bluuebunny/crossref_metadata_2025_split huggingface dataset:
A multiplier theorem for Fourier transforms
Authors: McCall James D. 
Published: July 2010

A function f analytic in the upper half-plane

Π
+

{\Pi ^ + }

is said to be of class

E
p

(

Π
+

)
(
0
>
p
>
∞
)

{E_p}({\Pi ^ + })(0 > p > \infty )

if there exists a constant C such that

∫

−
∞

∞

|

f
(
x
+
i
y
)

|

p

d
x
≤
C
>
∞

\smallint _{ - \infty }^\infty |f(x + iy){|^p}dx \leq C > \infty

for all

y
>
0

y > 0

. These classes are an extension of the

H
p

{H_p}

spaces of the unit disc U. For f belonging to

E
p

(

Π
+

)
(
0
>
p
≤
2
)

{E_p}({\Pi ^ + })(0 > p \leq 2)

, there exists a Fourier transform f with the property that

f
(
z
)
=
2

(
π

)

−
1

∫
0
∞

f
^

(
t
)

e

i
z
t

d
t

f(z) = 2{(\pi )^{ - 1}}\smallint _0^\infty \hat f(t){e^{izt}}dt

. This makes it possible to give a definition for the multiplication of

E
p

(

Π
+

)
(
0
>
p
≤
2
)

{E_p}({\Pi ^ + })(0 > p \leq 2)

into

L
q

(
0
,
∞
)

{L_q}(0,\infty )

that is analogous to the multiplication of

H
p

(
U
)

{H_p}(U)

into

l
q

{l_q}

. In this paper, we consider the case

0
>
p
>
1

0 > p > 1

and

p
≤
q

p \leq q

and derive a necessary and sufficient condition for multiplying

E
p

(

Π
+

)

{E_p}({\Pi ^ + })

into

L
q

(
0
,
∞
)

{L_q}(0,\infty )

.
The Fredholm spectrum of the sum and product of two operators
Authors: Shapiro Jack , Snow Morris 
Published: July 2010

Let

C
(
X
)

C(X)

denote the set of closed operators with dense domain on a Banach space X, and

L
(
X
)

L(X)

the set of all bounded linear operators on X. Let

Φ

(
X
)

{\mathbf {\Phi }}(X)

denote the set of all Fredholm operators on X, and

σ

Φ

(
A
)

{\sigma _{\mathbf {\Phi }}}(A)

the set of all complex numbers

λ

{\mathbf {\lambda }}

such that

(

λ

−
A
)
∉

Φ

(
X
)

({\mathbf {\lambda }} - A) \notin {\mathbf {\Phi }}(X)

. In this paper we establish conditions under which

σ

Φ

(
A
+
B
)
⊆

σ

Φ

(
A
)
+

σ

Φ

(
B
)
,

σ

Φ

(

B
A

¯

)
⊆

σ

Φ

(
A
)
⋅

σ

Φ

(
B
)

{\sigma _{\mathbf {\Phi }}}(A + B) \subseteq {\sigma _{\mathbf {\Phi }}}(A) + {\sigma _{\mathbf {\Phi }}}(B),{\sigma _{\mathbf {\Phi }}}(\overline {BA} ) \subseteq {\sigma _{\mathbf {\Phi }}}(A) \cdot {\sigma _{\mathbf {\Phi }}}(B)

, and

σ
Φ

(
A
B
)
⊆

σ
Φ

(
A
)

σ
Φ

(
B
)

{\sigma _\Phi }(AB) \subseteq {\sigma _\Phi }(A){\sigma _\Phi }(B)

.
Two weight function norm inequalities for the Poisson integral
Authors: Muckenhoupt Benjamin 
Published: July 2010

Let

f
(
x
)

f(x)

denote a complex valued function with period

2
π

2\pi

, let
\[

P
r

(
f
,
x
)
=

1

2
π

∫

−
π

π

(
1
−

r
2

)
f
(
y
)
d
y

1
−
2
r
cos
⁡
(
x
−
y
)
+

r
2

{P_r}(f,x) = \frac {1}{{2\pi }}\int _{ - \pi }^\pi {\frac {{(1 - {r^2})f(y)dy}}{{1 - 2r\cos (x - y) + {r^2}}}}

\]
be the Poisson integral of

f
(
x
)

f(x)

and let

|

I

|

|I|

denote the length of an interval

I
I

. For

1
⩽
p
>
∞

1 \leqslant p > \infty

and nonnegative

U
(
x
)

U(x)

and

V
(
x
)

V(x)

with period

2
π

2\pi

it is shown that there is a

C
C

, independent of

f
f

, such that
\[

sup

0
⩽
r
>
1

∫

−
π

π

|

P
r

(
f
,
x
)

|

p

U
(
x
)
d
x
⩽
C

∫

−
π

π

|

f
(
x
)

|

p

V
(
x
)
d
x

\sup \limits _{0 \leqslant r > 1} \int _{ - \pi }^\pi {|{P_r}(f,x){|^p}U(x)dx \leqslant C\int _{ - \pi }^\pi {|f(x){|^p}V(x)dx} }

\]
if and only if there is a

B
B

such that for all intervals

I
I

\[

[

1

|

I

|

∫
I

U
(
x
)
d
x

]

[

1

|

I

|

∫
I

[
V
(
x
)
]

−
1

/

(
p
−
1
)

d
x

]

p
−
1.

⩽
B
.

\left [ {\frac {1}{{|I|}}\int _I {U(x)dx} } \right ]{\left [ {\frac {1}{{|I|}}\int _I {{{[V(x)]}^{ - 1/(p - 1)}}dx} } \right ]^{p - 1.}} \leqslant B.

\]
Similar results are obtained for the nonperiodic case and in the case where

U
(
x
)
d
x

U(x)dx

and

V
(
x
)
d
x

V(x)dx

are replaced by measures.
Duality for Hadamard products with applications to extremal problems for functions regular in the unit disc
Authors: Ruscheweyh Stephan 
Published: July 2010

Let

A
A

be the set of functions regular in the unit disc

U

\mathcal {U}

and

A
0

{A_0}

the set of all functions

f
∈
A

f \in A

which satisfy

f
(
0
)
=
1

f(0) = 1

. For

V
⊂

A
0

V \subset {A_0}

define the dual set

V
∗

=
{
f
∈

A
0

|

f
∗
g
≠
0

for all

g
∈
V
,
z
∈

U

}
,

V

∗
∗

=

(

V
∗

)
∗

{V^ \ast } = \{ f \in {A_0}|f \ast g \ne 0{\text { for all }}g \in V,z \in \mathcal {U}\} ,{V^{ \ast \ast }} = {({V^ \ast })^ \ast }

. Here

f
∗
g

f \ast g

denotes the Hadamard product. THEOREM. Let

V
⊂

A
0

V \subset {A_0}

have the following properties: (i)

V
V

is compact, (ii)

f
∈
V

f \in V

implies

f
(
x
z
)
∈
V

f(xz) \in V

for all

|

x

|

⩽
1

|x| \leqslant 1

. Then

λ
(
V
)
=
λ
(

V

∗
∗

)

\lambda (V) = \lambda ({V^{ \ast \ast }})

for all continuous linear functionals

λ
\lambda

on

A
A

. This theorem has many applications to functions in

A
A

which are defined by properties like bounded real part, close-to-convexity, univalence etc.
Mutual existence of product integrals in normed rings
Authors: Helton Jon C. 
Published: July 2010

Definitions and integrals are of the subdivision-refinement type, and functions are from

R
×
R

R \times R

to N, where R denotes the set of real numbers and N denotes a ring which has a multiplicative identity element represented by 1 and a norm

|

⋅

|

| \cdot |

with respect to which N is complete and

|

1

|

=
1

|1| = 1

. If G is a function from

R
×
R

R \times R

to N, then

G
∈
O

M
∗

G \in O{M^\ast }

on [a, b] only if (i)

x

Π
y

(
1
+
G
)

_x{\Pi ^y}(1 + G)

exists for

a
≤
x
>
y
≤
b

a \leq x > y \leq b

and (ii) if

ε
>
0

\varepsilon > 0

, then there exists a subdivision D of [a, b] such that, if

{

x
i

}

i
=
0

n

\{ {x_i}\} _{i = 0}^n

is a refinement of D and

0
≤
p
>
q
≤
n

0 \leq p > q \leq n

, then
\[

|

x

p

∏

x
q

(
1
+
G
)
−

∏

i
=
p
+
1

q

(
1
+

G
i

)

|

>
ε
;

\left |{}_{x_{p}}\prod ^{x_q} (1 + G) - \prod \limits _{i = p + 1}^q {(1 + {G_i})} \right | > \varepsilon ;

\]
and

G
∈
O

M
∘

G \in O{M^ \circ }

on [a, b] only if (i)

x

Π
y

(
1
+
G
)

_x{\Pi ^y}(1 + G)

exists for

a
≤
x
>
y
≤
b

a \leq x > y \leq b

and (ii) the integral

∫
a
b

|

1
+
G
−
Π
(
1
+
G
)

|

\smallint _a^b|1 + G - \Pi (1 + G)|

exists and is zero. Further,

G
∈
O

P
∘

G \in O{P^ \circ }

on [a, b] only if there exist a-subdivision D of [a, b] and a number B such that, if

{

x
i

}

i
=
0

n

\{ {x_i}\} _{i = 0}^n

is a refinement of D and

0
>
p
≤
q
≤
n

0 > p \leq q \leq n

, then

|

Π

i
=
p

q

(
1
+

G
i

)

|

>
B

|\Pi _{i = p}^q(1 + {G_i})| > B

. If F and G are functions from

R
×
R

R \times R

to N,

F
∈
O

P
∘

F \in O{P^ \circ }

on [a, b], each of

lim

x
,
y
→

p
+

F
(
x
,
y
)

{\lim _{x,y \to {p^ + }}}F(x,y)

and

lim

x
,
y
→

p
−

F
(
x
,
y
)

{\lim _{x,y \to {p^ - }}}F(x,y)

exists and is zero for

p
∈
[
a
,
b
]

p \in [a,b]

, each of

lim

x
→

p
+

F
(
p
,
x
)
,

lim

x
→

p
−

F
(
x
,
p
)
,

lim

x
→

p
+

G
(
p
,
x
)

{\lim _{x \to {p^ + }}}F(p,x),{\lim _{x \to {p^ - }}}F(x,p),{\lim _{x \to {p^ + }}}G(p,x)

and

lim

x
→

p
−

G
(
x
,
p
)

{\lim _{x \to {p^ - }}}G(x,p)

exists for

p
∈
[
a
,
b
]

p \in [a,b]

, and G has bounded variation on [a, b], then any two of the following statements imply the other: (1)

F
+
G
∈
O

M
∗

F + G \in OM^\ast

on [a, b], (2)

F
∈
O

M
∗

F \in OM^\ast

on [a, b], and (3)

G
∈
O

M
∗

G \in OM^\ast

on [a, b]. In addition, with the same restrictions on F and G, any two of the following statements imply the other: (1)

F
+
G
∈
O

M
∘

F + G \in OM^\circ

on [a, b], (2)

F
∈
O

M
∘

F \in OM^\circ

on [a, b], and (3)

G
∈
O

M
∘

G \in OM^\circ

on [a, b]. The results in this paper generalize a theorem contained in a previous paper by the author [Proc. Amer. Math. Soc. 42 (1974), 96-103]. Additional background on product integration can be obtained from a paper by B. W. Helton [Pacific J. Math. 16 (1966), 297-322].
Spectral analysis of finite convolution operators
Authors: Frankfurt Richard 
Published: July 2010

In this paper the similarity problem for operators of the form

(
∗
)

T
:
f
(
x
)
→

∫
0
x

k
(
x
−
t
)
f
(
t
)
d
t

( \ast )\;T:f(x) \to \smallint _0^xk(x - t)f(t)dt

on

L
2

(
0
,
1
)

{L^2}(0,1)

is studied. Let

K
(
z
)
=

∫
0
1

k
(
t
)

e

i
t
z

d
t

K(z) = \smallint _0^1\;k(t){e^{itz}}dt

. A function

C
(
z
)

C(z)

is called a symbol for T if

C
(
z
)

C(z)

can be written in the form

C
(
z
)
=
K
(
z
)
+

e

i
z

G
(
z
)

C(z) = K(z) + {e^{iz}}G(z)

, where

G
(
z
)

G(z)

is a function bounded and analytic in a half plane

y
>
δ

y > \delta

, for some real number

δ
\delta

. Under suitable restrictions, it is shown that two operators of the form

(
∗
)

( \ast )

will be similar if they possess symbols which are asymptotically close together as

z
→
∞

z \to \infty

in some half plane

y
>
δ

y > \delta

.
Pointwise bounds on eigenfunctions and wave packets in 𝑁-body quantum systems. III
Authors: Simon Barry 
Published: July 2010

We provide a number of bounds of the form

|

ψ

|

⩽
O
(
exp
⁡
(
−
α

|

x

|

α

)
)
,
α
>
1

|\psi | \leqslant O(\exp ( - \alpha |x{|^\alpha })),\alpha > 1

, for

L
2

{L^2}

-eigenfunctions

ψ
\psi

of

−
Δ
+
V

- \Delta + V

with

V
→
∞

V \to \infty

rapidly as

|

x

|

→
∞

|x| \to \infty

. Our strongest results assert that if

|

V
(
x
)

|

⩾
c

x

2
m

|V(x)| \geqslant c{x^{2m}}

near infinity, then

|

ψ
(
x
)

|

⩽

D
ε

exp
⁡
(
−

(
c
−
ε

)

1

/

2

(
m
+
1

)

−
1

x

m
+
1

)

|\psi (x)| \leqslant {D _\varepsilon }\exp ( - {(c - \varepsilon )^{1/2}}{(m + 1)^{ - 1}}{x^{m + 1}})

, and if

|

V
(
x
)

|

⩽
c

x

2
m

|V(x)| \leqslant c{x^{2m}}

neat infinity, then for the ground state eigenfunction,

Ω
,
Ω
(
x
)
⩾

E
ε

exp
⁡
(
−

(
c
+
ε

)

1

/

2

(
m
+
1

)

−
1

x

m
+
1

)

\Omega ,\Omega (x) \geqslant {E _\varepsilon }\exp ( - {(c + \varepsilon )^{1/2}}{(m + 1)^{ - 1}}{x^{m + 1}})

.
An asymptotic formula for an integral in starlike function theory
Authors: London R. R. , Thomas D. K. 
Published: July 2010

The paper is concerned with the integral
\[

H
=

∫
0

2
π

|

f

|

σ

|

F

|

τ

(
Re
⁡
F

)
κ

d
θ

H = \int _0^{2\pi }|f{|^\sigma }|F{|^\tau }{(\operatorname {Re} F)^\kappa }\;d\theta

\]
in which f is a function regular and starlike in the unit disc,

F
=
z

f
′

/

f

F = zf’/f

, and the parameters

σ
,
τ
,
κ

\sigma ,\tau ,\kappa

are real. A study of H is of interest since various well-known integrals in the theory, such as the length of

f
(

|

z

|

=
r
)

f(|z| = r)

, the area of

f
(

|

z

|

⩽
r
)

f(|z| \leqslant r)

, and the integral means of f, are essentially obtained from it by suitably choosing the parameters. An asymptotic formula, valid as

r
→
1

r \to 1

, is obtained for H when f is a starlike function of positive order

α
\alpha

, and the parameters satisfy

α
σ
+
τ
+
κ
>
1
,
τ
+
κ
⩾
0
,
κ
⩾
0
,
σ
>
0

\alpha \sigma + \tau + \kappa > 1,\tau + \kappa \geqslant 0,\kappa \geqslant 0,\sigma > 0

. Several easy applications of this result are made; some to obtaining old results, two others in proving conjectures of Holland and Thomas.
Partitions of large multipartites with congruence conditions. I
Authors: Robertson M. M. , Spencer D. 
Published: July 2010

Let

p
(

n
1

,
…
,

n
j

:

A
1

,
…
,

A
j

)

p({n_1}, \ldots ,{n_j}:{A_1}, \ldots ,{A_j})

be the number of partitions of

(

n
1

,
…
,

n
j

)

({n_1}, \ldots ,{n_j})

where, for

1
⩽
l
⩽
j

1 \leqslant l \leqslant j

, the lth component of each part belongs to the set

A
l

=

⋃

h
(
l
)
=
1

q
(
l
)

{

a

l
h
(
l
)

+
M
v
:
v
=
0
,
1
,
2
,
…
}

{A_l} = \bigcup \nolimits _{h(l) = 1}^{q(l)} {\{ {a_{lh(l)}} + Mv :v = 0,1,2, \ldots \} }

and

M
,
q
(
l
)

M,q(l)

and the

a

l
h
(
l
)

{a_{lh(l)}}

are positive integers such that

0
>

a

l
1

>
⋯
>

a

l
q
(
l
)

⩽
M

0 > {a_{l1}} > \cdots > {a_{lq(l)}} \leqslant M

. Asymptotic expansions for

p
(

n
1

,
…
,

n
j

:

A
1

,
…
,

A
j

)

p({n_1}, \ldots ,{n_j}:{A_1}, \ldots ,{A_j})

are derived, when the

n
l

→
∞

{n_l} \to \infty

subject to the restriction that

n
1

⋯

n
j

⩽

n
l

j
+
1
−
∈

{n_1} \cdots {n_j} \leqslant n_l^{j + 1 - \in }

for all l, where

∈
\in

is any fixed positive number. The case

M
=
1

M = 1

and arbitrary j was investigated by Robertson [10] while several authors between 1940 and 1960 investigated the case

j
=
1

j = 1

for different values of M.
𝐿-functions of a quadratic form
Authors: Callahan T. , Smith R. A. 
Published: July 2010

Let Q be a positive definite integral quadratic form in n variables, with the additional property that the adjoint form

Q
†

{Q^\dagger }

is also integral. Using the functional equation of the Epstein zeta function, we obtain a symmetric functional equation of the L-function of Q with a primitive character

ω
mod
q

\omega \bmod q

(additive or multiplicative) defined by

Σ
ω
(
Q
(

x

)
)
Q

(

x

)

−
s

,
Re
⁡
(
s
)
>
n

/

2

\Sigma \omega (Q({\text {x}}))Q{({\text {x}})^{ - s}},\operatorname {Re} (s) > n/2

, where the summation extends over all

x

∈

Z
n

,

x

≠
0

{\text {x}} \in {Z^n},{\text {x}} \ne 0

; our result does not depend upon the usual restriction that q be relatively prime to the discriminant of Q, but rather on a much milder restriction.
Asymmetric maximal ideals in 𝑀(𝐺)
Authors: Saeki Sadahiro 
Published: July 2010

Let G be a nondiscrete LCA group,

M
(
G
)

M(G)

the measure algebra of G, and

M
0

(
G
)

{M_0}(G)

the closed ideal of those measures in

M
(
G
)

M(G)

whose Fourier transforms vanish at infinity. Let

Δ
G

,

Σ
G

{\Delta _G},{\Sigma _G}

and

Δ
0

{\Delta _0}

be the spectrum of

M
(
G
)

M(G)

, the set of all symmetric elements of

Δ
G

{\Delta _G}

, and the spectrum of

M
0

(
G
)

{M_0}(G)

, respectively. In this paper this is shown: Let

Φ
\Phi

be a separable subset of

M
(
G
)

M(G)

. Then there exist a probability measure

τ
\tau

in

M
0

(
G
)

{M_0}(G)

and a compact subset X of

Δ
0

∖

Σ
G

{\Delta _0}\backslash {\Sigma _G}

such that for each

|

c

|

⩽
1

|c| \leqslant 1

and each
\[

ν
∈
Φ

Card

{
f
∈
X
:

τ
^

(
f
)
=
c

and

|

ν
^

(
f
)

|

=
r
(
ν
)
}
⩾

2

c

.

\nu \in \Phi \;{\text {Card}}\;\{ f \in X:\hat \tau (f) = c\;{\text {and}}\;|\hat \nu (f)| = r(\nu )\} \geqslant {2^{\text {c}}}.

\]
Here

r
(
ν
)
=
sup
{

|

ν
^

(
f
)

|

:
f
∈

Δ
G

∖

G
^

}

r(\nu ) = \sup \{ |\hat \nu (f)|:f \in {\Delta _G}\backslash \hat G\}

. As immediate consequences of this result, we have (a) every boundary for

M
0

(
G
)

{M_0}(G)

is a boundary for

M
(
G
)

M(G)

(a result due to Brown and Moran), (b)

Δ
G

∖

Σ
G

{\Delta _G}\backslash {\Sigma _G}

is dense in

Δ
G

∖

G
^

{\Delta _G}\backslash \hat G

, (c) the set of all peak points for

M
(
G
)

M(G)

is

G
^

\hat G

if G is

σ
\sigma

-compact and is empty otherwise, and (d) for each

μ
∈
M
(
G
)

\mu \in M(G)

the set

μ
^

(

Δ
0

∖

Σ
G

)

\hat \mu ({\Delta _0}\backslash {\Sigma _G})

contains the topological boundary of

μ
^

(

Δ
G

∖

G
^

)

\hat \mu ({\Delta _G}\backslash \hat G)

in the complex plane.
Positive definite measures with applications to a Volterra equation
Authors: Staffans Olof J. 
Published: July 2010

We study the asymptotic behavior of the solutions of the nonlinear Volterra integrodifferential equation

(

∗

)

x
′

(
t
)
+

∫
0
t

g
(
x
(
t
−
τ
)
)

d
μ
(
τ
)
=
f
(
t
)
,

\begin{equation}\tag {$\ast $} x’(t) + \int _0^t {g(x(t - \tau ))\;d\mu (\tau ) = f(t),}\end{equation}

with a positive definite kernel

μ
\mu

. In particular, we give new sufficient conditions on the kernel

μ
\mu

, which together with standard assumptions on f and g yield results on boundedness and asymptotic behavior of the solutions of

(
∗
)

(\ast )

. Our proofs are based on the theory of distribution Fourier transforms.
A law of the iterated logarithm for stable summands
Authors: Pakshirajan R. P. , Vasudeva R. 
Published: July 2010

Let

X
1

,

X
2

,
…

{X_1},{X_2}, \ldots

be a sequence of independent indentically distributed stable random variables with parameters

α

(
0
>
α
>
2
)

\alpha \;(0 > \alpha > 2)

and

β
(

|

β

|

⩽
1
)

\beta (|\beta | \leqslant 1)

. Let

S
n

=

∑

i
=
1

n

X
i

{S_n} = \sum \nolimits _{i = 1}^n {{X_i}}

. Suppose that

(

S

1
,
n

)

({S_{1,n}})

and

(

S

2
,
n

)

({S_{2,n}})

are independent copies of the sequence

(

S
n

)

({S_n})

. In this paper we obtain the set of all limit points in the plane of the sequence
\[

{

|

n

−
1

/

α

(

S

1
,
n

−

a
n

)

|

1

/

(
log
⁡
log
⁡
n
)

,

|

n

−
1

/

α

(

S

2
,
n

−

a
n

)

|

1

/

(
log
⁡
log
⁡
n
)

}

\left \{ {|{n^{ - 1/\alpha }}({S_{1,n}} - {a_n}){|^{1/(\log \log n)}},|{n^{ - 1/\alpha }}({S_{2,n}} - {a_n}){|^{1/(\log \log n)}}} \right \}

\]
where

(

a
n

)

({a_n})

is zero if

α
≠
1

\alpha \ne 1

and is

(
2
β
n
log
⁡
n
)

/

π

(2\beta n\log n)/\pi

if

α
=
1

\alpha = 1

.
A superposition theorem for unbounded continuous functions
Authors: Doss Raouf 
Published: July 2010

Let

R
n

{R^n}

be the n-dimensional Euclidean space. We prove that there are 4n real functions

φ
q

{\varphi _q}

continuous on

R
n

{R^n}

with the following property: Every real function f, not necessarily bounded, continuous on

R
n

{R^n}

, can be written

f
(
x
)
=

Σ

q
=
1

2
n
+
1

g
(

φ
q

(
x
)
)
+

Σ

q
=
2
n
+
2

4
n

h
(

φ
q

(
x
)
)
,
x
∈

R
n

f(x) = \Sigma _{q = 1}^{2n + 1}g({\varphi _q}(x)) + \Sigma _{q = 2n + 2}^{4n}h({\varphi _q}(x)),x \in {R^n}

, where g, h are 2 real continuous functions of one variable, depending on f.
On the number of real zeros of a random trigonometric polynomial
Authors: Sambandham M. 
Published: July 2010

For the random trigonometric polynomial
\[

∑

n
=
1

N

g
n

(
t
)
cos
⁡
n
θ
,

\sum \limits _{n = 1}^N {{g_n}(t)\cos n\theta ,}

\]
where

g
n

(
t
)
,
0
⩽
t
⩽
1

{g_n}(t),0 \leqslant t \leqslant 1

, are dependent normal random variables with mean zero, variance one and joint density function
\[

|

M

|

1

/

2

(
2
π

)

−
N

/

2

exp
⁡
[
−
(
1

/

2
)

a
¯

′

M

a
¯

]

|M{|^{1/2}}{(2\pi )^{ - N/2}}\exp [ - (1/2)\bar a’M\bar a]

\]
where

M

−
1

{M^{ - 1}}

is the moment matrix with

ρ

i
j

=
ρ
,
0
>
ρ
>
1
,
i
≠
j
,
i
,
j
=
1
,
2
,
…
,
N

{\rho _{ij}} = \rho ,0 > \rho > 1,i \ne j,i,j = 1,2, \ldots ,N

and

a
¯

\bar a

is the column vector, we estimate the probable number of zeros.
On the growth of the integral means of subharmonic functions of order less than one
Authors: Abi-Khuzam Faruk F. 
Published: July 2010

Let u be a subharmonic function of order

λ
(
0
>
λ
>
1
)

\lambda (0 > \lambda > 1)

, and let

m
s

(
r
,
u
)

=

{

(
1

/

2
π
)

∫

−
π

π

|

u
(
r

e

i
θ

)

|

s

d
θ

}

1

/

s

{m_s}(r,u)\, = \,{\left \{ {(1/2\pi )\int _{ - \pi }^\pi {{{\left | {u(r{e^{i\theta }})} \right |}^s}} d\theta } \right \}^{1/s}}

. We compare the growth of

m
s

(
r
,
u
)

{m_s}(r,u)

with that of the Riesz mass of u as measured by

N

(
r
,
u
)

=
(
1

/

2
π
)

∫

−
π

π

u
(
r

e

i
θ

)
d
θ

N\,(r,u)\, = (1/2\pi )\int _{ - \pi }^\pi {u(r{e^{i\theta }})d\theta }

. A typical result of this paper states that the following inequality is sharp:

(

∗

)

lim

inf

x
→
∞

m

s

(
r
,

u
)

N

(
r
,

u
)

⩽

m

s

(

ψ

λ

)

\begin{equation} \tag {$\ast $} \underset {x\to \infty }{\lim \,\inf }\,\,\frac {{{m}_{s}}\left ( r,\,u \right )}{N\left ( r,\,u \right )}\,\leqslant \,{{m}_{s}}\left ( {{\psi }_{\lambda }} \right )\end{equation}

where

ψ
λ

(
θ
)

=

(
π
λ

/

sin

λ
)
cos

λ
θ

\psi _\lambda (\theta )\, = \,(\pi \lambda /\sin \,\lambda )\cos \,\lambda \theta

. The case

s

=

1

s\, = \,1

is due to Edrei and Fuchs, the case

s

=

2

s\, = \,2

is due to Miles and Shea and the case

s

=

∞

s\, = \,\infty

is due to Valiron.
Periodic solutions for a differential equation in Banach space
Authors: Lightbourne James H. 
Published: July 2010

Suppose X is a Banach space,

Ω
⊂
X

\Omega \subset X

is closed and convex, and

A
:
[
0
,
∞
)
×
Ω
→
X

A:[0,\infty ) \times \Omega \to X

is continuous. Then if
\[

lim

h
→
0

|

x
+
h
A
(
t
,
x
)
;
Ω

|

/

h
=
0

for

all

(
t
,
x
)
∈
[
0
,
∞
)
×
Ω
,

\lim \limits _{h \to 0} |x + hA(t,x);\Omega |/h = 0\quad {\text {for}}\;{\text {all}}\;(t,x) \in [0,\infty ) \times \Omega ,

\]
there exist approximate solutions to the initial value problem

(

I
V
P

)

u
′

(
t
)
=
A
(
t
,
u
(
t
)
)
,

u
(
0
)
=
x
∈
Ω
.

\begin{equation}\tag {$IVP$}u’(t) = A(t,u(t)),\quad u(0) = x \in \Omega .\end{equation}

In the case that

A
(
t
,
x
)
=
B
(
t
,
x
)
+
C
(
t
,
x
)

A(t,x) = B(t,x) + C(t,x)

, where B satisfies a dissipative condition and C is compact, we obtain a growth estimate on the measure of noncompactness of trajectories for a class of approximate solutions. This estimate is employed to obtain existence of periodic solutions to (IVP).
The order and symbol of a distribution
Authors: Weinstein Alan 
Published: July 2010

A definition is given, for an arbitrary distribution g on a manifold X, of the order and symbol of g at a point

(

χ
,
ξ

)

({\chi ,\xi })

of the cotangent bundle

T

∗

X

T^{\ast }X

. If

X
=

R

n

X = \textbf {R}^n

, the order of g at

(

0
,
ξ

)

({0,\xi })

is the growth order as

τ
→
∞

\tau \to \infty

of the distributions

g
τ

(
x
)
=

e

−
i

τ

⟨
x
,
ξ
⟩

g

(

x

/

τ

)

{g^\tau }(x) = {e^{ - i\sqrt \tau \langle x,\xi \rangle }}g\left ( {x /\sqrt \tau } \right )

; if the order is less than or equal to N, the N-symbol of g is the family

g
τ

{g^\tau }

modulo

O
(

τ

N
−
1

/

2

)

O({{\tau ^{N - 1/2}}})

. It is shown that the order and symbol behave in a simple way when g is acted upon by a pseudo-differential operator. If g is a Fourier integral distribution, suitable identifications can be made so that the symbol defined here agrees with the bundle-valued symbol defined by Hörmander.
An integral Riemann-Roch formula for induced representations of finite groups
Authors: Evens Leonard , Kahn Daniel S. 
Published: July 2010

Let H be a subgroup of the finite group G,

ξ
\xi

a finite dimensional complex representation of H and

ρ
\rho

the induced representation of G. If

s
k

(
ρ
)
∈

H

2
k

(
G
,

Z

)

{s_k}(\rho ) \in {H^{2k}}(G,\textbf {Z})

,

k
⩾
1

k \geqslant 1

denote the characteristic classes bearing the same relation to power sums that Chern classes bear to elementary symmetric functions, then we prove the following,

N
¯

(
k
)
(

s
k

(
ρ
)
−

T

r

H
→
G

(

s
k

(
ξ
)
)

)
=
0
,

\begin{equation} \bar N (k)( {{s_k}(\rho ) - {\text {T}}{{\text {r}}_{H \to G}}({s_k}(\xi ))}) = 0, \end{equation}

where

N
¯

(
k
)
=

∏

p

|

N
(
k
)

p

prime

p

\begin{equation} \bar N(k) = {\prod _{\begin {array}{*{20}{c}} {p|N(k)} \\ {p{\text {prime}}} \\ \end{array}}}p \end{equation}

and

N
(
k
)
=

(

∏

p

prime

p

[
k

/

p
−
1
]

)

/

k
!
.

\begin{equation} N(k) = \left ( {\begin {array}{*{20}{c}} {\prod \limits _{p{\text {prime}}} {{p^{[k/p - 1]}}}} \end{array} } \right )/k!. \end{equation}

(Tr denotes transfer.) Moreover,

N
¯

(
k
)

\bar N (k)

is the least integer with this property. This settles a question originally raised in a paper of Knopfmacher in which it was conjectured that the required bound was N(k).
A new approach to the limit theory of recurrent Markov chains
Authors: Athreya K. B. , Ney P. 
Published: July 2010

Let

{

X
n

;

n
⩾
0
}

\{ {X_n};\,n \geqslant 0\}

be a Harris-recurrent Markov chain on a general state space. It is shown that there is a sequence of random times

{

N
i

;

i
⩾
1
}

\{ {N_i};\,i \geqslant 1\}

such that

{

X

N
i

;

i
⩾
1
}

\{ {X_{{N_i}}};{\text { }}i \geqslant 1\}

are independent and identically distributed. This idea is used to show that

{

X
n

}

\{ {X_n}\}

is equivalent to a process having a recurrence point, and to develop a regenerative scheme which leads to simple proofs of the ergodic theorem, existence and uniqueness of stationary measures.