bluuebunny/crossref_metadata_2025_split huggingface dataset:
The method of negative curvature: the Kobayashi metric on 𝑃₂ minus 4 lines
Authors: Cowen Michael J. 
Published: June 2012

Bloch, and later H. Cartan, showed that if

H
1

,
…
,

H

n
+
2

{H_1}, \ldots ,{H_{n + 2}}

are

n
+
2

n + 2

hyperplanes in general position in complex projective space

P

n

{{\mathbf {P}}_n}

, then

P

n

−

H
1

∪
⋯
∪

H

n
+
2

{{\mathbf {P}}_n} - {H_1} \cup \cdots \cup {H_{n + 2}}

is (in current terminology) hyperbolic modulo

Δ
\Delta

, where

Δ
\Delta

is the union of the hyperplanes

(

H

1

∩
⋯
∩

H
k

)
⊕
(

H

k
+
1

∩
⋯
∩

H

n
+
2

)

({H_{^1}} \cap \cdots \cap {H_k}) \oplus ({H_{k + 1}} \cap \cdots \cap {H_{n + 2}})

for

2
⩽
k
⩽
n

2 \leqslant k \leqslant n

and all permutations of the

H
i

{H_i}

. Their results were purely qualitative. For

n
=
1

n = 1

, the thrice-punctured sphere, it is possible to estimate the Kobayashi metric, but no estimates were known for

n
⩾
2

n \geqslant 2

. Using the method of negative curvature, we give an explicit model for the Kobayashi metric when

n
=
2

n = 2

.
Weighted inequalities for one-sided maximal functions
Authors: Martín-Reyes F. J. , Ortega Salvador P. , de la Torre A. 
Published: June 2012

Let

M
g
+

M_g^ +

be the maximal operator defined by
\[

M
g
+

f
(
x
)
=

sup

h
>
0

(

∫
x

x
+
h

|

f
(
t
)

|

g
(
t
)
d
t

)

(

∫
x

x
+
h

g
(
t
)
d
t

)

−
1

,

M_g^ + f(x) = \sup \limits _{h > 0} \left ( {\int _x^{x + h} {|f(t)|g(t)dt} } \right ){\left ( {\int _x^{x + h} {g(t)dt} } \right )^{ - 1}},

\]
where

g
g

is a positive locally integrable function on

R

{\mathbf {R}}

. We characterize the pairs of nonnegative functions

(
u
,
v
)

(u,v)

for which

M
g
+

M_g^ +

applies

L
p

(
v
)

{L^p}(v)

in

L
p

(
u
)

{L^p}(u)

or in weak-

L
p

(
u
)

{L^p}(u)

. Our results generalize Sawyer’s (case

g
=
1

g = 1

) but our proofs are different and we do not use Hardy’s inequalities, which makes the proofs of the inequalities self-contained.
Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term
Authors: García Azorero J. , Peral Alonso I. 
Published: June 2012

We study the existence of solutions for the following nonlinear degenerate elliptic problems in a bounded domain

Ω
⊂

R

N

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

\[

−
div
⁡
(

|

∇
u

|

p
−
2

∇
u
)
=

|

u

|

p

∗

−
2

u
+
λ

|

u

|

q
−
2

u
,

λ
>
0
,

- \operatorname {div} (|\nabla u{|^{p - 2}}\nabla u) = |u{|^{{p^{\ast }} - 2}}u + \lambda |u{|^{q - 2}}u,\qquad \lambda > 0,

\]
where

p

∗

{p^{\ast }}

is the critical Sobolev exponent, and

u

|

δ
Ω

≡
0

u{|_{\delta \Omega }} \equiv 0

. By using critical point methods we obtain the existence of solutions in the following cases: If

p
>
q
>

p

∗

p > q > {p^{\ast }}

, there exists

λ
0

>
0

{\lambda _0} > 0

such that for all

λ
>

λ
0

\lambda > {\lambda _0}

there exists a nontrivial solution. If

max
(
p
,

p

∗

−
p

/

(
p
−
1
)
)
>
q
>

p

∗

\max (p,{p^{\ast }} - p/(p - 1)) > q > {p^{\ast }}

, there exists nontrivial solution for all

λ
>
0

\lambda > 0

. If

1
>
q
>
p

1 > q > p

there exists

λ
1

{\lambda _1}

such that, for

0
>
λ
>

λ
1

0 > \lambda > {\lambda _1}

, there exist infinitely many solutions. Finally, we obtain a multiplicity result in a noncritical problem when the associated functional is not symmetric.
Entropy for canonical shifts
Authors: Choda Marie 
Published: June 2012

For a

∗

^{\ast }

-endomorphism

σ
\sigma

of an injective finite von Neumann algebra

A
A

, we investigate the relations among the entropy

H
(
σ
)

H(\sigma )

for

σ
\sigma

, the relative entropy

H
(
A

|

σ
(
A
)
)

H(A|\sigma (A))

of

σ
(
A
)

\sigma (A)

for

A
A

, the generalized index

λ
(
A
,
σ
(
A
)
)

\lambda (A,\sigma (A))

, and the index for subfactors. As an application, we have the following relations for the canonical shift

Γ
\Gamma

for the inclusion

N
⊂
M

N \subset M

of type

II

1

\text {II}_{1}

factors with the finite index

[
M
:
N
]

[M:N]

,
\[

H
(
A

|

Γ
(
A
)
)
≤
2
H
(
Γ
)
≤
log
⁡
λ

(
A
,
Γ
(
A
)

)

−
1

=
2
log
⁡
[
M
:
N
]
,

H(A|\Gamma (A)) \leq 2H(\Gamma ) \leq \log \lambda {(A,\Gamma (A))^{ - 1}} = 2\log [M:N],

\]
where

A
A

is the von Neumann algebra generated by the two of the relative commutants of

M
M

. In the case of that

N
⊂
M

N \subset M

has finite depth, then all of them coincide.
Rational fibrations in differential homological algebra
Authors: Murillo Aniceto 
Published: June 2012

In this paper, a result of [6] is generalized as follows: Given a fibration

F
→
E

→

p

B

F \to E\xrightarrow {p}B

of simply connected spaces in which either, the fibre has finite dimensional rational cohomology, or, it has finite dimensional rational homotopy and

ρ
\rho

induces a surjection in rational homotopy, we construct an explicit isomorphism,
\[

φ
:

Ext

C
∗

(
B
,

Q

)

⁡
(

Q

,

C
∗

(
B
;

Q

)
)

⊗
^

Ext

C
∗

(
F
;

Q

)

⁡
(

Q

,

C
∗

(
F
,

Q

)
)

→

≅

Ext

C
∗

(
E
;

Q

)

⁡
(
Q
,

C
∗

(
E
;

Q

)
)
.

\begin {array}{*{20}{c}} {\varphi :\operatorname {Ext}_{{C^\ast }(B,{\mathbf {Q}})}({\mathbf {Q}},{C^\ast }(B;{\mathbf {Q}}))\hat \otimes \operatorname {Ext}_{{C^\ast }(F;{\mathbf {Q}})}({\mathbf {Q}},{C^\ast }(F,{\mathbf {Q}}))} \\ {\xrightarrow { \cong }\operatorname {Ext}_{{C^\ast }(E;{\mathbf {Q}})}(Q,{C^\ast }(E;{\mathbf {Q}})).} \\ \end {array}

\]
This is deduced from its "algebraic translation," a more general result in the environment of graded differential homological algebra.
Mountain impasse theorem and spectrum of semilinear elliptic problems
Authors: Tintarev Kyril 
Published: June 2012

This paper studies a minimax problem for functionals in Hilbert space in the form of

G
(
u
)
=

1
2

ρ

|

|

u

|

|

2

−
g
(
u
)

G(u) = \frac {1} {2}\rho ||u|{|^2} - g(u)

, where

g
(
u
)

g(u)

is Fréchet differentiable with weakly continuous derivative. If

G
G

has a "mountain pass geometry" it does not necessarily have a critical point. Such a case is called, in this paper, a "mountain impasse". This paper states that in a case of mountain impasse, there exists a sequence

u
j

∈
H

{u_j} \in H

such that
\[

g
′
(

u
j

)
=

ρ
j

u
j

,

ρ
j

→
ρ
,

|

|

u
j

|

|

→
∞
,

g\prime ({u_j}) = {\rho _j}{u_j},\quad {\rho _j} \to \rho ,||{u_j}|| \to \infty ,

\]
and

G
(

u
j

)

G({u_j})

approximates the minimax value from above. If
\[

γ
(
t
)
=

sup

|

|

u

|

|

2

=
t

g
(
u
)

\gamma (t) = \sup \limits _{||u|{|^2} = t} \;g(u)

\]
and
\[

J
0

=

(

2

inf

t
2

>

t
1

>
0

γ
(

t
2

)
−
γ
(

t
1

)

t
2

−

t
1

,
2

sup

t
2

>

t
1

>
0

γ
(

t
2

)
−
γ
(

t
1

)

t
2

−

t
1

)

,

{J_0} = \left ( {2\inf \limits _{{t_2} > {t_1} > 0} \frac {{\gamma ({t_2}) - \gamma ({t_1})}} {{{t_2} - {t_1}}},2\sup \limits _{{t_2} > {t_1} > 0} \frac {{\gamma ({t_2}) - \gamma ({t_1})}} {{{t_2} - {t_1}}}} \right ),

\]
then

g
′
(
u
)
=
ρ
u

g\prime (u) = \rho u

has a nonzero solution

u
u

for a dense subset of

ρ
∈

J
0

\rho \in {J_0}

.
Harmonic analysis and ultracontractivity
Authors: Cowling Michael , Meda Stefano 
Published: June 2012

Let

(

T
t

)

t
>
0

{({T_t})_{t > 0}}

be a symmetric contraction semigroup on the spaces

L
p

(
M
)

(
1
≤
p
≤
∞
)

{L^p}(M)\;(1 \leq p \leq \infty )

, and let the functions

ϕ
\phi

and

ψ
\psi

be "regularly related". We show that

(

T
t

)

t
>
0

{({T_t})_{t > 0}}

is

ϕ
\phi

-ultracontractive, i.e., that

(

T
t

)

t
>
0

{({T_t})_{t > 0}}

satisfies the condition

‖

T
t

f

‖

∞

≤
C
ϕ

(
t

)

−
1

‖
f
‖

1

{\left \| {{T_t}f} \right \|_\infty } \leq C\phi {(t)^{ - 1}}{\left \| f \right \|_1}

for all

f
f

in

L
1

(
M
)

{L^1}(M)

and all

t
t

in

R

+

{{\mathbf {R}}^ + }

, if and only if the infinitesimal generator

G

\mathcal {G}

has Sobolev embedding properties, namely,

‖

ψ

(

G

)

−
α

f

‖

q

≤
C

‖
f
‖

p

{\left \| {\psi {{(\mathcal {G})}^{ - \alpha }}f} \right \|_q} \leq C{\left \| f \right \|_p}

for all

f
f

in

L
p

(
M
)

{L^p}(M)

, whenever

1
>
p
>
q
>
∞

1 > p > q > \infty

and

α
=
1

/

p
−
1

/

q

\alpha = 1/p - 1/q

. We establish some new spectral multiplier theorems and maximal function estimates. In particular, we give sufficient conditions on

m
m

for

m
(

G

)

m(\mathcal {G})

to map

L
p

(
M
)

{L^p}(M)

to

L
q

(
M
)

{L^q}(M)

, and for the example where there exists

μ
\mu

in

R

+

{{\mathbf {R}}^ + }

such that

ϕ
(
t
)
=

t
μ

\phi (t) = {t^\mu }

for all

t
t

in

R

+

{{\mathbf {R}}^ + }

, we give conditions which ensure that the maximal function

sup

t
>
0

|

t
α

T
t

f
(
∙
)

|

{\sup _{t > 0}}|{t^\alpha }{T_t}f( \bullet )|

is bounded.
Cohomology classes associated to anomalies
Authors: Peterson Gregory Lambros 
Published: June 2012

One of the proposed settings for the description of anomalies in the setting of gauge field theories is a local bicomplex associated to a principal fiber bundle

G
→
P
→
M

G \to P \to M

. It is a bigraded algebra with two differentials which is invariantly defined, i.e. independent of local coordinates of

M
M

. I denote it by

S
M

∙
,
∙

S_M^{ \bullet , \bullet }

. Briefly,

S
M

p
,
q

S_M^{p,q}

consists of local

p
p

-multilinear operators from the gauge algebra into

q
q

-forms on

M
M

which depend on a connection

A
A

in a local manner; local means that the operators decrease supports. The gauge algebra is the Lie algebra of the gauge group, which consists of diffeomorphisms of

P
P

that respect the action of

G
G

and cover the identity diffeomorphism of

M
M

. In this setting, the anomalies are described as integrals over

M
M

whose integrands can be shown to be representatives of total cohomology classes in

H
1

(

S
M

∙
,
∙

)

{H^1}(S_M^{ \bullet , \bullet })

. The main reason for restricting to a local bicomplex is due to Peetre’s theorem. It states that local operators are differential operators over open sets

U
⊂
M

U \subset M

. This property is both mathematically natural and required by physical considerations. This paper explores the computation of the total cohomology of the local bicomplex by beginning with the coordinate description of the differential operators and then determining which of these differential operators can be used to construct invariantly defined objects. What is accomplished is the description of the differential operators which are invariant under the action of the local diffeomorphisms of

R

n

{\mathbb {R}^n}

and the computation of their total cohomology over open sets

U
⊂
M

U \subset M

. The main result is that
\[

H
d
∙

(

(

S
U

∙
,
∙

)

Diff

loc

(

R

n

)

)
≃

H
d
∙

(
W

(

g

)

[

n
2

]

)
,

H_d^ \bullet ({(S_U^{ \bullet , \bullet })^{{\operatorname {Diff}_{{\text {loc}}}}({\mathbb {R}^n})}}) \simeq H_d^ \bullet (W{(\mathfrak {g})_{[\tfrac {n} {2}]}}),

\]
where

(

S
U

∙
,
∙

)

Diff

loc

(

R

n

)

{(S_U^{ \bullet , \bullet })^{{{\operatorname {Diff}}_{{\text {loc}}}}({\mathbb {R}^n})}}

denotes the invariant differential operators over the open set

U
U

and

W

(

g

)

[

n
2

]

W{(\mathfrak {g})_{[\tfrac {n} {2}]}}

is the Weil algebra of

g

\mathfrak {g}

, the Lie algebra of

G
G

truncated at

[

n
2

]

[\tfrac {n} {2}]

, the greatest integer less than or equal to half the dimension of

M
M

. This shows that the cohomology groups over open sets are nonzero only in the range

n
≤
q
≤
n
+
r

n \leq q \leq n + r

where

r
r

is the dimension of the Lie algebra

g

\mathfrak {g}

, and in this range they are all finite dimensional. This result is globalized in the special case that the associated fiber bundle

ad
∗

P

\operatorname {ad}^\ast \;P

is trivializable.
Classification of singularities for blowing up solutions in higher dimensions
Authors: Velázquez J. J. L. 
Published: June 2012

Consider the Cauchy problem (P)\[{ut−Δu=upamp;whenx∈RN,amp;t>0,N≥1,u(x,0)=u0(x)amp;whenx∈RN,amp;\left \{ {\begin {array}{*{20}{c}} {{u_t} - \Delta u = {u^p}} \hfill & {{\text {when}}\;x \in {\mathbb {R}^N},} \hfill & {t > 0,N \geq 1,} \hfill \\ {u(x,0) = {u_0}(x)} \hfill & {{\text {when}}\;x \in {\mathbb {R}^N},} \hfill & {} \hfill \\ \end {array} } \right .\]wherep>1p > 1, andu0(x){u_0}(x)is a continuous, nonnegative and bounded function. It is known that, under fairly general assumptions onu0(x){u_0}(x), the unique solution of(P)({\text {P}}),u(x,t)u(x,t), blows up in a finite time, by which we mean that\[limsupt↑T(supx∈RNu(x,t))=+∞.\lim \sup \limits _{t \uparrow T} \left ( {\sup \limits _{x \in {\mathbb {R}^N}} \;u(x,t)} \right ) = + \infty .\]In this paper we shall assume thatu(x,t)u(x,t)blows up atx=0x = 0,t=T>+∞t = T > + \infty, and derive the possible asymptotic behaviours ofu(x,t)u(x,t)as(x,t)→(0,T)(x,t) \to (0,T), under general assumptions on the blow-up rate.
Entropy solutions for diffusion-convection equations with partial diffusivity
Authors: Escobedo M. , Vázquez J. L. , Zuazua Enrike 
Published: June 2012

We consider the Cauchy problem for the following scalar conservation law with partial viscosity
\[

u
t

=

Δ
x

u
+

∂
y

(
f
(
u
)
)
,

(
x
,
y
)
∈

R

N

,
t
>
0.

{u_t} = {\Delta _x}u + {\partial _y}(f(u)),\quad (x,y) \in {{\mathbf {R}}^N},t > 0.

\]
The existence of solutions is proved by the vanishing viscosity method. By introducing a suitable entropy condition we prove uniqueness of solutions. This entropy condition is inspired by the entropy criterion introduced by Kruzhkov for hyperbolic conservation laws but it takes into account the effect of diffusion.
On the solvability of systems of inclusions involving noncompact operators
Authors: Nistri P. , Obukhovskiĭ V. V. , Zecca P. 
Published: June 2012

We consider the solvability of a system
\[

{

y
∈

F
¯

(
x
,
y
)
,

x
∈

G
¯

(
x
,
y
)

\left \{ {\begin {array}{*{20}{c}} {y \in \bar F(x,y),} \\ {x \in \bar G(x,y)} \\ \end {array} } \right .

\]
of set-valued maps in two different cases. In the first one, the map

(
x
,
y
)
−
∘

F
¯

(
x
,
y
)

(x,y) - \circ \bar F(x,y)

is supposed to be closed graph with convex values and condensing in the second variable and

(
x
,
y
)
−
∘

G
¯

(
x
,
y
)

(x,y) - \circ \bar G(x,y)

is supposed to be a permissible map (i.e. composition of an upper semicontinuous map with acyclic values and a continuous, single-valued map), satisfying a condensivity condition in the first variable. In the second case

F
¯

\bar F

is as before with compact, not necessarily convex, values and

G
¯

\bar G

is an admissible map (i.e. it is composition of upper semicontinuous acyclic maps). In the latter case, in order to apply a fixed point theorem for admissible maps, we have to assume that the solution set

x
−
∘
S
(
x
)

x - \circ S(x)

of the first equation is acyclic. Two examples of applications of the abstract results are given. The first is a control problem for a neutral functional differential equation on a finite time interval; the second one deals with a semilinear differential inclusion in a Banach space and sufficient conditions are given to show that it has periodic solutions of a prescribed period.
Binary forms, hypergeometric functions and the Schwarz-Christoffel mapping formula
Authors: Bean Michael A. 
Published: June 2012

In a previous paper, it was shown that if

F
F

is a binary form with complex coefficients having degree

n
⩾
3

n \geqslant 3

and discriminant

D
F

≠
0

{D_F} \ne 0

, and if

A
F

{A_F}

is the area of the region

|

F
(
x
,
y
)

|

⩽
1

\left | {F(x,y)} \right | \leqslant 1

in the real affine plane, then

|

D
F

|

1

/

n
(
n
−
1
)

A
F

⩽
3
B
(

1
3

,

1
3

)

{\left | {{D_F}} \right |^{1/n(n - 1)}}{A_F} \leqslant 3B(\frac {1} {3},\frac {1} {3})

, where

B
(

1
3

,

1
3

)

B(\frac {1} {3},\frac {1} {3})

denotes the Beta function with arguments of 1/3. This inequality was derived by demonstrating that the sequence

{

M
n

}

\{ {M_n}\}

defined by

M
n

=
max

|

D
F

|

1

/

n
(
n
−
1
)

A
F

{M_n} = \max |{D_F}{|^{1/n(n - 1)}}{A_F}

, where the maximum is taken over all forms of degree

n
n

with

D
F

≠
0

{D_F} \ne 0

, is decreasing, and then by showing that

M
3

=
3
B
(

1
3

,

1
3

)

{M_3} = 3B(\frac {1} {3},\frac {1} {3})

. The resulting estimate,

A
F

⩽
3
B
(

1
3

,

1
3

)

{A_F} \leqslant 3B(\frac {1} {3},\frac {1} {3})

for such forms with integer coefficients, has had significant consequences for the enumeration of solutions of Thue inequalities. This paper examines the related problem of determining precise values for the sequence

{

M
n

}

\{ {M_n}\}

. By appealing to the theory of hypergeometric functions, it is shown that

M
4

=

2

7

/

6

B
(

1
4

,

1
2

)

{M_4} = {2^{7/6}}B(\frac {1} {4},\frac {1} {2})

and that

M
4

{M_4}

is attained for the form

X
Y
(

X
2

−

Y
2

)

XY({X^2} - {Y^2})

. It is also shown that there is a correspondence, arising from the Schwarz-Christoifel mapping formula, between a particular collection of binary forms and the set of equiangular polygons, with the property that

A
F

{A_F}

is the perimeter of the polygon corresponding to

F
F

. Based on this correspondence and a representation theorem for

|

D
F

|

1

/

n
(
n
−
1
)

A
F

|{D_F}{|^{1/n(n - 1)}}{A_F}

, it is conjectured that

M
n

=

D

F
n
∗

1

/

n
(
n
−
1
)

A

F
n
∗

{M_n} = D_{F_n^ * }^{1/n(n - 1)}{A_{F_n^*}}

, where

F
n
∗

(
X
,
Y
)
=

∏

k
=
1

n

(
X
sin
⁡

(

k
π

n

)

−
Y
cos
⁡

(

k
π

n

)

)

F_n^*(X,Y) = \prod _{k = 1}^n \left (X \sin \left (\frac {k\pi }{n}\right ) - Y \cos \left (\frac {k\pi }{n}\right )\right )

, and that the limiting value of the sequence

{

M
n

}

\{ {M_n}\}

is

2
π

2\pi

.
Quasilinear elliptic equations with VMO coefficients
Authors: Palagachev Dian K. 
Published: June 2012

Strong solvability and uniqueness in Sobolev space

W

2
,
n

(
Ω
)

{W^{2,n}}(\Omega )

are proved for the Dirichlet problem
\[

{

u
=
φ

on

∂
Ω
.

a

i
j

(
x
,
u
)

D

i
j

u
+
b
(
x
,
u
,
D
u
)
=
0

a

.e

.

Ω

\left \{ {_{u = \varphi \quad {\text {on}}\partial \Omega .}^{{a^{ij}}(x,u){D_{ij}}u + b(x,u,Du) = 0\quad {\text {a}}{\text {.e}}{\text {.}}\Omega }} \right .

\]
It is assumed that the coefficients of the quasilinear elliptic operator satisfy Carathéodory’s condition, the

a

i
j

{a^{ij}}

are

V

M

O

V\, M\, O

functions with respect to

x
x

, and structure conditions on

b
b

are required. The main results are derived by means of the Aleksandrov-Pucci maximum principle and Leray-Schauder’s fixed point theorem via a priori estimate for the

L

2
n

{L^{2n}}

-norm of the gradient.
Classifying spaces and Dirac operators coupled to instantons
Authors: Sanders Marc 
Published: June 2012

LetM(k,SU(l))M(k,SU(l))denote the moduli space of based gauge equivalence classes ofSU(l)SU(l)instantons on principal bundles overS4{S^4}with second Chern class equal tokk. In this paper we use Dirac operators coupled to such connections to study the topology of these moduli spaces asllincreases relative tokk. This "coupling" procedure produces maps∂u:M(k,SU(l))→BU(k){\partial _u}:M(k,SU(l)) \to BU(k), and we prove that in the limit overllsuch maps recover Kirwan’s[K][\text {K}]homotopy equivalenceM(k,SU)≃BU(k)M(k,SU) \simeq BU(k). We also compute, for anykkandll, the image of the homology map(∂u)∗:H∗(M(k,SU(l));Z)→H∗(BU(k);Z){({\partial _u})_ * }:{H_ * }(M(k,SU(l));Z) \to {H_ * }(BU(k);Z). Finally, we prove all the analogous results forSp(l)Sp(l)instantons.
On minimal sets of scalar parabolic equations with skew-product structures
Authors: Shen Wen Xian , Yi Yingfei 
Published: June 2012

Skew-product semi-flow

Π
t

:
X
×
Y
→
X
×
Y

{\Pi _t}:X \times Y \to X \times Y

which is generated by
\[

{

u
t

=

u

x
x

+
f
(
y
⋅

t
,
x
,
u
,

u
x

)
,

t
>
0
,

0
>
x
>
1
,

y
∈
Y
,

D

or

N

boundary conditions

\left \{ \begin {gathered} {u_t} = {u_{xx}} + f(y \cdot \,t,x,u,{u_x}),\qquad t > 0,\;0 > x > 1,\;y \in Y, \hfill \\ D\;{\text {or }}N\;{\text {boundary conditions}} \hfill \\ \end {gathered} \right .

\]
is considered, where

X
X

is an appropriate subspace of

H
2

(
0
,
1
)
,

(
Y
,

R

)

{H^2}(0,1),\;(Y,\,\mathbb {R})

is a minimal flow with compact phase space. It is shown that a minimal set

E
⊂
X
×
Y

E \subset X \times Y

of

Π
t

{\Pi _t}

is an almost

1

-

1

1{\text { - }}1

extension of

Y
Y

, that is, set

Y
0

=
{
y
∈
Y

|

card
⁡
(
E
⊂

P

−
1

(
y
)
)
=
1
}

{Y_0} = \{ y \in Y|\operatorname {card} (E \subset {P^{ - 1}}(y)) = 1\}

is a residual subset of

Y
Y

, where

P
:
X
×
Y
→
Y

P:X \times Y \to Y

is the natural projection. Consequently, if

(
Y
,

R

)

(Y,\mathbb {R})

is almost periodic minimal, then any minimal set

E
⊂
X
×
Y

E \subset X \times Y

of

Π
t

{\Pi _t}

is an almost automorphic minimal set. It is also proved that dynamics of

Π
t

{\Pi _t}

is closed in the category of almost automorphy, that is, a minimal set

E
⊂
X
×
Y

E \subset X \times Y

of

Π
t

{\Pi _t}

is almost automorphic minimal if and only if

(
Y
,

R

)

(Y,\mathbb {R})

is almost automorphic minimal. Asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed. Examples of nonalmost periodic almost automorphic minimal sets are provided.
NetCmpt: a network-based tool for calculating the metabolic competition between bacterial species
Authors: Kreimer Anat , Doron-Faigenboim Adi , Borenstein Elhanan , Freilich Shiri 
Published: June 2012

Abstract
Summary: NetCmpt is a tool for calculating the competitive potential between pairs of bacterial species. The score describes the effective metabolic overlap (EMO) between two species, derived from analyzing the topology of the corresponding metabolic models. NetCmpt is based on the EMO algorithm, developed and validated in previous studies. It takes as input lists of species-specific enzymatic reactions (EC numbers) and generates a matrix of the potential competition scores between all pairwise combinations.
Availability and implementation: NetCmpt is provided as both a web tool and a software package, designed for the use of non-computational biologists. The NetCmpt web tool, software, examples, and documentation are freely available online at http://app.agri.gov.il/shiri/NetComp.php.
Contact: anat.kreimer@gmail.com; shiri.freilich@gmail.com
Supplementary Information: Supplementary data are available at Bioinformatics online.
SYNCSA—R tool for analysis of metacommunities based on functional traits and phylogeny of the community components
Authors: Debastiani Vanderlei J. , Pillar Valério D. 
Published: June 2012

Abstract
Summary: SYNCSA is an R package for the analysis of metacommunities based on functional traits and phylogeny of the community components. It offers tools to calculate several matrix correlations that express trait-convergence assembly patterns, trait-divergence assembly patterns and phylogenetic signal in functional traits at the species pool level and at the metacommunity level.
Availability and implementation: SYNCSA is a package for the R environment, under a GPL-2 open-source license and freely available on CRAN official web server for R (http://cran.r-project.org).
Contact: vanderleidebastiani@yahoo.com.br