bluuebunny/crossref_metadata_2025 huggingface dataset:
Stable constant mean curvature surfaces with circular boundary
Authors: Alías Luis , López Rafael , Palmer Bennett
Published: July 2002

In this paper we study stable constant mean curvature surfaces in the Euclidean space

R

3

\mathbf {R}^3

with circular boundary. We show that in the case of genus zero, the only such surfaces are the spherical caps and the flat discs. We also extend this result to the case of surfaces in the other space forms, namely the sphere

S

3

\mathbf {S}^3

and the hyperbolic space

H

3

\mathbf {H}^3

.
Finite rank perturbations and distribution theory
Authors: Albeverio S. , Kurasov P.
Published: July 2002

Perturbations

A
T

A_T

of a selfadjoint operator

A
A

by symmetric finite rank operators

T
T

from

H

2

(
A
)

\mathcal {H}_2 (A)

to

H

−
2

(
A
)

\mathcal {H}_{-2} (A)

are studied. The finite dimensional family of selfadjoint extensions determined by

A
T

A_T

is given explicitly.
Average root numbers in families of elliptic curves
Authors: Rizzo Ottavio
Published: July 2002

We introduce a height measure on

Q

\mathbf {Q}

to count rational numbers. Through it, we prove a density result on the average value of the root numbers of families of twists of elliptic curves.
Periodic solutions of a periodic delay predator-prey system
Authors: Yongkun Li
Published: July 2002

The existence of a positive periodic solution for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row StartFraction normal d upper H left-parenthesis t right-parenthesis Over normal d t EndFraction equals r left-parenthesis t right-parenthesis upper H left-parenthesis t right-parenthesis left-bracket 1 minus StartFraction upper H left-parenthesis t minus tau left-parenthesis t right-parenthesis right-parenthesis Over upper K left-parenthesis t right-parenthesis EndFraction right-bracket minus alpha left-parenthesis t right-parenthesis upper H left-parenthesis t right-parenthesis upper P left-parenthesis t right-parenthesis comma StartFraction normal d upper P left-parenthesis t right-parenthesis Over normal d t EndFraction equals minus b left-parenthesis t right-parenthesis upper P left-parenthesis t right-parenthesis plus beta left-parenthesis t right-parenthesis upper P left-parenthesis t right-parenthesis upper H left-parenthesis t minus sigma left-parenthesis t right-parenthesis right-parenthesis EndLayout">

{

d

H
(
t
)

d

t

=
r
(
t
)
H
(
t
)

[
1
−

H
(
t
−
τ
(
t
)
)

K
(
t
)

]

−
α
(
t
)
H
(
t
)
P
(
t
)
,
 

d

P
(
t
)

d

t

=
−
b
(
t
)
P
(
t
)
+
β
(
t
)
P
(
t
)
H
(
t
−
σ
(
t
)
)

\begin{equation*} \begin {cases} \frac {\mathrm {d}H(t)}{\mathrm {d}t}=r(t)H(t) \left [1-\frac {H(t-\tau (t))}{K(t)}\right ] -\alpha (t)H(t) P(t),\ \frac {\mathrm {d}P(t)}{\mathrm {d}t}=-b(t)P(t)+\beta (t)P(t)H(t-\sigma (t)) \end{cases} \end{equation*}

is established, where

r
r

,

K
K

,

α
\alpha

,

b
b

,

β
\beta

are positive periodic continuous functions with period

ω
>
0

\omega >0

, and

τ
\tau

,

σ
\sigma

are periodic continuous functions with period

ω
\omega

.
(𝑍₂)^{𝑘}-actions with fixed point set of constant codimension 2^{𝑘}+1
Authors: Yanying Wang , Zhende Wu , Kai Ma
Published: July 2002

The groups

J

n
,
k

2
k

+
1

J_{n,k}^{2^k+1}

of cobordism classes in the unoriented cobordism group

M

O
n

MO_n

containing a representative

M
n

M^n

admitting a

(

Z
2

)
k

(Z_2)^k

-action with fixed point set of constant codimension

2
k

+
1

2^k+1

are determined.
The Weierstrass approximation theorem and a characterization of the unit circle
Authors: Bochnak J. , Kucharz W.
Published: July 2002

We study real algebraic morphisms from nonsingular real algebraic varieties

X
X

with

dim
⁡
X
≥
1

\dim X \geq 1

into nonsingular real algebraic curves

C
C

. We show, among other things, that the set of real algebraic morphisms from

X
X

into

C
C

is never dense in the space of all

C

∞

\mathcal C^\infty

maps from

X
X

into

C
C

, unless

C
C

is biregularly isomorphic to a Zariski open subset of the unit circle.
Weak ergodicity of stationary pairwise independent processes
Authors: Landers D. , Rogge L.
Published: July 2002

It is proven that a stationary process of pairwise independent random variables with values in a separable metric space is weakly ergodic, i.e. each random variable is independent of the system of invariant sets of the process. An example shows that a process of identically distributed pairwise independent random variables is in general, however, not weakly ergodic.
Toeplitz algebras on discrete abelian quasily ordered groups
Authors: Qingxiang Xu
Published: July 2002

In this note, Toeplitz operators on discrete abelian quasily ordered groups are studied, and a theorem of R. Douglas is generalized.
Generalized Dedekind symbols associated with the Eisenstein series
Authors: Fukuhara Shinji
Published: July 2002

We have shown recently that the space of modular forms, the space of generalized Dedekind sums, and the space of period polynomials are all isomorphic. In this paper, we will prove, under these isomorphisms, that the Eisenstein series correspond to the Apostol generalized Dedekind sums, and that the period polynomials are expressed in terms of Bernoulli numbers. This gives us a new more natural proof of the reciprocity law for the Apostol generalized Dedekind sums. Our proof yields as a by-product new polylogarithm identities.
Finite generation properties for fuchsian group von Neumann algebras tensor 𝐵(𝐻)
Authors: Rădulescu Florin
Published: July 2002

We prove that the algebra

A

=

L

(

F

N

)
⊗
B
(
H
)

\mathcal {A}=\mathcal {L}(F_{N})\otimes B(H)

,

F

N

F_{N}

a free group with finitely many generators, contains a subnormal operator

J
J

such that the linear span of the set

{
(

J

∗

)

n

J

m

|
n
,
m
=
0
,
1
,
2
,
.
.
.
}

\{(J^{*})^{n}J^{m}\vert n,m=0,1,2,...\}

is weakly dense in

A

\mathcal {A}

. This is the analogue for the

I

I

∞

II_{\infty }

factor

L

(

F

N

)
⊗
B
(
H
)

\mathcal {L}(F_{N})\otimes B(H)

,

N
N

finite, of a well known fact about the unilateral shift

S
S

on a Hilbert space

K
K

: the linear span of all the monomials

(

S

∗

)

n

S

m

(S^{*})^{n} S^{m}

is weakly dense in

B
(
K
)

B(K)

. We also show that for a suitable space

H

2

H^{2}

of square summable analytic functions, if

P
P

is the projection from the Hilbert space

L

2

L^{2}

of all square summable functions onto

H

2

H^{2}

and

M

j
¯

M_{\overline {j}}

is the unbounded operator of multiplication by

j
¯

\overline {j}

on

L

2

L^{2}

, then the (unbounded) operator

P

M

j
¯

(
I
−
P
)

PM_{\overline {j}}(I-P)

is nonzero (with nonzero domain).
Multiscale decompositions on bounded domains
Authors: Cohen A. , Dahmen W. , DeVore R.
Published: July 2002

A construction of multiscale decompositions relative to domains

Ω
⊂

R

d

\Omega \subset \mathbb {R}^d

is given. Multiscale spaces are constructed on

Ω
\Omega

which retain the important features of univariate multiresolution analysis including local polynomial reproduction and locally supported, stable bases.
Examples of torsion points on genus two curves
Authors: Boxall John , Grant David
Published: July 2002

We describe a method that sometimes determines all the torsion points lying on a curve of genus two defined over a number field and embedded in its Jacobian using a Weierstrass point as base point. We then apply this to the examples

y

2

=

x

5

+
x

y^{2}=x^{5}+x

,

y

2

=

x

5

+
5

x

3

+
x

y^{2}=x^{5}+5\,x^{3}+x

, and

y

2

−
y
=

x

5

y^{2}-y=x^{5}

.
Partial differential equations with matricial coefficients and generalized translation operators
Authors: Mahmoud N.
Published: July 2002

Let

Δ

α

\Delta _{\alpha }

be the Bessel operator with matricial coefficients defined on

(
0
,
∞
)

(0,\infty )

by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta Subscript alpha Baseline upper U left-parenthesis t right-parenthesis equals upper U left-parenthesis t right-parenthesis plus StartFraction 2 alpha plus upper I Over t EndFraction upper U prime left-parenthesis t right-parenthesis">

Δ

α

U
(
t
)
=
U
(
t
)
+

2
α
+
I

t

U
′

(
t
)

\begin{equation*}\Delta _{\alpha }U(t)=U(t)+\frac {2\alpha +I}{t}U’(t)\end{equation*}

where

α
\alpha

is a diagonal matrix and let

q
q

be an

n
×
n

n\times n

matrix-valued function. In this work, we prove that there exists an isomorphism

X
X

on the space of even

C

∞

{\mathcal C}^{\infty }

,

C

n

\mathbb {C}^n

-valued functions which transmutes

Δ

α

\Delta _{\alpha }

and

(

Δ

α

+
q
)

(\Delta _{\alpha }+q)

. This allows us to define generalized translation operators and to develop harmonic analysis associated with

(

Δ

α

+
q
)

(\Delta _{\alpha }+q)

. By use of the Riemann method, we provide an integral representation and we deduce more precise information on these operators.
The geography problem for irreducible spin four-manifolds
Authors: Park B. Doug , Szabó Zoltán
Published: July 2002

We study the geography problem for smooth irreducible simply-connected spin four-manifolds. For a large class of homotopy types, we exhibit both symplectic and non-symplectic representatives. We also compute the Seiberg-Witten invariants of all the four-manifolds we construct.
Specializations of Brauer classes over algebraic function fields
Authors: Fein Burton , Schacher Murray
Published: July 2002

Let

F
F

be either a number field or a field finitely generated of transcendence degree

≥
1

\ge 1

over a Hilbertian field of characteristic 0, let

F
(
t
)

F(t)

be the rational function field in one variable over

F
F

, and let

α
∈
Br
⁡
(
F
(
t
)
)

\alpha \in \operatorname {Br} (F(t))

. It is known that there exist infinitely many

a
∈
F

a\in F

such that the specialization

t
→
a

t\to a

induces a specialization

α
→

α
¯

∈
Br
⁡
(
F
)

\alpha \to \overline {\alpha }\in \operatorname {Br} (F)

, where

α
¯

\overline {\alpha }

has exponent equal to that of

α
\alpha

. Now let

K
K

be a finite extension of

F
(
t
)

F(t)

and let

β
=

res

K

/

F
(
t
)

⁡
(
α
)

\beta =\operatorname {res} _{K/F(t)}(\alpha )

. We give sufficient conditions on

α
\alpha

and

K
K

for there to exist infinitely many

a
∈
F

a\in F

such that the specialization

t
→
a

t\to a

has an extension to

K
K

inducing a specialization

β
→

β
¯

∈
Br
⁡
(

K
¯

)

\beta \to \overline {\beta }\in \operatorname {Br} (\overline {K})

,

K
¯

\overline {K}

the residue field of

K
K

, where

β
¯

\overline {\beta }

has exponent equal to that of

β
\beta

. We also give examples to show that, in general, such

a
∈
F

a\in F

need not exist.
An equivariant Brauer semigroup and the symmetric imprimitivity theorem
Authors: an Huef Astrid , Raeburn Iain , Williams Dana
Published: July 2002

Suppose that

(
X
,
G
)

(X,G)

is a second countable locally compact transformation group. We let

S
G

⁡
(
X
)

\operatorname {S}_G(X)

denote the set of Morita equivalence classes of separable dynamical systems

(
A
,
G
,
α
)

(A,G,\alpha )

where

A
A

is a

C

0

(
X
)

C_{0}(X)

-algebra and

α
\alpha

is compatible with the given

G
G

-action on

X
X

. We prove that

S

G

⁡
(
X
)

\operatorname {S}_{G}(X)

is a commutative semigroup with identity with respect to the binary operation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper A comma upper G comma alpha right-bracket left-bracket upper B comma upper G comma beta right-bracket equals left-bracket upper A circled-times Subscript upper X Baseline upper B comma upper G comma alpha circled-times Subscript upper X Baseline beta right-bracket">

[
A
,
G
,
α
]
[
B
,
G
,
β
]
=
[
A

⊗

X

B
,
G
,
α

⊗

X

β
]

[A,G,\alpha ][B,G,\beta ]=[A\otimes _{X}B,G,\alpha \otimes _{X}\beta ]

for an appropriately defined balanced tensor product on

C

0

(
X
)

C_{0}(X)

-algebras. If

G
G

and

H
H

act freely and properly on the left and right of a space

X
X

, then we prove that

S

G

⁡
(
X

/

H
)

\operatorname {S}_{G}(X/H)

and

S

H

⁡
(
G
∖
X
)

\operatorname {S}_{H}(G\setminus X)

are isomorphic as semigroups. If the isomorphism maps the class of

(
A
,
G
,
α
)

(A,G,\alpha )

to the class of

(
B
,
H
,
β
)

(B,H,\beta )

, then

A

⋊

α

G

A\rtimes _{\alpha }G

is Morita equivalent to

B

⋊

β

H

B\rtimes _{\beta }H

.
Symplectic 4-manifolds with Hermitian Weyl tensor
Authors: Apostolov Vestislav , Armstrong John
Published: July 2002

It is proved that any compact almost Kähler, Einstein 4-manifold whose fundamental form is a root of the Weyl tensor is necessarily Kähler.
Primes in short arithmetic progressions with rapidly increasing differences
Authors: Elliott P.
Published: July 2002

Primes are, on average, well distributed in short segments of arithmetic progressions, even if the associated moduli grow rapidly.
Polytopal linear retractions
Authors: Bruns Winfried , Gubeladze Joseph
Published: July 2002

We investigate graded retracts of polytopal algebras (essentially the homogeneous rings of affine cones over projective toric varieties) as polytopal analogues of vector spaces. In many cases we show that these retracts are again polytopal algebras and that codimension

1
1

retractions factor through retractions preserving the semigroup structure. We expect that these results hold in general. This paper is a part of the project started by the authors in 1999, where we investigate the graded automorphism groups of polytopal algebras. Part of the motivation comes from the observation that there is a reasonable ‘polytopal’ generalization of linear algebra (and, subsequently, that of algebraic

K
K

-theory).
On the shellability of the order complex of the subgroup lattice of a finite group
Authors: Shareshian John
Published: July 2002

We show that the order complex of the subgroup lattice of a finite group

G
G

is nonpure shellable if and only if

G
G

is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups.
Bi-Lipschitz homogeneous curves in ℝ² are quasicircles
Authors: Bishop Christopher
Published: July 2002

We show that a bi-Lipschitz homogeneous curve in the plane must satisfy the bounded turning condition, and that this is false in higher dimensions. Combined with results of Herron and Mayer this gives several characterizations of such curves in the plane.
Degree-one maps between hyperbolic 3-manifolds with the same volume limit
Authors: Soma Teruhiko
Published: July 2002

Suppose that

f
n

:

M
n

⟶

N
n

f_n:M_n\longrightarrow N_n

(
n
∈

N

)

(n\in {\mathbf N})

are degree-one maps between closed hyperbolic 3-manifolds with
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow normal infinity Endscripts upper V o l left-parenthesis upper M Subscript n Baseline right-parenthesis equals limit Underscript n right-arrow normal infinity Endscripts upper V o l left-parenthesis upper N Subscript n Baseline right-parenthesis greater-than normal infinity period">

lim

n
→
∞

Vol
⁡
(

M
n

)
=

lim

n
→
∞

Vol

(

N
n

)
>
∞
.

\lim _{n\rightarrow \infty } \operatorname {Vol} (M_n)=\lim _{n\rightarrow \infty }{\operatorname {Vol}}(N_n) >\infty .

\]
Then, our main theorem, Theorem 2, shows that, for all but finitely many

n
∈

N

n\in {\mathbf N}

,

f
n

f_n

is homotopic to an isometry. A special case of our argument gives a new proof of Gromov-Thurston’s rigidity theorem for hyperbolic 3-manifolds without invoking any ergodic theory. An example in §3 implies that, if the degree of these maps is greater than 1, the assertion corresponding to our theorem does not hold.
Higher type adjunction inequalities for Donaldson invariants
Authors: Muñoz Vicente
Published: July 2002

We prove new adjunction inequalities for embedded surfaces in four-manifolds with non-negative self-intersection number using the Donaldson invariants. These formulas are completely analogous to the ones obtained by Ozsváth and Szabó using the Seiberg-Witten invariants. To prove these relations, we give a fairly explicit description of the structure of the Fukaya-Floer homology of a surface times a circle. As an aside, we also relate the Floer homology of a surface times a circle with the cohomology of some symmetric products of the surface.
Spectral lifting in Banach algebras and interpolation in several variables
Authors: Popescu Gelu
Published: July 2002

Let

A

{\mathcal {A}}

be a unital Banach algebra and let

J
J

be a closed two-sided ideal of

A

{\mathcal {A}}

. We prove that if any invertible element of

A

/

J

{\mathcal {A}}/J

has an invertible lifting in

A

{\mathcal {A}}

, then the quotient homomorphism

Φ
:

A

→

A

/

J

\Phi :{\mathcal {A}}\to {\mathcal {A}}/J

is a spectral interpolant. This result is used to obtain a noncommutative multivariable analogue of the spectral commutant lifting theorem of Bercovici, Foiaş, and Tannenbaum. This yields spectral versions of Sarason, Nevanlinna–Pick, and Carathéodory type interpolation for

F

n

∞

⊗
¯

B
(

K

)

F_{n}^{\infty }\bar \otimes B({\mathcal {K}})

, the WOT-closed algebra generated by the spatial tensor product of the noncommutative analytic Toeplitz algebra

F

n

∞

F_{n}^{\infty }

and

B
(

K

)

B({\mathcal {K}})

, the algebra of bounded operators on a finite dimensional Hilbert space

K

{\mathcal {K}}

. A spectral tangential commutant lifting theorem in several variables is considered and used to obtain a spectral tangential version of the Nevanlinna-Pick interpolation for

F

n

∞

⊗
¯

B
(

K

)

F_{n}^{\infty }\bar \otimes B({\mathcal {K}})

.

In particular, we obtain interpolation theorems for matrix-valued bounded analytic functions on the open unit ball of

C

n

\mathbb {C}^{n}

, in which one bounds the spectral radius of the interpolant and not the norm.
Discrete Series Characters as Lifts from Two-structure Groups
Authors: Herb Rebecca
Published: July 2002

Let

G
G

be a connected reductive Lie group with a relatively compact Cartan subgroup. Then it has relative discrete series representations. The main result of this paper is a formula expressing relative discrete series characters on

G
G

as “lifts” of relative discrete series characters on smaller groups called two-structure groups for

G
G

. The two-structure groups are connected reductive Lie groups which are locally isomorphic to the direct product of an abelian group and simple groups which are real forms of

S
L
(
2
,

C

)

SL(2, \mathbf {C})

or

S
O
(
5
,

C

)

SO(5, \mathbf {C} )

. They are not necessarily subgroups of

G
G

, but they “share” the relatively compact Cartan subgroup and certain other Cartan subgroups with

G
G

. The character identity is similar to formulas coming from endoscopic lifting, but the two-structure groups are not necessarily endoscopic groups, and the characters lifted are not stable. Finally, the formulas are valid for non-linear as well as linear groups.
The multiplier ideals of a sum of ideals
Authors: Mustaţǎ Mircea
Published: July 2002

We prove that if

a

_

\underline {\mathbf {a}}

,

b

_

⊆

O

X

\underline {\mathbf {b}}\subseteq \mathcal {O}_X

are nonzero sheaves of ideals on a complex smooth variety

X
X

, then for every

γ
∈

Q

+

\gamma \in {\mathbb Q}_+

we have the following relation between the multiplier ideals of

a

_

\underline {\mathbf {a}}

,

b

_

\underline {\mathbf {b}}

and

a

_

+

b

_

\underline {\mathbf {a}}+\underline {\mathbf {b}}

:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper I left-parenthesis upper X comma gamma dot left-parenthesis bold a underbar plus bold b underbar right-parenthesis right-parenthesis subset-of-or-equal-to sigma-summation Underscript alpha plus beta equals gamma Endscripts script upper I left-parenthesis upper X comma alpha dot bold a underbar right-parenthesis dot script upper I left-parenthesis upper X comma beta dot bold b underbar right-parenthesis period">

I

(
X
,
γ
⋅
(

a

_

+

b

_

)
)

⊆

∑

α
+
β
=
γ

I

(
X
,
α
⋅

a

_

)
⋅

I

(
X
,
β
⋅

b

_

)
.

\begin{equation*}\mathcal {I}\left (X,\gamma \cdot (\underline {\mathbf {a}}+ \underline {\mathbf {b}})\right )\subseteq \sum _{\alpha +\beta =\gamma } \mathcal {I}(X,\alpha \cdot \underline {\mathbf {a}})\cdot \mathcal {I}(X,\beta \cdot \underline {\mathbf {b}}).\end{equation*}

A similar formula holds for the asymptotic multiplier ideals of the sum of two graded systems of ideals.

We use this result to approximate at a given point arbitrary multiplier ideals by multiplier ideals associated to zero dimensional ideals. This is applied to compare the multiplier ideals associated to a scheme in different embeddings.
Existence of curves with prescribed topological singularities
Authors: Keilen Thomas , Tyomkin Ilya
Published: July 2002

Throughout this paper we study the existence of irreducible curves

C
C

on smooth projective surfaces

Σ
\Sigma

with singular points of prescribed topological types

S

1

,
…
,

S

r

\mathcal S_1,\ldots ,\mathcal S_r

. There are necessary conditions for the existence of the type

∑

i
=
1

r

μ
(

S

i

)
≤
α

C
2

+
β
C
.
K
+
γ

\sum _{i=1}^r \mu (\mathcal S_i)\leq \alpha C^2+\beta C.K+\gamma

for some fixed divisor

K
K

on

Σ
\Sigma

and suitable coefficients

α
\alpha

,

β
\beta

and

γ
\gamma

, and the main sufficient condition that we find is of the same type, saying it is asymptotically proper. Ten years ago general results of this quality were not known even for the case

Σ
=

P

C

2

\Sigma =\mathbb P_{\mathbb C}^2

. An important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up

Σ
\Sigma

of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O Subscript normal upper Sigma overTilde Baseline left-parenthesis pi Superscript asterisk Baseline upper D minus sigma-summation Underscript i equals 1 Overscript r Endscripts m Subscript i Baseline upper E Subscript i Baseline right-parenthesis">

O

Σ
~

(

π
∗

D
−

∑

i
=
1

r

m
i

E
i

)

\mathcal O_{\widetilde {\Sigma }}(\pi ^*D-\sum _{i=1}^rm_iE_i)

, deduced from the Kawamata-Vieweg Vanishing Theorem. Its proof covers the first part of the paper, while the middle part is devoted to the existence theorems. In the last part we investigate our conditions on ruled surfaces, products of elliptic curves, surfaces in

P

C

3

\mathbb P_{\mathbb C}^3

, and K3-surfaces.
Second order Lagrangian Twist systems: simple closed characteristics
Authors: Van den Berg J. , Vandervorst R.
Published: July 2002

We consider a special class of Lagrangians that play a fundamental role in the theory of second order Lagrangian systems: Twist systems. This subclass of Lagrangian systems is defined via a convenient monotonicity property that such systems share. This monotonicity property (Twist property) allows a finite dimensional reduction of the variational principle for finding closed characteristics in fixed energy levels. This reduction has some similarities with the method of broken geodesics for the geodesic variational problem on Riemannian manifolds. On the other hand, the monotonicity property can be related to the existence of local Twist maps in the associated Hamiltonian flow. The finite dimensional reduction gives rise to a second order monotone recurrence relation. We study these recurrence relations to find simple closed characteristics for the Lagrangian system. More complicated closed characteristics will be dealt with in future work. Furthermore, we give conditions on the Lagrangian that guarantee the Twist property.
Extremal problems for quasiconformal maps of punctured plane domains
Authors: Marković Vladimir
Published: July 2002

The main goal of this paper is to give an affirmative answer to the long-standing conjecture which asserts that the affine map is a uniquely extremal quasiconformal map in the Teichmüller space of the complex plane punctured at the integer lattice points. In addition we derive a corollary related to the geometry of the corresponding Teichmüller space. Besides that we consider the classical dual extremal problem which naturally arises in the tangent space of the Teichmüller space. In particular we prove the uniqueness of Hahn-Banach extension of the associated linear functional given on the Bergman space of the integer lattice domain. Several useful estimates related to the local and global properties of integrable meromorphic functions and the delta functional (see the definition below) are also obtained. These estimates are intended to study the behavior of integrable functions near singularities and they are valid in general settings.
Determinacy and weakly Ramsey sets in Banach spaces
Authors: Bagaria Joan , López-Abad Jordi
Published: July 2002

We give a sufficient condition for a set of block subspaces in an infinite-dimensional Banach space to be weakly Ramsey. Using this condition we prove that in the Levy-collapse of a Mahlo cardinal, every projective set is weakly Ramsey. This, together with a construction of W. H. Woodin, is used to show that the Axiom of Projective Determinacy implies that every projective set is weakly Ramsey. In the case of

c
0

c_0

we prove similar results for a stronger Ramsey property. And for hereditarily indecomposable spaces we show that the Axiom of Determinacy plus the Axiom of Dependent Choices imply that every set is weakly Ramsey. These results are the generalizations to the class of projective sets of some theorems from W. T. Gowers, and our paper “Weakly Ramsey sets in Banach spaces.”
Powers in Finitely Generated Groups
Authors: Hrushovski E. , Kropholler P. , Lubotzky A. , Shalev A.
Published: July 2002

In this paper we study the set

Γ
n

\Gamma ^n

of

n

t
h

n^{th}

-powers in certain finitely generated groups

Γ
\Gamma

. We show that, if

Γ
\Gamma

is soluble or linear, and

Γ
n

\Gamma ^n

contains a finite index subgroup, then

Γ
\Gamma

is nilpotent-by-finite. We also show that, if

Γ
\Gamma

is linear and

Γ
n

\Gamma ^n

has finite index (i.e.

Γ
\Gamma

may be covered by finitely many translations of

Γ
n

\Gamma ^n

), then

Γ
\Gamma

is soluble-by-finite. The proof applies invariant measures on amenable groups, number-theoretic results concerning the

S
S

-unit equation, the theory of algebraic groups and strong approximation results for linear groups in arbitrary characteristic.
Partially hyperbolic fixed points with constraints
Authors: Bonckaert Patrick
Published: July 2002

We investigate the local conjugacy, at a partially hyperbolic fixed point, of a diffeomorphism (vector field) to its normally linear part in the presence of constraints, where the change of variables also must satisfy the constraints. The main result is applied to vector fields respecting a singular foliation, encountered, by F. Dumortier and R. Roussarie, in the desingularization of families of vector fields.
Immersed 𝑛-manifolds in 𝐑²ⁿ and the double points of their generic projections into 𝐑²ⁿ⁻¹
Authors: Saeki Osamu , Sakuma Kazuhiro
Published: July 2002

We give two congruence formulas concerning the number of non-trivial double point circles and arcs of a smooth map with generic singularities—the Whitney umbrellas—of an

n
n

-manifold into

R

2
n
−
1

\mathbf {R}^{2n-1}

, which generalize the formulas by Szücs for an immersion with normal crossings. Then they are applied to give a new geometric proof of the congruence formula due to Mahowald and Lannes concerning the normal Euler number of an immersed

n
n

-manifold in

R

2
n

\mathbf {R}^{2n}

. We also study generic projections of an embedded

n
n

-manifold in

R

2
n

\mathbf {R}^{2n}

into

R

2
n
−
1

\mathbf {R}^{2n-1}

and prove an elimination theorem of Whitney umbrella points of opposite signs, which is a direct generalization of a recent result of Carter and Saito concerning embedded surfaces in

R

4

\mathbf {R}^{4}

. The problem of lifting a map into

R

2
n
−
1

\mathbf {R}^{2n-1}

to an embedding into

R

2
n

\mathbf {R}^{2n}

is also studied.
Structural properties of the one-dimensional drift-diffusion models for semiconductors
Authors: Alabau Fatiha
Published: July 2002

This paper is devoted to the analysis of the one-dimensional current and voltage drift-diffusion models for arbitrary types of semiconductor devices and under the assumption of vanishing generation recombination. We show in the course of this paper, that these models satisfy structural properties, which are due to the particular form of the coupling of the involved systems. These structural properties allow us to prove an existence and uniqueness result for the solutions of the current driven model together with monotonicity properties with respect to the total current

I
I

, of the electron and hole current densities and of the electric field at the contacts. We also prove analytic dependence of the solutions on

I
I

. These results allow us to establish several qualitative properties of the current voltage characteristic. In particular, we give the nature of the (possible) bifurcation points of this curve, we show that the voltage function is an analytic function of the total current and we characterize the asymptotic behavior of the currents for large voltages. As a consequence, we show that the currents never saturate as the voltage goes to

±
∞

\pm \infty

, contrary to what was predicted by numerical simulations by M. S. Mock (Compel. 1 (1982), pp. 165–174). We also analyze the drift-diffusion models under the assumption of quasi-neutral approximation. We show, in particular, that the reduced current driven model has at most one solution, but that it does not always have a solution. Then, we compare the full and the reduced voltage driven models and we show that, in general, the quasi-neutral approximation is not accurate for large voltages, even if no saturation phenomenon occurs. Finally, we prove a local existence and uniqueness result for the current driven model in the case of small generation recombination terms.
On the Poles of Rankin-Selberg Convolutions of Modular Forms
Authors: Li Xian-jin
Published: July 2002

The Rankin-Selberg convolution is usually normalized by the multiplication of a zeta factor. One naturally expects that the non-normalized convolution will have poles where the zeta factor has zeros, and that these poles will have the same order as the zeros of the zeta factor. However, this will only happen if the normalized convolution does not vanish at the zeros of the zeta factor. In this paper, we prove that given any point inside the critical strip, which is not equal to

1
2

\frac {1}{2}

and is not a zero of the Riemann zeta function, there exist infinitely many cusp forms whose normalized convolutions do not vanish at that point.
Extremal problems and symmetrization for plane ring domains
Authors: Solynin A. , Vuorinen M.
Published: July 2002

We show that Teichmüller’s classical lower bound for the capacity of a ring domain, obtained by circular symmetrization, can be replaced by an explicit one which is almost always better. The proof is based on a duplication formula for the solution of an associated extremal problem. Some inequalities are obtained for conformal invariants.
Propagation of Gevrey Regularity for a Class of Hypoelliptic Equations
Authors: Bove Antonio , Tartakoff David
Published: July 2002

We prove results on the propagation of Gevrey and analytic wave front sets for a class of

C
∞

C^\infty

hypoelliptic equations with double characteristics.
On complete nonorientable minimal surfaces with low total curvature
Authors: Lopez Francisco
Published: July 2002

We classify complete nonorientable minimal surfaces in

R

\mathbb {R}

with total curvature

−
8
π

-8\pi

.
On biunitary permutation matrices and some subfactors of index 9
Authors: Krishnan Uma , Sunder V.
Published: July 2002

This paper is devoted to a study of the subfactors arising from vertex models constructed out of ‘biunitary’ matrices which happen to be permutation matrices. After a discussion on the computation of the higher relative commutants of the associated subfactor (in the members of the tower of Jones’ basic construction), we discuss the principal graphs of these subfactors for small sizes (

N
=
k
≤
3

N=k \leq 3

) of the vertex model. Of the 18 possibly inequivalent such biunitary matrices when

N
=
3

N = 3

, we compute the principal graphs completely in 15 cases, all of which turn out to be finite. In the last section, we prove that two of the three remaining cases lead to subfactors of infinite depth and discuss their principal graphs.
An Application of Convex Integration to Contact Geometry
Authors: Geiges Hansjörg , Gonzalo Jesús
Published: July 2002

We prove that every closed, orientable

3
3

-manifold

M
M

admits a parallelization by the Reeb vector fields of a triple of contact forms with equal volume form. Our proof is based on Gromov’s convex integration technique and the

h
h

-principle. Similar methods can be used to show that

M
M

admits a parallelization by contact forms with everywhere linearly independent Reeb vector fields. We also prove a generalization of this latter result to higher dimensions. If

M
M

is a closed

(
2
n
+
1
)

(2n+1)

-manifold with contact form

ω
\omega

whose contact distribution

ker
⁡
ω

\ker \omega

admits

k
k

everywhere linearly independent sections, then

M
M

admits

k
+
1

k+1

linearly independent contact forms with linearly independent Reeb vector fields.
Boundary and Lens Rigidity of Lorentzian Surfaces
Authors: Andersson Lars , Dahl Mattias , Howard Ralph
Published: July 2002

Let

g
g

be a Lorentzian metric on the plane

R

2

\mathbb {R}^2

that agrees with the standard metric

g
0

=
−
d

x
2

+
d

y
2

g_0=-dx^2+dy^2

outside a compact set and so that there are no conjugate points along any time-like geodesic of

(

R

2

,
g
)

(\mathbb {R}^2,g)

. Then

(

R

2

,
g
)

(\mathbb {R}^2,g)

and

(

R

2

,

g
0

)

(\mathbb {R}^2,g_0)

are isometric. Further, if

(
M
,
g
)

(M,g)

and

(

M
∗

,

g
∗

)

(M^*,g^*)

are two dimensional compact time oriented Lorentzian manifolds with space–like boundaries and so that all time-like geodesics of

(
M
,
g
)

(M,g)

maximize the distances between their points and

(
M
,
g
)

(M,g)

and

(

M
∗

,

g
∗

)

(M^*,g^*)

are “boundary isometric”, then there is a conformal diffeomorphism between

(
M
,
g
)

(M,g)

and

(

M
∗

,

g
∗

)

(M^*,g^*)

and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.
𝑝-adic Power Series which Commute under Composition
Authors: Li Hua-Chieh
Published: July 2002

When two noninvertible series commute to each other, they have same set of roots of iterates. Most of the results of this paper will be concerned with the problem of which series commute with a given noninvertible series. Our main theorem is a generalization of Lubin’s result about isogenies of formal groups.
Sums of Three or More Primes
Authors: Friedlander J. , Goldston D.
Published: July 2002

It has long been known that, under the assumption of the Riemann Hypothesis, one can give upper and lower bounds for the error

∑

p
≤
x

log
⁡
p
−
x

\sum _{p \le x} \log p - x

in the Prime Number Theorem, such bounds being within a factor of

(
log
⁡
x

)

2

(\log x)^{2}

of each other and this fact being equivalent to the Riemann Hypothesis. In this paper we show that, provided “Riemann Hypothesis” is replaced by “Generalized Riemann Hypothesis”, results of similar (often greater) precision hold in the case of the corresponding formula for the representation of an integer as the sum of

k
k

primes for

k
≥
4

k \ge 4

, and, in a mean square sense, for

k
≥
3

k \ge 3

. We also sharpen, in most cases to best possible form, the original estimates of Hardy and Littlewood which were based on the assumption of a “Quasi-Riemann Hypothesis”. We incidentally give a slight sharpening to a well-known exponential sum estimate of Vinogradov-Vaughan.
Menas’ Result is Best Possible
Authors: Apter Arthur , Shelah Saharon
Published: July 2002

Generalizing some earlier techniques due to the second author, we show that Menas’ theorem which states that the least cardinal

κ
\kappa

which is a measurable limit of supercompact or strongly compact cardinals is strongly compact but not

2

κ

2^{\kappa }

supercompact is best possible. Using these same techniques, we also extend and give a new proof of a theorem of Woodin and extend and give a new proof of an unpublished theorem due to the first author.
Decomposition of Birational Toric Maps in Blow-Ups and Blow-Downs
Authors: Wlodarczyk Jaroslaw
Published: July 2002

We prove that a toric birational map between two complete smooth toric varieties of the same dimension can be decomposed in a sequence of equivariant blow-ups and blow-downs along smooth centers.
Hilbert-Kunz functions and Frobenius functors
Authors: Chang Shou-Te
Published: July 2002

We study the asymptotic behavior as a function of

e
e

of the lengths of the cohomology of certain complexes. These complexes are obtained by applying the

e
e

-th iterated Frobenius functor to a fixed finite free complex with only finite length cohomology and then tensoring with a fixed finitely generated module. The rings involved here all have positive prime characteristic. For the zeroth homology, these functions also contain the class of Hilbert-Kunz functions that a number of other authors have studied. This asymptotic behavior is connected with certain intrinsic dimensions introduced in this paper: these are defined in terms of the Krull dimensions of the Matlis duals of the local cohomology of the module. There is a more detailed study of this behavior when the given complex is a Koszul complex.
On the Faber coefficients of functions univalent in an ellipse
Authors: Haliloglu E.
Published: July 2002

Let

E
E

be the elliptical domain
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E equals StartSet x plus i y colon StartFraction x squared Over left-parenthesis 5 slash 4 right-parenthesis squared EndFraction plus StartFraction y squared Over left-parenthesis 3 slash 4 right-parenthesis squared EndFraction greater-than 1 EndSet period">

E
=
{
x
+
i
y
:

x

2

(
5

/

4

)

2

+

y

2

(
3

/

4

)

2

>
1
}
.

E=\{x+iy: \frac {x^{2}}{(5/4)^{2}}+ \frac {y^{2}}{(3/4)^{2}}>1 \}.

\]
Let

S
(
E
)

S(E)

denote the class of functions

F
(
z
)

F(z)

analytic and univalent in

E
E

and satisfying the conditions

F
(
0
)
=
0

F(0)=0

and

F
′

(
0
)
=
1

F’(0)=1

. In this paper, we obtain global sharp bounds for the Faber coefficients of the functions

F
(
z
)

F(z)

in certain related classes and subclasses of

S
(
E
)
.

S(E).
On roots of random polynomials
Authors: Ibragimov Ildar , Zeitouni Ofer
Published: July 2002

We study the distribution of the complex roots of random polynomials of degree

n
n

with i.i.d. coefficients. Using techniques related to Rice’s treatment of the real roots question, we derive, under appropriate moment and regularity conditions, an exact formula for the average density of this distribution, which yields appropriate limit average densities. Further, using a different technique, we prove limit distribution results for coefficients in the domain of attraction of the stable law.
On the second adjunction mapping. The case of a 1-dimensional image
Authors: Beltrametti Mauro , Sommese Andrew
Published: July 2002

Let

L
^

\widehat {L}

be a very ample line bundle on an

n
n

-dimensional projective manifold

X
^

\widehat {X}

, i.e., assume that

L
^

≈

i
∗

O

P

N

(
1
)

\widehat {L}\approx i^*\mathcal {O}_{\mathbb {P}_ N}(1)

for some embedding

i
:

X
^

↪

P

N

i:\widehat {X}\hookrightarrow \mathbb {P}_ N

. In this article, a study is made of the meromorphic map,

φ
^

:

X
^

→
Σ

\widehat {\varphi } : \widehat {X}\to \Sigma

, associated to

|

K

X
^

+
(
n
−
2
)

L
^

|

|K_{\widehat {X}}+(n-2)\widehat {L}|

in the case when the Kodaira dimension of

K

X
^

+
(
n
−
2
)

L
^

K_{\widehat {X}}+(n-2)\widehat {L}

is

≥
3

\ge 3

, and

φ
^

\widehat {\varphi }

has a

1
1

-dimensional image. Assume for simplicity that

n
=
3

n=3

. The first main result of the paper shows that

φ
^

\widehat \varphi

is a morphism if either

h
0

(

K

X
^

+

L
^

)
≥
7

h^0(K_{\widehat X}+\widehat L)\geq 7

or

κ
(

X
^

)
≥
0

\kappa (\widehat {X})\geq 0

. The second main result of this paper shows that if

κ
(

X
^

)
≥
0

\kappa (\widehat X)\ge 0

, then the genus,

g
(
f
)

g(f)

, of a fiber,

f
f

, of the map induced by

φ
^

\widehat \varphi

on hyperplane sections is

≤
6

\leq 6

. Moreover, if

h
0

(

K

X
^

+

L
^

)
≥
21

h^0(K_{\widehat X}+\widehat L)\ge 21

then

g
(
f
)
≤
5

g(f)\leq 5

, a connected component

F
F

of a general fiber of

φ
^

\widehat \varphi

is either a

K
3

K3

surface or the blowing up at one point of a

K
3

K3

surface, and

h
1

(

O

X
^

)
≤
1

h^1(\mathcal {O}_{\widehat X})\le 1

. Finally the structure of the finite to one part of the Remmert-Stein factorization of

φ
^

\widehat \varphi

is worked out.
Nonexistence and uniqueness of positive solutions of Yamabe type equations on nonpositively curved manifolds
Authors: Bianchini Bruno , Rigoli Marco
Published: July 2002

We prove nonexistence and uniqueness of positive

C

2

C^{2}

–solutions of the elliptic equation

Δ
u
+
a
(
x
)
u
−
K
(
x
)

u

σ

=
0

\Delta u +a(x)u - K(x)u^{\sigma }=0

,

σ
>
1

\sigma >1

, on a nonpositively curved, complete manifold

(
M
,
g
)

(M,g)

.
The image of the 𝐵𝑃 Thom map for Eilenberg-MacLane spaces
Authors: Tamanoi Hirotaka
Published: July 2002

Fundamental classes in

B
P

BP

cohomology of Eilenberg–Mac Lane spaces are defined. The image of the Thom map from

B
P

BP

cohomology to mod-

p
p

cohomology is determined for arbitrary Eilenberg–Mac Lane spaces. This image is a polynomial subalgebra generated by infinitely many elements obtained by applying a maximum number of Milnor primitives to the fundamental class in mod-

p
p

cohomology. This subalgebra in mod

p
p

cohomology is invariant under the action of the Steenrod algebra, and it is annihilated by all Milnor primitives. We also show that

B
P

BP

cohomology determines Morava

K
K

cohomology for Eilenberg–Mac Lane spaces.
Contractions on a manifold polarized by an ample vector bundle
Authors: Andreatta Marco , Mella Massimiliano
Published: July 2002

A complex manifold

X
X

of dimension

n
n

together with an ample vector bundle

E
E

on it will be called a generalized polarized variety. The adjoint bundle of the pair

(
X
,
E
)

(X,E)

is the line bundle

K
X

+
d
e
t
(
E
)

K_X + det(E)

. We study the positivity (the nefness or ampleness) of the adjoint bundle in the case

r
:=
r
a
n
k
(
E
)
=
(
n
−
2
)

r := rank (E) = (n-2)

. If

r
≥
(
n
−
1
)

r\geq (n-1)

this was previously done in a series of papers by Ye and Zhang, by Fujita, and by Andreatta, Ballico and Wisniewski.

If

K
X

+
d
e
t
E

K_X+detE

is nef then, by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map

π
:
X
⟶
W

\pi :X \longrightarrow W

from

X
X

onto a normal projective variety

W
W

with connected fiber and such that

K
X

+
d
e
t
(
E
)
=

π
∗

H

K_X + det(E) = \pi ^*H

, for some ample line bundle

H
H

on

W
W

. We describe those contractions for which

d
i
m
F
≤
(
r
−
1
)

dimF \leq (r-1)

. We extend this result to the case in which

X
X

has log terminal singularities. In particular this gives Mukai’s conjecture 1 for singular varieties. We consider also the case in which

d
i
m
F
=
r

dimF = r

for every fiber and

π
\pi

is birational.
A Lie theoretic Galois theory for the spectral curves of an integrable system. II
Authors: McDaniel Andrew , Smolinsky Lawrence
Published: July 2002

In the study of integrable systems of ODE’s arising from a Lax pair with a parameter, the constants of the motion occur as spectral curves. Many of these systems are algebraically completely integrable in that they linearize on the Jacobian of a spectral curve. In an earlier paper the authors gave a classification of the spectral curves in terms of the Weyl group and arranged the spectral curves in a hierarchy. This paper examines the Jacobians of the spectral curves, again exploiting the Weyl group action. A hierarchy of Jacobians will give a basis of comparison for flows from various representations. A construction of V. Kanev is generalized and the Jacobians of the spectral curves are analyzed for abelian subvarieties. Prym-Tjurin varieties are studied using the group ring of the Weyl group

W
W

and the Hecke algebra of double cosets of a parabolic subgroup of

W
.

W.

For each algebra a subtorus is identified that agrees with Kanev’s Prym-Tjurin variety when his is defined. The example of the periodic Toda lattice is pursued.
The Mizohata complex
Authors: Meziani Abdelhamid
Published: July 2002

This paper deals with the local solvability of systems of first order linear partial differential equations defined by a germ

ω
\omega

at

0
∈

R

n
+
1

0\in \mathbb {R}^{n+1}

of a

C

\mathbb {C}

-valued, formally integrable (

ω
∧
d
ω
=
0

\omega \wedge d\omega =0

), 1-form with nondegenerate Levi form. More precisely, the size of the obstruction to the solvability, for

(
q
−
1
)

(q-1)

-forms

u
u

, of the equation

d
u
∧
ω
=
η
∧
ω
,

\begin{equation*}du\wedge \omega =\eta \wedge \omega ,\end{equation*}

where

η
\eta

is a given

q
q

-form satisfying

d
η
∧
ω
=
0

d\eta \wedge \omega =0

is estimated in terms of the De Rham cohomology relative to

ω
\omega
An index formula for elliptic systems in the plane
Authors: Rowley B.
Published: July 2002

An index formula is proved for elliptic systems of P.D.E.’s with boundary values in a simply connected region

Ω
\Omega

in the plane. Let

A

\mathcal {A}

denote the elliptic operator and

B

\mathcal {B}

the boundary operator. In an earlier paper by the author, the algebraic condition for the Fredholm property, i.e. the Lopatinskii condition, was reformulated as follows. On the boundary, a square matrix function

Δ

B

+

\Delta ^{+}_{{\mathcal {B}}}

defined on the unit cotangent bundle of

∂
Ω

\partial \Omega

was constructed from the principal symbols of the coefficients of the boundary operator and a spectral pair for the family of matrix polynomials associated with the principal symbol of the elliptic operator. The Lopatinskii condition is equivalent to the condition that the function

Δ

B

+

\Delta ^{+}_{{\mathcal {B}}}

have invertible values. In the present paper, the index of

(

A

,

B

)

({\mathcal {A}},{\mathcal {B}})

is expressed in terms of the winding number of the determinant of

Δ

B

+

\Delta ^{+}_{{\mathcal {B}}}

.
Order evaluation of products of subsets in finite groups and its applications. II
Authors: Arad Z. , Muzychuk M.
Published: July 2002

In this paper we give a new estimate of the cardinality of the product of subsets

A
B

AB

in a finite non-abelian simple group, where

A
A

is normal and

B
B

is arbitrary. This estimate improves the one given in J. Algebra 182 (1996), 577–603.
The transfer and symplectic cobordism
Authors: Bakuradze Malkhaz
Published: July 2002

The main result of this paper is the nilpotency fomulaϕi4=0\phi _{i}^{4} =0,∀i≥1\forall i\geq 1for N. Ray classesϕi\phi _{i}in the torsion of the symplectic bordism ringMSp∗MSp_{*}
𝐶*-extreme points in the generalised state spaces of a 𝐶*-algebra
Authors: Farenick Douglas , Morenz Phillip
Published: July 2002

In this paper we study the space

S

H

(
A
)

S_{H}(A)

of unital completely positive linear maps from a

C

∗

C^{*}

-algebra

A
A

to the algebra

B
(
H
)

B(H)

of continuous linear operators on a complex Hilbert space

H
H

. The state space of

A
A

, in this notation, is

S

C

(
A
)

S_{\mathbb {C}}(A)

. The main focus of our study concerns noncommutative convexity. Specifically, we examine the

C

∗

C^{*}

-extreme points of the

C

∗

C^{*}

-convex space

S

H

(
A
)

S_{H}(A)

. General properties of

C

∗

C^{*}

-extreme points are discussed and a complete description of the set of

C

∗

C^{*}

-extreme points is given in each of the following cases: (i) the cases

S

C

2

(
A
)

S_{{\mathbb {C}}^{2}}(A)

, where

A
A

is arbitrary ; (ii) the cases

S

C

r

(
A
)

S_{{\mathbb {C}}^{r}}(A)

, where

A
A

is commutative; (iii) the cases

S

C

r

(

M

n

)

S_{{\mathbb {C}}^{r}}(M_{n})

, where

M

n

M_{n}

is the

C

∗

C^{*}

-algebra of

n
×
n

n\times n

complex matrices. An analogue of the Krein-Milman theorem will also be established.