bluuebunny/crossref_metadata_2025_split huggingface dataset:
Stabilization of solutions of weakly singular quenching problems
Authors: Fila Marek , Levine Howard A. , Vázquez Juan L. 
Published: July 2010

In this paper we prove that if

0
>
β
>
1
,

D
⊂

R
N

0 > \beta > 1,\;D \subset {R^N}

is bounded, and

λ
>
0

\lambda > 0

, then every element of the

ω
\omega

-limit set of weak solutions of
\[

u
t

−
Δ
u
+
λ

u

−
β

χ

u
>
0

=
0

in

D
×
[
0
,
∞
)
,

u
=

{

1

on

∂
D
×
(
0
,
∞
)
,

u
0

>
0

on

D
¯

×
{
0
}

\begin {array}{*{20}{c}} {{u_t} - \Delta u + \lambda {u^{ - \beta }}{\chi _{u > 0}} = 0\quad {\text {in}}\;D \times [0,\infty ),} \\ {u = \left \{ \begin {gathered} 1\qquad \quad {\text {on}}\;\partial D \times (0,\infty ), \hfill \\ {u_0} > 0\quad {\text {on}}\;\bar D \times \{ 0\} \hfill \\ \end {gathered} \right .} \\ \end {array}

\]
is a weak stationary solution of this problem. A consequence of this is that if

D
D

is a ball,

λ
\lambda

is sufficiently small, and

u
0

{u_0}

is a radial, then the set

{
(
x
,
t
)

|

u
=
0
}

\{ (x,t)|u = 0\}

is a bounded subset of

D
×
[
0
,
∞
)

D \times [0,\infty )

.
Asymptotic behavior of solutions of Poincaré difference equations
Authors: Trench William F. 
Published: July 2010

It is shown that if the zeros

λ
1

,

λ
2

,
…
,

λ
n

{\lambda _1},{\lambda _2}, \ldots ,{\lambda _n}

of the polynomial
\[

q
(
λ
)
=

λ
n

+

a
1

λ

n
−
1

+
⋯
+

a
n

q(\lambda ) = {\lambda ^n} + {a_1}{\lambda ^{n - 1}} + \cdots + {a_n}

\]
are distinct and

r
r

is an integer in

{
1
,
2
,
…
,
n
}

\{ 1,2, \ldots ,n\}

such that

|

λ
s

|

≠

|

λ
r

|

|{\lambda _s}| \ne |{\lambda _r}|

if

s
≠
r

s \ne r

, then the Poincaré difference equation
\[

y
(
n
+
m
)
+
(

a
1

+

p
1

(
m
)
)
y
(
n
+
m
−
1
)
+
⋯
+
(

a
n

+

p
n

(
m
)
)
y
(
m
)
=
0

y(n + m) + ({a_1} + {p_1}(m))y(n + m - 1) + \cdots + ({a_n} + {p_n}(m))y(m) = 0

\]
has a solution

y
r

{y_r}

such that (A)

y
r

(
m
)
=

λ
r
m

(
1
+
o
(
1
)
)

{y_r}(m) = \lambda _r^m(1 + o(1))

as

m
→
∞

m \to \infty

, provided that the sums

∑

j
=
m

∞

p
i

(
j
)

(
1
⩽
i
⩽
n
)

\sum \nolimits _{j = m}^\infty {{p_i}(j)\;(1 \leqslant i \leqslant n)}

converge sufficiently rapidly. Our results improve over previous results in that these series may converge conditionally, and we give sharper estimates of the

o
(
1
)

o(1)

terms in (A).
Comfort types of ultrafilters
Authors: García-Ferreira Salvador 
Published: July 2010

Let

α
\alpha

be an infinite cardinal. Comfort pre-order on

β
(
α
)
∖
α

\beta (\alpha )\backslash \alpha

is defined as follows: for

p
,
q
∈
β
(
α
)
∖
α
,

p

⩽

c

q

p,q \in \beta (\alpha )\backslash \alpha ,\;p{ \leqslant _{\text {c}}}q

if every

q
q

-compact space is

p
p

-compact. For

p
∈
U
(
α
)

p \in U(\alpha )

, we let

T

RK

(
p
)

{T_{\operatorname {RK} }}(p)

be the type of

p
p

and

T

c

(
p
)
=
{
q
∈
U
(
α
)
:
q

⩽

c

p

⩽

c

q
}

{T_{\text {c}}}(p) = \{ q \in U(\alpha ):q{ \leqslant _{\text {c}}}p{ \leqslant _{\text {c}}}q\}

. Since

T

c

(
p
)

{T_{\text {c}}}(p)

is a union of types, it is natural to define

c
p

=

|

{

T

RK

(
q
)
:
q
∈

T

c

(
p
)
}

|

{c_p} = |\{ {T_{\operatorname {RK} }}(q):q \in {T_{\text {c}}}(p)\} |

. It is evident that

ω
⩽

c
p

⩽

2
α

\omega \leqslant {c_p} \leqslant {2^\alpha }

for

p
∈
U
(
α
)

p \in U(\alpha )

. We show that if

p
∈
U
(
α
)

p \in U(\alpha )

then

|

T

RK

(
p
)

|

=

|

α

α

/

p

|

|{T_{\operatorname {RK} }}(p)| = {|^\alpha }\alpha /p|

, and we use this equality to prove that

c
p

=

2
α

{c_p} = {2^\alpha }

whenever

p
p

is decomposable. We also note that if

p
p

is countably incomplete then

2
ω

⩽

c
p

⩽

2
α

{2^\omega } \leqslant {c_p} \leqslant {2^\alpha }

; if

p
p

is RK-minimal (selective) and

ω
>
α

\omega > \alpha

then

c
p

=
ω

{c_p} = \omega

and

c
p

=

2

ℵ
n

{c_p} = {2^{{\aleph _n}}}

for each

p
∈
U
(

ℵ
n

)

p \in U({\aleph _n})

and for

n
>
ω

n > \omega

. Finally, we prove that, if

α
\alpha

is a strong limit and

p
∈
U
(
α
)

p \in U(\alpha )

is indecomposable, then

β
p

(
α
)

{\beta _p}(\alpha )

is a

p
p

-compact, noninitially

α
\alpha

-compact space, where

β
p

(
α
)

{\beta _p}(\alpha )

is the

p
p

-compactification of

α
\alpha

.
The existence question in the calculus of variations: a density result
Authors: Cellina Arrigo , Mariconda Carlo 
Published: July 2010

We show the existence of a dense subset

D

\mathcal {D}

of

C

(

R

)

\mathcal {C}(\mathbb {R})

such that, for

g
g

in it, the problem
\[

minimum

∫
0
T

g
(
x
(
t
)
)
d
t
+

∫
0
T

h
(

x
′

(
t
)
)
d
t
,

x
(
0
)
=
a
,

x
(
T
)
=
b

{\text {minimum}}\;\int _0^T {g(x(t))dt + \int _0^T {h(x’(t))dt,\;x(0) = a,\;x(T) = b} }

\]
admits a solution for every lower semicontinuous

h
h

satisfying growth conditions
On the metrical theory of continued fractions
Authors: Nair R. 
Published: July 2010

Suppose

b
k

{b_k}

denotes either

ϕ
(
k
)

\phi (k)

or

ϕ
(

p
k

)

(
k
=
1
,
2
,
…
)

\phi ({p_k})\;(k = 1,2, \ldots )

where the polynomial

ϕ
\phi

maps

N

\mathbb {N}

to

N

\mathbb {N}

and

p
k

{p_k}

denotes the

k
k

th rational prime. Suppose

(

c
k

(
x
)

)

k
=
1

∞

({c_k}(x))_{k = 1}^\infty

denotes the sequences of partial quotients of the continued function expansion of the real number

x
x

. Then for certain functions

F
:

R

⩾
0

→

R

F:{\mathbb {R}_{ \geqslant 0}} \to \mathbb {R}

we show that
\[

lim

N
→
∞

F

−
1

[

F
(

c

b
1

(
x
)
)
+
⋯
+
F
(

c

b
k

(
x
)
)

N

]

=

F

−
1

[

1

(
log
⁡
2
)

∫
0
1

F
(

c
1

(
x
)
)

1
+
x

d
x

]

\lim \limits _{N \to \infty } {F^{ - 1}}\left [ {\frac {{F({c_{{b_1}}}(x)) + \cdots + F({c_{{b_k}}}(x))}} {N}} \right ] = {F^{ - 1}}\left [ {\frac {1} {{(\log 2)}}\int _0^1 {\frac {{F({c_1}(x))}} {{1 + x}}dx} } \right ]

\]
almost everywhere with respect to Lebesgue measure. This result with

b
k

=
k

{b_k} = k

is classical and due to Ryll-Nardzewski.
On the behavior of the constant in a decoupling inequality for martingales
Authors: Hitczenko Paweł 
Published: July 2010

Let

(

f
n

)

({f_n})

and

(

g
n

)

({g_n})

be two martingales with respect to the same filtration

(

F

n

)

({\mathcal {F}_n})

such that their difference sequences

(

d
n

)

({d_n})

and

(

e
n

)

({e_n})

satisfy
\[

P
(

d
n

≥
λ

|

F

n
−
1

)
=
P
(

e
n

≥
λ

|

F

n
−
1

)

P({d_n} \geq \lambda |{\mathcal {F}_{n - 1}}) = P({e_n} \geq \lambda |{\mathcal {F}_{n - 1}})

\]
for all real

λ
\lambda

’s and

n
≥
1

n \geq 1

. It is known that
\[

‖

f
∗

‖

p

≤

K
p

‖

g
∗

‖

p

,

1
≤
p
>
∞
,

{\left \| {{f^ \ast }} \right \|_p} \leq {K_p}{\left \| {{g^ \ast }} \right \|_p},\quad 1 \leq p > \infty ,

\]
for some constant

K
p

{K_p}

depending only on p. We show that

K
p

=
O
(
p
)

{K_p} = O(p)

. This will be obtained via a new version of Rosenthal’s inequality which generalizes a result of Pinelis and which may be of independent interest.
On Dirichlet series associated with polynomials
Authors: Carletti E. , Monti Bragadin G. 
Published: July 2010

Let

P
(
X
)

P(X)

be a polynomial of degree N with complex coefficients and

d
1

,

d
2

{d_1},{d_2}

two complex numbers with real part greater then

−
1

-1

. Consider the Dirichlet series associated with the triple

(
P
(
X
)
,

d
1

,

d
2

)

(P(X),{d_1},{d_2})

\[

L
(
s
)
=

∑

n
=
1

∞

P
(
n
)

(
n
+

d
1

)

s

(
n
+

d
2

)

s

.

L(s) = \sum \limits _{n = 1}^\infty {\frac {{P(n)}}{{{{(n + {d_1})}^s}{{(n + {d_2})}^s}}}.}

\]
In this paper we get an explicit formula for

L
(
s
)

L(s)

in terms of special functions which gives meromorphic continuation of

L
(
s
)

L(s)

with at most simple poles at

s
=
(
N
+
1
−
k
)

/

2
,
k
=
0
,
1
,
…

s = (N + 1 - k)/2,k = 0,1, \ldots

Finally we apply our explicit formula to Minakshisundaram’s zeta function of the three-dimensional sphere.
Exact covering systems and the Gauss-Legendre multiplication formula for the gamma function
Authors: Beebee John 
Published: July 2010

The Gauss-Legendre multiplication formula for the gamma function is

(
2
π

)

(
m
−
1
)

/

2

m

1

/

2
−
m
z

Γ
(
m
z
)
=
Γ
(
z
)
Γ
(
z
+

1
m

)
⋯
Γ
(
z
+

m
−
1

m

)

{(2\pi )^{(m - 1)/2}}{m^{1/2 - mz}}\Gamma (mz) = \Gamma (z)\Gamma (z + \tfrac {1} {m}) \cdots \Gamma (z + \tfrac {{m - 1}} {m})

. Let

{

a
i

(
mod

b
i

)
:
1
⩽
i
⩽
m
}

\{ {a_i}(\bmod {b_i}):1 \leqslant i \leqslant m\}

be an exact covering system with standardized offsets. Then
\[

Γ
(
z
)
=

Γ
(
z

/

b
1

)

b
1

1
−
z

/

b
1

∏

i
=
2

m

Γ
(
(
z
+

a
i

)

/

b
i

)

b
i

−
z

/

b
i

Γ
(

a
i

/

b
i

)

.

\Gamma (z) = \frac {{\Gamma (z/{b_1})}} {{{b_1}^{1 - z/{b_1}}}}\prod \limits _{i = 2}^m {\frac {{\Gamma ((z + {a_i})/{b_i})}} {{b_i^{ - z/{b_i}}\Gamma ({a_i}/{b_i})}}}.

\]
Conversely, if the above identity holds, then

{

a
i

(
mod

b
i

)
:
1
⩽
i
⩽
m
}

\{ {a_i}(\bmod {b_i}):1 \leqslant i \leqslant m\}

is an exact covering system with standardized offsets. The Gauss-Legendre multiplication formula is a special case of this identity.
A note on subcontinua of 𝛽[0,∞)-[0,∞)
Authors: Zhu Jian-Ping 
Published: July 2010

Let

M
=

⊕

n
∈
ω

I
n

M = { \oplus _{n \in \omega }}{I_n}

be the topological sum of countably many copies of the unit interval

I
I

. For any ultrafilter

u
∈

ω

∗

u \in {\omega ^{\ast }}

, we let

M
u

=
∩
{

cl

β
M

(
∪
{

I
n

:
n
∈
A
}
)
:
A
∈
u
}

{M^u} = \cap \{ {\operatorname {cl} _{\beta M}}( \cup \{ {I_n}:n \in A\} ):A \in u\}

. It is well known that

M
u

{M^u}

is a decomposable continuum with a very nice internal structure. In this paper, we show: (1) every nondegenerate subcontinuum of

β
[
0
,
∞
)
−
[
0
,
∞
)

\beta [0,\infty ) - [0,\infty )

contains a copy of

M
u

{M^u}

for some

u
∈

ω

∗

u \in {\omega ^{\ast }}

. (2) there is no nontrivial simple point in Laver’s model for the Borel conjecture. The second answers a question posed by Baldwin and Smith negatively.
A counterexample concerning the maximum and minimum of a subharmonic function
Authors: Fryntov Alexander 
Published: July 2010

For every

Δ
>
0

\Delta > 0

a function u subharmonic in the plane is constructed such that u has the order

ρ
=
1
+
Δ

\rho = 1 + \Delta

and satisfies the condition
\[

min
φ

u
(
r

e

i
φ

)

/

max
φ

u
(
r

e

i
φ

)
≤
−
(
C
+
1
)

for every

r
>
0
,

\min \limits _\varphi u(r{e^{i\varphi }})/\max \limits _\varphi u(r{e^{i\varphi }}) \leq - (C + 1)\quad \text {for every}\,r > 0,

\]
where

C
=
C
(
ρ
)
>
0

C = C(\rho ) > 0

. This example answers a question of W. K. Hayman.
On the medians of gamma distributions and an equation of Ramanujan
Authors: Choi K. P. 
Published: July 2010

For

n
≥
0

n \geq 0

, let

λ
(
n
)

\lambda (n)

denote the median of the

Γ
(
n
+
1
,
1
)

\Gamma (n + 1,1)

distribution. We prove that

n
+

2
3

>
λ
(
n
)
≤
min
(
n
+
log
⁡
2
,
n
+

2
3

+

(
2
n
+
2

)

−
1

)

n + \tfrac {2}{3} > \lambda (n) \leq \min (n + \log 2, n + \tfrac {2}{3} + {(2n + 2)^{ - 1}})

. These bounds are sharp. There is an intimate relationship between

λ
(
n
)

\lambda (n)

and an equation of Ramanujan. Based on this relationship, we derive the asymptotic expansion of

λ
(
n
)

\lambda (n)

as follows:
\[

λ
(
n
)
=
n
+

2
3

+

8

405
n

−

64

5103

n
2

+

2
7

⋅
23

3
9

⋅

5
2

n
3

+
⋯
.

\lambda (n) = n + \frac {2}{3} + \frac {8}{{405n}} - \frac {{64}}{{5103{n^2}}} + \frac {{{2^7} \cdot 23}}{{{3^9} \cdot {5^2}{n^3}}} + \cdots .

\]
Let median

(

Z
μ

)

({Z_\mu })

denote the median of a Poisson random variable with mean

μ
\mu

, where the median is defined to be the least integer m such that

P
(

Z
μ

≤
m
)
≥

1
2

P({Z_\mu } \leq m) \geq \tfrac {1}{2}

. We show that the bounds on

λ
(
n
)

\lambda (n)

imply
\[

μ
−
log
⁡
2
≤

median

(

Z
μ

)
>
μ
+

1
3

.

\mu - \log 2 \leq {\text {median}}({Z_\mu }) > \mu + \frac {1}{3}.

\]
This proves a conjecture of Chen and Rubin. These inequalities are sharp.
Higher-order nonlinear dispersive equations
Authors: Kenig Carlos E. , Ponce Gustavo , Vega Luis 
Published: July 2010

We study nonlinear dispersive equations of the form
\[

∂
t

u
+

∂
x

2
j
+
1

u
+
P
(
u
,

∂
x

u
,
…
,

∂
x

2
j

u
)
=
0
,

x
,
t
∈

R

,

j
∈

Z

+

,

{\partial _t}u + \partial _x^{2j + 1}u + P(u,{\partial _x}u, \ldots ,\partial _x^{2j}u) = 0,\qquad x,t \in \mathbb {R},\quad j \in {\mathbb {Z}^ + },

\]
where

P
(
⋅
)

P( \cdot )

is a polynomial having no constant or linear terms. It is shown that the associated initial value problem is locally well posed in weighted Sobolev spaces. The method of proof combines several sharp estimates for solutions of the associated linear problem and a change of dependent variable which allows us to consider data of arbitrary size.
On the size of finite Sidon sequences
Authors: Chen Sheng 
Published: July 2010

Let

h
≥
2

h \geq 2

be an integer. A set of positive integers B is called a

B
h

{B_h}

-sequence, or a Sidon sequence of order h, if all sums

a
1

+

a
2

+
⋯
+

a
h

{a_1} + {a_2} + \cdots + {a_h}

, where

a
i

∈
B
(
i
=
1
,
2
,
…
,
h
)

{a_i} \in B (i = 1,2, \ldots ,h)

, are distinct up to rearrangements of the summands. Let

F
h

(
n
)

{F_h}(n)

be the size of the maximum

B
h

{B_h}

-sequence contained in

{
1
,
2
,
…
,
n
}

\{ 1,2, \ldots ,n\}

. We prove that
\[

F

2
r
−
1

(
n
)
≤

(

(
r
!

)
2

n

)

1

/

(
2
r
−
1
)

+
O
(

n

1

/

(
4
r
−
2
)

)
.

{F_{2r - 1}}(n) \leq {({(r!)^2}n)^{1/(2r - 1)}} + O({n^{1/(4r - 2)}}).

\]
On the distance of the Riemann-Liouville operator from compact operators
Authors: Opic Bohumír 
Published: July 2010

We consider generalized Hardy operators
\[

T
f
(
x
)
=

∫
a
x

φ
(
x
,
y
)
f
(
y
)

d
y
,

x
∈
(
a
,
b
)
⊂

R

,

Tf(x) = \int _a^x {\varphi (x,y)f(y)\;dy,\quad x \in (a,b) \subset \mathbb {R},}

\]
acting between two weighted Lebesgue spaces

X
=

L
p

(
a
,
b
;
v
)

X = {L^p}(a,b;v)

and

Y
=

L
q

(
a
,
b
;
w
)
,
1
>
p
≤
q
>
∞

Y = {L^q}(a,b;w), 1 > p \leq q > \infty

, and present lower and upper bounds on the distance of T from the space of all compact linear operators P,

P
:
X
→
Y

P:X \to Y

. The conditions on the kernel

φ
(
x
,
y
)

\varphi (x,y)

are patterned in such a way that the above mentioned class of operators T contains the Riemann-Liouville fractional operators of orders equal to or greater than one.
Cesàro means of Fourier transforms and multipliers on 𝐿¹(𝑅)
Authors: Giang Dăng Vũ , Móricz Ferenc 
Published: July 2010

We prove that the Cesàro mean

σ
\sigma

of a multiplier

λ
\lambda

on

L
1

(

R

)

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

is also a multiplier on

L
1

(

R

)

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

. In the particular cases when (i)

λ
\lambda

is odd, we prove that

σ
\sigma

is the Fourier transform of an odd function in the Hardy space

H
1

(

R

)

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

, and (ii)

λ
\lambda

is even, we give a necessary and sufficient condition in order that

σ
\sigma

be a Fourier transform of an even function in

L
1

(

R

)

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

. As a corollary, we obtain a nontrivial condition for

λ
\lambda

in order to be a multiplier on

L
1

(

R

)

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

; namely,
\[

∫
0
∞

|

1
t

∫
0
t

{
λ
(
ξ
)
−
λ
(
−
ξ
)
}

d
ξ

|

d
t

t

>
∞
.

\int _0^\infty {\left | {\frac {1}{t}\int _0^t {\{ \lambda (\xi ) - \lambda ( - \xi )\} \,d\xi } } \right |} \frac {{dt}}{t} > \infty .

\]
We also prove Hardy type inequalities for multipliers and Hilbert transforms.
Absolute summability factors of type (𝑋,𝑌)
Authors: Mishra K. N. 
Published: July 2010

In Theorem 1 a set of sufficient conditions is investigated for the summability factor of type

θ
n

∈
(

|

N
¯

,
q

|

k

,

|

N
¯

,
p

|

k

)
,
k
≥
1

{\theta _n} \in (|\bar N ,q{|_k},|\bar N ,p{|_k}),k \geq 1

. In the special case when

k
=
1

k = 1

three different sets of necessary and sufficient conditions are established in Theorem 2 for the factor

θ
n

{\theta _n}

. Various known and some new results are also deduced as special cases.
An extremal problem for polynomials
Authors: Erëmenko A. , Lempert L. 
Published: July 2010

Let

f
(
z
)
=

z
n

+
⋯

f(z) = {z^n} + \cdots

be a polynomial such that the level set

E
=
{
z
:

|

f
(
z
)

|

≤
1
}

E = \{ z:|f(z)| \leq 1\}

is connected. Then

max
{

|

f
′

(
z
)

|

:
z
∈
E
}
≤

2

(
1

/

n
)
−
1

n
2

\max \{ |f’ (z)|:z \in E\} \leq {2^{(1/n) - 1}}{n^2}

, and this estimate is the best possible.
The Hankel transformation on 𝑀’_{𝜇} and its representation
Authors: Koh E. L. , Li C. K. 
Published: July 2010

The Hankel transformation was extended by Zemanian to certain generalized functions of slow growth through a generalization of Parseval’s equation as

(

(
1
)

)

⟨

h
μ

f
,
φ
⟩
=
⟨
f
,

h
μ

φ
⟩

\begin{equation}\tag {$(1)$} \langle {h_\mu }f,\varphi \rangle = \langle f,{h_\mu }\varphi \rangle \end{equation}

where

φ
,

h
μ

φ
∈

H
μ

,
f
∈

H
μ
′

\varphi , {h_\mu }\varphi \in {H_\mu }, f \in {H’_\mu }

. Later, Koh and Zemanian defined the generalized complex Hankel transformation on

J
μ

=

⋃

ν
=
1

∞

J

a
ν

,
μ

{J_\mu } = {\bigcup }_{\nu = 1}^\infty \,{J_{{a_\nu },\mu }}

, where

J

a
ν

,
μ

{J_{{a_\nu },\mu }}

is the testing function space which contains the kernel function,

x
y

J
μ

(
x
y
)

\sqrt {xy} {J_\mu }(xy)

. A transformation was defined directly as the application of a generalized function to the kernel function, i.e., for

f
∈

J
μ
′

f \in {J’_\mu }

,

(

(
2
)

)

(

h
μ

f
)
(
y
)
=
⟨
f
(
x
)
,

x
y

J
μ

(
x
y
)
⟩
.

\begin{equation}\tag {$(2)$} ({h_\mu }f)(y) = \langle f(x),\sqrt {xy} {J_\mu }(xy)\rangle .\end{equation}

In this paper, we extend definition (2) to a larger space of generalized functions. We first introduce the test function space

M

a
,
μ

{M_{a,\mu }}

which contains the kernel function and show that

H
μ

⊂

M

a
,
μ

⊂

J

a
,
μ

{H_\mu } \subset {M_{a,\mu }} \subset {J_{a,\mu }}

. We then form the countable union space

M
μ

=

⋃

ν
=
1

∞

M

a
ν

,
μ

{M_\mu } = {\bigcup }_{\nu = 1}^\infty \,{M_{{a_\nu },\mu }}

whose dual

M
μ
′

{M’_\mu }

has

J
μ
′

{J’_\mu }

as a subspace. Our main result is an inversion theorem stated as follows. Let

F
(
y
)
=
(

h
μ

f
)
(
y
)
=
⟨
f
(
x
)
,

x
y

J
μ

(
x
y
)
⟩
,
f
∈

M
μ
′

F(y) = ({h_\mu }f)(y) = \langle f(x),\sqrt {xy} {J_\mu }(xy)\rangle ,f \in {M’_\mu }

, where y is restricted to the positive real axis. Let

μ
≥
−

1
2

\mu \geq - \frac {1}{2}

. Then, in the sense of convergence in

H
μ
′

{H’_\mu }

,
\[

f
(
x
)
=

lim

r
→
∞

∫
0
r

F
(
y
)

x
y

J
μ

(
x
y
)
d
y
.

f(x) = \lim \limits _{r \to \infty } \int _0^r {F(y)} \sqrt {xy} {J_\mu }(xy)dy.

\]
This convergence gives a stronger result than the one obtained by Koh and Zemanian (1968). Secondly, we prove that every generalized function belonging to

M

a
,
μ

′

{M’_{a,\mu }}

can be represented by a finite sum of derivatives of measurable functions. This proof is analogous to the method employed in structure theorems for Schwartz distributions (Edwards, 1965), and similar to one by Koh (1970).
Characterizations of bounded sets in spaces of ultradistributions
Authors: Pilipović Stevan 
Published: July 2010

We characterize bounded sets in ultradistributions spaces

D

L
t

′

(

M
p

)

,

t
∈
[
1
,
∞
]
,

S

′

{

M
p

}

\mathcal {D}_{{L^t}}^{’({M_p})},\,t \in [1,\infty ],\,S{’^{\{ {M_p}\} }}

, and

S

′

(

M
p

)

S{’^{({M_p})}}

and bounded sets and convergent sequences in

D

′

(

M
p

)

\mathcal {D}{’^{({M_p})}}

and

D

′

{

M
p

}

\mathcal {D}{’^{\{ {M_p}\} }}

via the convolution by corresponding test functions. The structural theorems for

D

L
t

′

{

M
p

}

\mathcal {D}_{{L^t}}^{’\{ {M_p}\} }

and

D
~

L
t

′

{

M
p

}

,

t
∈
[
1
,
∞
]

\widetilde D_{{L^t}}^{’\{ {M_p}\} },\;t \in [1,\infty ]

, are also given.
A factorization constant for 𝑙ⁿ_{𝑝}, 0<𝑝<1
Authors: Peck N. T. 
Published: July 2010

We prove that if PT is a factorization of the identity operator on

ℓ
p
n

\ell _p^n

through

ℓ
∞
k

,
0
>
p

\ell _\infty ^k,0 > p

, then

‖
P
‖

‖
T
‖

≥
C

n

1

/

p
−
1

/

2

(
log
⁡
n

)

−
1

/

2

\left \| P \right \| \left \| T \right \| \geq C{n^{1/p - 1/2}}{(\log n)^{ - 1/2}}

. This is a corollary of a more general result on factoring the identity operator on a quasi-normed space X through

ℓ
∞
k

\ell _\infty ^k

.
Primes in arithmetic progression and uniform distribution
Authors: Wodzak M. A. 
Published: July 2010

Let f be an entire, non-polynomial function with real coefficients. Let

p
~

\tilde p

run through the primes such that

p
~

≡
h
mod
k
,
(
h
,
k
)
=
1

\tilde p \equiv h \bmod k,(h,k) = 1

, and let

M
>
1

M > 1

. If there exists

α
∈
(
1
,
4

/

3
)

\alpha \in (1,4/3)

such that

|

f
(
z
)

|

≤
exp
⁡
(

(
log
⁡

|

z

|

)
α

)

|f(z)| \leq \exp ({(\log |z|)^\alpha })

for all

|

z

|

>
M

|z| > M

, then the sequence

{
f
(

p
~

)
}

\{ f(\tilde p)\}

is uniformly distributed modulo 1.
The minimum norm of certain completely positive maps
Authors: Suen Ching Yun 
Published: July 2010

Let L be a completely bounded linear map from a unital

C
∗

{C^ \ast }

-algebra to the algebra of all bounded linear operators on a Hilbert space. Then
\[

min

{

‖
ϕ
‖

cb

:

(

ϕ

a
m
p
;

a
m
p
;
L

a
m
p
;
0

a
m
p
;
⋯

a
m
p
;
0

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;
⋮

L
∗

a
m
p
;

a
m
p
;
ϕ

a
m
p
;
L

a
m
p
;

a
m
p
;
0

0

a
m
p
;

a
m
p
;

a
m
p
;
⋱

a
m
p
;

a
m
p
;

⋮

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;

a
m
p
;
L

0

a
m
p
;
⋯

a
m
p
;
0

a
m
p
;

a
m
p
;

L
∗

a
m
p
;
ϕ

)

n
×
n

is completely

positive

for all

n

}

=
2
S
(
L
)
,

\min \left \{ {{{\left \| \phi \right \|}_{{\text {cb}}}}:{{\left ( {\begin {array}{*{20}{c}} \phi \hfill & {} \hfill & L \hfill & 0 \hfill & \cdots \hfill & 0 \hfill \\ {} \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & \vdots \hfill \\ {{L^ \ast }} \hfill & {} \hfill & \phi \hfill & L \hfill & {} \hfill & 0 \hfill \\ 0 \hfill & {} \hfill & {} \hfill & \ddots \hfill & {} \hfill & {} \hfill \\ \vdots \hfill & {} \hfill & {} \hfill & {} \hfill & {} \hfill & L \hfill \\ 0 \hfill & \cdots \hfill & 0 \hfill & {} \hfill & {{L^ \ast }} \hfill & \phi \hfill \\ \end {array} } \right )}_{n \times n}}\begin {array}{*{20}{c}} {{\text {is completely}}} \hfill \\ {{\text {positive}}} \hfill \\ {{\text {for all}}\;n} \hfill \\ \end {array} } \right \} = 2S(L),

\]
where

S
(
L
)
=
min
{

‖
ϕ
‖

cb

:
ϕ
±
Re
⁡
λ
L

S(L) = \min \{ {\left \| \phi \right \|_{{\text {cb}}}}:\phi \pm \operatorname {Re} \lambda L

is completely positive for all

|

λ

|

=
1
}

|\lambda | = 1\}

.
On the 𝐶^{∞} wave-front set of solutions of first-order nonlinear PDEs
Authors: Asano Claudio Hirofume 
Published: July 2010

Let

Ω
⊂

R

m
+
1

\Omega \subset {{\mathbf {R}}^{m + 1}}

be a neighborhood of the origin and assume

u
∈

C
2

(
Ω
)

u \in {C^2}(\Omega )

is a solution of the nonlinear PDE
\[

u
t

=
f
(
x
,
t
,
u
,

u
x

)
,

{u_t} = f(x,t,u,{u_x}),

\]
where

f
(
x
,
t
,

ζ
0

,
ζ
)

f(x,t,{\zeta _0},\zeta )

is

C
∞

{C^\infty }

in the variables

(
x
,
t
)
∈

R

m

×

R

(x,t) \in {{\mathbf {R}}^m} \times {\mathbf {R}}

and holomorphic in the variables

(

ζ
0

,
ζ
)
∈

C

×

C

m

({\zeta _0},\zeta ) \in {\mathbf {C}} \times {{\mathbf {C}}^m}

. We present a proof that
\[

W
F
(
u
)
⊂
char
⁡
(

L
u

)
,

WF(u) \subset \operatorname {char}({L^u}),

\]
where

W
F
(
u
)

WF(u)

denotes the

C
∞

{C^\infty }

wave-front set of u and

char
⁡
(

L
u

)

\operatorname {char}({L^u})

is the characteristic set of the linearized operator
\[

L
u

=

∂

∂
t

−

∑

j
=
1

m

(

∂
f

∂

ζ
j

)

(
x
,
t
,
u
,

u
x

)

∂

∂

x
j

.

{L^u} = \frac {\partial }{{\partial t}} - \sum \limits _{j = 1}^m {\left ( {\frac {{\partial f}}{{\partial {\zeta _j}}}} \right )} (x,t,u,{u_x})\frac {\partial }{{\partial {x_j}}}.

\]