bluuebunny/crossref_metadata_2025_split huggingface dataset:
A tool in establishing total variation convergence
Authors: Parthasarathy K. R. , Steerneman Ton 
Published: June 2010

Let

X
0

,

X
1

,

X
2

,
…

and

Y
0

,

Y
1

,

Y
2

,
…

{X_0},{X_1},{X_2}, \ldots {\text { and }}{Y_0},{Y_1},{Y_2}, \ldots

be sequences of random variables where

X
n

{X_n}

and

Y
n

{Y_n}

are independent,

L

X
n

→
L

X
0

L{X_n} \to L{X_0}

in total variation and

L

Y
n

→
L

Y
0

L{Y_n} \to L{Y_0}

in distribution. For certain mappings

T
T

sufficient conditions are given in order that

L
T

(

X
n

,

Y
n

)

→
L
T

(

X
0

,

Y
0

)

LT\left ( {{X_n},{Y_n}} \right ) \to LT\left ( {{X_0},{Y_0}} \right )

in total variation. For example, if

(

R

k

,

B
k

)

\left ( {{{\mathbf {R}}^k},{B_k}} \right )

is the outcome space of the

X
n

{X_n}

and

Y
n

{Y_n}

, and if

L

X
0

L{X_0}

is absolutely continuous (with respect to Lebesgue measure), then

L

(

X
n

+

Y
n

)

→
L

(

X
0

+

Y
0

)

L\left ( {{X_n} + {Y_n}} \right ) \to L\left ( {{X_0} + {Y_0}} \right )

in total variation.
On a factorization problem for convergent sequences and on Hankel forms in bounded sequences
Authors: Eggermont P. P. B. , Leung Y. J. 
Published: June 2010

We solve in the negative the following factorization problem of S. Mazur: Can every convergent sequence be written as

z
(
n
)
=

(
n
+
1

)

−
1

∑

i
=
0

n

x
(
i
)
y
(
n
−
i
)
,
n
=
0
,
1
,
…

z(n) = {(n + 1)^{ - 1}}\sum \nolimits _{i = 0}^n {x(i)y(n - i),n = 0,1, \ldots }

, with convergent sequences

x
x

and

y
y

? This problem also yields the solution of another problem of S. Mazur regarding bounded Hankel forms on the space of all bounded sequences.
Properties of endomorphism rings of modules and their duals
Authors: Khuri Soumaya Makdissi 
Published: June 2010

Let

R

M

_RM

be a nonsingular left

R
R

-module whose Morita context is nondegenerate, let

B
=

End

R

⁡
M

B = \operatorname {End}_{R}M

and let

M
∗

=

Hom

R

⁡
(
M
,
R
)

{M^ * } = \operatorname {Hom}_{R}(M,R)

. We show that

B
B

is left (right) strongly modular if and only if any element of

B
B

which has zero kernel in

R

M
(

M
R
∗

)

_RM(M_R^ * )

has essential image in

R

M
(

M
R
∗

)

_RM(M_R^ * )

, and that

B
B

is a left (right) Utumi ring if and only if every submodule

R

U

o

f

R

M
(

U
R
∗

of

M
R
∗

)

_RU{\text {o}}{{\text {f}}_R}M(U_R^ * {\text {of }}M_R^ * )

such that

U
⊥

=
0

(
⊥

U
∗

=
0
)

{U^ \bot } = 0{(^ \bot }{U^ * } = 0)

is essential in

R

M
(

M
R
∗

)

_RM(M_R^ * )

.
A new inequality for complex-valued polynomial functions
Authors: Rassias Themistocles M. 
Published: June 2010

Let

f
1

,

f
2

,
…
,

f
n

:

C

→

C

{f_1},{f_2}, \ldots ,{f_n}:{\mathbf {C}} \to {\mathbf {C}}

be complex-valued polynomial functions of degrees

d
1

,

d
2

,
…
,

d
n

{d_1},{d_2}, \ldots ,{d_n}

, respectively, of a complex variable

z
z

. Then
\[

M

f
1

M

f
2

⋯

M

f
n

≥

M

f
1

f
2

⋯

f
n

≥
k

M

f
1

M

f
2

⋯

M

f
n

{M_{{f_1}}}{M_{{f_2}}} \cdots {M_{{f_n}}} \geq {M_{{f_1}{f_2} \cdots {f_n}}} \geq k{M_{{f_1}}}{M_{{f_2}}} \cdots {M_{{f_n}}}

\]
where
\[

k
=

(

sin
⁡

2
n

π

8

d
1

)

d
1

(

sin
⁡

2
n

π

8

d
2

)

d
2

⋯

(

sin
⁡

2
n

π

8

d
n

)

d
n

.

k = {\left ( {\sin \frac {2}{n}\frac {\pi } {{8{d_1}}}} \right )^{{d_1}}}{\left ( {\sin \frac {2} {n}\frac {\pi }{{8{d_2}}}} \right )^{{d_2}}} \cdots {\left ( {\sin \frac {2}{n}\frac {\pi }{{8{d_n}}}} \right )^{{d_n}}}.

\]
Class numbers of pure fields
Authors: Mollin R. A. 
Published: June 2010

Necessary and sufficient conditions are given for the class number

h

K
i

{h_{{K_i}}}

of a pure field

K
=
Q
(

m

1

/

p
i

)

(

for

i
=
1
,
2
)

K = Q({m^{1/{p^i}}}){\text { }}({\text {for }}i = 1,2)

to be divisible by

p
r

{p^r}

for a given positive integer

r
r

and prime

p
p

. Moreover the divisibility of

h

K
i

{h_{{K_i}}}

by

p
p

is linked with the

p
p

-rank of the class group of the

K
(
ς
)

K(\varsigma )

and prime divisors of

m
m

, where

ς
\varsigma

is a primitive

p
p

th root of unity. Finally we prove in an easy fashion that for a given odd prime

p
p

and any natural number

t
t

there exist infinitely many non-Galois algebraic number fields (in fact pure fields) of degree

p
i

(
i
=
1
,
2
)

{p^i}(i = 1,2)

over

Q
Q

whose class numbers are all divisible by

p
t

{p^t}

.
Minimal degrees of faithful characters of finite groups with a T.I. Sylow 𝑝-subgroup
Authors: Berger T. R. , Landrock P. , Michler G. O. 
Published: June 2010

Using the classification of the finite simple groups we show in this article that a faithful complex character

χ
\chi

of a finite group

G
G

with a nonnormal T.I. Sylow

p
p

-subgroup

P
P

has degree

χ
(
1
)

>

|
P
|

−
1

\chi (1){\text { > }}\sqrt {\left | P \right |} - 1

. This result verifies a conjecture of H. S. Leonard [10].
On biduals of 𝐶*-tensor products
Authors: Quigg John C. 
Published: June 2010

Huruya [4] has proven that, for

C
∗

{C^*}

-algebras

A
1

{A_1}

and

A
2

{A_2}

,
\[

(

A
1

⊗

A
2

)

∗
∗

=

A
1

∗
∗

⊗
¯

A
2

∗
∗

{({A_1} \otimes {A_2})^{**}} = A_1^{**}\overline \otimes A_2^{**}

\]
for every

A
2

{A_2}

if and only if

A
1

{A_1}

is scattered. We strengthen this by proving that

(

A
1

⊗

A
2

)

∗
∗

=

A
1

∗
∗

⊗
¯

A
2

∗
∗

{({A_1} \otimes {A_2})^{**}} = A_1^{**}\overline \otimes A_2^{**}

if and only if

A
1

{A_1}

or

A
2

{A_2}

is scattered. We discuss ramifications to representation theory and related questions regarding normal representations of

W
∗

{W^*}

-tensor products.
On the singularities of the continuous Jacobi transform when 𝛼+𝛽=0
Authors: Zayed Ahmed I. 
Published: June 2010

Let

α
,
β
>
−
1

\alpha ,\beta > - 1

and

P

λ

(
α
,
β
)

(
x
)
=

(
1
−
x

)
α

(
1
+
x

)
β

P
λ

(
α
,
β
)

(
x
)

\mathcal {P}_\lambda ^{(\alpha ,\beta )}(x) = {(1 - x)^\alpha }{(1 + x)^\beta }P_\lambda ^{(\alpha ,\beta )}(x)

, where

P
λ

(
α
,
β
)

(
x
)

P_\lambda ^{(\alpha ,\beta )}(x)

is the Jacobi function of the first kind,

λ
≥
−
(
α
+
β
+
1
)

/

2

\lambda \geq - (\alpha + \beta + 1)/2

, and

−
1
>
x
≤
1

- 1 > x \leq 1

. Let
\[

F

(
α
,
β
)

(
λ
)
=

1

2

α
+
β
+
1

⟨

f
(
x
)
,

P

λ

(
α
,
β
)

(
x
)

⟩

=

1

2

α
+
β
+
1

∫

−
1

1

f
(
x
)

P

λ

(
α
,
β
)

(
x
)
d
x

{F^{(\alpha ,\beta )}}(\lambda ) = \frac {1} {{{2^{\alpha + \beta + 1}}}}\left \langle {f(x),\mathcal {P}_\lambda ^{(\alpha ,\beta )}(x)} \right \rangle = \frac {1} {{{2^{\alpha + \beta + 1}}}}\int _{ - 1}^1 {f(x)\mathcal {P}_\lambda ^{(\alpha ,\beta )}(x)dx}

\]
whenever the integral exists. It is known that for

α
+
β
=
0

\alpha + \beta = 0

, we have (*)
\[

f
(
x
)
=

lim

n
→
∞

4

∫
0
n

F

(
α
,
β
)

(

λ
−

1
2

)

P

λ
−
1

/

2

(
β
,
α
)

(
−
x
)
λ
×
sin
⁡
π
λ

Γ
2

(
λ
+
1

/

2
)

Γ
(
λ
+
α
+
1

/

2
)
Γ
(
λ
+
β
+
1

/

2
)

d
λ

f(x) = \lim \limits _{n \to \infty } 4\int _0^n {{F^{(\alpha ,\beta )}}\left ( {\lambda - \frac {1}{2}} \right )} P_{\lambda - 1/2}^{(\beta ,\alpha )}( - x)\lambda \times \sin \pi \lambda \frac {{{\Gamma ^2}(\lambda + 1/2)}}{{\Gamma (\lambda + \alpha + 1/2)\Gamma (\lambda + \beta + 1/2)}}d\lambda

\]
almost everywhere in

[
−
1
,
1
]

[-1,1]

. In this paper, we devise a technique to continue

f
(
x
)

f(x)

analytically to the complex

z
z

-plane and locate the singularities of

f
(
z
)

f(z)

by relating them to the singularities of
\[

g
(
t
)
=

∫
0
∞

e

−
λ
t

F

(
α
,
β
)

(
λ
)

d
λ

Γ
(
λ
+
α
+
1
)

.

g(t) = \int _0^\infty {{e^{ - \lambda t}}{F^{(\alpha ,\beta )}}(\lambda )} \frac {{d\lambda }}{{\Gamma (\lambda + \alpha + 1)}}.

\]
However, this will be done in the more general case where the limit in (*) exists in the sense of Schwartz distributions and defines a generalized function

f
(
x
)

f(x)

. In this case, we pass from

f
(
x
)

f(x)

to its analytic representation
\[

f
^

(
z
)
=

1

2
π
i

⟨

f
(
x
)
,

1

x
−
z

⟩

,

z
∉
supp
⁡
f
,

\hat f(z) = \frac {1} {{2\pi i}}\left \langle {f(x),\frac {1} {{x - z}}} \right \rangle ,\quad z \notin \operatorname {supp} f,

\]
and then relate the singularities of

f
^

(
z
)

\hat f(z)

to those of

g
(
t
)

g(t)

.
On the Fekete-Szegő problem for close-to-convex functions
Authors: Koepf Wolfram 
Published: June 2010

Let

S
S

be the familiar class of normalized univalent functions in the unit disk. Fekete and Szegö proved the well-known result
\[

max

f
∈
S

|

a
3

−
λ

a
2
2

|

=
1
+
2

e

−
2
λ

/

1
−
λ
)

{\max _{f \in S}}{\text {|}}{a_3} - \lambda a_2^2| = 1 + 2{e^{ - 2\lambda /1 - \lambda )}}

\]
for

λ
∈
[
0
,
1
]

\lambda \in [0,1]

. We consider the corresponding problem for the family

C
C

of close-to-convex functions and get
\[

max

f
∈
C

|

a
3

−
λ

a
2
2

=

{

3
−
4
λ

a
m
p
;

if

λ
∈
[
0
,
1

/

3
]
,

1

/

3
+
4

/

(
9
λ
)

a
m
p
;

if

λ
∈

[1/3,2/3],

1

a
m
p
;

if

λ
∈

[2/3,1]

.

\max \limits _{f \in C} |{a_3} - \lambda a_2^2 = \left \{ {\begin {array}{*{20}{c}} {3 - 4\lambda } & {{\text {if}}\lambda \in [0,1/3],} \\ {1/3 + 4/(9\lambda )} & {{\text {if}}\lambda \in {\text {[1/3,2/3],}}} \\ 1 & {{\text {if}}\lambda \in {\text {[2/3,1]}}{\text {.}}} \\ \end {array} } \right .

\]
As an application it is shown that

|

|

a
3

|

−

|

a
2

|

|

≤
1

||{a_3}| - |{a_2}|| \leq 1

for close-to-convex functions, in contrast to the result in

S
S

\[

max

f
∈
s

||

a
3

|

−

|

a
2

|

|

=
1.029....

\max \limits _{f \in s} {\text {||}}{a_3}| - |{a_2}|| = 1.029....

\]
Galois endomorphisms of the torsion subgroup of one-parameter generic formal groups
Authors: Zimmermann Karl 
Published: June 2010

Let

Z

p

{{\mathbf {Z}}_p}

be the ring of

p
p

-adic integers and let

Γ
\Gamma

be a one-parameter generic formal group of finite height

h
h

defined over

Z

p

[

[

t
1

,
…
,

t

h
−
1

]

]

=
A

{{\mathbf {Z}}_p}\left [ {\left [ {{t_1}, \ldots ,{t_{h - 1}}} \right ]} \right ] = A

. Let

K
K

be the field of fractions of

A
,
G
=
Gal
⁡

(

K
¯

/

K

)

A,G = \operatorname {Gal}\left ( {\bar K/K} \right )

and

T

(
Γ
)

T\left ( \Gamma \right )

the Tate module of

Γ
\Gamma

. The purpose of this paper is to give an elementary proof that the

map
⁡

End

A

⁡

(
Γ
)

→

End

G

⁡

(

T

(
Γ
)

)

\operatorname {map}\operatorname {End}_{A}\left ( \Gamma \right ) \to \operatorname {End}_{G}\left ( {T\left ( \Gamma \right )} \right )

is a surjection.
A Tauberian theorem for Hausdorff methods
Authors: Kuttner Brian , Parameswaran Mangalam R. 
Published: June 2010

Let

H
=
(
H
,
χ
)

H = (H,\chi )

be a regular Hausdorff summability method defined by the function

χ
∈

BV

[

0
,
1

]

\chi \in {\text {BV}}\left [ {0,1} \right ]

. It is shown that if

χ
\chi

is absolutely continuous on

[

0
,
1

]

\left [ {0,1} \right ]

, then the methods

H
H

and

V
⋅
H

V \cdot H

are equivalent for bounded sequences, where

V
V

belongs to a certain class of summability methods which includes the Cesàro methods

C
α

(
α

>

0
)

{C_\alpha }(\alpha {\text { > }}0)

, the Abel method

A
A

, and the methods

A
⋅

C
α

(
α

>

−
1
)

A \cdot {C_\alpha }(\alpha {\text { > }} - 1)

.
Almost commuting matrices need not be nearly commuting
Authors: Choi Man Duen 
Published: June 2010

Let

M

n

{\mathcal {M}_n}

be the collection of

n
×
n

n \times n

complex matrices with the Hilbert-space-operator norm. There exist two concrete matrices

A
,
B
∈

M

n

A,B \in {\mathcal {M}_n}

with

|

|

A

|

|

=
1
−
1

/

n
,

|

|

B

|

|

≤
1
,

|

|

A
B
−
B
A

|

|

≤
2

/

n

||A|| = 1 - 1/n,||B|| \leq 1,||AB - BA|| \leq 2/n

, but

|

|

A
−
R

|

|

+

|

|

B
−
S

|

|

≥
1
−
1

/

n

||A - R|| + ||B - S|| \geq 1 - 1/n

for all commuting pairs

R
,
S
∈

M

n

R,S \in {\mathcal {M}_n}

. It is shown explicitly that there is a natural obstruction which prevents almost commuting matrices to get close to any commuting pairs.
Convergence and integrability of double trigonometric series with coefficients of bounded variation
Authors: Móricz Ferenc 
Published: June 2010

We prove that if

c

(

j
,
k

)

→
0

c\left ( {j,k} \right ) \to 0

as

max

(

|

j

|

,

|

k

|

)

→
∞

\max \left ( {|j|,|k|} \right ) \to \infty

and
\[

∑

j
=
−
∞

∞

∑

k
=
−
∞

∞

|

Δ

11

c

(

j
,
k

)

|

>
∞
,

\sum \limits _{j = - \infty }^\infty {\sum \limits _{k = - \infty }^\infty {\left | {{\Delta _{11}}c\left ( {j,k} \right )} \right | > \infty ,} }

\]
then the series

∑

j
=
−
∞

∞

∑

k
=
−
∞

∞

c

(

j
,
k

)

e

i
(
j
x
+
k
y
)

\sum \nolimits _{j = - \infty }^\infty {\sum \nolimits _{k = - \infty }^\infty {c\left ( {j,k} \right ){e^{i(jx + ky)}}} }

converges both pointwise for every

(

x
,
y

)

∈

(

T
∖

{
0
}

)

2

\left ( {x,y} \right ) \in {\left ( {T\backslash \left \{ 0 \right \}} \right )^2}

and in the

L
p

(

T
2

)

{L^p}\left ( {{T^2}} \right )

-metric for

0
>
p
>
1

0 > p > 1

, where

T
T

is the one-dimensional torus. Both convergence statements remain valid for the three conjugate series under these same coefficient conditions.
Estimates for Green’s functions
Authors: Ramm A. G. , Li Lige 
Published: June 2010

Let

l
q

=
−

∇
2

+
q
(
x
)
,
x
∈

R
3

,
0
≤
q
≤
c

(
1
+

|
x
|

)

−
a

,
a

>

2
,

l
q

G
q

(
x
,
y
)
=
δ
(
x
−
y
)

{l_q} = - {\nabla ^2} + q(x),x \in {R^3},0 \leq q \leq c{(1 + \left | x \right |)^{ - a}},a{\text { > }}2,{l_q}{G_q}(x,y) = \delta (x - y)

. If

q
≥
p
≥
0
,
q
≢
p

q \geq p \geq 0,q \not \equiv p

, then

c

|

x
−
y

|

−
1

>

G

q

(
x
,
y
)

>

G
p

(
x
,
y
)
≤

(
4
π

|

x
−
y

|

)

−
1

,
x
≠
y

\begin {gathered} c\left | {x - y\left | {^{ - 1}} \right .{\text { > }}{{\text {G}}_q}(x,y){\text { > }}} \right .{G_p}(x,y) \leq {(4\pi \left | {x - y} \right |)^{ - 1}},x \ne y \end {gathered}

, for some positive

c
=
c
(
q
)

c = c(q)

. If

p
≢
0

p \not \equiv 0

then

G
p

>

(
4
π

|

x
−
y

|

)

−
1

,
x
≠
y

{G_p}{\text { > }}{(4\pi \left | {x - y} \right |)^{ - 1}},x \ne y

.
On the almost everywhere existence of the ergodic Hilbert transform
Authors: Gallardo Diego , Martín-Reyes F. J. 
Published: June 2010

Let

(
X
,

M

,
μ
)

(X,\mathfrak {M},\mu )

be a finite measure space,

T
T

an invertible measure-preserving transformation and

υ
\upsilon

a positive measurable function. For

p
=
1

p = 1

, we prove that the ergodic Hubert transform

H
f
(
X
)
=

li

m

n
→
∞

∑

i
=
−
n

n

′

f
(

T
i

x
)

/

i

Hf(X) = {\text {li}}{{\text {m}}_{n \to \infty }}\sum \nolimits _{i = - n}^n {’f({T^i}x)/i}

exists a.e. for every

f
f

in

L
1

(
υ
d
μ
)

{L^1}(\upsilon d\mu )

if and only if

in

f

i
≥
0

υ
(

T
i

x
)
>
0

{\text {in}}{{\text {f}}_{i \geq 0}}\upsilon ({T^i}x) > 0

a.e. We also solve the problem for

1
>
p
≤
2

1 > p \leq 2

. In this case the condition is

su

p

k
≥
1

k

−
1

∑

i
−
0

k
−
1

υ

−
1

/

(
p
−
1
)

(

T
i

x
)
>
∞

{\text {su}}{{\text {p}}_{k \geq 1}}{k^{ - 1}}\sum \nolimits _{i - 0}^{k - 1} {{\upsilon ^{ - 1/(p - 1)}}} ({T^i}x) > \infty

a.e. If the transformation

T
T

is ergodic, the characterizing conditions become that

1

/

υ
∈

L
∞

1/\upsilon \in {L^\infty }

and

υ

−
1

/

(
p
−
1
)

∈

L
1

(
μ
)

{\upsilon ^{ - 1/(p - 1)}} \in {L^1}(\mu )

, respectively. These characterizations, together with some recent results, give, for

1
≤
p
≤
2

1 \leq p \leq 2

, that

H
f
(
x
)

Hf(x)

exists a.e. for every

f
f

in

L
p

(
υ
d
μ
)

{L^p}(\upsilon d\mu )

if and only if the sequence of the Césàro-averages

k

−
1

(
f
(
x
)
+
f
(
T
x
)
+
…
f
(

T

k
−
1

x
)
)

{k^{ - 1}}(f(x) + f(Tx) + \ldots f({T^{k - 1}}x))

converge a.e. for every

f
f

in

L
p

(
υ
d
μ
)

{L^p}(\upsilon d\mu )

. This equivalence has recently been obtained by Jajte for a unitary operator, not necessarily positive, acting on

L
2

{L^2}

.
Nonnegative solutions for a class of radially symmetric nonpositone problems
Authors: Castro Alfonso , Shivaji R. 
Published: June 2010

We consider the existence of radially symmetric non-negative solutions for the boundary value problem
\[

−
Δ
u
(
x
)
=
λ
f
(
u
(
x
)
)

‖
x
‖

≤
1
,
x
∈

R
N

(
N
≥
2
)

u
(
x
)
=
0

‖
x
‖

=
1

\begin {array}{*{20}{c}} { - \Delta u(x) = \lambda f(u(x))\quad \left \| x \right \| \leq 1,x \in {R^N}(N \geq 2)} \\ {u(x) = 0\quad \left \| x \right \| = 1} \\ \end {array}

\]
where

λ
>
0
,
f
(
0
)
>
0

\lambda > 0,f(0) > 0

(non-positone),

f
′

≥
0

f’ \geq 0

and

f
f

is superlinear. We establish existence of non-negative solutions for

λ
\lambda

small which extends some work of our previous paper on non-positone problems, where we considered the case

N
=
1

N = 1

. Our work also proves a recent conjecture by Joel Smoller and Arthur Wasserman.
Simple near-rings associated with meromorphic products
Authors: Maxson C. J. , Smith K. C. 
Published: June 2010

Let

H
H

be a subgroup of

G
2

{G^2}

and let
\[

M
0

(

G
,
2
,
H

)

=

{

f
∈

M
0

(
G
)

|

f
(
H
)
⊆
H

}

.

{M_0}\left ( {G,2,H} \right ) = \left \{ {f \in {M_0}\left ( G \right )|f(H) \subseteq H} \right \}.

\]
In this paper we characterize in terms of properties of

H
H

when

M
0

(

G
,
2
,
H

)

{M_0}\left ( {G,2,H} \right )

is a simple near-ring.
Global preservation of nodal structure in coupled systems of nonlinear Sturm-Liouville boundary value problems
Authors: Cantrell Robert Stephen 
Published: June 2010

In this paper, we examine the solution set to the coupled system (

∗
*

)
\[

{

−
(

p
2

(
x
)

υ
′

(
x
)

)
′

+

q
2

(
x
)
υ
(
x
)
=
μ
υ
(
x
)
+
υ
(
x
)
⋅
g
(
u
(
x
)
,
υ
(
x
)
)
,

−
(

p
1

(
x
)

u
′

(
x
)

)
′

+

q
1

(
x
)
u
(
x
)
=
λ
u
(
x
)
+
u
(
x
)
⋅
f
(
u
(
x
)
,
υ
(
x
)
)

\left \{ {_{ - ({p_2}(x)\upsilon ’(x))’ + {q_2}(x)\upsilon (x) = \mu \upsilon (x) + \upsilon (x) \cdot g(u(x),\upsilon (x)),}^{ - ({p_1}(x)u’(x))’ + {q_1}(x)u(x) = \lambda u(x) + u(x) \cdot f(u(x),\upsilon (x))}} \right .

\]
where

λ
,
μ
∈
R
,
x
∈
[
a
,
b
]

\lambda ,\mu \in R,x \in [a,b]

, and the system (

∗
*

) is subject to zero Dirichlet boundary data on

u
u

and

υ
\upsilon

. We determine conditions on

f
f

and

g
g

which permit us to assert the existence of continua of solutions to (

∗
*

) characterized by

u
u

having

n
−
1

n - 1

simple zeros in

(
a
,
b
)
,
υ

(a,b),\upsilon

having

m
−
1

m - 1

simple zeros in

(
a
,
b
)

(a,b)

, where

n
n

and

m
m

are positive but not necessarily equal integers. Moreover, we also determine conditions under which these continua link solutions to (

∗
*

) of the form

(
λ
,
μ
,
u
,
0
)

(\lambda ,\mu ,u,0)

with

u
u

having

n
−
1

n - 1

simple zeros in

(
a
,
b
)

(a,b)

to solutions of (

∗
*

) of the form

(
λ
,
μ
,
0
,
υ
)

(\lambda ,\mu ,0,\upsilon )

with

υ
\upsilon

having

m
−
1

m - 1

simple zeros in

(
a
,
b
)

(a,b)

.
An entropy inequality for the bi-multivariate hypergeometric distribution
Authors: Kochman Fred , Murray Alan , West Douglas B. 
Published: June 2010

Given parameters

r
¯

=

r
1

,
…
,

r
m

\bar r = {r_1}, \ldots ,{r_m}

and

c
¯

=

c
1

,
…
,

c
n

\bar c = {c_1}, \ldots ,{c_n}

with

∑

r
i

=
∑

c
j

=
N

\sum {{r_i}} = \sum {{c_j}} = N

, the bi-multivariate hypergeometric distribution is the distribution on nonnegative integer

m
×
n

m \times n

matrices with row sums

r
¯

\bar r

and column sums

c
¯

\bar c

defined by

Prob

(
A
)

=
∏

r
i

!
∏

c
j

!

/

(

N
!
∏

a

i
j

!

)

{\text {Prob}}\left ( A \right ) = \prod {{r_i}} !\prod {{c_j}} ! / \left ( {N!\prod {{a_{ij}}!} } \right )

. It is shown that the entropy of this distribution is a Schur-concave function of the block-size parameters.
Singularly continuous measures in Nevai’s class 𝑀
Authors: Lubinsky D. S. 
Published: June 2010

Letdνd\nube a nonnegative Borel measure on[−π,π][ - \pi ,\pi ], with0>∫−ππdν>∞0 > \smallint _{ - \pi }^\pi d\nu > \inftyand with support of Lebesgue measure zero. We show that there exist{ηj}j=1∞⊂(0,∞)\{ {\eta _j}\} _{j = 1}^\infty \subset (0,\infty )and{tj}j=1∞⊂(−π,π)\{ {t_j}\} _{j = 1}^\infty \subset ( - \pi ,\pi )such that if\[dμ(θ):=∑j=1∞ηjdν(θ+tj),θ∈[−π,π],d\mu (\theta ): = \sum \limits _{j = 1}^\infty {{\eta _j}d\nu (\theta + {t_j}),\quad \theta \in [ - \pi ,\pi ],}\](with the usual periodic extensiondν(θ±2π)=dν(θ)d\nu (\theta \pm 2\pi ) = d\nu (\theta )), then the leading coefficients{κn(dμ)}n=0∞\{ {\kappa _n}(d\mu )\} _{n = 0}^\inftyof the orthonormal polynomials fordμd\musatisfy\[limn→∞κn(dμ)/κn+1(dμ)=1.\lim \limits _{n \to \infty } {\kappa _n}(d\mu )/{\kappa _{n + 1}}(d\mu ) = 1.\]As a consequence, we obtain pure singularly continuous measuresdαd\alphaon[−1,1][ - 1,1]lying in Nevai’s classMM.
On a differential equation for Koornwinder’s generalized Laguerre polynomials
Authors: Koekoek J. , Koekoek R. 
Published: June 2010

Koornwinder’s generalized Laguerre polynomials

{

L
n

α
,
N

(
x
)

}

n
=
0

∞

\left \{ {L_n^{\alpha ,N}(x)} \right \}_{n = 0}^\infty

are orthogonal on the interval

[
0
,
∞
)

[0,\infty )

with respect to the weight function

1

Γ
(
α
+
1
)

x
α

e

−
x

+
N
δ
(
x
)
,
α
>
−
1
,
N
≥
0

\frac {1}{{\Gamma (\alpha + 1)}}{x^\alpha }{e^{ - x}} + N\delta (x),\alpha > - 1,N \geq 0

. We show that these polynomials for

N
>
0

N > 0

satisfy a unique differential equation of the form
\[

N

∑

i
=
0

∞

a
i

(
x
)

y

(
i
)

(
x
)
+
x
y
(
x
)
+
(
α
+
1
−
x
)

y
′

(
x
)
+
n
y
(
x
)

=
0
,

N\sum \limits _{i = 0}^\infty {{a_i}(x){y^{(i)}}(x) + xy(x) + (\alpha + 1 - x)y’(x) + ny(x)} = 0,

\]
where

{

a
i

(
x
)

}

i
=
0

∞

\left \{ {{a_i}(x)} \right \}_{i = 0}^\infty

are continuous functions on the real line and

{

a
i

(
x
)

}

i
=
1

∞

\left \{ {{a_i}(x)} \right \}_{i = 1}^\infty

are independent of the degree

n
n

. If

N
>
0

N > 0

, only in the case of nonnegative integer values of

α
\alpha

this differential equation is of finite order.
Boundary values of holomorphic semigroups
Authors: Boyadzhiev Khristo , deLaubenfels Ralph 
Published: June 2010

Suppose

A
A

generates a bounded strongly continuous holomorphic semigroup of angle

π

/

2

\pi /2

. We show that

i
A

iA

generates a

(
1
−
A

)

−
r

{(1 - A)^{ - r}}

regularized group, which is

O
(
1
+

|

s

|

r

)

∀
r
>
γ
⩾
0

O(1 + |s{|^r})\;\forall r > \gamma \geqslant 0

, if and only if

|

|

e

z
A

|

|

||{e^{zA}}||

is

O
(

(
(
1
+

|

z

|

)

/

Re
⁡
(
z
)

)
r

)
∀
r
>
γ

O({((1 + |z|)/\operatorname {Re} (z))^r})\forall r > \gamma

and

i
A

iA

generates a bounded

(
1
−
A

)

−
r

{(1 - A)^{ - r}}

regularized group

∀
r
>
γ
⩾
0

\forall r > \gamma \geqslant 0

if and only if

|

|

e

z
A

|

|

||{e^{zA}}||

is

O
(

(
1

/

Re
⁡
(
z
)

)
r

)

∀
r
>
γ

O({(1/\operatorname {Re} (z))^r})\;\forall r > \gamma

. We apply this to the Schrödinger operator

i
(
Δ
−
V
)

i(\Delta - V)

.
Inequalities in the most simple Sobolev space and convolutions of 𝐿₂ functions with weights
Authors: Saitoh Saburou 
Published: June 2010

For the most simple Sobolev space on

R

\mathbb {R}

composed of real-valued and absolutely continuous functions

f
(
x
)

f(x)

on

R

\mathbb {R}

with finite norms
\[

{

∫

−
∞

∞

(

a
2

f
′

(
x
)

2

+

b
2

f

(
x
)

2

)

d
x

}

1

/

2

(
a
,
b
>
0
)
,

{\left \{ {\int _{ - \infty }^\infty {({a^2}{f’}{{(x)}^2} + {b^2}f{{(x)}^2})\,dx} } \right \}^{1/2}}\qquad (a,b > 0),

\]
we shall apply the theory of reproducing kernels, and derive natural norm inequalities in the space and the related inequalities for convolutions of

L
2

{L_2}

functions with weights.
The evolution operator approach to functional-differential equations with delay
Authors: Ruess Wolfgang M. 
Published: June 2010

The nonlinear nonautonomous functional differential equation

x
˙

(
t
)
∈
B
(
t
)

x
(
t
)
+
F
(
t
,

x
t

)
,

t
⩾
s
,

x
s

=
φ

\dot x(t) \in B(t)\,x(t) + F(t,{x_t}),\;t \geqslant s,\,{x_s} = \varphi

, is considered. The representation of the solution to this equation via the associated evolution operator is extended from the single-valued case to the general case of multivalued operators

B
(
t
)

B(t)

.