bluuebunny/crossref_metadata_2025 huggingface dataset:
On the sparsity of representations of rings of pure global dimension zero
Authors: Zimmermann-Huisgen Birge , Zimmermann Wolfgang
Published: June 2012

It is shown that the rings

R
R

all of whose left modules are direct sums of finitely generated modules satisfy the following finiteness condition: For each positive integer

n
n

there are only finitely many isomorphism types of (a) indecomposable left

R
R

-modules of length

n
n

; (b) finitely presented indecomposable right

R
R

-modules of length

n
n

; (c) indecomposable right

R
R

-modules having minimal projective resolutions with

n
n

relations. Moreover, our techniques yield a very elementary proof for the fact that the presence of the above decomposability hypothesis for both left and right

R
R

-modules entails finite representation type.
Discontinuous “viscosity” solutions of a degenerate parabolic equation
Authors: Bertsch Michiel , Dal Passo Roberta , Ughi Maura
Published: June 2012

We study a nonlinear degenerate parabolic equation of the second order. Regularizing the equation by adding some artificial viscosity, we construct a generalized solution. We show that this solution is not necessarily continuous at all points.
Maximal polynomials and the Ilieff-Sendov conjecture
Authors: Miller Michael J.
Published: June 2012

In this paper, we consider those complex polynomials which have all their roots in the unit disk, one fixed root, and all the roots of their first derivatives as far as possible from a fixed point. We conjecture that any such polynomial has all the roots of its derivative on a circle centered at the fixed point, and as many of its own roots as possible on the unit circle. We prove a part of this conjecture, and use it to define an algorithm for constructing some of these polynomials. With this algorithm, we investigate the 1962 conjecture of Sendov and the 1969 conjecture of Goodman, Rahman and Ratti and (independently) Schmeisser, obtaining counterexamples of degrees

6
6

,

8
8

,

10
10

, and

12
12

for the latter.
Endomorphism rings of formal 𝐴₀-modules
Authors: Yamagata Shuji
Published: June 2012

Let

A
0

{A_0}

be the valuation ring of a finite extension

K
0

{K_0}

of

Q
p

{Q_p}

and

A
⊃

A
0

A \supset {A_0}

be a complete discrete valuation ring with the perfect residue field. We consider the endomorphism rings of

n
n

-dimensional formal

A
0

{A_0}

-modules

Γ
\Gamma

over

A
A

of finite

A
0

{A_0}

-height with reduction absolutely simple up to isogeny. Especially we prove commutativity of

End

A
,

A
0

(
Γ
)

{\operatorname {End} _{A,{A_0}}}(\Gamma )

. Given an arbitrary finite unramified extension

K
1

{K_1}

of

K
0

{K_0}

, a variety of examples (different dimensions and different

A
0

{A_0}

-heights) is constructed whose absolute endomorphism rings are isomorphic to the valuation ring of

K
1

{K_1}

.
Comparison of commuting one-parameter groups of isometries
Authors: Bratteli Ola , Kurose Hideki , Robinson Derek W.
Published: June 2012

Let

α
,

β

\alpha ,\;\beta

be two commuting strongly continuous one-parameter groups of isometries on a Banach space

A

\mathcal {A}

with generators

δ
α

{\delta _\alpha }

and

δ
β

{\delta _\beta }

, and analytic elements

A

ω
α

,

A

ω
β

\mathcal {A}_\omega ^\alpha ,\;\mathcal {A}_\omega ^\beta

, respectively. Then it is easy to show that if

δ
α

{\delta _\alpha }

is relatively bounded by

δ
β

{\delta _\beta }

, then

A

ω
β

⊆

A

ω
α

\mathcal {A}_\omega ^\beta \subseteq \mathcal {A}_\omega ^\alpha

, and in this paper we establish the inverse implication for unitary one-parameter groups on Hilbert spaces and for one-parameter groups of

∗

^{\ast }

-automorphisms of abelian

C

∗

{C^{\ast }}

-algebras. It is not known in general whether the inverse implication holds or not, but it does not hold for one-parameter semigroups of contractions.
Baer modules over domains
Authors: Eklof Paul C. , Fuchs László , Shelah Saharon
Published: June 2012

For a commutative domain

R
R

with

1
1

, an

R
R

-module

B
B

is called a Baer module if

Ext
R
1

⁡
(
B
,
T
)
=
0

\operatorname {Ext} _R^1(B,T) = 0

for all torsion

R
R

-modules

T
T

. The structure of Baer modules over arbitrary domains is investigated, and the problem is reduced to the case of countably generated Baer modules. This requires a general version of the singular compactness theorem. As an application we show that over

h
h

-local Prüfer domains, Baer modules are necessarily projective. In addition, we establish an independence result for a weaker version of Baer modules.
Nonmetrizable topological dynamics and Ramsey theory
Authors: Bergelson Vitaly , Hindman Neil
Published: June 2012

Applying ideas from topological dynamics in compact metric spaces to the Stone-Cěch compactification of a discrete semigroup, several new proofs of old results and some new results in Ramsey Theory are obtained. In particular, two ultrafilter proofs of van der Waerden’s Theorem are given. An ultrafilter approach to "central" sets (sets which are combinatorially rich) is developed. This enables us to show that for any partition of the positive integers one cell is both additively and multiplicatively central. Also, a fortuitous answer to a question of Ellis is obtained.
Weakly almost periodic functions and thin sets in discrete groups
Authors: Chou Ching
Published: June 2012

A subset

E
E

of an infinite discrete group

G
G

is called (i) an

R
W

{R_W}

-set if any bounded function on

G
G

supported by

E
E

is weakly almost periodic, (ii) a weak

p
p

-Sidon set

(
1
≤
p
>
2
)

(1 \leq p > 2)

if on

l
1

(
E
)

{l^1}(E)

the

l
p

{l^p}

-norm is bounded by a constant times the maximal

C
∗

{C^*}

-norm of

l
1

(
G
)

{l^1}(G)

, (iii) a

T
T

-set if

x
E
∩
E

xE \cap E

and

E
x
∩
E

Ex \cap E

are finite whenever

x
≠
e

x \ne e

, and (iv) an

F
T

FT

-set if it is a finite union of

T
T

-sets. In this paper, we study relationships among these four classes of thin sets. We show, among other results, that (a) every infinite group

G
G

contains an

R
W

{R_W}

-set which is not an

F
T

FT

-set; (b) countable weak

p
p

-Sidon sets,

1
≤
p
>
4

/

3

1 \leq p > 4/3

are

F
T

FT

-sets.
Weighted inequalities for one-sided maximal functions
Authors: Martín-Reyes F. J. , Ortega Salvador P. , de la Torre A.
Published: June 2012

Let

M
g
+

M_g^ +

be the maximal operator defined by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript g Superscript plus Baseline f left-parenthesis x right-parenthesis equals sup Underscript h greater-than 0 Endscripts left-parenthesis integral Subscript x Superscript x plus h Baseline StartAbsoluteValue f left-parenthesis t right-parenthesis EndAbsoluteValue g left-parenthesis t right-parenthesis d t right-parenthesis left-parenthesis integral Subscript x Superscript x plus h Baseline g left-parenthesis t right-parenthesis d t right-parenthesis Superscript negative 1 Baseline comma">

M
g
+

f
(
x
)
=

sup

h
>
0

(

∫
x

x
+
h

|

f
(
t
)

|

g
(
t
)
d
t

)

(

∫
x

x
+
h

g
(
t
)
d
t

)

−
1

,

M_g^ + f(x) = \sup \limits _{h > 0} \left ( {\int _x^{x + h} {|f(t)|g(t)dt} } \right ){\left ( {\int _x^{x + h} {g(t)dt} } \right )^{ - 1}},

\]
where

g
g

is a positive locally integrable function on

R

{\mathbf {R}}

. We characterize the pairs of nonnegative functions

(
u
,
v
)

(u,v)

for which

M
g
+

M_g^ +

applies

L
p

(
v
)

{L^p}(v)

in

L
p

(
u
)

{L^p}(u)

or in weak-

L
p

(
u
)

{L^p}(u)

. Our results generalize Sawyer’s (case

g
=
1

g = 1

) but our proofs are different and we do not use Hardy’s inequalities, which makes the proofs of the inequalities self-contained.
Time-delay operators in semiclassical limit. II. Short-range potentials
Authors: Wang Xue Ping
Published: June 2012

This work is a continuation of [27]. We prove that quantum time-delay operator localized in a nontrapping energy interval is in fact an

h
h

-pseudodifferential operator with the

h
h

-principal symbol given by the classical time-delay function composed with the incoming wave operator in classical mechanics. The classical limit of time-delay operator is also given.
On the integrability of singular differential forms in two complex variables
Authors: Meziani A.
Published: June 2012

In this work we study the integrability of a germ at

0
∈

C

2

0 \in {{\mathbf {C}}^2}

of a singular differential form for which the closure of the integral curves are analytic varieties that pass through

0
0

. The focii of this paper are the existence of pure meromorphic integrals, the linearization and the nonexistence of a topological criterion for transcendental integrability.
The orderability and closed images of scattered spaces
Authors: Purisch S.
Published: June 2012

A (totally) orderable scattered space and a space homeomorphic to a subspace of an ordinal space are characterized in terms of a neighborhood subbase for each of their points plus what corresponds to a neighborhood base for each of their non-

Q
Q

-gaps. These generalize the characterizations in [ P

1

_{1}

] of an orderable compact scattered space and in [B] of a space homeomorphic to a compact ordinal space. Generalizing a result in [M] it is shown that a space is orderable and scattered iff it is the

2
2

to

1
1

image under a closed map of a subspace of an ordinal space. In response to a question of Telgarsky [T] a simple description is given of a closed map with discrete fibers from an orderable scattered space onto an orderable perfect space. Maps that preserve length conditions on a scattered space are touched upon.
Hyperbolicity properties of 𝐶² multi-modal Collet-Eckmann maps without Schwarzian derivative assumptions
Authors: Nowicki Tomasz , van Strien Sebastian
Published: June 2012

In this paper we study the dynamical properties of general

C
2

{C^2}

maps

f
:
[
0
,
1
]
→
[
0
,
1
]

f:[0,1] \to [0,1]

with quadratic critical points (and not necessarily unimodal). We will show that if such maps satisfy the well-known Collet-Eckmann conditions then one has (a) hyperbolicity on the set of periodic points; (b) nonexistence of wandering intervals; (c) sensitivity on initial conditions; and (d) exponential decay of branches (intervals of monotonicity) of

f
n

{f^n}

as

n
→
∞
;

n \to \infty ;

For these results we will not make any assumptions on the Schwarzian derivative

f
f

. We will also give an estimate of the return-time of points that start near critical points.
Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type
Authors: Garofalo Nicola , Lanconelli Ermanno
Published: June 2012

In this paper we establish a uniform Harnack inequality for a class of degenerate equations whose prototype is Kolmogorov’s equations in

R

3

:

D

yy

u
−

y

D
z

u
−

D
t

u
=
0

{{\mathbf {R}}^3}:{D_{{\text {yy}}}}u - {\text {y}}{D_z}u - {D_t}u = 0

. Our approach is based on mean value formulas for solutions of the equation under consideration on the level sets of the fundamental solution.
Local behavior of solutions of quasilinear elliptic equations with general structure
Authors: Rakotoson J.-M. , Ziemer William P.
Published: June 2012

This paper is motivated by the observation that solutions to certain variational inequalities involving partial differential operators of the form

div
⁡
A
(
x
,
u
,
∇
u
)
+
B
(
x
,
u
,
∇
u
)

\operatorname {div} A(x,u,\nabla u) + B(x,u,\nabla u)

, where

A
A

and

B
B

are Borel measurable, are solutions to the equation

div
⁡
A
(
x
,
u
,
∇
u
)
+
B
(
x
,
u
,
∇
u
)
=
μ

\operatorname {div} A(x,u,\nabla u) + B(x,u,\nabla u) = \mu

for some nonnegative Radon measure

μ
\mu

. Among other things, it is shown that if

u
u

is a Hölder continuous solution to this equation, then the measure

μ
\mu

satisfies the growth property

μ
[
B
(
x
,
r
)
]
⩽
M

r

n
−
p
+
ε

\mu [B(x,r)] \leqslant M{r^{n - p + \varepsilon }}

for all balls

B
(
x
,
r
)

B(x,r)

in

R

n

{{\mathbf {R}}^n}

. Here

ε
\varepsilon

depends on the Hölder exponent of

u
u

while

p
>
1

p > 1

is given by the structure of the differential operator. Conversely, if

μ
\mu

is assumed to satisfy this growth condition, then it is shown that

u
u

satisfies a Harnack-type inequality, thus proving that

u
u

is locally bounded. Under the additional assumption that

A
A

is strongly monotonic, it is shown that

u
u

is Hölder continuous.
Polynomial flows on 𝐶ⁿ
Authors: Coomes Brian A.
Published: June 2012

We show that polynomial flows on

R

n

{\mathbb {R}^n}

extend to functions holomorphic on

C

n
+
1

{\mathbb {C}^{n + 1}}

and that the group property holds after this extension. Then we give some methods, based on power series, for determining when a vector field has a polynomial flow.
Topological entropy of fixed-point free flows
Authors: Thomas Romeo F.
Published: June 2012

Topological entropy was introduced as an invariant of topological conjugacy and also as an analogue of measure theoretic entropy. Topological entropy for one parameter flows on a compact metric spaces is defined by Bowen. General statements are proved about this entropy, but it is not easy to calculate the topological entropy, and to show it is invariant under conjugacy. For all this I would like to try to pose a new direction and study a definition for the topological entropy that involves handling the technical difficulties that arise from allowing reparametrizations of orbits. Some well-known results are proved as well using this definition. These results enable us to prove some results which seem difficult to prove using Bowen’s definition. Also we show here that this definition is equivalent to Bowen’s definition for any flow without fixed points on a compact metric space. Finally, it is shown that the topological entropy of an expansive flow can be defined globally on a local cross sections.
Gel′fer functions, integral means, bounded mean oscillation, and univalency
Authors: Yamashita Shinji
Published: June 2012

A Gelfer function

f
f

is a holomorphic function in

D
=
{

|
z
|

>
1
}

D = \{ \left | z \right | > 1\}

such that

f
(
0
)
=
1

f(0) = 1

and

f
(
z
)
≠
−
f
(
w
)

f(z) \ne - f(w)

for all

z
z

,

w
w

in

D
D

. The family

G
G

of Gelfer functions contains the family

P
P

of holomorphic functions

f
f

in

D
D

with

f
(
0
)
=
1

f(0) = 1

and Re

f
>
0

f > 0

in

D
D

. If

f
f

is holomorphic in

D
D

and if the

L
2

{L^2}

mean of

f
′

f’

on the circle

{

|
z
|

=
r
}

\{ \left | z \right | = r\}

is dominated by that of a function of

G
G

as

r
→
1
−
0

r \to 1 - 0

, then

f
∈
B
M
O
A

f \in BMOA

. This has two recent and seemingly different results as corollaries. A core of the proof is the fact that

log

f
∈
B
M
O
A

{\operatorname {log}}f \in BMOA

if

f
∈
G

f \in G

. Besides the properties obtained concerning

f
∈
G

f \in G

itself, we shall investigate some families of functions where the roles played by

P
P

in Univalent Function Theory are replaced by those of

G
G

. Some exact estimates are obtained.
Addendum to the paper: “Existence of weak solutions for the Navier-Stokes equations with initial data in 𝐿^{𝑝}” [Trans. Amer. Math. Soc. 318 (1990), no. 1, 179–200; MR0968416 (90k:35199)]
Authors: Calderón Calixto P.
Published: June 2012

This paper considers the existence of global weak solutions for the Navier-Stokes equations in the infinite cylinder

R

n

×

R

+

{{\mathbf {R}}^n} \times {{\mathbf {R}}_ + }

with initial data in

L
r

{L^r}

,

n
≥
3

n \geq 3

,

1
>
r
>
∞

1 > r > \infty

. An imbedding theorem as well as related initial value problems are also studied, thus completing results in [2].
Infinitely many co-existing sinks from degenerate homoclinic tangencies
Authors: Davis Gregory J.
Published: June 2012

The evolution of a horseshoe is an interesting and important phenomenon in Dynamical Systems as it represents a change from a nonchaotic state to a state of chaos. As we are interested in determining how this transition takes place, we are studying certain families of diffeomorphisms. We restrict our attention to certain one-parameter families

{

F
t

}

\{ {F_t}\}

of diffeomorphisms in two dimensions. It is assumed that each family has a curve of dissipative periodic saddle points,

P
t

;

F
t
n

(

P
t

)
=

P
t

{P_t};\;F_t^n({P_t}) = {P_t}

, and

|

det
D

F
t
n

(

P
t

)

|

>
1

|\det DF_t^n({P_t})| > 1

. We also require the stable and unstable manifolds of

P
t

{P_t}

to form homoclinic tangencies as the parameter

t
t

varies through

t
0

{t_0}

. Our emphasis is the exploration of the behavior of families of diffeomorphisms for parameter values

t
t

near

t
0

{t_0}

. We show that there are parameter values

t
t

near

t
0

{t_0}

at which

F
t

{F_t}

has infinitely many co-existing periodic sinks.
A characterization of the complemented translation-invariant subspaces of 𝐻¹(𝑅)
Authors: Alspach Dale E.
Published: June 2012

The purpose of this paper is to characterize the complemented translation-invariant subspaces of

H
1

(

R

)

{H^1}({\mathbf {R}})

in terms of the zero set of the Fourier transform. It is shown that if

X
X

is such a subspace then

X
=
I
(
A
)

X = I(A)

where

A
A

is in the ring generated by arithmetic progressions and lacunary sequences and

A
A

is

ε
\varepsilon

-separated for some

ε
>
0

\varepsilon > 0

. This proves a conjecture of the author and D. Ullrich.
Some model theory of compact Lie groups
Authors: Nesin Ali , Pillay Anand
Published: June 2012

We consider questions of first order definability in a compact Lie group

G
G

. Our main result is that if such

G
G

is simple (and centerless) then the Lie group structure of

G
G

is first order definable from the abstract group structure. Along the way we also show (i) if

G
G

is non-Abelian and connected then a copy of the field

R

\mathbb {R}

is interpretable. in

(
G
,
⋅
)

(G, \cdot )

, and (ii) any "

1
1

-dimensional" field interpretable in

(

R

,
+
,
⋅
)

(\mathbb {R}, +, \cdot )

is definably (i.e., semialgebraically) isomorphic to the ground field

R

\mathbb {R}

.
Finite group actions on the moduli space of self-dual connections. I
Authors: Cho Yong Seung
Published: June 2012

Let

M
M

be a smooth simply connected closed

4
4

-manifold with positive definite intersection form. Suppose a finite group

G
G

acts smoothly on

M
M

. Let

π
:
E
→
M

\pi :E \to M

be the instanton number one quaternion line bundle over

M
M

with a smooth

G
G

-action such that

π
\pi

is an equivariant map. We first show that there exists a Baire set in the

G
G

-invariant metrics on

M
M

such that the moduli space

M

∗
G

\mathcal {M}_ * ^G

of

G
G

-invariant irreducible self-dual connections is a manifold. By utilizing the

G
G

-transversality theory of

T
\text {T}

. Petrie, we then identify cohomology obstructions to globally perturbing the full space

M

∗

{\mathcal {M}_ * }

of irreducible self-dual connections to a

G
G

-manifold when

G
=

Z

2

G = {{\mathbf {Z}}_2}

and the fixed point set of the

Z

2

{\mathbf {Z}}_2

action on

M
M

is a nonempty collection of isolated points and Riemann surfaces.
𝑞-tensor space and 𝑞-Weyl modules
Authors: Dipper Richard , James Gordon
Published: June 2012

We obtain the irreducible representations of the

q
q

-Schur algebra, motivated by the fact that these representations give all the irreducible representations of

G

L
n

(
q
)

G{L_n}(q)

in the nondescribing characteristic. The irreducible polynomial representations of the general linear groups in the describing characteristic are a special case of this construction.
On subordinated holomorphic semigroups
Authors: Carasso Alfred S. , Kato Tosio
Published: June 2012

If

[

e

−
t
A

]

[{e^{ - tA}}]

is a uniformly bounded

C
0

{C_0}

semigroup on a complex Banach space

X
X

, then

−

A
α

,

- {A^\alpha },

,

0
>
α
>
1

0 > \alpha > 1

, generates a holomorphic semigroup on

X
X

, and

[

e

−
t

A
α

]

[{e^{ - t{A^\alpha }}}]

is subordinated to

[

e

−
t
A

]

[{e^{ - tA}}]

through the Lévy stable density function. This was proved by Yosida in 1960, by suitably deforming the contour in an inverse Laplace transform representation. Using other methods, we exhibit a large class of probability measures such that the subordinated semigroups are always holomorphic, and obtain a necessary condition on the measure’s Laplace transform for that to be the case. We then construct probability measures that do not have this property.
Chebyshev polynomials in several variables and the radial part of the Laplace-Beltrami operator
Authors: Beerends R. J.
Published: June 2012

Chebyshev polynomials of the first and the second kind in

n
n

variables

z
1

,

z
2

,
…
,

z
n

{z_1},{z_2}, \ldots ,{z_n}

are introduced. The variables

z
1

,

z
2

,
…
,

z
n

{z_1},{z_2}, \ldots ,{z_n}

are the characters of the representations of

S
L
(
n
+
1
,

C

)

SL(n + 1,{\mathbf {C}})

corresponding to the fundamental weights. The Chebyshev polynomials are eigenpolynomials of a second order linear partial differential operator which is in fact the radial part of the Laplace-Beltrami operator on certain symmetric spaces. We give an explicit expression of this operator in the coordinates

z
1

,

z
2

,
…
,

z
n

{z_1},{z_2}, \ldots ,{z_n}

and then show how many results in the literature on differential equations satisfied by Chebyshev polynomials in several variables follow immediately from well-known results on the radial part of the Laplace-Beltrami operator. Related topics like orthogonality, symmetry relations, generating functions and recurrence relations are also discussed. Finally we note that the Chebyshev polynomials are a special case of a more general class of orthogonal polynomials in several variables.
On the range of the Radon 𝑑-plane transform and its dual
Authors: Gonzalez Fulton B.
Published: June 2012

We present direct, group-theoretic proofs of the range theorem for the Radon

d
d

-plane transform

f
→

f
^

f \to \hat f

on

S

(

R

n

)

\mathcal {S}({\mathbb {R}^n})

. (The original proof, by Richter, involves extensive use of local coordinate calculations on

G
(
d
,
n
)

G(d,n)

, the Grassmann manifold of affine

d
d

-planes in

R

n

{\mathbb {R}^n}

.) We show that moment conditions are not sufficient to describe this range when

d
>
n
−
1

d > n - 1

, in contrast to the compactly supported case. Finally, we show that the dual

d
d

-plane transform maps

E

(
G
(
d
,
n
)
)

\mathcal {E}(G(d,n))

surjectively onto

E

(

R

n

)

\mathcal {E}({\mathbb {R}^n})

.
On the existence and uniqueness of positive solutions for competing species models with diffusion
Authors: Dancer E. N.
Published: June 2012

In this paper, we consider strictly positive solutions of competing species systems with diffusion under Dirichlet boundary conditions. We obtain a good understanding of when strictly positive solutions exist, obtain new nonuniqueness results and a number of other results, showing how complicated these equations can be. In particular, we consider how the shape of the underlying domain affects the behaviour of the equations.
On the relative reflexivity of finitely generated modules of operators
Authors: Magajna Bojan
Published: June 2012

Let

R

\mathcal {R}

be a von Neumann algebra on a Hilbert space

H

\mathcal {H}

with commutant

R

′

\mathcal {R}’

and centre

C

\mathcal {C}

. For each subspace

S

\mathcal {S}

of

R

\mathcal {R}

let

ref

R

(

S

)

\operatorname {ref}_\mathcal {R}\,(\mathcal {S})

be the space of all

B
∈

R

B \in \mathcal {R}

such that

X
B
Y
=
0

XBY= 0

for all

X
,
Y
∈

R

X,Y \in \mathcal {R}

satisfying

X

S

Y
=
0

X\,\mathcal {S}\,Y = 0

. If

ref

R

(

S

)
=

S

\operatorname {ref}_\mathcal {R}\,(\mathcal {S})= \mathcal {S}

, the space

S

\mathcal {S}

is called

R

\mathcal {R}

-reflexive. (If

R

=

B

(

H

)

\mathcal {R}= \mathcal {B}(\mathcal {H})

and

S

\mathcal {S}

is an algebra containing the identity operator,

R

\mathcal {R}

-reflexivity reduces to the usual reflexivity in operator theory.) The main result of the paper is the following: if

S

\mathcal {S}

is one-dimensional, or if

S

\mathcal {S}

is arbitrary finite-dimensional but

R

\mathcal {R}

has no central portions of type

I

n

{{\text {I}}_n}

for

n
>
1

n > 1

, then the space

C

S

¯

\overline {\mathcal {C}\mathcal {S}}

is

R

\mathcal {R}

-reflexive and the space

R

′

S

¯

\overline {\mathcal {R}’\,\mathcal {S}}

is

B

(

H

)

\mathcal {B}(\mathcal {H})

-reflexive, where the bar denotes the closure in the ultraweak operator topology. If

R

\mathcal {R}

is a factor, then

R

′

S

\mathcal {R}’\,\mathcal {S}

is closed in the weak operator topology for each finite-dimensional subspace

S

\mathcal {S}

of

R

\mathcal {R}

.
Quadratic models for generic local 3-parameter bifurcations on the plane
Authors: Dumortier Freddy , Fiddelaers Peter
Published: June 2012

The first chapter deals with singularities occurring in quadratic planar vector fields. We make distinction between singularities which as a general system are of finite codimension and singularities which are of infinite codimension in the sense that they are nonisolated, or Hamiltonian, or integrable, or that they have an axis of symmetry after a linear coordinate change or that they can be approximated by centers. In the second chapter we provide quadratic models for all the known versal

k
k

-parameter unfoldings with

k
=
1
,
2
,
3

k = 1,2,3

, except for the nilpotent focus which cannot occur as a quadratic system. We finally show that a certain type of elliptic points of codimension

4
4

does not have a quadratic versal unfolding.
Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term
Authors: García Azorero J. , Peral Alonso I.
Published: June 2012

We study the existence of solutions for the following nonlinear degenerate elliptic problems in a bounded domain

Ω
⊂

R

N

\Omega \subset {{\mathbf {R}}^N}

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus d i v left-parenthesis StartAbsoluteValue nabla u EndAbsoluteValue Superscript p minus 2 Baseline nabla u right-parenthesis equals StartAbsoluteValue u EndAbsoluteValue Superscript p Super Superscript asterisk Superscript minus 2 Baseline u plus lamda StartAbsoluteValue u EndAbsoluteValue Superscript q minus 2 Baseline u comma lamda greater-than 0 comma">

−
div
⁡
(

|

∇
u

|

p
−
2

∇
u
)
=

|

u

|

p

∗

−
2

u
+
λ

|

u

|

q
−
2

u
,

λ
>
0
,

- \operatorname {div} (|\nabla u{|^{p - 2}}\nabla u) = |u{|^{{p^{\ast }} - 2}}u + \lambda |u{|^{q - 2}}u,\qquad \lambda > 0,

\]
where

p

∗

{p^{\ast }}

is the critical Sobolev exponent, and

u

|

δ
Ω

≡
0

u{|_{\delta \Omega }} \equiv 0

. By using critical point methods we obtain the existence of solutions in the following cases: If

p
>
q
>

p

∗

p > q > {p^{\ast }}

, there exists

λ
0

>
0

{\lambda _0} > 0

such that for all

λ
>

λ
0

\lambda > {\lambda _0}

there exists a nontrivial solution. If

max
(
p
,

p

∗

−
p

/

(
p
−
1
)
)
>
q
>

p

∗

\max (p,{p^{\ast }} - p/(p - 1)) > q > {p^{\ast }}

, there exists nontrivial solution for all

λ
>
0

\lambda > 0

. If

1
>
q
>
p

1 > q > p

there exists

λ
1

{\lambda _1}

such that, for

0
>
λ
>

λ
1

0 > \lambda > {\lambda _1}

, there exist infinitely many solutions. Finally, we obtain a multiplicity result in a noncritical problem when the associated functional is not symmetric.
First steps in descriptive theory of locales
Authors: Isbell John
Published: June 2012

F. Hausdorff and D. Montgomery showed that a subspace of a completely metrizable space is developable if and only if it is

F
σ

{F_\sigma }

and

G
δ

{G_\delta }

. This extends to arbitrary metrizable locales when "

F
σ

{F_\sigma }

" and "

G
δ

{G_\delta }

" are taken in the localic sense (countable join of closed, resp. meet of open, sublocales). In any locale, the developable sublocales are exactly the complemented elements of the lattice of sublocales. The main further results of this paper concern the strictly pointless relative theory, which exists because—always in metrizable locales— there exist nonzero pointless-absolute

G
δ

′

s

{G_\delta }’{\text {s}}

,

G
δ

{G_\delta }

in every pointless extension. For instance, the pointless part

pl

(

R

)

{\text {pl}}({\mathbf {R}})

of the real line is characterized as the only nonzero zero-dimensional separable metrizable pointless-absolute

G
δ

{G_\delta }

. There is no nonzero pointless-absolute

F
σ

{F_\sigma }

. The pointless part of any metrizable space is, if not zero, second category, i.e. not a countable join of nowhere dense sublocales.
Homotopie de l’espace des équivalences d’homotopie
Authors: Didierjean Geneviève
Published: June 2012

The spectral sequence of the self-fiber-homotopy-equivalences of a fibration provides a method to compute the homotopy groups of the space of self-equivalences of a space.
The connectivity of multicurves determined by integral weight train tracks
Authors: Haas Andrew , Susskind Perry
Published: June 2012

An integral weighted train track on a surface determines the isotopy class of an embedded closed

1
1

-manifold. We are interested in the connectivity of the resulting

1
1

-manifold. In general there is an algorithm for determining connectivity, and in the simplest case of a

2
2

-parameter train track on a surface of genus one there is an explicit formula. We derive a formula for the connectivity of the closed

1
1

-manifold determined by a

4
4

-parameter train track on a surface of genus two which is computable in polynomial time. We also give necessary and sufficient conditions on the parameters for the resulting

1
1

-manifold to be connected.
Homotopy invariants of nonorientable 4-manifolds
Authors: Kim Myung Ho , Kojima Sadayoshi , Raymond Frank
Published: June 2012

We define a

Z

4

{{\mathbf {Z}}_4}

-quadratic function on

π
2

{\pi _2}

for nonorientable

4
4

-manifolds and show that it is a homotopy invariant. We then use it to distinguish homotopy types of certain manifolds that arose from an analysis of toral action on nonorientable

4
4

-manifolds.
Intersection theory of moduli space of stable 𝑛-pointed curves of genus zero
Authors: Keel Sean
Published: June 2012

We give a new construction of the moduli space via a composition of smooth codimension two blowups and use our construction to determine the Chow ring.
Hessenberg varieties
Authors: De Mari F. , Procesi C. , Shayman M. A.
Published: June 2012

Numerical algorithms involving Hessenberg matrices correspond to dynamical systems which evolve on the subvariety of complete flags

S
1

⊂

S
2

⊂
⋯
⊂

S

n
−
1

{S_1} \subset {S_2} \subset \cdots \subset {S_{n - 1}}

in

C

n

{\mathbb {C}^n}

satisfying the condition

s
(

S
i

)
⊂

S

i
+
1

s({S_i}) \subset {S_{i + 1}}

,

∀
i

\forall i

, where

s
s

is an endomorphism of

C

n

{\mathbb {C}^n}

. This paper describes the basic topological features of the generalization to subvarieties of

G

/

B

G/B

,

G
G

a complex semisimple algebraic group, which are indexed by certain subsets of negative roots. In the special case where the subset consists of the negative simple roots, the variety coincides with the torus embedding associated to the decomposition into Weyl chambers.
Generalized Szegő theorems and asymptotics of cumulants by graphical methods
Authors: Avram Florin
Published: June 2012

We obtain some general asymptotics results about a class of deterministic sums called "sums with dependent indices," which generalize a classical theorem of Szegö. The above type of sums is encountered when establishing convergence to the Gaussian distribution of sums of Wick products by the method of cumulants. Our asymptotic results reduce in this situation the proof of the central limit theorem to the study of the connectivity of a family of associated graphs.
Affine 3-spheres with constant affine curvature
Authors: Magid Martin A. , Ryan Patrick J.
Published: June 2012

We classify the affine hyperspheres in

R
4

{R^4}

which have constant curvature in the affine metric

h
h

and whose Pick invariant is nonzero. In particular, the metric

h
h

must be flat.
A simple proof of the fundamental theorem of Kirby calculus on links
Authors: Lu Ning
Published: June 2012

In this paper, we relate surgeries on links and Heegaard decompositions, relate framed links and surface mapping classes, and give a simple proof of the fundamental theorem of Kirby calculus on links by the presentation of the surface mapping class groups.
Log concave sequences of symmetric functions and analogs of the Jacobi-Trudi determinants
Authors: Sagan Bruce E.
Published: June 2012

We prove that various sequences of elementary and complete homogeneous symmetric functions are log concave or PF. As corollaries we show that certain sequences of

q
q

-binomial coefficients and

q
q

-Stirling numbers have these properties. The principal technique used is a combinatorial interpretation of determinants using lattice paths due to Gessel and Viennot [G-V 85].
Transfinite graphs and electrical networks
Authors: Zemanian A. H.
Published: June 2012

All prior theories of infinite electrical networks assume that such networks are finitely connected, that is, between any two nodes of the network there is a finite path. This work establishes a theory for transfinite electrical networks wherein some nodes are not connected by finite paths but are connected by transfinite paths. Moreover, the voltages at those nodes may influence each other. The main difficulty to surmount for this extension is the construction of an appropriate generalization of the concept of connectedness. This is accomplished by extending the idea of a node to encompass infinite extremities of a graph. The construction appears to be novel and leads to a hierarchy of transfinite graphs indexed by the finite and infinite ordinals. Two equivalent existence and uniqueness theorems are established for transfinite resistive electrical networks based upon Tellegen’s equation, one using currents and the other using voltages as the fundamental quantities. Kirchhoff’s laws do not suffice for this purpose and indeed need not hold everywhere in infinite networks. Although transfinite countable electrical networks have in general an uncountable infinity of extremities, called "tips," the number of different tip voltages may be radically constrained by both the graph of the network and its resistance values. Conditions are established herein under which various tip voltages are compelled to be the same. Furthermore, a theorem of Shannon-Hagelbarger on the concavity of resistance functions is extended to the driving-point resistance between any two extremities of arbitrary ranks. This is based upon an extension of Thomson’s least power principle to transfinite networks.
Entropy for canonical shifts
Authors: Choda Marie
Published: June 2012

For a

∗

^{\ast }

-endomorphism

σ
\sigma

of an injective finite von Neumann algebra

A
A

, we investigate the relations among the entropy

H
(
σ
)

H(\sigma )

for

σ
\sigma

, the relative entropy

H
(
A

|

σ
(
A
)
)

H(A|\sigma (A))

of

σ
(
A
)

\sigma (A)

for

A
A

, the generalized index

λ
(
A
,
σ
(
A
)
)

\lambda (A,\sigma (A))

, and the index for subfactors. As an application, we have the following relations for the canonical shift

Γ
\Gamma

for the inclusion

N
⊂
M

N \subset M

of type

II

1

\text {II}_{1}

factors with the finite index

[
M
:
N
]

[M:N]

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H left-parenthesis upper A vertical-bar normal upper Gamma left-parenthesis upper A right-parenthesis right-parenthesis less-than-or-equal-to 2 upper H left-parenthesis normal upper Gamma right-parenthesis less-than-or-equal-to log lamda left-parenthesis upper A comma normal upper Gamma left-parenthesis upper A right-parenthesis right-parenthesis Superscript negative 1 Baseline equals 2 log left-bracket upper M colon upper N right-bracket comma">

H
(
A

|

Γ
(
A
)
)
≤
2
H
(
Γ
)
≤
log
⁡
λ

(
A
,
Γ
(
A
)

)

−
1

=
2
log
⁡
[
M
:
N
]
,

H(A|\Gamma (A)) \leq 2H(\Gamma ) \leq \log \lambda {(A,\Gamma (A))^{ - 1}} = 2\log [M:N],

\]
where

A
A

is the von Neumann algebra generated by the two of the relative commutants of

M
M

. In the case of that

N
⊂
M

N \subset M

has finite depth, then all of them coincide.
𝐻^{𝑝}- and 𝐿^{𝑝}-variants of multiparameter Calderón-Zygmund theory
Authors: Carbery Anthony , Seeger Andreas
Published: June 2012

We consider Calderón-Zygmund operators on product domains. Under certain weak conditions on the kernel a singular integral operator can be proved to be bounded on

H
p

(

R

×

R

×
⋯
×

R

)
,
0
>
p
≤
1

{H^p}(\mathbb {R} \times \mathbb {R} \times \cdots \times \mathbb {R}), 0 > p \leq 1

, if its behaviour on

L
2

{L^2}

and on certain scalar-valued and vector-valued rectangle atoms is known. Another result concerns an extension of the authors’ results on

L
p

{L^p}

-variants of Calderón-Zygmund theory [1,23] to the product-domain-setting. As an application, one obtains estimates for Fourier multipliers and pseudo-differential operators.
Holomorphic flows in 𝐶³,0 with resonances
Authors: Canille Martins Júlio Cesar
Published: June 2012

The topological classification, by conjugacy, of the germs of holomorphic diffeomorphisms

f
:

C

2

,
0
→

C

2

,
0

f:\,{{\mathbf {C}}^2},0 \to {{\mathbf {C}}^2},0

with

d
f
(
0
)
=
diag
⁡
(

λ
1

,

λ
2

)

df(0) = \operatorname {diag} ({\lambda _1},{\lambda _2})

, where

λ
1

{\lambda _1}

is a root of unity and

|

λ
2

|

≠
1

|{\lambda _2}| \ne 1

is given. This type of diffeomorphism appears as holonomies of singular foliations

F

X

{\mathcal {F}_X}

induced by holomorphic vector fields

X
:

C

3

,
0
→

C

3

,
0

X:{{\mathbf {C}}^3},0 \to {{\mathbf {C}}^3},0

normally hyperbolic and resonant. An explicit example of a such vector field without holomorphic invariant center manifold is presented. We prove that there are no obstructions in the holonomies for

F

X

{\mathcal {F}_X}

to be topologically equivalent to a product type foliation.
On compactly supported spline wavelets and a duality principle
Authors: Chui Charles K. , Wang Jian-zhong
Published: June 2012

Let

⋯
⊂

V

−
1

⊂

V
0

⊂

V
1

⊂
⋯

\cdots \subset {V_{ - 1}} \subset {V_0} \subset {V_1} \subset \cdots

be a multiresolution analysis of

L
2

{L^2}

generated by the

m
m

th order

B
B

-spline

N
m

(
x
)

{N_m}(x)

. In this paper, we exhibit a compactly supported basic wavelet

ψ
m

(
x
)

{\psi _m}(x)

that generates the corresponding orthogonal complementary wavelet subspaces

⋯
,

W

−
1

,

W
0

,

W
1

,
…

\cdots ,{W_{ - 1}},{W_0},{W_1}, \ldots

. Consequently, the two finite sequences that describe the two-scale relations of

N
m

(
x
)

{N_m}(x)

and

ψ
m

(
x
)

{\psi _m}(x)

in terms of

N
m

(
2
x
−
j
)
,
j
∈

Z

{N_m}(2x - j),j \in \mathbb {Z}

, yield an efficient reconstruction algorithm. To give an efficient wavelet decomposition algorithm based on these two finite sequences, we derive a duality principle, which also happens to yield the dual bases

{

N
~

m

(
x
−
j
)
}

\{ {\tilde N_m}(x - j)\}

and

{

ψ
~

m

(
x
−
j
)
}

\{ {\tilde \psi _m}(x - j)\}

, relative to

{

N
m

(
x
−
j
)
}

\{ {N_m}(x - j)\}

and

{

ψ
m

(
x
−
j
)
}

\{ {\psi _m}(x - j)\}

, respectively.
Harnack estimates and extinction profile for weak solutions of certain singular parabolic equations
Authors: DiBenedetto E. , Kwong Y. C.
Published: June 2012

We establish an intrinsic Harnack estimate for nonnegative weak solutions of the singular equation

(
1.1
)

(1.1)

below, for

m
m

in the optimal range

(
(
N
−
2

)
+

/

N
,
1
)

((N - 2)_+/N,1)

. Intrinsic means that, due to the singularity, the space-time dimensions in the parabolic geometry must be rescaled by a factor determined by the solution itself. Consequences are, sharp supestimates on the solutions and decay rates as

t
t

approaches the extinction time. Analogous results are shown for

p
p

-laplacian type equations.
Rational fibrations in differential homological algebra
Authors: Murillo Aniceto
Published: June 2012

In this paper, a result of [6] is generalized as follows: Given a fibration

F
→
E

→

p

B

F \to E\xrightarrow {p}B

of simply connected spaces in which either, the fibre has finite dimensional rational cohomology, or, it has finite dimensional rational homotopy and

ρ
\rho

induces a surjection in rational homotopy, we construct an explicit isomorphism,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row phi colon upper E x t Subscript upper C Sub Superscript asterisk Subscript left-parenthesis upper B comma bold upper Q right-parenthesis Baseline left-parenthesis bold upper Q comma upper C Superscript asterisk Baseline left-parenthesis upper B semicolon bold upper Q right-parenthesis right-parenthesis ModifyingAbove circled-times With caret upper E x t Subscript upper C Sub Superscript asterisk Subscript left-parenthesis upper F semicolon bold upper Q right-parenthesis Baseline left-parenthesis bold upper Q comma upper C Superscript asterisk Baseline left-parenthesis upper F comma bold upper Q right-parenthesis right-parenthesis 2nd Row ModifyingAbove right-arrow With approximately-equals upper E x t Subscript upper C Sub Superscript asterisk Subscript left-parenthesis upper E semicolon bold upper Q right-parenthesis Baseline left-parenthesis upper Q comma upper C Superscript asterisk Baseline left-parenthesis upper E semicolon bold upper Q right-parenthesis right-parenthesis period EndLayout">

φ
:

Ext

C
∗

(
B
,

Q

)

⁡
(

Q

,

C
∗

(
B
;

Q

)
)

⊗
^

Ext

C
∗

(
F
;

Q

)

⁡
(

Q

,

C
∗

(
F
,

Q

)
)

→

≅

Ext

C
∗

(
E
;

Q

)

⁡
(
Q
,

C
∗

(
E
;

Q

)
)
.

\begin {array}{*{20}{c}} {\varphi :\operatorname {Ext}_{{C^\ast }(B,{\mathbf {Q}})}({\mathbf {Q}},{C^\ast }(B;{\mathbf {Q}}))\hat \otimes \operatorname {Ext}_{{C^\ast }(F;{\mathbf {Q}})}({\mathbf {Q}},{C^\ast }(F,{\mathbf {Q}}))} \\ {\xrightarrow { \cong }\operatorname {Ext}_{{C^\ast }(E;{\mathbf {Q}})}(Q,{C^\ast }(E;{\mathbf {Q}})).} \\ \end {array}

\]
This is deduced from its "algebraic translation," a more general result in the environment of graded differential homological algebra.
Tensor products and Grothendieck type inequalities of operators in 𝐿_{𝑝}-spaces
Authors: Carl Bernd , Defant Andreas
Published: June 2012

Several results in the theory of

(
p
,
q
)

(p,q)

-summing operators are improved by a unified but elementary tensor product concept.
Noetherian ring extensions with trace conditions
Authors: Warfield Robert B.
Published: June 2012

Finite ring extensions of Noetherian rings with certain restrictions on the corresponding trace ideals are studied. This setting includes finite free extensions and extensions arising from actions of finite groups when the order of the group is invertible. In this setting we establish the following results which were previously obtained (for finite extensions without trace conditions) only under strong restrictions on the rings involved. Let

R
⊂
S

R \subset S

be an extension of Noetherian rings such that

S
S

is finitely generated as a left

R
R

-module and such that the left trace ideal of

S
S

in

R
R

is equal to

R
R

. If

S
S

is right fully bounded, or is a Jacobson ring, then

R
R

has the same property; furthermore,

R
R

and

S
S

have the same classical Krull dimension. If

S
S

is finitely generated as both a right and a left

R
R

-module, if both trace ideals of

S
S

in

R
R

are equal to

R
R

, and if

S
S

satisfies the strong second layer condition, then this condition also holds in

R
R

. Finally, we compare the link graphs of

R
R

and

S
S
An inverse problem for circle packing and conformal mapping
Authors: Carter Ithiel , Rodin Burt
Published: June 2012

Figure 1.1 illustrates the fact that if a region is almost packed with circles of radius

ε
\varepsilon

in the hexagonal pattern and if the unit disk is packed in an isomorphic pattern with circles of varying radii then, after suitable normalization, the correspondence of circles converges to the Riemann mapping function as

ε
→
0

\varepsilon \to 0

(see [15]). In the present paper an inverse of this result is obtained as illustrated by Figure 1.2; namely, if the unit disk is almost packed with

ε
\varepsilon

-circles there is an isomorphic circle packing almost filling the region such that, after suitable normalization, the circle correspondence converges to the conformal map of the disk onto the region as

ε
→
0

\varepsilon \to 0

. Note that this set up yields an approximate triangulation of the region by joining the centers of triples of mutually tangent circles. Since this triangulation is intimately related to the Riemann mapping it may be useful for grid generation [18].
A Fatou theorem for the solution of the heat equation at the corner points of a cylinder
Authors: Hui Kin Ming
Published: June 2012

In this paper the author proves existence and uniqueness of the initial-Dirichlet problem for the heat equation in a cylindrical domain

D
×
(
0
,
∞
)

D \times (0,\infty )

where

D
D

is a bounded smooth domain in

R
n

{R^n}

with zero lateral values. A unique representation of the strong solution is given in terms of measures

μ
\mu

on

D
D

and

λ
\lambda

on

∂
D

\partial D

. We also show that the strong solution

u
(
x
,
t
)

u(x,t)

of the heat equation in a cylinder converges a.e.

x
0

∈
∂
D
×
{
0
}

{x_0} \in \partial D \times \{ 0\}

as

(
x
,
t
)

(x,t)

converges to points on

∂
D
×
{
0
}

\partial D \times \{ 0\}

along certain nontangential paths.
Band sums of links which yield composite links. The cabling conjecture for strongly invertible knots
Authors: Eudave Muñoz Mario
Published: June 2012

We consider composite links obtained by bandings of another link. It is shown that if a banding of a split link yields a composite knot then there is a decomposing sphere crossing the band in one arc, unless there is such a sphere disjoint from the band. We also prove that if a banding of the trivial knot yields a composite knot or link then there is a decomposing sphere crossing the band in one arc. The last theorem implies, via double branched covers, that the only way we can get a reducible manifold by surgery on a strongly invertible knot is when the knot is cabled and the surgery is via the slope of the cabling annulus.
The monotonicity of the entropy for a family of degree one circle maps
Authors: Alsedà Lluís , Mañosas Francesc
Published: June 2012

For the natural biparametric family of piecewise linear circle maps with two pieces we show that the entropy increases when any of the two parameters increases. We also describe the regions of the parameter space where the monotonicity is strict.
Cohen-Macaulayness of semi-invariants for tori
Authors: Van den Bergh Michel
Published: June 2012

In this paper we give a new method, in terms of one-parameter subgroups, to study semi-invariants for algebraic tori. In some cases we obtain extensions to results in [7]. In other cases we obtain different proofs.
Remarks on prescribing Gauss curvature
Authors: Xu Xingwang , Yang Paul C.
Published: June 2012

We study the nonlinear partial differential equation for the problem of prescribing Gauss curvature

K
K

on

S
2

{S^2}

. We give an example of a rotationally symmetric

K
K

for which the Kazdan-Warner obstruction is satisfied but the equation has no rotationally symmetric solution. On the other hand, we give a simple sufficient condition for solvability of the equation when

K
K

is rotationally symmetric. Finally we give a sufficient condition for solvability when

K
K

is not necessarily rotationally symmetric.