bluuebunny/crossref_metadata_2025_split huggingface dataset:
A short proof of an inequality of Carleson’s
Authors: Neville Charles W. 
Published: June 2010

We give a simple proof that if

a
i

,
i
=
1
,
2
,
…

{a_i},i = 1,2, \ldots

, is a uniformly separated sequence in the unit disk, then

Σ
(
1
−

|

a
i

|

2

)

|

f
(

a
i

)

|

p

⩽
K

‖
f
‖

p
p

\Sigma (1 - |{a_i}{|^2})|f({a_i}){|^p} \leqslant K\left \|f\right \|_p^p

, for all

f
∈

H
p

f \in {H^p}

and

1
⩽
p
>
∞

1 \leqslant p > \infty

.
An inequality for functions of exponential type not vanishing in a half-plane
Authors: Govil N. K. 
Published: June 2010

Let

f
(
z
)

f(z)

be an entire function of order 1, type

τ
\tau

having no zero in

Im

z
>
0

\operatorname {Im} \;z > 0

. If

h
f

(
−
π

/

2
)
=
τ
,

h
f

(
π

/

2
)
⩽
0

{h_f}( - \pi /2) = \tau , {h_f}(\pi /2) \leqslant 0

then it is known that

sup

−
∞
>
x
>
∞

|

f
′

(
x
)

|

⩾
(
τ

/

2
)

sup

−
∞
>
x
>
∞

|

f
(
x
)

|

{\sup _{ - \infty > x > \infty }}|f’(x)| \geqslant (\tau /2){\sup _{ - \infty > x > \infty }}|f(x)|

. In this paper we consider the case when

f
(
z
)

f(z)

has no zero in

Im

z
>
k
,
k
⩽
0

\operatorname {Im} \;z > k, k \leqslant 0

and obtain a sharp result.
Reproducing kernels for 𝑞-Jacobi polynomials
Authors: Al-Salam Waleed A. , Ismail Mourad E. H. 
Published: June 2010

We derive a family of reproducing kernels for the q-Jacobi polynomials

Φ
n

(
α
,
β
)

(
x
)

=
2

Φ
1

(

q

−
n

,

q

n
−
1
+
β

;

q
α

;
q
,
q
x
)

\Phi _n^{(\alpha ,\beta )}(x){ = _2}{\Phi _1}({q^{ - n}},{q^{n - 1 + \beta }};{q^\alpha };q,qx)

. This is achieved by proving that the polynomials

Φ
n

(
α
,
β
)

(
x
)

\Phi _n^{(\alpha ,\beta )}(x)

satisfy a discrete Fredholm integral equation of the second kind with a positive symmetric kernel, then applying Mercer’s theorem.
A boundary value problem for 𝐻^{∞}(𝐷)
Authors: Cahill Rotraut Goubau 
Published: June 2010

Let

W
=

⋃

n
=
1

∞

W
n

W = \bigcup \nolimits _{n = 1}^\infty {{W_n}}

be an

F
σ

{F_\sigma }

subset of the unit circle of measure 0 and let

{

q
n

}
,
n
⩾
1

\{ {q_n}\} ,n \geqslant 1

, be a decreasing sequence with

q
1

⩽
1

{q_1} \leqslant 1

and

lim
n

q
n

=
0

{\lim _n}{q_n} = 0

. There exists an H in

H
∞

(
D
)

{H^\infty }(D)

of norm

q
1

{q_1}

whose modulus has radial limit along every radius which has radial limit of modulus

q
1

{q_1}

on

W
1

{W_1}

and

q

n
+
1

{q_{n + 1}}

on

W

n
+
1

∖

⋃

k
=
1

n

W
k

{W_{n + 1}}\backslash \bigcup \nolimits _{k = 1}^n {{W_k}}

. If W is simultaneously a

G
δ

{G_\delta }

set, H may be chosen to have no zeros on C. It follows that for W countable, say

W
=
{

e

i

w
n

}
,
n
⩾
1

W = \{ {e^{i{w_n}}}\} ,n \geqslant 1

, there is such an H of norm 1 for which

lim

r
→
1

H
(
r

e

i

w
n

)
=
1

/

n

{\lim _{r \to 1}}H(r{e^{i{w_n}}}) = 1/n

. The proof of the theorem depends on the existence of a special collection of closed sets

{

S
λ

}
,
λ
⩾
1

\{ {S_\lambda }\} ,\lambda \geqslant 1

, real, for which the function h, defined by

h
(
x
)
=

a
n

+
[
(
inf
{
λ

|

x
∈
S
}
)
−
n
]
(

a

n
+
1

−

a
n

)
,

a
n

=
−
ln

q
n

h(x) = {a_n} + [(\inf \{ \lambda |x \in S\} ) - n]({a_{n + 1}} - {a_n}),{a_n} = - \ln \;{q_n}

, is such that the function

H
(
w
)
=
exp
⁡
(
−
1

/

2
π
)
∫
[
(
w
+

e

i
u

)

/

(

e

i
u

−
w
)
]
h
(
u
)

H(w) = \exp ( - 1/2\pi )\smallint [(w + {e^{iu}})/({e^{iu}} - w)]h(u)

du has the required properties. Some of the techniques used are similar to those developed in an earlier paper [1].
On the innovation theorem
Authors: Lin T. F. 
Published: June 2010

Let

z
(
t
)
,
0
⩽
t
⩽
T

z(t),0 \leqslant t \leqslant T

, be the signal process and

y
(
t
)
=

∫
0
t

z
(
r
)

d
r
+
w
(
t
)

y(t) = \smallint _0^tz(r)\;dr + w(t)

be the observation process where

w
(
t
)

w(t)

is a process of independent increments. It is shown that, under certain conditions, the innovation process

v
(
t
)
=
y
(
t
)
−

∫
0
t

E
(
z
(
r
)

|

y
(
u
)
,
0
⩽
u
⩽
r
)

d
r

v(t) = y(t) - \smallint _0^tE(z(r)|y(u), 0 \leqslant u \leqslant r)\;dr

, has the same probability law as

w
(
t
)

w(t)

.
Solvable assosymmetric rings are nilpotent
Authors: Pokrass David , Rodabaugh David 
Published: June 2010

Assosymmetric rings are ones which satisfy the law

(
x
,
y
,
z
)
=
(
P
(
x
)
,
P
(
y
)
,
P
(
z
)
)

(x,y,z) = (P(x),P(y),P(z))

for each permutation P of x, y, z. Let A be an assosymmetric ring having characteristic different from 2 or 3. We show that if A is solvable then A is nilpotent. Also, if each subring generated by a single element is nilpotent, and if A has D.C.C. on right ideals, then A is nilpotent. We also give an example showing that the Wedderburn Principal Theorem fails for assosymmetirc rings.
A unique continuation theorem involving a degenerate parabolic operator
Authors: Lair Alan V. 
Published: June 2010

We consider the degenerate parabolic operator

L
[
u
]
=
γ
B
[
u
]
−

u
t

L[u] = \gamma B[u] - {u_t}

on a domain

D
=
Ω
×
(
0
,
T
]

D = \Omega \times (0,T]

where

B
[
u
]
=

Σ

i
,
j
=
1

n

(

a

i
j

(
x
)

u

x
j

)

x
i

B[u] = \Sigma _{i,j = 1}^n{({a_{ij}}(x){u_{{x_j}}})_{{x_i}}}

and

γ
\gamma

is an arbitrary complex number. Classically,

γ
=
1

\gamma = 1

and the real-valued matrix

(

a

i
j

)

({a_{ij}})

is positive definite. We assume

(

a

i
j

)

({a_{ij}})

is a real-valued symmetric matrix but not necessarily definite. We prove that any complex-valued function u which satisfies the inequality

|

L
[
u
]

|

⩽
c

|

u

|

|L[u]| \leqslant c|u|

for some nonnegative constant c and vanishes initially as well as on the boundary of

Ω
\Omega

must vanish on all of D. The theorem is particularly useful in studying uniqueness for many systems which are not parabolic.
Subnormal generalized Hausdorff operators
Authors: Ghosh B. K. , Rhoades B. E. , Trutt D. 
Published: June 2010

One type of generalized Hausdorff matrix is the lower triangular matrix with entries

h

n
k

=

(

n
−
k

n
+
α

)

Δ

n
−
k

ν
k

{h_{nk}} = (_{n - k}^{n + \alpha }){\Delta ^{n - k}}{\nu _k}

, where

Δ

ν
n

=

ν
n

−

ν

n
+
1

,

ν
n

=

∫
0
1

t

n
+
α

d
β
(
t
)

\Delta {\nu _n} = {\nu _n} - {\nu _{n + 1}},{\nu _n} = \smallint _0^1{t^{n + \alpha }}d\beta (t)

for some

β
(
t
)

\beta (t)

of bounded variation on [0, 1] and for some

α
⩾
0

\alpha \geqslant 0

. The matrix

H
α

{H_\alpha }

generated by

ν
n

=

(
n
+
α
+
1

)

−
1

{\nu _n} = {(n + \alpha + 1)^{ - 1}}

is shown to be a subnormal operator on

l
2

{l^2}

if

α
\alpha

is a nonnegative integer.
A class of one-parameter nonlinear semigroups with differentiable approximating semigroups
Authors: Parker G. Edgar 
Published: June 2010

Suppose that T is a strongly continuous semigroup of transformations on a subset C of a Banach space X. For

δ
>
0

\delta > 0

, consider

U
δ

(
t
)
=
{
(

δ

−
1

∫
0
δ

g
x

,

δ

−
1

∫
0
δ

g

T
(
t
)
x

)
:
x
∈
C
}

{U_\delta }(t) = \{ ({\delta ^{ - 1}}\smallint _0^\delta {g_x},{\delta ^{ - 1}}\smallint _0^\delta {g_{T(t)x}}):x \in C\}

where

g
x

g_x

denotes the trajectory of T from x. The class H of semigroups for which

U
δ

(
t
)

{U_\delta }(t)

is a function for

δ
>
0

\delta > 0

and

t
⩾
0

t \geqslant 0

contains all strongly continuous linear semigroups and Webb’s nonlinear nonexpansive example with no dense set of differentiability. If

T
∈
H
,

U
δ

=
{
(
t
,

U
δ

(
t
)
)
:
t
⩾
0
}

T \in H,{U_\delta } = \{ (t,{U_\delta }(t)):t \geqslant 0\}

is a semigroup on

{

δ

−
1

∫
0
δ

g
x

:
x
∈
C
}

\{ {\delta ^{ - 1}}\smallint _0^\delta {g_x}:x \in C\}

with continuously differentiable trajectories. Also, as

{

δ
n

}

n
=
1

∞

\{ {\delta _n}\} _{n = 1}^\infty

converges to 0, the trajectories of

{

U

δ
n

}

n
=
1

∞

\{ {U_{{\delta _n}}}\} _{n = 1}^\infty

uniformly approximate the trajectories of T.
A remark on meromorphic solutions of differential equations
Authors: Strelitz Sh. 
Published: June 2010

The following assertion is proven. Suppose the functions F, P, Q of the differential equation

F
(
z
,
w
,

w
′

,

w

(
n
)

)
=
P
(
z
,
w
)

/

Q
(
z
,
w
)

F(z,w,w’,{w^{(n)}}) = P(z,w)/Q(z,w)

to be polynomials of all their corresponding variables. If the considered equation has a transcendental meromorphic solution in

|

z

|

>
∞

|z| > \infty

, then

Q
(
z
,
w
)

Q(z,w)

does not depend on w. An example of possible applications is stated.
The solution by iteration of nonlinear equations in Hilbert spaces
Authors: Măruşter Şt. 
Published: June 2010

The weak and strong convergence of the iterates generated by

x

k
+
1

=
(
1
−

t
k

)

x
k

+

t
k

T

x
k

(

t
k

∈
R
)

{x_{k + 1}} = (1 - {t_k}){x_k} + {t_k}T{x_k}({t_k} \in R)

to a fixed point of the mapping

T
:
C
→
C

T:C \to C

are investigated, where C is a closed convex subset of a real Hilbert space. The basic assumptions are that T has at least one fixed point in C, and that

I
−
T

I - T

is demiclosed at 0 and satisfies a certain condition of monotony. Some applications are given.
On a necessary conditon for the Erdős-Rényi law of large numbers
Authors: Steinebach Josef 
Published: June 2010

For a sequence

{

X
i

}

i
=
1
,
2
,
…

{\{ {X_i}\} _{i = 1,2, \ldots }}

of independent, identically distributed random variables with existing moment-generating function

φ
(
t
)
=
E

exp
⁡
(
t

X
i

)

\varphi (t) = E\;\exp (t{X_i})

in some nondegenerate interval, Erdös and Rényi (1970) studied the maximum

D
(
N
,
K
)

D(N,K)

of the

N
−
K
+
1

N - K + 1

sample means

K

−
1

(

S

n
+
K

−

S
n

)
,

0
⩽
n
⩽
N
−
K

{K^{ - 1}}({S_{n + K}} - {S_n}),\;0 \leqslant n \leqslant N - K

, where

S
0

=
0
,

S
n

=

X
1

+
⋯
+

X
n

{S_0} = 0,\;{S_n} = {X_1} + \cdots + {X_n}

. They showed that for a certain range of numbers a there exist positive constants

C
(

a

)

C({\mathbf {a}})

such that

lim

N
→
∞

D
(
N
,
[
C
(

a

)
log
⁡
N
]
)
=

a

{\lim _{N \to \infty }}D(N,[C({\mathbf {a}})\log N]) = {\mathbf {a}}

with probability one. In the present paper it is shown that the existence of the moment-generating function is also a necessary condition, i.e. that

lim

sup

N
→
∞

D
(
N
,
[
C
log
⁡
N
]
)
=
∞

\lim {\sup _{N \to \infty }}D(N,[C\log N]) = \infty

for every positive constant C, if the moment-generating function does not exist for any positive number t.
Multiplication rules for polynomials
Authors: Nathanson Melvyn B. 
Published: June 2010

It is proved that the polynomial solutions of the functional equation
\[

F
(
z
)
F
(

z
+
1

/

a

)
=
F
(

a

z
2

+
(
b

/

a

+
1
)
z
+
c

/

a

)

F(z)F( {z + 1/\sqrt a } ) = F( {\sqrt a {z^2} + (b/\sqrt a + 1)z + c/\sqrt a } )

\]
are precisely

(
a

z
2

+
b
z
+
c

)
n

{(a{z^2} + bz + c)^n}

if

b
2

−
4
a
c
≠
0

{b^2} - 4ac \ne 0

and

(

a

z
+
b

/

2

a

)
n

{(\sqrt a z + b/2\sqrt a )^n}

if

b
2

−
4
a
c
=
0

{b^2} - 4ac = 0

.
A note on Volterra equations in a Hilbert space
Authors: Londen Stig-Olof , Staffans Olof J. 
Published: June 2010

An existence and uniqueness result is given on the nonlinear Volterra equation
\[

u
(
t
)
+
(

a
∗

A
u
)
(
t
)
∍
f
(
t
)
,

t
∈

R
+

,

u(t) + (a ^\ast Au)(t) \backepsilon f(t),\quad t \in {R^ + },

\]
when considered in a real Hilbert space. The result extends earlier work by Barbu and Londen.
A factorization theorem for groups and Lie algebras
Authors: Schenkman Eugene 
Published: June 2010

A proof is given for a generalization of a theorem of Mennicke that each member of a certain family of groups defined by generators and relations is finite. This leads to the following theorem on factorization of groups. Theorem. Let G be generated by abelian subgroups A, B, C, such that

[
A
,
B
]
⩽
A
,
[
B
,
C
]
⩽
B
,
[
C
,
A
]
⩽
C

[A,B] \leqslant A,[B,C] \leqslant B,[C,A] \leqslant C

; then the second derived group,

G
G

is nilpotent of class at most 3. Also proved is the analogue of the above theorem for Lie algebras.
A necessary and sufficient condition for Bloch functions
Authors: Timoney Richard M. 
Published: June 2010

A necessary and sufficient condition is given for a subset

E
⊆

C

E \subseteq {\mathbf {C}}

to satisfy
\[

Sup
⁡
{

|

f
′

(
z
)

|

(
1
−

|

z

|

2

)

|

z
∈

f

−
1

(
E
)
}
>
∞

⇒
Sup
⁡
{

|

f
′

(
z
)

|

(
1
−

|

z

|

2

)

|

z
∈
D
}
>
∞

\begin {array}{*{20}{c}} {\operatorname {Sup}\{ |f’(z)|(1 - |z{|^2})|z \in {f^{ - 1}}(E)\} > \infty } \\ { \Rightarrow \operatorname {Sup}\{ |f’(z)|(1 - |z{|^2})|z \in D\} > \infty } \\ \end {array}

\]
when

f
:
D
→

C

f:D \to {\mathbf {C}}

is analytic. The condition is that the complement of E should not contain large discs.
Double commutants of 𝐶_{⋅0} contractions
Authors: Uchiyama Mitsuru 
Published: June 2010

D. Sarason has shown that an operator in the double commutant of a contraction T of class

C
0

(
1
)

{C_0}(1)

is interpolated by a function in

H
∞

{H^\infty }

, that is,

{
T

}
′

=
{
ϕ
(
T
)
;
ϕ
∈

H
∞

}

\{ T\} ’ = \{ \phi (T);\phi \in {H^\infty }\}

, [3]. Generally, in [4] Sz.-Nagy and C. Foiaş have shown that an operator in the double commutant of a contraction T of class

C
0

(
n
)

{C_0}(n)

is interpolated by a function in

N
T

{N_T}

, that is,

{
T
}
=
{

ϕ
1

(
T

)

−
1

ϕ
2

(
T
)
:

ϕ
2

/

ϕ
1

∈

N
T

}

\{ T\} = \{ {\phi _1}{(T)^{ - 1}}{\phi _2}(T):{\phi _2}/{\phi _1} \in {N_T}\}

. In this note we shall show that an operator in the double commutant of a

C

⋅
0

{C_{ \cdot 0}}

contraction T with finite defect indices

δ

T
∗

>

δ
T

{\delta _{{T^ \ast }}} > {\delta _T}

is interpolated by a function in

H
∞

{H^\infty }

.
On the Alexander polynomials of certain three-component links
Authors: Kidwell Mark E. 
Published: June 2010

Let L be a three-component link all of whose linking numbers are zero. Write the Alexander polynomial of L as

Δ
(
x
,
y
,
z
)
=
(
1
−
x
)
(
1
−
y
)
(
1
−
z
)
f
(
x
,
y
,
z
)

\Delta (x,y,z) = (1 - x)(1 - y)(1 - z)f(x,y,z)

. Then the integer

|

f
(
1
,
1
,
1
)

|

|f(1,1,1)|

is a perfect square.
Lie triple derivations of von Neumann algebras
Authors: Miers C. Robert 
Published: June 2010

A Lie triple derivation of an associative algebra M is a linear map

L
:
M
→
M

L:M \to M

such that
\[

L
[
[
X
,
Y
]
,
Z
]
=
[

[
L
(
X
)
,
Y
]
,
Z

]
+
[

[
X
,
L
(
Y
)
]
,
Z

]
+
[

[
X
,
Y
]
,
L
(
Z
)

]

L[[X,Y],Z] = [ {[L(X),Y],Z} ] + [ {[X,L(Y)],Z} ] + [ {[X,Y],L(Z)} ]

\]
for all

X
,
Y
,
Z
∈
M

X,Y,Z \in M

. (Here

[
X
,
Y
]
=
X
Y
−
Y
X

[X,Y] = XY - YX

and [M, M] is the linear subspace of M generated by such terms.) We show that if M is a von Neumann algebra with no central abelian summands then there exists an operator

A
∈
M

A \in M

such that

L
(
X
)
=
[
A
,
X
]
+
λ
(
X
)

L(X) = [A,X] + \lambda (X)

where

λ
:
M
→

Z
M

\lambda :M \to {Z_M}

is a linear map which annihilates brackets of operators in M.
A criterion for Perron integrability
Authors: Sarkhel D. N. 
Published: June 2010

It is shown that a measurable function

f
:
I
=
[
a
,
b
]
→

R
e

f:I = [a,b] \to {R_e}

is necessarily Perron integrable if there exists at least one pair of functions

u
,
l
:
I
→
R

u,l:I \to R

such that (i)

u
(
x
−
)
⩽
u
(
x
)
⩽
u
(
x
+
)

u(x - ) \leqslant u(x) \leqslant u(x + )

and

l
(
x
−
)
⩾
l
(
x
)
⩾
l
(
x
+
)

l(x - ) \geqslant l(x) \geqslant l(x + )

on I, (ii)

I
∖
(

E
1

∪

E
2

)

I\backslash ({E_1} \cup {E_2})

is countable, where

E
1

=
{
x

|

D
−

u
(
x
)
>
−
∞
,

D
−

l
(
x
)
>
∞
}

{E_1} = \{ x|{D_ - }u(x) > - \infty ,{D^ - }l(x) > \infty \}

and

E
2

=
{
x

|

D
+

u
(
x
)
>
−
∞
,

D
+

l
(
x
)
>
∞
}

{E_2} = \{ x|{D_ + }u(x) > - \infty ,{D^ + }l(x) > \infty \}

, and (iii)

max
{

D
−

u
(
x
)
,

D
+

u
(
x
)
}
⩾
f
(
x
)
⩾
min
{

D
−

l
(
x
)
,

D
+

l
(
x
)
}

\max \{ {D_ - }u(x),{D_ + }u(x)\} \geqslant f(x) \geqslant \min \{ {D^ - }l(x),{D^ + }l(x)\}

a.e. on I. In the special case when u and l are respectively major and minor functions of f in the sense of H. Bauer, the result was proved by J. Marcinkiewicz.
𝐿²-boundedness for pseudo-differential operators with unbounded symbols
Authors: Childs Gary 
Published: June 2010

Kato has proven

L
2

{L^2}

-boundedness if the symbol

a
(
x
,
z
)

a(x,z)

is such that

|

D
x
β

D
z
α

a
(
x
,
z
)

|

⩽
(

constant

)

(
1
+

|

z

|

)

(

|

β

|

−

|

α

|

)
ρ

|D_x^\beta D_z^\alpha a(x,z)|\; \leqslant ({\text {constant}}){(1 + |z|)^{(|\beta | - |\alpha |)\rho }}

for

|

α

|

⩽
[
n

/

2
]
+
1
,

|

β

|

⩽
[
n

/

2
]
+
2

|\alpha | \leqslant [n/2] + 1,|\beta | \leqslant [n/2] + 2

and

0
>
ρ
>
1

0 > \rho > 1

. In this paper,

L
2

{L^2}

-boundedness is shown for a corresponding Hölder continuity condition which requires slightly less smoothness for

a
(
x
,
z
)

a(x,z)

.
Minimal 𝐻² interpolation in the Carathéodory class
Authors: Beller E. , Pinchuk B. 
Published: June 2010

For

(

c
1

,
…
,

c
n

)

({c_1}, \ldots ,{c_n})

in

C

n

{{\mathbf {C}}^n}

, let

C
(

c
1

,
…
,

c
n

)

C({c_1}, \ldots ,{c_n})

denote the class of functions

f
(
z
)
=
1
+

c
1

z
+
⋯
+

c
n

z
n

+

Σ

k
=
n
+
1

∞

a
k

z
k

f(z) = 1 + {c_1}z + \cdots + {c_n}{z^n} + \Sigma _{k = n + 1}^\infty {a_k}{z^k}

which are analytic and satisfy

Re
⁡
f
(
z
)
>
0

\operatorname {Re} f(z) > 0

in the unit disc. The unique function of least

H
2

{H^2}

norm in

C
(

c
1

,
…
,

c
n

)

C({c_1}, \ldots ,{c_n})

is explicitly determined.
An extremal problem for functions of positive real part with application to a radius of convexity problem
Authors: Hamilton D. H. , Tuan P. D. 
Published: June 2010

The functional

Re
⁡
{
z

p
′

(
z
)

/

(
p
(
z
)
+
β
+
i
t
)
}
,
β
>
−
1
,

|

z

|

⩽
r
,
0
>
r
>
1

\operatorname {Re} \{ zp’(z)/(p(z) + \beta + it)\} ,\beta > - 1,|z| \leqslant r,0 > r > 1

, is minimized for all real t over the class of functions of positive real part. The result is applied to obtain the radius of convexity for a family of regular functions.
On finite simple groups with a self-centralization system of type (2(𝑛))
Authors: Ferguson Pamela A. 
Published: June 2010

Let G denote a simple group with a self-centralization system of type

(
2
(
n
)
)

(2(n))

, where

n
>
3

n > 3

. Let

X
1

{X_1}

denote an exceptional character of G, then

X
1

(
1
)
=
k
n
+
2
ε

{X_1}(1) = kn + 2\varepsilon

where

ε
=
±
1

\varepsilon = \pm 1

. It is known that
\[

|

G

|

=
n

X

1

(
1
)
(

X

1

(
1
)
−
ε
)
(
l
n
+
1
)

|G| = nX_{1}(1)(X_{1}(1)-\varepsilon )(ln + 1)

\]
where l is a nonnegative integer. In this paper G is classified if

l
=
0
,
ε
=
1

l = 0,\varepsilon = 1

and

X
1

(
1
)

{X_1}(1)

is odd.
On iterated similarities of operators
Authors: Herrero Domingo A. 
Published: June 2010

Let

L

(

H

)

\mathcal {L}(\mathcal {H})

be the algebra of operators on a complex Hilbert space

H

\mathcal {H}

. If

A
∈

L

(

H

)

A \in \mathcal {L}(\mathcal {H})

is invertible and

{
X
∈

L

(

H

)
:

‖

A
k

X

A

−
k

‖

⩽
C
(
X
)
>
∞
,
k
=
0
,
1
,
2
,
…
}

\{ X \in \mathcal {L}(\mathcal {H}):\left \|{A^k}X{A^{ - k}}\right \| \leqslant C(X) > \infty ,k = 0,1,2, \ldots \}

coincides with

L

(

H

)

\mathcal {L}(\mathcal {H})

, then A is a multiple of a similarity of a unitary operator.
On 𝐸𝑥𝑡_{𝐾_{*}(𝐾)}(𝜋_{*}(𝐾), 𝐾_{*}(𝑀𝑈))
Authors: Clarke Francis 
Published: June 2010

In this note a spectral sequence is constructed which is used to prove the existence of elements of nonzero homological degree in

Ext

K
∗

(
K
)

(

π
∗

(
K
)
,

K
∗

(
M
U
)
)

{\operatorname {Ext}_{{K_ \ast }(K)}}({\pi _ \ast }(K), {K_ \ast }(MU))

. Examples of specific elements in terms of representatives in the cobar construction are given.
The real Brauer group of a finite CW-complex
Authors: Strickland Elisabetta 
Published: June 2010

In this paper a complete description in homological terms of the real Brauer group

Br
R

(
X
)

{\text {Br}_R}(X)

of a finite CW-complex X is obtained, in analogy with Jean Pierre Serre’s result on the complex Brauer group

Br
C

(
X
)

{\text {Br}_C}(X)

and the author’s one for the orthogonal case. Precisely it is shown that
\[

Br
R

(
X
)
≅

H
2

(
X
,

Z
2

)
⊕

H
0

(
X
,

Z
2

)
.

{\text {Br}_R}(X) \cong {H^2}(X,{Z_2}) \oplus {H^0}(X,{Z_2}).

\]
A maximal inequality for 𝐻¹-functions on a generalized Walsh-Paley group
Authors: Fujii Nobuhiko 
Published: June 2010

Let

G
=

∏

i
=
0

∞

Z
(

p
i

)

G = \prod _{i = 0}^\infty Z({p_i})

be the countable product of discrete cyclic groups of order

p
i

{p_i}

. We assume that

sup

i
⩾
0

p
i

>
∞

{\sup _{i \geqslant 0}}{p_i} > \infty

. We consider Walsh-Fourier series on G and define

H
1

{H^1}

-functions on G by the Coifman-Weiss atoms. Let

K
n

(
x
)

{K_n}(x)

be the nth (C, 1)-kernel. We prove that

∫
G

sup

n
⩾
1

|

(

K
n

∗
f
)
(
x
)

|

d
μ
⩽
C

‖
f
‖

H
1

{\smallint _G {{{\sup }_{n \geqslant 1}}|({K_n} \ast f)(x)|d\mu \leqslant C\left \| f \right \|} _{{H^1}}}

. Here

d
μ

d\mu

is the normalized Haar measure,

‖
‖

H
1

{\left \| \right \|_{{H^1}}}

is the

H
1

{H^1}

-norm and C is a constant independent of f.
Pseudolattice properties of the star-orthogonal partial ordering for star-regular rings
Authors: Hartwig Robert E. 
Published: June 2010

It is shown that a star-regular ring R forms a pseudo upper semilattice under the star-orthogonal partial ordering. That is, for every a, b in R, the set

{
c

|

c
⩾
a
,
c
⩾
b
}

\{ c|c \geqslant a,c \geqslant b\}

is nonempty if and only if

a
∨
b

a \vee b

exists in R, in which case
\[

a
∨
b
=
a
+
(
1
−
a

a
†

)
b

b
∗

[
(
1
−

a
†

a
)

b
∗

]
†

.

a \vee b = a + (1 - a{a^\dagger })b{b^ \ast }{[(1 - {a^\dagger }a){b^ \ast }]^\dagger }.

\]
A Cohen type inequality for compact Lie groups
Authors: Giulini Saverio , Soardi Paolo M. , Travaglini Giancarlo 
Published: June 2010

The following theorem is proved: let G denote a compact connected semisimple Lie group. There exists

θ
=
θ
(
G
)
(
3
⩽
θ
>
4
)

\theta = \theta (G)(3 \leqslant \theta > 4)

such that, if

χ
1

,
…
,

χ
N

{\chi _1}, \ldots ,{\chi _N}

are N distinct characters of G,

d
1

,
…
,

d
N

{d_1}, \ldots ,{d_N}

their dimensions,

c
1

,
…
,

c
N

{c_1}, \ldots ,{c_N}

complex numbers of modulus greater than or equal to one, then, for all

p
>
θ
,

|

|

|

Σ

j
=
1

N

c
j

d
j

χ
j

|

|

|

p

⩾

const
p

N

α
p

p > \theta ,|||\Sigma _{j = 1}^N{c_j}{d_j}{\chi _j}||{|_p} \geqslant \text {const}_p N^{\alpha _p}

where

|

|

|

⋅

|

|

|

p

||| \cdot ||{|_p}

denotes the

L
p

(
G
)

{L^p}(G)

convolutor norm and

const
p

\text {const}_p

and

α
p

=

α
p

(
G
)

{\alpha _p} = {\alpha _p}(G)

are positive constants. Results on divergence of Fourier series on compact Lie groups are deduced.
Sumsets of measurable sets
Authors: Nathanson Melvyn B. 
Published: June 2010

Let

A

1

,

A

2

,
…
,

A

n

{\mathcal {A}_1},{\mathcal {A}_2}, \ldots ,{\mathcal {A}_n}

be Lebesgue measurable sets of positive real numbers such that

inf

A

i

=
0

{\inf \mathcal {A}_i} = 0

for all i. Let

μ
\mu

denote Lebesgue measure and let

μ
∗

{\mu _ \ast }

denote inner Lebesgue measure. If

∑

i
=
1

n

μ
(

A

i

∩
[
0
,
t
]
)
⩾
γ
t

\sum \nolimits _{i = 1}^n \mu ({\mathcal {A}_i} \cap [0,t]) \geqslant \gamma t

for some

γ
⩽
1

\gamma \leqslant 1

and all

t
⩽
x

t \leqslant x

, then
\[

μ
∗

(
(

A

1

+

A

2

+
⋯
+

A

n

)
∩
[
0
,
x
]
)
⩾
γ
x
.

{\mu _ \ast }(({\mathcal {A}_1} + {\mathcal {A}_2} + \cdots + {\mathcal {A}_n}) \cap [0,x]) \geqslant \gamma x.

\]
This generalizes results of Dyson and Macbeath.