bluuebunny/crossref_metadata_2025 huggingface dataset:
Function spaces of completely metrizable spaces
Authors: Baars Jan , de Groot Joost , Pelant Jan
Published: June 2012

Let

X
X

and

Y
Y

be metric spaces and let

ϕ
:

C
p

(
X
)
→

C
p

(
Y
)

\phi :{C_p}(X) \to {C_p}(Y)

(resp.

ϕ
:

C
p
∗

(
X
)
→

C
p
∗

(
Y
)

\phi :C_p^\ast (X) \to C_p^\ast (Y)

) be a continuous linear surjection. We prove that

Y
Y

is completely metrizable whenever

X
X

is. As a corollary we obtain that complete metrizability is preserved by

l
p

{l_p}

(resp.

l
p
∗

l_p^\ast

-equivalence) in the class of all metric spaces. This solves Problem 35 in [2] (raised by Arhangel’skiĭ).
The Ehrenfeucht-Fraïssé-game of length 𝜔₁
Authors: Mekler Alan , Shelah Saharon , Väänänen Jouko
Published: June 2012

Let

A

\mathfrak {A}

and

B

\mathfrak {B}

be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fraïssé-game of length

ω
1

{\omega _1}

of

A

\mathfrak {A}

and

B

\mathfrak {B}

which we denote by

G

ω
1

(

A

,

B

)

{\mathcal {G}_{{\omega _1}}}(\mathfrak {A},\mathfrak {B})

. This game is like the ordinary Ehrenfeucht-Fraïssé-game of

L

ω
ω

{L_{\omega \omega }}

except that there are

ω
1

{\omega _1}

moves. It is clear that

G

ω
1

(

A

,

B

)

{\mathcal {G}_{{\omega _1}}}(\mathfrak {A},\mathfrak {B})

is determined if

A

\mathfrak {A}

and

B

\mathfrak {B}

are of cardinality

≤

ℵ
1

\leq {\aleph _1}

. We prove the following results: Theorem 1. If

V
=
L

V = L

, then there are models

A

\mathfrak {A}

and

B

\mathfrak {B}

of cardinality

ℵ
2

{\aleph _2}

such that the game

G

ω
1

(

A

,

B

)

{\mathcal {G}_{{\omega _1}}}(\mathfrak {A},\mathfrak {B})

is nondetermined. Theorem 2. If it is consistent that there is a measurable cardinal, then it is consistent that

G

ω
1

(

A

,

B

)

{\mathcal {G}_{{\omega _1}}}(\mathfrak {A},\mathfrak {B})

is determined for all

A

\mathfrak {A}

and

B

\mathfrak {B}

of cardinality

≤

ℵ
2

\leq {\aleph _2}

. Theorem 3. For any

κ
≥

ℵ
3

\kappa \geq {\aleph _3}

there are

A

\mathfrak {A}

and

B

\mathfrak {B}

of cardinality

κ
\kappa

such that the game

G

ω
1

(

A

,

B

)

{\mathcal {G}_{{\omega _1}}}(\mathfrak {A},\mathfrak {B})

is nondetermined.
Δ-sets
Authors: Knight R. W.
Published: June 2012

A model of

ZFC

{\text {ZFC}}

is constructed in which there exists a subset of the Moore plane that is countably paracompact but not normal. The method used in the construction is forcing using uncountable sets of finite partial functions,

ω
1

{\omega _1}

and

ω
2

{\omega _2}

are shown to be preserved using a fusion lemma.
An atriodic simple-4-od-like continuum which is not simple-triod-like
Authors: Minc Piotr
Published: June 2012

The paper contains an example of a continuum

K
K

such that

K
K

is the inverse limit of simple

4
4

-ods,

K
K

cannot be represented as the inverse limit of simple triods and each proper subcontinuum of

K
K

is an arc.
Unlinking via simultaneous crossing changes
Authors: Scharlemann Martin
Published: June 2012

Given two distinct crossings of a knot or link projection, we consider the question: Under what conditions can we obtain the unlink by changing both crossings simultaneously? More generally, for which simultaneous twistings at the crossings is the genus reduced? Though several examples show that the answer must be complicated, they also suggest the correct necessary conditions on the twisting numbers.
Solutions containing a large parameter of a quasi-linear hyperbolic system of equations and their nonlinear geometric optics approximation
Authors: Yoshikawa Atsushi
Published: June 2012

It is well known that a quasi-linear first order strictly hyperbolic system of partial differential equations admits a formal approximate solution with the initial data
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Superscript negative 1 Baseline a 0 left-parenthesis lamda x bullet eta comma x right-parenthesis r 1 left-parenthesis eta right-parenthesis comma lamda greater-than 0 comma x comma eta element-of bold upper R Superscript n Baseline comma eta not-equals 0">

λ

−
1

a
0

(
λ
x
∙
η
,
x
)

r
1

(
η
)
,
λ
>
0
,
x
,
η
∈

R

n

,
η
≠
0

{\lambda ^{ - 1}}{a_0}(\lambda x \bullet \eta ,x){r_1}(\eta ),\lambda > 0,x,\eta \in {{\mathbf {R}}^n}, \eta \ne 0

. Here

r
1

(
η
)

{r_1}(\eta )

is a characteristic vector, and

a
0

(
σ
,
x
)

{a_0}(\sigma ,x)

is a smooth scalar function of compact support. Under the additional requirements that

n
=
2

n = 2

or

3
3

and that

a
0

(
σ
,
x
)

{a_0}(\sigma ,x)

have the vanishing mean with respect to

σ
\sigma

, it is shown that a genuine solution exists in a time interval independent of

λ
\lambda

, and that the formal solution is asymptotic to the genuine solution as

λ
→
∞

\lambda \to \infty

.
Perverse sheaves and finite-dimensional algebras
Authors: Vilonen K.
Published: June 2012

We study finite dimensional algebras which arise from categories of perverse sheaves on complex algebraic varieties.
Invariant subspaces of the Dirichlet shift and pseudocontinuations
Authors: Richter Stefan , Sundberg Carl
Published: June 2012

In this paper we study extremal functions for invariant subspaces

M

\mathcal {M}

of the Dirichlet shift, i.e., solutions

φ
\varphi

of the extremal problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup left-brace StartAbsoluteValue f Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis 0 right-parenthesis EndAbsoluteValue slash double-vertical-bar f double-vertical-bar Subscript upper D Baseline colon f element-of script upper M comma f not-equals 0 right-brace">

sup
{

|

f

(
n
)

(
0
)

|

/

‖
f
‖

D

:
f
∈

M

,
f
≠
0
}

\sup \{ |{f^{(n)}}(0)|/{\left \| f \right \|_D}:f \in \mathcal {M},f \ne 0\}

\]
. Here

n
n

is the smallest nonnegative integer such that the sup is positive. It is known that such a function

φ
\varphi

generates

M

\mathcal {M}

. We show that the derivative

(
z
φ
)
′

(z\varphi )\prime

has a pseudocontinuation to the exterior disc. This pseudocontinuation is an analytic continuation exactly near those points of the unit circle where

φ
\varphi

is bounded away from zero. We also show that the radial limit of the absolute value of an extremal function exists at every point of the unit circle. Some of our results are valid for all functions that are orthogonal to a nonzero invariant subspace.
Carleson measures on spaces of homogeneous type
Authors: Gadbois Steven C. , Sledd William T.
Published: June 2012

Let

X
X

be a space of homogeneous type in the sense of Coifman and Weiss

[

CW

2
]

[{\text {CW}}2]

and let

X
+

=
X
×

R

+

{X^ + } = X \times {{\mathbf {R}}^ + }

. A positive function

F
F

on

X
+

{X^ + }

is said to have horizontal bounded ratio

(

HBR

)

({\text {HBR}})

on

X
+

{X^ + }

if there is a constant

A
F

{A_F}

so that

F
(
x
,
t
)
≤

A
F

F
(
y
,
t
)

F(x,t) \leq {A_F}F(y,t)

whenever

ρ
(
x
,
y
)
>
t

\rho (x,y) > t

. (By Harnack’s inequality, a well-known example is any positive harmonic function in the upper half plane.)

HBR

{\text {HBR}}

is a rich class that is closed under a wide variety of operations and it provides useful majorants for many classes of functions that are encountered in harmonic analysis. We are able to prove theorems in spaces of homogeneous type for functions in

HBR

{\text {HBR}}

which are analogous to the classical Carleson measure theorems and to extend these results to the functions which they majorize. These results may be applied to obtain generalizations of the original Carleson measure theorem, and of results of Flett’s which contain the Hardy-Littlewood theorems on intermediate spaces of analytic functions. Hörmander’s generalization of Carleson’s theorem is included and it is possible to extend those results to the atomic

H
p

{H^p}

spaces of Coifman and Weiss.
A maximally pathological Brouwer homeomorphism
Authors: Daw Edward Warwick
Published: June 2012

This paper constructs a Brouwer homeomorphism which does not act properly discontinuously on any nonempty invariant closed connected set in

R

2

{\mathbb {R}^2}

.
Bounded holomorphic functions on bounded symmetric domains
Authors: Cohen Joel M. , Colonna Flavia
Published: June 2012

Let D be a bounded homogeneous domain in

C

n

{\mathbb {C}^n}

, and let

Δ
\Delta

denote the open unit disk. If

z
∈
D

z \in D

and

f
:
D
→
Δ

f:D \to \Delta

is holomorphic, then

β
f

(
z
)

{\beta _f}(z)

is defined as the maximum ratio

|

∇
z

(
f
)
x

|

/

H
z

(
x
,

x
¯

)

1

/

2

|{\nabla _z}(f)x|/{H_z}{(x,\bar x)^{1/2}}

, where x is a nonzero vector in

C

n

{\mathbb {C}^n}

and

H
z

{H_z}

is the Bergman metric on D. The number

β
f

(
z
)

{\beta _f}(z)

represents the maximum dilation of f at z. The set consisting of all

β
f

(
z
)

{\beta _f}(z)

for

z
∈
D

z \in D

and

f
:
D
→
Δ

f:D \to \Delta

holomorphic, is known to be bounded. We let

c
D

{c_D}

, be its least upper bound. In this work we calculate

c
D

{c_D}

for all bounded symmetric domains having no exceptional factors and give indication on how to handle the general case. In addition we describe the extremal functions (that is, the holomorphic functions f for which

β
f

=

c
D

{\beta _f} = {c_D}

) when D contains

Δ
\Delta

as a factor, and show that the class of extremal functions is very large when

Δ
\Delta

is not a factor of D.
Ulam-Zahorski problem on free interpolation by smooth functions
Authors: Olevskiĭ A.
Published: June 2012

Let f be a function belonging to

C
n

[
0
,
1
]

{C^n}[0,1]

. Is it possible to find a smoother function

g
∈

C

n
+
1

g \in {C^{n + 1}}

(or at least

C

n
+
ε

{C^{n + \varepsilon }}

) which has infinitely many points of contact of maximal order n with f (or at least arbitrarily many such points with fixed norm

‖
g
‖

C

n
+
ε

{\left \| g \right \|_{{C^{n + \varepsilon }}}}

)? It turns out that for n = 0 and 1 the answer is positive, but if

n
≥
2

n \geq 2

, it is negative. This gives a complete solution to the Ulam-Zahorski question on free interpolation on perfect sets.
A controlled plus construction for crumpled laminations
Authors: Daverman R. J. , Tinsley F. C.
Published: June 2012

Given a closed n-manifold M

(
n
>
4
)

(n > 4)

and a finitely generated perfect subgroup P of

π
1

(
M
)

{\pi _1}(M)

, we previously developed a controlled version of Quillen’s plus construction, namely a cobordism (W, M, N) with the inclusion

j
:
N
↦
W

j:N \mapsto W

a homotopy equivalence and kernel of

i
#

:

π
1

(
M
)
↦

π
1

(
W
)

{i_\# }:{\pi _1}(M) \mapsto {\pi _1}(W)

equalling the smallest normal subgroup of

π
1

(
M
)

{\pi _1}(M)

containing P together with a closed map

p
:
W
↦
[
0
,
1
]

p:W \mapsto [0,1]

such that

p

−
1

(
t
)

{p^{ - 1}}(t)

is a closed n-manifold for every

t
∈
[
0
,
1
]

t \in [0,1]

and, in particular,

M
=

p

−
1

(
0
)

M = {p^{ - 1}}(0)

and

N
=

p

−
1

(
1
)

N = {p^{ - 1}}(1)

. We accomplished this by constructing an acyclic map of manifolds

f
:
M
↦
N

f:M \mapsto N

having the right fundamental groups, and W arose as the mapping cylinder of f with a collar attached along N. The main result here presents a condition under which the desired controlled plus construction can still be accomplished in many cases even when

π
1

(
M
)

{\pi _1}(M)

contains no finitely generated perfect subgroups. By-products of these results include a new method for constructing wild embeddings of codimension one manifolds and a better understanding of perfect subgroups of finitely presented groups.
Pseudocircles in dynamical systems
Authors: Kennedy Judy A. , Yorke James A.
Published: June 2012

We construct an example of a

C
∞

{C^\infty }

map on a 3-manifold which has an invariant set with an uncountable number of components, each of which is a pseudocircle. Furthermore, any map which is sufficiently close (in the

C
1

{C^1}

-metric) to the constructed map has a similar set.
The theory of Jacobi forms over the Cayley numbers
Authors: Eie M. , Krieg A.
Published: June 2012

As a generalization of the classical theory of Jacobi forms we discuss Jacobi forms on

H

×

C

8

\mathcal {H} \times {\mathbb {C}^8}

, which are related with integral Cayley numbers. Using the Selberg trace formula we give a simple explicit formula for the dimension of the space of Jacobi forms. The orthogonal complement of the space of cusp forms is shown to be spanned by certain types of Eisenstein series.
Flows in fibers
Authors: Tanaka Jun-ichi
Published: June 2012

Let

H
∞

(
Δ
)

{H^\infty }(\Delta )

be the algebra of all bounded analytic functions on the open unit disc

Δ
\Delta

, and let

M

(

H
∞

(
Δ
)
)

\mathfrak {M}({H^\infty }(\Delta ))

be the maximal ideal space of

H
∞

(
Δ
)

{H^\infty }(\Delta )

. Using a flow, we represent a reasonable portion of a fiber in

M

(

H
∞

(
Δ
)
)

\mathfrak {M}({H^\infty }(\Delta ))

. This indicates a relation between the corona theorem and the individual ergodic theorem. As an application, we answer a question of Forelli [3] by showing that there exists a minimal flow on which the induced uniform algebra is not a Dirichlet algebra. The proof rests on the fact that the closure of a nonhomeomorphic part in

M

(

H
∞

(
Δ
)
)

\mathfrak {M}({H^\infty }(\Delta ))

may contain a homeomorphic copy of

M

(

H
∞

(
Δ
)
)

\mathfrak {M}({H^\infty }(\Delta ))

. Taking suitable factors, we may construct a lot of minimal flows which are not strictly ergodic.
Nonlinear quantum fields in ≥4 dimensions and cohomology of the infinite Heisenberg group
Authors: Pedersen J. , Segal I. E. , Zhou Z.
Published: June 2012

Aspects of the cohomology of the infinite-dimensional Heisenberg group as represented on the free boson field over a given Hilbert space are treated. The 1-cohomology is shown to be trivial in certain spaces of generalized vectors. From this derives a canonical quantization mapping from classical (unquantized) forms to generalized operators on the boson field. An example, applied here to scalar relativistic fields, is the quantization of a given classical interaction Lagrangian or Hamiltonian, i.e., the establishment and characterization of corresponding boson field operators. For example, if

ϕ
\phi

denotes the free massless scalar field in d-dimensional Minkowski space (

d
≥
4

d \geq 4

, even) and if q is an even integer greater than or equal to 4, then

∫

M

0

:
ϕ

(
X

)
q

:
d
X

{\smallint _{{{\mathbf {M}}_0}}}:\phi {(X)^q}:dX

exists as a nonvanishing, Poincaré invariant, hermitian, selfadjointly extendable operator, where :

ϕ

(
X

)
q

\phi {(X)^q}

: denotes the Wick power. Applications are also made to the rigorous establishment of basic symbolic operators in heuristic quantum field theory, including certain massive field theories; to a class of pseudo-interacting fields obtained by substituting the free field into desingularized expressions for the total Hamiltonian in the conformally invariant case

d
=
q
=
4

d = q = 4

and to corresponding scattering theory.
Intersections of analytically and geometrically finite subgroups of Kleinian groups
Authors: Anderson James W.
Published: June 2012

We consider the intersection of pairs of subgroups of a Kleinian group of the second kind K whose limit sets intersect, where one subgroup G is analytically finite and the other J is geometrically finite, possibly infinite cyclic. In the case that J is infinite cyclic generated by M, we show that either some power of M lies in G or there is a doubly cusped parabolic element Q of G with the same fixed point as M. In the case that J is nonelementary, we show that the intersection of the limit sets of G and J is equal to the limit set of the intersection

G
∩
J

G \cap J

union with a subset of the rank 2 parabolic fixed points of K. Hence, in both cases, the limit set of the intersection is essentially equal to the intersection of the limit sets. The main facts used in the proof are results of Beardon and Pommerenke [4] and Canary [6], which yield that the Poincaré metric on the ordinary set of an analytically finite Kleinian group G is comparable to the Euclidean distance to the limit set of G.
Generalised Castelnuovo inequalities
Authors: Donohoe Liam A.
Published: June 2012

Given a Riemann surface of genus p, denoted by

X
p

{X_p}

, admitting j linear series of dimension r and degree n Accola derived a polynomial function

f
(
j
,
n
,
r
)

f(j,n,r)

so that

p
≤
f
(
j
,
n
,
r
)

p \leq f(j,n,r)

and exhibited plane models of Riemann surfaces attaining equality in the inequality. In this paper we provide a classification of all such

X
p

{X_p}

when

r
≥
6

r \geq 6

. In addition we classify curves,

X
p

{X_p}

, of maximal genus when

X
p

{X_p}

admits two linear series which have a common dimension but different degrees.
Positive harmonic functions vanishing on the boundary for the Laplacian in unbounded horn-shaped domains
Authors: Ioffe Dimitry , Pinsky Ross
Published: June 2012

Denote points

x
¯

∈

R

d
+
1

\bar x \in {R^{d + 1}}

,

d
≥
2

d \geq 2

, by

x
¯

=
(
ρ
,
θ
,
z
)

\bar x = (\rho ,\theta ,z)

, where

ρ
>
0

\rho > 0

,

θ
∈

S

d
−
1

\theta \in {S^{d - 1}}

, and

z
∈
R

z \in R

. Let

a
:
[
0
,
∞
)
→
(
0
,
∞
)

a:[0,\infty ) \to (0,\infty )

be a nondecreasing

C
2

{C^2}

-function and define the "horn-shaped" domain

Ω
=
{

x
¯

=
(
ρ
,
θ
,
z
)
:

|

z

|

>
a
(
ρ
)
}

\Omega = \{ \bar x = (\rho ,\theta ,z):|z| > a(\rho )\}

and its unit "cylinder"

D
=
{

x
¯

=
(
ρ
,
θ
,
z
)
∈
Ω
:
ρ
>
1
}

D = \{ \bar x = (\rho ,\theta ,z) \in \Omega :\rho > 1\}

. Under appropriate regularity conditions on a, we prove the following theorem: (i) If

∫
∞

a
(
ρ
)

/

ρ
2

d
ρ
=
∞

{\smallint ^\infty }a(\rho )/{\rho ^2}d\rho = \infty

, then the Martin boundary at infinity for

1
2

Δ

\frac {1}{2}\Delta

in

Ω
\Omega

is a single point, (ii) If

∫
∞

a
(
ρ
)

/

ρ
2

d
ρ
>
∞

{\smallint ^\infty }a(\rho )/{\rho ^2}d\rho > \infty

, then the Martin boundary at infinity for

1
2

Δ

\frac {1}{2}\Delta

in

Ω
\Omega

is homeomorphic to

S

d
−
1

{S^{d - 1}}

. More specifically, a sequence

{
(

ρ
n

,

θ
n

,

z
n

)

}

n
=
1

∞

⊂
Ω

\{ ({\rho _n},{\theta _n},{z_n})\} _{n = 1}^\infty \subset \Omega

satisfying

lim

n
→
∞

ρ
n

=
∞

{\lim _{n \to \infty }}{\rho _n} = \infty

is a Martin sequence if and only if

lim

n
→
∞

θ
n

{\lim _{n \to \infty }}{\theta _n}

exists on

S

d
−
1

{S^{d - 1}}

. From (i), it follows that the cone of positive harmonic functions in

Ω
\Omega

vanishing continuously on

∂
Ω

\partial \Omega

is one-dimensional. From (ii), it follows easily that the cone of positive harmonic functions on

Ω
\Omega

vanishing continuously on

∂
Ω

\partial \Omega

is generated by a collection of minimal elements which is homeomorphic to

S

d
−
1

{S^{d - 1}}

. In particular, the above result solves a problem stated by Kesten, who asked what the Martin boundary is for

1
2

Δ

\frac {1}{2}\Delta

in

Ω
\Omega

in the case

a
(
ρ
)
=
1
+

ρ
γ

a(\rho ) = 1 + {\rho ^\gamma }

,

0
>
γ
>
1

0 > \gamma > 1

. Our method of proof involves an analysis as

ρ
→
∞

\rho \to \infty

of the exit distribution on

∂
D

\partial D

for Brownian motion starting from

(
ρ
,
θ
,
z
)
∈
Ω

(\rho ,\theta ,z) \in \Omega

and conditioned to hit D before exiting

Ω
\Omega

.
Braid groups and left distributive operations
Authors: Dehornoy Patrick
Published: June 2012

The decidability of the word problem for the free left distributive law is proved by introducing a structure group which describes the underlying identities. This group is closely connected with Artin’s braid group

B
∞

{B_\infty }

. Braid colourings associated with free left distributive structures are used to show the existence of a unique ordering on the braids which is compatible with left translation and such that every generator

σ
i

{\sigma _i}

is preponderant over all

σ
k

{\sigma _k}

with

k
>
i

k > i

. This ordering is a linear ordering.
Semirigid spaces
Authors: Trnková Věra
Published: June 2012

Semirigid spaces are introduced and used as a means to construct two metrizable spaces with isomorphic monoids of continuous self-maps and nonisomorphic clones; this resolves Problem 1 in [13]. The clone of any free variety of a given type with sufficiently many constants is shown to be isomorphic to the clone of a metrizable (semirigid) space.
Multiplier Hopf algebras
Authors: Van Daele A.
Published: June 2012

In this paper we generalize the notion of Hopf algebra. We consider an algebra A, with or without identity, and a homomorphism

Δ
\Delta

from A to the multiplier algebra

M
(
A
⊗
A
)

M(A \otimes A)

of

A
⊗
A

A \otimes A

. We impose certain conditions on

Δ
\Delta

(such as coassociativity). Then we call the pair

(
A
,
Δ
)

(A,\Delta )

a multiplier Hopf algebra. The motivating example is the case where A is the algebra of complex, finitely supported functions on a group G and where

(
Δ
f
)
(
s
,
t
)
=
f
(
s
t
)

(\Delta f)(s,t) = f(st)

with

s
,
t
∈
G

s,t \in G

and

f
∈
A

f \in A

. We prove the existence of a counit and an antipode. If A has an identity, we have a usual Hopf algebra. We also consider the case where A is a

∗
\ast

-algebra. Then we show that (a large enough) subspace of the dual space can also be made into a

∗
\ast

-algebra.
Théorème de Ney-Spitzer sur le dual de 𝑆𝑈(2)
Authors: Biane Philippe
Published: June 2012

Let

ϕ
\phi

be a central, noneven, positive type function on

SU

(
2
)

{\text {SU}}(2)

with

ϕ
(
e
)
>
1

\phi (e) > 1

. For any polynomial function p on

SU

(
2
)

{\text {SU}}(2)

, let

V
(
p
)

V(p)

be the left convolution operator by

p

/

(
1
−
ϕ
)

p/(1 - \phi )

on

L
2

(

SU

(
2
)
)

{L^2}({\text {SU}}(2))

, we prove that

V
(
p
)

/

V
(
1
)

V(p)/V(1)

is a pseudodifferential operator of order 0 and give an explicit formula for its principal symbol. This is interpreted in terms of Martin compactification of a quantum random walk.
Interpolation of weighted and vector-valued Hardy spaces
Authors: Kisliakov Serguei V. , Xu Quan Hua
Published: June 2012

Real and complex interpolation methods, when applied to the couple

(

H

p
0

(

E
0

;

w
0

)
,

H

p
1

(

E
1

;

w
1

)
)

({H^{{p_0}}}({E_0};{w_0}),{H^{{p_1}}}({E_1};{w_1}))

, give what is expected if

E
0

{E_0}

and

E
1

{E_1}

are quasi-Banach lattices of measurable functions satisfying certain mild conditions and if

log
⁡
(

w
0

1

/

p
0

w
1

−
1

/

p
1

)
∈

BMO

(

w
0

,

w
1

\log (w_0^{1/{p_0}}w_1^{ - 1/{p_1}}) \in {\text {BMO}}\,({w_0},{w_1}

being weights on the unit circle). The last condition is in fact necessary. (It is expected, of course, that the resulting spaces coincide with the subspaces of analytic functions in the corresponding interpolation spaces for the couple
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper L Superscript p 0 Baseline left-parenthesis upper E 0 semicolon w 0 right-parenthesis comma upper L Superscript p 1 Baseline left-parenthesis upper E 1 semicolon w 1 right-parenthesis right-parenthesis period right-parenthesis">

(

L

p
0

(

E
0

;

w
0

)
,

L

p
1

(

E
1

;

w
1

)
)
.
)

({L^{{p_0}}}({E_0};{w_0}),{L^{{p_1}}}({E_1};{w_1})).)
On the additive formulae of the theta functions and a collection of Lambert series pertaining to the modular equations of degree 5
Authors: Shen Li-Chien
Published: June 2012

We examine the connection between the additive formulae of the theta functions, the Fourier series expansion of the 12 elliptic functions, and the logarithmic derivatives of the theta functions. As an application, we study the Lambert series related to the modular equations of degree 5 and many interesting identities of Ramanujan are found in this process.
Geometric invariants for Seifert fibred 3-manifolds
Authors: Ouyang Ming Qing
Published: June 2012

In this paper, we obtain a formula for the

η
\eta

-invariant of the signature operator for some circle bundles over Riemannian

2
2

-orbifolds. We then apply it to Seifert fibred

3
3

-manifolds endowed with one of the six Seifert geometries. By using a relation between the Chern-Simons invariant and the

η
\eta

-invariant, we also derive some elementary formulae for the Chern-Simons invariant of these manifolds. As applications, we show that some families of these manifolds cannot be conformally immersed into the Euclidean space

E

4

{{\mathbf {E}}^4}

.
Porous sets and null sets for elliptic harmonic measures
Authors: Wu Jang-Mei
Published: June 2012

We give a genuinely

n
n

-dimensional construction of uniformly elliptic operators

L
L

in

R

+
n

\mathbb {R}_ + ^n

(of divergence form, and of nondivergence form), which have positive

L
L

-harmonic measures on a class of porous sets on

∂

R

+
n

\partial \mathbb {R}_ + ^n

with zero surface measure. The porosity condition given is sharp. The earlier methods were all two dimensional.
The nilpotency class of finite groups of exponent 𝑝
Authors: Vaughan-Lee Michael
Published: June 2012

We investigate the properties of Lie algebras of characteristic

p
p

which satisfy the Engel identity

x

y
n

=
0

x{y^n} = 0

for some

n
>
p

n > p

. We establish a criterion which (when satisfied) implies that if

a
a

and

b
b

are elements of an Engel-

n
n

Lie algebra

L
L

then

a

b

n
−
2

a{b^{n - 2}}

generates a nilpotent ideal of

L
L

. We show that this criterion is satisfied for

n
=
6
,

p
=
7

n = 6,\,p = 7

, and we deduce that if

G
G

is a finite

m
m

-generator group of exponent

7
7

then

G
G

is nilpotent of class at most

51

m
8

51{m^8}

.
Arithmetic calculus of Fourier transforms by Igusa local zeta functions
Authors: Kimura Tatsuo
Published: June 2012

We show the possibility of explicit calculation of the Fourier transforms of complex powers of relative invariants of some prehomogeneous vector spaces over

R

\mathbb {R}

by using the explicit form of

p
p

-adic Igusa local zeta functions.
On the local regularity of solutions in linear viscoelasticity of several space dimensions
Authors: Kim Jong Uhn
Published: June 2012

In this paper we discuss the local regularity of solutions of a nonlocal system of equations which describe the motion of a viscoelastic medium in several space dimensions. Our main tool is the microlocal analysis combined with MacCamy’s trick and the argument of the classical energy method.
Upper bound for distortion of capacity under conformal mapping
Authors: Thurman Robert E.
Published: June 2012

For a finitely-connected domain

Ω
\Omega

containing

∞
\infty

, with boundary

Γ
\Gamma

, the logarithmic capacity

d
(
Γ
)

d(\Gamma )

is invariant under normalized conformal maps of

Ω
\Omega

. But the capacity of a subset

A
⊂
Γ

A \subset \Gamma

will likely be distorted by such a map. Duren and Schiffer showed that the sharp lower bound for the distortion of the capacity of such a set is the so-called "Robin capacity" of the set

A
A

. We present here the sharp upper bound for the distortion, in terms of conformal invariants of

Ω
\Omega

: the harmonic measures of the boundary components of

Ω
\Omega

and the periods of their harmonic conjugates (the Riemann matrix), and the capacity of

Γ
\Gamma

. In particular, the upper bound depends only on knowing which components of

Γ
\Gamma

contain parts of

A
A

, not on the specific distribution of

A
A

. An extremal configuration is described explicitly for a special case.
The Brown-Peterson homology of Mahowald’s 𝑋_{𝑘} spectra
Authors: Yan Dung Yung
Published: June 2012

We compute the Brown-Peterson homology of Mahowald’s

X
k

{X_k}

spectrum which is the Thom spectrum induced from

Ω

J

2
k

−
1

S
2

→

Ω
2

S
3

−

BO

\Omega {J_{{2^k} - 1}}{S^2} \to {\Omega ^2}{S^3} - {\text {BO}}

, and the edge homomorphism of the Adams-Novikov spectral sequence for

π
∗

(

X
k

)

{\pi _\ast }({X_k})

. We then compute the nonnilpotent elements of

π
∗

(

X
k

)

{\pi _\ast }({X_k})

.
Singular polynomials for finite reflection groups
Authors: Dunkl C. F. , de Jeu M. F. E. , Opdam E. M.
Published: June 2012

The Dunkl operators involve a multiplicity function as parameter. For generic values of this function the simultaneous kernel of these operators, acting on polynomials, is equal to the constants. For special values, however, this kernel is larger. We determine these singular values completely and give partial results on the representations of

G
G

that occur in this kernel.
Algebras associated to the Young-Fibonacci lattice
Authors: Okada Soichi
Published: June 2012

The algebra

F

n

{\mathcal {F}_n}

generated by

E
1

,

…

,

E

n
−
1

{E_1},\; \ldots \;,\;{E_{n - 1}}

subject to the defining relations
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript i Superscript 2 Baseline equals x Subscript i Baseline upper E Subscript i Baseline left-parenthesis i equals 1 comma ellipsis comma n minus 1 right-parenthesis comma upper E Subscript i plus 1 Baseline upper E Subscript i Baseline upper E Subscript i plus 1 Baseline equals y Subscript i Baseline upper E Subscript i plus 1 Baseline left-parenthesis i equals 1 comma ellipsis comma n minus 2 right-parenthesis comma upper E Subscript i Baseline upper E Subscript j Baseline equals upper E Subscript j Baseline upper E Subscript i Baseline left-parenthesis StartAbsoluteValue i minus j EndAbsoluteValue greater-than-or-slanted-equals 2 right-parenthesis">

E
i
2

=

x
i

E
i

(
i
=
1
,

…

,

n
−
1
)
,

E

i
+
1

E
i

E

i
+
1

=

y
i

E

i
+
1

(
i
=
1
,

…

,

n
−
2
)
,

E
i

E
j

=

E
j

E
i

(

|

i
−
j

|

⩾
2
)

E_i^2 = {x_i}{E_i}\;(i = 1,\; \ldots \;,\;n - 1),\;{E_{i + 1}}{E_i}{E_{i + 1}} = {y_i}{E_{i + 1}}\;(i = 1,\; \ldots \;,\;n - 2),\;{E_i}{E_j} = {E_j}{E_i}\;(|i - j| \geqslant 2)

is shown to be a semisimple algebra of dimension

n
!

n!

if the parameters

x
1

,

…

,

x

n
−
1

,

y
1

,

…

,

y

n
−
2

{x_1},\; \ldots \;,\;{x_{n - 1}},\;{y_1},\; \ldots \;,\;{y_{n - 2}}

are generic. We also prove that the Bratteli diagram of the tower

(

F

n

)

n
⩾
0

{({\mathcal {F}_n})_{n \geqslant 0}}

of these algebras is the Hasse diagram of the Young-Fibonacci lattice, which is an interesting example, as well as Young’s lattice, of a differential poset introduced by

R
\operatorname {R}

. Stanley. A Young-Fibonacci analogue of the ring of symmetric functions is given and studied.
The Mackey obstruction and the coadjoint orbits
Authors: Li Zongyi
Published: June 2012

This paper studies the Mackey obstruction representation theory at the coadjoint orbit level. It shows how to get rid of such obstructions and to get orbits of the "little groups". Such little group data is essential for inductive construction of coadjoint orbits of general Lie groups.
Algebraic field theory on curved manifolds
Authors: Olesen Martin
Published: June 2012

In this paper we set up an algebraic framework for the study of quantum field theory in a class of manifolds, which includes Minkowski space and the Kruskal spacetime. The formalism provides a unifying framework for studying problems of Bisognano-Wichmann type, e.g., Hawking radiation in black hole geometries. Analogously to flat spacetime, we establish a correspondence between isometries of certain wedge domains of spacetime and the modular structure of the local algebras. Under an ergodic hypothesis, the wedge algebras are shown to be type III factors as expected, and we derive a result concerning factorization of the equilibrium state. This result generalizes a similar one obtained by Sewell in [Ann. Phys. 141 (1982), 201-224]. Finally an example of a quantum field theory satisfying the basic axioms is constructed. The local algebras are field algebras of bosonic free field solutions to the Klein-Gordon equation twisted through a PCT-like conjugation, and we show that this model realizes the abstract properties developed on the axiomatic basis.
On the existence of global Tchebychev nets
Authors: Samelson Sandra L. , Dayawansa W. P.
Published: June 2012

Let

S
S

be a complete, open simply connected surface. Suppose that the integral of the Gauss curvature over arbitrary measurable sets is less than

π

/

2

\pi /2

in magnitude. We show that the surface admits a global Tchebychev net.
Bounded point evaluation in 𝐶ⁿ
Authors: Range R. M. , Stessin M. I.
Published: June 2012

A positive Borel measure

μ
\mu

on a domain

Ω
∈

C

n

\Omega \in {{\mathbf {C}}^n}

is said to be in

R

(
Ω
)

\mathcal {R}(\Omega )

, if point evaluations at every

p
∈
Ω

p \in \Omega

are locally uniformly bounded in

L
2

(
μ
)

{L^2}(\mu )

-norm. It is proved that the multiplication of a measure in

R

(
Ω
)

\mathcal {R}(\Omega )

by a function decreasing no faster than a power of a holomorphic function produces a measure in

R

(
Ω
)

\mathcal {R}(\Omega )

. Some applications to classical Hardy and Bergman spaces are given.
The exposed points of the set of invariant means
Authors: Miao Tianxuan
Published: June 2012

Let

G
G

be a

σ
\sigma

-compact infinite locally compact group, and let

L
I
M

LIM

be the set of left invariant means on

L
∞

(
G
)

{L^\infty }(G)

. We prove in this paper that if

G
G

is amenable as a discrete group, then

L
I
M

LIM

has no exposed points. We also give another proof of the Granirer theorem that the set

L
I
M
(
X
,
G
)

LIM(X,G)

of

G
G

-invariant means on

L
∞

(
X
,
β
,
p
)

{L^\infty }(X,\beta ,p)

has no exposed points, where

G
G

is an amenable countable group acting ergodically as measure-preserving transformations on a nonatomic probability space

(
X
,
β
,
p
)

(X,\beta ,p)

.
Singularities produced in conormal wave interactions
Authors: Holt Linda M.
Published: June 2012

Three problems on the interactions of conormal waves are considered. Two are examples which demonstrate that nonlinear spreading of singularities can occur when the waves are conormal. In one case, two of the waves are tangential, and the other wave is transversal to the first two. The third result is a noninteraction theorem. It is shown that under certain conditions, no nonlinear spreading of the singularities will occur.
Inequalities for zero-balanced hypergeometric functions
Authors: Anderson G. D. , Barnard R. W. , Richards K. C. , Vamanamurthy M. K. , Vuorinen M.
Published: June 2012

The authors study certain monotoneity and convexity properties of the Gaussian hypergeometric function and those of the Euler gamma function.
Power regular operators
Authors: Atzmon Aharon
Published: June 2012

We show that for a wide class of operators

T
T

on a Banach space, including the class of decomposable operators, the sequence

{

‖

T
n

x

‖

1

/

n

}

n
=
1

∞

\left \{ {{{\left \| {{T^n}x} \right \|}^{1/n}}} \right \}_{n = 1}^\infty

converges for every

x
x

in the space to the spectral radius of the restriction of

T
T

to the subspace

∨

n
=
0

∞

{

T
n

x
}

\vee _{n = 0}^\infty \{ {T^n}x\}

.
Asymptotic stability in functional-differential equations by Liapunov functionals
Authors: Zhang Bo
Published: June 2012

We consider the asymptotic stability in a system of functional differential equations

x
′

(
t
)
=
F
(
t
,

x
t

)

x’(t) = F(t,{x_t})

by Liapunov functionals

V
V

. The work generalizes some well-known results in the literature in that we only require the derivative of

V
V

to be negative definite on a sequence of intervals

I
n

=
[

S
n

,

t
n

]

{I_n} = [{S_n},{t_n}]

. We also show that it is not necessary to require a uniform upper bound on

V
V

for nonuniform asymptotic stability.
Polar 𝜎-ideals of compact sets
Authors: Debs Gabriel
Published: June 2012

Let

E
E

be a metric compact space. We consider the space

K

(
E
)

\mathcal {K}(E)

of all compact subsets of

E
E

endowed with the topology of the Hausdorff metric and the space

M

(
E
)

\mathcal {M}(E)

of all positive measures on

E
E

endowed with its natural

w

∗

{w^{\ast }}

-topology. We study

σ
\sigma

-ideals of

K

(
E
)

\mathcal {K}(E)

of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I equals upper I Subscript upper P Baseline equals left-brace upper K element-of script upper K left-parenthesis upper E right-parenthesis colon mu left-parenthesis upper K right-parenthesis equals 0 comma for-all mu element-of upper P right-brace">

I
=

I
P

=
{
K
∈

K

(
E
)
:
μ
(
K
)
=
0
,

∀
μ
∈
P
}

I = {I_P} = \{ K \in \mathcal {K}(E):\mu (K) = 0,\;\forall \mu \in P\}

where

P
P

is a given family of positive measures on

E
E

. If

M
M

is the maximal family such that

I
=

I
M

I = {I_M}

, then

M
M

is a band. We prove that several descriptive properties of

I
I

: being Borel, and having a Borel basis, having a Borel polarity-basis, can be expressed by properties of the band

M
M

or of the orthogonal band

M
′

M’

.
Bifurcation of minimal surfaces in Riemannian manifolds
Authors: Jost Jürgen , Li-Jost Xianqing , Peng Xiao Wei
Published: June 2012

We study the bifurcation of closed minimal surfaces in Riemannian manifolds through higher order variations of the area functional and relate it to elementary catastrophes.
Differential operators, 𝑛-branch curve singularities and the 𝑛-subspace problem
Authors: Cannings R. C. , Holland M. P.
Published: June 2012

Let

R
R

be the coordinate ring of a smooth affine curve over an algebraically closed field of characteristic zero

k
k

. For

S
S

a subalgebra of

R
R

with integral closure

R
R

denote by

D

(
S
)

\mathcal {D}(S)

the ring of differential operators on

S
S

and by

H
(
S
)

H(S)

the finite-dimensional factor of

D

(
S
)

\mathcal {D}(S)

by its unique minimal ideal. The theory of diagonal

n
n

-subspace systems is introduced. This is used to show that if

A
A

is a finite-dimensional

k
k

-algebra and

t
⩾
1

t \geqslant 1

is any integer there exists such an

S
S

with
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H left-parenthesis upper S right-parenthesis approximately-equals Start 2 By 2 Matrix 1st Row 1st Column upper A 2nd Column a m p semicolon asterisk 2nd Row 1st Column 0 2nd Column a m p semicolon upper M Subscript t Baseline left-parenthesis k right-parenthesis EndMatrix period">

H
(
S
)
≅

(

A

a
m
p
;

∗

0

a
m
p
;

M
t

(
k
)

)

.

H(S) \cong \left ( {\begin {array}{*{20}{c}} A & {\ast } \\ 0 & {{M_t}(k)} \\ \end {array} } \right ).

\]
Further, the Morita classes of

H
(
S
)

H(S)

are classified for curves with few branches, and it is shown how to lift Morita equivalences from

H
(
S
)

H(S)

to

D

(
S
)

\mathcal {D}(S)

.
A new functional equation of Pexider type related to the complex exponential function
Authors: Haruki Hiroshi , Rassias Themistocles M.
Published: June 2012

The purpose of this paper is to solve a new functional equation, characteristic for the complex exponential function, which contains four unknown entire functions and to solve, as an application, three further functional equations.
Entire functions, in the classification of differentiable germs tangent to the identity, in one or two variables
Authors: Ahern Patrick , Rosay Jean-Pierre
Published: June 2012

This paper presents a survey and some (hopefully) new facts on germs of maps tangent to the identity (in

R

,

C

,

\mathbb {R},\mathbb {C},

or

R

2

{\mathbb {R}^2}

), (maps

f
f

defined near

0
0

, such that

f
(
0
)
=
0

f(0) = 0

, and

f
′

(
0
)

f’(0)

is the identity). Proofs are mostly original. The paper is mostly oriented towards precise examples and the questions of descriptions of members in the conjugacy class, flows,

k
k

th root. It happened that entire functions provide clear and easy examples. However they should be considered just as a tool, not as the main topic. For example in Proposition

2
2

the function

z
↦
z
+

z
2

z \mapsto z + {z^2}

should be better thought of as the map

(
x
,
y
)
→
(
x
+

x
2

−

y
2

,
y
+
2
x
y
)

(x,y) \to (x + {x^2} - {y^2},y + 2xy)

.
Multiple viscous solutions for systems of conservation laws
Authors: Azevedo A. V. , Marchesin D.
Published: June 2012

We exhibit an example of mechanism responsible for multiple solutions in the Riemann problem for a mixed elliptic-hyperbolic type system of two quadratic polynomial conservation laws. In this example, multiple solutions result from folds in the set of Riemann solutions. The multiple solutions occur despite the fact that they all satisfy the viscous profile entropy criterion. The failure of this criterion to provide uniqueness is evidence in support of a need for conceptual change in the theory of shock waves for a system of conservation laws.
Free ideals of one-relator graded Lie algebras
Authors: Labute John P.
Published: June 2012

In this paper we show that a one-relator graded Lie algebra

g

=
L

/

(
r
)

\mathfrak {g} = L/(r)

, over a principal ideal domain

K
K

, has a homogeneous ideal

h

\mathfrak {h}

with

g

/

h

\mathfrak {g}/\mathfrak {h}

a free

K
K

-module of finite rank if the relator

r
r

is not a proper multiple of another element in the free Lie algebra

L
L

. As an application, we deduce that the center of a one-relator Lie algebra over

K
K

is trivial if the rank of

L
L

is greater than two. As another application, we find a new class of one-relator pro-

p
p

-groups which are of cohomological dimension

2
2

.
On a quadratic-trigonometric functional equation and some applications
Authors: Chung J. K. , Ebanks B. R. , Ng C. T. , Sahoo P. K.
Published: June 2012

Our main goal is to determine the general solution of the functional equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row f 1 left-parenthesis x y right-parenthesis plus f 2 left-parenthesis x y Superscript negative 1 Baseline right-parenthesis equals f 3 left-parenthesis x right-parenthesis plus f 4 left-parenthesis y right-parenthesis plus f 5 left-parenthesis x right-parenthesis f 6 left-parenthesis y right-parenthesis comma 2nd Row f Subscript i Baseline left-parenthesis t x y right-parenthesis equals f Subscript i Baseline left-parenthesis t y x right-parenthesis left-parenthesis i equals 1 comma 2 right-parenthesis EndLayout">

f
1

(
x
y
)
+

f
2

(
x

y

−
1

)
=

f
3

(
x
)
+

f
4

(
y
)
+

f
5

(
x
)

f
6

(
y
)
,

f
i

(
t
x
y
)
=

f
i

(
t
y
x
)

(
i
=
1
,
2
)

\begin {array}{*{20}{c}} {{f_1}(xy) + {f_2}(x{y^{ - 1}}) = {f_3}(x) + {f_4}(y) + {f_5}(x){f_6}(y),} \\ {{f_i}(txy) = {f_i}(tyx)\qquad (i = 1,2)} \\ \end {array}

\]
where

f
i

{f_i}

are complex-valued functions defined on a group. This equation contains, among others, an equation of H. Swiatak whose general solution was not known until now and an equation studied by K.S. Lau in connection with a characterization of Rao’s quadratic entropies. Special cases of this equation also include the Pexider, quadratic, d’Alembert and Wilson equations.
Bounded geodesics of Riemann surfaces and hyperbolic manifolds
Authors: Fernández J. L. , Melián M. V.
Published: June 2012

We study the set of bounded geodesics of hyperbolic manifolds. For general Riemann surfaces and for hyperbolic manifolds with some finiteness assumption on their geometry we determine its Hausdorff dimension. Some applications to diophantine approximation are included.
The 7-connected cobordism ring at 𝑝=3
Authors: Hovey Mark A. , Ravenel Douglas C.
Published: June 2012

In this paper, we study the cobordism spectrum

M
O

⟨
8
⟩

MO\left \langle 8 \right \rangle

at the prime

3
3

. This spectrum is important because it is conjectured to play the role for elliptic cohomology that Spin cobordism plays for real

K
K

-theory. We show that the torsion is all killed by

3
3

, and that the Adams-Novikov spectral sequence collapses after only

2
2

differentials. Many of our methods apply more generally.
Focusing at a point and absorption of nonlinear oscillations
Authors: Joly J.-L. , Métivier G. , Rauch J.
Published: June 2012

Several recent papers give rigorous justifications of weakly nonlinear geometric optics. All of them consider oscillating wave trains on domains where focusing phenomena do not exist, either because the space dimension is equal to one, or thanks to a coherence assumption on the phases. This paper is devoted to a study of some nonlinear effects of focusing. In a previous paper, the authors have given a variety of examples which show how focusing in nonlinear equations can spoil even local existence in the sense that the domain of existence shrinks to zero as the wavelength decreases to zero. On the other hand, there are many problems for which global existence is known and in those cases it is natural to ask what happens to oscillations as they pass through a focus. The main goal of this paper is to present such a study for some strongly dissipative semilinear wave equations and spherical wavefronts which focus at the origin. We show that the strongly nonlinear phenomenon which is produced is that oscillations are killed by the simultaneous action of focusing and dissipation. Our study relies on the analysis of Young measures and two-scale Young measures associated to sequences of solutions. The main step is to prove that these measures satisfy appropriate transport equations. Then, their variances are shown to satisfy differential inequalities which imply a propagation result for their support.
Banach spaces with the 2-summing property
Authors: Arias A. , Figiel T. , Johnson W. B. , Schechtman G.
Published: June 2012

A Banach space

X
X

has the

2
2

-summing property if the norm of every linear operator from

X
X

to a Hilbert space is equal to the

2
2

-summing norm of the operator. Up to a point, the theory of spaces which have this property is independent of the scalar field: the property is self-dual and any space with the property is a finite dimensional space of maximal distance to the Hilbert space of the same dimension. In the case of real scalars only the real line and real

ℓ
∞
2

\ell _\infty ^2

have the

2
2

-summing property. In the complex case there are more examples; e.g., all subspaces of complex

ℓ
∞
3

\ell _\infty ^3

and their duals.
Multiple solutions for a semilinear elliptic equation
Authors: del Pino Manuel A. , Felmer Patricio L.
Published: June 2012

Let

Ω
\Omega

be a bounded, smooth domain in

R

N

{\mathbb {R}^N}

,

N
⩾
1

N \geqslant 1

. We consider the problem of finding nontrivial solutions to the elliptic boundary value problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row normal upper Delta u plus lamda u equals h left-parenthesis x right-parenthesis StartAbsoluteValue u EndAbsoluteValue Superscript p minus 1 Baseline u in normal upper Omega 2nd Row u equals 0 on partial-differential normal upper Omega EndLayout">

Δ
u
+
λ
u
=
h
(
x
)

|

u

|

p
−
1

u

in 

Ω

u
=
0

on

∂
Ω

\begin {array}{*{20}{c}} {\Delta u + \lambda u = h(x)|u{|^{p - 1}}u\quad {\text {in }}\Omega } \\ {u = 0\quad {\text {on}}\partial \Omega } \\ \end {array}

\]
where

h
⩾
0

h \geqslant 0

,

h
≢
0

h\not \equiv 0

is Hölder continuous on

Ω
¯

\overline \Omega

and

p
>
1

p > 1

,

λ
\lambda

are constants. Let

Ω
0

{\Omega _0}

denote the interior of the set where

h
h

vanishes in

Ω
\Omega

. We assume

h
>
0

h > 0

a.e. on

Ω
∖

Ω
0

\Omega \backslash {\Omega _0}

and consider the eigenvalues

λ
i

(
Ω
)

{\lambda _i}(\Omega )

and

λ
i

(

Ω
0

)

{\lambda _i}({\Omega _0})

of the Dirichlet problem in

Ω
\Omega

and

Ω
0

{\Omega _0}

respectively. We prove that no nontrivial solution of the equation exists if

λ
\lambda

satisfies, for some

k
⩾
1

k \geqslant 1

,
\[

λ
k

(

Ω
0

)
⩽
λ
⩽

λ

k
+
1

(
Ω
)

{\lambda _k}({\Omega _0}) \leqslant \lambda \leqslant {\lambda _{k + 1}}(\Omega )

\]
On the other hand, if, for some nonnegative integers

s
s

,

k
k

with

s
⩾
k
+
1

s \geqslant k + 1

,

λ
\lambda

satisfies
\[

λ
s

(
Ω
)
>
λ
>

λ

k
+
1

(

Ω
0

)

{\lambda _s}(\Omega ) > \lambda > {\lambda _{k + 1}}({\Omega _0})

\]
then the equation above possesses at least

s
−
k

s - k

pairs of nontrivial solutions. For the proof of these results we use a variational approach. In particular, the existence result takes advantage of the even character of the associated functional.
A note on singularities in semilinear problems
Authors: Guedda Mohammed , Kirane Mokhtar
Published: June 2012

We consider the equation

Δ
u
−

1
2

x
.
Δ
u
−

u

q
−
1

+

u
q

=
0
,

for

q
>
1

\Delta u - \frac {1} {2}x.\Delta u - \frac {u} {{q - 1}} + {u^q} = 0,{\text {for}}q > 1

. We study the isolated singularities and present a nonlinear technique, and give a complete classification.