bluuebunny/crossref_metadata_2025 huggingface dataset:
Subadditivity of homogeneous norms on certain nilpotent Lie groups
Authors: Cygan Jacek
Published: June 2010

Let

N
N

be a Lie group with its Lie algebra generated by the left-invariant vector fields

X
1

,
…
,

X
k

{X_1}, \ldots ,{X_k}

on

N
N

. An explicit fundamental solution for the (hypoelliptic) operator

L
=

X
1
2

+
⋯
+

X
k
2

L = X_1^2 + \cdots + X_k^2

on

N
N

has been obtained for the Heisenberg group by Folland [1] and for the nilpotent (Iwasawa) groups of isometries of rank-one symmetric spaces by Kaplan and Putz [2]. Recently Kaplan [3] introduced a (still larger) class of step-

2
2

nilpotent groups

N
N

arising from Clifford modules for which similar explicit solutions exist. As in the case of

L
L

being the ordinary Laplacian on

N
=

R

k

N = {{\mathbf {R}}^k}

, these solutions are of the form

g
↦

const

‖
g
‖

2
−
m

g \mapsto {\text {const}}{\left \| g \right \|^{2 - m}}

,

g
∈
N

g \in N

, where the "norm" function

‖

‖

\left \| {} \right \|

satisfies a certain homogeneity condition. We prove that the above norm is also subadditive.
Norming nil algebras
Authors: Dales H. G.
Published: June 2010

A commutative nil algebra with countable basis is normable, but a commutative nilpotent algebra is not necessarily normable.
Boundary behavior of univalent functions satisfying a Hölder condition
Authors: Essén Matts
Published: June 2010

Let

f
f

be univalent in the unit disk

U
U

and continuous in

U
∪
T

U \cup T

, where

T
=
∂
U

T = \partial U

. We prove that if

f
f

satisfies a Hölder condition, then each point in

f
(
T
)

f(T)

is the image of at most finitely many points on

T
T

. The bound for the number of preimages depends in a sharp way on the Hölder exponent.
A note on disconjugate differential equations and growth estimates
Authors: Erbe L.
Published: June 2010

Subfunctions and differential inequality techniques are applied to certain classes of second and third order nonlinear equations to obtain growth estimates on the solutions.
Extension of Bernšteĭn’s theorem
Authors: Tung S. H.
Published: June 2010

A well-known theorem of Bernstein states that if a polynomial of degree

n
n

of a complex variable has its modulus no larger than one on the unit disk then the modulus of its derivative will not exceed

n
n

on the unit disk. Here we extend the theorem to polynomials on the unit ball in several complex variables.
Abelian 𝑝-group actions on homology spheres
Authors: Dotzel Ronald M.
Published: June 2010

The Borel formula is extended to an identity covering actions of arbitrary Abelian

p
p

-groups. Specifically, suppose

G
G

is an Abelian

p
p

-group which acts on a finite

CW

{\text {CW}}

-complex

X
X

which is a

Z
p

{Z_p}

-homology

n
n

-sphere. Each

X
H

{X^H}

must be a

Z
p

{Z_p}

-homology

n
(
H
)

n(H)

-sphere and then
\[

n
−
n
(
G
)
=
∑

(
n
(
K
)

−
n
(
K

/

p
)
)

n - n(G) = \sum {(n(K)} - n(K/p))

\]
where the sum is over

A
0

=
{

K
|

G

/

K

is

cyclic

}

{A_0} = \{ \left . K \right |G/K\;{\text {is}}\;{\text {cyclic}}\}

and the group

K

/

p

K/p

is defined by
\[

K

/

p
=
{
g
∈

G
|

p
g
∈
K
}
.

K/p = \{ g \in \left . G \right |pg \in K\} .

\]
This result is an immediate corollary of Theorem 2, whose converse Theorem 1, is also proven. Thus actions of Abelian

p
p

-groups on homology spheres resemble linear representations.
A refinement on Michael’s characterization of paracompactness
Authors: Hung H. H.
Published: June 2010

It is proved that a

T
1

{T_1}

space is paracompact (and Hausdorff) if (and only if) every well (but arbitrarily) ordered open cover has an open refinement in the form of a countable union of families, (each of which being well ordered by the naturally induced well order) the initial segments of each of which are cushioned in the corresponding initial segments of the cover, i.e., the closures of the unions of the initial segments of each of which are contained within the unions of the corresponding initial segments of the cover.
A simple proof that the concordance group of algebraically slice knots is infinitely generated
Authors: Jiang Bo Ju
Published: June 2010

A simple proof of the result stated in the title is obtained by making the Casson-Gordon invariant additive.
On the dimension of 𝜇-spaces
Authors: Mizokami T.
Published: June 2010

The main results are as follows: Theorem 1. Let

C

\mathcal {C}

be the class of all

μ
\mu

-spaces. Then the following are equivalent for every

X
∈

C

X \in \mathcal {C}

: (1) dim

X
⩽
n

X \leqslant n

, (2) there exists a closed mapping

f
f

of

Z
∈

C

Z \in \mathcal {C}

with dim

Z
⩽
0

Z \leqslant 0

onto

X
X

such that ord

f
⩽
n
+
1

f \leqslant n + 1

, (3)

X
=

∪

i
=
1

n
+
1

X
i

X = \cup _{i = 1}^{n + 1}{X_i}

, where dim

X
i

⩽
n

{X_i} \leqslant n

for each

i
i

and (4) Ind

X
⩽
n

X \leqslant n

. Theorem 2. A space

X
X

is a

μ
\mu

-space with dim

X
⩽
n

X \leqslant n

if and only if

X
X

is the inverse limit of an inverse sequence

{

X
i

,

g
j
i

}

\left \{ {{X_i},g_j^i} \right \}

of paracompact

σ
\sigma

-metric spaces

X
i

{X_i}

such that dim

X
i

⩽
n

{X_i} \leqslant n

for every

i
∈
N

i \in N

. As the applications of them, some product theorems for covering dimension are given.
A note on ℎ-proximities
Authors: Ori Ramesh , Thron W. J.
Published: June 2010

It is shown that regularity and local bicompactness of the space

(
X
,
c
)

(X,c)

is a necessary condition for the existence of a largest

h
h

-proximity on

(
X
,
c
)

(X,c)

. That these conditions are sufficient was shown earlier by the authors.
An order theoretic characterization of nested spaces
Authors: Smithson R. E.
Published: June 2010

A characterization of nested spaces in terms of their order theoretic properties is given. This characterization is analogous to the order theoretic characterization of trees given by L. E. Ward, Jr.
A characterization of manifold decompositions satisfying the disjoint triples property
Authors: Garity Dennis J.
Published: June 2010

A metric space

X
X

satisfies the Disjoint Triples Property

(

D

D

3

)

({\text {D}}{{\text {D}}_{\text {3}}})

if maps

f
1

{f_1}

,

f
2

{f_2}

and

f
3

{f_3}

from

B
2

{B^2}

into

X
X

are approximable by maps

f
~

1

{\tilde f_1}

,

f
~

2

{\tilde f_2}

and

f
~

3

{\tilde f_3}

with

∩

i
=
1

3

f
~

i

(

B
2

)
=
∅

\cap _{i = 1}^3{\tilde f_i}({B^2}) = \emptyset

. Those CE decompositions of manifolds satisfying

D

D

3

{\text {D}}{{\text {D}}_3}

and yielding finite-dimensional nonmanifold decomposition spaces are shown to be precisely those intrinsically

0
0

-dimensional decompositions which yield nonshrinkable null cellular decompositions under amalgamation. This characterization results in another proof of the fact that

E
n

/

G
×

E
1

{E^n}/G \times {E^1}

is secretly

0
0

-dimensional where

G
G

is a CE usc decomposition of

E
n

{E^n}

,

n
⩾
4

n \geqslant 4

, with

E
n

/

G

{E^n}/G

finite dimensional.
Inverse elementary divisor problem for nonnegative matrices
Authors: Minc Henryk
Published: June 2010

Given a diagonalizable positive matrix

A
A

, there exists a positive matrix with the same spectrum as

A
A

, and with arbitrarily prescribed elementary divisors, provided that elementary divisors corresponding to nonreal eigenvalues occur in conjugate pairs. It is also shown that a similar result holds for doubly stochastic matrices.
On the absolute convergence of lacunary Fourier series
Authors: Patadia J. R. , Shah V. M.
Published: June 2010

In 1958, P. B. Kennedy [2] proved a two-part theorem on the order of magnitude of the coefficients and the absolute convergence of lacunary Fourier series, and he conjectured that his conclusions remained valid under weaker conditions. The conjecture on the order of magnitude of the coefficients was established by M. and S. I. Izumi [1]. In this note the second half of the conjecture, concerning absolute convergence, is deduced from a recent result of Patadia [3].
The distribution function in the Morrey space
Authors: Álvarez Alonso Josefina
Published: June 2010

For

1
⩽
p
⩽
∞

1 \leqslant p \leqslant \infty

, we consider

p
p

-integrable functions on a finite cube

Q
0

{Q_0}

in

R

n

{{\mathbf {R}}^n}

, satisfying
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis StartFraction 1 Over StartAbsoluteValue upper Q EndAbsoluteValue EndFraction integral Underscript upper Q Endscripts StartAbsoluteValue f left-parenthesis x right-parenthesis minus f Subscript upper Q Baseline EndAbsoluteValue Superscript p Baseline d x right-parenthesis Superscript 1 slash p Baseline less-than-or-slanted-equals upper C phi left-parenthesis StartAbsoluteValue upper Q EndAbsoluteValue right-parenthesis">

(

1

|

Q

|

∫
Q

|

f
(
x
)
−

f
Q

|

p

d
x

)

1

/

p

⩽
C
φ
(

|

Q

|

)

{\left ( {\frac {1} {{|Q|}}\int _Q {|f(x) - {f_Q}{|^p}dx} } \right )^{1/p}} \leqslant C\varphi (|Q|)

\]
for every parallel subcube

Q
Q

of

Q
0

{Q_0}

, where

|

Q

|

|Q|

denotes the volume of

Q
Q

,

f
Q

{f_Q}

is the mean value of

f
f

over

Q
Q

and

φ
(
t
)

\varphi (t)

is a nonnegative function defined in

(
0
,
∞
)

(0,\infty )

, such that

φ
(
t
)

\varphi (t)

is nonincreasing near zero,

φ
(
t
)
→
∞

\varphi (t) \to \infty

as

t
→
0

t \to 0

, and

t

φ
p

(
t
)

t{\varphi ^p}(t)

is nondecreasing near zero. The constant

C
C

does not depend on

Q
Q

. Let

g
g

be a nonnegative

p
p

-integrable function

g
:
(
0
,
1
)
→

R

g:(0,1) \to {\mathbf {R}}

such that

g
g

is nonincreasing and

g
(
t
)
→
∞

g(t) \to \infty

as

t
→
0

t \to 0

. We prove here that there exist a cube

Q
0

{Q_0}

and a function

f
f

satisfying condition

(
1
)

(1)

for every parallel subcube

Q
Q

of

Q
0

{Q_0}

, such that

δ
f

(
λ
)
⩾

C
1

δ
g

(
λ
)

{\delta _f}(\lambda ) \geqslant {C_1}{\delta _g}(\lambda )

for

λ
⩾

λ
0

\lambda \geqslant {\lambda _0}

,

C
1

>
0

{C_1} > 0

, where

δ
(
λ
)

\delta (\lambda )

denotes the distribution function.
On weak* continuous operators on ℬ(ℋ)
Authors: Weber Robert E.
Published: June 2010

If

Δ
\Delta

is a weak* continuous bounded linear operator on

B

(

H

)

\mathcal {B}(\mathcal {H})

that fixes the ideal of compact operators

K

\mathcal {K}

and

Δ
0

{\Delta _0}

and

δ
\delta

are the induced maps on

K

\mathcal {K}

and

B

(

H

)

/

K

\mathcal {B}(\mathcal {H})/\mathcal {K}

then it is shown that

Δ
\Delta

has closed range, has dense range, is bounded below, or is onto if and only if both

Δ
0

{\Delta _0}

and

δ
\delta

have the same property. These results are then applied to the operator

X
→
A
X
B

X \to AXB

.
Contractive commutants and invariant subspaces
Authors: Moore R. L.
Published: June 2010

Let

T
T

be a bounded operator on a Banach space

X

\mathfrak {X}

and let

K
K

be a nonzero compact operator. In [1] and [4] it is shown that if

λ
\lambda

is a complex number and if

T
K
=
λ
K
T

TK=\lambda KT

, then

T
T

has a hyperinvariant subspace. In [1], S. Brown goes on to show that if

X

\mathfrak {X}

is reflexive and if

T
K
=
λ
K
T

TK = \lambda KT

and

T
B
=
μ
B
T

TB = \mu BT

for some

λ
\lambda

,

μ
\mu

with

|
λ
|

≠
1

\left | \lambda \right | \ne 1

and

(
1
−

|
μ
|

)

/

(
1
−

|
λ
|

)
⩾
0

(1 - \left | \mu \right |)/(1 - \left | \lambda \right |) \geqslant 0

, then

B
B

has an invariant subspace. Below we extend both these results by showing that the entire class of operators satisfying the above conditions on

B
B

has an invariant subspace.
On the divergence of extension procedures in isol theory
Authors: McLaughlin T. G.
Published: June 2010

We show that the Myhill and Nerode extensions begin to disagree on the domain of the Nerode extension at a point in the arithmetical hierarchy

⩽

Δ
3
0

\leqslant \Delta _3^0

. This disagreement, at level

Δ
3
0

\Delta _3^0

, goes hand in hand with a certain way in which the Myhill extension fails, at

Δ
3
0

\Delta _3^0

, to commute with composition.
A refinement of Cantor’s theorem
Authors: Kirmayer G.
Published: June 2010

It is shown that there is no surjection from the one-element subsets of a set containing an infinite co-infinite set to the infinite co-infinite subsets of that set. It is also shown that there is no surjection from the one-element subsets of an infinite set to the infinite subsets of that set. The proof can be formalized in a subtheory of both Zermelo Set Theory and New Foundations (and thus makes no use of the Axiom of Choice).
Metrizability and the Fréchet-Urysohn property in topological groups
Authors: Nyikos Peter J.
Published: June 2010

A question of Arhangel’skii, whether weakly first countable topological groups are metrizable, is answered in two ways: if the Hausdorff axiom is assumed, the answer is yes, but in general a weakly first countable topological group need not be pseudometrizable. The former result is obtained as a corollary of a more general sufficient condition for a sequential group to be Fréchet-Urysohn. A general necessary and sufficient condition for a sequential group to be Fréchet-Urysohn is given, and a number of questions are raised. Examples are given to show in what respect the theorems of the paper are the "best possible".
Strong collectionwise normality and M. E. Rudin’s Dowker space
Authors: Hart K. P.
Published: June 2010

stract. We investigate the relationship between strong collectionwise normality and some other separation properties. The conclusion is that in general there is none. In addition some properties of M. E. Rudin’s Dowker space are found.
Which connected metric spaces are compact?
Authors: Beer Gerald
Published: June 2010

A metric space

X
X

is called chainable if for each

ε
>
0

\varepsilon > 0

each two points in

X
X

can be joined

ε
\varepsilon

-chain.

X
X

is called uniformly chainable if for

ε
\varepsilon

there exists an integer

n
n

such that each two points can be joined

ε
\varepsilon

-chain of length at most

n
n

. Theorem. A chainable metric space

X
X

is a continuum if and only if

X
X

is uniformly chainable and there exists

δ
>
0

\delta > 0

such that each closed

δ
\delta

-ball is compact. Using Ramsey’s Theorem a sequential characterization of uniformly chainable metric spaces is obtained, paralleling the one for totally bounded spaces.
Banach bundles and a theorem of J. M. G. Fell
Authors: Seda Anthony Karel
Published: June 2010

We apply a theorem of J. M. G. Fell to construct the regular representation of a topological groupoid.
Countable injective modules are sigma injective
Authors: Megibben Charles
Published: June 2010

In this note we show that a countable injective module is

∑
\sum

-injective and consequently a ring

R
R

is left noetherian if the category of left

R
R

-modules has a countable injective cogenerator. Our proof can be modified to establish the corresponding result for quasi-injective modules. We also give an example of a nonnoetherian commutative ring

R
R

such that the category of

R
R

-modules has a countable cogenerator.
Derivations in prime rings
Authors: Felzenszwalb B.
Published: June 2010

Let

R
R

be a ring and

d
≠
0

d \ne 0

a derivation of

R
R

such that

d
(

x
n

)
=
0

d({x^n}) = 0

,

n
=
n
(
x
)
⩾
1

n = n(x) \geqslant 1

, for all

x
∈
R

x \in R

. It is shown that if

R
R

is primitive then

R
R

is an infinite field of characteristic

p
>
0

p > 0

and

p

|

n
(
x
)

p|n(x)

if

d
(
x
)
≠
0

d(x) \ne 0

. Moreover, if

R
R

is prime and the set of integers

n
(
x
)

n(x)

is bounded, the same conclusion holds.
Subharmonic functions outside a compact set in 𝑅ⁿ
Authors: Anandam Victor
Published: June 2010

Let

u
u

be a subharmonic function defined outside a compact set in

R

2

{{\mathbf {R}}^2}

. Then

u
u

is of the form

u
(
x
)
=
s
(
x
)
−
α
log
⁡

|
x
|

u(x) = s(x) - \alpha \log \left | x \right |

outside a disc where

s
(
x
)

s(x)

is a nonconstant subharmonic function in

R

2

{{\mathbf {R}}^2}

and

α
⩾
0

\alpha \geqslant 0

. Some applications and the analogues in

R

n

{{\mathbf {R}}^n}

,

n
⩾
3

n \geqslant 3

, are given.
Derivations on commutative Banach algebras
Authors: Khosravi A.
Published: June 2010

Let

A
A

be a commutative Banach algebra with radical

R
R

and

D
D

be a derivation on

A
A

. If

K
=
{
x
∈
R
:

for every 

n
⩾
1
,

D
n

x
∈
R
}

K = \{ x \in R:\;{\text {for every }}n \geqslant 1,\;{D^n}x \in R\}

, then

D
A
⊆
R

DA \subseteq R

if and only if

K
K

is closed.
Two UC-sets whose union is not a UC-set
Authors: Fournier John J. F.
Published: June 2010

It is shown that the union of two sets of uniform convergence need not be a set of uniform convergence.
Compactness criteria for Riemannian manifolds
Authors: Galloway Gregory J.
Published: June 2010

Ambrose, Calabi and others have obtained Ricci curvature conditions (weaker than Myers’ condition) which ensure the compactness of a complete Riemannian manifold. Using standard index form techniques we relate the problem of finding such Ricci curvature criteria to that of establishing the conjugacy of the scalar Jacobi equation. Using this relationship we obtain a Ricci curvature condition for compactness which is weaker than that of Ambrose and, in fact, which is best among a certain class of conditions.
An algebraic analogue of a conjecture of G. W. Whitehead
Authors: Miller Haynes
Published: June 2010

Using a theorem of W. M. Singer, an algebraic analogue of a conjecture of G. W. Whitehead is proved, generalizing the algebraic Kahn-Priddy theorem of W. H. Lin.
The cohesive property
Authors: Jungck Gerald
Published: June 2010

We introduce the concept of cohesive families of neighborhood bases. We thereby obtain conditions necessary and sufficient to ensure that a separable space be second countable, and sufficiency conditions for complete collectionwise normality. As by-products we obtain metrizability criteria. We prove, e.g., that a

T
1

{T_1}

space is metrizable iff it has a refined development

{

G
n

:
n
∈
N
}

\{ {G_n}:n \in N\}

such that

{

B
p

:
p
∈
X
}

\{ {B_p}:p \in X\}

with

B
p

=
{

St

(
p
,

G
n

)
:
n
∈
N
}

{B_p} = \{ {\text {St}}(p,{G_n}):n \in N\}

is cohesive.
Strong local homogeneity does not imply countable dense homogeneity
Authors: van Mill Jan
Published: June 2010

We give an example of a connected and locally connected subspace of the plane which is Baire and strongly locally homogeneous (as a consequence, the example is homogeneous) but which is not countable dense homogeneous.
Finite-dimensional right duo algebras are duo
Authors: Courter R. C.
Published: June 2010

Available examples of right (but not left) duo rings include rings without unity element which are two dimensional algebras over finite prime fields. We prove that a right duo ring with unity element which is a finite dimensional algebra over an arbitrary field

K
K

is a duo ring. This result is obtained as a corollary of a theorem on right duo, right artinian rings

R
R

with unity: left duo-ness is equivalent to each right ideal of

R
R

having equal right and left composition lengths, which is equivalent to the same property on

R
R

alone. Another result concerns algebras over a field which are semiprimary right duo rings: such an algebra is left duo provided (1) the algebra is finite dimensional modulo its radical and (2) the square of the radical is zero. These two provisions are shown to be essential by examples which are local algebras, duo on one side only.
Strict local rings
Authors: Herzog J.
Published: June 2010

In this paper we introduce the notion of a strict local ring. A local Cohen-Macaulay ring

(
B
,
m
)

(B,m)

is called strict if, whenever a local ring

(
A
,
n
)

(A,n)

specializes by a regular sequence to

B
B

, then the associated graded ring

g

r

n

(
A
)

{\text {g}}{{\text {r}}_n}(A)

is Cohen-Macaulay. We show that an artinian graded algebra

B
B

is strict if for the graded cotangent module we have

T
1

(
B

/

k
,
B

)
r

=
0

for 

ν
>
−
1

{T^1}{(B/k,B)_r} = 0{\text {for }}\nu > - 1

. Various examples are considered where this condition holds. In particular, with this method we reprove a result of J. Sally [6].
On the divisible part of the Brauer group of a field
Authors: Würfel Tilmann
Published: June 2010

For a field

k
k

and an odd prime

p
≠
char
⁡
(
k
)

p \ne \operatorname {char} (k)

such that the

p
p

-primary component

B

(
k

)

(
p
)

B{(k)_{(p)}}

of the Brauer group

B
(
k
)

B(k)

of

k
k

is not zero there exists a finite extension

k

/

k

k/k

such that

B

(
k

)

(
p
)

B{(k)_{(p)}}

contains a nontrivial divisible subgroup.
Subspace maps of operators on Hilbert space
Authors: Longstaff W. E.
Published: June 2010

An operator

A
A

acting on a Hilbert space

H
H

gives rise to a map

φ
A

{\varphi _A}

on the set of subspaces of

H
H

given by

φ
A

(
M
)
=

A
M

¯

{\varphi _A}(M) = \overline {AM}

, where

’

−

’

{\text {’}} - {\text {’}}

denotes norm closure. This map is called the subspace map of

A
A

. By identifying subspaces with projections in the usual way it is shown that for

A
≠
0

A \ne 0

,

φ
A

{\varphi _A}

is uniformly (respectively, strongly) continuous if and only if the approximate point spectrum of

A
A

does not contain 0. In this case it is proved that

φ
A

{\varphi _A}

preserves the property of being uniformly (respectively, strongly, weakly) closed and its effect on reflexivity is described.
Uniqueness and quasimeasures on the group of integers of a 𝑝-series field
Authors: Wade William R. , Yoneda Kaoru
Published: June 2010

Let

G
G

be the group of integers of a

p
p

-series field and suppose that

S
S

is a character series on

G
G

. If

N
1

{N_1}

,

N
2

,
…

{N_2}, \ldots

is any sequence of integers and if

S

p

N
j

→
0

{S_{{p^{{N_j}}}}} \to 0

a.e. on

G
G

, as

j
→
∞

j \to \infty

, then

S
S

will be the zero series provided

S
S

never diverges unboundedly.
A Radon-Nikodým theorem for natural cones associated with von Neumann algebras
Authors: Kosaki Hideki
Published: June 2010

The natural cone associated with a von Neumann algebra admitting a cyclic and separating vector

ξ
0

{\xi _0}

is considered. For any vector

ξ
\xi

in the cone, there always exists a closed operator

t
t

affiliated with the algebra satisfying

ξ
=
t
J
t
J

ξ
0

\xi = tJtJ{\xi _0}

.
Shape operators of Einstein hypersurfaces in indefinite space forms
Authors: Magid Martin A.
Published: June 2010

The possible shape operators for an Einstein hypersurface in an indefinite space form are classified algebraically. If the shape operator

A
A

is not diagonalizable then either

A
2

=
0

{A^2} = 0

or

A
2

=
−

b
2

Id

{A^2} = - {b^2}{\text {Id}}

.
Totally real minimal immersions of 𝑛-dimensional real space forms into 𝑛-dimensional complex space forms
Authors: Ejiri Norio
Published: June 2010

n
n

-dimensional totally real minimal submanifolds of constant sectional curvature in

n
n

-dimensional complex space forms are totally geodesic or flat.
Glidebending of general caps: an infinitesimal treatment
Authors: Kann Edgar
Published: June 2010

We prove that two

C
3

{C^3}

isometric surfaces

S
′

S’

and

S
S

with (not necessarily planar) boundary, immersed in

E
3

{E^3}

, with

K
>
0

K > 0

, having their spherical images in the same open hemisphere, and which are oriented so that the mean curvatures have the same sign, are congruent if they satisfy the glidebending boundary condition: Let

{
n
:
e
⋅
n
>
0

\{ n:e \cdot n > 0

,

e
=

const

. vector} 

e = {\text {const}}{\text {. vector\} }}

be the open hemisphere. Then

e
⋅

X
′

=
e
⋅
X

e \cdot X’ = e \cdot X

at corresponding boundary points, where

X
′

X’

and

X
X

are position vectors of

S
′

S’

and

S
S

. The method uses the fact that the surface

1
2

(
X
−

X
′

)

\tfrac {1} {2}(X - X’)

is an infinitesimal bending field for the mean surface

1
2

(
X
+

X
′

)

\tfrac {1} {2}(X + X’)

and is elementary in that it uses the rotation vector of classical infinitesimal bending theory but no integral formulas, maximum principles for elliptic operators or index theorems. The surfaces considered need not have a simple projection on a plane, be convex in the large (an example is given) nor be simply connected. We use the method to prove the finite and infinitesimal rigidity of general caps and caps under glidebending.
A characterization of the uniform closure of the set of homeomorphisms of a compact totally disconnected metric space into itself
Authors: Miles Frank B.
Published: June 2010

The limit index

λ
(
x
)

\lambda (x)

of a point

x
x

in a compact metric space is defined. (Roughly: Isolated points have index 0, limit points have index 1, limit points of limit points have index 2, and so forth.) Then the following theorem is proved. Theorem 1. Let

E
E

be a compact, totally disconnected metric space. Then the uniform closure of the set of homeomorphisms of

E
E

into itself is the set

C
λ

{C_\lambda }

of continuous functions

f
f

from

E
E

to

E
E

satisfying (1)

λ
(
x
)
⩽
λ
(
f
(
x
)
)

f
o
r

a
l
l

x
∈
E

\lambda (x) \leqslant \lambda (f(x))\;for\;all\;x \in E

, and (2) if

y
y

is not a condensation point of

E
E

, then

f

−
1

(
y
)

{f^{ - 1}}(y)

contains at most one

x
x

such that

λ
(
x
)
=
λ
(
y
)

\lambda (x) = \lambda (y)

. Further, the set of homeomorphisms of

E
E

into

E
E

is a dense

G
δ

{G_\delta }

subset of the complete metric space

C
λ

{C_\lambda }

.
On finitely dominated CW complexes
Authors: Dydak Jerzy
Published: June 2010

Let

D

\mathcal {D}

be the class of all CW complexes homotopy dominated by finite CW complexes. In this paper we prove the following theorem. Theorem. Suppose a connected CW complex

X
∈

D

X \in \mathcal {D}

is a union of two connected subcomplexes

X
1

{X_1}

,

X
2

{X_2}

with

X
1

∩

X
2

=

X
0

∈

D

{X_1} \cap {X_2} = {X_0} \in \mathcal {D}

. Then

X
1

{X_1}

,

X
2

∈

D

{X_2} \in \mathcal {D}

if one of the following conditions is satisfied: (i)

π
1

(

X
0

,
x
)
→

π
1

(
X
,
x
)

{\pi _1}({X_0},x) \to {\pi _1}(X,x)

is a monomorphism for each

x
∈

X
0

x \in {X_0}

, (ii)

π
1

(

X
i

)
→

π
1

(
X
)

{\pi _1}({X_i}) \to {\pi _1}(X)

is a monomorphism for

i
=
1
,
2

i = 1,2

and

π
1

(

X
1

)

{\pi _1}({X_1})

,

π
1

(

X
2

)

{\pi _1}({X_2})

are finitely presented.
Is every GO-topology a join of two orderable topologies?
Authors: Meyer Paul R. , Neumann Lara Victor , Wilson Richard G.
Published: June 2010

As a generalization of

GO

{\text {GO}}

-topologies (

GO = generalized ordered

{\text {GO = generalized ordered}}

), we are interested in those topologies (here called

JO

{\text {JO}}

-topologies) on a set

X
X

which can be expressed as a join of orderable topologies (the join being taken in the lattice of all topologies on

X
X

). If a topology

t
t

is the join of

m
m

orderable topologies we say

t
t

is

JO

[
m
]

{\text {JO}}[m]

. It is not difficult to prove that every

GO

{\text {GO}}

-topology is a

JO

{\text {JO}}

-topology, but the question (raised in [M 71]) as to whether every

GO

{\text {GO}}

-topology is

JO

[
2
]

{\text {JO}}[2]

seems much more difficult). We show that

X
X

is

JO[2]

{\text {JO[2]}}

if

X
X

is a subspace of an orderable space

D
D

, where

D
D

is either metrizable and locally separable, or connected with countable cellularity. (The theorem is actually more general than is stated here.) We give an example to show that for any positive integer

n
n

there is a finite join of order topologies which is not

JO

[
n
]

{\text {JO}}[n]

, but these are not

GO

{\text {GO}}

-topologies.
Strongly reversible groups
Authors: Tamura Takayuki
Published: June 2010

Following Thierrin [9], a group

G
G

is called strongly reversible if for every

x
x

,

y
∈
G

y \in G

there are positive integers

l
l

,

m
m

,

n
n

such that

(
x
y

)
l

=

x
m

y
n

=

y
n

x
m

{(xy)^l} = {x^m}{y^n} = {y^n}{x^m}

. This paper studies the structure of strongly reversible groups.
Extension of a result of Beachy and Blair
Authors: Desale G. B. , Varadarajan K.
Published: June 2010

Let

α
:
R
→
R

\alpha :R \to R

be an automorphism of a ring

R
R

. If

R
R

is an

α
\alpha

-reduced ring in which every faithful left ideal is cofaithful, then the same is true for the

α
\alpha

-twisted power series ring

R
∗

[
[
x
]
]

{R^* }[[x]]

.
Spectral properties of compact lattice homomorphisms
Authors: Wickstead Anthony W.
Published: June 2010

Given two nonzero eigenvalues of a lattice homomorphism on a relatively uniformly complete vector lattice, of different moduli and with at least one isolated in the set of all eigenvalues, we show that corresponding eigenvectors must be disjoint. The analogous result for the approximate point spectrum of a lattice homomorphism on a Banach lattice is deduced. We give an infinite spectral decomposition for a lattice homomorphism, on a Banach lattice with order continuous norm, which is compact and has an adjoint which is also a lattice homomorphism. From this we deduce that if it has nonnegative spectrum, then it is the direct sum of a nilpotent lattice homomorphism and one that is central.
A generalization of the Ahlfors-Schwarz lemma
Authors: Wolpert Scott
Published: June 2010

Consider, for a compact surface, two conformal metrics

d

s
2

d{s^2}

and

d

σ
2

d{\sigma ^2}

of negative Gauss curvature. Assume the curvatures

K
(
d

s
2

)

K(d{s^2})

and

K
(
d

σ
2

)

K(d{\sigma ^2})

satisfy

K
(
d

s
2

)

K(d{s^2})

. It is concluded that

d

s
2

⩽
d

σ
2

d{s^2} \leqslant d{\sigma ^2}

. In particular if the curvature is pinched,

−

c
1

⩽
K
(
d

s
2

)
⩽
−

c
2

>
0

- {c_1} \leqslant K(d{s^2}) \leqslant - {c_2} > 0

, then the inequality

c
1

−
1

/

2

d

γ
2

⩽
d

s
2

⩽

c
2

−
1

/

2

d

γ
2

{c_1}^{ - 1/2}d{\gamma ^2} \leqslant d{s^2} \leqslant {c_2}^{ - 1/2}d{\gamma ^2}

follows for

d

γ
2

d{\gamma ^2}

the constant curvature

−
1

-1

metric.
La pathologie des relèvements invariants
Authors: Talagrand Michel
Published: June 2010

Sur un groupe abélien localement compact

G
G

, il existe une fonction mesurable

f
f

telle que pour tout relèvement

ρ
\rho

de

L
∞

(
G
)

{L^\infty }(G)

invariant par translation, la fonction

ρ
(
f
)

\rho (f)

ne soit pas universellement mesurable, et ne possède pas la propriété de Baire forte.
The structure of a critical set of a complete intersection singularity
Authors: Yomdin Yosef
Published: June 2010

Let

f
:
(

C
n

,
0
)
→
(

C
k

,
0
)

f:({C^n},0) \to ({C^k},0)

be the germ of an isolated complete intersection singularity. The structure of strata

μ
=
const

\mu = \operatorname {const}

in a critical set of

f
f

is studied. The main result is the following: if the dimension of the stratum

μ
=
μ
(
0
)

\mu = \mu (0)

is

k
−
1

k - 1

, then

f
f

coincides with a family of hypersurface singularities with constant Milnor number.
A note on ideals in the disc algebra
Authors: Forelli Frank
Published: June 2010

We offer an elementary theorem on ideals in the disc algebra

A
(

D

)

A({\mathbf {D}})

, which by way of a corollary, one, identifies the maximal ideals of

A
(

D

)

A({\mathbf {D}})

, and two, provides a proof, which avoids the axiom of choice, that every proper ideal in

A
(

D

)

A({\mathbf {D}})

is contained in a maximal ideal.
On Jordan representations of unbounded operator algebras
Authors: Bhatt Subhash J.
Published: June 2010

Every closed Jordan

∗

^*

-representation of an

E

C
∗

E{C^*}

-algebra is the sum of a closed

∗

^*

-representation and a closed

∗

^*

-antirepresentation.
The symmetry group of a curve preserves a plane
Authors: Gulliver Robert , Morgan Frank
Published: June 2010

It is shown that the symmetries of a closed curve without self-intersections in euclidean or hyperbolic

n
n

-space have an invariant plane in common. This allows a complete characterization of the symmetry group.
Fixed points of nonexpansive condensing multivalued mappings on metric spaces
Authors: Yen Ch’i Lin
Published: June 2010

In this paper we consider the existence of fixed points of nonexpansive condensing multi-valued mappings from a certain kind of metric space into itself; the spaces, here, are neither linear nor compact. Our result generalizes a theorem of Dotson and also a theorem of Bose and Mukherjee in some respect.
Compact extensions of compactly generated nilpotent groups are pro-Lie
Authors: Hofmann Karl H. , Liukkonen John R. , Mislove Michael W.
Published: June 2010

We show that compact extensions of compactly generated (locally compact) pro-Lie groups are pro-Lie, and that compactly generated (locally compact) nilpotent groups are pro-Lie. As a consequence compact extensions of compactly generated nilpotents are pro-Lie. We give examples indicating limitations to extending our results.
The cotypeset of a torsion free abelian group of rank two
Authors: Vinsonhaler C. , Wickless W. J.
Published: June 2010

The cotypeset (set of types of rank one factors) of a torsion free abelian group of rank two is characterized.
Spherical harmonics generating bounded biharmonics
Authors: Beaver Bradley , Sario Leo , Wang Cecilia
Published: June 2010

Let

H
2

B
(
R
)

{H^2}B(R)

be the family of bounded nonharmonic biharmonic functions on a Riemannian manifold

R
R

. On the punctured Euclidean

N
N

-space
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E 0 Superscript upper N Baseline equals StartSet x equals left-parenthesis x Superscript 1 Baseline comma ellipsis comma x Superscript upper N Baseline right-parenthesis StartAbsoluteValue 0 greater-than EndAbsoluteValue x vertical-bar greater-than normal infinity EndSet">

E
0
N

=
{
x
=
(

x
1

,
…
,

x
N

)

|

0
>

|

x

|

>
∞
}

E_0^N = \{ x = ({x^1}, \ldots ,{x^N})|0 > |x| > \infty \}

,

H
2

B

{H^2}B

is void for

N
>
3

N > 3

, whereas for

N
=
2
,
3

N = 2,3

, it is generated by certain fundamental spherical harmonics. It is also known that

H
2

B

{H^2}B

remains void on the Riemannian manifold

E
α
N

E_\alpha ^N

,

N
>
3

N > 3

, obtained by endowing

E
0
N

E_0^N

with the non-Euclidean metric

d

s
α

=

r
α

|

d
x

|

d{s_\alpha } = {r^\alpha }|dx|

,

α
∈
R

\alpha \in R

. The purpose of the present paper is to show that the fundamental spherical harmonics continue generating

H
2

B
(

E
α
3

)

{H^2}B(E_\alpha ^3)

, despite the distorting metric

d

s
α

d{s_\alpha }

. An analogous result holds for

E
α
2

E_\alpha ^2

.
On a characterization of invariant subspace lattices of weighted shifts
Authors: Yadav B. S. , Chatterjee S.
Published: June 2010

The paper concerns itself with the characterization of invariant subspace lattices of weighted shift operators on the Hilbert space

l
2

{l^2}

with suitable conditions on their weights. This characterization is also extended to the case of Banach spaces

l
p

{l^p}

,

1
>
p
>
∞

1 > p > \infty

.
Absolute total-effectiveness of a total effective (𝑁,𝑝_{𝑛}) method
Authors: Kumar Arun
Published: June 2010

In the present paper we investigate absolute total effectiveness of a

(
N
,

p
n

)

(N,{p_n})

method. It is established that a method known to be total effective is also absolute total effective.
Lie and Jordan ideals of operators on Hilbert space
Authors: Fong C. K. , Miers C. R. , Sourour A. R.
Published: June 2010

A linear manifold

L

\mathfrak {L}

in

B

(

H

)

\mathfrak {B}(\mathfrak {H})

is a Lie ideal in

B

(

H

)

\mathfrak {B}(\mathfrak {H})

if and only if there is an associative ideal

J

\mathfrak {J}

such that

[

J

,

B

(

H

)
]
⊆

L

⊆

J

+

C

I

[\mathfrak {J},\mathfrak {B}(\mathfrak {H})] \subseteq \mathfrak {L} \subseteq \mathfrak {J} + {\mathbf {C}}I

. Also

L

\mathfrak {L}

is a Lie ideal if and only if it contains the unitary orbit of every operator in it. On the other hand, a subset of

B

(

H

)

\mathfrak {B}(\mathfrak {H})

is a Jordan ideal if and only if it is an associative ideal.
Renorming dual Banach lattices
Authors: Ghoussoub N.
Published: June 2010

We show that a Banach lattice is order continuous as well as its dual if and only if it admits an equivalent Fréchet differentiable and locally uniformly convex lattice norm such that its dual norm is also locally uniformly convex.
A simple condition ensuring the Arens regularity of bilinear mappings
Authors: Arikan Nilgün
Published: June 2010

We give a simple criterion for certain Banach algebras to be Arens regular, which applies in particular to the algebras

l
1

{l^1}

with pointwise multiplication,

L
∞

(
G
)

{L^\infty }(G)

, where

G
G

is a compact group with convolution, and the trace-class algebra. This criterion is best established in the more general context of the regularity of bilinear maps, and depends on the existence of extensions of such maps.
Ultrafilter limits and finitely additive probability
Authors: Sibley Thomas Q.
Published: June 2010

Ultrafilter limits provide the natural convergence notion for finitely additive probability. The finitely additive infinitely divisible laws are closed under ultrafilter limits. The characteristic function of any convolution of finitely additive probability measures is the product of their characteristic functions.
The nonhomogeneity of the 𝐸-tree—answer to a problem raised by D. Jensen and A. Ehrenfeucht
Authors: Misercque Didier
Published: June 2010

We prove that the ordered system of all

C
1

E

P
′

s

{C^1}EP’{\text {s}}

, under the order "admits embedding in" is not homogeneous. This answers a problem raised in [2].
Inner points and breadth in certain compact semilattices
Authors: Brown D. R. , Stepp J. W.
Published: June 2010

A point

x
∈
X

x \in X

is inner if there exists an open set

U
U

containing

x
x

such that for each open set

V
V

with

x
∈
V
⊆
U

x \in V \subseteq U

, the inclusion homomorphism

i
∗

:

{i^* }:

:

H
∗

(
X
,
X
∖
V
)
→

H
∗

(
X
,
X
∖
U
)

{H^*}(X,X \setminus V) \to {H^*}(X,X \setminus U)

is nontrivial. In this note it is proved that, if

X
X

is a compact, chainwise connected topological semilattice of codimension

n
n

, and

x
x

is a point of breadth

n
+
1

n + 1

, then

x
x

is an inner point.
On the covering dimension of inverse limits
Authors: Katuta Yûkiti
Published: June 2010

We give sufficient conditions for an inverse system of spaces of covering dimension

⩽
n

\leqslant n

to have inverse limit of covering dimension

⩽
n

\leqslant n

.
Cardinal functions on modifications of uniform spaces and fine uniform spaces
Authors: Ku̇rková Věra
Published: June 2010

The paper studies the question for which modifications

r
r

of Unif the following theorem can be generalized by substituting a precompact modification

p
p

by

r
r

: A uniform space has the finest uniformity inducing its proximity if and only if each proximally continuous mapping from this space to any other uniform space is uniformly continuous. By means of two cardinal functions defined on the class of all modifications of Unif there is shown that this is possible only for cardinal modifications

p
α

{p^\alpha }

. Assuming GCH, the problem for cardinal modifications

p
α

{p^\alpha }

is solved for uniform spaces of a limited point-character (in dependence on

α
\alpha

).
A duality principle
Authors: Sander Wolfgang
Published: June 2010

With the aid of the Baire category theory we prove an extension of Erdös’ well-known duality principle concerning sets of Lebesgue measure zero and sets of first category.
On holomorphic functions satisfying 𝑓(𝑧)(1-𝑧²)≤1 in the unit disc
Authors: Wirths Karl-Joachim
Published: June 2010

Let

f
f

be holomorphic in

D
=
{

z
|

|
z
|

>
1
}

D = \{ \left . z \right |\left | z \right | > 1\}

,

|

f
(
z
)

|

(
1
−

|
z
|

2

)
⩽
1

\left | {f(z)} \right |(1 - {\left | z \right |^2}) \leqslant 1

in

D
D

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Overscript bar Endscripts Underscript StartAbsoluteValue z EndAbsoluteValue right-arrow 1 Endscripts StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue left-parenthesis 1 minus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis greater-than 1">

lim
¯

|
z
|

→
1

|

f
(
z
)

|

(
1
−

|
z
|

2

)
>
1

{\overline {\lim } _{\left | z \right | \to 1}}\left | {f(z)} \right |(1 - {\left | z \right |^2}) > 1

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L left-parenthesis f right-parenthesis colon equals left-brace z vertical-bar StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue left-parenthesis 1 minus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis equals 1 right-brace">

L
(
f
)
:=
{

z
|

|

f
(
z
)

|

(
1
−

|
z
|

2

)
=
1
}

L(f): = \{ \left . z \right |\left | {f(z)} \right |(1 - {\left | z \right |^2}) = 1\}

. It is shown that the set

L
(
f
)

L(f)

consists of one simple closed curve

γ
\gamma

and a finite number of points in the bounded component of

C

∖
γ

{\mathbf {C}}\backslash \gamma

if

L
(
f
)

L(f)

is an infinite set.
Centers and nearest points of sets
Authors: Szeptycki P. , Van Vleck F. S.
Published: June 2010

For a Banach space

X
X

and a subset

A
A

of

X
X

,

c
A

{c_A}

denotes the Čebyšev center of

A
A

and

P
A

x

{P_A}x

denotes the nearest point in

A
A

to the point

x
x

in

X
X

. The space of all subsets of

X
X

is furnished with the Hausdorff metric. The modulus of continuity of the function

A
→

c
A

A \to {c_A}

is computed in the case when

X
X

is a Hilbert space and the sets

A
A

are compact; the same is done for the function

A
→

P
A

x

A \to {P_A}x

, for fixed

x
x

, in the case when

X
X

is uniformly convex and the sets

A
A

are convex and closed.
A fixed point theorem for the sum of two mappings
Authors: Hadžić Olga
Published: June 2010

A generalization of a fixed point theorem of Rzepecki is proved and it is shown that in a paranormed space

E
E

this result yields, under certain circumstances, solutions to the equation

x
=
T
x
+
S
x

x = Tx + Sx

for

T
:
E
→
E

T:E \to E

either continuous and affine or a generalized contraction, and

S
:
K
⊆
E
→
E

S:K \subseteq E \to E

compact.
Some expansion formulas for a class of singular partial differential equations
Authors: Altin Abdullah
Published: June 2010

We obtain a generalization of the Almansi’s expansion and the Lord Kelvin principle for solutions of a class of iterated elliptic or ultrahyperbolic equations. We also obtain some homogeneous function expansions for solutions of the same equations.
Sur le volume des zéros des fonctions holomorphes et bornées dans la boule de 𝐶ⁿ
Authors: Amar Éric
Published: June 2010

We give an example of a zero set of a holomorphic bounded function in the unit ball of

C

n

{{\mathbf {C}}^n}

,

n
⩾
2

n \geqslant 2

, with infinite area. This generalizes a previous work of W. Rudin done in the case

n
=
2

n = 2

.
Peak sets for the real part of a function algebra
Authors: Briem Eggert
Published: June 2010

We show that if

A
A

is a function algebra with the property that every peak set for re

A
A

is an interpolation set for

A
A

then

A
=
C
(
X
)

A = C(X)

.
Lie products, closed geodesics and Fuchsian groups
Authors: Beardon Alan F.
Published: June 2010

Two results concerning closed geodesics on a Riemann surface are proved. The method of proof indirectly involves the Lie product of two complex matrices.
A Plücker equation for curves in real projective space
Authors: Quine J. R.
Published: June 2010

For smooth closed curves in real projective space we write an equation relating the Plücker characteristics, various winding numbers, and a new characteristic involving pairs of points on the line at infinity. This theorem is a generalization of the Umlaufsatz and also relates directly to Plücker’s equations for algebraic curves.
A formula for deviation from commutativity: the transfer and Steenrod squares
Authors: Kubelka Richard P.
Published: June 2010

The ordinary cohomology transfer associated to the orbit space projection of a finite group action need not commute with stable cohomology operations. In particular, if an even group acts on a space, the resulting transfer

τ
\tau

will not generally commute with the Steenrod squares,

S

q

i

{\text {S}}{{\text {q}}^i}

. This paper contains a formula for the deviation from commutativity

(

S

q

i

τ
−
τ

S

q

i

)
x

({\text {S}}{{\text {q}}^i}\tau - \tau {\text {S}}{{\text {q}}^i})x

in the case of an involution. The formula involves the restriction of

x
x

to the cohomology of the fixed point set, as well as certain naturally occurring characteristic classes.
A new proof of a theorem of Solomon-Tits
Authors: Curtis C. W. , Lehrer G. I.
Published: June 2010

Let

Δ
\Delta

be the combinatorial building of a finite group of Lie type

G
G

. A new proof is given of the theorem of Solomon-Tits on the

G
G

-module structure of the rational homology

H
∗

(
Δ
)

{H_* }(\Delta )

of

Δ
\Delta

.
Combinatorial equivalence between group presentations
Authors: Jajodia Sushil
Published: June 2010

Let

P

=
(

x
1

,
…
,

x
n

:

W
1

,
…
,

W
m

)

\mathcal {P} = ({x_1}, \ldots ,{x_n}:{W_1}, \ldots ,{W_m})

and

R

=
(

x
1

,
…
,

x
n

:

R
1

,
…
,

R
m

)

\mathcal {R} = ({x_1}, \ldots ,{x_n}:{R_1}, \ldots ,{R_m})

be two presentations, with the same generators, for a group

π
\pi

. In this note, we give a necessary and sufficient criterion which insures the existence of a combinatorial equivalence between

P

\mathcal {P}

and

R

\mathcal {R}

requiring only replacement operations.
Rational nonlinear characters of metabelian groups
Authors: Basmaji B. G.
Published: June 2010

Let

G
G

be a finite metabelian group with all nonlinear irreducible characters rational. Then the exponent of the commutator group

G
′

G’

is a prime or divides 16, 24, or 40. If

G
′

G’

is also cyclic, then its order is a prime or divides 12.
On functions in the ball algebra
Authors: Wojtaszczyk P.
Published: June 2010

We show that there exists a function in a ball algebra such that almost every slice function has a series of Taylor coefficients divergent with every power

p
>
2

p > 2

.
Nomographic functions are nowhere dense
Authors: Buck R. Creighton
Published: June 2010

A function

f
f

of

n
n

variables is nomographic if it can be represented in the format
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis equals h left-parenthesis phi 1 left-parenthesis x 1 right-parenthesis plus midline-horizontal-ellipsis plus phi Subscript n Baseline left-parenthesis x Subscript n Baseline right-parenthesis right-parenthesis">

f
(

x
1

,
…
,

x
n

)
=
h
(

ϕ
1

(

x
1

)
+
⋯
+

ϕ
n

(

x
n

)
)

f({x_1}, \ldots ,{x_n}) = h({\phi _1}({x_1}) + \cdots + {\phi _n}({x_n}))

\]
where the

ϕ
i

{\phi _i}

and

h
h

are continuous. Every continuous function of

n
n

variables has a representation as a sum of not more than

2
n
+
1

2n + 1

nomographic functions [textbf9]. This paser gives a constructive proof that the nomographic functions form a nowhere dense subset of the space

C
[

I
n

]

C[{I^n}]

.
Group theoretic remarks on Riesz systems on balls
Authors: Korányi A. , Vági S.
Published: June 2010

The space of boundary values of Riesz systems on the unit ball of

R

n

{{\mathbf {R}}^n}

is decomposed under the action of the special orthogonal group

S
O
(
n
)

SO(n)

. The corresponding irreducible representations are explicitly determined. Two applications of this result are given.