bluuebunny/crossref_metadata_2025 huggingface dataset:
Behaviour of the velocity of one-dimensional flows in porous media
Authors: Vázquez Juan Luis
Published: June 2010

The one-dimensional flow of gas of density

u
u

through a porous medium obeys the equation

u
t

=

(

u
m

)

x
x

{u_t} = {({u^m})_{xx}}

, where

m
>
1
,
x
∈

R

m > 1,x \in {\mathbf {R}}

and

t
>
0

t > 0

. We prove that the local velocity of the gas, given by

υ
=
−
m

u

m
−
2

u
x

\upsilon = - m{u^{m - 2}}{u_x}

, not only is bounded for

t
⩾
τ
>
0

t \geqslant \tau > 0

but approaches an

N
N

-wave profile as

t
→
∞

t \to \infty

.

N
N

-waves are the typical asymptotic profiles for some first-order conservation laws, a class of nonlinear hyperbolic equations. The case

m
⩽
1

m \leqslant 1

is also studied: there are solutions with unbounded velocity while others have bounded velocity.
Period doubling and the Lefschetz formula
Authors: Franks John
Published: June 2010

This article gives an application of the Lefschetz fixed point theorem to prove, under certain hypotheses, the existence of a family of periodic orbits for a smooth map. The family has points of periods

2
k

p

{2^k}p

for some

p
p

and all

k
≥
0

k \geq 0

. There is a version of the result for a parametrized family

f
t

f_t

which shows that these orbits are "connected" in parametrized space under appropriate hypotheses.
Deductive varieties of modules and universal algebras
Authors: Hogben Leslie , Bergman Clifford
Published: June 2010

A variety of universal algebras is called deductive if every subquasivariety is a variety. The following results are obtained: (1) The variety of modules of an Artinian ring is deductive if and only if the ring is the direct sum of matrix rings over local rings, in which the maximal ideal is principal as a left and right ideal. (2) A directly representable variety of finite type is deductive if and only if either (i) it is equationally complete, or (ii) every algebra has an idempotent element, and a ring constructed from the variety is of the form (1) above.
The typical structure of the sets {𝑥:𝑓(𝑥)=ℎ(𝑥)} for 𝑓 continuous and ℎ Lipschitz
Authors: Wójtowicz Zygmunt
Published: June 2010

Let

R
R

be the space of real numbers and

C
C

the space of continuous functions

f
:
[
0
,
1
]
→
R

f:[0,1] \to R

with the uniform norm. Bruckner and Garg prove that there exists a residual set

B
B

in

C
C

such that for every function

f
∈
B

f \in B

there exists a countable dense set

Λ
f

{\Lambda _f}

in

R
R

such that: for

λ
∉

Λ
f

\lambda \notin {\Lambda _f}

the top and bottom levels in the direction

λ
\lambda

of

f
f

are singletons, in between these levels there are countably many levels in the direction

λ
\lambda

of

f
f

that consist of a nonempty perfect set together with a single isolated point, and the remaining levels in the direction

λ
\lambda

of

f
f

are all perfect; for

λ
∈

Λ
f

\lambda \in {\Lambda _f}

the level structure in the direction

λ
\lambda

of

f
f

is the same except that one (and only one) of the levels has two isolated points instead of one. In this paper we show that the analogue of the above theorem holds: if we replace the family of straight lines

{
λ
x
+
c
}

\{ \lambda x + c\}

by a

2
2

-parameter family

H
H

that is almost uniformly Lipschitz; and if we replace

{
λ
x
+
c
}

\{ \lambda x + c\}

by a homeomorphical image of a certain

2
2

-parameter family

H
H

that is almost uniformly Lipschitz.
Equivariant diffeomorphisms with simple recurrences on two-manifolds
Authors: de Melo W. , dos Reis G. L. , Mendes P.
Published: June 2010

We consider the class of diffeomorphisms, on compact two-dimensional manifolds, which are invariant under the action of a compact Lie group

G
G

and whose nonwandering set consists of a finite number of

G
G

-orbits. We describe the modulus of stability of almost all diffeomorphisms in this class.
On the spectra of compact nilmanifolds
Authors: Fox Jeffrey S.
Published: June 2010

We show the equivalence of the Howe-Richardson multiplicity formula for compact nilmanifolds and the formula obtained by Corwin and Greenleaf using the Selberg trace formula.
On the group 𝑆𝑆𝐹(𝐺),𝐺 a cyclic group of prime order
Authors: Maller M. , Whitehead J.
Published: June 2010

We extend the definition of the obstruction group

SSF

(
G
)

{\text {SSF}}(G)

in the case where

G
G

is a cyclic group of prime order. We show that an endomorphism of a free

Z
G

ZG

-module is a direct summand of a virtual permutation if its characteristic polynomial has the appropriate form. Among these endomorphisms the virtual permutations are detected by

K
0

{K_0}

. The main application is in detecting Morse-Smale isotopy classes.
Abelian rank of normal torsion-free finite index subgroups of polyhedral groups
Authors: Lee Youn W.
Published: June 2010

Suppose that

P
P

is a convex polyhedron in the hyperbolic

3
3

-space with finite volume and

P
P

has integer

(
>
1
)

( > 1)

submultiples of

π
\pi

as dihedral angles. We prove that if the rank of the abelianization of a normal torsion-free finite index subgroup of the polyhedral group

G
G

associated to

P
P

is one, then

P
P

has exactly one ideal vertex of type

(
2
,
2
,
2
,
2
)

(2,2,2,2)

and

G
G

has an index two subgroup which does not contain any one of the four standard generators of the stabilizer of the ideal vertex.
Volumes of tubes about complex submanifolds of complex projective space
Authors: Gray Alfred
Published: June 2010

Simple formulas in terms of Chern classes are given for the volume of a tube about a Kähler submanifold of a space of constant holomorphic sectional curvature. A comparison theorem which generalizes these formulas is also given. Tubes about complete intersections in complex projective space are studied in detail.
On the existence and classification of extensions of actions on submanifolds of disks and spheres
Authors: Assadi Amir , Browder William
Published: June 2010

Given a

G
G

-action

ψ
:
G
×
W
→
W

\psi :G \times W \to W

and an embedding

W
⊂

D
n

W \subset {D^n}

, when is it possible to find a

G
G

-action

ϕ
:
G
×

D
n

→
D

\phi :G \times {D^n} \to D

such that

D
n

−
W

{D^n} - W

is

G
G

-free? Sufficient conditions of cohomological nature for the existence of such extensions are given and the extensions are classified. This leads to the characterization of the stationary point sets and classification of semifree actions on disks up to

G
G

-diffeomorphism under suitable dimension hypotheses.
On the decomposition numbers of the finite general linear groups. II
Authors: Dipper Richard
Published: June 2010

Let

q
q

be a prime power,

G
=

GL
n

(
q
)

G = {\operatorname {GL} _n}(q)

and let

r
r

be a prime not dividing

q
q

. Using representations of Hecke algebras associated with symmetric groups over arbitrary fields, the

r
r

-modular irreducible

G
G

-modules are classified. The decomposition matrix

D
D

of

G
G

(with respect to

r
r

) is partly described in terms of decomposition matrices of Hecke algebras, and it is shown that

D
D

is lower unitriangular, provided the irreducible characters and irreducible Brauer characters of

G
G

are suitably ordered.
Boundary behavior of 𝐵𝑀𝑂(𝐵_{𝑛})
Authors: Russo Paula A.
Published: June 2010

If

f
f

is a holomorphic function of bounded mean oscillation in the unit ball of

C

n

{{\mathbf {C}}^n}

, then it has radial limits at almost every point of the boundary of the ball. The question remains as to how nicely one can expect this function to behave on subsets of the boundary of zero measure. For example, there is a holomorphic BMO function in the ball that has a finite radial limit at no point of the

n
n

-torus. We show here that this is not an isolated phenomenon; there exists at least one other

n
n

-dimensional submanifold of the boundary of the ball with this same behavior.
Multivariate rational approximation
Authors: DeVore Ronald A. , Yu Xiang Ming
Published: June 2010

We estimate the error in approximating a function

f
f

by rational functions of degree

n
n

in the norm of

L
q

(
Ω
)
,

Ω
:=

[
0
,

1

]
d

{L_q}(\Omega ),\,\Omega : = {[0,\,1]^d}

. Among other things, we prove that if

f
f

is in the Sobolev space

W
p
k

(
Ω
)

W_p^k(\Omega )

and if

k

/

d
−
1

/

p
+
1

/

q
>
0

k/d - 1/p + 1/q > 0

, then

f
f

can be approximated by rational functions of degree

n
n

to an order

O
(

n

−
k

/

d

)

O({n^{ - k/d}})

.
The Gauss map for surfaces. I. The affine case
Authors: Weiner Joel L.
Published: June 2010

Let

M
M

be a connected oriented surface and let

G
2
c

G_2^c

be the Grassmannian of oriented

2
2

-planes in Euclidean

(
2
+
c
)

(2 + c)

-space,

E

2
+
c

{{\mathbf {E}}^{2 + c}}

. Smooth maps

t
:
M
→

G
2
c

t:M \to G_2^c

are studied to determine whether or not they are Gauss maps. Both local and global results are obtained. If

t
t

is a Gauss map of an immersion

X
:

M
→

E

2
+
c

X:\;M \to {{\mathbf {E}}^{2 + c}}

, we study the extent to which

t
t

uniquely determines

X
X

under certain circumstances.
The Gauss map for surfaces. II. The Euclidean case
Authors: Weiner Joel L.
Published: June 2010

We study smooth maps

t
:

M
→

G
2
c

t:\;M \to G_2^c

of a Riemann surface

M
M

into the Grassmannian

G
2
c

G_2^c

of oriented

2
2

-planes in

E

2
+
c

{{\mathbf {E}}^{2 + c}}

and determine necessary and sufficient conditons on

t
t

in order that it be the Gauss map of a conformal immersion

X
:

M
→

E

2
+
c

X:\;M \to {{\mathbf {E}}^{2 + c}}

. We sometimes view

t
t

as an oriented riemannian vector bundle; it is a subbundle of

E

M

2
+
c

{\mathbf {E}}_M^{2 + c}

, the trivial bundle over

M
M

with fibre

E

2
+
c

{{\mathbf {E}}^{2 + c}}

. The necessary and sufficient conditions obtained for simply connected

M
M

involve the curvatures of

t
t

and

t
⊥

{t^ \bot }

, the orthogonal complement of

t
t

in

E

M

2
+
c

{\mathbf {E}}_M^{2 + c}

, as well as certain components of the tension of

t
t

viewed as a map

t
:

M
→

S
C

(
1
)

t:\;M \to {S^C}(1)

, where

S
C

(
1
)

{S^C}(1)

is a unit sphere of dimension

C
C

that contains

G
2
c

G_2^c

as a submanifold in a natural fashion. If

t
t

satisfies a particular necessary condition, then the results take two different forms depending on whether or not

t
t

is the Gauss map of a conformal minimal immersion. The case

t
:

M
→

G
2
2

t:\;M \to G_2^2

is also studied in some additional detail.
The least 𝑟-free number in an arithmetic progression
Authors: McCurley Kevin S.
Published: June 2010

Let

n
r

(
a
,
q
)

{n_r}(a,q)

be the least

r
r

-free number in the arithmetic progession

a
a

modulo

q
q

. Several results are proved that give lower bounds for

n
r

(
a
,
q
)

{n_r}(a,q)

, improving on previous results due to Erdös and Warlimont. In addition, a heuristic argument is given, leading to two conjectures that would imply that the results of the paper are close to best possible.
Extensions of a theorem of Wintner on systems with asymptotically constant solutions
Authors: Trench William F.
Published: June 2010

A theorem of Wintner concerning sufficient conditions for a system

y
′

=
A
(
t
)
y

y’ = A(t)y

to have linear asymptotic equilibrium is extended to a system

x
′

=
A
(
t
)
x
+
f
(
t
,
x
)

x’ = A(t)x + f(t,x)

. The integrability conditions imposed on

f
f

permit conditional convergence of some of the improper integrals that occur. The results improve on Wintner’s even if

f
=
0

f = 0

.
Properties of relatively free inverse semigroups
Authors: Reilly N. R. , Trotter P. G.
Published: June 2010

The objective of this paper is to study structural properties of relatively free inverse semigroups in varieties of inverse semigroups. It is shown, for example, that if

S
S

is combinatorial (i.e.,

H

\mathcal {H}

is trivial), completely semisimple (i.e., every principal factor is a Brandt semigroup or, equivalently,

S
S

does not contain a copy of the bicyclic semigroup) or

E
E

-unitary (i.e.,

E
(
S
)

E(S)

is the kernel of the minimum group congruence) then the relatively free inverse semigroup

F

V

X

F{\mathcal {V}_X}

on the set

X
X

in the variety

V

\mathcal {V}

generated by

S
S

is also combinatorial, completely semisimple or

E
E

-unitary, respectively. If

S
S

is a fundamental (i.e., the only congruence contained in

H

\mathcal {H}

is the identity congruence) and

|

X

|

⩾

ℵ
0

|X| \geqslant {\aleph _0}

, then

F

V

X

F{\mathcal {V}_X}

is also fundamental.

F

V

X

F{\mathcal {V}_X}

may not be fundamental if

|

X

|

>

ℵ
0

|X| > {\aleph _0}

. It is also shown that for any variety of groups

U

\mathcal {U}

and for

|

X

|

⩾

ℵ
0

|X| \geqslant {\aleph _0}

, there exists a variety of inverse semigroups

V

\mathcal {V}

which is minimal with respect to the properties (i)

F

V

X

F{\mathcal {V}_X}

is fundamental and (ii)

V

∩

G

=

U

\mathcal {V} \cap \mathcal {G} = \mathcal {U}

, where

G

\mathcal {G}

is the variety of groups. In the main result of the paper it is shown that there exists a variety

V

\mathcal {V}

for which

F

V

X

F{\mathcal {V}_X}

is not completely semisimple, thereby refuting a long standing conjecture.
Boundary uniqueness theorems in 𝐶ⁿ
Authors: Cima Joseph A. , Straube Emil J.
Published: June 2010

Let

n
n

-dimensional manifolds

Γ
k

,

k
=
1
,
2
,
…

{\Gamma _k},\,k = 1,2, \ldots

, be given in a smoothly bounded domain

Ω
⊂

C

n

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

. Assume that the

Γ
k

{\Gamma _k}

"converge" to an

n
n

-dimensional, totally real manifold

Γ
⊆
∂
Ω

\Gamma \subseteq \partial \Omega

and that a function

f
f

analytic in

Ω
\Omega

has the property that its traces

f
k

{f_k}

on

Γ
k

{\Gamma _k}

have distributional limit zero as

k
→
∞

k \to \infty

(or assume that

f
k

→
0

{f_k} \to 0

pointwise). Then under the assumption that

f
f

is polynomially bounded near

P
∈
Γ

P \in \Gamma

by

(
dist
⁡
(
z
,
∂
Ω
)

)

−
1

{(\operatorname {dist} (z,\partial \Omega ))^{ - 1}}

we conclude that

f
f

is identically zero.
Lower bounds of the gap between the first and second eigenvalues of the Schrödinger operator
Authors: Yu Qi Huang , Zhong Jia Qing
Published: June 2010

In this paper the authors prove the following theorem: Let

Ω
\Omega

be a smooth strictly convex bounded domain in

R
n

{R^n}

and

V
:
Ω
→
R

V:\Omega \to R

a nonnegative convex function. Suppose

λ
1

{\lambda _1}

and

λ
2

{\lambda _2}

are the first and second nonzero eigenvalues of the equation
\[

−
Δ
f
+
V
f
=
λ
f
,

f

|

∂
Ω

≡
0.

- \Delta f + Vf = \lambda f,\qquad f{|_{\partial \Omega }} \equiv 0.

\]
Then

λ
2

−

λ
1

⩾

π
2

/

d
2

{\lambda _2} - {\lambda _1} \geqslant {\pi ^2}/{d^2}

, where

d
d

is the diameter of

Ω
\Omega

.
General convergence of continued fractions
Authors: Jacobsen Lisa
Published: June 2010

We introduce a new concept of convergence of continued fractions—general convergence. Moreover, we compare it to the ordinary convergence concept and to strong convergence. Finally, we prove some properties of general convergence.
Stability of minimal orbits
Authors: Brothers John E.
Published: June 2010

Let

G
G

be a compact Lie group of isometries of a riemannian manifold

M
M

. It is well known that the minimal principal orbits are those on which the volume function

v

{\mathbf {v}}

, which assigns to

p
∈
M

p \in M

the volume of the orbit of

p
p

, is critical. It is shown that stability of a minimal orbit on which the hessian of

v

{\mathbf {v}}

is nonnegative is determined by the degree of involutivity of the distribution of normal planes to the orbits. Specifically, if the lengths of the tangential components of Lie brackets of pairs of orthonormal normal vector fields are sufficiently small relative to the hessian of

v

{\mathbf {v}}

, then the minimal orbit is stable, and conversely. Computable lower bounds are obtained for the values of these parameters at which stability turns to instability. These lower bounds are positive even in the case where

v

{\mathbf {v}}

is constant, and are finite unless the normal distribution is involutive. Several examples in which

M
M

is a compact classical Lie group and

G
G

is a subgroup of

M
M

are discussed, showing in particular that the above estimates are sharp.
Boundary behavior of positive solutions of the heat equation on a semi-infinite slab
Authors: Mair B. A.
Published: June 2010

In this paper, the abstract Fatou-Naim-Doob theorem is used to investigate the boundary behavior of positive solutions of the heat equation on the semi-infinite slab

X
=

R

n
−
1

×

R

+

×
(
0
,
T
)

X = {{\mathbf {R}}^{n - 1}} \times {{\mathbf {R}}_ + } \times (0,T)

. The concept of semifine limit is introduced, and relationships are obtained between fine, semifine, parabolic, one-sided parabolic and two-sided parabolic limits at points on the parabolic boundary of

X
X

. A Carleson-Calderón-type local Fatou theorem is also obtained for solutions on a union of two-sided parabolic regions.
Connected locally connected toposes are path-connected
Authors: Moerdijk I. , Wraith G. C.
Published: June 2010

A conjecture of A. Joyal is proved, which states that, in contrast to topological spaces, toposes which are connected and locally connected are also path-connected. The reason for this phenomenon is the triviality of cardinality considerations in the topos-theoretic setting; any inhabited object pulls back to an enumerable object under some open surjective geometric morphism. This result points towards a homotopy theory for toposes.
Equivariant bundles and cohomology
Authors: Kozlowski A.
Published: June 2010

Let

G
G

be a topological group,

A
A

an abelian topological group on which

G
G

acts continuously and

X
X

a

G
G

-space. We define "equivariant cohomology groups" of

X
X

with coefficients in

A
A

,

H
G
i

(
X
;
A
)

H_G^i(X;A)

, for

i
≥
0

i \geq 0

which generalize Graeme Segal’s continuous cohomology of the topological group

G
G

with coefficients in

A
A

. In particular we have

H
G
1

(
X
;
A
)
≃

H_G^1(X;A) \simeq

equivalence classes of principal

(
G
,
A
)

(G,A)

-bundles over

X
X

. We show that when

G
G

is a compact Lie group and

A
A

an abelian Lie group we have for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i greater-than 1 upper H Subscript upper G Superscript i Baseline left-parenthesis upper X semicolon upper A right-parenthesis asymptotically-equals upper H Superscript i Baseline left-parenthesis upper E upper G times Subscript upper G Baseline upper X semicolon tau upper A right-parenthesis">

i
>
1

H
G
i

(
X
;
A
)
≃

H
i

(
E
G

×
G

X
;
τ
A
)

i > 1\;H_G^i(X;A) \simeq {H^i}(EG{ \times _G}X;\tau A)

where

τ
A

\tau A

is the sheaf of germs of sections of the bundle

(
X
×
E
G
×
A
)

/

G
→
(
X
×
E
G
)

/

G

(X \times EG \times A)/G \to (X \times EG)/G

. For

i
=
1

i = 1

and the trivial action of

G
G

on

A
A

this is a theorem of Lashof, May and Segal.
The Fefferman metric and pseudo-Hermitian invariants
Authors: Lee John M.
Published: June 2010

C. Fefferman has shown that a real strictly pseudoconvex hypersurface in complex

n
n

-space carries a natural conformal Lorentz metric on a circle bundle over the manifold. This paper presents two intrinsic constructions of the metric, valid on an abstract

CR

{\text {CR}}

manifold. One is in terms of tautologous differential forms on a natural circle bundle; the other is in terms of Webster’s pseudohermitian invariants. These results are applied to compute the connection and curvature forms of the Fefferman metric explicitly.
Peano arithmetic and hyper-Ramsey logic
Authors: Schmerl James H.
Published: June 2010

It is known that

PA

(

Q
2

)

{\text {PA}}({Q^2})

, Peano arithmetic in a language with the Ramsey quantifier, is complete and compact and that its first-order consequences are the same as those of

Π
1
1

-CA
0

\Pi _1^1{\text {-CA}_0}

. A logic

H

R

ω

\mathcal {H}{\mathcal {R}_\omega }

, called hyper-Ramsey logic, is defined; it is the union of an increasing sequence

H

R

1

⊆

H

R

2

⊆

H

R

3

⊆
⋯

\mathcal {H}{\mathcal {R}_1} \subseteq {\mathcal {H}_{\mathcal {R}2}} \subseteq \mathcal {H}{\mathcal {R}_3} \subseteq \cdots

of sublogics, and

H

R

1

\mathcal {H}{\mathcal {R}_1}

contains

L
(

Q
2

)

L({Q^2})

. It is proved that

PA

(

H

R

n

)

{\text {PA}}(\mathcal {H}{\mathcal {R}_n})

, which is Peano arithmetic in the context of

H

R

n

\mathcal {H}{\mathcal {R}_n}

, has the same first-order consequences as

Π
n
1

-CA
0

\Pi _n^1{\text {-CA}_0}

. A by-product and ingredient of the proof is, for example, the existence of a model of

CA

{\text {CA}}

having the form

(

N

,

Class

(

N

)
)

(\mathcal {N}, {\text {Class}}(\mathcal {N}))

.
Optimal-partitioning inequalities for nonatomic probability measures
Authors: Elton John , Hill Theodore P. , Kertz Robert P.
Published: June 2010

Suppose

μ
1

,
…
,

μ
n

{\mu _1}, \ldots ,{\mu _n}

are nonatomic probability measures on the same measurable space

(
S
,

B

)

(S,\mathcal {B})

. Then there exists a measurable partition

{

S
i

}

i
=
1

n

\{ {S_i}\} _{i = 1}^n

of

S
S

such that

μ
i

(

S
i

)
≥

(
n
+
1
−
M

)

−
1

{\mu _i}({S_i}) \geq {(n + 1 - M)^{ - 1}}

for all

i
=
1
,
…
,
n

i = 1, \ldots ,n

, where

M
M

is the total mass of

∨

i
=
1

n

μ
i

\vee _{i = 1}^n\,{\mu _i}

(the smallest measure majorizing each

μ
i

{\mu _i}

). This inequality is the best possible for the functional

M
M

, and sharpens and quantifies a well-known cake-cutting theorem of Urbanik and of Dubins and Spanier. Applications are made to

L
1

{L_1}

-functions, discrete allocation problems, statistical decision theory, and a dual problem.
A regularity theorem for minimizing hypersurfaces modulo 𝜈
Authors: Morgan Frank
Published: June 2010

It is proved that an

(
n
−
1
)

(n - 1)

-dimensional, area-minimizing flat chain modulo

ν
\nu

in

R

n

{{\mathbf {R}}^n}

, with smooth extremal boundary of at most

ν

/

2

\nu /2

components, has an interior singular set of Hausdorff dimension at most

n
−
8

n - 8

. Similar results hold for more general integrands.
On maximal functions and Poisson-Szegő integrals
Authors: Sueiro Juan
Published: June 2010

We study a class of maximal functions of Hardy-Littlewood type defined on spaces of homogeneous type and we give necessary and sufficient conditions for the corresponding maximal operators to be of weak type

(
1
,
1
)

(1,1)

. As a consequence we show that Poisson-Szegö integrals of

L
p

{L^p}

functions possess certain boundary limits which are not implied by Korányi’s theorem.
Some results on locally finitely presentable categories
Authors: Makkai M. , Pitts A. M.
Published: June 2010

We prove that any full subcategory of a locally finitely presentable (l.f.p.) category having small limits and filtered colimits preserved by the inclusion functor is itself l.f.p. Here "full" may be weakened to "full with respect to isomorphisms." Further, we characterize those left exact functors

I
:

C

→

D

I:{\mathbf {C}} \to {\mathbf {D}}

between small categories with finite limits for which the functor
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Superscript asterisk Baseline colon bold upper L bold upper E bold upper X left-parenthesis bold upper D comma Set right-parenthesis right-arrow bold upper L bold upper E bold upper X left-parenthesis bold upper C comma Set right-parenthesis">

I

∗

:

L
E
X

(

D

,

Set)

→

L
E
X

(

C

,Set)

{I^{\ast }}:{\mathbf {LEX}}({\mathbf {D}},{\text {Set)}} \to {\mathbf {LEX}}{\text {(}}{\mathbf {C}}{\text {,Set)}}

induced by composition is full and faithful. As an application, we prove a theorem on sheaf representations, a consequence of which is that, for any site

C

=
(

C

,
J
)

\mathcal {C} = ({\mathbf {C}},J)

on a category

C

{\mathbf {C}}

with finite limits, defined by a subcanonical Grothendieck topology

J
J

, the closure in

L
E
X

(

C

,

Set)

{\mathbf {LEX}}({\mathbf {C}},{\text {Set)}}

under small limits and filtered colimits of the models of

C

\mathcal {C}

is the whole of

L
E
X

(

C

,

Set)

{\mathbf {LEX}}({\mathbf {C}},{\text {Set)}}

.
Parallel translation of curvature along geodesics
Authors: Hebda James J.
Published: June 2010

According to the Cartan-Ambrose-Hicks Theorem, two simply-connected, complete Riemannian manifolds are isometric if, given a certain correspondence between all the broken geodesics emanating from a point in one manifold, and all those emanating from a point in the other, the parallel translates of the curvature tensor agree along corresponding broken geodesics. For generic metrics on a surface, the hypothesis can be refined so that it is enough to compare curvature along corresponding unbroken geodesics in order to obtain the isometry.
Application of a theorem of M. G. Kreĭn to singular integrals
Authors: Wittmann Rainer
Published: June 2010

We give Hölder and

L
2

{L^2}

estimates for singular integrals on homogeneous spaces in the sense of Coifman and Weiss. The fundamental tool which allows us to pass from Hölder to

L
2

{L^2}

estimates, is a theorem of M. G. Krein.
Completely reducible operators that commute with compact operators
Authors: Rosenoer Shlomo
Published: June 2010

It is shown that if

T
T

is a completely reducible operator on a Banach space and

T
K
=
K
T

TK = KT

, where

K
K

is an injective compact operator with a dense range, then

T
T

is a scalar type spectral operator. Other related results are also obtained.
Construction of group actions on four-manifolds
Authors: Edmonds Allan L.
Published: June 2010

It is shown that any cyclic group of odd prime order acts on any closed, simply connected topological

4
4

-manifold, inducing the identity on integral homology. The action is locally linear except perhaps at one isolated fixed point. In the case of primes greater than three a more careful argument is used to show that the action can be constructed to be locally linear.
Whitney continua of curves
Authors: Kato Hisao
Published: June 2010

In this paper, we prove several theorems relating shape properties of Whitney continua of curves. In particular, we investigate the fundamental dimension and the shape type of Whitney continua of curves.
Uniform distribution of two-term recurrence sequences
Authors: Vélez William Yslas
Published: June 2010

Letu0,u1,A,B{u_0},\,{u_1},\,A,\,Bbe rational integers and forn⩾2n \geqslant 2defineun=Aun−1+Bun−2{u_n} = A{u_{n - 1}} + B{u_{n - 2}}. The sequence(un)({u_n})is clearly periodic modulommand we say that(un)({u_n})is uniformly distributed modulommif for everyss, every residue modulommoccurs the same number of times in the sequence of residuesus,us+1,…,us+N−1{u_s},\,{u_{s + 1}},\, \ldots ,\,{u_{s + N - 1}}, whereNNis the period of(un)({u_n})modulomm. If(un)({u_n})is uniformly distributed modulommthenmmdividesNN, so we writeN=mfN = mf. Several authors have characterized thosemmfor which(un)({u_n})is uniformly distributed modulomm. In fact in this paper we will show that a much stronger property holds whenm=pk,pm = {p^k},\,p, a prime. Namely, if(un)({u_n})is uniformly distributed modulopk{p^k}with periodpkf{p^k}f, then every residue modulopk{p^k}appears exactly once in the sequenceus,us+f,…,us+(pk−1)f{u_s},\,{u_{s + f}},\, \ldots ,\,{u_{s + ({p^k} - 1)f}}, for everyss. We also characterize those compositemmfor which this more stringent property holds.
Cohomology theories on spaces
Authors: Spanier E.
Published: June 2010

In this paper a previously proven uniqueness theorem for nonnegative cohomology theories on the same space is extended to cohomology theories on the same finite-dimensional space. In this form it is applicable to extraordinary cohomology theories. An example is given to show that the theorem does not hold without finite dimensionality.
The étale cohomology of 𝑝-torsion sheaves. I
Authors: Hawkins William Anthony
Published: June 2010

This paper generalizes a formula of Grothendieck, Ogg, and Shafarevich that expresses the Euler-Poincaré characteristic of a constructible sheaf of

F
l

{F_l}

-modules on a smooth, proper curve, over an algebraically closed field

k
k

of characteristic

p
>
0

p > 0

, as a sum of local and global terms, where

l
≠
p

l \ne p

. The primary focus is on removing the restriction on

l
l

. We begin with calculations for

p
p

-torsion sheaves trivialized by

p
p

-extensions, but using etale cohomology to give a unified proof for all primes

l
l

. In the remainder of this work, only

p
p

-torsion sheaves are considered. We show the existence on

X

et

{X_{{\text {et}}}}

,

X
X

a scheme of characteristic

p
p

, of a short exact sequence of sheaves, involving the tangent space at the identity of a finite, flat, height 1, commutative group scheme, and the subsheaf fixed by the

p
p

th power endomorphism; the latter turns out to be an etale group scheme. A corollary gives complete results on the Euler-Poincaré characteristic of a constructible sheaf of

F
p

{F_p}

-modules on a smooth, proper curve, over an algebraically closed field

k
k

of characteristic

p
>
0

p > 0

, when the generic stalk has rank

p
p

. Explicit computations are given for the Euler characteristics of such

p
p

-torsion sheaves on

P
1

{P^1}

and a result on elliptic surfaces is included. A study is made of the comparison of the

p
p

-ranks of abelian extensions of curves. Several examples of

p
p

-ranks for nonhyperelliptic curves are discussed. The paper concludes with a brief sketch of results on certain constructible sheaves of

F
q

{F_q}

-modules,

q
=

p
r

,

r
≥
1

q={p^r},\,r \ge 1

.
The space of framed functions
Authors: Igusa Kiyoshi
Published: June 2010

We define the notion of a “framed function” on a compact smooth manifold

N
N

and we show that the space of all framed functions on

N
N

is

(
dim

N
−
1
)

(\operatorname {dim} \,N - 1)

-connected. A framed function on

N
N

is essentially a smooth function

N
→

R

N \to \mathbf {R}

with only Morse and birth-death singularities together with certain additional structure.
Infinitely many traveling wave solutions of a gradient system
Authors: Terman David
Published: June 2010

We consider a system of equations of the form

u
t

=

u

x
x

+
∇
F
(
u
)

{u_t} = {u_{xx}} + \nabla F(u)

. A traveling wave solution of this system is one of the form

u
(
x
,

t
)
=
U
(
z
)
,

z
=
x
+
θ
t

u(x,\,t) = U(z),\,z = x + \theta t

. Sufficient conditions on

F
(
u
)

F(u)

are given to guarantee the existence of infinitely many traveling wave solutions.
Koszul homology and the structure of low codimension Cohen-Macaulay ideals
Authors: Vasconcelos Wolmer V.
Published: June 2010

The relationship between the properties of the Koszul homology modules of two ideals connected by linkage is studied. If the ideal

I
I

is either (i) a Cohen-Macaulay ideal of codimension 3, or (ii) a Gorenstein ideal of codimension 4, the one-dimensional Koszul module carries considerable information on the structural nature of the linkage class of

I
I

in case (i), or on the conormal module of

I
I

in case (ii). Emphasis is given to the verification of the properties by computation.
Convex subcones of the contingent cone in nonsmooth calculus and optimization
Authors: Ward Doug
Published: June 2010

The tangential approximants most useful in nonsmooth analysis and optimization are those which lie "between" the Clarke tangent cone and the Bouligand contigent cone. A study of this class of tangent cones is undertaken here. It is shown that although no convex subcone of the contingent cone has the isotonicity property of the contingent cone, there are such convex subcones which are more "accurate" approximants than the Clarke tangent cone and possess an associated subdifferential calculus that is equally strong. In addition, a large class of convex subcones of the contingent cone can replace the Clarke tangent cone in necessary optimality conditions for a nonsmooth mathematical program. However, the Clarke tangent cone plays an essential role in the hypotheses under which these calculus rules and optimality conditions are proven. Overall, the results obtained here suggest that the most complete theory of nonsmooth analysis combines a number of different tangent cones.
Character table and blocks of finite simple triality groups ³𝐷₄(𝑞)
Authors: Deriziotis D. I. , Michler G. O.
Published: June 2010

Based on recent work of Spaltenstein [14] and the Deligne-Lusztig theory of irreducible characters of finite groups of Lie type, in this paper the character table of the finite simple groups

3

D
4

(
q
)

{}^3{D_4}(q)

is given. As an application we obtain a classification of the irreducible characters of

3

D
4

(
q
)

{}^3{D_4}(q)

into

r
r

-blocks for all primes

r
>
0

r > 0

. This enables us to verify Brauer’s height zero conjecture, his conjecture on the bound of irreducible characters belonging to a give block, and the Alperin-McKay conjecture for the simple triality groups

3

D
4

(
q
)

{}^3{D_4}(q)

. It also follows that for every prime

r
r

there are blocks of defect zero in

3

D
4

(
q
)

{}^3{D_4}(q)

.
Braids and the Jones polynomial
Authors: Franks John , Williams R. F.
Published: June 2010

An important new invariant of knots and links is the Jones polynomial, and the subsequent generalized Jones polynomial or two-variable polynomial. We prove inequalities relating the number of strands and the crossing number of a braid with the exponents of the variables in the generalized Jones polynomial which is associated to the link formed from the braid by connecting the bottom ends to the top ends. We also relate an exponent in the polynomial to the number of components of this link.
On a property of Castelnuovo varieties
Authors: Ciliberto Ciro
Published: June 2010

Castelnuovo varieties are those irreducible complete varieties in a projective space whose geometric genus is maximal according to their dimension, degree and embedding dimension. In this paper, extending results by Severi and Accola, we prove that, under suitable conditions, such varieties are birational if and only if they are projectively equivalent.
Group actions on the complex projective plane
Authors: Wilczyński Dariusz M.
Published: June 2010

Let

G
G

be a finite or compact Lie group. It is shown that

G
G

acts on the complex projective plane (resp. on the Chern manifold) if and only if

G
G

is isomorphic to a subgroup (resp. a pseudofree subgroup) of

P
U
(
3
)

PU(3)

. All actions considered are effective, locally smooth, and trivial on homology.
On strongly summable ultrafilters and union ultrafilters
Authors: Blass Andreas , Hindman Neil
Published: June 2010

We prove that union ultrafilters are essentially the same as strongly summable ultrafilters but ordered-union ultrafilters are not. We also prove that the existence of ultrafilters of these sorts implies the existence of

P
P

-points and therefore cannot be established in ZFC.
Crossed products of continuous-trace 𝐶*-algebras by smooth actions
Authors: Raeburn Iain , Rosenberg Jonathan
Published: June 2010

We study in detail the structure of

C

∗

{C^{\ast }}

-crossed products of the form

A
⋊

α

G

A \rtimes {}_\alpha G

, where

A
A

is a continuous-trace algebra and

α
\alpha

is an action of a locally compact abelian group

G
G

on

A
A

, especially in the case where the action of

G
G

on

A
^

\hat A

has a Hausdorff quotient and only one orbit type. Under mild conditions, the crossed product has continuous trace, and we are often able to compute its spectrum and Dixmier-Douady class. The formulae for these are remarkably interesting even when

G
G

is the real line.
Infinite rank Butler groups
Authors: Dugas Manfred , Rangaswamy K. M.
Published: June 2010

A torsion-free abelian group

G
G

is said to be a Butler group if

Bext
⁡
(
G
,

T
)

\operatorname {Bext} (G,\,T)

for all torsion groups

T
T

. It is shown that Butler groups of finite rank satisfy what we call the torsion extension property (T.E.P.). A crucial result is that a countable Butler group

G
G

satisfies the T.E.P. over a pure subgroup

H
H

if and only if

H
H

is decent in

G
G

in the sense of Albrecht and Hill. A subclass of the Butler groups are the so-called

B
2

{B_2}

-groups. An important question left open by Arnold, Bican, Salce, and others is whether every Butler group is a

B
2

{B_2}

-group. We show under

(
V
=
L
)

(V = L)

that this is indeed the case for Butler groups of rank

ℵ
1

{\aleph _1}

. On the other hand it is shown that, under ZFC, it is undecidable whether a group

B
B

for which

Bext
⁡
(
B
,

T
)
=
0

\operatorname {Bext} (B,\,T) = 0

for all countable torsion groups

T
T

is indeed a

B
2

{B_2}

-group.
Counting semiregular permutations which are products of a full cycle and an involution
Authors: Jackson D. M.
Published: June 2010

Character theoretic methods and the group algebra of the symmetric group are used to derive properties of the number of permutations, with only

p
p

-cycles, for an arbitrary but fixed

p
p

, which are expressible as the product of a full cycle and a fixed point free involution. This problem has application to single face embeddings of

p
p

-regular graphs on surfaces of given genus.
The action of a solvable group on an infinite set never has a unique invariant mean
Authors: Krasa Stefan
Published: June 2010

Theorem 1 of the paper proves a conjecture of J. Rosenblatt on nonuniqueness of invariant means for the action of a solvable group

G
G

on an infinite set

X
X

. The same methods used in this proof yield even a more general result: Nonuniqueness still holds if

G
G

is an amenable group containing a solvable subgroup

H
H

such that

card
⁡
(
G

/

H
)
⩽
card
⁡
(
H
)

\operatorname {card} (G/H) \leqslant \operatorname {card} (H)

.
The first case of Fermat’s last theorem is true for all prime exponents up to 714,591,416,091,389
Authors: Granville Andrew , Monagan Michael B.
Published: June 2010

We show that if the first case of Fermat’s Last Theorem is false for prime exponent

p
p

then

p
2

{p^2}

divides

q
p

−
q

{q^p} - q

for all primes

q
⩽
8
q

q \leqslant 8q

. As a corollary we state the theorem of the title.
Fixed points of arc-component-preserving maps
Authors: Hagopian Charles L.
Published: June 2010

The following classical problem remains unsolved: If

M
M

is a plane continuum that does not separate the plane and

f
f

is a map of

M
M

into

M
M

, must

f
f

have a fixed point? We prove that the answer is yes if

f
f

maps each arc-component of

M
M

into itself. Since every deformation of a space preserves its arc-components, this result establishes the fixed-point property for deformations of nonseparating plane continua. It also generalizes the author’s theorem [10] that every arcwise connected nonseparating plane continuum has the fixed-point property. Our proof shows that every arc-component-preserving map of an indecomposable plane continuum has a fixed point. We also prove that every tree-like continuum that does not contain uncountably many disjoint triods has the fixed-point property for arc-component-preserving maps.
The minimal model of the complement of an arrangement of hyperplanes
Authors: Falk Michael
Published: June 2010

In this paper the methods of rational homotopy theory are applied to a family of examples from singularity theory. Let

A

{\mathbf {A}}

be a finite collection of hyperplanes in

C

l

{{\mathbf {C}}^l}

, and let

M
=

C

l

−

⋃

H
∈

A

H

M = {{\mathbf {C}}^l} - \bigcup \nolimits _{H \in {\mathbf {A}}} H

. We say

A

{\mathbf {A}}

is a rational

K
(
π
,

1
)

K(\pi ,\,1)

arrangement if the rational completion of

M
M

is aspherical. For these arrangements an identity (the LCS formula) is established relating the lower central series of

π
1

(
M
)

{\pi _1}(M)

to the cohomology of

M
M

. This identity was established by group-theoretic means for the class of fiber-type arrangements in previous work. We reproduce this result by showing that the class of rational

K
(
π
,

1
)

K(\pi ,\,1)

arrangements contains all fiber-type arrangements. This class includes the reflection arrangements of types

A
l

{A_l}

and

B
l

{B_l}

. There is much interest in arrangements for which

M
M

is a

K
(
π
,

1
)

K(\pi ,\,1)

space. The methods developed here do not apply directly because

M
M

is rarely a nilpotent space. We give examples of

K
(
π
,

1
)

K(\pi ,\,1)

arrangements which are not rational

K
(
π
,

1
)

K(\pi ,\,1)

for which the LCS formula fails, and

K
(
π
,

1
)

K(\pi ,\,1)

arrangements which are not rational

K
(
π
,

1
)

K(\pi ,\,1)

where the LCS formula holds. It remains an open question whether rational

K
(
π
,

1
)

K(\pi ,\,1)

arrangements are necessarily

K
(
π
,

1
)

K(\pi ,\,1)

.
A construction of pseudo-Anosov homeomorphisms
Authors: Penner Robert C.
Published: June 2010

We describe a generalization of Thurston’s original construction of pseudo-Anosov maps on a surface

F
F

of negative Euler characteristic. In fact, we construct whole semigroups of pseudo-Anosov maps by taking appropriate compositions of Dehn twists along certain families of curves; our arguments furthermore apply to give examples of pseudo-Anosov maps on nonorientable surfaces. For each self-map

f
:
F
→
F

f:F \to F

arising from our recipe, we construct an invariant "bigon track" (a slight generalization of train track) whose incidence matrix is Perron-Frobenius. Standard arguments produce a projective measured foliation invariant by

f
f

. To finally prove that

f
f

is pseudo-Anosov, we directly produce a transverse invariant projective measured foliation using tangential measures on bigon tracks. As a consequence of our argument, we derive a simple criterion for a surface automorphism to be pseudo-Anosov.
On the theory of biorthogonal polynomials
Authors: Iserles A. , Nørsett S. P.
Published: June 2010

Let

φ
(
x
,

μ
)

\varphi (x,\,\mu )

be a distribution in

x
∈

R

x \in {\mathbf {R}}

for every

μ
\mu

in a real parameter set

Ω
\Omega

. Subject to additional technical conditions, we study

m
m

th degree monic polynomials

p
m

{p_m}

that satisfy the biorthogonality conditions
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript negative normal infinity Superscript normal infinity Baseline p Subscript m Baseline left-parenthesis x right-parenthesis d phi left-parenthesis x comma mu Subscript l Baseline right-parenthesis equals 0 comma l equals 1 comma 2 comma ellipsis comma m comma m greater-than-or-slanted-equals 1">

∫

−
∞

∞

p
m

(
x
)

d
φ
(
x
,

μ
l

)
=
0
,

l
=
1
,

2
,
…
,

m
,

m
⩾
1

\int _{ - \infty }^\infty {{p_m}(x)\,d\varphi (x,{\mu _l}) = 0,} \qquad l = 1,\,2, \ldots ,\,m,\;m \geqslant 1

\]
, for a distinct sequence

μ
1

,

μ
2

,

…

∈
Ω

{\mu _1},\,{\mu _2},\, \ldots \; \in \Omega \,

. Necessary and sufficient conditions for existence and uniqueness are established, as well as explicit determinantal and integral representations. We also consider loci of zeros, existence of Rodrigues-type formulae and reducibility to standard orthogonality. The paper is accompanied by several examples of biorthogonal systems.
General gauge theorem for multiplicative functionals
Authors: Chung K. L. , Rao K. M.
Published: June 2010

We generalize our previous work on the gauge theorem and its various consequences and complements, initiated in [8] and somewhat extended by subsequent investigations (see [6]). The generalization here is two-fold. First, instead of the Brownian motion, the underlying process is now a fairly broad class of Markov processes, not necessarily having continuous paths. Second, instead of the Feynman-Kac functional, the exponential of a general class of additive functionals is treated. The case of Schrödinger operator

Δ

/

2
+
ν

\Delta /2 + \nu

, where

ν
\nu

is a suitable measure, is a simple special case. The most general operator, not necessarily a differential one, which may arise from our potential equations is briefly discussed toward the end of the paper. Concrete instances of applications in this case should be of great interest.
The connection map for attractor-repeller pairs
Authors: McCord Christopher
Published: June 2010

In the Conley index theory, the connection map of the homology attractor-repeller sequence provides a means of detecting connecting orbits between a repeller and attractor in an isolated invariant set. In this work, the connection map is shown to be additive: under suitable decompositions of the connecting orbit set, the connection map of the invariant set equals the sum of the connection maps of the decomposition elements. This refines the information provided by the homology attractor-repeller sequence. In particular, the properties of the connection map lead to a characterization of isolated invariant sets with hyperbolic critical points as an attractor-repeller pair.
Harmonically immersed surfaces of 𝑅ⁿ
Authors: Jensen Gary R. , Rigoli Marco
Published: June 2010

Some generalizations of classical results in the theory of minimal surfaces

f
:
M
→

R

n

f:M \to {{\mathbf {R}}^n}

are shown to hold in the more general case of harmonically immersed surfaces.
Applications of nonstandard models and Lebesgue measure to sequences of natural numbers
Authors: Leth Steven C.
Published: June 2010

By use of a nonstandard model, sequences of natural numbers are associated with a collection of closed subsets of reals in a natural way. The topological and measure-theoretic properties of the associated closed sets are used to prove standard theorems and define new density functions on sequences.
Asymptotic periodicity of the iterates of positivity preserving operators
Authors: Miklavčič M.
Published: June 2010

Assume that (A1)

X
X

is a real Banach space. (A2)

X
+

{X^ + }

is a closed subset of

X
X

with the following properties: (i) if

x
∈

X
+

x \in {X^ + }

,

y
∈

X
+

y \in {X^ + }

,

α
∈
[
0
,

∞
)

\alpha \in [0,\,\infty )

then

x
+
y
∈

X
+

x + y \in {X^ + }

and

α
x
∈

X
+

\alpha x \in {X^ + }

; (ii) there exists

M
0

∈
(
0
,

∞
)

{M_0} \in (0,\,\infty )

such that for each

x
∈
X

x \in X

there exist

x
+

∈

X
+

{x_ + } \in {X^ + }

and

x
−

∈

X
+

{x_ - } \in {X^ + }

which satisfy
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x equals x Subscript plus Baseline minus x Subscript minus Baseline comma StartAbsoluteValue EndAbsoluteValue x Subscript plus Baseline StartAbsoluteValue EndAbsoluteValue less-than-or-slanted-equals upper M 0 StartAbsoluteValue EndAbsoluteValue x StartAbsoluteValue EndAbsoluteValue comma StartAbsoluteValue EndAbsoluteValue x Subscript minus Baseline StartAbsoluteValue EndAbsoluteValue less-than-or-slanted-equals upper M 0 StartAbsoluteValue EndAbsoluteValue x StartAbsoluteValue EndAbsoluteValue">

x
=

x
+

−

x
−

,

|

|

x
+

|

|

⩽

M
0

|

|

x

|

|

,

|

|

x
−

|

|

⩽

M
0

|

|

x

|

|

x = {x_ + } - {x_ - },\qquad ||{x_ + }|| \leqslant {M_0}||x||,\qquad ||{x_ - }|| \leqslant {M_0}||x||

\]
and if

x
=

y
+

−

y
−

x = {y_ + } - {y_ - }

for some

y
+

∈

X
+

{y_ + } \in {X^ + }

,

y
−

∈

X
+

{y_ - } \in {X^ + }

then

y
+

−

x
+

∈

X
+

{y_ + } - {x_ + } \in {X^ + }

; (iii) if

x
∈

X
+

x \in {X^ + }

,

y
∈

X
+

y \in {X^ + }

then

|

|

x

|

|

⩽

|

|

x
+
y

|

|

||x|| \leqslant ||x + y||

. (A3)

B
B

is a bounded linear operator on

X
X

. (A4)

B

X
+

⊂

X
+

B{X^ + } \subset {X^ + }

. (A5)

F
0

{F_0}

is a nonempty compact subset of

X
X

and

lim

n
→
∞

dist
⁡
(

B
n

x
,

F
0

)
=
0

{\lim _{n \to \infty }}\operatorname {dist} ({B^n}x,\,{F_0}) = 0

whenever

x
∈

X
+

x \in {X^ + }

and

|

|

x

|

|

=
1

||x|| = 1

. Then

B
n

x

{B^n}x

is asymptotically periodic for every

x
∈
X

x \in X

. This, and other properties of

B
B

, are proven in the paper.
Automorphisms of hyperbolic dynamical systems and 𝐾₂
Authors: Zizza Frank
Published: June 2010

Let

σ
:
Σ
→
Σ

\sigma :\Sigma \to \Sigma

be a subshift of finite type and

Aut
⁡
(
σ
)

\operatorname {Aut} (\sigma )

be the group of homeomorphisms of

Σ
\Sigma

which commute with

σ
\sigma

. In [Wl], Wagoner constructs an invariant for the group

Aut
⁡
(
σ
)

\operatorname {Aut} (\sigma )

using

K
K

-theoretic methods. Smooth hyperbolic dynamical systems can be modeled by subshifts of finite type over the nonwandering sets. In this paper we extend Wagoner’s construction to produce an invariant on the group of homeomorphisms of a smooth manifold which commute with a fixed hyperbolic diffeomorphism. We then proceed to show that this dynamical invariant can be calculated (at least

mod
2

\bmod 2

) from the homology groups of the manifold and the action of the diffeomorphism and the homeomorphisms on the homology groups.