bluuebunny/crossref_metadata_2025_split huggingface dataset:
Estimates of positive linear operators on 𝐿^{𝑝}
Authors: Assani I. 
Published: June 2010

Let

0
>
α
>
1

0 > \alpha > 1

and

T
T

a positive linear operator on

L
p

,
1
>
p
>
+
∞

{L^p},1 > p > + \infty

, such that

|

|

(
1
−
α
)
I
+
α
T

|

|

p

≤
1

{|| {(1 - \alpha )I + \alpha T} ||_p} \leq 1

(

I
I

= identity). For such operators, (which do not necessarily satisfy (i)

|

|

T

|

|

p

≤
1

{|| T ||_p} \leq 1

(contraction), (ii)

sup

n
≥
1

‖

(
I
+
T
+
⋯
+

T

n
−
1

)

/

n

‖

p

≤
1

{\sup _{n \geq 1}}{\left \| {(I + T + \cdots + {T^{n - 1}})/n} \right \|_p} \leq 1

(Cesàro mean bounded by one)) [1] we show, using M. A. Akcoglu’s estimate, that
\[

‖

sup

n
≥
1

n
∈
N

f
+
T
f
+
⋯
+

T

n
−
1

f

n

‖

p

≤
γ
(
α
)

|

|

f

|

|

p

for any

f
∈

L
p

.

{\left \| {\sup \limits _{\begin {array}{*{20}{c}} {n \geq 1} \\ {n \in N} \\ \end {array} } \frac {{f + Tf + \cdots + {T^{n - 1}}f}}{n}} \right \|_p} \leq \gamma (\alpha )||f|{|_p}\;{\text {for any }}f \in {L^p}.

\]
We also obtain the pointwise ergodic theorem in

L
p

{L^p}

.
Toeplitz spectral inclusion and generalized in modulus property
Authors: Janas J. 
Published: June 2010

Let

A
⊂

C
b

(
X
)

A \subset {C_b}(X)

be a function algebra and let

μ
\mu

be a Borel measure on

X
X

and

H
2

(
μ
)
=

A
¯

{H^2}(\mu ) = \bar A

—the closure in

L
2

(
μ
)

{L^2}(\mu )

. It turns out that the spectral inclusion theorem for Toeplitz operators, defined in the above context, implies the density of finite sums

∑
i

|

u
i

|

2

,

u
i

∈
A

\sum \nolimits _i {{{\left | {{u_i}} \right |}^2},{u_i} \in A}

, in (i) the cone of positive functions in

L
1

(
μ
)

{L^1}(\mu )

, (ii) the cone of positive functions in

L
p

(
μ
)
,
p
≤
2

{L^p}(\mu ),p \leq 2

, if

μ
(
X
)

>

∞

\mu (X){\text { > }}\infty

, (iii) the cone of positive functions in

C
(
X
)

C(X)

, if

X
X

is compact.
Almost Jordan rings
Authors: Hentzel Irvin Roy , Peresi Luiz Antonio 
Published: June 2010

It is well known that any Jordan ring satisfies the identity:

2
(
(
a
x
)
x
)
x
+
a
(
(
x
x
)
x
)
=
3
(
a
(
x
x
)
)
x

2((ax)x)x + a((xx)x) = 3(a(xx))x

. We show that this identity along with commutativity implies the Jordan identity in any semiprime ring. The proof requires characteistic

≠
2
,
3

\ne 2,3

.
The conjugate property for Diophantine approximation of continued fractions
Authors: Tong Jing Cheng 
Published: June 2010

Let

ξ
\xi

be an irrational number with simple continued fraction expansion

ξ
=
[

a
0

;

a
1

,
…
,

a
i

,
…
]

\xi = [{a_0};{a_1}, \ldots ,{a_i}, \ldots ]

, and

p
i

/

q
i

{p_i}/{q_i}

be its

i
i

th convergent. In this paper we first prove the duality of some inequalities, and then prove the following conjugate properties for symmetric and asymmetric Diophantine approximations. (i) Among any three consecutive convergents

p
i

/

q
i

(
i
=
n
−
1
,
n
,
n
+
1
)

{p_i}/{q_i}(i = n - 1,n,n + 1)

, at least one satisfies
\[

ξ
−

p
i

/

q
i

|

>
1

/

(

a

n
+
1

2

+
4

q
i
2

)

,

\xi - {p_i}/{q_i}| > 1/\left ( {\sqrt {a_{^{n + 1}}^2 + 4q_i^2} } \right ),

\]
and at least one does not satisfy this inequality. (ii) Let

τ
\tau

be a positive real number. Among any four consecutive convergents

p
i

/

q
i

(
i
=
n
−
1
,
n
,
n
+
1
,
n
+
2
)

{p_i}/{q_i}(i = n - 1,n,n + 1,n + 2)

, at least one satisfies
\[

−
1

/

(

c

n

2

+
4
τ

q
i
2

)

>
ξ
−

p
i

/

q
i

>
τ

/

(

c
n
2

+
4
τ

q
i
2

)

,

- 1/\left ( {\sqrt {c_{^n}^2 + 4\tau q_i^2} } \right ) > \xi - {p_i}/{q_i} > \tau /\left ( {\sqrt {c_n^2 + 4\tau q_i^2} } \right ),

\]
and at least one does not satisfy this inequality, where

c
n

=

a

n
+
1

{c_n} = {a_{n + 1}}

if

n
n

is odd,

c
n

=

a

n
+
2

{c_n} = {a_{n + 2}}

if

n
n

is even.
Finitely nonreflexive Banach spaces
Authors: Wójtowicz Marek 
Published: June 2010

A Banach space

X
X

is said to be finitely nonreflexive if for some

n
≥
1

n \geq 1

the space

H
n

(
X
)

{H_n}(X)

is reflexive with

dim
⁡

H
n

(
X
)
≥
1

\dim {H_n}(X) \geq 1

, where

H
0

(
X
)
=
X

{H_0}(X) = X

, and

H

n
+
1

(
X
)
=

H
n

(
X

)

∗
∗

/

H
n

(
X
)

{H_{n + 1}}(X) = {H_n}{(X)^{**}}/{H_n}(X)

. A simple construction of (pairwise nonisomorphic) finitely nonreflexive separable Banach spaces which are both nonisomorphic to their Cartesian squares and isomorphic to their biduals is presented.
𝐿²-boundedness of spherical maximal operators with multidimensional parameter sets
Authors: Ha Young-Hwa 
Published: June 2010

For

s
>
0

s > 0

, let

M
s

f
(
x
)
=

∫

|

y

|

=
1

f
(
x
−
s
y
)
d
σ
(
y
)

{M_s}f(x) = \int _{|y| = 1} {f(x - sy)d\sigma (y)}

be the spherical mean operator on

R
n

{R^n}

. For a certain class of surfaces

S
S

in

R
+

n
+
1

R_ + ^{n + 1}

with

dim
⁡
S
=
n
−
2

\dim S = n - 2

or

dim
⁡
S
=
n
−
1

\dim S = n - 1

with an additional condition, the maximal operator
\[

M

f
(
x
)
=

sup

(
u
,
s
)
∈
S

|

M
s

f
(
x
−
u
)

|

\mathcal {M}f(x) = \sup \limits _{(u,s) \in S} |{M_s}f(x - u)|

\]
is shown to be bounded on

L
2

(

R
n

)

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

. This extends (on

L
2

(

R
n

)

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

) the theorem of Stein [7], where

S
=
{
(
0
,
s
)
:
s
>
0
}

S = \{ (0,s):s > 0\}

, and its generalizations to

dim
⁡
S
=
1

\dim S = 1

in Greenleaf [2] and Sogge and Stein [6].
On the homotopy type of the spectrum representing elliptic cohomology
Authors: Baker Andrew 
Published: June 2010

In this paper we analyse the homotopy type at primes

p
>
3

p > 3

of the ring spectrum

E
ℓ
ℓ

E\ell \ell

representing a version of elliptic cohomology whose coefficient ring

E
ℓ

ℓ
∗

E\ell {\ell _ * }

agrees with the ring of modular forms for

S

L
2

(

Z

)

S{L_2}(\mathbb {Z})

. For any prime (=maximal) graded ideal

P

◃
E
ℓ

ℓ
∗

\mathcal {P} \triangleleft E\ell {\ell _*}

containing the Eisenstein function

E

p
−
1

{E_{p - 1}}

as well as

p
p

, we show that there is a morphism of ring spectra
\[

E
(
2
)

^

→
(
E
ℓ
ℓ

)

P

^

\widehat {E(2)} \to (E\ell \ell )_{\hat {\mathcal {P}}}

\]
and a corresponding splitting
\[

(
E
ℓ
ℓ

)

P

^

≃

⋁
i

Σ

2
θ
(
i
)

E
(
2
)

^

(E\ell \ell )_{\hat {\mathcal {P}}} \simeq \bigvee \limits _i {\Sigma ^{2\theta (i)}}\widehat {E(2)}

\]
of algebra spectra over

E
(
2
)

^

\widehat {E(2)}

(the

I
2

{I_2}

-adic completion of

E
(
2
)

E(2)

); here

(

)

P

^

(\;)_{\hat {\mathcal {P}}}

denotes the

P

\mathcal {P}

-adic completion of the spectrum

E
ℓ
ℓ

E\ell \ell

. Moreover, there is a multiplicative reduction

(
E
ℓ
ℓ

/

P

)
∗

(

)

{(E\ell \ell /\mathcal {P})^ * }(\;)

and we similarly show that there is a splitting of

K
(
2
)

K(2)

algebra spectra
\[

E
ℓ
ℓ

/

P

≃

⋁
i

Σ

2

θ
′

(
i
)

K
(
2
)
.

E\ell \ell /\mathcal {P} \simeq \bigvee \limits _i {\Sigma ^{2\theta ’(i)}}K(2).

\]
In each case the indexing

i
i

ranges over a finite set.
Index of faithful normal conditional expectations
Authors: Tsui Sze-Kai 
Published: June 2010

Let

E
E

be a faithful normal conditional expectation of a factor

M
M

onto its subfactor

N
N

, and the index of

E
E

be denoted by

IND
E

{\operatorname {IND}_E}

. We investigate the question: For two such faithful normal conditional expectations

E
1

,

E
2

{E_1},{E_2}

of

M
M

onto

N
N

, when does

IND

E
1

=

IND

E
2

{\operatorname {IND}}_{{E_1}} = {\operatorname {IND}}_{{E_2}}

hold? In this paper we answer this question completely for type

I
I

factor

M
M

. We also derive a tensor product formula for index, i.e.,

IND

E
1

⊗

E
2

=
(

IND

E
1

)
(

IND

E
2

)

{\operatorname {IND}}_{{E_1} \otimes {E_2}} = ({\operatorname {IND}}_{{E_1}})({\operatorname {IND}}_{{E_2}})

. For any

α
>
9

\alpha > 9

we construct uncountable nonisomorphic faithful normal conditional expectations

E
E

of a factor

M
M

onto its subfactor

N
N

with

IND
E

=
α

{\operatorname {IND}_E} = \alpha

such that both of

M
M

and

N
N

, are of type

I
I

or

I
I

II

or

I
I

I
λ

,
0
≤
λ
≤
1

II{I_\lambda },0 \leq \lambda \leq 1

. For each

β
∈
{
4

cos
2

π

/

n
,

|

n
≥
3
}
∪
[
4
,
∞
)

\beta \in \{ 4{\cos ^2}\pi /n,|n \geq 3\} \cup [4,\infty )

we exhibit a type

I
I

I
λ

II{I_\lambda }

factor

M
M

and its subfactor

N
N

and a faithful normal conditional expectation

E
E

such that

IND

E

=
β

{\operatorname {IND}}_E = \beta

.
On the clustering conjecture for Bernoulli factors of Bernoulli shifts
Authors: Keller G. 
Published: June 2010

We give sufficient algebraic conditions on the probabilities

p
i

{p_i}

of a Bernoulli shift

B
(

p

)

=

B
(

p
1

,
…
,

p
M

)

B({\mathbf {p}}){\text { = }}B({p_1}, \ldots ,{p_M})

which imply that if

B
(

q

)
=
B
(

q
1

,
…
,

q
N

)

B({\mathbf {q}}) = B({q_1}, \ldots ,{q_N})

is a continuous factor of

B
(

p

)

B({\mathbf {p}})

, then

q

{\mathbf {q}}

is a clustering of

p

{\mathbf {p}}

.
The Weyl transform and 𝐿^{𝑝} functions on phase space
Authors: Simon Barry 
Published: June 2010

This is primarily a negative paper showing that a bound of the form

|

|

W
(
f
)

|

|

operator norm

≤
c

|

|

f

|

|

p

||W(f)||{\text {operator norm}} \leq c||f|{|_p}

fails for the Weyl transform if

p
>
2

p > 2

.

L
p

{L^p}

properties of Wigner distribution functions are discussed as well as Cwikel’s theorem.
A nonsolvable complex vector field with Hölder coefficients
Authors: Jacobowitz Howard 
Published: June 2010

It is known that the equation
\[

∂
u

∂
t

−
α
(
ξ
,
t
)

∂
u

∂
ξ

=
f
(
ξ
,
τ
)

\frac {{\partial u}}{{\partial t}} - \alpha (\xi ,t)\frac {{\partial u}}{{\partial \xi }} = f(\xi ,\tau )

\]
is solvable in a neighborhood of the origin provided Im

α
\alpha

does not change sign and

α
\alpha

is at least Lipschitz smooth. An example is given where solvability fails although

α
\alpha

is of Hölder class

λ
\lambda

for all

0
>
λ
>
1

0 > \lambda > 1

. Further, the only solutions to
\[

∂
u

∂
t

−
α
(
ξ
,
t
)

∂
u

∂
ξ

=
0

\frac {{\partial u}}{{\partial t}} - \alpha (\xi ,t)\frac {{\partial u}}{{\partial \xi }} = 0

\]
are the constant functions.
Topological types of quasi-ordinary singularities
Authors: Oh Kyungho 
Published: June 2010

A germ

(
X
,
x
)

(X,x)

of a complex analytic hypersurface in

C

d
+
1

{\mathbb {C}^{d + 1}}

is quasi-ordinary if it can be represented as the image of an open neighborhood of

0
0

in

C

d

{\mathbb {C}^d}

under the map

(

s
1

,
…
,

s
d

)
↦
(

s
1
n

,
…
,

s
d
n

,
ζ
(

s
1

,
…
,

s
d

)
)
,

n
>
0

({s_1}, \ldots ,{s_d}) \mapsto (s_1^n, \ldots ,s_d^n,\zeta ({s_1}, \ldots ,{s_d})),\;n > 0

, where

ζ
\zeta

is a convergent power series. It is shown that the topological type of the singularity

(
X
,
x
)
⊂
(

C

d
+
1

,
0
)

(X,x) \subset ({\mathbb {C}^{d + 1}},0)

is determined by a certain set of fractional monomials, called the characteristic monomials, appearing in the fractional power series

ζ
(

t
1

1

/

n

,
…
,

t
d

1

/

n

)

\zeta (t_1^{1/n}, \ldots ,t_d^{1/n})

.
Near-rings of invariants. II
Authors: Maxson C. J. 
Published: June 2010

Let

E
E

be the equivalence relation determined by the orbits of a group

G
G

acting on a set

X
X

and let

K
K

be a group. Using a "sandwich" function

φ
:
X

/

E
→
K
,

M
(
X
,
G
,
K
,
φ
)
=
{
f
:
X
→
K

|

f

is constant on each

E

-class

}

\varphi :X/E \to K,\;M(X,G,K,\varphi ) = \{ f:X \to K|f\;{\text {is constant on each}}\;E{\text {-class}}\}

is a near-ring, the near-ring of invariants determined by

(
X
,
G
,
K
,
φ
)

(X,G,K,\varphi )

. In this paper we continue the study of the transfer of information between the structure of the near-ring of invariants and properties of the group action

(
G
,
X
)

(G,X)

, the group

K
K

, and the function

φ
:
K
→
X

/

E

\varphi :K \to X/E

.
An elementary simultaneous approximation theorem
Authors: Kilgore Theodore 
Published: June 2010

We will give an elementary and direct proof that for

f
∈

C
q

[
−
1
,
1
]

f \in {C^q}[ - 1,1]

there exists a sequence of polynomials

P
n

{P_n}

of degree at most

n

(
n
>
2
q
)

n\;(n > 2q)

such that for

k
=
0
,
…
,
q

k = 0, \ldots ,q

\[

|

f

(
k
)

(
x
)
−

P
n

(
k
)

(
x
)

|

⩽

M

q
,
k

(

1
−

x
2

n

)

q
−
k

E

n
−
q

(

f

(
q
)

)
,

|{f^{(k)}}(x) - P_n^{(k)}(x)| \leqslant {M_{q,k}}{\left ( {\frac {{\sqrt {1 - {x^2}} }} {n}} \right )^{q - k}}{E_{n - q}}({f^{(q)}}),

\]
with

M

q
,
k

{M_{q,k}}

depending only upon

q
q

and

k
k

. Moreover

f

(
q
)

(
±
1
)
=

P
n

(
q
)

(
±
1
)

{f^{(q)}}( \pm 1) = P_n^{(q)}( \pm 1)

.
Nonresonance conditions on the potential for a second-order periodic boundary value problem
Authors: Omari Pierpaolo , Zanolin Fabio 
Published: June 2010

We consider the periodic problem\[−u=f(u)+h(t),u(0)=u(2π),u′(0)=u′(2π),\begin {array}{*{20}{c}} { - u = f(u) + h(t),} \\ {u(0) = u(2\pi ),\qquad u’(0) = u’(2\pi ),} \\ \end {array}\]and prove its solvability for any givenhh, under new assumptions on the asymptotic behaviour of the potential of the nonlinearityff, with respect to two consecutive eigenvalues of the associated linear problem.
Embeddings and immersions of a 2-sphere in 4-manifolds
Authors: Gan Dan Yan , Guo Jian Han 
Published: June 2010

Let

M
M

be

C

P
2

#
(
−
C

P
2

)
#

P
1

#
⋯
#

P

m
+
k

C{P^2}\# ( - C{P^2})\# {P_1}\# \cdots \# {P_{m + k}}

, where

P
1

,
…
,

P

m
+
k

{P_1}, \ldots ,{P_{m + k}}

are copies of

−
C

P
2

- C{P^2}

. Let

h
,
g
,

g
1

,
…
,

g

m
+
k

h,g,{g_1}, \ldots ,{g_{m + k}}

be the images of the standard generators of

H
2

(
C

P
2

;
Z
)
,

H
2

(
−
C

P
2

;
Z
)
,

H
2

(

P
1

;
Z
)
,
…
,

H
2

(

P

m
+
k

;
Z
)

{H_2}(C{P^2};Z),\;{H_2}( - C{P^2};Z),\;{H_2}({P_1};Z), \ldots ,{H_2}({P_{m + k}};Z)

in

H
2

(
M
;
Z
)

{H_2}(M;Z)

respectively. Let

ξ
=
p
h
+
q
g
+

∑

i
=
1

m

r
i

g
i

\xi = ph + qg + \sum \nolimits _{i = 1}^m {{r_i}{g_i}}

be an element of

H
2

(
M
;
Z
)

{H_2}(M;Z)

. Suppose

ξ
2

=
l
>
0
,

p
2

−

q
2

⩾
8
,

|

p

|

−

|

q

|

⩾
2

{\xi ^2} = l > 0,\,{p^2} - {q^2} \geqslant 8,\,|p| - |q| \geqslant 2

, and

r
i

≠
0

{r_i} \ne 0

. If

2
(
m
+
l
−
2
)
⩾

p
2

−

q
2

2(m + l - 2) \geqslant {p^2} - {q^2}

, then

ξ
\xi

cannot be represented by a smoothly embedded

2
2

-sphere. If

2
(
m
+
r
+
[
(
l
−
r
−
1
)

/

4
]
−
1
)
⩾

p
2

−

q
2

2(m + r + [(l - r - 1)/4] - 1) \geqslant {p^2} - {q^2}

for some

r
r

with

0
⩽
r
⩽
l
−
1

0 \leqslant r \leqslant l - 1

, then for a normal immersion

f
f

of a

2
2

-sphere representing

ξ
\xi

the number of points of positive self-intersection

d
f

⩾
[
(
l
−
r
−
1
)

/

4
]
+
1

{d_f} \geqslant [(l - r - 1)/4] + 1

.
A transitivity theorem for algebras of elementary operators
Authors: Magajna Bojan 
Published: June 2010

Let

A

\mathcal {A}

be a

C

∗

{C^{\ast }}

-algebra and

E

\mathcal {E}

the algebra of all elementary operators on

A

\mathcal {A}

, and let

a
→

=
(

a
1

,
…
,

a
n

)
,

b
→

=
(

b
1

,
…
,

b
n

)
∈

A

n

\vec a = ({a_1}, \ldots ,{a_n}),\;\vec b = ({b_1}, \ldots ,{b_n}) \in {\mathcal {A}^n}

. It is proved that

b
→

\vec b

is contained in the closure of the set

{
(
E

a
1

,
…
,
E

a
n

)
:
E
∈

E

}

\{ (E{a_1}, \ldots ,E{a_n}):E \in \mathcal {E}\}

if and only if each complex linear combination

∑

j
=
1

n

λ
j

b
j

\sum \nolimits _{j = 1}^n {{\lambda _j}} {b_j}

is contained in the closed two-sided ideal generated by

∑

j
=
1

n

λ
j

a
j

\sum \nolimits _{j = 1}^n {{\lambda _j}} {a_j}

. In particular, a bounded linear operator on

A

\mathcal {A}

preserves all closed two-sided ideals if and only if it is in the strong closure of

E

\mathcal {E}

.
A functional equation from probability theory
Authors: Baker John A. 
Published: June 2010

The functional equation

(

(
1
)

)

f
(
x
)
=

∏

j
=
1

N

[
f
(

β
j

x
)
]

γ
j

\begin{equation}\tag {$(1)$} f(x) = \prod \limits _{j = 1}^N {{{[f({\beta _j}x)]}^{{\gamma _j}}}}\end{equation}

has been used by Laha and Lukacs (Aequationes Math. 16 (1977), 259-274) to characterize normal distributions. The aim of the present paper is to study (1) under somewhat different assumptions than those assumed by Laha and Lukacs by using techniques which, in the author’s opinion, are simpler than those employed by the afore-mentioned authors. We will prove, for example, that if

0
>

β
j

>
1

0 > {\beta _j} > 1

and

γ
j

>
0

{\gamma _j} > 0

for

1
≤
j
≤
N
,
∑

j
=
1

N

β
j
k

γ
j

=
1

1 \leq j \leq N, \sum {_{j = 1}^N\beta _j^k{\gamma _j} = 1}

, where k is a natural number,

f
:

R

→
[
0
,
+
∞
)

f:\mathbb {R} \to [0, + \infty )

, (1) holds for

x
∈

R

x \in \mathbb {R}

and

f

(
k
)

(
0
)

{f^{(k)}}(0)

exists then either

f
≡
0

f \equiv 0

or there exists a real constant c such that

f
(
x
)
=
exp
⁡
(
c

x
k

)

f(x) = \exp (c{x^k})

for all

x
∈

R

x \in \mathbb {R}

.
The Takaga operator, Bernoulli sequences, smoothness conditions and fractal curves
Authors: Deliu Anca , Wingren Peter 
Published: June 2010

We map Lipschitz spaces and functions of bounded variation by the operator

σ
:
φ
(
x
)
→

∑
0
∞

2

−
n

φ
(

2
n

x
)
,
x
∈
[
0
,
1
]

\sigma :\varphi (x) \to \sum \nolimits _0^\infty {{2^{ - n}}} \varphi ({2^n}x),x \in [0,1]

, and we estimate the Hausdorff measure of

σ
(
φ
)

\sigma (\varphi )

. We furthermore introduce a class of continuous and nowhere differentiable functions on [0,1] which we call

T

\mathcal {T}

. We make a refined analysis of the fractal and smoothness properties of the functions in

T

\mathcal {T}

and study the relationship between the two. We show that all the functions in

T

\mathcal {T}

have box dimension equal to

1
2

\frac {1}{2}

, with respect to the dimension family

{
t

/

(
log
⁡

1
t

)
s

:
s
∈

R

+

}

\{ t/{(\log \frac {1}{t})^s}:s \in {\mathbb {R}^ + }\}

, but that their order of smoothness covers a wide range.
Triangular UHF algebras over arbitrary fields
Authors: Baker R. L. 
Published: June 2010

Let K be an arbitrary field. Let

(

q
n

)

({q_n})

be a sequence of positive integers, and let there be given a family

{

Ψ

n
m

|

n
≥
m
}

\{ {\Psi _{nm}}|n \geq m\}

of unital K-monomorphisms

Ψ

n
m

:

T

q
m

(
K
)
→

T

q
n

(
K
)

{\Psi _{nm}}:{T_{{q_m}}}(K) \to {T_{{q_n}}}(K)

such that

Ψ

n
p

Ψ

p
m

=

Ψ

n
m

{\Psi _{np}}{\Psi _{pm}} = {\Psi _{nm}}

whenever

m
≤
n

m \leq n

, where

T

q
n

(
K
)

{T_{{q_n}}}(K)

is the K-algebra of all

q
n

×

q
n

{q_n} \times {q_n}

upper triangular matrices over K. A triangular UHF (TUHF) K-algebra is any K-algebra that is K-isomorphic to an algebraic inductive limit of the form

T

=

lim
→

(

T

q
n

(
K
)
;

Ψ

n
m

)

\mathcal {T} = \lim \limits _ \to ({T_{{q_n}}}(K);{\Psi _{nm}})

. The first result of the paper is that if the embeddings

Ψ

n
m

{\Psi _{nm}}

satisfy certain natural dimensionality conditions and if

S

=

lim
→

(

T

p
n

(
K
)
;

Φ

n
m

)

\mathcal {S} = \lim \limits _ \to ({T_{{p_n}}}(K);{\Phi _{nm}})

is an arbitrary TUHF K-algebra, then

S

\mathcal {S}

is K-isomorphic to

T

\mathcal {T}

, only if the supernatural number,

N
[
(

p
n

)
]

N[({p_n})]

, of

(

q
n

)

({q_n})

is less than or equal to the supernatural number,

N
[
(

p
n

)
]

N[({p_n})]

, of

(

p
n

)

({p_n})

. Thus, if the embeddings

Φ

n
m

{\Phi _{nm}}

also satisfy the above dimensionality conditions, then

S

\mathcal {S}

is K-isomorphic to

T

\mathcal {T}

, only if

N
[
(

p
n

)
]
=
N
[
(

q
n

)
]

N[({p_n})] = N[({q_n})]

. The second result of the paper is a nontrivial "triangular" version of the fact that if p, q are positive integers, then there exists a unital K-monomorphism

Φ
:

M
q

(
K
)
→

M
p

(
K
)

\Phi :{M_q}(K) \to {M_p}(K)

, only if

q

|

p

q|p

. The first result of the paper depends directly on the second result.
Gaussian estimates and holomorphy of semigroups
Authors: Ouhabaz El-Maati 
Published: June 2010

We show that if a selfadjoint semigroup T on

L
2

(
Ω
)

{L^2}(\Omega )

satisfies a Gaussian estimate

|

T
(
t
)
f

|

≤
M
G
(
b
t
)

|

f

|

,
0
≤
t
≤
1
,
f
∈

L
2

(
Ω
)

|T(t)f| \leq MG(bt)|f|,0 \leq t \leq 1,f \in {L^2}(\Omega )

(where

G
=
G

(
t

)

t
≥
0

G = G{(t)_{t \geq 0}}

is the Gaussian semigroup on

L
2

(

R
N

)

{L^2}({R^N})

and

Ω
\Omega

is an open set of

R
N

{R^N}

), then T defines a holomorphic semigroup of angle

π
2

\frac {\pi }{2}

on

L
p

(
Ω
)

{L^p}(\Omega )

. We obtain by duality the same result on

C
0

(
Ω
)

{C_0}(\Omega )

. Applications to uniformly elliptic operators and Schrödinger operators are given.
Oscillation and nonoscillation criteria for delay differential equations
Authors: Elbert Á. , Stavroulakis I. P. 
Published: June 2010

Oscillation and nonoscillation criteria for the first-order delay differential equation
\[

x
′

(
t
)
+
p
(
t
)
x
(
τ
(
t
)
)
=
0
,

t
≥

t
0

,
τ
(
t
)
>
t
,

x’(t) + p(t)x(\tau (t)) = 0,\quad t \geq {t_0},\tau (t) > t,

\]
are established in the case where
\[

∫

τ
(
t
)

t

p
(
s
)
d
s
≥

1
e

and

lim

t
→
∞

∫

τ
(
t
)

t

p
(
s
)
d
s
=

1
e

.

\int _{\tau (t)}^t {p(s)ds \geq \frac {1}{e}\quad {\text {and}}\quad \lim \limits _{t \to \infty } \int _{\tau (t)}^t {p(s)ds = \frac {1}{e}.} }

\]
Approximating topological metrics by Riemannian metrics
Authors: Ferry Steven C. , Okun Boris L. 
Published: June 2010

We study the relation between (topological) inner metrics and Riemannian metrics on smoothable manifolds. We show that inner metrics on smoothable manifolds can be approximated by Riemannian metrics. More generally, if

f
:
M
→
X

f:M \to X

is a continuous surjection from a smooth manifold to a compact metric space with

f

−
1

(
x
)

{f^{ - 1}}(x)

connected for every

x
∈
X

x \in X

, then there is a metric d on X and a sequence of Riemannian metrics

{

ψ
i

}

\{ {\psi _i}\}

on M so that

(
M
,

ψ
i

)

(M,{\psi _i})

converges to (X, d) in Gromov-Hausdorff space. This is used to obtain a (fixed) contractibility function

ρ
\rho

and a sequence of Riemannian manifolds with

ρ
\rho

as contractibility function so that

lim
(
M
,

ψ
i

)

\lim (M,{\psi _i})

is infinite dimensional. Using results of Dranishnikov and Ferry, this also gives examples of nonhomeomorphic manifolds M and N and a contractibility function

ρ
\rho

so that for every

ε
>
0

\varepsilon > 0

there are Riemannian metrics

ϕ
ε

{\phi _\varepsilon }

and

ψ
ε

{\psi _\varepsilon }

on M and N so that

(
M
,

ϕ
ε

)

(M,{\phi _\varepsilon })

and

(
N
,

ψ
ε

)

(N,{\psi _\varepsilon })

have contractibility function

ρ
\rho

and

d

G
H

(
(
M
,

ϕ
ε

)
,
(
N
,

ψ
ε

)
)
>
ε

{d_{GH}}((M,{\phi _\varepsilon }),(N,{\psi _\varepsilon })) > \varepsilon

.
Four-point affine Lie algebras
Authors: Bremner Murray 
Published: June 2010

We consider Lie algebras of the form

g

⊗
R

\mathfrak {g} \otimes R

where

g

\mathfrak {g}

is a simple complex Lie algebra and

R
=

C

[
s
,

s

−
1

,

(
s
−
1

)

−
1

,

(
s
−
a

)

−
1

]

R = \mathbb {C}[s,{s^{ - 1}},{(s - 1)^{ - 1}},{(s - a)^{ - 1}}]

for

a
∈

C

−
{
0
,
1
}

a \in \mathbb {C} - \{ 0,1\}

. After showing that R is isomorphic to a quadratic extension of the ring

C

[
t
,

t

−
1

]

\mathbb {C}[t,{t^{ - 1}}]

of Laurent polynomials, we prove that

g
⊗
R

g \otimes R

is a quasi-graded Lie algebra with a triangular decomposition. We determine the universal central extension of

g

⊗
R

\mathfrak {g} \otimes R

and show that the cocycles defining it are closely related to ultraspherical (Gegenbauer) polynomials.
Local minimizers of integral functionals are global minimizers
Authors: Giner E. 
Published: June 2010

We show that local minimizers of integral functionals associated with a measurable integrand

f
:
Ω
×
E
→

R

∪
{
±
∞
}

f:\Omega \times E \to \mathbb {R} \cup \{ \pm \infty \}

are actually global minimizers. Here

(
Ω
,

S

,
μ
)

(\Omega , \mathcal {S},\mu )

is a measured space with an atomless

σ
\sigma

-finite positive measure, E is a separable Banach space, and the integral functional

I
f

(
x
)
=

∫
Ω
∗

f
(
ω
,
x
(
ω
)
)
d
μ

{I_f}(x) = \smallint _\Omega ^ \ast f(\omega ,x(\omega ))d\mu

is defined on

L
p

(
Ω
,
E
)

{L_p}(\Omega ,E)

or, more generally, on some decomposable set of measurable mappings x from

Ω
\Omega

into E.