bluuebunny/crossref_metadata_2025_split huggingface dataset:
On a domain characterization of Schrödinger operators with gradient magnetic vector potentials and singular potentials
Authors: Goldstein Jerome A. , Svirsky Roman 
Published: June 2010

Of concern are the minimal and maximal operators on

L
2

(

R

n

)

{L^2}({{\mathbf {R}}^n})

associated with the differential expression
\[

τ
Q

=

∑

j
=
1

n

(
i
∂

/

∂

x
j

+

q
j

(
x
)

)
2

+
W
(
x
)

{\tau _Q} = \sum \limits _{j = 1}^n {(i\partial /\partial {x_j}} + {q_j}(x){)^2} + W(x)

\]
where

(
q
,
…
,

q
n

)
=
grad
⁡
Q

(q, \ldots ,{q_n}) = \operatorname {grad}Q

for some real function

W
W

on

R

n

{{\mathbf {R}}^n}

and

W
W

satisfies

c

|
x
|

−
2

≤
W
(
x
)
≤
C

|
x
|

−
2

c{\left | x \right |^{ - 2}} \leq W(x) \leq C{\left | x \right |^{ - 2}}

. In particular, for

Q
=
0

Q = 0

,

τ
Q

{\tau _Q}

reduces to the singular Schrödinger operator

−
Δ
+
W
(
x
)

- \Delta + W(x)

. Among other results, it is shown that the maximal operator (associated with the

τ
Q

{\tau _Q}

) is the closure of the minimal operator, and its domain is precisely
\[

Dom
⁡

(

∑

j
=
1

n

(
i
∂

/

∂

x
j

+

q
j

(
x
)
)

2

)

∩
Dom
⁡
(
W
)
,

\operatorname {Dom}\left ( {\sum \limits _{j = 1}^n {{{(i\partial /\partial {x_j} + {q_j}(x))}^2}} } \right ) \cap \operatorname {Dom}(W),

\]
provided that

C
≥
c
>
−
n
(
n
−
4
)

/

4

C \geq c > - n(n - 4)/4

.
Surjective isometries of weighted Bergman spaces
Authors: Kolaski Clinton J. 
Published: June 2010

Let

Ω
\Omega

be a bounded, simply connected domain in

C

n

=

R

2
n

{{\mathbf {C}}^n} = {R^{2n}}

, let

F
∈

L
1

(

m
Ω

)

F \in {L^1}({m_\Omega })

be positive and continuo on

Ω
\Omega

, and let

B
F
P

(
Ω
)
=

L
p

(
F
d
m
)
∩
H
(
Ω
)
(
0
>
p
>
∞
)

B_F^P(\Omega ) = {L^p}(Fdm) \cap H(\Omega )(0 > p > \infty )

denote the weighted Bergman space over

Ω
\Omega

. We characterize those automorphisms

Φ
\Phi

of

Ω
\Omega

such that the map

f
→
g
⋅
(
f
∘
Φ
)

f \to g \cdot (f \circ \Phi )

is a surjective isometry of

B
F
P

(
Ω
)

B_F^P(\Omega )

, including an explicit description of

|

g

|

|g|

.
The Odlyzko conjecture and O’Hara’s unimodality proof
Authors: Stanton Dennis , Zeilberger Doron 
Published: June 2010

We observe that Andrew Odlyzko’s conjecture that the Maclaurin coefficients of

1

/

[
(
1
+
q
)
(
1
+
q
+

q
2

)
⋯
(
1
+
q
+
⋯
+

q

k
−
1

)
]

1/[(1 + q)(1 + q + {q^2}) \cdots (1 + q + \cdots + {q^{k - 1}})]

have alternating signs is an almost immediate consequence of an identity that is implied by Kathy O’Hara’s recent magnificent combinatorial proof of the unimodality of the Gaussian coefficients.
Oscillation of neutral differential equations with periodic coefficients
Authors: Huang Qing Guang , Chen Shao Zhu 
Published: June 2010

We establish a necessary and sufficient condition for the oscillation of all solutions to the neutral differential equation
\[

[
x
(
t
)
−
p
(
x
)
(
t
−
r
)

]
′

+

∑

i
=
1

n

q
i

(
t
)
x
(
t
−
i
r
)
=
0

,

[x(t) - p(x)(t - r)]’ + \sum \limits _{i = 1}^n {{q_i}(t)x(t - ir) = 0} ,

\]
where

0
≤
p
≤
1
,
r
>
0

0 \leq p \leq 1,r > 0

are constants and

q
i

(
t
)
≥
0
,
i
=
1
,
…
,
n

{q_i}(t) \geq 0,i = 1, \ldots ,n

, are continuous

r
r

-periodic functions.
Instability of nonnegative solutions for a class of semipositone problems
Authors: Brown K. J. , Shivaji R. 
Published: June 2010

We consider the boundary value problem\[−Δu(x)=λf(u(x)),x∈ΩBu(x)=0,x∈∂Ω\begin {gathered} - \Delta u(x) = \lambda f(u(x)),\quad x \in \Omega \hfill \\ Bu(x) = 0,\quad x \in \partial \Omega \hfill \\ \end {gathered}\]whereΩ\Omegais a bounded region inRN{R^N}with smooth boundary,Bu=αh(x)u+(1−α)∂u/∂nBu = \alpha h(x)u + (1 - \alpha )\partial u/\partial nwhereα∈[0,1]h:∂Ω→R+\alpha \in [0,1]h:\partial \Omega \to {R^ + }withh=1h = 1whenα=1\alpha = 1,λ>0,f\lambda > 0,fis a smooth function withf(0)>0f(0) > 0(semipositone),f′(u)>0f’(u) > 0foru>0u > 0andf(u)≥0f(u) \geq 0foru>0u > 0. We prove that every nonnegative solution isunstable.
Spin characteristic classes and reduced 𝐾𝑆𝑝𝑖𝑛 group of a low-dimensional complex
Authors: Li Bang He , Duan Hai Bao 
Published: June 2010

This note studies relations between Spin bundles, over a CW-complex of dimension

≤
9

\leq 9

, and their first two Spin characteristic classes. In particular by taking Spin characteristic classes, it is proved that the stable classes of Spin bundles over a manifold

M
M

with dimension

≤
8

\leq 8

are in one to one correspondence with the pairs of cohomology classes

(

q
1

,

q
2

)
∈

H
4

(
M
;

Z

)
×

H
8

(
M
;

Z

)

({q_1},{q_2}) \in {H^4}(M;\mathbb {Z}) \times {H^8}(M;\mathbb {Z})

satisfying
\[

(

q
1

∪

q
2

+

q
2

)
mod
3
+

U
3
1

∪
(

q
1

mod
3
)
≡
0

({q_1} \cup {q_2} + {q_2})\bmod 3 + U_3^1 \cup ({q_1}\bmod 3) \equiv 0

\]
, where

U
3
1

∈

H
4

(
M
;

Z

3

)

U_3^1 \in {H^4}(M;{\mathbb {Z}_3})

is the indicated Wu-class of

M
M

. As an application a computation is made for

K
Spin

~

(
M
)

\widetilde {K\operatorname {Spin} }(M)

, where

M
M

is an eight-dimensional manifold with understood cohomology rings over

Z

,

Z

2

,

\mathbb {Z},{\mathbb {Z}_2},

, and

Z

3

{\mathbb {Z}_3}

.
Ball covering on manifolds with nonnegative Ricci curvature near infinity
Authors: Liu Zhong-dong 
Published: June 2010

Let

M
M

be a complete open Riemannian manifold with nonnegative Ricci curvature outside a compact set

B
B

. We show that the following ball covering property (see [LT]) is true provided that the sectional curvature has a lower bound: For a fixed

p
0

∈
M

{p_0} \in M

, there exist

N
>
0

N > 0

and

r
0

>
0

{r_0} > 0

such that for

r
≥

r
0

r \geq {r_0}

, there exist

p
1

p_1

, …,

p
k

∈
∂

B

p
0

(
2
r
)
,
k
≤
N

p_k \in \partial {B_{p0}}(2r),k \leq N

, with
\[

⋃

j
=
1

k

B

p
j

(
r
)
⊃
∂

B

p
0

(
2
r
)
.

\bigcup \limits _{j = 1}^k {{B_{{p_j}}}(r) \supset \partial {B_{{p_0}}}(2r).}

\]
Furthermore

N
N

and

r
0

{r_0}

depend only on the dimension, the lower bound on the sectional curvature, and the radius of the ball at

p
0

{p_0}

that contains

B
B

.
ℎ₀-torsion bounds in the cohomology of the Steenrod algebra
Authors: Monks Kenneth G. 
Published: June 2010

In this paper we use a technique of M. Hopkins to prove that the cohomology of the finite Hopf subalgebra of the

mod
2

\bmod 2

Steenrod algebra generated by

Sq
⁡

(

2
i

)

\operatorname {Sq}\left ( {{2^i}} \right )

with

i
≤
n

i \leq n

, has

h
0

{h_0}

-torsion bound

2

n
+
1

−
n
−
2

for

n
≥
1

{2^{n + 1}} - n - 2{\text { for }}n \geq 1

.
A disconjugacy criterion of W. T. Reid for difference equations
Authors: Peterson Allan , Ridenhour Jerry 
Published: June 2010

Our main result is a disconjugacy criterion for the selfadjoint vector difference equation

L
y
(
t
)
≡
Δ
[
P
(
t
−
1
)
Δ
y
(
t
−
1
)
]
+
Q
(
t
)
y
(
t
)
=
0

Ly(t) \equiv \Delta [P(t - 1)\Delta y(t - 1)] + Q(t)y(t) = 0

. This result is the analogue of a famous result of W. T. Reid for the corresponding differential equations case. Unlike the differential equations case we will see there is an exceptional case in which, as we will show by counterexample, the conclusion of the main result is no longer valid. A disfocality criterion is also given. We believe these results are new even in the scalar case.
Certain averages on the 𝑎-adic numbers
Authors: Asmar Nakhlé H. , Nair Radhakrishnan 
Published: June 2010

For

L
p

∩

L
2

{L^p} \cap {L^2}

functions

f
f

, with

p
p

greater than one, defined on the

a

{\mathbf {a}}

-adic numbers

Ω

a

{\Omega _{\mathbf {a}}}

, we consider averages like
\[

A
N

(
1
)

f

(

x

)

=

1
N

∑

n
=
1

N

f

(

x

+

n
2

α

)

,

and

A
N

(
2
)

f

(

x

)

=

1
N

∑

n
=
1

N

f

(

x

+

p
n

α

)

,

A_N^{\left ( 1 \right )}f\left ( {\mathbf {x}} \right ) = \frac {1}{N}\sum \limits _{n = 1}^N {f\left ( {{\mathbf {x}} + {n^2}\alpha } \right ),{\text { and }}A_N^{\left ( 2 \right )}f\left ( {\mathbf {x}} \right ) = \frac {1}{N}\sum \limits _{n = 1}^N {f\left ( {{\mathbf {x}} + {p_n}\alpha } \right ),} }

\]
where

x

{\mathbf {x}}

and

α
\alpha

are in

Ω

a

{\Omega _{\mathbf {a}}}

. Here

p
n

{p_n}

denotes the

n
n

th prime. These averages are known to converge for almost all

x

{\mathbf {x}}

. We describe explicitly these limits, which possibly contrary to expectation, turn out in general not to be the integral of

f
f

.
Furuta’s inequality and a generalization of Ando’s theorem
Authors: Fujii Masatoshi , Kamei Eizaburo 
Published: June 2010

As a continuation of our preceding notes, we discuss Furuta’s inequality under the chaotic order defined by

log
⁡
A
≥
log
⁡
B

\log A \geq \log B

for positive invertible operators

A
A

and

B
B

. We prove that Furuta’s type inequalities

(

B
r

A
p

B
r

)

2
r

/

(

p
+
2
r

)

≥

B

2
r

{\left ( {{B^r}{A^p}{B^r}} \right )^{2r/\left ( {p + 2r} \right )}} \geq {B^{2r}}

and

A

2
r

≥

(

A
r

B
p

A
r

)

2
r

/

(

p
+
2
r

)

{A^{2r}} \geq {\left ( {{A^r}{B^p}{A^r}} \right )^{2r/\left ( {p + 2r} \right )}}

hold true for

p
≥
r
≥
0

p \geq r \geq 0

, which is a generalization of an inequality due to Ando (Math. Ann. 279 (1987), 157-159).
Saeki’s improvement of the Vitali-Hahn-Saks-Nikodým theorem holds precisely for Banach spaces having cotype
Authors: Abraham Paul 
Published: June 2010

We prove that a Banach space

X
X

has nontrivial cotype if and only if given any

σ
\sigma

-field

Σ
\Sigma

and any sequence

μ
n

:
Σ
→
X

{\mu _n}:\Sigma \to X

of strongly additive vector measures such that for some

γ
≥
1
,
lim

sup

n
→
∞

‖

μ
n

(
E
)

‖

≤
γ
lim

inf

n
→
∞

‖

μ
n

(
E
)

‖

>
∞

\gamma \geq 1,\lim {\sup _{n \to \infty }}\left \| {{\mu _n}\left ( E \right )} \right \| \leq \gamma \lim {\inf _{n \to \infty }}\left \| {{\mu _n}\left ( E \right )} \right \| > \infty

for each

E
∈
Σ

E \in \Sigma

then

{

μ
n

:
n
∈

N

}

\left \{ {{\mu _n}:n \in \mathbb {N}} \right \}

is uniformly strongly additive.
Projections in Kac-Moody Lie algebras
Authors: Misra Kailash C. , Putcha Mohan S. 
Published: June 2010

Let

g

\mathfrak {g}

be a Kac-Moody Lie algebra and

B

\mathcal {B}

its set of positive Borel subalgebras. If

b

∈

B

\mathfrak {b} \in \mathcal {B}

and

p

\mathfrak {p}

is a parabolic subalgebra, let

proj

p

(

b

)
=

p

∩

b

+

r

n

(

p

)

{\operatorname {proj} _\mathfrak {p}}(\mathfrak {b}) = \mathfrak {p} \cap \mathfrak {b} + {\mathfrak {r}_n}(\mathfrak {p})

where

r

n

(

p

)

{\mathfrak {r}_n}(\mathfrak {p})

denotes the nilradical of

p

\mathfrak {p}

. In this paper we consider the idempotent maps

E

p

,

p

−

=

proj

p

∘

proj

p

−

:

B

→

B

{E_{\mathfrak {p},{\mathfrak {p}^ - }}} = {\operatorname {proj} _\mathfrak {p}} \circ {\operatorname {proj} _{{\mathfrak {p}^ - }}}:\mathcal {B} \to \mathcal {B}

, where

p

\mathfrak {p}

and

p

−

{\mathfrak {p}^ - }

are opposite parabolic subalgebras with

p

\mathfrak {p}

being of positive type. We consider the semigroup

M
=
M
(

g

)

M = M(\mathfrak {g})

generated (with respect to composition) by the maps

E

p

,

p

−

{E_{\mathfrak {p},{\mathfrak {p}^ - }}}

. In particular we show that the maximal subgroups of

M
M

are closely related to proper Levi subgroups of the Kac-Moody group associated with

g

\mathfrak {g}

.
Remarks on a multiplier conjecture for univalent functions
Authors: Fournier Richard , Ruscheweyh Stephan 
Published: June 2010

In this paper we study some aspects of a conjecture on the convolution of univalent functions in the unit disk

D

\mathbb {D}

, which was recently proposed by Grünberg, Rønning, and Ruscheweyh (Trans. Amer. Math. Soc. 322 (1990), 377-393) and is as follows: let

D

:=
{
f

analytic in

D

:

|

f
(
z
)

|

≤
Re
⁡

f
′

(
z
)
,
z
∈

D

}

\mathcal {D}: = \{ f{\text { analytic in }}\mathbb {D}:\left | {f(z)} \right | \leq \operatorname {Re} f’(z),z \in \mathbb {D}\}

and

g
,
h
∈

S

g,h \in \mathcal {S}

(the class of normalized univalent functions in

D

\mathbb {D}

. Then

Re
⁡
(
f
∗
g
∗
h
)
(
z
)

/

z
>
0

\operatorname {Re} (f*g*h)(z)/z > 0

in

D

\mathbb {D}

. We discuss several special cases, which lead to interesting, more specific statements about functions in

S

\mathcal {S}

, determine certain extreme points of

D

\mathcal {D}

, and note that the former conjectures of Bieberbach and Sheil-Small are contained in this one. It is an interesting matter of fact that the functions in

D

\mathcal {D}

, which are "responsible" for the Bieberbach coefficient estimates are not extreme points in

D

\mathcal {D}

.
Finite exponential series and Newman polynomials
Authors: Goddard Bart 
Published: June 2010

A Newman polynomial is a sum of powers of

z
z

, with constant term 1. The Newman polynomial of four terms whose minimum modulus on the unit circle is as large as possible is found by examining the expression
\[

f
(
4
)
=

sup

x
1

>
⋯
>

x
4

inf

α
∈
ℜ

|

∑

j
=
1

4

e

i

x
j

α

|

f(4) = \sup \limits _{{x_1} > \cdots > {x_4}} \inf \limits _{\alpha \in \Re } \left | {\sum \limits _{j = 1}^4 {{e^{i{x_j}\alpha }}} } \right |

\]
and determining an extremal system

(

x
1

,
…
,

x
4

)

({x_1}, \ldots ,{x_4})

using a technique that reduces the problem to a finite search.
Conjugacy separability of a class of 1-relator products
Authors: Allenby R. B. J. T. 
Published: June 2010

We prove the conjugacy separability of groups of the form

G
=
⟨

a
1

,
…

a
r

,

b
1

,
…
,

b
s

:

a
i

e
i

=

b
j

f
j

=
(
U
(

a
1

,
…
,

a
r

)
V
(

b
1

,
…
,

b
s

)

)
m

=
1
⟩

G = \langle {a_1}, \ldots {a_r},{b_1}, \ldots ,{b_s}:a_i^{{e_i}} = b_j^{{f_j}} = (U({a_1}, \ldots ,{a_r})V({b_1}, \ldots ,{b_s}) {)^m} = 1\rangle

, where

m
>
1

m > 1

.
The commutant of a certain compression
Authors: Ross William T. 
Published: June 2010

Let

G
G

be any bounded region in the complex plane and

K
⊂
G

K \subset G

be a simple compact arc of class

C
1

{C^1}

. Let

A
2

(
G
∖
K
)

{A^2}(G\backslash K)

(resp.

A
2

(
G
)

{A^2}(G)

) be the Bergman space on

G
∖
K

G\backslash K

(resp.

G
G

). Let

S
S

be the operator multiplication by

z
z

on

A
2

(
G
∖
K
)

{A^2}(G\backslash K)

and

C
=

P

N

S

|

N

C = {P_\mathcal {N}}S{|_\mathcal {N}}

be the compression of

S
S

to the semi-invariant subspace

N

=

A
2

(
G
∖
K
)
⊖

A
2

(
G
)

\mathcal {N} = {A^2}(G\backslash K) \ominus {A^2}(G)

. We show that the commutant of

C

∗

{C^{\ast }}

is the set of all operators of the form

A

−
1

M
h

A

{A^{ - 1}}{M_h}A

, where

h
h

is a multiplier on a certain Sobolev space of functions on

K
K

and

(
A
f
)
(
w
)
=

∫
G

f
(
z
)

(

z
¯

−

w
¯

)

−
1

d
A
(
z
)
(
w
∈
K
)

(Af)(w) = \int _G {f(z){{(\overline z - \overline w )}^{ - 1}}dA(z)(w \in K)}

. We also use multiplier theory in fractional order Sobolev spaces to obtain further information about

C
C

.
Boundary behavior of generalized Poisson integrals for the half-space and the Dirichlet problem for the Schrödinger operator
Authors: Kheifits Alexander I. 
Published: June 2010

The boundary properties are investigated for the generalized Poisson integral
\[

u
(
X
)
=

∫

R

n

k
(
X
,
y
)
f
(
y
)
d
y
,

u(X) = \int _{{\mathbb {R}^n}} {k(X,y)f(y)dy,}

\]
where

X
X

is a point of the upper half-space

R

+

n
+
1

,

f
∈

L

p

(

R

n

)
,

1
⩽

p

⩽
∞

\mathbb {R}_ + ^{n + 1},\;f \in {L^{\mathbf {p}}}({\mathbb {R}^n}),\;1 \leqslant {\mathbf {p}} \leqslant \infty

and the kernel

k
k

has some special properties. Our results imply the known boundary properties of the harmonic Poisson integrals on the half-space. As an application we derive a solution of the Dirichlet problem for the operator

−
Δ
+
c
(
X
)
,

X
∈

R

+

n
+
1

- \Delta + c(X),\;X \in \mathbb {R}_ + ^{n + 1}

, with boundary data

f
∈

L

p

(

R

n

)

f \in {L^{\mathbf {p}}}({\mathbb {R}^n})

.
Special values of the Lerch zeta function and the evaluation of certain integrals
Authors: Williams Kenneth S. , Zhang Nan Yue 
Published: June 2010

The Lerch zeta function

Φ
(
x
,
a
,
s
)

\Phi (x,a,s)

is defined by the series
\[

Φ
(
x
,
a
,
s
)
=

∑

n
=
0

∞

e

2
n
π
i
x

(
n
+
a
)

s

,

\Phi (x,a,s) = \sum \limits _{n = 0}^\infty {\frac {{{e^{2n\pi ix}}}} {{{{(n + a)}^s}}}} ,

\]
where

x
x

is real,

0
>
a
⩽
1

0 > a \leqslant 1

, and

σ
=
Re
⁡
(
s
)
>
1

\sigma = \operatorname {Re} (s) > 1

if

x
x

is an integer and

σ
>
0

\sigma > 0

otherwise. In this paper we study the function

J

(

s
,
a

)

=
Φ
(

1
2

,
a
,
s
)

J\left ( {s,a} \right ) = \Phi (\tfrac {1} {2},a,s)

. We use its integral representation
\[

J

(

s
,
a

)

=

a

−
s

2

+
2

∫
0
∞

(

a
2

+

y
2

)

−
s

/

2

sin
⁡

(

s

tan

−
1

y
a

)

e

π
y

d
y

e

2
π
y

−
1

J\left ( {s,a} \right ) = \frac {{{a^{ - s}}}} {2} + 2\int _0^\infty {{{({a^2} + {y^2})}^{ - s/2}}\sin \left ( {s\,{{\tan }^{ - 1}}\frac {y} {a}} \right )} \frac {{{e^{\pi y}}dy}} {{{e^{2\pi y}} - 1}}

\]
to obtain the values of certain definite integrals; for example, we show that
\[

∫
0
∞

cosh
⁡
x
log
⁡
x

cosh
⁡
2
x
−
cos
⁡
2
π
a

d
x

=

π

2
sin
⁡
π
a

{

log
⁡

Γ
(
(
1
+
a
)

/

2
)

Γ
(
a

/

2
)

+

1
2

log
⁡

(

2
π
cot
⁡

π
a

2

)

}

,

0
>
a
>
1.

\begin {gathered} \int _0^\infty {\frac {{\cosh x\log x}} {{\cosh 2x - \cos 2\pi a}}} dx \hfill \\ \qquad = \frac {\pi } {{2\sin \pi a}}\left \{ {\log \frac {{\Gamma ((1 + a)/2)}} {{\Gamma (a/2)}} + \frac {1} {2}\log \left ( {2\pi \cot \frac {{\pi a}} {2}} \right )} \right \},\qquad 0 > a > 1. \hfill \\ \end {gathered}

\]
A characterization of the second dual of 𝐶₀(𝑆,𝐴)
Authors: Choy Stephen T. L. , Wong James C. S. 
Published: June 2010

Let

S
S

be a locally compact Hausdorff space, and let

A
A

be a Banach space. The space of the continuous functions from

S
S

to

A
A

vanishing at infinity is denoted by

C
0

(
S
,
A
)

{C_0}(S,A)

. Let

M
W
(
S
,

A

∗

)

MW(S,{A^{\ast }})

be the space of the representing measures of all the bounded linear functionals on

C
0

(
S
,
A
)

{C_0}(S,A)

. For

μ
∈
M
W
(
S
,

A

∗

)

\mu \in MW(S,{A^{\ast }})

let
\[

L
∞

(

|

μ

|

,

A

∗

∗

,

A

∗

)
=
{
f
:
S
→

A

∗

∗

|

f
(
⋅
)

x

∗

∈

L
∞

(

|

μ

|

)
∀

x

∗

∈

A

∗

}
.

{L_\infty }(|\mu |,{A^{{\ast }{\ast }}},{A^{\ast }}) = \{ f:S \to {A^{{\ast }{\ast }}}|f( \cdot ){x^{\ast }} \in {L_\infty }(|\mu |)\forall {x^{\ast }} \in {A^{\ast }}\}.

\]
The second dual of

C
0

(
S
,
A
)

{C_0}(S,A)

is characterized in the general case by means of certain elements in the product linear space

∏

{

L
∞

(

|

μ

|

,

A

∗

∗

,

A

∗

)
:
μ
∈
M
W
(
S
,

A

∗

)
}

\prod {\{ {L_\infty }(|\mu |,{A^{{\ast }{\ast }}},{A^{\ast }}):\mu \in MW(S,{A^{\ast }})\} }

.
Discrepancy of behavior of perturbed sequences in 𝐿^{𝑝} spaces
Authors: Reinhold-Larsson Karin 
Published: June 2010

Given

p
∈
[
1
,
∞
)

p \in [1,\infty )

, examples of sequences

{

n
k

}

k
⊂

N

{\{ {n_k}\} _{k \subset \mathbb {N}}}

such that for any ergodic dynamical system

(
X
,
β
,
m
,
T
)

(X,\beta ,m,T)

the averages
\[

A
N

f
(
x
)
=

1
N

∑

k
=
1

N

f
(

T

n
k

x
)

{A_N}f(x) = \frac {1} {N}\sum \limits _{k = 1}^N {f({T^{{n_k}}}x)}

\]
converge almost everywhere in all

L
q

(
X
)
,

q
>
p

{L^q}(X),\,q > p

, but fail to have a finite limit for some function in

L
p

(
X
)

{L^p}(X)

are shown. Also, sequences such that for all ergodic dynamical systems the averages

A
N

f
(
x
)

{A_N}f(x)

do not converge for some function

f
∈

L
p

(
X
)

f \in {L^p}(X)

for all

1
⩽
p
>
∞

1 \leqslant p > \infty

but do converge for all functions in

L
∞

(
X
)

{L^\infty }(X)

are shown.
𝜀-isometric embeddings
Authors: Qian Songwei 
Published: June 2010

In this paper we study into

ε
\varepsilon

-isometries. We prove that if

φ
\varphi

is an

ε
\varepsilon

-isometry from

L
p

(

Ω
1

,

Σ
1

,

μ
1

)

{L^p}({\Omega _1},{\Sigma _1},{\mu _1})

into

L
p

(

Ω
2

,

Σ
2

,

μ
2

)

{L^p}({\Omega _2},{\Sigma _2},{\mu _2})

(for some

p
,
1
>
p
>
∞

p, 1 > p > \infty

), then there is a linear operator

T
:

L
p

(

Ω
2

,

Σ
2

,

μ
2

)
↦

L
p

(

Ω
1

,

σ
1

,

μ
1

)

T:{L^p}({\Omega _2},{\Sigma _2},{\mu _2}) \mapsto {L^p}({\Omega _1},{\sigma _1},{\mu _1})

with

‖
T
‖

=
1

\left \| T \right \| = 1

such that

‖

T
∘
φ
(
f
)
−
f

‖

≤
6
ε

\left \| {T \circ \varphi (f) - f} \right \| \leq 6\varepsilon

for each

f
∈

L
p

(

Ω
1

,

Σ
1

,

μ
1

)

f \in {L^p}({\Omega _1},{\Sigma _1},{\mu _1})

. This forms a link between an into isometry result of Figiel and a surjective

ε
\varepsilon

-isometry result of Gevirtz in the case of

L
p

{L^p}

spaces.
Jacobi’s bound for the order of systems of first order differential equations
Authors: Lando Barbara A. 
Published: June 2010

Let

A
1

,
…
,

A
n

{A_1}, \ldots ,{A_n}

be a system of differential polynomials in the differential indeterminates

y

(
1
)

,
…
,

y

(
n
)

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

, and let

M

\mathcal {M}

be an irreducible component of the differential variety

M

(

A
1

,
…
,

A
n

)

\mathcal {M}({A_1}, \ldots ,{A_n})

. If

dim
⁡

M

=
0

\dim \mathcal {M} = 0

, there arises the question of securing an upper bound for the order of

M

\mathcal {M}

in terms of the orders

r

i
j

{r_{ij}}

of the polynomials

A
i

{A_i}

in

y

(
j
)

{y^{(j)}}

. It has been conjectured that the Jacobi number
\[

J
=
J
(

r

i
j

)
=
max
{

∑

i
=
1

n

r

i

j
i

:

j
1

,
…
,

j
n

is a permutation of 1,

…
,
n
}

J = J({r_{ij}}) = \max \{\sum \limits _{i = 1}^n {{r_{i{j_i}}}} :{j_1}, \ldots ,{j_n}{\text { is a permutation of 1,}} \ldots ,n \}

\]
provides such a bound. In this paper

J
J

is obtained as a bound for systems consisting of first order polynomials. Differential kernels are employed in securing the bound, with the theory of kernels obtained in a manner analogous to that of difference kernels as given by R. M. Cohn.
Uniqueness criteria for solutions of singular boundary value problems
Authors: Dunninger D. R. , Levine Howard A. 
Published: June 2010

In this paper we consider the equation

(

1

)

u
(
t
)
+
(
k

/

t
)

u
′

(
t
)
+
A
u
(
t
)
=
0
,

0
>
t
>
T
,

u
(
T
)
=
0
,

\begin{equation}\tag {$1$} u(t) + (k/t)u’(t) + Au(t) = 0,\quad 0 > t > T,\quad u(T) = 0,\end{equation}

where

u
:
(
0
,
T
)
→
D
(
A
)
⊂
B

u:(0,T) \to D(A) \subset B

is a Banach space valued function taking values in a dense subdomain

D
(
A
)

D(A)

of the Banach space B. Here A is a closed (possibly unbounded) linear operator on

D
(
A
)

D(A)

while k is a real constant. The differential equation is an abstract Euler-Poisson-Darboux equation. We give necessary and sufficient conditions on the point spectrum of A to insure uniqueness of the strong solution

u
≡
0

u \equiv 0

as well as sufficient conditions on the point spectrum to insure uniqueness of weak solutions. u is only required to satisfy (a)

t
k

‖

u
′

(
t
)

‖

→

0
+

{t^k}\left \| {u’(t)} \right \| \to {0^ + }

as

t
→

0
+

t \to {0^ + }

if

k
>
1

k > 1

, (b)

t
k

‖

u
′

(
t
)

‖

+

t

k
+
1

‖

u
(
t
)

‖

→
0

{t^k}\left \| {u’(t)} \right \| + {t^{k + 1}}\left \| {u(t)} \right \| \to 0

as

t
→

0
+

,
0
>
k
⩽
1

t \to {0^ + },0 > k \leqslant 1

, (c)

t

‖

u
′

(
t
)

‖

+

‖

u
(
t
)

‖

→
0

t\left \| {u’(t)} \right \| + \left \| {u(t)} \right \| \to 0

as

t
→

0
+

,
k
>
0

t \to {0^ + },k > 0

. The operator A need not possess a complete set of eigenvectors nor need one have a backward uniqueness theorem available for (1) for the Cauchy final value problem. Our techniques extend to the n-axially symmetric abstract equation

(

2

)

∑

i
=
1

n

[

∂
2

u

/

∂

t
i
2

+
(

k
i

/

t
i

)
∂
u

/

∂

t
i

]
+
A
u
=
0.

\begin{equation}\tag {$2$} \sum \limits _{i = 1}^n {[{\partial ^2}u/\partial t_i^2 + ({k_i}/{t_i})\partial u/\partial {t_i}] + Au = 0.} \end{equation}

The proofs rest upon an application of the Hahn-Banach Theorem and the consequent separation properties of

B
∗

{B^\ast }

, the dual of B, as well as the completeness properties of the eigenfunctions of certain Bessel equations associated with (1).