bluuebunny/crossref_metadata_2025 huggingface dataset:
Nanoporosity induced by ion implantation in deposited amorphous Ge thin films
Authors: Romano L. , Impellizzeri G. , Bosco L. , Ruffino F. , Miritello M. , Grimaldi M. G.
Published: June 2012

The formation of a nano-porous structure in amorphous Ge thin film (sputter-deposited on SiO2) during ion irradiation at room temperature with 300 keV Ge+ has been observed. The porous film showed a sponge-like structure substantially different from the columnar structure reported for ion implanted bulk Ge. The voids size and structure resulted to be strongly affected by the material preparation, while the volume expansion turned out to be determined only by the nuclear deposition energy. In SiGe alloys, the swelling occurs only if the Ge concentration is above 90%. These findings rely on peculiar characteristics related to the mechanism of voids nucleation and growth, but they are crucial for future applications of active nanostructured layers such as low cost chemical and biochemical sensing devices or electrodes in batteries.
A deep acceptor defect responsible for the yellow luminescence in GaN and AlGaN
Authors: Kamyczek P. , Placzek-Popko E. , Kolkovsky Vl. , Grzanka S. , Czernecki R.
Published: June 2012

In the present study, the electrical and optical properties of deep defects in p-i-n GaN junction and AlGaN/GaN heterojunction are investigated by means of the deep level transient spectroscopy (DLTS), Laplace DLTS, and electroluminescence (EL) techniques. We demonstrate that in both structures the yellow luminescence (YL) is a dominant band in the EL spectra recorded at room temperature. We correlate the YL band with the minority DLTS peaks observed at about 370 K. A gallium vacancy-related defect seems to be a probable candidate as to the origin of the defect. Another dominant majority peak observed in the DLTS studies was concluded to be linked with a donor-like defect in the upper half of the bandgap. The origin of the defect is discussed.
Evidences of electrochemical graphene functionalization and substrate dependence by Raman and scanning tunneling spectroscopies
Authors: Daniels Kevin M. , Daas B. K. , Srivastava N. , Williams C. , Feenstra R. M. , Sudarshan T. S. , Chandrashekhar M. V. S.
Published: June 2012

Electrochemical functionalization and possible hydrogenation of treated epitaxial graphene samples on 6H-SiC are presented. To attract H+ ions to react with the exposed working cathode, a 10% sulfuric acid electrolyte was used with a Pt counter anode. Functionalization was determined using Raman spectroscopy and measured by a marked increase in I(D)/I(G) ratio and introduction of C-H bond peak at ∼2930 cm−1. There was also a marked increase in fluorescence background, which clearly differentiates functionalization from lattice damage in the graphene. Quantifying the fluorescence, we estimate that H-incorporation as high as 50% was achieved based on results on hydrocarbons, although other functional groups cannot be excluded. We further distinguished these functionalization signatures from lattice damage through measurements on nanocrystalline graphene on a and m plane SiC, which displayed very different surface morphologies and no measureable fluorescence. Finally, we show that the extent of functionalization is strongly substrate dependent by using samples cut from three semi-insulating 6H-SiC substrates with similar resistivity but orientations varying from on-axis (∼0.02°), 0.5° to 1.0° off-axis. This functionalization was found to be thermally reversible at ∼1000 °C. Scanning tunneling spectroscopy indicates the presence of sp3-like localized states not present in the starting graphene, further supporting the assertion that functionalization has occurred.
Spleen serves as a reservoir of osteoclast precursors through vitamin D-induced IL-34 expression in osteopetrotic
op/op
mice
Authors: Nakamichi Yuko , Mizoguchi Toshihide , Arai Atsushi , Kobayashi Yasuhiro , Sato Masahiro , Penninger Josef M. , Yasuda Hisataka , Kato Shigeaki , DeLuca Hector F. , Suda Tatsuo , Udagawa Nobuyuki , Takahashi Naoyuki
Published: June 2012

Osteoclasts are generated from monocyte/macrophage-lineage precursors in response to colony-stimulating factor 1 (CSF-1) and receptor activator of nuclear factor-κB ligand (RANKL). CSF-1–mutated

CSF-1
op/op

mice as well as

RANKL
−/−

mice exhibit osteopetrosis (OP) caused by osteoclast deficiency. We previously identified RANKL receptor (RANK)/CSF-1 receptor (CSF-1R) double-positive cells as osteoclast precursors (OCPs), which existed in bone in

RANKL
−/−

mice. Here we show that OCPs do not exist in bone but in spleen in

CSF-1
op/op

mice, and spleen acts as their reservoir. IL-34, a newly discovered CSF-1R ligand, was highly expressed in vascular endothelial cells in spleen in

CSF-1
op/op

mice. Vascular endothelial cells in bone also expressed IL-34, but its expression level was much lower than in spleen, suggesting a role of IL-34 in the splenic generation of OCPs. Splenectomy (SPX) blocked CSF-1–induced osteoclastogenesis in

CSF-1
op/op

mice. Osteoclasts appeared in aged

CSF-1
op/op

mice with up-regulation of IL-34 expression in spleen and bone. Splenectomy blocked the age-associated appearance of osteoclasts. The injection of 2-methylene-19-nor-(20
S
)-1α,25(OH)
2
D
3
(2MD), a potent analog of 1α,25-dihidroxyvitamin D
3
, into

CSF-1
op/op

mice induced both hypercalcemia and osteoclastogenesis. Administration of 2MD enhanced IL-34 expression not only in spleen but also in bone through a vitamin D receptor-mediated mechanism. Either splenectomy or siRNA-mediated knockdown of IL-34 suppressed 2MD-induced osteoclastogenesis. These results suggest that IL-34 plays a pivotal role in maintaining the splenic reservoir of OCPs, which are transferred to bone in response to diverse stimuli, in

CSF-1
op/op

mice. The present study also suggests that the IL-34 gene in vascular endothelial cells is a unique target of vitamin D.
Hölder estimates for degenerate elliptic equations with coercive Hamiltonians
Authors: Capuzzo Dolcetta I. , Leoni F. , Porretta A.
Published: June 2012

We prove a priori estimates and regularity results for some quasilinear degenerate elliptic equations arising in optimal stochastic control problems. Our main results show that strong coerciveness of gradient terms forces bounded viscosity subsolutions to be globally Hölder continuous, and solutions to be locally Lipschitz continuous. We also give an existence result for the associated Dirichlet problem.
The word and geodesic problems in free solvable groups
Authors: Myasnikov A. , Roman’kov V. , Ushakov A. , Vershik A.
Published: June 2012

We study the computational complexity of the Word Problem (WP) in free solvable groups

S

r
,
d

S_{r,d}

, where

r
≥
2

r \geq 2

is the rank and

d
≥
2

d \geq 2

is the solvability class of the group. It is known that the Magnus embedding of

S

r
,
d

S_{r,d}

into matrices provides a polynomial time decision algorithm for WP in a fixed group

S

r
,
d

S_{r,d}

. Unfortunately, the degree of the polynomial grows together with

d
d

, so the uniform algorithm is not polynomial in

d
d

. In this paper we show that WP has time complexity

O
(
r
n

log
2

⁡
n
)

O(r n \log _2 n)

in

S

r
,
2

S_{r,2}

, and

O
(

n
3

r
d
)

O(n^3 r d)

in

S

r
,
d

S_{r,d}

for

d
≥
3

d \geq 3

. However, it turns out, that a seemingly close problem of computing the geodesic length of elements in

S

r
,
2

S_{r,2}

is

N
P

NP

-complete. We prove also that one can compute Fox derivatives of elements from

S

r
,
d

S_{r,d}

in time

O
(

n
3

r
d
)

O(n^3 r d)

; in particular, one can use efficiently the Magnus embedding in computations with free solvable groups. Our approach is based on such classical tools as the Magnus embedding and Fox calculus, as well as on relatively new geometric ideas; in particular, we establish a direct link between Fox derivatives and geometric flows on Cayley graphs.
Corestrictions of algebras and splitting fields
Authors: Krashen Daniel
Published: June 2012

Given a field

F
F

, an étale extension

L

/

F

L/F

and an Azumaya algebra

A

/

L

A/L

, one knows that there are extensions

E

/

F

E/F

such that

A

⊗
F

E

A \otimes _F E

is a split algebra over

L

⊗
F

E

L \otimes _F E

. In this paper we bound the degree of a minimal splitting field of this type from above and show that our bound is sharp in certain situations, even in the case where

L

/

F

L/F

is a split extension. This gives in particular a number of generalizations of the classical fact that when the tensor product of two quaternion algebras is not a division algebra, the two quaternion algebras must share a common quadratic splitting field.

In another direction, our constructions combined with results of Karpenko (1995) also show that for any odd prime number

p
p

, the generic algebra of index

p
n

p^n

and exponent

p
p

cannot be expressed nontrivially as the corestriction of an algebra over any extension field if

n
>

p
2

n > p^2

.
Structure of Zariski-closed algebras
Authors: Belov-Kanel Alexei , Rowen Louis , Vishne Uzi
Published: June 2012

The objective of this paper is to describe the structure of Zariski-closed algebras, which provide a useful generalization to finite dimensional algebras in the study of representable algebras over finite fields. Our results include a version of Wedderburn’s principal theorem as well as a more explicit description using representations, in terms of “gluing” in Wedderburn components. Finally, we construct “generic” Zariski-closed algebras, whose description is considerably more complicated than the description of generic algebra of finite dimensional algebras.

Special attention is given to infinite dimensional algebras over finite fields.
Counting primitive points of bounded height
Authors: Widmer Martin
Published: June 2012

Let

k
k

be a number field and

K
K

a finite extension of

k
k

. We count points of bounded height in projective space over the field

K
K

generating the extension

K

/

k

K/k

. As the height gets large we derive asymptotic estimates with a particularly good error term respecting the extension

K

/

k

K/k

. In a future paper we will use these results to get asymptotic estimates for the number of points of fixed degree over

k
k

. We also introduce the notion of an adelic Lipschitz height generalizing that of Masser and Vaaler. This will lead to further applications involving points of fixed degree on linear varieties and algebraic numbers of fixed degree satisfying certain subfield conditions.
Almost periodic operators in 𝑉𝑁(𝐺)
Authors: Chou Ching
Published: June 2012

Let

G
G

be a locally compact group,

A
(
G
)

A(G)

the Fourier algebra of

G
G

,

B
(
G
)

B(G)

the Fourier-Stieltjes algebra of

G
G

and

VN

(
G
)

{\text {VN}}(G)

the von Neumann algebra generated by the left regular representation

λ
\lambda

of

G
G

. Then

A
(
G
)

A(G)

is the predual of

VN

(
G
)

{\text {VN}}(G)

;

VN

(
G
)

{\text {VN}}(G)

is a

B
(
G
)

B(G)

-module and

A
(
G
)

A(G)

is a closed ideal of

B
(
G
)

B(G)

. Let

AP

(

G
^

)
=
{
T
∈

VN

(
G
)
:
u
↦
u
⋅
T

{\text {AP}}(\hat G) = \{ T \in {\text {VN}}(G):u \mapsto u \cdot T

is a compact operator from

A
(
G
)

A(G)

into

VN

(
G
)
}

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

, the space of almost periodic operators in

VN

(
G
)

{\text {VN}}(G)

. Let

C
δ
∗

(
G
)

C_\delta ^*(G)

be the

C
∗

{C^*}

-algebra generated by

{
λ
(
x
)
:
x
∈
G
}

\{ \lambda (x):x \in G\}

. Then

C
δ
∗

(
G
)
⊂

AP

(

G
^

)

C_\delta ^*(G) \subset {\text {AP}}(\hat G)

. For a compact

G
G

, let

E
E

be the rank one operator on

L
2

(
G
)

{L^2}(G)

that sends

h
∈

L
2

(
G
)

h \in {L^2}(G)

to the constant function

∫

h
(
x
)
d
x

\int {h(x)dx}

. We have the following results: (1) There exists a compact group

G
G

such that

E
∈
AP
(

G
^

)
∖

C
δ
∗

(
G
)

E \in \text {AP}(\hat G)\backslash C_\delta ^*(G)

. (2) For a compact Lie group

G
G

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E element-of AP left-parenthesis ModifyingAbove upper G With caret right-parenthesis left right double arrow upper E element-of upper C Subscript delta Superscript asterisk Baseline left-parenthesis upper G right-parenthesis left right double arrow upper L Superscript normal infinity Baseline left-parenthesis upper G right-parenthesis">

E
∈

AP(

G
^

)

⇔
E
∈

C
δ
∗

(
G
)
⇔

L
∞

(
G
)

E \in {\text {AP(}}\hat G{\text {)}} \Leftrightarrow E \in C_\delta ^*(G) \Leftrightarrow {L^\infty }(G)

has a unique left invariant mean

⇔
G

\Leftrightarrow G

is semisimple. (3) If

G
G

is an extension of a locally compact abelian group by an amenable discrete group then

AP

(

G
^

)
=

C
δ
∗

(
G
)

{\text {AP}}(\hat G) = C_\delta ^*(G)

. (4) Let

G
=

F

r

G = {{\mathbf {F}}_r}

, the free group with

r
r

generators,

1
>
r
>
∞

1 > r > \infty

. If

T
∈

VN

(
G
)

T \in {\text {VN}}(G)

and

u
↦
u
⋅
T

u \mapsto u \cdot T

is a compact operator from

B
(
G
)

B(G)

into

VN

(
G
)

{\text {VN}}(G)

then

T
∈

C
δ
∗

(
G
)

T \in C_\delta ^*(G)

.
Epicomplete Archimedean 𝑙-groups and vector lattices
Authors: Ball Richard N. , Hager Anthony W.
Published: June 2012

An object

G
G

in a category is epicomplete provided that the only morphisms out of

G
G

which are simultaneously epi and mono are the isomorphisms. We characterize the epicomplete objects in the category

A
r
c
h

{\mathbf {Arch}}

, whose objects are the archimedean lattice-ordered groups (archimedean

ℓ
\ell

-groups) and whose morphisms are the maps preserving both group and lattice structure (

ℓ
\ell

-homomorphisms). Recall that a space is basically disconnected if the closure of each cozero subset is open. Theorem. The following are equivalent for

G
∈

A
r
c
h

G \in {\mathbf {Arch}}

. (a)

G
G

is

A
r
c
h

{\mathbf {Arch}}

epicomplete. (b)

G
G

is an

A
r
c
h

{\mathbf {Arch}}

extremal suboject of

D
(
Y
)

D(Y)

for some basically disconnected compact Hausdorff space

Y
Y

. Here

D
(
Y
)

D(Y)

denotes the continuous extended real-valued functions on

Y
Y

which are finite on a dense subset. (c)

G
G

is conditionally and laterally

σ
\sigma

-complete (meaning each countable subset of positive elements of

G
G

which is either bounded or pairwise disjoint has a supremum), and

G
G

is divisible. The analysis of

A
r
c
h

{\mathbf {Arch}}

rests on an analysis of the closely related category

W

{\mathbf {W}}

, whose objects are of the form

(
G
,
u
)

(G,u)

, where

G
∈

A
r
c
h

G \in {\mathbf {Arch}}

and

u
u

is a weak unit (meaning

g
∧
u
=
0

g \wedge u = 0

implies

g
=
0

g = 0

for all

g
∈
G

g \in G

), and whose morphisms are the

ℓ
\ell

-homomorphism preserving the weak unit. Theorem. The following are equivalent for

(
G
,
u
)
∈

W

(G,u) \in {\mathbf {W}}

. (a)

(
G
,
u
)

(G,u)

is

W

{\mathbf {W}}

epicomplete. (b)

(
G
,
u
)

(G,u)

is

W

{\mathbf {W}}

isomorphic to

(
D
(
Y
)
,
1
)

(D(Y),1)

. (c)

(
G
,
u
)

(G,u)

is conditionally and laterally

σ
\sigma

-complete, and

G
G

is divisible.
Mean-square approximation by polynomials on the unit disk
Authors: Kriete Thomas L. , MacCluer Barbara D.
Published: June 2012

We investigate

P
2

(
μ
)

{P^2}(\mu )

, the closure in

L
2

(
μ
)

{L^2}(\mu )

of the complex polynomials, for certain measures

μ
\mu

on the closed unit disk in the complex plane. Specifically, we present a condition on

μ
\mu

which guarantees that

P
2

(
μ
)

{P^2}(\mu )

decomposes into a natural direct sum.
Complex interpolation of normed and quasinormed spaces in several dimensions. II. Properties of harmonic interpolation
Authors: Slodkowski Zbigniew
Published: June 2012

This paper is a continuation of the study of harmonic interpolation families of normed or quasinormed spaces parametrized by points of a domain in

C

k

{{\mathbf {C}}^k}

. It is shown, among other things, that each of the following properties holds for all the intermediate quasinormed spaces, if it holds for all given boundary spaces: (1) being a normed space; (2) being a Hilbert space; (3) satisfying the triangle inequality by the

r
r

th power of the quasinorm; (4) being uniformly convex; and (5) being uniformly smooth. As a principal tool, the notion of a harmonic set valued function (a generalization of analytic multifunction) is introduced and studied.
Hamilton-Jacobi equations with state constraints
Authors: Capuzzo-Dolcetta I. , Lions P.-L.
Published: June 2012

In the present paper we consider Hamilton-Jacobi equations of the form

H
(
x
,
u
,
∇
u
)
=
0
,

x
∈
Ω

H(x,u,\nabla u) = 0,\;x \in \Omega

, where

Ω
\Omega

is a bounded open subset of

R
n

,
H

{R^n},H

is a given continuous real-valued function of

(
x
,
s
,
p
)
∈
Ω
×
R
×

R
n

(x,s,p) \in \Omega \times R \times {R^n}

and

∇
u

\nabla u

is the gradient of the unknown function

u
u

. We are interested in particular solutions of the above equation which are required to be supersolutions, in a suitable weak sense, of the same equation up to the boundary of

Ω
\Omega

. This requirement plays the role of a boundary condition. The main motivation for this kind of solution comes from deterministic optimal control and differential games problems with constraints on the state of the system, as well from related questions in constrained geodesics.
Hausdorff dimension of harmonic measures on negatively curved manifolds
Authors: Kifer Yuri , Ledrappier François
Published: June 2012

We show by probabilistic means that harmonic measures on manifolds, whose curvature is sandwiched between two negative constants have positive Hausdorff dimensions. A lower bound for harmonic measures of open sets is derived, as well. We end with the results concerning the Hausdorff dimension of harmonic measures on universal covers of compact negatively curved manifolds.
Almost split sequences and Zariski differentials
Authors: Martsinkovsky Alex
Published: June 2012

Let

R
R

be a complete two-dimensional integrally closed analytic

k
k

-algebra. Associated with

R
R

is the Auslander module

A
A

from the fundamental sequence

0
→

ω
R

→
A
→
R
→
k
→
0

0 \to {\omega _R} \to A \to R \to k \to 0

and the module of Zariski differentials

D
k

(
R

)

∗
∗

{D_k}{(R)^{ * * }}

. We conjecture that these modules are isomorphic if and only if

R
R

is graded. We prove this conjecture for (a) hypersurfaces

f
=

X
3
n

+

g

(

X
1

,

X
2

)

f = X_3^n + {\text {g}}({X_1},{X_2})

, (b) quotient singularities, and (c)

R
R

graded Gorenstein.
Local approximation by certain spaces of exponential polynomials, approximation order of exponential box splines, and related interpolation problems
Authors: Dyn N. , Ron A.
Published: June 2012

Local approximation order to smooth complex valued functions by a finite dimensional space

H

\mathcal {H}

, spanned by certain products of exponentials by polynomials, is investigated. The results obtained, together with a suitable quasi-interpolation scheme, are used for the derivation of the approximation order attained by the linear span of translates of an exponential box spline. The analysis of a typical space

H

\mathcal {H}

is based here on the identification of its dual with a certain space

P

\mathcal {P}

of multivariate polynomials. This point of view allows us to solve a class of multivariate interpolation problems by the polynomials from

P

\mathcal {P}

, with interpolation data characterized by the structure of

H

\mathcal {H}

, and to construct bases of

P

\mathcal {P}

corresponding to the interpolation problem.
A conformal inequality related to the conditional gauge theorem
Authors: McConnell Terry R.
Published: June 2012

We prove the inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h left-parenthesis x right-parenthesis Superscript negative 1 Baseline upper G left-parenthesis x comma y right-parenthesis h left-parenthesis y right-parenthesis less-than-or-slanted-equals c upper G left-parenthesis x comma y right-parenthesis plus c">

h

(
x

)

−
1

G
(
x
,
y
)
h
(
y
)
⩽
c
G
(
x
,
y
)
+
c

h{(x)^{ - 1}}G(x,y)h(y) \leqslant cG(x,y) + c

, where

G
G

is the Green function of a plane domain

D
,

h

D,\;h

is positive and harmonic on

D
D

, and

c
c

is a constant whose value depends on the topological nature of the domain. In particular, for the class of proper simply connected domains

c
c

may be taken to be an absolute constant. As an application, we prove the Conditional Gauge Theorem for plane domains of finite area for which the constant

c
c

in the above inequality is finite.
An invariant of regular isotopy
Authors: Kauffman Louis H.
Published: June 2012

This paper studies a two-variable Laurent polynomial invariant of regular isotopy for classical unoriented knots and links. This invariant is denoted

L
K

{L_K}

for a link

K
K

, and it satisfies the axioms: 1. Regularly isotopic links receive the same polynomial. 2.

L

[

unk

]

=
1

{L_{[{\text {unk}}]}} = 1

. 3.

L

[

unk

]

=
a
L
,

L

[

unk

]

=

a

−
1

L

{L_{[{\text {unk}}]}} = aL,\qquad {L_{[{\text {unk}}]}} = {a^{ - 1}}L

. 4.
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript left-bracket unk right-bracket Baseline plus upper L Subscript left-bracket unk right-bracket Baseline equals z left-parenthesis upper L Subscript left-bracket unk right-bracket Baseline plus upper L Subscript left-bracket unk right-bracket Baseline right-parenthesis">

L

[

unk

]

+

L

[

unk]

=
z
(

L

[

unk]

+

L

[

unk]

)

{L_{[{\text {unk}}]}} + {L_{[{\text {unk]}}}} = z({L_{[{\text {unk]}}}} + {L_{[{\text {unk]}}}})

. Small diagrams indicate otherwise identical parts of larger diagrams. Regular isotopy is the equivalence relation generated by the Reidemeister moves of type II and type III. Invariants of ambient isotopy are obtained from

L
L

by writhe-normalization.
Analysis of a class of probability preserving measure algebras on compact intervals
Authors: Connett William C. , Schwartz Alan L.
Published: June 2012

The measure algebras of the title are those which are also hypergroups with some regularity conditions. Examples include the convolutions associated with Jacobi polynomial series and Fourier Bessel series. It is shown here that there is a one-to-one correspondence between these hypergroups and a class of Sturm-Liouville problems which have the characters of the hypergroup as eigenfunctions. The interplay between these two characterizations allows a detailed analysis which includes a Hilb-type formula for the characters and asymptotic estimates for the Plancherel measure and the eigenvalues of the associated Sturm-Liouville problem.
A topological characterization of 𝑅-trees
Authors: Mayer John C. , Oversteegen Lex G.
Published: June 2012

R

\mathbf {R}

-trees arise naturally in the study of groups of isometries of hyperbolic space. An

R

\mathbf {R}

-tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals

R

\mathbf {R}

. Actions on

R

\mathbf {R}

-trees can be viewed as ideal points in the compactification of groups of isometries. As such they have applications to the study of hyperbolic manifolds. Our concern in this paper, however, is with the topological characterization of

R

\mathbf {R}

-trees. Our main theorem is the following: Let

(
X
,
p
)

(X,p)

be a metric space. Then

X
X

is uniquely arcwise connected and locally arcwise connected if, and only if,

X
X

admits a compatible metric

d
d

such that

(
X
,
d
)

(X,d)

is an

R

\mathbf {R}

-tree. Essentially, we show how to put a convex metric on a uniquely arcwise connected, locally arcwise connected, metrizable space.
Using subnormality to show the simple connectivity at infinity of a finitely presented group
Authors: Profio Joseph S.
Published: June 2012

A CW-complex

X
X

is simply connected at infinity if for each compact

C
C

in

X
X

there exists a compact

D
D

in

X
X

such that loops in

X
−
D

X - D

are homotopically trivial in

X
−
C

X - C

. Let

G
G

be a finitely presented group and

X
X

a finite CW-complex with fundamental group

G
G

.

G
G

is said to be simply connected at infinity if the universal cover of

X
X

is simply connected at infinity. B. Jackson and C. M. Houghton have independently shown that if

G
G

and a normal subgroup

H
H

are infinite finitely presented groups with

G

/

H

G/H

infinite and either

H
H

or

G

/

H

G/H

1
1

-ended, then

G
G

is simply connected at infinity. In the case where

H
H

is

1
1

-ended, we exhibit a class of groups showing that the "finitely presented" hypothesis on

H
H

cannot be reduced to "finitely generated." We address the question: if

N
N

is normal in

H
H

and

H
H

is normal in

G
G

and these are infinite groups with

N
N

and

G
G

finitely presented and either

N
N

or

G

/

H

G/H

is

1
1

-ended, is

G
G

simply connected at infinity? In the case that

N
N

is

1
1

-ended, the answer is shown to be yes. In the case that

G

/

H

G/H

is

1
1

-ended, we exhibit a class of such groups that are not simply connected at infinity.
On a theorem of Stein
Authors: Krantz Steven G.
Published: June 2012

In this paper the Kobayashi metric on a domain in

C

n

{{\mathbf {C}}^n}

is used to define a new function space. Elements of this space belong to a nonisotropic Lipschitz class. It is proved that if

f
f

is holomorphic on the domain and in the classical Lipschitz space

Λ
α

{\Lambda _\alpha }

then in fact

f
f

is in the new function space. The result contains the original result of Stein on this subject and provides the optimal result adapted to any domain. In particular, it recovers the Hartogs extension phenomenon in the category of Lipschitz spaces.
The nonstandard treatment of Hilbert’s fifth problem
Authors: Hirschfeld Joram
Published: June 2012

We give a nonstandard proof that every locally Euclidean group is a Lie group. The heart of the proof is a strong nonstandard variant of Gleason’s lemma for a class of groups that includes all locally Euclidean groups.
The Dirichlet problem for radially homogeneous elliptic operators
Authors: Bass Richard F.
Published: June 2012

The Dirichlet problem in the unit ball is considered for the strictly elliptic operator

L
=
∑

a

i
j

D

i
j

L = \sum {{a_{ij}}{D_{ij}}}

, where the

a

i
j

{a_{ij}}

, are smooth away from the origin and radially homogeneous:

a

i
j

(
r
x
)
=

a

i
j

(
x
)
,

r
>
0
,

x
≠
0

{a_{ij}}(rx) = {a_{ij}}(x),\;r > 0,\;x \ne 0

. Existence and uniqueness are proved for solutions in a certain space of functions. Necessary and sufficient conditions are given for an extended maximum principle to hold.
Existence of weak solutions for the Navier-Stokes equations with initial data in 𝐿^{𝑝}
Authors: Calderón Calixto P.
Published: June 2012

The existence of weak solutions for the Navier-Stokes equations for the infinite cylinder with initial data in

L
p

{L^p}

is considered in this paper. We study the case of initial data in

L
p

(

R
n

)

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

,

2
>
p
>
n

2 > p > n

, and

n
=
3
,
4

n = 3,4

. An existence theorem is proved covering these important cases and therefore, the "gap" between the Hopf-Leray theory

(
p
=
2
)

(p = 2)

and that of Fabes-Jones-Riviere

(
p
>
n
)

(p > n)

is bridged. The existence theorem gives a new method of constructing global solutions. The cases

p
=
n

p = n

are treated at the end of the paper.
On solvable groups of finite Morley rank
Authors: Nesin Ali
Published: June 2012

We investigate solvable groups of finite Morley rank. We find conditions on

G
G

for

G
′

G’

to split in

G
G

. In particular, if

G
′

G’

is abelian and

Z
(
G
)
=
1

Z(G) = 1

we prove that

G
=

G
′

×
T

G = G’ \times T

for some

T
T

and the ring

Z

[
T
]

/

ann

G
′

{\mathbf {Z}}[T]/\text {ann}G’

is intepretable in

G
G

. We exploit the methods used in proving these results to find more information about solvable groups.
Composite ribbon number one knots have two-bridge summands
Authors: Bleiler Steven A. , Eudave Muñoz Mario
Published: June 2012

A composite ribbon knot which can be sliced with a single band move has a two-bridge summand.
Tiled orders of finite global dimension
Authors: Fujita Hisaaki
Published: June 2012

We define a projective link between maximal ideals, with respect to which an idealizer preserves being of finite global dimension. Let

D
D

be a local Dedekind domain with the quotient ring

K
K

. We show that for

2
≤
n
≤
5

2 \leq n \leq 5

, every tiled

D
D

-order of finite global dimension in

(
K

)
n

{(K)_n}

is obtained by iterating idealizers w.r.t. projective links from a hereditary order. For

n
≥
6

n \geq 6

, we give a tiled

D
D

-order in

(
K

)
n

{(K)_n}

without this property, which is also a counterexample to Tarsy’s conjecture, saying that the maximum finite global dimension of such an order is

n
−
1

n - 1

.
A regularity theory for variational problems with higher order derivatives
Authors: Clarke F. H. , Vinter R. B.
Published: June 2012

We consider problems in the calculus of variations in one independent variable and where the Lagrangian involves derivatives up to order

N
N

,

N
≥
1

N \ge 1

. Existence theory supplies mild hypotheses under which there are minimizers for such problems, but they need to be strengthened for standard necessary conditions to apply. For problems with

N
>
1

N > 1

, this paper initiates investigation of regularity properties, and associated necessary conditions, which obtain strictly under the hypotheses of existence theory. It is shown that the

N
N

th derivative of a minimizer is locally essentially bounded off a closed set of zero measure, the set of "points of bad behaviour". Additional hypotheses are shown to exclude occurrence of points of bad behaviour. Finally a counter example suggests respects in which problems with

N
>
1

N > 1

exhibit pathologies not present in the

N
=
1

N = 1

case.
Link homotopy with one codimension two component
Authors: Kirk Paul A.
Published: June 2012

Link maps with one codimension two component are studied and an invariant of link maps modulo link homotopy is constructed using ideas from knot theory and immersion theory. This invariant is used to give examples of nontrivial link homotopy classes and to show that there are infinitely many distinct link homotopy classes in many dimensions. A link map with the codimension two component embedded is shown to be nullhomotopic. These ideas are applied to the special case of

2
2

-spheres in

S
4

{S^4}

to give simple examples of the failure of the Whitney trick in dimension

4
4

.
Varieties of group representations and Casson’s invariant for rational homology 3-spheres
Authors: Boyer S. , Nicas A.
Published: June 2012

Andrew Casson’s

Z

{\mathbf {Z}}

-valued invariant for

Z

{\mathbf {Z}}

-homology

3
3

-spheres is shown to extend to a

Q

{\mathbf {Q}}

-valued invariant for

Q

{\mathbf {Q}}

-homology

3
3

-spheres which is additive with respect to connected sums. We analyze conditions under which the set of abelian

SL
2

(

C

)

{\operatorname {SL} _2}({\mathbf {C}})

and

SU
⁡
(
2
)

\operatorname {SU} (2)

representations of a finitely generated group is isolated. A formula for the dimension of the Zariski tangent space to an abelian

SL
2

(

C

)

{\operatorname {SL} _2}({\mathbf {C}})

or

SU
⁡
(
2
)

\operatorname {SU} (2)

representation is obtained. We also derive a sum theorem for Casson’s invariant with respect to toroidal splittings of a

Z

{\mathbf {Z}}

-homology

3
3

-sphere.
Unusually large gaps between consecutive primes
Authors: Maier Helmut , Pomerance Carl
Published: June 2012

Let

G
(
x
)

G(x)

denote the largest gap between consecutive primes below

x
x

. In a series of papers from 1935 to 1963, Erdàs, Rankin, and Schànhage showed that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis x right-parenthesis greater-than-or-equal-to left-parenthesis c plus o left-parenthesis 1 right-parenthesis right-parenthesis log x l o g l o g x l o g l o g l o g l o g x left-parenthesis l o g l o g l o g x right-parenthesis Superscript negative 2">

G
(
x
)
≥
(
c
+
o
(
1
)
)

log

x

loglog

x

loglogloglog

x

(

logloglog

x

)

−
2

G(x) \geq (c + o(1)){\operatorname {log}}x{\operatorname {loglog}}x{\operatorname {loglogloglog}}x{({\operatorname {logloglog}}x)^{ - 2}}

\]
, where

c
=

e
γ

c = {e^\gamma }

and

γ
\gamma

is Euler’s constant. Here, this result is shown with

c
=

c
0

e
γ

c = {c_0}{e^\gamma }

where

c
0

=
1.31256
…

{c_0} = 1.31256 \ldots

is the solution of the equation

4

/

c
0

−

e

−
4

/

c
0

=
3

4/{c_0} - {e^{ - 4/{c_0}}} = 3

. The principal new tool used is a result of independent interest, namely, a mean value theorem for generalized twin primes lying in a residue class with a large modulus.
The connection matrix in Morse-Smale flows
Authors: Reineck James F.
Published: June 2012

In a Morse-Smale flow with no periodic orbits, it is shown that the connection matrix is unique. In the case of periodic orbits, nonuniqueness can occur. We show that on

2
2

-manifolds, with some technical assumptions, given a connection matrix for the flow, one can replace the periodic orbits with doubly-connected rest points and obtain a new flow with no periodic orbits having the given connection matrix.
A class of algebras similar to the enveloping algebra of 𝑠𝑙(2)
Authors: Smith S. P.
Published: June 2012

Fix

f
∈

C

[
X
]

f \in {\mathbf {C}}[X]

. Define

R
=

C

[
A
,
B
,
H
]

R = {\mathbf {C}}[A,B,H]

subject to the relations
\[

H
A
−
A
H
=
A
,

H
B
−
B
H
=
−
B
,

A
B
−
B
A
=
f
(
H
)

HA - AH = A,\quad HB - BH = - B,\quad AB - BA = f(H)

\]
. We study these algebras (for different

f
f

) and in particular show how they are similar to (and different from)

U
(

sl

(
2
)
)

U({\text {sl}}(2))

, the enveloping algebra of

sl

(
2
,

C

)

{\text {sl}}(2,{\mathbf {C}})

. There is a notion of highest weight modules and a category

O

\mathcal {O}

for such

R
R

. For each

n
>
0

n > 0

, if

f
(
x
)
=

(
x
+
1

)

n
+
1

−

x

n
+
1

f(x) = {(x + 1)^{n + 1}} - {x^{n + 1}}

, then

R
R

has precisely

n
n

simple modules in each finite dimension, and every finite-dimensional

R
R

-module is semisimple.
Symmetry properties of the solutions to Thomas-Fermi-Dirac-von Weizsäcker type equations
Authors: Benguria Rafael D. , Yarur Cecilia
Published: June 2012

We consider a semilinear elliptic equation with a spherically symmetric potential (specifically, Thomas-Fermi-Dirac-von Weizsäcker type equations without electronic repulsion). Assuming some regularity properties of the solutions at the origin and at infinity, we prove that the solutions have spherical symmetry.
The domain spaces of quasilogarithmic operators
Authors: Cwikel M. , Jawerth B. , Milman M.
Published: June 2012

The construction of intermediate Banach spaces in interpolation theory and the study of commutator inequalities in this context are closely related to certain nonlinear operators

Ω
\Omega

. Here an explicit characterization of the domain spaces of these operators

Ω
\Omega

is obtained, and the characterization is related to logarithmic Sobolev inequalities.
Metric transforms and Euclidean embeddings
Authors: Deza M. , Maehara H.
Published: June 2012

It is proved that if

0
⩽
c
⩽
0.72

/

n

0 \leqslant c \leqslant 0.72/n

then for any

n
n

-point metric space

(
X
,
d
)

(X,d)

, the metric space

(
X
,

d
c

)

(X,{d^c})

is isometrically embeddable into a Euclidean space. For

6
6

-point metric space,

c
=

1
2

log
2

3
2

c = \tfrac {1} {2}{\log _2}\tfrac {3} {2}

is the largest exponent that guarantees the existence of isometric embeddings into a Euclidean space. Such largest exponent is also determined for all

n
n

-point graphs with "truncated distance".
Solving Beltrami equations by circle packing
Authors: He Zheng-Xu
Published: June 2012

We use Andreev-Thurston’s theorem on the existence of circle packings to construct approximating solutions to the Beltrami equations on Riemann surfaces. The convergence of the approximating solutions on compact subsets will be shown. This gives a constructive proof of the existence theorem for Beltrami equations.
Continuous spatial semigroups of *-endomorphisms of 𝔅(ℌ)
Authors: Powers Robert T. , Price Geoffrey
Published: June 2012

To each continuous semigroup of

∗
*

-endomorphisms

α
\alpha

of

B

(

H

)

\mathfrak {B}\left ( \mathfrak {H} \right )

with an intertwining semigroup of isometries there is associated a

∗
*

-representation

π
\pi

of the domain

O

(
δ
)

\mathfrak {O}(\delta )

of the generator of

α
\alpha

. It is shown that the Arveson index

d
∗

(
α
)

{d_ * }(\alpha )

is the number of times the representation

π
\pi

contains the identity representation of

O

(
δ
)

\mathfrak {O}(\delta )

. This result is obtained from an analysis of the relation between two semigroups of isometries,

U
U

and

S
S

, satisfying the condition

S

(
t

)
∗

U
(
t
)
=

e

−
λ
t

I

S{(t)^*}U(t) = {e^{ - \lambda t}}I

for

t
≥
0

t \geq 0

and

λ
>
0

\lambda > 0

.
Generalized balanced tournament designs
Authors: Lamken E. R.
Published: June 2012

A generalized balanced tournament design,

G
B
T
D
(
n
,
k
)

GBTD(n,k)

, defined on a

k
n

kn

-set

V
V

, is an arrangement of the blocks of a

(
k
n
,
k
,
k
−
1
)

(kn,k,k - 1)

-

B
I
B
D

BIBD

defined on

V
V

into an

n
×
(
k
n
−
1
)

n \times (kn - 1)

array such that (1) every element of

V
V

is contained in precisely one cell of each column, and (2) every element of

V
V

is contained in at most

k
k

cells of each row. In this paper, we introduce

G
B
T
D
(
n
,
k
)
s

GBTD(n,k)s

and describe connections between these designs and several other types of combinatorial designs. We also show how to use

G
B
T
D
s

GBTDs

to construct resolvable, near resolvable, doubly resolvable and doubly near resolvable

B
I
B
D
s

BIBDs

.
Some global results on extension of CR-objects in complex manifolds
Authors: Lupacciolu Guido
Published: June 2012

We prove some results concerning the holomorphic extendability of CR-objects defined on real hypersurfaces of a complex manifold. After a preliminary generalization of the classic theorem on the extendability from the boundary of a relatively compact domain, we discuss the extendability from a part of the boundary of such a domain, the one side extendability from a piece of hypersurface and the extendability from the boundary of an unbounded domain.
Complete localization of domains with noncompact automorphism groups
Authors: Kim Kang-Tae
Published: June 2012

We prove a characterization of the domains in

C

n

{{\mathbf {C}}^n}

with an automorphism orbit accumulating at a boundary point at which the boundary is real analytic and convex up to a biholomorphic change of local coordinates. This result generalizes the well-known Wong-Rosay theorem on strongly pseudoconvex domains to the case of locally convex domains with real analytic boundaries.
The stochastic mechanics of the Pauli equation
Authors: Wallstrom Timothy C.
Published: June 2012

In stochastic mechanics, the Bopp-Haag-Dankel diffusions on

R

3

×
SO
⁡
(
3
)

{\mathbb {R}^3} \times \operatorname {SO} (3)

are used to represent particles with spin. Bopp and Haag showed that in the limit as the particle’s moment of inertia

I
I

goes to zero, the solutions of the Bopp-Haag equations converge to that of the regular Pauli equation. Nelson has conjectured that in the same limit, the projections of the Bopp-Haag-Dankel diffusions onto

R

3

{\mathbb {R}^3}

converge to a Markovian limit process. In this paper, we prove this conjecture for spin

spin

1
2

\operatorname {spin} \;\tfrac {1} {2}

and regular potentials, and identify the limit process as the diffusion naturally associated with the solution to the regular Pauli equation.
The 27-dimensional module for 𝐸₆. III
Authors: Aschbacher Michael
Published: June 2012

This is the third in a series of five papers investigating the subgroup structure of the universal Chevalley group

G
=

E
6

(
F
)

G = {E_6}(F)

of type

E
6

{E_6}

over a field

F
F

and the geometry induced on the

27
27

-dimensional

F
G

FG

-module

V
V

by the symmetric trilinear form

f
f

preserved by

G
G

. The series uses the geometry on

V
V

to describe and enumerate (up to a small list of ambiguities) all closed maximal subgroups of

G
G

when

F
F

is finite or algebraically closed.
Hall-Littlewood functions, plane partitions, and the Rogers-Ramanujan identities
Authors: Stembridge John R.
Published: June 2012

We apply the theory of Hall-Littlewood functions to prove several multiple basic hypergeometric series identities, including some previously known generalizations of the Rogers-Ramanujan identities due to G. E. Andrews and D. M. Bressoud. The techniques involve the adaptation of a method due to I. G. Macdonald for calculating partial fraction expansions of certain types of symmetric formal power series. Macdonald originally used this method to prove a pair of generating function identities for plane partitions conjectured by MacMahon and Bender-Knuth. We show that this method can also be used to prove another pair of plane partition identities recently obtained by R. A. Proctor.
Matroid representations and free arrangements
Authors: Ziegler Günter M.
Published: June 2012

We show that Terao’s Conjecture ("Freeness of the module of logarithmic forms at a hyperplane arrangement is determined by its abstract matroid") holds over fields with at most four elements. However, an example demonstrates that the field characteristic has to be fixed for this.
The heat equation for Riemannian foliations
Authors: Nishikawa Seiki , Ramachandran Mohan , Tondeur Philippe
Published: June 2012

Let

F

\mathcal {F}

be a Riemannian foliation on a closed oriented manifold

M
M

, with the transversal Laplacian

Δ
B

{\Delta _B}

acting on the basic forms

Ω
B
r

(

F

)

\Omega _B^r(\mathcal {F})

of degree

r
⩾
0

r \geqslant 0

. We construct the fundamental solution

e
B
r

(
x
,
y
,
t
)

e_B^r(x,y,t)

for the basic heat operator

∂

/

∂
t
+

Δ
B

\partial /\partial t + {\Delta _B}

, and prove existence and uniqueness for the solution of the heat equation on

Ω
B
r

(

F

)

\Omega _B^r(\mathcal {F})

. As an application we give a new proof for the deRham-Hodge decomposition theorem for

Δ
B

{\Delta _B}

in

Ω
B
r

(

F

)

\Omega _B^r(\mathcal {F})

, generalizing the approach to the classical deRham-Hodge theorem pioneered by Milgram and Rosenbloom.
The extreme projections of the regular simplex
Authors: Filliman P.
Published: June 2012

The largest and smallest projections of the regular

n
n

-dimensional simplex into a

k
k

-dimensional subspace are determined for certain values of

n
n

and

k
k

. These results suggest that the smallest

k
k

-dimensional projection and the largest

(
n
−
k
)

(n - k)

-dimensional projection occur in orthogonal subspaces of

R
n

{R^n}

.
The diffeotopy group of the twisted 2-sphere bundle over the circle
Authors: Kim M. Ho , Raymond Frank
Published: June 2012

The diffeotopy group of the nontrivial

2
2

-sphere bundle over the circle is shown to be isomorphic to

Z

2

⊕

Z

2

{\mathbb {Z}_2} \oplus {\mathbb {Z}_2}

. The first generator is induced by a reflection across the base circle, while a second generator comes from rotating the

2
2

-sphere fiber as one travels around the base circle. The technique employed also shows that homotopic diffeomorphisms are diffeotopic.
Bounded polynomial vector fields
Authors: Cima Anna , Llibre Jaume
Published: June 2012

We prove that, for generic bounded polynomial vector fields in

R

n

{{\mathbf {R}}^n}

with isolated critical points, the sum of the indices at all their critical points is

(
−
1

)
n

{( - 1)^n}

. We characterize the local phase portrait of the isolated critical points at infinity for any bounded polynomial vector field in

R

2

{{\mathbf {R}}^2}

. We apply this characterization to show that there are exactly seventeen different behaviours at infinity for bounded cubic polynomial vector fields in the plane.
Existence and uniqueness of algebraic curvature tensors with prescribed properties and an application to the sphere theorem
Authors: Seaman Walter
Published: June 2012

An existence and uniqueness theorem is proved for algebraic curvature tensors and then applied to yield a global geometric theorem for locally weakly quarter pinched Riemannian manifolds whose second Betti number is nonzero.
Power series space representations of nuclear Fréchet spaces
Authors: Vogt Dietmar
Published: June 2012

Let

E
E

be a nuclear graded Fréchet space such that the norms satisfy inequalities
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript k Superscript 2 Baseline less-than-or-equal-to upper C Subscript k Baseline StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript k minus 1 Baseline StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript k minus 1">

|

|

|

|

k
2

≤

C
k

|

|

|

|

k
−
1

|

|

|

|

k
−
1

||||_k^2 \leq {C_k}|||{|_{k - 1}}|||{|_{k - 1}}

for all

k
k

, let

F
F

be a graded Fréchet space such that the dual (extended real valued) norms satisfy inequalities
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript k Superscript asterisk 2 Baseline less-than-or-equal-to upper D Subscript k Baseline StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript k minus 1 Superscript asterisk Baseline StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript k plus 1 Superscript asterisk">

|

|

|

|

k

∗
2

≤

D
k

|

|

|

|

k
−
1

∗

|

|

|

|

k
+
1

∗

||||_k^{*2} \leq {D_k}||||_{k - 1}^*||||_{k + 1}^*

for all

k
k

, and let

A
A

be a tame (resp. linearly tame) linear map from

F
F

to

E
E

. Then there exists a tame (resp. linearly tame) factorization of

A
A

through a power series space

Λ
∞
2

(
α
)

\Lambda _\infty ^2(\alpha )

. In the case of a tame quotient map,

E
E

is tamely equivalent to a power series space of infinite type. This applies in particular to the range of a tame (resp. linearly tame) projection in a power series space

Λ
∞
2

(
α
)

\Lambda _\infty ^2(\alpha )

. In this case one does not need nuclearity. It also applies to the tame spaces in the sense of the various implicit function theorems. If they are nuclear, they are tamely equivalent to power series spaces

Λ
∞

(
α
)

{\Lambda _\infty }(\alpha )

.
The fixed point theorem in equivariant cohomology
Authors: Jones J. D. S. , Petrack S. B.
Published: June 2012

In this paper we study the

S
1

{S^1}

-equivariant de Rham cohomology of infinite dimensional

S
1

{S^1}

-manifolds. Our main example is the free loop space

L
X

LX

where

X
X

is a finite dimensional manifold with the circle acting by rotating loops. We construct a new form of equivariant cohomology

h
T
∗

h_T^*

which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces

L
X

LX

. This gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of Witten [16], Atiyah [2], and Bismut [5]. In particular we show, following Atiyah in [2], that the

A
^

\hat A

-polynomial of

X
X

arises as an equivariant characteristic class, in the theory

h
T
∗

h_T^*

, of the normal bundle to

X
X

, considered as the space of constant loops, in

L
X

LX

.
Defect relations for degenerate meromorphic maps
Authors: Chen Wan Xi
Published: June 2012

Using a concept called subgeneral position and adapting a weight function created by E. I. Nochka, this work proves the Cartan’s conjecture on defect relations for a degenerate meromorphic map from a parabolic manifold into a projective space.
Character theory and rooted maps in an orientable surface of given genus: face-colored maps
Authors: Jackson D. M. , Visentin T. I.
Published: June 2012

The character theoretic approach [5] to the enumeration of rooted maps in an orientable surface of arbitrary genus is extended to

2
2

-face-colorable rooted maps. In particular, we show that there exists, for each genus, a correspondence between the set of

2
2

-colored triangulations and a set of

2
2

-colored rooted maps of all lower genera with a distinguished subset of vertices.