bluuebunny/crossref_metadata_2025 huggingface dataset:
𝜆-continuous Markov chains. II
Authors: Moy Shu-teh C.
Published: June 2010

Continuing the investigation in [8] we study a

λ
\lambda

-continuous Markov operator

P
P

. It is shown that, if

P
P

is conservative and ergodic,

P
P

is indeed “periodic” as is the case when the state space is discrete; there is a positive integer

δ
\delta

, called the period of

P
P

, such that the state space may be decomposed into

δ
\delta

cyclically moving sets

C
0

,
⋯
,

C

δ
−
1

{C_0}, \cdots ,{C_{\delta - 1}}

and, for every positive integer

n
,

P

n
δ

n,{P^{n\delta }}

acting on each

C
i

{C_i}

alone is ergodic. It is also shown that

P
P

maps

L
q

(
μ
)

{L_q}(\mu )

into

L
q

(
μ
)

{L_q}(\mu )

where

μ
\mu

is the nontrivial invariant measure of

P
P

and

1
≦
q
≦
∞

1 \leqq q \leqq \infty

. If

μ
\mu

is finite and normalized then it is shown that (1) if

f
∈

L
∞

(
λ
)

f \in {L_\infty }(\lambda )

, then

{

P

n
δ
+
k

f
}

\{ {P^{n\delta + k}}f\}

converges a.e.

(
λ
)

(\lambda )

to

g
k

=

∑

i
=
0

δ
−
1

c

i
+
k

1

C
i

{g_k} = \sum \nolimits _{i = 0}^{\delta - 1} {{c_{i + k}}} {1_{{C_i}}}

where

c
j

=
δ

∫

C
j

f
d
μ

{c_j} = \delta {\smallint _{{C_j}}}fd\mu

if

0
≦
j
≦
δ
−
1

0 \leqq j \leqq \delta - 1

and

c
j

=

c
i

{c_j} = {c_i}

if

j
=
m
δ
+
i
,
0
≦
i
≦
δ
−
1

j = m\delta + i,0 \leqq i \leqq \delta - 1

, (2)

{

P

n
δ
+
k

f
}

\{ {P^{n\delta + k}}f\}

converges in

L
q

(
μ
)

{L_q}(\mu )

to

g
k

{g_k}

if

f
∈

L
q

(
μ
)

f \in {L_q}(\mu )

, and(3)

lim

inf

n
→
∞

P

n
δ
+
k

f
=

g
k

\lim {\inf _{n \to \infty }}{P^{n\delta + k}}f = {g_k}

a.e.

(
λ
)

(\lambda )

if

f
∈

L
1

(
μ
)

f \in {L_1}(\mu )

and

f
≧
0

f \geqq 0

. If

μ
\mu

is infinite, then it is shown that (1) if

f
≧
0
,
f
∈

L
q

(
μ
)

f \geqq 0,f \in {L_q}(\mu )

for some

1
≦
q
>
∞

1 \leqq q > \infty

, then

lim

inf

n
→
∞

P
n

f
=
0

\lim {\inf _{n \to \infty }}{P^n}f = 0

a.e.

(
λ
)

(\lambda )

, (2) there exists a sequence

{

E
k

}

\{ {E_k}\}

of sets such that

X
=

∪

k
=
1

∞

E
k

X = \cup _{k = 1}^\infty {E_k}

and

lim

n
→
∞

P

n
δ
+
i

1

E
k

=
0

{\lim _{n \to \infty }}{P^{n\delta + i}}{1_{{E_k}}} = 0

a.e.

(
λ
)

(\lambda )

for

i
=
0
,
1
,
⋯
,
δ
−
1

i = 0,1, \cdots ,\delta - 1

and

k
=
1
,
2
,
⋯

k = 1,2, \cdots

.
Some spectral properties of an operator associated with a pair of nonnegative matrices
Authors: Menon M. V.
Published: June 2010

An operator—in general nonlinear—associated with a pair of non-negative matrices, is defined and some of its spectral properties studied. If the pair of matrices are a square matrix A and the identity matrix of the same order, the operator reduces to the linear operator A. The results obtained include generalizations of one of the principal conclusions of the theorem of Perron-Frobenius.
Automorphism groups on compact Riemann surfaces
Authors: Kiley W. T.
Published: June 2010

For

g
≧
2

g \geqq 2

, let

N
(
g
)

N(g)

be the order of the largest automorphism group on a Riemann surface of genus

g
g

. In this paper, lower bounds for

N
(
g
)

N(g)

for various sequences of

g
g

’s are obtained. Sequences of appropriate groups are constructed. Each of these groups is then realized as a group of cover transformations of a surface covering the Riemann sphere. The genus of the resulting surface is then found by using the Riemann-Hurwitz formula and the automorphism group of the surface contains the given group. Each lower bound which is found is also shown to be sharp. That is, there are infinitely many

g
g

’s in the sequence to which the bound applies for which

N
(
g
)

N(g)

does not exceed the bound.
Packing and reflexivity in Banach spaces
Authors: Kottman Clifford A.
Published: June 2010

A measure of the “massiveness” of the unit ball of a Banach space is introduced in terms of an efficiency of the tightest packing of balls of equal size in the unit ball. This measure is computed for the

l
p

{l_p}

-spaces, and spaces with distinct measures are shown to be not nearly isometric. A new convexity condition, which is compared to

B
B

-convexity, uniform smoothness, and uniform convexity, is introduced in terms of this measure, and is shown to be a criterion of reflexivity. The property dual to this convexity condition is also exposed and examined.
On the radius of convexity and boundary distortion of Schlicht functions
Authors: Tepper David E.
Published: June 2010

Let

w
=
f
(
z
)
=
z
+

∑

n
=
2

∞

a
n

z
n

w = f(z) = z + \sum \nolimits _{n = 2}^\infty {{a_n}{z^n}}

be regular and univalent for

|

z

|

>
1

|z| > 1

and map

|

z

|

>
1

|z| > 1

onto a region which is starlike with respect to

w
=
0

w = 0

. If

r
0

{r_0}

denotes the radius of convexity of

w
=
f
(
z
)

w = f(z)

,

d
0

=
min

|

f
(
z
)

|

d_0 = \min |f(z)|

for

|

z

|

=

r
0

|z| = {r_0}

, and

d
∗

=
inf

|

β

|

{d^ \ast } = \inf |\beta |

for

f
(
z
)
≠
β

f(z) \ne \beta

, then it has been conjectured that

d
0

/

d
∗

≧
2

/

3

{d_0}/{d^ \ast } \geqq 2/3

. It is shown here that

d
0

/

d
∗

≧
0.343
…

{d_0}/{d^ \ast } \geqq 0.343 \ldots

, which improves the old estimate

d
0

/

d
∗

≧
0.268
…

{d_0}/{d^ \ast } \geqq 0.268 \ldots

. In addition, sharp estimates for

r
0

{r_0}

are given which depend on the value of

|

a
2

|

|{a_2}|

.
Compact directed spaces
Authors: Ward L. E.
Published: June 2010

A directed space is a partially ordered topological space in which each two elements have a common predecessor. It is a consequence of a theorem of A. D. Wallace that a compact directed space is acyclic if each of its principal ideals is acyclic. This result is extended by considering the situation where at most finitely many principal ideals are not acyclic. It turns out that some of the elements which generate nonacyclic principal ideals must be maximal and that the

p
p

th cohomology group of the space must contain the

p
p

th cohomology group of such a principal ideal as a direct summand. In the concluding sections it is shown that these spaces can be made acyclic by dividing out a closed ideal which contains all of the nonacyclic principal ideals, and some results on the acyclicity properties of minimal partial orders on compact spaces are proved.
Erratum to: “Isolated invariant sets for flows on vector bundles” (Trans. Amer. Math. Soc. 203 (1975), 359–390)
Authors: Selgrade James F.
Published: June 2010

Let (F, f) be a flow on a vector bundle (E, M). If f is minimal on M, then the exponential growth rate intervals associated with the factors of an invariant splitting of E, corresponding to a Morse decomposition of PE, are disjoint intervals.
Dyadic methods in the measure theory of numbers
Authors: Baker R. C.
Published: June 2010

Some new theorems in metric diophantine approximation are obtained by dyadic methods. We show for example that if

m
1

,

m
2

,
…

{m_1},{m_2}, \ldots

, are distinct integers with

m
n

=
O
(

n
p

)

{m_n} = O({n^p})

then

Σ

n
⩽
N

e
(

m
n

x
)
=
O
(

N

1
−
q

)

{\Sigma _{n \leqslant N}}e({m_n}x) = O({N^{1 - q}})

except for a set of x of Hausdorff dimension at most

(
p
+
4
q
−
1
)

/

(
p
+
2
q
)

(p + 4q - 1)/(p + 2q)

; and that for any sequence of intervals

I
1

,

I
2

,
…

{I_1},{I_2}, \ldots

in [0, 1) the number of solutions of

{

x
n

}
∈

I
n

(
n
⩽
N
)

\{ {x^n}\} \in {I_n}\;(n \leqslant N)

is a.e. asymptotic to

Σ

n
⩽
N

|

I
n

|

(
x
>
1
)

{\Sigma _{n \leqslant N}}|{I_n}|(x > 1)

.
Almost sure behavior of linear functionals of supercritical branching processes
Authors: Asmussen Søren
Published: June 2010

The exact a.s. behavior of any linear functional

Z
n

⋅
a

{Z_n} \cdot a

of a supercritical positively regular p-type

(
1
>
p
>
∞
)

(1 > p > \infty )

Galton-Watson process

{

Z
n

}

\{ {Z_n}\}

is found under a second moment hypothesis. The main new results are of iterated logarithm type, with normalizing constants depending on the decomposition of a according to the Jordan canonical form of the offspring mean matrix.
Extremal arcs and extended Hamiltonian systems
Authors: Clarke Frank H.
Published: June 2010

A general variational problem is considered; it involves the minimization of an integrand L of a very general nature. The Lagrangian L is allowed to assume the value

+
∞

+ \infty

, and need satisfy no differentiability or convexity conditions. A Hamiltonian corresponding to the problem is defined via the conjugate function of convex analysis, and it is shown how one obtains necessary conditions in the form of an extended Hamiltonian system. This system is expressed in terms of certain “generalized gradients” previously developed by the author. A further result is given which has the feature that the principal hypotheses required, as well as the ensuing conclusions, are entirely in terms of H. This allows the treatment of classes of problems in which H is more amenable to direct analysis than L. The approach also sheds light on the relation between existence theory and the theory of necessary conditions, since the results may easily be compared with R. T. Rockafellar’s recent work on existence theory, in which H also plays a central role. As an example of its application the main result is specialized to a differential inclusion problem. A specific example of its use is also given, an unorthodox optimal control problem with a discontinuous cost functional.
The two-generator subgroups of one-relator groups with torsion
Authors: Pride Stephen J.
Published: June 2010

The main aim of this paper is to show that every two-generator subgroup of any one-relator group with torsion is either a free product of cyclic groups or is a one-relator group with torsion. This result is proved by using techniques for reducing pairs of elements in certain HNN groups. These techniques not only apply to one-relator groups with torsion but also to a large number of other groups, and some additional applications of the techniques are included in the paper. In particular, examples are given to show that the following result of K. Honda is no longer true for infinite groups: if g is a commutator in a finite group G then every generator of

sgp

{
g
}

{\text {sgp}}\{ g\}

is a commutator in G. This confirms a conjecture of B. H. Neumann.
Spectra and measure inequalities
Authors: Putnam C. R.
Published: June 2010

Let T be a bounded operator on a Hilbert space

H

\mathfrak {H}

and let

T
z

=
T
−
z
I

{T_z} = T - zI

. Then the operators
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript z Baseline upper T Subscript z Superscript asterisk Baseline comma upper T Subscript z Baseline upper T Subscript t Baseline left-parenthesis upper T Subscript z Baseline upper T Subscript t Baseline right-parenthesis Superscript asterisk">

T
z

T
z
∗

,

T
z

T
t

(

T
z

T
t

)
∗

{T_z}T_z^\ast ,{T_z}{T_t}{({T_z}{T_t})^\ast }

, and

T
z

T
t

T
s

(

T
z

T
t

T
s

)
∗

{T_z}{T_t}{T_s}{({T_z}{T_t}{T_s})^\ast }

are nonnegative for all complex numbers z, t, and s. We shall obtain some norm estimates for nonnegative lower bounds of these operators, when z, t, and s are restricted to certain sets, in terms of certain capacities or area measures involving the spectrum and point spectrum of T. A typical such estimate is the following special case of Theorem 4 below: Let

H

\mathfrak {H}

be separable and suppose that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript z Baseline upper T Subscript t Baseline left-parenthesis upper T Subscript z Baseline upper T Subscript t Baseline right-parenthesis Superscript asterisk Baseline greater-than-or-slanted-equals upper D greater-than-or-slanted-equals 0">

T
z

T
t

(

T
z

T
t

)
∗

⩾
D
⩾
0

{T_z}{T_t}{({T_z}{T_t})^\ast } \geqslant D \geqslant 0

for all z and t not belonging to the closure of the interior of the point spectrum of T. In addition, suppose that the boundary of the interior of the point spectrum of T has Lebesgue planar measure 0. Then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper D double-vertical-bar Superscript 1 slash 2 Baseline less-than-or-slanted-equals pi Superscript negative 1 Baseline meas Subscript 2 Baseline left-parenthesis sigma Subscript p Baseline left-parenthesis upper T right-parenthesis right-parenthesis">

‖
D
‖

1

/

2

⩽

π

−
1

meas
2

(

σ
p

(
T
)
)

{\left \| D \right \|^{1/2}} \leqslant {\pi ^{ - 1}}\;{\text {meas}_2}\;({\sigma _p}(T))

. If T is the adjoint of the simple unilateral shift, then equality holds with

D
=
I
−

T
∗

T

D = I - {T^\ast }T

.
Small zeros of additive forms in many variables
Authors: Schmidt Wolfgang M.
Published: June 2010

It is shown that if s is large as a function of k and of

ε
>
0

\varepsilon > 0

, then the diophantine equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a 1 x 1 Superscript k Baseline plus midline-horizontal-ellipsis plus a Subscript s Baseline x Subscript s Superscript k Baseline equals b 1 y 1 Superscript k Baseline plus midline-horizontal-ellipsis plus b Subscript s Baseline y Subscript s Superscript k">

a
1

x
1

k

+
⋯
+

a
s

x
s
k

=

b
1

y
1
k

+
⋯
+

b
s

y
s
k

{a_1}{x_1}^k + \cdots + {a_s}x_s^k = {b_1}y_1^k + \cdots + {b_s}y_s^k

with positive coefficients

a
1

,
…
,

a
s

{a_1}, \ldots ,{a_s}

,

b
1

,
…
,

b
s

{b_1}, \ldots ,{b_s}

has a nontrivial solution in nonnegative integers

x
1

,
…
,

x
s

{x_1}, \ldots ,{x_s}

,

y
1

,
…
,

y
s

{y_1}, \ldots ,{y_s}

not exceeding

m

(

1

/

k

)

+
ε

{m^{\left ( {1/k} \right ) + \varepsilon }}

, where m is the maximum of the coefficients.
On a case of extensions of group schemes
Authors: Weisfeiler B.
Published: June 2010

The extensions of a smooth connected commutative group scheme whose generic fiber is

G
m

{G_m}

by the additive group scheme are studied. The results are most explicit in the case when the basic scheme is the spectrum of an integral domain containing a field.
Duality theory for covariant systems
Authors: Landstad Magnus B.
Published: June 2010

If

(
A
,
ρ
,
G
)

(A,\rho ,G)

is a covariant system over a locally compact group G, i.e.

ρ
\rho

is a homomorphism from G into the group of

∗

^{\ast }

-automorphisms of an operator algebra A, there is a new operator algebra

A

\mathfrak {A}

called the covariance algebra associated with

(
A
,
ρ
,
G
)

(A,\rho ,G)

. If A is a von Neumann algebra and

ρ
\rho

is

σ
\sigma

-weakly continuous,

A

\mathfrak {A}

is defined such that it is a von Neumann algebra. If A is a

C

∗

{C^{\ast }}

-algebra and

ρ
\rho

is norm-continuous

A

\mathfrak {A}

will be a

C

∗

{C^{\ast }}

-algebra. The following problems are studied in these two different settings: 1. If

A

\mathfrak {A}

is a covariance algebra, how do we recover A and

ρ
\rho

? 2. When is an operator algebra

A

\mathfrak {A}

the covariance algebra for some covariant system over a given locally compact group G?
Some new constructions and estimates in the problem of least area
Authors: Parks Harold
Published: June 2010

Surfaces of least k dimensional area in

R

n

{\textbf {R}^n}

are constructed by minimization of the n dimensional volume of suitably thickened sets subject to a homological constraint. Specifically, let

1

⩽

k

⩽

n

1 \,\, \leqslant \,\,k\,\, \leqslant \,n

be integers and

B

⊂

R

n

B\, \subset \,{\textbf {R}^n}

be compact and

k

−

1

k\, - \,1

rectifiable. Let G be a compact abelian group and L be a subgroup of the Čech homology group

H

k
−
1

(

B
;

G

)

{H_{k - 1}}\left ( {B;\,\,G} \right )

(in case

k
=

1

k = \,1

, suppose, additionally, L is contained in the kernel of the usual augmentation map). J. F. Adams has defined what it means for a compact set

X

⊂

R

n

\textrm {X}\, \subset \,{\textbf {R}^n}

to span L. Using also a natural notion of what it means for a compact set to be

ε
\varepsilon

-thick, we show that, for each

ε

>

0

\varepsilon \, > \,0

, there exists an

ε
\varepsilon

-thick set which minimizes n dimensional volume subject to the requirement that it span L. Our main result is that as

ε
\varepsilon

approaches 0 a subsequence of the above volume minimizing sets converges in the Hausdorff distance topology to a set, X, which minimizes k dimensional area subject to the requirement that it span L. It follows, of course, from the regularity results of Reifenberg or Almgren that, except for a compact singular set of zero k dimensional measure, X is a real analytic minimal submanifold of

R

n

{\textbf {R}^n}

.
Nonselfadjoint crossed products (invariant subspaces and maximality)
Authors: McAsey Michael , Muhly Paul S. , Saito Kichi-Suke
Published: June 2010

Let

L

\mathcal {L}

be the von Neumann algebra crossed product determined by a finite von Neumann algebra M and a trace preserving automorphism. In this paper we investigate the invariant subspace structure of the subalgebra

L

+

{\mathcal {L}_ + }

of

L

\mathcal {L}

consisting of those operators whose spectrum with respect to the dual automorphism group on

L

\mathcal {L}

is nonnegative, and we determine conditions under which

L

+

{\mathcal {L}_ + }

is maximal among the

σ
\sigma

-weakly closed subalgebras of

L

\mathcal {L}

. Our main result asserts that the following statements are equivalent: (1) M is a factor; (2)

L

+

{\mathcal {L}_ + }

is a maximal

σ
\sigma

-weakly closed subalgebra of

L

\mathcal {L}

; and (3) a version of the Beurling, Lax, Halmos theorem is valid for

L

+

{\mathcal {L}_ + }

. In addition, we prove that if

A

\mathfrak {A}

is a subdiagonal algebra in a von Neumann algebra

B

\mathcal {B}

and if a form of the Beurling, Lax, Halmos theorem holds for

A

\mathfrak {A}

, then

B

\mathcal {B}

is isomorphic to a crossed product of the form

L

\mathcal {L}

and

A

\mathfrak {A}

is isomorphic to

L

+

{\mathcal {L}_ + }

.
Noncollision singularities in the four-body problem
Authors: Shelton Robert Orrin
Published: June 2010

It is shown that if there is a singularity in a solution of the four-body problem which is not a collision then the motion of the bodies near the singularity is nearly one-dimensional. This is established by grouping the bodies into natural clusters and showing the angular momentum of each cluster with respect to its center of mass tends to zero near the singularity. This is related to Sperling’s proof of von Zeipel’s theorem.
Focal points for a linear differential equation whose coefficients are of constant signs
Authors: Elias Uri
Published: June 2010

The differential equation considered is

y

(
n
)

+
Σ

p
i

(
x
)

y

(
i
)

=
0

{y^{(n)}} + \Sigma {{p_i}(x){y^{(i)}}} = 0

, where

σ
i

p
i

(
x
)
⩾
0
,
i
=
0
,
…
,
n
−
1
,

σ
i

=
±
1

{\sigma _i}{p_i}(x) \geqslant 0,i = 0,\ldots ,n - 1,{\sigma _i} = \pm 1

. The focal point

ζ
(
a
)

\zeta (a)

is defined as the least value of s,

s
>
a

s > a

, such that there exists a nontrivial solution y which satisfies

y

(
i
)

(
a
)
=
0
,

σ
i

σ

i
+
1

>
0

{y^{(i)}}(a) = 0,{\sigma _i}{\sigma _{i + 1}} > 0

and

y

(
i
)

(
s
)
=
0

{y^{(i)}}(s) = 0

,

σ
i

σ

i
+
1

>
0

{\sigma _i}{\sigma _{i + 1}} > 0

. Our method is based on a characterization of

ζ
(
a
)

\zeta (a)

by solutions which satisfy

σ
i

y

(
i
)

>
0
,
i
=
0
,
…
,
n
−
1

{\sigma _i}{y^{(i)}} > 0,i = 0,\ldots ,n - 1

, on

[
a
,
b
]

[a,b]

,

b
>
ζ
(
a
)

b > \zeta (a)

. We study the behavior of the function

ζ
\zeta

and the dependence of

ζ
(
a
)

\zeta (a)

on

p
0

,
…
,

p

n
−
1

{p_0},\ldots ,{p_{n - 1}}

when at least a certain

p
i

(
x
)

{p_i}(x)

does not vanish identically near a or near

ζ
(
a
)

\zeta (a)

. As an application we prove the existence of an eigenvalue of a related boundary value problem.
Milnor’s 𝜇̄-invariants and Massey products
Authors: Porter Richard
Published: June 2010

The main result of this paper gives an interpretation of Milnor’s

μ
¯

\overline \mu

-invariants of a link in terms of Massey products in the complement of the link. The approach presented here can be used to give topological proofs of results about the

μ
¯

\overline \mu

-invariants obtained by Milnor using different methods.
A theorem on free envelopes
Authors: John Chester C.
Published: June 2010

The free envelope of a finite commutative semigroup was defined by Grillet [Trans. Amer. Math. Soc. 149 (1970), 665-682] to be a finitely generated free commutative semigroup

F
(
S
)

F(S)

with identity and a homomorphism

α
:

S

→

F
(
S
)

\alpha :\,S\, \to \,F(S)

endowed with certain properties. Grillet raised the following question: does

α
(
S
)

\alpha (S)

always generate a pure subgroup of the free Abelian group with the same basis as

F
(
S
)

F(S)

? We prove this is indeed the case. It follows as a result of two lemmas. Lemma 1: Given a full rank proper subgroup H of a finitely generated free Abelian group F and a basis X of F there exists a surjective homomorphism

f
:

F

→

Z

f:\,F\, \to \,{\textbf {Z}}

such that f is positive on X and

f

|
H

{f_{\left | H \right .}}

is not surjective. Lemma 2: A finitely generated totally cancellative reduced subsemigroup of a finitely generated free Abelian group F is contained in the positive cone of some basis of F. The following duality theorem is also proved. Let

S

∗

≅

Hom
⁡
(
S
,

N
)

{S^{\ast }}\, \cong \,\operatorname {Hom} (S,\,N)

where N is the nonnegative integers under addition. Then

S

≅

S

∗

∗

S\, \cong \,{S^{{\ast }{\ast }}}

if and only if S is isomorphic to a unitary subsemigroup of a finitely generated free commutative semigroup with identity.
On positive contractions in 𝐿^{𝑝}-spaces
Authors: Schaefer H. H.
Published: June 2010

Let T denote a positive contraction

(
T

⩾

0
,

‖
T
‖

⩽

1
)

(T\, \geqslant \,0,\,\left \| T \right \|\, \leqslant \,1)

on a space

L
p

(
μ
)

(
1

>

p

>

+

∞
)

{L^p}(\mu )\,(1\, > \,p\, > \, + \,\infty )

. A primitive nth root of unity

ε
\varepsilon

is in the point spectrum

P
σ
(
T
)

P\sigma (T)

iff it is in

P
σ
(

T
′

)

P\sigma (T’)

; if so, the unimodular group generated by

ε
\varepsilon

is in both

P
σ
(
T
)

P\sigma (T)

and

P
σ
(

T
′

)

P\sigma (T’)

. In turn, this is equivalent to the existence of n-dimensional Riesz subspaces of

L
p

{L^p}

and

L
q

(

p

−

1

+

q

−

1

=

1
)

{L^q}({p^{ - \,1}}\, + \,{q^{ - \,1}}\, = \,1)

which are in canonical duality and on which T (resp.,

T
′

T’

) acts as an isometry. If, in addition, T is quasi-compact then the spectral projection associated with the unimodular spectrum of T (resp.,

T
′

T’

) is a positive contraction onto a Riesz subspace of

L
p

{L^p}

(resp.,

L
q

{L^q}

) on which T (resp.,

T
′

T’

) acts as an isometry.
Equivariant fibrations and transfer
Authors: Waner Stefan
Published: June 2010

The basic properties of equivariant fibrations are described, including an equivariant version of the Ďold Theorem. The foundations of equivariant stable homotopy theory are described, and the theory of equivariant transfer is developed.
Zeros of successive derivatives of entire functions of the form ℎ(𝑧)𝑒𝑥𝑝(-𝑒^{𝑧})
Authors: Edrei Albert
Published: June 2010

Consider

f
(
z
)

=

h
(
z
)
exp
⁡
(
−

e
z

)

(
z

=

x

+

i
y
)

f(z)\, = \,h(z)\exp ( - {e^z})\,(z\, = \,x\, + \,iy)

where

h
(
z
)

h(z)

is a real entire function of finite order having no zeros in some strip

{
x

+

i
y
:

|

y
−
π

|

>

η
1

,

x

>

x
0

}

(
0

>

η
1

)

\{ x\, + \,iy:\,\left | {y - \pi } \right |\, > \,{\eta _1},\,x\, > \,{x_0}\} \,(0\, > \,{\eta _1})

. The author studies the power series (1)

f
(
τ

+

z
)

=

Σ

n
=
0

∞

a
n

z
n

f(\tau \, + \,z)\, = \,\Sigma _{n = 0}^\infty \,{a_n}{z^n}

(

τ
\tau

real) and the number

N
(

τ
1

,

τ
2

;

n
)

N({\tau _1},\,{\tau _2};\,n)

of real zeros of

f

(
n
)

(
z
)

{f^{(n)}}(z)

which lie in the interval

[

τ
1

,

τ
2

]

[{\tau _1},\,{\tau _2}]

. He proves (2)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N left-parenthesis tau 1 comma tau 2 semicolon n right-parenthesis tilde left-parenthesis tau 2 minus tau 1 right-parenthesis n slash left-parenthesis log n right-parenthesis squared left-parenthesis n right-arrow normal infinity right-parenthesis">

N
(

τ
1

,

τ
2

;

n
)

∼

(

τ
2

−

τ
1

)
n

/

(
log

n

)
2

(
n
→
∞
)

N({\tau _1},{\tau _2};\,n)\, \sim \,({\tau _2} - {\tau _1})n/{(\log \,n)^2}\,(n \to \infty )

. With regard to the expansion (1) he determines a positive, strictly increasing, unbounded sequence

{

v
k

}

k
=
1

∞

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

such that

(

v

k
+
1

−

v
k

)

/

log

v
k

→
1

(
k
→
∞
)

({v_{k + 1}} - {v_k})/\log \,{v_k} \to 1\,(k \to \infty )

and having the following properties: (i) if

v
k

>

n

>

v

k
+
1

{v_k}\, > \,n\, > \,{v_{k + 1}}

, then

a
n

≠

0

{a_n}\, \ne \,0

and all the

a
n

{a_n}

have the same sign; (ii) if in addition

v

k
+
1

>

m

>

v

k
+
2

{v_{k + 1}}\, > \,m\, > \,{v_{k + 2}}

, tnen

a
m

a
n

>
0

{a_m}{a_n}\, > 0

. It is possible to deduce from (2) the complete characterization of the final set (in the sense of Pólya) of

exp
⁡
(
−

e
z

)

\exp ( - {e^z})

.
Generation of analytic semigroups by strongly elliptic operators under general boundary conditions
Authors: Stewart H. Bruce
Published: June 2010

Strongly elliptic operators are shown to generate analytic semigroups of evolution operators in the topology of uniform convergence, when realized under general boundary conditions on (possibly) unbounded domains. An application to the existence and regularity of solutions to parabolic initial-boundary value problems is indicated.
The fundamental theorem on torsion classes of lattice-ordered groups
Authors: Martinez Jorge
Published: June 2010

This paper generalizes the earlier notion of a torsion class to a setting where its significance can be fully realized. The dual notion of a torsion-free class is herein defined and the fundamental Connection Theorem is proved. In addition, a few restrictions are considered, in particular, how to view the application of the main theorem to the hereditary classes.
On linear algebraic semigroups
Authors: Putcha Mohan S.
Published: June 2010

Let K be an algebraically closed field. By an algebraic semigroup we mean a Zariski closed subset of

K
n

{K^n}

along with a polynomially defined associative operation. Let S be an algebraic semigroup. We show that S has ideals

I
0

,

…

,

I
t

{I_0},\, \ldots \,,\,{I_t}

such that

S

=

I
t

⊇

⋯

⊇

I
0

S\, = \,{I_t}\, \supseteq \, \cdots \, \supseteq \,{I_0}

,

I
0

{I_0}

is the completely simple kernel of S and each Rees factor semigroup

I
k

/

I

k
−
1

{I_k}/{I_{k - 1}}

is either nil or completely 0-simple

(
k

=

1
,

…

,

t
)

(k\, = \,1,\, \ldots \,,\,t)

. We say that S is connected if the underlying set is irreducible. We prove the following theorems (among others) for a connected algebraic semigroup S with idempotent set

E
(
S
)

E(S)

. (1) If

E
(
S
)

E(S)

is a subsemigroup, then S is a semilattice of nil extensions of rectangular groups. (2) If all the subgroups of S are abelian and if for all

a

∈

S

a\, \in \,S

, there exists

e

∈

E
(
S
)

e\, \in \,E(S)

such that

e
a

=

a
e

=
a

ea\, = \,ae\, = a

, then S is a semilattice of nil extensions of completely simple semigroups. (3) If all subgroups of S are abelian and if S is regular, then S is a subdirect product of completely simple and completely 0-simple semigroups. (4) S has only trivial subgroups if and only if S is a nil extension of a rectangular band.
Every contractible fan is locally connected at its vertex
Authors: Oversteegen Lex G.
Published: June 2010

We prove that each contractible fan is locally connected at its vertex. It follows that every contractible fan is embeddable in the plane. This gives a solution to a problem raised by J. J. Charatonik and C. A. Eberhart.
Composition series for analytic continuations of holomorphic discrete series representations of 𝑆𝑈(𝑛,𝑛)
Authors: Ørsted Bent
Published: June 2010

We study a certain family of holomorphic discrete series representations of the semisimple Lie group

G

=

S
U
(
n
,

n
)

G\, = \,SU(n,\,n)

and the corresponding analytic continuation in the inducing parameter

λ
\lambda

. At the values of

λ
\lambda

where the representations become reducible, we compute the composition series in terms of a Peter-Weyl basis on the Shilov boundary of the Hermitian symmetric space for G.
Stationary logic and ordinals
Authors: Seese D. G.
Published: June 2010

The

L
(

a
a

)

L({\mathbf {aa}})

-theory of ordinals is investigated. It is proved that this theory is decidable and that each ordinal is finitely determinate.
Orientation-reversing Morse-Smale diffeomorphisms on the torus
Authors: Batterson Steve
Published: June 2010

For orientation-reversing diffeomorphisms on the torus necessary and sufficient conditions are given for an isotopy class to admit a Morse-Smale diffeomorphism with a specified periodic behavior.
On spectral theory and convexity
Authors: Fong C. K. , Lam Louisa
Published: June 2010

A compact convex set

K
K

in a locally convex algebra is said to be a spectral carrier if, for all

x
x

,

y
∈
K

y \in K

, we have

x
y
=
y
x
∈
K

xy = yx \in K

and

x
+
y
−
x
y
∈
K

x + y - xy \in K

. We show that if a compact convex set

K
K

is a spectral carrier, then the idempotents in

K
K

are exactly the extreme points of

K
K

and form a complete lattice. Conversely, if a compact set

K
K

is a closed convex hull of a lattice of commuting idempotents, then

K
K

is a spectral carrier. Furthermore, a metrizable spectral carrier is a Choquet simplex if and only if its extreme points form a chain of idempotents.
Homotopy groups of the space of self-homotopy-equivalences
Authors: McCullough Darryl
Published: June 2010

Let

M
M

be a connected sum of

r
r

closed aspherical manifolds of dimension

n
⩾
3

n \geqslant 3

, and let

E
M

EM

denote the space of self-homotopy-equivalences of

M
M

, with basepoint the identity map of

M
M

. Using obstruction theory, we calculate

π
q

(
E
M
)

{\pi _q}(EM)

for

1
⩽
q
⩽
n
−
3

1 \leqslant q \leqslant n - 3

and show that

π

n
−
1

(
E
M
)

{\pi _{n - 1}}(EM)

is not finitely-generated. As an application, for the case

n
=
3

n = 3

and

r
⩾
3

r \geqslant 3

we show that infinitely many generators of

π
1

(
E

M
3

,

i

d

M

)

{\pi _1}(E{M^3},{\text {i}}{{\text {d}}_M})

can be realized by isotopies, to conclude that

π
1

(

Homeo

(

M
3

)
,

i

d

M

)

{\pi _1}({\text {Homeo}}({M^3}),{\text {i}}{{\text {d}}_M})

is not finitely-generated.
Characterization of some zero-dimensional separable metric spaces
Authors: van Mill Jan
Published: June 2010

Let

X
X

be a separable metric zero-dimensional space for which all nonempty clopen subsets are homeomorphic. We show that, up to homeomorphism, there is at most one space

Y
Y

which can be written as an increasing union

∪

n
=
1

∞

F
n

\cup _{n = 1}^\infty {F_n}

of closed sets so that for all

n
∈

N

n \in {\mathbf {N}}

,

F
n

{F_n}

is a copy of

X
X

which is nowhere dense in

F

n
+
1

{F_{n + 1}}

. If moreover

X
X

contains a closed nowhere dense copy of itself, then

Y
Y

is homeomorphic to

Q

×
X

{\mathbf {Q}} \times X

where

Q

{\mathbf {Q}}

denotes the space of rational numbers. This gives us topological characterizations of spaces such as

Q

×

C

{\mathbf {Q}} \times {\mathbf {C}}

and

Q

×

P

{\mathbf {Q}} \times {\mathbf {P}}

.
A unicity theorem for meromorphic mappings between algebraic varieties
Authors: Drouilhet S. J.
Published: June 2010

Using the techniques of value distribution theory in several complex variables, we obtain a theorem which can be used to determine whether two nondegenerate meromorphic mappings from an affine algebraic variety to a projective algebraic variety of the same or lower dimension are identical. The theorem generalizes a result of

R
R

. Nevanlinna in one complex variable.
Unflat connections in 3-sphere bundles over 𝑆⁴
Authors: Derdziński Andrzej , Rigas A.
Published: June 2010

The paper concerns connections in

3
3

-sphere bundles over

4
4

-manifolds having the property of unflatness, which is a necessary condition in order that a natural construction give a Riemannian metric of positive sectional curvature in the total space. It is shown that, as conjectured by A. Weinstein, the only

3
3

-sphere bundle over

S
4

{S^4}

with an unflat connection is the Hopf bundle.
Multiplicatively invariant subspaces of Besov spaces
Authors: Nilsson Per
Published: June 2010

We study subspaces of Besov spaces

B
p

s
,
q

B_p^{s,q}

which are invariant under pointwise multiplication by characters. The case

s
>
0

s > 0

is completely described, and for the case

s
⩽
0

s \leqslant 0

we extend known results.
Some properties of measure and category
Authors: Miller Arnold W.
Published: June 2010

Several elementary cardinal properties of measure and category on the real line are studied. For example, one property is that every set of real numbers of cardinality less than the continuum has measure zero. All of the properties are true if the continuum hypothesis is assumed. Several of the properties are shown to be connected with the properties of the set of functions from integers to integers partially ordered by eventual dominance. Several, but not all, combinations of these properties are shown to be consistent with the usual axioms of set theory. The main technique used is iterated forcing.
On the convergence of closed-valued measurable multifunctions
Authors: Salinetti Gabriella , Wets Roger J.-B.
Published: June 2010

In this paper we study the convergence almost everywhere and in measure of sequences of closed-valued multifunctions. We first give a number of criteria for the convergence of sequences of closed subsets. These results are used to obtain various characterizations for the convergence of measurable multifunctions. In particular we are interested in the convergence properties of (measurable) selections.
The topology on the primitive ideal space of transformation group 𝐶*-algebras and C.C.R. transformation group 𝐶*-algebras
Authors: Williams Dana P.
Published: June 2010

If

(
G
,
Ω
)

(G,\Omega )

is a second countable transformation group and the stability groups are amenable then

C
∗

(
G
,
Ω
)

{C^ \ast }(G,\Omega )

is C.C.R. if and only if the orbits are closed and the stability groups are C.C.R. In addition, partial results relating closed orbits to C.C.R. algebras are obtained in the nonseparable case. In several cases, the topology of the primitive ideal space is calculated explicitly. In particular, if the stability groups are all contained in a fixed abelian subgroup

H
H

, then the topology is computed in terms of

H
H

and the orbit structure, provided

C
∗

(
G
,
Ω
)

{C^ \ast }(G,\Omega )

and

C
∗

(
H
,
Ω
)

{C^ \ast }(H,\Omega )

are

E
H

EH

-regular. These conditions are automatically met if

G
G

is abelian and

(
G
,
Ω
)

(G,\Omega )

is second countable.
The cyclic connectivity of homogeneous arcwise connected continua
Authors: Bellamy David P. , Lum Lewis
Published: June 2010

A continuum is cyclicly connected provided each pair of its points lie together on some simple closed curve. In 1927, G. T. Whyburn proved that a locally connected plane continuum is cyclicly connected if and only if it contains no separating points. This theorem was fundamental in his original treatment of cyclic element theory. Since then numerous authors have obtained extensions of Whyburn’s theorem. In this paper we characterize cyclic connectedness in the class of all Hausdorff continua. Theorem. The Hausdorff continuum

X
X

is cyclicly connected if and only if for each point

x
∈
X

x \in X

,

x
x

lies in the relative interior of some arc in

X
X

and

X
−
{
x
}

X - \{ x\}

is arcwise connected. We then prove that arcwise connected homogeneous metric continua are cyclicly connected.
The local Kronecker-Weber theorem
Authors: Lubin Jonathan
Published: June 2010

The extension of a local field generated by adjoining the torsion points on a suitable formal group is essentially the maximal abelian extension of the field. This fact is proven by appealing to the functorial properties of the Herbrand transition function of higher ramification theory.
Refinement properties and extensions of filters in Boolean algebras
Authors: Balcar Bohuslav , Simon Petr , Vojtáš Peter
Published: June 2010

We consider the question, under what conditions a given family

A
A

in a Boolean algebra

B

\mathcal {B}

has a disjoint refinement. Of course,

A
A

cannot have a disjoint refinement if

A
A

is a dense subset of an atomless

B

\mathcal {B}

, or if

B

\mathcal {B}

is complete and

A
A

generates an ultrafilter on

B

\mathcal {B}

. We show in the first two sections that these two counterexamples can be the only possible ones. The third section is concerned with the question, how many sets must necessarily be added to a given filter in order to obtain an ultrafilter base.
The structure of tensor products of semilattices with zero
Authors: Grätzer G. , Lakser H. , Quackenbush R.
Published: June 2010

If

A
A

and

B
B

are finite lattices, then the tensor product

C
C

of

A
A

and

B
B

in the category of join semilattices with zero is a lattice again. The main result of this paper is the description of the congruence lattice of

C
C

as the free product (in the category of bounded distributive lattices) of the congruence lattice of

A
A

and the congruence lattice of

B
B

. This provides us with a method of constructing finite subdirectly irreducible (resp., simple) lattices: if

A
A

and

B
B

are finite subdirectly irreducible (resp., simple) lattices then so is their tensor product. Another application is a result of E. T. Schmidt describing the congruence lattice of a bounded distributive extension of

M
3

{M_3}

.
On the weak behaviour of partial sums of Legendre series
Authors: Chanillo S.
Published: June 2010

We show that the partial sum operator associated with the Legendre series is restricted weak type, but not weak type, on the

L
p

{L^p}

spaces when

p
=
4

p = 4

.
A family of links and the Conway calculus
Authors: Giller Cole A.
Published: June 2010

In 1969, J. H. Conway gave efficient methods of calculating abelian invariants of classical knots and links. The present paper includes a detailed exposition (with new proofs) of these methods and extensions in several directions. The main application given here is as follows. A link

L
L

of two unknotted components in

S
3

{S^3}

has the distinct lifting property for

p
p

if the lifts of each component to the

p
p

-fold cover of

S
3

{S^3}

branched along the other are distinct. The

p
p

-fold covers of these lifts are homeomorphic, and so

L
L

gives an example of two distinct knots with the same

p
p

-fold cover. The above machinery is then used to construct an infinite family of links, each with the distinct lifting property for all

p
⩾
2

p \geqslant 2

.
Boolean algebras with no rigid or homogeneous factors
Authors: Štěpánek Petr
Published: June 2010

A simple construction of Boolean algebras with no rigid or homogeneous factors is described. It is shown that for every uncountable cardinal

κ
\kappa

there are

2
κ

{2^\kappa }

isomorphism types of Boolean algebras of power

κ
\kappa

with no rigid or homogeneous factors. A similar result is obtained for complete Boolean algebras for certain regular cardinals. It is shown that every Boolean algebra can be completely embedded in a complete Boolean algebra with no rigid or homogeneous factors in such a way that the automorphism group of the smaller algebra is a subgroup of the automorphism group of the larger algebra. It turns out that the cardinalities of antichains in both algebras are the same. It is also shown that every

κ
\kappa

-distributive complete Boolean algebra can be completely embedded in a

κ
\kappa

-distributive complete Boolean algebra with no rigid or homogeneous factors.
Congruences between systems of eigenvalues of modular forms
Authors: Jochnowitz Naomi
Published: June 2010

We modify and generalize proofs of Tate and Serre in order to show that there are only a finite number of systems of eigenvalues for the Hecke operators with respect to

Γ
0

(
N
)
mod
l

{\Gamma _0}(N)\bmod l

. We also summarize results for

Γ
1

(
N
)

{\Gamma _1}(N)

. Using these results, we show that an arbitrary prime divides the discriminant of the classical Hecke ring to a power which grows linearly with

k
k

. In this way, we find a lower bound for the discriminant of the Hecke ring. After limiting ourselves to cusp forms, we also find an upper bound. Lastly we use the constructive nature of Tate and Serre’s result to describe the structure and dimensions of the generalized eigenspaces for the Hecke operators

mod
l

\bmod l

.
Measurable parametrizations of sets in product spaces
Authors: Srivatsa V. V.
Published: June 2010

Various parametrization theorems are proved. In particular the following is shown: Let

B
B

be a Borel subset of

I
×
I

I \times I

(where

I
=
[
0
,

1
]

I = [0,\,1]

) with uncountable vertical sections. Let

∑

∪
˙

N

\sum \dot \cup N

be the discrete (topological) union of

∑
\sum

, the space of irrationals, and

N
N

, the set of natural numbers with discrete topology. Then there is a map

f
:
I
×
(
∑

∪
˙

N
)
→
I

f:I \times (\sum \dot \cup N) \to I

measurable with respect to the product of the analytic

σ
\sigma

-field on

I
I

(that is, the smallest

σ
\sigma

-field on

I
I

containing the analytic sets) and the Borel

σ
\sigma

-field on

∑

∪
˙

N

\sum \dot \cup N

such that

f
(
t
,

⋅
)
:

∑

∪
˙

N
→
I

f(t,\,\, \cdot ):\,\sum \dot \cup N \to I

is a one-one continuous map of

∑

∪
˙

N

\sum \dot \cup N

onto

{
x
:
(
t
,

x
)
∈
B
}

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

for each

t
∈
T

t \in T

. This answers a question of Cenzer and Mauldin.
Saturation properties of ideals in generic extensions. I
Authors: Baumgartner James E. , Taylor Alan D.
Published: June 2010

We consider saturation properties of ideals in models obtained by forcing with countable chain condition partial orderings. As sample results, we mention the following. If

M
[
G
]

M[G]

is obtained from a model

M
M

of GCH via any

σ
\sigma

-finite chain condition notion of forcing (e.g. add Cohen reals or random reals) then in

M
[
G
]

M[G]

every countably complete ideal on

ω
1

{\omega _1}

is

ω
3

{\omega _3}

-saturated. If "

σ
\sigma

-finite chain condition" is weakened to "countable chain condition," then the conclusion no longer holds, but in this case one can conclude that every

ω
2

{\omega _2}

-generated countably complete ideal on

ω
1

{\omega _1}

(e.g. the nonstationary ideal) is

ω
3

{\omega _3}

-saturated. Some applications to

P

ω
1

(

ω
2

)

{\mathcal {P}_{{\omega _1}}}({\omega _2})

are included and the role played by Martin’s Axiom is discussed. It is also shown that if these weak saturation requirements are combined with some cardinality constraints (e.g.

2

ℵ
1

>

(

2

ℵ
0

)
+

)

{2^{{\aleph _1}}} > {({2^{{\aleph _0}}})^ + })

), then the consistency of some rather large cardinals becomes both necessary and sufficient.
The Bergman kernel function and proper holomorphic mappings
Authors: Bell Steven R.
Published: June 2010

It is proved that a proper holomorphic mapping

f
f

between bounded complete Reinhardt domains extends holomorphically past the boundary and that if, in addition,

f

−
1

(
0
)
=
{
0
}

{f^{ - 1}}(0) = \{ 0\}

, then

f
f

is a polynomial mapping. The proof is accomplished via a transformation rule for the Bergman kernel function under proper holomorphic mappings.
Permutation-partition pairs. II. Bounds on the genus of the amalgamation of graphs
Authors: Stahl Saul
Published: June 2010

Bounds are derived on the extent to which the parameter

μ
(
P
,

∏
)

\mu (P,\,\prod )

can fail to be additive over disjoint permutations. This is done by associating an Eulerian digraph to each such pair and relating the maximum orbiticity

μ
(
P
,

∏
)

\mu (P,\,\prod )

to the decompositions of this digraph’s arc set into arc disjoint cycles. These bounds are then applied to obtain information about the genus of the amalgamation of graphs.
Growth of solutions of linear differential equations at a logarithmic singularity
Authors: Adolphson A. , Dwork B. , Sperber S.
Published: June 2010

We consider differential equations

Y
′

=
A
Y

Y’ = AY

with a regular singular point at the origin, where

A
A

is an

n
×
n

n \times n

matrix whose entries are

p
p

-adic meromorphic functions. If the solution matrix at the origin is of the form

Y
=
P
exp
⁡
(
θ
log
⁡
x
)

Y = P\exp (\theta \log x)

, where

P
P

is an

n
×
n

n \times n

matrix of meromorphic functions and

θ
\theta

is an

n
×
n

n \times n

constant matrix whose Jordan normal form consists of a single block, then we prove that the entries of

P
P

have logarithmic growth of order

n
−
1

n - 1

.
Generation of finite groups of Lie type
Authors: Seitz Gary M.
Published: June 2010

Let

p
p

be an odd prime and

G
G

a finite group of Lie type in characteristic other than

p
p

. Fix an elementary abelian

p
p

-subgroup of

Aut
⁡
(
G
)

\operatorname {Aut} (G)

. It is shown that in most cases

G
G

is generated by the centralizers of the maximal subgroups of

E
E

. Results are established concerning the notions of layer generation and balance, and the strongly

p
p

-embedded subgroups of

Aut
⁡
(
G
)

\operatorname {Aut} (G)

are determined.
Star-finite representations of measure spaces
Authors: Anderson Robert M.
Published: June 2010

In nonstandard analysis,

∗

^{\ast }

-finite sets are infinite sets which nonetheless possess the formal properties of finite sets. They permit a synthesis of continuous and discrete theories in many areas of mathematics, including probability theory, functional analysis, and mathematical economics.

∗

^{\ast }

-finite models are particularly useful in building new models of economic or probabilistic processes. It is natural to ask what standard models can be obtained from these

∗

^{\ast }

-finite models. In this paper, we show that a rich class of measure spaces, including the Radon spaces, are measure-preserving images of

∗

^{\ast }

-finite measure spaces, using a construction introduced by Peter A. Loeb [15]. Moreover, we show that a number of measure-theoretic constructs, including integrals and conditional expectations, are naturally expressed in these models. It follows that standard models which can be expressed in terms of these measure spaces and constructs can be obtained from

∗

^{\ast }

-finite models.
Interior and boundary continuity of weak solutions of degenerate parabolic equations
Authors: Ziemer William P.
Published: June 2010

In this paper we consider degenerate parabolic equations of the form

(

∗

)

({\ast })

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta left-parenthesis u right-parenthesis Subscript t minus d i v upper A left-parenthesis x comma t comma u comma u Subscript x Baseline right-parenthesis plus upper B left-parenthesis x comma t comma u comma u Subscript x Baseline right-parenthesis contains-as-member 0">

β

(
u

)
t

−
div
⁡
A
(
x
,

t
,

u
,

u
x

)
+
B
(
x
,

t
,

u
,

u
x

)
∋
0

\beta {(u)_t} - \operatorname {div} A(x,\,t,\,u,\,{u_x}) + B(x,\,t,\,u,\,{u_x}) \ni 0

\]
where

A
A

and

B
B

are, respectively, vector and scalar valued Baire functions defined on

U
×

R
1

×

R
n

U \times {R^1} \times {R^n}

, where

U
U

is an open subset of

R

n
+
1

(
x
,

t
)

{R^{n + 1}}(x,\,t)

. The functions

A
A

and

B
B

are subject to natural structural inequalities. Sufficiently general conditions are allowed on the relation

β
⊂

R
1

×

R
1

\beta \subset {R^1} \times {R^1}

so that the porus medium equation and the model for the two-phase Stefan problem can be considered. The main result of the paper is that weak solutions of

(

∗

)

({\ast })

are continuous throughout

U
U

. In the event that

U
=
Ω
×
(
0
,

T
)

U = \Omega \times (0,\,T)

where

Ω
\Omega

is an open set of

R
n

{R^n}

, it is also shown that a weak solution is continuous at

(

x
0

,

t
0

)
∈
∂
Ω
×
(
0
,

T
)

({x_0},{t_0}) \in \partial \Omega \times (0,\,T)

provided

x
0

{x_0}

is a regular point for the Laplacian on

Ω
\Omega

.
Generalization of continuous posets
Authors: Novak Dan
Published: June 2010

In this paper we develop a general theory of continuity in partially ordered sets. Among the interesting special cases of this theory is the theory of continuous lattices developed by D. Scott, J. Lawson and others.