bluuebunny/crossref_metadata_2025_split huggingface dataset:
Matrix presentations of braids and applications
Authors: Lee Sang 
Published: July 2002

We show that there exists a one-to-one correspondence between the class of certain block tridiagonal matrices with the entries

−
1
,
0
,

-1, 0,

or

1
1

and the free monoid generated by

2
n

2n

generators

σ

1

,
⋯
,

σ

n

,

σ

1

−
1

,
⋯
,

σ

n

−
1

\sigma _{1}, \cdots ,\sigma _{n}, \sigma _{1}^{-1},\cdots , \sigma _{n}^{-1}

and relation

σ

i

±
1

σ

j

±
1

=

σ

j

±
1

σ

i

±
1

(

|

i
−
j

|

≥
2
)

\sigma _{i}^{\pm 1}\sigma _{j}^{\pm 1} = \sigma _{j}^{\pm 1}\sigma _{i}^{\pm 1}~ (|i-j| \geq 2)

and give some applications for braids. In particular, we give new formulation of the reduced Alexander matrices for closed braids.
Bounded composition operators with closed range on the Dirichlet space
Authors: Luecking Daniel 
Published: July 2002

For composition operators on spaces of analytic functions it is well known that norm estimates can be converted to Carleson measure estimates. The boundedness of the composition operator becomes equivalent to a Carleson measure inequality. The measure corresponding to a composition operator

C
φ

C_\varphi

on the Dirichet space

D

\mathcal D

is

d

ν
φ

=

n
φ

d
A

d\nu _\varphi = n_\varphi \,dA

, where

n
φ

(
z
)

n_\varphi (z)

is the cardinality of the preimage

φ

−
1

(
z
)

\varphi ^{-1}(z)

. The composition operator will have closed range if and only if the corresponding measure satisfies a “reverse Carleson measure” theorem:

‖
f

‖

D

2

≤
∫

|

f
′

|

2

d

ν
φ

\| f \|_{\mathcal {D}}^2 \le \int |f’|^2 \,d\nu _\varphi

for all

f
∈

D

f\in \mathcal D

. Assuming

C
φ

C_\varphi

is bounded, a necessary condition for this inequality is a reverse of the Carleson condition: (C)

ν
φ

(
S
)
≥
c

|

S

|

\nu _\varphi (S) \ge c |S|

for all Carleson squares

S
S

. It has long been known that this is not sufficient for a completely general measure. Here we show that it is also not sufficient for the special measures

ν
φ

\nu _\varphi

. That is, we construct a function

φ
\varphi

such that

C
φ

C_\varphi

is bounded and

ν
φ

\nu _\varphi

satisfies (C) but the composition operator

C
φ

C_\varphi

does not have closed range.
A bound on the reduction number of a primary ideal
Authors: Rossi M. 
Published: July 2002

Let

(
A
,

M

)

( A,\mathcal {M})

be a local ring of positive dimension

d
d

and let

I
I

be an

M

\mathcal {M}

-primary ideal. We denote the reduction number of

I
I

by

r
(
I
)

r(I)

, which is the smallest integer

r
r

such that

I

r
+
1

=
J

I
r

I^{r+1}=JI^r

for some reduction

J
J

of

I
.

I.

In this paper we give an upper bound on

r
(
I
)

r(I)

in terms of numerical invariants which are related with the Hilbert coefficients of

I
I

when

A
A

is Cohen-Macaulay. If

d
=
1

d=1

, it is known that

r
(
I
)
≤
e
(
I
)
−
1

r(I) \le e(I) -1

where

e
(
I
)

e(I)

denotes the multiplicity of

I
.

I.

If

d
≤
2
,

d \le 2,

in Corollary 1.5 we prove

r
(
I
)
≤

e
1

(
I
)
−
e
(
I
)
+
λ
(
A

/

I
)
+
1

r(I) \le e_1(I) - e(I) + \lambda (A/I) + 1

where

e
1

(
I
)

e_1(I)

is the first Hilbert coefficient of

I
.

I.

From this bound several results follow. Theorem 1.3 gives an upper bound on

r
(
I
)

r(I)

in a more general setting.
A classification of one dimensional almost periodic tilings arising from the projection method
Authors: Mingo James 
Published: July 2002

For each irrational number

α
\alpha

, with continued fraction expansion

[
0
;

a
1

,

a
2

,

a
3

,
…
]

[0; a_1, a_2,a_3, \dots ]

, we classify, up to translation, the one dimensional almost periodic tilings which can be constructed by the projection method starting with a line of slope

α
\alpha

. The invariant is a sequence of integers in the space

X
α

=
{
(

x
i

)

i
=
1

∞

∣

x
i

∈
{
0
,
1
,
2
,
…
,

a
i

}

X_\alpha = \{(x_i)_{i=1}^\infty \mid x_i \in \{0,1,2, \dots ,a_i\}

and

x

i
+
1

=
0

x_{i+1} = 0

whenever

x
i

=

a
i

}

x_i = a_i\}

modulo the equivalence relation generated by tail equivalence and

(

a
1

,
0
,

a
3

,
0
,
…
)
∼
(
0
,

a
2

,
0
,

a
4

,
…
)
∼
(

a
1

−
1
,

a
2

−
1
,

a
3

−
1
,
…
)

(a_1, 0, a_3, 0, \dots ) \sim (0, a_2, 0, a_4, \dots ) \sim (a_1 -1, a_2 - 1, a_3 - 1, \dots )

. Each tile in a tiling

T

\textsf {T}

, of slope

α
\alpha

, is coded by an integer

0
≤
x
≤
[
α
]

0 \leq x \leq [\alpha ]

. Using a composition operation, we produce a sequence of tilings

T

1

=

T

,

T

2

,

T

3

,
…

\textsf {T}_1 = \textsf {T}{}, \textsf {T}_2, \textsf {T}_3, \dots

. Each tile in

T

i

\textsf {T}_i

gets absorbed into a tile in

T

i
+
1

\textsf {T}_{i+1}

. A choice of a starting tile in

T

1

\textsf {T}_1

will thus produce a sequence in

X
α

X_\alpha

. This is the invariant.
𝑆-integral points of ℙⁿ-{2𝕟+1 hyperplanes in general position over number fields and function fields}
Authors: Wang Julie 
Published: July 2002

For the number field case we will give an upper bound on the number of the

S
S

-integral points in

P

n

(
K
)
−
{
2
n
+
1
hyperplanes in general

\mathbb {P}^n(K)-\{ 2n+1\text { hyperplanes in general}

position
}

\text {position}\}

. The main tool here is the explicit upper bound of the number of solutions of

S
S

-unit equations (Invent. Math. 102 (1990), 95–107). For the function field case we will give a bound on the height of the

S
S

-integral points in

P

n

(
K
)
−
{
2
n
+
1
hyperplanes in general position
}

\mathbb {P}^n(K)-\{ 2n+1\text { hyperplanes in general position}\}

. We will also give a bound for the number of “generators" of those

S
S

-integral points. The main tool here is the

S
S

-unit Theorem by Brownawell and Masser (Proc. Cambridge Philos. Soc. 100 (1986), 427–434).
Global (and local) analyticity for second order operators constructed from rigid vector fields on products of tori
Authors: Tartakoff David 
Published: July 2002

We prove global analytic hypoellipticity on a product of tori for partial differential operators which are constructed as rigid (variable coefficient) quadratic polynomials in real vector fields satisfying the Hörmander condition and where

P
P

satisfies a “maximal” estimate. We also prove an analyticity result that is local in some variables and global in others for operators whose prototype is
\[

P
=

(

∂

∂

x
1

)

2

+

(

∂

∂

x
2

)

2

+

(
a
(

x
1

,

x
2

)

∂

∂
t

)

2

P=\left ( \frac \partial {\partial x_1}\right ) ^2+\left ( \frac \partial { \partial x_2}\right ) ^2+\left ( a(x_1,x_2)\frac \partial {\partial t}\right )^2

\]
(with analytic

a
(
x
)
,
a
(
0
)
=
0

a(x),a(0)=0

, naturally, but not identically zero). The results, because of the flexibility of the methods, generalize recent work of Cordaro and Himonas in [Global analytic hypoellipticity of a class of degenerate elliptic operators on the torus, Math. Res. Lett. 1 (1994), 501–510] and Himonas in [On degenerate elliptic operators of infinite type, Math. Z. (to appear)] which showed that certain operators known not to be locally analytic hypoelliptic (those of Baouendi and Goulaouic [Analyticity for degenerate elliptic equations and applications, Proc. Sympos. Pure Math., vol. 23, Amer. Math. Soc., Providence, RI, 1971, pp. 79–84], Hanges and Himonas [Singular solutions for sums of squares of vector fields, Comm. Partial Differential Equations 16 (1991), 1503–1511], and Christ [Certain sums of squares of vector fields fail to be analytic hypoelliptic, Comm. Partial Differential Equations 10 (1991), 1695–1707]) were globally analytic hypoelliptic on products of tori.
Finite-dimensional lattice-subspaces of 𝐶(Ω) and curves of ℝⁿ
Authors: Polyrakis Ioannis 
Published: July 2002

Let

x
1

,
…
,

x
n

x_1,\dotsc ,x_n

be linearly independent positive functions in

C
(
Ω
)

C(\Omega )

, let

X
X

be the vector subspace generated by the

x
i

x_i

and let

β
\beta

denote the curve of

R

n

\mathbb R^n

determined by the function

β
(
t
)
=

1

z
(
t
)

(

x
1

(
t
)
,

x
2

(
t
)
,
…
,

x
n

(
t
)
)

\beta (t)=\frac {1}{z(t)} (x_1(t),x_2(t),\dotsc ,x_n(t))

, where

z
(
t
)
=

x
1

(
t
)
+

x
2

(
t
)
+
⋯
+

x
n

(
t
)

z(t)=x_1(t)+x_2(t)+\dotsb +x_n(t)

. We establish that

X
X

is a vector lattice under the induced ordering from

C
(
Ω
)

C(\Omega )

if and only if there exists a convex polygon of

R

n

\mathbb R^n

with

n
n

vertices containing the curve

β
\beta

and having its vertices in the closure of the range of

β
\beta

. We also present an algorithm which determines whether or not

X
X

is a vector lattice and in case

X
X

is a vector lattice it constructs a positive basis of

X
X

. The results are also shown to be valid for general normed vector lattices.
Gross spaces
Authors: Shelah Saharon , Spinas Otmar 
Published: July 2002

AGross spaceis a vector spaceEEof infinite dimension over some fieldFF, which is endowed with a symmetric bilinear formΦ:E2→F\Phi :E^{2} \rightarrow Fand has the property that every infinite dimensional subspaceU⊆EU\subseteq Esatisfies dim U⊥>U^{\perp }>dim EE. Gross spaces over uncountable fields exist (in certain dimensions) (see [H. Gross and E. Ogg,Quadratic spaces with few isometries, Comment. Math. Helv. 48 (1973), 511-519]). The existence of a Gross space over countable or finite fields (in a fixed dimension not above the continuum) is independent of the axioms of ZFC. Here we continue the investigation of Gross spaces. Among other things, we show that if the cardinal invariantbequalsω1\omega _{1}, a Gross space in dimensionω1\omega _{1}exists over every infinite field, and that it is consistent that Gross spaces exist over every infinite field but not over any finite field. We also generalize the notion of a Gross space and construct generalized Gross spaces in ZFC.
Exact controllability and stabilizability of the Korteweg-de Vries equation
Authors: Russell David , Zhang Bing-Yu 
Published: July 2002

In this paper, we consider distributed control of the system described by the Korteweg-de Vries equation

(i)

∂
t

u
+
u

∂
x

u
+

∂
x
3

u
=
f

\begin{equation*} \partial _t u + u \partial _x u + \partial _x^3 u = f \tag {i} \end{equation*}

on the interval

0
≤
x
≤
2
π
,

t
≥
0

0\leq x\leq 2\pi , \, t\geq 0

, with periodic boundary conditions

(ii)

∂
x
k

u
(
2
π
,
t
)
=

∂
x
k

u
(
0
,
t
)
,

k
=
0
,
1
,
2
,

\begin{equation*} \partial ^k_x u(2\pi , t ) = \partial ^k_x u(0,t) , \quad k=0,1,2, \tag {ii} \end{equation*}

where the distributed control

f
≡
f
(
x
,
t
)

f\equiv f(x,t)

is restricted so that the “volume”

∫
0

2
π

u
(
x
,
t
)
d
x

\int ^{2\pi }_0 u(x,t) dx

of the solution is conserved. Both exact controllability and stabilizibility questions are studied for the system. In the case of open loop control, if the control

f
f

is allowed to act on the whole spatial domain

(
0
,
2
π
)

(0,2\pi )

, it is shown that the system is globally exactly controllable, i.e., for given

T
>
0

T> 0

and functions

ϕ
(
x
)

\phi (x)

,

ψ
(
x
)

\psi (x)

with the same “volume”, one can alway find a control

f
f

so that the system (i)–(ii) has a solution

u
(
x
,
t
)

u(x,t)

satisfying
\[

u
(
x
,
0
)
=
ϕ
(
x
)
,

u
(
x
,
T
)
=
ψ
(
x
)
.

u(x,0) = \phi (x) , \qquad \quad u(x,T) = \psi (x) .

\]
If the control

f
f

is allowed to act on only a small subset of the domain

(
0
,
2
π
)

(0,2\pi )

, then the same result still holds if the initial and terminal states,

ψ
\psi

and

ϕ
\phi

, have small “amplitude” in a certain sense. In the case of closed loop control, the distributed control

f
f

is assumed to be generated by a linear feedback law conserving the “volume” while monotonically reducing

∫
0

2
π

u
(
x
,
t

)
2

d
x

\int ^{2\pi }_0 u(x,t)^2 dx

. The solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state. As in the open loop control case, a small amplitude assumption is needed if the control is allowed to act on only a small subdomain. The smoothing property of the periodic (linear) KdV equation discovered recently by Bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper.
On extension of cocycles to normalizer elements, outer conjugacy, and related problems
Authors: Danilenko Alexandre , Golodets Valentin 
Published: July 2002

Let

T
T

be an ergodic automorphism of a Lebesgue space and

α
\alpha

a cocycle of

T
T

with values in an Abelian locally compact group

G
G

. An automorphism

θ
\theta

from the normalizer

N
[
T
]

N[T]

of the full group

[
T
]

[T]

is said to be compatible with

α
\alpha

if there is a measurable function

φ
:
X
→
G

\varphi : X \to G

such that

α
(
θ
x
,
θ
T

θ

−
1

)
=
−
φ
(
x
)
+
α
(
x
,
T
)
+
φ
(
T
x
)

\alpha (\theta x, \theta T\theta ^{-1}) = - \varphi (x) + \alpha (x, T) + \varphi (Tx)

at a.e.

x
x

. The topology on the set

D
(
T
,
α
)

D(T, \alpha )

of all automorphisms compatible with

α
\alpha

is introduced in such a way that

D
(
T
,
α
)

D(T , \alpha )

becomes a Polish group. A complete system of invariants for the

α
\alpha

-outer conjugacy (i.e. the conjugacy in the quotient group

D
(
T
,
α
)

/

[
T
]
)

D(T, \alpha )/[T])

is found. Structure of the cocycles compatible with every element of

N
[
T
]

N[T]

is described.
Bloch constants of bounded symmetric domains
Authors: Zhang Genkai 
Published: July 2002

Let

D

1

D_{1}

and

D

2

D_{2}

be two irreducible bounded symmetric domains in the complex spaces

V

1

V_{1}

and

V

2

V_{2}

respectively. Let

E
E

be the Euclidean metric on

V

2

V_{2}

and

h
h

the Bergman metric on

V

1

V_{1}

. The Bloch constant

b
(

D

1

,

D

2

)

b(D_{1}, D_{2})

is defined to be the supremum of

E
(

f

′

(
z
)
x
,

f

′

(
z
)
x

)

1
2

/

h

z

(
x
,
x

)

1

/

2

E(f^{\prime }(z)x, f^{\prime }(z)x)^{\frac {1}{2}}/h_{z}(x, x)^{1/2}

, taken over all the holomorphic functions

f
:

D

1

→

D

2

f: D_{1}\to D_{2}

and

z
∈

D

1

z\in D_{1}

, and nonzero vectors

x
∈

V

1

x\in V_{1}

. We find the constants for all the irreducible bounded symmetric domains

D

1

D_{1}

and

D

2

D_{2}

. As a special case we answer an open question of Cohen and Colonna.
Hecke algebras, 𝑈_{𝑞}𝑠𝑙_{𝑛}, and the Donald-Flanigan conjecture for 𝑆_{𝑛}
Authors: Gerstenhaber Murray , Schaps Mary 
Published: July 2002

The Donald–Flanigan conjecture asserts that the integral group ring

Z

G

\mathbb {Z}G

of a finite group

G
G

can be deformed to an algebra

A
A

over the power series ring

Z

[
[
t
]
]

\mathbb {Z}[[t]]

with underlying module

Z

G
[
[
t
]
]

\mathbb {Z}G[[t]]

such that if

p
p

is any prime dividing

#
G

\#G

then

A

⊗

Z

[
[
t
]
]

F

p

(
(
t
)
)

¯

A\otimes _{\mathbb {Z}[[t]]}\overline {\mathbb {F}_{p}((t))}

is a direct sum of total matric algebras whose blocks are in natural bijection with and of the same dimensions as those of

C

G
.

\mathbb {C}G.

We prove this for

G
=

S

n

G = S_{n}

using the natural representation of its Hecke algebra

H

\mathcal {H}

by quantum Yang-Baxter matrices to show that over

Z

[
q
]

\mathbb {Z}[q]

localized at the multiplicatively closed set generated by

q
q

and all

i

q

2

=
1
+

q

2

+

q

4

+
⋯
+

q

2
(
i
−
1
)

,
i
=
1
,
2
,
…
,
n

i_{q^{2}} = 1+q^{2} + q^{4} + \dots + q^{2(i-1)}, i = 1,2,\dots , n

, the Hecke algebra becomes a direct sum of total matric algebras. The corresponding “canonical" primitive idempotents are distinct from Wenzl’s and in the classical case (

q
=
1

q=1

), from those of Young.
A bound on the geometric genus of projective varieties verifying certain flag conditions
Authors: Di Gennaro Vincenzo 
Published: July 2002

Fix integersn,r,s1,...,sln,r,s_{1},...,s_{l}and letS(n,r;s1,...,sl)\mathcal {S}(n,r;s_{1},...,s_{l})be the set of all integral, projective and nondegenerate varietiesVVof degrees1s_{1}and dimensionnnin the projective spacePr\mathbf {P}^{r}, such that, for alli=2,...,li=2,...,l,VVdoes not lie on any variety of dimensionn+i−1n+i-1and degree>si>s_{i}. We say that a varietyVVsatisfies aflag conditionof type(n,r;s1,...,sl)(n,r;s_{1},...,s_{l})ifVVbelongs toS(n,r;s1,...,sl)\mathcal {S}(n,r;s_{1},...,s_{l}). In this paper, under the hypothesess1>>...>>sls_{1}>>...>>s_{l}, we determine an upper boundGh(n,r;s1,...,sl)G^{h}(n,r;s_{1},...,s_{l}), depending only onn,r,s1,...,sln,r,s_{1},...,s_{l}, for the numberG(n,r;s1,...,sl):=max{pg(V):V∈S(n,r;s1,...,sl)}G(n,r;s_{1},...,s_{l}):= {max} {\{} p_{g}(V) : V\in \mathcal {S}(n,r;s_{1},...,s_{l}){\}}, wherepg(V)p_{g}(V)denotes the geometric genus ofVV. In casen=1n=1andl=2l=2, the study of an upper bound for the geometric genus has a quite long history and, forn≥1n\geq 1,l=2l=2ands2=r−ns_{2}=r-n, it has been introduced by Harris. We exhibit sharp results for particular ranges of our numerical datan,r,s1,...,sln,r,s_{1},...,s_{l}. For instance, we extend Halphen’s theorem for space curves to the case of codimension two and characterize the smooth complete intersections of dimensionnninPn+3\mathbf {P}^{n+3}as the smooth varieties of maximal geometric genus with respect to appropriate flag condition. This result applies to smooth surfaces inP5\mathbf {P}^{5}. Next we discuss how farGh(n,r;s1,...,sl)G^{h}(n,r;s_{1},...,s_{l})is fromG(n,r;s1,...,sl)G(n,r;s_{1},...,s_{l})and show a sort of lifting theorem which states that, at least in certain cases, the varietiesV∈S(n,r;s1,...,sl)V\in \mathcal {S}(n,r;s_{1},...,s_{l})of maximal geometric genusG(n,r;s1,...,sl)G(n,r;s_{1},...,s_{l})must in fact lie on a flag such asV=Vs1n⊂Vs2n+1⊂...⊂Vsln+l−1⊂PrV=V_{s_{1}}^{n}\subset V_{s_{2}}^{n+1}\subset ...\subset V_{s_{l}}^{n+l-1}\subset {\mathbf {P}^{r}}, whereVsjV^{j}_{s}denotes a subvariety ofPr\mathbf {P}^{r}of degreessand dimensionjj. We also discuss further generalizations of flag conditions, and finally we deduce some bounds for Castelnuovo’s regularity of varieties verifying flag conditions.
The approximate functional formula for the theta function and Diophantine Gauss sums
Authors: Coutsias E. , Kazarinoff N. 
Published: July 2002

We consider the polygonal lines in the complex plane

C

\Bbb {C}

whose

N
N

-th vertex is defined by

S
N

=

∑

n
=
0

N

′

exp
⁡
(
i
ω
π

n
2

)

S_N = \sum _{n=0}^{N\,’} \exp (i\omega \pi n^2)

(with

ω
∈

R

\omega \in \Bbb {R}

), where the prime means that the first and last terms in the sum are halved. By introducing the discrete curvature of the polygonal line, and by exploiting the similarity of segments of the line, for small

ω
\omega

, to Cornu spirals (C-spirals), we prove the precise renormalization formula

a
m
p
;

|

∑

k
=
0

N

′

exp
⁡
(
i
ω
π

k
2

)
−

exp
⁡
(
s
g
n
(
ω
)
i
π

/

4
)

|

ω

|

∑

k
=
0

n

′

exp
⁡
(
−
i

π
ω

k
2

)
|

a
m
p
;

≤
C

|

ω
N
−
n

ω

|

,
0
>

|

ω

|

>
1
,

\begin{equation} \begin {split} &\left | \sum _{k=0}^{N}\,’ \exp (i\omega \pi k^2) -\frac {\exp (sgn(\omega )i\pi /4)}{\sqrt {|\omega |}} \sum _{k=0}^n \,’ \exp (-i\frac {\pi }{\omega } k^2)\right |\\ &\qquad \leq C \left |\frac {\omega N - n}{\omega }\right |, 0>|\omega | >1, \end{split} \end{equation}

where

N
=
[
[
n

/

ω
]
]

N=[[n/\omega ]]

, the nearest integer to

n

/

ω

n/\omega

and

1
>
C
>
3.14

1>C>3.14

. This formula, which sharpens Hardy and Littlewood’s approximate functional formula for the theta function, generalizes to irrationals, as a Diophantine inequality, the well-known sum formula of Gauss. The geometrical meaning of the relation between the two limits is that the first sum is taken to a point of inflection of the corresponding C-spirals. The second sum replaces whole C-spirals of the first by unit vectors times scale and phase factors. The block renormalization procedure implied by this replacement is governed by the circle map

ω
→
−

1
ω

(
mod

2
)
,
ω
∈
]
−
1
,
+
1
[
∖
{
0
}
,

\begin{equation} \omega \rightarrow -\frac {1}{\omega } \pmod 2 , \omega \in ]-1,+1[ \setminus \{0\}, \end{equation}

whose orbits are analyzed by expressing

ω
\omega

as an even continued fraction.
A new degree bound for vector invariants of symmetric groups
Authors: Fleischmann P. 
Published: July 2002

Let

R
R

be a commutative ring,

V
V

a finitely generated free

R
R

-module and

G
≤
G

L
R

(
V
)

G\le GL_R(V)

a finite group acting naturally on the graded symmetric algebra

A
=
S
(
V
)

A=S(V)

. Let

β
(
V
,
G
)

\beta (V,G)

denote the minimal number

m
m

, such that the ring

A
G

A^G

of invariants can be generated by finitely many elements of degree at most

m
m

.

For

G
=

Σ
n

G=\Sigma _n

and

V
(
n
,
k
)

V(n,k)

, the

k
k

-fold direct sum of the natural permutation module, one knows that

β
(
V
(
n
,
k
)
,

Σ
n

)
≤
n

\beta (V(n,k),\Sigma _n) \le n

, provided that

n
!

n!

is invertible in

R
R

. This was used by E. Noether to prove

β
(
V
,
G
)
≤

|

G

|

\beta (V,G) \le |G|

if

|

G

|

!
∈

R
∗

|G|! \in R^*

.

In this paper we prove

β
(
V
(
n
,
k
)
,

Σ
n

)
≤
m
a
x
{
n
,
k
(
n
−
1
)
}

\beta (V(n,k),\Sigma _n) \le max\{n,k(n-1)\}

for arbitrary commutative rings

R
R

and show equality for

n
=

p
s

n=p^s

a prime power and

R
=

Z

R = \mathbb {Z}

or any ring with

n
⋅

1
R

=
0

n\cdot 1_R=0

. Our results imply

β
(
V
,
G
)
≤
m
a
x
{

|

G

|

,
rank
⁡
(
V
)
(

|

G

|

−
1
)
}

\begin{equation*} \beta (V,G)\le max\{|G|, \operatorname {rank}(V)(|G|-1)\}\end{equation*}

for any ring with

|

G

|

∈

R
∗

|G| \in R^*

.
Green’s function, harmonic transplantation, and best Sobolev constant in spaces of constant curvature
Authors: Bandle C. , Brillard A. , Flucher M. 
Published: July 2002

We extend the method of harmonic transplantation from Euclidean domains to spaces of constant positive or negative curvature. To this end the structure of the Green’s function of the corresponding Laplace-Beltrami operator is investigated. By means of isoperimetric inequalities we derive complementary estimates for its distribution function. We apply the method of harmonic transplantation to the question of whether the best Sobolev constant for the critical exponent is attained, i.e. whether there is an extremal function for the best Sobolev constant in spaces of constant curvature. A fairly complete answer is given, based on a concentration-compactness argument and a Pohozaev identity. The result depends on the curvature.
Operator valued weights without structure theory
Authors: Falcone Tony , Takesaki Masamichi 
Published: July 2002

A result of Haagerup, generalizing a theorem of Takesaki, states the following: If

N

⊂

M

{\mathcal {N}}\subset {\mathcal {M}}

are von Neumann algebras, then there exists a faithful, normal and semi-finite (fns) operator valued weight

T
:

M

+

→

N

+

^

T \colon {\mathcal {M}}_{+} \rightarrow \widehat {{\mathcal {N}}_{+}}

if and only if there exist fns weights

φ
~

\tilde \varphi

on

M

{\mathcal {M}}

and

φ
\varphi

on

N

{\mathcal {N}}

satisfying

σ

t

φ

(
x
)
=

σ

t

φ
~

(
x
)

∀
x
∈

N

,
t
∈

R

\sigma ^{\varphi }_{t}(x) = \sigma ^{\tilde \varphi }_{t}(x) \, \forall x \in {\mathcal {N}} , t \in \mathbb {R}

. In fact,

T
T

can be chosen such that

φ
~

=
φ
∘
T

\tilde \varphi = \varphi \circ T

;

T
T

is then uniquely determined by this condition. We present a proof of the above which does not use any structure theory.