bluuebunny/crossref_metadata_2025_split huggingface dataset:
A new Castelnuovo bound for two codimensional subvarieties of 𝑃^{𝑟}
Authors: Alzati Alberto 
Published: June 2010

Let

X
X

be a smooth

n
n

-dimensional projective subvariety of

P

r

(

C

)
,
(
r
≥
3
)

{\mathbb {P}^r}(\mathbb {C}),(r \geq 3)

. For any positive integer

k
,
X

k,X

is said to be

k
k

-normal if the natural map

H
0

(

P

r

,

O

P

r

(
k
)
)
→

H
0

(
X
,

O

X

(
k
)
)

{H^0}({\mathbb {P}^r},{\mathcal {O}_{\mathbb {P}r}}(k)) \to {H^0}(X,{\mathcal {O}_X}(k))

is surjective. Mumford and Bayer showed that

X
X

is

k
k

-normal if

k
≥
(
n
+
1
)
(
d
−
2
)
+
1

k \geq (n + 1)(d - 2) + 1

where

d
=
deg
⁡
(
X
)

d = \deg (X)

. Better inequalities are known when

n
n

is small (Gruson-Peskine, Lazarsfeld, Ran). In this paper we consider the case

n
=
r
−
2

n = r - 2

, which is related to Hartshorne’s conjecture on complete intersections, and we show that if

k
≥
d
+
1
+
(
1

/

2
)
r
(
r
−
1
)
−
2
r

k \geq d + 1 + (1/2)r(r - 1) - 2r

then

X
X

is

k
k

-normal and

I
X

{I_X}

, the ideal sheaf of

X
X

in

P

r

{\mathbb {P}^r}

, is

(
k
+
1
)

(k + 1)

-regular. About these problems Lazarsfeld developed a technique based on generic projections of

X
X

in

P

n
+
1

{\mathbb {P}^{n + 1}}

; our proof is an application of some recent results of Ran’s (on the secants of

X
X

): we show that in our case there exists a projection such generic as Lazarsfeld requires. When

r
≥
6

r \geq 6

we also give a better inequality:

k
≥
d
−
1
+
(
1

/

2
)
r
(
r
−
1
)
−
(
r
−
1
)
[
(
r
+
4
)

/

2
]

k \geq d - 1 + (1/2)r(r - 1) - (r - 1)[(r + 4)/2]

([] means: integer part); it is obtained by refining Lazarsfeld’s technique with the help of some results of ours about

k
k

-normality.
A two-generator presentation for the Picard group
Authors: Brunner A. M. 
Published: June 2010

The following

2
2

-generator

5
5

-relator presentation is given for the Picard group

PSL
⁡
(
2
,
Z
(
i
)
)

\operatorname {PSL} (2,Z(i))

:
\[

l
a
n
g
l
e
a
,
w
,
b
;
b
=
a

w
2

a

−
1

w

−
2

a

w
2

,

(

a
2

w
a

w

−
1

)
2

=

(
a
w
a

w

−
1

)
3

=

(
w
b

)
2

=

(
a
b

)
2

=

b
2

=
1
⟩
.

langle a,w,b;b = a{w^2}{a^{ - 1}}{w^{ - 2}}a{w^2},\ {({a^2}wa{w^{ - 1}})^2} = {(awa{w^{ - 1}})^3} = {(wb)^2} = {(ab)^2} = {b^2} = 1\rangle .

\]
Quadric representation of a submanifold
Authors: Dimitrić Ivko 
Published: June 2010

If

x
:

M
n

→

E
m

x: M^n \to E^m

is an isometric immersion of a smooth manifold into a Euclidean space then the map

x
~

=
x

x

t

\tilde {x} = x x^{\mathrm {t}}

]> (t denotes transpose) is called the quadric representation of

M
M

.

x
~

\tilde {x}

is said to be of finite type (

k
k

-type) if it can be decomposed into a sum of finitely many

(
k
)

(k)

eigenfunctions of Laplacian from different eigenspaces. We study map

x
~

\tilde {x}

in general, especially as related to the condition of being of finite type. Certain classification results are obtained for manifolds with

1
1

-and

2
2

-type quadric representation.
Stabilization in a gradient system with a conservation law
Authors: Pego Robert L. 
Published: June 2010

Suppose

∑

μ
j

=
1

\sum {{\mu _j} = 1}

and

F
:

R

↦

R

F:{\mathbf {R}} \mapsto {\mathbf {R}}

is

C
1

{C^1}

with

F
′

F’

piecewise

C
1

{C^1}

. For the finite system of ordinary differential equations
\[

u
˙

i

=

F
′

(

u
i

)
−

∑
j

μ
j

F
′

(

u
j

)
=
0

,

{\dot u_i} = F’({u_i}) - \sum \limits _j {{\mu _j}F’({u_j}) = 0} ,

\]
I prove that every bounded solution stabilizes to some equilibrium as

t
→
∞

t \to \infty

. For this system,

∑

μ
j

u
j

\sum {{\mu _j}{u_j}}

is conserved and the quantity

∑

μ
j

F
(

u
j

)

\sum {{\mu _j}F({u_j})}

is nonincreasing and serves as a Lyapunov function, but the set of equilibria can be connected and degenerate. Essential use is made of a result related to one of Hale and Massat that an

ω
\omega

-limit set that lies in a

C
1

{C^1}

hyperbolic manifold of equilibria must be a singleton.
Asymptotic behaviour and oscillation of classes of integrodifferential equations
Authors: Nasr A. H. 
Published: June 2010

Under some conditions on the integrodifferential equations
\[

y
¨

(
t
)

+

∫
0
t

k

(

t
−
s

)

y

(
s
)

d
s
+
φ

(
t
)

∫
0
t

K

(

t
−
s

)

y
˙

(
s
)

d
s
=
f

[

t
,
y

(
t
)

,

y
˙

(
t
)

,

∫
0
t

g

(

t
,
s
,
y

(
s
)

,

y
˙

(
s
)

)

d
s

]

,

t
≥
0
,

\ddot y\left ( t \right ) + \int _0^t {k\left ( {t - s} \right )y\left ( s \right )ds + \varphi \left ( t \right )} \int _0^t {K\left ( {t - s} \right )\dot y\left ( s \right )ds = f\left [ {t,y\left ( t \right ),\dot y\left ( t \right ),\int _0^t {g\left ( {t,s,y\left ( s \right ),\dot y\left ( s \right )} \right )ds} } \right ]} ,\quad t \geq 0,

\]
,
\[

y
¨

(
t
)

+

∫
1
t

k

(

t
s

)

y

(
s
)

1
s

d
s
+
φ

(
t
)

∫
1
t

K

(

t
s

)

y
˙

(
s
)

d
s
=
f

[

t
,
y

(
t
)

,

y
˙

(
t
)

,

∫
1
t

g

(

t
,
s
,
y

(
s
)

,

y
˙

(
s
)

)

d
s

]

,

t
≥
1
,

\ddot y\left ( t \right ) + \int _1^t {k\left ( {\frac {t}{s}} \right )y\left ( s \right )} \frac {1}{s}ds + \varphi \left ( t \right )\int _1^t {K\left ( {\frac {t}{s}} \right )\dot y\left ( s \right )ds = f\left [ {t,y\left ( t \right ),\dot y\left ( t \right ),\int _1^t {g\left ( {t,s,y\left ( s \right ),\dot y\left ( s \right )} \right )ds} } \right ],\quad t \geq 1,}

\]
, the explicit asymptote of solutions is proved to be

y

(
t
)

=
A
sin
⁡

(

ω
t
+
δ

)

y\left ( t \right ) = A\sin \left ( {\omega t + \delta } \right )

as

t
→
∞

t \to \infty

. From this asymptote, the oscillatory behavior of the equations, the limit of the amplitudes, and the limit of the distance between consecutive zeros of the solutions are evident. Their definite values are also determined.
𝑅-type summability methods, Cauchy criteria, 𝑃-sets and statistical convergence
Authors: Connor Jeff 
Published: June 2010

A summability method

S
S

is called an

R
R

-type summability method if

S
S

is regular and

x
y

xy

is strongly

S
S

-summable to 0 whenever

x
x

is strongly

S
S

-summable to 0 and

y
y

is a bounded sequence. Associated with each

R
R

-type summability method

S
S

are the following two methods: convergence in

μ
\mu

-density and

μ
\mu

-statistical convergence where

μ
\mu

is a measure generated by

S
S

. In this note we extend the notion of statistically Cauchy to

μ
\mu

-Cauchy and show that a sequence is

μ
\mu

-Cauchy if and only if it is

μ
\mu

-statistically convergent. Let

W

(
A
)

=

A
¯

β

N

∩
β

N

∖

N

W\left ( A \right ) = {\overline A ^{\beta \mathbb {N}}} \cap \beta \mathbb {N}\backslash \mathbb {N}

for

A
⊂

N

A \subset \mathbb {N}

and

K

=

∩

{

W

(
A
)

:
A
⊆

N

,

χ
A

is

strongly

S
−

summable

to

1

}

\mathcal {K}{\text { = }} \cap \left \{ {W\left ( A \right ):A \subseteq \mathbb {N}{\text {,}}{\chi _A}\;{\text {is}}\;{\text {strongly}}\;S - {\text {summable}}\;{\text {to}}\;1} \right \}

. Then

μ
\mu

-Cauchy is equivalent to convergence in

μ
\mu

-density if and only if every

G
δ

{G_\delta }

that contains

K

\mathcal {K}

in

β

N

∖

N

\beta \mathbb {N}\backslash \mathbb {N}

is a neighborhood of

K

\mathcal {K}

in

β

N

∖

N

\beta \mathbb {N}\backslash \mathbb {N}

. As an application, we show that the bounded strong summability field of a nonnegative regular matrix admits a Cauchy criterion.
Second order ergodic theorems for ergodic transformations of infinite measure spaces
Authors: Aaronson Jon , Denker Manfred , Fisher Albert M. 
Published: June 2010

For certain pointwise dual ergodic transformations

T
T

we prove almost sure convergence of the log-averages
\[

1

log
⁡
N

∑

n
=
1

N

1

n
a

(
n
)

∑

k
=
1

n

f
∘

T
k

(

f
∈

L
1

)

\frac {1}{{\log N}}\sum \limits _{n = 1}^N {\frac {1}{{na\left ( n \right )}}\sum \limits _{k = 1}^n {f \circ {T^k}\left ( {f \in {L_1}} \right )} }

\]
and the Chung-Erdös averages
\[

1

log
⁡
a

(
N
)

∑

k
=
1

N

1

a

(
k
)

f
∘

T
k

(

f
∈

L
1
+

)

\frac {1}{{\log a\left ( N \right )}}\sum \limits _{k = 1}^N {\frac {1}{{a\left ( k \right )}}f \circ {T^k}} \left ( {f \in L_1^ + } \right )

\]
towards

∫
f

\smallint f

, where

a

(
n
)

a\left ( n \right )

denotes the return sequence of

T
T

.
Essential maps and manifolds
Authors: Mertens Jean-François 
Published: June 2010

Let

(
M
,
∂
M
)

(M,\partial M)

be a compact

n
n

-manifold with boundary, orientable over a field

K
K

with characteristic

q
q

. For

f
:
(
Y
,
∂
Y
)
→
(
M
,
∂
M
)

f:(Y,\partial Y) \to (M,\partial M)

, with

Y
Y

compact, and

(
X
,
∂
X
)

(X,\partial X)

a compact pair,

g
:
X
→
M

g:X \to M

, let

(
P
,
∂
P
)
=
{
(
y
,
x
)
∈
Y
×
(
X
,
∂
X
)

|

f
(
y
)
=
g
(
x
)
}

(P,\partial P) = \{ (y,x) \in Y \times (X,\partial X)|f(y) = g(x)\}

denote the fibered product, with

p
p

as the projection to

(
X
,
∂
X
)

(X,\partial X)

. In Čech-cohomology with coefficients

K
K

, we show that if

[
u
n
k

]
n

(
f
)

{[unk]^n}(f)

is injective then so is

[
u
n
k

]
∗

(
p
)

{[unk]^*}(p)

—and a number of strengthenings, which point to a concept of

q
q

-essential map from one compact space to another.
On invertible hypercyclic operators
Authors: Herrero Domingo A. , Kitai Carol 
Published: June 2010

Let

A
A

be an invertible (bounded linear) operator acting on a complex Banach space

X

\mathcal {X}

.

A
A

is called hypercyclic if there is a vector

y
y

in

X

\mathcal {X}

such that the orbit

Orb
⁡
(
A
;
y
)
:=
{
y
,
A
y
,

A
2

,
y
,
…
}

\operatorname {Orb}(A;y): = \{ y,Ay,{A^2},y, \ldots \}

is dense in

X

\mathcal {X}

. (

X

\mathcal {X}

is necessarily separable and infinite dimensional.) Theorem 1. The following are equivalent for an invertible operator A acting on

X

:
(

i

)
A

\mathcal {X}:({\text {i}})A

or

A

−
1

{A^{ - 1}}

is hypercyclic; (ii)

A
A

and

A

−
1

{A^{ - 1}}

are hypercyclic; (iii) there is a vector

z
z

such that

Orb
⁡
(
A
;
z

)
−

=
Orb
⁡
(

A

−
1

;z

)
−

=

X

\operatorname {Orb}(A;z)^ - = \operatorname {Orb}({A^{ - 1}}{\text {;z}})^ - = \mathcal {X}

(the upper bar denotes norm-closure); (iv) there is a vector

y
y

in

X

\mathcal {X}

such that
\[

[
Orb
⁡
(
A
;
y
)
∪
Orb
⁡
(

A

−
1

;
y
)

]
−

=

X

.

[\operatorname {Orb}(A;y) \cup \operatorname {Orb}({A^{ - 1}};y)]^ - = \mathcal {X}.

\]
. Theorem 2. If

A
A

is not hypercyclic, then

A
A

and

A

−
1

{A^{ - 1}}

have a common nontrivial invariant closed subset.
Cyclically presented groups embedded in one-relator products of cyclic groups
Authors: Campbell Colin M. , Heggie Patricia M. , Robertson Edmund F. , Thomas Richard M. 
Published: June 2010

We consider the groups defined by the presentations
\[

⟨
a
,
b
:

a
2

=

b
n

=
a

b

−
1

a
b

(
a
b
a

b

−
1

)

α
−
1

a

b
2

a

b

−
2

=
1
⟩

\langle a,b:{a^2} = {b^n} = a{b^{ - 1}}ab{(aba{b^{ - 1}})^{\alpha - 1}}a{b^2}a{b^{ - 2}} = 1\rangle

\]
and investigate their structure for small values of

α
\alpha

. This forms part of a general investigation into the structure of groups defined by presentations of the form
\[

⟨
a
,
b
:

a
2

=

b
n

=
w
(
a
,
b
)
=
1
⟩
.

\langle a,b:{a^2} = {b^n} = w(a,b) = 1\rangle .

\]
Connections between these groups and the Fibonacci groups are also explored.
Existence of a nontrivial solution to a strongly indefinite semilinear equation
Authors: Buffoni B. , Jeanjean L. , Stuart C. A. 
Published: June 2010

Under general hypotheses, we prove the existence of a nontrivial solution for the equation

L
u
=
N
(
u
)

Lu = N(u)

, where

u
u

belongs to a Hilbert space

H
H

,

L
L

is an invertible continuous selfadjoint operator, and

N
N

is superlinear. We are particularly interested in the case where

L
L

is strongly indefinite and

N
N

is not compact. We apply the result to the Choquard-Pekar equation
\[

−
Δ
u
(
x
)
+
p
(
x
)
u
(
x
)
=
u
(
x
)

∫

R

3

u
2

(
y
)

|

x
−
y

|

d
y
,

u
∈

H
1

(

R

3

)
,

u
≠
0
,

- \Delta u(x) + p(x)u(x) = u(x)\int _{{\mathbb {R}^3}} {\frac {{{u^2}(y)}} {{|x - y|}}dy,\qquad u \in {H^1}({\mathbb {R}^3}),\quad u \ne 0,}

\]
where

p
∈

L
∞

(

R

3

)

p \in {L^\infty }({\mathbb {R}^3})

is a periodic function.
Axiomatization and undecidability results for metrizable betweenness relations
Authors: Mendris Robert , Zlatoš Pavol 
Published: June 2010

Let d be a metric on a nonempty set A. The ternary betweenness relation

T
d

{T_d}

induced by d on A is defined by
\[

T
d

(
x
,
y
,
z
)
⇔
d
(
x
,
y
)
+
d
(
y
,
z
)
=
d
(
x
,
z
)

{T_d}(x,y,z) \Leftrightarrow d(x,y) + d(y,z) = d(x,z)

\]
for

x
,
y
,
z
∈
A

x,y,z \in A

. Allowing the range of d to vary over some "reasonable" ordered additive algebraic structures (not just the real numbers), we will prove that the class

M

\mathcal {M}

of all metrizable ternary structures, i.e., the class of all structures

(
A
,

T
d

)

(A,{T_d})

, where d is some metric on A, is an elementary class which can be axiomatized by a set of universal Horn sentences. Further, using an algorithm of linear programming, we will show that the first-order theory of

M

\mathcal {M}

is recursively axiomatizable and its universal part is decidable. On the other hand, the theory of

M

\mathcal {M}

is not finitely axiomatizable and the theory of finite members of

M

\mathcal {M}

is hereditarily undecidable.
A congruence for primes
Authors: Sun Zhi Wei 
Published: June 2010

With the help of the Pell sequence we obtain the following new congruence for odd primes:
\[

∑

k
=
1

(
p
−
1
)

/

2

1

k
⋅

2
k

≡

∑

k
=
1

[
3
p

/

4
]

(
−
1
)

k
−
1

k

(
mod

p
)
.

\sum \limits _{k = 1}^{(p - 1)/2} {\frac {1}{{k \cdot {2^k}}} \equiv \sum \limits _{k = 1}^{[3p/4]} {\;\frac {{{{( - 1)}^{k - 1}}}}{k}} \quad \pmod p.}

\]
Existence and nonexistence of positive eigenfunctions for the 𝑝-Laplacian
Authors: Binding P. A. , Huang Y. X. 
Published: June 2010

We study the relation between (i) the principal eigencurve, i.e., the graph ofμ1{\mu _1}satisfying(∗)−div(|∇u|p−2∇u)+(q(x)−λw(x))|u|p−2u=μ1(λ)|u|p−2μ\begin{equation}\tag {$ \ast $} - {\operatorname {div}}(|\nabla u{|^{p - 2}}\nabla u) + (q(x) - \lambda w(x))|u{|^{p - 2}}u = {\mu _1}(\lambda )|u{|^{p - 2}}\mu \end{equation}on a smooth bounded domainΩ\OmegainRN{{\mathbf {R}}^N}withp>1p > 1, and (ii) existence and nonexistence of positive solutions of(∗)( \ast )withμ1(λ)=0{\mu _1}(\lambda ) = 0. Eigencurve arguments are used extensively.
Entire solutions of the abstract Cauchy problem in a Hilbert space
Authors: deLaubenfels Ralph , Yao Fuyuan 
Published: June 2010

We show that, whenever the linear operator A is symmetric and densely defined, on a Hilbert space, then the abstract Cauchy problem
\[

d

d
z

u
(
z
)
=

A
∗

(
u
(
z
)
)

(
z
∈

C

)
,

u
(
0
)
=
x

\frac {d}{{dz}}u(z) = {A^ \ast }(u(z))\quad (z \in {\mathbf {C}}),\qquad u(0) = x

\]
has an entire solution, for all initial data x in the image of

e

−

A
¯

A
∗

{e^{ - \bar A{A^ \ast }}}

, which is a dense set.
Another generalization of Anderson’s theorem
Authors: Du Hong Ke 
Published: June 2010

In this paper, we prove that if A and B are normal operators on a Hilbert space H, then, for every operator S satisfying

A
S
B
=
S
,

‖

A
X
B
−
X
+
S

‖

≥

‖
A
‖

−
1

‖
B
‖

−
1

‖
S
‖

ASB = S, \left \| {AXB - X + S} \right \| \geq {\left \| A \right \|^{ - 1}}{\left \| B \right \|^{ - 1}}\left \| S \right \|

for all operators

X
∈
B
(
H
)

X \in B(H)

, and that if A and B are contractions, then, for every operator S satisfying

A
S
B
=
S

ASB = S

and

A
∗

S

B
∗

=
S
,

‖

A
X
B
−
X
+
S

‖

≥

‖
S
‖

{A^ \ast }S{B^ \ast } = S,\left \| {AXB - X + S} \right \| \geq \left \| S \right \|

for all operators

X
∈
B
(
H
)

X \in B(H)

, where

B
(
H
)

B(H)

denotes the set of all bounded linear operators on H.
Spectra and measure inequalities
Authors: Putnam C. R. 
Published: June 2010

Let T be a bounded operator on a Hilbert space

H

\mathfrak {H}

and let

T
z

=
T
−
z
I

{T_z} = T - zI

. Then the operators

T
z

T
z
∗

,

T
z

T
t

(

T
z

T
t

)
∗

{T_z}T_z^\ast ,{T_z}{T_t}{({T_z}{T_t})^\ast }

, and

T
z

T
t

T
s

(

T
z

T
t

T
s

)
∗

{T_z}{T_t}{T_s}{({T_z}{T_t}{T_s})^\ast }

are nonnegative for all complex numbers z, t, and s. We shall obtain some norm estimates for nonnegative lower bounds of these operators, when z, t, and s are restricted to certain sets, in terms of certain capacities or area measures involving the spectrum and point spectrum of T. A typical such estimate is the following special case of Theorem 4 below: Let

H

\mathfrak {H}

be separable and suppose that

T
z

T
t

(

T
z

T
t

)
∗

⩾
D
⩾
0

{T_z}{T_t}{({T_z}{T_t})^\ast } \geqslant D \geqslant 0

for all z and t not belonging to the closure of the interior of the point spectrum of T. In addition, suppose that the boundary of the interior of the point spectrum of T has Lebesgue planar measure 0. Then

‖
D
‖

1

/

2

⩽

π

−
1

meas
2

(

σ
p

(
T
)
)

{\left \| D \right \|^{1/2}} \leqslant {\pi ^{ - 1}}\;{\text {meas}_2}\;({\sigma _p}(T))

. If T is the adjoint of the simple unilateral shift, then equality holds with

D
=
I
−

T
∗

T

D = I - {T^\ast }T

.
Small zeros of additive forms in many variables
Authors: Schmidt Wolfgang M. 
Published: June 2010

It is shown that if s is large as a function of k and of

ε
>
0

\varepsilon > 0

, then the diophantine equation

a
1

x
1

k

+
⋯
+

a
s

x
s
k

=

b
1

y
1
k

+
⋯
+

b
s

y
s
k

{a_1}{x_1}^k + \cdots + {a_s}x_s^k = {b_1}y_1^k + \cdots + {b_s}y_s^k

with positive coefficients

a
1

,
…
,

a
s

{a_1}, \ldots ,{a_s}

,

b
1

,
…
,

b
s

{b_1}, \ldots ,{b_s}

has a nontrivial solution in nonnegative integers

x
1

,
…
,

x
s

{x_1}, \ldots ,{x_s}

,

y
1

,
…
,

y
s

{y_1}, \ldots ,{y_s}

not exceeding

m

(

1

/

k

)

+
ε

{m^{\left ( {1/k} \right ) + \varepsilon }}

, where m is the maximum of the coefficients.
Zeros of successive derivatives of entire functions of the form ℎ(𝑧)𝑒𝑥𝑝(-𝑒^{𝑧})
Authors: Edrei Albert 
Published: June 2010

Consider

f
(
z
)

=

h
(
z
)
exp
⁡
(
−

e
z

)

(
z

=

x

+

i
y
)

f(z)\, = \,h(z)\exp ( - {e^z})\,(z\, = \,x\, + \,iy)

where

h
(
z
)

h(z)

is a real entire function of finite order having no zeros in some strip

{
x

+

i
y
:

|

y
−
π

|

>

η
1

,

x

>

x
0

}

(
0

>

η
1

)

\{ x\, + \,iy:\,\left | {y - \pi } \right |\, > \,{\eta _1},\,x\, > \,{x_0}\} \,(0\, > \,{\eta _1})

. The author studies the power series (1)

f
(
τ

+

z
)

=

Σ

n
=
0

∞

a
n

z
n

f(\tau \, + \,z)\, = \,\Sigma _{n = 0}^\infty \,{a_n}{z^n}

(

τ
\tau

real) and the number

N
(

τ
1

,

τ
2

;

n
)

N({\tau _1},\,{\tau _2};\,n)

of real zeros of

f

(
n
)

(
z
)

{f^{(n)}}(z)

which lie in the interval

[

τ
1

,

τ
2

]

[{\tau _1},\,{\tau _2}]

. He proves (2)

N
(

τ
1

,

τ
2

;

n
)

∼

(

τ
2

−

τ
1

)
n

/

(
log

n

)
2

(
n
→
∞
)

N({\tau _1},{\tau _2};\,n)\, \sim \,({\tau _2} - {\tau _1})n/{(\log \,n)^2}\,(n \to \infty )

. With regard to the expansion (1) he determines a positive, strictly increasing, unbounded sequence

{

v
k

}

k
=
1

∞

\{ {v_k}\} _{k = 1}^\infty

such that

(

v

k
+
1

−

v
k

)

/

log

v
k

→
1

(
k
→
∞
)

({v_{k + 1}} - {v_k})/\log \,{v_k} \to 1\,(k \to \infty )

and having the following properties: (i) if

v
k

>

n

>

v

k
+
1

{v_k}\, > \,n\, > \,{v_{k + 1}}

, then

a
n

≠

0

{a_n}\, \ne \,0

and all the

a
n

{a_n}

have the same sign; (ii) if in addition

v

k
+
1

>

m

>

v

k
+
2

{v_{k + 1}}\, > \,m\, > \,{v_{k + 2}}

, tnen

a
m

a
n

>
0

{a_m}{a_n}\, > 0

. It is possible to deduce from (2) the complete characterization of the final set (in the sense of Pólya) of

exp
⁡
(
−

e
z

)

\exp ( - {e^z})

.
Interior and boundary continuity of weak solutions of degenerate parabolic equations
Authors: Ziemer William P. 
Published: June 2010

In this paper we consider degenerate parabolic equations of the form

(

∗

)

({\ast })

\[

β

(
u

)
t

−
div
⁡
A
(
x
,

t
,

u
,

u
x

)
+
B
(
x
,

t
,

u
,

u
x

)
∋
0

\beta {(u)_t} - \operatorname {div} A(x,\,t,\,u,\,{u_x}) + B(x,\,t,\,u,\,{u_x}) \ni 0

\]
where

A
A

and

B
B

are, respectively, vector and scalar valued Baire functions defined on

U
×

R
1

×

R
n

U \times {R^1} \times {R^n}

, where

U
U

is an open subset of

R

n
+
1

(
x
,

t
)

{R^{n + 1}}(x,\,t)

. The functions

A
A

and

B
B

are subject to natural structural inequalities. Sufficiently general conditions are allowed on the relation

β
⊂

R
1

×

R
1

\beta \subset {R^1} \times {R^1}

so that the porus medium equation and the model for the two-phase Stefan problem can be considered. The main result of the paper is that weak solutions of

(

∗

)

({\ast })

are continuous throughout

U
U

. In the event that

U
=
Ω
×
(
0
,

T
)

U = \Omega \times (0,\,T)

where

Ω
\Omega

is an open set of

R
n

{R^n}

, it is also shown that a weak solution is continuous at

(

x
0

,

t
0

)
∈
∂
Ω
×
(
0
,

T
)

({x_0},{t_0}) \in \partial \Omega \times (0,\,T)

provided

x
0

{x_0}

is a regular point for the Laplacian on

Ω
\Omega

.
Linking numbers and the elementary ideals of links
Authors: Traldi Lorenzo 
Published: June 2010

Let

L
=

K
1

∪

⋯
∪

K
μ

⊆

S
3

L = {K_1} \cup \, \cdots \cup {K_\mu } \subseteq {S^3}

be a tame link of

μ
⩾
2

\mu \geqslant 2

components, and

H
H

the abelianization of

G
=

π
1

(

S
3

−
L
)

G = {\pi _1}({S^3} - L)

. Let

L

=
(

L

i
j

)

\mathcal {L} = ({\mathcal {L}_{ij}})

be the

μ
×
μ

\mu \times \mu

matrix with entries in

Z

H

\mathbf {Z}H

given by

L

i
i

=

∑

k
≠
i

l
(

K
i

,

K
k

)
⋅
(

t
k

−
1
)

\mathcal {L}{_{ii}} = \sum \nolimits _{k \ne i} {l({K_i},{K_k}) \cdot ({t_k} - 1)}

and for

i
≠
j

L

i
j

=
l
(

K
i

,

K
j

)
⋅
(
1
−

t
i

)

i \ne j\,{\mathcal {L}_{ij}} = l({K_i},{K_j}) \cdot (1 - {t_i})

. Then if

0
>
k
>
μ

0 > k > \mu

\[

∑

i
=
0

k
−
1

E

μ
−
k
+
i

(
L
)
⋅

(
I
H
)

2
i

+

(
I
H
)

2
k

=

∑

i
=
0

k
−
1

E

μ
−
k
+
i

(

L

)
⋅

(
I
H
)

2
i

+

(
I
H
)

2
k

\sum \limits _{i = 0}^{k - 1} {{E_{\mu - k + i}}(L) \cdot {{(IH)}^{2i}} + {{(IH)}^{2k}} = \sum \limits _{i = 0}^{k - 1} {{E_{\mu - k + i}}(\mathcal {L}) \cdot {{(IH)}^{2i}} + {{(IH)}^{2k}}} }

\]
Various consequences of this equality are derived, including its application to the reduced elementary ideals. These results are used to give several different characterizations of links in which all the linking numbers are zero.
Semistability at the end of a group extension
Authors: Mihalik Michael L. 
Published: June 2010

A

1
1

-ended

CW

{\text {CW}}

-complex,

Q
Q

, is semistable at

∞
\infty

if all proper maps

r
:

[
0
,
∞
)
→
Q

r:\ [0,\infty ) \to Q

are properly homotopic. If

X
1

{X_1}

and

X
2

{X_2}

are finite

CW

{\text {CW}}

-complexes with isomorphic fundamental groups, then the universal cover of

X
1

{X_1}

is semistable at

∞
\infty

if and only if the universal cover of

X
2

{X_2}

is semistable at

∞
\infty

. Hence, the notion of a finitely presented group being semistable at

∞
\infty

is well defined. We prove Main Theorem. Let

1
→
H
→
G
→
K
→
1

1 \to H \to G \to K \to 1

be a short exact sequence of finitely generated infinite groups. If

G
G

is finitely presented, then

G
G

is semistable at

∞
\infty

. Theorem. If

A
A

and

B
B

are locally compact, connected noncompact

C
W

CW

-complexes, then

A
×
B

A \times B

is semistable at

∞
\infty

. Theorem.

⟨

x
,
y
:
x

y
b

x

−
1

=

y
c

;
b

a
n
d

c

n
o
n
z
e
r
o

i
n
t
e
g
e
r
s

⟩

\langle \;x,y:x{y^b}{x^{ - 1}} = {y^c};b\; and \; c \; nonzero\; integers\; \rangle

is semistable at

∞
\infty

. The proofs are geometrical in nature and the main tool is covering space theory.