bluuebunny/crossref_metadata_2025 huggingface dataset:
On maximal rearrangement inequalities for the Fourier transform
Authors: Jurkat W. B. , Sampson G.
Published: June 2010

Suppose that

w
w

is a measurable function on

R

n

{{\mathbf {R}}^n}

and denote by

W
=

w
∗

W = {w^ \ast }

the decreasing rearrangement of

|
w
|

\left | w \right |

(provided that it exists). We show that the

n
n

-dimensional Fourier transform

f
^

\hat f

satisfies (1)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar w ModifyingAbove f With caret double-vertical-bar Subscript q Baseline less-than-or-slanted-equals double-vertical-bar upper W left-parenthesis ModifyingAbove f With caret right-parenthesis Superscript asterisk Baseline double-vertical-bar Subscript q Baseline less-than-or-slanted-equals upper C double-vertical-bar upper W left-parenthesis t right-parenthesis integral Subscript 0 Superscript 1 slash t Baseline f Superscript asterisk Baseline double-vertical-bar left-parenthesis upper C absolute constant right-parenthesis comma">

‖

w

f
^

‖

q

⩽

‖

W

(

f
^

)

∗

‖

q

⩽
C

‖

W
(
t
)

∫
0

1

/

t

f
∗

‖

(
C
 

absolute constant

)
,

{\left \| {w\hat f} \right \|_q} \leqslant {\left \| {W{{(\hat f)}^ \ast }} \right \|_q} \leqslant C\left \| {W(t)\int _0^{1/t} {{f^ \ast }} } \right \|\quad (C\ {\text {absolute constant}}),

\]
if

1
>
q
>
∞

1 > q > \infty

and

t

2

/

q
−
1

W
(
t
)
↘

{t^{2/q - 1}}W(t) \searrow

for

t
>
0

t > 0

. We also show that (2)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar w ModifyingAbove f With caret double-vertical-bar Subscript q Baseline greater-than-or-slanted-equals c Subscript n comma q Baseline double-vertical-bar w left-parenthesis x right-parenthesis integral Underscript StartAbsoluteValue v EndAbsoluteValue less-than-or-slanted-equals 1 slash StartAbsoluteValue x EndAbsoluteValue Endscripts f left-parenthesis y right-parenthesis d y double-vertical-bar Subscript q Baseline left-parenthesis f nonnegative right-parenthesis comma">

‖

w

f
^

‖

q

⩾

c

n
,
q

‖

w
(
x
)

∫

|
v
|

⩽
1

/

|
x
|

f
(
y
)

d
y

‖

q

(
f
 

nonnegative),

{\left \| {w\hat f} \right \|_q} \geqslant {c_{n,q}}{\left \| {w(x)\int _{\left | v \right | \leqslant 1/\left | x \right |} {f(y)} dy} \right \|_q}\quad (f\ {\text {nonnegative),}}

\]
if

1
>
q
>
∞

1 > q > \infty

and

w
w

is nonnegative and symmetrically decreasing. Inequality (2) implies that (1) is maximal in the sense that the left side reaches the right side if

f
f

is nonnegative and symmetrically decreasing. Hence, (1) implies all other possible estimates in terms of

W
W

and

f
∗

{f^ \ast }

. The cases

q
≠
2

q \ne 2

of (1) can be derived from the case

q
=
2

q = 2

(and same

f
f

) by a convexity principle which does not involve interpolation. The analogue of (1) for Fourier series is due to H. L. Montgomery if

q
⩾
2

q \geqslant 2

(then the extra condition on

W
W

is automatically satisfied).
On cyclic trigonal Riemann surfaces. I
Authors: Accola Robert D. M.
Published: June 2010

Definition. Call the Riemann surfaces for the equation

y
3

=
P
(
x
)

{y^3} = P(x)

cyclic trigonal. For one case of genus

4
4

(

2
2

distinct

g
3
1

g_3^1

’s) and all genera greater than

4
4

, cyclic trigonal Riemann surfaces are characterized by the vanishing properties of the theta function at certain

(
1

/

6
)

(1/6)

-periods of the Jacobian. Also for trigonal Riemann surfaces of genera

5
5

,

6
6

, and

7
7

, homogeneous theta relations are derived using the fact that Prym varieties for trigonal Riemann surfaces are Jacobians.
Commuting analytic functions
Authors: Cowen Carl C.
Published: June 2010

Let

f
f

and

g
g

(not conformal automorphisms of the unit disk) be analytic mappings of the unit disk into itself. We say

f
f

and

g
g

commute if

f
∘
g
=
g
∘
f

f \circ g = g \circ f

. This paper characterizes those functions

g
g

that commute with a given function

f
f

. Several corollaries of this characterization give qualitative information about

g
g

given similar information about

f
f

, and examples are given in each case to show the limitations of the conclusions. Some of the qualitative properties considered are univalence, fixed point sets, and whether two such

g
g

must commute with each other.
Variational invariants of Riemannian manifolds
Authors: Siegel Jerrold , Williams Frank
Published: June 2010

This paper treats higher-dimensional analogues to the minimum geodesic distance in a compact Riemannian manifold

M
M

with finite fundamental group. These invariants are based on the concept of homotopy distance in

M
M

. This defines a parametrized variational problem which is approached by globalizing the Morse theory of the spaces of paths between two points of

M
M

to the space of all paths in

M
M

. We develop machinery that we apply to calculate the invariants for numerous examples. In particular, we shall observe that knowledge of the invariants for the standard spheres determines the question of the existence of elements of Hopf invariant one.
Homomorphisms and endomorphisms in varieties of pseudocomplemented distributive lattices (with applications to Heyting algebras)
Authors: Adams M. E. , Koubek V. , Sichler J.
Published: June 2010

According to a result by K. B. Lee, the lattice of varieties of pseudocomplemented distributive lattices is the

ω
+
1

\omega + 1

chain

B

−
1

⊂

B
0

⊂

B
1

⊂
⋯
⊂

B
n

⊂
⋯
⊂

B
ω

{B_{ - 1}} \subset {B_0} \subset {B_1} \subset \cdots \subset {B_n} \subset \cdots \subset {B_\omega }

in which the first three varieties are formed by trivial, Boolean, and Stone algebras respectively. In the present paper it is shown that any Stone algebra is determined within

B
1

{B_1}

by its endomorphism monoid, and that there are at most two nonisomorphic algebras in

B
2

{B_2}

with isomorphic monoids of endomorphisms; the pairs of such algebras are fully characterized both structurally and in terms of their common endomorphism monoid. All varieties containing

B
3

{B_3}

are shown to be almost universal. In particular, for any infinite cardinal

κ
\kappa

there are

2
κ

{2^\kappa }

nonisomorphic algebras of cardinality

κ
\kappa

in

B
3

{B_3}

with isomorphic endomorphism monoids. The variety of Heyting algebras is also almost universal, and the maximal possible number of nonisomorphic Heyting algebras of any infinite cardinality with isomorphic endomorphism monoids is obtained.
On the realization of invariant subgroups of 𝜋_{∗}(𝑋)
Authors: Zabrodsky A.
Published: June 2010

Let

p
p

be a prime and

T
:
X
→
X

T:X \to X

a self map. Let

A
A

be a multiplicatively closed subset of the algebraic closure of

F
p

{F_p}

. Denote by

V

T
,
A

{V_{T,A}}

the set of characteristic values of

π

∗

(
T
)
⊗

F
p

{\pi _{\ast } }(T) \otimes {F_p}

lying in

A
A

. It is proved that under certain conditions

V

T
,
A

{V_{T,A}}

is realizable by a pair

X
~

,

T
~

\tilde X,\tilde T

: There exist a space

X
~

\tilde X

, maps

T
~

:

X
~

→

X
~

\tilde T:\tilde X \to \tilde X

and

f
:

X
~

→

X

f:\tilde X\: \to \:X

so that

f

∘

T
~

∼
T

∘

f
,

π
∗

(
F
)

f\,\circ \,\tilde T\sim T\,\circ \,f,{\pi _ * }(F)

is

mod

p

\bmod \, p

injective and

i
m

(

π

∗

(
f
)
⊗

F
p

)
=

V

T
,
A

{\rm {im}}({\pi _{\ast } }(f) \otimes {F_p}) = {V_{T,A}}

. This theorem yields, among others, examples of spaces whose

mod

p

\bmod \, p

cohomology rings are polynomial algebras.
Infinite-to-one codes and Markov measures
Authors: Boyle Mike , Tuncel Selim
Published: June 2010

We study the structure of infinite-to-one continuous codes between subshifts of finite type and the behaviour of Markov measures under such codes. We show that if an infinite-to-one code lifts one Markov measure to a Markov measure, then it lifts each Markov measure to uncountably many Markov measures and the fibre over each Markov measure is isomorphic to any other fibre. Calling such a code Markovian, we characterize Markovian codes through pressure. We show that a simple condition on periodic points, necessary for the existence of a code between two subshifts of finite type, is sufficient to construct a Markovian code. Several classes of Markovian codes are studied in the process of proving, illustrating and providing contrast to the main results. A number of examples and counterexamples are given; in particular, we give a continuous code between two Bernoulli shifts such that the defining vector of the image is not a clustering of the defining vector of the domain.
Approximating groups of bundle automorphisms by loop spaces
Authors: Bencivenga Roberto
Published: June 2010

D. H. Gottlieb proved in 1972 that the group of automorphisms of a numerable

G
G

-bundle

p
:
X
→
B

p:X \to B

is weakly homotopy equivalent to

Ω

Map
⁡
(
B
,

B
G

;
k
)

\Omega \;\operatorname {Map}(B,{B_G};k)

, where

k
:
B
→

B
G

k:B \to {B_G}

is a classifying map for

p
p

. We refine this classical result by constructing a genuine homotopy equivalence between these two spaces which is natural with respect to numerable bundle morphisms, can be generalized to fibre bundles, and can be interpreted as a natural isomorphism between two suitably defined functors.
Generic reducing fields of Jordan pairs
Authors: Petersson Holger P.
Published: June 2010

Generic reducing fields of Jordan pairs, generalizing at the same time generic splitting fields of associative algebras and generic zero fields of quadratic forms, are intrinsically defined and constructed. The most elementary properties are derived, and the relationship with other generic constructions, particularly those linked to Brauer-Severi varieties, are investigated. As an application it is shown that there exist nonisomorphic exceptional Jordan division algebras having the same splitting fields.
On hypersingular integrals and anisotropic Bessel potential spaces
Authors: Dappa H. , Trebels W.
Published: June 2010

In this paper we characterize anisotropic potential spaces in terms of hypersingular integrals of mixed homogeneity with respect to a general dilation matrix.
The Godbillon-Vey invariant of a transversely homogeneous foliation
Authors: Brooks Robert , Goldman William
Published: June 2010

A real projective foliation is a foliation

F

\mathfrak {F}

with a system of local coordinates transverse to

F

\mathfrak {F}

modelled on

R

P
1

{\mathbf {R}}{P^1}

(so that the coordinate changes are real linear fractional transformations). Given a closed manifold

M
M

, there is but a finite set of values in

H
3

(
M
;

R

)

{H^3}(M;{\mathbf {R}})

which the Godbillon-Vey invariant of such foliations may assume. A bound on the possible values, in terms of the fundamental group, is computed. For

M
M

an oriented circle bundle over a surface, this finite set is explicitly computed.
The role of countable dimensionality in the theory of cell-like relations
Authors: Ancel Fredric D.
Published: June 2010

Consider only metrizable spaces. The notion of a slice-trivial relation is introduced, and Theorem 3.2 is proved. This theorem sets forth sufficient conditions for a continuous relation with compact

U

V
∞

U{V^\infty }

point images to be slice-trivial. Theorem 4.5 posits a number of necessary and sufficient conditions for a map to be a hereditary shape equivalence. Several applications of these two theorems are made, including the following. Theorem 5.1. A cell-like map

f
:
X
→
Y

f:X \to Y

is a hereditary shape equivalence if there is a sequence

{

K
n

}

\{ {K_n}\}

of closed subsets of

Y
Y

such that (1)

Y
−

⋃

n
=
1

∞

K
n

Y - \bigcup \nolimits _{n = 1}^\infty {{K_n}}

is countable dimensional, and (2)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f vertical-bar f Superscript negative 1 Baseline left-parenthesis upper K Subscript n Baseline right-parenthesis colon f Superscript negative 1 Baseline left-parenthesis upper K Subscript n Baseline right-parenthesis right-arrow upper K Subscript n Baseline">

f

|

f

−
1

(

K
n

)
:

f

−
1

(

K
n

)
→

K
n

f|{f^{ - 1}}({K_n}):{f^{ - 1}}({K_n}) \to {K_n}

is a hereditary shape equivalence for each

n
≥
1

n \geq 1

. Theorem 5.9. If

f
:
X
→
Y

f:X \to Y

is a proper onto map whose point inverses are

U

V
∞

U{V^\infty }

sets, then

Y
Y

is an absolute neighborhood extensor for the class of countable dimensional spaces. Furthermore, if

Y
Y

is countable dimensional, then

Y
Y

is an absolute neighborhood retract. Theorem 5.9 is of particular interest when specialized to the identity map of a locally contractible space.
Minimal leaves in foliations
Authors: Cass Daniel M.
Published: June 2010

The paper defines a property of open Riemannian manifolds, called quasi-homogeneity. This property is quasi-isometry invariant and is shown to hold for any manifold which appears as a minimal leaf in a foliation. Examples are given of surfaces which are not quasi-homogeneous. One such is the well-known noncompact leaf of Reeb’s foliation of

S
3

{S^3}

. These surfaces have bounded geometry.
The duration of transients
Authors: Pelikan S.
Published: June 2010

A transformation

T
T

defined on

X
⊂

R

n

X \subset {{\mathbf {R}}^n}

for which

T
(
X
)
⊃
X

T(X) \supset X

is considered. A transient in

X
X

is a trajectory

x
,
T
x
,
…
,

T
m

x
⊂
X

x,Tx, \ldots ,{T^m}x \subset X

so that

T

m
+
1

x
∉
X

{T^{m + 1}}x \notin X

. In this case,

m
m

is the duration of the transient. A method for estimating the average duration of transients is given, and an example of a transformation with exceedingly long transients is described.
On the restriction of the Fourier transform to curves: endpoint results and the degenerate case
Authors: Christ Michael
Published: June 2010

For smooth curves

Γ
\Gamma

in

R

n

{{\mathbf {R}}^n}

with certain curvature properties it is shown that the composition of the Fourier transform in

R

n

{{\mathbf {R}}^n}

followed by restriction to

Γ
\Gamma

defines a bounded operator from

L
p

(

R

n

)

{L^p}({{\mathbf {R}}^n})

to

L
q

(
Γ
)

{L^q}(\Gamma )

for certain

p
,
q

p,q

. The curvature hypotheses are the weakest under which this could hold, and

p
p

is optimal for a range of

q
q

. In the proofs the problem is reduced to the estimation of certain multilinear operators generalizing fractional integrals, and they are treated by means of rearrangement inequalities and interpolation between simple endpoint estimates.
General defect relations of holomorphic curves
Authors: Niino Kiyoshi
Published: June 2010

Let

x
:

C

→

P
n

C

x:{\mathbf {C}} \to {P_n}{\mathbf {C}}

be a holomorphic curve of finite lower order

μ
\mu

, and let

A
=
{
α
}

A = \{ \alpha \}

be an arbitrary finite family of holomorphic curves

α
:

C

→

(

P
n

C

)
∗

\alpha :{\mathbf {C}} \to {({P_n}{\mathbf {C}})^\ast }

satisfying

T
(
r
,
α
)
=
o
(
T
(
r
,
x
)
)

(
r
→
∞
)

T(r,\alpha ) = o(T(r,x))\;(r \to \infty )

. Suppose

x
x

is nondegenerate with respect to

A
A

, and

A
A

is in general position. We show the following general defect relations: (1)

x
x

has at most

n
n

deficient curves in

A
A

if

μ
=
0

\mu = 0

. (2)

∑

α
∈
A

δ
(
α
)
≤
n

if

0
>
μ
≤
1

/

2

\sum \nolimits _{\alpha \in A} {\delta (\alpha ) \leq n\;{\text {if}}\;0 > \mu \leq 1/2}

. (3)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript alpha element-of upper A Endscripts delta left-parenthesis alpha right-parenthesis less-than-or-equal-to left-bracket 2 n mu right-bracket plus 1 if 1 slash 2 greater-than mu greater-than plus normal infinity">

∑

α
∈
A

δ
(
α
)
≤
[
2
n
μ
]
+
1

if

1

/

2
>
μ
>
+
∞

\sum \nolimits _{\alpha \in A} {\delta (\alpha ) \leq [2n\mu ] + 1\;{\text {if}}\;1/2 > \mu > + \infty }

.
Uniform operators
Authors: Wang Hsiao Lan , Stampfli Joseph G.
Published: June 2010

A general representation theorem for uniform operators is obtained which enables one to verify a conjecture of Cowen and Douglas in the presence of a mild additional restriction.
Chaos, periodicity, and snakelike continua
Authors: Barge Marcy , Martin Joe
Published: June 2010

The results of this paper relate the dynamics of a continuous map

f
f

of the interval and the topology of the inverse limit space with bonding map

f
f

. These inverse limit spaces have been studied by many authors, and are examples of what Bing has called "snakelike continua". Roughly speaking, we show that when the dynamics of

f
f

are complicated, the inverse limit space contains indecomposable subcontinua. We also establish a partial converse.
Invariant regions for systems of conservation laws
Authors: Hoff David
Published: June 2010

We describe necessary and sufficient conditions for a region in

R

n

{{\mathbf {R}}^n}

to be invariant for (Glimm) solutions of the system of

n
n

conservation laws

u
t

+
f

(
u

)
x

=
0

{u_t} + f{(u)_x} = 0

. We also make some observations about the invariance of such regions for certain finite difference approximations of solutions of systems of conservation laws.
Propagation of 𝐶^{∞} regularity for fully nonlinear second order strictly hyperbolic equations in two variables
Authors: Godin Paul
Published: June 2010

It is shown that if

u
u

is a

C
3

{C^3}

solution of a fully nonlinear second order strictly hyperbolic equation in two variables, then

u
u

is

C
∞

{C^\infty }

at a point

m
m

as soon as it is

C
∞

{C^\infty }

at some point of each of the two bicharacteristic curves through

m
m

. For semilinear equations, such a result was obtained before by Rauch and Reed if

u
∈

C
1

u \in {C^1}
Separation in countably paracompact spaces
Authors: Watson W. Stephen
Published: June 2010

We study the question "Are discrete families of points separated in countably paracompact spaces?" in the class of first countable spaces and the class of separable spaces.
Multiparameter maximal functions along dilation-invariant hypersurfaces
Authors: Carlsson Hasse , Sjögren Peter , Strömberg Jan-Olov
Published: June 2010

Consider the hypersurface

x

n
+
1

=

Π
1
n

x
i

α
i

{x_{n + 1}} = \Pi _1^nx_i^{{\alpha _i}}

in

R

n
+
1

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

. The associated maximal function operator is defined as the supremum of means taken over those parts of the surface lying above the rectangles

{
0
⩽

x
i

⩽

h
i

,

i
=
1
,
…
,
n
}

\{ 0 \leqslant {x_i} \leqslant {h_i},\;i = 1, \ldots ,n\}

. We prove that this operator is bounded on

L
p

{L^p}

for

p
>
1

p > 1

. An analogous result is proved for a quadratic surface in

R

3

{{\mathbf {R}}^3}

.
Existence in the large for Riemann problems for systems of conservation laws
Authors: Sever Michael
Published: June 2010

An existence theorem in the large is obtained for the Riemann problem for nonlinear systems of conservation laws. Our principal assumptions are strict hyperbolicity, genuine nonlinearity in the strong sense, and the existence of a convex entropy function. The entropy inequality is used to obtain an a priori estimate of the strengths of the shocks and refraction waves forming a solution; existence of such a solution then follows by an application of finite-dimensional degree theory. The case of a single degenerate field is also included, with an additional assumption on the existence of Riemann invariants.
Compensated compactness and general systems of conservation laws
Authors: DiPerna Ronald J.
Published: June 2010

We outline a general program and present some new results dealing with oscillations in weakly convergent solution sequences to systems of conservation laws. The analysis employs the Young measure and the Tartar-Murat theory of compensated compactness and deals with systems of hyperbolic and elliptic type.
Linearizing certain reductive group actions
Authors: Bass H. , Haboush W.
Published: June 2010

Is every algebraic action of a reductive algebraic group

G
G

on affine space

C

n

{{\mathbf {C}}^n}

equivalent to a linear action? The "normal linearization theorem" proved below implies that, if each closed orbit of

G
G

is a fixed point, then

C

n

{{\mathbf {C}}^n}

is

G
G

-equivariantly isomorphic to

(

C

n

)
G

×

C

m

{({{\mathbf {C}}^n})^G} \times {{\mathbf {C}}^m}

for some linear action of

G
G

on

C

m

{{\mathbf {C}}^m}

.
Graphic flows and multiple disjointness
Authors: Auslander Joseph , Markley Nelson
Published: June 2010

A graphic flow is a totally minimal flow for which the only minimal subsets of the product flow are the graphs of the powers of the generating homeomorphism. The POD flows of Furstenberg, Keynes, and Shapiro [5] are examples of graphic flows. Graphic flows are in some ways analogous to ergodic systems with minimal self-joinings [11]. Various disjointness results concerning graphic flows and their powers are obtained, and their regular factors are determined.
Nonmonotoneity of Picard principle
Authors: Nakai Mitsuru , Tada Toshimasa
Published: June 2010

Two nonnegative

C
∞

{C^\infty }

functions

P
(
z
)

P(z)

and

Q
(
z
)

Q(z)

on the punctured unit disk

0
>

|

z

|

⩽
1

0 > |z| \leqslant 1

are constructed such that

Q
(
z
)
⩽
P
(
z
)

Q(z) \leqslant P(z)

and there exists only one Martin minimal boundary point for the equation

Δ
u
=
P
u

\Delta u = Pu

over

z
=
0

z = 0

and, nevertheless, there exist exactly two Martin minimal boundary points for the equation

Δ
u
=
Q
u

\Delta u = Qu

over

z
=
0

z = 0

.
Derivation, 𝐿^{Ψ}-bounded martingales and covering conditions
Authors: Talagrand Michel
Published: June 2010

Let

(
Ω
,

Σ
,

P
)

(\Omega ,\,\Sigma ,\,P)

be a complete probability space. Let

(

Σ
t

)

t
∈
J

{({\Sigma _t})_{t \in J}}

be a directed family of sub-

σ
\sigma

-algebras of

Σ
\Sigma

. Let

(
Φ
,

Ψ
)

(\Phi ,\,\Psi )

be a pair of conjugate Young functions. We investigate the covering conditions that are equivalent to the essential convergence of

L
Ψ

{L^\Psi }

-bounded martingales. We do not assume that either

Φ
\Phi

or

Ψ
\Psi

satisfy the

Δ
2

{\Delta _2}

condition. We show that when

Φ
\Phi

satisfies condition Exp, that is when there exists an

a
>
0

a > 0

such that

Φ
(
u
)
≤
exp

a
u

\Phi (u) \leq \operatorname {exp} \,au

for each

u
≥
0

u \ge 0

, the essential convergence of

L
Ψ

{L^\Psi }

-bounded martingales is equivalent to the classical covering condition

V
Φ

{V_\Phi }

. This covers in particular the classical case

Ψ
(
t
)
=
t

(
log

t

)
+

\Psi (t) = t{(\operatorname {log} \,t)^ + }

. The growth condition Exp on

Φ
\Phi

cannot be relaxed. When

J
J

contains a countable cofinite set, we show that the essential convergence of

L
Ψ

{L^\Psi }

-bounded martingales is equivalent to a covering condition

D
Φ

{D_\Phi }

(that is weaker than

V
Φ

{V_\Phi }

). When

Φ
\Phi

fails condition Exp, condition

D
Φ

{D_\Phi }

is optimal. Roughly speaking, in the case of

L
1

{L^1 }

-bounded martingales, condition

D
Φ

{D_\Phi }

means that, locally, the Vitali condition with finite overlap holds. We also investigate the case where

J
J

does not contain a countable cofinal set and

Φ
\Phi

fails condition Exp. In this case, it seems impossible to characterize the essential convergence of

L
Ψ

{L^\Psi }

-bounded martingales by a covering condition. Using the Continuum Hypothesis, we also produce an example where all equi-integrable

L
1

{L^1 }

-bounded martingales, but not all

L
1

{L^1 }

-bounded martingales, converge essentially. Similar results are also established in the derivation setting.
Topological properties of Souslin subsets
Authors: Hansell R. W.
Published: June 2010

Let

X
X

be a subparacompact regular space such that the projection map

p
:
X
×

P

→
X

p:X \times {\mathbf {P}} \to X

, where

P

{\mathbf {P}}

is the space of irrational numbers, preserves collections of sets having a

σ
\sigma

-locally finite refinement. It is shown that

p
p

then preserves generalized

F
σ

{F_\sigma }

-sets. It follows that, if

X
X

has any tpological property which is hereditary with respect to generalized

F
σ

{F_\sigma }

-sets, then every Souslin subset of

X
X

will also have this property in the relative topology. Such topological properties include nearly all covering properties (paracompactness, metacompactness, etc.), as well as normality, collectionwise normality, and the Lindelöf property. We show that the above mapping property will hold whenever

X
X

is a

P
(
ω
)

P(\omega )

-space, thus, in particular, when

X
X

is any Souslin (hence any Baire) subset of a compact space crossed with a metrizable space. Additional topological properties of Souslin subsets, such as topological completeness, realcompactness,

z
z

-embeddedness, and the properties of being a

P
P

-space or

Σ
\Sigma

-space, are also considered.
𝐻^{𝑝}-classes on rank one symmetric spaces of noncompact type. I. Nontangential and probabilistic maximal functions
Authors: Cifuentes Patricio
Published: June 2010

Two kinds of

H
p

{H^p}

-classes of harmonic functions are defined on a general rank one symmetric space of noncompact type. The first one is introduced by using a nontangential maximal function. The second is related to the diffusion generated by the Laplace-Beltrami operator. The equivalence of the two classes is proven for

0
>
p
>
∞

0 > p > \infty

.
The law of the iterated logarithm in uniformly convex Banach spaces
Authors: Ledoux Michel
Published: June 2010

We give necessary and sufficient conditions for a random variable

X
X

with values in a uniformly convex Banach space

B
B

to satisfy the law of the iterated logarithm. Precisely, we show that a

B
B

-valued random variable

X
X

satisfies the (compact) law of the iterated logarithm if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-brace StartAbsoluteValue EndAbsoluteValue upper X StartAbsoluteValue EndAbsoluteValue squared slash upper L 2 StartAbsoluteValue EndAbsoluteValue upper X StartAbsoluteValue EndAbsoluteValue right-brace greater-than normal infinity">

E
{

|

|

X

|

|

2

/

L
2

|

|

X

|

|

}
>
∞

E\{ ||X|{|^2}/{L_2}||X||\} > \infty

, the family
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace StartAbsoluteValue x Superscript asterisk Baseline left-parenthesis upper X right-parenthesis EndAbsoluteValue squared semicolon x Superscript asterisk Baseline element-of upper B Superscript asterisk Baseline comma StartAbsoluteValue EndAbsoluteValue x Superscript asterisk Baseline StartAbsoluteValue EndAbsoluteValue equals 1 right-brace">

{

|

x

∗

(
X
)

|

2

;

x

∗

∈

B

∗

,

|

|

x

∗

|

|

=
1
}

\{ |{x^{\ast }}(X){|^2};\,{x^{\ast }} \in {B^{\ast }},\,||{x^{\ast }}|| = 1\}

is uniformly integrable and

S
n

/

a
n

→
0

{S_n}/{a_n} \to 0

in probability.
Ω-stable limit set explosions
Authors: Patterson S. E.
Published: June 2010

Certain diffeomorphisms of two-dimensional manifolds are considered. These diffeomorphisms have a finite hyperbolic limit set which contains a limit set cycle. The only nontransverse cycle connection in these cycles is a complete coincidence of one component of the unstable manifold of one periodic point with one component of the stable manifold of some other periodic point. A one-parameter family of diffeomorphisms containing the original diffeomorphism is described. It is shown that for parameter values arbitrarily near the parameter value corresponding to the original map these diffeomorphisms have a much enlarged limit set and are

Ω
\Omega

-stable.
Blow up near higher modes of nonlinear wave equations
Authors: Sternberg Natalia
Published: June 2010

This paper is concerned with the instability properties of higher modes of the nonlinear wave equation

u

t
t

−
Δ
u
−
f
(
u
)
=
0

{u_{tt}} - \Delta u - f(u) = 0

defined on a smoothly bounded domain with Dirichlet boundary conditions. It is shown that they are unstable in the sense that in any neighborhood of a higher mode there exists a solution of the given equation which blows up in finite time.
A critical set with nonnull image has large Hausdorff dimension
Authors: Norton Alec
Published: June 2010

The question of how complicated a critical set must be to have a nonnull image is answered by relating its Hausdorff dimension to the (Hölder) differentiability of the map. This leads to a new extension of the Morse-Sard Theorem. The main tool is an extended version of Morse’s Lemma.
On starshaped rearrangement and applications
Authors: Kawohl Bernhard
Published: June 2010

A radial symmetrization technique is investigated and new properties are proven. The method transforms functions

u
u

into new functions

u
∗

{u^\ast }

with starshaped level sets and is therefore called starshaped rearrangement. This rearrangement is in general not equimeasurable, a circumstance with some surprising consequences. The method is then applied to certain variational and free boundary problems and yields new results on the geometrical properties of solutions to these problems. In particular, the Lipschitz continuity of free boundaries can now be easily obtained in a new fashion.
The Bergman projection of 𝐿^{∞} in tubes over cones of real, symmetric, positive-definite matrices
Authors: Békollé David
Published: June 2010

We determine a defining kernel for the Bergman projection of

L
∞

{L^\infty }

in tubes over cones of real, symmetric, positive-definite matrices.
Word maps, isotopy and entropy
Authors: Fried David
Published: June 2010

We find diffeomorphisms of low entropy in each isotopy class on

S
3

×

S
3

{S^3} \times {S^3}

. These arise as word maps, a nonabelian analogue of toral automorphisms. Hyperbolic examples of equal entropy are also found. The group

π
0

Diff

(

S
3

×

S
3

)

{\pi _0}\;\operatorname {Diff}\,({S^3} \times {S^3})

is computed.
The Pontryagin maximum principle from dynamic programming and viscosity solutions to first-order partial differential equations
Authors: Barron Emmanuel Nicholas , Jensen Robert
Published: June 2010

We prove the Pontryagin Maximum Principle for the Lagrange problem of optimal control using the fact that the value function of the problem is the viscosity solution of the associated Hamilton-Jacobi-Bellman equation. The proof here makes rigorous the formal proof of Pontryagin’s principle known for at least three decades.
Persistence of form and the value group of reducible cubics
Authors: Elliott P. D. T. A.
Published: June 2010

It is proved that the values of

x
(

x
2

+
c
)

x({x^2} + c)

,

c
≠
0

c \ne 0

, at positive integers, multiplicatively generate the positive rationals. Analogs in rational function fields are obtained.
Wrappings of permutations
Authors: Stahl Saul
Published: June 2010

A theory of wrappings of permutations is constructed which is analogous to the well-known concept of branched coverings of Riemann surfaces. It is shown that this theory is strong enough to contain combinatorial definitions of such well-known groups as Fuchsian groups of the first kind and triangle groups.
The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data
Authors: Venakides Stephanos
Published: June 2010

We study the initial value problem for the Korteweg-de Vries equation
\[

(

i

)

u
t

−
6
u

u
x

+

ε
2

u

x
x
x

=
0

({\text {i}})\quad {u_t} - 6u{u_x} + {\varepsilon ^2}{u_{xxx}} = 0

\]
in the limit of small dispersion, i.e.,

ε
→
0

\varepsilon \to 0

. When the unperturbed equation
\[

(

ii

)

u
t

−
6
u

u
x

=
0

({\text {ii}})\quad {u_t} - 6u{u_x} = 0

\]
develops a shock, rapid oscillations arise in the solution of the perturbed equation (i) In our study: a. We compute the weak limit of the solution of (i) for periodic initial data as

ε
→
0

\varepsilon \to 0

. b. We show that in the neighborhood of a point

(
x
,

t
)

(x,\,t)

the solution

u
(
x
,

t
,

ε
)

u(x,\,t,\,\varepsilon )

can be approximated either by a constant or by a periodic or by a quasiperiodic solution of equation (i). In the latter case the associated wavenumbers and frequencies are of order

O
(
1

/

ε
)

O(1/\varepsilon )

. c. We compute the number of phases and the wave parameters associated with each phase of the approximating solution as functions of

x
x

and

t
t

. d. We explain the mechanism of the generation of oscillatory phases. Our computations in a and c are subject to the solution of the Lax-Levermore evolution equations (7.7). Our results in b-d rest on a plausible averaging assumption.
Global solvability on compact nilmanifolds of three or more steps
Authors: Cygan Jacek M. , Richardson Leonard F.
Published: June 2010

We apply the methods of representation theory of nilpotent Lie groups to study the convergence of Fourier series of smooth global solutions to first order invariant partial differential equations

D
f
=
g

Df = g

in

C
∞

{C^\infty }

of a compact nilmanifold of three or more steps. We investigate which algebraically well-defined conditions on

D
D

in the complexified Lie algebra imply that smooth infinite-dimensional irreducible solutions, when they exist, satisfy estimates strong enough to guarantee uniform convergence of the irreducible (or primary) Fourier series to a smooth global solution. This extends and improves the results of an earlier two step paper.
Stationary configurations of point vortices
Authors: O’Neil Kevin Anthony
Published: June 2010

The motion of point vortices in a plane of fluid is an old problem of fluid mechanics, which was given a Hamiltonian formulation by Kirchhoff. Stationary configurations are those which remain self-similar throughout the motion. Results of two types are presented. Configurations which are in equilibrium or which translate uniformly are counted using methods of algebraic geometry, which establish necessary and sufficient conditions for existence. Relative equilibria (rigidly rotating configurations) which lie on a line are studied using a topological construction applicable to other power-law systems. Upper and lower bounds for such configurations are found for vortices with mixed circulations. Arrangements of three vortices which collide in finite time are well known. One-dimensional families of such configurations are shown to exist for more than three vortices. Stationary configurations of four vortices are examined in detail.
Bounded weak solutions of an elliptic-parabolic Neumann problem
Authors: Hulshof J.
Published: June 2010

In this paper we establish existence and uniqueness for bounded weak solutions of an elliptic-parabolic Neumann problem. We also describe the asymptotic behavior as

t
→
∞

t \to \infty

.
A GCH example of an ordinal graph with no infinite path
Authors: Larson Jean A.
Published: June 2010

It is hard to find nontrivial positive partition relations which hold for many ordinals in ordinary set theory, or even ordinary set theory with the additional assumption of the Generalized Continuum Hypothesis. Erdös, Hajnal and Milner have proved that limit ordinals

α
>

ω
1

ω
+
2

\alpha > \omega _1^{\omega + 2}

satisfy a positive partition relation that can be expressed in graph theoretic terms. In symbols one writes

α
→

(
α
,

infinite
⁡
path

)
2

\alpha \to {(\alpha ,\,\operatorname {infinite} \operatorname {path} )^2}

to mean that every graph on an ordinal

α
\alpha

either has a subset order isomorphic to

α
\alpha

in which no two points are joined by an edge or has an infinite path. This positive result generalizes to ordinals of cardinality

ℵ
m

{\aleph _m}

for

m
m

a natural number. However, the argument, based on a set mapping theorem, works only on the initial segment of the limit ordinals of cardinality

ℵ
m

{\aleph _m}

for which the set mapping theorem is true. In this paper, the Generalized Continuum Hypothesis is used to construct counterexamples for a cofinal set of ordinals of cardinality

ℵ
m

{\aleph _m}

, where

m
m

is a natural number at least two.
The asymptotic behavior of the solutions of degenerate parabolic equations
Authors: Bandle Catherine , Pozio M. A. , Tesei Alberto
Published: June 2010

Existence of stationary states is established by means of the method of upper and lower solutions. The structure of the solution set is discussed and a uniqueness property for certain classes is proved by a generalized maximum principle. It is then shown that all solutions of the parabolic equation converge to a stationary state.
Pure subgroups of torsion-free groups
Authors: Hill Paul , Megibben Charles
Published: June 2010

In this paper, we show that certain new notions of purity stronger than the classical concept are relevant to the study of torsion-free abelian groups. In particular, implications of

∗

{\ast }

-purity, a concept introduced in one of our recent papers, are investigated. We settle an open question (posed by Nongxa) by proving that the union of an ascending countable sequence of

∗

{\ast }

-pure subgroups is completely decomposable provided the subgroups are. This result is false for ordinary purity. The principal result of the paper, however, deals with

Σ
\Sigma

-purity, a concept stronger than

∗

{\ast }

-purity but weaker than the usual notion of strong purity. Our main theorem, which has a number of corollaries including the recent result of Nongxa that strongly pure subgroups of separable groups are again separable, states that a

Σ
\Sigma

-pure subgroup of a

k
k

-group is itself a

k
k

-group. Among other results is the negative resolution of the conjecture (valid in the countable case) that a strongly pure subgroup of a completely decomposable group is again completely decomposable.
Holomorphic kernels and commuting operators
Authors: Athavale Ameer
Published: June 2010

Necessary and sufficient conditions in terms of operator polynomials are obtained for an

m
m

-tuple

T
=
(

T
1

,
…
,

T
m

)

T = ({T_1}, \ldots ,{T_m})

of commuting bounded linear operators on a separable Hilbert space

H

\mathcal {H}

to extend to an

m
˙

\dot m

-tuple

S
=
(

S
1

,
…
,

S
m

)

S = ({S_1}, \ldots ,{S_m})

of operators on some Hilbert space

K

\mathcal {K}

, where each

S
i

{S_i}

is realized as a

∗

{\ast }

-representation of the adjoint of a multiplication operator on the tensor product of a special type of functional Hilbert spaces. Also, necessary and sufficient conditions in terms of operator polynomials are obtained for

T
T

to have a commuting normal extension.
Manifolds on which only tori can act
Authors: Lee Kyung Bai , Raymond Frank
Published: June 2010

A list of various types of connected, closed oriented manifolds are given. Each of the manifolds support some of the well-known compact transformation group properties enjoyed by aspherical manifolds. We list and describe these classes and their transformation group properties in increasing generality. We show by various examples that these implications can never be reversed. This establishes a hierarchy in terms of spaces in one direction and the properties they enjoy in the opposite direction.
Nonlinear stability of vortex patches
Authors: Tang Yun
Published: June 2010

To establish the nonlinear (Liapunov) stability of both circular and elliptical vortex patches in the plane for the nonlinear dynamical system generated by the two-dimensional Euler equations of incompressible, inviscid hydrodynamics. This is accomplished by using a relative variational principle in terms of energy function. A counterexample shows that our result in the case of an elliptical vortex patch is the best one that can be attained by applying the energy estimate.
Infinitesimally rigid polyhedra. II. Modified spherical frameworks
Authors: Whiteley Walter
Published: June 2010

In the first paper, Alexandrov’s Theorem was studied, and extended, to show that convex polyhedra form statically rigid frameworks in space, when built with plane-rigid faces. This second paper studies two modifications of these polyhedral frameworks: (i) block polyhedral frameworks, with some discs as open holes, other discs as space-rigid blocks, and the remaining faces plane-rigid; and (ii) extended polyhedral frameworks, with individually added bars (shafts) and selected edges removed. Inductive methods are developed to show the static rigidity of particular patterns of holes and blocks and of extensions, in general realizations of the polyhedron. The methods are based on proof techniques for Steinitz’s Theorem, and a related coordinatization of the proper realizations of a

3
3

-connected spherical polyhedron. Sample results show that: (a) a single

k
k

-gonal block and a

k
k

-gonal hole yield static rigidity if and only if the block and hole are

k
k

-connected in a vertex sense; and (b) a

4
4

-connected triangulated sphere, with one added bar, is a statically rigid circuit (removing any one bar leaves a minimal statically rigid framework). The results are also interpreted as a description of which dihedral angles in a triangulated sphere will flex when one bar is removed.
Elliptic and parabolic BMO and Harnack’s inequality
Authors: Aimar Hugo
Published: June 2010

We give a generalization of the John-Nirenberg lemma which can be applied to prove

A
2

{A_2}

type conditions for small powers of positive solutions of elliptic and parabolic, degenerate and nondegenerate operators.
Spinor bundles on quadrics
Authors: Ottaviani Giorgio
Published: June 2010

We define some stable vector bundles on the complex quadric hypersurfaceQn{Q_n}of dimensionnnas the natural generalization of the universal bundle and the dual of the quotient bundle onQ4≃Gr⁡(1,3){Q_4} \simeq \operatorname {Gr} (1,\,3). We call them spinor bundles. Whenn=2k−1n = 2k - 1there is one spinor bundle of rank2k−1{2^{k - 1}}. Whenn=2kn = 2kthere are two spinor bundles of rank2k−1{2^{k - 1}}. Their behavior is slightly different according asn≡0(mod4)n \equiv 0\;(\bmod 4)orn≡2(mod4)n \equiv 2\;(\bmod 4). As an application, we describe some moduli spaces of rank33vector bundles onQ5{Q_5}andQ6{Q_6}.
Pseudodifferential operators with coefficients in Sobolev spaces
Authors: Marschall Jürgen
Published: June 2010

Pseudo-differential operators with coefficients in Sobolev spaces

H

r
,
q

,
1
⩽
q
⩽
∞

{H^{r,q}},1 \leqslant q \leqslant \infty

, and their adjoints are studied on Hardy-Sobolev spaces

H

s
,
p

,

0
>
p
⩽
∞

{H^{s,p}},\;0 > p \leqslant \infty

. A symbolic calculus for these operators is developed, and the microlocal properties are studied. Finally, the invariance under coordinate transformations is proved.
Embedding graphs into colored graphs
Authors: Hajnal A. , Komjáth P.
Published: June 2010

If

X
X

is a graph,

κ
\kappa

a cardinal, then there is a graph

Y
Y

such that if the vertex set of

Y
Y

is

κ
\kappa

-colored, then there exists a monocolored induced copy of

X
X

; moreover, if

X
X

does not contain a complete graph on

α
\alpha

vertices, neither does

Y
Y

. This may not be true, if we exclude noncomplete graphs as subgraphs. It is consistent that there exists a graph

X
X

such that for every graph

Y
Y

there is a two-coloring of the edges of

Y
Y

such that there is no monocolored induced copy of

X
X

. Similarly, a triangle-free

X
X

may exist such that every

Y
Y

must contain an infinite complete graph, assuming that coloring

Y
Y

’s edges with countably many colors a monocolored copy of

X
X

always exists.
The dynamics of rotating waves in scalar reaction diffusion equations
Authors: Angenent S. B. , Fiedler B.
Published: June 2010

The maximal compact attractor for the RDE

u
t

=

u

x
x

+
f
(
u
,

u
x

)

{u_t} = {u_{xx}} + f(u,\,{u_x})

with periodic boundary conditions is studied. It is shown that any

ω
\omega

-limit set contains a rotating wave, i.e., a solution of the form

U
(
x
−
c
t
)

U(x - ct)

. A number of heteroclinic orbits from one rotating wave to another are constructed. Our main tool is the Nickel-Matano-Henry zero number. The heteroclinic orbits are obtained via a shooting argument, which relies on a generalized Borsuk-Ulam theorem.
Strong homology is not additive
Authors: Mardešić S. , Prasolov A. V.
Published: June 2010

Using the continuum hypothesis (CH) we show that strong homology groups

H
p

¯

(
X
)

\overline {{H_p}} (X)

do not satisfy Milnor’s additivity axiom. Moreover, CH implies that strong homology does not have compact supports and that

H
p

¯

(
X
)

\overline {{H_p}} (X)

need not vanish for

p
>
0

p > 0

.
Deformations of finite-dimensional algebras and their idempotents
Authors: Schaps M.
Published: June 2010

Let

B
B

be a finite dimensional algebra over an algebraically closed field

K
K

. If we represent primitive idempotents by points and basis vectors in

e
i

B

e
j

{e_i}B{e_j}

by "arrows" from

e
j

{e_j}

to

e
i

{e_i}

, then any specialization of the algebra acts on this directed graph by coalescing points. This implies that the number of irreducible components in the scheme parametrizing

n
n

-dimensional algebras is no less than the number of loopless directed graphs with a total of

n
n

vertices and arrows. We also show that the condition of having a distributive ideal lattice is open.
Groups acting on affine algebras
Authors: Farkas Daniel R.
Published: June 2010

General actions of groups on commutative affine domains are studied. We prove a finiteness theorem for orbits of ideals and an ergodic theorem inspired by results from the theories of group algebras and universal enveloping algebras.
Definable sets in ordered structures. III
Authors: Pillay Anand , Steinhorn Charles
Published: June 2010

We show that any

o
o

-minimal structure has a strongly

o
o

-minimal theory.
Uniformly fat sets
Authors: Lewis John L.
Published: June 2010

In this paper we study closed sets

E
E

which are "locally uniformly fat" with respect to a certain nonlinear Riesz capacity. We show that

E
E

is actually "locally uniformly fat" with respect to a weaker Riesz capacity. Two applications of this result are given. The first application is concerned with proving Sobolev-type inequalities in domains whose complements are uniformly fat. The second application is concerned with the Fekete points of

E
E

.
Factoring operators satisfying 𝑝-estimates
Authors: Byrd Stan
Published: June 2010

Necessary and sufficient conditions for a positive operator to factor through a Banach lattice satisfying upper and lower estimates are presented. These conditions are then combined to give a necessary condition for a positive operator to factor through a super-reflexive Banach lattice. An example is given to show that, in spite of the name given by Beauzamy, uniformly convexifying operators need not factor through any uniformly convex lattice
Complex interpolation of normed and quasinormed spaces in several dimensions. I
Authors: Slodkowski Zbigniew
Published: June 2010

A variety of complex interpolation methods for families of normed or quasi-normed spaces, parametrized by points of domains in complex homogeneous spaces, parametrized by points of domains in complex homogeneous spaces, is developed. Results on existence, continuity, uniqueness, reiteration and duality for interpolation are proved, as well as on interpolation of operators. A minimum principle for plurisubharmonic functions is obtained and used as a tool for the duality theorem.
Pseudoconvex classes of functions. III. Characterization of dual pseudoconvex classes on complex homogeneous spaces
Authors: Slodkowski Zbigniew
Published: June 2010

Invariant classes of functions on complex homogeneous spaces, with properties similar to those of the class of plurisubharmonic functions, are studied. The main tool is a regularization method for these classes, and the main theorem characterizes dual classes of functions (where duality is defined in terms of the local maximum property). These results are crucial in proving a duality theorem for complex interpolation of normed spaces, which is given elsewhere.
Volumes of small balls on open manifolds: lower bounds and examples
Authors: Croke Christopher B. , Karcher Hermann
Published: June 2010

Question: "Under what curvature assumptions on a complete open manifold is the volume of balls of a fixed radius bounded below independent of the center point?" Two theorems establish such assumptions and two examples sharply limit their weakening. In particular we give an example of a metric onR4{{\mathbf {R}}^4}(extending to higher dimensions) of positive Ricci curvature, whose sectional curvatures decay to00, and such that the volume of balls goes uniformly to00as the center goes to infinity.