bluuebunny/crossref_metadata_2025 huggingface dataset:
Adapted probability distributions
Authors: Hoover Douglas N. , Keisler H. Jerome
Published: June 2010

We introduce a family of notions of equivalence for stochastic processes on spaces with an underlying filtration. These refine the notion of having the same distribution by taking account of the relation of the processes to their underlying filtrations. The weakest of these notions is the same as the notion of synonymity introduced by Aldous. Analysis of the strongest equivalence property leads to spaces with a strong universality property for adapted stochastic processes, which we call saturation. Spaces having this property contain ’strong’ solutions to a large class of stochastic integral equations.
Nonvanishing local cohomology classes
Authors: Moskowitz Ira
Published: June 2010

We discuss the nonvanishing of a top-dimensional canonical cohomology class of the space

B
¯

D

i
f

f
ω

M

\bar B\mathcal {D}if{f_\omega }\;M

. We treat parallelizable and odd-dimensional stably parallelizable manifolds.
Strong laws of large numbers for products of random matrices
Authors: Pincus Steve
Published: June 2010

This work, on products of random matrices, is inspired by papers of Furstenberg and Kesten (Ann. Math. Statist. 31 (1960), 457-469) and Furstenberg (Trans. Amer. Math. Soc. 108 (1963), 377-428). In particular, a formula was known for almost sure limits for normalized products of random matrices in terms of a stationary measure. However, no explicit computational techniques were known for these limits, and little was known about the stationary measures. We prove two main theorems. The first assumes that the random matrices are upper triangular and computes the almost sure limits in question. For the second, we assume the random matrices are

2
×
2

2 \times 2

and Bernoulli, i.e., random matrices whose support is two points. Then the second theorem gives an asymptotic result for the almost sure limits, with rates of convergence in some cases.
Periodic solutions of Hamilton’s equations and local minima of the dual action
Authors: Clarke Frank H.
Published: June 2010

The dual action is a functional whose extremals lead to solutions of Hamilton’s equations. Up to now, extremals of the dual action have been obtained either through its global minimization or through application of critical point theory. A new methodology is introduced in which local minima of the dual action are found to exist. Applications are then made to the existence of Hamiltonian trajectories having prescribed period.
Local operators and derivations on 𝐶*-algebras
Authors: Batty C. J. K.
Published: June 2010

The variations on a theme of locality for a pair of operators

(
H
,
K
)

(H,K)

on a

C
∗

{C^\ast }

-algebra

A

\mathfrak {A}

are expressed algebraically. If

K
K

is a

∗
\ast

-derivation generating an action of

R

\mathbb {R}

on

A

\mathfrak {A}

, and

H
H

is

∗
\ast

-linear and

K
K

-local, then, under certain restrictions,

H
H

is shown to be very closely related to

K
K

.
Compact group actions and maps into 𝐾(𝜋,1)-spaces
Authors: Gottlieb Daniel H. , Lee Kyung B. , Özaydin Murad
Published: June 2010

Let

G
G

act on an aspherical manifold

M
M

. If

G
G

is a compact Lie group acting effectively and homotopically trivially then

G
G

must be abelian. We prove a much more general form of this result, thus extending results of Donnelly and Schultz. Our method gives us a splitting result for torus actions complementing a result of Conner and Raymond. We also generalize a theorem of Schoen and Yau on homotopy equivariance.
Degrees of indiscernibles in decidable models
Authors: Kierstead H. A. , Remmel J. B.
Published: June 2010

We show that the problem of finding an infinite set of indiscernibles in an arbitrary decidable model of a first order theory is essentially equivalent to the problem of finding an infinite path through a recursive

ω
\omega

-branching tree. Similarly, we show that the problem of finding an infinite set of indiscernibles in a decidable model of an

ω
\omega

-categorical theory with decidable atoms is essentially equivalent to finding an infinite path through a recursive binary tree.
On the index of a number field
Authors: Nart Enric
Published: June 2010

Arithmetic invariants are found which determine the indexi(K)i(K)of a number fieldKK. They are used to obtain an explicit formula under certain restrictions onKK. They provide also a complete explanation of a phenomenon conjectured by Ore [8{\mathbf {8}}] and showed by Engstrom in a particular case [2{\mathbf {2}}].
Some structure theory for a class of triple systems
Authors: Hopkins Nora C.
Published: June 2010

This paper deals with a class of triple systems satisfying two generalized five linear identities and having nondegenerate bilinear forms with certain properties. If

(
M
,
{
,
,
}
)

(M,\{ ,,\} )

is such a triple system with bilinear form

ϕ
(
,
)

\phi (,)

, it is shown that if

M
M

is semisimple, then

M
M

is the direct sum of simple ideals if

ϕ
\phi

is symmetric or symplectic or if

M
M

is completely reducible as a module for its right multiplication algebra

L

\mathcal {L}

. It is also shown that if

M
M

is a completely reducible

L

\mathcal {L}

-module,

M
M

is the direct sum of a semisimple ideal and the center of

M
M

. Such triple systems can be embedded into certain nonassociative algebras and the results on the triple systems are extended to these algebras.
Filtering cohomology and lifting vector bundles
Authors: Evans E. Graham , Griffith Phillip
Published: June 2010

For a module

M
M

over a local Cohen-Macaulay ring

R
R

we develop a (finite) sequence of presentations of

M
M

which facilitates the study of invariants arising from the cohomology modules of

M
M

. As an application we use this data, in case

R
R

is regular and

M
M

represents a vector bundle on the punctured spectrum of

R
R

with a vanishing cohomology module, to obtain bounds on how far

M
M

can be lifted as a vector bundle.
Monge-Ampère measures associated to extremal plurisubharmonic functions in 𝐶ⁿ
Authors: Levenberg Norman
Published: June 2010

We consider the extremal plurisubharmonic functions

L
E
∗

L_E^\ast

and

U
E
∗

U_E^\ast

associated to a nonpluripolar compact subset

E
E

of the unit ball

B
⊂

C

n

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

and show that the corresponding Monge-Ampère measures

(
d

d
c

L
E
∗

)
n

{(d{d^c}L_E^\ast )^n}

and

(
d

d
c

U
E
∗

)
n

{(d{d^c}U_E^\ast )^n}

are mutually absolutely continuous. We then discuss the polynomial growth condition

(

L
∗

)

({L^\ast })

, a generalization of Leja’s polynomial condition in the plane, and study the relationship between the asymptotic behavior of the orthogonal polynomials associated to a measure on

E
E

and the

(

L
∗

)

({L^\ast })

condition.
A kernel approach to the local solvability of the tangential Cauchy Riemann equations
Authors: Boggess A. , Shaw M.-C.
Published: June 2010

An integral kernel approach is given for the proof of the theorem of Andreotti and Hill which states that the

Y
(
q
)

Y(q)

condition of Kohn is a sufficient condition for local solvability of the tangential Cauchy Riemann equations on a real hypersurface in

C

n

{{\mathbf {C}}^n}

. In addition, we provide an integral kernel approach to nonsolvability for a certain class of real hypersurfaces in the case when

Y
(
q
)

Y(q)

is not satisfied.
𝑃-points with countably many constellations
Authors: Rosen Ned I.
Published: June 2010

If the continuum hypothesis

(

CH

)

({\text {CH}})

is true, then for any

P
P

point ultrafilter

D
D

(on the set of natural numbers) there exist initial segments of the Rudin-Keisler ordering, restricted to (isomorphism classes of)

P
P

points which lie above

D
D

, of order type

ℵ
1

{\aleph _1}

. In particular, if

D
D

is an

RK

{\text {RK}}

-minimal ultrafilter, then we have

(

CH

)

({\text {CH}})

that there exist

P
P

-points with countably many constellations.
Solving semilinear partial differential equations with probabilistic potential theory
Authors: Glover Joseph , McKenna P. J.
Published: June 2010

Techniques of probabilistic potential theory are applied to solve

−
L
u
+
f
(
u
)
=
μ

- Lu + f(u) = \mu

, where

μ
\mu

is a signed measure,

f
f

a (possibly discontinuous) function and

L
L

a second order elliptic or parabolic operator on

R
d

{R^d}

or, more generally, the infinitesimal generator of a Markov process. Also formulated are sufficient conditions guaranteeing existence of a solution to a countably infinite system of such equations.
Homology and cell structure of nilpotent spaces
Authors: Lewis Robert H.
Published: June 2010

Let

A
A

and

X
X

denote finitely dominated nilpotent

CW

{\text {CW}}

complexes. We are interested in questions relating the homology groups of such spaces to their cell structure and homotopy type. We solve a problem posed by Brown and Kahn, that of constructing nilpotent complexes of minimal dimension. When the fundamental group is finite, the three-dimensional complex we construct may not be finite; we then construct a finite six-dimensional complex. We investigate the set of possible cofibers of maps

A
→
X

A \to X

, and find a severe restriction. When it is met and the fundamental group is finite,

X
X

can be constructed from

A
A

by attaching cells in a natural way. The restriction implies that the classical notion of homology decomposition has no application to nilpotent complexes. We show that the Euler characteristic of

X
X

must be zero. Several corollaries are derived to the theory of finitely dominated nilpotent complexes. Several of these results depend upon a purely algebraic theorem that we prove concerning the vanishing of homology of nilpotent modules over nilpotent groups.
On derivations annihilating a maximal abelian subalgebra
Authors: Price Geoffrey L.
Published: June 2010

Let

A

\mathcal {A}

be an

AF

C
∗

{\text {AF}}\;{C^\ast }

-algebra, and let

δ
\delta

be a closed

∗
\ast

-derivation which annihilates the maximal abelian subalgebra

C

\mathcal {C}

of diagonal elements of

A

\mathcal {A}

. Then we show that

δ
\delta

generates an approximately inner

C
∗

{C^\ast }

-dynamics on

A

\mathcal {A}

, and that

δ
\delta

is a commutative

∗
\ast

-derivation. Any two closed

∗
\ast

-derivations vanishing on

C

\mathcal {C}

are shown to be strongly commuting. More generally, if

δ
\delta

is a semiderivation on

A

\mathcal {A}

which vanishes on

C

\mathcal {C}

, we prove that

δ
\delta

is a generator of a semigroup of strongly positive contractions of

A

\mathcal {A}

.
Completely unstable dynamical systems
Authors: Goel Sudhir K. , Neumann Dean A.
Published: June 2010

We associate with the

C
r

(
r

≥

1
)

{C^r}\,(r\, \ge \,1)

dynamical system

ϕ
\phi

on an

m
m

-manifold

M
M

, the orbit space

M

/

ϕ

M/\phi

, defined to be the set of orbits of

ϕ
\phi

with the quotient topology. If

ϕ
\phi

is completely unstable,

M

/

ϕ

M/\phi

turns out to be a

C
r

(
m

−

1
)

{C^r}\,(m\, - \,1)

-nonseparated manifold. It is known that for a completely unstable flow

ϕ
\phi

on a contractible manifold

M
,

M

/

ϕ

M,\,M/\phi

is Hausdorff if and only if

ϕ
\phi

is parallelizable. In general, we place an order on the non-Hausdorff points of

M

/

ϕ

M/\phi

(essentially) by setting

p
¯

>

q
¯

\bar p > \bar q

if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Superscript negative 1 Baseline left-parenthesis q overbar right-parenthesis subset-of-or-equal-to upper J Superscript plus Baseline left-parenthesis pi Superscript negative 1 Baseline left-parenthesis p overbar right-parenthesis right-parenthesis">

π

−
1

(

q
¯

)
⊆

J
+

(

π

−
1

(

p
¯

)
)

{\pi ^{ - 1}}(\bar q) \subseteq {J^ + }({\pi ^{ - 1}}(\bar p))

. Our result is that

(
M
,

ϕ
)

(M,\,\phi )

is topologically equivalent to

(

M
′

,

ϕ
′

)

(M’,\,\phi ’)

if and only if

M

/

ϕ

M/\phi

is order isomorphic to

M
′

/

ϕ
′

M’/\phi ’

.
Maps between surfaces
Authors: Skora Richard
Published: June 2010

The Uniqueness Conjecture states if

ϕ
,

ψ
:

M
→
N

\phi ,\,\psi :\,M \to N

are

d
d

-fold, simple, primitive, branched coverings between closed, connected surfaces, then

ϕ
\phi

and

ψ
\psi

are equivalent. The Uniqueness Conjecture is proved in the case that

M
M

and

N
N

are nonorientable and

N
=

R

P
2

N = \mathbf {R}{P^2}

or Klein bottle. It is also proved in the case that

M
M

and

N
N

are nonorientable and

d

/

2
>
d
χ
(
N
)
−
χ
(
M
)

d/2 > d\chi (N) - \chi (M)

. As an application it is shown that two

d
d

-fold, branched coverings

ϕ
:

M
1

→
N
,

ψ
:

M
2

→
N

\phi :{M_1} \to N,\,\psi :{M_2} \to N

between closed, connected surfaces are branched cobordant.
Differential identities in prime rings with involution
Authors: Lanski Charles
Published: June 2010

Let

R
R

be a prime ring with involution. Using work of V. K. Kharchenko it is shown that any generalized identity for

R
R

involving derivations of

R
R

and the involution of

R
R

is a consequence of the generalized identities with involution which

R
R

satisfies. In obtaining this result, a generalization, to rings satisfying a GPI, of the classical theorem characterizing inner derivations of finite-dimensional simple algebras is required. Consequences of the main theorem are that in characteristic zero no outer derivation of

R
R

can act algebraically on the set of symmetric elements of

R
R

, and if the images of the set of symmetric elements under the derivations of

R
R

satisfy a polynomial relation, then

R
R

must satisfy a generalized polynomial identity.
(𝑛-1)-axial 𝑆𝑂(𝑛) and 𝑆𝑈(𝑛) actions on homotopy spheres
Authors: Ball R. D.
Published: June 2010

Let

G
(
n
)
=
O
(
n
)

G(n) = O(n)

or

U
(
n
)

U(n)

and

S
G
(
n
)
=
S
O
(
n
)

SG(n) = SO(n)

or

S
U
(
n
)

SU(n)

. For each integer

m
⩾
1

m \geqslant 1

a family

{

S

γ
,
σ

:
γ
∈
H
,
σ
∈
K
}

\{ {S_{\gamma ,\sigma }}:\gamma \in H,\sigma \in K\}

of

(
n
−
1
)

(n - 1)

-axial

S
G
(
n
)

SG(n)

homotopy spheres

S

γ
,
σ

{S_{\gamma ,\sigma }}

is constructed. Each

S

γ
,
σ

{S_{\gamma ,\sigma }}

has fixed point set of dimension

(
m
−
1
)
⩾
0

(m - 1) \geqslant 0

and orbit space of dimension

r
=

1
2

n
(
n
−
1
)
+
(
m
−
1
)

r = \tfrac {1} {2}n(n - 1) + (m - 1)

(resp.

r
=

(
n
−
1

)
2

+
m
−
1

r = {(n - 1)^2} + m - 1

) if

S
G
(
n
)
=
S
O
(
n
)

SG(n) = SO(n)

(resp.

S
U
(
n
)

SU(n)

).

H
H

is

π

r
−
1

(
S
G
(
n
)

/

G
(
n
−
1
)
)

{\pi _{r - 1}}(SG(n)/G(n - 1))

.

K
K

is trivial if

S
G
(
n
)
=
S
O
(
n
)

SG(n) = SO(n)

and is a homotopy theoretically defined subgroup of sections of an

S
2

{S^2}

bundle depending only on

m
m

and

n
n

if

S
G
(
n
)
=
S
U
(
n
)

SG(n) = SU(n)

. Assume that

m
m

and

n
n

satisfy the mild restriction

§
5

\S 5

, (1). It is shown that the above family is universal for

(
n
−
1
)

(n - 1)

-axial

S
G
(
n
)

SG(n)

homotopy spheres and provides a classification analogous to the classification of fibre bundles: for each

(
n
−
1
)

(n - 1)

-axial

S
G
(
n
)

SG(n)

homotopy sphere

Σ
\Sigma

there is a

S

γ
,
σ

{S_{\gamma ,\sigma }}

and a unique equivariant stratified map

Σ
→

S

γ
,
σ

\Sigma \to {S_{\gamma ,\sigma }}

.

Σ
\Sigma

is equivariantly diffeomorphic to the pullback of

S

γ
,
σ

{S_{\gamma ,\sigma }}

via the map

B
(
Σ
)
→
B
(

S

γ
,
σ

)

B(\Sigma ) \to B({S_{\gamma ,\sigma }})

of orbit spaces. If

S
G
(
n
)
=
S
O
(
n
)

SG(n) = SO(n)

then

γ
\gamma

is unique (and

σ
=
1

\sigma = 1

). If

S
G
(
n
)
=
S
U
(
n
)

SG(n) = SU(n)

then

γ
\gamma

is unique modulo the image of
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript r minus 1 Baseline upper S left-parenthesis upper U left-parenthesis n minus 2 right-parenthesis times upper U left-parenthesis 2 right-parenthesis right-parenthesis slash upper U left-parenthesis k minus 1 right-parenthesis times upper U left-parenthesis 1 right-parenthesis in upper H period">

π

r
−
1

S
(
U
(
n
−
2
)
×
U
(
2
)
)

/

U
(
k
−
1
)
×
U
(
1
)

in

H
.

{\pi _{r - 1}}S(U(n - 2) \times U(2))/U(k - 1) \times U(1)\quad {\text {in}}\;H.

\]
An example is given showing that the differentiable structure of the underlying smooth manifold of

S

γ
,
σ

{S_{\gamma ,\sigma }}

may be exotic.
Group-graded rings and duality
Authors: Quinn Declan
Published: June 2010

We give an alternative construction of the duality between finite group actions and group gradings on rings which was shown by Cohen and Montgomery in [1]. This duality is then used to extend known results on skew group rings to corresponding results for large classes of group-graded rings. Finally we modify the construction slightly to handle infinite groups.
Classification of semisimple algebraic monoids
Authors: Renner Lex E.
Published: June 2010

Let

X
X

be a semisimple algebraic monoid with unit group

G
G

. Associated with

E
E

is its polyhedral root system

(
X
,
Φ
,
C
)

(X,\Phi ,C)

, where

X
=
X
(
T
)

X = X(T)

is the character group of the maximal torus

T
⊆
G

T \subseteq G

,

Φ
⊆
X
(
T
)

\Phi \subseteq X(T)

is the set of roots, and

C
=
X
(

T
¯

)

C = X(\overline T )

is the character monoid of

T
¯

⊆
E

\overline T \subseteq E

(Zariski closure). The correspondence

E
→
(
X
,
Φ
,
C
)

E \to (X,\Phi ,C)

is a complete and discriminating invariant of the semisimple monoid

E
E

, extending the well-known classification of semisimple groups. In establishing this result, monoids with preassigned root data are first constructed from linear representations of

G
G

. That done, we then show that any other semisimple monoid must be isomorphic to one of those constructed. To do this we devise an extension principle based on a monoid analogue of the big cell construction of algebraic group theory. This, ultimately, yields the desired conclusions.
The extended module of indecomposables for mod 2 finite 𝐻-spaces
Authors: Kane R. M.
Published: June 2010

We construct a generalization of the usual module of indecomposables for the

mod
2

\bmod 2

cohomology of a finite

H
H

-space. Structure theorems are obtained regarding how the Steenrod algebra acts on this module.
On the restriction of the Fourier transform to a conical surface
Authors: Taberner Bartolome Barcelo
Published: June 2010

Let

Γ
\Gamma

be the surface of a circular cone in

R

3

{{\mathbf {R}}^3}

. We show that if

1
⩽
p
>
4

/

3

1 \leqslant p > 4/3

,

1

/

q
=
3
(
1
−
1

/

p
)

1/q = 3(1 - 1/p)

and

f
∈

L
p

(

R

3

)

f \in {L^p}({{\mathbf {R}}^3})

, then the Fourier transform of

f
f

belongs to

L
q

(
Γ
,
d
σ
)

{L^q}(\Gamma ,d\sigma )

for a certain natural measure

σ
\sigma

on

Γ
\Gamma

. Following P. Tomas we also establish bounds for restrictions of Fourier transforms to conic annuli at the endpoint

p
=
4

/

3

p = 4/3

, with logarithmic growth of the bound as the thickness of the annulus tends to zero.
Polyhedral resolutions of algebraic varieties
Authors: Carlson James A.
Published: June 2010

We give a method for constructing relatively small smooth simplicial resolutions of singular projective algebraic varieties. For varieties of dimension

n
n

, at most

n
n

applications of the basic process yields a resolution of combinatorial dimension at most

n
n

. The object so obtained may be used to compute the mixed Hodge stucture of the underlying variety.
Isometric homotopy in codimension two
Authors: Moore John Douglas
Published: June 2010

This article investigates the structure of the space of isometric immersions from a simply connected

n
n

-dimensional Riemannian manifold with positive sectional curvatures into

(
n
+
2
)

(n + 2)

-dimensional Euclidean space

E

n
+
2

{E^{n + 2}}

. It is proven that if

n
⩾
4

n \geqslant 4

and

M
n

{M^n}

is such a manifold which admits a

C
∞

{C^\infty }

isometric immersion as a hypersurface in

E

n
+
1

{E^{n + 1}}

, then any

C
∞

{C^\infty }

isometric immersion from

M
n

{M^n}

into

E

n
+
2

{E^{n + 2}}

is

C

2
n
−
4

{C^{2n - 4}}

homotopic through isometric immersions to an immersion whose image lies in some hyperplane.
Local and global envelopes of holomorphy of domains in 𝐶²
Authors: Bedford Eric
Published: June 2010

A criterion is given for a smoothly bounded domain

D
⊂

C

2

D \subset {{\mathbf {C}}^2}

to be locally extendible to a neighborhood of a point

z
0

∈
∂
D

{z_0} \in \partial D

. (This result may also be formulated in terms of extension of CR functions on

∂
D

\partial D

.) This is related to the envelope of holomorphy of the semitubular domain
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega left-parenthesis normal upper Phi right-parenthesis equals StartSet left-parenthesis z comma w right-parenthesis element-of bold upper C squared colon upper R e w plus r Superscript k Baseline normal upper Phi left-parenthesis theta right-parenthesis greater-than 0 EndSet comma">

Ω
(
Φ
)
=
{
(
z
,
w
)
∈

C

2

:
Re
⁡
w
+

r
k

Φ
(
θ
)
>
0
}
,

\Omega (\Phi ) = \{ (z,w) \in {{\mathbf {C}}^2}:\operatorname {Re} w + {r^k}\Phi (\theta ) > 0\} ,

\]
where

r
=

|

z

|

r = |z|

,

θ
=
arg
⁡
(
z
)

\theta = \arg (z)

. Necessary and sufficient conditions are given for the envelope of holomorphy of

Ω
(
Φ
)

\Omega (\Phi )

to be

C

2

{{\mathbf {C}}^2}

. These conditions are equivalent to the existence of a subharmonic minorant for

r
k

Φ
(
θ
)

{r^k}\Phi (\theta )

.
Rational homotopy of spaces of maps into spheres and complex projective spaces
Authors: Møller Jesper Michael , Raussen Martin
Published: June 2010

We investigate the rational homotopy classification problem for the components of some function spaces with

S
n

{S^n}

or

C

P
n

{\mathbf {C}}{P^n}

as target space.
On the existence and uniqueness of complex structure and spaces with “few” operators
Authors: Szarek Stanisław J.
Published: June 2010

We construct a

2
n

2n

-dimensional real normed space whose (Banach-Mazur) distance to the set of spaces admitting complex structure is of order

n

1

/

2

{n^{1/2}}

, and two complex

n
n

-dimensional normed spaces which are isometric as real spaces, but whose complex Banach-Mazur distance is of order

n
n

. Both orders of magnitude are the largest possible. We also construct finite-dimensional spaces with the property that all “well-bounded” operators on them are “rather small” (in the sense of some ideal norm) perturbations of multiples of identity. We also state some “metatheorem”, which can be used to produce spaces with various pathological properties.
Maximal functions associated with curves and the Calderón-Zygmund method of rotations
Authors: Sato Shuichi
Published: June 2010

Let

δ
t

(
t
>
0
)

{\delta _t}(t > 0)

be a dilation in

R

n

(
n
⩾
2
)

{{\mathbf {R}}^n}(n \geqslant 2)

defined by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta Subscript t Baseline x equals left-parenthesis t Superscript alpha 1 Baseline x 1 comma t Superscript alpha 2 Baseline x 2 comma ellipsis comma t Superscript alpha Super Subscript n Superscript Baseline x Subscript n Baseline right-parenthesis left-parenthesis x equals left-parenthesis x 1 comma x 2 comma ellipsis comma x Subscript n Baseline right-parenthesis right-parenthesis comma">

δ
t

x
=
(

t

α
1

x
1

,

t

α
2

x
2

,
…
,

t

α
n

x
n

)

(
x
=
(

x
1

,

x
2

,
…
,

x
n

)
)
,

{\delta _t}x = ({t^{{\alpha _1}}}{x_1},{t^{{\alpha _2}}}{x_2}, \ldots ,{t^{{\alpha _n}}}{x_n})\qquad (x = ({x_1},{x_2}, \ldots ,{x_n})),

\]
where

α
i

>
0
(
i
=
1
,
2
,
…
,
n
)

{\alpha _i} > 0(i = 1,2, \ldots ,n)

and

α
i

≠

α
j

{\alpha _i} \ne {\alpha _j}

if

i
≠
j

i \ne j

. For

ν
∈

R

n

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

with

|

ν

|

=
1

|\nu | = 1

, let

Γ
ν

:
(
0
,
∞
)
→

R

n

{\Gamma _\nu }:(0,\infty ) \to {{\mathbf {R}}^n}

be a curve defined by

Γ
ν

(
t
)
=

δ
t

ν

(
0
>
t
>
∞
)

{\Gamma _\nu }(t) = {\delta _t}\nu \,(0 > t > \infty )

. Using maximal functions associated with the curves

Γ
ν

{\Gamma _\nu }

, we define an operator

M
M

which is a nonisotropic analogue of the one studied in R. Fefferman [2]. It is proved that

M
M

is a bounded operator on

L
p

(

R

n

)

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

for some

p
p

with

1
>
p
>
2

1 > p > 2

. As its application we prove the

L
p

{L^p}

boundedness of operators of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Superscript asterisk Baseline left-parenthesis f right-parenthesis left-parenthesis x right-parenthesis equals sup Underscript epsilon greater-than 0 Endscripts StartAbsoluteValue upper T Subscript epsilon Baseline left-parenthesis f right-parenthesis left-parenthesis x right-parenthesis EndAbsoluteValue">

T

∗

(
f
)
(
x
)
=

sup

ε
>
0

|

T
ε

(
f
)
(
x
)

|

{T^{\ast }}(f)(x) = {\sup _{\varepsilon > 0}}|{T_\varepsilon }(f)(x)|

, where

T
ε

{T_\varepsilon }

is an integral operator associated with a variable kernel with mixed homogeneity.
Hyperreflexivity and a dual product construction
Authors: Larson David R.
Published: June 2010

We show that an example of a nonhyperreflexive CSL algebra recently constructed by Davidson and Power is a special case of a general and natural reflexive subspace construction. Completely different techniques of proof are needed because of absence of symmetry. It is proven that if

S

\mathcal {S}

and

I

\mathcal {I}

are reflexive proper linear subspaces of operators acting on a separable Hilbert space, then the hyperreflexivity constant of

(

S

⊥

⊗

I

⊥

)
⊥

{({\mathcal {S}_ \bot } \otimes {\mathcal {I}_ \bot })^ \bot }

is at least as great as the product of the constants of

S

\mathcal {S}

and

I

\mathcal {I}

.
Anosov diffeomorphisms and expanding immersions. II
Authors: Jones Lowell
Published: June 2010

This paper continues the study of hyperbolic attractors, expanding immersions, and quotient solenoids which was begun in a previous paper of the same title. The main result states that certain hyperbolic attractors are topologically conjugate to an Anosov diffeomorphism.
Existence and nonoscillation theorems for an Emden-Fowler equation with deviating argument
Authors: Trench William F.
Published: June 2010

Sufficient conditions are given for a generalized Emden-Fowler equation with deviating argument to have nonoscillatory solutions with prescribed asymptotic behavior as

t
→
∞

t \to \infty

. The integrability condition on the nonlinear term requires only conditional convergence, supplemented by a condition on the order of convergence, which is automatically satisfied in some important special cases. The exponent in the nonlinear term may be any real number. The deviating argument is not assumed to be purely advanced or retarded, and, in some cases, need not tend to infinity. Some of the results are global, in that the desired solution is shown to exist on a given interval, rather than only for sufficiently large

t
t

.
A convergent series expansion for hyperbolic systems of conservation laws
Authors: Harabetian Eduard
Published: June 2010

We consider the discontinuous piecewise analytic initial value problem for a wide class of conservation laws that includes the full three-dimensional Euler equations. The initial interaction at an arbitrary curved surface is resolved in time by a convergent series. Among other features the solution exhibits shock, contact, and expansion waves as well as sound waves propagating on characteristic surfaces. The expansion waves correspond to the one-dimensional rarefactions but have a more complicated structure. The sound waves are generated in place of zero strength shocks, and they are caused by mismatches in derivatives.
Conformally flat manifolds whose development maps are not surjective. I
Authors: Kamishima Yoshinobu
Published: June 2010

Let

M
M

be an

n
n

-dimensional conformally flat manifold. A universal covering of

M
,

M
~

M,\,\tilde M

admits a conformal development map into

S
n

{S^n}

. When a development map is not surjective, we can relate the boundary of the development image with the limit set of the holonomy group of

M
M

. In this paper, we study properties of closed conformally flat manifolds whose development maps are not surjective.
Algebraic meridians of knot groups
Authors: Tsau Chichen M.
Published: June 2010

We propose the conjecture that every automorphism of a knot group preserves the meridian up to inverse and conjugation. We establish the conjecture for all composite knots, all torus knots, most cable knots, and at most one exception for hyperbolic knots; moreover we prove that the Property P Conjecture implies our conjecture. We also investigate hyperbolic knots in more detail, and give an example of figure-eight knot group and its automorphisms.
Unicity of a holomorphic functional calculus in infinite dimensions
Authors: Galé José E.
Published: June 2010

L. Waelbroeck gives a holomorphic functional calculus for Banach algebras and analytic functions on Banach spaces. The properties of this calculus extend the well-known properties for the case of several complex variables. In this last situation, W. Zame has obtained a theorem of unicity where the famous condition of compatibility is dropped. We obtain a theorem analogous to Zame’s for Waelbroeck’s calculus restricted to a certain algebra of germs of functions. We consider Banach spaces whose topological duals have the bounded approximation property. Also, results of the same kind as above are given for bornological algebras.
Weakly definable types
Authors: Kirby L. A. S. , Pillay A.
Published: June 2010

We study some generalizations of the notion of a definable type, first in an abstract setting in terms of ultrafilters on certain Boolean algebras, and then as applied to model theory.
Algebraic relations among solutions of linear differential equations
Authors: Singer Michael F.
Published: June 2010

Using power series methods, Harris and Sibuya [3,4] recently showed that if

k
k

is an ordinary differential field of characteristic zero and

y
≠
0

y \ne 0

is an element of a differential extension of

k
k

such that

y
y

and

1

/

y

1/y

satisfy linear differential equations with coefficients in

k
k

, then

y
′

/

y

y\prime /y

is algebraic over

k
k

. Using differential galois theory, we generalize this and characterize those polynomial relations among solutions of linear differential equations that force these solutions to have algebraic logarithmic derivatives. We also show that if

f
f

is an algebraic function of genus

≥
1

\geq 1

and if

y
y

and

f
(
y
)

f(y)

or

y
y

and

e

∫
y

{e^{\int y}}

satisfy linear differential equations, then

y
y

is an algebraic function.
On algebras with convolution structures for Laguerre polynomials
Authors: Kanjin Yūichi
Published: June 2010

In this paper we treat the convolution algebra connected with Laguerre polynomials which was constructed by Askey and Gasper [1]. For this algebra, we study the maximal ideal space, Wiener’s general Tauberian theorem, spectral synthesis and Helson sets. We also study Sidon sets and idempotent measures for the algebras with dual convolution structures.
Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d’espace
Authors: Métivier Guy
Published: June 2010

The existence of shock front solutions to a system of conservation laws in several space variables has been proved by A. Majda, solving a Cauchy problem, with a suitable discontinuous Cauchy data. But, in general, the solution to such a Cauchy problem will present

N
N

singularities,

N
N

being the number of laws. In this paper we solve (locally) this Cauchy problem, with a Cauchy data which is piecewise smooth, in the case where all the singularities are expected to be shock waves. Actually the construction is written for a system of two laws, with two space variables and similarly, for such a system, the same method enables us to study the interaction of two shock waves. The key point, in the construction below, is the study of a nonlinear, free boundary Goursat problem.
The Riemann hypothesis and the Turán inequalities
Authors: Csordas George , Norfolk Timothy S. , Varga Richard S.
Published: June 2010

A solution is given to a fifty-eight year-old open problem of G Pólya, involving the Taylor coefficients of the Riemann

ξ
\xi

-function.
Playful Boolean algebras
Authors: Veličković Boban
Published: June 2010

We show that for an atomless complete Boolean algebra

B

\mathcal {B}

of density

≤

2

ℵ
0

\leq {2^{{\aleph _0}}}

, the Banach-Mazur, the split and choose, and the Ulam game on

B

\mathcal {B}

are equivalent. Moreover, one of the players has a winning strategy just in trivial cases: Empty wins iff

B

\mathcal {B}

adds a real; Nonempty wins iff

B

\mathcal {B}

has a

σ
\sigma

-closed dense set. This extends some previous results of Foreman, Jech, and Vojtáš
Brownian motion at a slow point
Authors: Barlow Martin T. , Perkins Edwin A.
Published: June 2010

If

c
>
1

c > 1

there are points

T
(
ω
)

T(\omega )

such that the piece of a Brownian path

B
,
X
(
t
)
=
B
(
T
+
t
)
−
B
(
T
)

B,X(t) = B(T + t) - B(T)

, lies within the square root boundaries

±
c

t

\pm c\sqrt t

. We study probabilistic and sample path properties of

X
X

. In particular, we show that

X
X

is an inhomogeneous Markov process satisfying a certain stochastic differential equation, and we analyze the local behaviour of its local time at zero.
Attracting orbits in Newton’s method
Authors: Hurley Mike
Published: June 2010

It is well known that the dynamical system generated by Newton’s Method applied to a real polynomial with all of its roots real has no periodic attractors other than the fixed points at the roots of the polynomial. This paper studies the effect on Newton’s Method of roots of a polynomial "going complex". More generally, we consider Newton’s Method for smooth real-valued functions of the formfμ(x)=g(x)+μ{f_\mu }(x) = g(x) + \mu,μ\mua parameter. Ifμ0{\mu _0}is a point of discontinuity of the mapμ→\mu \to(the number of roots offμ{f_\mu }), then, in the presence of certain nondegeneracy conditions, we show that there are values ofμ\munearμ0{\mu _0}for which the Newton function offμ{f_\mu }has nontrivial periodic attractors.
Absolute subretracts and weak injectives in congruence modular varieties
Authors: Davey Brian A. , Kovács L. G.
Published: June 2010

Absolute subretracts and weak injectives in congruence modular varieties of universal algebras are investigated by focusing attention on the directly indecomposibles. The proofs rely on a congruence modular version of generalized direct products (direct products with amalgamation) and on the generalized Jónsson Lemma for congruence modular varieties. The results have immediate application to varieties of groups or rings.
The Euler characteristic as an obstruction to compact Lie group actions
Authors: Hauschild Volker
Published: June 2010

Actions of compact Lie groups on spaces

X
X

with

H

∗

(
X
,

Q

)
≅

Q

[

x
1

,
…
,

x
n

]

/

I
0

{H^{\ast }}(X,{\mathbf {Q}}) \cong {\mathbf {Q}}[{x_1}, \ldots ,{x_n}]/{I_0}

,

Q
∈

I
0

Q \in {I_0}

a definite quadratic form,

deg
⁡

x
i

=
2

\deg {x_i} = 2

, are considered. It is shown that the existence of an effective action of a compact Lie group

G
G

on such an

X
X

implies

χ
(
X
)
≡
O
(

|

W
G

|

)

\chi (X) \equiv O(|WG|)

, where

χ
(
X
)

\chi (X)

is the Euler characteristic of

X
X

and

|

W
G

|

|WG|

means the order of the Weyl group of

G
G

. Moreover the diverse symmetry degrees of such spaces are estimated in terms of simple cohomological data. As an application it is shown that the symmetry degree

N
t

(
G

/

T
)

{N_t}(G/T)

is equal to

dim
⁡
G

\dim G

if

G
G

is a compact connected Lie group and

T
⊂
G

T \subset G

its maximal torus. Effective actions of compact connected Lie groups

K
K

on

G

/

T

G/T

with

dim
⁡
K
=
dim
⁡
G

\dim K = \dim G

are completely classified.
Doubly sliced knots which are not the double of a disk
Authors: Smolinsky L.
Published: June 2010

In this paper we show that double disk knots can be distinguished from general doubly sliced knots in dimensions

4
n
+
1

4n + 1

.
Singularly perturbed quadratically nonlinear Dirichlet problems
Authors: DeSanti Albert J.
Published: June 2010

The Dirichlet problem for singularly perturbed elliptic equations of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon normal upper Delta u equals upper A left-parenthesis bold x comma u right-parenthesis nabla u dot nabla u plus bold upper B left-parenthesis bold x comma u right-parenthesis dot nabla u plus upper C left-parenthesis bold x comma u right-parenthesis">

ε
Δ
u
=
A
(

x

,
u
)
∇
u
⋅
∇
u
+

B

(

x

,
u
)
⋅
∇
u
+
C
(

x

,
u
)

\varepsilon \Delta u = A({\mathbf {x}},u)\nabla u \cdot \nabla u + {\mathbf {B}}({\mathbf {x}},u) \cdot \nabla u + C({\mathbf {x}},u)

in

Ω
∈

E
n

\Omega \in {E^n}

is studied. Under explicit and easily checked conditions, solutions are shown to exist for

ε
\varepsilon

sufficiently small and to exhibit specified asymptotic behavior as

ε
→
0

\varepsilon \to 0

. The results are obtained using a method based on the theory of partial differential inequalities.
Applications of the covering lemma for sequences of measures
Authors: Mitchell W.
Published: June 2010

We present several applications of the covering lemma for the core model for sequences of measures, including characterizations of the large cardinal strength necessary to make the filter of closed, unbounded subsets of

ω
1

{\omega _1}

an ultrafilter or to change the cofinality of a regular cardinal, and a characterization of the minimal inner model containing an arbitrary elementary embedding.
Nonharmonic Fourier series and spectral theory
Authors: Benzinger Harold E.
Published: June 2010

We consider the problem of using functions

g
n

(
x
)
:=
e
x
p
(
i

λ
n

x
)

{g_n}(x): = exp(i{\lambda _n}x)

to form biorthogonal expansions in the spaces

L
p

(
−
π
,

π
)

{L^p}( - \pi ,\,\pi )

, for various values of

p
p

. The work of Paley and Wiener and of Levinson considered conditions of the form

|

λ
n

−
n

|

≤
Δ
(
p
)

\left | {{\lambda _n} - n} \right | \leq \Delta (p)

which insure that

{

g
n

}

\{ {g_n}\}

is part of a biorthogonal system and the resulting biorthogonal expansions are pointwise equiconvergent with ordinary Fourier series. Norm convergence is obtained for

p
=
2

p = 2

. In this paper, rather than imposing an explicit growth condition, we assume that

{

λ
n

−
n
}

\{ {\lambda _n} - n\}

is a multiplier sequence on

L
p

(
−
π
,

π
)

{L^p}( - \pi ,\,\pi )

. Conditions are given insuring that

{

g
n

}

\{ {g_n}\}

inherits both norm and pointwise convergence properties of ordinary Fourier series. Further,

λ
n

{\lambda _n}

and

g
n

{g_n}

are shown to be the eigenvalues and eigenfunctions of an unbounded operator

Λ
\Lambda

which is closely related to a differential operator,

i
Λ

i\Lambda

generates a strongly continuous group and

−

Λ
2

- {\Lambda ^2}

generates a strongly continuous semigroup. Half-range expansions, involving

cos

λ
n

x

{\text {cos}}{\lambda _n}x

or

sin

λ
n

x

{\text {sin}}{\lambda _n}x

on

(
0
,

π
)

(0,\,\pi )

are also shown to arise from linear operators which generate semigroups. Many of these results are obtained using the functional calculus for well-bounded operators.
Knapp-Wallach Szegő integrals. II. The higher parabolic rank case
Authors: Blank B. E.
Published: June 2010

Let

G
G

be a connected reductive linear Lie group with compact center and real rank

l
l

. For each integer

k
(
1
⩽
k
⩽
l
)

k(1 \leqslant k \leqslant l)

and each discrete series representation

π
\pi

of

G
G

, an explicit embedding of

π
\pi

into a generalized principal series representation induced from a parabolic subgroup of rank

k
k

is given. The existence of such embeddings was proved by W. Schmid. In this paper an explicit integral formula with Szegö kernel is given which provides these mappings.
𝐹-purity and rational singularity in graded complete intersection rings
Authors: Fedder Richard
Published: June 2010

A simple criterion is given for determining “almost completely” whether the positively graded complete intersection ring
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R equals upper K left-bracket upper X 1 comma ellipsis comma upper X Subscript n plus t Baseline right-bracket slash left-parenthesis upper G Subscript i Baseline comma ellipsis comma upper G Subscript t Baseline right-parenthesis">

R
=
K
[

X
1

,

…
,

X

n
+
t

]

/

(

G
i

,

…
,

G
t

)

R = K[{X_1},\, \ldots ,\,{X_{n + t}}]/({G_i},\, \ldots ,\,{G_t})

, of dimension

n
n

, has an

F
F

-pure type singularity at

m
=
(

X
1

,

…
,

X

n
+
t

)

m = ({X_1},\, \ldots ,\,{X_{n + t}})

. Specifically, if

deg
⁡
(

X
i

)
=

α
i

>
0

\operatorname {deg} ({X_i}) = {\alpha _i} > 0

for

1
≤
i
≤
n
+
t

1 \leq i \leq n + t

and

deg
⁡
(

G
i

)
=

d
i

>
0

\operatorname {deg} ({G_i}) = {d_i} > 0

for

i
≤
i
≤
t

i \leq i \leq t

, then there exists an integer

δ
\delta

determined by the singular locus of

R
R

such that: (1)

R
R

has

F
F

-pure type if

Σ

i
=
1

t

d
i

−

Σ

i
=
1

n
+
t

α
i

>
δ

\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} > \delta

. (2)

R
R

does not have

F
F

-pure type if

Σ

i
=
1

t

d
i

−

Σ

i
=
1

n
+
t

α
i

>
0

\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} > 0

. The characterization given by this theorem is particularly effective if the singularity of

R
R

at

m
m

is isolated. In that case,

δ
=
0

\delta = 0

so that only the condition

Σ

i
=
1

t

d
i

−

Σ

i
=
1

n
+
t

α
i

=
0

\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} = 0

is not solved by the above result. In particular, it follows from work of Kei-ichi Watanabe that if

R
R

has an isolated rational singularity, then

R
R

has

F
F

-pure type. The converse is also “almost true” with the only exception being the case where

Σ

i
=
1

t

d
i

−

Σ

i
=
1

n
+
t

α
i

=
0

\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} = 0

. In proving this criterion, a weak but more stable form of

F
F

-purity, called

F
F

-contractedness, is defined and explored.

R
R

is

F
F

-contracted (in characteristic

p
>
0

p > 0

) if every system of parameters for

m
m

is contracted with respect to the Frobenius map

F
:

R
→
R

F:\,R \to R

. Just as for

F
F

-purity, the notion of

F
F

-contracted type is defined in characteristic 0 by reduction to characteristic

p
p

. The two notions of

F
F

-pure (type) and

F
F

-contracted (type) coincide when

R
R

is Gorenstein; whence, in particular, when

R
R

is a complete intersection ring.
On root invariants of periodic classes in 𝐸𝑥𝑡_{𝐴}(𝑍/2,𝑍/2)
Authors: Shick Paul
Published: June 2010

We prove that if a class in the cohomology of the mod 2 Steenrod algebra is

mod

2

\operatorname {mod}\,2

-periodic in the sense of [10], then its root invariant must be

υ

n
+
1

{\upsilon _{n + 1}}

-periodic, where

υ

n

{\upsilon _{n}}

denotes the

n
n

th generator of

π
∗

(

BP

)

{\pi _ \ast }({\text {BP}})

.
The structure of 𝜎-ideals of compact sets
Authors: Kechris A. S. , Louveau A. , Woodin W. H.
Published: June 2010

Motivated by problems in certain areas of analysis, like measure theory and harmonic analysis, where

σ
\sigma

-ideals of compact sets are encountered very often as notions of small or exceptional sets, we undertake in this paper a descriptive set theoretic study of

σ
\sigma

-ideals of compact sets in compact metrizable spaces. In the first part we study the complexity of such ideals, showing that the structural condition of being a

σ
\sigma

-ideal imposes severe definability restrictions. A typical instance is the dichotomy theorem, which states that

σ
\sigma

-ideals which are analytic or coanalytic must be actually either complete coanalytic or else

G
δ

{G_\delta }

. In the second part we discuss (generators or as we call them here) bases for

σ
\sigma

-ideals and in particular the problem of existence of Borel bases for coanalytic non-Borel

σ
\sigma

-ideals. We derive here a criterion for the nonexistence of such bases which has several applications. Finally in the third part we develop the connections of the definability properties of

σ
\sigma

-ideals with other structural properties, like the countable chain condition, etc.
The moduli of compact continuations of an open Riemann surface of genus one
Authors: Shiba M.
Published: June 2010

Let

(
R
,

{
A
,

B
}
)

(R,\,\{ A,\,B\} )

be a marked open Riemann surface of genus one. Denote by

(
T
,

{

A
T

,

B
T

}
,
i
)

(T,\,\{ {A_T},\,{B_T}\} ,i)

a pair of a marked torus

(
T
,

{

A
T

,

B
T

}
)

(T,\,\{ {A_T},\,{B_T}\} )

and a conformal embedding

i
i

of

R
R

into

T
T

with

i
(
A
)

i(A)

and

i
(
B
)

i(B)

homotopic respectively to

A
T

{A_T}

and

B
T

{B_T}

. We say that

(
T
,

{

A
T

,

B
T

}
,
i
)

(T,\,\{ {A_T},\,{B_T}\} ,i)

and

(

T
′

,

{

A
T
′

,

B
T
′

}
,

i
′

)

(T’,\,\{ {A_T’},\,{B_T’}\} ,i’)

are equivalent if

i
′

∘

i

−
1

i’ \circ {i^{ - 1}}

extends to a conformal mapping of

T
T

onto

T
′

{T^\prime }

. The equivalence classes are called compact continuations of

(
R
,

{
A
,

B
}
)

(R,\,\{ A,\,B\} )

and the set of moduli of compact continuations of

(
R
,

{
A
,

B
}
)

(R,\,\{ A,\,B\} )

is denoted by

M
=
M
(
R
,

{
A
,

B
}
)

M = M(R,\,\{ A,\,B\} )

. Then

M
M

is a closed disk in the upper half plane. The radius of

M
M

represents the size of the ideal boundary of

R
R

and gives a generalization of Schiffer’s span for planar domains; in particular, it vanishes if and only if

R
R

belongs to the class

O

A
D

{O_{AD}}

. On the other hand, any holomorphic differential on

R
R

with distinguished imaginary part produces in a canonical manner a compact continuation of

(
R
,

{
A
,

B
}
)

(R,\,\{ A,\,B\} )

. Such a compact continuation is referred to as a hydrodynamic continuation of

(
R
,

{
A
,

B
}
)

(R,\,\{ A,\,B\} )

. The boundary of

M
M

parametrizes in a natural way the space of hydrodynamic continuations; i.e., the hydrodynamic continuations have extremal properties.
Positive solutions of systems of semilinear elliptic equations: the pendulum method
Authors: Glover Joseph
Published: June 2010

Conditions are formulated which guarantee the existence of positive solutions for systems of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row minus normal upper Delta u 1 plus f 1 left-parenthesis u 1 comma ellipsis comma u Subscript n Baseline right-parenthesis equals mu 1 comma 2nd Row minus normal upper Delta u 2 plus f 2 left-parenthesis u 1 comma ellipsis comma u Subscript n Baseline right-parenthesis equals mu 2 comma 3rd Row vertical-ellipsis vertical-ellipsis vertical-ellipsis vertical-ellipsis 4th Row minus normal upper Delta u Subscript n Baseline plus f Subscript n Baseline left-parenthesis u 1 comma ellipsis comma u Subscript n Baseline right-parenthesis equals mu Subscript n Baseline comma EndLayout">

−
Δ

u
1

+

f
1

(

u
1

,
…
,

u
n

)
=

μ
1

,

−
Δ

u
2

+

f
2

(

u
1

,
…
,

u
n

)
=

μ
2

,

⋮

⋮

⋮

⋮

−
Δ

u
n

+

f
n

(

u
1

,
…
,

u
n

)
=

μ
n

,

\begin {gathered} - \Delta {u_1} + {f_1}({u_1}, \ldots ,\,{u_n}) = {\mu _1}, \hfill \\ - \Delta {u_2} + {f_2}({u_1}, \ldots ,\,{u_n}) = {\mu _2}, \hfill \\ \vdots \quad \quad \quad \quad \quad \vdots \quad \quad \quad \vdots \quad \vdots \hfill \\ - \Delta {u_n} + {f_n}({u_1}, \ldots ,\,{u_n}) = {\mu _n}, \hfill \\ \end {gathered}

\]
, where

Δ
\Delta

is the Laplacian (with Dirichlet boundary conditions) on an open domain in

R

d

{\mathbf {R}^d}

, and where each

μ
i

{\mu _i}

is a positive measure. The main tools used are probabilistic potential theory, Markov processes, and an iterative scheme which is not a generalization of the one used for quasimonotone systems. Quasimonotonicity is not assumed and new results are obtained even for the case where

∂

f
k

/

∂

x
j

>
0

\partial {f_k}/\partial {x_j} > 0

for every

k
k

and

j
j

.
On the Möbius function
Authors: Maier Helmut
Published: June 2010

We investigate incomplete convolutions of the Möbius function of the form

∑

d

|

n
;
d
≤
z

μ
(
d
)

\sum \nolimits _{d|n;d \leq z} {\mu (d)}

. It is shown that for almost all integers

n
n

one can find

z
z

for which this sum is large.