bluuebunny/crossref_metadata_2025 huggingface dataset:
Leavable gambling problems with unbounded utilities
Authors: Maitra A. , Purves R. , Sudderth W.
Published: June 2012

The optimal return function

U
U

of a Borel measurable gambling problem with a positive utility function is known to be universally measurable. With a negative utility function, however,

U
U

may not be so measurable. As shown here, the measurability of

U
U

for all Borel gambling problems with negative utility functions is equivalent to the measurability of all PCA sets, a property of such sets known to be independent of the usual axioms of set theory. If the utility function is further required to satisfy certain uniform integrability conditions, or if the gambling problem corresponds to an optimal stopping problem, the optimal return function is measurable. Another return function

W
W

is introduced as an alternative to

U
U

. It is shown that

W
W

is always measurable and coincides with

U
U

when the utility function is positive.
Application of the generalized Weierstrass preparation theorem to the study of homogeneous ideals
Authors: Amasaki Mutsumi
Published: June 2012

The system of Weierstrass polynomials, defined originally for ideals in convergent power series rings, together with its sequence of degrees allows us to analyze a homogeneous ideal directly. Making use of it, we study local cohomology modules, syzygies, and then graded Buchsbaum rings. Our results give a formula which to some extent clarifies the connection among the matrices appearing in the free resolution starting from a system of Weierstrass polynomials, a rough classification of graded Buchsbaum rings in the general case and a complete classification of graded Buchsbaum integral domains of codimension two.
On infinite-dimensional manifold triples
Authors: Sakai Katsuro , Wong Raymond Y.
Published: June 2012

Let

Q
Q

denote the Hilbert cube

[
−
1
,
1

]
ω

,

s
=

(
−
1
,
1

)
ω

{[ - 1,1]^\omega },\;s = {( - 1,1)^\omega }

the pseudo-interior of

Q
,

Σ
=
{
(

x
i

)
∈
s

|

sup

|

x
i

|

>
1
}

Q,\;\Sigma = \{ ({x_i}) \in s|\sup |{x_i}| > 1\}

and

σ
=
{
(

x
i

)
∈
s

|

x
i

=
0

except for finitely many

i
}

\sigma = \{ ({x_i}) \in s|{x_i} = 0\;{\text {except for finitely many}}\;i\}

. A triple

(
X
,
M
,
N
)

(X,M,N)

of separable metrizable spaces is called a

(
Q
,
Σ
,
σ
)

(Q,\Sigma ,\sigma )

- (or

(
s
,
Σ
,
σ
)

(s,\Sigma ,\sigma )

-)manifold triple if it is locally homeomorphic to

(
Q
,
Σ
,
σ
)

(Q,\Sigma ,\sigma )

(or

(
s
,
Σ
,
σ
)

(s,\Sigma ,\sigma )

). In this paper, we study such manifold triples and give some characterizations.
Dirac manifolds
Authors: Courant Theodore James
Published: June 2012

A Dirac structure on a vector space

V
V

is a subspace of

V
V

with a skew form on it. It is shown that these structures correspond to subspaces of

V
⊕

V

∗

V \oplus {V^{\ast }}

satisfying a maximality condition, and having the property that a certain symmetric form on

V
⊕

V

∗

V \oplus {V^{\ast }}

vanishes when restricted to them. Dirac structures on a vector space are analyzed in terms of bases, and a generalized Cayley transformation is defined which takes a Dirac structure to an element of

O
(
V
)

O(V)

. Finally a method is given for passing a Dirac structure on a vector space to a Dirac structure on any subspace. Dirac structures on vector spaces are generalized to smooth Dirac structures on a manifold

P
P

, which are defined to be smooth subbundles of the bundle

T
P
⊕

T

∗

P

TP \oplus {T^{\ast }}P

satisfying pointwise the properties of the linear case. If a bundle

L
⊂
T
P
⊕

T

∗

P

L \subset TP \oplus {T^{\ast }}P

defines a Dirac structure on

P
P

, then we call

L
L

a Dirac bundle over

P
P

. A

3
3

-tensor is defined on Dirac bundles whose vanishing is the integrability condition of the Dirac structure. The basic examples of integrable Dirac structures are Poisson and presymplectic manifolds; in these cases the Dirac bundle is the graph of a bundle map, and the integrability tensors are

[
B
,
B
]

[B,B]

and

d
Ω

d\Omega

respectively. A function

f
f

on a Dirac manifold is called admissible if there is a vector field

X
X

such that the pair

(
X
,
d
f
)

(X,df)

is a section of the Dirac bundle

L
L

; the pair

(
X
,
d
f
)

(X,df)

is called an admissible section. The set of admissible functions is shown to be a Poisson algebra. A process is given for passing Dirac structures to a submanifold

Q
Q

of a Dirac manifold

P
P

. The induced bracket on admissible functions on

Q
Q

is in fact the Dirac bracket as defined by Dirac for constrained submanifolds.
Isometric isomorphisms between Banach algebras related to locally compact groups
Authors: Ghahramani F. , Lau A. T. , Losert V.
Published: June 2012

Let

G
1

{G_1}

,

G
2

{G_2}

be locally compact groups. We prove in this paper that if

T
T

is an isometric isomorphism from the Banach algebra

LUC

(

G
1

)
∗

{\text {LUC}}{({G_1})^*}

(the continuous dual of the Banach space of left uniformly continuous functions on

G
1

{G_1}

, equipped with Arens multiplication) onto

LUC

(

G
2

)
∗

{\text {LUC}}{({G_2})^*}

, then

T
T

maps

M
(

G
1

)

M({G_1})

onto

M
(

G
2

)

M({G_2})

and

L
1

(

G
1

)

{L^1}({G_1})

onto

L
1

(

G
2

)

{L^1}({G_2})

. We also prove that any isometric isomorphism from

L
1

(

G
1

)

∗
∗

{L^1}{({G_1})^{**}}

(second conjugate algebra of

L
1

(

G
1

)

{L^1}({G_1})

) onto

L
1

(

G
2

)

∗
∗

{L^1}{({G_2})^{**}}

maps

L
1

(

G
1

)

{L^1}({G_1})

onto

L
1

(

G
2

)

{L^1}({G_2})

.
Intrinsic formality and certain types of algebras
Authors: Lupton Gregory
Published: June 2012

In this paper, a type of algebra is introduced and studied from a rational homotopy point of view, using differential graded Lie algebras. The main aim of the paper is to establish whether or not such an algebra is the rational cohomology algebra of a unique rational homotopy type of spaces. That is, in the language of rational homotopy, whether or not such an algebra is intrinsically formal. Examples are given which show that, in general, this is not so—7.8 and 7.9. However, whilst it is true that not all such algebras are intrinsically formal, some of them are. The main results of this paper show a certain class of these algebras to be intrinsically formal—Theorem

2
2

(6.1); and a second, different type of algebra also to be intrinsically formal—Theorem

1
1

(5.2), which type of algebra overlaps with the first type in many examples of interest. Examples are given in

§
7

\S 7

.
Realization of the level one standard 𝐶̃_{2𝑘+1}-modules
Authors: Misra Kailash C.
Published: June 2012

In this paper we study the level one standard (or irreducible integrable highest weight) modules for the affine symplectic Lie algebras. In particular, we give concrete realizations of all level one standard modules for the affine symplectic Lie algebras of even rank.
Layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains
Authors: Lewis Jeff E.
Published: June 2012

The symbolic calculus of pseudodifferential operators of Mellin type is applied to study layer potentials on a plane domain

Ω
+

{\Omega ^ + }

whose boundary

∂

Ω
+

{\partial \Omega ^ + }

is a curvilinear polygon. A "singularity type" is a zero of the determinant of the matrix of symbols of the Mellin operators and can be used to calculate the "bad values" of

p
p

for which the system is not Fredholm on

L
p

(
∂

Ω
+

)

{L^p}(\partial {\Omega ^ + })

. Using the method of layer potentials we study the singularity types of the system of elastostatics
\[

L

u

=
μ
Δ

u

+
(
λ
+
μ
)
∇
div
⁡

u

=
0.

L{\mathbf {u}} = \mu \Delta {\mathbf {u}} + (\lambda + \mu )\nabla \operatorname {div} {\mathbf {u}} = 0.

\]
in a plane domain

Ω
+

{\Omega ^ + }

whose boundary

∂

Ω
+

{\partial \Omega ^ + }

is a curvilinear polygon. Here

μ
>
0

\mu > 0

and

−
μ
≤
λ
≤
+
∞

-\mu \le \lambda \le +\infty

. When

λ
=
+
∞

\lambda = +\infty

, the system is the Stokes system of hydrostatics. For the traction double layer potential, we show that all singularity types in the strip

0
>
Re
⁡
z
>
1

0 > \operatorname {Re} z > 1

lie in the interval

(

1
2

,
1

)

\left ( {\frac {1} {2},1} \right )

so that the system of integral equations is a Fredholm operator of index

0
0

on

L
p

(
∂

Ω
+

)

{L^p}(\partial {\Omega ^ + })

for all

p
p

,

2
≤
p
>
∞

2 \le p > \infty

. The explicit dependence of the singularity types on

λ
\lambda

and the interior angles

θ
\theta

of

∂

Ω
+

{\partial \Omega ^ + }

is calculated; the singularity type of each corner is independent of

λ
\lambda

iff the corner is nonconvex.
A one-phase hyperbolic Stefan problem in multi-dimensional space
Authors: Li De Ning
Published: June 2012

The hyperbolic heat transfer model is obtained by replacing the classical Fourier’s law with the relaxation relation

τ

q
→

t
+

q
→

=
−
k
∇
T

\tau \vec qt + \vec q = - k\nabla T

. The sufficient and necessary conditions are derived for the local existence and uniqueness of classical solutions for multi-

D

{\text {D}}

Stefan problem of hyperbolic heat transfer model where phase change is accompanied with delay of latent heat storage.
The 𝑣₁-periodic homotopy groups of an unstable sphere at odd primes
Authors: Thompson Robert D.
Published: June 2012

The

mod

p

\bmod \;p

v
1

{v_1}

-periodic homotopy groups of a space

X
X

are defined by considering the homotopy classes of maps of a Moore space into

X
X

and then inverting the Adams self map. In this paper we compute the

p
p

v
1

{v_1}

-periodic homotopy groups of an odd dimensional sphere, localized at an odd prime. This is done by showing that these groups are isomorphic to the stable

mod

p

\bmod \;p

v
1

{v_1}

-periodic homotopy groups of

B

Σ
p

2
(
p
−
1
)
n

B\Sigma _p^{2(p - 1)n}

, the

2
(
p
−
1
)
n

2(p - 1)n

skeleton of the classifying space for the symmetric group

Σ
p

{\Sigma _p}

. There is a map
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega Superscript 2 n plus 1 Baseline upper S Superscript 2 n plus 1 Baseline right-arrow normal upper Omega Superscript normal infinity Baseline left-parenthesis upper J logical-and upper B normal upper Sigma Subscript p Superscript 2 left-parenthesis p minus 1 right-parenthesis n Baseline right-parenthesis">

Ω

2
n
+
1

S

2
n
+
1

→

Ω
∞

(
J
∧
B

Σ
p

2
(
p
−
1
)
n

)

{\Omega ^{2n + 1}}{S^{2n + 1}} \to {\Omega ^\infty }(J \wedge B\Sigma _p^{2(p - 1)n})

, where

J
J

is a spectrum constructed from connective

K
K

-theory, and the image in homotopy is studied.
Second order theta functions and vector bundles over Jacobi varieties
Authors: Yuen David S.
Published: June 2012

We consider the Picard vector bundles defined over Jacobi varieties. The rank

g
+
1

g + 1

Picard bundle imbeds in the rank

2
g

{2^g}

Clifford bundle, so the second order theta functions, viewed appropriately, span the dual of the Picard bundle over each fiber. We prove a result on the minimum number of such second order theta functions required to span the whole bundle at each point. We give an application of using these functions to describe subvarieties of the Jacobian. There follow comments on which functions we could use, and generalizations to higher order theta functions.
Algebraic distance graphs and rigidity
Authors: Homma M. , Maehara H.
Published: June 2012

An algebraic distance graph is defined to be a graph with vertices in

E
n

{E^n}

in which two vertices are adjacent if and only if the distance between them is an algebraic number. It is proved that an algebraic distance graph with finite vertex set is complete if and only if the graph is "rigid". Applying this result, we prove that (1) if all the sides of a convex polygon

Γ
\Gamma

which is inscribed in a circle are algebraic numbers, then the circumradius and all diagonals of

Γ
\Gamma

are also algebraic numbers, (2) the chromatic number of the algebraic distance graph on a circle of radius

r
r

is

∞
\infty

or

2
2

accordingly as

r
r

is algebraic or not. We also prove that for any

n
>
0

n > 0

, there exists a graph

G
G

which cannot be represented as an algebraic distance graph in

E
n

{E^n}

.
Random Blaschke products
Authors: Cochran W. George
Published: June 2012

Let

{

θ
n

(
ω
)
}

\{ {\theta _n}(\omega )\}

be a sequence of independent random variables uniformly distributed on

[
0
,
2
π
]

[0,2\pi ]

, and let

z
n

(
ω
)
=

r
n

e

i

θ
n

(
ω
)

{z_n}(\omega ) = {r_n}{e^{i{\theta _n}(\omega )}}

for a fixed but arbitrary sequence of radii

r
n

{r_n}

satisfying the Blaschke condition

∑

(
1
−

r
n

)
>
∞

\sum {(1 - {r_n}) > \infty }

. We show that the random Blaschke product with zeros

z
n

(
ω
)

{z_n}(\omega )

is almost surely not in the little Bloch space, and we describe necessary conditions and sufficient conditions on the radii

r
n

{r_n}

so that

{

z
n

(
ω
)
}

\{ {z_n}(\omega )\}

is almost surely an interpolating sequence.
Large deviations in dynamical systems and stochastic processes
Authors: Kifer Yuri
Published: June 2012

The paper exhibits a unified approach to large deviations of dynamical systems and stochastic processes based on the existence of a pressure functional and on the uniqueness of equilibrium states for certain dense sets of functions. This enables us to generalize recent results from [OP, Y, and D] on large deviations for dynamical systems, as well, as to recover Donsker-Varadhan’s [DV2] large deviation estimates for Markov processes.
Smooth great circle fibrations and an application to the topological Blaschke conjecture
Authors: Yang C. T.
Published: June 2012

We study great smooth circle fibrations of round spheres and Blaschke manifolds of the homotopy type of complex projective spaces.
Univalent functions which map onto regions of given transfinite diameter
Authors: Duren P. L. , Schiffer M. M.
Published: June 2012

By a variational method, the sharp upper bound is obtained for the second coefficients of normalized univalent functions which map the unit disk onto regions of prescribed transfinite diameter, or logarithmic capacity.
Möbius invariant Hilbert spaces of holomorphic functions in the unit ball of 𝐶ⁿ
Authors: Zhu Ke He
Published: June 2012

We prove that there exists a unique Hilbert space of holomorphic functions in the open unit ball of

C

n

{\mathbb {C}^n}

whose (semi-) inner product is invariant under Möbius transformations.
A countably compact topological group 𝐻 such that 𝐻×𝐻 is not countably compact
Authors: Hart Klaas Pieter , van Mill Jan
Published: June 2012

Using

M

A

countable

{\mathbf {M}}{{\mathbf {A}}_{{\text {countable}}}}

we construct a topological group with the properties mentioned in the title.
Closed convex hulls of unitary orbits in von Neumann algebras
Authors: Hiai Fumio , Nakamura Yoshihiro
Published: June 2012

Let

M

\mathcal {M}

be a von Neumann algebra. The distance

dist
⁡
(
x
,
co
⁡

U

(
y
)
)

\operatorname {dist} (x,\operatorname {co} \mathcal {U}(y))

between

x
x

and

co
⁡

U

(
y
)

\operatorname {co} \mathcal {U}(y)

for selfadjoint operators

x
x

,

y
∈

M

y \in \mathcal {M}

and the distance

dist
⁡
(
φ
,
co
⁡

U

(
ψ
)
)

\operatorname {dist} (\varphi ,\operatorname {co} \mathcal {U}(\psi ))

between

φ
\varphi

and

co
⁡

U

(
ψ
)

\operatorname {co} \mathcal {U}(\psi )

for selfadjoint elements

φ
\varphi

,

ψ
∈

M

∗

\psi \in {\mathcal {M}_*}

are exactly estimated, where

co
⁡

U

(
y
)

\operatorname {co} \mathcal {U}(y)

and

co
⁡

U

(
ψ
)

\operatorname {co} \mathcal {U}(\psi )

are the convex hulls of the unitary orbits of

y
y

and

ψ
\psi

, respectively. This is done separately in the finite factor case, in the infinite semifinite factor case, and in the type III factor case. Simple formulas of distances between two convex hulls of unitary orbits are also given. When

M

\mathcal {M}

is a von Neumann algebra on a separable Hilbert space, the above cases altogether are combined under the direct integral decomposition of

M

\mathcal {M}

into factors. As a result, it is known that if

M

\mathcal {M}

is

σ
\sigma

-finite and

x
∈

M

x \in \mathcal {M}

is selfadjoint, then

co
¯

U

(
x
)
=

co
¯

w

U

(
x
)

\overline {\operatorname {co} } \mathcal {U}(x) = {\overline {\operatorname {co} } ^{\mathbf {w}}}\mathcal {U}(x)

where

co
¯

U

(
x
)

\overline {\operatorname {co} } \mathcal {U}(x)

and

co
¯

w

U

(
x
)

{\overline {\operatorname {co} } ^{\mathbf {w}}}\mathcal {U}(x)

are the closures of

co
⁡

U

(
x
)

\operatorname {co} \mathcal {U}(x)

in norm and in the weak operator topology, respectively.
Generating modules efficiently over noncommutative Noetherian rings
Authors: Coutinho S. C.
Published: June 2012

The Forster-Swan Theorem gives an upper bound on the number of generators of a module over a commutative ring in terms of local data. Stafford showed that this theorem could be generalized to arbitrary right and left noetherian rings. In this paper a similar result is proved for right noetherian rings with finite Krull dimension. A new dimension function—the basic dimension—is the main tool used in the proof of this result.
Strong shape for topological spaces
Authors: Dydak Jerzy , Nowak Sławomir
Published: June 2012

Strong shape equivalences for topological spaces are introduced in a way which generalizes easily to inverse systems of topological spaces. Each space is then mapped via a strong shape equivalence into a fibrant inverse system of ANRs. This leads naturally to defining the strong shape category SSh for topological spaces. Other descriptions of SSh are also provided.
Distal functions and unique ergodicity
Authors: Salehi Ebrahim
Published: June 2012

A. Knapp [5] has shown that the set,

D
(
S
)

D(S)

, of all distal functions on a group

S
S

is a norm closed subalgebra of

l
∞

(
S
)

{l^\infty }(S)

that contains the constants and is closed under the complex conjugation and left translation by elements of

S
S

. Also it is proved that [7] for any

k
∈

N

k \in \mathbb {N}

and any

λ
∈

R

\lambda \in \mathbb {R}

the function

f
:

Z

→

C

f:\mathbb {Z} \to \mathbb {C}

defined by

f
(
n
)
=

e

i
λ

n
k

f(n) = {e^{i\lambda {n^k}}}

is distal on

Z

\mathbb {Z}

. Now let

W

{\mathbf {W}}

be the norm closure of the algebra generated by the set of functions
\[

{
n
↦

e

i
λ

n
k

:
k
∈

N

,

λ
∈

R

}
,

\{ n \mapsto {e^{i\lambda {n^k}}}:k \in \mathbb {N},\;\lambda \in \mathbb {R}\} ,

\]
which will be called the Weyl algebra. According to the facts mentioned above, all members of the Weyl Algebra are distal functions on

Z

\mathbb {Z}

. In this paper, we will show that any element of

W

{\mathbf {W}}

is uniquely ergodic (Theorem 2.13) and that the set

W

{\mathbf {W}}

does not exhaust all the distal functions on

Z

\mathbb {Z}

(Theorem 2.14). The latter will answer the question that has been asked (to the best of my knowledge) by P. Milnes [6]. The term Weyl algebra is suggested by S. Glasner. I would like to express my warmest gratitude to S. Glasner for his helpful advise, and to my advisor Professor Namioka for his enormous helps and contributions.
Conformal automorphisms and conformally flat manifolds
Authors: Goldman William M. , Kamishima Yoshinobu
Published: June 2012

A geometric structure on a smooth

n
n

-manifold

M
M

is a maximal collection of distinguished charts modeled on a

1
1

-connected

n
n

-dimensional homogeneous space

X
X

of a Lie group

G
G

where coordinate changes are restrictions of transformations from

G
G

. There exists a developing map

d
e
v
:
w
m
→
X

dev:wm \to X

which is always locally a diffeomorphism. It is in general far from globally being a diffeomorphism. We study the rigid property of developing maps of

(
G
,
X
)

(G,X)

-manifolds. As an application we shall classify closed conformally flat manifolds

M
M

when the universal covering space

M
~

\tilde M

supports a one parameter group of conformal transformations.
Hamilton-Jacobi equations with singular boundary conditions on a free boundary and applications to differential games
Authors: Bardi Martino , Soravia Pierpaolo
Published: June 2012

A class of Hamilton-Jacobi equations arising in generalized timeoptimal control problems and differential games is considered. The natural global boundary value problem for these equations has a singular boundary condition on a free boundary. The uniqueness of the continuous solution and of the free boundary is proved in the framework of viscosity solutions. A local uniqueness theorem is also given, as well as some existence results and several applications to control and game theory. In particular a relaxation theorem (weak form of the bang-bang principle) is proved for a class of nonlinear differential games.
On completing unimodular polynomial vectors of length three
Authors: Rao Ravi A.
Published: June 2012

It is shown that if

R
R

is a local ring of dimension three, with

1
2

∈
R

\frac {1} {2} \in R

, then a polynomial three vector

(

v
0

(
X
)
,

v
1

(
X
)
,

v
2

(
X
)
)

({v_0}(X),{v_1}(X),{v_2}(X))

over

R
[
X
]

R[X]

can be completed to an invertible matrix if and only if it is unimodular. In particular, if

1

/

3
!
∈
R

1/3! \in R

, then every stably free projective

R
[

X
1

,
…
,

X
n

]

R[{X_1}, \ldots ,{X_n}]

-module is free.
The Maslov class of the Lagrange surfaces and Gromov’s pseudo-holomorphic curves
Authors: Polterovich L. V.
Published: June 2012

For an immersed Lagrange submanifold

W
⊂

T
∗

X

W \subset {T^\ast }X

, one can define a nonnegative integer topologic invariant

m
(
W
)

m(W)

such that the image of

H
1

(
W
;

Z

)

{H_1}(W;{\mathbf {Z}})

under the Maslov class is equal to

m
(
W
)
⋅

Z

m(W) \cdot {\mathbf {Z}}

. In this paper, the value of

m
(
W
)

m(W)

is calculated for the case of a two-dimensional oriented manifold

X
X

with the universal cover homeomorphic to

R

2

{{\mathbf {R}}^2}

and an embedded Lagrange torus

W
W

. It is proved that if

X
=

T

2

X = {{\mathbf {T}}^2}

and

W
W

is homologic to the zero section, then

m
(
W
)
=
0

m(W) = 0

. In all the other cases

m
(
W
)
=
2

m(W) = 2

. The last result is true also for a wide class of oriented properly embedded Lagrange surfaces in

T
∗

R

2

{T^\ast }{{\mathbf {R}}^2}

. The proof is based on the Gromov’s theory of pseudo-holomorphic curves. Some applications to the hamiltonian mechanics are mentioned.
Structure locale de l’espace des rétractions d’une surface
Authors: Cauty Robert
Published: June 2012

Let

Σ
\Sigma

be a compact connected

2
2

-manifold, and

R

(
Σ
)

\mathcal {R}(\Sigma )

the space of retractions of

Σ
\Sigma

. We prove that

R

(
Σ
)

\mathcal {R}(\Sigma )

is an

l
2

{l^2}

-manifold if the boundary of

Σ
\Sigma

is not empty, and is the union of an

l
2

{l^2}

-manifold and an isolated point

i

d

Σ

{\text {i}}{{\text {d}}_\Sigma }

if

Σ
\Sigma

is closed.
New results on the Pompeiu problem
Authors: Garofalo Nicola , Segàla Fausto
Published: June 2012

Let

p
N

(
w
)
=

∑

k
=
0

N

a
k

w
k

{p_N}(w) = \sum \nolimits _{k = 0}^N {{a_k}{w^k}}

,

w
∈

C

w \in \mathbb {C}

,

N
∈

N

N \in \mathbb {N}

, be a polynomial with complex coefficients. In this paper we prove that if

D
⊂

R

2

D \subset {\mathbb {R}^2}

is a simply-connected bounded open set whose boundary is a closed, simple curve parametrized by

x
(
s
)
=

x
1

(
s
)
+
i

x
2

(
s
)
=

p
N

(

e

i
s

)

x(s) = {x_1}(s) + i{x_2}(s) = {p_N}({e^{is}})

,

s
∈
[
−
π
,
π
]

s \in [ - \pi ,\pi ]

, then

D
D

has the Pompeiu property unless

N
=
1

N = 1

and

p
1

(
w
)
=

a
1

w
+

a
2

{p_1}(w) = {a_1}w + {a_2}

in which case

D
D

is a disk. This result supports the conjecture that modulo sets of zero two-dimensional Lebesgue measure, the disk is the only simply-connected, bounded open set which fails to have the Pompeiu property.
On the structure of certain locally compact topological groups
Authors: Wu Ta Sun
Published: June 2012

A locally compact topological group

G
G

is called an

(

H

)

({\text {H}})

group if

G
G

has a maximal compact normal subgroup with Lie factor. In this note, we study the problem when a locally compact group is an

(

H

)

({\text {H}})

group.
Crossed simplicial groups and their associated homology
Authors: Fiedorowicz Zbigniew , Loday Jean-Louis
Published: June 2012

We introduce a notion of crossed simplicial group, which generalizes Connes’ notion of the cyclic category. We show that this concept has several equivalent descriptions and give a complete classification of these structures. We also show how many of Connes’ results can be generalized and simplified in this framework.
Extensions of measures invariant under countable groups of transformations
Authors: Krawczyk Adam , Zakrzewski Piotr
Published: June 2012

We consider countably additive, nonnegative, extended real-valued measures vanishing on singletons. Given a group

G
G

of bijections of a set

X
X

and a

G
G

-invariant measure

m
m

on

X
X

we ask whether there exists a proper

G
G

-invariant extension of

m
m

. We prove, among others, that if

Q

\mathbb {Q}

is the group of rational translations of the reals, then there is no maximal

Q

\mathbb {Q}

-invariant extension of the Lebesgue measure on

R

\mathbb {R}

. On the other hand, if

2
ω

{2^\omega }

is real-valued measurable, then there exists a maximal

σ
\sigma

-finite

Q

\mathbb {Q}

-invariant measure defined on a proper

σ
\sigma

-algebra of subsets of

R

\mathbb {R}

.
Some results on the Šarkovskiĭ partial ordering of permutations
Authors: Jungreis Irwin
Published: June 2012

If

π
\pi

is a cyclic permutation and

x
x

is a periodic point of a continuous function

f
:

R

↦

R

f:{\mathbf {R}} \mapsto {\mathbf {R}}

with

period

(
x
)
=

order

(
π
)
=
n

{\text {period}}(x) = {\text {order}}(\pi ) = n

, then we say that

x
x

has type

π
\pi

if the orbit of

x
x

consists of points

x
1

>

x
2

>
⋯
>

x
n

{x_1} > {x_2} > \cdots > {x_n}

with

f
(

x
i

)
=

x

π
(
i
)

f({x_i}) = {x_{\pi (i)}}

. In analogy with Sarkovskii’s Theorem, we define a partial ordering on cyclic permutations by

θ
→
π

\theta \to \pi

if every continuous function with a periodic point of type

θ
\theta

also has a point of type

π
\pi

. In this paper we examine this partial order form the point of view of critical points, itineraries, and kneading sequences. We show that

θ
→
π

\theta \to \pi

if and only if the maxima of

θ
\theta

are "higher" and the minima "lower" than those of

π
\pi

, where "higher" and "lower" are precisely defined in terms of itineraries. We use this to obtain several results about

→
\to

: there are no minimal upper bounds; if

π
\pi

and

θ
\theta

have the same number of critical points (or if they differ by

1
1

or sometimes

2
2

), then

θ
→
π

\theta \to \pi

if and only if

θ
→

π
∗

\theta \to {\pi _\ast }

for some period double

π
∗

{\pi _\ast }

of

π
\pi

; and finally, we prove a conjecture of Baldwin that maximal permutations of size

n
n

have

n
−
2

n - 2

critical points, and obtain necessary and sufficient conditions for such a permutation to be maximal.
The solution of length four equations over groups
Authors: Edjvet Martin , Howie James
Published: June 2012

Let

G
G

be a group,

F
F

the free group generated by

t
t

and let

r
(
t
)
∈
G
∗
F

r(t) \in G \ast F

. The equation

r
(
t
)
=
1

r(t) = 1

is said to have a solution over

G
G

if it has a solution in some group that contains

G
G

. This is equivalent to saying that the natural map

G
→
⟨
G
∗
F

|

r
(
t
)
⟩

G \to \langle G \ast F|r(t)\rangle

is injective. There is a conjecture (attributed to M. Kervaire and F. Laudenbach) that injectivity fails only if the exponent sum of

t
t

in

r
(
t
)

r(t)

is zero. In this paper we verify this conjecture in the case when the sum of the absolute values of the exponent of

t
t

in

r
(
t
)

r(t)

is equal to four.
Concentrated cyclic actions of high periodicity
Authors: Berend Daniel , Katz Gabriel
Published: June 2012

The class of concentrated periodic diffeomorphisms

g
:
M
→
M

g:M \to M

is introduced. A diffeomorphism is called concentrated if, roughly speaking, its normal eigenvalues range in a small (with respect to the period of

g
g

and the dimension of

M
M

) arc on the circle. In many ways, the cyclic action generated by such a

g
g

behaves on the one hand as a circle action and on the other hand as a generic prime power order cyclic action. For example, as for circle actions,

Sign
⁡
(
g
,
M
)
=
Sign
⁡
(

M
g

)

\operatorname {Sign} (g,M) = \operatorname {Sign} ({M^g})

, provided that the left-hand side is an integer; as for prime power order actions,

g
g

cannot have a single fixed point if

M
M

is closed. A variety of integrality results, relating to the usual signatures of certain characteristic submanifolds of the regular neighbourhood of

M
g

{M^g}

in

M
M

to

Sign
⁡
(
g
,
M
)

\operatorname {Sign} (g,M)

via the normal

g
g

-representations, is established.
Towards a functional calculus for subnormal tuples: the minimal normal extension
Authors: Conway John B.
Published: June 2012

In this paper the study of a functional calculus for subnormal

n
n

-tuples is initiated and the minimal normal extension problem for this functional calculus is explored. This problem is shown to be equivalent to a mean approximation problem in several complex variables which is solved. An analogous uniform approximation problem is also explored. In addition these general results are applied together with The Area and the The Coarea Formula from Geometric Measure Theory to operators on Bergman spaces and to the tensor product of two subnormal operators. The minimal normal extension of the tensor product of the Bergman shift with itself is completely determined.
Lie flows of codimension 3
Authors: Gallego E. , Reventós A.
Published: June 2012

We study the following realization problem: given a Lie algebra of dimension

3
3

and an integer

q
,
0
≤
q
≤
3

q,0 \leq q \leq 3

, is there a compact manifold endowed with a Lie flow transversely modeled on

G

\mathcal {G}

and with structural Lie algebra of dimension

q
q

? We give here a quite complete answer to this problem but some questions remain still open

(

cf.

§
2

({\text {cf.}}\;\S 2

.
Linear series with an 𝑁-fold point on a general curve
Authors: Schubert David
Published: June 2012

A linear series

(
V
,

L

)

(V,\mathcal {L})

on a curve

X
X

has an

N
N

-fold point along a divisor

D
D

of degree

N
N

if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension left-parenthesis upper V intersection upper H Superscript 0 Baseline left-parenthesis upper X comma script upper L left-parenthesis negative upper D right-parenthesis right-parenthesis right-parenthesis greater-than-or-equal-to dimension upper V minus 1">

dim
⁡
(
V
∩

H
0

(
X
,

L

(
−
D
)
)
)
≥
dim

V
−
1

\dim (V \cap {H^0}\;(X,\mathcal {L}\,(- D))) \geq \dim \;V - 1

. The dimensions of the families of linear series with an

N
N

-fold point are determined for general curves.
A nonstandard resonance problem for ordinary differential equations
Authors: Ahmad Shair
Published: June 2012

Necessary and sufficient conditions are established for the existence of bounded solutions for a class of second order differential equations.
A cubic counterpart of Jacobi’s identity and the AGM
Authors: Borwein J. M. , Borwein P. B.
Published: June 2012

We produce exact cubic analogues of Jacobi’s celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Subscript n plus 1 Baseline colon equals StartFraction a Subscript n Baseline plus 2 b Subscript n Baseline Over 3 EndFraction and b Subscript n plus 1 Baseline colon equals RootIndex 3 StartRoot b Subscript n Baseline left-parenthesis StartFraction a Subscript n Superscript 2 Baseline plus a Subscript n Baseline b Subscript n Baseline plus b Subscript n Superscript 2 Baseline Over 3 EndFraction right-parenthesis period EndRoot">

a

n
+
1

:=

a
n

+
2

b
n

3

and

b

n
+
1

:=

b
n

(

a
n
2

+

a
n

b
n

+

b
n
2

3

)

.

3

{a_{n + 1}}: = \frac {{{a_n} + 2{b_n}}} {3}\quad {\text {and}}\quad {b_{n + 1}}: = \sqrt [3]{{{b_n}\left ( {\frac {{a_n^2 + {a_n}{b_n} + b_n^2}} {3}} \right ).}}

\]
The limit of this iteration is identified in terms of the hypergeometric function

2

F
1

(
1

/

3
,
2

/

3
;
1
;
⋅
)

{}_2{F_1}(1/3,2/3;1; \cdot )

, which supports a particularly simple cubic transformation.
Inner amenable locally compact groups
Authors: Lau Anthony To Ming , Paterson Alan L. T.
Published: June 2012

In this paper we study the relationship between amenability, inner amenability and property

P
P

of a von Neumann algebra. We give necessary conditions on a locally compact group

G
G

to have an inner invariant mean

m
m

such that

m
(
V
)
=
0

m(V) = 0

for some compact neighborhood

V
V

of

G
G

invariant under the inner automorphisms. We also give a sufficient condition on

G
G

(satisfied by the free group on two generators or an I.C.C. discrete group with Kazhdan’s property

T
T

, e.g.,

SL

(
n
,

Z

)

{\text {SL}}(n,\mathbb {Z})

,

n
≥
3

n \geq 3

) such that each linear form on

L
2

(
G
)

{L^2}(G)

which is invariant under the inner automorphisms is continuous. A characterization of inner amenability in terms of a fixed point property for left Banach

G
G

-modules is also obtained.
On the distance between unitary orbits of weighted shifts
Authors: Marcoux Laurent
Published: June 2012

In this paper, we consider invertible bilateral weighted shift operators acting on a complex separable Hilbert space

H

\mathcal {H}

. They have the property that there exist a constant

τ
>
0

\tau > 0

and an orthonormal basis

{

e
i

}

i
∈

Z

{\{ {{e_i}} \}_{i \in \mathbb {Z}}}

for

H

\mathcal {H}

with respect to which a shift

V
V

acts by

W

e
i

=

w
i

e

i
+
1

,
i
∈

Z

W{e_i}= {w_i}{e_{i + 1}},i \in \mathbb {Z}

and

|

w
i

|

≥
τ

{\mathbf {|}}{w_i}{\mathbf {|}} \geq \tau

. The equivalence class
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper U left-parenthesis upper W right-parenthesis equals left-brace upper U Superscript asterisk Baseline upper W upper U colon upper U element-of script upper B left-parenthesis script upper H right-parenthesis comma upper U unitary right-brace">

U

(
W
)
=
{

U

∗

W
U
:
U
∈

B

(

H

)
,
U

unitary

}

\mathcal {U}(W)= \{ {U^{\ast }}\;WU:U \in \mathcal {B}(\mathcal {H}),U\;{\text {unitary}}\}

of weighted shifts with weight sequence (with respect to the basis

{

U

∗

e
i

}

i
∈

Z

{\{ {U^{\ast }}{e_i}\} _{i \in \mathbb {Z}}}

for

H

)

\mathcal {H})

identical to that of

W
W

forms the unitary orbit of

W
W

. Given two shifts

W
W

and

V
V

, one can define a distance
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d left-parenthesis script upper U left-parenthesis upper V right-parenthesis comma script upper U left-parenthesis upper W right-parenthesis right-parenthesis equals inf left-brace right-brace comma element-of element-of parallel-to colon parallel-to colon minus minus parallel-to parallel-to XYX of UU left-parenthesis right-parenthesis upper V comma element-of element-of upper Y of UU left-parenthesis right-parenthesis upper W">

d
(

U

(
V
)
,

U

(
W
)
)
=
inf
{
∥

X
−
Y
∥:
X
∈

U

(
V
)
,
Y
∈

U

(
W
)
}

d(\mathcal {U}(V),\mathcal {U}(W))= \inf \{\parallel \,X - Y\parallel :X \in \mathcal {U}(V),Y \in \mathcal {U}(W)\}

between the unitary orbits of

W
W

and

V
V

. We establish numerical estimates for upper and lower bounds on this distance which depend upon information drawn from finite dimensional restrictions of these operators.
Minimal submanifolds of 𝐸²ⁿ⁺¹ arising from degenerate 𝑆𝑂(3) orbits on the Grassmannian
Authors: Landsberg J. M.
Published: June 2012

We give new examples of minimal submanifolds of

E

2
n
+
1

{{\mathbf {E}}^{2n + 1}}

characterised by having their Gauss map’s image lie in degenerate

S
O
(
3
)

SO(3)

orbits of

G

p
,
2
n
+
1

{G_{p,2n + 1}}

, the Grassmannian of

p
p

-planes in

E

2
n
+
1

{{\mathbf {E}}^{2n + 1}}

(where the action on

G

p
,
2
n
+
1

{G_{p,2n + 1}}

is induced from the irreducible

S
O
(
3
)

SO(3)

action on

R

2
n
+
1

{{\mathbf {R}}^{2n + 1}}

). These submanifolds are all given explicitly in terms of holomorphic data and are linearly full in

E

2
n
+
1

{{\mathbf {E}}^{2n + 1}}

.
Solving ordinary differential equations in terms of series with real exponents
Authors: Grigor′ev D. Yu. , Singer M. F.
Published: June 2012

We generalize the Newton polygon procedure for algebraic equations to generate solutions of polynomial differential equations of the form

∑

i
=
0

∞

α
i

x

β
i

\sum \nolimits _{i = 0}^\infty {{\alpha _i}{x^{{\beta _i}}}}

where the

α
i

{\alpha _i}

are complex numbers and the

β
i

{\beta _i}

are real numbers with

β
0

>

β
1

>
⋯

{\beta _0} > {\beta _1} > \cdots

. Using the differential version of the Newton polygon process, we show that any such a series solution is finitely determined and show how one can enumerate all such solutions of a given polynomial differential equation. We also show that the question of deciding if a system of polynomial differential equations has such a power series solution is undecidable.
On the spectral character of Toeplitz operators on multiply connected domains
Authors: Clancey Kevin F.
Published: June 2012

An explicit resolvent formula is given for selfadjoint Toeplitz operators acting on the least harmonic majorant Hardy spaces of a multiply connected planar domain. This formula is obtained by using theta functions associated with the double of the domain. Several consequences concerning the spectral resolutions of selfadjoint Toeplitz operators are deduced.
Convex optimization and the epi-distance topology
Authors: Beer Gerald , Lucchetti Roberto
Published: June 2012

Let

Γ
(
X
)

\Gamma (X)

denote the proper, lower semicontinuous, convex functions on a Banach space

X
X

, equipped with the completely metrizable topology

τ
\tau

of uniform convergence of distance functions on bounded sets. A function

f
f

in

Γ
(
X
)

\Gamma (X)

is called well-posed provided it has a unique minimizer, and each minimizing sequence converges to this minimizer. We show that well-posedness of

f
∈
Γ
(
X
)

f \in \Gamma (X)

is the minimal condition that guarantees strong convergence of approximate minima of

τ
\tau

-approximating functions to the minimum of

f
f

. Moreover, we show that most functions in

⟨
Γ
(
X
)
,

τ

a
w

⟩

\langle \Gamma (X),{\tau _{aw}}\rangle

are well-posed, and that this fails if

Γ
(
X
)

\Gamma (X)

is topologized by the weaker topology of Mosco convergence, whenever

X
X

is infinite dimensional. Applications to metric projections are also given, including a fundamental characterization of approximative compactness.
Continuity of translation in the dual of 𝐿^{∞}(𝐺) and related spaces
Authors: Graham Colin C. , Lau Anthony T. M. , Leinert Michael
Published: June 2012

Let

X
X

be a Banach space and

G
G

a locally compact Hausdorff group that acts as a group of isometric linear operators on

X
X

. The operation of

x
∈
G

x \in G

on

X
X

will be denoted by

L
x

{L_x}

. We study the set

X
c

{X_c}

of elements

μ
∈
X

\mu \in X

such that

x
↦

L
x

μ

x \mapsto {L_x}\mu

is continuous with respect to the topology on

G
G

and the norm-topology on

X
X

. The spaces

X
X

studied include
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M left-parenthesis upper G right-parenthesis Superscript asterisk Baseline comma LUC left-parenthesis upper G right-parenthesis Superscript asterisk Baseline comma upper L Superscript normal infinity Baseline left-parenthesis upper G right-parenthesis Superscript asterisk Baseline comma VN left-parenthesis upper G right-parenthesis">

M

(
G

)

∗

,

LUC

(
G

)

∗

,

L
∞

(
G

)

∗

,

VN

(
G
)

M{(G)^{\ast } },{\text {LUC}}{(G)^{\ast } },{L^\infty }{(G)^{\ast } },{\text {VN}}(G)

, and

VN

(
G

)

∗

{\text {VN}}{(G)^{\ast } }

. In most cases, characterizations of

X
c

{X_c}

do not appear to be possible, and we give constructions that illustrate this. We relate properties of

X
c

{X_c}

to properties of

G
G

. For example, if

X
c

{X_c}

is sufficiently small, then

G
G

is compact, or even finite, depending on the case. We give related results and open problems.
Cyclic Galois extensions and normal bases
Authors: Greither C.
Published: June 2012

A Kummer theory is presented which does not need roots of unity in the ground ring. For

R
R

commutative with

p

−
1

∈
R

{p^{ - 1}} \in R

we study the group of cyclic Galois extensions of fixed degree

p
n

{p^n}

in detail. Our theory is well suited for dealing with cyclic

p
n

{p^n}

-extensions of a number field

K
K

which are unramified outside

p
p

. We then consider the group

Gal
⁡
(

O

K

[

p

−
1

]
,

C

p
n

)

\operatorname {Gal}({\mathcal {O}_K}[{p^{ - 1}}],{C_{{p^n}}})

of all such extensions, and its subgroup

NB

(

O

K

[

p

−
1

]
,

C

p
n

)

{\text {NB}}({\mathcal {O}_K}[{p^{ - 1}}],{C_{{p^n}}})

of extensions with integral normal basis outside

p
p

. For the size of the latter we get a simple asymptotic formula

(
n
→
∞
)

(n \to \infty )

, and the discrepancy between the two groups is in some way measured by the defect

δ
\delta

in Leopoldt’s conjecture.
Inequalities for the integral means of holomorphic functions and their derivatives in the unit ball of 𝐂ⁿ
Authors: Shi Ji Huai
Published: June 2012

In this paper, the following two inequalities are proved:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript 1 Baseline left-parenthesis 1 minus r right-parenthesis Superscript a StartAbsoluteValue alpha EndAbsoluteValue plus b Baseline upper M Subscript p Superscript a Baseline left-parenthesis r comma upper D Superscript alpha Baseline f right-parenthesis d r less-than-or-equal-to upper K integral Subscript 0 Superscript 1 Baseline left-parenthesis 1 minus r right-parenthesis Superscript b Baseline upper M Subscript p Superscript a Baseline left-parenthesis r comma f right-parenthesis d r comma integral Subscript 0 Superscript 1 Baseline left-parenthesis 1 minus r right-parenthesis Superscript b Baseline upper M Subscript p Superscript a Baseline left-parenthesis r comma f right-parenthesis d r less-than-or-equal-to upper K left-brace sigma-summation Underscript StartAbsoluteValue alpha EndAbsoluteValue less-than-or-equal-to m minus 1 Endscripts StartAbsoluteValue left-parenthesis upper D Superscript alpha Baseline f right-parenthesis left-parenthesis 0 right-parenthesis EndAbsoluteValue Superscript a Baseline plus sigma-summation Underscript StartAbsoluteValue alpha EndAbsoluteValue equals m Endscripts integral Subscript 0 Superscript 1 Baseline left-parenthesis 1 minus r right-parenthesis Superscript a m plus b Baseline upper M Subscript p Superscript a Baseline left-parenthesis r comma upper D Superscript alpha Baseline f right-parenthesis d r right-brace">

∫
0
1

(
1
−
r
)

a

|

α

|

+
b

M
p
a

(
r
,

D

α

f
)

d
r
≤
K

∫
0
1

(
1
−
r
)

b

M
p
a

(
r
,
f
)

d
r
,

∫
0
1

(
1
−
r
)

b

M
p
a

(
r
,
f
)

d
r

≤
K

{

∑

|

α

|

≤
m
−
1

|

(

D
α

f
)
(
0
)

|

a

+

∑

|

α

|

=
m

∫
0
1

(
1
−
r
)

a
m
+
b

M
p
a

(
r
,

D

α

f
)

d
r
}

\int _0^1 {{(1 - r)}^{a|\alpha | + b}}M_p^a(r,D^{\alpha } f)\,dr \leq K\int _0^1 {{(1 - r)}^b}M_p^a(r,f)\,dr, \int _0^1 {{(1 - r)}^b}M_p^a(r,f)\,dr \\ \leq K\left \{ \sum \limits _{|\alpha | \leq m - 1} |({D^\alpha }f)(0)|^a + \sum \limits _{|\alpha | = m} \int _0^1 {(1 - r)}^{am + b}M_p^a(r,D^{\alpha }f)\,dr \right \}

\]
where

α
=
(

α
1

,
…
,

α
n

)

\alpha = ({\alpha _1}, \ldots ,{\alpha _n})

is multi-index,

0
>
p
>
∞
,
0
>
a
>
∞

0 > p > \infty ,0 > a > \infty

and

−
1
>
b
>
∞

- 1 > b > \infty

. These are a generalization of some classical results of Hardy and Littlewood. Using these two inequalities, we generalize a theorem in

[
9
]

[9]

. The methods used in the proof of Theorem 1 lead us to obtain Theorem 7, which enables us to generalize some theorems about the pluriharmonic conjugates in

[
8
]

[8]

and

[
2
]

[2]

.
Regularity properties of commutators and layer potentials associated to the heat equation
Authors: Lewis John L. , Murray Margaret A. M.
Published: June 2012

In recent years there has been renewed interest in the solution of parabolic boundary value problems by the method of layer potentials. In this paper we consider graph domains

D
=
{
(
x
,
t
)
:
x
>
f
(
t
)
}

D = \{ (x,t):x > f(t)\}

in

R

2

{\mathcal {R}^2}

, where the boundary function

f
f

is in

I

1

/

2

(

BMO

)

{I_{1/2}}({\text {BMO}})

. This class of domains would appear to be the minimal smoothness class for the solvability of the Dirichlet problem for the heat equation by the method of layer potentials. We show that, for

1
>
p
>
∞

1 > p > \infty

, the boundary single-layer potential operator for

D
D

maps

L
p

{L^p}

into the homogeneous Sobolev space

I

1

/

2

(

L
p

)

{I_{1/2}}({L^p})

. This regularity result is obtained by studying the regularity properties of a related family of commutators. Along the way, we prove

L
p

{L^p}

estimates for a class of singular integral operators to which the

T1

{\text {T1}}

Theorem of David and Journé does not apply. The necessary estimates are obtained by a variety of real-variable methods.
Multibasic Eulerian polynomials
Authors: Foata Dominique , Zeilberger Doron
Published: June 2012

Eulerian polynomials with several bases are defined. Their combinatorial interpretations are given as well as congruence properties modulo some ideals generated by cyclotomic polynomials.
Vanishing of 𝐻²_{𝑤}(𝑀,𝐾(𝐻)) for certain finite von Neumann algebras
Authors: Rădulescu Florin
Published: June 2012

We prove the vanishing of the second Hochschild cohomology group

H
w
2

(
M
,
K
(
H
)
)

H_w^2\,(M,K(H))

, whenever

M
⊂
B
(
H
)

M \subset B(H)

is a finite countably decomposable von Neumann algebra not containing a non

Γ
\Gamma

-factor or a factor without Cartan subalgebra as a direct summand. Here

H
H

is a Hubert space, and

K
(
H
)

K(H)

the compact operators.
Terms in the Selberg trace formula for 𝑆𝐿(3,𝒵)\𝒮ℒ(3,ℛ)/𝒮𝒪(3,ℛ) associated to Eisenstein series coming from a minimal parabolic subgroup
Authors: Wallace D. I.
Published: June 2012

In this paper we compute the contribution to the trace formula for

S
L
(
3
,

Z

)

SL(3,\mathcal {Z})

of the integrals associated to inner products of Eisenstein series. We show these reduce to corresponding integrals for a lower rank trace formula plus a few residual terms.
Shadows of convex bodies
Authors: Ball Keith
Published: June 2012

It is proved that if

C
C

is a convex body in

R

n

{\mathbb {R}^n}

then

C
C

has an affine image

C
~

\tilde C

(of nonzero volume) so that if

P
P

is any

1
1

-codimensional orthogonal projection,
\[

|

P

C
~

|

≥

|

C
~

|

(
n
−
1
)

/

n

.

|P\tilde C| \geq \,|\tilde C{|^{(n - 1)\,/\,n}}.

\]
It is also shown that there is a pathological body,

K
K

, all of whose orthogonal projections have volume about

n

\sqrt n

times as large as

|

K

|

(
n
−
1
)

/

n

|K{|^{(n - 1)\,/\,n}}

.
Definable singularity
Authors: Mitchell William J.
Published: June 2012

The main result of this paper is a characterization of singular cardinals in terms of the core model, assuming that there is no model of

∃
κ

o
(
κ
)
=

κ

+
+

\exists \kappa \,o(\kappa )= {\kappa ^{ + + }}

. This characterization is used to prove a result in infinitary Ramsey theory. In the course of the proof we develop a simplified statement of the covering lemma for sequences of measures which avoids the use of mice. We believe that this development will be capable of isolating almost all applications of the covering lemma from the detailed structure of the core model.
Weighted inequalities for maximal functions associated with general measures
Authors: Andersen Kenneth F.
Published: June 2012

For certain positive Borel measures

μ
\mu

on

R

{\mathbf {R}}

and for

T
μ

{T_\mu }

any of three naturally associated maximal function operators of Hardy-Littlewood type, the weight pairs

(
u
,
υ
)

(u,\upsilon )

for which

T
μ

{T_\mu }

is of weak type

(
p
,
p
)
,
1
≤
p
>
∞

(p,p),1 \leq p > \infty

, and of strong type

(
p
,
p
)
,
1
>
p
>
∞

(p,p),1 > p > \infty

, are characterized. Only minimal assumptions are placed on

μ
\mu

; in particular,

μ
\mu

need not satisfy a doubling condition nor need it be continuous.