bluuebunny/crossref_metadata_2025 huggingface dataset:
The Stefan problem with small surface tension
Authors: Friedman Avner , Reitich Fernando
Published: June 2012

The Stefan problem with small surface tension

ε
\varepsilon

is considered. Assuming that the classical Stefan problem (with

ε
=
0

\varepsilon = 0

) has a smooth free boundary

Γ
\Gamma

, we denote the temperature of the solution by

θ
0

{\theta _0}

and consider an approximate solution

θ
0

+
ε
u

{\theta _0} + \varepsilon u

for the case where

ε
≠
0

\varepsilon \ne 0

,

ε
\varepsilon

small. We first establish the existence and uniqueness of

u
u

, and then investigate the effect of

u
u

on the free boundary

Γ
\Gamma

. It is shown that small surface tension affects the free boundary

Γ
\Gamma

radically differently in the two-phase problem than in the one-phase problem.
A formal Mellin transform in the arithmetic of function fields
Authors: Goss David
Published: June 2012

The Mellin transform is a fundamental tool of classical arithmetic. We would also like such a tool in the arithmetic of function fields based on Drinfeld modules, although a construction has not yet been found. One formal approach to finding Mellin transforms in classical theory is through

p
p

-adic measures. It turns out that this approach also works for function fields. Thus this paper is devoted to exploring what can be learned this way. We will establish some very enticing connections with gamma functions and the Kummer-Vandiver conjecture for function fields.
Modèle minimal équivariant et formalité
Authors: Lambre Thierry
Published: June 2012

We study the rational equivariant homotopy type of a topological space

X
X

equipped with an action of the group of integers modulo

n
n

. For

n
=

p
k

n= {p^k}

(

p
p

prime,

k
k

a positive integer), we build an algebraic model which gives the rational equivariant homotopy type of

X
X

. The homotopical fixed-point set appears in the construction of a model of the fixed-points set. In general, this model is different from

G

{\text {G}}

. Triantafillou’s model

[

T1

]

[{\text {T1}}]

. For

n
=
p

n= p

(

p
p

prime), we then give a notion of equivariant formality. We prove that this notion is equivalent to the formalizability of the inclusion of fixed-points set

i
:

X

Z

p

→
X

i:{X^{{\mathbb {Z}_p}}} \to X

. Examples and counterexamples of

Z

p

{\mathbb {Z}_p}

-formal spaces are given.
Actions of loop groups on harmonic maps
Authors: Bergvelt M. J. , Guest M. A.
Published: June 2012

We describe a general framework in which subgroups of the loop group

Λ
G

l
n

C

\Lambda G{l_n}\mathbb {C}

act on the space of harmonic maps from

S
2

{S^2}

to

G

l
n

C

G{l_n}\mathbb {C}

. This represents a simplification of the action considered by Zakharov-Mikhailov-Shabat [ZM, ZS] in that we take the contour for the Riemann-Hilbert problem to be a union of circles; however, it reduces the basic ingredient to the well-known Birkhoff decomposition of

Λ
G

l
n

C

\Lambda G{l_n}\mathbb {C}

, and this facilitates a rigorous treatment. We give various concrete examples of the action, and use these to investigate a suggestion of Uhlenbeck [Uh] that a limiting version of such an action ("completion") gives rise to her fundamental process of "adding a uniton". It turns out that this does not occur, because completion preserves the energy of harmonic maps. However, in the special case of harmonic maps from

S
2

{S^2}

to complex projective space, we describe a modification of this completion procedure which does indeed reproduce "adding a uniton".
Equivariant fixed point index and fixed point transfer in nonzero dimensions
Authors: Prieto Carlos , Ulrich Hanno
Published: June 2012

Dold’s fixed point index and fixed point transfer are generalized for certain coincidence situations, namely maps which change the "equivariant dimension." Those invariants change the dimension correspondingly. A formula for the index of a situation over a space with trivial group action is exhibited. For the transfer, a generalization of Dold’s Lefschetz-Hopf trace formula is proved.
A measure of smoothness related to the Laplacian
Authors: Ditzian Z.
Published: June 2012

A

K
K

-functional on

f
∈
C

(

R
d

)

f \in C\,({R^d})

given by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper K With tilde left-parenthesis f comma t squared right-parenthesis equals inf left-parenthesis StartAbsoluteValue EndAbsoluteValue f minus g StartAbsoluteValue EndAbsoluteValue plus t squared StartAbsoluteValue EndAbsoluteValue normal upper Delta g StartAbsoluteValue EndAbsoluteValue semicolon g element-of upper C squared left-parenthesis upper R Superscript d Baseline right-parenthesis right-parenthesis">

K
~

(
f
,

t
2

)
=
inf
(

|

|

f
−
g

|

|

+

t
2

|

|

Δ
g

|

|

;
g
∈

C
2

(

R
d

)
)

\tilde K\,(f,{t^2})= \inf (||f - g|| + {t^2}||\Delta g||;g \in {C^2}\,({R^d}))

\]
will be shown to be equivalent to the modulus of smoothness
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove w With tilde left-parenthesis f comma t right-parenthesis equals sup Underscript 0 greater-than h less-than-or-equal-to t Endscripts double-vertical-bar 2 d f left-parenthesis x right-parenthesis minus sigma-summation Underscript i equals 1 Overscript d Endscripts left-bracket f left-parenthesis x plus h e Subscript i Baseline right-parenthesis plus f left-parenthesis x minus h e Subscript i Baseline right-parenthesis right-bracket double-vertical-bar period">

w
~

(
f
,
t
)
=

sup

0
>
h
≤
t

‖

2

d
f
(
x
)
−

∑

i
=
1

d

[
f
(
x
+
h

e
i

)
+
f
(
x
−
h

e
i

)
]

‖

.

\tilde w\,(f,t)= \sup \limits _{0 > h \leq t} \,\left \| {2\,df(x) - \sum \limits _{i = 1}^d {[f(x + h{e_i}) + f(x - h{e_i})]} } \right \|.

\]
The situation for other Banach spaces of functions on

R
d

{R^d}

will also be resolved.
Regular points for ergodic Sinaĭ measures
Authors: Tsujii Masato
Published: June 2012

Ergodic properties of smooth dynamical systems are considered. A point is called regular for an ergodic measure

μ
\mu

if it is generic for

μ
\mu

and the Lyapunov exponents at it coincide with those of

μ
\mu

. We show that an ergodic measure with no zero Lyapunov exponent is absolutely continuous with respect to unstable foliation

[
L
]

[\text {L}]

if and only if the set of all points which are regular for it has positive Lebesgue measure.
Varieties of periodic attractor in cellular automata
Authors: Hurley Mike
Published: June 2012

We apply three alternate definitions of "attractor" to cellular automata. Examples are given to show that using the different definitions can give different answers to the question "Does this cellular automaton have a periodic attractor?" The three definitions are the topological notion of attractor as used by C. Conley, a more measure-theoretic version given by J. Milnor, and a variant of Milnor’s definition that is based on the concept of the "center of attraction" of an orbit. Restrictions on the types of periodic orbits that can be periodic attractors for cellular automata are described. With any of these definitions, a cellular automaton has at most one periodic attractor. Additionally, if Conley’s definition is used, then a periodic attractor must be a fixed point. Using Milnor’s definition, each point on a periodic attractor must be fixed by all shifts, so the number of symbols used is an upper bound on the period; whether the actual upper bound is

1
1

is unknown. With the third definition this restriction is removed, and examples are given of onedimensional cellular automata on three symbols that have finite "attractors" of arbitrarily large size (with the third definition, a finite attractor is not necessarily a single periodic orbit).
Linear topological classifications of certain function spaces
Authors: Valov Vesko M.
Published: June 2012

Some linear classification results for the spaces

C
P

(
X
)

{C_P}(X)

and

C
P

∗

(
X
)

C_P^{\ast }\,(X)

are proved.
The duality between subsemigroups of Lie groups and monotone functions
Authors: Neeb Karl-Hermann
Published: June 2012

In this paper we give a characterization of those convex cones

W
W

in the Lie algebra

L

(
G
)

{\mathbf {L}}(G)

of a connected Lie group

G
G

which are global in

G
G

, i.e. for which there exists a closed subsemigroup

S
S

in

G
G

having

W
W

as its tangent wedge

L

(
S
)

{\mathbf {L}}(S)

. The main result is the Characterization Theorem II.12. We also prove in Corollary II.6 that each germ of a strictly

W
W

-positive function belongs to a global function if there exists at least one strictly

W
W

-positive function. We apply the Characterization Theorem to obtain some general conditions for globality and to give a complete description of the global cones in compact Lie algebras.
Invariant subspaces with finite codimension in Bergman spaces
Authors: Aleman Alexandru
Published: June 2012

For an arbitrary bounded domain in

C

\mathbb {C}

there are described those finite codimensional subspaces of the Bergman space that are invariant under multiplication by

z
z

.
Studying links via closed braids. V. The unlink
Authors: Birman Joan S. , Menasco William W.
Published: June 2012

The main result is a version of Markov’s Theorem which does not involve stabilization, in the special case of the

r
r

-component link. As a corollary, it is proved that the stabilization index of a closed braid representative of the unlink is at most

1
1

. To state the result, we need the concept of an "exchange move", which modifies a closed braid without changing its link type or its braid index. For generic closed braids exchange moves change conjugacy class. Theorem

1
1

shows that exchange moves are the only obstruction to reducing a closed

n
n

-braid representative of the

r
r

-component unlink to the standard closed

r
r

-braid representative, through a sequence of braids of nonincreasing braid index.
Stable and uniformly stable unit balls in Banach spaces
Authors: Suárez Granero Antonio
Published: June 2012

Let

X
X

be a Banach space with closed unit ball

B
X

{B_X}

and, for

x
∈
X

x \in X

,

r
≥
0

r \geq 0

, put

B
(
x
;
r
)
=
{
u
∈
X
:

|

|

u
−
x

|

|

≤
r
}

B(x;r)= \{ u \in X:||u - x|| \leq r\}

and

V
(
x
,
r
)
=

B
X

∩
B
(
x
;
r
)

V(x,r)= {B_X} \cap B(x;r)

. We say that

B
X

{B_X}

(or in general a convex set) is stable if the midpoint map

Φ

1

/

2

:

B
X

×

B
X

→

B
X

{\Phi _{1/2}}:{B_X} \times {B_X} \to {B_X}

, with

Φ

1

/

2

(
u
,
υ
)
=

1
2

(
u
+
υ
)

{\Phi _{1/2}}(u,\upsilon )= \frac {1}{2}(u + \upsilon )

, is open. We say that

B
X

{B_X}

is uniformly stable (US) if there is a map

α
:
(
0
,
2
]
→
(
0
,
2
]

\alpha :(0,2] \to (0,2]

, called a modulus of uniform stability, such that, for each

x
,
y
∈

B
X

x,y \in {B_X}

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r element-of left-parenthesis 0 comma 2 right-bracket comma upper V left-parenthesis one half left-parenthesis x plus y right-parenthesis semicolon alpha left-parenthesis r right-parenthesis right-parenthesis subset-of-or-equal-to one half left-parenthesis upper V left-parenthesis x semicolon r right-parenthesis plus upper V left-parenthesis y semicolon r right-parenthesis right-parenthesis">

r
∈
(
0
,
2
]
,
V
(

1
2

(
x
+
y
)
;
α
(
r
)
)
⊆

1
2

(
V
(
x
;
r
)
+
V
(
y
;
r
)
)

r \in (0,2],V(\frac {1} {2}(x + y);\alpha (r)) \subseteq \frac {1} {2}(V(x;r) + V(y;r))

. Among other things, we see: (i) if

dim
⁡
X
≥
3

\dim X \geq 3

, then

X
X

admits an equivalent norm such that

B
X

{B_X}

is not stable; (ii) if

dim
⁡
X
>
∞

\dim X > \infty

,

B
X

{B_X}

is stable iff

B
x

{B_x}

is US; (iii) if

X
X

is rotund,

X
X

is uniformly rotund iff

B
X

{B_X}

is US; (iv) if

X
X

is

3.2.

I.P

3.2.{\text {I.P}}

,

B
X

{B_X}

is US and

α
(
r
)
=
r

/

2

\alpha (r)= r/2

is a modulus of US; (v)

B
X

{B_X}

is US iff

B

X

∗
∗

{B_{{X^{ \ast \ast }}}}

is US and

X
X

,

X

∗
∗

{X^{ \ast \ast }}

have (almost) the same modulus of US; (vi)

B
X

{B_X}

is stable (resp. US) iff

B

C
(
K
,
X
)

{B_{C(K,X)}}

is stable (resp. US) for each compact

K
K

iff

B

A
(
K
,
X
)

{B_{A(K,X)}}

is stable (resp. US) for each Choquet simplex

K
K

; (vii)

B
X

{B_X}

is stable iff

B

L
p

(
μ
,
X
)

{B_{{L_p}(\mu ,X)}}

is stable for each measure

μ
\mu

and

1
≤
p
>
∞

1 \leq p > \infty

.
A sharp inequality for martingale transforms and the unconditional basis constant of a monotone basis in 𝐿^{𝑝}(0,1)
Authors: Choi K. P.
Published: June 2012

Let

1
>
p
>
∞

1 > p > \infty

. Let

d
=
(

d
1

,

d
2

,
…
)

d = ({d_1},{d_2}, \ldots )

be a real-valued martingale difference sequence,

θ
=
(

θ
1

,

θ
2

,
…
)

\theta = ({\theta _1},{\theta _2}, \ldots )

is a predictable sequence taking values in

[
0
,
1
]

[0,1]

. We show that the best constant of the inequality,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar sigma-summation Underscript k equals 1 Overscript n Endscripts theta Subscript k Baseline d Subscript k Baseline double-vertical-bar Subscript p Baseline less-than-or-equal-to c Subscript p Baseline double-vertical-bar sigma-summation Underscript k equals 1 Overscript n Endscripts d Subscript k Baseline double-vertical-bar Subscript p Baseline comma n greater-than-or-equal-to 1 comma">

‖

∑

k
=
1

n

θ
k

d
k

‖

p

≤

c
p

‖

∑

k
=
1

n

d
k

‖

p

,

n
≥
1
,

{\left \| {\sum \limits _{k = 1}^n {{\theta _k}{d_k}} } \right \|_p} \leq {c_p}{\left \| {\sum \limits _{k = 1}^n {{d_k}} } \right \|_p}, \quad n \geq 1,

\]
satisfies
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c Subscript p Baseline equals StartFraction p Over 2 EndFraction plus one half log left-parenthesis StartFraction 1 plus y Over 2 EndFraction right-parenthesis plus StartFraction alpha 2 Over p EndFraction plus midline-horizontal-ellipsis comma">

c
p

=

p
2

+

1
2

log

(

1
+
y

2

)

+

α
2

p

+
⋯
,

{c_p} = \frac {p}{2} + \frac {1}{2}\;\log \;\left ({\frac {{1 + y}}{2}} \right ) + \frac {{{\alpha _2}}}{p} + \cdots ,

\]
where

γ
=

e

−
2

\gamma = {e^{ - 2}}

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha 2 equals left-bracket one half log StartFraction 1 plus gamma Over 2 EndFraction right-bracket squared plus one half log StartFraction 1 plus gamma Over 2 EndFraction minus 2 left-parenthesis StartFraction gamma Over 1 plus gamma EndFraction right-parenthesis squared">

α
2

=

[

1
2

log

1
+
γ

2

]

2

+

1
2

log

1
+
γ

2

−
2

(

γ

1
+
γ

)

2

{\alpha _2} = {\left [ {\frac {1}{2}\;\log \;\frac {{1 + \gamma }}{2}} \right ]^2} + \frac {1}{2}\;\log \;\frac {{1 + \gamma }}{2} - 2{\left ({\frac {\gamma }{{1 + \gamma }}} \right )^2}

. The best constant equals the unconditional basis constant of a monotone basis of

L
p

(
0
,
1
)

{L^p}(0,1)

.
Analytic operator valued function space integrals as an ℒ(ℒ_{𝓅},ℒ_{𝓅’}) theory
Authors: Chang Kun Soo , Ryu Kun Sik
Published: June 2012

The existence of an analytic operator-valued function space integral as an

S

(

L
p

,

L

p
′

)

\mathcal {S}({L_p},{L_{p’}})

theory

(
1
≤
p
≤
2
)

(1 \leq p \leq 2)

has been established for certain functionals involving the Lebesgue measure. Recently, Johnson and Lapidus proved the existence of the integral as an operator on

L
2

{L_2}

for certain functionals involving any Borel measure. We establish the existence of the integral as an operator from

L
p

{L_p}

to

L

p
′

(

1
>
p
>
2

)

{L_{p’}}\;({1 > p > 2} )

for certain functionals involving some Borel measures.
Two-dimensional Cremona groups acting on simplicial complexes
Authors: Wright David
Published: June 2012

We show that the

2
2

-dimensional Cremona group
\[

Cr
2

=

Aut
k

k
(
X
,
Y
)

\operatorname {Cr}_2 = \operatorname {Aut}_k\;k(X,Y)

\]
acts on a

2
2

-dimensional simplicial complex

C
C

, which has as vertices certain models in the function field

k
(
X
,
Y
)

k(X,Y)

. The fundamental domain consists of one face

F
F

. This yields a structural description of

Cr
2

\operatorname {Cr}_2

as an amalgamation of three subgroups along pairwise intersections. The subgroup

GA

2

=

Aut
k

k
[
X
,
Y
]

{\text {GA}}_2 = \operatorname {Aut}_k\;k[X,Y]

(integral Cremona group) acts on

C
C

by restriction. The face

F
F

has an edge

E
E

such that the

GA

2

{\text {GA}}_2

translates of

E
E

form a tree

T
T

. The action of

GA

2

{\text {GA}}_2

on

T
T

yields the well-known structure theory for

GA

2

{\text {GA}}_2

as an amalgamated free product, using Serre’s theory of groups acting on trees.
Isomorphism invariants for abelian groups
Authors: Arnold D. M. , Vinsonhaler C. I.
Published: June 2012

Let

A
=
(

A
1

,
…
,

A
n

)

A= ({A_1},\ldots ,{A_n})

be an

n
n

-tuple of subgroups of the additive group,

Q
Q

, of rational numbers and let

G
(
A
)

G(A)

be the kernel of the summation map

A
1

⊕
⋯
⊕

A
n

→
∑

A
i

{A_1} \oplus \cdots \oplus {A_n} \to \sum \;{A_i}

and

G
[
A
]

G[A]

the cokernel of the diagonal embedding

∩

A
1

→

A
1

⊕
⋯
⊕

A
n

\cap \,{A_1} \to {A_1} \oplus \cdots \oplus {A_n}

. A complete set of isomorphism invariants for all strongly indecomposable abelian groups of the form

G
(
A
)

G(A)

, respectively,

G
[
A
]

G[A]

, is given. These invariants are then extended to complete sets of isomorphism invariants for direct sums of such groups and for a class of mixed abelian groups properly containing the class of Warfield groups.
Gauge invariant quantization on Riemannian manifolds
Authors: Liu Zhang Ju , Qian Min
Published: June 2012

For every pointwise polynomial function on each fiber of the cotangent bundle of a Riemannian manifold

M
M

, a family of differential operators is given, which acts on the space of smooth sections of a vector bundle on

M
M

. Such a correspondence may be considered as a rule to quantize classical systems moving in a Riemannian manifold or in a gauge field. Some applications of our construction are also given in this paper.
On the symmetric square: applications of a trace formula
Authors: Flicker Yuval Z.
Published: June 2012

In this paper we prove the existence of the symmetric-square lifting of admissible and of automorphic representations from the group

SL

(
2
)

{\text {SL}}(2)

to the group

PGL

(
3
)

{\text {PGL}}(3)

. Complete local results are obtained, relating the character of an

SL

(
2
)

{\text {SL}}(2)

-packet with the twisted character of self-contragredient

PGL

(
3
)

{\text {PGL}}(3)

-modules. Our global results relate packets of cuspidal representations of

SL

(
2
)

{\text {SL}}(2)

with a square-integrable component, and self-contragredient automorphic

PGL

(
3
)

{\text {PGL}}(3)

-modules with a component coming from a square-integrable one. The sharp results, which concern

SL

(
2
)

{\text {SL}}(2)

rather than

GL

(
2
)

{\text {GL}}(2)

, are afforded by the usage of the trace formula. The surjectivity and injectivity of the correspondence implies that any self-contragredient automorphic

PGL

(
3
)

{\text {PGL}}(3)

-module as above is a lift, and that the space of cuspidal

SL

(
2
)

{\text {SL}}(2)

-modules with a square-integrable component admits multiplicity one theorem and rigidity ("strong multiplicity one") theorem for packets (and not for individual representations). The techniques of this paper, based on the usage of regular functions to simplify the trace formula, are pursued in the sequel [

VI
\text {VI}

] to extend our results to all cuspidal

SL

(
2
)

{\text {SL}}(2)

-modules and self-contragredient

PGL

(
3
)

{\text {PGL}}(3)

-modules
On the symmetric square: definitions and lemmas
Authors: Flicker Yuval Z.
Published: June 2012

We define the symmetric square lifting for admissible and automorphic representations, from the group

H
=

H
0

=

SL

(
2
)

H = {H_0} = {\text {SL}}(2)

, to the group

G
=

PGL

(
3
)

G = {\text {PGL}}(3)

, and derive its basic properties. This lifting is defined by means of Shintani character relations. The definition is suggested by the computation of orbital integrals (stable and unstable) in our On the symmetric square: Orbital integrals, Math. Ann. 279 (1987), 173-193. It is compatible with dual group homomorphisms

λ
0

:

H
^

→

G
^

{\lambda _0}:\widehat {H} \to \widehat {G}

and

λ
1

:

H
^

1

→

G
^

{\lambda _1}:{\widehat {H}_1} \to \widehat {G}

, where

H
1

=

PGL

(
2
)

{H_1} = {\text {PGL}}(2)

. The lifting is proven for induced, trivial and special representations, and both spherical functions and orthogonality relations of characters are studied.
Frames associated with an abelian 𝑙-group
Authors: Madden James J.
Published: June 2012

Every archimedean

l
l

-group (lattice-ordered group) with weak unit is shown to be isomorphic to a sub-

l
l

-group of the

l
l

-group of continuous realvalued functions on a Tychonoff locale canonically associated with the

l
l

-group. This strengthens the classical Yosida representation theorem in a useful way. The proof uses methods from universal algebra and is constructive.
Almost periodic potentials in higher dimensions
Authors: Papanicolaou Vassilis G.
Published: June 2012

This work was motivated by the paper of R. Johnson and J. Moser (see [J-M] in the references) on the one-dimensional almost periodic potentials. Here we study the operator

L
=
−
Δ

/

2
−
q

L = - \Delta /2 - q

, where

q
q

is an almost periodic function in

R
d

{R^d}

. It is shown that some of the results of [J-M] extend to the multidimensional case (our approach includes the one-dimensional case as well). We start with the kernel

k
(
t
,
x
,
y
)

k(t,x,y)

of the semigroup

e

−
t
L

{e^{ - tL}}

. For fixed

t
>
0

t > 0

and

u
∈

R
d

u \in {R^d}

, it is known (we review the proof) that

k
(
t
,
x
,
x
+
u
)

k(t,x,x + u)

is almost periodic in

x
x

with frequency module not bigger than the one of

q
q

. We show that

k
(
t
,
x
,
y
)

k(t,x,y)

is, also, uniformly continuous on

[
a
,
b
]
×

R
d

×

R
d

[a,b] \times {R^d} \times {R^d}

. These results imply that, if we set

y
=
x
+
u

y = x + u

in the kernel

G
m

(
x
,
y
;
z
)

{G^m}(x,y;z)

of

(
L
−
z

)

−
m

{(L - z)^{ - m}}

it becomes almost periodic in

x
x

(for the case

u
=
0

u = 0

we must assume that

m
>
d

/

2

m > d/2

), which is a generalization of an old one-dimensional result of Scharf (see [S.G]). After this, we are able to define

w
m

(
z
)
=

M
x

[

G
m

(
x
,
x
;
z
)
]

{w_m}(z) = {M_x}[{G^m}(x,x;z)]

, and, by integrating this

m
m

times, an analog of the complex rotation number

w
(
z
)

w(z)

of [J-M]. We also show that, if

e
(
x
,
y
;
λ
)

e(x,y;\lambda )

is the kernel of the projection operator

E
λ

{E_\lambda }

associated to

L
L

, then the mean value

α
(
λ
)
=

M
x

[
e
(
x
,
x
;
λ
)
]

\alpha (\lambda ) = {M_x}[e(x,x;\lambda )]

exists. In one dimension, this (times

π
\pi

) is the rotation number. In higher dimensions (

d
=
1

d = 1

included), we show that

d
α
(
λ
)

d\alpha (\lambda )

is the density of states measure of [A-S] and it is related to

w
m

(
z
)

{w_m}(z)

in a nice way. Finally, we derive a formula for the functional derivative of

w
m

(
z
;
q
)

{w_m}(z;q)

with respect to

q
q

, which extends a result of [J-M].
Characterizations of algebras arising from locally compact groups
Authors: Patterson Paul L.
Published: June 2012

Two Banach

∗

^{\ast }

-algebras are naturally associated with a locally compact group,

G
G

: the group algebra,

L
1

(
G
)

{L^1}(G)

, and the measure algebra,

M
(
G
)

M(G)

. Either of these Banach algebras is a complete set of invariants for

G
G

. In any Banach

∗

^{\ast }

-algebra,

A
A

, the norm one unitary elements form a group,

S
S

. Using

S
S

we characterize those Banach

∗

^{\ast }

-algebras,

A
A

, which are isometrically

∗

^{\ast }

-isomorphic to

M
(
G
)

M(G)

. Our characterization assumes that

A
A

is the dual of some Banach space and that its operations are continuous in the resulting weak

∗

^{\ast }

topology. The other most important condition is that the convex hull of

S
S

must be weak

∗

^{\ast }

dense in the unit ball of

A
A

. We characterize Banach

∗

^{\ast }

-algebras which are isomerically isomorphic to

L
1

(
G
)

{L^1}(G)

for some

G
G

as those algebras,

A
A

, whose double centralizer algebra,

D
(
A
)

D(A)

, satisfies our characterization for

M
(
G
)

M(G)

. In addition we require

A
A

to consist of those elements of

D
(
A
)

D(A)

on which

S
S

(defined relative to

D
(
A
)

D(A)

) acts continuously with its weak

∗

^{\ast }

topology. Using another characterization of

L
1

(
G
)

{L^1}(G)

we explicitly calculate the above isomorphism between

A
A

and

L
1

(
G
)

{L^1}(G)

.
A Chern character in cyclic homology
Authors: Zamboni Luca Quardo
Published: June 2012

We show that inner derivations act trivially on the cyclic cohomology of the normalized cyclic complex

C

(
Ω
)

/

D

(
Ω
)

\mathcal {C}(\Omega )/\mathcal {D}(\Omega )

where

Ω
\Omega

is a differential graded algebra. This is then used to establish the fact that the map introduced in

[
GJ
]

[ \text {GJ} ]

defines a Chern character in

K
K

theory.
The structure of the space of coadjoint orbits of an exponential solvable Lie group
Authors: Currey Bradley N.
Published: June 2012

In this paper we address the problem of describing in explicit algebraic terms the collective structure of the coadjoint orbits of a connected, simply connected exponential solvable Lie group

G
G

. We construct a partition

℘
\wp

of the dual

g

∗

{\mathfrak {g}^{\ast } }

of the Lie algebra

g

\mathfrak {g}

of

G
G

into finitely many

Ad

∗

⁡
(
G
)

\operatorname {Ad}^{\ast } (G)

-invariant algebraic sets with the following properties. For each

Ω
∈
℘

\Omega \in \wp

, there is a subset

Σ
\Sigma

of

Ω
\Omega

which is a cross-section for the coadjoint orbits in

Ω
\Omega

and such that the natural mapping

Ω

/

Ad

∗

⁡
(
G
)
→
Σ

\Omega /\operatorname {Ad}^{\ast } (G) \to \Sigma

is bicontinuous. Each

Σ
\Sigma

is the image of an analytic

Ad

∗

⁡
(
G
)

\operatorname {Ad}^{\ast }(G)

-invariant function

P
P

on

Ω
\Omega

and is an algebraic subset of

g

∗

{\mathfrak {g}^{\ast }}

. The partition

℘
\wp

has a total ordering such that for each

Ω
∈
℘

\Omega \in \wp

,

∪
{
Ω
′
:
Ω
′
≤
Ω
}

\cup \{ \Omega \prime :\Omega \prime \leq \Omega \}

is Zariski open. For each

Ω
\Omega

there is a cone

W
⊂

g

∗

W \subset {\mathfrak {g}^{\ast } }

, such that

Ω
\Omega

is naturally a fiber bundle over

Σ
\Sigma

with fiber

W
W

and projection

P
P

. There is a covering of

Σ
\Sigma

by finitely many Zariski open subsets

O
O

such that in each

O
O

, there is an explicit local trivialization

Θ
:

P

−
1

(
O
)
→
W
×
O

\Theta :{P^{ - 1}}(O) \to W \times O

. Finally, we show that if

Ω
\Omega

is the minimal element of

℘
\wp

(containing the generic orbits), then its cross-section

Σ
\Sigma

is a differentiable submanifold of

g

∗

{\mathfrak {g}^{\ast } }

. It follows that there is a dense open subset

U
U

of

G

∅
^

G\hat \emptyset

such that

U
U

has the structure of a differentiable manifold and

G

∅
^

∼
U

G\widehat \emptyset \sim U

has Plancherel measure zero.
Earthquakes on Riemann surfaces and on measured geodesic laminations
Authors: Bonahon Francis
Published: June 2012

Let

S
S

be a closed orientable surface of genus at least

2
2

. We study properties of its Teichmüller space

T

(
S
)

\mathcal {T}(S)

, namely of the space of isotopy classes of conformal structures on

S
S

. W. P. Thurston introduced a certain compactification of

T

(
S
)

\mathcal {T}(S)

by what he called the space of projective measured geodesic laminations. He also introduced some transformations of Teichmüller space, called earthquakes, which are intimately related to the geometry of

T

(
S
)

\mathcal {T}(S)

. A general problem is to understand which geometric properties of Teichmüller space subsist at infinity, on Thurston’s boundary. In particular, it is natural to ask whether earthquakes continuously extend at certain points of Thurston’s boundary, and at precisely which points they do so. This is the principal question addressed in this paper.
Iterating maps on cellular complexes
Authors: Willson Stephen J.
Published: June 2012

Let

K
K

be a finite simplicial complex and

f
:
K
→
K

f:K \to K

be a "skeletal" map. A digraph

D
D

is defined whose vertices correspond to the simplexes of

K
K

and whose arcs give information about the behavior of

f
f

on the simplexes. For every walk in

D
D

there exists a point of

K
K

whose iterates under

f
f

mimic the walk. Periodic walks are mimicked by a periodic point. Digraphs with uncountably many infinite walks are characterized; the corresponding maps

f
f

exhibit complicated behavior.
Nonexistence of nodal solutions of elliptic equations with critical growth in ℝ²
Authors: Adimurthi  , Yadava S. L.
Published: June 2012

Let

f
(
t
)
=
h
(
t
)

e

b

t
2

f(t) = h(t){e^{b{t^2}}}

be a function of critical growth. Under a suitable assumption on

h
h

, we prove that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column minus normal upper Delta u equals f left-parenthesis u right-parenthesis 2nd Column a m p semicolon in upper B left-parenthesis upper R right-parenthesis subset-of double-struck upper R squared comma 2nd Row 1st Column u equals 0 2nd Column a m p semicolon on partial-differential upper B left-parenthesis upper R right-parenthesis comma EndLayout">

−
Δ
u
=
f
(
u
)

a
m
p
;

in

B
(
R
)
⊂

R

2

,

u
=
0

a
m
p
;

on

∂
B
(
R
)
,

\begin {array}{*{20}{c}} { - \Delta u = f(u)} \hfill & {{\text {in}}\;B(R) \subset {\mathbb {R}^2},} \hfill \\ {u = 0} \hfill & {{\text {on}}\;\partial B(R),} \hfill \\ \end {array}

\]
does not admit a radial solution which changes sign for sufficiently small

R
R

.
Nests of subspaces in Banach space and their order types
Authors: Arias Alvaro , Farmer Jeff
Published: June 2012

This paper addresses some questions which arise naturally in the theory of nests of subspaces in Banach space. The order topology on the index set of a nest is discussed, as well as the method of spatial indexing by a vector; sufficient geometric conditions for the existence of such a vector are found. It is then shown that a continuous nest exists in any Banach space. Applications and examples follow; in particular, an extension of the Volterra nest in

L
∞

[

0
,
1

]

{L^\infty }[ {0,1} ]

to a continuous one, a continuous nest in a Banach space having no two elements isomorphic to one another, and a characterization of separable

L

p

{\mathcal {L}_p}

-spaces in terms of nests.
A deformation of tori with constant mean curvature in 𝑅³ to those in other space forms
Authors: Umehara Masaaki , Yamada Kotaro
Published: June 2012

It is shown that tori with constant mean curvature in

R

3

{\mathbb {R}^3}

constructed by Wente

[
7
]

[7]

can be deformed to tori with constant mean curvature in the hyperbolic

3
3

-space or the

3
3

-sphere.
Classification of finite-dimensional universal pseudo-boundaries and pseudo-interiors
Authors: Dijkstra J. J. , van Mill J. , Mogilski J.
Published: June 2012

Let

n
n

and

k
k

be fixed integers such that

n
≥
1

n \geq 1

and

0
≤
k
≤
n

0 \leq k \leq n

. Let

B
k
n

B_k^n

and

s
k
n

s_k^n

denote the

k
k

-dimensional universal pseudo-boundary and the

k
k

-dimensional universal pseudo-interior in

R

n

{{\mathbf {R}}^n}

, respectively. The aim of this paper is to prove that

B
k
n

B_k^n

is homeomorphic to

B
k
m

B_k^m

if and only if

s
k
n

s_k^n

is homeomorphic to

s
k
m

s_k^m

if and only if

n
=
m

n = m

or

n
n

,

m
≥
2
k
+
1

m \geq 2k + 1

.
Degree one maps between geometric 3-manifolds
Authors: Rong Yong Wu
Published: June 2012

Let

M
M

and

N
N

be two compact orientable

3
3

-manifolds, we say that

M
≥
N

M \geq N

, if there is a degree one map from

M
M

to

N
N

. This gives a way to measure the complexity of

3
3

-manifolds. The main purpose of this paper is to give a positive answer to the following conjecture: if there is an infinite sequence of degree one maps between Haken manifolds, then eventually all the manifolds are homeomorphic to each other. More generally, we prove a theorem which says that any infinite sequence of degree one maps between the so-called "geometric

3
3

-manifolds" must eventually become homotopy equivalences.
Functors on the category of finite sets
Authors: Dougherty Randall
Published: June 2012

Given a covariant or contravariant functor from the category of finite sets to itself, one can define a function from natural numbers to natural numbers by seeing how the functor maps cardinalities. In this paper we answer the question: what numerical functions arise in this way from functors? The sufficiency of the conditions we give is shown by simple constructions of functors. In order to show the necessity, we analyze the way in which functions in the domain category act on members of objects in the range category, and define combinatorial objects describing this action; the permutation groups in the domain category act on these combinatorial objects, and the possible sizes of orbits under this action restrict the values of the numerical function. Most of the arguments are purely combinatorial, but one case is reduced to a statement about permutation groups which is proved by group-theoretic methods.
Cohomology of the symplectic group 𝑆𝑝₄(𝑍). I. The odd torsion case
Authors: Brownstein Alan , Lee Ronnie
Published: June 2012

Let

h
2

{h_2}

be the degree two Siegel space and

S
p
(
4
,

Z

)

Sp(4,\mathbb {Z})

the symplectic group. The quotient

S
p
(
4
,

Z

)
∖

h
2

Sp(4,\mathbb {Z})\backslash {h_2}

can be interpreted as the moduli space of stable Riemann surfaces of genus

2
2

. This moduli space can be decomposed into two pieces corresponding to the moduli of degenerate and nondegenerate surfaces of genus

2
2

. The decomposition leads to a Mayer-Vietoris sequence in cohomology relating the cohomology of

S
p
(
4
,

Z

)

Sp(4,\mathbb {Z})

to the cohomology of the genus two mapping class group

Γ
2
0

\Gamma _2^0

. Using this tool, the

3
3

- and

5
5

-primary pieces of the integral cohomology of

S
p
(
4
,

Z

)

Sp(4,\mathbb {Z})

are computed.
Global convexity properties of some families of three-dimensional compact Levi-flat hypersurfaces
Authors: Barrett David E.
Published: June 2012

We consider various examples of compact Levi-flat hypersurfaces in two-dimensional complex manifolds, exploring the interplay between geometric properties of the induced foliation, behavior of the tangential Cauchy-Riemann equations along the hypersurface, and pseudoconvexity properties of small neighborhoods of the hypersurface.
Existence of smooth solutions to the classical moment problems
Authors: Jorgensen Palle E. T.
Published: June 2012

Let

s
(
0
)
,
s
(
1
)
,
…

s(0),s(1), \ldots

be a given sequence, and define

s
(
n
)
=

s
(
−
n
)

¯

s(n) = \overline {s( - n)}

for

n
>
0

n > 0

. If

Σ
Σ

ξ
¯

n

ξ
m

s
(
m
−
n
)
≥
0

\Sigma \Sigma {\overline \xi _n}{\xi _m}s(m - n) \geq 0

holds for all finite sequences

(

ξ
n

)

n
∈

Z

{({\xi _n})_{n \in \mathbb {Z}}}

, then it is known that there is a positive Borel measure

μ
\mu

on the circle

T

\mathbb {T}

such that

s
(
n
)
=

∫

−
π

π

e

i
n
t

d
μ
(
t
)

s(n) = \smallint _{ - \pi }^\pi {{e^{int}}d\mu (t)}

, and conversely. Our main theorem provides a necessary and sufficient condition on the sequence

(
s
(
n
)
)

(s(n))

that the measure

μ
\mu

may be chosen to be smooth. A measure

μ
\mu

is said to be smooth if it has the same spectral type as the operator

i
d

/

d
t

id/dt

acting on

L
2

(

T

)

{L^2}(\mathbb {T})

with respect to Haar measure

d
t

dt

on

T

\mathbb {T}

: Equivalently,

μ
\mu

is a superposition (possibly infinite) of measures of the form

|

w
(
t
)

|

2

d
t

|w(t){|^2}dt

with

w
∈

L
2

(

T

)

w \in {L^2}(\mathbb {T})

such that

d
w

/

d
t
∈

L
2

(

T

)

dw/dt \in {L^2}(\mathbb {T})

. The condition is stated purely in terms of the initially given sequence

(
s
(
n
)
)

(s(n))

: We show that a smooth representation exists if and only if, for some

ε
∈

R

+

\varepsilon \in {\mathbb {R}_ + }

, the a priori estimate
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation sigma-summation s left-parenthesis m minus n right-parenthesis xi overbar Subscript n Baseline xi Subscript m Baseline greater-than-or-equal-to epsilon StartAbsoluteValue sigma-summation n s left-parenthesis n right-parenthesis xi Subscript n Baseline EndAbsoluteValue squared">

∑

∑

s
(
m
−
n
)

ξ
¯

n

ξ
m

≥
ε

|

∑

n
s
(
n
)

ξ
n

|

2

\sum {\sum {s(m - n){{\overline \xi }_n}{\xi _m} \geq \varepsilon {{\left | {\sum {ns(n){\xi _n}} } \right |}^2}} }

\]
is valid for all finite double sequences

(

ξ
n

)

({\xi _n})

. An analogous result is proved for the determinate (Hamburger) moment problem on the line. But the corresponding result does not hold for the indeterminate moment problem.
Connected simple systems, transition matrices, and heteroclinic bifurcations
Authors: McCord Christopher , Mischaikow Konstantin
Published: June 2012

Given invariant sets

A
A

,

B
B

, and

C
C

, and connecting orbits

A
→
B

A \to B

and

B
→
C

B \to C

, we state very general conditions under which they bifurcate to produce an

A
→
C

A \to C

connecting orbit. In particular, our theorem is applicable in settings for which one has an admissible semiflow on an isolating neighborhood of the invariant sets and the connecting orbits, and for which the Conley index of the invariant sets is the same as that of a hyperbolic critical point. Our proof depends on the connected simple system associated with the Conley index for isolated invariant sets. Furthermore, we show how this change in connected simple systems can be associated with transition matrices, and hence, connection matrices. This leads to some simple examples in which the nonuniqueness of the connection matrix can be explained by changes in the connected simple system.
Galois groups of maximal 𝑝-extensions
Authors: Ware Roger
Published: June 2012

Let

p
p

be an odd prime and

F
F

a field of characteristic different from

p
p

containing a primitive

p
p

th root of unity. Assume that the Galois group

G
G

of the maximal

p
p

-extension of

F
F

has a finite normal series with abelian factor groups. Then the commutator subgroup of

G
G

is abelian. Moreover,

G
G

has a normal abelian subgroup with pro-cyclic factor group. If, in addition,

F
F

contains a primitive

p
2

{p^2}

th root of unity then

G
G

has generators

{
x
,

y
i

}

i
∈
I

{\{ x,{y_i}\} _{i \in I}}

with relations

y
i

y
j

=

y
j

y
i

{y_i}{y_j} = {y_j}{y_i}

and

x

y
i

x

−
1

=

y
i

q
+
1

x{y_i}{x^{ - 1}} = y_i^{q + 1}

where

q
=
0

q = 0

or

q
=

p
n

q = {p^n}

for some

n
≥
1

n \geq 1

. This is used to calculate the cohomology ring of

G
G

, when

G
G

has finite rank. The field

F
F

is characterized in terms of the behavior of cyclic algebras (of degree

p
p

) over finite

p
p

-extensions.
Elementary proofs of the abstract prime number theorem for algebraic function fields
Authors: Zhang Wen-Bin
Published: June 2012

Elementary proofs of the abstract prime number theorem of the form

Λ
(
m
)
=

q
m

+
O
(

q
m

m

−
1

)

\Lambda (m) = {q^m} + O({q^m}{m^{ - 1}})

for algebraic function fields are given. The proofs use a refinement of a tauberian theorem of Bombieri.
Fourier multipliers on Lipschitz curves
Authors: McIntosh Alan , Qian Tao
Published: June 2012

We develop the theory of Fourier multipliers acting on

L
p

(
γ
)

{L_p}(\gamma )

where

γ
\gamma

is a Lipschitz curve of the form

γ
=
{
x
+
i
g
(
x
)
}

\gamma = \{ x + ig(x)\}

with

‖
g
‖

∞

>
∞

\left \| g\right \| _\infty > \infty

and

‖
g
′
‖

∞

>
∞

\left \| g\prime \right \| _\infty > \infty

. The aim is to better understand convolution singular integrals

B
B

defined naturally on such curves by
\[

B
u
(
z
)
=

p.v.

∫
γ

φ
(
z
−
ζ
)
u
(
ζ
)
d
ζ

Bu(z) = {\text {p.v.}}\int _\gamma {\varphi (z - \zeta )u(\zeta )d\zeta }

\]
for almost all

z
∈
γ

z \in \gamma

.
On the 𝑝-adic completions of nonnilpotent spaces
Authors: Bousfield A. K.
Published: June 2012

This paper deals with the

p
p

-adic completion

F

p
∞

X

{F_{p\infty }}X

developed by Bousfield-Kan for a space

X
X

and prime

p
p

. A space

X
X

is called

F
p

{F_p}

-good when the map

X
→

F

p
∞

X

X \to {F_{p\infty }}X

is a

mod
-
p

\bmod \text {-}p

homology equivalence, and called

F
p

{F_p}

-bad otherwise. General examples of

F
p

{F_p}

-good spaces are established beyond the usual nilpotent or virtually nilpotent ones. These include the polycyclic-by-finite spaces. However, the wedge of a circle with a sphere of positive dimension is shown to be

F
p

{F_p}

-bad. This provides the first example of an

F
p

{F_p}

-bad space of finite type and implies that the

p
p

-profinite completion of a free group on two generators must have nontrivial higher

mod
-
p

\bmod \text {-}p

homology as a discrete group. A major part of the paper is devoted to showing that the desirable properties of nilpotent spaces under the

p
p

-adic completion can be extended to the wider class of

p
p

-seminilpotent spaces.
Nonsingular affine 𝑘*-surfaces
Authors: Rynes Jean
Published: June 2012

Nonsingular affine

k

∗

{k^{\ast } }

-surfaces are classified as certain invariant open subsets of projective

k

∗

{k^{\ast }}

-surfaces. A graph is defined which is an equivariant isomorphism invariant of an affine

k

∗

{k^{\ast }}

-surface. Over the complex numbers, it is proved that the only acyclic affine surface which admits an effective action of the group

C

∗

{{\mathbf {C}}^{\ast } }

is

C

2

{{\mathbf {C}}^2}

which admits only linear actions of

C

∗

{{\mathbf {C}}^{\ast }}

.
Liouvillian first integrals of differential equations
Authors: Singer Michael F.
Published: June 2012

Liouvillian functions are functions that are built up from rational functions using exponentiation, integration, and algebraic functions. We show that if a system of differential equations has a generic solution that satisfies a liouvillian relation, that is, there is a liouvillian function of several variables vanishing on the curve defined by this solution, then the system has a liouvillian first integral, that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set. We can refine this result in special cases to show that the first integral must be of a very special form. For example, we can show that if the system

d
x

/

d
z
=
P
(
x
,
y
)

dx/dz = P(x,y)

,

d
y

/

d
z
=
Q
(
x
,
y
)

dy/dz = Q(x,y)

has a solution

(
x
(
z
)
,
y
(
z
)
)

(x(z),y(z))

satisfying a liouvillian relation then either

x
(
z
)

x(z)

and

y
(
z
)

y(z)

are algebraically dependent or the system has a liouvillian first integral of the form

F
(
x
,
y
)
=
∫
R
Q

d
x
−
R
P

d
y

F(x,y) = \smallint RQ\,dx - RP\,dy

where

R
=
exp
⁡
(
∫
U

d
x
+
V

d
y
)

R = \exp (\smallint U\,dx + V\,dy)

and

U
U

and

V
V

rational functions of

x
x

and

y
y

. We can also reprove an old result of Ritt stating that a second order linear differential equation has a nonconstant solution satisfying a liouvillian relation if and only if all of its solutions are liouvillian.
Complete nonorientable minimal surfaces in 𝑅³
Authors: Ishihara Tōru
Published: June 2012

We will study complete minimal immersions of nonorientable surfaces into

R
3

{R^3}

. Especially, we construct a nonorientable surface

P
2

{P_2}

which is homeomorphic to a Klein bottle and show that for any integer

m
≥
4

m \geq 4

, there are complete minimal immersion of

M
=

P
2

−
{
q
}

M = {P_2} - \{ q\}

,

q
∈

P
2

q \in {P_2}

in

R
3

{R^3}

with one end and total curvature

C
(
M
)
=
−
4
m
π

C(M) = - 4m\pi

.
Number of solutions with a norm bounded by a given constant of a semilinear elliptic PDE with a generic right-hand side
Authors: Nabutovsky Alexander
Published: June 2012

We consider a semilinear boundary value problem

−
Δ
u
+
f
(
u
,
x
)
=
0

- \Delta u + f(u,x) = 0

in

Ω
⊂

R

N

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

and

u
=
0

u = 0

on

∂
Ω

\partial \Omega

. We assume that

f
f

is a

C
∞

{C^\infty }

-smooth function and

Ω
\Omega

is a bounded domain with a smooth boundary. For any

C
α

{C^\alpha }

-smooth perturbation

h
(
x
)

h(x)

of the right-hand side of the equation we consider the function

N
h

(
S
)

{N_h}(S)

defined as the number of

C

2
+
α

{C^{2 + \alpha }}

-smooth solutions

u
u

such that

‖
u
‖

C
0

(
Ω
)

≤
S

\left \| u\right \| _{{C^0}(\Omega )} \leq S

of the perturbed problem. How "small"

N
h

(
S
)

{N_h}(S)

can be made by a perturbation

h
(
x
)

h(x)

such that

‖
h
‖

C
0

(
Ω
)

≤
ε
?

\left \| h\right \| _{{C^0}(\Omega )} \leq \varepsilon ?

We present here an explicit upper bound in terms of

ε
\varepsilon

,

S
S

and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="max Underscript StartAbsoluteValue u EndAbsoluteValue less-than-or-equal-to upper S comma x element-of normal upper Omega overbar Endscripts double-vertical-bar upper D Subscript u Superscript i Baseline f left-parenthesis u comma x right-parenthesis double-vertical-bar left-parenthesis i element-of StartSet 0 comma 1 comma 2 EndSet right-parenthesis period">

max

|

u

|

≤
S
,
x
∈

Ω
¯

‖

D
u
i

f
(
u
,
x
)
‖

(
i
∈
{
0
,
1
,
2
}
)
.

\max \limits _{|u| \leq S,x \in \bar \Omega } \left \| D_u^i f(u,x)\right \| \quad (i \in \{ 0,1,2\} ).

\]
If

S
S

is fixed then

h
h

can be chosen by such a way that the upper bound persists under small in

C
0

{C^0}

-topology perturbations of

h
h

. We present an explicit lower bound for the radius of the ball of such admissible perturbations.
Noncommutative matrix Jordan algebras
Authors: Brown Robert B. , Hopkins Nora C.
Published: June 2012

We consider noncommutative degree two Jordan algebras

J

\mathcal {J}

of two by two matrices whose off diagonal entries are from an anticommutative algebra

S

\mathcal {S}

. We give generators and relations for the automorphism group of

J

\mathcal {J}

and determine the derivation algebra Der

J

\mathcal {J}

in terms of mappings on

S

\mathcal {S}

. We also give an explicit construction of all

S

\mathcal {S}

for which Der

J

\mathcal {J}

does not kill the diagonal idempotents and give conditions for nonisomorphic

S

\mathcal {S}

’s to give isomorphic

J

\mathcal {J}

’s.
Eta invariants of Dirac operators on foliated manifolds
Authors: Perić Goran
Published: June 2012

We define the eta function of Dirac operators on foliated manifolds. We show that the eta functions are regular at the origin thus defining corresponding eta invariants of foliated manifolds. Under the hypothesis of invertibility of the operator in question, we prove the Atiyah-Singer relative index theorem for Dirac operators on foliated manifolds. Then we discuss the homotopy invariance of the index with respect to secondary data.
Analytic geometry of complex superspaces
Authors: Flenner H. , Sundararaman D.
Published: June 2012

A detailed account of the analytic geometry of complex superspaces is given in this paper. Several representability criteria and representability theorems are proved. In particular, the existence of a versal family of deformations (parametrized by a complex superspace) for any compact complex superspace is proved.
On the generalized Ramanujan-Nagell equation 𝑥²-𝐷=2ⁿ⁺²
Authors: Le Mao Hua
Published: June 2012

Let

D
D

be a positive integer which is odd. In this paper we prove that the equation

x
2

−
D
=

2

n
+
2

{x^2} - D = {2^{n + 2}}

has at most three positive integer solutions

(
x
,
n
)

(x,n)

except when

D
=

2

2
m

−
3
⋅

2

m
+
1

+
1

D = {2^{2m}} - 3 \cdot {2^{m + 1}} + 1

, where

m
m

is a positive integer with

m
≥
3

m \geq 3

.