bluuebunny/crossref_metadata_2025_split huggingface dataset:
Cardinal conditions for strong Fubini theorems
Authors: Shipman Joseph 
Published: June 2012

If

κ
1

,

κ
2

,
…
,

κ
n

{\kappa _1},{\kappa _2}, \ldots ,{\kappa _n}

are cardinals with

κ
1

{\kappa _1}

the cardinality of a nonmeasurable set, and for

i
=
2
,
3
,
…
,
n

i = 2,3, \ldots ,n

κ
i

{\kappa _i}

is the cardinality of a set of reals which is not the union of

κ

i
−
1

{\kappa _{i - 1}}

measure-

0
0

sets, then for any nonnegative function

f
:

R

n

→

R

f:{{\mathbf {R}}^n} \to {\mathbf {R}}

all of the iterated integrals
\[

I
σ

=
∬
⋯
∫

f
(

x
1

,

x
2

,
…
,

x
n

)
d

x

σ
(
1
)

d

x

σ
(
2
)

⋯
d

x

σ
(
n
)

,

σ
∈

S
n

{I_\sigma } = \iint \cdots \int {f({x_1},{x_2}, \ldots ,{x_n})d{x_{\sigma (1)}}d{x_{\sigma (2)}} \cdots d{x_{\sigma (n)}},\quad \sigma \in {S_n}}

\]
, which exist are equal. If all

n
!

n!

of the integrals exist, then the weaker condition of the case

n
=
2

n = 2

implies they are equal. These cardinal conditions are consistent with and independent of ZFC, and follow from the existence of a real-valued measure on the continuum. Other necessary conditions and sufficient conditions for the existence and equality of iterated integrals are also treated.
Change of variable results for 𝐴_{𝑝}- and reverse Hölder 𝑅𝐻ᵣ-classes
Authors: Johnson R. , Neugebauer C. J. 
Published: June 2012

We study conditions under which the map
\[

T

h
,
γ

w
(
x
)
=
w
(
h
(
x
)
)

|

h
′

(
x
)

|

γ

{T_{h,\gamma }}w(x)= w(h(x))|h’(x){|^\gamma }

\]
maps the Muckenhoupt class

A
p

{A_p}

into

A
q

{A_q}

and the reverse Hölder class

R

H

r
1

R{H_{{r_1}}}

into

R

H

r
2

R{H_{{r_2}}}

.
On lifting Hecke eigenforms
Authors: Walling Lynne H. 
Published: June 2012

A classical Hilbert modular form

f
∈

M

k

(

Γ
0

(

N

,

I

)
,

χ

N

)

f \in {\mathcal {M}_k}({\Gamma _0}(\mathcal {N},\mathcal {I}),{\chi _\mathcal {N}})

cannot be an eigenform for the full Hecke algebra. We develop a means of lifting a classical form to a modular form

F
∈

⊕
λ

M

k

(

Γ
0

(

N

,

I

λ

)
,

χ

N

)

F \in { \oplus _\lambda }{\mathcal {M}_k}({\Gamma _0}(\mathcal {N},{\mathcal {I}_\lambda }),{\chi _\mathcal {N}})

which is an eigenform for the full Hecke algebra. Using this lift, we develop the newform theory for a space of cusp forms

S

k

(

Γ
0

(

N

,

I

)
,

χ

N

)

{\mathcal {S}_k}({\Gamma _0}(\mathcal {N},\mathcal {I}),{\chi _\mathcal {N}})

; we also use theta series to construct eigenforms for the full Hecke algebra.
Adams’ cobar equivalence
Authors: Félix Yves , Halperin Stephen , Thomas Jean-Claude 
Published: June 2012

Let

F
F

be the homotopy fibre of a continuous map

Y

→

ω

X

Y\xrightarrow {\omega }X

, with

X
X

simply connected. We modify and extend a construction of Adams to obtain equivalences of DGA’s and DGA modules,
\[

Ω

C

∗

(
X
)

→

≃

C

U

∗

(
Ω
X
)
,

\Omega {C_{\ast }}(X)\xrightarrow { \simeq }C{U_{\ast }}(\Omega X),

\]
and
\[

Ω
(

C

∗

ω

(
Y
)
;

C

∗

(
X
)
)

→

≃

C

U

∗

(
F
)
,

\Omega (C_{\ast }^\omega (Y);{C_{\ast }}(X))\xrightarrow { \simeq }C{U_{\ast }}(F),

\]
where on the left-hand side

Ω
(
−
)

\Omega ( - )

denotes the cobar construction. Our equivalences are natural in

X
X

and

ω
\omega

. Using this result we show how to read off the algebra

H

∗

(
Ω
X
;
R
)

{H_{\ast }}(\Omega X;R)

and the

H

∗

(
Ω
X
;
R
)

{H_{\ast }}(\Omega X;R)

module,

H

∗

(
F
;
R
)

{H_{\ast }}(F;R)

, from free models for the singular cochain algebras

C

S

∗

(
X
)

C{S^{\ast }}(X)

and

C

S

∗

(
Y
)

C{S^{\ast }}(Y)

; here we assume

R
R

is a principal ideal domain and

X
X

and

Y
Y

are of finite

R
R

type.
A phenomenon of reciprocity in the universal Steenrod algebra
Authors: Lomonaco Luciano 
Published: June 2012

In this paper we compute the cohomology algebra of certain subalgebras

L
r

{L_r}

and certain quotients

K
s

{K_s}

of the

mod

2

\bmod \, 2

universal Steenrod algebra

Q
Q

, the algebra of cohomology operations for

H
∞

{H_\infty }

-ring spectra (see

[
M
]

[\text {M}]

). We prove that
\[

Ext

L
r

⁡
(

F
2

,

F
2

)
≅

K

−
k
+
1

,

Ext

K
s

⁡
(

F
2

,

F
2

)
≅

L

−
s
+
1

\operatorname {Ext}_{{L_r}}({F_2},{F_2}) \cong {K_{ - k + 1}}, \qquad \operatorname {Ext}_{{K_s}}({F_2},{F_2}) \cong {L_{ - s + 1}}

\]
with

r
r

,

s
s

integers and

r
≤
1

r \leq 1

,

s
≥
0

s \geq 0

. We also observe that some of the algebras

L
r

{L_r}

,

K
s

{K_s}

are well known objects in stable homotopy theory and in fact our computation generalizes the fact that

H

∗

(

A
L

)
≅

Λ

opp

{H^{\ast } }({A_L}) \cong \Lambda ^{{\text {opp}}}

and

H

∗

(

Λ

opp

)
≅

A
L

{H^{\ast } }({\Lambda ^{{\text {opp}}}}) \cong {A_L}

(see, for instance,

[
P
]

[\text {P}]

). Here

A
L

{A_L}

is the Steenrod algebra for simplicial restricted Lie algebras and

Λ
\Lambda

is the

E
1

{E_1}

-term of the Adams spectral sequence discovered in

[
B-S
]

[\text {B-S}]

.
A local Weyl’s law, the angular distribution and multiplicity of cusp forms on product spaces
Authors: Huntley Jonathan , Tepper David 
Published: June 2012

Let

Γ

/

H

\Gamma /\mathcal {H}

be a finite volume symmetric space with

H

\mathcal {H}

the product of half planes. Let

Δ
i

{\Delta _i}

be the Laplacian on the

i
i

th half plane, and assume that we have a cusp form

ϕ
\phi

, so we have

Δ
i

ϕ
=

λ
i

ϕ

{\Delta _i}\phi = {\lambda _i}\phi

for

i
=
1
,
2
,
…
,
n

i = 1,2, \ldots ,n

. Let

λ
→

=
(

λ
1

,
…
,

λ
n

)

\vec \lambda = ({\lambda _1}, \ldots ,{\lambda _n})

and let
\[

R
=

r
1
2

+
⋯
+

r
n
2

R = \sqrt {r_1^2 + \cdots + r_n^2}

\]
with

r
i
2

+

1
4

=

λ
i

r_i^2 + \frac {1} {4} = {\lambda _i}

. Letting

r
→

=
(

r
1

,
…
,

r
n

)

\vec r = ({r_1}, \ldots ,{r_n})

, we let

M
(

r
→

)

M(\vec r)

denote the dimension of the space of cusp forms with eigenvalue

λ
→

\vec \lambda

. More generally, let

M
(

r
→

,
a
)

M(\vec r,a)

denote the number of independent eigenfunctions such that the

r
→

\vec r

associated to an eigenfunction is inside the ball of radius

a
a

, centered at

r
→

\vec r

. We will define a function

f
(

r
→

)

f(\vec r)

, which is generally equal to a linear sum of products of the

r
i

{r_i}

. We prove the following theorems. Theorem 1.
\[

M
(

r
→

)
=
O

(

f
(

r
→

)

(
log
⁡
R

)
n

)

.

M(\vec r) = O\left (\frac {f(\vec r)} {(\log R)^n} \right ).

\]
Theorem 2.
\[

M
(

r
→

,
A
)
=

2
n

f
(

r
→

)
+
O

(

f
(

r
→

)

log
⁡
R

)

.

M (\vec {r}, A) = 2^n f(\vec {r})+O\left (\frac {f(\vec r)}{\log R} \right ).

\]
Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem
Authors: Lee Tzong-Yow , Ni Wei-Ming 
Published: June 2012

We investigate the behavior of the solution

u
(
x
,
t
)

u(x,t)

of
\[

{

∂
u

∂
t

=
Δ
u
+

u
p

a
m
p
;

in

R

n

×
(
0
,
T
)
,

u
(
x
,
0
)
=
φ
(
x
)

a
m
p
;

in

R

n

,

\left \{ {\begin {array}{*{20}{c}} {\frac {{\partial u}} {{\partial t}} = \Delta u + {u^p}} \hfill & {{\text {in}}\;{\mathbb {R}^n} \times (0,T),} \hfill \\ {u(x,0) = \varphi (x)} \hfill & {{\text {in}}\;{\mathbb {R}^n},} \hfill \\ \end {array} } \right .

\]
where

Δ
=

∑

i
=
1

n

∂
2

/

∂

x
i

2

\Delta = \sum \nolimits _{i = 1}^n {{\partial ^2}/\partial _{{x_i}}^2}

is the Laplace operator,

p
>
1

p > 1

is a constant,

T
>
0

T > 0

, and

φ
\varphi

is a nonnegative bounded continuous function in

R

n

{\mathbb {R}^n}

. The main results are for the case when the initial value

φ
\varphi

has polynomial decay near

x
=
∞

x = \infty

. Assuming

φ
∼
λ

(
1
+

|

x

|

)

−
a

\varphi \sim \lambda {(1 + |x|)^{ - a}}

with

λ
\lambda

,

a
>
0

a > 0

, various questions of global (in time) existence and nonexistence, large time behavior or life span of the solution

u
(
x
,
t
)

u(x,t)

are answered in terms of simple conditions on

λ
\lambda

,

a
a

,

p
p

and the space dimension

n
n

.
A Haar-type theory of best 𝐿₁-approximation with constraints
Authors: Kroó András , Schmidt Darrell 
Published: June 2012

A general setting for constrained

L
1

{L^1}

-approximation is presented. Let

U
n

{U_n}

be a finite dimensional subspace of

C
[
a
,
b
]

C[a,b]

and

L
L

be a linear operator from

U
n

{U_n}

to

C
r

(
K
)

(
r
=
0
,
1
)

{C^r}(K)\;(r = 0,1)

where

K
K

is a finite union of disjoint, closed, bounded intervals. For

υ
,
u
∈

C
r

(
K
)

\upsilon ,u \in {C^r}(K)

with

υ
>
u

\upsilon > u

, the approximating set is

U
~

n

(
υ
,
u
)
=
{
p
∈

U
n

:
υ
≤
L
p
≤
u

on

K
}

{\tilde U_n}(\upsilon ,u) = \{ p \in {U_n}:\upsilon \leq Lp \leq u\;{\text {on}}\;K\}

and the norm is

‖
f

‖
w

=

∫
a
b

|

f

|

w

d
x

\|f\|_w = \int _a^b {|f|w\,dx}

where

w
w

a positive continuous function on

[
a
,
b
]

[a,b]

. We obtain necessary and sufficient conditions for

U
~

n

(
υ
,
u
)

{\tilde U_n}(\upsilon ,u)

to admit unique best

‖

⋅

‖
w

\|\;\cdot \;\|_w

-approximations to all

f
∈
C
[
a
,
b
]

f \in C[a,b]

for all positive continuous

w
w

and all

υ
,
u
∈

C
r

(
K
)

(
r
=
0
,
1
)

\upsilon ,u \in {C^r}(K)\;(r = 0,1)

satisfying a nonempty interior condition. These results are applied to several

L
1

{L^1}

-approximation problems including polynomial and spline approximation with restricted derivatives, lacunary polynomial approximation with restricted derivatives, and others.
Justification of multidimensional single phase semilinear geometric optics
Authors: Joly Jean-Luc , Rauch Jeffrey 
Published: June 2012

For semilinear strictly hyperbolic systems

L
u
=
f
(
x
,
u
)

Lu= f(x,u)

, we construct and justify high frequency nonlinear asymptotic expansions of the form
\[

u
ε

(
x
)
∼

∑

j

≥

0

ε
j

U
j

(
x
,
φ

(
x
)

/

ε

,

L

u
ε

−
f
(
x
,

u
ε

)
∼
0
.

{u^\varepsilon }(x)\sim \sum \limits _{j\, \geq \,0} {{\varepsilon ^j}{U_j}(x,\varphi \,(x)/\varepsilon }, \quad L{u^\varepsilon } - f(x,{u^\varepsilon })\sim 0 .

\]
The study of the principal term of such expansions is called nonlinear geometric optics in the applied literature. We show (i) formal expansions with periodic profiles

U
j

{U_j}

can be computed to all orders, (ii) the equations for the profiles from (i) are solvable, and (iii) there are solutions of the exact equations which have the formal series as high frequency asymptotic expansion.
Local integrability of Mizohata structures
Authors: Hounie Jorge , Malagutti Pedro 
Published: June 2012

In this work we study the local integrability of strongly pseudoconvex Mizohata structures of rank

n
>
2

n > 2

(and co-rank

1
1

). These structures are locally generated in an appropriate coordinate system

(

t
1

,
…
,

t
n

,
x
)

({t_1}, \ldots ,{t_n},x)

by flat perturbations of Mizohata vector fields

M
j

=

∂

∂

t
j

−
i

t
j

∂

∂
x

{M_j} = \frac {\partial } {{\partial {t_j}}} - i{t_j}\frac {\partial } {{\partial x}}

,

j
=
1
,
…
,
n

j = 1, \ldots ,n

. For this, we first prove the global integrability of small perturbations of the structure generated by

∂

∂

z
¯

+

σ
1

∂

∂
z

\frac {\partial } {{\partial \bar z}} + {\sigma _1}\frac {\partial } {{\partial z}}

,

∂

∂

θ

n
−
1

+

σ
j

∂

∂
z

\frac {\partial } {{\partial {\theta _{n - 1}}}} + {\sigma _j}\frac {\partial } {{\partial z}}

,

j
=
2
,
…
,
n

j = 2, \ldots ,n

, defined over a manifold

C

×
S

{\mathbf {C}} \times S

, where

S
S

is simply connected.
Fragments of bounded arithmetic and bounded query classes
Authors: Krajíček Jan 
Published: June 2012

We characterize functions and predicates

Σ

i
+
1

b

\Sigma _{i + 1}^b

-definable in

S
2
i

S_2^i

. In particular, predicates

Σ

i
+
1

b

\Sigma _{i + 1}^b

-definable in

S
2
i

S_2^i

are precisely those in bounded query class

P

Σ
i
p

[
O
(
log
⁡
n
)
]

{P^{\Sigma _i^p}}[O(\log n)]

(which equals to

Log

Space

Σ
i
p

\operatorname {Log}\;{\text {Space}}^{\Sigma _i^p}

by [B-H, W]). This implies that

S
2
i

≠

T
2
i

S_2^i \ne T_2^i

unless

P

Σ
i
p

[
O
(
log
⁡
n
)
]
=

Δ

i
+
1

p

{P^{\Sigma _i^p}}[O(\log n)] = \Delta _{i + 1}^p

. Further we construct oracle

A
A

such that for all

i
≥
1

i \geq 1

:

P

Σ
i
p

(
A
)

[
O
(
log
⁡
n
)
]
≠

Δ

i
+
1

p

(
A
)

{P^{\Sigma _i^p(A)}}[O(\log n)] \ne \Delta _{i + 1}^p(A)

. It follows that

S
2
i

(
α
)
≠

T
2
i

(
α
)

S_2^i(\alpha ) \ne T_2^i(\alpha )

for all

i
≥
1

i \geq 1

. Techniques used come from proof theory and boolean complexity.
Gauss map of minimal surfaces with ramification
Authors: Ru Min 
Published: June 2012

We prove that for any complete minimal surface

M
M

immersed in

R
n

{R^n}

, if in

C

P

n
−
1

C{P^{n - 1}}

there are

q
>
n
(
n
+
1
)

/

2

q > n(n + 1)/2

hyperplanes

H
j

{H_j}

in general position such that the Gauss map of

M
M

is ramified over

H
j

{H_j}

with multiplicity at least

e
j

{e_j}

for each

j
j

and
\[

∑

j
=
1

q

(

1
−

(
n
−
1
)

e
j

)

>
n
(
n
+
1
)

/

2

\sum \limits _{j = 1}^q {\left ({1 - \frac {{(n - 1)}} {{{e_j}}}} \right ) > n(n + 1)/2}

\]
, then

M
M

must be flat.
𝐿^{𝑝}-boundedness of pseudo-differential operators of class 𝑆_{0,0}
Authors: Hwang I. L. , Lee R. B. 
Published: June 2012

We study the

L
p

{L^p}

-boundedness of pseudo-differential operators with the support of their symbols being contained in

E
×

R

n

E \times {{\mathbf {R}}^n}

, where

E
E

is a compact subset of

R

n

{{\mathbf {R}}^n}

, and their symbols have derivatives with respect to

x
x

only up to order

k
k

, in the Hölder continuous sense, where

k
>
n

/

2

k > n/2

(the case

1
>
p
⩽
2

1 > p \leqslant 2

) and

k
>
n

/

p

k > n/p

(the case

2
>
p
>
∞

2 > p > \infty

). We also give a new proof of the

L
p

{L^p}

-boundedness,

1
>
p
>
∞

1 > p > \infty

, of pseudo-differential operators of class

S

0
,
0

m

S_{0,0}^m

, where

m
=
m
(
p
)
=
−
n

|

1

/

p
−
1

/

2

|

m = m(p) = - n|1/p - 1/2|

, and

a
∈

S

0
,
0

m

a \in S_{0,0}^m

satisfies

|

∂
x
α

∂
ξ
β

a
(
x
,
ξ
)

|

⩽

C

α
,
β

⟨
ξ

⟩
m

|\partial _x^\alpha \partial _\xi ^\beta a(x,\xi )| \leqslant {C_{\alpha ,\beta }}{\langle \xi \rangle ^m}

for

(
x
,
ξ
)
∈

R

n

×

R

n

,

|

α

|

⩽
k

(x,\xi ) \in {{\mathbf {R}}^n} \times {{\mathbf {R}}^n},|\alpha | \leqslant k

and

|

β

|

⩽

k
′

|\beta | \leqslant k’

, in the Hölder continuous sense, where

k
>
n

/

2
,

k
′

>
n

/

p

k > n/2,k’ > n/p

(the case

1
>
p
⩽
2

1 > p \leqslant 2

) and

k
>
n

/

p
,

k
′

>
n

/

2

k > n/p,k’ > n/2

(the case

2
>
p
>
∞

2 > p > \infty

).
A family of real 2ⁿ-tic fields
Authors: Shen Yuan Yuan , Washington Lawrence C. 
Published: June 2012

We study the family of polynomials
\[

P
n

(
X
;
a
)
=
ℜ
(

(
X
+
i

)

2
n

)
−

a

2
n

ℑ
(

(
X
+
i

)

2
n

)

{P_n}(X;a) = \Re ({(X + i)^{{2^n}}}) - \frac {a}{{{2^n}}}\Im ({(X + i)^{{2^n}}})

\]
and determine when

P
n

(
X
;
a
)

{P_n}(X;a)

,

a
∈

Z

a \in \mathbb {Z}

, is irreducible. The roots are all real and are permuted cyclically by a linear fractional transformation defined over the real subfield of the

2
n

{2^n}

th cyclotomic field. The families of fields we obtain are natural extensions of those studied by M.-N. Gras and Y.-Y. Shen, but in general the present fields are non-Galois for

n
≥
4

n \geq 4

. From the roots we obtain a set of independent units for the Galois closure that generate an "almost fundamental piece" of the full group of units. Finally, we discuss the two examples where our fields are Galois, namely

a
=
±

2
n

a = \pm {2^n}

and

a
=
±

2
4

∙
239

a = \pm {2^4} \bullet 239

.
Remarks on some integrals and series involving the Stirling numbers and 𝜁(𝑛)
Authors: Shen Li-Chien 
Published: June 2012

From the perspective of the well-known identity
\[

2

F
1

(
a
,
b
;
c
;
1
)
=

Γ
(
c
)
Γ
(
c
−
a
−
b
)

Γ
(
c
−
a
)
Γ
(
c
−
b
)

,

{}_2{F_1}(a,b;c;1) = \frac {{\Gamma (c)\Gamma (c - a - b)}} {{\Gamma (c - a)\Gamma (c - b)}},

\]
we clarify the connections between the Stirling numbers

s
k
n

s_k^n

and the Riemann zeta function

ζ
(
n
)

\zeta (n)

. As a consequence, certain series and integrals can be evaluated in terms of

ζ
(
n
)

\zeta (n)

and

s
k
n

s_k^n

.
Some inequalities of algebraic polynomials with nonnegative coefficients
Authors: Chen Weiyu 
Published: June 2012

Let

S
n

{S_n}

be the collection of all algebraic polynomials of degree

⩽
n

\leqslant n

with nonnegative coefficients. In this paper we discuss the extremal problem
\[

sup

p
n

(
x
)
∈

S
n

∫
a
b

(

p
′

n

(
x
)
)

2

ω
(
x
)
d
x

∫
a
b

p
n
2

(
x
)
ω
(
x
)
d
x

\sup \limits _{{p_n}(x) \in {S_n}} \frac {{\int \limits _a^b {{{({{p’}_n}(x))}^2}\omega (x)dx} }} {{\int \limits _a^b {p_n^2(x)\omega (x)dx} }}

\]
where

ω
(
x
)

\omega (x)

is a positive and integrable function. This problem is solved completely in the cases
\[

(

i

)
[
a
,
b
]
=
[
−
1
,
1
]
,
ω
(
x
)
=

(
1
−

x
2

)
α

,
α
>
−
1
;

({\text {i}})[a,b] = [ - 1,1],\omega (x) = {(1 - {x^2})^\alpha },\alpha > - 1;

\]

\[

(

ii

)
[
a
,
b
)
=
[
0
,
∞
)
,
ω
(
x
)
=

x
α

e

−
x

,
α
>
−
1
;

({\text {ii}})[a,b) = [0,\infty ),\omega (x) = {x^\alpha }{e^{ - x}},\alpha > - 1;

\]

\[

(

iii

)
(
a
,
b
)
=
(
−
∞
,
∞
)
,
ω
(
x
)
=

e

−
α

x
2

,
α
>
0.

({\text {iii}})(a,b) = ( - \infty ,\infty ),\omega (x) = {e^{ - \alpha {x^2}}},\alpha > 0.

\]
The second case was solved by Varma for some values of

α
\alpha

and by Milovanović completely. We provide a new proof here in this case.
An optimal condition for the LIL for trigonometric series
Authors: Berkes I. 
Published: June 2012

By a classical theorem (Salem-Zygmund [6], Erdős-Gàl [3]), if

(

n
k

)

({n_k})

is a sequence of positive integers satisfying

n

k
+
1

/

n
k

⩾
q
>
1

(
k
=
1
,
2
,
…
)

{n_{k + 1}}/{n_k} \geqslant q > 1\;(k = 1,2, \ldots )

then

(
cos
⁡

n
k

x
)

(\cos {n_k}x)

obeys the law of the iterated logarithm, i.e., (1)
\[

lim

sup

N
→
∞

(
N
log
⁡
log
⁡
N

)

−
1

/

2

∑

k
⩽
N

cos
⁡

n
k

x
=
1

a

.e

.

\lim \sup \limits _{N \to \infty } {(N\log \log N)^{ - 1/2}}\sum \limits _{k \leqslant N} {\cos {n_k}x = 1\quad {\text {a}}{\text {.e}}{\text {.}}}

\]
It is also known (Takahashi [7, 8]) that the Hadamard gap condition

n

k
+
1

/

n
k

⩾
q
>
1

{n_{k + 1}}/{n_k} \geqslant q > 1

can be essentially weakened here but the problem of finding the precise gap condition for the LIL (1) has remained open. In this paper we find, using combinatorial methods, an optimal gap condition for the upper half of the LIL, i.e., the inequality

⩽
1

\leqslant 1

in (1).
The relative Burnside module and the stable maps between classifying spaces of compact Lie groups
Authors: Minami Norihiko 
Published: June 2012

Tom Dieck’s Burnside ring of compact Lie groups is generalized to the relative case: For anyG▹NG \triangleright N, a compact Lie group and its normal subgroupA(G▹N)A(G \triangleright N)is defined to be an appropriate set of the equivalence classes of compactGG-ENR’s with freeNN-action, in such a way thatψ:A(G▹N)≃πG/N0(S0;B(N,G)+)\psi :A(G \triangleright N) \simeq \pi _{G/N}^0({S^0};B{(N,G)_ + }), whereB(N,G)B(N,G)is the classifying space of principal(N,G)(N,G)-bundle. Under the "product" situation, i.e.G=F×K,N=K,A(F×K▹K)G = F \times K,\;N = K,\;A(F \times K \triangleright K)is also denoted byA(F,K)A(F,K), as it turns out to be the usualA(F,K)A(F,K)when bothFFandKKare finite. Then a couple of applications are given to the study of stable maps between classifying spaces of compact Lie groups: a conceptual proof of Feshbach’s double coset formula, and a density theorem on the mapαp∧:A(L,H)p∧→{BL+,BH+}p∧\alpha _p^ \wedge :A(L,H)_p^ \wedge \to \{ B{L_{ + ,}}B{H_ + }\} _p^ \wedgefor any compact Lie groupsL,KL,\;Kwhenppis odd. (Some restriction is applied toLLwhenp=2p = 2.) This latter result may be regarded as the pushout of Feshbach’s density theorem and the theorem of May-Snaith-Zelewski, over the celebrated Carlsson solution of Segal’s Burnside ring conjecture.
Equilibria and quasiequilibria for infinite collections of interacting Fleming-Viot processes
Authors: Dawson Donald A. , Greven Andreas , Vaillancourt Jean 
Published: June 2012

In this paper of infinite systems of interacting measure-valued diffusions each with state space

P

(

[
0
,
1
]

)

\mathcal {P}\left ( {[0,1]} \right )

, the set of probability measures on [0, 1], is constructed and analysed (Fleming-Viot systems). These systems arise as diffusion limits of population genetics models with infinitely many possible types of individuals (labelled by [0, 1]), spatially distributed over a countable collection of sites and evolving as follows. Individuals can migrate between sites and after an exponential waiting time a colony replaces its population by a new generation where the types are assigned by resampling from the empirical distribution of types at this site. It is proved that, depending on recurrence versus transience properties of the migration mechanism, the system either clusters as

t
→
∞

t \to \infty

, that is, converges in distribution to a law concentrated on the states in which all components are equal to some

δ
u

,

u
∈
[
0
,
1
]

{\delta _u},\ u \in [0,1]

, or the system approaches a nontrivial equilibrium state. The properties of the equilibrium states, respectively the cluster formation, are studied by letting a parameter in the migration mechanism tend to infinity and explicitly identifying the limiting dynamics in a sequence of different space-time scales. These limiting dynamics have stationary states which are quasi-equilibria of the original system, that is, change only in longer time scales. Properties of these quasi-equilibria are derived and related to the global equilibrium process for large

N
N

. Finally we establish that the Fleming-Viot systems are the unique dynamics which remain invariant under the associated space-time renormalization procedure.
Radially symmetric internal layers in a semilinear elliptic system
Authors: del Pino Manuel A. 
Published: June 2012

Let

B
B

denote the unit ball in

R
N

,

N
⩾
1

{R^N},\quad N \geqslant 1

. We consider the problem of finding nonconstant solutions to a class of elliptic systems including the Gierer and Meinhardt model of biological pattern formation,
\[

(
1.1
)

ε
2

Δ
u
−
u
+

u
2

1
+
k

u
2

+
p
=
0

in

B
,

(1.1)\qquad {\varepsilon ^2}\Delta u - u + \frac {{{u^2}}} {{1 + k{u^2}}} + p = 0\quad {\text {in}}B,

\]

\[

(
1.2
)

D
Δ
v
−
v
+

u
2

=
0

in

B
,

(1.2)\quad D\Delta v - v + {u^2} = 0\quad {\text {in}}B,

\]

\[

(
1.3
)

∂
u

∂
n

=
0
=

∂
v

∂
n

on

∂
B
,

(1.3)\quad \frac {{\partial u}} {{\partial n}} = 0 = \frac {{\partial v}} {{\partial n}}\quad {\text {on}}\partial B,

\]
where

ε
\varepsilon

,

D
D

,

k
k

and

ρ
\rho

denote positive constants and

n
n

the unit outer normal to

∂
B

\partial B

. Assuming that the parameters

ρ
\rho

,

k
k

are small and

D
D

large, we construct a family of radially symmetric solutions to (1.1)-(1.3) indexed by the parameter

ε
\varepsilon

, which exhibits an internal layer in

B
B

, as

ε
→
0

\varepsilon \to 0

.
Geometry of strictly convex domains and an application to the uniform estimate of the \overline∂-problem
Authors: Chen Ten Ging 
Published: June 2012

In this paper, we construct a nice defining function

ρ
\rho

for a bounded smooth strictly convex domain

Ω
\Omega

in

R
n

{R^n}

with explicit gradient and Hessian estimates near the boundary

∂
Ω

\partial \Omega

of

Ω
\Omega

. From the approach, we deduce that any two normals through

∂
Ω

\partial \Omega

do not intersect in any tubular neighborhood of

∂
Ω

\partial \Omega

with radius which is less than

1
K

\frac {1} {K}

, where

K
K

is the maximum principal curvature of

∂
Ω

\partial \Omega

. Finally, we apply such

ρ
\rho

to obtain an explicit upper bound of the constant

C
Ω

{C_\Omega }

in the Henkin’s estimate

‖

H
Ω

f

‖

L
∞

(
Ω
)

⩽

C
Ω

‖
f
‖

L
∞

(
Ω
)

{\left \| {{H_\Omega }f} \right \|_{{L^\infty }(\Omega )}} \leqslant {C_\Omega }{\left \| f \right \|_{{L^\infty }(\Omega )}}

of the

∂
\partial

-problem on strictly convex domains

Ω
\Omega

in

C

n

{{\mathbf {C}}^n}

.