bluuebunny/crossref_metadata_2025_split huggingface dataset:
The special values at negative integers of Dirichlet series associated with polynomials of several variables
Authors: Eie Min King 
Published: June 2010

Let

P
(
X
)

P(X)

be a product of

k
k

linear forms in

r
r

variables

X
1

,
…
,

X
r

{X_1}, \ldots ,{X_r}

as given by
\[

P
(

X
1

,
…
,

X
r

)
=

∏

j
=
1

k

(

a

j
1

X
1

+
⋯
+

a

j
r

X
r

+

δ
j

)
,

Re
⁡

a

j
i

>
0
,

Re
⁡

(

δ
j

+

∑

i
=
1

r

a

j
i

)

>
0.

\begin {array}{*{20}{c}} {P({X_1}, \ldots ,{X_r}) = \prod \limits _{j = 1}^k {({a_{j1}}{X_1} + \cdots + {a_{jr}}{X_r} + {\delta _j}),} } \\ {\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \operatorname {Re} {a_{ji}} > 0,\;\operatorname {Re} \left ( {{\delta _j} + \sum \limits _{i = 1}^r {{a_{ji}}} } \right ) > 0.} \\ \end {array}

\]
Suppose that

β
=
(

β
1

,
…
,

β
r

)

\beta = ({\beta _1}, \ldots ,{\beta _r})

is an

r
r

-tuple of nonnegative integers. Consider the zeta function
\[

Z
(
P
,
β
)
(
s
)
=

∑

n

1
=
1

∞

⋯

∑

n

r
=
1

∞

n
1

β
1

⋯

n
r

β
r

P

(

n
1

,
…
,

n
r

)

−
s

,

Re
⁡
s
>

r
+

|

β

|

k

,

Z(P,\beta )(s) = \sum \limits _{{n_{1 = 1}}}^\infty { \cdots \sum \limits _{{n_{r = 1}}}^\infty {n_1^{{\beta _1}} \cdots n_r^{{\beta _r}}P{{({n_1}, \ldots ,{n_r})}^{ - s}},\qquad \operatorname {Re} s > \frac {{r + |\beta |}} {k}} } ,

\]
where

|

β

|

=

β
1

+
⋯
+

β
r

|\beta | = {\beta _1} + \cdots + {\beta _r}

.

Z
(
P
,
β
)
(
s
)

Z(P,\beta )(s)

has an analytic continuation in the whole complex plane and it is regular at

s
=
0
,

−
1
,

−
2
,
…
,
−
m
,
…

s = 0,\; - 1,\; - 2, \ldots , - m, \ldots

. In this paper, we shall compute the explicit values of

Z
(
P
,
β
)
(
s
)

Z(P,\beta )(s)

at

s
=
0
,

−
1
,

−
2
,
…
,

−
m
,
…

s = 0,\; - 1,\; - 2, \ldots ,\; - m, \ldots

and express them in terms of finite sums of polynomials in Bernoulli numbers.
On groups related to the Hecke groups
Authors: Knopp Marvin I. , Newman Morris 
Published: June 2010

Let

[

1

a
m
p
;

λ
1

0

a
m
p
;
1

]

,

[

1

a
m
p
;
0

λ
2

a
m
p
;
1

]

\left [ {\begin {array}{*{20}{c}} 1 & {{\lambda _1}} \\ 0 & 1 \\ \end {array} } \right ],\left [ {\begin {array}{*{20}{c}} 1 & 0 \\ {{\lambda _2}} & 1 \\ \end {array} } \right ]

be parabolic elements of

PSL
⁡
(
2
,
R
)

\operatorname {PSL} (2,R)

, where

λ
1

,

λ
2

>
0

{\lambda _1},{\lambda _2} > 0

. The principal result shown here is that

K
(

λ
1

,

λ
2

)

K({\lambda _1},{\lambda _2})

, the group generated by these elements, is discrete if and only if

λ
1

λ
2

⩾
4

{\lambda _1}{\lambda _2} \geqslant 4

, or

λ
1

λ
2

=
4

cos
2

(
π

/

p
)

{\lambda _1}{\lambda _2} = 4{\cos ^2}(\pi /p)

, where

p
p

is an integer

⩾
3

\geqslant 3

. When

λ
1

λ
2

=
4

cos
2

(
π

/

p
)
,

K
(

λ

1
,

λ
2

)

{\lambda _1}{\lambda _2} = 4{\cos ^2}(\pi /p),\;K({\lambda _{1,}}{\lambda _2})

is conjugate to the classical Hecke group

H
(
2
cos
⁡
(
π

/

p
)
)

H(2\cos (\pi /p))

if

p
p

is odd; while if

p
p

is even,

K
(

λ
1

,

λ
2

)

K({\lambda _1},{\lambda _2})

is conjugate to a subgroup of

H
(
2
cos
⁡
(
π

/

p
)
)

H(2\cos (\pi /p))

of index

2
2

. When

λ
1

λ
2

⩾
4
,

K
(

λ
1

,

λ
2

)

{\lambda _1}{\lambda _2} \geqslant 4,\;K({\lambda _1},{\lambda _2})

is conjugate to a subgroup of

H
(

(

λ
1

λ
2

)

)

H(\sqrt {({\lambda _1}{\lambda _2})} )

of index

2
2

. In all of these cases

K
(

λ
1

,

λ
2

)

K({\lambda _1},{\lambda _2})

is the free product of two cyclic groups.
Perturbation of spectrums of 2×2 operator matrices
Authors: Du Hong Ke , Jin Pan 
Published: June 2010

In this paper, we study the perturbation of spectrums of

2
×
2

2 \times 2

operator matrices such as

M
C

=

[

A

a
m
p
;
C

0

a
m
p
;
B

]

{M_C} = \left [ {\begin {array}{*{20}{c}} A & C \\ 0 & B \\ \end {array} } \right ]

on the Hilbert space

H
⊕
K

H \oplus K

. For given A and B, we prove that
\[

⋂

C
∈
B
(
K
,
H
)

σ
(

M
C

)
=

σ
π

(
A
)
∪

σ
δ

(
B
)
∪
{
λ
∈
C
:
n
(
B
−
λ
)
≠
d
(
A
−

λ
)
}
,

\bigcap \limits _{C \in B(K,H)} {\sigma ({M_C}) = {\sigma _\pi }(A) \cup {\sigma _\delta }(B) \cup \{ \lambda \in C:n(B - \lambda ) \ne d(A - } \lambda )\} ,

\]
where

σ
(
T
)
,

σ
π

(
T
)
,

σ
δ

(
T
)
,
n
(
T
)

\sigma (T),{\sigma _\pi }(T),{\sigma _\delta }(T),n(T)

, and

d
(
T
)

d(T)

denote the spectrum of T, approximation point spectrum, defect spectrum, nullity, and deficiency, respectively. Some related results are obtained.
𝑊^{𝑝}-spaces and Fourier transform
Authors: Pathak R. S. , Upadhyay S. K. 
Published: June 2010

The spaces

W
M
p

,

W

M
,
a

p

,

W

Ω
,
p

,

W

Ω
,
b
,
p

,

W
M

Ω
,
p

,

W

M
,
a

Ω
,
b
,
p

W_M^p,W_{M,a}^p,{W^{\Omega ,p}},{W^{\Omega ,b,p}},W_M^{\Omega ,p},W_{M,a}^{\Omega ,b,p}

generalizing the spaces of type W due to Gurevich (also given by Friedman, and Gelfand and Shilov) are investigated. Here M,

Ω
\Omega

are certain continuous increasing convex functions, a, b are positive constants and

1
≤
p
>
∞

1 \leq p > \infty

. The Fourier transformation F is shown to be a continuous linear mapping as follows:

F
:

W

M
,
a

p

→

W

Ω
,
1

/

a
,
r

,
F
:

W

Ω
,
b
,
p

→

W

M
,
1

/

b

r

,
F
:

W

M
,
a

Ω
,
b
,
p

→

W

M
,
1

/

b

Ω
,
1

/

a
,
r

F:W_{M,a}^p \to {W^{\Omega ,1/a,r}},F:{W^{\Omega ,b,p}} \to W_{M,1/b}^r,F:W_{M,a}^{\Omega ,b,p} \to W_{M,1/b}^{\Omega ,1/a,r}

. These results will be used in investigating uniqueness classes of certain Cauchy problems in future work.
Descriptions of conditional expectations induced by non-measure preserving transformations
Authors: Lambert Alan , Weinstock Barnet M. 
Published: June 2010

Given a measure-preserving transformation T acting on a

σ
\sigma

-finite measure space

(
X
,

A

,
m
)

(X,\mathcal {A},m)

and a

σ
\sigma

-finite sigma algebra

B

⊂

A

\mathcal {B} \subset \mathcal {A}

, the conditional expectations

E
(
⋅

|

B

)

E( \cdot |\mathcal {B})

acting on

L
∞

(

A

)

{L^\infty }(\mathcal {A})

and

E
(
⋅

|

T

−
1

B

)

E( \cdot |{T^{ - 1}}\mathcal {B})

acting on

L
∞

(

T

−
1

A

)

{L^\infty }({T^{ - 1}}\mathcal {A})

are known to be related by the formula

[
E
(
f

|

B

)
]
∘
T
=
E
(
f
∘
T

|

T

−
1

B

)

[E(f|\mathcal {B})] \circ T = E(f \circ T|{T^{ - 1}}\mathcal {B})

. In this note the conditional expectation

E
(
⋅

|

T

−
1

B

)

E( \cdot |{T^{ - 1}}\mathcal {B})

is investigated in the non-measure-preserving case, and those transformations for which the above equation holds are characterized in terms of measurability conditions for

d
(
m
∘

T

−
1

)

/

d
m

d(m \circ {T^{ - 1}})/dm

. It is precisely in the non-measure-preserving case that the measurability of

d
(
m
∘

T

−
1

)

/

d
m

d(m \circ {T^{ - 1}})/dm

plays an important role. Relatedly, it is shown that if composition by T intertwines

E
(
⋅

|

B

)

E( \cdot |\mathcal {B})

and any mapping

Λ
\Lambda

, then

Λ
\Lambda

is a conditional expectation induced by a measure equivalent to m. These results were motivated by a result concerning induced conditional expectation operators on

C
∗

{C^ \ast }

-algebras, and the paper concludes with a brief description of this

C
∗

{C^\ast }

-algebra setting.
Some inequalities of algebraic polynomials
Authors: Varma A. K. 
Published: June 2010

Erdös and Lorentz showed that by considering the special kind of the polynomials better bounds for the derivative are possible. Let us denote by

H
n

{H_n}

the set of all polynomials whose degree is n and whose zeros are real and lie inside

[
−
1
,
1
)

[ - 1,1)

. Let

P
n

∈

H
n

{P_n} \in {H_n}

and

P
n

(
1
)
=
1

{P_n}(1) = 1

; then the object of Theorem 1 is to obtain the best lower bound of the expression

∫

−
1

1

|

P
n
′

(
x
)

|

p

d
x

\smallint _{ - 1}^1|P_n’(x){|^p}\,dx

for

p
≥
1

p \geq 1

and characterize the polynomial which achieves this lower bound. Next, we say that

P
n

∈

S
n

[
0
,
∞
)

{P_n} \in {S_n}[0,\infty )

if

P
n

{P_n}

is a polynomial whose degree is n and whose roots are all real and do not lie inside

[
0
,
∞
)

[0,\infty )

. In Theorem 2, we shall prove Markov-type inequality for such a class of polynomials belonging to

S
n

[
0
,
∞
)

{S_n}[0,\infty )

in the weighted

L
p

{L_p}

norm (p integer). Here

‖

P
n

‖

L
p

=

(

∫
0
∞

|

P
n

(
x
)

|

p

e

−
x

d
x

)

1

/

p

{\left \| {{P_n}} \right \|_{{L_p}}} = {(\smallint _0^\infty |{P_n}(x){|^p}{e^{ - x}}\,dx)^{1/p}}

. In Theorem 3 we shall consider another analogous problem as in Theorem 2.
Positive unstable periodic solutions for superlinear parabolic equations
Authors: Hirano Norimichi , Mizoguchi Noriko 
Published: June 2010

In this paper, we are concerned with a superlinear parabolic equation
\[

{

∂
u

∂
t

−
Δ
u
=

u
p

+
h
(
t
,
x
)
,

a
m
p
;

(
t
,
x
)
∈

R

+

×
Ω
,

u
=
0
,

a
m
p
;

(
t
,
x
)
∈

R

+

×
∂
Ω
,

u
>
0
,

a
m
p
;

(
t
,
x
)
∈

R

+

×
∂
Ω
,

\left \{ {\begin {array}{*{20}{c}} {\frac {{\partial u}}{{\partial t}} - \Delta u = {u^p} + h(t,x),} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \Omega ,} \hfill \\ {{u = 0,}} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \partial \Omega ,} \hfill \\ {{u > 0,}} \hfill & {(t,x) \in {{\mathbf {R}}_ + } \times \partial \Omega ,} \hfill \\ \end {array} } \right .

\]
where

Ω
⊂

R

N

\Omega \subset {{\mathbf {R}}^N}

is a bounded domain with smooth boundary

∂
Ω

\partial \Omega

, h is T-periodic with respect to the first variable, and

1
>
p
>

N
+
2

N
−
2

1 > p > \frac {{N + 2}}{{N - 2}}

if

N
≥
3

N \geq 3

and

1
>
p
>
+
∞

1 > p > + \infty

if

N
≤
2

N \leq 2

. It is shown that there exist a stable and an unstable positive T-periodic solution for this problem if h is sufficiently small in

L
∞

{L^\infty }

.
Mayer-Vietoris formula for the determinant of a Laplace operator on an even-dimensional manifold
Authors: Lee Yoonweon 
Published: June 2010

Let

Δ
\Delta

be a Laplace operator acting on differential p-forms on an even-dimensional manifold M. Let

Γ
\Gamma

be a submanifold of codimension 1. We show that if B is a Dirichlet boundary condition and R is a Dirichlet-Neumann operator on

Γ
\Gamma

, then

Det

(
Δ
+
λ
)
=

Det

(
Δ
+
λ
,
B
)

Det

(
R
+
λ
)

{\operatorname {Det}}(\Delta + \lambda ) = {\operatorname {Det}}(\Delta + \lambda ,B){\operatorname {Det}}(R + \lambda )

and

Det
∗

Δ
=

1

(
det
A
)

2

Det

(
Δ
,
B
)

Det
∗

R

{\operatorname {Det}^ \ast }\Delta = \frac {1}{{{{(\det A)}^2}}}{\operatorname {Det}}(\Delta ,B){\operatorname {Det}^ \ast }R

. This result was established in 1992 by Burghelea, Friedlander, and Rappeler for a 2-dimensional manifold with

p
=
0

p = 0

.
Some inequalities for entire functions of exponential type
Authors: Gardner Robert B. , Govil N. K. 
Published: June 2010

If

f
(
z
)

f(z)

is an asymmetric entire function of exponential type

τ
\tau

,
\[

‖
f
‖

=

sup

−
∞
>
x
>
∞

|

f
(
x
)

|

,

\left \| f \right \| = \sup \limits _{ - \infty > x > \infty } |f(x)|,

\]
then according to a well-known result of R. P. Boas,
\[

‖

f
′

‖

≤

τ
2

‖
f
‖

\left \| {f’} \right \| \leq \frac {\tau }{2}\left \| f \right \|

\]
and
\[

|

f
(
x
+
i
y
)

|

≤

(

e

τ

|

y

|

+
1
)

2

‖
f
‖

,

−
∞
>
x
>
∞
,
−
∞
>
y
≤
0.

|f(x + iy)| \leq \frac {{({e^{\tau |y|}} + 1)}}{2}\left \| f \right \|,\quad - \infty > x > \infty , - \infty > y \leq 0.

\]
Both of these inequalities are sharp. In this paper we generalize the above two inequalities of Boas by proving a sharp inequality which, besides giving as special cases the above two inequalities of Boas, yields some other results as well.
Spectral mapping theorems and perturbation theorems for Browder’s essential spectrum
Authors: Nussbaum Roger D. 
Published: June 2010

If

T
T

is a closed, densely defined linear operator in a Banach space, F. E. Browder has defined the essential spectrum of

T
,
ess
⁡
(
T
)

T,\operatorname {ess} (T)

[1]. We derive below spectral mapping theorems and perturbation theorems for Browder’s essential spectrum. If

T
T

is a bounded linear operator and

f
f

is a function analytic on a neighborhood of the spectrum of

T
T

, we prove that

f
(
ess
⁡
(
T
)
)
=
ess
⁡
(
f
(
T
)
)

f(\operatorname {ess} (T)) = \operatorname {ess} (f(T))

. If

T
T

is a closed, densely defined linear operator with nonempty resolvent set and

f
f

is a polynomial, the same theorem holds. For a closed, densely defined linear operator

T
T

and a bounded linear operator

B
B

which commutes with

T
T

, we prove that

ess
⁡
(
T
+
B
)
⊆
ess
⁡
(
T
)
+
ess
⁡
(
B
)
=
{
μ
+
v
:
μ
∈
ess
⁡
(
T
)
,
v
∈
ess
⁡
(
B
)
}

\operatorname {ess} (T + B) \subseteq \operatorname {ess} (T) + \operatorname {ess} (B) = \{ \mu + v:\mu \in \operatorname {ess} (T),v \in \operatorname {ess} (B)\}

. By making additional assumptions, we obtain an analogous theorem for

B
B

unbounded.
Application of the extremum principle to investigating certain extremal problems
Authors: Mikołajczyk L. , Walczak S. 
Published: June 2010

Denote by C, K, X, respectively, a complex plane, the disc

{
z

∈

C:

|
z
|

>

1
}

\{ z\, \in \,{\textbf {C:}}\,\left | z \right |\, > \,1\}

and any compact Hausdorff space. Denote by P a set of probabilistic measures defined on Borel subsets of the space X. For

μ

∈

P

\mu \, \in \,P

, let

f
(
z
)

=

∫
X

q
(
z
,

t
)

d
μ

,

z

∈

K

f(z)\, = \,\int _X {q(z,\,t)\,d\mu } ,\,z\, \in \,K

, and

F

=

{
f
:

μ

∈

P
}

{\mathcal {F}}\, = \,\{ f:\,\mu \, \in \,P\}

. Consider a finite sequence of real functions

F
0

,

F

1
,

…
,

F
m

{F_0},\,{F_{1,}}\, \ldots ,\,{F_m}

defined in the space

R

2
n

{R^{2n}}

. Let

ζ
1

,

…
,

ζ
k

{\zeta _1},\, \ldots ,\,{\zeta _k}

be fixed points of the disc K and

η
(
f
)

=

[
re

f

(
0
)

(
ζ
)
,

im

f

(
0
)

(

ζ
1

)
,

…
,

re

f

(

n
1

)

(

ζ
1

)
,

im

f

(

n
1

)

(

ζ
1

)
;

…
,

re

f

(
0
)

(

ζ
k

)
,

im

f

(
0
)

(

ζ
k

)
,

…
,

re

f

(

n
k

)

(

ζ
k

)
,

im

f

(

n
k

)

(

ζ
k

)
]

\eta (f)\, = \,[\operatorname {re} \,{f^{(0)}}(\zeta ),\,\operatorname {im} \,{f^{(0)}}({\zeta _1}),\, \ldots ,\,\operatorname {re} \,{f^{({n_1})}}({\zeta _1}),\,\operatorname {im} \,{f^{({n_1})}}({\zeta _1});\, \ldots ,\,\operatorname {re} \,{f^{(0)}}({\zeta _k}),\,\operatorname {im} \,{f^{(0)}}({\zeta _k}),\, \ldots ,\,\operatorname {re} \,{f^{({n_k})}}({\zeta _k}),\,\operatorname {im} \,{f^{({n_k})}}({\zeta _k})]

, where

f

∈

F

,

n

=

n
1

+

⋯

+

n
k

+

k

f\, \in \,{\mathcal {F}},\,n\, = \,{n_1}\, + \, \cdots \, + \,{n_k}\, + \,k

. Let

F
j

(
f
)

=

F
j

(
η
(
f
)
)
,

j

=

0
,

1
,

…
,

m

{F_j}(f)\, = \,{F_j}(\eta (f)),\,j\, = \,0,\,1,\, \ldots ,\,m

. We consider the following extremal problem. Determine a minimum of the functional

F
0

(
f
)

{F_0}(f)

under the conditions

F
j

(
f
)

⩽

0
,

j

=

1
,

2
,

…
,

m
,

f

∈

F

{F_j}(f)\, \leqslant \,0,\,j\, = \,1,\,2,\, \ldots ,\,m,\,f\, \in \,{\mathcal {F}}

. We apply the extremum principle to solve this problem. In the linear case this problem was investigated in [11].
On the contact between complex manifolds and real hypersurfaces in 𝐶³
Authors: Bloom Thomas 
Published: June 2010

Let

m
m

be a real

C

∞

{\mathcal {C}^\infty }

hypersurface of an open subset of

C

3

{{\mathbf {C}}^3}

and let

p
∈
M

p \in M

. Let

a
1

(
M
,
p
)

{a^1}(M,p)

denote the maximal order of contact of a one-dimensional complex submanifold of a neighborhood of

p
p

in

C

3

{{\mathbf {C}}^3}

with

M
M

at

p
p

. Let

c
1

(
M
,
p
)

{c^1}(M,p)

denote the

sup
{
m
∈

Z

|

\sup \{ m \in {\mathbf {Z}}|

for all tangential holomorphic vector fields

L
L

with

L
(
p
)
≠
0

L(p) \ne 0

then

L

i
0

L
¯

j
0

…

L

i
n

L
¯

j
n

(

L

M

(
L
)
)
(
p
)
=
0
}

{L^{{i_0}}}{\bar L^{{j_0}}} \ldots {L^{{i_n}}}{\bar L^{{j_n}}}({\mathfrak {L}_M}(L))(p) = 0\}

where

i
0

,
…
,

i
n

;

j
0

,
…
,

j
n

{i_0}, \ldots ,{i_n};{j_0}, \ldots ,{j_n}

are positive integers such that

∑

t
=
0

n

i
t

+

j
t

=
m
−
3

\sum \nolimits _{t = 0}^n {{i_t} + {j_t} = m - 3}

and

L

M

(
L
)

{\mathfrak {L}_M}(L)

denotes the Levi form of

M
M

evaluated on the vector field

L
L

. Theorem. If

M
M

is pseudoconvex near

p
∈
M

p \in M

then

a
1

(
M
,
p
)
=

c
1

(
M
,
p
)

{a^1}(M,p) = {c^1}(M,p)

.
On (𝐾_{∗}(𝑍/𝑝²𝑍) and related homology groups
Authors: Evens Leonard , Friedlander Eric M. 
Published: June 2010

It is shown that, for

p
⩾
5

p \geqslant 5

,
\[

R
=

Z

/

p
2

Z

,

K
3

(
R
)
=

Z

/

p
2

Z

+

Z

/

(

p
2

−
1
)

Z

R = {\mathbf {Z}} / {p^2}{\mathbf {Z}},\,{K_3}(R) = {\mathbf {Z}} / {p^2}{\mathbf {Z}} + {\mathbf {Z}} / ({p^2} - 1){\mathbf {Z}}

\]
and

K
4

(
R
)
=
0

{K_4}(R) = 0

. Similar calculations are made for

R
R

the ring of dual numbers over

Z

/

p

Z

{\mathbf {Z}} / p{\mathbf {Z}}

. The calculation reduces to finding homology groups of

Sl
⁡
(
R
)

\operatorname {Sl} (R)

. A key tool is the spectral sequence of the group extension of

Sl
⁡
(
n
,

p
2

)

\operatorname {Sl} (n,\,{p^2})

over

Sl
⁡
(
n
,

p
)

\operatorname {Sl} (n,\,p)

. The terms of this spectral sequence depend in turn on the homology of

Gl
⁡
(
n
,

p
)

\operatorname {Gl} (n,\,p)

with coefficients various multilinear modules. Calculation of the differentials uses the Charlap-Vasquez description of

d
2

{d^2}

.
Topological invariant means on the von Neumann algebra 𝑉𝑁(𝐺)
Authors: Chou Ching 
Published: June 2010

Let

V
N
(
G
)

VN(G)

be the von Neumann algebra generated by the left regular representation of a locally compact group

G
G

,

A
(
G
)

A(G)

the Fourier algebra of

G
G

and

T
I
M
(

G
^

)

TIM(\hat G)

the set of topological invariant means on

V
N
(
G
)

VN(G)

. Let

F

1

=
{

O

∈

(

l
∞

)
∗

}
:

O

⩾
0
,

|

|

O

|

|

=
1

{\mathcal {F}_1} = \{ \mathcal {O} \in {({l^\infty })^ \ast }\} :\mathcal {O} \geqslant 0,\;||\mathcal {O}|| = 1

and

O

(
f
)
=
0

if

f
∈

l
∞

\mathcal {O}(f) = 0\;{\text {if}}\;f \in {l^\infty }

and

f
(
n
)
→
0
}

f(n) \to 0\}

. We show that if

G
G

is nondiscrete then there exists a linear isometry

Λ
\Lambda

of

(

l
∞

)
∗

{({l^\infty })^ \ast }

into

V
N

(
G

)
∗

VN{(G)^ \ast }

such that

Λ
(

F

1

)
⊂
T
I
M
(

G
^

)

\Lambda ({\mathcal {F}_1}) \subset TIM(\hat G)

. When

G
G

is further assumed to be second countable then

F

1

{\mathcal {F}_1}

can be embedded into some predescribed subsets of

T
I
M
(

G
^

)

TIM(\hat G)

. To prove these embedding theorems for second countable groups we need the existence of a sequence of means

{

u
n

}

\{ {u_n}\}

in

A
(
G
)

A(G)

such that their supports in

V
N
(
G
)

VN(G)

are mutually orthogonal and

|

|

u

u
n

−

u
n

|

|

→
0

if

u

||u{u_n} - {u_n}|| \to 0\;{\text {if}}\;u

is a mean in

A
(
G
)

A(G)

. Let

F
(

G
^

)

F(\hat G)

be the space of all

T
∈
V
N
(
G
)

T \in VN(G)

such that

m
(
T
)

m(T)

is a constant as

m
m

runs through

T
I
M
(

G
^

)

TIM(\hat G)

and let

W
(

G
^

)

W(\hat G)

be the space of weakly almost periodic elements in

V
N
(
G
)

VN(G)

. We show that the following conditions are equivalent: (i)

G
G

is discrete, (ii)

F
(

G
^

)

F(\hat G)

is an algebra and (iii)

(
A
(
G
)
⋅
V
N
(
G
)
)
∩
F
(

G
^

)
⊂
W
(

G
^

)

(A(G) \cdot VN(G)) \cap F(\hat G) \subset W(\hat G)

.
A correction and some additions to: “Fundamental solutions for differential equations associated with the number operator”
Authors: Lee Yuh Jia 
Published: June 2010

Let

(
H
,
B
)

(H,B)

be an abstract Wiener pair and

N

\mathfrak {N}

the operator defined by

N

u
(
x
)
=
−

trace

H

D
2

u
(
x
)
+
(
x
,
D
u
(
x
)
)

\mathfrak {N}u(x) = - {\text {trace}}_H{D^2}u(x) + (x,Du(x))

, where

x
∈
B

x \in B

and

(
⋅
,
⋅
)

(\cdot , \cdot )

denotes the

B
-

B

∗

B\text {-}B^{\ast }

pairing. In this paper, we point out a mistake in the previous paper concerning the existence of fundamental solutions of

N

k

{\mathfrak {N}^k}

and intend to make a correction. For this purpose, we study the fundamental solution of the operator

(

N

+
λ
I

)
k

(
λ
>
0
)

{(\mathfrak {N} + \lambda I)^k}\,(\lambda > 0)

and investigate its behavior as

λ
→
0

\lambda \to 0

. We show that there exists a family

{

Q
λ

(
x
,
d
y
)
}

\{{Q_\lambda }(x,dy)\}

of measures which serves as the fundamental solution of

(

N

+
λ
I

)
k

{(\mathfrak {N} + \lambda I)^k}

and, for a suitable function

f
f

, we prove that the solution of

N

k

u
=
f

{\mathfrak {N}^k}u = f

can be represented by

u
(
x
)
=

lim

λ
→
0

∫
B

f
(
y
)

Q
λ

(
x
,
d
y
)
+
C

u(x) = {\lim _{\lambda \to 0}}\int _B f(y){Q_\lambda }(x,dy) + C

, where

C
C

is a constant.
Linear superpositions with mappings which lower dimension
Authors: Sternfeld Y. 
Published: June 2010

It is shown that for every

n
n

-dimensional compact metric space

X
X

, there exist

2
n
+
1

2n + 1

functions

{

φ
j

}

j
=
1

2
n
+
1

\{ {\varphi _j}\}_{j = 1}^{2n + 1}

in

C
(
X
)

C(X)

and

n
n

mappings

{

ψ
i

}

i
=
1

n

\{ {\psi _i}\}_{i = 1}^n

on

X
X

with

1
1

-dimensional range each, with the following property: for every

0
⩽
k
⩽
n

0 \leqslant k \leqslant n

, every

k
k

tuple

{

ψ

i
l

}

l
=
1

k

\{ {\psi _{i_l}}\}_{l = 1}^k

of the

ψ
i

{\psi _i}

’s and every

2
(
n
−
k
)
+
1

2(n - k) + 1

tuple

{

φ

j
m

}

m
=
1

2
(
n
−
k
)
+
1

\{ {\varphi _{{j_m}}}\}_{m = 1}^{2(n - k) + 1}

of the

φ
j

{\varphi _j}

’s, each

f
∈
C
(
X
)

f \in C(X)

can be represented as

f
(
x
)
=

Σ

l
=
1

k

g
l

(

ψ

i
l

(
x
)
)
+

Σ

m
=
1

2
(
n
−
k
)
+
1

h
m

(

φ

j
m

(
x
)
)

f(x) = \Sigma _{l = 1}^k{g_l}({\psi _{{i_l}}}(x)) + \Sigma _{m = 1}^{2(n - k) + 1}{h_m}({\varphi _{{j_m}}}(x))

, with

g
l

∈
C
(

ψ

i
l

(
X
)
)

{g_l} \in C({\psi _{{i_l}}}(X))

and

h
m

∈
C
(
R
)

{h_m} \in C(R)

. It is also shown that in many cases the number

2
(
n
−
k
)
+
1

2(n - k) + 1

is the smallest possible.
Homology cobordisms, link concordances, and hyperbolic 3-manifolds
Authors: Myers Robert 
Published: June 2010

Let

M
0
3

M_0^3

and

M
1
3

M_1^3

be compact, oriented

3
3

-manifolds. They are homology cobordant (respectively relative homology cobordant) if

∂

M
1
3

=
∅

(

resp.

∂

M
1
3

≠
∅
)

\partial M_1^3 = \emptyset \;({\text {resp.}}\;\partial M_1^3 \ne \emptyset )

and there is a smooth, compact oriented

4
4

-manifold

W
4

{W^4}

such that

∂

W
4

=

M
0
3

−

M
1
3

\partial {W^4} = M_0^3 - M_1^3

(resp.

∂

W
4

=

M
0
3

−

M
1
3

)
∪
(

M
i
3

×
[
0
,
1
]
)

\partial {W^4} = M_0^3 - M_1^3) \cup (M_i^3 \times [0,1])

and

H

∗

(

M
i
3

;

Z

)
→

H

∗

(

W
4

;

Z

)

{H_{\ast }}(M_i^3;{\mathbf {Z}}) \to {H_{\ast }}({W^4};{\mathbf {Z}})

are isomorphisms,

i
=
0
,
1

i = 0,1

. Theorem. Every closed, oriented

3
3

-manifold is homology cobordant to a hyperbolic

3
3

-manifold. Theorem. Every compact, oriented

3
3

-manifold whose boundary is nonempty and contains no

2
2

-spheres is relative homology cobordant to a hyperbolic

3
3

-manifold. Two oriented links

L
0

{L_0}

and

L
1

{L_1}

in a

3
3

-manifold

M
3

{M^3}

are concordant if there is a set

A
2

{A^2}

of smooth, disjoint, oriented annuli in

M
×
[
0
,
1
]

M \times [0,1]

such that

∂

A
2

=

L
0

−

L
1

\partial {A^2} = {L_0} - {L_1}

, where

L

i

⊆

M
3

×
{
i
}
,
i
=
0
,
1

{L_{i}} \subseteq \;{M^3} \times \{ i\} ,i = 0,1

. Theorem. Every link in a compact, oriented

3
3

-manifold

M
3

{M^3}

whose boundary contains no

2
2

-spheres is concordant to a link whose exterior is hyperbolic. Corollary. Every knot in

S
3

{S^3}

is concordant to a knot whose exterior is hyperbolic.