bluuebunny/crossref_metadata_2025 huggingface dataset:
Hyperbolic invariants of knots and links
Authors: Adams Colin , Hildebrand Martin , Weeks Jeffrey
Published: June 2012

Tables of values for the hyperbolic volume, number of symmetries, cusp volume and conformal invariants of the cusps are given for hyperbolic knots through ten crossings and hyperbolic links of

2
,
3

2, 3

and

4
4

components through

9
9

crossings. The horoball patterns and the canonical triangulations are displayed for knots through eight crossings and for particularly interesting additional examples of knots and links.
Resonance and the second BVP
Authors: Shapiro Victor L.
Published: June 2012

Let

Ω
⊂

R

N

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

be a bounded open connected set with the cone property, and let

1
>
p
>
∞

1 > p > \infty

. Also, let

Q
u

Qu

be the

2
m

2m

th order quasilinear differential operator in generalized divergence form:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q u equals sigma-summation Underscript 1 less-than-or-equal-to StartAbsoluteValue alpha EndAbsoluteValue less-than-or-equal-to m Endscripts left-parenthesis negative 1 right-parenthesis Superscript StartAbsoluteValue alpha EndAbsoluteValue Baseline upper D Superscript alpha Baseline upper A Subscript alpha Baseline left-parenthesis x comma xi Subscript m Baseline left-parenthesis u right-parenthesis right-parenthesis comma">

Q
u
=

∑

1
≤

|

α

|

≤
m

(
−
1
)

|

α

|

D
α

A
α

(
x
,

ξ
m

(
u
)
)

,

Qu = \sum \limits _{1 \leq |\alpha | \leq m} {{{(- 1)}^{|\alpha |}}{D^\alpha }{A_\alpha }(x,{\xi _m}(u))},

\]
where for

u
∈

W

m
,
p

u \in {W^{m,p}}

,

ξ
m

(
u
)
=
{

D
α

u
:

|

α

|

≤
m
}

{\xi _m}(u) = \{ {D^\alpha }u:|\alpha | \leq m\}

. (For

m
=
1

m = 1

,

Q
u
=
−

∑

i
=
1

N

A
i

(
x
,
u
,
D
u
)

Qu = - \sum \nolimits _{i = 1}^N {{A_i}(x,u,Du)}

.) Under four assumptions on

A
α

{A_\alpha }

—Carathéodory, growth, monotonicity for

|

α

|

=
m

|\alpha | = m

, and ellipticity—results at resonance are established for the equation

Q
u
=
G
+
f
(
x
,
u
)

Qu = G + f(x,u)

, where

G
∈

[

W

m
,
p

(
Ω
)

]
∗

G \in {[{W^{m,p}}(\Omega )]^\ast }

and

f
(
x
,
u
)

f(x,u)

satisfies a one-sided condition (plus others). For the case

m
=
1

m = 1

, these results are tantamount to generalized solutions of the second

BVP
\text {BVP}

.
Utility functions which ensure the adequacy of stationary strategies
Authors: Monticino Michael G.
Published: June 2012

Within a Dubins and Savage gambling framework, a stationary strategy is a strategy which selects a gamble at each time based solely on the gambler’s present fortune. We determine conditions upon the gambler’s utility function under which stationary strategies allow the gambler to maximize his return. The class of utility functions which satisfies these conditions, termed nearly leavable shift invariant functions, is large and contains many of the common gambling utility functions. Moreover, this class is closed under uniform limits. These results are obtained with the setting of an analytic gambling house.
Blow-up of straightening-closed ideals in ordinal Hodge algebras
Authors: Bruns Winfried , Simis Aron , Trung Ngô Viêt
Published: June 2012

We study a class of ideals

I
I

in graded ordinal Hodge algebras

A
A

. These ideals are distinguished by the fact that their powers have a canonical standard basis. This leads to Hodge algebra structures on the Rees ring and the associated graded ring. Furthermore, from a natural standard filtration one obtains a depth bound for

A

/

I
n

A/{I^n}

which, under certain conditions, is sharp for

n
n

large. Frequently one observes that

I
n

=

I

(
n
)

{I^n}= {I^{(n)}}

. Under suitable hypotheses it is possible to calculate the divisor class group of the Rees algebra. Our main examples are ideals of "virtual" maximal minors and ideals of maximal minors "fixing a submatrix".
Recursive linear orders with incomplete successivities
Authors: Downey Rodney G. , Moses Michael F.
Published: June 2012

A recursive linear order is said to have intrinsically complete successivities if, in every recursive copy, the successivities form a complete set. We show (Theorem 1) that there is a recursive linear order with intrinsically complete successivities but (Theorem 2) that this cannot be a discrete linear oder. We investigate the related issues of intrinsically non-low and non-semilow successivities in discrete linear orders. We show also (Theorem 3) that no recursive linear order has intrinsically

w
t
t

wtt

-complete successivities.
On the convergence of moment problems
Authors: Borwein J. M. , Lewis A. S.
Published: June 2012

We study the problem of estimating a nonnegative density, given a finite number of moments. Such problems arise in numerous practical applications. As the number of moments increases, the estimates will always converge weak

∗

^\ast

as measures, but need not converge weakly in

L
1

{L_1}

. This is related to the existence of functions on a compact metric space which are not essentially Riemann integrable (in some suitable sense). We characterize the type of weak convergence we can expect in terms of Riemann integrability, and in some cases give error bounds. When the estimates are chosen to minimize an objective function with weakly compact level sets (such as the Bolzmann-Shannon entropy) they will converge weakly in

L
1

{L_1}

. When an

L
p

{L_p}

norm

(
1
>
p
>
∞
)

(1 > p > \infty )

is used as the objective, the estimates actually converge in norm. These results provide theoretical support to the growing popularity of such methods in practice.
Almost tangent and cotangent structures in the large
Authors: Thompson G. , Schwardmann U.
Published: June 2012

We examine some global properties of integrable almost tangent and cotangent manifolds. In particular, we extend several results which essentially characterize tangent and cotangent bundles as, respectively, regular almost tangent and cotangent structures.
How porous is the graph of Brownian motion?
Authors: Cox J. T. , Griffin Philip S.
Published: June 2012

We prove that the graph of Brownian motion is almost surely porous, and determine the Hausdorff dimension of sets with a given porosity index. In particular we show that the porosity index of the graph is

γ
0

≐
0.6948

{\gamma _0} \doteq 0.6948

.
The generalized Dowling lattices
Authors: Hanlon Phil
Published: June 2012

In this paper we study a new class of lattices called the generalized Dowling lattices. These lattices are parametrized by a positive integer

n
n

, a finite group

G
G

, and a meet sublattice

K
K

of the lattice of subgroups of

G
G

. For an appropriate choice of

K
K

the generalized Dowling lattice

D
n

(
G
,
K
)

{D_n}(G,K)

agrees with the ordinary Dowling lattice

D
n

(
G
)

{D_n}(G)

. For a different choice of

K
K

, the generalized Dowling lattices are the lattice of intersections of a set of subspaces in complex space. The set of subspaces, defined in terms of a representation of

G
G

, generalizes the thick diagonal in

C

n

{\mathbb {C}^n}

. We compute the Möbius function and characteristic polynomial of the lattice

D
n

(
G
,
K
)

{D_n}(G,K)

along with the homology of

D
n

(
G
,
K
)

{D_n}(G,K)

in terms of the homology of

K
K

. We go on to compute the character of

G
G

wr

S
n

{S_n}

acting on the homology of

D
n

(
G
,
K
)

{D_n}(G,K)

. This computation provides a nontrivial generalization of a result due to Stanley concerning the character of

S
n

{S_n}

acting on the top homology of the partition lattice.
Complete coinductive theories. II
Authors: Lachlan A. H.
Published: June 2012

Let

T
T

be a complete theory over a relational language which has an axiomatization by

∃
∀

\exists \forall

-sentences. The properties of models of

T
T

are studied. It is shown that existential formulas are stable. A theory of forking and independence based on Boolean combinations of existential formulas in

∃
∀

\exists \forall

-saturated models of

T
T

is developed for which the independence relation is shown to satisfy a very strong triviality condition. It follows that

T
T

is tree-decomposable in the sense of Baldwin and Shelah. It is also shown that if the language is finite, then

T
T

has a prime model.
Adapted sets of measures and invariant functionals on 𝐿^{𝑝}(𝐺)
Authors: Nillsen Rodney
Published: June 2012

Let

G
G

be a locally compact group. If

G
G

is compact, let

L
0
p

(
G
)

L_0^p(G)

denote the functions in

L
p

(
G
)

{L^p}(G)

having zero Haar integral. Let

M
1

(
G
)

{M^1}(G)

denote the probability measures on

G
G

and let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P Superscript 1 Baseline left-parenthesis upper G right-parenthesis equals upper M Superscript 1 Baseline left-parenthesis upper G right-parenthesis intersection upper L Superscript 1 Baseline left-parenthesis upper G right-parenthesis">

P

1

(
G
)
=

M
1

(
G
)
∩

L
1

(
G
)

{\mathcal {P}^1}(G) = {M^1}(G) \cap {L^1}(G)

. If

S
⊆

M
1

(
G
)

S \subseteq {M^1}(G)

, let

Δ
(

L
p

(
G
)
,
S
)

\Delta ({L^p}(G),S)

denote the subspace of

L
p

(
G
)

{L^p}(G)

generated by functions of the form

f
−
μ
∗
f

f - \mu \ast f

,

f
∈

L
p

(
G
)

f \in {L^p}(G)

,

μ
∈
S

\mu \in S

. If

G
G

is compact,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta left-parenthesis upper L Superscript p Baseline left-parenthesis upper G right-parenthesis comma upper S right-parenthesis subset-of-or-equal-to upper L 0 Superscript p Baseline left-parenthesis upper G right-parenthesis">

Δ
(

L
p

(
G
)
,
S
)
⊆

L
0
p

(
G
)

\Delta ({L^p}(G),S) \subseteq L_0^p(G)

. When

G
G

is compact, conditions are given on

S
S

which ensure that for some finite subset

F
F

of

S
S

,

Δ
(

L
p

(
G
)
,
F
)
=

L
0
p

(
G
)

\Delta ({L^p}(G),F) = L_0^p(G)

for all

1
>
p
>
∞

1 > p > \infty

. The finite subset

F
F

will then have the property that every

F
F

-invariant linear functional on

L
p

(
G
)

{L^p}(G)

is a multiple of Haar measure. Some results of a contrary nature are presented for noncompact groups. For example, if

1
≤
p
≤
∞

1 \leq p \leq \infty

, conditions are given upon

G
G

, and upon subsets

S
S

of

M
1

(
G
)

{M^1}(G)

whose elements satisfy certain growth conditions, which ensure that

L
p

(
G
)

{L^p}(G)

has discontinuous,

S
S

-invariant linear functionals. The results are applied to show that for

1
≤
p
≤
∞

1 \leq p \leq \infty

,

L
p

(

R

)

{L^p}(\mathbb {R})

has an infinite, independent family of discontinuous translation invariant functionals which are not

P

1

(

R

)

{\mathcal {P}^1}(\mathbb {R})

-invariant.
Heegaard diagrams of 3-manifolds
Authors: Ochiai Mitsuyuki
Published: June 2012

For a

3
3

-manifold

M
(
L
)

M(L)

obtained by an integral Dehn surgery along an

n
n

-bridge link

L
L

with

n
n

-components we define a concept of planar Heegaard diagrams of

M
(
L
)

M(L)

using a link diagram of

L
L

. Then by using Homma-Ochiai-Takahashi’s theorem and a planar Heegaard diagram of

M
(
L
)

M(L)

we will completely determine if

M
(
L
)

M(L)

is the standard

3
3

-sphere in the case when

L
L

is a

2
2

-bridge link with

2
2

-components.
Complex representations of matrix semigroups
Authors: Okniński Jan , Putcha Mohan S.
Published: June 2012

Let

M
M

be a finite monoid of Lie type (these are the finite analogues of linear algebraic monoids) with group of units

G
G

. The multiplicative semigroup

M

n

(
F
)

{\mathcal {M}_n}(F)

, where

F
F

is a finite field, is a particular example. Using Harish-Chandra’s theory of cuspidal representations of finite groups of Lie type, we show that every complex representation of

M
M

is completely reducible. Using this we characterize the representations of

G
G

extending to irreducible representations of

M
M

as being those induced from the irreducible representations of certain parabolic subgroups of

G
G

. We go on to show that if

F
F

is any field and

S
S

any multiplicative subsemigroup of

M

n

(
F
)

{\mathcal {M}_n}(F)

, then the semigroup algebra of

S
S

over any field of characteristic zero has nilpotent Jacobson radical. If

S
=

M

n

(
F
)

S = {\mathcal {M}_n}(F)

, then this algebra is Jacobson semisimple. Finally we show that the semigroup algebra of

M

n

(
F
)

{\mathcal {M}_n}(F)

over a field of characteristic zero is regular if and only if

ch
⁡
(
F
)
=
p
>
0

\operatorname {ch} (F) = p > 0

and

F
F

is algebraic over its prime field.
An 𝐿²-cohomology construction of unitary highest weight modules for 𝑈(𝑝,𝑞)
Authors: Mantini Lisa A.
Published: June 2012

In this paper a geometric construction is given of all unitary highest weight modules of

G
=
U
⁡
(
p
,
q
)

G = \operatorname {U} (p,q)

. The construction is based on the unitary model of the

k
k

th tensor power of the metaplectic representation in a Bargmann-Segal-Fock space of square-integrable differential forms. The representations are constructed as holomorphic sections of certain vector bundles over

G

/

K

G/K

, and the construction is implemented via an integral transform analogous to the Penrose transform of mathematical physics.
Reflected Brownian motion in a cone with radially homogeneous reflection field
Authors: Kwon Y. , Williams R. J.
Published: June 2012

This work is concerned with the existence and uniqueness of a strong Markov process that has continuous sample paths and the following additional properties. (i) The state space is a cone in

d
d

-dimensions

(
d
≥
3
)

(d \geq 3)

, and the process behaves in the interior of the cone like ordinary Brownian motion. (ii) The process reflects instantaneously at the boundary of the cone, the direction of reflection being fixed on each radial line emanating from the vertex of the cone. (iii) The amount of time that the process spends at the vertex of the cone is zero (i.e., the set of times for which the process is at the vertex has zero Lebesgue measure). The question of existence and uniqueness is cast in precise mathematical terms as a submartingale problem in the style used by Stroock and Varadhan for diffusions on smooth domains with smooth boundary conditions. The question is resolved in terms of a real parameter

α
\alpha

which in general depends in a rather complicated way on the geometric data of the problem, i.e., on the cone and the directions of reflection. However, a criterion is given for determining whether

α
>
0

\alpha > 0

. It is shown that there is a unique continuous strong Markov process satisfying (i)-(iii) above if and only if

α
>
2

\alpha > 2

, and that starting away from the vertex, this process does not reach the vertex if

α
≤
0

\alpha \leq 0

and does reach the vertex almost surely if

0
>
α
>
2

0 > \alpha > 2

. If

α
≥
2

\alpha \geq 2

, there is a unique continuous strong Markov process satisfying (i) and (ii) above; it reaches the vertex of the cone almost surely and remains there. These results are illustrated in concrete terms for some special cases. The process considered here serves as a model for comparison with a reflected Brownian motion in a cone having a nonradially homogeneous reflection field. This is discussed in a subsequent work by Kwon.
Harmonic measure versus Hausdorff measures on repellers for holomorphic maps
Authors: Zdunik Anna
Published: June 2012

This paper is a continuation of a joint paper of the author with F. Przytycki and M. Urbański. We study a harmonic measure on a boundary of so-called repelling boundary domain; an important example is a basin of a sink for a rational map. Using the results of the above-mentioned paper we prove that either the boundary of the domain is an analytically embedded circle or interval, or else the harmonic measure is singular with respect to the Hausdorff measure corresponding to the function

ϕ
c

(
t
)
=
t

exp

(
c

log
⁡

1
t

log

log

log

1
t

)

{\phi _c}(t)= t\;\exp \,\left (c\sqrt {\log \frac {1} {t}\,\log \,\log \,\log } \frac {1} {t}\right )

for some

c
>
0

c > 0

.
On the homology of 𝑆𝑈(𝑛) instantons
Authors: Boyer Charles P. , Mann Benjamin M. , Waggoner Daniel
Published: June 2012

In this paper we study the homology of the moduli spaces of instantons associated to principal

S
U

(
n
)

{\mathbf {SU}}(n)

bundles over the four-sphere. This is accomplished by exploiting an "iterated loop space" structure implicit in the disjoint union of all moduli spaces associated to a fixed

S
U

(
n
)

{\mathbf {SU}}(n)

with arbitrary instanton number and relating these spaces to the known homology structure of the four-fold loop space on

B

S
U

(
n
)

B{\mathbf {SU}}(n)

.
An equivariant torus theorem for involutions
Authors: Holzmann W. H.
Published: June 2012

A complete classification is given for equivariant surgery on incompressible tori with respect to involutions with possible

1
1

- or

2
2

-dimensional fixed sets.
The adjoint arc in nonsmooth optimization
Authors: Loewen Philip D. , Rockafellar R. T.
Published: June 2012

We extend the theory of necessary conditions for nonsmooth problems of Bolza in three ways: first, we incorporate state constraints of the intrinsic type

x
(
t
)
∈
X
(
t
)

x(t) \in X(t)

for all

t
t

; second, we make no assumption of calmness or normality; and third, we show that a single adjoint function of bounded variation simultaneously satisfies the Hamiltonian inclusion, the Euler-Lagrange inclusion, and the Weierstrass-Pontryagin maximum condition, along with the usual transversality relations.
Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations
Authors: Franchi Bruno
Published: June 2012

In this paper we prove a Sobolev-Poincaré inequality for a class of function spaces associated with some degenerate elliptic equations. These estimates provide us with the basic tool to prove an invariant Harnack inequality for weak positive solutions. In addition, Hölder regularity of the weak solutions follows in a standard way.
Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface
Authors: Thomassen Carsten
Published: June 2012

We describe all regular tilings of the torus and the Klein bottle. We apply this to describe, for each orientable (respectively nonorientable) surface

S
S

, all (but finitely many) vertex-transitive graphs which can be drawn on

S
S

but not on any surface of smaller genus (respectively crosscap number). In particular, we prove the conjecture of Babai that, for each

g
⩾
3

g \geqslant 3

, there are only finitely many vertex-transitive graphs of genus

g
g

. In fact, they all have order

>

10

10

g

> {10^{10}}g

. The weaker conjecture for Cayley graphs was made by Gross and Tucker and extends Hurwitz’ theorem that, for each

g
⩾
2

g \geqslant 2

, there are only finitely many groups that act on the surface of genus

g
g

. We also derive a nonorientable version of Hurwitz’ theorem.
Flowbox manifolds
Authors: Aarts J. M. , Oversteegen L. G.
Published: June 2012

A separable and metrizable space

X
X

is called a flowbox manifold if there exists a base for the open sets each of whose elements has a product structure with the reals

Re
\operatorname {Re}

as a factor such that a natural consistency condition is met. We show how flowbox manifolds can be divided into orientable and nonorientable ones. We prove that a space

X
X

is an orientable flowbox manifold if and only if

X
X

can be endowed with the structure of a flow without restpoints. In this way we generalize Whitney’s theory of regular families of curves so as to include self-entwined curves in general separable metric spaces. All spaces under consideration are separable and metrizable.
Isotopy invariants of graphs
Authors: Jonish D. , Millett K. C.
Published: June 2012

The development of oriented and semioriented algebraic invariants associated to a class of embeddings of regular four valent graphs is given. These generalize the analogous invariants for classical knots and links, can be determined from them by means of a weighted averaging process, and define them by means of a new state model. This development includes the elucidation of the elementary spatial equivalences (generalizations of the classical Reidemeister moves), and the extension of fundamental concepts in classical knot theory, such as the linking number, to this class spatial graphs.
The structure of open continuous mappings having two valences
Authors: Lyzzaik A. K. , Stephenson Kenneth
Published: June 2012

The authors study open continuous functions which map the unit disc to compact Riemann surfaces and which assume each value in the range space (with a finite number of exceptions) either

p
p

or

q
q

times for some positive integers

p
p

,

q
q

. Although the questions here originated in efforts to understand mapping properties of locally univalent analytic functions, the authors remove analyticity assumptions and show that the underlying issues are topological and combinatoric in nature. The mappings are studied by embedding their image surfaces in compact covering spaces, a setting which allows the consideration of fairly general ranges and which accommodates branch and exceptional points. Known results are generalized and extended; several open questions are posed, particularly regarding the higher dimensional analogues of the results.
Computing the topological entropy of general one-dimensional maps
Authors: Góra P. , Boyarsky A.
Published: June 2012

A matrix-theoretic method for computing the topological entropy of continuous, piecewise monotonic maps of the interval is presented. The method results in a constructive procedure which is easily implemented on the computer. Examples for families of unimodal, nonunimodal and discontinuous maps are presented.
Separating points from closed convex sets over ordered fields and a metric for 𝑅̃ⁿ
Authors: Robson Robert O.
Published: June 2012

Let

R
R

be an arbitrary ordered field, let

R
¯

\bar R

be a real closure, and let

R
~

\tilde R

and

R
~

n

{\tilde R^n}

denote the real spectra of

R
¯

[
X
]

\bar R[X]

and

R
¯

[

X
1

,
…
,

X
n

]

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

. We prove that a closed convex subset in

R
n

{R^n}

may be separated from a point not in it via a continuous "linear" functional taking values in

R
~

\tilde R

and that there is a

R
~

\tilde R

-valued metric on

R
~

n

{\tilde R^n}

. The methods rely on the ultrafilter interpretation of points in

R
~

n

{\tilde R^n}

and on the existence of suprema and infima of sets in

R
~

\tilde R

.
Polynomial solutions to constant coefficient differential equations
Authors: Smith S. Paul
Published: June 2012

Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D 1 comma ellipsis comma upper D Subscript r Baseline element-of double-struck upper C left-bracket partial-differential slash partial-differential x 1 comma ellipsis comma partial-differential slash partial-differential x Subscript n Baseline right-bracket">

D
1

,
…
,

D
r

∈

C

[
∂

/

∂

x
1

,
…
,
∂

/

∂

x
n

]

{D_1}, \ldots ,{D_r} \in \mathbb {C}[\partial /\partial {x_1}, \ldots ,\partial /\partial {x_n}]

be constant coefficient differential operators with zero constant term. Let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S equals StartSet f element-of double-struck upper C left-bracket x 1 comma ellipsis comma x Subscript n Baseline right-bracket vertical-bar upper D Subscript j Baseline left-parenthesis f right-parenthesis equals 0 for all 1 less-than-or-slanted-equals j less-than-or-slanted-equals r EndSet">

S
=
{
f
∈

C

[

x
1

,
…
,

x
n

]

|

D
j

(
f
)
=
0

for all 

1
⩽
j
⩽
r
}

S = \{ f \in \mathbb {C}[{x_1}, \ldots ,{x_n}]|{D_j}(f) = 0\;{\text {for all }}1 \leqslant j \leqslant r\}

\]
be the space of polynomial solutions to the system of simultaneous differential equations

D
j

(
f
)
=
0

{D_j}(f) = 0

. It is proved that

S
S

is a module over

D

(
V
)

\mathcal {D}(V)

, the ring of differential operators on the affine scheme

V
V

with coordinate ring
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C left-bracket partial-differential slash partial-differential x 1 comma ellipsis comma partial-differential slash partial-differential x Subscript n Baseline right-bracket slash mathematical left-angle upper D 1 comma ellipsis comma upper D Subscript r Baseline mathematical right-angle">

C

[
∂

/

∂

x
1

,
…
,
∂

/

∂

x
n

]

/

⟨

D
1

,
…
,

D
r

⟩

\mathbb {C}[\partial /\partial {x_1}, \ldots ,\partial /\partial {x_n}]/\left \langle {{D_1}, \ldots ,{D_r}} \right \rangle

. If

V
V

is smooth and irreducible, then

S
S

is a simple

D

(
V
)

\mathcal {D}(V)

-module,

S
=
1.

D

(
V
)

S = 1.\mathcal {D}(V)

, and the generators for

D

(
V
)

\mathcal {D}(V)

yield an algorithm for obtaining a basis for

S
S

. If

V
V

is singular, then

S
S

need not be simple. However,

S
S

is still a simple

D

(
V
)

\mathcal {D}(V)

-module for certain curves

V
V

, and certain homogeneous spaces

V
V

, and this allows one to obtain a basis for

S
S

, through knowledge of

D

(
V
)

\mathcal {D}(V)

.
Prinjective modules, reflection functors, quadratic forms, and Auslander-Reiten sequences
Authors: de la Peña J. A. , Simson D.
Published: June 2012

Let

A
,

B

A,\;B

be artinian rings and let

A

M
B

_A{M_B}

be an

(
A
−
B
)

(A - B)

-bimodule which is a finitely generated left

A
A

-module and a finitely generated right

B
B

-module. A right

A

M
B

_A{M_B}

-prinjective module is a finitely generated module
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Subscript upper R Baseline equals left-parenthesis upper X prime Subscript upper A Baseline comma upper X Subscript upper B Baseline comma phi colon upper X prime Subscript upper A Baseline circled-times Subscript upper A Baseline upper M Subscript upper B Baseline right-arrow upper X Subscript upper B Baseline right-parenthesis">

X
R

=
(

X
A
′

,

X
B

,
φ
:

X
A
′

⊗
A

M
B

→

X
B

)

{X_R} = (X_A’, X_B, \varphi :X_A’ \otimes _A M_B \to X_B)

over the triangular matrix ring
\[

R
=

(

A

a
m
p
;

A

M
B

0

a
m
p
;
B

)

R = \left ( {\begin {array}{*{20}{c}} A & {_A{M_B}} \\ 0 & B \\ \end {array} } \right )

\]
such that

X
A
′

X_A’

is a projective

A
A

-module,

X
B

X_B

is an injective

B
B

-module, and

φ
\varphi

is a

B
B

-homomorphism. We study the category

prin
⁡
(
R

)
B
A

\operatorname {prin} (R)_B^A

of right

A

M
B

_A{M_B}

-prinjective modules. It is an additive Krull-Schmidt subcategory of

mod
(
R
)

\bmod (R)

closed under extensions. For every

X
,

Y

X,\;Y

in

prin
⁡
(
R

)
B
A

,

Ext
R
2

⁡
(
X
,

Y
)
=
0

\operatorname {prin} (R)_B^A,\;\operatorname {Ext} _R^2(X,\,Y) = 0

. When

R
R

is an Artin algebra, the category

prin
⁡
(
R

)
B
A

\operatorname {prin} (R)_B^A

has Auslander-Reiten sequences and they can be computed in terms of reflection functors. In the case that

R
R

is an algebra over an algebraically closed field we give conditions for

prin
⁡
(
R

)
B
A

\operatorname {prin} (R)_B^A

to be representation-finite or representation-tame in terms of a Tits form. In some cases we calculate the coordinates of the Auslander-Reiten translation of a module using a Coxeter linear transformation.
Finite codimensional subalgebras of Stein algebras and semiglobally Stein algebras
Authors: Khoái Hà Huy , Khuê Nguyen Văn
Published: June 2012

The following theorem is proved: For each finite codimensional subalgebra

A
A

of a Stein algebra

B
B

there exists a natural number

n
n

such that

B
B

is algebraically isomorphic to

A
⊕

C

n

A \oplus {{\mathbf {C}}^n}

.
An intrinsic description of shape
Authors: Sanjurjo José M. R.
Published: June 2012

We give in this paper a description of the shape category of compacta in terms of multivalued maps. We introduce the notion of a multi-net and prove that the shape category of compacta is isomorphic to the category

H
N

HN

whose objects are metric compacta and whose morphisms are homotopy classes of multi-nets. This description is intrinsic in the sense that it does not make use of external elements such as ANR-expansions or embeddings in appropriate AR-spaces. We present many applications of this new formulation of shape.
Equivalence of families of functions on the natural numbers
Authors: Laflamme Claude
Published: June 2012

We present some consequences of the inequality

u

>

g

\mathfrak {u} > \mathfrak {g}

among cardinal invariants of the continuum, which has previously been shown to be consistent relative to ZFC. We are interested in its effect on two orderings of families of functions on the natural numbers; in particular we show that, under

u

>

g

\mathfrak {u} > \mathfrak {g}

, there are exactly five equivalence classes for both orderings (excluding the families bounded by a fixed constant function). This implies, under the same hypothesis, the existence of exactly four classes of rarefaction of measure zero sets.
Théorie de Sullivan pour la cohomologie à coefficients locaux
Authors: Gómez-Tato Antonio
Published: June 2012

The classical moment map of symplectic geometry is used to canonically associate to a unitary representation of a Lie group

G
G

a

G
G

-invariant subset of the dual of the Lie algebra. This correspondence is in some sense dual to geometric quantization. The nature and convexity of this subset is investigated for

G
G

compact semisimple.
Lattice-ordered groups whose lattices determine their additions
Authors: Conrad Paul F. , Darnel Michael R.
Published: June 2012

In this paper it is shown that several large and important classes of lattice-ordered groups, including the free abelian lattice-ordered groups, have their group operations completely determined by the underlying lattices, or determined up to

l
l

-isomorphism.
The “Defektsatz” for central simple algebras
Authors: Gräter Joachim
Published: June 2012

LetQQbe a central simple algebra finite-dimensional over its centerFFand letVVbe a valuation ring ofFF. ThenVVhas an extension toQQ, i.e., there exists a Dubrovin valuation ringBBofQQsatisfyingV=F∩BV= F \cap B. Generally, the number of extensions ofVVtoQQis not finite and therefore the so-called intersection property of Dubrovin valuation ringsB1,…,Bn{B_1}, \ldots ,{B_n}is introduced. This property is defined in terms of the prime ideals and the valuation overrings of the intersectionB1∩⋯∩Bn{B_1} \cap \cdots \; \cap {B_n}. It is shown that there exists a uniquely determined natural numbernndepending only onVVand having the following property: IfB1,…,Bk{B_1}, \ldots ,{B_k}are extensions ofVVhaving the intersection property thenk≤nk \leq nandk=nk= nholds if and only ifB1∩⋯∩Bk{B_1} \cap \cdots \cap {B_k}is integral overVV. Letnnbe the extension number ofVVtoQQ. There exist extensionsB1,⋯,Bn{B_1}, \cdots ,{B_n}ofVVhaving the intersection property and ifR1,…,Rn{R_1}, \ldots ,{R_n}are also extensions ofVVhaving the intersection property thenB1∩⋯∩Bn{B_1} \cap \cdots \cap {B_n}andR1∩⋯∩Rn{R_1} \cap \cdots \cap {R_n}are conjugate. The main result regarding the extension number is the Defektsatz:<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper Q colon upper F right-bracket equals f Subscript upper B Baseline left-parenthesis upper Q slash upper F right-parenthesis e Subscript upper B Baseline left-parenthesis upper Q slash upper F right-parenthesis n squared p Superscript d">[Q:F]=fB(Q/F)eB(Q/F)n2pd[Q:F]= {f_B}(Q/F){e_B}(Q/F){n^2}{p^d}, wherefB(Q/F){f_B}(Q/F)is the residue degree,eB(Q/F){e_B}(Q/F)the ramification index,nnthe extension number,p=char⁡(V/J(V))p = \operatorname {char}(V/J(V)), anddda natural number.
The complete integral closure of 𝑅[𝑋]
Authors: Lucas Thomas G.
Published: June 2012

For a reduced ring

R
R

that is completely integrally closed it is not always the case that the corresponding polynomial ring

R
[
X
]

R[X]

is completely integrally closed. In this paper the question of when

R
[
X
]

R[X]

is completely integrally closed is shown to be related to the question of when

R
R

is completely integrally closed in

T
(
R
[
X
]
)

T(R[X])

the total quotient ring of

R
[
X
]

R[X]

. A characterization of the complete integral closure of

R
[
X
]

R[X]

is given in the main theorem and this result is used to characterize the complete integral closure of the semigroup ring

R
[
S
]

R[S]

when

S
S

is a torsion-free cancellative monoid.
The Gauss map for Kählerian submanifolds of 𝑅ⁿ
Authors: Rigoli Marco , Tribuzy Renato
Published: June 2012

We introduce a Gauss map for Kähler submanifolds of Euclidean space and study its geometry in relation to that of the given immersion. In particular we generalize a number of results of the classical theory of minimal surfaces in Euclidean space.
A global Łojasiewicz inequality for algebraic varieties
Authors: Ji Shanyu , Kollár János , Shiffman Bernard
Published: June 2012

Let

X
X

be the locus of common zeros of polynomials

f
1

,
…
,

f
k

{f_1}, \ldots ,{f_k}

in

n
n

complex variables. A global upper bound for the distance to

X
X

is given in the form of a Lojasiewicz inequality. The exponent in this inequality is bounded by

d

min
(
n
,
k
)

{d^{\min (n,k)}}

where

d
=
max
(
3
,
deg
⁡

f
i

)

d = \max (3,\deg {f_i})

. The estimates are also valid over an algebraically closed field of any characteristic.
On explosions of solutions to a system of partial differential equations modelling chemotaxis
Authors: Jäger W. , Luckhaus S.
Published: June 2012

A system of partial differential equations modelling chemotactic aggregation is analysed (Keller-Segel model). Conditions on the system of parameters are given implying global existence of smooth solutions. In two space dimensions and radially symmetric situations, explosion of the bacteria concentration in finite time is shown for a class of initial values.
𝐼𝐼₁ factors, their bimodules and hypergroups
Authors: Sunder V. S.
Published: June 2012

In this paper, we introduce a notion that we call a hypergroup; this notion captures the natural algebraic structure possessed by the set of equivalence classes of irreducible bifinite bimodules over a

II

1

\text {II}_{1}

factor. After developing some basic facts concerning bimodules over

II

1

\text {II}_{1}

factors, we discuss abstract hypergroups. To make contact with the problem of what numbers can arise as index-values of subfactors of a given

II

1

\text {II}_{1}

factor with trivial relative commutant, we define the notion of a dimension function on a hypergroup, and prove that every finite hypergroup admits a unique dimension function, we then give some nontrivial examples of hypergroups, some of which are related to the Jones subfactors of index

4

cos
2

π

/

(
2
n
+
1
)

4{\cos ^2}\pi /(2n + 1)

. In the last section, we study the hypergroup invariant corresponding to a bifinite module, which is used, among other things, to obtain a transparent proof of a strengthened version of what Ocneanu terms ’the crossed-product remembering the group.’
Estimates of the Caccioppoli-Schauder type in weighted function spaces
Authors: Troianiello Giovanni Maria
Published: June 2012

We deal with imbeddings of certain weighted function spaces as well as with the corresponding norm estimates for solutions to second order elliptic problems. We redemonstrate some results of Gilbarg and Hörmander by a technique, entirely different from theirs, which enables us to cover a range of parameters excluded by them.
Free 𝛼-extensions of an Archimedean vector lattice and their topological duals
Authors: Macula Anthony J.
Published: June 2012

Arch denotes the category of Archimedean vector lattices with vector lattice homomorphisms, and

α
\alpha

denotes an uncountable cardinal number or the symbol

∞
\infty

.

Arch
⁡
(
α
)

\operatorname {Arch}(\alpha )

denotes the category of Arch objects with

α
\alpha

-complete Arch morphisms.
On the Künneth formula for intersection cohomology
Authors: Cohen Daniel C. , Goresky Mark , Ji Lizhen
Published: June 2012

We find the natural perversity functions for which intersection cohomology satisfies the Künneth formula.
Convolution and hypergroup structures associated with a class of Sturm-Liouville systems
Authors: Connett William C. , Markett Clemens , Schwartz Alan L.
Published: June 2012

Product formulas of the type
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript k Baseline left-parenthesis theta right-parenthesis u Subscript k Baseline left-parenthesis phi right-parenthesis equals integral Subscript 0 Superscript pi Baseline u Subscript k Baseline left-parenthesis xi right-parenthesis upper D left-parenthesis xi comma theta comma phi right-parenthesis d xi">

u
k

(
θ
)

u
k

(
ϕ
)
=

∫
0
π

u
k

(
ξ
)
D
(

ξ
,
θ
,
ϕ
)

d
ξ

{u_k}(\theta ){u_k}(\phi ) = \int _0^\pi {{u_k}(\xi )D(} \xi ,\theta ,\phi )\;d\xi

\]
are obtained for the eigenfunctions of a class of second order regular and regular singular Sturm-Liouville problems on

[
0
,
π
]

[0,\pi ]

by using the Riemann integration method to solve a Cauchy problem for an associated hyperbolic differential equation. When

D
(
ξ
,
θ
,
ϕ
)

D(\xi ,\theta ,\phi )

is nonnegative (which can be guaranteed by a simple restriction on the differential operator of the Sturm-Liouville problem), it is possible to define a convolution with respect to which

M
[
0
,
π
]

M[0,\pi ]

becomes a Banach algebra with the functions

u
k

(
ξ
)

/

u
0

(
ξ
)

{u_k}(\xi )/{u_0}(\xi )

as its characters. In fact this measure algebra is a Jacobi type hypergroup. It is possible to completely describe the maximal ideal space and idempotents of this measure algebra.
Isometries of CSL algebras
Authors: Solel Baruch
Published: June 2012

We show that every Jordan isomorphism of

CSL
\text {CSL}

algebras, whose restriction to the diagonal of the algebra is a selfadjoint map, is the sum of an isomorphism and an anti-isomorphism. It follows that every surjective linear isometry of

CSL
\text {CSL}

algebras is the sum of an isomorphism and an anti-isomorphism, followed by a unitary multiplication.
Hadamard matrices and their designs: a coding-theoretic approach
Authors: Assmus E. F. , Key J. D.
Published: June 2012

To every finite dimensional algebraic coefficient system (defined below)

(
Θ
,
V
)

(\Theta ,V)

over the De Rham algebra

Ω
(
M
)

\Omega (M)

of a manifold

M
M

, Sullivan builds a local system

ρ
Θ

:

π
1

(
M
)
→
V

{\rho _\Theta }:{\pi _1}(M) \to V

, in the topological sense, such that the two cohomologies

H

ρ
Θ

∗

(
M
;
V
)

H_{{\rho _\Theta }}^{\ast }(M;V)

and

H
Θ

∗

(
Ω
(
M
)
;
V
)

H_\Theta ^{\ast }(\Omega (M);V)

are isomorphic. In this paper, if

K

{\mathbf {K}}

is a simplicial set and

(
Θ
,
V
)

(\Theta ,V)

an algebraic system over the

C
∞

{C^\infty }

forms

A
∞

(

K

)

{A_\infty }({\mathbf {K}})

, we prove a similar result. We use it to extend the Hirsch lemma to the case of fibration whose fiber is an Eilenberg-Mac Lane space with certain non nilpotent action of the fundamental group of the basis. We apply this to a model of the hyperbolic torus; different from the nilpotent one, this new model is a better mirror of the topology.
Global regularity on 3-dimensional solvmanifolds
Authors: Cygan Jacek M. , Richardson Leonard F.
Published: June 2012

Let

M
M

be any

3
3

-dimensional (nonabelian) compact solvmanifold. We apply the methods of representation theory to study the convergence of Fourier series of smooth global solutions to first order invariant partial differential equations

D
f
=
g

Df = g

in

C
∞

(
M
)

{C^\infty }(M)

. We show that smooth infinite-dimensional irreducible solutions, when they exist, satisfy estimates strong enough to guarantee uniform convergence of the irreducible (or primary) Fourier series to a smooth global solution.
On the Θ-function of a Riemannian manifold with boundary
Authors: Hsu Pei
Published: June 2012

Let

Ω
\Omega

be a compact Riemannian manifold of dimension

n
n

with smooth boundary. Let

λ
1

>

λ
2

≤
⋯

{\lambda _1} > {\lambda _2} \leq \cdots

be the eigenvalues of the Laplace-Beltrami operator with the boundary condition

[
∂

/

∂
n
+
γ
]
ϕ
=
0

[\partial /\partial n + \gamma ]\phi = 0

. The associated

Θ
\Theta

-function

Θ
γ

(
t
)
=

∑

n
=
1

∞

exp
⁡
[
−

λ
n

t
]

{\Theta _\gamma }(t) = \sum \nolimits _{n = 1}^\infty {\exp [ - {\lambda _n}t]}

has an asymptotic expansion of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 4 pi t right-parenthesis Superscript n slash 2 Baseline normal upper Theta Subscript gamma Baseline left-parenthesis t right-parenthesis equals a 0 plus a 1 t Superscript 1 slash 2 Baseline plus a 2 t plus a 3 t Superscript 3 slash 2 Baseline plus a 4 t squared plus midline-horizontal-ellipsis period">

(
4
π
t

)

n

/

2

Θ
γ

(
t
)
=

a
0

+

a
1

t

1

/

2

+

a
2

t
+

a
3

t

3

/

2

+

a
4

t
2

+
⋯
.

{(4\pi t)^{n/2}}{\Theta _\gamma }(t) = {a_0} + {a_1}{t^{1/2}} + {a_2}t + {a_3}{t^{3/2}} + {a_4}{t^2} + \cdots .

\]
The values of

a
0

{a_0}

,

a
1

{a_1}

are well known. We compute the coefficients

a
2

{a_2}

and

a
3

{a_3}

in terms of geometric invariants associated with the manifold by studying the parametrix expansion of the heat kernel

p
(
t
,
x
,
y
)

p(t,x,y)

near the boundary. Our method is a significant refinement and improvement of the method used in [McKean-Singer, J. Differential Geometry 1 (1969), 43-69].
Bounded analytic functions on two sheeted discs
Authors: Hayashi Mikihiro , Nakai Mitsuru , Segawa Shigeo
Published: June 2012

Results of qualitative nature of both positive and negative directions on the point separation by bounded analytic functions of smooth subregions of two sheeted discs are given when two sheeted discs themselves are not separated by bounded analytic functions. We are, in particular, concerned about roles of branch points in two sheeted discs played in the point separation by bounded analytic functions.
Radii of convergence and index for 𝑝-adic differential operators
Authors: Young Paul Thomas
Published: June 2012

We study the radii of

p
p

-adic convergence of solutions at a generic point of homogeneous linear differential operators whose coefficients are analytic elements. As an application we prove a conjecture of P. Robba (for a certain class of operators) concerning the relation between radii of convergence and index on analytic elements. We also give an explicit factorization theorem for

p
p

-adic differential operators, based on the radii of generic convergence and the slopes of the associated Newton polygon.
Sharp estimate of the Laplacian of a polyharmonic function and applications
Authors: Kounchev Ognyan Iv.
Published: June 2012

The classical sharp inequality of Markov estimates the values of the derivative of the polynomial of degree

n
n

in the interval

[
a
,
b
]

[a,b]

through the uniform norm of the polynomial in the same interval multiplied by

2

n
2

/

(
b
−
a
)

2{n^2}/(b - a)

. In the present paper we provide an exact estimate for the values of the Laplacian of a polyharmonic function of degree

m
m

by the uniform norm of the polyharmonic function multiplied by

2

(
m
−
1

)
2

/

R
2

(
x
)

2{(m - 1)^2}/{R^2}(x)

where

R
(
x
)

R(x)

is the distance from the point

x
x

to the boundary of the domain. The inequality of Markov (and the similar inequality of Bernstein about trigonometric polynomials) finds many applications in approximation theory for functions of one variable. We prove analogues to some of these results in the multivariate case.
Expansions of chromatic polynomials and log-concavity
Authors: Brenti Francesco
Published: June 2012

In this paper we present several results and open problems about logconcavity properties of sequences associated with graph colorings. Five polynomials intimately related to the chromatic polynomial of a graph are introduced and their zeros, combinatorial and log-concavity properties are studied. Four of these polynomials have never been considered before in the literature and some yield new expansions for the chromatic polynomial.
Boundary tangential convergence on spaces of homogeneous type
Authors: Cifuentes Patricio , Dorronsoro José R. , Sueiro Juan
Published: June 2012

We study tangential convergence of convolutions with approximate identities of functions defined on a homogeneous type space and having a certain regularity. Our results contain those already known for the Euclidean case and give new ones for stratified nilpotent Lie groups and for solutions of the Dirichlet problem on Lipschitz domains.
Normal form and linearization for quasiperiodic systems
Authors: Chow Shui-Nee , Lu Kening , Shen Yun Qiu
Published: June 2012

In this paper, we consider the following system of differential equations:
\[

θ
˙

=
ω
+
Θ
(
θ
,
z
)
,

z
˙

=
A
z
+
f
(
θ
,
z
)
,

\dot \theta = \omega + \Theta (\theta ,z), \quad \dot z = Az + f(\theta ,z),

\]
where

θ
∈

C
m

\theta \in {C^m}

,

ω
=
(

ω
1

,
…
,

ω
m

)
∈

R
m

\omega = ({\omega _1}, \ldots ,{\omega _m}) \in {R^m}

,

z
∈

C
n

z \in {C^n}

,

A
A

is a diagonalizable matrix,

f
f

and

Θ
\Theta

are analytic functions in both variables and

2
π

2\pi

-periodic in each component of the vector

θ
,
Θ
=
O
(

|

z

|

)

\theta ,\Theta = O(|z|)

and

f
=
O
(

|

z

|

2

)

f = O(|z{|^2})

as

z
→
0

z \to 0

. We study the normal form of this system of the equations and prove that this system can be transformed to a system of linear equations
\[

θ
˙

=
ω
,

z
˙

=
A
z

\dot \theta = \omega , \quad \dot z = Az

\]
by an analytic transformation provided that the eigenvalues of

A
A

and the frequency

ω
\omega

satisfy certain small-divisor conditions.
On the structure of twisted group 𝐶*-algebras
Authors: Packer Judith A. , Raeburn Iain
Published: June 2012

We first give general structural results for the twisted group algebras

C

∗

(
G
,
σ
)

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

of a locally compact group

G
G

with large abelian subgroups. In particular, we use a theorem of Williams to realise

C

∗

(
G
,
σ
)

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

as the sections of a

C

∗

{C^{\ast }}

-bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras. Next we use a device of Rosenberg to show that, when

Γ
\Gamma

is a discrete subgroup of a solvable Lie group

G
G

, the

K
K

-groups

K

∗

(

C

∗

(
Γ
,
σ
)
)

{K_ {\ast } }({C^{\ast } }(\Gamma ,\sigma ))

are isomorphic to certain twisted

K
K

-groups

K

∗

(
G

/

Γ
,
δ
(
σ
)
)

{K^{\ast } }(G/\Gamma ,\delta (\sigma ))

of the homogeneous space

G

/

Γ

G/\Gamma

, and we discuss how the twisting class

δ
(
σ
)
∈

H
3

(
G

/

Γ
,

Z

)

\delta (\sigma ) \in {H^3}(G/\Gamma ,\mathbb {Z})

depends on the cocycle

σ
\sigma

. For many particular groups, such as

Z

n

{\mathbb {Z}^n}

or the integer Heisenberg group,

δ
(
σ
)

\delta (\sigma )

always vanishes, so that

K

∗

(

C

∗

(
Γ
,
σ
)
)

{K_ {\ast } }({C^{\ast } }(\Gamma ,\sigma ))

is independent of

σ
\sigma

, but a detailed analysis of examples of the form

Z

n

⋊

Z

{\mathbb {Z}^n} \rtimes \mathbb {Z}

shows this is not in general the case.
Excluding subdivisions of infinite cliques
Authors: Robertson Neil , Seymour P. D. , Thomas Robin
Published: June 2012

For every infinite cardinalkkwe characterize graphs not containing a subdivision ofKk{K_k}.