bluuebunny/crossref_metadata_2025_split huggingface dataset:
On approximate antigradients
Authors: Gan Xiao-Xiong , Stromberg Karl R. 
Published: June 2010

For

n
∈

N

n \in \mathbb {N}

and

I
=
[
0
,
1
]

I = [0,1]

, let

I
n

{I^n}

be the unit cube and

λ
n

{\lambda ^n}

the Lebesgue measure in

R

n

{\mathbb {R}^n}

. It is proved that if

f
:

I
n

→

R

n

f:{I^n} \to {\mathbb {R}^n}

and

F
0

:

I
n

→

R

{F_0}:{I^n} \to \mathbb {R}

are continuous and

ε
>
0

\varepsilon > 0

, then there exist a continuous

F
:

I
n

→

R

F:{I^n} \to \mathbb {R}

and an open set

W
⊂

(

I
n

)
∘

W \subset {({I^n})^ \circ }

with

λ
n

(
W
)
=
1

{\lambda ^n}(W) = 1

such that (i)

∇
F

\nabla F

exists and is continuous on

W
W

, (ii)

|

|

∇
F
(
x
)
−
f
(
x
)

|

|

>
ε

∀
x
∈
W

||\nabla F(x) - f(x)|| > \varepsilon \;\forall x \in W

, and (iii)

|

F
(
x
)
−

F
0

(
x
)

|

>
ε

∀
x
∈

I
n

|F(x) - {F_0}(x)| > \varepsilon \;\forall x \in {I^n}

, where

|

|

y

|

|

=

(

∑

j
=
1

n

y
j
2

)

1

/

2

∀
y
∈

R

n

||y|| = {\left ( {\sum \nolimits _{j = 1}^n {y_j^2} } \right )^{1/2}}\;\forall y \in {\mathbb {R}^n}

.
An inequality for sections and projections of a convex set
Authors: Spingarn Jonathan E. 
Published: June 2010

Let

K
⊂

R

d

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

be a convex body,

γ
\gamma

its center of mass. For

Λ
⊂

R

d

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

a subspace of dimension

d
−
k

d - k

, we establish the inequality
\[

Vol
d

(
K
)
⩽

Vol

d
−
k

(
K

|

Λ
)

Vol
k

(
(
K
−
γ
)
∩

Λ
⊥

)

{\operatorname {Vol} _d}(K) \leqslant {\operatorname {Vol} _{d - k}}(K|\Lambda ){\operatorname {Vol} _k}((K - \gamma ) \cap {\Lambda ^ \bot })

\]
(where

K

|

Λ

K|\Lambda

denotes orthogonal projection of

K
K

onto

Λ
\Lambda

). Equality holds only if each

k
k

-dimensional section of

K
K

parallel to

Λ
⊥

{\Lambda ^ \bot }

is a translate of

(
K
−
γ
)
∩

Λ
⊥

(K - \gamma ) \cap {\Lambda ^ \bot }

.
An improved Poincaré inequality
Authors: Hurri-Syrjänen Ritva 
Published: June 2010

We show that a large class of domains

D
D

in

R

n

{\mathbb {R}^n}

including John domains satisfies the improved Poincaré inequality
\[

|

|

u
(
x
)
−

u
D

|

|

L
q

(
D
)

⩽
c

|

|

∇
u
(
x
)
d

(
x
,
∂
D

)
δ

|

|

L
p

(
D
)

||u(x) - {u_D}|{|_{{L^q}(D)}} \leqslant c||\nabla u(x)d{(x,\partial D)^\delta }|{|_{{L^p}(D)}}

\]
where

p
⩽
q
⩽

n
p

n
−
p
(
1
−
δ
)

,

p
(
1
−
δ
)
>
n
,

δ
∈
[
0
,
1
]
,

c
=
c
(
p
,
q
,
δ
,
D
)
>
∞

p \leqslant q \leqslant \tfrac {{np}} {{n - p(1 - \delta )}},\;p(1 - \delta ) > n,\;\delta \in [0,1],\;c = c(p,q,\delta ,D) > \infty

, and

u
u

is in an appropriate Sobolev class.
Attouch-Wets convergence and a differential operator for convex functions
Authors: Beer Gerald , Théra Michel 
Published: June 2010

The purpose of this note is to characterize Attouch-Wets convergence for sequences of proper lower semicontinuous convex functions defined on a Banach space X in terms of the behavior of an operator

Δ
\Delta

defined on the space of such functions with values in

X
×
R
×

X
∗

X \times R \times {X^ \ast }

, defined by

Δ
(
f
)
=
{
(
x
,
f
(
x
)
,
y
)
:
(
x
,
y
)
∈
∂
f
}

\Delta (f) = \{ (x,f(x),y):(x,y) \in \partial f\}

. We show that

⟨

f
n

⟩

\langle {f_n}\rangle

is Attouch-Wets convergent to f if and only if points of

Δ
(
f
)

\Delta (f)

lying in a fixed bounded set can be uniformly approximated by points of

Δ
(

f
n

)

\Delta ({f_n})

for large n. The operator

Δ
\Delta

is a natural carrier of the Borwein variational principle, which is a key tool in both directions of our proof.
Singular perturbations of integrodifferential equations in Banach space
Authors: Liu James H. 
Published: June 2010

Let

ε
>
0

\varepsilon > 0

and consider
\[

ε
2

u
(
t
;
ε
)
+

u
′

(
t
;
ε
)
=
A
u
(
t
;
ε
)
+

∫
0
t

K
(
t
−
s
)
A
u
(
s
;
ε
)

d
s
+
f
(
t
;
ε
)
,

t
≥
0
,

u
(
0
;
ε
)
=

u
0

(
ε
)
,

u
′

(
0
;
ε
)
=

u
1

(
ε
)
,

\begin {array}{*{20}{c}} {{\varepsilon ^2}u(t;\varepsilon ) + u’ (t;\varepsilon ) = Au(t;\varepsilon ) + \int _0^t {K(t - s)Au(s;\varepsilon )\;ds + f(t;\varepsilon ),\quad t \geq 0,} } \\ {u(0;\varepsilon ) = {u_0}(\varepsilon ),\quad u’ (0;\varepsilon ) = {u_1}(\varepsilon ),} \\ \end {array}

\]
and
\[

w
′

(
t
)
=
A
w
(
t
)
+

∫
0
t

K
(
t
−
s
)
A
w
(
s
)

d
s
+
f
(
t
)

t
≥
0
,
w
(
0
)
=

w
0

,

w’ (t) = Aw(t) + \int _0^t {K(t - s)Aw(s)\,ds + f(t)\quad t \geq 0,w(0) = {w_0},}

\]
in a Banach space X when

ε
→
0

\varepsilon \to 0

. Here A is the generator of a strongly continuous cosine family and a strongly continuous semigroup, and

K
(
t
)

K(t)

is a bounded linear operator for

t
≥
0

t \geq 0

. With some convergence conditions on initial data and

f
(
t
;
ε
)

f(t;\varepsilon )

and smoothness conditions on

K
(
∙
)

K( \bullet )

, we prove that if

ε
→
0

\varepsilon \to 0

, then

u
(
t
;
ε
)
→
w
(
t
)

u(t;\varepsilon ) \to w(t)

in X uniformly for

t
∈
[
0
,
T
]

t \in [0,T]

for any fixed

T
>
0

T > 0

. We will apply this to an equation in viscoelasticity.
The strong law of large numbers: a weak-𝑙₂ view
Authors: Heinkel Bernard 
Published: June 2010

Let

(

X
k

)

({X_k})

be a sequence of independent, centered, and square integrable real-valued random variables. To that sequence one associates
\[

∀
n
∈

N

,

ξ
n

=

‖

(

2

−
n

X
k

)
,

2
n

+
1
≤
k
≤

2

n
+
1

‖

2
,
∞

.

\forall n \in \mathbb {N},\quad {\xi _n} = {\left \| {({2^{ - n}}{X_k}),\;{2^n} + 1 \leq k \leq {2^{n + 1}}} \right \|_{2,\infty }}.

\]
When there exists

K
≥
1

K \geq 1

such that
\[

∑

n
≥
1

P
K

(

ξ
n

>

c
n

)
>
+
∞
,

\sum \limits _{n \geq 1} {{P^K}({\xi _n} > {c_n}) > + \infty ,}

\]
where

(

c
n

)

({c_n})

is a suitable sequence of positive constants, then the strong law of large numbers holds if and only if

(

X
k

/

k
)

({X_k}/k)

converges almost surely to 0.
Cycles in 𝐶^{𝑟}-twists
Authors: Carroll C. R. 
Published: June 2010

Let

f
t

,
0
≤
t
≤
1

{f_t},0 \leq t \leq 1

, be a continuous one-parameter family of

C
r

{C^r}

diffeomorphisms of the circle obtained by monotonically twisting away from

f
0

{f_0}

. It is well known that for a dense set of parameter values

t
,

f
t

{t, f_t}

has a periodic orbit. To what extent does the distribution in the circle of these periodic orbits reflect the degree of differentiability r? We show that if

r
≥
6
,

sup
t

‖

f
t

∘

f
0

−
1

‖

C
r

>
∞

r \geq 6,{\sup _t}{\left \| {{f_t} \circ f_0^{ - 1}} \right \|_{{C^r}}} > \infty

, and the rotation number of

f
0

{f_0}

is an irrational

α
\alpha

with bounded continued fraction expansion, then periodic orbits corresponding to small values of the parameter t echo the metric structure of

f
0

{f_0}

in the following sense: If the rotation number of the orbit is a convergent of

α
\alpha

, then the orbit divides the circle into intervals of nearly equal

μ
\mu

-measure, where

μ
\mu

is the invariant Borel probability measure of

f
0

{f_0}

. The corresponding result for low differentiability is false.
A Pythagorean inequality
Authors: Reid Russell M. 
Published: June 2010

Let

{

v
1

,

v
2

,

v
3

,
…
}

\{ {v_1},{v_2},{v_3}, \ldots \}

be a sequence of elements of a Hilbert space, and suppose that (one or both of) the inequalities

d
2

∑

a
i
2

≤

‖

∑

a
i

v
i

‖

2

≤

D
2

∑

a
i
2

{d^2}\sum {a_i^2 \leq {{\left \| {\sum {{a_i}{v_i}} } \right \|}^2} \leq {D^2}} \sum {a_i^2}

hold for every finite sequence of scalars

{

a
i

}

\{ {a_i}\}

. If an element

v
0

{v_0}

is adjoined to

{

v
i

}

\{ {v_i}\}

, then the resulting set satisfies (one or both of)

d
0
2

∑

a
i
2

≤

‖

∑

a
i

v
i

‖

2

≤

D
0
2

∑

a
i
2

d_0^2\sum {a_i^2 \leq {{\left \| {\sum {{a_i}{v_i}} } \right \|}^2} \leq D_0^2} \sum {a_i^2}

, where, denoting the norm of

v
0

{v_0}

by r and its distance from the closed linear span of the

v
i

{v_i}

by

δ
\delta

,
\[

d
0
2

=

d
2

+

1
2

(

r
2

−

d
2

−

(

r
2

+

d
2

)

2

−
4

d
2

δ
2

)

d_0^2 = {d^2} + \frac {1}{2}\left ( {{r^2} - {d^2} - \sqrt {{{({r^2} + {d^2})}^2} - 4{d^2}{\delta ^2}} } \right )

\]
and
\[

D
0
2

=

D
2

+

1
2

(

r
2

−

D
2

+

(

r
2

+

D
2

)

2

−
4

D
2

δ
2

)

.

D_0^2 = {D^2} + \frac {1}{2}\left ( {{r^2} - {D^2} + \sqrt {{{({r^2} + {D^2})}^2} - 4{D^2}{\delta ^2}} } \right ).

\]
Both bounds are best possible. If

v
0

{v_0}

is in the span of the original set, the expressions above simplify to

d
0

=
0

{d_0} = 0

and

D
0
2

=

D
2

+

r
2

D_0^2 = {D^2} + {r^2}

. If the original set is a single unit vector

v
1

{v_1}

, so

d
=
D
=
1

d = D = 1

, and if

v
0

⊥

v
1

{v_0} \bot {v_1}

is a unit vector so

δ
=
1

\delta = 1

, then the above is

(

a
2

+

b
2

)
≤

‖

a

v
0

+
b

v
1

‖

2

≤
(

a
2

+

b
2

)

({a^2} + {b^2}) \leq {\left \| {a{v_0} + b{v_1}} \right \|^2} \leq ({a^2} + {b^2})

, the Pythagorean Theorem. Several consequences are deduced. If

v
i

{v_i}

are unit vectors,

∑

a
i
2

=
1

\sum {a_i^2 = 1}

, and

δ
i

{\delta _i}

is the distance from

v
i

{v_i}

to the span of its predecessors (so that the volume of the parallelotope spanned by the

v
i

{v_i}

is

V
n

=

δ
1

δ
2

⋯

δ
n

{V_n} = {\delta _1}{\delta _2} \cdots {\delta _n}

), the above result is used to show that

‖

∑

i
=
0

n

a
i

v
i

‖

≥

V
n

/

2

n

/

2

\left \| {\sum \nolimits _{i = 0}^n {{a_i}{v_i}} } \right \| \geq {V_n}/{2^{n/2}}

.
Isomorphisms of standard operator algebras
Authors: Šemrl Peter 
Published: June 2010

Let X and Y be Banach spaces,

dim
⁡
X
=
∞

\dim X = \infty

, and let

A

\mathcal {A}

and

B

\mathcal {B}

be standard operator algebras on X and Y, respectively. Assume that

ϕ
:

A

→

B

\phi :\mathcal {A} \to \mathcal {B}

is a bijective mapping satisfying

‖

ϕ
(
A
B
)
−
ϕ
(
A
)
ϕ
(
B
)

‖

≤
ε
,
A
,
B
∈

A

\left \| {\phi (AB) - \phi (A)\phi (B)} \right \| \leq \varepsilon ,A,B \in \mathcal {A}

, where

ε
\varepsilon

is a given positive real number (no linearity or continuity of

ϕ
\phi

is assumed). Then

ϕ
\phi

is a spatially implemented linear or conjugate linear algebra isomorphism. In particular,

ϕ
\phi

is continuous.
Stolarsky’s inequality with general weights
Authors: Maligranda Lech , Pečarić Josip E. , Persson Lars Erik 
Published: June 2010

Recently Stolarsky proved that the inquality

(

∗

)

∫
0
1

g
(

x

1

/

(
a
+
b
)

)

d
x
≥

∫
0
1

g
(

x

1

/

a

)

d
x

∫
0
1

g
(

x

1

/

b

)

d
x

\begin{equation}\tag {$\ast $} \int _0^1 {g({x^{1/(a + b)}})\,dx \geq \int _0^1 {g({x^{1/a}})\,dx\int _0^1 {g({x^{1/b}})\,dx} } } \end{equation}

holds for every

a
,
b
>
0

a,b > 0

and every nonincreasing function on [0, 1] satisfying

0
≤
g
(
u
)
≤
1

0 \leq g(u) \leq 1

. In this paper we prove a weighted version of this inequality. Our proof is based on a generalized Chebyshev inequality. In particular, our result shows that the inequality

(
∗
)

( \ast )

holds for every function g of bounded variation. We also generalize another inequality by Stolarsky concerning the

Γ
\Gamma

-function.
On 𝑑-parameter pointwise ergodic theorems in 𝐿₁
Authors: Hasegawa Shigeru , Sato Ryotaro 
Published: June 2010

Let

P
1

,
…
,

P
d

{P_1}, \ldots ,{P_d}

be commuting positive linear contractions on

L
1

{L_1}

and let

T
1

,
…
,

T
d

{T_1}, \ldots ,{T_d}

be (not necessarily commuting) linear contractions on

L
1

{L_1}

such that

|

T
i

f

|

≤

P
i

|

f

|

|{T_i}f| \leq {P_i}|f|

for

1
≤
i
≤
d

1 \leq i \leq d

and

f
∈

L
1

f \in {L_1}

. In this paper we prove that if each

P
i

,
1
≤
i
≤
d

{P_i},1 \leq i \leq d

, satisfies the mean ergodic theorem, then the averages

A
n

(

T
1

,
…
,

T
d

)
f
=

A
n

(

T
1

)
⋯

A
n

(

T
d

)
f

{A_n}({T_1}, \ldots ,{T_d})f = {A_n}({T_1}) \cdots {A_n}({T_d})f

, where

A
n

(

T
i

)
=

n

−
1

∑

k
=
0

n
−
1

T
i
k

{A_n}({T_i}) = {n^{ - 1}}\sum \nolimits _{k = 0}^{n - 1} {T_i^k}

, converge a.e. for every

f
∈

L
1

f \in {L_1}

. When

T
1

,
…
,

T
d

{T_1}, \ldots ,{T_d}

commute, we further prove that the

L
1

{L_1}

-norm convergence of the averages

A
n

(

P
1

,
…
,

P
d

)
f

{A_n}({P_1}, \ldots ,{P_d})f

for every

f
∈

L
1

f \in {L_1}

implies the a.e. convergence of the averages

A
n

(

T
1

,
…
,

T
d

)
f

{A_n}({T_1}, \ldots ,{T_d})f

for every

f
∈

L
1

f \in {L_1}

. These improve Çömez and Lin’s ergodic theorem.
Operators with finite chain length and the ergodic theorem
Authors: Laursen K. B. , Mbekhta M. 
Published: June 2010

With a technical assumption (E-k), which is a relaxed version of the condition

T
n

/

n
→
0
,
n
→
∞

{T^n}/n \to 0,n \to \infty

, where T is a bounded linear operator on a Banach space, we prove a generalized uniform ergodic theorem which shows, inter alias, the equivalence of the finite chain length condition

(
X
=

(
I
−
T

)
k

X
⊕
ker
⁡

(
I
−
T

)
k

)

(X = {(I - T)^k}X \oplus \ker {(I - T)^k})

, of closedness of

(
I
−
T

)
k

X

{(I - T)^k}X

, and of quasi-Fredholmness of

I
−
T

I - T

. One consequence, still assuming (E-k), is that

I
−
T

I - T

is semi-Fredholm if and only if

I
−
T

I - T

is Riesz-Schauder. Other consequences are: a uniform ergodic theorem and conditions for ergodicity for certain classes of multipliers on commutative semisimple Banach algebras.
Solution of the Baire order problem of Mauldin
Authors: Balcerzak Marek , Rogowska Dorota 
Published: June 2010

Let X be an uncountable Polish space, and let I be a proper

σ
\sigma

-ideal of subsets of X such that

{
x
}
∈
I

\{ x\} \in I

for each

x
∈
X

x \in X

. Denote by

B
α

(
I
)
,
α
≤

ω
1

{B_\alpha }(I),\alpha \leq {\omega _1}

, the Baire system generated by the family of functions

f
:
X
→

R

f:X \to \mathbb {R}

continuous I almost everywhere. We prove that if

r
(
I
)
=
min
{
α
≤

ω
1

:

B

α
+
1

(
I
)
=

B
α

(
I
)
}

r(I) = \min \{ \alpha \leq {\omega _1}:{B_{\alpha + 1}}(I) = {B_\alpha }(I)\}

, then either

r
(
I
)
=
1

r(I) = 1

or

r
(
I
)
=

ω
1

r(I) = {\omega _1}

. This answers the problem raised by R. D. Mauldin in 1973.
Countably additive homomorphisms between von Neumann algebras
Authors: Bunce L. J. , Hamhalter J. 
Published: June 2010

Let M and N be von Neumann algebras where M has no abelian direct summand. A

∗
\ast

-homomorphism

π
:
M
→
N

\pi :M \to N

is said to be countably additive if

π
(

∑
1
∞

e
n

)
=

∑
1
∞

π
(

e
n

)

\pi (\sum \nolimits _1^\infty {{e_n}) = \sum \nolimits _1^\infty {\pi ({e_n})} }

, for every sequence

(

e
n

)

({e_n})

of orthogonal projections in M. We prove that a

∗
\ast

-homomorphism

π
:
M
→
N

\pi :M \to N

is countably additive if and only if

π
(
e
∨
f
)
=
π
(
e
)
∨
π
(
f
)

\pi (e \vee f) = \pi (e) \vee \pi (f)

for every pair of projections e and f of M. A corollary is that if, in addition, M has no Type

I

2

{{\text {I}}_2}

direct summands, then every lattice morphism from the projections of M into the projections of N is a

σ
\sigma

-lattice morphism.
Nonnegative solutions of the radial Laplacian with nonlinearity that changes sign
Authors: Cac N. P. , Fink A. M. , Gatica J. A. 
Published: June 2010

We find a solution to the radial Laplacian equation

y
+

N
−
1

x

y
′

+
λ
a
(
x
)
f
(
y
)
=
0
,

y
′

(
0
)
=
y
(
1
)
=
0

y + \frac {{N - 1}}{x}y’ + \lambda a(x)f(y) = 0,y’ (0) = y(1) = 0

when a may change sign and is "sufficiently positive". The function f is qualitatively like

e
y

{e^y}

, and we conclude solutions for

0
≤
λ
≤

λ
0

0 \leq \lambda \leq {\lambda _0}

.
Topological classification of irreducible plane curve singularities in terms of Weierstrass polynomials
Authors: Kang Chunghyuk 
Published: June 2010

Let

f
(
z
,
y
)

f(z,y)

be analytically irreducible at 0 and

f
(
0
)
=
0

f(0) = 0

. Then the plane curve singularity defined by f has the same topological type as the curve defined by

f

k
+
1

{f_{k + 1}}

for some

k
≥
0

k \geq 0

where

f
1

=

z
a

+

y
b

,

f
2

=

f
1

n

21

+

y

m

11

z

m

12

,

f
3

=

f
2

n

31

+

f
1

n

22

y

m

21

z

m

22

,
…

{f_1} = {z^a} + {y^b},{f_2} = {f_1}^{{n_{21}}} + {y^{{m_{11}}}}{z^{{m_{12}}}},{f_3} = {f_2}^{{n_{31}}} + {f_1}^{{n_{22}}}{y^{{m_{21}}}}{z^{{m_{22}}}}, \ldots

are defined by induction on k with distinct numerical conditions topologically invariant. Moreover, we give an easy alternate proof of Zariski’s topological classification theorem of irreducible plane curve singularities.
The local limit theorem and some related aspects of super-critical branching processes
Authors: Athreya Krishna B. , Ney Peter 
Published: June 2010

Let

{

Z
n

:
n
=
0
,
1
,
2
,
…
}

\{ {Z_n}:n = 0,1,2, \ldots \}

be a Galton-Watson branching process with offspring p.g.f.

f
(
s
)
=

Σ
0
∞

p
j

s
j

f(s) = \Sigma _0^\infty {p_j}{s^j}

. Assume (i)

1
>
m
=

f
′

(
1
−
)
=

Σ
1
∞

j

p
j

>
∞

1 > m = f’(1 - ) = \Sigma _1^\infty j{p_j} > \infty

, (ii)

Σ
1
∞

j
2

p
j

>
∞

\Sigma _1^\infty {j^2}{p_j} > \infty

and (iii)

γ
0

=

f
′

(
q
)
>
0

{\gamma _0} = f’(q) > 0

, where

q
q

is the extinction probability of the process. Let

w
(
x
)

w(x)

denote the density function of

W
W

, the almost sure limit of

Z
n

m

−
n

{Z_n}{m^{ - n}}

with

Z
0

=
1
,

w

(
i
)

(
x
)

{Z_0} = 1,{w^{(i)}}(x)

the

i
i

-fold convolution of

w
(
x
)
,

P
n

(
i
,
j
)
=
P
(

Z
n

=
j

|

Z
0

=
i
)
,

δ
0

=
(
log
⁡

γ
0

−
1

)

(
log
⁡
m

)

−
1

w(x),{P_n}(i,j) = P({Z_n} = j|{Z_0} = i),{\delta _0} = (\log \gamma _0^{ - 1}){(\log m)^{ - 1}}

and

β
0

=

m

δ
0

/

(
3
+

δ
0

)

{\beta _0} = {m^{{\delta _0}/(3 + {\delta _0})}}

. Then for any

0
>
β
>

β
0

0 > \beta > {\beta _0}

and

i
i

we can find a constant

C
=
C
(
i
,
β
)

C = C(i,\beta )

such that
\[

|

m
n

P
n

(
i
,
j
)
−

w

(
i
)

(

m

−
n

j
)

|

≦
C
[

β
0

−
n

(

m

−
n

j

)

−
1

+

β

−
n

]

|{m^n}{P_n}(i,j) - {w^{(i)}}({m^{ - n}}j)| \leqq C[\beta _0^{ - n}{({m^{ - n}}j)^{ - 1}} + {\beta ^{ - n}}]

\]
for all

j
≧
1

j \geqq 1

. Applications to the boundary theory of the associated space time process are also discussed.
The spectrum of partial differential operators on 𝐿^{𝑝}(𝑅ⁿ)
Authors: Iha Franklin T. , Schubert C. F. 
Published: June 2010

The purpose of this paper is to prove that if the polynomial

P
(
ξ
)

P(\xi )

associated with a partial differential operator

P
P

on

L
p

(

R
n

)

{L^p}({R^n})

, with constant coefficients, has the growth property,

|

P
(
ξ
)

|

−
1

=
O
(

|

ξ

|

−
r

)
,

|

ξ

|

→
∞

|P(\xi ){|^{ - 1}} = O(|\xi {|^{ - r}}),|\xi | \to \infty

for some

r
>
0

r > 0

, then the spectrum of

P
P

is either the whole complex plane or it is the numerical range of

P
(
ξ
)

P(\xi )

; and if

P
(
ξ
)

P(\xi )

has some additional property (all the coefficients of

P
(
ξ
)

P(\xi )

being real, for example), then the spectrum of

P
P

is the numerical range for those

p
p

sufficiently close to 2. It is further shown that the growth property alone is not sufficient to ensure that the spectrum of

P
P

is the numerical range of

P
(
ξ
)

P(\xi )

.
Logarithmic convexity, first order differential inequalities and some applications
Authors: Levine Howard Allen 
Published: June 2010

Let, for

t
∈
[
0
,
T
)
(
T
>
∞
)
,
D
(
t
)

t \in [0,T)(T > \infty ),D(t)

be a dense linear subspace of a Hilbert space

H
H

, and let

M
(
t
)

M(t)

and

N
(
t
)

N(t)

be linear operators (possibly unbounded) mapping

D
(
t
)

D(t)

into

H
H

. Let

f
:
[
0
,
T
)
×
H
→
H

f:[0,T) \times H \to H

. We give sufficient conditions on

M
,
N

M,N

and

f
f

in order to insure uniqueness and stability of solutions to
\[

(
1
)

M
(
t
)
d
u

/

d
t
=
N
(
t
)
u
+
f
(
t
,
u
)
,

u
(
0
)

given
.

(1)\quad M(t)du/dt = N(t)u + f(t,u),\quad u(0)\;\text {given}.

\]
This problem is not in general well posed in the sense of Hadamard. We cite some examples of (1) from the literature. We also give some examples of the problem
\[

(
2
)

M
(
t
)

d
2

u

d

t
2

=
N
(
t
)
u
+
f

(

t
,
u
,

d
u

d
t

)

,

u
(
0
)
,

d
u

d
u

(
0
)

prescribed
,

(2)\quad M(t)\frac {{{d^2}u}}{{d{t^2}}} = N(t)u + f\left ( {t,u,\frac {{du}}{{dt}}} \right ),\quad u(0),\frac {{du}}{{du}}(0)\;\text {prescribed},

\]
for which questions of uniqueness and stability were discussed in a previous paper.
On the order of a starlike function
Authors: Holland F. , Thomas D. K. 
Published: June 2010

It is shown that if

f
∈
S

f \in S

, the class of normalised starlike functions in the unit

disc
⁡
Δ

\operatorname {disc} \Delta

, then
\[

(

i

)

lim

r
→
1
−

log
⁡

P
λ

(
r
)

−
log
⁡
(
1
−
r
)

=
α
λ

for

λ
>
0
;

({\text {i}})\quad \quad \lim \limits _{r \to 1 - } \frac {{\log {P_\lambda }(r)}}{{ - \log (1 - r)}} = \alpha \lambda {\text { for }}\lambda > 0;

\]

\[

(

ii

)

lim

r
→
1
−

log
⁡

|

|

f
r

|

|

p

−
log
⁡
(
1
−
r
)

=
α
p
−
1

for

α
p
>
1
;

({\text {ii}})\quad \quad \lim \limits _{r \to 1 - } \frac {{\log ||{f_r}|{|_p}}}{{ - \log (1 - r)}} = \alpha p - 1{\text { for }}\alpha p > 1;

\]
and
\[

(

iii

)

lim

r
→
1
−

log
⁡

|

|

f
r
′

|

|

p

−
log
⁡
(
1
−
r
)

=
(
1
+
α
)
p
−
1

for (1 +

α
)
p
>
1
,

({\text {iii}})\quad \lim \limits _{r \to 1 - } \frac {{\log ||f’_r||_{p}}}{{ - \log (1 - r)}} = (1 + \alpha )p - 1\quad {\text {for (1 + }}\alpha )p > 1,

\]
where

P
λ

(
r
)
=

Σ

n
=
1

∞

n

λ
−
1

|

a
n

|

λ

r
n

,
(

a
n

)

{P_\lambda }(r) = \Sigma _{n = 1}^\infty {n^{\lambda - 1}}|{a_n}{|^\lambda }{r^n},({a_n})

is the sequence of coefficients and

α
\alpha

the order of

f
f

, and where
\[

|

|

f
r

|

|

p

=

1

2
π

∫
0

2
π

|

f
(
r

e

i
θ

)

|

p

d
θ
.

||{f_r}|{|_p} = \frac {1}{{2\pi }}\int _0^{2\pi } {|f(r{e^{i\theta }})} {|^p}d\theta .

\]
The results extend work of Pommerenke. The methods of the paper yield various other results, one in particular being
\[

lim

sup

n
→
∞

log

+

n

|

a
n

|

log
⁡
n

=
α

\lim \sup \limits _{n \to \infty } \frac {{{{\log }^ + }n|{a_n}|}}{{\log n}} = \alpha

\]
, a result which has an analogy in the theory of entire functions.
The existence of solutions of abstract partial difference polynomials.
Authors: Bentsen Irving 
Published: June 2010

A partial difference (p.d.) ring is a commutative ring

R
R

together with a (finite) set of isomorphisms (called transforming operators) of

R
R

into

R
R

which commute under composition. It is shown here that (contrary to the ordinary theory [R. M. Cohn, Difference algebra]) there exist nontrivial algebraically irreducible abstract p.d. polynomials having no solution and p.d. fields having no algebraically closed p.d. overfield. If

F
F

is a p.d. field with two transforming operators, then the existence of a p.d. overfield of

F
F

whose underlying field is an algebraic closure of that of

F
F

is a necessary and sufficient condition for every nontrivial algebraically irreducible abstract p.d. polynomial

P
P

in the p.d. polynomial ring

F
{

y

(
1
)

,

y

(
2
)

,
…
,

y

(
n
)

}

F\{ {y^{(1)}},{y^{(2)}}, \ldots ,{y^{(n)}}\}

to have a solution

η
\eta

(in some p.d. overfield of

F
F

) such that:

η
\eta

has

n
−
1

n - 1

transformal parameters,

η
\eta

is not a proper specialization over

F
F

of any other solution of

P
P

, and, if

Q
Q

is a p.d. polynomial whose indeterminates appear effectively in

P
P

and

Q
Q

is annulled by

η
\eta

, then

Q
Q

is a multiple of

P
.
P

P.P

has at most finitely many isomorphically distinct such solutions. Necessity holds if

F
F

has finitely many transforming operators.
The minimum modulus of certain small entire functions
Authors: Fenton P. C. 
Published: June 2010

Suppose that

f
(
z
)

f(z)

is an entire function satisfying
\[

min

|

z

|

=
r

|

f
(
z
)

|

⩽
C
(
σ
)

max

|

z

|

=
r

|

f
(
z
)

|

,

\min \limits _{|z| = r} |f(z)| \leqslant C(\sigma )\max \limits _{|z| = r} |f(z)|,

\]
for

r
⩾

ρ
0

>
0

r \geqslant {\rho _0} > 0

, where

σ
>
0

\sigma > 0

and
\[

C
(
σ
)
=

∏

k
=
1

∞

{

1
−
exp
⁡
(
−
(
2
k
−
1
)

/

4
σ
)

1
+
exp
⁡
(
−
(
2
k
−
1
)

/

4
σ
)

}

2

.

C(\sigma ) = \prod \limits _{k = 1}^\infty {{{\left \{ {\frac {{1 - \exp ( - (2k - 1) / 4\sigma )}} {{1 + \exp ( - (2k - 1) / 4\sigma )}}} \right \}}^2}.}

\]
It is shown that
\[

lim
_

r
→
∞

max

|

z

|

=
r

|

f
(
z
)

|

−
σ

(
log
⁡
r
)

2

log
⁡
r

⩾
−
2
σ
log
⁡
(
max
(

ρ
0

,

|

a
1

|

)
)
,

\underline {\lim } \limits _{r \to \infty } \frac {{{{\max }_{|z| = r}}|f(z)| - \sigma {{(\log r)}^2}}} {{\log r}} \geqslant - 2\sigma \log (\max ({\rho _0},|{a_1}|)),

\]
where

a
1

{a_1}

is the first nonzero zero of

f
f

.