bluuebunny/crossref_metadata_2025_split huggingface dataset:
On the order of the error function fo the 𝑘-free integers
Authors: Suryanarayana D. , Sitaramachandra Rao R. 
Published: June 2010

Let

Δ
k

(
x
)

{\Delta _k}(x)

and

Δ
k

′

(
x
)

{\Delta _k}’ (x)

be the error functions in the asymptotic formulae for the number and the sum of

k
k

-free integers

≦
x

\leqq x

. On the assumption of the Riemann hypothesis, we prove the following results by elementary methods:
\[

Δ
k

′

(
x
)
−
x

Δ
k

(
x
)
=
O
(

x

1
+
3

/

(
4
k
+
1
)
+
ε

)

{\Delta _k}’ (x) - x{\Delta _k}(x) = O({x^{1 + 3/(4k + 1) + \varepsilon }})

\]
and
\[

1
x

∫
1
x

Δ
k

(
t
)
d
t
=
O
(

x

3

/

(
4
k
+
1
)
ε

)
,

\frac {1}{x}\int _1^x {{\Delta _k}(t)dt = O({x^{3/(4k + 1)\varepsilon }}),}

\]
where

ε
>
0

\varepsilon > 0

.
A priori bounds for boundary sets
Authors: Bebernes J. W. , Fraker Ross 
Published: June 2010

Consider

y
=
f
(
t
,
y
,

y
′

)

y = f(t,y,y’)

with boundary conditions

(
0
,
y
(
0
)
,

y
′

(
0
)
)
∈

S
1

,
(
1
,
y
(
1
)
,

y
′

(
1
)
)
∈

S
2

(0,y(0),y’(0)) \in {S_1},(1,y(1),y’(1)) \in {S_2}

. It is shown that the boundary value problem has a solution for certain boundary sets

S
1

{S_1}

and

S
2

{S_2}

which depend on the assumed Nagumo condition for

f
(
t
,
y
,

y
′

)

f(t,y,y’)

.
On functions of bounded boundary rotation
Authors: Liu Ming-chit 
Published: June 2010

Let

U
=
{
z
=
r

e

i
θ

|

r
>
1
}

U = \{ z = r{e^{i\theta }}\left | {r > 1\} } \right .

. For

k
≧
2

k \geqq 2

let

V
k

{V_k}

be the class of normalized analytic functions

f
(
z
)

f(z)

such that the boundary rotation of

f
(
U
)

f(U)

is at most

k
π

k\pi

. Let

A
(
r
)

A(r)

be the integral
\[

∫
0

2
π

∫
0
r

|

f
′

(
ρ

e

i
θ

)

|

2

ρ
d

ρ
d

θ
,

\int _0^{2\pi } {\int _0^r {\left | {f’(\rho {e^{i\theta }})} \right |} } {^2}\rho d\,\rho d\,\theta ,

\]

L
(
r
)

L(r)

the length of the image of the circle

|
z
|

=
r

\left | z \right | = r

under the mapping

f
(
z
)

f(z)

. In this paper the author proves that for

z
∈
U

z \in U

if

f
(
z
)
∈

V
k

f(z) \in {V_k}

then
\[

lim sup

r
→
1

(

sup

f
∈

V
k

L
(
r
)
)

(
π
A
(
r
)
log
⁡

(

1
+
r

1
−
r

)

)

−
1

/

2

≦
k
.

\limsup _{r \to 1} \left ( \sup _{f \in {V_k}} L(r) \right ) \left ( \pi A(r)\log \left ( \frac {1 + r}{1 - r} \right ) \right )^{-1/2} \leqq k.

\]
This generalizes to arbitrary

k
≧
2

k \geqq 2

the recent result of Nunokawa for the case

k
=
2

k = 2

.
Unitary groups and commutators
Authors: Kauffman Robert M. 
Published: June 2010

If H is a possibly unbounded selfadjoint operator and A is a closed operator in a Hilbert space, the relation

(

U
t

−
1

A

U
t

f

)
′

=
i

U
t

−
1

(
A
H
−
H
A
)

U
t

f

(U_t^{ - 1}A{U_t}f)’ = iU_t^{ - 1}(AH - HA){U_t}f

can be shown to hold under relatively reasonable hypotheses on A and f, where

U
t

=

e

i
H
t

{U_t} = {e^{iHt}}

. This relation can then be used to relate properties of the commutator

A
H
−
H
A

AH - HA

to properties of A and H.
On an identity of Eckford Cohen
Authors: Subbarao M. V. , Suryanarayana D. 
Published: June 2010

We characterize all multiplicative arithmetical functions

f
k

(
r
)

{f_k}(r)

such that an identity of the form
\[

∑

r
=
1

∞

f
k

(
r
)

c
k

(
n
,
r
)
=

q
k

(
n
)
g
(
k
)
,

g
(
k
)
≠
0
,

\sum \limits _{r = 1}^\infty {{f_k}(r){c_k}(n,r) = {q_k}(n)g(k),\quad g(k) \ne 0,}

\]
holds for all n, where

q
k

(
n
)

{q_k}(n)

is the characteristic function of the set of k-free integers and

c
k

(
n
,
r
)

{c_k}(n,r)

is the generalized Ramanujan sum. This characterization yields several arithmetical identities of the above form including an identity of Eckford Cohen, which occurs as a special case of our theorem on taking

f
k

(
r
)
=
μ
(
r
)

/

J
k

(
r
)

{f_k}(r) = \mu (r)/{J_k}(r)

and

g
(
k
)
=
ζ
(
k
)

g(k) = \zeta (k)

.
Zeros of 𝜁^{’}(𝑠) in the critical strip
Authors: Spira Robert 
Published: June 2010

It is shown that the abscissa of convergence for the Dirichlet series

(
−
1

)
k

(
1
−

2

1
−
s

)

k
+
1

ζ

(
k
)

(
s
)

{( - 1)^k}{(1 - {2^{1 - s}})^{k + 1}}{\zeta ^{(k)}}(s)

is zero, where

ζ
(
s
)

\zeta (s)

is the Riemann zeta function. This implies the existence of infinitely many zeros of

ζ
′

(
s
)

\zeta ’(s)

in the critical strip.
On an integral formula for closed hypersurfaces of the sphere
Authors: Houh Chorng-shi 
Published: June 2010

In a compact oriented hypersurface

M
n

{M^n}

of the sphere

S

n
+
1

{S^{n + 1}}

the integral formula

∫

M
n

∇

K
r

d
V
=
n

∫

M
n

(

K
r

K
1

−

K

r
+
1

)
e
d
V

{\smallint _{{M^n}}}\nabla {K_r}dV = n{\smallint _{{M^n}}}({K_r}{K_1} - {K_{r + 1}})edV

is proved where

K
r

{K_r}

is the rth mean curvature, e is the unit normal of

M
n

{M^n}

in

S

n
+
1

{S^{n + 1}}

. Some applications are considered.
Geöcze area and a convergence property
Authors: Gariepy Ronald 
Published: June 2010

Suppose f is a continuous mapping with finite Lebesgue area from a polyhedral region

X
⊂

R
k

X \subset {R^k}

into

R
n

,
2
≦
k
≦
n

{R^n},2 \leqq k \leqq n

. Let

f
=
l
∘
m

f = l \circ m

be the monotone-light factorization of f with middle space M. If f satisfies a “cylindrical condition” considered by T. Nishiura, then a current valued measure T over M can be associated with f by means of the Cesari-Weierstrass integral, and if

{

f
i

}

\{ {f_i}\}

is any sequence of quasi-linear maps

f
i

:
X
→

R
n

{f_i}:X \to {R^n}

converging uniformly to f with bounded areas, then
\[

T
(
g
)
(
ϕ
)
=

lim

i
→
∞

∫
X

(
g
∘
m
)

f
i
#

ϕ

T(g)(\phi ) = \lim \limits _{i \to \infty } \int _X {(g \circ m)f_i^\# \phi }

\]
whenever

ϕ
\phi

is an infinitely differentiable k-form in

R
n

{R^n}

and g is a continuous real valued function on M which vanishes on

m
(
Bdry

X
)

m (\text {Bdry} \; X)

. The total variation measure of T, taken with respect to mass, coincides with the Geöcze area measure over M.
On the location of zeros of second-order differential equations
Authors: Komkov Vadim 
Published: June 2010

The paper considers the location of zeros of the equation

(
α
(
t
)

x
′

)
′

+
γ
(
t
)
x
=
0
,
t
∈
[

t
0

,

t
1

]

(\alpha (t)x’)’ + \gamma (t)x = 0,t \in [{t_0},{t_1}]

. The following theorem is proved. Let

[
a
,
a
+
T
]
,
T
=
n
a

[a,a + T],T = na

(n a positive integer), be a subset of

[

t
0

,

t
1

]

[{t_0},{t_1}]

. Denote

ω
=
π

/

T

\omega = \pi /T

. Let the coefficient functions obey the inequality

∫
a

a
+
T

{
γ
(
t
)
−

ω
2

α
(
t
)

sin
2

(
ω
t
)
}
d
t
>

ω
2

∫
a

a
+
T

{
α
cos
⁡
2
ω
t
}
d
t

\smallint _a^{a + T}\{ \gamma (t) - {\omega ^2}\alpha (t){\sin ^2}(\omega t)\} dt > {\omega ^2}\smallint _a^{a + T}\{ \alpha \cos 2\omega t\} dt

. Then every solution of this equation will have a zero on

[
a
,
a
+
T
]

[a,a + T]

. A more general form of this theorem is also proved.
On the arithmetic mean of Fourier-Stieltjes coefficients
Authors: Georgakis Constantine 
Published: June 2010

Let

{

a
n

}

n
=
0

∞

\{ {a_n}\} _{n = 0}^\infty

be the cosine Fourier-Stieltjes coefficients of the Borel measure

μ
\mu

and

{

a
0

,
(

a
1

+
⋯
+

a
n

)

/

n

}

n
=
1

∞

=
{

(
T
a

)
n

}

n
=
0

∞

\{ {a_0},({a_1} + \cdots + {a_n})/n\} _{n = 1}^\infty = \{ {(Ta)_n}\} _{n = 0}^\infty

be the sequence of their arithmetic means. Then

∑

n
=
0

∞

(
T
a
)

n

cos
⁡
n
x

\sum \nolimits _{n = 0}^\infty {{{(Ta)}_n}\cos nx}

is a Fourier-Stieltjes series. Moreover, (a)

∑

n
=
0

∞

(
T
a
)

n

cos
⁡
n
x

\sum \nolimits _{n = 0}^\infty {{{(Ta)}_n}\cos nx}

is a Fourier series if and only if

(
T
a

)
n

→
0

{(Ta)_n} \to 0

at infinity or, equivalently, the measure

μ
\mu

is continuous at the origin, (b)

∑

n
=
1

∞

(
T
a
)

n

sin
⁡
n
x

\sum \nolimits _{n = 1}^\infty {{{(Ta)}_n}\sin nx}

is a Fourier series if and only if the function

x

−
1

μ
(
[
0
,
x
)
)

{x^{ - 1}}\mu ([0,x))

is in

L
1

[
0
,
π
]

{L^1}[0,\pi ]

. These results form the best possible analogue of a theorem of G. Goes, concerning arithmetic means of Fourier-Stieltjes sine coefficients, and improve considerably the theorems of L. Fejér and N. Wiener on the inversion and quadratic variation of Fourier-Stieltjes coefficients.
On convolutions with the Möbius function
Authors: Segal S. L. 
Published: June 2010

By using the results of [6], it is proved that for an extensive class of increasing functions h,

(

∗

)

∑

1
≦
d
≦
x

μ
(
d
)

d

h

(

x
d

)

∼
x

h
′

(
x
)

as

x
→
∞

\begin{equation}\tag {$*$}\sum \limits _{1 \leqq d \leqq x} {\frac {{\mu (d)}}{d}h\left ( {\frac {x}{d}} \right )} \sim xh’(x)\quad {\text {as}}\;x \to \infty \end{equation}

where

μ
\mu

denotes the Möbius function. This result incidentally settles affirmatively Remark (iii) of [6], and refines the Tauberian Theorem 2 of that paper. It is also shown that one type of condition imposed in [6] is necessary to the truth of the cited Theorem 2, at least if some sort of quasi-Riemann hypothesis is true. Nevertheless, examples are given to show that on the one hand

(
∗

)

( ^\ast )

may be true for functions not covered by the first theorem of this paper, and on the other that some sort of nonnaïve condition on a function h is necessary to ensure the truth of

(
∗

)

(^\ast )

.
Representations of Euler classes
Authors: Osborn Howard 
Published: June 2010

For any endomorphism

K
K

of an oriented module

F
F

with inner product there is an element

pf

K

{\text {pf }}K

in the ground ring

R
R

, a constant multiple of the classical pfaffian in the case

F
=

R

2
n

F = {R^{2n}}

. If

R
R

is the algebra of even-dimensional differential forms on a smooth manifold, and if

F
F

is the tensor product of

R
R

and the module of sections of an oriented

2
n

2n

-plane bundle, then any connection in the bundle induces a curvature transformation

K
:
F
→
F

K:F \to F

for which

(
4
π

)

−
n

pf

K

{(4\pi )^{ - n}}{\text {pf }}K

represents the Euler class. Properties of Euler classes are immediate consequences of corresponding properties of

pf

{\text {pf}}

.
On the first boundary value problem for [ℎ(𝑥,𝑥^{’},𝑡)]^{’}= 𝑓(𝑥,𝑥^{’},𝑡)
Authors: Ford Wayne T. 
Published: June 2010

The boundary value problem for

[
h
(
x
,

x
′

,
t
)

]
′

=
f
(
x
,

x
′

,
t
)

[h(x,x’,t)]’ = f(x,x’,t)

is studied with

x
(
0
)
=
x
(
1
)
=
0

x(0) = x(1) = 0

. It is assumed that substitution of functions u and v in

L
2

(
0
,
1
)

{L_2}(0,1)

into h and f produces the functions

h
[
u
(
⋅
)
,
v
(
⋅
)
,
⋅
]

h[u( \cdot ),v( \cdot ), \cdot ]

and

f
[
u
(
⋅
)
,
v
(
⋅
)
,
⋅
]

f[u(\cdot ),v(\cdot ),\cdot ]

in

L
2

(
0
,
1
)

{L_2}(0,1)

such that this map from

L
2

(
0
,
1
)
×

L
2

(
0
,
1
)

{L_2}(0,1) \times {L_2}(0,1)

into

L
2

(
0
,
1
)
×

L
2

(
0
,
1
)

{L_2}(0,1) \times {L_2}(0,1)

is hemicontinuous. Existence and uniqueness are shown in

H
0
1

(
0
,
1
)

H_0^1(0,1)

under the assumption that constants

λ
\lambda

and

η
\eta

exist such that
\[

[
(
V
−
v
)
[
h
(
U
,
V
,
t
)
−
h
(
u
,
v
,
t
)
]
+
(
U
−
u
)
[
f
(
U
,
V
,
t
)
−
f
(
u
,
v
,
t
)
]
]
≧
λ

(
V
−
v

)
2

−
η

(
U
−
u

)
2

[(V - v)[h(U,V,t) - h(u,v,t)] + (U - u)[f(U,V,t) - f(u,v,t)]] \geqq \lambda {(V - v)^2} - \eta {(U - u)^2}

\]
whenever t lies between zero and one while u, v, U and V are arbitrary. Also, it is assumed that

λ
\lambda

and

λ

π
2

−
η

\lambda {\pi ^2} - \eta

are positive.
On the order of some error functions related to 𝑘-free integers
Authors: Joshi V. S. 
Published: June 2010

Let

Δ
k

(
x
)

{\Delta _k}(x)

and

Δ

k
′

(
x
)

\Delta {’_k}(x)

be the error functions in the asymptotic formulae for the number and the sum of k-free integers not exceeding x. We prove that on the assumption of Riemann hypothesis, we have
\[

Δ

k
′

(
x
)
−
x

Δ
k

(
x
)
=
O
(

x

1
+
1

/

2
k
+
ε

)

\Delta {’_k}(x) - x{\Delta _k}(x) = O({x^{1 + 1/2k + \varepsilon }})

\]
and
\[

1
x

∫
1
x

Δ
k

(
t
)
d
t
=
O
(

x

1

/

2
k
+
ε

)
,

\frac {1}{x}\int _1^x {{\Delta _k}(t)dt = O({x^{1/2k + \varepsilon }}),}

\]
for arbitrary

ε
>
0

\varepsilon > 0

.
A property of a class of nonlinear difference equations
Authors: Howard F. T. 
Published: June 2010

Let

g
(
n
)

g(n)

be a rational function of

n
n

whose denominator is divisible by the same power of 2 for each

n
n

and let

a
1

,

a
2

,
⋯

{a_1},{a_2}, \cdots

be any sequence of rational numbers such that for

a
n

=
g
(
n
)
(

a
1

a

n
−
1

+

a
2

a

n
−
2

+
⋯
+

a

n
−
1

a
1

)

{a_n} = g(n)({a_1}{a_{n - 1}} + {a_2}{a_{n - 2}} + \cdots + {a_{n - 1}}{a_1})

. In this paper we determine the exact power of 2 dividing the denominator of

a
n

{a_n}

for each

n
n

and prove congruences

(
mod

4
)

\pmod 4

and

(
mod

8
)

\pmod 8

.
Prime generalized alternative rings 𝐼 with nontrivial idempotent
Authors: Smith Harry F. 
Published: June 2010

A generalized alternative ring

I
I

is a nonassociative ring

R
R

in which the identities

(
w
x
,
y
,
z
)
+
(
w
,
x
,
[
y
,
z
]
)
−
w
(
x
,
y
,
z
)
−
(
w
,
y
,
z
)
x
;
(
[
w
,
x
]
,
y
,
z
)
+
(
w
,
x
,
y
z
)
−
y
(
w
,
x
,
z
)
−
(
w
,
x
,
y
)
z

(wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x;([w,x],y,z) + (w,x,yz) - y(w,x,z) - (w,x,y)z

; and

(
x
,
x
,
x
)

(x,x,x)

are identically zero. It is demonstrated here that if

R
R

is a ring of this type with characteristic different from two and three, then

R
R

semiprime with idempotent

e
e

implies that

R
R

has a Peirce decomposition relative to

e
e

. Furthermore, if

R
R

is prime and

e
≠
0
,
1

e \ne 0,1

; then

R
R

must be alternative.
Equations which characterize inner product spaces
Authors: Senechalle David A. 
Published: June 2010

It is shown that if

N
N

is a normed linear space and there is a point

y
y

of norm 1 such that an inequality of the type

a
2

|

|

x

|

|

2

≦

lim

u
→
0

G
(
{

|

|

b
i

u
x
+

c
i

y

|

|

}

i
=
1

n

)
≦

b
2

|

|

x

|

|

2

{a^2}||x|{|^2} \leqq {\lim _{u \to 0}}G(\{ ||{b_i}ux + {c_i}y||\} _{i = 1}^n) \leqq {b^2}||x|{|^2}

holds for all

x
x

in

N
N

(where

0
>
a
≦
b

0 > a \leqq b

, the

c
i

{c_i}

’s are nonzero and

G
G

and

|

|

⋅

|

|

|| \cdot ||

satisfy a certain twice-differentiability condition), then

N
N

is isomorphic to an inner product space and

inf

|

|

T

|

|

⋅

|

|

T

−
1

|

|

≦
b

/

a

\inf ||T|| \cdot ||{T^{ - 1}}|| \leqq b/a

, where the infimum is taken over all linear homeomorphisms

T
T

between

N
N

and an inner product space. In the event that

a
=
b
=
1

a = b = 1

, the inequality reduces to an equation which characterizes inner product spaces. An example shows that these results do not follow without the twice-differentiability condition on

G
G

.
On the derivative of a polynomial
Authors: Govil N. K. 
Published: June 2010

If

p
(
z
)
=

Σ

v
=
0

n

a
v

z
v

p(z) = \Sigma _{v = 0}^n {{a_v}{z^v}}

is a polynomial of degree

n
n

having all its zeros in

|

z

|

≦
K
≦
1

|z| \leqq K \leqq 1

, then it is known that

max

|

z

|

=
1

|

p
′

(
z
)

|

≧
(
n

/

(
1
+
K
)
)

max

|

z

|

=
1

|

p
(
z
)

|

{\max _{|z| = 1}}|p’(z)| \geqq (n/(1 + K)){\max _{|z| = 1}}|p(z)|

. In this paper we consider the case when

K
>
1

K > 1

and obtain a sharp result.
Holomorphic solutions of functional differential systems near singular points
Authors: Grimm L. J. , Hall L. M. 
Published: June 2010

Functional analysis techniques are used to prove a theorem, analogous to the Harris-Sibuya-Weinberg theorem for ordinary differential equations, which yields as corollaries a number of existence theorems for holomorphic solutions of linear functional differential systems of the form

z
D

y
′

(
z
)
=
A
(
z
)
y
(
z
)
+
B
(
z
)
y
(
α
z
)
+
C
(
z
)

y
′

(
α
z
)

{z^D}y’(z) = A(z)y(z) + B(z)y(\alpha z) + C(z)y’(\alpha z)

in the neighborhood of the singularity at

z
=
0

z = 0

.
A simple alternative problem for finding periodic solutions of second order ordinary differential systems
Authors: Bebernes J. W. 
Published: June 2010

Existence of solutions for

x
=
f
(
t
,
x
,

x
′

)
,
x
(
0
)
=
x
(
1
)
,

x
′

(
0
)
=

x
′

(
1
)

x = f(t,x,x’),x(0) = x(1),x’(0) = x’(1)

are proven by considering a simple alternative problem to which Leray-Schauder degree arguments can be directly applied.
Starlike functions
Authors: McCarty Carl P. 
Published: June 2010

Let

S

∗

[
α
]

{\mathcal {S}^\ast }[\alpha ]

denote the class of functions

f
(
z
)
=
z
+

∑

n
=
2

∞

a
n

z
n

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

analytic in

|

z

|

>
1

|z| > 1

and for which

|

z

f
′

(
z
)

/

f
(
z
)
−
1

|

>
1
−
α

|zf’(z)/f(z) - 1| > 1 - \alpha

for

|

z

|

>
1

|z| > 1

and

α
∈
[
0
,
1
)

\alpha \in [0,1)

. Sharp results concerning coefficients, distortion, and the radius of convexity are obtained. Furthermore, it is shown that

∑

n
=
2

∞

[
(
n
−
α
)

/

(
1
−
α
)
]

|

a
n

|

>
1

\sum \nolimits _{n = 2}^\infty {[(n - \alpha )/(1 - \alpha )]|{a_n}| > 1}

is a sufficient condition for

f
(
z
)
∈

S

∗

[
α
]

f(z) \in {\mathcal {S}^\ast }[\alpha ]

.
On the dominated ergodic theorem in 𝐿₂ space
Authors: Akcoglu M. A. , Sucheston L. 
Published: June 2010

Let

T
T

be a contraction on

L
2

{L_2}

of a

σ
\sigma

-finite measure space,

A
n

(
T
)

{A_n}(T)

the operator

(
1

/

n
)
(

T
0

+
⋯
+

T
n

)
,
S
(
T
)
f

(1/n)({T^0} + \cdots + {T^n}),S(T)f

the function

sup
n

|

A
n

(
T
)
f

|

{\sup _n}|{A_n}(T)f|

. Theorem 1. Assume that, whatever be the measure space,

S
(
U
)
f
∈

L
2

S(U)f \in {L_2}

for each unitary operator

U
U

on

L
2

{L_2}

and each function

f
∈

L
2

f \in {L_2}

. Then there exists a universal constant

K
K

such that

|

|

S
(
T
)
f

|

|

≦
K

|

|

f

|

|

||S(T)f|| \leqq K||f||

for each contraction

T
T

on

L
2

{L_2}

and each

f
∈

L
2

f \in {L_2}

. Theorem 2. Let

T
T

be a contraction on

L
2

{L_2}

and let

U
U

be a unitary dilation of

T
T

acting on a Hilbert space

H
H

containing

L
2

{L_2}

. If all expressions of the form

Σ

n
=
1

∞

P
n

A
n

(
U
)

\Sigma _{n = 1}^\infty {{P_n}{A_n}(U)}

, where

P
n

{P_n}

are mutually orthogonal projections, are bounded operators on

H
H

, then for each

f
∈

L
2

,
S
(
T
)
f
∈

L
2

f \in {L_2}, S(T)f \in {L_2}

and

A
n

(
T
)
f

{A_n}(T)f

converges a.e.
Mutual existence of product integrals
Authors: Helton Jon C. 
Published: June 2010

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

R
×
R

R \times R

to R, where R represents the real numbers. Let

O

M
∘

O{M^ \circ }

be the class of functions G such that

x

∏

y

(
1
+
G
)

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

exists for

a
≦
x
>
y
≦
b

a \leqq x > y \leqq b

and

∫
a
b

|

1
+
G
−
∏

(
1
+
G
)

|

=
0

\smallint _a^b|1 + G - \prod {(1 + G)| = 0}

. Let

O

P
∘

O{P^ \circ }

be the class of functions G such that

|

∏

q
=
i

j

(
1
+

G
q

)

|

|\prod \nolimits _{q = i}^j {(1 + {G_q})|}

is bounded for refinements

{

x
q

}

q
=
0

n

\{ {x_q}\} _{q = 0}^n

of a suitable subdivision of [a, b]. If F and G are functions from

R
×
R

R \times R

to R such that

F
∈
O

P
∘

F \in O{P^ \circ }

on [a, b],

lim

x
,
y
→

p
+

F
(
x
,
y
)

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

and

lim

x
,
y
→

p
−

F
(
x
,
y
)

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

exist and are zero for

p
∈
[
a
,
b
]

p \in [a,b]

, each of

lim

x
→

p
+

F
(
p
,
x
)
,

lim

x
→

p
−

F
(
x
,
p
)
,

lim

x
→

p
+

G
(
p
,
x
)

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

and

lim

x
→

p
−

G
(
x
,
p
)

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

exist for

p
∈
[
a
,
b
]

p \in [a,b]

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

F
+
G
∈
O

M
∘

F + G \in O{M^ \circ }

on [a, b], (2)

F
∈
O

M
∘

F \in O{M^ \circ }

on [a, b], and (3)

G
∈
O

M
∘

G \in O{M^\circ }

on [a, b].
An ergodic super-property of Banach spaces defined by a class of matrices
Authors: Brunel A. , Fong H. , Sucheston L. 
Published: June 2010

A matrix

(

a

n
i

)

({a_{ni}})

is called an

R
R

-matrix if (A)

Σ
i

a

n
i

↛
0

{\Sigma _i}{a_{ni}} \nrightarrow 0

, and (B)

lim
n

a

n
i

=
0

{\lim _n}{a_{ni}} = 0

for each

i
i

. A Banach space

X
X

is called

R
R

-ergodic if for each isometry

T
T

and each

x
∈
X

x \in X

, there is an

R
R

-matrix

(

a

n
i

)

({a_{ni}})

such that

Σ
i

a

n
i

T
i

x

→

W

{\Sigma _i}{a_{ni}}{T^i}x\xrightarrow {{\text {W}}}

(converges weakly). Given two Banach spaces

F
F

and

X
X

, write

F

fr

X

F{\text { fr }}X

if for each finite-dimensional subspace

F
′

F’

of

F
F

and

ϵ
>
0

\epsilon > 0

, there is an isomorphism

V
V

from

F
′

F’

onto a subspace of

X
X

such that

|

|

|

x

|

|

−

|

|

V
x

|

|

|

>
ϵ

\left | {||x|| - ||Vx||} \right | > \epsilon

for each

x
∈

F
′

x \in F’

with

|

|

x

|

|

≤
1

||x|| \leq 1

.

X
X

is called super-

R
R

-ergodic if

F
F

is

R
R

-ergodic for each

F

fr

X

F{\text { fr }}X

. Theorem.

X
X

is super-

R
R

-ergodic if and only if

X
X

is super-reflexive. The proof is based on the following: Theorem. Let

T
T

be a linear operator on

X
,
(

a

n
i

)

X,({a_{ni}})

a matrix satisfying (A),

x
∈
X

x \in X

such that

Σ
i

a

n
i

T
i

x

→

W

x
¯

{\Sigma _i}{a_{ni}}{T^i}x\xrightarrow {{\text {W}}}\bar x

. Then there is a constant

α
\alpha

such that

(
x
−
a

x
¯

)
∈

(
I
−
T
)
X

¯

(x - a\bar x) \in \overline {(I - T)X}

.
A correction to the paper: “Semi-open sets and semi-continuity in topological spaces” (Amer. Math. Monthly 70 (1963), 36–41) by Norman Levine
Authors: Hamlett T. R. 
Published: June 2010

A subset

A
A

of a topological space is said to be semi-open if there exists an open set

U
U

such that

U
⊆
A
⊆
Cl
⁡
(
U
)

U \subseteq A \subseteq \operatorname {Cl} (U)

where

Cl
⁡
(
U
)

\operatorname {Cl} (U)

denotes the closure of

U
U

. The class of semi-open sets of a given topological space

(
X
,

T

)

(X,\mathcal {T})

is denoted

S

.O

.

(
X
,

T

)

{\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T})

. In the paper Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41, Norman Levine states in Theorem 10 that if

T

\mathcal {T}

and

T

∗

{\mathcal {T}^ \ast }

are two topologies for a set

X
X

such that

S

.O

.

(
X
,

T

)
⊆

S

.O

.

(
X
,

T

∗

)

{\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) \subseteq {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast })

, then

T

⊆

T

∗

\mathcal {T} \subseteq {\mathcal {T}^ \ast }

. In a corollary to this theorem, Levine states that if

S

.O

.

(
X
,

T

)
=

S

.O

.

(
X
,

T

∗

)

{\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) = {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast })

, then

T

=

T

∗

\mathcal {T} = {\mathcal {T}^ \ast }

. An example is given which shows the above-mentioned theorem and its corollary are false. This paper shows how different topologies on a set which determine the same class of semi-open subsets can arise from functions, and points out some of the implications of two topologies being related in this manner.
Weighted norm inequalities for fractional integrals
Authors: Welland G. V. 
Published: June 2010

A simpler proof of an inequality of Muckenhoupt and Wheeden is given. Let

T
α

f
(
x
)
=
∫
f
(
y
)

|

x
−
y

|

α
−
d

d
y

{T_\alpha }f(x) = \smallint f(y)|x - y{|^{\alpha - d}}dy

be given for functions defined in

R

d

{{\mathbf {R}}^d}

. Let

υ
\upsilon

be a weight function which satisfies
\[

(

|

Q

|

−
1

∫
Q

[
υ
(
x
)
]

q

d
x

)

1

/

q

(

|

Q

|

−
1

∫
Q

[
υ
(
x
)
]

−

p
′

d
x

)

1

/

p
′

≤
K

(|Q{|^{ - 1}}\int _Q {{{[\upsilon (x)]}^q}dx{)^{1/q}}(|Q{|^{ - 1}}\int _Q {{{[\upsilon (x)]}^{ - p’}}dx{)^{1/p’}} \leq K} }

\]
for each cube,

Q
Q

, with sides parallel to a standard system of axes and

|

Q

|

|Q|

is the measure of such a cube. Suppose

1

/

q
=
1

/

p
−
α

/

d

1/q = 1/p - \alpha /d

and

0
>
α
>
d
,
1
>
p
>
d

/

α

0 > \alpha > d,1 > p > d/\alpha

. Then there exists a constant such that

|

|

(

T
α

f
)
υ

|

|

q

≤
C

|

|

f
υ

|

|

p

||({T_\alpha }f)\upsilon |{|_q} \leq C||f\upsilon |{|_p}

. Certain results for

p
=
1

p = 1

and

q
=
∞

q = \infty

are also given.