bluuebunny/crossref_metadata_2025_split huggingface dataset:
Capacitability theorem in measurable gambling theory
Authors: Maitra A. , Purves R. , Sudderth W. 
Published: June 2012

A player in a measurable gambling house

Γ
\Gamma

defined on a Polish state space

X
X

has available, for each

x
∈
X

x \in X

, the collection

Σ
(
x
)

\Sigma (x)

of possible distributions

σ
\sigma

for the stochastic process

x
1

,

x
2

,
…

{x_1},{x_2}, \ldots

of future states. If the object is to control the process so that it will lie in an analytic subset

A
A

of

H
=
X
×
X
×
⋯

H = X \times X \times \cdots

, then the player’s optimal reward is
\[

M
(
A
)
(
x
)
=
sup
{
σ
(
A
)
:
σ
∈
Σ
(
x
)
}
.

M(A)(x) = \sup \{ \sigma (A):\sigma \in \Sigma (x)\}.

\]
The operator

M
(
∙
)
(
x
)

M( \bullet )(x)

is shown to be regular in the sense that
\[

M
(
A
)
(
x
)
=
inf
M
(
{
τ
>
∞
}
)
(
x
)
,

M(A)(x) = \inf M(\{ \tau > \infty \} )(x),

\]
where the infimum is over Borel stopping times

τ
\tau

such that

A
⊆
{
τ
>
∞
}

A \subseteq \{ \tau > \infty \}

. A consequence of this regularity property is that the value of

M
(
A
)
(
x
)

M(A)(x)

is unchanged if, as in the gambling theory of Dubins and Savage, the player is allowed to use nonmeasurable strategies. This last result is seen to hold for bounded, Borel measurable payoff functions including that of Dubins and Savage.
Self-similar measures and their Fourier transforms. II
Authors: Strichartz Robert S. 
Published: June 2012

A self-similar measure on

R

n

{{\mathbf {R}}^n}

was defined by Hutchinson to be a probability measure satisfying

(

∗

)

({\ast })

\[

μ
=

∑

j
=
1

m

a
j

μ
∘

S
j

−
1

\mu = \sum \limits _{j = 1}^m {{a_j}\mu \circ S_j^{ - 1}}

\]
, where

S
j

x
=

ρ
j

R
j

x
+

b
j

{S_j}x = {\rho _j}{R_j}x + {b_j}

is a contractive similarity

(
0
>

ρ
j

>
1
,

R
j

(0 > {\rho _j} > 1,{R_j}

orthogonal) and the weights

a
j

{a_j}

satisfy

0
>

a
j

>
1
,

∑

j
=
1

m

a
j

=
1

0 > {a_j} > 1,\sum \nolimits _{j = 1}^m {{a_j} = 1}

. By analogy, we define a self-similar distribution by the same identity

(

∗

)

( {\ast } )

but allowing the weights

a
j

{a_j}

to be arbitrary complex numbers. We give necessary and sufficient conditions for the existence of a solution to

(

∗

)

( {\ast } )

among distributions of compact support, and show that the space of such solutions is always finite dimensional. If

F
F

denotes the Fourier transformation of a self-similar distribution of compact support, let
\[

H
(
R
)
=

1

R

n
−
β

∫

|

x

|

≤
R

|

F
(
x
)

|

2

d
x
,

H(R) = \frac {1}{{{R^{n - \beta }}}}\int _{|x| \leq R} {|F(x){|^2}dx,}

\]
where

β
\beta

is defined by the equation

∑

j
=
1

m

ρ
j

−
β

|

a
j

|

2

=
1

\sum \nolimits _{j = 1}^m {\rho _j^{ - \beta }|{a_j}{|^2} = 1}

. If

ρ
j

ν
j

=
ρ

\rho _j^{{\nu _j}} = \rho

for some fixed

ρ
\rho

and

ν
j

{\nu _j}

positive integers we say the

{

ρ
j

}

\{ {\rho _j}\}

are exponentially commensurable. In this case we prove (under some additional hypotheses) that

H
(
R
)

H(R)

is asymptotic (in a suitable sense) to a bounded function

H
~

(
R
)

\tilde H(R)

that is bounded away from zero and periodic in the sense that

H
~

(
ρ
R
)
=

H
~

(
R
)

\tilde H(\rho R) = \tilde H(R)

for all

R
>
0

R > 0

. If the

{

ρ
j

}

\{ {\rho _j}\}

are exponentially incommensurable then

lim

R
→
∞

H
(
R
)

{\lim _{R \to \infty }}H(R)

exists and is nonzero.
𝑊^{2,𝑝}-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients
Authors: Chiarenza Filippo , Frasca Michele , Longo Placido 
Published: June 2012

We prove a well-posedness result in the class

W

2
,
p

∩

W
0

1
,
p

{W^{2,p}} \cap W_0^{1,p}

for the Dirichlet problem
\[

{

L
u
=
f

a
m
p
;

a.e.

in

Ω

,

u
=
0

a
m
p
;

on

∂
Ω

.

\left \{ {\begin {array}{*{20}{c}} {Lu = f} & {{\text {a.e.}}\;{\text {in}}\;\Omega }, \\ {u = 0} & {{\text {on}}\;\partial \Omega }. \\ \end {array} } \right .

\]
We assume the coefficients of the elliptic nondivergence form equation that we study are in

VMO

∩

L
∞

{\text {VMO}} \cap {L^\infty }

.
Continuous dependence of nonmonotonic discontinuous differential equations
Authors: Biles Daniel C. 
Published: June 2012

Continuous dependence of solutions for a class of nonmonotonic, discontinuous differential equations is studied. First, a local existence theorem due to

Z
Z

. Wu is extended to a larger class. Then, a result concerning continuous dependence for this larger class is proven. This employs a type of convergence similar to Gihman’s Convergence Criterion, which is defined to be
\[

For all

a
,
b

and

y

lim

n
→
∞

∫
a
b

f
n

(
s
,
y
)
d
s
=

∫
a
b

f
∞

(
s
,
y
)

d
s

.

{\text {For all}}\;a,b\;{\text {and}}\;y\quad \lim \limits _{n \to \infty } \int _a^b {{f_n}(s,y)ds = } \int _a^b {{f_\infty }(s,y)\,ds}.

\]
The significance of Gihman’s Convergence Criterion is that for certain classes of differential equations it has been found to be necessary and sufficient for continuous dependence. Finally, examples are presented to motivate and clarify this continuous dependence result.
The Koebe semigroup and a class of averaging operators on 𝐻^{𝑝}(𝐷)
Authors: Siskakis Aristomenis G. 
Published: June 2012

We study on the Hardy space

H
p

{H^p}

the operators

T
F

{T_F}

given by
\[

T
F

(
f
)
(
z
)
=

1
z

∫
0
z

f
(
ζ
)

1

F
(
ζ
)

d
ζ

{T_F}(f)(z) = \frac {1} {z}\int _0^z {f(\zeta )\frac {1} {{F(\zeta )}}\;d\zeta }

\]
where

F
(
z
)

F(z)

is analytic on the unit disc

D

\mathbb {D}

and has

Re
⁡
F
(
z
)
≥
0

\operatorname {Re} F(z) \geq 0

. Each such operator is closely related to a strongly continuous semigroup of weighted composition operators. By studying first an extremal such semigroup (the Koebe semigroup) we are able to obtain the upper bound

‖

T
F

‖

p

≤
2
p
Re
⁡
(
1

/

F
(
0
)
)
+

|

Im
⁡
(
1

/

F
(
0
)
)

|

{\left \| {{T_F}} \right \|_p} \leq 2p\operatorname {Re} (1/F(0)) + |\operatorname {Im} (1/F(0))|

for the norm. We also show that

T
F

{T_F}

is compact on

H
p

{H^p}

if and only if the measure

μ
\mu

in the Herglotz representation of

1

/

F

1/F

is continuous.
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

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

.
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
\[

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.
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

(

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})).)
Asymptotics for orthogonal rational functions
Authors: Bultheel A. , González-Vera P. , Hendriksen E. , Njåstad O. 
Published: June 2012

Let{αn}\{ {\alpha _n}\}be a sequence of (not necessarily distinct) points in the open unit disk, and let\[B0=1,Bn(z)=∏m=1nαm¯|αm|(αm−z)(1−αm¯z),n=1,2,…,{B_0} = 1,\quad {B_n}(z) = \prod \limits _{m = 1}^n {\frac {{\overline {{\alpha _m}} }} {{|{\alpha _m}|}}\frac {{({\alpha _m} - z)}} {{(1 - \overline {{\alpha _m}} z}}),\qquad n = 1,2, \ldots ,}\](αn¯|αn|=−1\frac {{\overline {{\alpha _n}} }} {{|{\alpha _n}|}} = - 1whenαn=0{\alpha _n} = 0). Letμ\mube a finite (positive) Borel measure on the unit circle, and let{φn(z)}\{ {\varphi _n}(z)\}be orthonormal functions obtained by orthogonalization of{Bn:n=0,1,2,…}\{ {B_n}:n = 0,1,2, \ldots \}with respect toμ\mu. Boundedness and convergence properties of the reciprocal orthogonal functionsφn∗(z)=Bn(z)φn(1/z¯)¯\varphi _n^*(z) = {B_n}(z)\overline {{\varphi _n}(1/\overline z )}and the reproducing kernelskn(z,w)=∑m=0nφm(z)φm(w)¯{k_n}(z,w) = \sum \nolimits _{m = 0}^n {{\varphi _m}(z)\overline {{\varphi _m}(w)} }are discussed in the situation|αn|⩽R>1|{\alpha _n}| \leqslant R > 1for allnn, in particular their relationship to the Szegö condition∫−ππln⁡μ′(θ)dθ>−∞\int _{ - \pi }^\pi {\ln \mu ’(\theta )d\theta > - \infty }and noncompleteness inL2(μ){L_2}(\mu )of the system{φn(z):n=0,1,2,…}\{ {\varphi _n}(z):n = 0,1,2, \ldots \}. Limit functions ofφn∗(z)\varphi _n^{\ast }(z)andkn(z,w){k_n}(z,w)are obtained. In particular, if a subsequence{αn(s)}\{ {\alpha _{n(s)}}\}converge toα\alpha, then the subsequence{φn(s)∗(z)}\{ \varphi _{n(s)}^{\ast }(z)\}converges to\[eiλ1−|α|21−α¯z1σμ(z),λ∈R,{e^{i\lambda }}\frac {{\sqrt {1 - |\alpha {|^2}} }} {{1 - \overline \alpha z}}\frac {1} {{{\sigma _{\mu (z)}}}},\qquad \lambda \in {\mathbf {R}},\]where\[σμ(z)=2πexp⁡[14π∫−ππeiθ+zeiθ−zln⁡μ′(θ)dθ].{\sigma _\mu }(z) = \sqrt {2\pi } \exp \left [ {\frac {1} {{4\pi }}\int _{ - \pi }^\pi {\frac {{{e^{i\theta }} + z}} {{{e^{i\theta }} - z}}} \ln \mu ’(\theta )d\theta } \right ].\]The kernels{kn(z,w)}\{ {k_n}(z,w)\}converge to1/(1−zw¯)σμ(z)σμ(w)¯1/(1 - z\overline w ){\sigma _\mu }(z)\overline {{\sigma _\mu }(w)}. The results generalize corresponding results from the classical Szegö theory (concerned with the polynomial situationαn=0{\alpha _n} = 0for allnn).
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

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.
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

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’

.
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
\[

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)

.
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
\[

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.
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
\[

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.
Sobolev orthogonal polynomials and spectral differential equations
Authors: Jung I. H. , Kwon K. H. , Lee D. W. , Littlejohn L. L. 
Published: June 2012

We find necessary and sufficient conditions for a spectral differential equation
\[

L
N

[
y
]
(
x
)
=

∑

i
=
1

N

ℓ
i

(
x
)

y

(
i
)

(
x
)
=

λ
n

y
(
x
)

{L_N}[y](x) = \sum \limits _{i = 1}^N {{\ell _i}(x){y^{(i)}}(x) = {\lambda _n}y(x)}

\]
to have Sobolev orthogonal polynomials of solutions, which are orthogonal relative to the Sobolev (pseudo-) inner product
\[

ϕ
(
p
,
q
)
=

∫

R

p
q
d
μ
+

∫

R

p
′

q
′

d
v
,

\phi (p,q) = \int _\mathbb {R}^{} {pqd\mu + \int _\mathbb {R}^{} {p’q’dv,} }

\]
where

d
μ

d\mu

and

d
v

dv

are signed Borel measures having finite moments. This result generalizes a result by H. L. Krall, which handles the case when

d
v
=
0

dv = 0

.
Equivalence relations induced by actions of Polish groups
Authors: Solecki Sławomir 
Published: June 2012

We give an algebraic characterization of those sequences

(

H
n

)

({H_n})

of countable abelian groups for which the equivalence relations induced by Borel (or, equivalently, continuous) actions of

H
0

×

H
1

×

H
2

×
⋯

{H_0} \times {H_1} \times {H_2} \times \cdots

are Borel. In particular, the equivalence relations induced by Borel actions of

H
ω

{H^\omega }

,

H
H

countable abelian, are Borel iff

H
≃

⊕
p

(

F
p

×

Z

(

p
∞

)

n
p

)

H \simeq { \oplus _p}({F_p} \times \mathbb {Z}{({p^\infty })^{{n_p}}})

, where

F
p

{F_p}

is a finite

p
p

-group,

Z

(

p
∞

)

\mathbb {Z}({p^\infty })

is the quasicyclic

p
p

-group,

n
p

∈
ω

{n_p} \in \omega

, and

p
p

varies over the set of all primes. This answers a question of R. L. Sami by showing that there are Borel actions of Polish abelian groups inducing non-Borel equivalence relations. The theorem also shows that there exist non-locally compact abelian Polish groups all of whose Borel actions induce only Borel equivalence relations. In the process of proving the theorem we generalize a result of Makkai on the existence of group trees of arbitrary height.
On the ideal class group of real biquadratic fields
Authors: Sime Patrick J. 
Published: June 2012

We discuss the structure of the ideal class group of real biquadratic fieldsKK, concentrating on the case that the44-rank of the ideal class groups of the quadratic subfields ofKKis00. In this case, we give estimates for the44-rank of the ideal class group ofKK. As an example, letK=Q(p,627)K = \mathbb {Q}(\sqrt p ,\sqrt {627} ), whereppis a prime satisfying certain congruence conditions. The22-primary part of the ideal class group ofKKis then isomorphic to(Z/4Z)2,Z/4Z×(Z/2Z)2{(\mathbb {Z}/4\mathbb {Z})^2},\mathbb {Z}/4\mathbb {Z} \times {(\mathbb {Z}/2\mathbb {Z})^2}, or(Z/2Z)4{(\mathbb {Z}/2\mathbb {Z})^4}. Further, each of the above occurs infinitely often.
First Report of Squash leaf curl virus in Cucurbits in Lebanon
Authors: Sobh H. , Samsatly J. , Jawhari M. , Najjar C. , Haidar A. , Abou-Jawdah Y. 
Published: June 2012

During the second squash cropping season, which coincides with high whitefly populations, a high incidence of plants with severe leaf curl symptoms was observed. Many farmers reported yield losses ranging from 70 to 80%. Surveys were conducted over five cropping seasons (2008 to 2010) and covered the coastal areas of Lebanon. A total of 675 samples were collected, including cucumber (Cucumis sativus), squash (Cucurbita sp.), melon (Cucumis melo), and watermelon (Citrullus lanatus). All squash samples had leaf curl symptoms, whereas 75 to 85% of cucumber, melon, and watermelon samples showed yellowing symptoms. The remaining 15 to 25% were asymptomatic. Total nucleic acids were extracted according to a small-scale CTAB protocol (4). PCR assays were initially conducted using the universal degenerate primers PAL1v1978 and PAR1c496, designed to detect DNA-A of several begomoviruses (3). Following sequencing of 22 randomly selected amplicons, BLASTN analysis showed that 19 samples were infected with Squash leaf curl virus (SLCV). SLCV specific primers: (SqA1R: 5′AGCTGTATCTTGGGCAACAGA3′ and SqA2F: 5′TATCTCCCATCTTGGCAAGG3′; amplicon size: 601 bp) were used for detection in the 675 samples. SLCV was detected in 223/249 (89%), 83/145 (57%), 129/229 (56%), and 25/52 (48%) of squash, cucumber, melon, and watermelon samples, respectively. The SLCV genome from a symptomatic squash plant collected from Akkar, North Lebanon, was amplified by rolling circle amplification (RCA) using the TempliPhi Amplification Kit (GE Healthcare). The product was used for biolistic inoculation of squash and cucumber as described (2). Severe leaf curl symptoms were observed on 7/10 of the squash seedlings (cv. Camelia F1) within 2 weeks of inoculation. However, no symptoms were observed on cucumber (cv. Beit alpha) 1 month after inoculation, even though 6/11 (54%) of the plants were positive for SLCV in PCR assays. Several primer sets were used for sequencing the full SLCV genome using the RCA product as template. The sequences were submitted to GenBank under accession numbers HM368373 and HM368374 (SLCV DNA A and B, respectively). Phylogenetic analysis showed that SLCV DNA A was most closely related to SLCV isolates from Egypt (DQ285019) and Israel (HQ184436) with 99% nucleotide identity; SLCV DNA B was most closely related to the same SLCV isolate from Israel (HQ184437) with 99% nucleotide identity. SLCV was first observed on squash in California in 1977, but was introduced during the last decade to the Mediterranean region (1) and currently is widespread all over Lebanon, posing a great threat to squash production. References: (1) Antignus et al. Phytoparasitica 31:415, 2003. (2) Guenoune-Gelbart et al. J. Virol. Methods 168:87, 2010. (3) Rojas et al. Plant Dis. 77:340, 1993. (4) Zhang et al. J. Virol. Methods 71:45, 1998.