bluuebunny/crossref_metadata_2025_split huggingface dataset:
On classical quotients of polynomial identity rings with involution
Authors: Rowen Louis Halle 
Published: June 2010

Let

(
R
,
∗
)

(R, \ast )

denote a ring

R
R

with involution

(
∗
)

( \ast )

, where “involution” means “

anti - automorphism of order

≦

two

{\text {anti - automorphism of order}} \leqq {\text { two}}

". We can specialize many ring-theoretical concepts to rings with involution; in particular an ideal of

(
R
,
∗
)

(R, \ast )

is an ideal of

R
R

stable under

(
∗
)

( \ast )

, and the center of

(
R
,
∗
)

(R, \ast )

is the set of central elements of

R
R

which are fixed under

(
∗
)

( \ast )

. Then we say

(
R
,
∗
)

(R, \ast )

is prime when the product of any two nonzero ideals of

(
R
,
∗
)

(R, \ast )

is nonzero; similarly

(
R
,
∗
)

(R, \ast )

is semiprime when any power of a nonzero ideal of

(
R
,
∗
)

(R, \ast )

is nonzero. The main result of this paper is a strong analogue to Posner’s theorem [5], namely that any prime

(
R
,
∗
)

(R, \ast )

with polynomial identity has a ring of quotients

R
T

{R_T}

, formed merely by adjoining inverses of nonzero elements of the center of

(
R
,
∗
)

(R, \ast )

. This quotient ring

(

R
T

,
∗
)

({R_T}, \ast )

is simple and finite dimensional over its center. An extension of these results to semiprime Goldie rings with polynomial identity is given.
Nonlinear oscillation of a sublinear delay equation of arbitrary order
Authors: Kusano Takaŝi , Onose Hiroshi 
Published: June 2010

The equations considered generalize
\[

x

(
n
)

(
t
)
+
p
(
t
)

|

x
(
g
(
t
)
)

|

α

sgn
⁡
x
(
g
(
t
)
)
=
0
,

0
>
α
>
1.

{x^{(n)}}(t) + p(t)|x(g(t)){|^\alpha }\operatorname {sgn} x(g(t)) = 0,\quad 0 > \alpha > 1.

\]
A necessary and sufficient condition is established that all solutions are oscillatory when

n
n

is even and are either oscillatory or strongly monotone when

n
n

is odd. The result makes clear a difference in oscillatory property between sublinear delay equations and the corresponding ordinary differential equations.
A tauberian group algebra
Authors: Mueller-Roemer Peter R. 
Published: June 2010

Let G be the group of real matrices
\[

(
x
,
y
)
=

(

e
x

a
m
p
;
0

y

a
m
p
;
1

)

(
x
,
y
∈
R
)
.

(x,y) = \left ( {\begin {array}{*{20}{c}} {{e^x}} & 0 \\ y & 1 \\ \end {array} } \right )\quad (x,y \in R).

\]
Every proper closed two-sided ideal of

L
1

(
G
)

{L^1}(G)

is contained in a maximal modular two-sided ideal. The strong radical of

L
1

(
G
)

{L^1}(G)

is the set of all

f
∈

L
1

(
G
)

f \in {L^1}(G)

with

∫
f
(
x
,
y
)

d
y
=
0

\smallint f(x,y)\;dy = 0

for almost all

x
∈
R

x \in R

. The strong structure spaces of

L
1

(
G
)

{L^1}(G)

and

L
1

(
R
)

{L^1}(R)

are homeomorphic.
Certain elements in quotients of measure algebras
Authors: Saeki Sadahiro 
Published: June 2010

Let

G
G

be a locally compact group, and

M
(
G
)

M(G)

the convolution algebra with unit

δ
\delta

of all bounded Radon measures on

G
G

. Let

I
I

be a left ideal in

M
(
G
)

M(G)

and let

C
∩

I
⊥

C \cap {I^ \bot }

be the space of all bounded continuous functions

P
P

on

G
G

with

∫
G

P
d
μ
=
0

\int _GPd\mu = 0

for all

μ
\mu

in

I
I

. Suppose that there exists a function

P
0

{P_0}

in

C
∩

I
⊥

C \cap {I^ \bot }

such that

|

|

P
0

|

|

∞

>
lim

sup

x
→
∞

|

P
0

(
x
)

|

||{P_0}|{|_\infty } > \lim {\sup _{x \to \infty }}|{P_0}(x)|

. Let also

λ
\lambda

and

μ
\mu

be two measures in

M
(
G
)

M(G)

such that

|

|

λ
+
I

|

|

=

|

|

μ
+
I

|

|

=
1

||\lambda + I|| = ||\mu + I|| = 1

, and

(
λ
+
I
)
∗
μ
⊂
δ
+
1

(\lambda + I) \ast \mu \subset \delta + 1

. In this paper we prove under these conditions that there exist a complex number

c
c

of modulus one and a point

x
0

{x_0}

in

G
G

such that

∫
G

P
d
μ
=
c
P
(

x
0

)

\int _G Pd\mu = cP({x_0})

for all functions

P
P

in

C
∩

I
⊥

C \cap {I^ \bot }

with

|

|

P

|

|

∞

≦

|

∫
G

P
d
μ

|

=
1

||P|{|_\infty } \leqq |\int _G Pd\mu | = 1

. This generalizes a theorem of K. deLeeuw and Y. Katznelson.
A characterization of thin operators in a von Neumann algebra
Authors: Olsen Catherine L. 
Published: June 2010

Let

A

\mathcal {A}

be a von Neumann algebra,

J

\mathcal {J}

a uniformly closed, weakly dense, two-sided ideal in

A

,

L

\mathcal {A},\mathcal {L}

the center of

A

\mathcal {A}

, and

P

\mathcal {P}

the lattice of projections in

J

\mathcal {J}

. An operator

A
∈

A

A \in \mathcal {A}

is thin relative to

J

\mathcal {J}

if

A
=
Z
+
K

A = Z + K

, for some

Z
∈

L

,
K
∈

J

Z \in \mathcal {L},K \in \mathcal {J}

. The thin operators relative to

J

\mathcal {J}

are characterized as those

A
∈

A

A \in \mathcal {A}

satisfying

lim

P
∈

P

|

|

A
P
−
P
A

|

|

=
0

{\lim _{P \in \mathcal {P}}}||AP - PA|| = 0

. It is also shown that
\[

lim

sup

P
∈

P

|

|

P
A
P
−
A
P

|

|

=
lim

sup

P
∈

P

|

|

P
A
P
−
P
A

|

|

.

\lim \sup \limits _{P \in \mathcal {P}} ||PAP - AP|| = \lim \sup \limits _{P \in \mathcal {P}} ||PAP - PA||.

\]
A generalization of a theorem by D. K. Faddeev
Authors: Behnen Konrad 
Published: June 2010

In this paper we give a simple proof of the statement

lim

n
→
∞

∫

K
n

(
x
,
y
)
f
(
y
)
d
μ
(
y
)
=
f
(
x
)

{\lim _{n \to \infty }}\int {{K_n}(x,y)f(y)d\mu (y) = f(x)}

for

μ
\mu

-almost all

x
x

under weaker and more general assumptions than those of the usual Faddeev theorems.
Convexity of vector-valued functions
Authors: Hsu Ih Ching , Kuller Robert G. 
Published: June 2010

Let

(

B

,
≪
)

(\mathcal {B}, \ll )

be a Banach lattice, and

(
a
,
b
)

(a,b)

be an open interval on the real line. A function

F
:
(
a
,
b
)
→

B

F:(a,b) \to \mathcal {B}

is defined to be weakly convex if there exists a nonnegative nondecreasing continuous function

G
:
(
a
,
b
)
→

B

G:(a,b) \to \mathcal {B}

such that

p
[
F
(
S
)
]
+
t
p
[
G
(
s
)
]
≤
p
[
F
(
s
+
t
)
]

p[F(S)] + tp[G(s)] \leq p[F(s + t)]

, whenever

s
s

and

s
+
t

s + t

are in

(
a
,
b
)

(a,b)

for each positive linear functional

p
p

on

B

\mathcal {B}

. A representation theorem is proved as follows: If

F
F

is weakly convex on

(
a
,
b
)

(a,b)

and is bounded on an interval contained in

(
a
,
b
)

(a,b)

, then

(
B
)

∫

a
+
ϵ

x

G
(
s
)
d
m
=
F
(
x
)
−
F
(
a
+
ϵ
)

(B)\int _{a + \epsilon }^x {G(s)dm = F(x) - F(a + \epsilon )}

, where

(
B
)

∫

a
+
ϵ

x

G
(
s
)
d
m

(B)\int _{a + \epsilon }^x {G(s)dm}

is the Bochner integral of

G
G

on

[
a
,
ϵ
,
x
]

[a, \epsilon ,x]

with

0
>
ϵ

0 > \epsilon

and

a
>
ϵ
>
x
>
b

a > \epsilon > x > b

.
Spectra of compact composition operators
Authors: Caughran James G. , Schwartz Howard J. 
Published: June 2010

Let

ϕ
\phi

be holomorphic and map the open unit disk into itself, and let

C
ϕ

:
f
→
f
∘
ϕ

{C_\phi }:f \to f \circ \phi

be the composition operator on

H
2

{H^2}

generated by

ϕ
\phi

. If

C
ϕ

{C_\phi }

is a compact operator then

(
1
)
ϕ
(

z
0

)
=

z
0

(1)\phi ({z_0}) = {z_0}

for some

z
0

ϵ
D

{z_0} \epsilon D

;

(
2
)
σ
(

C
ϕ

)
=
{

ϕ
′

(

z
0

)
n

:
n
=
0
,
1
,
2
,
…
}
∪
{
0
}

(2)\sigma ({C_\phi }) = \{ \phi ’{({z_0})^n}:n = 0,1,2, \ldots \} \cup \{ 0\}

.
A note on a basis problem
Authors: Anderson J. M. 
Published: June 2010

It is shown that the functions

{
exp
−

λ
ν

x

}

ν
=
1

∞

\{ \exp - {\lambda _\nu }x\} _{\nu = 1}^\infty

form a basis for the subspace of

L

2

(
0
,
∞
)

{\mathcal {L}_2}(0,\infty )

which they span if and only if
\[

inf
μ

∏

ν
=
1
;
ν
≠
μ

∞

|

λ
ν

−

λ
μ

λ
¯

ν

+

λ
μ

|

=
δ
>
0.

\inf \limits _\mu \prod \limits _{\nu = 1;\nu \ne \mu }^\infty {|\frac {{{\lambda _\nu } - {\lambda _\mu }}}{{{{\bar \lambda }_\nu } + {\lambda _\mu }}}| = \delta > 0.}

\]
The proof uses certain estimates concerning interpolation in

H
2

{H_2}

due to Shapiro and Shields. The proof makes explicit a construction embedded in a paper of Binmore [1, Theorems 9-12.].
The Lévy-Baxter theorem for Gaussian random fields: a sufficient condition
Authors: Kawada Takayuki 
Published: June 2010

For a function

F
F

of some class and for a Gaussian random field

{
X
(

t

)
;

t

ϵ

[
0
,

1

]
N

}

\{ X({\text {t}});\;{\text {t}}\;\epsilon {[0,\;1]^N}\}

,

F
F

-variation

V

F
,
n

(
X
)

{V_{F,n}}(X)

is defined as

V

F
,
n

(
X
)
=

∑

i

F
(

X

i

,
n

)

{V_{F,n}}(X) = {\sum _{\text {i}}}F({X_{{\text {i}},n}})

, where

X

i

,
n

=

Δ
N
s

⋯

Δ
1
s

X
(
s

i

)
,

s
=

2

−
n

{X_{{\text {i}},n}} = \Delta _N^s \cdots \Delta _1^sX(s{\text {i}}),\;s = {2^{ - n}}

, and

Δ
k
s

\Delta _k^s

is the difference operator in the

k
th

k\operatorname {th}

component of

i

=
(

i
1

,
⋯
,

i
N

)

{\text {i}} = ({i_1}, \cdots ,{i_N})

,

(
1
⩽

i
k

⩽

2
n

;
k
=
1
,
…
,
N
)

(1 \leqslant {i_k} \leqslant {2^n};k = 1, \ldots ,N)

. Here is presented a sufficient condition for the existence with probability

1
1

of the limit of the normalization of

V

F
,
n

(
X
)

{V_{F,n}}(X)

as

n
→
∞

n \to \infty

.
Operators satisfying certain growth conditions
Authors: Patel S. M. , Gupta B. C. 
Published: June 2010

Let

T
T

be an operator on a complex Hilbert space

H
H

. Some growth conditions on operator radius of the resolvent of

T
T

are studied. Moreover, it is shown that the conjecture, due to V. Istrătescu, that for operators

T
T

satisfying growth condition

(

G

1

)

({{\text {G}}_1})

\[

sup

|

|

x

|

|

=
1

{

|

|

T
x

|

|

2

−

|

(
T
x
,

x
)

|

2

}
=

R
T
2

,

\sup \limits _{||x|| = 1} \;\{ ||Tx|{|^2} - |(Tx,\;x){|^2}\} = R_T^2,

\]
where

R
T

{R_T}

is the radius of the smallest circular disk containing the spectrum

σ
(
T
)

\sigma (T)

, turns out to be false.
Rational approximation to solutions of linear differential equations with algebraic coefficients
Authors: Schmidt Wolfgang M. 
Published: June 2010

Let

K
K

be the field of formal series

α
=

a
k

t
k

+
⋯
+

a
0

+

a

−
1

t

−
1

+
⋯

\alpha = {a_k}{t^k} + \cdots + {a_0} + {a_{ - 1}}{t^{ - 1}} + \cdots

and let

|

|

|\;\;|

be the valuation with

|

α

|

=

2
k

|\alpha | = {2^k}

if

a
k

≠
0

{a_k} \ne 0

. Suppose

α
ϵ
K

\alpha \epsilon K

satisfies an

m

th

m{\text {th}}

order linear differential equation whose coefficients are algebraic functions of

t
t

. Then for

ϵ
>
0

\epsilon > 0

there are only finitely many rational functions

p
(
t
)

/

q
(
t
)

p(t)/q(t)

with

|

α
−
p
(
t
)

/

q
(
t
)

|

>

|

q
(
t
)

|

−
2
−
4
m
−
ϵ

|\alpha - p(t)/q(t)| > |q(t){|^{ - 2 - 4m - \epsilon }}

.
Sum of a double series
Authors: Sharma B. L. 
Published: June 2010

In this paper we obtain the sum of a double series

F
(
1
,
1
)

F(1,1)

and, in a particular case, we get a new formula

4

F
3

(
1
)

_4{F_3}(1)

,
\[

4

F
3

[

α
,
β
−
α
,
1

/

2
ρ
,
1

/

2
ρ
+
1

/

2
;
1

1

/

2
β
,
1

/

2
(
1
+
β
)
,
1
+
ρ
;

]

=

Γ
(
β
−
ρ
−
α
)
Γ
(
β
)

Γ
(
β
−
ρ
)
Γ
(
β
−
α
)

,

_4{F_3}\left [ {\begin {array}{*{20}{c}} {\alpha ,\beta - \alpha ,1/2\rho ,1/2\rho + 1/2;1} \\ {1/2\beta ,1/2(1 + \beta ),1 + \rho ;} \\ \end {array} } \right ] = \tfrac {{\Gamma (\beta - \rho - \alpha )\Gamma (\beta )}} {{\Gamma (\beta - \rho )\Gamma (\beta - \alpha )}},

\]
provided that

R
(
β
−
α
)
>
0

R(\beta - \alpha ) > 0

,

R
(
β
−
ρ
−
α
)
>
0

R(\beta - \rho - \alpha ) > 0

and

R
(
β
−
ρ
)
>
0

R(\beta - \rho ) > 0

. If

α
=
−
n

\alpha = - n

, the formula reduces to a known result due to Bailey [2].
A change of rings theorem and the Artin-Rees property
Authors: Boratyński M. 
Published: June 2010

A two-sided ideal

A

\mathfrak {A}

of the ring

R
R

is said to have the left

A
R

AR

property if for every left ideal

I
I

and every

k
k

there exists an

n
n

such that

A

n

∩
I
⊂

A

k

I

{\mathfrak {A}^n} \cap I \subset {\mathfrak {A}^k}I

. Let

R
R

be a left noetherian ring and

A

\mathfrak {A}

a two-sided ideal contained in its Jacobson radical. If

A

\mathfrak {A}

has the

AR
\operatorname {AR}

property then

l

gl

dim

R
⩽

p

dim

R

/

A

+

l

gl dim

R

/

A

{\text {l}}\;{\text {gl}}\;{\text {dim}}\;R \leqslant {\text {p}}\;{\text {dim}}\;R/\mathfrak {A} + {\text {l}}\;{\text {gl dim }}R/\mathfrak {A}

, where

p

dim

R

/

A

{\text {p}}\;{\text {dim}}\;R/\mathfrak {A}

denotes the (left) projective dimension of the module

R

/

A

R/\mathfrak {A}

.
Some remarks on uniqueness for a class of singular abstract Cauchy problems
Authors: Donaldson James A. , Goldstein Jerome A. 
Published: June 2010

Of concern is the Cauchy problem for equations of the form

u
(
t
)
+
α
(
t
)

u
′

(
t
)
+

S
2

u
(
t
)
=
0

(
′

=
d

/

d
t
)

u(t) + \alpha (t)u’(t) + {S^2}u(t) = 0(’ = d/dt)

on a complex Hilbert space

X
X

.

S
S

is a selfadjoint operator on

X
X

while

α
\alpha

is a continuous function on

(
0
,
∞
)

(0,\infty )

which can be unbounded at

t
=
0

t = 0

. Uniqueness results are obtained for these equations by applying a uniqueness theorem for nonlinear equations. Furthermore, nonuniqueness examples for the linear abstract Euler-Poisson-Darboux equation, which is contained in this class, show that the uniqueness theorem is best possible.
A new proof for an inequality of Jenkins
Authors: Leeman George B. 
Published: June 2010

A new proof of Jenkins’ inequality
\[

Re
⁡
(

e

2
i
θ

a
3

−

e

2
i
θ

a
2
2

−
τ

e

i
θ

a
2

)
⩽
1
+

3
8

τ
2

−

1
4

τ
2

log
⁡
(
τ

/

4
)
,

0
⩽
τ
⩽
4
,

\operatorname {Re} ({e^{2i\theta }}{a_3} - {e^{2i\theta }}a_2^2 - \tau {e^{i\theta }}{a_2}) \leqslant 1 + \tfrac {3} {8}{\tau ^2} - \tfrac {1} {4}{\tau ^2}\log (\tau /4),\quad 0 \leqslant \tau \leqslant 4,

\]
for univalent functions

f
(
z
)
=
z
+

∑

n
=
2

∞

a
n

z
n

f(z) = z + \sum \nolimits _{n = 2}^\infty {{a_n}{z^n}}

is presented.
Separation for differential operators and the 𝐿^{𝑝} spaces
Authors: Zettl Anton 
Published: June 2010

The differential operator
\[

M
y
=

y

(
n
)

+

q

n
−
1

y

(
n
−
1
)

+
⋯
+

q
1

y
′

+

q
0

y

My = {y^{(n)}} + {q_{n - 1}}{y^{(n - 1)}} + \cdots + {q_1}y’ + {q_0}y

\]
is said to be separated in

L
p

(
0
,
∞
)

{L^p}(0,\infty )

if

y
∈

L
p

(
0
,
∞
)
,

y

(
n
−
1
)

y \in {L^p}(0,\infty ),{y^{(n - 1)}}

absolutely continuous, and

M
y
∈

L
p

(
0
,
∞
)

My \in {L^p}(0,\infty )

imply that

q
i

y

(
i
)

∈

L
p

(
0
,
∞
)

{q_i}{y^{(i)}} \in {L^p}(0,\infty )

for all

i
=
0
,
1
,
…
,
n
−
1

i = 0,1, \ldots ,n - 1

. As a special case of our main result we obtain:

M
M

is separated in

L
p

(
0
,
∞
)

{L^p}(0,\infty )

if

q
i

=

b
i

+

s
i

{q_i} = {b_i} + {s_i}

where

b
i

{b_i}

is essentially bounded and

s
i

∈

L

p
i

(
0
,
∞
)

{s_i} \in {L^{pi}}(0,\infty )

for some

p
i

⩾
p
,
i
=
0
,
1
,
…
,
n
−
1

{p_i} \geqslant p,i = 0,1, \ldots ,n - 1

. The case

n
=
2
,
p
=
2
,

q
1

≡
0

n = 2,p = 2,{q_1} \equiv 0

is due to Everitt-Giertz-Weidmann [2]. In the same paper these authors show that this result is best possible in the sense that for any

p
p

satisfying

1
>
p
>
2

1 > p > 2

a function

q
q

exists in

L
p

(
0
,
∞
)

{L^p}(0,\infty )

such that

y
+
q
y

y + qy

is not separated in

L
2

(
0
,
∞
)

{L^2}(0,\infty )

.
The asymptotic expansion of the zeta-function of a compact semisimple Lie group
Authors: Cahn Robert S. 
Published: June 2010

If

G
G

is a connected, simply connected, semisimple Lie group with metric given by the negative of the Killing form and zeta-function

Z
(
t
)

Z(t)

, then
\[

Z
(
t
)
=

Vol
⁡
G

(
4
π
t
)

dim
⁡
G

/

2

exp
⁡

|

δ

|

2

t
+

exponentially small error as

t
↓
0.

Z(t) = \frac {{\operatorname {Vol} G}} {{{{(4\pi t)}^{\dim G/2}}}}\exp |\delta {|^2}t + {\text {exponentially small error as }}t \downarrow 0.

\]
Cyclically monotone linear operators
Authors: Shiu Elias S. W. 
Published: June 2010

A linear operator on a complex Hilbert space

H

\mathcal {H}

is called

n
n

-cyclically monotone if for each sequence

x
0

,

x
1

,
…
,

x

n
−
1

,

x
n

=

x
0

{x_0},{x_1}, \ldots ,{x_{n - 1}},{x_n} = {x_0}

of

n
n

elements in

H

,

Σ

j
=
0

n
−
1

Re
⁡
(
T

x
j

−

x

j
+
1

)
⩾
0

\mathcal {H},\Sigma _{j = 0}^{n - 1}\operatorname {Re} (T{x_j} - {x_{j + 1}}) \geqslant 0

. We show that

T
T

is

n
n

-cyclically monotone if and only if

|

Arg
⁡
(
T
x
,
x
)

|

⩽
π

/

n
,
∀
x
∈

H

|\operatorname {Arg} (Tx,x)| \leqslant \pi /n,\forall x \in \mathcal {H}

. If

T
m

{T_m}

and

T
n

{T_n}

are

m
m

- and

n
n

-cyclically monotone operators, then the spectrum of the product

T
m

T
n

{T_m}{T_n}

lies in the sector

{
z
∈

C

:

|

Arg

z

|

⩽
π

/

m
+
π

/

n
}

\{ z \in {\mathbf {C}}:|\operatorname {Arg} \;z| \leqslant \pi /m + \pi /n\}

.
Representation of numbers by cascades
Authors: Chen C. C. , Daykin D. E. 
Published: June 2010

A cascade

C
C

is defined as a sum of binomial coefficients
\[

C
=

(

a
h

h

)

+

(

a

h
−
1

h
−
1

)

+
⋯
+

(

a
t

t

)

C = \left ( {\begin {array}{*{20}{c}} {{a_h}} \\ h \\ \end {array} } \right ) + \left ( {\begin {array}{*{20}{c}} {{a_{h - 1}}} \\ {h - 1} \\ \end {array} } \right ) + \cdots + \left ( {\begin {array}{*{20}{c}} {{a_t}} \\ t \\ \end {array} } \right )

\]
where

a
h

>

a

h
−
1

>
⋯
>

a
t

{a_h} > {a_{h - 1}} > \cdots > {a_t}

. In this expression, we assume that

(
h
a

)
=
0

(_h^a) = 0

whenever

a
>
h

a > h

. Given a cascade

C
C

and a sequence

ε
=
⟨

ε
h

,

ε

h
−
1

,
…
,

ε
t

⟩

\varepsilon = \langle {\varepsilon _h},{\varepsilon _{h - 1}}, \ldots ,{\varepsilon _t}\rangle

of signs (i.e.

ε
i

=
+
1

or

−
1

{\varepsilon _i} = + 1\;{\text {or}}\; - 1

for each

i
i

), we define
\[

ε
C
=

ε
h

(

a
h

h

)

+
⋯
+

ε
t

(

a
t

t

)

.

\varepsilon C = {\varepsilon _h}\left ( {\begin {array}{*{20}{c}} {{a_h}} \\ h \\ \end {array} } \right ) + \cdots + {\varepsilon _t}\left ( {\begin {array}{*{20}{c}} {{a_t}} \\ t \\ \end {array} } \right ).

\]
Also, we put
\[

α
C
=

(

a
h

h
+
1

)

+

(

a

h
−
1

h

)

+
⋯
+

(

a
t

t
+
1

)

.

\alpha C = \left ( {\begin {array}{*{20}{c}} {{a_h}} \\ {h + 1} \\ \end {array} } \right ) + \left ( {\begin {array}{*{20}{c}} {{a_{h - 1}}} \\ h \\ \end {array} } \right ) + \cdots + \left ( {\begin {array}{*{20}{c}} {{a_t}} \\ {t + 1} \\ \end {array} } \right ).

\]
In the paper, we prove that for any sequence

⟨

n
0

,

n
1

,
…
,

n
s

⟩

\langle {n_0},\,{n_1}, \ldots ,{n_s}\rangle

of integers, there exist a cascade

C
C

and a corresponding sequence

ε
\varepsilon

of signs such that

n
i

=
ε

α
i

C

{n_i} = \varepsilon {\alpha ^i}C

for

i
=
0
,

1
,
…
,
s

i = 0,\;1, \ldots ,s

where

α
0

C
=
C
,

α
1

C
=
α
C
,

α
2

C
=
α
(

α
1

C
)

{\alpha ^0}C = C,\;{\alpha ^1}C = \alpha C,\;{\alpha ^2}C = \alpha ({\alpha ^1}C)

, and recursively,

α
n

C
=
α
(

α

n
−
1

C
)

{\alpha ^n}C = \alpha ({\alpha ^{n - 1}}C)

.
A limit theorem for measurable random processes and its applications
Authors: Grinblat L. Š. 
Published: June 2010

Let the measurable random processes

ξ
1

(
t
)
,
…
,

ξ
n

(
t
)
,
…

{\xi _1}(t), \ldots ,{\xi _n}(t), \ldots

and

ξ
(
t
)

\xi (t)

be defined on

[
0
,

1
]

[0,\;1]

. There exists

C
C

such that for all

n
n

and

t
t

we have

E

|

ξ
n

(
t
)

|

p

⩽
C
,

p
⩾
1

E|{\xi _n}(t){|^p} \leqslant C,\;p \geqslant 1

. The following assertion is valid: if for any finite set of points

t
1

,
…
,

t
k

⊂
[
0
,

1
]

{t_1}, \ldots ,{t_k} \subset [0,\;1]

the joint distribution of

ξ
n

(

t
1

)
,
…
,

ξ
n

(

t
k

)

{\xi _n}({t_1}), \ldots ,{\xi _n}({t_k})

converges to the joint distribution of

ξ
(

t
1

)
,
…
,
ξ
(

t
k

)

\xi ({t_1}), \ldots ,\xi ({t_k})

, and if

E

|

ξ
n

(
t
)

|

p

→
E

|

ξ
(
t
)

|

p

E|{\xi _n}(t){|^p} \to E|\xi (t){|^p}

for all

t
∈
[
0
,

1
]

t \in [0,\;1]

, then for any continuous functional

f
f

on

L
p

[
0
,

1
]

{L_p}[0,\;1]

the distribution of

f
(

ξ
n

(
t
)
)

f({\xi _n}(t))

converges to the distribution of

f
(
ξ
(
t
)
)

f(\xi (t))

. This statement immediately implies the convergence of distributions in some limit theorems for the sums of independent random variables (for example, in one of the theorems of P. Erdös and M. Kac) and in some statistical criteria (for example, in the

ω
2

{\omega ^2}

-criterion of Cramér and von Mises).
Scalar curvatures on 𝑂(𝑀),𝐺₂(𝑀)
Authors: Rigas Alcibiades 
Published: June 2010

We show that every

C
∞

f
:

G
2

(
M
)
→

R

,

M
n

{C^\infty }f:{G_2}(M) \to {\mathbf {R}},{M^n}

a compact connected riemannian manifold

n
⩾
3

n \geqslant 3

, is the scalar curvature function of some complete riemannian metric on

G
2

(
M
)

{G_2}(M)

, the grassmann bundle of

2
2

planes over

M
M

, except possibly when

K
=

constant

⩾
0

K = {\text {constant }} \geqslant 0

. A similar result holds for

O
(
M
)

O(M)

bundle of orthonormal frames on

M
M

.
A note on unconditionally converging series in 𝐿_{𝑝}
Authors: Ørno Peter 
Published: June 2010

Theorem. A series

∑

f
i

\sum {{f_i}}

in

L
p

[
0
,

1
]
(
1
⩽
p
⩽
2
)

{L_p}[0,\,1](1 \leqslant p \leqslant 2)

is unconditionally convergent if and only if for each

i
i

and for all

t
∈
[
0
,

1
]
,

f
i

(
t
)
=

α
i

g
(
t
)

w
i

(
t
)

t \in [0,\,1],\;{f_i}(t) = {\alpha _i}g(t){w_i}(t)

where

(

α
i

)
∈

l
2

,

g
∈

L
2

[
0
,

1
]

({\alpha _i}) \in {l_2},\;g \in {L_2}[0,\,1]

and

(

w
i

)

({w_i})

is an orthonormal sequence in

L
2

[
0
,

2
]

{L_2}[0,\,2]

. This characterization allows the generalization (to u.c. series in

L
p

[
0
,

1
]

{L_p}[0,\,1]

) of several classical theorems concerning almost everywhere convergence of orthogonal series in

L
2

{L_2}

.
On odd dimensional surgery with finite fundamental group
Authors: Hausmann Jean-Claude 
Published: June 2010

One proves that, for any finite group G and homomorphism

ω
:
G
→

Z

/

2

Z

\omega :G \to {\mathbf {Z}}/2{\mathbf {Z}}

, the natural homomorphism

L

2
k
+
1

h

(

Z

G
,
ω
)
→

L

2
k
+
1

h

(

Q

G
,
ω
)

L_{2k + 1}^h({\mathbf {Z}}G,\omega ) \to L_{2k + 1}^h({\mathbf {Q}}G,\omega )

between Wall surgery groups is identically zero. Some results concerning the exponent of

L

2
k
+
1

h

(

Z

G
;
ω
)

L_{2k + 1}^h({\mathbf {Z}}G;\omega )

are deduced.
On the nonexistence of a projection from functions of 𝑥 to functions of 𝑥ⁿ
Authors: Milman P. 
Published: June 2010

The subspace

ϕ
∗

C
∞

(

R

1

)
⊂

C
∞

(

R

1

)

{\phi ^ \ast }{C^\infty }({{\mathbf {R}}^1}) \subset {C^\infty }({{\mathbf {R}}^1})

of all

C
∞

{C^\infty }

functions of

ϕ
(
x
)
=

x
n

,
n
=
1
,
2
,
3
,
…

\phi (x) = {x^n},n = 1,2,3, \ldots

, is a closed subspace of

C
∞

(

R

1

)

{C^\infty }({{\mathbf {R}}^1})

by Glaeser’s Composition Theorem. We prove that for

n
>
2

n > 2

there does not exist a linear continuous projection

π
\pi

from

C
∞

(

R

1

)

{C^\infty }({{\mathbf {R}}^1})

onto

ϕ
∗

C
∞

(

R

1

)

{\phi ^ \ast }{C^\infty }({{\mathbf {R}}^1})

.
A mapping theorem for logarithmic and integration-by-parts operators
Authors: Appling William D. L. 
Published: June 2010

Suppose U is a set, F is a field of subsets of U,

p

A
B

{\mathfrak {p}_{AB}}

is the set of all bounded real-valued finitely additive functions defined on F, and W is a collection of functions from F into

exp
⁡
(

R

)

\exp ({\mathbf {R}})

, closed under multiplication, each element of which has range union bounded and bounded away from 0. Let

P

\mathcal {P}

denote the set to which T belongs iff T is a function from W into

p

A
B

{\mathfrak {p}_{AB}}

such that if each of

α
\alpha

and

β
\beta

is in W and V is in F, then the following integrals exist and the following “integration-by-parts” equation holds:
\[

∫
V

α
(
I
)
T
(
β
)
(
I
)
+

∫
V

β
(
I
)
T
(
α
)
(
I
)
=
T
(
α
β
)
(
V
)
.

\int _V \alpha (I)T(\beta )(I) + \int _V {\beta (I)T(\alpha )(I) = T(\alpha \beta )(V).}

\]
Let

L

\mathfrak {L}

denote the set to which S belongs iff S is a function from W into

p

A
B

{\mathfrak {p}_{AB}}

such that if each of

α
\alpha

and

β
\beta

is in W, then the integral

∫
U

α
(
I
)
S
(
β
)
(
I
)

\smallint _U {\alpha (I)S(\beta )(I)}

exists and the following “logarithmic” equation holds:

S
(
α
β
)
=
S
(
α
)
+
S
(
β
)

S(\alpha \beta ) = S(\alpha ) + S(\beta )

. It is shown that

{
(
T
,
S
)
:
T

in

P

,

S
=
{
(
α
,

∫

(
1

/

α
)
T
(
α
)
)
:

α

in

W
}
}

\{ (T,S):T\;{\text {in}}\;\mathcal {P},\;S = \{ (\alpha ,\;\smallint {(1/\alpha )T(\alpha )):\;\alpha \;{\text {in}}\;W\} \} }

is a one-one mapping from

P

\mathcal {P}

onto

L

\mathfrak {L}

.