bluuebunny/crossref_metadata_2025 huggingface dataset:
Perturbed dynamical systems with an attracting singularity and weak viscosity limits in Hamilton-Jacobi equations
Authors: Perthame B.
Published: June 2012

We give a new PDE proof of the Wentzell-Freidlin theorem concerning small perturbations of a dynamical system
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row upper L Subscript epsilon Baseline u Subscript epsilon Baseline equals minus StartFraction epsilon Over 2 EndFraction normal upper Delta u Subscript epsilon Baseline minus b dot nabla u 3 equals 0 in normal upper Omega comma 2nd Row u Subscript epsilon Baseline equals phi on partial-differential normal upper Omega period EndLayout">

L
ε

u
ε

=
−

ε
2

Δ

u
ε

−
b
⋅
∇

u
3

=
0

in

Ω
,

u
ε

=
φ

on

∂
Ω
.

\begin {gathered} {L_\varepsilon }{u_\varepsilon } = - \tfrac {\varepsilon } {2}\Delta {u_\varepsilon } - b \cdot \nabla {u_3} = 0\quad {\text {in}}\;\Omega , \hfill \\ {u_\varepsilon } = \varphi \quad {\text {on}}\;\partial \Omega . \hfill \\ \end {gathered}

\]
We prove that, if

b
b

has a single attractive singular point,

u
ε

{u_\varepsilon }

converges uniformly on compact subsets of

Ω
\Omega

, and with an exponential decay, to a constant

μ
\mu

, and we determine

μ
\mu

. We also treat the case of Neumann boundary condition. In order to do so, we perform the asymptotic analysis for some ergodic measure which leads to a study of the viscosity limit of a Hamilton-Jacobi equation. This is achieved under very general assumptions by using a weak formulation of the viscosity limits of these equations. Résumé. Nous donnons une nouvelle preuve, par des méthodes EDP, du théorème de Wentzell-Freidlin concernant les petites perturbations d’un système dynamique:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row upper L Subscript epsilon Baseline u Subscript epsilon Baseline equals minus StartFraction epsilon Over 2 EndFraction normal upper Delta u Subscript epsilon Baseline minus b dot nabla u Subscript epsilon Baseline equals 0 dans normal upper Omega comma 2nd Row u Subscript epsilon Baseline equals phi sur partial-differential normal upper Omega period EndLayout">

L
ε

u
ε

=
−

ε
2

Δ

u
ε

−
b
⋅
∇

u
ε

=
0

dans

Ω
,

u
ε

=
φ

sur

∂
Ω
.

\begin {gathered} {L_\varepsilon }{u_\varepsilon } = - \tfrac {\varepsilon } {2}\Delta {u_\varepsilon } - b \cdot \nabla {u_\varepsilon } = 0\quad {\text {dans}}\;\Omega , \hfill \\ {u_\varepsilon } = \varphi \quad {\text {sur}}\;\partial \Omega . \hfill \\ \end {gathered}

\]
Nous prouvons que, si

b
b

a un seul point singulier attractif, alors

u
ε

{u_\varepsilon }

converge vers une constant

μ
\mu

, uniformément sur tout compact, et avec une vitesse exponentielle. Nous déterminons

μ
\mu

. Nous traitons aussi le cas de conditions aux limites de Neuman. Pour cela, nous faisons l’analyse asymptotique d’une mesure ergodique intervenant naturellement dans le problème, ce qui revient à étudier la limite par viscosité évanescente dans une équation de Hamilton-Jacobi. Ceci est réalisé sous des hypothèses très générales gâce à un passage à la limite faible dans cette équation.
Prime ideals in differential operator rings. Catenarity
Authors: Brown K. A. , Goodearl K. R. , Lenagan T. H.
Published: June 2012

Let

R
R

be a commutative algebra over the commutative ring

k
k

, and let

Δ
=
{

δ
1

,
…
,

δ
n

}

\Delta = \{ {\delta _1}, \ldots ,{\delta _n}\}

be a finite set of commuting

k
k

-linear derivations from

R
R

to

R
R

. Let

T
=
R
[

θ
1

,
…
,

θ
n

;

δ
1

,
…
,

δ
n

]

T = R[{\theta _1}, \ldots ,{\theta _n};{\delta _1}, \ldots ,{\delta _n}]

be the corresponding ring of differential operators. We define and study an isomorphism of left

R
R

-modules between

T
T

and its associated graded ring

R
[

x
1

,
…
,

x
n

]

R[{x_1}, \ldots ,{x_n}]

, a polynomial ring over

R
R

. This isomorphism is used to study the prime ideals of

T
T

, with emphasis on the question of catenarity. We prove that

T
T

is catenary when

R
R

is a commutative noetherian universally catenary

k
k

-algebra and one of the following cases occurs: (A)

k
k

is a field of characteristic zero and

Δ
\Delta

acts locally finitely; (B)

k
k

is a field of positive characteristic; (C)

k
k

is the ring of integers,

R
R

is affine over

k
k

, and

Δ
\Delta

acts locally finitely.
Extensions of valuation rings in central simple algebras
Authors: Brungs H.-H. , Gräter J.
Published: June 2012

Certain subrings

R
R

of simple algebras

Q
Q

, finite dimensional over their center

K
K

, are studied. These rings are called

Q
Q

-valuation rings since they share many properties with commutative valuation rings. Let

V
V

be a valuation ring of

K
K

, the center of

Q
Q

, and let

R

\mathcal {R}

be the set of

Q
Q

-valuation rings

R
R

in

Q
Q

with

R
∩
K
=
V

R \cap K = V

, then

|

R

|

≥
1

\left | \mathcal {R} \right | \geq 1

. This extension theorem, which does not hold if one considers only total valuation rings, was proved by N. I. Dubrovin. Here, first a somewhat different proof of this result is given and then information about the set

R

\mathcal {R}

is obtained. Theorem. The elements in

R

\mathcal {R}

are conjugate if

V
V

has finite rank. Theorem. The elements in

R

\mathcal {R}

are total valuation rings if

R

\mathcal {R}

contains one total valuation ring. In this case

Q
Q

is a division ring. Theorem.

R

\mathcal {R}

if

R

\mathcal {R}

contains an invariant total valuation ring.
A classification of Baire class 1 functions
Authors: Kechris A. S. , Louveau A.
Published: June 2012

We study in this paper various ordinal ranks of (bounded) Baire class

1
1

functions and we show their essential equivalence. This leads to a natural classification of the class of bounded Baire class

1
1

functions

B

1

{\mathcal {B}_1}

in a transfinite hierarchy

B

1
ξ

(
ξ
>

ω
1

)

\mathcal {B}_1^\xi (\xi > {\omega _1})

of "small" Baire classes, for which (for example) an analysis similar to the Hausdorff-Kuratowski analysis of

Δ
2
0

\Delta _2^0

sets via transfinite differences of closed sets can be carried out. The notions of pseudouniform convergence of a sequence of functions and optimal convergence of a sequence of continuous functions to a Baire class

1
1

function

f
f

are introduced and used in this study.
A Hurewicz spectral sequence for homology
Authors: Blanc David A.
Published: June 2012

For any connected space

X

{\mathbf {X}}

and ring

R
R

, we describe a first-quadrant spectral sequence converging to

H
~

∗

(

X

;
R
)

{\tilde H_*}({\bf {X}};R)

, whose

E
2

{E^2}

-term depends only on the homotopy groups of

X

{\mathbf {X}}

and the action of the primary homotopy operations on them. We show that (for simply connected

X

{\mathbf {X}}

) the

E
2

{E^2}

-term vanishes below a line of slope

1

/

2

1/2

; computing part of the

E
2

{E^2}

-term just above this line, we find a certain periodicity, which shows, in particular, that this vanishing line is best possible. We also show how the differentials in this spectral sequence can be used to compute certain Toda brackets.
Modular representation theory of finite groups with T.I. Sylow 𝑝-subgroups
Authors: Blau H. I. , Michler G. O.
Published: June 2012

Let

p
p

be a fixed prime, and let

G
G

be a finite group with a T.I. Sylow

p
p

-subgroup

P
P

. Let

N
=

N
G

(
P
)

N = {N_G}(P)

and let

k
(
G
)

k(G)

be the number of conjugacy classes of

G
G

. If

z
(
G
)

z(G)

denotes the number of

p
p

-blocks of defect zero, then we show in this article that

z
(
G
)
=
k
(
G
)
−
k
(
N
)

z(G) = k(G) - k(N)

. This result confirms a conjecture of J. L. Alperin. Its proof depends on the classification of the finite simple groups. Brauer’s height zero conjecture and the Alperin-McKay conjecture are also verified for finite groups with a T.I. Sylow

p
p

-subgroup.
The Jacobian module of a Lie algebra
Authors: Brennan J. P. , Pinto M. V. , Vasconcelos W. V.
Published: June 2012

There is a natural way to associate to the commuting variety

C
(
A
)

C(A)

of an algebra

A
A

a module over a polynomial ring. It serves as a vehicle to study the arithmetical properties of

C
(
A
)

C(A)

, particularly Cohen-Macaulayness. The focus here is on Lie algebras and some of their representations.
A notion of rank for unitary representations of general linear groups
Authors: Scaramuzzi Roberto
Published: June 2012

A notion of rank for unitary representations of general linear groups over a locally compact, nondiscrete field is defined. Rank measures how singular a representation is, when restricted to the unipotent radical of a maximal parabolic subgroup. Irreducible representations of small rank are classified. It is shown how rank determines to a large extent the asymptotic behavior of matrix coefficients of the representations.
Massey products in the cohomology of groups with applications to link theory
Authors: Stein David
Published: June 2012

Invariants of links in

S
3

{S^3}

are developed using a modification of the Massey product of one-dimensional classes in the cohomology of certain groups. The theory yields two types of invariants, invariants which depend upon a collection of meridians, or basing, of a link, and invariants which do not. The invariants, which are independent of the basing, are compared with John Milnor’s

μ
¯

\overline \mu

-invariants. For two component links, a collection of ostensibly based invariants is shown to be independent of the basing. If the linking number of the components of such a link is zero, the resulting invariants may be equivalent to the Sato-Levine-Cochran invariants.
Minimal identities of symmetric matrices
Authors: Ma Wen Xin , Racine Michel L.
Published: June 2012

Let

H
n

(
F
)

{H_n}(F)

denote the subspace of symmetric matrices of

M
n

(
F
)

{M_n}(F)

, the full matrix algebra with coefficients in a field

F
F

. The subspace

H
n

(
F
)
⊂

M
n

(
F
)

{H_n}(F)\subset {M_n}(F)

does not have any polynomial identity of degree less than

2
n

2n

. Let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript k Superscript i Baseline left-parenthesis x 1 comma ellipsis comma x Subscript k Baseline right-parenthesis equals sigma-summation Underscript StartLayout 1st Row alpha element-of script upper S Subscript k Baseline 2nd Row 1 less-than-or-equal-to i less-than-or-equal-to k comma sigma Superscript negative 1 Baseline left-parenthesis i right-parenthesis identical-to 1 comma 2 mod 4 EndLayout Endscripts left-parenthesis negative 1 right-parenthesis Superscript sigma Baseline x Subscript sigma left-parenthesis 1 right-parenthesis Baseline x Subscript sigma left-parenthesis 2 right-parenthesis Baseline midline-horizontal-ellipsis x Subscript sigma left-parenthesis k right-parenthesis Baseline comma">

T
k
i

(

x
1

,
…
,

x
k

)
=

∑

α
∈

S

k

1
≤
i
≤
k
,

σ

−
1

(
i
)
≡
1
,
2
mod
⁡
4

(
−
1
)

σ

x

σ
(
1
)

x

σ
(
2
)

⋯

x

σ
(
k
)

,

T_k^i({x_1}, \ldots ,{x_k}) = \sum \limits _{\begin {array}{*{20}{c}} {\alpha \in {\mathcal {S}_k}} \\ {1 \leq i \leq k,{\sigma ^{ - 1}}(i) \equiv 1,2\operatorname {mod} 4} \\ \end {array} } {{{( - 1)}^\sigma }{x_{\sigma (1)}}} {x_{\sigma (2)}} \cdots {x_{\sigma (k)}},

\]
, and

e
(
n
)
=
n

e(n) = n

if

n
n

is even,

n
+
1

n + 1

if

n
n

is odd. For all

n
≥
1
,

T

2
n

i

n \geq 1,T_{2n}^i

is an identity of

H
n

(
F
)

{H_n}(F)

. If the characteristic of

F
F

does not divide

e
(
n
)
!

e(n)!

and if

n
≠
3

n \ne 3

, then any homogeneous polynomial identity of

H
n

(
F
)

{H_n}(F)

of degree

2
n

2n

is a consequence of

T

2
n

i

T_{2n}^i

. The case

n
=
3

n = 3

is also dealt with. The proofs are algebraic, but an equivalent formulation of the first result in graph-theoretical terms is given.
A complete classification of the piecewise monotone functions on the interval
Authors: Baldwin Stewart
Published: June 2012

We define two functions

f
f

and

g
g

on the unit interval

[
0
,
1
]

[0,1]

to be strongly conjugate iff there is an order-preserving homeomorphism

h
h

of

[
0
,
1
]

[0,1]

such that

g
=

h

−
1

f
h

g = {h^{ - 1}}fh

(a minor variation of the more common term "conjugate", in which

h
h

need not be order-preserving). We provide a complete set of invariants for each continuous (strictly) piecewise monotone function such that two such functions have the same invariants if and only if they are strongly conjugate, thus providing a complete classification of all such strong conjugacy classes. In addition, we provide a criterion which decides whether or not a potential invariant is actually realized by some piecewise monotone continuous function.
The influence of boundary data on the number of solutions of boundary value problems with jumping nonlinearities
Authors: Harris Greg A.
Published: June 2012

This paper contains results concerning the number of solutions, as a function of the boundary data, for boundary value problems with jumping nonlinearities. An example seems to indicate that boundary data has a different influence on this number than does forcing data. Through approximating techniques this example leads to lower bounds on solution numbers for the more general case
Geometrical implications of certain infinite-dimensional decompositions
Authors: Ghoussoub N. , Maurey B. , Schachermayer W.
Published: June 2012

We investigate the connections between the "global" structure of a Banach space (i.e. the existence of certain finite and infinite dimensional decompositions) and the geometrical properties of the closed convex bounded subsets of such a space (i.e. the existence of extremal and other topologically distinguished points). The global structures of various—supposedly pathological— examples of Banach spaces constructed by R. C. James turn out to be more "universal" than expected. For instance James-tree-type (resp. James-matrix-type) decompositions characterize Banach spaces with the Point of Continuity Property (resp. the Radon-Nikodým Property). Moreover, the Convex Point of Continuity Property is stable under the formation of James-infinitely branching tree-type "sums" of infinite dimensional factors. We also give several counterexamples to various questions relating some topological and geometrical concepts in Banach spaces.
Amenability of weighted convolution algebras on locally compact groups
Authors: Grønbæk Niels
Published: June 2012

We give a direct transition from the existence of a bounded right approximate identity in the diagonal ideal for a weighted convolution algebra on a locally compact group to the existence of translation invariant means on an associated weighted

L
∞

{L^\infty }

-space, thus giving a characterization of amenability for such an algebra.
Classification of crossed-product 𝐶*-algebras associated with characters on free groups
Authors: Yin Hong Sheng
Published: June 2012

We study the classification problem of crossed-product

C
∗

{C^ * }

-algebras of the form

C
r
∗

(
G
)

×

α
χ

Z

C_r^ * (G){ \times _{{\alpha _\chi }}}{\mathbf {Z}}

, where

G
G

is a discrete group,

χ
\chi

is a one-dimensional character of

G
G

, and

α
χ

{\alpha _\chi }

is the unique

∗
*

-automorphism of

C
r
∗

(
G
)

C_r^ * (G)

such that if

U
U

is the left regular representation of

G
G

, then

α

χ

(

U

g

)
=
χ
(
g
)

U

g

{\alpha _{\chi }(U_{g})=\chi (g)U_{g}}

,

g
∈
G

g \in G

. When

G
=

F

n

{G = F_{n}}

, the free group on

n
n

generators, we have a complete classification of these crossed products up to

∗
*

-isomorphism which generalizes the classification of irrational and rational rotation

C
∗

{C^ * }

-algebras. We show that these crossed products are determined by two

K
K

-theoretic invariants, that these two invariants correspond to two orbit invariants of the characters under the natural

Aut
⁡
(

F
n

)

\operatorname {Aut} ({F_n})

-action, and that these two orbit invariants completely classify the characters up to automorphisms of

F
n

{F_n}

. The classification of crossed products follows from these results. We also consider the same problem for

G
G

some other groups.
Weight strings in nonstandard representations of Kac-Moody algebras
Authors: Dillon Meighan I.
Published: June 2012

We consider the weights which occur in arbitrary irreducible highest weight representations of Kac-Moody algebras and determine conditions under which certain weights may or may not occur.
A topological persistence theorem for normally hyperbolic manifolds via the Conley index
Authors: Floer Andreas
Published: June 2012

We prove that the cohomology ring of a normally hyperbolic manifold of a diffeomorphism

f
f

persists under perturbation of

f
f

. We do not make any quantitative assumptions on the expansion and contraction rates of

D
f

Df

on the normal and the tangent bundles of

N
N

.
Curvatures and similarity of operators with holomorphic eigenvectors
Authors: Uchiyama Mitsuru
Published: June 2012

The curvature of the holomorphic vector bundle generated by eigenvectors of operators is estimated, and the necessary and sufficient conditions for contractions to be similar or quasi-similar with unilateral shifts are given.
The Schubert calculus, braid relations, and generalized cohomology
Authors: Bressler Paul , Evens Sam
Published: June 2012

Let

X
X

be the flag variety of a compact Lie group and let

h

∗

{h^{\ast }}

be a complex-oriented generalized cohomology theory. We introduce operators on

h

∗

(
X
)

{h^{\ast }}(X)

which generalize operators introduced by Bernstein, Gel’fand, and Gel’fand for rational cohomology and by Demazure for

K
K

-theory. Using the Becker-Gottlieb transfer, we give a formula for these operators, which enables us to prove that they satisfy braid relations only for the two classical cases, thereby giving a topological interpretation of a theorem proved by the authors and extended by Gutkin.
Leray functor and cohomological Conley index for discrete dynamical systems
Authors: Mrozek Marian
Published: June 2012

We introduce the Leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological Conley index of an isolated invariant set of a homeomorphism on a locally compact metric space. We prove the homotopy and additivity properties for this index and compute the index in some examples. As one of applications we prove the existence of nonconstant, bounded solutions of the Euler approximation of a certain system of ordinary differential equations.
The Mourre estimate for dispersive 𝑁-body Schrödinger operators
Authors: Dereziński Jan
Published: June 2012

We prove the Mourre estimate for a certain class of dispersive

N
N

-body Schrödinger operators. Using this estimate we derive some properties of those operators such as the local finiteness of the finite spectrum and the absence of the singular continuous spectrum.
Some weighted inequalities on product domains
Authors: Lin Henry
Published: June 2012

We extend the results of R. Fefferman [3] on the bidisc to higher product domains via induction. As an application, we extend the weighted inequality for Calderon-Zygmund operators on the bidisc to higher product domains, and we also extend the result of the Littlewood-Paley operator corresponding to the arbitrary disjoint rectangles to the weighted case.
Asymptotic stability of planar rarefaction waves for viscous conservation laws in several dimensions
Authors: Xin Zhou Ping
Published: June 2012

This paper concerns the large time behavior toward planar rarefaction waves of the solutions for scalar viscous conservation laws in several dimensions. It is shown that a planar rarefaction wave is nonlinearly stable in the sense that it is an asymptotic attractor for the viscous conservation law. This is proved by using a stability result of rarefaction wave for scalar viscous conservation laws in one dimension and an elementary

L
2

{L^2}

-energy method.
A new proof of the strong partition relation on 𝜔₁
Authors: Jackson Steve
Published: June 2012

Assuming the axiom of determinacy, we give a new proof of the strong partition relation on

ω
1

{\omega _1}

. The proof is direct and avoids appeal to complicated set-theoretic machinery.
Local orders whose lattices are direct sums of ideals
Authors: Haefner Jeremy
Published: June 2012

LetRRbe a complete local Dedekind domain with quotient fieldKKand letΛ\Lambdabe a localRR-order in a separableKK-algebra. This paper classifies those ordersΛ\Lambdasuch that every indecomposableRR-torsionfreeΛ\Lambda-module is isomorphic to an ideal ofΛ\Lambda. These results extend to the noncommutative case some results for commutative rings found jointly by this author and L. Levy.
The spectra and commutants of some weighted composition operators
Authors: Carlson James W.
Published: June 2012

An operator

T

u
g

{T_{ug}}

on a Hilbert space

H
H

of functions on a set

X
X

defined by

T

u
g

(
f
)
=
u
(
f
∘
g
)

{T_{ug}}(f) = u(f \circ g)

, where

f
f

is in

H
,

u
:
X
→

C

H,\;u:X \to {\mathbf {C}}

and

g
:
X
→
X

g:X \to X

, is called a weighted composition operator. In this paper

X
X

is the set of integers and

H
=

L
2

(

Z

,
μ
)

H = {L^2}({\mathbf {Z}},\mu )

, where

μ
\mu

is a measure whose sigma-algebra is the power set of

Z

{\mathbf {Z}}

. One distinguished space is

l
2

=

L
2

(

Z

,
μ
)

{l^2} = {L^2}({\mathbf {Z}},\mu )

, where

μ
\mu

is counting measure. The most important results given here are the determination of the spectrum of

T

u
g

{T_{ug}}

on

l
2

{l^2}

and a characterization of the commutant of

T
g

{T_g}

on

L
2

(

Z

,
μ
)

{L^2}({\mathbf {Z}},\mu )

. To obtain many of the results it was necessary to assume the function

g
g

to be one-to-one except on a finite subset of the integers.
A spanning set for 𝒞(ℐⁿ)
Authors: Bloom Thomas
Published: June 2012

C

(

I
n

)

\mathcal {C}({I^n})

denotes the Banach space of continuous functions on the unit

n
n

-cube,

I
n

{I^n}

, in

R

n

{{\mathbf {R}}^n}

. Let

{

a
i

}

\{ {a^i}\}

,

i
=
0
,
1
,
2
,
…
,

i = 0,1,2, \ldots ,

, be a countable collection of

n
n

-tuples of positive real numbers satisfying

lim
i

a
j
i

=
+
∞

{\operatorname {lim}_i}a_j^i = + \infty

for

j
=
1
,
…
,
n

j = 1, \ldots ,n

. We canonically enlarge the family of monomials

{

x

a
i

}

\{ {x^{{a^i}}}\}

to a family of functions

F

(
A
)

\mathcal {F}(A)

. Conjecture. The linear span of

F

(
A
)

\mathcal {F}(A)

is dense in

C

(

I
n

)

\mathcal {C}({I^n})

if and only if

Σ

i
=
0

∞

1

/

|

a
i

|

=
+
∞

\Sigma _{i = 0}^\infty 1/\left | {{a^i}} \right | = + \infty

. For

n
=
1

n = 1

this is equivalent to the Müntz-Szasz theorem. For

n
>
1

n > 1

we prove the necessity in general and the sufficiency under the additional hypothesis that there exist constants

G
G

,

N
>
1

N > 1

such that

|

a
i

|

≤
G

exp

(

i
N

)

\left | {{a^i}} \right | \leq G{\operatorname {exp}}({i^N})

for all

i
i

.
Ergodicity of finite-energy diffusions
Authors: Wallstrom Timothy C.
Published: June 2012

Recently, the existence of a class of diffusion processes with highly singular drift coefficients has been established under the assumption of "finite energy." The drift singularities of these diffusions greatly complicate their ergodicity properties; indeed, they can render the diffusion nonergodic. In this paper, a method is given for estimating the relaxation time of a finite-energy diffusion, when it is ergodic. These results are applied to show that the set of

spin
−

1
2

\operatorname {spin} - \tfrac {1} {2}

diffusions of stochastic mechanics is uniformly ergodic.
𝐿¹-approximation with constraints
Authors: Pinkus Allan , Strauss Hans
Published: June 2012

In this paper we study problems of best

L
1

{L^1}

-approximation to continuous functions from finite-dimensional subspaces under a variety of constraints. Included are problems of bounded coefficient approximation, approximation with interpolation, restricted range approximation, and restricted range and derivative approximation. Emphasis is placed on problems of uniqueness.
A short proof of principal kinematic formula and extensions
Authors: Rother W. , Zähle M.
Published: June 2012

Federer’s extension of the classical principal kinematic formula of integral geometry to sets with positive reach is proved in a direct way by means of generalised unit normal bundles, associated currents, and the coarea theorem. This enables us to extend the relation to more general sets. At the same time we get a short proof for the well-known variants from convex geometry and differential geometry.
Iterated spinning and homology spheres
Authors: Suciu Alexander I.
Published: June 2012

Given a closed

n
n

-manifold

M
n

{M^n}

and a tuple of positive integers

P
P

, let

σ
P

M

{\sigma _P}M

be the

P
P

-spin of

M
M

. If

M
n

⋍̸

S
n

{M^n} \not \backsimeq {S^n}

and

P
≠
Q

P \ne Q

(as unordered tuples), it is shown that

σ
P

M
⋍̸

σ
Q

M

{\sigma _P}M\not \backsimeq {\sigma _Q}M

if either (1)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript asterisk Baseline left-parenthesis upper M Superscript n Baseline right-parenthesis neither-approximately-nor-actually-equals upper H Subscript asterisk Baseline left-parenthesis upper S Superscript n Baseline right-parenthesis">

H
∗

(

M
n

)
≇

H
∗

(

S
n

)

{H_*}({M^n})\not \cong {H_*}({S^n})

, (2)

π
1

M

{\pi _1}M

finite, (3)

M
M

aspherical, or (4)

n
=
3

n = 3

. Applications to the homotopy classification of homology spheres and knot exteriors are given.
Invariant tori for the billiard ball map
Authors: Kovachev Valery , Popov Georgi
Published: June 2012

For an

n
n

-dimensional domain

Ω
(
n
≥
3
)

\Omega (n \geq 3)

with a smooth boundary which is strictly convex in a neighborhood of an elliptic closed geodesic

O

\mathcal {O}

, the existence of a family of invariant tori for the billiard ball map with a positive measure is proved under the assumptions of nondegeneracy and

N
N

-elementarity,

N
≥
5

N \geq 5

, of the corresponding to

O

\mathcal {O}

Poincaré map. Moreover, the conjugating diffeomorphism constructed is symplectic. An analogous result is obtained in the case

n
=
2

n=2

. It is shown that the lengths of the periodic geodesics determine uniquely the invariant curves near the boundary and the billiard ball map on them up to a symplectic diffeomorphism.
Complex interpolation for normed and quasi-normed spaces in several dimensions. III. Regularity results for harmonic interpolation
Authors: Slodkowski Zbigniew
Published: June 2012

The paper continues the study of one of the complex interpolation methods for families of finite-dimensional normed spaces

{

C

n

,

|

|

⋅

|

|

z

}

z
∈
G

{\{ {{\mathbf {C}}^n},|| \cdot |{|_z}\} _{z \in G}}

, where

G
G

is open and bounded in

C

k

{{\mathbf {C}}^k}

. The main result asserts that (under a mild assumption on the datum) the norm function

(
z
,
w
)
→

|

|

w

|

|

z
2

(z,w) \to ||w||_z^2

belongs to some anisotropic Sobolew class and is characterized by a nonlinear PDE of second order. The proof uses the duality theorem for the harmonic interpolation method (obtained earlier by the author). A new, simpler proof of this duality relation is also presented in the paper.
Domain-independent upper bounds for eigenvalues of elliptic operators
Authors: Hook Stephen M.
Published: June 2012

Let

Ω
⊆

R

m

\Omega \subseteq {\mathbb {R}^m}

be a bounded open set,

∂
Ω

\partial \Omega

its boundary and

Δ
\Delta

the Laplacian on

R

m

{\mathbb {R}^m}

. Consider the elliptic differential equation: (1)
\[

−
Δ
u
=
λ
u

in

Ω
;

u
=
0

on

∂
Ω
.

- \Delta u = \lambda u\quad {\text {in}}\;\Omega ;\qquad u = 0\quad {\text {on}}\;\partial \Omega .

\]
It is known that the eigenvalues,

λ
i

{\lambda _i}

, of (1) satisfy (2)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript i equals 1 Overscript n Endscripts StartFraction lamda Subscript i Baseline Over lamda Subscript n plus 1 Baseline minus lamda Subscript i Baseline EndFraction greater-than-or-slanted-equals StartFraction m n Over 4 EndFraction">

∑

i
=
1

n

λ
i

λ

n
+
1

−

λ
i

⩾

m
n

4

\sum \limits _{i = 1}^n {\frac {{{\lambda _i}}} {{{\lambda _{n + 1}} - {\lambda _i}}}} \geqslant \frac {{mn}} {4}

\]
provided that

λ

n
+
1

>

λ
n

{\lambda _{n + 1}} > {\lambda _n}

. In this paper we abstract the method used by Hile and Protter [2] to establish (2) and apply the method to a variety of second-order elliptic problems, in particular, to all constant coefficient problems. We then consider a variety of higher-order problems and establish an extension of (2) for problem (1) where the Laplacian is replaced by a more general operator in a Hilbert space.
Zeta functions of formal languages
Authors: Berstel Jean , Reutenauer Christophe
Published: June 2012

Motivated by symbolic dynamics and algebraic geometry over finite fields, we define cyclic languages and the zeta function of a language. The main result is that the zeta function of a cyclic language which is recognizable by a finite automation is rational.
Abelian and nondiscrete convergence groups on the circle
Authors: Hinkkanen A.
Published: June 2012

A group

G
G

of homeomorphisms of the unit circle onto itself is a convergence group if every sequence of elements of

G
G

contains a subsequence, say

g

n

{{\text {g}}_n}

, such that either (i)

g

n

→

g

{{\text {g}}_n} \to {\text {g}}

and

g

n

−
1

→

g

−
1

{\text {g}}_n^{ - 1} \to {{\text {g}}^{ - 1}}

uniformly on the circle where

g

{\text {g}}

is a homeomorphism, or (ii)

g

n

→

x

0

{{\text {g}}_n} \to {{\text {x}}_0}

and

g

n

−
1

→

y

0

{\text {g}}_n^{ - 1} \to {{\text {y}}_0}

uniformly on compact subsets of the complements of

{

y

0

}

\{ {{\text {y}}_0}\}

and

{

x

0

}

\{ {{\text {x}}_0}\}

, respectively, for some points

x

0

{{\text {x}}_0}

and

y

0

{{\text {y}}_0}

of the circle (possibly

x

0

 = 

y

0

{{\text {x}}_0}{\text { = }}{{\text {y}}_0}

). For example, a group of

K
K

-quasisymmetric maps, for a fixed

K
K

, is a convergence group. We show that if

G
G

is an abelian or nondiscrete convergence group, then there is a homeomorphism

f
f

such that

f
∘
G
∘

f

−
1

f \circ G \circ {f^{ - 1}}

is a group of Màbius transformations.
Weighted norm estimates for the Fourier transform with a pair of weights
Authors: Strömberg Jan-Olov , Wheeden Richard L.
Published: June 2012

We prove weighted norm inequalities of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar ModifyingAbove f With caret double-vertical-bar Subscript upper L Sub Subscript u Sub Superscript q Subscript Baseline less-than-or-equal-to upper C double-vertical-bar f double-vertical-bar Subscript upper H Sub Subscript upsilon Sub Superscript p Subscript Baseline comma 0 greater-than p less-than-or-equal-to q greater-than normal infinity comma">

‖

f
^

‖

L
u
q

≤
C

‖
f
‖

H
υ
p

,

0
>
p
≤
q
>
∞
,

{\left \| {\hat f} \right \|_{L_u^q}} \leq C{\left \| f \right \|_{H_\upsilon ^p}},\quad 0 > p \leq q > \infty ,

\]
for the Fourier transform on

R

n

{{\mathbf {R}}^n}

. For some weight functions

υ
\upsilon

, the Hardy space

H
υ
p

H_\upsilon ^p

on the right can be replaced by

L
υ
p

L_\upsilon ^p

. The proof depends on making an atomic decomposition of

f
f

and using cancellation properties of the atoms.
On the existence of central sequences in subfactors
Authors: Bisch Dietmar H.
Published: June 2012

We prove a relative version of [Co1, Theorem 2.1] for a pair of type

I

I

1

{\text {I}}{{\text {I}}_1}

-factors

N
⊂
M

N \subset M

. This gives a list of necessary and sufficient conditions for the existence of nontrivial central sequences of

M
M

contained in the subfactor

N
N

. As an immediate application we obtain a result by Bédos [Be, Theorem A], showing that if

N
N

has property

Γ
\Gamma

and

G
G

is an amenable group acting freely on

N
N

via some action

σ
\sigma

, then the crossed product

N

×
σ

G

N{ \times _\sigma }G

has property

Γ
\Gamma

. We also include a proof of a relative Mc Duff-type theorem (see [McD, Theorems

1
1

,

2
2

and

3
3

]), which gives necessary and sufficient conditions implying that the pair

N
⊂
M

N \subset M

is stable.
Boundary behavior of the fast diffusion equation
Authors: Kwong Y. C.
Published: June 2012

The fast diffusion equation

Δ

υ
m

=

υ
t

\Delta {\upsilon ^m} = {\upsilon _t}

,

0
>
m
>
1

0 > m > 1

, is a degenerate nonlinear parabolic equation of which the existence of a unique continuous weak solution has been established. In this paper we are going to obtain a Lipschitz growth rate of the solution at the boundary of

Ω
\Omega

and estimate that in terms of the various data.
Weak Chebyshev subspaces and 𝐴-subspaces of 𝐶[𝑎,𝑏]
Authors: Li Wu
Published: June 2012

In this paper we show some very interesting properties of weak Chebyshev subspaces and use them to simplify Pinkus’s characterization of

A
A

subspaces of

C
[
a
,
b
]

C[a,b]

. As a consequence we obtain that if the metric projection

P
G

{P_G}

from

C
[
a
,
b
]

C[a,b]

onto a finite-dimensional subspace

G
G

has a continuous selection and elements of

G
G

have no common zeros on

(
a
,
b
)

(a,b)

, then

G
G

is an

A
A

-subspace.
Basic dual homotopy invariants of Riemannian foliations
Authors: Pang Peter Y. H.
Published: June 2012

In this paper, we use the Sullivan minimal model construction to produce invariants for Riemannian foliations. Existence and vanishing results are proved for these invariants.
Homotopy theory of minimal simplicial spaces
Authors: Pritt Mark D.
Published: June 2012

Various aspects of homotopy theory in the category of minimal simplicial spaces are studied. It is shown that the usual results of homotopy theory hold in this category, and necessary and sufficient conditions are given under which a simplicial space has the homotopy type of a minimal simplicial space. Continuous cohomology in this category is also studied.
The Arf and Sato link concordance invariants
Authors: Sturm Beiss Rachel
Published: June 2012

The Kervaire-Arf invariant is a

Z

/

2

Z/2

valued concordance invariant of knots and proper links. The

β
\beta

invariant (or Sato’s invariant) is a

Z
Z

valued concordance invariant of two component links of linking number zero discovered by J. Levine and studied by Sato, Cochran, and Daniel Ruberman. Cochran has found a sequence of invariants

{

β
i

}

\{ {\beta _i}\}

associated with a two component link of linking number zero where each

β
i

{\beta _i}

is a

Z
Z

valued concordance invariant and

β
0

=
β

{\beta _0} = \beta

. In this paper we demonstrate a formula for the Arf invariant of a two component link

L
=
X
∪
Y

L = X \cup Y

of linking number zero in terms of the

β
\beta

invariant of the link:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a r f left-parenthesis upper X union upper Y right-parenthesis equals a r f left-parenthesis upper X right-parenthesis plus a r f left-parenthesis upper Y right-parenthesis plus beta left-parenthesis upper X union upper Y right-parenthesis left-parenthesis mod 2 right-parenthesis period">

arf
⁡
(
X
∪
Y
)
=
arf
⁡
(
X
)
+
arf
⁡
(
Y
)
+
β
(
X
∪
Y
)

(
mod
2
)
.

\operatorname {arf} (X \cup Y) = \operatorname {arf} (X) + \operatorname {arf} (Y) + \beta (X \cup Y)\quad (\bmod 2).

\]
This leads to the result that the Arf invariant of a link of linking number zero is independent of the orientation of the link’s components. We then establish a formula for

|

β

|

|\beta |

in terms of the link’s Alexander polynomial

Δ
(
x
,
y
)
=
(
x
−
1
)
(
y
−
1
)
f
(
x
,
y
)

\Delta (x,y) = (x - 1)(y - 1)f(x,y)

:
\[

|

β
(
L
)

|

=

|

f
(
1
,
1
)

|

.

|\beta (L)| = |f(1,1)|.

\]
Finally we find a relationship between the

β
i

{\beta _i}

invariants and linking numbers of lifts of

X
X

and

Y
Y

in a

Z

/

2

Z/2

cover of the compliment of

X
∪
Y

X \cup Y

.
The structure of Legendre foliations
Authors: Pang Myung-Yull
Published: June 2012

The local and global structure of Legendre foliations of contact manifolds is analysed. The main invariant of a Legendre foliation is shown to be a quadratic form on the tangent bundle to the foliation—the fundamental quadratic form. The equivalence problem is solved in the case when the fundamental quadratic form is nondegenerate and a generalization of Chern’s solution to the equivalence problem for Finsler manifolds is obtained. A normal form for Legendre foliations is given which is closely related to Weinstein’s structure theorem for Lagrangian foliations. It is shown that every compact, simply connected leaf of a Legendre foliation is diffeomorphic to a sphere.
Stability of individual elements under one-parameter semigroups
Authors: Batty Charles J. K. , Vù Quôc Phóng
Published: June 2012

Let

{
T
(
t
)
:
t
⩾
0
}

\{ T(t):t \geqslant 0\}

be a

C
0

{C_0}

-semigroup on a Banach space

X
X

with generator

A
A

, and let

x
∈
X

x \in X

. If

σ
(
A
)
∩
i

R

\sigma (A) \cap i{\mathbf {R}}

is empty and

t
↦
T
(
t
)
x

t \mapsto T(t)x

is uniformly continuous, then

|

|

T
(
t
)
x

|

|

→
0

||T(t)x|| \to 0

as

t
→
∞

t \to \infty

. If the semigroup is sun-reflexive,

σ
(
A
)
∩
i

R

\sigma (A) \cap i{\mathbf {R}}

is countable,

P
σ
(
A
)
∩
i

R

P\sigma (A) \cap i{\mathbf {R}}

is empty, and

t
↦
T
(
t
)
x

t \mapsto T(t)x

is uniformly weakly continuous, then

T
(
t
)
x
→
0

T(t)x \to 0

weakly as

t
→
∞

t \to \infty

. Questions of almost periodicity and of stabilization of contraction semigroups on Hilbert space are also discussed.
Upper bounds for ergodic sums of infinite measure preserving transformations
Authors: Aaronson Jon , Denker Manfred
Published: June 2012

For certain conservative, ergodic, infinite measure preserving transformations

T
T

we identify increasing functions

A
A

, for which
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit sup StartFraction 1 Over upper A left-parenthesis n right-parenthesis EndFraction sigma-summation Underscript k equals 1 Overscript n Endscripts f ring upper T Superscript k Baseline equals integral Underscript upper X Endscripts f d mu a period e period">

lim sup

n
→
∞

1

A
(
n
)

∑

k
=
1

n

f
∘

T
k

=

∫
X

f
d
μ

a

.e

.

\limsup \limits _{n \to \infty } \frac {1} {{A(n)}}\sum \limits _{k = 1}^n {f \circ } {T^k} = \int _X {fd\mu } \quad {\text {a}}{\text {.e}}{\text {.}}

\]
holds for any nonnegative integrable function

f
f

. In particular the results apply to some Markov shifts and number-theoretic transformations, and include the other law of the iterated logarithm.
Eventual finite order generation for the kernel of the dimension group representation
Authors: Wagoner J. B.
Published: June 2012

The finite order generation problem (FOG) in symbolic dynamics asks whether every element in the kernel of the dimension group representation of a subshift of finite type

(

X
A

,

σ
A

)

({X_A},{\sigma _A})

is a product of elements of finite order in the group

Aut

(

σ
A

)

{\operatorname {Aut}}({\sigma _A})

of homeomorphisms of

X
A

{X_A}

commuting with

σ
A

{\sigma _A}

. We study the space of strong shift equivalences over the nonnegative integers, and the first application is to prove Eventual FOG which says that every inert symmetry of

σ
A

{\sigma _A}

is a product of finite order homeomorphisms of

X
A

{X _A}

commuting with sufficiently high powers of

σ
A

{\sigma _A}

. Then we discuss the relation of FOG to Williams’ lifting problem (LIFT) for symmetries of fixed points. In particular, either FOG or LIFT is false. Finally, we also discuss

p
p

-adic convergence and other implications of Eventual FOG for gyration numbers.
Generalized second derivatives of convex functions and saddle functions
Authors: Rockafellar R. T.
Published: June 2012

The theory of second-order epi-derivatives of extended-real-valued functions is applied to convex functions on

R

n

{\mathbb {R}^n}

and shown to be closely tied to proto-differentiation of the corresponding subgradient multifunctions, as well as to second-order epi-differentiation of conjugate functions. An extension is then made to saddle functions, which by definition are convex in one argument and concave in another. For this case a concept of epi-hypo-differentiability is introduced. The saddle function results provide a foundation for the sensitivity analysis of primal and dual optimal solutions to general finite-dimensional problems in convex optimization, since such solutions are characterized as saddlepoints of a convex-concave Lagrangian function, or equivalently as subgradients of the saddle function conjugate to the Lagrangian.
Cells and the reflection representation of Weyl groups and Hecke algebras
Authors: Douglass J. Matthew
Published: June 2012

Let

H

\mathcal {H}

be the generic algebra of the finite crystallographic Coxeter group

W
W

, defined over the ring

Q

[

u

1

/

2

,

u

−
1

/

2

]

\mathbb {Q}[{u^{1/2}},{u^{ - 1/2}}]

. First, the two-sided cell corresponding to the reflection representation of

H

\mathcal {H}

is shown to consist of the nonidentity elements of

W
W

having a unique reduced expression. Next, the matrix entries of this representation are computed in terms of certain Kazhdan-Lusztig polynomials. Finally, the Kazhdan-Lusztig polynomials just mentioned are described in case

W
W

is of type

A

l
−
1

{{\text {A}}_{l - 1}}

or

B

l

{{\text {B}}_l}

.
An asymptotic formula for hypo-analytic pseudodifferential operators
Authors: Berhanu S.
Published: June 2012

An asymptotic expansion formula for hypo-analytic pseudodifferential operators is proved and applications are given.
Stability of Newton boundaries of a family of real analytic singularities
Authors: Suzuki Masahiko
Published: June 2012

Let

f
t

(
x
,
y
)

{f_t}(x,y)

be a real analytic

t
t

-parameter family of real analytic functions defined in a neighborhood of the origin in

R

2

{\mathbb {R}^2}

. Suppose that

f
t

(
x
,
y
)

{f_t}(x,y)

admits a blow analytic trivilaization along the parameter

t
t

(see the definition in

§
1

\S 1

of this paper). Under this condition, we prove that there is a real analytic

t
t

-parameter family

σ
t

(
x
,
y
)

{\sigma _t}(x,y)

with

σ
0

(
x
,
y
)
=
(
x
,
y
)

{\sigma _0}(x,y)=(x,y)

and

σ
t

(
0
,
0
)
=
(
0
,
0
)

{\sigma _t}(0,0)=(0,0)

of local coordinates in which the Newton boundaries of

f
t

(
x
,
y
)

{f_t}(x,y)

are stable. This fact claims that the blow analytic equivalence among real analytic singularities is a fruitful relationship since the Newton boundaries of singularities contains a lot of informations on them.
Inner functions and cyclic vectors in the Bloch space
Authors: Anderson J. M. , Fernández J. L. , Shields A. L.
Published: June 2012

In this paper we construct a singular inner function whose polynomial multiples are dense in the little Bloch space

B

0

{\mathcal {B}_0}

. To do this we construct a singular measure on the unit circle with "best possible" control of both the first and second differences.
Circuit partitions and the homfly polynomial of closed braids
Authors: Jaeger François
Published: June 2012

We present an expansion of the homfly polynomial

P
(
D
,
z
,
a
)

P(D,z,a)

of a braid diagram

D
D

in terms of its circuit partitions. Another aspect of this result is an expression of

P
(
D
,
z
,
a
)

P(D,z,a)

as the trace of a matrix associated to

D
D

in a simple way. We show how certain degree properties of the homfly polynomial can be derived easily from this model. In particular we obtain that if

D
D

is a positive braid diagram on

n
n

strings with

w
w

crossings, the maximum degree of

P
(
D
,
z
,
a
)

P(D,z,a)

in the variable

a
a

equals

n
−
1
−
w

n - 1 - w

. Nous présentons une expansion pour le polynôme homfly

P
(
D
,
z
,
a
)

P(D,z,a)

d’un diagramme de tresse

D
D

en termes de ses partitions en circuits. Un autre aspect de ce résultat consiste en une expression de

P
(
D
,
z
,
a
)

P(D,z,a)

comme trace d’une matrice associee de façon simple à

D
D

. Nous montrons comment certaines propriétés de degré du polynôme homfly dérivent simplement de ce modèle. En particulier nous obtenons que pour un diagramme de tresse positif

D
D

à

n
n

brins et

w
w

croisements, le degré maximum de

P
(
D
,
z
,
a
)

P(D,z,a)

en la variable

a
a

est égal à

n
−
1
−
w

n - 1- w

.
Roots of unity and the Adams-Novikov spectral sequence for formal 𝐴-modules
Authors: Johnson Keith
Published: June 2012

The cohomology of a Hopf algebroid related to the Adams-Novikov spectral sequence for formal

A
A

-modules is studied in the special case in which

A
A

is the ring of integers in the field obtained by adjoining

p
p

th roots of unity to

Q

^

p

{\widehat {\mathbb {Q}}_p}

, the

p
p

-adic numbers. Information about these cohomology groups is used to give new proofs of results about the

E
2

{E_2}

term of the Adams spectral sequence based on

2
2

-local complex

K
K

-theory, and about the odd primary Kervaire invariant elements in the usual Adams-Novikov spectral sequence.
Random products of contractions in Banach spaces
Authors: Dye J. , Khamsi M. A. , Reich S.
Published: June 2012

We show that the random product of a finite number of

(
W
)

(W)

contractions converges weakly in all smooth reflexive Banach spaces. If one of the contractions is compact, then the convergence is uniform.
Strictly cyclic operator algebras
Authors: Froelich John
Published: June 2012

We prove several results about the lattice of invariant subspaces of general strictly cyclic and strongly strictly cyclic operator algebras. A reflexive operator algebra

A
A

with a commutative subspace lattice is strictly cyclic iff

Lat
⁡

(
A

)
⊥

\operatorname {Lat}{(A)^ \bot }

contains a finite number of atoms and each nonzero element of

Lat
⁡

(
A

)
⊥

\operatorname {Lat}{(A)^ \bot }

contains an atom. This leads to a characterization of the

n
n

-strictly cyclic reflexive algebras with a commutative subspace lattice as well as an extensive generalization of D. A. Herrero’s result that there are no triangular strictly cyclic operators. A reflexive operator algebra

A
A

with a commutative subspace lattice is strongly strictly cyclic iff

Lat
⁡
(
A
)

\operatorname {Lat}(A)

satisfies A.C.C. The distributive lattices which are attainable by strongly strictly cyclic reflexive algebras are the complete sublattices of

{
0
,
1
]
×
{
0
,
1
}
×
⋯

\{ 0,1] \times \{ 0,1\} \times \cdots

which satisfy A.C.C. We also show that if

Alg
⁡
(

L

)

\operatorname {Alg}(\mathcal {L})

is strictly cyclic and

L

⊆

\mathcal {L} \subseteq

atomic m.a.s.a. then

Alg
⁡
(

L

)

\operatorname {Alg}(\mathcal {L})

contains a strictly cyclic operator.
Subelliptic estimates for the \overline∂-Neumann problem for 𝑛-1 forms
Authors: Ho Lop-Hing
Published: June 2012

In this note we deal with the problem of the subelliptic estimates of the

∂
¯

\bar \partial

-Neumann problem on nonpseudoconvex domains. In the first part we give a necessary condition for

n
−
1

n - 1

forms in a class of domains. In the second part we give the exact estimate for a class of domains where the Levi form of a vector field

L
L

is bounded below by a certain function.
Plane autonomous systems with rational vector fields
Authors: Benzinger Harold E.
Published: June 2012

The differential equation

z
˙

=
R
(
z
)

\dot z= R(z)

is studied, where

R
R

is an arbitrary rational function. It is shown that the Riemann sphere is decomposed into finitely many open sets, on each of which the flow is analytic and, in each time direction, there is common long-term behavior. The boundaries of the open sets consist of those points for which the flow fails to be analytic in at least one time direction. The main idea is to express the differential equation as a continuous Newton method

z
˙

=
−
f
(
z
)

/

f
′

(
z
)

\dot z = - f(z)/f’\;(z)

, where

f
f

is an analytic function which can have branch points and essential singularities. A method is also given for the computer generation of phase plane portraits which shows the correct time parametrization and which is noniterative, thereby avoiding the problems associated with the iteration of rational functions.