

h
h

on

Ω
\Omega

which is uniformly continuous on

E
E

, there is a sequence of harmonic polynomials which converges to

h
h

uniformly on

E
E

. As an immediate corollary we obtain a characterization of Mergelyan pairs for harmonic functions.
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.
Completely contractive representations for some doubly generated antisymmetric operator algebras
Authors: Power S.
Published: July 2002

Contractive weak star continuous representations of the Fourier binest algebra

A

\mathcal A

(of Katavolos and Power) are shown to be completely contractive. The proof depends on the approximation of

A

\mathcal A

by semicrossed product algebras

A
(

D

)
×

Z

+

A(\mathbb D)\times \mathbb Z_+

and on the complete contractivity of contractive representations of such algebras. The latter result is obtained by two applications of the Sz.-Nagy–Foias lifting theorem. In the presence of an approximate identity of compact operators it is shown that an automorphism of a general weakly closed operator algebra is necessarily continuous for the weak star topology and leaves invariant the subalgebra of compact operators. This fact and the main result are used to show that isometric automorphisms of the Fourier binest algebra are unitarily implemented.
Notes on homology cobordisms of plumbed homology 3-spheres
Authors: Saveliev Nikolai
Published: July 2002

The gauge-theoretical invariants of Donaldson and Seiberg-Witten are used to detect some infinite order elements in the homology cobordism group of integral homology 3-spheres.
Hypergroups with invariant metric
Authors: Voit Michael
Published: July 2002

The purpose of this note is to extend the following classical result from groups to hypergroups in the sense of C.F. Dunkl, R.I. Jewett, and R. Spector: If a hypergroup has a countable neighborhood base of its identity, then

K
K

admits a left- or a right-invariant metric. Moreover, it admits an invariant metric if and only if there exists a countable conjugation-invariant neighborhood base of the identity.
A compact HL-space need not have a monolithic hyperspace
Authors: Brandsma Henno , van Mill Jan
Published: July 2002

We show that Kunen’s example of a compact L-space, constructed under CH, can be modified so that it has a non-monolithic hyperspace. This answers a question of Bell’s. This example is also relevant to a question of Arhangel’skiĭ’s.
Logarithmic Sobolev inequalities and the growth of 𝐿^{𝑝} norms
Authors: Rothaus O.
Published: July 2002

We show that many of the recent results on exponential integrability of Lip 1 functions, when a logarithmic Sobolev inequality holds, follow from more fundamental estimates of the growth of

L

p

L^{p}

norms under the same hypotheses.
Quasipositive plumbing (constructions of quasipositive knots and links, V)
Authors: Rudolph Lee
Published: July 2002

A Seifert surface

S
⊂

S

3

=
∂

D

4

S\subset S^{3}=\partial D^{4}

is a fiber surface if a push-off

S
→

S

3

∖
S

S\to S^{3}\setminus S

induces a homotopy equivalence; roughly,

S
S

is quasipositive if pushing

Int
⁡
S

\operatorname {Int} S

into

Int
⁡

D

4

⊂

C

2

\operatorname {Int} D^{4}\subset \mathbb {C}^{2}

produces a piece of complex plane curve. A Murasugi sum (or plumbing) is a way to fit together two Seifert surfaces to build a new one. Gabai proved that a Murasugi sum is a fiber surface iff both its summands are; we prove the analogue for quasipositive Seifert surfaces. The slice (or Murasugi) genus

g

s

(
L
)

g_{s}(L)

of a link

L
⊂

S

3

L\subset S^{3}

is the least genus of a smooth surface

S
⊂

D

4

S\subset D^{4}

bounded by

L
L

. By the local Thom Conjecture,

g

s

(
∂
S
)
=
g
(
S
)

g_{s}(\partial S)=g(S)

if

S
⊂

S

3

S\subset S^{3}

is quasipositive; we derive a lower bound for

g

s

(
∂
S
)

g_{s}(\partial S)

for any Seifert surface

S
S

, in terms of quasipositive subsurfaces of

S
S

.
Collapsible polyhedra and median spaces
Authors: van de Vel M.
Published: July 2002

It is shown that a collapsible, compact, connected, simplicial polyhedron admits a cubical subdivision and a median convexity, such that all cubes are convex subspaces with a convexity of subcubes. Conversely, a compact, connected, cubical polyhedron with a convexity as described admits a collapsible simplicial subdivision. Such a convexity, when it exists, is uniquely determined by the corresponding cubical presentation. Some related open problems have been formulated.
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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x 1 Superscript alpha 1 minus 1 slash 2 Baseline midline-horizontal-ellipsis x Subscript d Superscript alpha Super Subscript d Superscript minus 1 slash 2 Baseline left-parenthesis 1 minus StartAbsoluteValue bold x EndAbsoluteValue Subscript 1 Baseline right-parenthesis Superscript alpha Super Subscript d plus 1 Superscript minus 1 slash 2">

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

.
Once more nice equations for nice groups
Authors: Abhyankar Shreeram , Loomis Paul
Published: July 2002

In a previous paper, nice quintinomial equations were given for unramified coverings of the affine line in nonzero characteristic

p
p

with the projective symplectic isometry group PSp

(
2
m
,
q
)

(2m,q)

and the (vectorial) symplectic isometry group Sp

(
2
m
,
q
)

(2m,q)

as Galois groups where

m
>
2

m>2

is any integer and

q
>
1

q>1

is any power of

p
p

. Here we deform these equations to get nice quintinomial equations for unramified coverings of the once punctured affine line in characteristic

p
p

with the projective symplectic similitude group PGSp

(
2
m
,
q
)

(2m,q)

and the (vectorial) symplectic similitude group GSp

(
2
m
,
q
)

(2m,q)

as Galois groups.
Constructing ultraweakly continuous functionals on ℬ(ℋ)
Authors: Bridges D. , Ward N.
Published: July 2002

In this paper we give a constructive characterisation of ultraweakly continuous linear functionals on the space of bounded linear operators on a separable Hilbert space.
A bound for the derived and Frattini subgroups of a prime-power group
Authors: Ellis Graham
Published: July 2002

This paper is based on the seemingly new observation that the Schur multiplier

M
(
G
)

M(G)

of a

d
d

-generator group of prime-power order

p
n

p^n

has order

|

M
(
G
)

|

≤

p

d
(
2
n
−
d
−
1
)

/

2

|M(G)|\le p^{d(2n-d-1)/2}

. We prove several related results, including sufficient conditions for a sharper bound on

|

M
(
G
)

|

|M(G)|

to be an equality.
On self-intersections of immersed surfaces
Authors: Li Gui-Song
Published: July 2002

A daisy graph is a union of immersed circles in 3-space which intersect only at the triple points. It is shown that a daisy graph can always be realized as the self-intersection set of an immersed closed surface in 3-space and the surface may be chosen to be orientable if and only if the daisy graph has an even number of edges on each immersed circle.
Moduli of plane curve singularities with a single characteristic exponent
Authors: Peraire Rosa
Published: July 2002

This paper studies the moduli space corresponding to irreducible germs of plane analytic curve with a single characteristic exponent. We stratify the moduli space corresponding to such germs using an analytical invariant introduced by Zariski. Then, we compute the minimum Tjurina number on each stratum as well as the dimension of the strata.
Centralizers in residually finite torsion groups
Authors: Shalev Aner
Published: July 2002

Let

G
G

be a residually finite torsion group. We show that, if

G
G

has a finite 2-subgroup whose centralizer is finite, then

G
G

is locally finite. We also show that, if

G
G

has no

2
2

-torsion, and

Q
Q

is a finite 2-group acting on

G
G

in such a way that the centralizer

C
G

(
Q
)

C_G(Q)

is soluble, or of finite exponent, then

G
G

is locally finite.
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

.
Borel automorphisms with no finite invariant measure
Authors: Eigen S. , Hajian A. , Weiss B.
Published: July 2002

An uncountable family of non-isomorphic Borel automorphisms which do not preserve any finite measure is presented.
Bourgain’s analytic projection revisited
Authors: Kislyakov S.
Published: July 2002

For a positive function

w
w

on the unit circle with

log
⁡
w
∈

L
1

\log w\in L^1

, the following two statements are equivalent: (a)

log
⁡
w
∈
B
M
O

\log w\in BMO

; (b) there is an operator

Q
Q

projecting

L
p

(
w
)

L^p(w)

onto

H
p

(
w
)

H^p(w)

for all

1
>
p
>
∞

1>p>\infty

at once and having weak type (1,1) with respect to

w
w

.
The exposed points of the set of invariant means on an ideal
Authors: Miao Tianxuan
Published: July 2002

Let

G
G

be a

σ
\sigma

-compact locally compact nondiscrete group and let

Q
Q

be a

G
G

-invariant ideal of

L

∞

(
G
)

L^{\infty }(G)

. We denote the set of left invariant means

m
m

on

L

∞

(
G
)

L^{\infty }(G)

that are zero on

Q
Q

(i.e.

m
(
f
)
=
0

m(f) = 0

for all

f
∈
Q

f\in Q

) by

L
I

M

Q

LIM_{Q}

. We show that, when

G
G

is amenable as a discrete group and the closed

G
G

-invariant subset of the spectrum of

L

∞

(
G
)

L^{\infty }(G)

corresponding to

Q
Q

is a

G

δ

G_{\delta }

-set,

L
I

M

Q

LIM_{Q}

is very large in the sense that every nonempty

G

δ

G_{\delta }

-subset of

L
I

M

Q

LIM_{Q}

contains a norm discrete copy of

β

N

\beta \mathbb {N}

, where

β

N

\beta \mathbb {N}

is the Stone-

C
ˇ

e
c
h

\mathrm {\check {C}ech}

compactification of the set

N

\mathbb {N}

of positive integers with the discrete topology. In particular, we prove that

L
I

M

Q

LIM_{Q}

has no exposed points in this case and every nonempty

G

δ

G_{\delta }

-subset of the set of left invariant means on

L

∞

(
G
)

L^{\infty }(G)

contains a norm discrete copy of

β

N

\beta \mathbb {N}

.
The existence and uniqueness of injectors for Fitting sets of solvable groups
Authors: Hawthorn Ian
Published: July 2002

A short alternative proof is given of the existence and uniqueness of injectors in a Fitting set which avoids use of the Carter subgroup, a concept from the dual theory of projectors.
On the projectivity of module coalgebras
Authors: Ng Siu-Hung
Published: July 2002

In this paper, we derive some criteria for the projectivity of a module coalgebra over a finite dimensional Hopf algebra. In particular, we show that any Hopf algebra over a field of characteristic zero is faithfully flat over its group-like subHopf algebra. Finally, we prove that if

B
B

is a finite dimensional subHopf algebra of a Hopf algebra

A
A

, then

B
B

is normal in

A
A

if and only if

A

B
+

=

B
+

A

AB^+ = B^+A

. This improves a result by S. Montgomery (1993).
Fixed points and random fixed points for weakly inward approximable maps
Authors: O’Regan Donal
Published: July 2002

In this paper we obtain new fixed point and random fixed point theory for approximable maps.
Projective modules and Hilbert spaces with a Nevanlinna-Pick kernel
Authors: Clancy Robert , McCullough Scott
Published: July 2002

In this paper we solve a mapping problem for a particular class of Hilbert modules over an algebra multipliers of a diagonal Nevanlinna-Pick (NP) kernel. In this case, the regular representation provides a multiplier norm which induces the topology on the algebra. In particular, we show that, in an appropriate category, a certain class of Hilbert modules are projective. In addition, we establish a commutant lifting theorem for diagonal NP kernels.
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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript p comma q comma t Baseline left-parenthesis upper A comma upper B comma r comma s right-parenthesis equals upper A Superscript negative r slash 2 Baseline left-brace upper A Superscript r slash 2 Baseline left-parenthesis upper A Superscript negative t slash 2 Baseline upper B Superscript p Baseline upper A Superscript negative t slash 2 Baseline right-parenthesis Superscript s Baseline upper A Superscript r slash 2 Baseline right-brace Superscript left-parenthesis q minus t plus r right-parenthesis slash left-bracket left-parenthesis p minus t right-parenthesis s plus r right-bracket Baseline upper A Superscript negative r slash 2">

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.
One-way intervals of circle maps
Authors: Ancel Lauren , Hero Michael
Published: July 2002

An interval in the circle

S
1

S^1

is one-way with respect to a map

f
:

S
1

→

S
1

f:S^1\to S^1

if under repeated applications of

f
f

all points of the interval move in the same direction. The main result is that every locally one-way interval is either one-way or is the union of two overlapping one-way subintervals. An example is given which illustrates that the latter case can occur; however, it is proved that the latter case cannot occur if the interval is covered by the image of the map. As a corollary, it is shown that if

f
f

has periodic points, then every interval which contains no periodic points is either one-way or is the union of two overlapping one-way subintervals.
A short proof of a characterization of reflexivity of James
Authors: Oja Eve
Published: July 2002

A short direct proof is given to a well-known intrinsic characterization of reflexivity due to R. C. James.
A counterexample to the Bartle-Graves selection theorem for multilinear maps
Authors: Fernandez Cecília
Published: July 2002

We present an example showing that the multilinear version of the Bartle-Graves Selection Theorem is false, even on finite dimensional spaces.
On rationality of the cogrowth series
Authors: Kouksov Dmitri
Published: July 2002

The cogrowth series of a group

G
G

depends on the presentation of the group. We show that the cogrowth series of a non-empty presentation is a rational function not equal to 1 if and only if

G
G

is finite. Except for the trivial group, this property is independent of presentation.
Bounds on the order of cross characteristic subgroups of the finite simple groups of Lie type
Authors: Asparouhov Tihomir
Published: July 2002

Let

X
(
r
)

X(r)

and

G
(
q
)

G(q)

be finite groups of Lie type and

r
r

and

q
q

be coprime. If

G
(
q
)

G(q)

is embedded in

X
(
r
)

X(r)

, then the Landazuri-Seitz-Zalesskii theorem implies that

G
(
q
)

G(q)

is small relative to

X
(
r
)

X(r)

. We formalize this observation and illustrate how it can be used with some applications.
The Kaplansky test problems for ℵ₁-separable groups
Authors: Eklof Paul , Shelah Saharon
Published: July 2002

We answer a long-standing open question by proving in ordinary set theory, ZFC, that the Kaplansky test problems have negative answers for

ℵ
1

\aleph _1

-separable abelian groups of cardinality

ℵ
1

\aleph _1

. In fact, there is an

ℵ
1

\aleph _1

-separable abelian group

M
M

such that

M
M

is isomorphic to

M
⊕
M
⊕
M

M\oplus M\oplus M

but not to

M
⊕
M

M\oplus M

. We also derive some relevant information about the endomorphism ring of

M
M

.
Periodic solutions of the second order differential equations with Lipschitzian condition
Authors: Wang Zaihong
Published: July 2002

In this paper, we prove an infinity of periodic solutions to the periodically forced nonlinear Duffing equation

x
¨

+
g
(
x
)
=
p
(
t
)

\ddot {x}+g(x)=p(t)

.
Every group has a terminating transfinite automorphism tower
Authors: Hamkins Joel
Published: July 2002

Iteratively taking the automorphism group of any group leads, transfinitely, to a fixed point.
The moduli of substructures of a compact complex space
Authors: Schuster Peter
Published: July 2002

We construct a space

W
X

W_X

of fine moduli for the substructures of an arbitrary compact complex space

X
X

. A substructure

(
X
,

A

)

(X,\mathcal {A})

of

X
X

is given by a subalgebra

A

\mathcal {A}

of the structure sheaf

O

X

\mathcal {O}_X

with the additional feature that

(
X
,

A

)

(X,\mathcal {A})

is also a complex space;

(
X
,

A

)

(X,\mathcal {A})

and

(
X
,

A
′

)

(X,\mathcal {A’})

are called equivalent if and only if

A

\mathcal {A}

and

A
′

\mathcal {A’}

are isomorphic as subalgebras of

O

X

\mathcal {O}_X

.

Since substructures are quotients, it is only natural to start with the fine moduli space

Q
X

Q_X

of all complex-analytic quotients of

X
X

. In order to obtain a representable moduli functor of substructures, we are forced to concentrate on families of quotients which satisfy some flatness condition for relative differential modules of higher order. Considering the corresponding flatification of

Q
X

Q_X

, we realize that its open subset

W
X

W_X

consisting of all substructures turns out to be a complex space which has the required universal property.
A ZFC Dowker space in ℵ_{𝜔+1}: An application of pcf theory to topology
Authors: Kojman Menachem , Shelah Saharon
Published: July 2002

The existence of a Dowker space of cardinalityℵω+1\aleph _{\omega +1}and weightℵω+1\aleph _{\omega +1}is proved in ZFC using pcf theory.
Strong ergodic theorems for commutative semigroups of operators
Authors: Kada O.
Published: July 2002

We prove strong mean convergence theorems and the existence of ergodic projection and retraction for commutative semigroups of operators which is Eberlein-weakly almost periodic.
A variant of the diamond principle for combinatorial ideals
Authors: Abe Y.
Published: July 2002

We use a variant of the diamond principle to show many ideals on

κ
\kappa

are not

2

κ

2^{\kappa }

-saturated if

κ
\kappa

is large. For instance, the

Π
1
1

\Pi ^1_1

-indescribable ideal is not

2

κ

2^{\kappa }

-saturated if

κ
\kappa

is almost ineffable.
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:\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove q With dot Subscript i Baseline equals minus sigma-summation Underscript j equals 1 Overscript n Endscripts StartFraction a Subscript i j Baseline Over q Subscript j Superscript k Baseline EndFraction comma q Subscript i Baseline left-parenthesis t right-parenthesis greater-than 0 comma i equals 1 comma ellipsis comma n comma">q˙i=−∑j=1naijqjk,qi(t)>0,i=1,…,n,\dot q_i=-\sum _{j=1}^n \frac {a_{ij}}{q_j^k},\quad q_i(t)>0,\qquad i=1,\dots , n,\]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.
Anticommuting derivations
Authors: Pedersen Steen
Published: July 2002

We show that there are no non-trivial closable derivations of a

C

∗

C^{*}

-algebra anticommuting with an ergodic action of a compact group, supposing that the set of squares is dense in the group. We also show that there are no non-trivial closable densely defined rank one derivations on any

C

∗

C^{*}

-algebra.
The moduli space of 𝑆𝑈(3)-flat connections and the fusion rules
Authors: Hayashi Masato
Published: July 2002

The aim of this paper is to determine the existence condition of the moduli space of

S
U
(
3
)

SU(3)

-flat connections on

3
3

-holed

2
2

-sphere

D
D

, the so-called pair of pants, and to study its relationship to the

s
l

^

(
3
;

C

)

\widehat {\mathfrak {sl}}(3;\mathbb {C})

fusion rules. The existence condition can be expressed by a system of inequalities with the entries of highest weights with respect to the fundamental weights. This gives a necessary condition for the fusion coefficents to be nontrivial. We also find that the fusion coefficient of a triplet of extremal highest weights equals one. This can be considered a quantum counterpart of the PRV-conjecture.