bluuebunny/crossref_metadata_2025 huggingface dataset:
Connected algebraic monoids
Authors: Putcha Mohan S.
Published: June 2010

Let

S
S

be a connected algebraic monoid with group of units

G
G

and lattice of regular

J

\mathcal {J}

-classes

U

(
S
)

\mathcal {U}(S)

. The connection between the solvability of

G
G

and the semilattice decomposition of

S
S

into archimedean semigroups is further elaborated. If

S
S

has a zero and if

U

(
S
)
≤
7

\mathcal {U}(S)\le 7

, then it is shown that

G
G

is solvable if and only if

U

(
S
)

\mathcal {U}(S)

is relatively complemented. If

J
∈

U

(
S
)

J\in \mathcal {U}(S)

, then we introduce two basic numbers

θ
(
J
)

\theta (J)

and

δ
(
J
)

\delta (J)

and study their properties. Crucial to this process is the theorem that for any indempotent

e
e

of

S
S

, the centralizer of

e
e

in

G
G

is connected. Connected monoids with central idempotents are also studied. A conjecture about their structure is forwarded. It is pointed out that the maximal connected submonoids of

S
S

with central idempotents need not be conjugate. However special maximal connected submonoids with central idempotents are conjugate. If

S
S

is regular, then

S
S

is a Clifford semigroup if and only if for all

f
∈
E
(
S
)

f\in E(S)

, the set

{
e

|

e
∈
E
(
S
)
,

e
≥
f
}

\{ e|e \in E(S),\,e \geq f\}

is finite. Finally the maximal semilattice image of any connected monoid is determined.
A symplectic Banach space with no Lagrangian subspaces
Authors: Kalton N. J. , Swanson R. C.
Published: June 2010

In this paper we construct a symplectic Banach space

(
X
,
Ω
)

(X,\Omega )

which does not split as a direct sum of closed isotropic subspaces. Thus, the question of whether every symplectic Banach space is isomorphic to one of the canonical form

Y
×

Y
∗

Y \times {Y^ \ast }

is settled in the negative. The proof also shows that

L

(
X
)

\mathfrak {L}(X)

admits a nontrivial continuous homomorphism into

L

(
H
)

\mathfrak {L}(H)

where

H
H

is a Hilbert space.
Convergence acceleration for continued fractions 𝐾(𝑎_{𝑛}/1)
Authors: Jacobsen Lisa
Published: June 2010

A known method for convergence acceleration of limit periodic continued fractions

K
(

a
n

/

1
)
,

a
n

→
a

K({a_n}/1),{a_n} \to a

, is to replace the approximants

S
n

(
0
)

{S_n}(0)

by "modified approximants"

S
n

(

f

∗

)

{S_n}({f^{\ast }})

, where

f

∗

=
K
(
a

/

1
)

f^{\ast } = K(a/1)

. The present paper extends this idea to a larger class of converging continued fractions. The "modified approximants" will then be

S
n

(

f

(
n

)
′

)

{S_n}({f^{(n)’}})

, where

K
(

a
n
′

/

1
)

K({a’_n}/1)

is a converging continued fraction whose tails

f

(
n
)
′

{f^{(n)\prime }}

are all known, and where

a
n

−

a
n
′

→
0

{a_n} - a_n^\prime \to 0

. As a measure for the improvement obtained by this method, upper bounds for the ratio of the two truncation errors are found.
Linking numbers and the elementary ideals of links
Authors: Traldi Lorenzo
Published: June 2010

Let

L
=

K
1

∪

⋯
∪

K
μ

⊆

S
3

L = {K_1} \cup \, \cdots \cup {K_\mu } \subseteq {S^3}

be a tame link of

μ
⩾
2

\mu \geqslant 2

components, and

H
H

the abelianization of

G
=

π
1

(

S
3

−
L
)

G = {\pi _1}({S^3} - L)

. Let

L

=
(

L

i
j

)

\mathcal {L} = ({\mathcal {L}_{ij}})

be the

μ
×
μ

\mu \times \mu

matrix with entries in

Z

H

\mathbf {Z}H

given by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L Subscript i i Baseline equals sigma-summation Underscript k not-equals i Endscripts l left-parenthesis upper K Subscript i Baseline comma upper K Subscript k Baseline right-parenthesis dot left-parenthesis t Subscript k Baseline minus 1 right-parenthesis">

L

i
i

=

∑

k
≠
i

l
(

K
i

,

K
k

)
⋅
(

t
k

−
1
)

\mathcal {L}{_{ii}} = \sum \nolimits _{k \ne i} {l({K_i},{K_k}) \cdot ({t_k} - 1)}

and for

i
≠
j

L

i
j

=
l
(

K
i

,

K
j

)
⋅
(
1
−

t
i

)

i \ne j\,{\mathcal {L}_{ij}} = l({K_i},{K_j}) \cdot (1 - {t_i})

. Then if

0
>
k
>
μ

0 > k > \mu

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript i equals 0 Overscript k minus 1 Endscripts upper E Subscript mu minus k plus i Baseline left-parenthesis upper L right-parenthesis dot left-parenthesis upper I upper H right-parenthesis Superscript 2 i Baseline plus left-parenthesis upper I upper H right-parenthesis Superscript 2 k Baseline equals sigma-summation Underscript i equals 0 Overscript k minus 1 Endscripts upper E Subscript mu minus k plus i Baseline left-parenthesis script upper L right-parenthesis dot left-parenthesis upper I upper H right-parenthesis Superscript 2 i Baseline plus left-parenthesis upper I upper H right-parenthesis Superscript 2 k">

∑

i
=
0

k
−
1

E

μ
−
k
+
i

(
L
)
⋅

(
I
H
)

2
i

+

(
I
H
)

2
k

=

∑

i
=
0

k
−
1

E

μ
−
k
+
i

(

L

)
⋅

(
I
H
)

2
i

+

(
I
H
)

2
k

\sum \limits _{i = 0}^{k - 1} {{E_{\mu - k + i}}(L) \cdot {{(IH)}^{2i}} + {{(IH)}^{2k}} = \sum \limits _{i = 0}^{k - 1} {{E_{\mu - k + i}}(\mathcal {L}) \cdot {{(IH)}^{2i}} + {{(IH)}^{2k}}} }

\]
Various consequences of this equality are derived, including its application to the reduced elementary ideals. These results are used to give several different characterizations of links in which all the linking numbers are zero.
Skewness in Banach spaces
Authors: Fitzpatrick Simon , Reznick Bruce
Published: June 2010

Let

E
E

be a Banach space. One often wants to measure how far

E
E

is from being a Hilbert space. In this paper we define the skewness

s
(
E
)

s(E)

of a Banach space

E
E

,

0
⩽
s
(
E
)
⩽
2

0 \leqslant s(E) \leqslant 2

, which describes the asymmetry of the norm. We show that

s
(
E
)
=
s
(

E

∗

)

s(E) = s({E^{\ast }})

for all Banach spaces

E
E

. Further,

s
(
E
)
=
0

s(E) = 0

if and only if

E
E

is a (real) Hilbert space and

s
(
E
)
=
2

s(E) = 2

if and only if

E
E

is quadrate, so

s
(
E
)
>
2

s(E) > 2

implies

E
E

is reflexive. We discuss the computation of

s
(

L
p

)

s({L^p})

and describe its asymptotic behavior near

p
=
1
,
2

p = 1,2

and

∞
\infty

. Finally, we discuss a higher-dimensional generalization of skewness which gives a characterization of smooth Banach spaces.
Dehn surgery and satellite knots
Authors: Gordon C. McA.
Published: June 2010

For certain kinds of

3
3

-manifolds, the question whether such a manifold can be obtained by nontrivial Dehn surgery on a knot in

S
3

{S^3}

is reduced to the corresponding question for hyperbolic knots. Examples are, whether one can obtain

S
3

{S^3}

, a fake

S
3

{S^3}

, a fake

S
3

{S^3}

with nonzero Rohlin invariant,

S
1

×

S
2

{S^1} \times {S^2}

, a fake

S
1

×

S
2

,

S
1

×

S
2

#
M

{S^1} \times {S^2}, {S^1} \times {S^2} \# M

with

M
M

nonsimply-connected, or a fake lens space.
AF algebras with directed sets of finite-dimensional *-subalgebras
Authors: Lazar Aldo J.
Published: June 2010

We characterize the unital

A
F

AF

algebras whose families of finite dimensional

∗

^{\ast }

-subalgebras are directed by inclusion. A representation theorem for the algebras of this class allows us to classify them up to

∗

^{\ast }

-isomorphisms.
Measurable representations of preference orders
Authors: Mauldin R. Daniel
Published: June 2010

A continuous preference order on a topological space

Y
Y

is a binary relation

≼
\preccurlyeq

which is reflexive, transitive and complete and such that for each

x
,
{
y
:
x
≼
y
}

x,\{y:x \preccurlyeq y\}

and

{
y
:
y
≼
x
}

\{y:y \preccurlyeq x\}

are closed. Let

T
T

and

X
X

be complete separable metric spaces. For each

t
t

in

T
T

, let

B
t

{B_t}

be a nonempty subset of

X
X

, let

≼
t

{ \preccurlyeq _t}

be a continuous preference order on

B
t

{B_t}

and suppose

E
=
{
(
t
,
x
,
y
)
:
x

≼
t

y
}

E = \{(t,x,y): x{ \preccurlyeq _t}y\}

is a Borel set. Let

B
=
{
(
t
,
x
)
:
x
∈

B
t

}

B = \{(t,x):x \in {B_t}\}

. Theorem 1. There is an

S

(
T
)
⊗

B

(
X
)

\mathcal {S}(T) \otimes \mathcal {B}(X)

-measurable map

g
g

from

B
B

into

R
R

so that for each

t
,
g
(
t
,
⋅
)

t,g(t,\cdot )

is a continuous map of

B
t

{B_t}

into

R
R

and

g
(
t
,
x
)
⩽
g
(
t
,
y
)

g(t,x) \leqslant g(t,y)

if and only if

x

≼
t

y

x{ \preccurlyeq _t}y

. (Here

S

(
T
)

\mathcal {S}(T)

forms the

C
C

-sets of Selivanovskii and

B

(
X
)

\mathcal {B}(X)

is a Borel field on

X
X

.) Theorem 2. If for each

t
,

B
t

t,{B_t}

is a

σ
\sigma

-compact subset of

Y
Y

, then the map

g
g

of the preceding theorem may be chosen to be Borel measurable. The following improvement of a theorem of Wesley is proved using classical methods. Theorem 3. Let

g
g

be the map constructed in Theorem 1. If

μ
\mu

is a probability measure defined on the Borel subsets of

T
T

, then there is a Borel set

N
N

such that

μ
(
N
)
=
0

\mu (N) = 0

and such that the restriction of

g
g

to

B
∩
(
(
T
−
N
)
×
X
)

B \cap ((T - N) \times X)

is Borel measurable.
Rigidity of pseudoconformal connections
Authors: Markowitz Michael , Schlafly Roger
Published: June 2010

Let

M

2
n
−
1

(
n
⩾
3
)

{M^{2n - 1}}(n \geqslant 3)

be a strictly pseudoconvex abstract

CR

{\text {CR}}

-hypersurface

CR

{\text {CR}}

-immersed in the unit sphere in

C

N

{{\mathbf {C}}^N}

. We show that the pseudoconformal connection induced on

M
M

by the standard flat connection agrees with the intrinsic normal connection of Cartan-Chern-Tanaka if and only if

M
M

is pseudoconformally flat. In this case

M
M

is a piece of the transverse intersection of

S

2
N
−
1

{S^{2N - 1}}

with a complex

n
n

-plane in

C

N

{{\mathbf {C}}^N}

.
Biholomorphic invariants of a hyperbolic manifold and some applications
Authors: Fridman B. L.
Published: June 2010

A biholomorphically invariant real function

h
x

{h_x}

is defined for a hyperbolic manifold

X
X

. Properties of such functions are studied. These properties are applied to prove the following theorem. If a hyperbolic manifold

X
X

can be exhausted by biholomorphic images of a strictly pseudoconvex domain

D
⊂

C

n

D \subset {{\mathbf {C}}^n}

with

∂
D

∈

C
3

\partial D\; \in \;{C^3}

, then

X
X

is biholomorphically equivalent either to

D
D

or to the unit ball in

C

n

{{\mathbf {C}}^n}

. The properties of

h
D

{h_D}

are also applied to some questions concerning the group of analytical automorphisms of a strictly pseudoconvex domain and to similar questions concerning polyhedra.
Constructing approximate fibrations
Authors: Chapman T. A. , Ferry Steve
Published: June 2010

In this paper two results concerning the construction of approximate fibrations are established. The first shows that there are approximate fibrations

p
:
M
→

S
2

p:M \to S^2

which are homotopic to bundle maps but which cannot be approximated by bundle maps. Here

M
M

can be a compact

Q
Q

-manifold or some topological

n
n

-manifold,

n
⩾
5

n \geqslant 5

. The second shows how to construct approximate fibrations

p
:
M
→
B

p:M \to B

whose fibers do not have finite homotopy type, for any

B
B

of Euler characteristic zero. Here

M
M

can be a compact

Q
Q

-manifold and

B
B

only has to be an ANR, or

M
M

can be an

n
n

-manifold,

n
⩾
6

n \geqslant 6

, and

B
B

must then also be a topological manifold.
On derivations of certain algebras related to irreducible triangular algebras
Authors: Solel Baruch
Published: June 2010

This paper deals with derivations on algebras that are generated by a maximal abelian selfadjoint algebra of operators

A

\mathcal {A}

on a Hilbert space and a group of unitary operators acting on it. A necessary and sufficient condition for such a derivation to be implemented by an operator affiliated with

A

\mathcal {A}

is given. The results are related to the study of derivations on a certain class of irreducible triangular algebras.
Semistability at the end of a group extension
Authors: Mihalik Michael L.
Published: June 2010

A

1
1

-ended

CW

{\text {CW}}

-complex,

Q
Q

, is semistable at

∞
\infty

if all proper maps

r
:
 
[
0
,
∞
)
→
Q

r:\ [0,\infty ) \to Q

are properly homotopic. If

X
1

{X_1}

and

X
2

{X_2}

are finite

CW

{\text {CW}}

-complexes with isomorphic fundamental groups, then the universal cover of

X
1

{X_1}

is semistable at

∞
\infty

if and only if the universal cover of

X
2

{X_2}

is semistable at

∞
\infty

. Hence, the notion of a finitely presented group being semistable at

∞
\infty

is well defined. We prove Main Theorem. Let

1
→
H
→
G
→
K
→
1

1 \to H \to G \to K \to 1

be a short exact sequence of finitely generated infinite groups. If

G
G

is finitely presented, then

G
G

is semistable at

∞
\infty

. Theorem. If

A
A

and

B
B

are locally compact, connected noncompact

C
W

CW

-complexes, then

A
×
B

A \times B

is semistable at

∞
\infty

. Theorem.

⟨

x
,
y
:
x

y
b

x

−
1

=

y
c

;
b

a
n
d

c

n
o
n
z
e
r
o

i
n
t
e
g
e
r
s

⟩

\langle \;x,y:x{y^b}{x^{ - 1}} = {y^c};b\; and \; c \; nonzero\; integers\; \rangle

is semistable at

∞
\infty

. The proofs are geometrical in nature and the main tool is covering space theory.
Meromorphic functions that share four values
Authors: Gundersen Gary G.
Published: June 2010

An old theorem of

R

{\text {R}}

. Nevanlinna states that if two distinct nonconstant meromorphic functions share four values counting multiplicities, then the functions are Möbius transformations of each other, two of the shared values are Picard values for both functions, and the cross ratio of a particular permutation of the shared values equals -1. In this paper we show that if two nonconstant meromorphic functions share two values counting multiplicities and share two other values ignoring multiplicities, then the functions share all four values counting multiplicities.
Stable orbits of differentiable group actions
Authors: Stowe Dennis
Published: June 2010

We prove that a compact orbit of a smooth Lie group action is stable provided the first cohomology space vanishes for the normal representation at some (equivalently, every) point of the orbit. When the orbit is a single point, the acting group need only be compactly generated and locally compact for this conclusion to hold. Applied to foliations, this provides a sufficient condition for the stability of a compact leaf and includes the stability theorems of Reeb and Thurston and of Hirsch as cases.
Gauss sums and Fourier analysis on multiplicative subgroups of 𝑍_{𝑞}
Authors: Diamond Harold G. , Gerth Frank , Vaaler Jeffrey D.
Published: June 2010

LetG(q)G(q)denote the multiplicative group of invertible elements inZq{{\mathbf {Z}}_q}, the ring of integers moduloqq. LetH⊆G(q)H \subseteq G(q)be a multiplicative subgroup with cosetsaHaHandbHbH. Iff: Zq→Cf:\ {\mathbf {Z}}_q \to {\mathbf {C}}is supported inaHaHwe show thatffcan be recovered from the values off^\hat frestricted tobHbHif and only if Gauss sums forHHare nonvanishing. Heref^\hat fis the (finite) Fourier transform offfwith respect to the additive groupZq{{\mathbf {Z}}_q}. The main result is a simple criterion for deciding when these Gauss sums are nonvanishing. IfH=G(q)H = G(q)thenffcan be recovered fromf^\hat frestricted toG(q)G(q)by a particularly elementary formula. This formula provides some inequalities and extremal functions.
On the rate of convergence of moments in the central limit theorem for lattice distributions
Authors: Hall Peter
Published: June 2010

We derive sharp asymptotic expressions for moments of the type

E
{
b
(

|

S
n

/

n

1

/

2

|

)
}

E\{ b(|{S_n}/{n^{1/2}}|)\}

, where

S
n

{S_n}

is a sum of independent lattice-valued random variables with finite variance, and

b
b

is a concave function. It is shown that the behaviour of

b
b

at the origin has a profound effect on the behaviour of such moments, and that this influence accounts for the major difference between the properties of moments of lattice and nonlattice sums. Asymptotic expansions for moments of sums of lattice-valued variables are also derived.
Bounds for integral solutions of diagonal cubic equations
Authors: Leung Ka Hin
Published: June 2010

It was proved by Davenport [3] that for the nonzero integral

λ
i

{\lambda _i}

the equation

λ
1

x
1
3

+
⋯
+

λ
8

x
8
3

=
0

{\lambda _1}x_1^3 + \cdots + {\lambda _8}x_8^3 = 0

always has a nontrivial integral solution. In this paper, we investigate the bounds of nontrivial integral solutions in terms of

λ
1

,
…
,

λ
8

{\lambda _1}, \ldots ,{\lambda _8}

.
Equivariant vector fields on spheres
Authors: Namboodiri Unni
Published: June 2010

We address the following question: If

G
G

is a compact Lie group and

S
(
M
)

S(M)

is the unit sphere of an

R
[
G
]

R[G]

-module

M
M

, then how many orthonormal

G
G

-invariant vector fields can be found on

S
(
M
)

S(M)

? We call this number the

G
G

-field number of

M
M

. Under reasonable hypotheses on

M
M

, we reduce this question to determining when the difference of two

G
G

-vector bundles vanishes in a certain subquotient of the

K

O
G

K{O_G}

-theory of a real projective space. In this general setting, we solve the problem for

2
2

-groups, for odd-order groups, and for abelian groups. If

M
M

also has "enough" orbit types (for example, all of them), then we solve the problem for arbitrary finite groups. We also show that under mild hypotheses on

M
M

, the

G
G

-field number depends only on the dimensions of the fixed point sets of

M
M

.
𝐹-purity and rational singularity
Authors: Fedder Richard
Published: June 2010

We investigate singularities which are

F
F

-pure (respectively

F
F

-pure type). A ring

R
R

of characteristic

p
p

is

F
F

-pure if for every

R
R

-module

M
,
0
→
M
⊗
R
→
M
⊗

1

R

M,0 \to M \otimes R \to M \otimes \, ^1R

is exact where

1

R

^1R

denotes the

R
R

-algebra structure induced on

R
R

via the Frobenius map (if

r
∈
R

r \in R

and

s
∈

1

R

s \in \, ^{1}R

, then

r
⋅
s
=

r
p

s

r \cdot s = {r^p}s

in

1

R

^1R

).

F
F

-pure type is defined in characteristic

0
0

by reducing to characteristic

p
p

. It is proven that when

R
=
S

/

I

R = S/I

is the quotient of a regular local ring

S
S

,

R
R

is

F
F

-pure at the prime ideal

Q
Q

if and only if

(

I

[
p
]

:
I
)
⊄

Q

[
p
]

({I^{[p]}}:I) \not \subset {Q^{[p]}}

. Here,

J

[
p
]

{J^{[p]}}

denotes the ideal

{

a
p

|

a
∈
J
}

\{ {a^p}|a \in J\}

. Several theorems result from this criterion. If

f
f

is a quasihomogeneous hypersurface having weights

(

r
1

,
…
,

r
n

)

({r_1},\ldots ,{r_n})

and an isolated singularity at the origin: (1)

∑

i
=
1

n

r
i

>
1

\sum \nolimits _{i = 1}^n {{r_i} > 1}

implies

K
[

X
1

,
…
,

X
n

]

/

(
f
)

K[{X_1},\ldots ,{X_n}]/(f)

has

F
F

-pure type at

m
=
(

X
1

,
…
,

X
n

)

m = ({X_1},\ldots ,{X_n})

. (2)

∑

i
=
1

n

r
i

>
1

\sum \nolimits _{i = 1}^n {{r_i} > 1}

implies

K
[

X
1

,
…
,

X
n

]

/

(
f
)

K[{X_1},\ldots ,{X_n}]/(f)

does not have

F
F

-pure type at

m
m

. (3)

∑

i
=
1

n

r
i

=
1

\sum \nolimits _{i = 1}^n {{r_i} = 1}

remains unsolved, but does connect with a problem that number theorists have studied for many years. This theorem parallels known results about rational singularities. It is also proven that classifying

F
F

-pure singularities for complete intersection ideals can be reduced to classifying such singularities for hypersurfaces, and that the

F
F

-pure locus in the maximal spectrum of

K
[

X
1

,
…
,

X
n

]

/

I

K[{X_1},\ldots ,{X_n}]/I

, where

K
K

is a perfect field of characteristic

P
P

, is Zariski open. An important conjecture is that

R

/

f
R

R/fR

is

F
F

-pure (type) should imply

R
R

is

F
F

-pure (type) whenever

R
R

is a Cohen-Macauley, normal local ring. It is proven that

Ext
1

⁡

(
1

R
,
R
)
=
0

\operatorname {Ext}^1{(^1}R,R) = 0

is a sufficient, though not necessary, condition. A local ring

(
R
,
m
)

(R,m)

of characteristic

p
p

is

F
F

-injective if the Frobenius map induces an injection on the local cohomology modules

H
m
i

(
R
)
→

H
m
i

(
1

R
)

H_m^i(R) \to H_m^i{(^1}R)

. An example is constructed which is

F
F

-injective but not

F
F

-pure. From this a counterexample to the conjecture that

R

/

f
R

R/fR

is

F
F

-pure implies

R
R

is

F
F

-pure is constructed. However, it is not a domain, much less normal. Moreover, it does not lead to a counterexample to the characteristic

0
0

version of the conjecture.
Diophantine approximation properties of certain infinite sets
Authors: Schmidt Wolfgang M.
Published: June 2010

We exhibit various infinite sets of reals whose finite subsets do not have good simultaneous rational approximations. In particular there is an infinite set such that each finite subset is "badly approximable" in the sense that Dirichlet’s theorem is best possible up to a multiplicative constant.
The sufficiency of the Matkowsky condition in the problem of resonance
Authors: Lin Ching Her
Published: June 2010

We consider the sufficiency of the Matkowsky condition concerning the differential equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon y plus f left-parenthesis x comma epsilon right-parenthesis y Superscript prime Baseline plus g left-parenthesis x comma epsilon right-parenthesis y equals 0 left-parenthesis negative a less-than-or-slanted-equals x less-than-or-slanted-equals b right-parenthesis">

ε
y
+
f
(
x
,
ε
)

y
′

+
g
(
x
,
ε
)
y
=
0

(
−
a
⩽
x
⩽
b
)

\varepsilon y + f(x,\varepsilon )y’ + g(x,\varepsilon )y = 0\;( - a \leqslant x \leqslant b)

under the assumption that

f
(
0
,
ε
)
=
0

f(0,\varepsilon ) = 0

identically in

ε
,

f
x

(
0
,
ε
)
≠
0

\varepsilon ,{f_x}(0,\varepsilon ) \ne 0

with

f
>
0

f > 0

for

x
>
0

x > 0

and

f
>
0

f > 0

for

x
>
0

x > 0

. Y. Sibuya proved that the Matkowsky condition implies resonance in the sense of N. Kopell if

f
f

and

g
g

are convergent power series for

|

ε

|

>
ρ

(
ρ
>
0
)
,
f
(
x
,
0
)
=
−
2
x

|\varepsilon | > \rho \;(\rho > 0),f(x,0)=-2x

and the interval

[
−
a
,
b
]

[ - a,b]

is contained in a disc

D
D

with center at

0
0

. The main problem in this work is to remove from Sibuya’s result the assumption that

D
D

is a disc.
Hereditarily additive families in descriptive set theory and Borel measurable multimaps
Authors: Hansell Roger W.
Published: June 2010

A family

B

\mathcal {B}

of Borel subsets of a space

X
X

is (boundedly) Borel additive if, for some countable ordinal

α
\alpha

, the union of every subfamily of

B

\mathcal {B}

is a Borel set of class

α
\alpha

in

X
X

. A problem which arises frequently in nonseparable descriptive set theory is to find conditions under which this property is "hereditary" in the sense that any selection of a Borel subset from each member of

B

\mathcal {B}

(of uniform bounded class) will again be a Borel additive family. Similar problems arise for other classes of projective sets; in particular, for Souslin sets and their complements. Positive solutions to the problem have previously been obtained by the author and others when

X
X

is a complete metric space or under additional set-theoretic axioms. We give here a fairly general solution to the problem, without any additional axioms or completeness assumptions, for an abstract "descriptive class" in the setting of generalized metric spaces (e.g., spaces with a

σ
\sigma

-point-finite open base). A typical corollary states that any point-finite (co-) Souslin additive family in (say) a metrizable space is hereditarily (co-) Souslin additive. (There exists a point-countable

F
σ

{F_\sigma }

additive family of subsets of the real line which has a point selection which is not even Souslin additive.) Two structure theorems for "hereditarily additive" families are proven, and these are used to obtain a nonseparable extension of the fundamental measurable selection theorem of Kuratowski and Ryll-Nardzewski, and a complete solution to the problem of Kuratowski on the Borel measurability of complex and product mappings for nonseparable metric spaces.
Criteria for solvability of left invariant operators on nilpotent Lie groups
Authors: Corwin Lawrence
Published: June 2010

We define a special nilpotent Lie group

N
N

to be one which has a

1
1

-dimensional center, dilations, square-integrable representations, and a maximal subordinate algebra common to almost all functionals on the Lie algebra

N

\mathfrak {N}

. Every nilpotent Lie group with dilations imbeds in such a special group so that the dilations extend. Let

L
L

be a homogeneous left invariant differential operator on

N
N

. We give a representation-theoretic condition on

L
L

which

L
L

must satisfy if it has a tempered fundamental solution and which implies global solvability of

L
L

. (The sufficiency is a corollary of a more general theorem, valid on all nilpotent

N
N

.) For the Heisenberg group, the condition is equivalent to having a tempered fundamental solution.
On the oscillation of differential transforms of eigenfunction expansions
Authors: Prather C. L. , Shaw J. K.
Published: June 2010

This paper continues the study of Pólya and Wiener, Hille and Szegö into the connections between the oscillation of derivatives of a real function and its analytic character. In the present paper, a Sturm-Liouville operator

L
L

is applied successively to an infinitely differentiable function which admits a certain eigenfunction expansion. The eigenfunction expansion is assumed to be "conservative", in the sense of Hille. Several theorems are given which link the frequency of oscillation of

(

L
k

f
)
(
x
)

({L^k}f)(x)

to the size of the coefficients of

f
(
x
)

f(x)

, and thus to its analytic character.
Geometric condition for universal interpolation in ℰ̂’
Authors: Squires William A.
Published: June 2010

It is known that if

h
h

is an entire function of exponential type and

Z
(
h
)
=

{

z
k

}

k
=
1

Z(h) = {\{ {z_k}\} _{k = 1}}

with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue h prime left-parenthesis z Subscript k Baseline right-parenthesis EndAbsoluteValue greater-than-or-slanted-equals epsilon exp left-parenthesis minus c StartAbsoluteValue z Subscript k Baseline EndAbsoluteValue right-parenthesis">

|

h
′

(

z
k

)

|

⩾
ε
exp
⁡
(
−
c

|

z
k

|

)

|h’({z_k})| \geqslant \varepsilon \exp (- c|{z_k}|)

for constants

ϵ
\epsilon

,

C
C

independent of

k
k

, then

{

z
k

}

k
=
1

∞

\{ {z_k}\} _{k = 1}^\infty

is a universal interpolation sequence. That is, given any sequence of complex numbers

{

a
k

}

k
=
1

∞

\{ {a_k}\} _{k = 1}^\infty

such that

|

a
k

|

⩽
A

exp
⁡
(
B

|

z
k

|

)

|{a_k}| \leqslant A\,\exp (B|{z_k}|)

for constants

A
,
B

A,B

independent of

K
K

then there exists

g
g

of exponential type such that

g
(

z
k

)
=

a
k

g({z_k}) = {a_k}

. This note is concerned with finding geometric conditions which make

{

z
k

}

k
=
1

∞

\{ {z_k}\} _{k = 1}^\infty

a universal interpolation sequence for various spaces of entire functions. For the space of entire functions of exponential type a necessary and sufficient condition for

{

z
k

}

k
=
1

∞

\{ {z_k}\} _{k = 1}^\infty

to be a universal interpolation sequence is that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript StartAbsoluteValue z Subscript k Baseline EndAbsoluteValue Baseline n left-parenthesis z Subscript k Baseline comma t right-parenthesis d t slash t less-than-or-slanted-equals upper C StartAbsoluteValue z Subscript k Baseline EndAbsoluteValue plus upper D comma k equals 1 comma 2 comma ellipsis">

∫
0

|

z
k

|

n
(

z
k

,
t
)

d
t

/

t
⩽
C

|

z
k

|

+
D
,
k
=
1

,
2
,
…

\int _0^{|{z_k}|} {n({z_k},t)\,dt/t \leqslant C|{z_k}| + D,k = 1} , 2,\ldots

, where

n
(

z
k

,
t
)

n({z_k},t)

is the number of points of

{

z
k

}

k
=
1

∞

\{ {z_k}\} _{k = 1}^\infty

in the disc of radius

t
t

about

z
k

{z_k}

, excluding

z
k

{z_k}

, and

C
,
D

C,D

are constants independent of

k
k

. Results for the space
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove script upper E With caret prime equals StartSet f entire StartAbsoluteValue EndAbsoluteValue f left-parenthesis z right-parenthesis vertical-bar less-than-or-slanted-equals upper A exp left-bracket upper B StartAbsoluteValue upper I m z EndAbsoluteValue plus upper B log left-parenthesis 1 plus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis right-bracket EndSet">

E

^

′

=
{
f

entire

|

|

f
(
z
)

|

⩽
A

exp
⁡
[
B

|

Im
⁡
z

|

+
B
log
⁡
(
1
+

|

z

|

2

)
]
}

\hat {\mathcal {E}}^\prime = \{ f\;{\text {entire}}||f(z)| \leqslant A\;\exp [B|\operatorname {Im} z| + B\log (1 + |z|^{2})]\}

are given but the theory is not as complete as for the above example.
Szegő limit theorems for the harmonic oscillator
Authors: Janssen A. J. E. M. , Zelditch Steven
Published: June 2010

Let

H
=
−

1
2

d
2

/

d

x
2

+

1
2

x
2

H = - \frac {1}{2}{d^2}/d{x^2} + \frac {1}{2}{x^2}

be the harmonic oscillator Hamiltonian on

L
2

(

R

)

{L^2}( {\mathbf {R}})

, and let

A
A

be a selfadjoint

D
O

DO

of order

O
O

in the Beals-Fefferman class with weights
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi equals 1 comma normal upper Phi left-parenthesis x comma xi right-parenthesis equals left-parenthesis 1 plus StartAbsoluteValue xi EndAbsoluteValue squared plus StartAbsoluteValue x EndAbsoluteValue squared right-parenthesis Superscript 1 slash 2">

φ
=
1
,
Φ
(
x
,
ξ
)
=

(
1
+

|

ξ

|

2

+

|

x

|

2

)

1

/

2

\varphi = 1,\Phi (x,\xi ) = {(1 + |\xi \,{|^2} + |x\,{|^2})^{1/2}}

. Form the measure
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu left-parenthesis f right-parenthesis equals limit Underscript lamda right-arrow normal infinity Endscripts left-parenthesis 1 slash rank pi Subscript lamda Baseline right-parenthesis trace f left-parenthesis pi Subscript lamda Baseline upper A pi Subscript lamda Baseline right-parenthesis">

μ
(
f
)
=

lim

λ
→
∞

(
1

/

rank

π
λ

)

tr

f
(

π
λ

A

π
λ

)

\mu (f) = {\lim _{\lambda \to \infty }}(1/{\text {rank}}\;{\pi _\lambda })\,{\text {tr}}\,f({\pi _\lambda }\,A{\pi _\lambda })

where

π
λ

A

π
λ

{\pi _\lambda }\,A{\pi _\lambda }

is the compression of

A
A

onto the span of the Hermite functions with eigenvalue less than or equal to

λ
\lambda

. Then one has the following Szegö limit theorem:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu left-parenthesis f right-parenthesis equals limit Underscript upper T right-arrow normal infinity Endscripts StartFraction 1 Over 2 pi upper T EndFraction integral Underscript upper H left-parenthesis x comma xi right-parenthesis less-than-or-slanted-equals upper T Endscripts f left-parenthesis a left-parenthesis x comma xi right-parenthesis right-parenthesis d x d xi for f element-of upper C left-parenthesis bold upper R right-parenthesis period">

μ
(
f
)
=

lim

T
→
∞

1

2

π

T

∫

H
(
x
,
ξ
)
⩽
T

f
(
a
(
x
,
ξ
)
)

d
x

d
ξ

for

 
f
∈
C
(

R

)
.

\mu (f) = \lim \limits _{T \to \infty } \;\frac {1} {{2\,\pi \,T}}\;\int _{H(x,\xi ) \leqslant T} {f(a(x,\xi ))\;dx} \;d\xi \qquad {\text {for}}\ f \in C({\mathbf {R}}).

\]
For the special case where

f
(
x
)
=
x

f(x) = x

, this will be proved for a considerably wider class of operators by employing the Weyl correspondence. Moreover, by using estimates on Wigner functions of Hermite functions we are able to prove the full Szegö theorem for a fairly general class of multiplication operators.
Inverses and parametrices for right-invariant pseudodifferential operators on two-step nilpotent Lie groups
Authors: Miller Kenneth G.
Published: June 2010

Let

P
P

be a right-invariant pseudodifferential operator with principal part

P
0

{P_0}

on a simply connected two-step nilpotent Lie group

G
G

of type

H
H

. It will be shown that if

π
(

P
0

)

\pi (P_0)

is injective in

S

π

{\mathcal {S}_\pi }

for every nontrivial irreducible unitary representation

π
\pi

of

G
G

, then

P
P

has a pseudodifferential left parametrix. For such groups this generalizes the Rockland-Helffer-Nourrigat criterion for the hypoellipticity of a homogeneous right-invariant partial differential operator on

G
G

. If, in addition,

π
(
P
)

\pi (P)

is injective in

S

π

{\mathcal {S}_\pi }

for every irreducible unitary representation of

G
G

, it will be shown that

P
P

has a pseudodifferential left inverse. The constructions of the inverse and parametrix make use of the Kirillov theory, their symbols being obtained on the orbits individually and then pieced together.
Length dependence of solutions of FitzHugh-Nagumo equations
Authors: Collins Clyde
Published: June 2010

We investigate the behavior of the solutions of the problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column u Subscript t Baseline equals u Subscript x x Baseline minus alpha u minus upsilon plus u squared left-parenthesis 1 plus alpha minus u right-parenthesis comma 2nd Column a m p semicolon upsilon Subscript t Baseline equals eta upsilon Subscript x x Baseline plus sigma u minus gamma upsilon comma 2nd Row 1st Column u left-parenthesis 0 comma t right-parenthesis equals g left-parenthesis t right-parenthesis comma upsilon left-parenthesis 0 comma t right-parenthesis equals h left-parenthesis t right-parenthesis comma 2nd Column a m p semicolon u Subscript x Baseline left-parenthesis upper L comma t right-parenthesis equals upsilon Subscript x Baseline left-parenthesis upper L comma t right-parenthesis equals 0 EndLayout">

u
t

=

u

x
x

−
α
u
−
υ
+

u
2

(
1
+
α
−
u
)
,

a
m
p
;

υ
t

=
η

υ

x
x

+
σ
u
−
γ
υ
,

u
(
0
,
t
)
=
g
(
t
)
,

υ
(
0
,
t
)
=
h
(
t
)
,

a
m
p
;

u
x

(
L
,
t
)
=

υ
x

(
L
,
t
)
=
0

\begin {array}{*{20}{c}} {{u_t} = {u_{xx}} - \alpha u - \upsilon + {u^2}(1 + \alpha - u),} & {{\upsilon _t} = \eta {\upsilon _{xx}} + \sigma u - \gamma \upsilon ,} \\ {u(0,t) = g(t),\quad \upsilon (0,t) = h(t),} & {{u_x}(L,t) = {\upsilon _x}(L,t) = 0} \\ \end {array}

\]
where

t
⩾
0

t \geqslant 0

and

0
>
x
>
L
⩽
∞

0 > x > L \leqslant \infty

. Solutions of the above equations are considered a qualitative model of conduction of nerve axon impulses. Using explicit constructions and semigroup methods, we obtain decay results on the norms of differences between the solution for

L
L

infinite and the solutions when

L
L

is large but finite. We conclude that nerve impulses for long finite nerves become uniformly close to those of the semi-infinite nerves away from the right endpoint of the finite nerve.
Harish-Chandra modules with the unique embedding property
Authors: Collingwood David H.
Published: June 2010

Let

G
G

be a connected semisimple real matrix group. In view of Casselman’s subrepresentation theorem, every irreducible admissible representation of

G
G

may be realized as a submodule of some principal series representation. We give a classification of representations with a unique embedding into principal series, in the case of regular infinitesimal character. Our basic philosophy is to link the theory of asymptotic behavior of matrix coefficients with the theory of coherent continuation of characters. This is accomplished by using the "Jacquet functor" and the Kazhdan-Lusztig conjectures.
Homomorphisms of cocompact Fuchsian groups on 𝑃𝑆𝐿₂(𝑍_{𝑝ⁿ}[𝑥]/(𝑓(𝑥)))
Authors: Cohen Jeffrey
Published: June 2010

We obtain conditions under which

PSL

2

(

Z

p
n

[
x
]

/

(
f
(
x
)
)
)

{\text {PSL}}_2({Z_{{p^n}}}[x]/(f(x)))

is a factor of

(
l
,
m
,
n
)

(l,m,n)

. Using this, certain results about factors of cocompact Fuchsian groups are obtained. For example, it is shown that: (i)

Γ
\Gamma

has infinitely many simple nonabelian factors. (ii)

Γ
\Gamma

has factors with nontrivial center. (iii) For each

n
n

, there exists

m
m

such that

Γ
\Gamma

has at least

n
n

factors of order

m
m

. Further, all factored normal subgroups can be taken torsion-free. Also, new Hurwitz groups and noncongruence subgroups of the modular group are obtained.
A minimal model for ¬𝐶𝐻: iteration of Jensen’s reals
Authors: Abraham Uri
Published: June 2010

A model of

ZFC

+

2

ℵ
0

=

ℵ
2

{\text {ZFC}} + {2^{\aleph _0}} = {\aleph _2}

is constructed which is minimal with respect to being a model of

¬

CH

\neg {\text {CH}}

. Any strictly included submodel of

ZF

{\text {ZF}}

(which contains all the ordinals) satisfies

CH

{\text {CH}}

. In this model the degrees of constructibility have order type

ω
2

{\omega _2}

. A novel method of using the diamond is applied here to construct a countable-support iteration of Jensen’s reals: In defining the

α

th

\alpha {\text {th}}

stage of the iteration the diamond "guesses" possible

β
>
α

\beta > \alpha

stages of the iteration.
Asymptotic Dirichlet problems for harmonic functions on Riemannian manifolds
Authors: Choi Hyeong In
Published: June 2010

We define the asymptotic Dirichlet problem and give a sufficient condition for solving it. This proves an existence of nontrivial bounded harmonic functions on certain classes of noncompact complete Riemannian manifolds.
Some applications of Nevanlinna theory to mathematical logic: identities of exponential functions
Authors: Henson C. Ward , Rubel Lee A.
Published: June 2010

In this paper we study identities between certain functions of many variables that are constructed by using the elementary functions of addition

x
+
y

x+y

, multiplication

x
⋅
y

x \cdot y

, and two-place exponentiation

x
y

x^y

. For a restricted class of such functions, we show that every true identity follows from the natural set of eleven axioms. The rates of growth of such functions, in the case of a single independent variable

x
x

, as

x
→
∞

x \to \infty

, are also studied, and we give an algorithm for the Hardy relation of eventual domination, again for a restricted class of functions. Value distribution of analytic functions of one and of several complex variables, especially the Nevanlinna characteristic, plays a major role in our proofs.
Decompositions of the maximal ideal space of 𝐿^{∞}
Authors: Gorkin Pamela
Published: June 2010

In this paper we show the existence of one point maximal antisymmetric sets for

H
∞

+
C

{H^\infty } + C

.
Brauer’s height conjecture for 𝑝-solvable groups
Authors: Gluck David , Wolf Thomas R.
Published: June 2010

We complete the proof of the height conjecture for

p
p

-solvable groups, using the classification of finite simple groups.
Group-graded rings, smash products, and group actions
Authors: Cohen M. , Montgomery S.
Published: June 2010

Let

A
A

be a

k
k

-algebra graded by a finite group

G
G

, with

A
1

{A_1}

the component for the identity element of

G
G

. We consider such a grading as a “coaction” by

G
G

, in that

A
A

is a

k

[
G

]
∗

k{[G]^ \ast }

-module algebra. We then study the smash product

A
#
k

[
G

]
∗

A\# k{[G]^ \ast }

; it plays a role similar to that played by the skew group ring

R

∗

G

R\, \ast \,G

in the case of group actions, and enables us to obtain results relating the modules over

A
,

A
1

A,\,{A_1}

, and

A
#
k

[
G

]
∗

A\# k{[G]^ \ast }

. After giving algebraic versions of the Duality Theorems for Actions and Coactions (results coming from von Neumann algebras), we apply them to study the prime ideals of

A
A

and

A
1

{A_1}

. In particular we generalize Lorenz and Passman’s theorem on incomparability of primes in crossed products. We also answer a question of Bergman on graded Jacobson radicals.
Variational problems with two phases and their free boundaries
Authors: Alt Hans Wilhelm , Caffarelli Luis A. , Friedman Avner
Published: June 2010

The problem of minimizing

∫

[
∇
υ

|

2

+

q
2

(
x
)

λ
2

(
υ
)
]
d
x

\int {[\nabla \upsilon {|^2}} + {q^2}(x){\lambda ^2}(\upsilon )]dx

in an appropriate class of functions

υ
\upsilon

is considered. Here

q
(
x
)
≠
0

q(x) \ne 0

and

λ
2

(
υ
)
=

λ
1
2

{\lambda ^2}(\upsilon ) = \lambda _1^2

if

υ
>
0
,
=

λ
2
2

\upsilon > 0, = \lambda _2^2

if

υ
>
0

\upsilon > 0

. Any minimizer

u
u

is harmonic in

{
u
≠
0
}

\{ u \ne 0\}

and

|

∇
u

|

2

|\nabla u{|^2}

has a jump
\[

q
2

(
x
)

(

λ
1
2

−

λ
2
2

)

{q^2}(x)\left ( {\lambda _1^2 - \lambda _2^2} \right )

\]
across the free boundary

{
u
≠
0
}

\{ u \ne 0\}

. Regularity and various properties are established for the minimizer

u
u

and for the free boundary.
Moduli of continuity in 𝑅ⁿ and 𝐷⊂𝑅ⁿ
Authors: Ditzian Z.
Published: June 2010

The

r
r

modulus of continuity for

f
∈
C
(

R
n

)

f \in C({R^n})

is expressed in terms of

r
r

moduli of continuity in

n
n

independent directions. Generalizations to other spaces of functions on

R
n

{R^n}

or

D
⊂

R
n

D \subset {R^n}

are also given.
Realizability of branched coverings of surfaces
Authors: Edmonds Allan L. , Kulkarni Ravi S. , Stong Robert E.
Published: June 2010

A branched covering

M
→
N

M \to N

of degree

d
d

between closed surfaces determines a collection

D

\mathfrak {D}

of partitions of

d
d

—its "branch data"—corresponding to the set of branch points. The collection of partitions must satisfy certain obvious conditions implied by the Riemann-Hurwitz formula. This paper investigates the extent to which any such finite collection

D

\mathfrak {D}

of partitions of

d
d

can be realized as the branch data of a suitable branched covering. If

N
N

is not the

2
2

-sphere, such data can always be realized. If

D

\mathfrak {D}

contains sufficiently many elements compared to

d
d

, then it can be realized. And whenever

d
d

is nonprime, examples are constructed of nonrealizable data.
Strong martingale convergence of generalized conditional expectations on von Neumann algebras
Authors: Hiai Fumio , Tsukada Makoto
Published: June 2010

Accardi and Cecchini generalized the concept of conditional expectations on von Neumann algebras. In this paper we give some conditions for strong convergence of increasing or decreasing martingales of Accardi and Cecchini’s conditional expectations.
On the proper holomorphic equivalence for a class of pseudoconvex domains
Authors: Landucci M.
Published: June 2010

A complete and explicit description of the holomorphic proper mappings between weakly pseudoconvex domains of the class

Δ
p

{\Delta _p}

(see ( •) below) is given.
Hecke modular forms and the Kac-Peterson identities
Authors: Andrews George E.
Published: June 2010

The identity of certain Hecke modular forms with well-known infinite products is derived in an elementary manner. New identities and applications are discussed.
Eisenstein series of weight 3\over2. II
Authors: Pei Ting Yi
Published: June 2010

In a previous paper we proved that for some special levels, in the space of elliptic modular forms with weight

3

/

2

3/2

the orthogonal complement of the subspace of cusp forms with respect to the Petersson inner product is generated by the Eisenstein series. In this paper we prove that this fact is true for any level.
Concatenations applied to analytic hypoellipticity of operators with double characteristics
Authors: Kwon Kil Hyun
Published: June 2010

We use the method of concatenations to get a sufficient condition for a class of analytic pseudodifferential operators with double characteristics to be analytic hypoelliptic.
Milnor’s invariants and the completions of link modules
Authors: Traldi Lorenzo
Published: June 2010

Let

L
L

be a tame link of

μ
⩾
2

\mu \geqslant 2

components in

S
3

{S^3}

,

H
H

the abelianization of its group

π
1

(

S
3

−
L
)

{\pi _1}({S^3} - L)

, and

I
H

IH

the augmentation ideal of the integral group ring

Z

H

{\mathbf {Z}}H

. The

I
H

IH

-adic completions of the Alexander module and Alexander invariant of

L
L

are shown to possess presentation matrices whose entries are given in terms of certain integers

μ
(

i
1

,
…
,

i
q

)

\mu ({i_1}, \ldots ,{i_q})

introduced by J. Milnor. Various applications to the theory of the elementary ideals of these modules are given, including a condition on the Alexander polynomial necessary for the linking numbers of the components of

L
L

with each other to all be zero. In the special case

μ
=
2

\mu = 2

, it is shown that the various Milnor invariants

μ
¯

(
[
r
+
1
,
s
+
1
]
)

\bar \mu ([r + 1,s + 1])

are determined (up to sign) by the Alexander polynomial of

L
L

, and that this Alexander polynomial is

0
0

iff

μ
¯

(
[
r
+
1
,
s
+
1
]
)
=
0

\bar \mu ([r + 1,s + 1]) = 0

for all

r
,
s
⩾
0

r,s \geqslant 0

with

r
+
s

r + s

even; also, the Chen groups of

L
L

are determined (up to isomorphism) by those nonzero

μ
¯

(
[
r
+
1
,
s
+
1
]
)

\bar \mu ([r + 1,s + 1])

with

r
+
s

r + s

minimal. In contrast, it is shown by example that for

μ
⩾
3

\mu \geqslant 3

the Alexander polynomials of a link and its sublinks do not determine its Chen groups.
Countable homogeneous tournaments
Authors: Lachlan A. H.
Published: June 2010

A tournament

T
T

is called homogeneous just in case every isomorphism of subtournaments of smaller cardinality can be lifted to an automorphism of

T
T

. It is shown that there are precisely three homogeneous tournaments of power

ℵ
0

{\aleph _0}

. Some analogous results for

2
2

-tournaments are obtained.
An interface tracking algorithm for the porous medium equation
Authors: DiBenedetto E. , Hoff David
Published: June 2010

We study the convergence of a finite difference scheme for the Cauchy problem for the porous medium equation

u
t

=

(

u
m

)

x

x

,
m
>
1

{u_t} = {({u^m})_{x\,x}},m > 1

. The scheme exhibits the following two features. The first is that it employs a discretization of the known interface condition for the propagation of the support of the solution. We thus generate approximate interfaces as well as an approximate solution. The second feature is that it contains a vanishing viscosity term. This term permits an estimate of the form

∥

(

u

m
−
1

)

x

x

∥

1
,

R

⩽
c

/

t

\parallel {({u^{m - 1}})_{x\,x}}\;\parallel _{1,{\mathbf {R}}} \leqslant c/t

. We prove that both the approximate solution and the approximate interfaces converge to the correct ones. Finally error bounds for both solution and free boundaries are proved in terms of the mesh parameters.
Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems
Authors: Beals Michael , Reed Michael
Published: June 2010

The authors develop a calculus of pseudodifferential operators with nonsmooth coefficients in order to study the regularity of solutions to linear equations

P

(
x
,
D
)

u
=
f

P\,(x,D)\,u = f

. The regularity theorems are similar to those of Bony, but the calculus and the methods of proof are quite different. We apply the linear results to study the regularity properties of solutions to quasilinear partial differential equations.
Codimension 1 orbits and semi-invariants for the representations of an equioriented graph of type 𝐷_{𝑛}
Authors: Abeasis S.
Published: June 2010

We consider the Dynkin diagram

D
n

{D_n}

equioriented and the variety
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H o m left-parenthesis upper V 1 comma upper V 3 right-parenthesis times normal upper Pi Subscript 1 equals 2 Superscript n Baseline upper H o m left-parenthesis upper V Subscript i Baseline comma upper V Subscript i plus 1 Baseline right-parenthesis">

Hom
⁡
(

V
1

,

V
3

)
×

Π

1
=
2

n

Hom
⁡
(

V
i

,

V

i
+
1

)

\operatorname {Hom}({V_1},{V_3}) \times \Pi _{1 = 2}^n \operatorname {Hom} ({V_i},{V_{i + 1}})

,

V
j

{V_j}

a vector space over

K
K

, on which the group

G
=

∏

i
=
1

n

GL

(

V
i

)

G = \prod \nolimits _{i = 1}^n {{\text {GL}}} ({V_i})

acts. We determine the maximal orbit and the codim.

1
1

orbits of this action, giving their decomposition in terms of the irreducible representations of

D
n

{D_n}

. We also deduce a set of algebraically independent semi-invariant polynomials which generate the ring of semi-invariants.