bluuebunny/crossref_metadata_2025 huggingface dataset:
Note on nilpotent derivations
Authors: Lee P. H. , Lee T. K.
Published: July 2010

Let

R
R

be a prime ring with center

Z
Z

. Suppose that

d
d

is a derivation on

R
R

such that

d
n

(
x
)
∈
Z

{d^n}(x) \in Z

for all

x
x

, where

n
n

is a fixed integer. It is shown that either

d
n

(
x
)
=
0

{d^n}(x) = 0

for all

x
∈
R

x \in R

or

R
R

is a commutative integral domain. Moreover, the same conclusion holds even if we assume that

d
n

(
x
)
∈
Z

{d^n}(x) \in Z

merely for all

x
∈
I

x \in I

, where

I
I

is a nonzero ideal of

R
R

.
Class groups, totally positive units, and squares
Authors: Edgar H. M. , Mollin R. A. , Peterson B. L.
Published: July 2010

Given a totally real algebraic number field

K
K

, we investigate when totally positive units,

U
K
+

U_K^ +

, are squares,

U
K
2

U_K^2

. In particular, we prove that the rank of

U
K
+

/

U
K
2

U_K^ + /U_K^2

is bounded above by the minimum of (1) the

2
2

-rank of the narrow class group of

K
K

and (2) the rank of

U
L
+

/

U
L
2

U_L^ + /U_L^2

as

L
L

ranges over all (finite) totally real extension fields of

K
K

. Several applications are also provided.
A note on a theorem of Ahern and Sarason
Authors: Dudziak James
Published: July 2010

A strong weak* maximality theorem due to Ahern and Sarason is simply deduced from a more easily obtained and weaker maximality theorem of Garnett.
Orthogonality and the Hausdorff dimension of the maximal measure
Authors: Lopes Artur Oscar
Published: July 2010

In this paper the orthogonality properties of iterated polynomials are shown to remain valid in some cases for rational maps. Using a functional equation fulfilled by the generating function, the author shows that the Hausdorff dimension of the maximal measure is a real analytical function of the coefficients of an Axiom A rational map satisfying the property that all poles of

f
f

and zeros of

f
′

(
z
)

f’(z)

have multiplicity one.
Uniqueness for singular backward parabolic inequalities
Authors: Lair Alan V.
Published: July 2010

We prove that the only solution of the partial differential inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis u Subscript t Baseline plus sigma-summation Underscript i equals 1 Overscript n Endscripts left-bracket u Subscript x Sub Subscript i Subscript Baseline Subscript x Sub Subscript i Subscript Baseline plus left-parenthesis StartFraction k Subscript i Baseline Over x Subscript i Baseline EndFraction right-parenthesis u Subscript x Sub Subscript i Subscript Baseline right-bracket right-parenthesis squared less-than-or-slanted-equals c left-bracket u squared plus StartAbsoluteValue u Subscript x Baseline EndAbsoluteValue squared right-bracket">

(

u
t

+

∑

i
=
1

n

[

u

x
i

x
i

+

(

k
i

x
i

)

u

x
i

]

)

2

⩽
c

[

u
2

+

|

u
x

|

2

]

{\left ( {{u_t} + \sum \limits _{i = 1}^n {\left [ {{u_{{x_i}}}_{{x_i}} + \left ( {\frac {{{k_i}}}{{{x_i}}}} \right ){u_{{x_i}}}} \right ]} } \right )^2} \leqslant c\left [ {{u^2} + |{u_x}{|^2}} \right ]

\]
on a bounded region with homogeneous initial and boundary conditions is the trivial one.
Uniform ergodic theorems for locally integrable semigroups and pseudoresolvents
Authors: Shaw Sen-Yen
Published: July 2010

We study uniform ergodicity (at

∞
\infty

) of a locally integrable operator semigroup

T
(
⋅
)

T( \cdot )

of type

w
0

{w_0}

under a suitable condition which is weaker than the usual one

w
0

⩽
0

{w_0} \leqslant 0

. We also give a precise characterization of the uniform Cesàro-ergodicity for semigroups of class

(
0
,
A
)

(0,A)

. To prove the part of Abel-ergodicity we first prove a general uniform ergodic theorem for pseudo-resolvents.
A note on two summability methods
Authors: Bor Hüseyin
Published: July 2010

In this paper we have established a relation between the

|

N
,

p
n

|

k

|N,{p_n}{|_k}

and

|

C
,
1

|

k

|C,1{|_k}

summability methods, which is the converse of the result of Bor [2].
The eigenfunctions of compact weighted endomorphisms of 𝐶(𝑋)
Authors: Uhlig Harald
Published: July 2010

In this note we characterize the eigenmanifolds of compact operators

u

C
Φ

:
f
→
u
⋅
f
∘
Φ

u{C_\Phi }:f \to u \cdot f \circ \Phi

on

C
(
X
)

C(X)

and determine their ascents. As an application we show an easy method for computing the eigenmanifolds of a matrix with at most one nonzero element in each row.
On a question of N. Salinas
Authors: Chō Muneo
Published: July 2010

In [5], Salinas asked the following question: If

T
=
(

T
1

,
…
,

T
n

)

T = ({T_1}, \ldots ,{T_n})

consists of commuting hyponormal operators, is it true that (1)

δ
(
T
−
λ
)
=
d
(
λ
,

σ
π

(
T
)
)

\delta (T - \lambda ) = d(\lambda ,{\sigma _\pi }(T))

and (2)

r
π

(
T
)
=

|

|

D
T

|

|

{r_\pi }(T) = ||{D_T}||

? He proved that, for a doubly commuting

n
n

-tuple of quasinormal operators, (2) was true and (1) was true for

λ
=
0

\lambda = 0

. In this paper we shall show that (2) holds for a doubly commuting

n
n

-tuple of hyponormal operators and give an example of a subnormal operator which does not satisfy (1).
Some maximum principles in semilinear elliptic equations
Authors: Schaefer Philip W.
Published: July 2010

We develop maximum principles for functions defined on the solutions to a class of semilinear, second order, uniformly elliptic partial differential equations. These principles are related to recent theorems of Protter and Protter and Weinberger and to a technique initiated by Payne for the determination of gradient bounds on the solution of the equation.
Compact Lorentzian manifolds without closed nonspacelike geodesics
Authors: Galloway Gregory J.
Published: July 2010

We prove that every compact two-dimensional Lorentzian manifold contains a closed timelike or null geodesic. We then construct a two-dimensional example without any closed timelike geodesies and a three-dimensional example without any closed timelike or null geodesics.
Precomplete equivalence relations in dominical categories
Authors: Bernardi Claudio
Published: July 2010

In this paper precomplete equivalence relations are introduced in dominical and recursion categories. After recalling definitions in

§
2

\S 2

, in

§
3

\S 3

an analogue of Eršov’s Fixed Point Theorem is proved, as well as the recursive inseparability of two disjoint (unions of) equivalence classes. In

§
4

\S 4

"morphisms" between equivalence relations are discussed, while

§
5

\S 5

is devoted to the construction of an example (a new characterization of precomplete equivalence relations is also supplied in Lemma 5).
On the generic nonexistence of first integrals
Authors: Hurley Mike
Published: July 2010

The property of having no

C
n

{C^n}

first integrals other than constants is shown to be generic in

Diff
r

(
M
)

{\operatorname {Diff} ^r}(M)

for each

r
=
1
,
2
,
…

r = 1,2, \ldots

, where

n
n

is the dimension of

M
M

.
A local strong 𝑈𝑉^{∞}-property of the homeomorphism groups of 𝑅^{∞}- (𝑄^{∞}-) manifolds
Authors: Liem Vo Thanh
Published: July 2010

We will show in this note that if

M
M

is an

R
∞

 - (or 

Q
∞

 - )

{R^\infty }{\text { - (or }}{Q^\infty }{\text { - )}}

manifold having the homotopy type of a finite complex, then the homeomorphism group of

M
M

, endowed with the compact-open topology, has the local strong

U

V
∞

U{V^\infty }

-property with respect to the classes of pseudo CW complexes

C

W

(

C

)

\mathcal {C}\mathcal {W}(\mathcal {C})

and

C

W

(

M

)

\mathcal {C}\mathcal {W}(\mathcal {M})

.
A group-theoretic property of the Euclidean metric
Authors: Williamson Robert , Janos Ludvik
Published: July 2010

Let

d
d

denote a metric on

R

n

(
n
∈

N

)

{{\mathbf {R}}^n}(n \in {\mathbf {N}})

compatible with its Euclidean toplogy, and let

I
(
d
)

I(d)

be the group of isometries on

R

n

{{\mathbf {R}}^n}

relative to this metric. We show that whenever

I
(
d
)

I(d)

includes the group of motions then

I
(
d
)

I(d)

is identical with it.
Steenrod homology and local connectedness
Authors: Dydak Jerzy
Published: July 2010

Steenrod homology is used to explain results concerning

L

C

n

{\text {L}}{{\text {C}}^n}

-divisors and one-point compactifications of

L

C

n

{\text {L}}{{\text {C}}^n}

-spaces. It is shown that the one-point compactification

w
X

wX

of a locally compact metrizable space

X
X

is

hl

c

n

{\text {hl}}{{\text {c}}^n}

iff

X
X

is

hl

c

n

{\text {hl}}{{\text {c}}^n}

and its Steenrod

k
k

th homology is finite generated for

k
⩽
n

k \leqslant n

.
The set of balanced orbits of maps of 𝑆¹ and 𝑆³ actions
Authors: Jaworowski Jan
Published: July 2010

Suppose that the group

G
=

S
1

G = {S^1}

or

G
=

S
3

G = {S^3}

acts freely on a space

X
X

and on a representation space

V
V

for

G
G

. Let

f
:
X
→
V

f:X \to V

. The paper studies the size of the subset of

X
X

consisting of orbits over which the average of

f
f

is zero. The result can be viewed as an extension of the Borsuk-Ulam theorem.
Numerical invariants for semiconjugacy of homeomorphisms of the circle
Authors: Matsumoto Shigenori
Published: July 2010

We give numerical invariants for deciding when two actions of a given group on the circle are semiconjugate. We give conditions for vanishing of an invariant called bounded real Euler class. We also determine endomorphisms of the discrete group

PSL

(
2
,

R

)

{\text {PSL}}(2,{\mathbf {R}})

.
A remark on companionship and property P
Authors: Scharlemann Martin
Published: July 2010

If some nontrivial Dehn surgery on

γ
\gamma

yields a homotopy

3
3

-sphere and

κ
\kappa

is a companion of

γ
\gamma

, then some nontrivial Dehn surgery on

κ
\kappa

yields a homotopy

3
3

-sphere.
Multiple disjointness for weakly mixing regular minimal flows
Authors: McMahon Douglas
Published: July 2010

We show that pairwise disjointness implies multiple disjointness for metric, weakly mixing regular minimal flows with an abelian phase group. A result on the disjointness of graphic minimal flows is also included.
Prabir Roy’s space Δ as a counterexample in shape theory
Authors: Alonso Moron M.
Published: July 2010

In this note we use the space

Δ
\Delta

in order to prove that a result concerning movability and mutational retractions cannot be transferred from the compact to the arbitrary metrizable case.
On the 𝑝-adic heights of some abelian varieties
Authors: Imai Hideo
Published: July 2010

For an abelian variety defined over an algebraic number field, different definitions of

p
p

-adic heights have been given by several authors. In this note, we shall prove that the

p
p

-adic height defined by A. Néron and that by P. Schneider coincide.
Two-generator groups with perfect Frattini subgroups
Authors: Evans M. J.
Published: July 2010

In this paper,

2
2

-generator groups with perfect Frattini subgroups are constructed. The construction also shows that every countable group can be embedded in the Frattini subgroup of a

2
2

-generator group.
Isolation theorem for products of linear forms
Authors: Cusick T. W.
Published: July 2010

A theorem of Cassels and Swinnerton-Dyer about products of three linear forms with real coefficients is generalized to products of any number of linear forms.
An inequality for eigenvalues of Sturm-Liouville problems
Authors: Bandle Catherine , Philippin Gérard
Published: July 2010

By means of simple transformations, inequalities for eigenvalues corresponding to different boundary conditions are derived.
A remark on 𝐴₁-weights for the strong maximal function
Authors: Soria Fernando
Published: July 2010

If

g
g

is a locally integrable function and

M
M

is the Hardy-Littlewood maximal function then

(
M
g

)
δ

{(Mg)^\delta }

represents an

A
1

{A_1}

-weight of

M
M

for every

0
>
δ
>
1

0 > \delta > 1

; that is,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M left-parenthesis upper M g right-parenthesis Superscript delta Baseline left-parenthesis x right-parenthesis less-than-or-equal-to upper C Subscript delta Baseline left-parenthesis upper M g left-parenthesis x right-parenthesis right-parenthesis Superscript delta">

M

(
M
g

)
δ

(
x
)
≤

C
δ

(
M
g
(
x
)

)
δ

M{(Mg)^\delta }(x) \leq {C_\delta }{(Mg(x))^\delta }

a.e. In this paper we show that this result does not hold in general if we replace

M
M

by the strong maximal operator.
Compact composition operators on spaces of boundary-regular holomorphic functions
Authors: Shapiro Joel H.
Published: July 2010

We consider holomorphic functionsϕ\phitaking the unit discUUinto itself, and Banach spacesXXconsisting of functions holomorphic inUUand continuous on its closure; and show that under some natural hypotheses onXX:ifϕ\phiinduces a compact composition operator onXX,thenϕ(U)\phi (U)must be a relatively compact subset ofUU. SpacesXXwhich satisfy the hypotheses of this theorem include the disc algebra, "heavily" weighted Dirichlet spaces, spaces of holomorphic Lipschitz functions, and the space of functions with derivative in a Hardy spaceHp(p≥1){H^p}(p \geq 1). It is well known that the theorem isnottrue for "large" spaces such as the Hardy and Bergman spaces. Surprisingly, it also fails in "very small spaces," such as the Hilbert space of holomorphic functionsf(z)=∑anznf(z) = \sum {{a_n}{z^n}}determined by the condition∑|an|2exp⁡(n)>∞\sum {|{a_n}{|^2}\exp \left ( {\sqrt n } \right ) > \infty }. The property of Möbiusinvariance plays a crucial and mysterious role in these matters.
An elementary proof of the Grothendieck inequality
Authors: Blei Ron C.
Published: July 2010

An elementary proof of the Grothendieck inequality is given.
Stability property of Möbius mappings
Authors: Iwaniec T.
Published: July 2010

Let

F
F

be an arbitrary class of continuous mappings acting and ranging on domains in

R

n

{{\mathbf {R}}^n}

which is invariant under similarity transformations of

R

n

{{\mathbf {R}}^n}

and the restriction of a map to any subdomain. The class Möb of Möbius mappings acting in

R

n

{{\mathbf {R}}^n}

is of particular interest. Assume that the class

F
F

is "

c
c

-uniformly close" to Möb. Then we show that any map in

F
F

is either constant or a local quasiconformal homeomorphism. As a corollary we obtain a distinctly elementary proof of the Local Injectivity Theorem for quasiregular mappings.
The asymptotic behavior of univalent functions
Authors: Hu Ke , Dong Xin Han
Published: July 2010

We generalize Bazilevic’s Theorem and provide an alternative proof of an important theorem of Hamilton.
Orthogonality preserving maps and the Laguerre functional
Authors: Allaway William R.
Published: July 2010

Let

ℜ
[
x
]

\Re [x]

be the usual algebra of all polynomials in the indeterminate

x
x

over the field of real numbers

ℜ
\Re

, and let

φ
\varphi

be a linear operator mapping

ℜ
[
x
]

\Re [x]

into

ℜ
[
x
]

\Re [x]

. In this paper we show that if

φ
\varphi

maps every orthogonal polynomial sequence into an orthogonal polynomial sequence, then

φ
\varphi

is defined by

φ
(

x
n

)
=
s

(
a
x
+
b

)
n

,
n
=
0
,
1
,
2
,
…

\varphi ({x^n}) = s{(ax + b)^n},n = 0,1,2, \ldots

, where

s
,
a
,

s,a,

, and

b
b

belong to

ℜ
\Re

,

s
≠
0

s \ne 0

, and

a
≠
0

a \ne 0

.
On the almost everywhere convergence to 𝐿^{𝑝} data for higher order hyperbolic operators
Authors: Sogge Christopher D.
Published: July 2010

First we prove a sharp maximal Fourier integral theorem for

L
p

(

R

n

)
,

1
>
p
≤
2

{L^p}({{\mathbf {R}}^n}),\;1 > p \leq 2

, using the techniques of [4-6]. Then we apply the maximal theorem to prove a sharp result concerning the almost everywhere convergence to

L
p

{L^p}

-initial data for the Cauchy problem for smooth variable coefficient strictly hyperbolic linear partial differential operators of order

m
>
2

m > 2

.
Some consequences of left invertibility
Authors: Hochwald Scott H. , Morrel Bernard B.
Published: July 2010

Let

A
A

be a noncommutative Banach algebra with identity

e
e

. Let

L
L

be a multiplicative semigroup of left-invertible elements of

A
A

which properly contains the invertible elements of

A
A

. Then there does not exist a function

g
:
L
→
A

g:L \to A

such that

g
(
a
b
)
=
g
(
b
)
g
(
a
)

g(ab) = g(b)g(a)

and

g
(
a
)
a
=
e

g(a)a = e

for all elements

a
a

and

b
b

of

L
L

. This paper contains an elementary proof of this result, and thereby answers a question posed by G. R. Allan.
On the support of quasi-invariant measures on infinite-dimensional Grassmann manifolds
Authors: Pickrell Doug
Published: July 2010

One antisymmetric analogue of Gaussian measure on a Hilbert space is a certain measure on an infinite-dimensional Grassmann manifold. The main purpose of this paper is to show that the characteristic function of this measure is continuous in a weighted norm for graph coordinates. As a consequence the measure is supported on a thickened Grassmann manifold. The action of certain unitary transformations, in particular smooth loops

S
1

→
U
(
n
,

C

)

{S^1} \to U(n,{\mathbf {C}})

, extends to this thickened Grassmannian, and the measure is quasiinvariant with respect to these point transformations.
Some functional equations in Banach algebras and an application
Authors: Vukman J.
Published: July 2010

In this paper some results concerning certain functional equations in complex Banach algebras are presented. One of these results is used to prove an abstract generalization of the classical Jordan-Neumann characterization of pre-Hilbert space.
Similarity of a linear strict set-contraction and the radius of the essential spectrum
Authors: Shih Mau-Hsiang
Published: July 2010

If

A
A

is a bounded linear operator on a Hilbert space, define

r
e

(
A
)

{r_e}(A)

, the essential spectral radius of

A
A

, by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r Subscript e Baseline left-parenthesis upper A right-parenthesis colon equals sup left-brace right-brace colon vertical-bar vertical-bar lamda colon separator separator equals equals element-of element-of lamda of essess left-parenthesis right-parenthesis upper A times times essentialspectrumofA period">

r
e

(
A
)
:=
sup
{

|

λ

|

:
λ
∈
ess
⁡
(
A
)
=
essential

spectrum

of

A
}
.

{r_e}(A): = \sup \{ |\lambda |:\lambda \in \operatorname {ess}(A) = \operatorname {essential}\;\operatorname {spectrum}\;{\text {of}}\;A\} .

\]
It is shown that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r Subscript e Baseline left-parenthesis upper A right-parenthesis equals inf left-brace alpha left-parenthesis upper S Superscript negative 1 Baseline upper A upper S right-parenthesis vertical-bar upper S colon upper H right-arrow upper H is a bounded invertible linear m a p right-brace comma">

r
e

(
A
)
=
inf
⁡
{
α
(

S

−
1

A
S
)

|

S
:
H
→
H

is

a

bounded

invertible

linear

map
}
,

{r_e}(A) = \operatorname {inf}\{ \alpha ({S^{ - 1}}AS)|S:H \to H\;{\text {is}}\;{\text {a}}\;{\text {bounded}}\;{\text {invertible}}\;{\text {linear}}\;\operatorname {map}\} ,

\]
where

α
\alpha

is the Kuratowski measure of noncompactness. As a consequence, a charcterization of the similarity of a linear strict set-contraction is obtained.
A small arithmetic hyperbolic three-manifold
Authors: Chinburg Ted
Published: July 2010

The hyperbolic three-manifold which results from

(
5
,
1
)

(5,1)

Dehn surgery on the complement of a figure-eight knot in

S
3

{S^3}

is arithmetic.
A closed surface of genus one in 𝐸³ cannot contain seven circles through each point
Authors: Takeuchi Nobuko
Published: July 2010

There exists a closed surface of genus one in

E
3

{E^3}

which contains six cirlces through each point, but any closed surface of genus one in

E
3

{E^3}

cannot contain seven circles through each point.
A LIL for occupation times of stable processes
Authors: Shieh Narn-Rueih
Published: July 2010

We prove a Strassen-type law of iterated logarithms for the occupation times of an

R
d

{R^d}

-valued

(
d
≥
1
)

(d \geq 1)

stable process with the scaling property and positive density functions. An immediate application of our result is to obtain the asymptotic behavior of the occupation times of a path occupied in large spheres.
Orientation preserving actions of finite abelian groups on spheres
Authors: Dotzel Ronald M.
Published: July 2010

If

G
G

is a finite Abelian group acting as a

Z

(

P

)

{{\mathbf {Z}}_{(\mathcal {P})}}

-homology

n
n

-sphere

X
X

(where

P

\mathcal {P}

is the set of primes dividing

|

G

|

)

|G|)

, then there is an integer valued function

n
(
,
G
)

n(,G)

defined on the prime power subgroups

H
H

of

G
G

such that

X
H

{X^H}

has the

Z

(
p
)

{{\mathbf {Z}}_{(p)}}

-homology of a sphere

S

n
(
H
,
G
)

{S^{n(H,G)}}

. We prove here that there exists a real representation

R
R

of

G
G

such that for any prime power subgroup

H
H

of

G
,
dim
⁡
(
S
(

R
H

)
)
=
n
(
H
,
G
)

G,\dim (S({R^H})) = n(H,G)

where

S
(

R
H

)

S({R^H})

is the unit sphere of

R
H

{R^H}

, provided that

n
−
n
(
H
,
G
)

n - n(H,G)

is even whenever

H
H

is a

2
2

-subgroup of

G
G

.
Metrization of additive 𝜅-metric spaces
Authors: Isiwata Takesi
Published: July 2010

We prove that an additive

κ
\kappa

-metric space

X
X

is metrizable if

X
X

is one of the following: (1)

X
X

is pseudocompact, (2)

X
X

is a

w
M

wM

-space, (3)

X
X

is a paracompact

β
\beta

-space, especially stratifiable.
Some homotopy properties of the homeomorphism groups of 𝑅^{∞} (𝑄^{∞})-manifolds
Authors: Liem Vo Thanh
Published: July 2010

In this note we will prove that, given an

R
∞

(

Q
∞

)

{R^\infty }({Q^\infty })

-manifold

M
M

, there is a deformation of

Homo
⁡
(
M
)

\operatorname {Homo}(M)

into

Homeo
⁡
(
M
)

\operatorname {Homeo}(M)

whose final stage is a weak homotopy equivalence, and that if

M
M

has the homotopy type of a finite simplicial complex, then

Homeo
⁡
(
M
)

\operatorname {Homeo}(M)

is an

ANE
⁡
(

C

W

(

M

)
)

\operatorname {ANE}(\mathcal {C}\mathcal {W}(\mathcal {M}))

and an

ANE
⁡
(

C

W

(

C

)
,

G
δ

)

\operatorname {ANE}(\mathcal {C}\mathcal {W}(\mathcal {C}),{G_\delta })

.
A note on Hausdorff measures of quasi-self-similar sets
Authors: McLaughlin John
Published: July 2010

Sullivan has demonstrated that quasi-self-similarity provides a useful point of view for the study of expanding dynamical systems. In [4, p. 57] he posed the question: Is the Hausdorff measure of a quasi-self-similar set positive and finite in its Hausdorff dimension? This paper answers both parts of this question. In

§
1

\S 1

the positivity is established for compact sets, and a lower bound is given for their Hausdorff measure. However, in

§
2

\S 2

the finiteness is disproved. In fact, a quasi-self-similar set is constructed for which the Hausdorff measure is actually

σ
\sigma

-infinite.
Locally Lipschitz continuous perturbations of linear dissipative operators and nonlinear semigroups
Authors: Oharu Shinnosuke , Takahashi Tadayasu
Published: July 2010

Locally Lipschitz continuous perturbations of linear

m
m

-dissipative operators in Banach spaces are considered from the point of view of the nonlinear semigroup theory. A necessary and sufficient condition is given for a semilinear operator

A
+
F

A + F

to be the infinitesimal generator of a nonlinear semigroup which provides mild solutions (with exponential growth) of the semilinear evolution equation

u
′

(
t
)
=
(
A
+
F
)
u
(
t
)

u’(t) = (A + F)u(t)

. It turns out that a characterization of Hille-Yosida type for nonlinearly perturbed contraction semigroups is obtained.
Minimal sets in recurrent discrete flows
Authors: Knight Ronald A.
Published: July 2010

Each orbit closure of a recurrent discrete flow on a locally compact space is shown to contain at most one compact minimal subset. In particular, such an orbit closure contains a unique compact minimal subset if and only if the orbit closure is compact.
Embedding and unknotting of some polyhedra
Authors: Sarkaria K. S.
Published: July 2010

If a compact polyhedron

X
n

,
n
≥
3

{X^n},n \geq 3

(resp.

n
≥
2

n \geq 2

), has the property that any two of its nonsingular points can be joined by an arc containing at most one singular point, then

X
n

{X^n}

embeds in

R

2
n

{{\mathbf {R}}^{2n}}

(resp. unknots in

R

2
n
+
1

{{\mathbf {R}}^{2n + 1}}

).
Chebyshev type estimates for Beurling generalized prime numbers
Authors: Zhang Wen-Bin
Published: July 2010

We consider a Beurling generalized prime system for which the distribution function

N
(
x
)

N(x)

of the integers satisfies
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 1 Superscript normal infinity Baseline x Superscript negative 1 Baseline left-brace sup Underscript x less-than-or-slanted-equals y Endscripts StartFraction StartAbsoluteValue upper N left-parenthesis y right-parenthesis minus upper A y EndAbsoluteValue Over y EndFraction right-brace d x greater-than normal infinity">

∫
1
∞

x

−
1

{

sup

x
⩽
y

|

N
(
y
)
−
A
y

|

y

}

d
x
>
∞

\int _1^\infty {{x^{ - 1}}} \left \{ {\sup \limits _{x \leqslant y} \frac {{\left | {N(y) - Ay} \right |}} {y}} \right \}dx > \infty

\]
with constant

A
>
0

A > 0

. We shall prove that the Chebyshev type estimates
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than limit inf Underscript x right-arrow normal infinity Endscripts StartFraction psi left-parenthesis x right-parenthesis Over x EndFraction comma limit sup Underscript x right-arrow normal infinity Endscripts StartFraction psi left-parenthesis x right-parenthesis Over x EndFraction greater-than normal infinity">

0
>
lim

inf

x
→
∞

ψ
(
x
)

x

,

lim

sup

x
→
∞

ψ
(
x
)

x

>
∞

0 > \lim \inf \limits _{x \to \infty } \frac {{\psi (x)}} {x},\quad \lim \sup \limits _{x \to \infty } \frac {{\psi (x)}} {x} > \infty

\]
hold for the system. This gives a partial proof of one of Diamond’s conjectures.
A characterization of inner automorphisms
Authors: Schupp Paul E.
Published: July 2010

It turns out that one can characterize inner automorphisms without mentioning either conjugation or specific elements. We prove the following Theorem Let

G
G

be a group and let

α
\alpha

be an automorphism of

G
G

. The automorphism

α
\alpha

is an inner automorphism of

G
G

if and only if

α
\alpha

has the property that whenever

G
G

is embedded in a group

H
H

, then

α
\alpha

extends to some automorphism of

H
H

.
Reduction exponent and degree bound for the defining equations of graded rings
Authors: Trung Ngô Viêt
Published: July 2010

The paper gives upper degree bounds for the defining equations of certain graded rings in terms of the reduction exponent and the multiplicity.
Semisimple representations and affine rings
Authors: Farkas Daniel R.
Published: July 2010

If all finite-dimensional representations of an affine algebra are semisimple, then there are only finitely many representations of each degree.
The Frattini argument and 𝑡-groups
Authors: Brewster Ben , Sehgal Surinder
Published: July 2010

If enough subgroups of a group satisfy the Frattini argument in the group, then normality is a transitive relation within the group. Subgroup functors are used to specify what enough is.
The asymptotic behavior of firmly nonexpansive mappings
Authors: Reich Simeon , Shafrir Itai
Published: July 2010

We present several new results on the asymptotic behavior of firmly nonexpansive mappings in Banach spaces and in the Hilbert ball.
Iterations of Hardy-Littlewood maximal functions
Authors: Neugebauer C. J.
Published: July 2010

It is shown that the iteration of two maximal functions is essentially no larger than the inner maximal function.
Noncommuting unitary groups and local boundedness
Authors: Rusinek Jan
Published: July 2010

We exhibit two unitary strongly continuous one-parameter groups

(

e

A
1

t

)

t
∈

R

{({e^{{A_1}t}})_{t \in {\mathbf {R}}}}

and

(

e

A
2

t

)

t
∈

R

{({e^{{A_2}t}})_{t \in {\mathbf {R}}}}

acting in a Hilbert space

H
H

, a dense subspace

D
D

of

H
H

contained in the domains of

A
1

{A_1}

and

A
2

{A_2}

such that

(

A
1

(
D
)
∪

A
2

(
D
)
)
⊂
D

({A_1}(D) \cup {A_2}(D)) \subset D

and

(

e

A
1

t

(
D
)
∪

e

A
2

t

(
D
)
)
⊂
D

({e^{{A_1}t}}(D) \cup {e^{{A_2}t}}(D)) \subset D

for each

t
∈

R

t \in {\mathbf {R}}

, and an element

x
x

of

D
D

such that the function

t
→

‖

A
1

e

A
2

t

x

‖

t \to \left \| {{A_1}{e^{{A_2}t}}x} \right \|

is not locally bounded.
Nilpotent automorphism groups of bordered Klein surfaces
Authors: May Coy L.
Published: July 2010

Let

X
X

be a compact bordered Klein surface of algebraic genus

g
⩾
2

g \geqslant 2

, and let

G
G

be a group of automorphisms of

X
X

. Then the order of

G
G

is at most

12
(
g
−
1
)

12(g - 1)

. Here we improve this general bound in an important special case. We show that if

G
G

is nilpotent, then the order of

G
G

is at most

8
(
g
−
1
)

8(g - 1)

. This bound is the best possible. We construct infinite families of surfaces that have a nilpotent automorphism group of order

8
(
g
−
1
)

8(g - 1)

. The nilpotent groups of maximum possible order must be

2
2

-groups. We prove that if the nilpotent group

G
G

acts on a bordered surface of genus

g
g

such that

o
(
G
)
=
8
(
g
−
1
)

o(G) = 8(g - 1)

, then

g
−
1

g - 1

is a power of 2. Further, our examples show that for each nonnegative integer

t
t

there is a bordered surface with genus

g
=

2
t

+
1

g = {2^t} + 1

and a group of automorphisms of order

8
(
g
−
1
)

8(g - 1)

.
Convergence in measure for semigroup-valued integrals
Authors: Fleischer Isidore
Published: July 2010

The semigroup-valued integral of M. Sion [S] is reformulated for a general notion of approximation by sums of values taken by a set function integrand. A convergence in measure theorem is established, which yields both his pointwise dominated convergence theorem as well as an integrability criterion which specializes to his existence theorem.
An elementary proof of a theorem of S. Mardešić
Authors: Nikiel Jacek
Published: July 2010

A simple proof is given of the following theorem of S. Mardešić: Each arc is an inverse limit of some inverse system of copies of

[
0
,
1
]

[0,1]

with monotone bonding maps.
Solving weighted norm inequalities using the Rubio de Francia algorithm
Authors: Bloom Steve
Published: July 2010

Given an operator

T
T

bounded on a weighted

L
p

{L^p}

space, the factorization technique of Rubio de Francia forces strong conditions on the weight. This algorithm is extended to two weight problems, and is shown to yield not just necessary but sufficient conditions in a wide range of settings.
Zeros of derivatives of entire functions
Authors: Craven Thomas , Csordas George , Smith Wayne
Published: July 2010

It is shown that if a real entire function of genus one has only finitely many nonreal zeros, then, its derivatives, from a certain one onward, have only real zeros.
On the derivative with respect to a point
Authors: Goodman A. W.
Published: July 2010

The derivative of a polynomial

p
(
z
)

p(z)

with respect to a point

ς
\varsigma

is defined by the formula

A
ς

p
(
z
)
=
(
ς
−
z
)

p
′

(
z
)
+
n
p
(
z
)

{A_\varsigma }p(z) = (\varsigma - z)p’(z) + np(z)

, where

n
n

is the degree of the polynomial. Let

p
(
z
)

p(z)

have all its zeros in the unit disk and one zero at

z
=
1

z = 1

. We determine a minimal region that must contain at least one zero of

A
ς

p
(
z
)

{A_\varsigma }p(z)

.
Cut-set sums and tree processes
Authors: Falconer K. J.
Published: July 2010

Suppose that an infinite tree has a value assigned to each vertex. We obtain estimates for the sums of such values over cut-sets of the tree. For certain tree processes, where the values are given by random variables, we investigate the almost sure behavior of such cut-set sums. Processes of this type arise in problems concerning random fractals and flows in random networks.
Determining the cohomological dimension of certain compacta
Authors: Rubin Leonard R.
Published: July 2010

In previous work of this author and of John J. Walsh it was shown that many of the known examples of hereditarily infinite dimensional compacta have infinite cohomological dimension. In this paper, the class of compacta whose cohomological dimension is known to be infinite will be enlarged.
𝛽([0,∞)) does not contain nondegenerate hereditarily indecomposable continua
Authors: Smith Michel
Published: July 2010

Bellamy has shown that if

A
=

[

0
,
∞

)

A = \left [ {0,\infty } \right )

, then

β
A
−
A

\beta A - A

is an indecomposable continuum and every nondegenerate subcontinuum of

β
A
−
A

\beta A - A

can be mapped onto every metric continuum. Thus, it follows that every nondegenerate subcontinuum of

β
A
−
A

\beta A - A

contains a nondegenerate indecomposable continuum. We show, however, that no nondegenerate subcontinuum of

β
A
−
A

\beta A - A

is hereditarily indecomposable. Thus, every nondegenerate subcontinuum of

β
A
−
A

\beta A - A

contains a decomposable continuum as well as a nondegenerate indecomposable continuum.
𝐾-to-1 functions on arcs for 𝐾 even
Authors: Heath Jo W.
Published: July 2010

For exactly

k
k

-to-

1
1

functions from

[
0
,
1
]

[0,1]

into

[
0
,
1
]

[0,1]

: (1) at least one discontinuity is required (Harrold), (2) if

k
=
2

k = 2

, then infinitely many discontinuities are needed, for any Hausdorff image space (Heath), (3) if

k
=
4

k = 4

, or if

k
k

is odd, then there is such a function with only one discontinuity (Katsuura and Kellum), and, it is shown here that (4) if

k
k

is even and

k
>
4

k > 4

, then there is such a function with only two discontinuities, and no such function exists with fewer discontinuities.
Fourier series with positive coefficients
Authors: Ash J. Marshall , Rains Michael , Vági Stephen
Published: July 2010

Extending a result of N. Wiener, it is shown that functions on the circle with positive Fourier coefficients that are

p
p

th power integrable near

0
,
1
>
p
≤
2

0,1 > p \leq 2

, have Fourier coefficients in

l

p
′

{l^{p’}}

.
Scalar-reflexive rings
Authors: Hadwin Don , Kerr Jeanne Wald
Published: July 2010

A module over a commutative ring is scalar-reflexive if the only endormorphisms that leave invariant every submodule are scalars. We investigate the class of rings for which every module is scalar-reflexive and the class of rings for which every finitely generated module is scalar-reflexive. For a certain class of rings we show that these properties are equivalent to every finitely generated module being a direct sum of cyclic modules.
On continued fractions of given period
Authors: Friesen Christian
Published: July 2010

A direct proof is given of the fact that, for any

k
∈

Z

+

k \in {{\mathbf {Z}}^ + }

, there are infinitely many squarefree integers

N
N

, where the continued fraction expansion of

N

\sqrt N

has period equal to

k
k

.
Embedding of 𝑈_{𝜔}-groups in 𝐷_{𝜔}-groups
Authors: Cassidy Charles
Published: July 2010

The purpose of this paper is to give certain new conditions under which it is possible to show that a group in which the equation

x
p

=
a

{x^p} = a

has at most one solution for every

a
a

, can be embedded in another group in which the equation

x
p

=
a

{x^p} = a

has exactly one solution for every

a
a

. It has been known for a long time that locally nilpotent groups satisfy the above property but some other sufficient conditions have also been found by G. Baumslag in his thesis. Conditions related to Baumslag’s are examined.
Faithful abelian groups of infinite rank
Authors: Albrecht Ulrich
Published: July 2010

Let

B
B

be a subgroup of an abelian group

G
G

such that

G

/

B

G/B

is isomorphic to a direct sum of copies of an abelian group

A
A

. For

B
B

to be a direct summand of

G
G

, it is necessary that

G
G

be generated by

B
B

and all homomorphic images of

A
A

in

G
G

. However, if the functor

Hom
⁡
(
A
,
−
)

\operatorname {Hom} (A, - )

preserves direct sums of copies of

A
A

, then this condition is sufficient too if and only if

M

⊗

E
(
A
)

A

M{ \otimes _{E(A)}}A

is nonzero for all nonzero right

E
(
A
)

E(A)

-modules

M
M

. Several examples and related results are given.
Algebraic monoids whose nonunits are products of idempotents
Authors: Putcha Mohan S.
Published: July 2010

Let

M
M

be a connected regular linear algebraic monoid with zero and group of units

G
G

. Suppose

G
G

is nearly simple, i.e. the center of

G
G

is one dimensional and the derived group

G
′

G’

is a simple algebraic group. Then it is shown that

S
=
M
∖
G

S = M\backslash G

is an idempotent generated semigroup. If

M
M

has a unique nonzero minimal ideal, the converse is also proved. It follows that if

G
0

{G_0}

is any simple algebraic group defined over an algebraically closed field

K
K

and if

Φ
:

G
0

→
G
L
(
n
,
K
)

\Phi :{G_0} \to GL(n,K)

is any representation of

G
0

{G_0}

, then the nonunits of the monoid

M
(
Φ
)
=

K
Φ
(

G
0

)

¯

M(\Phi ) = \overline {K\Phi ({G_0})}

form an idempotent generated semigroup.
Existence of generalized homoclinic orbits for one parameter families of flows
Authors: Mischaikow Konstantin
Published: July 2010

We consider a one parameter family of flows generated by

x
˙

=
f

(

x
,
λ

)

\dot x = f\left ( {x,\lambda } \right )

, where

λ
∈

[

0
,
1

]

\lambda \in \left [ {0,1} \right ]

. We, also, assume that there exists an isolated invariant set,

S
S

, which continues across the interval

[

0
,
1

]

\left [ {0,1} \right ]

, and that we know the connection matrices for

x
˙

=
f

(

x
,
1

)

\dot x = f\left ( {x,1} \right )

and

x
˙

=
f

(

x
,
0

)

\dot x = f\left ( {x,0} \right )

. We then give conditions under which a Morse decomposition of

S
S

cannot continue across

[

0
,
1

]

\left [ {0,1} \right ]

. Furthermore, using the language of chain recurrence we define a generalized homoclinic orbit and give conditions under which such objects exist.
The spaces 𝐻^{𝑝}(𝐵_{𝑛}),0<𝑝<1, and 𝐵_{𝑝𝑞}(𝐵_{𝑛}),0<𝑝<𝑞<1, are not locally convex
Authors: Shi Ji Huai
Published: July 2010

In this note, we use the Ryll-Wojtaszczyk polynomials to prove that the spaces

H
p

(

B
n

)

,
0
>
p
>
1

{H^p}\left ( {{B_n}} \right ),0 > p > 1

, and

B

p
q

(

B
n

)

,
0
>
p
>
q
>
1

{B_{pq}}\left ( {{B_n}} \right ),0 > p > q > 1

, fail to be locally convex.
The Fenchel-Moreau theorem for set functions
Authors: Lai Hang-Chin , Lin Lai-Jui
Published: July 2010

The Fenchel-Moreau theorem for set functions is proved, and some properties of subdifferential and conjugate functional of set functions are investigated.
Small systems convergence and metrizability
Authors: Johnson Roy A. , Niewiarowski Jerzy , Świątkowski Tadeusz
Published: July 2010

The abstract notion of convergence of functions with respect to a small system has its roots in the concept of convergence of functions in measure. The second author has shown that convergence of functions with respect to a small system generates a Fréchet topology. In this paper we show that convergence with respect to a small system is equivalent to convergence with respect to a certain complete pseudometric (or metric if we consider equivalence classes of functions with respect to the small system).
Strong unicity of best approximations in 𝐿_{∞}(𝑆,Σ,𝜇)
Authors: Smarzewski Ryszard
Published: July 2010

We prove that the set of functions in a Banach space

L
∞

(
S
,
∑
,
μ
)

{L_\infty }(S,\sum ,\mu )

, which have a strongly unique best approximation in a finite dimensional subspace, is dense in the set of functions having a unique best approximation in the subspace.
Complex interpolating polynomials
Authors: Varma A. K.
Published: July 2010

Let

I

n
,
m

(

f
,
z

)

{I_{n,m}}\left ( {f,z} \right )

be the unique interpolatory polynomial of degree

≤
2
n
−
1

\leq 2n - 1

satisfying the conditions given by (1.1) where the

z

k
n

{z_{kn}}

’s are the zeros of the polynomial

z
n

−
1

{z^n} - 1

. The object of this paper is to consider the rate of convergence of

I

m
,
n

(

f
,
z

)

{I_{m,n}}\left ( {f,z} \right )

to

f

(
z
)

f\left ( z \right )

in the

L
p

{L_p}

norm where

f
∈
C

[

|

z

|

≤
1

]

f \in C\left [ {|z| \leq 1} \right ]

. This problem was initially raised by P. Turán in the case

p
=
2

p = 2

and in this case the solution was obtained by J. Szabados and A. K. Varma in [7].
Weak-star generators of 𝑍ⁿ,𝑛≥1, and transitive operator algebras
Authors: Ansari Mohamad A.
Published: July 2010

A function

f
f

in

H
∞

{H^\infty }

is said to be a weak-star generator (

W

∗

{{\text {W}}^*}

-gen.) of the function

e
n

(
z
)

=

z
n

,

|

z

|

>
1
,

n
≥
1

{e_n}\left ( z \right ) = {z^n},\,|z| > 1,\,n \geq 1

, if

lim
α

p
α

∘
f
=

e
n

{\lim _\alpha }{p_\alpha } \circ f = {e_n}

(

W

∗

{{\text {W}}^*}

-topology), for some net (

p
α

{p_\alpha }

) of complex polynomials. For the case

n
=
1

n = 1

,

f
f

is called a

W

∗

{{\text {W}}^*}

-gen. of

H
∞

{H^\infty }

. The

W

∗

{{\text {W}}^*}

-generators of

H
∞

{H^\infty }

have been defined and characterized by Sarason. It is the purpose of the present paper to give necessary and sufficient conditions for a function to generate

e
n

{e_n}

. As a result, it follows from our characterization that certain analytic Toeplitz operators have the transitive algebra property.
More on the differentiability of convex functions
Authors: Verona Maria Elena
Published: July 2010

Let

C
C

be a closed, convex set in a topological vector space

X
X

such that

N
S
(
C
)

NS(C)

, the set of its nonsupport points, is nonempty (this is always the case if

X
X

is Banach separable; if

X
X

is Fréchet,

N
S

(
C
)

NS\left ( C \right )

is residual in

C
C

). If

X
X

is normed, we prove that any locally Lipschitz, convex real function

f
f

on

C
C

is subdifferentiable on

N
S

(
C
)

NS\left ( C \right )

. If in addition

X
X

is Banach separable, we prove that

f
f

is smooth on a residual subset of

N
S

(
C
)

NS\left ( C \right )

.
Approximation by operators with fixed nullity
Authors: Bouldin Richard
Published: July 2010

Let

T
T

be a fixed operator on a complex separable Hilbert space

H
H

. The distance from

T
T

to the operators with nullity equal to

n
n

, for each possible value of

n
n

, is determined.
Solutions with compact support of the porous medium equation in arbitrary dimensions
Authors: Watanabe Michiaki
Published: July 2010

Compactness of the support is discussed of a solution

u
u

to the Cauchy problem for the porous medium equation

u
t

=
Δ
ϕ
(
u
)
,
t
>
0

{u_t} = \Delta \phi (u),t > 0

, in

R
N

{R^N}

of arbitrary dimension

N
≥
1

N \geq 1

, where

ϕ
\phi

is a nondecreasing function on

R
1

{R^1}

. It is shown that if

u
(
0
,
x
)
=
0

u(0,x) = 0

for

|
x
|

≥
R
,
R
>
0

\left | x \right | \geq R,R > 0

, then for all

t
≥
0

t \geq 0

\[

u
(
t
,
x
)
=
0

a

.e

.

|
x
|

≥
R
+
C

t

1

/

2

u(t,x) = 0\quad {\text {a}}{\text {.e}}{\text {.}}\left | x \right | \geq R + C{t^{1/2}}

\]
with a constant

C
C

depending on

ϕ
\phi

and

u
(
0
,
⋅
)

u(0, \cdot )

. The result is well known when

N
=
1

N = 1

, but the study for

N
>
1

N > 1

has somehow been neglected.
Weighted norm inequalities for Bochner-Riesz spherical summation multipliers
Authors: Andersen Kenneth F.
Published: July 2010

Sufficient conditions to be satisfied by nonnegative weight functions

ω

(

|
x
|

)

\omega \left ( {\left | x \right |} \right )

are given in order that the Bochner-Riesz spherical summation multiplier operators restricted to radial functions of

R
n

{R^n}

be bounded on

L
p

(

R
n

;
ω

(

|
x
|

)

d
x

)

{L^p}\left ( {{R^n};\omega \left ( {\left | x \right |} \right )dx} \right )

. For a certain class of weights these conditions are also necessary.
Integral inequalities of Hardy and Poincaré type
Authors: Boas Harold P. , Straube Emil J.
Published: July 2010

The Poincaré inequality

|

|

u

|

|

p

≤
C

|

|

∇
u

|

|

p

||u|{|_p} \leq C||\nabla u|{|_p}

in a bounded domain holds, for instance, for compactly supported functions, for functions with mean value zero and for harmonic functions vanishing at a point. We show that it can be improved to
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue u StartAbsoluteValue EndAbsoluteValue Subscript p Baseline less-than-or-equal-to upper C StartAbsoluteValue EndAbsoluteValue delta Superscript beta Baseline nabla u StartAbsoluteValue EndAbsoluteValue Subscript p">

|

|

u

|

|

p

≤
C

|

|

δ
β

∇
u

|

|

p

||u|{|_p} \leq C||{\delta ^\beta }\nabla u|{|_p}

, where

δ
\delta

is the distance to the boundary, and the positive exponent

β
\beta

depends on the smoothness of the boundary.
Degrees of constant-to-one factor maps
Authors: Trow Paul
Published: July 2010

Let

f
f

be a constant-to-one endomorphism of degree

d
d

, of a sub-shift of finite type

Σ
A

{\Sigma _A}

. If

p
p

is a prime dividing

d
d

, then

p
p

divides every nonleading coefficient of

χ
A

{\chi _A}

, the characteristic polynomial for

A
A

. Further constraints are given for the possible degrees of a constant-to-one factor map between subshifts of finite type.
On blow up of solutions of nonlinear evolution equations
Authors: Korman Philip
Published: July 2010

We give a complete description of domains of blow up for general second order inequalities, which allows us to obtain some new results on nonexistence of global solutions for nonlinear hyperbolic equations, both in

R
n

{R^n}

and bounded domains.
On the Marchaud-type inequality
Authors: Ditzian Z.
Published: July 2010

Marchaud’s inequality, which is valid in many function spaces, was strengthened by M. F. Timan [3] for

L
p

,
1
>
p
>
∞

{L_p},1 > p > \infty

, following a technique of A. Zygmund [5]. In both the above-mentioned articles the powerful Littlewood-Paley theorem is used. In the present paper a direct and, I believe much simpler, proof is given for that stronger Marchaud-type inquality. Moreover, the result will apply to a more general class of function spaces. It will be shown that it is sufficient that the "modulus of smoothness" of the norm is of "power-type"

p
p

and that translation is an isometry.
The density of extreme points in complex polynomial approximation
Authors: Kroó András , Saff E. B.
Published: July 2010

Let

K
K

be a compact set in the complex plane having connected and regular complement, and let

f
f

be any function continuous on

K
K

and analytic in the interior of

K
K

. For the polynomials

p
n
∗

(
f
)

p_n^*(f)

of respective degrees at most

n
n

of best uniform approximation to

f
f

on

K
K

, we investigate the density of the sets of extreme points
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript n Baseline left-parenthesis f right-parenthesis colon equals StartSet z element-of upper K colon StartAbsoluteValue f left-parenthesis z right-parenthesis minus p Subscript n Superscript asterisk Baseline left-parenthesis f right-parenthesis left-parenthesis z right-parenthesis EndAbsoluteValue equals StartAbsoluteValue EndAbsoluteValue f minus p Subscript n Superscript asterisk Baseline left-parenthesis f right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript upper K Baseline EndSet">

A
n

(
f
)
:=
{
z
∈
K
:

|

f
(
z
)
−

p
n
∗

(
f
)
(
z
)

|

=

|

|

f
−

p
n
∗

(
f
)

|

|

K

}

{A_n}(f): = \{ z \in K:|f(z) - p_n^*(f)(z)| = ||f - p_n^*(f)|{|_K}\}

\]
in the boundary of

K
K

.
Total paracompactness and Banach spaces
Authors: Lupiañez Francisco Gallego
Published: July 2010

In this paper, we study some problems related to the Corson theorem. In particular we prove that

c
0

{c_0}

does not fulfil such a theorem; hence this theorem is not valid for all infinite-dimensional Banach spaces. We give also generalizations of Corson’s theorem for some infinite-dimensional normed spaces.