bluuebunny/crossref_metadata_2025_split huggingface dataset:
Tauberian theorems for the Laplace-Stieltjes transform
Authors: Batty C. J. K. 
Published: June 2012

Let

α
:
[
0
,
∞
)
→

C

\alpha :[0,\infty ) \to {\mathbf {C}}

be a function of locally bounded variation, with

α
(
0
)
=
0

\alpha (0) = 0

, whose Laplace-Stieltjes transform

g
(
z
)
=

∫
0
∞

e

−
z
t

d
α
(
t
)

g(z) = \int _0^\infty {{e^{ - zt}}d\alpha (t)}

is absolutely convergent for

Re
⁡
z
>
0

\operatorname {Re} z > 0

. Let

E
E

be the singular set of

g
g

in

i

R

i{\mathbf {R}}

, and suppose that

0
∉
E

0 \notin E

. Various estimates for

lim

sup

t
→
∞

|

α
(
t
)
−
g
(
0
)

|

\lim {\sup _{t \to \infty }}|\alpha (t) - g(0)|

are obtained. In particular,

α
(
t
)
→
g
(
0
)

\alpha (t) \to g(0)

as

t
→
∞

t \to \infty

if
\[

(

i)

E

is null,

(ii)

sup

y
∈
E

sup

t
>
0

|

∫
0
t

e

−
i
y
s

d
α
(
s
)

|

>
∞
,

(

iii)

lim

δ
↓
0

lim

sup

t
→
∞

sup

t
−
δ
⩽
s
⩽
t

|

α
(
s
)
−
α
(
t
)

|

=
0.

\begin {gathered} ({\text {i)}}\quad E\,{\text {is null,}} \hfill \\ {\text {(ii)}}\quad \sup \limits _{y \in E} \sup \limits _{t > 0} \left | {\int _0^t {{e^{ - iys}}\,d\alpha (s)} } \right | > \infty , \hfill \\ ({\text {iii)}}\quad \lim \limits _{\delta \downarrow 0} \lim \sup \limits _{t \to \infty } \sup \limits _{t - \delta \leqslant s \leqslant t} |\alpha (s) - \alpha (t)| = 0. \hfill \\ \end {gathered}

\]
This result contains Tauberian theorems for Laplace transforms, power series, and Dirichlet series.
The nonlinear geometry of linear programming. III. Projective Legendre transform coordinates and Hilbert geometry
Authors: Lagarias J. C. 
Published: June 2012

This paper studies projective scaling trajectories, which are the trajectories obtained by following the infinitesimal version of Karmarkar’s linear programming algorithm. A nonlinear change of variables, projective Legendre transform coordinates, is introduced to study these trajectories. The projective Legendre transform mapping has a coordinate-free geometric interpretation in terms of the notion of "centering by a projective transformation." Let

H

{\mathsf {H}}

be a set of linear programming constraints

{
⟨

a

j

,

x

⟩
≥

b
j

:
1
≤
j
≤
m
}

\{ \langle {{\mathbf {a}}_j},{\mathbf {x}}\rangle \geq {b_j}:1 \leq j \leq m\}

on

R

n

{{\mathbf {R}}^n}

such that its polytope of feasible solutions

P

H

{P_{\mathsf {H}}}

is bounded and contains

0

{\mathbf {0}}

in its interior. The projective Legendre transform mapping

ψ

H

{\psi _{\mathsf {H}}}

is given by
\[

ψ

H

(

x

)
=

ϕ

H

(

x

)

m
+
⟨

ϕ

H

(

x

)
,

x

⟩

,

w
h
e
r
e

ϕ

H

(

x

)
=
−

∑

j
=
1

m

a

j

⟨

a

j

,

x

⟩
−

b
j

.

{\psi _{\mathsf {H}}}({\mathbf {x}}) = \frac {{{\phi _{\mathsf {H}}}({\mathbf {x}})}} {{m + \langle {\phi _{\mathsf {H}}}({\mathbf {x}}),{\mathbf {x}}\rangle }},\quad {\mathsf {where}}{\phi _{\mathsf {H}}}({\mathbf {x}}) = - \sum \limits _{j = 1}^m {\frac {{{{\mathbf {a}}_j}}} {{\langle {{\mathbf {a}}_j},{\mathbf {x}}\rangle - {b_j}}}.}

\]
Here

ϕ

H

(
x
)

{\phi _{\mathsf {H}}}(x)

is the Legendre transform coordinate mapping introduced in part II.

ψ

H

(

x

)

{\psi _{\mathsf {H}}}({\mathbf {x}})

is a one-to-one and onto mapping of the interior of the feasible solution polytope

Int
⁡
(

P

H

)

\operatorname {Int} ({P_{\mathsf {H}}})

to the interior of its polar polytope

Int
⁡
(

P

H

∘

)

\operatorname {Int} (P_{\mathsf {H}}^\circ )

. The set of projective scaling trajectories with objective function

⟨

c

,

x

⟩
−

c
0

\langle {\mathbf {c}},{\mathbf {x}}\rangle - {c_0}

are mapped under

ψ

H

{\psi _{\mathsf {H}}}

to the set of straight line segments in

Int
⁡
(

P

H

∘

)

\operatorname {Int} (P_{\mathsf {H}} ^\circ )

passing through the boundary point

−

c

/

c
0

- {\mathbf {c}}/{c_0}

of

P

H

∘

P_{\mathsf {H}} ^\circ

. As a consequence the projective scaling trajectories (for all objective functions) can be interpreted as the complete set of "geodesics" (actually distinguished chords) of a projectively invariant metric geometry on

Int
⁡
(

P

H

)

\operatorname {Int} ({P_{\mathsf {H}}})

, which is isometric to Hilbert geometry on the interior of the polar polytope

P

H

∘

P_{\mathsf {H}}^\circ

.
Sobolev interpolation inequalities with weights
Authors: Gutiérrez Cristian E. , Wheeden Richard L. 
Published: June 2012

We study weighted local Sobolev interpolation inequalities of the form
\[

1

w
2

(
B
)

∫
B

|

u
(
x
)

|

p
h

w
2

(
x
)
d
x
≤
c

(

1

v
(
B
)

∫
B

|

u
(
x
)

|

p

v
(
x
)
d
x

)

h
−
1

×

(

|

B

|

p

/

n

w
1

(
B
)

∫
B

|

∇
u
(
x
)

|

p

w
1

(
x
)
d
x
+

1

v
(
B
)

∫
B

|

u
(
x
)

|

p

v
(
x
)
d
x

)

,

\begin {gathered} \frac {1} {{{w_2}(B)}}{\int \limits _B {|u(x){|^{ph}}{w_2}(x)dx \leq c\left ( {\frac {1} {{v(B)}}\int \limits _B {|u(x){|^p}v(x)dx} } \right )} ^{h - 1}} \hfill \\ \times \left ( {\frac {{|B{|^{p/n}}}} {{{w_1}(B)}}\int \limits _B {|\nabla u(x){|^p}{w_1}(x)dx + \frac {1} {{v(B)}}} \int \limits _B {|u(x){|^p}v(x)dx} } \right ), \hfill \\ \end {gathered}

\]
, where

1
>
p
>
∞
,
h
>
1
,
B

1 > p > \infty ,h > 1, B

is a ball in

R

n

{{\mathbf {R}}^n}

, and

v
v

,

w
1

{w_1}

, and

w
2

{w_2}

are weight functions. The case

p
=
2

p = 2

is of special importance in deriving regularity results for solutions of degenerate parabolic equations. We also study the analogous inequality without the second summand on the right in the case

u
u

has compact support in

B
B

, and we derive global Landau inequalities

‖

∇
u

‖

L
w
q

≤
c

‖

∇
u

‖

L
v
p

1
−
a

‖

∇
2

u

‖

L
v
p

a

,
0
>
a
>
1
,
1
>
p
≤
q
>
∞

{\left \| {\nabla u} \right \|_{L_w^q}} \leq c\left \| {\nabla u} \right \|_{L_v^p}^{1 - a}\left \| {{\nabla ^2}u} \right \|_{L_v^p}^a,0 > a > 1,1 > p \leq q > \infty

, when

u
u

has compact support.
Parity and generalized multiplicity
Authors: Fitzpatrick P. M. , Pejsachowicz Jacobo 
Published: June 2012

Assuming that

X
X

and

Y
Y

are Banach spaces and

α
:
[
a
,
b
]
→

L

(
X
,
Y
)

\alpha :[a,b] \to \mathcal {L}(X,Y)

is a path of linear Fredholm operators with invertible endpoints, in

[

F

−
P
1
]

[{\text {F}} - \text {P}1]

we defined a homotopy invariant of

α
,
σ
(
α
,
I
)
∈

Z

2

\alpha ,\sigma (\alpha ,I) \in {{\mathbf {Z}}_2}

, the parity of

α
\alpha

on

I
I

. The parity plays a fundamental role in bifurcation problems, and in degree theory for nonlinear Fredholm-type mappings. Here we prove (a) that, generically, the parity is a

mod

2

\bmod \, 2

count of the number of transversal intersections of

α
(
I
)

\alpha (I)

with the set of singular operators, (b) that if

λ
0

{\lambda _0}

is an isolated singular point of

α
\alpha

, then the local parity
\[

σ
(
α
,

λ
0

)
≡

lim

ε
→
0

σ
(
α
,
[

λ
0

−
ε
,

λ
0

+
ε
]
)

\sigma (\alpha ,{\lambda _0}) \equiv \lim \limits _{\varepsilon \to 0} \sigma (\alpha ,[{\lambda _0} - \varepsilon ,{\lambda _0} + \varepsilon ])

\]
remains invariant under Lyapunov-Schmidt reduction, and (c) that

σ
(
α
,

λ
0

)
=

(
−
1

)

M
G

(

λ
0

)

\sigma (\alpha ,{\lambda _0}) = {(- 1)^{{M_G}({\lambda _0})}}

, where

M
G

(

λ
0

)

{M_G}({\lambda _0})

is any one of the various concepts of generalized multiplicity which have been defined in the context of linearized bifurcation data.
Control of degenerate diffusions in 𝑅^{𝑑}
Authors: Hijab Omar 
Published: June 2012

An optimal regularity result is established for the viscosity solution of the degenerate elliptic equation
\[

−
A
v
+
F
(
x
,
υ
,
D
υ
)
=
0
,

- Av + F(x,\upsilon ,D\upsilon )= 0,

\]

A
=

1
2

∑

a

i
j

(
x
)

∂
2

/

∂

x

i

∂

x
j

,
x
∈

R

d

A= \frac {1}{2}\sum {{a_{ij}}(x){\partial ^2}/\partial \,x{_i}\,\partial \,{x_j}}, x \in {{\mathbf {R}}^d}

. We assume the equation is of Bellman type, i.e.

F
(
x
,
υ
,
p
)
=

sup

u
∈
U

[
b
(
x
,
u
)
⋅
p
+
c
(
x
,
u
)
υ
−
f
(
x
,
u
)
]

F(x,\upsilon ,p)= {\sup _{u \in U}}[b(x,u) \cdot p + c(x,u)\upsilon - f(x,u)]

,

U
⊂

R

d

U \subset {{\mathbf {R}}^d}

. If we set

λ
≡

inf

x
,
u

c
(
x
,
u
)

\lambda \equiv {\inf _{x,u}}c(x,u)

, then there exists

λ
0

≥
0

{\lambda _0} \geq 0

such that

0
>
λ
>

λ
0

0 > \lambda > {\lambda _0}

implies

υ
\upsilon

is Hölder, while

λ
>

λ
0

\lambda > {\lambda _0}

implies

υ
\upsilon

is Lipschitz. The following is established: Suppose the equation is also of Lipschitz type, i.e. suppose there is a Lipschitz function

u
(
x
,
υ
,
p
)

u(x,\upsilon ,p)

such that the supremum in

F

(
x
,
υ
,
p
)

F\,(x,\upsilon ,p)

is uniquely attained at

u
=
u

(
x
,
υ
,
p
)

u= u\,(x,\upsilon ,p)

; then there exists

λ
1

>

λ
0

{\lambda _1} > {\lambda _0}

such that

λ
>

λ
1

\lambda > {\lambda _1}

implies

υ
\upsilon

is

C

1
,
1

,

{C^{1,1}},

i.e.

D
υ

D\upsilon

exists and is Lipschitz.
L’espace des arcs d’une surface
Authors: Cauty Robert 
Published: June 2012

We prove that, for any surface

M
M

, the space of arcs contained in

M
M

, with the topology induced by the Hausdorff distance, is homeomorphic to

M
×

∑

∞

M \times {\sum }^\infty

, where

∑
=
{
(

x
i

)
∈

l
2

/

∑

i
=
1

∞

(
i

x
i

)

2

>
∞
}

\sum = \{ ({x_i}) \in {l^2}/\sum \nolimits _{i = 1}^\infty {{{(i{x_i})}^2} > \infty \} }

.
𝐿^{𝑝} estimates for the X-ray transform restricted to line complexes of Kirillov type
Authors: Wang Hann Tzong 
Published: June 2012

Let there be given a piecewise continuous rectifiable curve

ϕ
:

R

→

R

n

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

. Let

G

1
,
n

(

M

1
,
n

)

{G_{1,n}}({M_{1,n}})

be the usual Grassmannian (bundle) in

R

n

{{\mathbf {R}}^n}

. Define an

n
n

-dimensional submanifold

M
ϕ

(

R

n

)

{M_\phi }({{\mathbf {R}}^n})

of

M

1
,
n

{M_{1,n}}

as the set of all copies of

G

1
,
n

{G_{1,n}}

along the curve

ϕ
\phi

. Following Kirillov, we know that a nice function

f
(
x
)

f(x)

can be recovered from its X-ray transform

R

1
,
n

f

{R_{1,n}}f

on

M
ϕ

(

R

n

)

{M_\phi }({{\mathbf {R}}^n})

if and only if the curve

ϕ
\phi

intersects almost every affine hyperplane. Define a measure on

M
ϕ

(

R

n

)

{M_\phi }({{\mathbf {R}}^n})

by

d
μ
=
d

μ
x

(
π
)
d
λ
(
x
)

d\mu = d{\mu _x}(\pi )d\lambda (x)

, where

d

μ
x

d{\mu _x}

is the probability measure on

M

1
,
n

{M_{1,n}}

carried by the set of lines passing through the point

x
x

and invariant under the stabilizer of

x
x

in

O
(
n
)

O(n)

and

d
λ

d\lambda

is the usual measure on

ϕ
\phi

. We show that, if

n
>
2

n > 2

and

ϕ
\phi

is unbounded, then

‖

R

1
,
n

f
‖

L
q

(

M
ϕ

(

R

n

)
,
d
μ
)

≤
C

‖
f
‖

L
p

(

R

n

)

\left \| {R_{1,n}}f\right \|_{{L^q}({M_\phi }({{\mathbf {R}}^n}),d\mu )} \leq C\left \| f\right \| _{{L^p}({{\mathbf {R}}^n})}

if and only if

p
=
q
=
n
−
1

p = q = n - 1

and

ϕ
\phi

is line-like, that is,

λ
(
ϕ
∩
B
(
0
;
R
)
)
=
O
(
R
)

\lambda (\phi \cap B(0;R)) = O(R)

. This result gives a classification of Kirillov line complexes in terms of

L
p

{L^p}

estimates.
The regular module problem. I
Authors: Berger T. R. , Hargraves B. B. , Shelton C. 
Published: June 2012

In the study of induced representations the following problem arises: Let

H
=
A
G

H = AG

be a finite solvable group and

k

{\mathbf {k}}

a field with

char
⁡

k

∤

|

A

|

\operatorname {char}{\mathbf {k}}\nmid \; |A|

. Let

V
V

be an irreducible, faithful, primitive

k

[
A
G
]

{\mathbf {k}}[AG]

-module. Suppose

H
H

contains a normal extraspecial

r
r

-subgroup

R
R

with

Z
(
R
)
≤
Z
(
H
)

Z(R) \leq Z(H)

and that

A
A

acts faithfully on

R
R

. Under what conditions does

A
A

have a regular direct summand in

V
V

? In this paper we consider this question under the hypotheses that

G
=
M
R

G = MR

, where

1
≠
M

1 \ne M

is normal abelian in

A
M

AM

,

A
A

is nilpotent,

(

|

A

|

,

|

M
R

|

)
=
(

|

M

|

,

|

R

|

)
=
1

(|A|,|MR|) = (|M|,|R|) = 1

, and

R

/

Z
(
R
)

R/Z(R)

is a faithful, irreducible

A
M

AM

-module. We show that

A
A

has at least three regular direct summands in

V
V

unless

|

A

|

|A|

,

exp
⁡
(
M
)

\exp (M)

, and

r
r

satisfy certain very restrictive conditions.
Examples of capacity for some elliptic operators
Authors: Wu Jang-Mei 
Published: June 2012

We study

L
L

-capacities for uniformly elliptic operators of nondivergence form
\[

L
=

∑

i
,
j

a

i
j

(
x
)

∂
2

∂

x
i

∂

x
j

+

∑
j

a
j

(
x
)

∂

∂

x
j

;

L = \sum \limits _{i,j} {{a_{ij}}(x)\frac {{{\partial ^2}}} {{\partial {x_i}\partial {x_j}}} + } \sum \limits _j {{a_j}(x)\frac {\partial } {{\partial {x_j}}};}

\]
and construct examples of large sets having zero

L
L

-capacity for some

L
L

, and small sets having positive

L
L

-capacity. The relations between ellipticity constants of the coefficients and the sizes of these sets are also considered.
An analogue of Siegel’s 𝜙-operator for automorphic forms for 𝐺𝐿_{𝑛}(𝑍)
Authors: Grenier Douglas 
Published: June 2012

If

S

P
n

\mathcal {S}{P_n}

is the symmetric space of

n
×
n

n \times n

positive matrices,

Y
∈

S

P
n

Y \in \mathcal {S}{P_n}

can be decomposed into
\[

Y
=

(

1

a
m
p
;
0

x

a
m
p
;
I

)

(

v

−
1

a
m
p
;
0

0

a
m
p
;

v

−
1

/

(
n
−
1
)

W

)

(

1

a
m
p
;

T
x

0

a
m
p
;
I

)

,

Y = \left ( {\begin {array}{*{20}{c}} 1 & 0 \\ x & I \\ \end {array} } \right )\left ( {\begin {array}{*{20}{c}} {{v^{ - 1}}} & 0 \\ 0 & {{v^{ - 1/(n - 1)}}W} \\ \end {array} } \right )\left ( {\begin {array}{*{20}{c}} 1 & {{T_x}} \\ 0 & I \\ \end {array} } \right ),

\]
where

W
∈

S

P

n
−
1

W \in \mathcal {S}{P_{n - 1}}

. By letting

v
→
∞

v \to \infty

we obtain the

ϕ
\phi

-operator that attaches to every automorphic form for

G

L
n

(

Z

)

G{L_n}(\mathbb {Z})

,

f
(
Y
)

f(Y)

, an automorphic form for

G

L

n
−
1

(

Z

)

G{L_{n - 1}}(\mathbb {Z})

,

f

|

ϕ
(
W
)

f|\phi (W)

.
Fundamental solutions for hypoelliptic differential operators depending analytically on a parameter
Authors: Mantlik Frank 
Published: June 2012

Let

P
(
λ
,
D
)
=

∑

|

α

|

≤
m

a
α

(
λ
)

D
α

P(\lambda ,D) = \sum \nolimits _{|\alpha | \leq m} {{a_\alpha }(\lambda ){D^\alpha }}

be a differential operator with constant coefficients

a
α

{a_\alpha }

depending analytically on a parameter

λ
\lambda

. Assume that each

P
(
λ
,
D
)

P(\lambda ,D)

is hypoelliptic and that the strength of

P
(
λ
,
D
)

P(\lambda ,D)

is independent of

λ
\lambda

. Under this condition we show that there exists a regular fundamental solution of

P
(
λ
,
D
)

P(\lambda ,D)

which also depends analytically on

λ
\lambda

.
On Lagrange interpolation at disturbed roots of unity
Authors: Chui Charles K. , Shen Xie Chang , Zhong Le Fan 
Published: June 2012

Letznk=eitnk{z_{nk}} = {e^{i{t_{nk}}}},0≤tn0>⋯>tnn>2π0 \leq {t_{n0}} > \cdots > {t_{nn}} > 2\pi,ffa function in the disc algebraAA, andμn=max{|tnk−2kπ/(n+1)|:0≤k≤n}{\mu _n} = \max \{ |{t_{nk}} - 2k\pi /(n + 1)|:0 \leq k \leq n\}. Denote byLn(f;⋅){L_n}(f;\; \cdot )the polynomial of degreennthat agrees withffat{znk:k=0,…,n}\{ {z_{nk}}:k = 0, \ldots ,n\}. In this paper, we prove that for everypp,0>p>∞0 > p > \infty, there exists aδp>0{\delta _p} > 0, such that||Ln(f;⋅)−f||p=O(ω(f;1n))||{L_n}(f;\cdot ) - f|{|_p} = O(\omega (f;\frac {1} {n}))wheneverμn≤δp/(n+1){\mu _n} \leq {\delta _p}/(n + 1). It must be emphasized thatδp{\delta _p}necessarily depends onpp, in the sense that there exists a family{znk:k=0,…,n}\{ {z_{nk}}:k = 0, \ldots ,n\}withμn=δ2/(n+1){\mu _n} = {\delta _2}/(n + 1)and such that||Ln(f;⋅)−f||2=O(ω(f;1n))||{L_n}(f;\cdot ) - f|{|_2} = O(\omega (f;\frac {1} {n}))for allf∈Af \in A, butsup{||Ln(f;⋅)||p:f∈A,||f||∞=1}\sup \{ ||{L_n}(f;\cdot )|{|_p}:f \in A,||f|{|_\infty } = 1\}diverges for sufficiently large values ofpp. In establishing our estimates, we also derive a Marcinkiewicz-Zygmund type inequality for{znk}\{ {z_{nk}}\}.
Chebyshev type estimates for Beurling generalized prime numbers. II
Authors: Zhang Wen-Bin 
Published: June 2012

Let

N
(
x
)

N(x)

be the distribution function of the integers in a Beurling generalized prime system. The Chebyshev type estimates for Beurling generalized prime numbers in the general case
\[

N
(
x
)
=
x

∑

ν
=
1

n

A
ν

log

ρ
ν

−
1

x
+
O
(
x

log

−
γ

x
)

N(x) = x\sum \limits _{\nu = 1}^n {{A_\nu }} {\log ^{{\rho _\nu } - 1}}x + O(x{\log ^{ - \gamma }}x)

\]
is a long standing question. In this paper we shall give an affirmative answer to the question by proving that the Chebyshev type estimates
\[

0
>
lim

inf

x
→
∞

ψ
(
x
)

x

,

lim

sup

x
→
∞

ψ
(
x
)

x

>
∞

0 > \lim \inf \limits _{x \to \infty } \frac {{\psi (x)}}{x},\quad \lim \sup \limits _{x \to \infty } \frac {{\psi (x)}}{x} > \infty

\]
hold even under weaker condition
\[

∫
1
∞

x

−
1

{

sup

x
>
∞

y

−
1

|

N
(
y
)
−
y

∑

ν
=
1

n

A
ν

log

ρ
ν

−
1

y

|

}

d
x
>
∞

\int _1^\infty {{x^{ - 1}}} \left \{ {\sup \limits _{x > \infty } {y^{ - 1}}\left | {N(y) - y\sum \limits _{\nu = 1}^n {{A_\nu }} {{\log }^{{\rho _\nu } - 1}}y} \right |} \right \}\,dx > \infty

\]
with

ρ
n

=
τ
≥
1

\rho _n=\tau \geq 1

,

0
>

ρ
1

>

ρ
2

>
⋯
>

ρ
n

0>\rho _1>\rho _2 >\cdots > \rho _n

, and

A
n

>
0

A_n > 0

. This generalizes a result of Diamond and a result of the present author.
Loop space homology of spaces of small category
Authors: Félix Yves , Thomas Jean-Claude 
Published: June 2012

Only little is known concerning

H
∗

(
Ω
X
;

k

)

{H_\ast }(\Omega X;{\mathbf {k}})

, the loop space homology of a finite

CW

{\text {CW}}

complex

X
X

with coefficients in a field

k

{\mathbf {k}}

. A space

X
X

is called an

r
r

-cone if there exists a filtration

∗
=

X
0

⊂

X
1

⊂
⋯
⊂

X
r

=
X

\ast = {X_0} \subset {X_1} \subset \cdots \subset {X_r} = X

, such that

X
i

{X_i}

has the homotopy type of the cofibre of a map from a wedge of sphere into

X

i
−
1

{X_{i - 1}}

. Denote by

A
X

{A_X}

the sub-Hopf algebra image of

H
∗

(
Ω

X
1

)

{H_\ast }(\Omega {X_1})

. We prove then that for a graded

r
r

-cone,

r
≤
3

r \leq 3

, there exists an isomorphism

A
X

⊗
T
(
U
)

→

≅

H
∗

(
Ω
X
)

{A_X} \otimes T(U)\xrightarrow { \cong }{H_\ast }(\Omega X)

.
On the Cauchy problem for reaction-diffusion equations
Authors: Wang Xuefeng 
Published: June 2012

The simplest model of the Cauchy problem considered in this paper is the following

(
∗
)

(\ast )

\[

u
t

=
Δ
u
+

u
p

,

a
m
p
;

x
∈

R
n

,
t
>
0
,
u
≥
0
,
p
>
1
,

u

|

t
=
0

=
ϕ
∈

C
B

(

R
n

)
,

a
m
p
;

ϕ
≥
0
,
ϕ

≢

0.

\begin {array}{*{20}{c}} {{u_t} = \Delta u + {u^p},} \hfill & {x \in {R^n},t > 0,u \geq 0,p > 1,} \hfill \\ {u{|_{t = 0}} = \phi \in {C_B}({R^n}),} \hfill & {\phi \geq 0,\phi \,\not \equiv \,0.} \hfill \\ \end {array} \;

\]
It is well known that when

1
>
p
≤
(
n
+
2
)

/

n

1 > p \leq (n + 2)/n

, the local solution of

(
∗
)

(\ast )

blows up in finite time as long as the initial value

ϕ
\phi

is nontrivial; and when

p
>
(
n
+
2
)

/

n

p > (n + 2)/n

, if

ϕ
\phi

is "small",

(
∗
)

(\ast )

has a global classical solution decaying to zero as

t
→
+
∞

t \to + \infty

, while if

ϕ
\phi

is "large", the local solution blows up in finite time. The main aim of this paper is to obtain optimal conditions on

ϕ
\phi

for global existence and to study the asymptotic behavior of those global solutions. In particular, we prove that if

n
≥
3

n \geq 3

,

p
>
n

/

(
n
−
2
)

p > n/(n - 2)

,
\[

0
≤
ϕ
(
x
)
≤
λ

u
s

(
x
)
=
λ

(

2
(
n
−
2
)

(
p
−
1
)

2

(

p
−

n

n
−
2

)

)

1

/

(
p
−
1
)

|

x

|

−
2

/

(
p
−
1
)

0 \leq \phi (x) \leq \lambda {u_s}(x) = \lambda {\left ( {\frac {{2(n - 2)}} {{{{(p - 1)}^2}}}\left ( {p - \frac {n} {{n - 2}}} \right )} \right )^{1/(p - 1)}}|x{|^{ - 2/(p - 1)}}

\]
(

u
s

{u_s}

is a singular equilibrium of

(
∗
)

(\ast )

) where

0
>
λ
>
1

0 > \lambda > 1

, then

(
∗
)

(\ast )

has a (unique) global classical solution

u
u

with

0
≤
u
≤
λ

u
s

0 \leq u \leq \lambda {u_s}

and
\[

u
(
x
,
t
)
≤

(
(

λ

1
−
p

−
1
)
(
p
−
1
)
t

)

−
1

/

(
p
−
1
)

.

u(x,t) \leq {(({\lambda ^{1 - p}} - 1)(p - 1)t)^{ - 1/(p - 1)}}.

\]
(This result implies that

u
0

≡
0

{u_0} \equiv 0

is stable w.r.t. to a weighted

L
∞

{L^\infty }

topology when

n
≥
3

n \geq 3

and

p
>
n

/

(
n
−
2
)

p > n/(n - 2)

.) We also obtain some sufficient conditions on

ϕ
\phi

for global nonexistence and those conditions, when combined with our global existence result, indicate that for

ϕ
\phi

around

u
s

{u_s}

, we are in a delicate situation, and when

p
p

is fixed,

u
0

≡
0

{u_0} \equiv 0

is "increasingly stable" as the dimension

n
↑
+
∞

n \uparrow + \infty

. A slightly more general version of

(
∗
)

(\ast )

is also considered and similar results are obtained.
On a conjecture of Lin-Ni for a semilinear Neumann problem
Authors: Adimurthi  , Yadava S. L. 
Published: June 2012

Let

Ω
\Omega

be a bounded domain in

R

n

{\mathbb {R}^n}

(
n
≥
3
)

(n \geq 3)

and

λ
>
0

\lambda > 0

. We consider
\[

−
Δ
u
+
λ
u
=

u

(
n
+
2
)

/

(
n
−
2
)

a
m
p
;

in

Ω
,

u
>
0

a
m
p
;

in

Ω
,

∂
u

∂
ν

=
0

a
m
p
;

on

∂
Ω
,

\begin {array}{*{20}{c}} { - \Delta u + \lambda u = {u^{(n + 2)/(n - 2)}}} & {{\text {in}}} \; {\Omega ,} \\ {u > 0} & {{\text {in}}} \; {\Omega ,} \\ {\frac {{\partial u}} {{\partial \nu }} = 0} & {{\text {on}}} \; {\partial \Omega ,} \\ \end {array}

\]
and show that for

λ
\lambda

sufficiently small, the minimal energy solutions are only constants.
The structure of a random graph at the point of the phase transition
Authors: Łuczak Tomasz , Pittel Boris , Wierman John C. 
Published: June 2012

Consider the random graph models

G
(
n
,
#

edges

=
M
)

G(n,\# \;{\text {edges}} = M)

and

G
(
n
,
Prob
⁡
(

edge

)
=
p
)

G(n,\operatorname {Prob}({\text {edge}}) = p)

with

M
=
M
(
n
)
=
(
1
+
λ

n

−
1

/

3

)
n

/

2

M = M(n) = (1 + \lambda {n^{ - 1/3}})n/2

and

p
=
p
(
n
)
=
(
1
+
λ

n

−
1

/

3

)

/

n

p = p(n) = (1 + \lambda {n^{ - 1/3}})/n

. For

l
≥
−
1

l \geq - 1

define an

l
l

-component of a random graph as a component which has exactly

l
l

more edges than vertices. Call an

l
l

-component with

l
≥
1

l \geq 1

a complex component. For both models, we show that when

λ
\lambda

is constant, the expected number of complex components is bounded, almost surely (a.s.) each of these components (if any exist) has size of order

n

2

/

3

{n^{2/3}}

, and the maximum value of

l
l

is bounded in probability. We prove that a.s. the largest suspended tree in each complex component has size of order

n

2

/

3

{n^{2/3}}

, and deletion of all suspended trees results in a "smoothed" graph of size of order

n

1

/

3

{n^{1/3}}

, with the maximum vertex degree

3
3

. The total number of branching vertices, i.e., of degree

3
3

, is bounded in probability. Thus, each complex component is almost surely topologically equivalent to a

3
3

-regular multigraph of a uniformly bounded size. Lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order

n

1

/

3

{n^{1/3}}

. We find a relatively simple integral formula for the limit distribution of the numbers of complex components, which implies, in particular, that all values of the "complexity spectrum" have positive limiting probabilities. We also answer questions raised by Erdös and Rényi back in 1960. It is proven that there exists

p
(
λ
)

p(\lambda )

, the limiting planarity probability, with

0
>
p
(
λ
)
>
1

0 > p(\lambda ) > 1

,

p
(
−
∞
)
=
1

p( - \infty ) = 1

,

p
(
∞
)
=
0

p(\infty ) = 0

. In particular,

G
(
n
,
M
)

(
G
(
n
,
p
)
,

resp

.
)

G(n,M)\quad (G(n,p),{\text {resp}}.)

is almost surely nonplanar iff

(
M
−
n

/

2
)

n

−
2

/

3

→
∞

(
(
n
p
−
1
)

n

−
1

/

3

)
→
∞
,

resp

.
)

(M - n/2){n^{ - 2/3}} \to \infty \;((np - 1){n^{ - 1/3}}) \to \infty ,{\text {resp}}.)

.