bluuebunny/crossref_metadata_2025_split huggingface dataset:
On weighted norm inequalities for positive linear operators
Authors: Kerman R. , Sawyer E. 
Published: June 2010

Let

T
T

be a positive linear operator defined for nonnegative functions on a

σ
\sigma

-finite measure space

(

X
,
m
,
μ

)

\left ( {X,m,\mu } \right )

. Given

1
>
p
>
∞

1 > p > \infty

and a nonnegative weight function

w
w

on

X
X

, it is shown that there exists a nonnegative weight function

v
v

, finite

μ
\mu

-almost everywhere on

X
X

, such that (1)
\[

∫
X

(

T
f

)

p

w
d
μ
≤

∫
X

f
p

v
d
μ

,

for all

f
≤
0

\int _X {{{\left ( {Tf} \right )}^p}wd\mu \leq \int _X {{f^p}vd\mu } } ,\quad {\text {for all }}f\leq 0

\]
, if and only if there exists

ϕ
\phi

positive

μ
\mu

-almost everywhere on

X
X

with (2)
\[

∫
X

(

T
ϕ

)

p

w
d
μ
>
∞
.

\int \limits _X {{{\left ( {T\phi } \right )}^p}wd\mu > \infty .}

\]
In case (2) holds, we may take

v
=

ϕ

1
−
p

T
∗

[

(

T
ϕ

)

p
−
1

w

]

v = {\phi ^{1 - p}}{T^*}\left [ {{{\left ( {T\phi } \right )}^{p - 1}}w} \right ]

in (1). This partially answers a question of B. Muckenhoupt in [5]. Applications to some specific operators are also given.
There can be 𝐶*-embedded dense proper subspaces in 𝛽𝜔-𝜔
Authors: van Douwen Eric K. , Kunen Kenneth , van Mill Jan 
Published: June 2010

Fine and Gillman have shown that CH implies that if

X
X

is a dense proper subspace of

ω
∗

=
β
ω
−
ω

{\omega ^ * } = \beta \omega - \omega

, then

β
X
≠

ω
∗

\beta X \ne {\omega ^ * }

. Here it is shown to be consistent with

MA + c =

ω
2

{\text {MA + c = }}{\omega _2}

that for every

p
∈

ω
∗

p \in {\omega ^ * }

we have

β
(

ω
∗

−
{
p
}
)
=

ω
∗

\beta ({\omega ^ * } - \{ p\} ) = {\omega ^ * }

and also that

ω
∗

{\omega ^ * }

has a dense subspace

X
X

with dense complement such that

β
X
=

ω
∗

\beta X = {\omega ^ * }

.
On composition of four-symbol 𝛿-codes and Hadamard matrices
Authors: Yang C. H. 
Published: June 2010

It is shown that key instruments for composition of four-symbol

δ
\delta

-codes are the Lagrange identity for polynomials, a certain type of quasisymmetric sequences (i.e., a set of normal or near normal sequences) and base sequences. The following is proved: If a set of base sequences for length

t
t

and a set of normal (or near normal) sequences for length

n
n

exist then four-symbol

δ
\delta

-codes of length

(

2
n
+
1

)

t

(

or

n
t

)

\left ( {2n + 1} \right )t\left ( {{\text {or }}nt} \right )

can be composed by application of the Lagrange identity. Consequently a new infinite family of Hadamard matrices of order

4
u
w

4uw

can be constructed, where

w
w

is the order of Williamson matrices and

u
=

(

2
n
+
1

)

t

(

or

n
t

)

u = \left ( {2n + 1} \right )t\left ( {{\text {or }}nt} \right )

. Other related topics are also discussed.
Selfadjointness of the *-representation generated by the sum of two positive linear functionals
Authors: Inoue Atsushi 
Published: June 2010

Let

ϕ
\phi

and

ψ
\psi

be positive linear functionals on a

∗
*

-algebra

A

\mathcal {A}

. When the closed

∗
*

-representations

π
ϕ

{\pi _\phi }

and

π
ψ

{\pi _\psi }

of

A

\mathcal {A}

generated by the GNS-construction for

ϕ
\phi

and

ψ
\psi

are self-adjoint, we shall show that

π

ϕ
+
ψ

{\pi _{\phi + \psi }}

is self-adjoint if and only if

π

ϕ
+
ψ

(

A

)

′

w

D

(

π

ϕ
+
ψ

)

⊂

D

(

π

ϕ
+
ψ

)

{\pi _{\phi + \psi }}{\left ( \mathcal {A} \right )’}_w\mathcal {D}\left ( {{\pi _{\phi + \psi }}} \right ) \subset \mathcal {D}\left ( {{\pi _{\phi + \psi }}} \right )

; and there exists a self-adjoint extension

ρ
\rho

of

π

ϕ
+
ψ

{\pi _{\phi + \psi }}

suchthat

ρ

(

A

)

′

w

=

π

ϕ
+
ψ

(

A

)

′

w

\rho {\left ( \mathcal {A} \right )’}_w = {\pi _{\phi + \psi }}{\left ( \mathcal {A} \right )’}_w

if and only if

π

ϕ
+
ψ

(

A

)

′

w

{\pi _{\phi + \psi }}{\left ( \mathcal {A} \right )’}_w

is a von Neumann algebra.
Selfadjointness of the momentum operator with a singular term
Authors: Watanabe Michiaki , Watanabe Shuji 
Published: June 2010

Self-adjointness is shown of the momentum operator

{
u
∈

H
1

(

R

1

)
:
u

/

x
∈

L
2

(

R

1

)
}

\{ u \in {H^1}({{\mathbf {R}}^1}):u/x \in {L^2}({{\mathbf {R}}^1})\}

, with domain

{

u
∈

H
1

(

R

1

)
:
u

/

x
∈

L
2

(

R

1

)

}

\left \{ {u \in {H^1}({{\mathbf {R}}^1}):u/x \in {L^2}({{\mathbf {R}}^1})} \right \}

when

c
>
1

c > 1

or

c
>
−
1

c > - 1

. This operator appears in a harmonic oscillator system with the generalized commutation relations by Wigner:

i
p
=
[
x
,
H
]

ip = [x,H]

and

−
i
x
=
[
p
,
H
]

- ix = [p,H]

for the Hamiltonian

H
H

and the multiplication operator

x
x

. The proof is carried out by generation of a unitary group in terms of ip, based on the Hille-Yosida theorem and Stone’s theorem. The result is applied to the self-adjoitness of

H
=
(

p
2

+

x
2

)

/

2

H = ({p^2} + {x^2})/2

.
Vector-valued Hausdorff summability methods and ergodic theorems
Authors: Yoshimoto Takeshi 
Published: June 2010

Suppose

X
X

and

Y
Y

are two general Banach spaces. Let

H
=
(

Λ

n
,
k

)

H = ({\Lambda _{n,k}})

be a general

B

[
X
,
Y
]

{\mathbf {B}}[X,Y]

-operator valued Hausdorff summability method:

Λ

n
,
k

=

(
k
n

)

Δ

n
−
k

U
k

{\Lambda _{n,k}} = (_k^n){\Delta ^{n - k}}{U_k}

for

k
≤
n

k \leq n

and

Λ

n
,
k

=

θ

X
,
Y

{\Lambda _{n,k}} = {\theta _{X,Y}}

for

k
>
n

k > n

, where

{

U
k

}

k
=
0

∞

\{ {U_k}\} _{k = 0}^\infty

is a sequence of operators in

B

[
X
,
Y
]

{\mathbf {B}}[X,Y]

and

Δ
\Delta

denotes the backward difference (operator) and

θ

X
,
Y

(
x
)
=

0
Y

{\theta _{X,Y}}(x) = {0_Y}

(the zero element in

Y
Y

) for all

x
∈

x \in

. Then some necessary and sufficient conditions are given for the mean and uniform convergence of the averages
\[

∑

k
=
0

n

(
k
n

)

Δ

n
−
k

U
k

(

T
k

x
)

(
x
∈
X
,
T
∈

B

[
X
]
)
.

\sum \limits _{k = 0}^n {(_k^n){\Delta ^{n - k}}{U_k}({T^k}x)} \quad (x \in X,T \in {\mathbf {B}}[X]).

\]
A uniqueness result for a semilinear reaction-diffusion system
Authors: Escobedo M. , Herrero M. A. 
Published: June 2010

Let

(
u
(
t
,
x
)
,
v
(
t
,
x
)
)

(u(t,x),v (t,x))

and

(

u
¯

(
t
,
x
)
,

v
¯

(
t
,
x
)
)

(\bar u(t,x),\bar v (t,x))

be two nonnegative classical solutions of (S)
\[

{

u
t

=
Δ
u
+

v
p

,

a
m
p
;

p
>
0

v
t

=
Δ
v
+

u
q

,

a
m
p
;

q
>
0

\left \{ {\begin {array}{*{20}{c}} {{u_t} = \Delta u + {v ^p},} & {p > 0} \\ {{v _t} = \Delta v + {u^q},} & {q > 0} \\ \end {array} } \right .

\]
in some strip

S
T

=
(
0
,
T
)
×

R

N

{S_T} = (0,T) \times {\mathbb {R}^N}

, where

0
>
T
≤
∞

0 > T \leq \infty

, and suppose that
\[

u
(
0
,
x
)
=

u
¯

(
0
,
x
)
,

v
(
0
,
x
)
=

v
¯

(
0
,
x
)
,

u(0,x) = \bar u(0,x),\quad v (0,x) = \bar v (0,x),

\]
where

u
(
0
,
x
)

u(0,x)

and

v
(
0
,
x
)

v (0,x)

are continuous, nonnegative, and bounded real functions, one of which is not identically zero. Then one has
\[

u
(
t
,
x
)
=

u
¯

(
t
,
x
)
,

v
(
t
,
x
)
=

v
¯

(
t
,
x
)

in

S
T

.

u(t,x) = \bar u(t,x),\quad v (t,x) = \bar v (t,x)\quad {\text {in}}\;{S_T}.

\]
If

p
q
≥
1

pq \geq 1

, the result is also true if

u
(
0
,
x
)
=
v
(
0
,
x
)
=
0

u(0,x) = v (0,x) = 0

. On the other hand, when

0
>
p
q
>
1

0 > pq > 1

, the set of solutions of (S) with zero initial values is given by
\[

u
(
t
;
s
)
=

c
1

(
t
−
s

)
+

(
p
+
1
)

/

(
1
−
p
q
)

,

v
(
t
;
s
)
=

c
2

(
t
−
s

)
+

(
q
+
1
)

/

(
1
−
p
q
)

,

u(t;s) = {c_1}(t - s)_ + ^{(p + 1)/(1 - pq)},\quad v (t;s) = {c_2}(t - s)_ + ^{(q + 1)/(1 - pq)},

\]
where

0
≤
s
≤
t
,

c
1

0 \leq s \leq t,{c_1}

and

c
2

{c_2}

are two positive constants depending only on

p
p

and

q
q

, and

(
ξ

)
+

=
max
{
ξ
,
0
}

(\xi )_+ = \max \{ \xi , 0 \}

.
Blow-up of 𝐻¹ solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity
Authors: Ogawa Takayoshi , Tsutsumi Yoshio 
Published: June 2010

We investigate the blow-up of solutions in

H
1

(

R

)

{H^1}({\mathbf {R}})

with negative energy for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity:
\[

i

u
t

=
−

u

x
x

−

|

u

|

4

u
,

t
>
0
,
x
∈

R

,

u
(
0
,
x
)
=

u
0

(
x
)
,

x
∈

R

.

.

\begin {gathered} i{u_t} = - {u_{xx}} - |u{|^4}u,\quad t > 0,x \in {\mathbf {R}}, \hfill \\ u(0,x) = {u_0}(x),\quad x \in {\mathbf {R}}. \hfill \\ \end {gathered} .

\]
In our result we remove the weight condition

x

u
0

∈

L
2

(

R

)

x{u_0} \in {L^2}({\mathbf {R}})

, which was always assumed to show the blow-up of solutions in the previous papers.
Vietoris-Begle theorem and spectra
Authors: Dydak Jerzy , Kozlowski George 
Published: June 2010

The following generalization of the Vietoris-Begle Theorem is proved: Suppose

{

E
k

}

k
≥
1

{\left \{ {{E_k}} \right \}_{k \geq 1}}

is a CW spectrum and

f
:

X
′

→
X

f:X’ \to X

is a closed surjective map of paracompact Hausdorff spaces such that

Ind
⁡
X
=
m
>
∞

\operatorname {Ind} X = m > \infty

. If

f
∗

:

E
k

(
x
)
→

E
k

(

f

−
1

(
x
)
)

{f^*}:{E^k}(x) \to {E^k}({f^{ - 1}}(x))

is an isomorphism for all

x
∈
X

x \in X

and

k
=

m
0

,

m
0

+
1
,
…
,

m
0

+
m

k = {m_0},{m_0} + 1, \ldots ,{m_0} + m

, then

f
∗

:

E
n

(
X
)
→

E
n

(

X
′

)

{f^*}:{E^n}(X) \to {E^n}(X’)

is an isomorphism and

f
∗

:

E

n
+
1

(
X
)
→

E

n
+
1

(

X
′

)

{f^*}:{E^{n + 1}}(X) \to {E^{n + 1}}(X’)

is a monomorphism for

n
=

m
0

+
m

n = {m_0} + m

.
On the duals of Lebesgue-Bochner 𝐿^{𝑝} spaces
Authors: Cengiz Bahattin 
Published: June 2010

Let

(
X
,

A

,
μ
)

(X,\mathcal {A},\mu )

be an arbitrary positive measure space. We prove that there exist an extremally disconnected (locally) compact Hausdorff space

Y
Y

and a perfect (regular) Borel measure

ν
\nu

on

Y
Y

such that

L
p

(
μ
,

E

)
≃

L
p

(
ν
,
E
)

{L^p}(\mu ,\textrm {E}) \simeq {L^p}(\nu ,E)

for all

1
≤
p
>
∞

1 \leq p > \infty

and any Banach space

E
E

. If

E
∗

{E^*}

is separable, then

L
p

(
μ
,

E

)
∗
≃

L
q

(
μ
,

E

∗

)

{L^p}(\mu ,\textrm {E})* \simeq {L^q}(\mu ,{\textrm {E}^*})

for all

1
>
p
>
∞
,

1
p

+

1
q

=
1

1 > p > \infty ,\;\frac {1}{p} + \frac {1}{q} = 1

, and

L
1

(
μ
,

E

)
∗
≃

L
∞

(
ν
,

E

∗

)
≃
C
(
β
Y
,

E

∗
∗

)

{L^1}(\mu ,\textrm {E})* \simeq {L^\infty }(\nu ,{\textrm {E}^*}) \simeq C(\beta Y,\textrm {E}_*^*)

, where

E
∗
∗

E_*^*

denotes

E
∗

{E^*}

endowed with the weak* topology. In particular

L
1

(
μ

)
∗

≃

L
∞

(
ν
)

{L^1}{(\mu )^*} \simeq {L^\infty }(\nu )

.
Periodic potentials with minimal energy bands
Authors: Ashbaugh Mark S. , Svirsky Roman 
Published: June 2010

We consider the problem of minimizing the width of the lowest band in the spectrum of Hill’s equation,

−
u
+
q

(
x
)

u
=
λ
u

- u + q\left ( x \right )u = \lambda u

on

R

\mathbb {R}

with

q

(

x
+
1

)

=
q

(
x
)

q\left ( {x + 1} \right ) = q\left ( x \right )

for all

x
∈

R

x \in \mathbb {R}

, when the potential function

q
q

is allowed to vary over a ball of radius

M
>
0

in

L
∞

M > 0{\text { in }}{L^\infty }

. We show that minimizing potentials

q
∗

{q_ * }

exist and that, when considered as functions on the circle, they must have exactly one well on which

q
∗

(
x
)

{q_ * }\left ( x \right )

must equal

−
M

- M

and one barrier on which

q
∗

(
x
)

{q_ * }\left ( x \right )

must equal

M
M

; these are the only values that

q
∗

{q_ * }

can assume (up to changes on sets of measure zero). That is, on the circle there is a single interval where

q
∗

(
x
)

=
M

{q_ * }\left ( x \right ) = M

and on the complementary interval

q
∗

(
x
)

=
−
M

{q_ * }\left ( x \right ) = - M

. These results can be used to solve the problem of minimizing the gap between the lowest Neumann eigenvalue and either the lowest Dirichlet eigenvalue or the second Neumann eigenvalue for the same equation restricted to the interval

[
0
,
1
]

[0,1]

.
Products of Poincaré domains
Authors: Stanoyevitch Alexander 
Published: June 2010

A domain

Ω
⊆

R

N

\Omega \subseteq {\mathbb {R}^N}

of finite

N
N

-dimensional Lebesgue measure is a

p
p

-Poincaré domain

(
1
⩽
p
⩽
∞
)

(1 \leqslant p \leqslant \infty )

if there exists a positive constant

K
K

such that the

p
p

-Poincaré inequality

|

|

u

|

|

L
p

(
Ω
)

⩽
K

|

|

∇
u

|

|

L
p

(
Ω
)

||u|{|_{{L^p}(\Omega )}} \leqslant K||\nabla u|{|_{{L^p}(\Omega )}}

is valid for all Sobolev functions

u
∈

W

1
,
p

(
Ω
)

u \in {W^{1,p}}(\Omega )

that integrate to zero. Define

K
p

(
Ω
)

{K_p}(\Omega )

to be the smallest such

K
K

if

Ω
\Omega

is a

p
p

-Poincaré domain and to be infinity otherwise. We obtain comparability relations between

K
p

(

Ω
1

×

Ω
2

)

{K_p}({\Omega _1} \times {\Omega _2})

and the pair

K
p

(

Ω
1

)
,

K
p

(

Ω
2

)

{K_p}({\Omega _1}),\;{K_p}({\Omega _2})

. In particular, our results show that

p
p

-Poincaré domains are closed under cartesian products (for all

p
p

), and that in case

p
p

equals

2
2

, we have

K
2

(

Ω
1

×

Ω
2

)
=
max
{

K
2

(

Ω
1

)
,

K
2

(

Ω
2

)
}

{K_2}({\Omega _1} \times {\Omega _2}) = \max \{ {K_2}({\Omega _1}),\;{K_2}({\Omega _2})\}

.
Rational elasticity of factorizations in Krull domains
Authors: Anderson D. D. , Anderson David F. , Chapman Scott T. , Smith William W. 
Published: June 2010

For an atomic domain

R
R

, we define the elasticity of

R
R

as

ρ
(
R
)
=
sup
(
m

/

n

|

x
1

⋯

x
m

=

y
1

⋯

y
n

,

for

x
i

,

y
j

∈
R

irreducibles}

\rho (R) = \sup (m/n|{x_1} \cdots {x_m} = {y_1} \cdots {y_n},\;{\text {for}}\;{x_i},{y_j} \in R\;{\text {irreducibles\} }}

and let

l
R

(
x
)

{l_R}(x)

and

L
R

(
x
)

{L_R}(x)

denote, respectively, the inf and sup of the lengths of factorizations of a nonzero nonunit

x
∈
R

x \in R

into the product of irreducible elements. We answer affirmatively two rationality conjectures about factorizations. First, we show that

ρ
(
R
)

\rho (R)

is rational when

R
R

is a Krull domain with finite divisor class group. Secondly, we show that when

R
R

is a Krull domain, the two limits

l
R

(

x
n

)

/

n

{l_R}({x^n})/n

and

L
R

(

x
n

)

/

n

{L_R}({x^n})/n

, as

n
n

goes to infinity, are positive rational numbers. These answer, respectively, conjectures of D. D. Anderson and D. F. Anderson, and D. F. Anderson and P. Pruis. (The second question has also been solved by A. Geroldinger and F. Halter-Koch.)
Flat core properties associated to the 𝑝-Laplace operator
Authors: Kamin Shoshana , Véron Laurent 
Published: June 2010

We study the formation of a flat hat pattern in the profile of the positive solution of an equation of the type:

ε

Δ
p

u
−

u

p
−
1

(
1
−
u

)
θ

=
0

(
0
>
θ
>
p
−
1
)

\varepsilon {\Delta _p}u - {u^{p - 1}}{(1 - u)^\theta } = 0\;(0 > \theta > p - 1)

in a bounded domain

Ω
\Omega

. When

ε
\varepsilon

tends to

0
+

{0^ + }

, the growth of the zone where

u
=

u
ε

u = {u_\varepsilon }

takes the value

1
1

in

Ω
\Omega

is studied.
Polar decompositions of bounded linear functionals on operator subalgebras
Authors: Kusuda Masaharu 
Published: June 2010

Let

M
M

be a von Neumann algebra and let

φ
\varphi

be a normal linear functional on a strongly closed

C

∗

{C^{\ast }}

-subalgebra

N
N

of

M
M

. Denote by

F

φ

{\mathcal {F}_\varphi }

the set of normal linear functionals

ψ
\psi

on

M
M

extending

φ
\varphi

with

|

|

ψ

|

|

=

|

|

φ

|

|

||\psi || = ||\varphi ||

. It is shown that there exists a partial isometry

v
v

in

N
N

such that
\[

φ
=

|

φ

|

(
v
⋅
)
,

|

φ

|

=
φ
(

v

∗

⋅
)
,

|

|

φ

|

|

=

|

|

|

φ

|

|

|

\varphi = |\varphi |(v \cdot ),\qquad |\varphi | = \varphi ({v^{\ast }} \cdot ),\qquad ||\varphi || = |||\varphi |||

\]
and
\[

ψ
=

|

ψ

|

(
v
⋅
)
,

|

ψ

|

=
ψ
(

v

∗

⋅
)
,

|

|

ψ

|

|

=

|

|

|

ψ

|

|

|

\psi = |\psi |(v\cdot ),\qquad |\psi | = \psi ({v^{\ast }}\cdot ),\qquad ||\psi || = |||\psi |||

\]
for all

ψ
\psi

in

F

φ

{\mathcal {F}_\varphi }

, where

|

φ

|

|\varphi |

and

|

ψ

|

|\psi |

denote the absolute values of

φ
\varphi

and

ψ
\psi

respectively. Let

A
A

be a

C

∗

{C^{\ast }}

-algebra and let

B
B

be a

C

∗

{C^{\ast }}

-subalgebra of

A
A

. As a consequence of this result, we obtain that every state on

B
B

has a unique state extension to

A
A

if and only if every bounded linear functional on

B
B

has a unique norm-preserving extension to a bounded linear functional on

A
A

.
On generalized maximal functions
Authors: Schröder Bernd S. W. 
Published: June 2010

In this paper we study the question of under what circumstances the quantity

|

|

sup

t
>
∞
,

a
∈

R

|

∫
0
t

f
(
a
,

M
s

)

d

M
s

|

|

|

p

||{\sup _{t > \infty ,\,a \in \mathbb {R}}}|\int _0^t f (a,{M_s})\,d{M_s}|\;|{|_p}

is comparable to

|

|

M
∞

∗

|

|

p

||M_\infty ^{\ast }|{|_p}

, where

M
t

{M_t}

is a continuous martingale and

f
f

is a bounded Borel-measurable function.
Sobolev and quasiconformal extension domains
Authors: Ghamsari Manouchehr 
Published: June 2010

A domain

D
⊂

R

n

D \subset {\mathbb {R}^n}

has the quasiconformal extension property if each quasiconformal self-map of

D
D

extends to a quasiconformal self-map of

R

n

;

D

{\mathbb {R}^n};\;D

has the Sobolev extension property if there is a bounded linear operator

Λ
:

W

1
,
n

(
D
)
→

W

1
,
n

(

R

n

)

\Lambda :{W^{1,n}}(D) \to {W^{1,n}}({\mathbb {R}^n})

. We consider the relation between the above extension properties for

n
⩾
3

n \geqslant 3

. We show that for domains quasiconformally equivalent to a ball the quasiconformal extension property implies the Sobolev extension property. We show that this is not true in general. Next the Sobolev extension property does not imply the extension property for quasiconformal maps which extend as homeomorphisms. Finally if

G
⊂

R

2

G \subset {\mathbb {R}^2}

and if

D
=
G
×

R

D = G \times \mathbb {R}

is quasiconformally equivalent to a ball, then

D
D

has the quasiconformal extension property if and only if

D
D

is a quasiball.
Associative and Jordan shift algebras
Authors: Loos Ottmar , Neher Erhard 
Published: June 2010

Let

R
R

be the shift algebra, i.e., the associative algebra presented by generators

u
,
v

u,v

and the relation

u
v
=
1

uv = 1

. As N. Jacobson showed,

R
R

contains an infinite family of matrix units. In this paper, we describe the Jordan algebra

R
+

{R^ + }

and its unital special universal envelope by generators and relations. Moreover, we give a presentation for the Jordan triple system defined on

R
R

by

P
x

y
=
x

y

∗

x

{P_x}y = x{y^{\ast }}x

where

∗

^{\ast }

is the involution on

R
R

with

u

∗

=
v

{u^{\ast }} = v

. As a consequence, we prove the existence of an infinite rectangular grid in a Jordan triple system

V
V

containing tripotents

c
c

and

d
d

with

V
2

(
c
)
=

V
2

(
d
)
⊕
(

V
2

(
c
)
∩

V
1

(
d
)
)

{V_2}(c) = {V_2}(d) \oplus ({V_2}(c) \cap {V_1}(d))

and

V
2

(
c
)
∩

V
1

(
d
)
≠
0

{V_2}(c) \cap {V_1}(d) \ne 0

.
A suspension theorem for continuous trace 𝐶*-algebras
Authors: Dădărlat Marius 
Published: June 2010

Let

B

\mathcal {B}

be a stable continuous trace

C

∗

{C^{\ast }}

-algebra with spectrum

Y
Y

. We prove that the natural suspension map

S

∗

:
[

C
0

(
X
)
,

B

]
→
[

C
0

(
X
)
⊗

C
0

(

R

)
,

B

⊗

C
0

(

R

)
]

{S_{\ast }}:[{C_0}(X),\mathcal {B}] \to [{C_0}(X) \otimes {C_0}({\mathbf {R}}),\mathcal {B} \otimes {C_0}({\mathbf {R}})]

is a bijection, provided that both

X
X

and

Y
Y

are locally compact connected spaces whose one-point compactifications have the homotopy type of a finite CW-complex and

X
X

is noncompact. This is used to compute the second homotopy group of

B

\mathcal {B}

in terms of

K
K

-theory. That is,

[

C
0

(

R

2

)
,

B

]
=

K
0

(

B

0

)

[{C_0}({{\mathbf {R}}^2}),\mathcal {B}] = {K_0}({\mathcal {B}_0})

, where

B

0

{\mathcal {B}_0}

is a maximal ideal of

B

\mathcal {B}

if

Y
Y

is compact, and

B

0

=

B

{\mathcal {B}_0} = \mathcal {B}

if

Y
Y

is noncompact.
An operator-valued Yeh-Wiener integral and a Kac-Feynman Wiener integral equation
Authors: Park Chull , Skoug David 
Published: June 2010

Let

C
[
0
,
T
]

C[0,T]

denote Wiener space, i.e., the space of all continuous functions

η
(
t
)

\eta (t)

on

[
0
,
T
]

[0,T]

such that

η
(
0
)
=
0

\eta (0) = 0

. For

Q
=
[
0
,
S
]
×
[
0
,
T
]

Q = [0,S] \times [0,T]

let

C
(
Q
)

C(Q)

denote Yeh-Wiener space, i.e., the space of all

R

\mathbb {R}

-valued continuous functions

x
(
s
,
t
)

x(s,t)

on

Q
Q

such that

x
(
0
,
t
)
=
x
(
s
,
0
)
=
0

x(0,t) = x(s,0) = 0

for all

(
s
,
t
)

(s,t)

in

Q
Q

. For

h
∈

L
2

(
Q
)

h \in {L_2}(Q)

let

Z
(
x
;
s
,
t
)

Z(x;s,t)

be the Gaussian process defined by the stochastic integral
\[

Z
(
x
;
s
,
t
)
=

∫
0
t

∫
0
s

h
(
u
,
v
)
d
x
(
u
,
v
)
.

Z(x;s,t) = \int _0^t {\int _0^s {h(u,v)dx(u,v).} }

\]
Then for appropriate functionals

F
F

and

ψ
\psi

we show that the operator-valued function space integral
\[

(

I
λ
h

(
F
)
ψ
)
(
η
(
⋅
)
)
=

E
x

[
F
(

λ

−
1

/

2

Z
(
x
;
⋅
,
⋅
)
+
η
(
⋅
)
)
ψ
(

λ

−
1

/

2

Z
(
x
;
S
,
⋅
)
+
η
(
⋅
)
)
]

(I_\lambda ^h(F)\psi )(\eta ( \cdot )) = {E_x}[F({\lambda ^{ - 1/2}}Z(x; \cdot , \cdot ) + \eta ( \cdot ))\psi ({\lambda ^{ - 1/2}}Z(x;S, \cdot ) + \eta ( \cdot ))]

\]
is the unique solution of a Kac-Feynman Wiener integral equation. We also use this integral equation to evaluate various Yeh-Wiener integrals.
Upper bounds for the derivative of exponential sums
Authors: Borwein Peter , Erdélyi Tamás 
Published: June 2010

The equality
\[

sup
p

|

p
′

(
a
)

|

‖
p
‖

[
a
,
b
]

=

2

n
2

b
−
a

\sup \limits _p \frac {{|p’(a)|}}{{{{\left \| p \right \|}_{[a,b]}}}} = \frac {{2{n^2}}}{{b - a}}

\]
is shown, where the supremum is taken for all exponential sums p of the form
\[

p
(
t
)
=

a
0

+

∑

j
=
1

n

a
j

e

λ
j

t

,

a
j

∈

R

,

p(t) = {a_0} + \sum \limits _{j = 1}^n {{a_j}{e^{{\lambda _j}t}},\quad {a_j} \in {\mathbf {R}},}

\]
with nonnegative exponents

λ
j

{\lambda _j}

. The inequalities
\[

‖

p
′

‖

[
a
+
δ
,
b
−
δ
]

≤
4

(
n
+
2

)
3

δ

−
1

‖
p
‖

[
a
,
b
]

{\left \| {p’} \right \|_{[a + \delta ,b - \delta ]}} \leq 4{(n + 2)^3}{\delta ^{ - 1}}{\left \| p \right \|_{[a,b]}}

\]
and
\[

‖

p
′

‖

[
a
+
δ
,
b
−
δ
]

≤
4

2

(
n
+
2

)
3

δ

−
3

/

2

‖
p
‖

L
2

[
a
,
b
]

{\left \| {p’} \right \|_{[a + \delta ,b - \delta ]}} \leq 4\sqrt 2 {(n + 2)^3}{\delta ^{ - 3/2}}{\left \| p \right \|_{{L_2}[a,b]}}

\]
are also proved for all exponential sums of the above form with arbitrary real exponents. These results improve inequalities of Lorentz and Schmidt and partially answer a question of Lorentz.
A simple proof of a remarkable continued fraction identity
Authors: Anderson P. G. , Brown T. C. , Shiue P. J.-S. 
Published: June 2010

We give a simple proof of a generalization of the equality
\[

∑

n
=
1

∞

1

2

[
n

/

τ
]

=
[
0
,

2
0

,

2
1

,

2
1

,

2
2

,

2
3

,

2
5

,
…
]
,

\sum \limits _{n = 1}^\infty {\frac {1}{{{2^{[n/\tau ]}}}} = [0,{2^0},{2^1},{2^1},{2^2},{2^3},{2^5}, \ldots ],}

\]
where

τ
=
(
1
+

5

)

/

2

\tau = (1 + \sqrt 5 )/2

and the exponents of the partial quotients are the Fibonacci numbers, and some closely related results.