bluuebunny/crossref_metadata_2025 huggingface dataset:
Dominated permutations of subsequences of random variables
Authors: Meyerowitz Aaron , Schwartz Mark
Published: June 2010

The generalized strong law of large numbers of Komlós may be extended to include certain dominated permutations of the random variables. A further extension to larger classes of permutations is obtained through decompositions of sequences of positive integers.
Nonexistence of stable harmonic maps to and from certain homogeneous spaces and submanifolds of Euclidean space
Authors: Howard Ralph , Wei S. Walter
Published: June 2010

Call a compact Riemannian manifold

M
M

a strongly unstable manifold if it is not the range or domain of a nonconstant stable harmonic map and also the homotopy class of any map to or from

M
M

contains elements of arbitrarily small energy. If

M
M

is isometrically immersed in Euclidean space, then a condition on the second fundamental form of

M
M

is given which implies

M
M

is strongly unstable. As compact isotropy irreducible homogeneous spaces have "standard" immersions into Euclidean space this allows a complete list of the strongly unstable compact irreducible symmetric spaces to be made.
A rigidity property for the set of all characters induced by valuations
Authors: Bieri Robert , Groves John R. J.
Published: June 2010

If

K
K

is a field and

G
G

a finitely generated multiplicative subgroup of

K
K

then every real valuation on

K
K

induces a character

G
→

R

G \to {\mathbf {R}}

. It is known that the set

Δ
(
G
)
⊆

R

n

\Delta (G) \subseteq {{\mathbf {R}}^n}

of all characters induced by valuations is polyhedral. We prove that

Δ
(
G
)

\Delta (G)

satisfies a certain rigidity property and apply this to give a new and conceptual proof of the Brewster-Roseblade result [4] on the group of automorphisms of

K
K

stabilizing

G
G

.
Spherical polynomials and the periods of a certain modular form
Authors: Kramer David
Published: June 2010

The space of cusp forms on

S

L

2

(

Z

)

{\text {S}}{{\text {L}}_2}({\mathbf {Z}})

of weight

2
k

2k

is spanned by certain modular forms with rational periods.
Definable sets in ordered structures. I
Authors: Pillay Anand , Steinhorn Charles
Published: June 2010

This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the

O

\mathcal {O}

-minimal structures. The definition of this class and the corresponding class of theories, the strongly

O

\mathcal {O}

-minimal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories. Theorems 2.1 and 2.3, respectively, provide characterizations of

O

\mathcal {O}

-minimal ordered groups and rings. Several other simple results are collected in

§
3

\S 3

. The primary tool in the analysis of

O

\mathcal {O}

-minimal structures is a strong analogue of "forking symmetry," given by Theorem 4.2. This result states that any (parametrically) definable unary function in an

O

\mathcal {O}

-minimal structure is piecewise either constant or an order-preserving or reversing bijection of intervals. The results that follow include the existence and uniqueness of prime models over sets (Theorem 5.1) and a characterization of all

ℵ
0

{\aleph _0}

-categorical

O

\mathcal {O}

-minimal structures (Theorem 6.1).
Axiom 3 modules
Authors: Hill Paul , Megibben Charles
Published: June 2010

By introducing the concept of a knice submodule, a refinement of the notion of nice subgroup, we are able to formulate a version of the third axiom of countability appropriate to the study of

p
p

-local mixed groups in the spirit of the well-known characterization of totally projective

p
p

-groups. Our Axiom

3
3

modules, in fact, form a class of

Z

p

{{\mathbf {Z}}_{\mathbf {p}}}

-modules, encompassing the totally projectives in the torsion case, for which we prove a uniqueness theorem and establish closure under direct summands. Indeed Axiom

3
3

modules turn out to be precisely the previously classified Warfield modules. But with the added power of the third axiom of countability characterization, we derive numerous new results, including the resolution of a long-standing problem of Warfield and theorems in the vein of familiar criteria due to Kulikov and Pontryagin.
Torsion free groups
Authors: Hill Paul , Megibben Charles
Published: June 2010

In this paper we introduce the class of torsion free

k
k

-groups and the notion of a knice subgroup. Torsion free

k
k

-groups form a class of groups more extensive than the separable groups of Baer, but they enjoy many of the same closure properties. We establish a role for knice subgroups of torsion free groups analogous to that played by nice subgroups in the study of torsion groups. For example, among the torsion free groups, the balanced projectives are characterized by the fact that they satisfy the third axiom of countability with respect to knice subgroups. Separable groups are characterized as those torsion free

k
k

-groups with the property that all finite rank, pure knice subgroups are direct summands. The introduction of these new classes of groups and subgroups is based on a preliminary study of the interplay between primitive elements and

∗
\ast

-valuated coproducts. As a by-product of our investigation, new proofs are obtained for many classical results on separable groups. Our techniques lead naturally to the discovery that a balanced subgroup of a completely decomposable group is itself completely decomposable provided the corresponding quotient is a separable group of cardinality not exceeding

ℵ
1

{\aleph _1}

; that is, separable groups of cardinality

ℵ
1

{\aleph _1}

have balanced projective dimension

≤
1

\leq 1

.
Factorial property of a ring of automorphic forms
Authors: Tsuyumine Shigeaki
Published: June 2010

A ring of automorphic forms is shown to be factorial under some conditions on the domain and on the Picard group. As an application, we show that any divisor on the moduli space

M

g

{\mathfrak {M}_g}

of curves of genus

g
⩾
3

g \geqslant 3

is defined by a single element, and that the Satake compactification of

M

g

{\mathfrak {M}_g}

is written as a projective spectrum of a factorial graded ring. We find a single element which defines the closure of

M

′
4

{\mathfrak {M}\prime _4}

in

M

4

{\mathfrak {M}_4}

where

M

′
4

{\mathfrak {M}\prime _4}

is the moduli of curves of genus four whose canonical curves are exhibited as complete intersections of quadric cones and of cubics in

P

3

{\mathbb {P}^3}

.
Spanier-Whitehead duality in étale homotopy
Authors: Joshua Roy
Published: June 2010

We construct a

(
mod

-

l
)

(\bmod {\text {-}}l)

Spanier-Whitehead dual for the etale homotopy type of any geometrically unibranched and projective variety over an algebraically closed field of arbitrary characteristic. The Thom space of the normal bundle to imbedding any compact complex manifold in a large sphere as a real submanifold provides a Spanier-Whitehead dual for the disjoint union of the manifold and a base point. Our construction generalises this to any characteristic. We also observe various consequences of the existence of a

(
mod

-

l
)

(\bmod {\text {-}}l)

Spanier-Whitehead dual.
On integers free of large prime factors
Authors: Hildebrand Adolf , Tenenbaum Gérald
Published: June 2010

The number

Ψ
(
x
,
y
)

\Psi (x,y)

of integers

≤
x

\leq x

and free of prime factors

>
y

> y

has been given satisfactory estimates in the regions

y
≤

(
log
⁡
x

)

3

/

4
−
ε

y \leq {(\log x)^{3/4 - \varepsilon }}

and

y
>
exp
⁡
{

(
log
⁡
log
⁡
x

)

5

/

3
+
ε

}

y > \exp \{ {(\log \log x)^{5/3 + \varepsilon }}\}

. In the intermediate range, only very crude estimates have been obtained so far. We close this "gap" and give an expression which approximates

Ψ
(
x
,
y
)

\Psi (x,y)

uniformly for

x
≥
y
≥
2

x \geq y \geq 2

within a factor

1
+
O
(
(
log
⁡
y
)

/

(
log
⁡
x
)
+
(
log
⁡
y
)

/

y
)

1 + O((\log y)/(\log x) + (\log y)/y)

. As an application, we derive a simple formula for

Ψ
(
c
x
,
y
)

/

Ψ
(
x
,
y
)

\Psi (cx,y)/\Psi (x,y)

, where

1
≤
c
≤
y

1 \leq c \leq y

. We also prove a short interval estimate for

Ψ
(
x
,
y
)

\Psi (x,y)

.
On the covering dimension of the set of solutions of some nonlinear equations
Authors: Fitzpatrick P. M. , Massabò I. , Pejsachowicz J.
Published: June 2010

We prove an abstract theorem whose sole hypothesis is that the degree of a certain map is nonzero and whose conclusions imply sharp, multidimensional continuation results. Applications are given to nonlinear partial differential equations.
Equivariant Morse theory for flows and an application to the 𝑁-body problem
Authors: Pacella Filomena
Published: June 2010

In this paper, using Conley’s index and equivariant cohomology, some Morse type inequalities are deduced for a flow equivariant with respect to the action of a compact topological group. In the case of a gradient flow induced by a nondegenerate smooth function these inequalities coincide with those described by R. Bott. The theory is applied to the study of the central configurations of

N
N

-bodies.
Unknotting information from 4-manifolds
Authors: Cochran T. D. , Lickorish W. B. R.
Published: June 2010

Results of S. K. Donaldson, and others, concerning the intersection forms of smooth

4
4

-manifolds are used to give new information on the unknotting numbers of certain classical knots. This information is particularly sensitive to the signs of the knot crossings changed in an unknotting process.
Vector fields in the vicinity of a circle of critical points
Authors: Mattei J.-F. , Teixeira M. A.
Published: June 2010

In this paper the

C
S

{C^S}

-conjugacy between vector fields on

R

2

{{\mathbf {R}}^2}

having a circle of critical points is studied.
Crossed products and inner actions of Hopf algebras
Authors: Blattner Robert J. , Cohen Miriam , Montgomery Susan
Published: June 2010

This paper develops a theory of crossed products and inner (weak) actions of arbitrary Hopf algebras on noncommutative algebras. The theory covers the usual examples of inner automorphisms and derivations, and in addition is general enough to include "inner" group gradings of algebras. We prove that if

π
:
H
→

H
¯

\pi :H \to \overline H

is a Hopf algebra epimorphism which is split as a coalgebra map, then

H
H

is algebra isomorphic to

A

#
σ

H

A{\# _\sigma }H

, a crossed product of

H
H

with the left Hopf kernel

A
A

of

π
\pi

. Given any crossed product

A

#
σ

H

A{\# _\sigma }H

with

H
H

(weakly) inner on

A
A

, then

A

#
σ

H

A{\# _\sigma }H

is isomorphic to a twisted product

A
τ

[
H
]

{A_\tau }[H]

with trivial action. Finally, if

H
H

is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of

A
A

implies that of

A

#
σ

H

A{\# _\sigma }H

; in particular this is true if the (weak) action of

H
H

is inner.
Minimal submanifolds of a sphere with bounded second fundamental form
Authors: Gauchman Hillel
Published: June 2010

Let

h
h

be the second fundamental form of an

n
n

-dimensional minimal submanifold

M
M

of a unit sphere

S

n
+
p

(
p
⩾
2
)

{S^{n + p}}(p \geqslant 2)

,

S
S

be the square of the length of

h
h

, and

σ
(
u
)
=

|

|

h
(
u
,
u
)

|

|

2

\sigma (u) = ||h(u,u)|{|^2}

for any unit vector

u
∈
T
M

u \in TM

. Simons proved that if

S
⩽
n

/

(
2
−
1

/

p
)

S \leqslant n/(2 - 1/p)

on

M
M

, then either

S
≡
0

S \equiv 0

, or

S
≡
n

/

(
2
−
1

/

p
)

S \equiv n/(2 - 1/p)

. Chern, do Carmo, and Kobayashi determined all minimal submanifolds satisfying

S
≡
n

/

(
2
−
1

/

p
)

S \equiv n/(2 - 1/p)

. In this paper the analogous results for

σ
(
u
)

\sigma (u)

are obtained. It is proved that if

σ
(
u
)
⩽

1
3

\sigma (u) \leqslant \tfrac {1} {3}

, then either

σ
(
u
)
≡
0

\sigma (u) \equiv 0

, or

σ
(
u
)
≡

1
3

\sigma (u) \equiv \tfrac {1} {3}

. All minimal submanifolds satisfying

σ
(
u
)

\sigma (u)

are determined. A stronger result is obtained if

M
M

is odd-dimensional.
On the Hausdorff dimension of some graphs
Authors: Mauldin R. Daniel , Williams S. C.
Published: June 2010

Consider the functions
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Subscript b Baseline left-parenthesis x right-parenthesis equals sigma-summation Underscript n equals negative normal infinity Overscript normal infinity Endscripts b Superscript minus alpha n Baseline left-bracket normal upper Phi left-parenthesis b Superscript n Baseline x plus theta Subscript n Baseline right-parenthesis minus normal upper Phi left-parenthesis theta Subscript n Baseline right-parenthesis right-bracket comma">

W
b

(
x
)
=

∑

n
=
−
∞

∞

b

−
α
n

[
Φ
(

b
n

x
+

θ
n

)
−
Φ
(

θ
n

)
]
,

{W_b}(x) = \sum \limits _{n = - \infty }^\infty {{b^{ - \alpha n}}[\Phi ({b^n}x + {\theta _n}) - \Phi ({\theta _n})],}

\]
where

b
>
1

b > 1

,

0
>
α
>
1

0 > \alpha > 1

, each

θ
n

{\theta _n}

is an arbitrary number, and

Φ
\Phi

has period one. We show that there is a constant

C
>
0

C > 0

such that if

b
b

is large enough, then the Hausdorff dimension of the graph of

W
b

{W_b}

is bounded below by

2
−
α
−
(
C

/

ln
⁡
b
)

2 - \alpha - (C/\ln b)

. We also show that if a function

f
f

is convex Lipschitz of order

α
\alpha

, then the graph of

f
f

has

σ
\sigma

-finite measure with respect to Hausdorff’s measure in dimension

2
−
α

2 - \alpha

. The convex Lipschitz functions of order

α
\alpha

include Zygmund’s class

Λ
α

{\Lambda _\alpha }

. Our analysis shows that the graph of the classical van der Waerden-Tagaki nowhere differentiable function has

σ
\sigma

-finite measure with respect to

h
(
t
)
=
t

/

ln
⁡
(
1

/

t
)

h(t) = t/\ln (1/t)

.
On the theory of fundamental norming bounded biorthogonal systems in Banach spaces
Authors: Terenzi Paolo
Published: June 2010

Let

X
X

and

Y
Y

be quasi complementary subspaces of a separable Banach space

B
B

and let

(

z
n

)

({z_n})

be a sequence complete in

X
X

. Then (a) there exists a uniformly minimal norming

M
M

-basis

(

x
n

)

({x_n})

of

X
X

with

x
m

∈
span
⁡

(

z
n

)

n
⩾

q
m

{x_m} \in \operatorname {span} {({z_n})_{n \geqslant {q_m}}}

for every

m
m

,

q
m

→
∞

{q_m} \to \infty

; (b) if

(

x
n

)

({x_n})

is a uniformly minimal norming

M
M

-basis of

X
X

, there exists a uniformly minimal norming

M
M

-basis of

B
B

which is an extension of

(

x
n

)

({x_n})

; (c) there exists a uniformly minimal norming

M
M

-basis

(

x
n

)
∪
(

y
n

)

({x_n}) \cup ({y_n})

of

B
B

with

(

x
n

)
⊂
X

({x_n}) \subset X

and

(

y
n

)
⊂
Y

({y_n}) \subset Y

.
Porous sets and quasisymmetric maps
Authors: Väisälä Jussi
Published: June 2010

A set

A
A

in

R
n

{R^n}

is called porous if there is

α
>
0

\alpha > 0

such that every ball

B
¯

(
x
,
r
)

\overline B (x,r)

contains a point whose distance from

A
A

is at least

α
r

\alpha r

. We show that porosity is preserved by quasisymmetric maps, in particular, by bilipschitz maps. Local versions are also given.
The adelic zeta function associated with the space of binary cubic forms with coefficients in a function field
Authors: Datskovsky Boris A.
Published: June 2010

In this paper we study the adelic zeta function associated with the prehomogeneous vector space of binary cubic forms, defined over a function field. We establish its rationality, find its poles and residues and a simple functional equation that this zeta function satisfies.
Countably generated Douglas algebras
Authors: Izuchi Keiji
Published: June 2010

Under a certain assumption of

f
f

and

g
g

in

L
∞

{L^\infty }

which is considered by Sarason, a strong separation theorem is proved. This is available to study a Douglas algebra

[

H
∞

,

f
]

[{H^\infty },\,f]

generated by

H
∞

{H^\infty }

and

f
f

. It is proved that (1) ball

(
B

/

H
∞

+
C
)

(B/{H^\infty } + C)

does not have exposed points for every Douglas algebra

B
B

, (2) Sarason’s three functions problem is solved affirmatively, (3) some characterization of

f
f

for which

[

H
∞

,

f
]

[{H^\infty },\,f]

is singly generated, and (4) the

M
M

-ideal conjecture for Douglas algebras is not true.
Forbidden intersections
Authors: Frankl Peter , Rödl Vojtěch
Published: June 2010

About ten years ago P. Erdös conjectured that if

F

\mathcal {F}

is a family of subsets of

{
1
,
2
,
…
,
n
}

\{ 1,2, \ldots ,n\}

without

F
F

,

F
′

∈

F

F’ \in \mathcal {F}

,

|

F
∩

F
′

|

=
[
n

/

4
]

|F \cap F’| = [n/4]

, then

|

F

|

>

(
2
−
ε

)
n

|\mathcal {F}| > {(2 - \varepsilon )^n}

holds for some positive absolute constant

ε
\varepsilon

. Here this conjecture is proved in a stronger form (Theorem 1.1), which solves a

\mathdollar 250 problem of Erdös. Suppose

C

\mathcal {C}

is a code (i.e., a collection of sequences of length

n
n

) over an alphabet of

q
q

elements, where

1
2

>
δ
>
0

\tfrac {1} {2} > \delta > 0

is arbitrary. Suppose further that there are no two codewords at Hamming distance

d
d

where

d
d

is a fixed integer,

δ
n
>
d
>
(
1
−
δ
)
n

\delta n > d > (1 - \delta )n

, and

d
d

is even if

q
=
2

q = 2

. Then

|

C

|

>

(
q
−
ε

)
n

|\mathcal {C}| > {(q - \varepsilon )^n}

, where

ε
>
0

\varepsilon > 0

depends only on

q
q

and

δ
\delta

. The following conjecture of Erdös and Szemerédi is also proved: If

F

\mathcal {F}

is a family of subsets of

{
1
,
2
,
…
,
n
}

\{ 1,2, \ldots ,n\}

not containing a weak

Δ
\Delta

-system of size

r
r

(cf. Definition 1.8), then

|

F

|

>

(
2
−

ε
r

)
n

|\mathcal {F}| > {(2 - {\varepsilon _r})^n}

,

ε
r

>
0

{\varepsilon _r} > 0

holds. An old conjecture of Larman and Rogers is established in the following stronger form: Let

A

\mathcal {A}

be a collection of

4
n

4n

-dimensional

(
±
1
)

( \pm 1)

-vectors,

r
⩾
2

r \geqslant 2

is a fixed integer. Suppose that

A
A

does not contain

r
r

pairwise orthogonal vectors. Then

|

A

|

>

(
2
−
ε

)

4
n

|\mathcal {A}| > {(2 - \varepsilon )^{4n}}

. All these results can be deduced from our most general result (Theorem 1.16) which concerns the intersection pattern of families of partitions. This result has further implications in Euclidean Ramsey theory as well as for isometric embeddings into the Hamming space

H
(
n
,
q
)

H(n,q)

(cf. Theorem 9.1).
Matrices whose powers are 𝑀-matrices or 𝑍-matrices
Authors: Friedland Shmuel , Hershkowitz Daniel , Schneider Hans
Published: June 2010

A matrix

A
A

all of whose (positive) powers are

Z
Z

-matrices is called here a

Z
M

ZM

-matrix. A matrix is called a

Z
M
A

ZMA

-matrix if all powers of

A
A

are irreducible

Z
Z

-matrices. We prove that the spectrum of a

Z
M
A

ZMA

-matrix is real and only the eigenvalue minimal in absolute value may be negative. By means of an operation called inflation which generalizes the Kronecker product of two matrices, we determine the class of

Z
M
A

ZMA

-matrices of order

n
n

in terms of the classes of

Z
M
A

ZMA

-matrices of smaller orders. We use this result to show that a

Z
M
A

ZMA

-matrix is positively diagonally similar to a symmetric matrix. Similar results hold for

M
M
A

MMA

-matrices which are defined in analogy with

Z
M
A

ZMA

-matrices in terms of

M
M

-matrices, and for

Z
M
O

ZMO

-matrices which are defined to be

Z
M

ZM

-matrices such that all odd powers are irreducible and all even powers reducible. We also prove that a matrix is a

Z
M
A

ZMA

-,

Z
M
O

ZMO

- or

M
M
A

MMA

-matrix under apparently weaker conditions. If

A
A

is a real matrix such that all sufficiently large powers of

A
A

are

Z
Z

-matrices, then

A
A

is a

Z
M
A

ZMA

-matrix if

A
2

{A^2}

is irreducible,

A
A

is a

Z
M
O

ZMO

-matrix if

A
A

is irreducible and

A
2

{A^2}

is reducible, and

A
A

is an

M
M
A

MMA

-matrix if

A
A

is an irreducible

Z
Z

-matrix and some odd power of

A
A

is an

M
M

-matrix.
Fredholm, Hodge and Liouville theorems on noncompact manifolds
Authors: Lockhart Robert
Published: June 2010

Fredholm, Liouville, Hodge, and

L
2

{L^2}

-cohomology theorems are proved for Laplacians associated with a class of metrics defined on manifolds that have finitely many ends. The metrics are conformal to ones that are asymptotically translation invariant. They are not necessarily complete. The Fredholm results are, of necessity, with respect to weighted Sobolev spaces. Embedding and compact embedding theorems are also proved for these spaces.
New results on automorphic integrals and their period functions
Authors: Cavaliere Richard A.
Published: June 2010

Automorphic integrals, being generalizations of automorphic forms on discrete subgroups of

S
L
(
2
,

R

)

SL(2,\,\mathbf {R})

, share properties similar to those of forms. In this article I obtain a natural boundary result for integrals which is similar to that which holds for forms. If an automorphic integral on a given group behaves like a form on a subgroup of finite index (i.e., the period functions are identically zero), then in fact the integral must be a form on the whole group. Specializing to modular integrals with integer dimension I obtain a lower bound on the number of poles of the period functions which, of necessity, lie in quadratic extensions of the rationals.
Produced representations of Lie algebras and Harish-Chandra modules
Authors: Heumos Michael J.
Published: June 2010

The comultiplication of the universal enveloping algebra of a Lie algebra is used to give modules produced from a subalgebra, an additional compatible structure of a module over an algebra of formal power series. When only the

k

\mathfrak {k}

-finite elements of this algebra act on a module, conditions are given that insure that it is the Harish-Chandra module of a produced module. The results are then applied to Zuckerman derived functor modules for reductive Lie algebras. The main application describes a setting where the Zuckerman functors and production from a subalgebra commute.
The structure of groups which are almost the direct sum of countable abelian groups
Authors: Mekler Alan H.
Published: June 2010

The notion of being in standard form is defined for the groups described in the title of the paper which are of cardinality

ω
1

{\omega _1}

. Being in "standard form" is a structural description of the group. The consequences of being in standard form are explored, sometimes with the use of additional set-theoretic axioms. It is shown that it is consistent that a large class of these groups, including every weakly

ω
1

{\omega _1}

-separable

ω
1

{\omega _1}

-

Σ
\Sigma

-cyclic group of cardinality

ω
1

{\omega _1}

, can be put in standard form.
Riccati techniques and variational principles in oscillation theory for linear systems
Authors: Butler G. J. , Erbe L. H. , Mingarelli A. B.
Published: June 2010

We consider the seond order differential system

(
1
)

Y
+
Q
(
t
)
Y
=
0

(1)\,Y + Q(t)Y = 0

, where

Q
Q

,

Y
Y

are

n
×
n

n \times n

matrices with

Q
=
Q
(
t
)

Q = Q(t)

a continuous symmetric matrix-valued function,

t
∈
[
a
,

+
∞
]

t \in [a,\, + \infty ]

. We obtain a number of sufficient conditions in order that all prepared solutions

Y
(
t
)

Y(t)

of

(
1
)

(1)

are oscillatory. Two approaches are considered, one based on Riccati techniques and the other on variational techniques, and involve assumptions on the behavior of the eigenvalues of

Q
(
t
)

Q(t)

(or of its integral). These results extend some well-known averaging techniques for scalar equations to system

(
1
)

(1)

.
A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions
Authors: Borwein J. M. , Preiss D.
Published: June 2010

We show that, typically, lower semicontinuous functions on a Banach space densely inherit lower subderivatives of the same degree of smoothness as the norm. In particular every continuous convex function on a space with a Gâteaux (weak Hadamard, Fréchet) smooth renorm is densely Gâteaux (weak Hadamard, Fréchet) differentiable. Our technique relies on a more powerful analogue of Ekeland’s variational principle in which the function is perturbed by a quadratic-like function. This "smooth" variational principle has very broad applicability in problems of nonsmooth analysis.
Conjugacy classes in algebraic monoids
Authors: Putcha Mohan S.
Published: June 2010

Let

M
M

be a connected linear algebraic monoid with zero and a reductive group of units

G
G

. The following theorem is established. Theorem. There exist affine subsets

M
1

,
…
,

M
k

{M_1}, \ldots ,{M_k}

of

M
M

, reductive groups

G
1

,
…
,

G
k

{G_1}, \ldots ,{G_k}

with antiautomorphisms

∗

^{\ast }

, surjective morphisms

θ
i

:

M
i

→

G
i

{\theta _i}:{M_i} \to {G_i}

, such that: (1) Every element of

M
M

is conjugate to an element of some

M
i

{M_i}

, and (2) Two elements

a
a

,

b
b

in

M
i

{M_i}

are conjugate in

M
M

if and only if there exists

x
∈

G
i

x \in {G_i}

such that

x

θ
i

(
a
)

x

∗

=

θ
i

(
b
)

x{\theta _i}(a){x^{\ast }} = {\theta _i}(b)

. As a consequence, it is shown that

M
M

is a union of its inverse submonoids.
Towers and injective cohomology algebras
Authors: Goerss Paul , Smith Larry
Published: June 2010

Let

Y
Y

be a space of finite type such that

H

∗

Y

{H^{\ast }}Y

is injective as an unstable algebra over the Steenrod algebra

A
A

and such that

H
¯

∗

Y

{\overline H ^{\ast }}Y

is

A
A

-unbounded. Let

X
X

be a simply connected

p
p

-complete space. Then any map of

A
A

-algebras

f
:

H

∗

Ω
X
→

H

∗

Y

f:{H^{\ast }}\Omega X \to {H^{\ast }}Y

can be realized as a map of spaces.
An operator-theoretic formulation of asynchronous exponential growth
Authors: Webb G. F.
Published: June 2010

A strongly continuous semigroup of bounded linear operators

T
(
t
)

T(t)

,

t
⩾
0

t \geqslant 0

, in the Banach space

X
X

has asynchronous exponential growth with intrinsic growth constant

λ
0

{\lambda _0}

provided that there is a nonzero finite rank operator

P
0

{P_0}

in

X
X

such that

lim

t
→
∞

e

−

λ
0

t

T
(
t
)
=

P
0

{\lim _{t \to \infty }}{e^{ - {\lambda _0}t}}T(t) = {P_0}

. Necessary and sufficient conditions are established for

T
(
t
)

T(t)

,

t
⩾
0

t \geqslant 0

, to have asynchronous exponential growth. Applications are made to a maturity-time model of cell population growth and a transition probability model of cell population growth.
The Szegő kernel as a singular integral kernel on a family of weakly pseudoconvex domains
Authors: Diaz Katharine Perkins
Published: June 2010

The Szegö kernels on the weakly pseudoconvex domains

{
Im
⁡

z
2

>

|

z
1

|

2
k

}

\{ \operatorname {Im} {z_2} > |{z_1}{|^{2k}}\}

,

k
∈

Z
+

k \in {Z^ + }

, have been computed by Greiner and Stein. After constructing a global, nonisotropic pseudometric suitable for Calderón-Zygmund singular integral theory on the boundaries of the domains, we study principal value operators associated to these Szegö kernels. We prove that the principal value operators are bounded on

L
p

{L^p}

for

1
>
p
>
∞

1 > p > \infty

, and that they preserve certain nonisotropic Lipschitz classes. We then derive a Plemelj formula that relates the principal value operators to the Szegö projections. From this formula we deduce that the Szegö projections are also bounded on

L
p

{L^p}

, for

1
>
p
>
∞

1 > p > \infty

, and that they preserve the same nonisotropic Lipschitz classes.
Comparison between analytic capacity and the Buffon needle probability
Authors: Murai Takafumi
Published: June 2010

We show that analytic capacity and the Buffon needle probability are not comparable.
Complementation in Kreĭn spaces
Authors: de Branges Louis
Published: June 2010

A generalization of the concept of orthogonal complement is introduced in complete and decomposable complex vector spaces with scalar product.
Weyl groups and the regularity properties of certain compact Lie group actions
Authors: Straume Eldar
Published: June 2010

The geometric weight system of a

G
G

-manifold

X
X

(acyclic or spherical) is the nonlinear analogue of the weight system of a linear representation. We study the possible realization of a given

G
G

-weight pattern, via the interaction between roots, weights and the Weyl group, together with various fixed point results of P. A. Smith type. If the orbit structure is reasonably simple, then the

G
G

-weight pattern must in fact coincide with that of a simple representation. This in turn implies that

X
X

is (orthogonally) modeled on the linear

G
G

-space, e.g., with the same orbit types. In particular, complete results in this direction are obtained for a certain family of

G
G

-manifolds,

G
G

a classical group. In this family the weight patterns are of

2
2

-parametric type, and it includes essentially all cases where the principal isotropy type is nontrivial. This family also covers many cases with trivial principal isotropy type.
A Brouwer translation theorem for free homeomorphisms
Authors: Slaminka Edward E.
Published: June 2010

We prove a generalization of the Brouwer Translation Theorem which applies to a class of homeomorphisms (free homeomorphisms) which admit fixed points, but retain a dynamical property of fixed point free orientation preserving homeomorphsims. That is, if

h
:

M
2

→

M
2

h:{M^2} \to {M^2}

is a free homeomorphism where

M
2

{M^2}

is a surface, then whenever

D
D

is a disc and

h
(
D
)
∩
D
=
∅

h(D) \cap D = \emptyset

, we have that

h
n

(
D
)
∩
D
=
∅

{h^n}(D) \cap D = \emptyset

for all

n
≠
0

n \ne 0

. Theorem. Let

h
h

be a free homeomorphism of

S
2

{S^2}

, the two-sphere, with finite fixed point set

F
F

. Then each

p
∈

S
2

−
F

p \in {S^2} - F

lies in the image of an embedding

ϕ
p

:
(

R
2

,

0
)
→
(

S
2

−
F
,

p
)

{\phi _p}:({R^2},\,0) \to ({S^2} - F,\,p)

such that: (i)

h

ϕ
p

=

ϕ
p

τ

h{\phi _p} = {\phi _p}\tau

, where

τ
(
z
)
=
z
+
1

\tau (z) = z + 1

is the canonical translation of the plane, and (ii) the image of each vertical line under

ϕ
p

{\phi _p}

is closed in

S
2

−
F

{S^2} - F

.
Testing analyticity on rotation invariant families of curves
Authors: Globevnik Josip
Published: June 2010

Let

Γ
⊂
C

\Gamma \subset C

be a piecewise smooth Jordan curve, symmetric with respect to the real axis, which contains the origin in its interior and which is not a circle centered at the origin. Let

Ω
\Omega

be the annulus obtained by rotating

Γ
\Gamma

around the origin. We characterize the curves

Γ
\Gamma

with the property that if

f
∈
C
(
Ω
)

f \in C(\Omega )

is analytic on

s
Γ

s\Gamma

for every

s
s

,

|

s

|

=
1

|s| = 1

, then

f
f

is analytic in Int

Ω
\Omega

.
Eigensharp graphs: decomposition into complete bipartite subgraphs
Authors: Kratzke Thomas , Reznick Bruce , West Douglas
Published: June 2010

Let

τ
(
G
)

\tau (G)

be the minimum number of complete bipartite subgraphs needed to partition the edges of

G
G

, and let

r
(
G
)

r(G)

be the larger of the number of positive and number of negative eigenvalues of

G
G

. It is known that

τ
(
G
)
⩾
r
(
G
)

\tau (G) \geqslant r(G)

; graphs with

τ
(
G
)
=
r
(
G
)

\tau (G) = r(G)

are called eigensharp. Eigensharp graphs include graphs, trees, cycles

C
n

{C_n}

with

n
=
4

n = 4

or

n
≠
4
k

n \ne 4k

, prisms

C
n

◻

K
2

{C_n}\square {K_2}

with

n
≠
3
k

n \ne 3k

, "twisted prisms" (also called "Möbius ladders")

M
n

{M_n}

with

n
=
3

n = 3

or

n
≠
3
k

n \ne 3k

, and some Cartesian products of cycles. Under some conditions, the weak (Kronecker) product of eigensharp graphs is eigensharp. For example, the class of eigensharp graphs with the same number of positive and negative eigenvalues is closed under weak products. If each graph in a finite weak product is eigensharp, has no zero eigenvalues, and has a decomposition into

τ
(
G
)

\tau (G)

stars, then the product is eigensharp. The hypotheses in this last result can be weakened. Finally, not all weak products of eigensharp graphs are eigensharp.
A characterization of two weight norm inequalities for fractional and Poisson integrals
Authors: Sawyer Eric T.
Published: June 2010

For

1
>
p
⩽
q
>
∞

1 > p \leqslant q > \infty

and

w
(
x
)

w(x)

,

v
(
x
)

v(x)

nonnegative functions on

R

n

{{\mathbf {R}}^n}

, we show that the weighted inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis integral StartAbsoluteValue upper T f EndAbsoluteValue Superscript q Baseline w right-parenthesis Superscript 1 slash q Baseline less-than-or-slanted-equals upper C left-parenthesis integral f Superscript p Baseline v right-parenthesis Superscript 1 slash p">

(

∫

|

T
f

|

q

w

)

1

/

q

⩽
C

(

∫

f
p

v

)

1

/

p

{\left ( {\int {|Tf{|^q}w} } \right )^{1/q}} \leqslant C{\left ( {\int {{f^p}v} } \right )^{1/p}}

\]
holds for all

f
⩾
0

f \geqslant 0

if and only if both
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral left-bracket upper T left-parenthesis chi Subscript upper Q Baseline v Superscript 1 minus p prime Baseline right-parenthesis right-bracket Superscript q Baseline w less-than-or-slanted-equals upper C 1 left-parenthesis integral Underscript upper Q Endscripts v Superscript 1 minus p prime Baseline right-parenthesis Superscript q slash p Baseline greater-than normal infinity">

∫

[
T
(

χ
Q

v

1
−

p
′

)
]

q

w
⩽

C
1

(

∫
Q

v

1
−

p
′

)

q

/

p

>
∞

\int {{{[T({\chi _Q}{v^{1 - p’}})]}^q}w \leqslant {C_1}{{\left ( {\int _Q {{v^{1 - p’}}} } \right )}^{q/p}} > \infty }

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral left-bracket upper T left-parenthesis chi Subscript upper Q Baseline w right-parenthesis right-bracket Superscript p prime Baseline v Superscript 1 minus p prime less-than-or-slanted-equals upper C 2 left-parenthesis integral Underscript upper Q Endscripts w right-parenthesis Superscript p prime slash q prime Baseline greater-than normal infinity">

∫

[
T
(

χ
Q

w
)
]

p
′

v

1
−

p
′

⩽

C
2

(

∫
Q

w

)

p
′

/

q
′

>
∞

{\int {{{[T({\chi _Q}w)]}^{p’}}{v^{1 - p’}} \leqslant {C_2}\left ( {\int _Q w } \right )} ^{p’/q’}} > \infty

\]
hold for all dyadic cubes

Q
Q

. Here

T
T

denotes a fractional integral or, more generally, a convolution operator whose kernel

K
K

is a positive lower semicontinuous radial function decreasing in

|

x

|

|x|

and satisfying

K
(
x
)
⩽
C
K
(
2
x
)

K(x) \leqslant CK(2x)

,

x
∈

R

n

x \in {{\mathbf {R}}^n}

. Applications to degenerate elliptic differential operators are indicated. In addition, a corresponding characterization of those weights

v
v

on

R

n

{{\mathbf {R}}^n}

and

w
w

on

R

+

n
+
1

{\mathbf {R}}_ + ^{n + 1}

for which the Poisson operator is bounded from

L
p

(
v
)

{L^p}(v)

to

L
q

(
w
)

{L^q}(w)

is given.
On certain fibred ribbon disc pairs
Authors: Aitchison Iain R. , Silver Daniel S.
Published: June 2010

We prove that for any free group automorphism

ϕ

∗

{\phi ^{\ast }}

having a specified form there exists an invertible ribbon disc pair

(

B
4

,

D
2

)

({B^4},\,{D^2})

such that the closure of

B
4

−
nbd
⁡
(

D
2

)

{B_4} - \operatorname {nbd} ({D^2})

fibres over the circle with fibre a handlebody and monodromy equal to

ϕ

∗

{\phi ^{\ast }}

. We apply this to obtain results about ribbon

1
1

- and

2
2

-knots.
The Casson-Gordon invariants in high-dimensional knot theory
Authors: Ruberman Daniel
Published: June 2010

The Casson-Gordon invariants of knots in all dimensions are interpreted in terms of surgery theory. Applications are given to finding non-doubly slice knots, and doubly slice knots which are not the double of a disk knot. In even dimensions, the property of being doubly slice is shown to be largely homotopy theoretic, while in odd dimensions the surgery-theoretic method shows such properties to depend on more than the homotopy type.
A strong contractivity property for semigroups generated by differential operators
Authors: Kauffman Robert M.
Published: June 2010

Frequently, nonconservative semigroups generated by partial differential operators in

L

2
,
ρ

(

R
k

)

{L_{2,\rho }}({R^k})

have the property that initial conditions which are large at

|

x

|

=
∞

|x| = \infty

become immediately small at infinity for all

t
>
0

t > 0

. This property is related to the rate of decay of eigenfunctions of the differential operator. In this paper this phenomenon is investigated for a large class of differential operators of second and higher order. New estimates on the rate of decay of the eigenfunctions are included, which are related in special cases to those of Agmon.
Regularity of solutions of two-dimensional Monge-Ampère equations
Authors: Schulz Friedmar , Liao Liang Yuan
Published: June 2010

In the paper we investigate the regularity of solutionsz(x,y)∈C1,1(Ω)z(x,\,y) \in {C^{1,1}}(\Omega ), resp.C1,1(Ω¯){C^{1,1}}(\overline \Omega )of elliptic Monge-Ampére equations of the form\[Ar+2Bs+Ct+(rt−s2)=E.Ar + 2Bs + Ct + (rt - {s^2}) = E.\]It is shown thatz(x,y)∈C2,α(Ω)z(x,\,y) \in {C^{2,\alpha }}(\Omega ), resp.C2,α(Ω¯){C^{2,\alpha }}(\overline \Omega ), with corresponding a priori estimates, ifA,B,C,E∈Cα(Ω×R3)A,\,B,\,C,\,E \in {C^\alpha }(\Omega \times {{\mathbf {R}}^3}). The results are deduced via the Campanato technique for equations of variational structure invoking a Legendre-like transformation.
Area-minimizing integral currents with boundaries invariant under polar actions
Authors: Lander Julian C.
Published: June 2010

Let

G
G

be a compact, connected subgroup of

S
O
(
n
)

SO(n)

acting on

R

n

{{\mathbf {R}}^n}

, and let the action of

G
G

be polar. (Polar actions include the adjoint action of a Lie group

H
H

on the tangent space to the symmetric space

G

/

H

G/H

at the identity coset.) Let

B
B

be an

(
m
−
1
)

(m - 1)

-dimensional submanifold without boundary, invariant under the action of

G
G

, and lying in the union of the principal orbits of

G
G

. It is shown that, if

S
S

is an area-minimizing integral current with boundary

B
B

, then

S
S

is invariant under the action of

G
G

. This result is extended to a larger class of boundaries, and to a class of parametric integrands including the area integrand.
Trace identities and 𝑍/2𝑍-graded invariants
Authors: Berele Allan
Published: June 2010

We prove Razmyslov’s theorem on trace identities for

M

k
,

l

{M_{k,\,l}}

using the invariant theory of

pl
⁡
(
k
,

l
)

\operatorname {pl} (k,\,l)

.
Total linking number modules
Authors: Neto Oziride Manzoli
Published: June 2010

Given a codimension two link

L
L

in a sphere

S
k

{S^k}

with complement

X
=

S
k

−
L

X = {S^k} - L

, the total linking number covering of

L
L

is the covering

X
^

→
X

\hat X \to X

associated to the surjection

π
1

(
X
)
→
Z

{\pi _1}(X) \to Z

defined by sending the meridians to

1
1

. The homology

H

∗

(

X
^

)

{H_{\ast }}(\hat X)

define weaker invariants than the homology of the universal abelian covering of

L
L

. The groups

H
i

(

X
^

)

{H_i}(\hat X)

are modules over

Z

[

t
,

t

−
1

]

Z\left [ {t,\,{t^{ - 1}}} \right ]

and this work gives an algebraic characterization of these modules for

k
⩾
4

k \geqslant 4

except for the pseudo null part of

H
1

(

X
^

)

{H_1}(\hat X)

.
Remarks on Grassmannian supermanifolds
Authors: Sánchez-Valenzuela Oscar Adolfo
Published: June 2010

This paper studies some aspects of a particular class of examples of supermanifolds; the supergrassmannians, introduced in [Manin]. Their definition, in terms of local data and glueing isomorphisms, is reviewed. Explicit formulas in local coordinates are given for the Lie group action they come equipped with. It is proved that, for those supergrassmannians whose underlying manifold is an ordinary grassmannian, their structural sheaf can be realized as the sheaf of sections of the exterior algebra bundle of some canonical vector bundle. This realization holds true equivariantly for the Lie group action in question, thus making natural in these cases the identification given in [Batchelor]. The proof depends on the computation of the transition functions of the supercotangent bundle as defined in a previous work [OASV 2]. Finally, it is shown that there is a natural supergroup action involved (in the sense of [OASV 3]) and hence, the supergrassmannians may be regarded as examples of superhomogeneous spaces—a notion first introduced in [Kostant]. The corresponding Lie superalgebra action can be realized as superderivations of the structural sheaf; explicit formulas are included for those supergrassmannians identifiable with exterior algebra vector bundles.
Cauchy problem of hyperbolic conservation laws in multidimensional space with intersecting jump initial data
Authors: Li De Ning
Published: June 2010

Cauchy problem of hyperbolic conservation laws in multidimensional space is considered, where the initial data have several jump discontinuity surfaces which develop into shock fronts intersecting at a common submanifold. Local existence is proved, assuming compatible conditions and uniform stability. For isentropic flow in

2
2

-dimensional space, the interaction of two shock fronts and the nonexistence of three intersecting shock fronts are discussed.
A remark on a theorem of Vo Van Tan
Authors: Colţoiu Mihnea
Published: June 2010

In this paper we consider the following problem: Let

(
X
,

S
)

(X,\,S)

be a

1
1

-convex manifold with

1
1

-dimensional exceptional set

S
S

. Does it follow that

X
X

is a Kähler manifold? Although this was answered in the affirmative by Vo Van Tan in two papers, we show that his proofs are wrong. It is also shown that the Kähler condition implies that any strongly pseudoconvex domain

D
⋐
X

D \Subset X

is embeddable, i.e. can be realized as a closed analytic submanifold in some

C

N

×

P

M

{{\mathbf {C}}^N} \times {{\mathbf {P}}_M}

. On the other hand it is known that under some additional assumptions on

S
S

(

S
S

is not rational or

S
≃

P

1

S \simeq {{\mathbf {P}}^1}

and

dim
⁡
X
≠
3

\operatorname {dim} X \ne 3

) it follows that

X
X

is embeddable, in particular it is Kählerian.
Finite basis theorems for relatively congruence-distributive quasivarieties
Authors: Pigozzi Don
Published: June 2010

Q

\mathcal {Q}

is any quasivariety. A congruence relation

Θ
\Theta

on a member

A

{\mathbf {A}}

of

Q

\mathcal {Q}

is a

Q

\mathcal {Q}

-congruence if

A

/

Θ
∈

Q

{\mathbf {A}}/\Theta \in \mathcal {Q}

. The set

C
o

n

Q

A

Co{n_\mathcal {Q}}{\mathbf {A}}

of all

Q

\mathcal {Q}

-congruences is closed under arbitrary intersection and hence forms a complete lattice

C
o

n

Q

A

{\mathbf {Co}}{{\mathbf {n}}_\mathcal {Q}}{\mathbf {A}}

.

Q

\mathcal {Q}

is relatively congruence-distributive if

C
o

n

Q

A

{\mathbf {Co}}{{\mathbf {n}}_\mathcal {Q}}{\mathbf {A}}

is distributive for every

A

∈

Q

{\mathbf {A}} \in \mathcal {Q}

. Relatively congruence-distributive quasivarieties occur naturally in the theory of abstract data types.

Q

\mathcal {Q}

is finitely generated if it is generated by a finite set of finite algebras. The following generalization of Baker’s finite basis theorem is proved. Theorem I. Every finitely generated and relatively congruence-distributive quasivariety is finitely based. A subquasivariety

R

\mathcal {R}

of an arbitrary quasivariety

Q

\mathcal {Q}

is called a relative subvariety of

Q

\mathcal {Q}

if it is of the form

V

∩

Q

\mathcal {V} \cap \mathcal {Q}

for some variety

V

\mathcal {V}

, i.e., a base for

R

\mathcal {R}

can be obtained by adjoining only identities to a base for

Q

\mathcal {Q}

. Theorem II. Every finitely generated relative subvariety of a relatively congruence-distributive quasivariety is finitely based. The quasivariety of generalized equality-test algebras is defined and the structure of its members studied. This gives rise to a finite algebra whose quasi-identities are finitely based while its identities are not. Connections with logic and the algebraic theory of data types are discussed.
Local uncertainty inequalities for locally compact groups
Authors: Price John F. , Sitaram Alladi
Published: June 2010

Let

G
G

be a locally compact unimodular group equipped with Haar measure

m
m

,

G
^

\hat G

its unitary dual and

μ
\mu

the Plancherel measure (or something closely akin to it) on

G
^

\hat G

. When

G
G

is a euclidean motion group, a non-compact semisimple Lie group or one of the Heisenberg groups we prove local uncertainty inequalities of the following type: given

θ
∈

[

0
,

1
2

)

\theta \in \left [ {0,\tfrac {1} {2}} \right .)

there exists a constant

K
θ

{K_\theta }

such that for all

f
f

in a certain class of functions on

G
G

and all measurable

E
⊆

G
^

E \subseteq \hat G

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis integral Underscript upper E Endscripts trace left-parenthesis pi left-parenthesis f right-parenthesis Superscript asterisk Baseline pi left-parenthesis f right-parenthesis right-parenthesis d mu left-parenthesis pi right-parenthesis right-parenthesis Superscript 1 slash 2 Baseline less-than-or-slanted-equals upper K Subscript theta Baseline mu left-parenthesis upper E right-parenthesis Superscript theta Baseline StartAbsoluteValue EndAbsoluteValue phi Subscript theta Baseline f StartAbsoluteValue EndAbsoluteValue Subscript 2">

(

∫
E

Tr
⁡
(
π

(
f
)

∗

π
(
f
)
)

d
μ
(
π
)

)

1

/

2

⩽

K
θ

μ

(
E

)
θ

|

|

ϕ
θ

f

|

|

2

{\left ( {\int _E {\operatorname {Tr} (\pi {{(f)}^{\ast }}\pi (f))\,d\mu (\pi )} } \right )^{1/2}} \leqslant {K_\theta }\mu {(E)^\theta }||{\phi _\theta }f|{|_2}

\]
where

ϕ
θ

{\phi _\theta }

is a certain weight function on

G
G

(for which an explicit formula is given). When

G
=

R

k

G = {{\mathbf {R}}^k}

the inequality has been established with

ϕ
θ

(
x
)
=

|

x

|

k
θ

{\phi _\theta }(x) = |x{|^{k\theta }}

.
The closing lemma for generalized recurrence in the plane
Authors: Peixoto Maria Lúcia Alvarenga
Published: June 2010

We prove a version of the Closing Lemma for

C
r

{C^r}

vector fields on the plane,

r
⩾
1

r \geqslant 1

, and for a kind of recurrence obtained using the concept of prolongational limit sets. We call it generalized recurrence. Given a nonperiodic point

p
p

in the generalized recurrent set we perturb the vector field in order to get a new vector field arbitrarily close to it, with a closed orbit through

p
p

.
Stable manifolds in the method of averaging
Authors: Schecter Stephen
Published: June 2010

Consider the differential equation

z
˙

=
ε
f
(
z
,

t
,

ε
)

\dot z = \varepsilon f(z,\,t,\,\varepsilon )

, where

f
f

is

T
T

periodic in

t
t

and

ε
>
0

\varepsilon > 0

is a small parameter, and the averaged equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove z With dot equals ModifyingAbove f With bar left-parenthesis z right-parenthesis colon equals left-parenthesis 1 slash upper T right-parenthesis integral Subscript 0 Superscript upper T Baseline f left-parenthesis z comma t comma 0 right-parenthesis d t">

z
˙

=

f
¯

(
z
)
:=
(
1

/

T
)

∫
0
T

f
(
z
,

t
,

0
)

d
t

\dot z = \overline f (z): = (1/T)\,\int _0^T {\,f(z,\,t,\,0)\,dt}

. Suppose the averaged equation has a hyperbolic equilibrium at

z
=
0

z = 0

with stable manifold

W
¯

\overline W

. Let

β
ε

(
t
)

{\beta _\varepsilon }(t)

denote the hyperbolic

T
T

-periodic solution of

z
˙

=
ε
f
(
z
,

t
,

ε
)

\dot z = \varepsilon f(z,\,t,\,\varepsilon )

near

z
≡
0

z \equiv 0

. We prove a result about smooth convergence of the stable manifold of

β
ε

(
t
)

{\beta _\varepsilon }(t)

to

W
¯

×

R

\overline W \times {\mathbf {R}}

as

ε
→
0

\varepsilon \to 0

. The proof uses ideas of Vanderbauwhede and van Gils about contractions on a scale of Banach spaces.
Smoothness up to the boundary for solutions of the nonlinear and nonelliptic Dirichlet problem
Authors: Xu C. J. , Zuily C.
Published: June 2010

For the Dirichlet problem associated with a general real second order p.d.e.

F
(
x
,

u
,

∇
u
,

∇
2

u
)
=
0

F(x,\,u,\,\nabla u,\,{\nabla ^2}u) = 0

in a smooth open set

Ω
\Omega

of

R

n

{{\mathbf {R}}^n}

, we prove smoothness up to the boundary of the solution

u
u

for which the linearized characteristic form is nonnegative and satisfies Hörmander’s brackets condition, the boundary of

Ω
\Omega

being noncharacteristic.
A Riemannian geometric invariant and its applications to a problem of Borel and Serre
Authors: Chen Bang-Yen , Nagano Tadashi
Published: June 2010

A new geometric invariant will be introduced, studied and determined on compact symmetric spaces.