bluuebunny/crossref_metadata_2025 huggingface dataset:
Banach spaces that have normal structure and are isomorphic to a Hilbert space
Authors: Bernal Javier , Sullivan Francis
Published: June 2010

We prove that given a Hilbert space

(

E
,

|

|

⋅

|

|

)

\left ( {E,|| \cdot ||} \right )

, and

|

⋅

|

| \cdot |

a norm on

E
E

such that for all

x
∈
E

x \in E

,

1

/

β

|
x
|

⩽

‖
x
‖

⩽

|
x
|

1/\beta \left | x \right | \leqslant \left \| x \right \| \leqslant \left | x \right |

for some

β
\beta

, if

1
⩽
β
>

2

1 \leqslant \beta > \sqrt 2

, then

(

E
,

|

⋅

|

)

\left ( {E,| \cdot |} \right )

satisfies a convexity property from which normal structure follows.
A two weight inequality for the fractional integral when 𝑝=𝑛/𝛼
Authors: Harboure Eleonor , Macías Roberto A. , Segovia Carlos
Published: June 2010

Let

I
α

{I_\alpha }

be the fractional integral operator defined as
\[

I
α

f
(
x
)
=
∫

f
(
y
)

|

x
−
y

|

α
−
n

d
y
.

{I_\alpha }f(x) = \int {f(y){{\left | {x - y} \right |}^{\alpha - n}}dy.}

\]
Given a weight

w
w

(resp.

υ
\upsilon

), necessary and sufficient conditions are given for the existence of a nontrivial weight

υ
\upsilon

(resp.

w
w

) such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upsilon chi Subscript upper B Baseline double-vertical-bar Subscript normal infinity Baseline StartFraction 1 Over StartAbsoluteValue upper B EndAbsoluteValue EndFraction integral Underscript upper B Endscripts StartAbsoluteValue upper I Subscript alpha Baseline f left-parenthesis x right-parenthesis minus m Subscript upper B Baseline left-parenthesis upper I Subscript alpha Baseline f right-parenthesis EndAbsoluteValue d x less-than-or-slanted-equals upper C left-parenthesis integral StartAbsoluteValue f EndAbsoluteValue Superscript n slash alpha Baseline w right-parenthesis Superscript alpha slash n">

‖

υ

χ
B

‖

∞

1

|
B
|

∫
B

|

I
α

f
(
x
)
−

m
B

(

I
α

f
)

|

d
x
⩽
C

(

∫

|
f
|

n

/

α

w

)

α

/

n

{\left \| {\upsilon {\chi _B}} \right \|_\infty }\frac {1}{{\left | B \right |}}\int _B {\left | {{I_\alpha }f(x) - {m_B}({I_\alpha }f)} \right |} dx \leqslant C{\left ( {\int {{{\left | f \right |}^{n/\alpha }}w} } \right )^{\alpha /n}}

\]
holds for any ball

B
B

such that

‖

v

χ
B

‖

∞

>
0

{\left \| {v{\chi _B}} \right \|_\infty } > 0

.
Uniform order-convergence for complete lattices
Authors: Greco Gabriele H.
Published: June 2010

We introduce a purely lattice-theoretical definition of uniform order-convergence of a net of functions with values in a complete lattice. We will show that for completely distributive lattices the uniform order-convergence is induced by a uniformity.
Higher Whitehead groups of certain bundles over Seifert manifolds
Authors: Nicas A. J. , Stark C. W.
Published: June 2010

Vanishing results for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W h Subscript j Baseline left-parenthesis pi 1 upper M right-parenthesis circled-times upper R left-parenthesis upper R equals bold upper Z comma bold upper Q comma or bold upper Z left-bracket 1 slash 2 right-bracket right-parenthesis">

W

h

j

(

π
1

M
)
⊗
R
(
R
=

Z

,

Q

,

 or 

Z

[
1

/

2
]
)

{\text {W}}{{\text {h}}_j}({\pi _1}M) \otimes R(R = {\mathbf {Z}},{\mathbf {Q}},{\text { or }}{\mathbf {Z}}[1/2])

are obtained when

M
M

is a closed aspherical manifold which is the total space of a bundle over an insufficiently large Seifert manifold with infinite fundamental group of hyperbolic type. Allowable fibers include Riemannian flat manifolds and closed aspherical manifolds with poly-

Z

{\mathbf {Z}}

fundamental groups. Corollaries. concern the homotopy groups of the group

TOP

(
M
)

{\text {TOP}}(M)

of self-homeomorphisms of

M
M

.
Cubic subfields of exceptional simple Jordan algebras
Authors: Petersson H. P. , Racine M. L.
Published: June 2010

Let

E

/

k

E/k

be a cubic field extension and

J
J

a simple exceptional Jordan algebra of degree 3 over

k
k

. Then

E
E

is a reducing field of

J
J

if and only if

E
E

is isomorphic to a (maximal) subfield of some isotope of

J
J

. If

k
k

has characteristic not 2 or 3 and contains the third roots of unity then every simple exceptional Jordan division algebra of degree 3 over

k
k

contains a cyclic cubic subfield.
Restricted Lie algebras with semilinear 𝑝-mappings
Authors: Farnsteiner Rolf
Published: June 2010

Restricted Lie algebras

(
L
,
[
p
]
)

(L,[p])

having the property that some power of the

p
p

-mapping

[
p
]

[p]

is semilinear are investigated.
Path derivatives and growth control
Authors: Bruckner A. M. , Johnson K. G.
Published: June 2010

We show that a standard theorem relating the growth of a function on a measurable set to constraints on its Dini derivates extends to a class of generalized derivatives. More precisely, we show that if

|

F
¯

E
′

|

⩽
M

\left | \bar {F}’_E \right | \leqslant M

and

|

F
_

E
′

|

⩽
M

\left | \underline {F}’_E \right | \leqslant M

on a measurable set

A
A

, then

λ
(
F
(
A
)
)
⩽
M
λ
(
A
)

\lambda (F(A)) \leqslant M\lambda (A)

. Here

λ
\lambda

denotes Lebesgue measure and

F
¯

E
′

{\bar F’_E}

and

F
_

E
′

\underline {F}’_E

are the extreme derivates of

F
F

relative to a system of paths which satisfy the Intersection Condition [1]. In particular, the result holds in the setting of unilateral differentiation, approximate differentiation, preponderant differentiation and qualitative differentiation.
The Lorentz space as a dual space
Authors: Ghatage Pratibha G.
Published: June 2010

If

(
X
,
S
,
μ
)

(X,S,\mu )

is a finite, completely nonatomic measure space and

ϕ
(
t
)
=

t

1

/

p

(
p
>
1
)

\phi (t) = {t^{1/p}}(p > 1)

then the Lorentz space

N
ϕ

{N_\phi }

is the dual space of the closed span of simple functions in

M
ϕ

(
=

N
ϕ
∗

)

{M_\phi }( = N_\phi ^ * )

.
A fixed point theorem for inverse limits of fans
Authors: Marsh M. M.
Published: June 2010

In [1] David Bellamy gave an example of a tree-like continuum without the fixed point property. Another example of a tree-like continuum which admits a fixed point free mapping was given in [5] by L. G. Oversteegen and J. T. Rogers, Jr. We prove in this paper that a certain inverse limit of fans has the fixed point property. Certain possible methods for simplifying Bellamy’s and Oversteegen and Roger’s examples are eliminated by our theorem.
The conjugacy problem for graph products with central cyclic edge groups
Authors: Horadam K. J.
Published: June 2010

A graph product is the fundamental group of a graph of groups. Amongst the simplest examples are HNN extensions and free products with amalgamation. Graph products with cyclic edge groups inherit a solvable conjugacy problem from their vertex groups under certain conditions, the most important of which imposed here is that all the edge group generators in each vertex group are powers of a common central element. Under these conditions the conjugacy problem is solvable for any two elements not both of zero reduced length in the graph product, and for arbitrary pairs of elements in HNN extensions, tree products and many graph products over finite-leaf roses. The conjugacy problem is not solvable in general for elements of zero reduced length in graph products over graphs with infinitely many circuits.
Nilpotency of derivations. II
Authors: Chung L. O. , Luh Jiang
Published: June 2010

The authors recently proved that for a semiprime ring without

2
2

-torsion, a nilpotent derivation must have odd nilpotency. In this paper, we show the intriguing phenomenon that for a semiprime ring with characteristic 2, the nilpotency of a nilpotent derivation must be of the form

2
n

{2^n}

. Combining these two results, we show that for a general semiprime ring with no torsion condition, the nilpotency of a nilpotent derivation is either odd or a power of 2.
Minimal cyclotomic splitting fields for group characters
Authors: Mollin R. A.
Published: June 2010

Let

F
F

be a finite Galois extension of the rational number field

Q
Q

, and let

G
G

be a finite group of exponent

n
n

with absolutely irreducible character

χ
\chi

. This paper provides sufficient conditions for the existence of a minimal degree splitting field

L
L

with

F

(
χ
)

⊆
L
⊆
F

(

ε
n

)

F\left ( \chi \right ) \subseteq L \subseteq F\left ( {{\varepsilon _n}} \right )

, where

ε
n

{\varepsilon _n}

is a primitive

n
n

th root of unity. We obtain as immediate corollaries known results pertaining to this question in the literature. Moreover we obtain necessary and sufficient conditions for the existence of a minimal splitting field

L
L

as above which is cyclic over

F

(
χ
)

F\left ( \chi \right )

. The machinery we use to achieve the above results are certain genus numbers of

F

(
χ
)

F\left ( \chi \right )

.
On symmetry of Banach Jordan algebras
Authors: Aupetit B. , Youngson M. A.
Published: June 2010

Using a very simple subharmonic argument we prove that a Banach Jordan algebra is Hermitian if and only if the sum of two positive elements is positive. We apply this result to give a characterization of Banach Jordan algebras with involution which are

J

B
∗

J{B^ * }

-algebras for an equivalent norm.
On maxima of Takagi-van der Waerden functions
Authors: Baba Yoshikazu
Published: June 2010

Generalizing Takagi’s function

F
2

(
x
)

{F_2}\left ( x \right )

and van der Waerden’s function

F

10

(
x
)

{F_{10}}\left ( x \right )

, we introduce a class of nowhere differentiable continuous functions

F
r

(
x
)

{F_r}\left ( x \right )

,

r
⩾
2

r \geqslant 2

. Some properties of

F
r

(
x
)

{F_r}\left ( x \right )

concerning especially maxima are discussed. When

r
r

is even, the Hausdorff dimension of the set of

x
,

{x^,}

’s giving the maxima of

F
r

(
x
)

{F_r}\left ( x \right )

is proved to be

1

/

2

1/2

.
𝑃²(𝜇) and bounded point evaluations
Authors: Trent Tavan T. , Wang James L.
Published: June 2010

It is shown that if

g
∈

C
c
1

(

C

)

g \in C_c^1({\mathbf {C}})

with

∂
¯

g

\bar \partial g

nonvanishing on the support of

μ
\mu

and if

P
2

(
μ
)

{P^2}(\mu )

has no bounded point evaluations, then

sp

{

P
2

(
μ
)
+
g

P
2

(
μ
)

}
−

=

L
2

(
μ
)

{\text {sp}}{\{ {P^2}(\mu ) + g{P^2}(\mu )\} ^ - } = {L^2}(\mu )

. Similar theorems stating that in the absence of bounded point evaluations

P
2

(
μ
)

{P^2}(\mu )

is "almost"

L
2

(
μ
)

{L^2}(\mu )

are derived. As a consequence, to show that

P
2

(
μ
)
=

L
2

(
μ
)

{P^2}(\mu ) = {L^2}(\mu )

in the absence of bounded point evaluations, one need only show that, for example,

z
−
λ

∈

P
2

(
μ
)

\sqrt {z - \lambda } \in {P^2}(\mu )

for complex

λ
\lambda

’s.
Examples of centralizers of automorphism groups
Authors: Turull Alexandre
Published: June 2010

If

A
G

AG

is a finite solvable group with

(

|
A
|

,

|
G
|

)
=
1

(\left | A \right |,\left | G \right |) = 1

and

G
◃
A
G

G \triangleleft AG

, we explore the relationship between the Fitting height of

G
G

, the composition length of

A
A

and the structure of

C
G

(
A
)

{C_G}(A)

. For any

A
A

, we give examples where the difference between the Fitting height of

G
G

and that of

C
G

(
A
)

{C_G}(A)

is exactly twice the composition length of

A
A

.
Inversion and representation for the Poisson-Laguerre transform
Authors: Haimo Deborah Tepper
Published: June 2010

The Poisson-Laguerre transform of a function

ϕ
\phi

is given by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis n comma t right-parenthesis equals sigma-summation Underscript m equals 0 Overscript normal infinity Endscripts g left-parenthesis n comma m semicolon t right-parenthesis phi left-parenthesis m right-parenthesis StartFraction m factorial Over normal upper Gamma left-parenthesis m plus alpha plus 1 right-parenthesis EndFraction">

u
(
n
,
t
)
=

∑

m
=
0

∞

g
(
n
,
m
;
t
)
ϕ
(
m
)

m
!

Γ
(
m
+
α
+
1
)

u(n,t) = \sum \limits _{m = 0}^\infty {g(n,m;t)\phi (m)\frac {{m!}}{{\Gamma (m + \alpha + 1)}}}

\]
where

g
g

, defined by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g left-parenthesis n comma m semicolon t right-parenthesis equals StartFraction normal upper Gamma left-parenthesis n plus m plus alpha plus 1 right-parenthesis Over n factorial m factorial EndFraction StartFraction t Superscript m plus m Baseline Over left-parenthesis 1 plus t right-parenthesis Superscript n plus m plus alpha plus 1 Baseline EndFraction dot Subscript 2 Baseline upper F 1 left-parenthesis negative n comma negative m semicolon negative n minus m minus alpha semicolon 1 minus StartFraction 1 Over t squared EndFraction right-parenthesis comma">

g
(
n
,
m
;
t
)
=

Γ
(
n
+
m
+
α
+
1
)

n
!
m
!

t

m
+
m

(
1
+
t
)

n
+
m
+
α
+
1

⋅
2

F
1

(

−
n
,
−
m
;
−
n
−
m
−
α
;
1
−

1

t
2

)

,

g(n,m;t) = \frac {{\Gamma (n + m + \alpha + 1)}}{{n!m!}}\frac {{{t^{m + m}}}}{{{{(1 + t)}^{n + m + \alpha + 1}}}}{ \cdot _2}{F_1}\left ( { - n, - m; - n - m - \alpha ;1 - \frac {1}{{{t^2}}}} \right ),

\]
s the associated function of the source solution

g
(
n
;
t
)
=
g
(
n
,
0
;
t
)

g(n;t) = g(n,0;t)

of the Laguerre difference heat equation
\[

∇
n

u
(
n
,
t
)
=

u
t

(
n
,
t
)
,

{\nabla _n}u(n,t) = {u_t}(n,t),

\]
with
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nabla Subscript n Baseline f left-parenthesis n right-parenthesis equals left-parenthesis n plus 1 right-parenthesis f left-parenthesis n plus 1 right-parenthesis equals left-parenthesis 2 n plus alpha plus 1 right-parenthesis f left-parenthesis n right-parenthesis plus left-parenthesis n plus alpha right-parenthesis f left-parenthesis n minus 1 right-parenthesis period">

∇
n

f
(
n
)
=
(
n
+
1
)
f
(
n
+
1
)
=
(
2
n
+
α
+
1
)
f
(
n
)
+
(
n
+
α
)
f
(
n
−
1
)
.

{\nabla _n}f(n) = (n + 1)f(n + 1) = (2n + \alpha + 1)f(n) + (n + \alpha )f(n - 1).

\]
A simple algorithm for the inversion of the transform

(
∗
)

(*)

is derived. For

m
=
0

m = 0

, the transform

(
∗
)

(*)

is basically a power series so that the inversion algorithm leads to a useful identity involving binomial coefficients. In addition, a subclass of functions is characterized that is representable by a Poisson-Laguerre transform

(
∗
)

(*)

.
Convolution inequalities on finite cyclic groups and the pseudomeasure norm
Authors: Ritter David L.
Published: June 2010

A characterization of the probability measures that define hypercontractive convolution operators on finite cyclic groups has been given in terms of the pseudomeasure norm. Here the pseudomeasure norm is shown to be a poor quantitative predictor of hypercontractiveness in an asymptotic sense.
Clear visibility and sets which are almost starshaped
Authors: Breen Marilyn
Published: June 2010

Let

S
S

be a nonempty compact connected set in

R
2

{R^2}

. Assume that every two points of local nonconvexity of

S
S

are clearly visible from a common point of

S
S

. Then

S
S

is "almost" starshaped. In fact, for any point

y
y

in

R
2

{R^2}

there is a line

L
L

through

y
y

such that

L
∩
S

L \cap S

is convex and every point of

S
S

sees via

S
S

a point of

L
L

. If "clearly visible" is replaced by the weaker term "visible", then the result fails.
Open retractions on locally connected continua
Authors: Miklos S.
Published: June 2010

It is shown that a locally connected continuum

X
X

admits open retractions onto each of its subcontinua if and only if

X
X

is either an arc or a simple closed curve.
Homotopy equivalences on 3-manifolds
Authors: Venema Gerard A.
Published: June 2010

Suppose

M
3

{M^3}

is a

3
3

-manifold and

f
:

M
3

→
X

f:{M^3} \to X

is a homotopy equivalence onto an ANR

X
X

. In this paper the cellularity properties of point preimages under

f
f

are studied. It is shown that for every open cover

α
\alpha

of

X
X

there exists an open cover

β
\beta

of

X
X

such that if

f
f

is a

β
\beta

-equivalence then each

f

−
1

(
x
)

{f^{ - 1}}\left ( x \right )

is

α
\alpha

-cellular in

M
+
3

×

R

1

M_ + ^3 \times {{\mathbf {R}}^1}

. In fact, the (open) cellularity occurs in a continuous fashion and so the map

f
f

can be approximated by a Euclidean bundle map.
On cohomology automorphisms of complex flag manifolds
Authors: Hoffman Michael , Homer William
Published: June 2010

We conjecture a classification of the automorphisms of the rational cohomology ring of

U

(
n
)

/

H

U\left ( n \right )/H

for

H
H

a closed connected subgroup of maximal rank in

U

(
n
)

U\left ( n \right )

, and prove a partial result.
Coefficient bounds for the inverse of a function with derivative in 𝒫. II
Authors: Libera Richard J. , Złotkiewicz Eligiusz J.
Published: June 2010

Sharp upper bounds are given for the coefficients of the inverse of a normalized univalent function whose derivative has positive real part. This resolves a question raised by these authors in Proc. Amer. Math. Soc. 87 (1983), 251-257.
The 𝐾-functional for 𝐻¹ and BMO
Authors: Jawerth Björn
Published: June 2010

Peetre’s

K
K

-functional for the Hardy space

H
1

{H^1}

and the space BMO of functions of bounded mean oscillation is explicitly characterized in terms of truncated square functions.
Smooth group actions on sphere bundles over spheres and on Brieskorn manifolds
Authors: Masuda Mikiya
Published: June 2010

This paper solves one of the problems in [2] proposed by M. Davis. Our result asserts that natural smooth actions on certain sphere bundles over spheres are distinct from those on certain Brieskorn manifolds. Besides this an alternative proof to a result of Davis is given which seems rather more direct than his proof.
𝐺-projective groups
Authors: Vinsonhaler C. , Wickless W.
Published: June 2010

Let TF be the category of torsion free abelian groups of finite rank and homomorphisms. For

G
G

in TF let

PC

(
G
)

{\text {PC}}\left ( G \right )

be the projective class in TF generated by

{
G
}

\left \{ G \right \}

. Theorem.

PC

(
G
)

{\text {PC}}\left ( G \right )

consists exactly of groups of the form

P
⊕
F

P \oplus F

, where

F
F

is finite rank free and

P
P

is

G
G

-projective

P
⊕

P
′

≅

G
n

P \oplus P’ \cong {G^n}

for some positive integer

n
n

).
Smooth solutions of initial-value problems in regions with corners
Authors: Svendsen E. C.
Published: June 2010

The method of images is used to show the existence and uniqueness of smooth solutions of initial-value problems in regions with corners.
A note on state splitting
Authors: Friedman Joel
Published: June 2010

Using state splitting methods, we construct nice matrix representatives for arbitrary subshifts of finite type.
Some remarks on locally nilpotent groups of finite rank
Authors: Smith Howard
Published: June 2010

The main subclasses of locally nilpotent groups are considered in conjunction with the extra hypothesis of finite rank or an allied property. These classes are ordered by inclusion, examples being indicated wherever inclusion is proper.
On integral inequalities associated with a linear operator equation
Authors: Subrahmanyam M. B.
Published: June 2010

In this paper we apply control theoretic concepts to formulate and solve a generalized problem for the determination of best possible constants in integral inequalities. We investigate the problem of existence of functions for which the best constant is attained, and also the conditions satisfied by these functions.
Noncontractive uniformly Lipshitzian semigroups in Hilbert space
Authors: Tingley Daryl
Published: June 2010

It is shown that any

k
k

-Lipshitzian,

k
>
π

/

2

k > \pi /2

, noncontractive commutative semigroup acting on a closed bounded convex set in Hilbert space has a common fixed point.
Finitely additive measures on 𝑁 and the additive property
Authors: Mekler Alan H.
Published: June 2010

Finitely additive measures on

N

{\mathbf {N}}

satisfying an approximation of countable additivity, called (AP), are studied. These measures are generalizations of

p
p

-points. From a

p
p

-point a translation invariant measure with (AP) is constructed. It is consistent that no measure with (AP) exists.
Un résultat d’uniformisation borélienne
Authors: Debs Gabriel
Published: June 2010

On montré que si

B
B

est un borélien du produit

X
×
Y

X \times Y

de deux espaces polonais tel que (i) pour tout

x
∈
X

x \in X

la coupe

B
(
x
)

B(x)

est la réunion d’un

K
σ

{K_\sigma }

et d’un

G

δ

{\mathcal {G}_\delta }

de

Y
Y

et (ii) pour tout ouvert

V
V

de

Y
Y

l’ensemble

{

x
∈
X
:
B
(
x
)
∩
V
≠
∅

}

\left \{ {x \in X:B(x) \cap V \ne \emptyset } \right \}

est borélien, alors

B
B

admet une uniformisation borélienne.
Homeomorphism groups and homogeneous spaces
Authors: van Mill Jan
Published: June 2010

We show that the

2
2

-sphere

S
2

{S^2}

can be decomposed into two sets

A
A

and

B
B

such that (1) both

A
A

and

B
B

are connected, locally connected, dense and Baire, (2) both

A
A

and

B
B

are (much more than) topologically homogeneous, (3)

A
A

is not homeomorphic to

B
B

, and (4) the autohomeomorphism groups of

A
A

and

B
B

are (algebraically) isomorphic.
Sequential coreflections of stratifiable spaces
Authors: Foged L.
Published: June 2010

We prove that the sequential coreflection of a stratifiable space which is the union of countably many closed

k
k

-spaces is stratifiable.
On pairs of coefficients of bounded polynomials
Authors: Brickman L. , Rahman Q. I. , Ruscheweyh St.
Published: June 2010

Given a polynomial of degree at most

n
n

, we estimate the sum of the moduli of any two of its coefficients in terms of its supremum norm on the unit disk.
Symbolic powers of prime ideals and their topology
Authors: Schenzel Peter
Published: June 2010

For a prime ideal

P
P

of a commutative Noetherian ring

R
R

a necessary and sufficient condition is given to determine when the

P
P

-adic topology is equivalent, resp. linearly equivalent, to the

P
P

-symbolic topology. The last means that the symbolic Rees ring is a finitely generated module over the ordinary Rees ring of

P
P

. Then it is considered when the integral closure of all the powers of

P
P

are primary.
Topological equivalence in the space of integrable vector-valued functions
Authors: Gutman Semion
Published: June 2010

The Banach space

L
1

(
0
,
T
;
X
)

{L^1}(0,T;X)

is retopologized by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue f EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue equals max StartAbsoluteValue EndAbsoluteValue integral Subscript a Superscript b Baseline f d t StartAbsoluteValue EndAbsoluteValue">

|

|

|

f

|

|

|

=
max

|

|

∫
a
b

f
d
t

|

|

|||f||| = \max ||\int _a^bfdt||

,

0
⩽
a
⩽
b
⩽
T

0 \leqslant a \leqslant b \leqslant T

, where

|

|

.

|

|

||.||

is the norm in the given Banach space

X
X

. It is shown here that this topology coincides with the usual weak topology of

L
1

(
0
,
T
;
X
)

{L^1}(0,T;X)

on a wide class of weakly compact subsets.
Köthe’s example of an incomplete LB-space
Authors: Kucera J. , McKennon K.
Published: June 2010

The inductive limit of reflexive Banach spaces is fast complete. But the inductive limit of reflexive Fréchet spaces may not be fast complete. Also, an example of a complete, reflexive, nuclear space with a quotient space which is not fast complete is given.
Contractions with the bicommutant property
Authors: Takahashi Katsutoshi
Published: June 2010

It is shown that if

T
T

is a contraction for which there is an operator

W
W

with dense range such that

W
T
=
S
W

WT = SW

for some unilateral shift

S
S

, then

T
T

has the bicommutant property, that is, the double commutant of

T
T

is the weakly closed algebra generated by

T
T

and the identity. As an example of such a contraction we have a contraction

T
T

such that

I
−

T
∗

T

I - {T^ * }T

is of trace class and the spectrum of

T
T

fills the unit disc.
On spherical submanifolds with nullity
Authors: Dajczer Marcos , Gromoll Detlef
Published: June 2010

We derive a new condition for spherical submanifolds with positive relative nullity. As an application we obtain that the Ricci curvature is bounded from above by the curvature of the ambient space.
Stability of harmonic maps and minimal immersions
Authors: Pan Y. L. , Shen Y. B.
Published: June 2010

It was proved by J. Simons [10] that there does not exist any stable minimal submanifold in the Euclidean sphere

S
n

{S^n}

, and P. F. Leung proved that any stable harmonic map from any Riemannian manifold to

S
n

{S^n}

, where

n
⩾
3

n \geqslant 3

, is a constant. In this paper, we generalize their results and indicate that there are many manifolds having such properties as

S
n

{S^n}

.
A characterization of Clarke’s strict tangent cone via nonlinear semigroups
Authors: Penot Jean-Paul
Published: June 2010

Clarke’s strict tangent cone

T
X
↑

(
a
)

T_X^ \uparrow (a)

at

a
∈
X

a \in X

to a closed subset of a Banach space

E
E

is shown to contain the limit inferior of tangent cones

T
X

(
x
)

{T_X}(x)

to

X
X

at

x
x

as

x
→
a

x \to a

,

x
∈
X

x \in X

. Several characterizations of

T
X
↑

(
a
)

T_X^ \uparrow (a)

are presented. As a consequence various tangential and subtangential conditions for continuous vector fields on

X
X

are shown to be equivalent.
Totally real minimal submanifolds of a complex projective space
Authors: Urbano Francisco
Published: June 2010

An

n
n

-dimensional positively curved compact totally real minimal submanifold of an

n
n

-dimensional complex projective space is totally geodesic.
A spurious Brownian motion
Authors: Hardin Clyde D.
Published: June 2010

There exists an

R

d

{{\mathbf {R}}^d}

-valued mean-zero Gaussian process, all of whose projections agree with the projections of standard Brownian motion, yet which is not standard Brownian motion.
A remark about separation of 𝐾-analytic sets in the product spaces
Authors: Piątkiewicz Leszek
Published: June 2010

A direct proof, using only "classical" notions, of a well-known theorem about approximation of coanalytic sets in the plane with "large" sections by Borel sets is given. A generalization of this theorem for hereditarily

K
K

-analytic spaces is also considered.
Continuity of homomorphisms on a Baire group
Authors: Fleischer Isidore , Traynor Tim
Published: June 2010

A pointwise converging sequence of continuous homomorphisms is equicontinuous.
Gorenstein algebras and the Cayley-Bacharach theorem
Authors: Davis E. D. , Geramita A. V. , Orecchia F.
Published: June 2010

This paper is an examination of the connection between the classical Cayley-Bacharach theorem for complete intersections in

P

2

{{\mathbf {P}}^2}

and properties of graded Gorenstein algebras.
The convexity of a domain and the superharmonicity of the signed distance function
Authors: Armitage D. H. , Kuran Ü.
Published: June 2010

Let

D
D

be a domain in

R

N

{{\mathbf {R}}^N}

with nonempty boundary

∂
D

\partial D

and let

u
u

be the signed distance function from

∂
D

\partial D

, i.e.

u
=
±

u = \pm

dist according as we are in or outside

D
¯

\overline D

. We prove that, for any

N
⩾
2
,
u

N \geqslant 2,u

is superharmonic in

R

N

{{\mathbf {R}}^N}

if and only if

D
D

is convex. When

N
=
2

N = 2

, this criterion requires the superharmonicity of

u
u

in

D
D

only.
Criteria for a Blaschke quotient to be of uniformly bounded characteristic
Authors: Yamashita Shinji
Published: June 2010

Criteria for a quotient

B
1

/

B
2

{B_1}/{B_2}

of Blaschke products

B
1

{B_1}

and

B
2

{B_2}

to be of uniformly bounded characteristic are proposed in terms of interpolating sequences.
On the kernel of a Markov projection on 𝐶(𝑋)
Authors: Atalla Robert E.
Published: June 2010

Let

X
X

be a compact metric space and

L
L

a closed linear subspace of

C
(
X
)

C(X)

, the real valued continuous functions on

X
X

. We give necessary and sufficient conditions of an algebraic nature for

L
L

to be the kernel of a Markov projection

P
P

on

C
(
X
)

C(X)

. We also characterize compact spaces for which our result holds as those for which the Borsuk-Dugundji simultaneous extension theorem holds.
Integer invariants of certain even-dimensional knots
Authors: Kearton C.
Published: June 2010

Integer invariants of certain simple

Z

{\mathbf {Z}}

-torsion-free

2
q

2q

-knots,

q
⩾
4

q \geqslant 4

, are defined. It is shown that for

q
⩾
5

q \geqslant 5

, certain of these invariants must vanish,

mod

2

\mod 2

, if the knot is doubly-null-concordant.
A group-theoretic characterization of 𝑀-groups
Authors: Parks Alan E.
Published: June 2010

Groups having the property that all their complex irreducible characters are monomial are characterized in terms of the embedding of cyclic sections of the group.
Complex powers of the Laplace operator on the circle
Authors: Morava Jack
Published: June 2010

The classical zeta function of Lerch has an analytic continuation as a distribution on the circle which seems to be very different from its usual analytic continuation: for example, the Bernoulli polynomials come out upside down.
The homotopy type of hyperplane posets
Authors: Edelman Paul H. , Walker James W.
Published: June 2010

Previously, Edelman had defined a partial order on the regions of a euclidean space dissected by hyperplanes. The goal of this paper is to compute the homotopy type of open intervals in these posets. Techniques from the theory of shellable posets are used.
The factorization of a linear conjugate symmetric involution in Hilbert space
Authors: Moeller James W.
Published: June 2010

Let

X
X

be a closed linear transformation whose domain is dense in the complex separable Hilbert space

H
H

and whose adjoint is denoted by

X
∗

{X^ * }

. The operator

X
X

is said to be conjugate symmetric if

Γ
(
X
)
⊂
Γ
(
Q

X
∗

Q
)

\Gamma (X) \subset \Gamma (Q{X^ * }Q)

, where

Γ
(
X
)

\Gamma (X)

represents the graph of

X
X

in

H
⊗
H

H \otimes H

and

Q
Q

is a conjugation on

H
H

. The main theorem in this note states that a conjugate symmetric linear involution

X
X

satisfies the operator equation

X
=
Q

X
∗

Q

X = Q{X^ * }Q

.
Total curvatures and minimal areas of complete surfaces
Authors: Shiohama Katsuhiro
Published: June 2010

Minimal areas for certain classes of finitely connected complete open surfaces are obtained by using a Bonnesen-style isoperimetric inequality for large balls on the surfaces. In particular, the minimal area of Riemannian planes whose Gaussian curvatures are bounded above by 1 is

4
π

4\pi

.
Homology of integral upper-triangular matrices
Authors: Dwyer W. G.
Published: June 2010

We calculate the homology of the multiplicative group of integral upper-triangular

n
×
n

n \times n

matrices at all primes

p
⩾
n
−
1

p \geqslant n - 1

.
A characterization of the invariant subspaces of direct sums of strictly cyclic algebras
Authors: Rosenthal Erik
Published: June 2010

Two characterizations of the invariant subspace lattice of

A

(
n
)

{A^{(n)}}

for a strictly cyclic operator algebra

A
A

on a separable Hilbert space are proven.
The topological spaces that support Haar systems
Authors: McCullough Scott , Wulbert Daniel
Published: June 2010

A topological space

X
X

supports a nontrivial Haar system if and only if there is a continuous one-to-one map of

X
X

into the unit circle. This gives an elementary proof of Mairhuber’s classical theorem for compact

X
X

. It answers a 1960 question asked by R. R. Phelps when

X
X

is locally compact.
On nonrigidity of harmonic maps into spheres
Authors: Tóth Gábor
Published: June 2010

This note studies nonrigidity of equivariant harmonic maps

f
:
M
→

S
n

f:M \to {S^n}

of a Riemannian homogeneous space

M
M

into the Euclidean

n
n

-sphere

S
n

{S^n}

via representation theory applied to the induced module structure on

R

n

{{\mathbf {R}}^n}

and, for specific

M
M

, produces (divergence-free) Jacobi fields along

f
f

which do not come from isometric deformations of

f
f

on the range.
On random approximations and a random fixed point theorem for set valued mappings
Authors: Sehgal V. M. , Singh S. P.
Published: June 2010

We prove a random fixed point theorem in a Banach space for set valued mappings and then derive a corollary that yields a fixed point theorem of Bharucha-Reid and Mukherjea, as a special case.
Isomorphisms of trees
Authors: Franek Frantisek
Published: June 2010

Let

κ
\kappa

,

λ
\lambda

be cardinals,

κ
≥

ℵ
1

\kappa \geq {\aleph _1}

and regular, and

2
≤
λ
≤
κ

2 \leq \lambda \leq \kappa

. If

κ
>

ℵ
1

\kappa > {\aleph _1}

and

λ
>
κ

\lambda > \kappa

, and if there is a

κ
\kappa

-Suslin (

κ
\kappa

-Aronszajn,

κ
\kappa

-Kurepa) tree, then there are

2
κ

{2^\kappa }

normal

λ
\lambda

-ary rigid nonisomorphic

κ
\kappa

-Suslin (

κ
\kappa

-Aronszajn,

κ
\kappa

-Kurepa) trees. If there is a Suslin (Aronszajn, Kurepa) tree, then there is a normal rigid Suslin (Aronszajn, Kurepa) tree. If there is a

κ
\kappa

-Canadian tree, then there are

2
κ

{2^\kappa }

normal

λ
\lambda

-ary rigid nonisomorphic

κ
\kappa

-Canadian trees.
Examples on harmonic measure and normal numbers
Authors: Wu Jang-Mei
Published: June 2010

Suppose that

F
F

is a bounded set in

R

m

{{\mathbf {R}}^m}

,

m
⩾
2

m \geqslant 2

, with positive capacity. Add to

F
F

a disjoint set

E
E

so that

E
∪
F

E \cup F

is closed, and let

D
=

R

m

∖
(
E
∪
F
)

D = {{\mathbf {R}}^m}\backslash (E \cup F)

. Under what conditions on the added set

E
E

do we have harmonic measure

ω
(
F
,
D
)
=
0

\omega (F,D) = 0

? It turns out that besides the size of

E
E

near

F
F

, the location of

E
E

relative to

F
F

also plays an important role. Our example, based on normal numbers, stresses this fact.
Holomorphic quasiregular mappings
Authors: Poletsky E. A.
Published: June 2010

Holomorphic quasiregular mappings in bounded domains in

C

n

{{\mathbf {C}}^n}

are studied. It is shown that the growth of the Jacobian of these mappings depends on the behavior of the boundary of a domain. In particular, the Jacobian is bounded when the boundary is smooth. Some applications to the theory of quasiregular mappings between Hermitian manifolds are given.
Weighted reverse weak type inequalities for the ergodic maximal function and the classes 𝐿𝑙𝑜𝑔⁺𝐿
Authors: Andersen Kenneth F. , Young Wo-Sang
Published: June 2010

D. Ornstein proved that the ergodic maximal function satisfies a reverse weak type inequality, and from this he deduced that the integrability of the maximal function

f
∗

{f^ * }

implies that

f
f

belongs to

L

log
+

L

L{\log ^ + }L

. Weighted analogues of these results are proved.
Examples of compact non-Kähler almost Kähler manifolds
Authors: Cordero Luis A. , Fernández M. , de León Manuel
Published: June 2010

A parametrized family of compact non-Kähler almost Kähler manifolds

M

2
r
+
2

{M^{2r + 2}}

,

r
⩾
1

r \geqslant 1

, is constructed. For

r
r

odd, these manifolds are shown to have odd first Betti number, so they cannot be Kählerian.
On the Blackwell property of Luzin sets
Authors: Jasiński Jakub
Published: June 2010

We prove under the CH (Continuum Hypothesis) that there are strongly Blackwell Luzin sets and not Blackwell Luzin sets. We also indicate how these results can be generalized under MA (Martin’s Axiom).
On a conjecture of Balog
Authors: Hildebrand Adolf
Published: June 2010

A conjecture of A. Balog is proved which gives a sufficient condition on a set

A
A

of positive integers such that

A
∩
(
A
+
1
)

A \cap (A + 1)

is infinite. A consequence of this result is that, for every

ε
>
0

\varepsilon > 0

, there are infinitely many integers

n
n

such that both

n
n

and

n
+
1

n + 1

have a prime factor

>

n

1
−
ε

> {n^{1 - \varepsilon }}

.
The completely mixed single-controller stochastic game
Authors: Filar Jerzy A.
Published: June 2010

We consider a zero-sum stochastic game with finitely many states and actions. Further we assume that the transition probabilities depend on the actions of only one player (player II, in our case), and that the game is completely mixed. That is, every optimal stationary strategy for either player assigns a positive probability to every action in every state. For these games, properties analogous to those derived by Kaplansky [4] for the completely mixed matrix games, are established in this paper. These properties lead to the counterintuitive conclusion that the controller need not know the law of motion in order to play optimally, but his opponent does not have this luxury.
The transitive property of parallel lines is a characteristic property of real strictly convex Banach spaces
Authors: Valentine J. E.
Published: June 2010

In a recent paper Freese and Murphy said a complete, convex, externally convex metric space has the vertical angle property provided for each four of its distinct points

p
p

,

q
q

,

r
r

,

s
s

, if

m
m

is a midpoint of

p
p

and

q
q

and of

r
r

and

s
s

, then

p
r
=
q
s

pr = qs

. In this paper we say a line

L
L

is parallel to a line

N
N

in such a space provided

L
L

and

N
N

contain points

p
p

,

r
r

, and

q
q

,

s
s

, respectively, such that the segments

S

(

p
,
q

)

S\left ( {p,q} \right )

and

S

(

r
,
s

)

S\left ( {r,s} \right )

have a common midpoint

m
m

. We further assume that if line

L
L

is parallel to line

N
N

and line

N
N

is parallel to line

R
R

, then

L
L

is parallel to

R
R

. The main result of this paper is that such a space is a real strictly convex Banach space. Since real strictly convex Banach spaces have all of the above properties, the characterization is then complete.
Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance
Authors: Beer Gerald
Published: June 2010

Atsuji has internally characterized those metric spaces

X
X

for which each real-valued continuous function on

X
X

is uniformly continuous as follows: (1) the set

X
′

X’

of limit points of

X
X

is compact, and (2) for each

ε
>
0

\varepsilon > 0

, the set of points in

X
X

whose distance from

X
′

X’

exceeds

ε
\varepsilon

is uniformly discrete. We obtain these new characterizations: (a) for each metric space

Y
Y

, the Hausdorff metric on

C

(

X
,
Y

)

C\left ( {X,Y} \right )

, induced by a metric on

X
×
Y

X \times Y

compatible with the product uniformity, yields the topology of uniform convergence; (b) there exists a metric space

Y
Y

containing an arc for which the Hausdorff metric on

C

(

X
,
Y

)

C\left ( {X,Y} \right )

yields the topology of uniform convergence; (c) the Hausdorff metric topology on

CL

(
X
)

{\text {CL}}\left ( X \right )

is at least as strong as the Vietoris topology. We also characterize those metric spaces whose hyperspace is such a space.
Normal subgroups of the general linear groups over von Neumann regular rings
Authors: Vaserstein L. N.
Published: June 2010

Let

A
A

be a von Neumann regular ring or, more generally, let

A
A

be an associative ring with 1 whose reduction modulo its Jacobson radical is von Neumann regular. We obtain a complete description of all subgroups of

G

L

n

A
,
n
≥
3

{\text {G}}{{\text {L}}_n}A,n \geq 3

, which are normalized by elementary matrices.
On the punctual and local ergodic theorem for nonpositive power bounded operators in 𝐿^{𝑝}_{𝐶}[0,1],1<𝑝<+∞
Authors: Assani I.
Published: June 2010

We show in this note that there exists a function

f
∈

∩
1

>
p
>
+
∞

L

C

p

[
0
,
1
]

f \in { \cap _1}_{ > p > + \infty }L_{\mathbf {C}}^p[0,1]

and for each

p
p

an isomorphism

T
:

L

C

p

→

L

C

p

T:L_{\mathbf {C}}^p \to L_{\mathbf {C}}^p

such that

su

p

n
∈

Z

‖

T
n

‖

>
+
∞

{\text {su}}{{\text {p}}_{n \in {\mathbf {Z}}}}\left \| {{T^n}} \right \| > + \infty

and

T
T

does not satisfy the punctual ergodic theorem. We give also an example of a one-parameter semigroup

(

T
t

,
t
⩾
0
)

({T_t},t \geqslant 0)

of power bounded operators in each

L

C

p

(
1
>
p
>
+
∞
)

L_{\mathbf {C}}^p(1 > p > + \infty )

for which the assertion of the local ergodic theorem

(
(
1

/

t
)

∫
0
t

T
s

f
d
s

((1/t)\smallint _0^t{T_s}fds

converge almost everywhere as

t
→

0
+

t \to {0_ + }

for all

f
∈

L
p

f \in {L^p}

fails to be true.
The geodesic flow for discrete groups of infinite volume
Authors: Nicholls Peter J.
Published: June 2010

Let

Γ
\Gamma

be a discrete group acting in the unit ball

B
B

of euclidean

n
n

-space and

T
(
B
)

T(B)

the unit tangent space of

B
B

. We define the geodesic flow

g
t

{g_t}

on the quotient space

Ω
=
T
(
B
)

/

Γ

\Omega = T(B)/\Gamma

and show that for discrete groups of infinite volume the flow is of zero type—namely, for measurable subsets

A
,
B

A,B

of

Ω
\Omega

which are of finite measure,

lim

t
→
∞

g
t

(
A
)
∩
B
=
0

{\lim _{t \to \infty }}{g_t}(A) \cap B = 0

. Using this result, we give a new and elementary proof of the fact that for a discrete group of infinite volume,

N
(
r
)
=
o
(
V
{
x
:

|
x
|

>
r
}
)

N(r) = o(V\{ x:\left | x \right | > r\} )

as

r
→
1

r \to 1

, where

N
(
r
)

N(r)

is the orbital counting function and

V
V

denotes hyperbolic volume.
On linking coefficients
Authors: Habegger Nathan
Published: June 2010

The possible values of linking coefficients for two component links are studied. An example of a link

S
3

∪

S
2

{S^3} \cup {S^2}

in

S
5

{S^5}

having linking coefficient in

π
3

(

S
2

)

{\pi _3}({S^2})

of Hopf invariant two is constructed. A generalization to links

S

m
−
2

∪

S
p

{S^{m - 2}} \cup {S^p}

in

S
m

{S^m}

is obtained in the metastable range. Applications to embeddings of two cell complexes are cited.
A new characterization for 𝑝-local balanced projective groups
Authors: Lane Mark
Published: June 2010

By introducing the notion of a

K

{\text {K}}

-nice submodule, we obtain a characterization of

p
p

-local balanced projectives perfectly analogous to the familiar third axiom of countability characterization of totally projective

p
p

-groups. We use this new characterization to prove that if a

p
p

-local group

G
G

satisfies the third axiom of countability with respect to nice submodules and has a

K

{\text {K}}

-basis, then

G
G

is a balanced projective.
Second derivative 𝐿^{𝑝}-estimates for elliptic equations of nondivergent type
Authors: Lin Fang-Hua
Published: June 2010

We obtain an a priori estimate of second derivatives in

L
p

{L^p}

, for some

p
>
0

p > 0

, for solutions of nondivergent, uniformly elliptic P.D.E.’s of second order.
Approximation and Besov spaces on stratified groups
Authors: Giulini Saverio
Published: June 2010

This paper treats analogues of the classical Bernstein and Jackson theorems in the nonclassical context of the stratified groups. Besov spaces on stratified groups are characterized by their approximation theoretic properties.
The essential boundary of certain sets
Authors: Bruckner A. M. , Davies Roy O. , Goffman C.
Published: June 2010

The essential boundary of a measurable set is related to the de Giorgi perimeter and was introduced by Vol’pert in his "improvement" of Federer’s work. For a totally disconnected compact set of positive measure in

n
n

space the essential boundary can be of Hausdorff

n
−
1

n - 1

dimension but cannot have

σ
\sigma

finite

(
n
−
1
)

(n - 1)

-measure.
A remark concerning a mechanical characterization of the sphere
Authors: Pamfilos Paris
Published: June 2010

It is shown that if a (sufficiently) small ball rolls freely without sliding on a compact surface in such a way that its center moves with constant in magnitude velocity for every initial condition, then the surface must be a sphere.
The depth of tranches in 𝜆-dendroids
Authors: Mohler Lee
Published: June 2010

According to the well-known theory of Kuratowski, any hereditarily decomposable chainable continuum admits a decomposition into tranches. These tranches are themselves chainable and thus admit decompositions into their own tranches. We may thus define nested sequences

{

T
α

}

\{ {T_\alpha }\}

of tranches-within-tranches, indexed by countable ordinals

α
\alpha

, and finally terminating in a singleton set. E. S. Thomas, Jr. has asked whether, for a given continuum

C
C

, there is a countable ordinal bound on the length of all such nests

{

T
α

}

\{ {T_\alpha }\}

in

C
C

. We answer Thomas’s question in the affirmative. By generalizing the definitions, we obtain the same result for

λ
\lambda

-dendroids. We also answer, for chainable continua, a related question of Illiadis.
A note on a theorem of Perron
Authors: Prasad K. C. , Lari M.
Published: June 2010

Given an infinite simple continued fraction

[

a
0

,

a
1

,
…
,

a
n

,
…

]

\left [ {{a_0},{a_1}, \ldots ,{a_n}, \ldots } \right ]

, let

M
n

{M_n}

denote

[

0
,

a
n

,

a

n
−
1

,
…
,

a
1

]

+

[

a

n
+
1

,

a

n
+
2

,
…

]

\left [ {0,{a_n},{a_{n - 1}}, \ldots ,{a_1}} \right ] + \left [ {{a_{n + 1}},{a_{n + 2}}, \ldots } \right ]

. A well-known result due to Perron [1, III, 212] states: If

a

n
+
2

=
m

{a_{n + 2}} = m

, then there is a

k
k

in

{

n
,
n
+
1
,
n
+
2

}

\left \{ {n,n + 1,n + 2} \right \}

for which

M
k

>

m
2

+
4

{M_k} > \sqrt {{m^2} + 4}

. In this note we give a new proof for this result and add that there is a

j
j

in

{

n
,
n
+
1
,
n
+
2

}

\left \{ {n,n + 1,n + 2} \right \}

for which

M
j

>

m
2

+
4

{M_j} > \sqrt {{m^2} + 4}

.
On a question of Feit
Authors: Ferguson Pamela A. , Turull Alexandre
Published: June 2010

The following theorem is proved: Assume

χ
\chi

is an irreducible complex character of the finite group

G
G

and

G
G

is

π
\pi

-solvable where

π
\pi

is the set of prime divisors of

χ

(
1
)

\chi \left ( 1 \right )

. Then

f

(
χ
)

f\left ( \chi \right )

contains an element of order

f

(
χ
)

f\left ( \chi \right )

.
On Birkhoff quadrature formulas
Authors: Varma A. K.
Published: June 2010

In an earlier work the author has obtained new quadrature formulas (see (1.3)) based on function values and second derivatives on the zeros of

∏
n

(
x
)

{\prod _n}\left ( x \right )

as defined by (1.2). The proof given earlier was quite long. The object of this paper is to provide a proof of this quadrature formula which is extremely simple and indeed does not even require the use of fundamental polynomials of (0,2) interpolation.
A note on elementary operators on the Calkin algebra
Authors: Gravner Janko
Published: June 2010

Various parts of the spectrum of elementary operators on the Calkin algebra are characterised by means of the joint (Harte) spectra of

n
n

-tuples of operators.
Partitions and diamond
Authors: Matet Pierre
Published: June 2010

We restate the diamond principle in terms of partitions, and we show that a weakening of diamond follows from the generalized continuum hypothesis.
Another counterexample in ANR theory
Authors: van Mill Jan
Published: June 2010

We answer an old question due to Kuratowski by constructing a (separable metric) space

X
X

having the following properties: (1)

X
X

is not an ANR, and (2) for every space

Y
Y

and for every compact

A
⊆
Y

A \subseteq Y

, every continuous map

f
:
A
→
X

f:A \to X

can be continuously extended to a map

f
¯

:
Y
→
X

\bar f:Y \to X

.
Perfect pre-images of collectionwise normal spaces
Authors: Daniels Peg
Published: June 2010

We show that the following are true for perfect maps: (1) collectionwise normality with respect to compact (Lindelöf) sets is preserved inversely; (2) collectionwise normality with respect to countably compact (

ω
1

{\omega _1}

-compact) sets is preserved inversely if and only if the domain space is normal with respect to countably compact (

ω
1

{\omega _1}

-compact) sets; and (3) if

P
P

is any property such that (i)

P
P

is preserved by perfect maps, (ii) the free union of spaces satisfying

P
P

also satisfies

P
P

, (iii)

P
P

is closed hereditary, and (iv)

P
P

plus collectionwise normality implies countable metacompactness, then collectionwise normality with respect to closed

P
P

-sets is preserved inversely if the domain space is normal with respect to closed

P
P

-sets. Examples of such a property

P
P

are paracompactness, submetacompactness, stratifiability and countable metacompactness.
The equations defining a curve of genus 4
Authors: Kempf George R.
Published: June 2010

For most curves of genus 4 and characteristic

⩾
3

\geqslant 3

the second osculating cone of the theta divisor is the cone over the canonical curve.