bluuebunny/crossref_metadata_2025 huggingface dataset:
Properties of standard maps
Authors: Huckabay Gary M.
Published: June 2010

Let X and Y be compact metric spaces. Let

S
(
X
,
Y
)

S(X,Y)

denote the collection of standard maps of X onto Y. We establish that

S
(
C
,
Y
)

S(C,Y)

is a dense subset of

C
(
C
,
Y
)

C(C,Y)

, where C is the Cantor set. If f is a standard map and

G
(
f
,
Y
)
{
A
(
f
,
Y
)
}

G(f,Y)\{ A(f,Y)\}

denotes the subgroup of

H
(
X
)

H(X)

which preserves {interchanges} the point-inverses of f, then there is a continuous homomorphism of

A
(
f
,
Y
)

A(f,Y)

into

H
(
Y
)

H(Y)

with kernel

G
(
f
,
Y
)

G(f,Y)

. We also show that

G
(
f
,
Y
)

G(f,Y)

and

A
(
f
,
Y
)

A(f,Y)

are closed subsets of

H
(
X
)

H(X)

.
On the measurability of orbits in Borel actions
Authors: Miller Douglas E.
Published: June 2010

We replace measure with category in an argument of G. W. Mackey to characterize closed subgroups H of a totally nonmeager, 2nd countable topological group G in terms of the quotient Borel structure G/H. As a corollary, we obtain an improved version of a theorem of C. Ryll-Nardzewski on the Borel measurability of orbits in continuous actions by Polish groups.
Homology tori with nontrivial Serre spectral sequence
Authors: Grossman Jerrold W.
Published: June 2010

An example is constructed of a fibration over the torus which induces isomorphisms on all homology groups but which has a nontrivial Serre spectral sequence.
A geometrical procedure for killing the middle dimensional homology groups of algebraic hypersurfaces
Authors: Libgober A. S.
Published: June 2010

Explicit construction for decomposition of algebraic hypersurfaces into a connected sum of handles and a homological projective space is discussed. Also a connection is provided between Levine’s results about the Arf invariant in the theory of knots and the computation of Arf invariant of hypersurfaces by S. Morita and J. Wood.
Flat or open implies going down
Authors: Dobbs David E. , Papick Ira J.
Published: June 2010

Let R, T be commutative rings with identity, and

f
:
R
→
T

f:R \to T

a unital ring homomorphism. We give an elementary, unified proof of the fact that f has the going down property, if T is flat as an R-module or if the induced map

F
:

Spec

(
T
)
→

Spec

(
R
)

F:{\text {Spec}}(T) \to {\text {Spec}}(R)

is open.
Coincident pairs of continuous sections in profinite groups
Authors: Appelgate H. , Onishi H.
Published: June 2010

Given a profinite group G and a closed subgroup H, we show there exist continuous sections

s
:

(
G

/

H

)
1

→
G

s:{(G/H)_1} \to G

and

s
′

:

(
G

/

H

)
r

→
G

s’:{(G/H)_r} \to G

of the projections

G
→

(
G

/

H

)
1

G \to {(G/H)_1}

and

G
→

(
G

/

H

)
r

G \to {(G/H)_r}

onto the left and right coset spaces, respectively, such that

im

(
s
)
=

im

(

s
′

)

{\text {im}}(s) = {\text {im}}(s’)

.
Lower bounds for the zeros of Bessel functions
Authors: McCann Roger C.
Published: June 2010

Let

j

p
,
n

{j_{p,n}}

denote the nth positive zero of

J
p

,
p
⩾
0

{J_p},p \geqslant 0

. Then
\[

j

p
,
n

⩾

(

j

0
,
n

2

+

p
2

)

1

/

2

.

{j_{p,n}} \geqslant {( {j_{0,n}^2 + {p^2}} )^{1/2}}.

\]
Discontinuous semilinear differential equations and multiple valued maps
Authors: Rauch Jeffrey
Published: June 2010

To treat boundary value problems with nonlinearity of the form

g
(
u
)

g(u)

where

g
∈

L
∞

loc

(

R

)

g \in L_\infty ^{{\text {loc}}}({\mathbf {R}})

one must extend g to be multiple valued. Once this is done weak solutions are obtained.
Embedding in topological vector spaces
Authors: Richardson Gary
Published: June 2010

Let LX denote the set of all continuous linear functionals on the locally convex topological vector space X. The space

L

co

X

{L_{{\text {co}}}}X

denotes LX endowed with the compact-open topology. We investigate the spaces, X, which have the property that the natural map from X into

L

co

(

L

co

X
)

{L_{{\text {co}}}}({L_{{\text {co}}}}X)

is an embedding.
Unique Hahn-Banach extensions and simultaneous extensions
Authors: Hasumi Morisuke
Published: June 2010

In this note we mention certain connections existing between unique Hahn-Banach extensions and simultaneous extensions. We also describe an application of a continuous selection theorem to simultaneous extensions.
The norm of the sum of two projections
Authors: Vidav Ivan
Published: June 2010

Let e and

f

,

e

+

f

≠
0

{\mathbf {f}},{\mathbf {e}} + {\mathbf {f}} \ne 0

, be two projections of a

C
∗

{C^\ast }

-algebra A. J. Duncan and P. J. Taylor have shown that

‖

e

+

f

‖

=
1
+

‖

e
f

‖

\left \| {{\mathbf {e}} + {\mathbf {f}}} \right \| = 1 + \left \| {{\mathbf {ef}}} \right \|

. In this paper an algebraic proof of this equality is given.
On Calabi’s inhomogeneous Einstein-Kaehler manifolds
Authors: Wolf Joseph A.
Published: June 2010

We use some information on Lie groups to replace a long computation of Calabi, proving that certain complete Einstein-Kaehler manifolds are not locally homogeneous, and finding their isometry groups.
A note on monotonic ortho-bases
Authors: Phillips Thomas M.
Published: June 2010

At the 1974 Topology Conference at Charlotte, North Carolina, Peter Nyikos introduced the concept of an ortho-base and announced that a

T
2

{T_2}

paracompact first-countable

β
\beta

-space having an ortho-base is metrizable. The purpose of this paper is to introduce an obvious monotonic generalization of ortho-bases and to prove the following theorem. Theorem. If S is a regular

T
0

{T_0}

space having a monotonic ortho-base, then each of the following implies that S has a base of countable order: (1) S is connected; (2) S is a

β
c

{\beta _c}

-space; (3) S is a first-countable monotonic

β
\beta

-space. Nyikos’ theorem is a corollary to (3) and Arhangel’skiǐ’s theorem that a

T
2

{T_2}

paracompact space having a base of countable order is metrizable.
Spheres in 𝐸³ which are homogeneous over a 0-dimensional set
Authors: Burgess C. E.
Published: June 2010

A 2-sphere S in Euclidean 3-space

E
3

{E^3}

is defined to be homogeneous over the subset X of S if for each p,

q
∈
X

q \in X

there is a homeomorphism

h
:

E
3

→

E
3

h:{E^3} \to {E^3}

such that

h
(
S
)
=
S

h(S) = S

and

h
(
p
)
=
q

h(p) = q

. It is shown that a 2-sphere S in

E
3

{E^3}

is tame from one side provided S is locally tame modulo a tame 0-dimensional set C such that S is homogeneous over C. An example is described to show that it is necessary to require that C be tame.
A complex which cannot be pushed around in 𝐸³
Authors: Starbird Michael
Published: June 2010

This paper contains an example of a finite complex C with triangulation T which admits two linear embeddings f and g into

E
3

{E^3}

so that although there is an isotopy of

E
3

{E^3}

taking the embedding f to g there is no continuous family of linear embeddings of C starting at f and ending at g. No such example can exist in

E
2

{E^2}

.
Poisson brackets and commutator brackets. II
Authors: Prosser Reese T.
Published: June 2010

A previous result, that for certain problems in Hamiltonian mechanics the commutator bracket is determined by the Poisson bracket, is here extended to include certain problems defined in terms of current algebras.
Linear representations of semigroups of Boolean matrices
Authors: Kim Ki Hang , Roush Fred W.
Published: June 2010

Let

B
n

{B_n}

be the multiplicative semigroup of

n
×
n

n \times n

matrices over the semiring 0, 1 under the operations “or” and “and". We show that the least possible degree of a faithful representation of

B
n

{B_n}

over a field is

2
n

−
1

{2^n} - 1

by studying representations of a subsemigroup of

B
n

{B_n}

. By different methods we answer the same question for the subsemigroups of Boolean matrices greater than or equal to some permutation matrix (Hall matrices) and greater than or equal to the identity (reflexive Boolean matrices). We prove every representation of the latter semigroup can be triangularized.
The domain covered by a typically-real function
Authors: Goodman A. W.
Published: June 2010

We find the largest possible domain that is covered by

f
(
E
)

f(E)

for every typically-real function

f
(
z
)

f(z)

. In the process we obtain a set of universal typically-real functions.
A short proof of an inequality of Carleson’s
Authors: Neville Charles W.
Published: June 2010

We give a simple proof that if

a
i

,
i
=
1
,
2
,
…

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

, is a uniformly separated sequence in the unit disk, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma left-parenthesis 1 minus StartAbsoluteValue a Subscript i Baseline EndAbsoluteValue squared right-parenthesis StartAbsoluteValue f left-parenthesis a Subscript i Baseline right-parenthesis EndAbsoluteValue Superscript p Baseline less-than-or-slanted-equals upper K double-vertical-bar f double-vertical-bar Subscript p Superscript p">

Σ
(
1
−

|

a
i

|

2

)

|

f
(

a
i

)

|

p

⩽
K

‖
f
‖

p
p

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

, for all

f
∈

H
p

f \in {H^p}

and

1
⩽
p
>
∞

1 \leqslant p > \infty

.
On a Hill’s equation with double eigenvalues
Authors: Hochstadt Harry
Published: June 2010

Let

Δ
(
λ
)

\Delta (\lambda )

be the discriminant of a Hill’s equation with a

π
\pi

-periodic potential

q
(
x
)

q(x)

. It is shown that if

2
+
Δ
(
λ
)

2 + \Delta (\lambda )

has only double zeros then

q
(
x
)

q(x)

is necessarily

π

/

2

\pi /2

-periodic.
Almost periodic operators of inner automorphism groups
Authors: Thayer F. Javier
Published: June 2010

We examine the almost periodic part of the inner automorphism group of

B

(
H
)

\mathcal {B}(H)

arising from a representation of a locally compact abelian group.
On the duality between smoothability and dentability
Authors: Lewis Ted
Published: June 2010

By renorming

l
1

{l_1}

with an equivalent dual norm it is shown that smoothability of the unit ball of a conjugate Banach space

E
∗

{E^\ast }

does not imply dentability of the unit ball of either E or

E

∗
∗

{E^{\ast \ast }}

. It is also shown that the unit ball may be smoothable yet fail to be smooth at any point.
On analyticity of local resolvents and existence of spectral subspaces
Authors: Fong Che Kao
Published: June 2010

We present some sufficient conditions for a function from an open set in C into a Hilbert space H such that

(
T
−
z
)
f
(
z
)
=
x
(
T
∈
B
(
H
)

(T - z)f(z) = x(T \in B(H)

and

x
∈
H

x \in H

) to be analytic. As an application we show that hyperinvariant subspaces exist for certain class of operators.
Connected and connectivity multifunctions
Authors: Smithson R. E.
Published: June 2010

Conditions are given under which multifunctions which preserve connected sets will have a connected graph.
Preferred sets in topological dynamics
Authors: Elaydi Saber , Lam Ping Fun
Published: June 2010

The second author introduced a set of dynamical invariants called indivisibilities in one of his previous papers [3]. Following the same idea but from a different point of view this note concerns a class of invariants in transformation groups called the ’preferred sets’. For a special class of transformation groups, the number of permissible preferred sets is determined.
Determining automorphisms of the recursively enumerable sets
Authors: Shore Richard A.
Published: June 2010

We answer two questions of A. Nerode and give information about how the structure of

E

∗

{\mathcal {E}^\ast }

, the lattice of r.e. sets modulo finite sets, is determined by various subclasses. Theorem. If

C

∗

{\mathcal {C}^\ast }

is any nontrivial recursively invariant subclass of

E

∗

{\mathcal {E}^\ast }

, then any automorphism of

E

∗

{\mathcal {E}^\ast }

is determined uniquely by its action on

C

∗

{\mathcal {C}^\ast }

. Theorem. If

C

∗

{\mathcal {C}^\ast }

is the class of recursive sets modulo finite sets or

M

∗

⊆

C

∗

⊆

S

∗

{\mathcal {M}^\ast } \subseteq {\mathcal {C}^\ast } \subseteq {\mathcal {S}^\ast }

(

M

∗

{\mathcal {M}^\ast }

= maximal sets,

S

∗

{\mathcal {S}^\ast }

= simple sets) then there is an automorphism of (the lattice generated by)

C

∗

{\mathcal {C}^\ast }

which does not extend to one of

E

∗

{\mathcal {E}^\ast }

.
On subgroups of the multiplicative group of skew fields
Authors: Lichtman A.
Published: June 2010

We give an example of a skew field D for which the multiplicative group

D
∗

{D^ \ast }

of D contains a finitely generated subgroup which satisfies a nontrivial identity and has no solvable normal subgroup of finite index. It gives a negative answer to a question of S. Bachmuth.
On generic asymptotic stability of differential equations in Banach space
Authors: De Blasi F. S. , Myjak J.
Published: June 2010

The asymptotic stability of the zero solution of the differential equation

(
∗
)

x
′

=
A
x
+
f
(
x
)

( \ast )\;x’ = Ax + f(x)

is studied, when the pertubation f is in a given complete metric space

M

\mathfrak {M}

. It is known that the zero solution of

(
∗
)

( \ast )

is asymptotically stable whenever f is in a certain proper subset

N

⊂

M

\mathfrak {N} \subset \mathfrak {M}

. It is shown that, while

N

\mathfrak {N}

is of Baire first category in

M

\mathfrak {M}

, on the contrary the set

M

0

{\mathfrak {M}_0}

of all those f for which the zero solution of

(
∗
)

( \ast )

is asymptotically stable is a proper residual subset of

M

\mathfrak {M}

.
Matrix maps of bounded sequences in a Banach space
Authors: Maddox I. J.
Published: June 2010

Matrices of bounded linear operators are applied to bounded sequences in a Banach space. The main result is related to Knopp’s core theorem for scalar sequences and matrices. From the main result, necessary and sufficient conditions are given for a matrix of operators to map bounded sequences into null sequences in a Banach space.
Isometries of quasitriangular operator algebras
Authors: Hopenwasser Alan , Plastiras Joan
Published: June 2010

Let

(

P
n

)

({P_n})

be an increasing sequence of finite rank projections on a separable Hilbert space. Assume

P
n

{P_n}

converges strongly to the identity operator I. The quasitriangular operator algebra determined by

(

P
n

)

({P_n})

is defined to be the set of all bounded linear operators T for which
\[

lim

x
→
∞

‖

(
I
−

P
n

)
T

P
n

‖

=
0.

\lim \limits _{x \to \infty } \left \| {(I - {P_n})T{P_n}} \right \| = 0.

\]
In this note we prove that two quasitriangular algebras are unitarily equivalent if, and only if, there exists a unital linear isometry mapping one algebra onto the other.
A note on the extension of compact operators
Authors: Lima Ȧsvald
Published: June 2010

We prove that for a real Banach space A the following properties are equivalent: (1) For every pair X, Y of Banach spaces such that

X
⊆
Y

X \subseteq Y

, every compact linear operator

T
:
X
→
A

T:X \to A

admits an almost normpreserving extension

T
~

:
Y
→
A

\tilde T:Y \to A

. (2) The same as (1) but with

dim

X
=
2

\dim \, X = 2

and

Y
=

l
∞
3

Y = l_\infty ^3

.
Generic Morse-Smale diffeomorphisms have only trivial symmetries
Authors: Togawa Yoshio
Published: June 2010

The purpose of this paper is to prove that for a

C
1

{C^1}

-generic Morse-Smale diffeomorphism f, the set of symmetries of f,

Z
(
f
)

Z(f)

, is equal to

{

f
k

|

k
∈

Z

}

\{ {f^k}|k \in {\mathbf {Z}}\}

.
Large minimal realizers of a partial order
Authors: Maurer Stephen B. , Rabinovitch I.
Published: June 2010

The size of a minimum realizer of a partial order is called the dimension of that partial order. Here we initiate the study of minimal realizers which are not minimum. As an aid to the study of such realizers, we associate to each minimal realizer certain critical digraphs. We characterize all such critical digraphs for the antichain on n elements, and consequently deduce that for

n
⩾
4

n \geqslant 4

, the maximum size of a minimal realizer is

[

n
2

/

4
]

[{n^2}/4]

.
An uncountable Noetherian ring with free additive group
Authors: O’Neill John D.
Published: June 2010

An example is given of an uncountable Noetherian ring whose additive group is free. This answers a question posed indirectly by L. Fuchs.
A metrizable cancellative semigroup without translation invariant metric
Authors: Kranz Przemysław , Guo Mai
Published: June 2010

This note gives an example of a topological cancellative abelian metrizable semigroup without any translation invariant metric, hence showing that semigroups behave in this respect differently from groups [1].
A reciprocity theorem for tensor products of group representations
Authors: Moore Calvin C. , Repka Joe
Published: June 2010

Let G be a type I separable locally compact group. By studying a representation of

G
×
G
×
G

G \times G \times G

we show that a measure class

λ
\lambda

on

G
×
G
×
G

G \times G \times G

which describes the decompositions of tensor products is invariant under permutations, and that the multiplicity

n
(

π
1

,

π
2

,

π
3

)

n({\pi _1},{\pi _2},{\pi _3})

of

π
¯

3

{\bar \pi _3}

in

π
1

⊗

π
2

{\pi _1} \otimes {\pi _2}

is a symmetric function of its variables up to a

λ
\lambda

null set.
A necessary and sufficient condition that a function on the maximal ideal space of a Banach algebra be a multiplier
Authors: Wood James A.
Published: June 2010

Consider a regular commutative, semisimple Banach algebra with a bounded approximate identity whose Gelfand transforms have compact support. A necessary and sufficient condition is given for a complex valued function defined on the maximal ideal space to determine a multiplier of the algebra. This theorem is similar to one proved by F. T. Birtel, but omits Birtel’s assumption that the algebra be topologically embeddable in its second dual.
A stability theorem on quasi-reflexive operators
Authors: Wu Tsu-Chih
Published: June 2010

A range-closed bounded linear operator between Banach spaces is quasi-reflexive if both its kernel and cokernel are quasi-reflexive spaces. Under suitable conditions, if an operator is sufficiently close to a quasi-reflexive operator, it is itself quasi-reflexive.
More on connectedness im kleinen and local connectedness in 𝐶(𝑋)
Authors: Goodykoontz Jack T.
Published: June 2010

Let X be a compact connected metric space and

2
X

(
C
(
X
)
)

{2^X}(C(X))

denote the hyperspace of closed subsets (subcontinua) of X. Let

M
∈
C
(
X
)

M \in C(X)

. If

2
X

{2^X}

is connected im kleinen at M, then

C
(
X
)

C(X)

is locally arcwise connected at M. A characterization of connectedness im kleinen in

C
(
X
)

C(X)

is given. Indecomposability of X is related to an absence of local connectedness in

2
X

{2^X}

and

C
(
X
)

C(X)

. An example is given of a continuum X and a subcontinuum M such that

C
(
X
)

C(X)

is connected im kleinen at M but not locally connected at M.
Power maps and principal bundles
Authors: Noakes J. L.
Published: June 2010

Let G be a path connected topological group. We investigate the integers m for which the mth power map on G extends to an overmap of principal G-bundles.
The image of 𝐺 bordism in 𝑍₂ bordism
Authors: Beem R. Paul
Published: June 2010

For finite cyclic groups G of even order, the image of unoriented G bordism in Z/2Z bordism and the kernel of the extension homomorphism from Z/2Z to G bordism depend only on whether or not the order of G is divisible by four. If so, then these sets are equal and are equal to the image of circle bordism in Z/2Z bordism and the kernel of extension to circle bordism. If not, then extension is a monomorphism and restriction is an epimorphism.
States on quantum logics and their connection with a theorem of Alexandroff
Authors: Béaver O. R. , Cook T. A.
Published: June 2010

We generalize the notion of regularity of measures to quantum logics and then prove that each regular finitely additive state on a quantum logic is countably additive. Examples are given from measure theory and quantum mechanics.
Minimal injective resolutions under flat base change
Authors: Foxby Hans-Bjørn , Thorup Anders
Published: June 2010

For a flat morphism

φ
:
A
→
B

\varphi :A \to B

of noetherian rings, the minimal injective resolution of the B-module

M

⊗
A

B

M{ \otimes _A}B

is described in terms of the minimal injective resolution of the finitely generated A-module M and the minimal injective resolutions of the fibers of

φ
\varphi

.
Class groups of rings of invariants
Authors: Waterhouse William C.
Published: June 2010

Let G be a connected algebraic group acting on a Krull domain S with ring of invariants R. Then an extension E of

Cl

(
R
)

{\text {Cl}}(R)

by a free abelian group (related to blowing up) maps to

Cl

(
S
)

{\text {Cl}}(S)

with kernel related to characters of G.
On the nonexistence of hypercommuting polynomials
Authors: Kovacs Amos
Published: June 2010

If

f
(

x
1

,
…
,

x
n

)

f({x_1}, \ldots ,{x_n})

is not central for R, then the additive group generated by all specializations of f in R contains a noncentral Lie ideal of R. This is used, among other things, to prove: Theorem. Let R be a semiprime algebra over an infinite field,

f
1

,
…
,

f
t

{f_1}, \ldots ,{f_t}

polynomials in disjoint sets of variables all noncentral for R. Then, if R satisfies

S
t

[

f
1

,
…
,

f
t

]

{S_t}[{f_1}, \ldots ,{f_t}]

, R must satisfy

S
t

[

x
1

,
…
,

x
t

]

{S_t}[{x_1}, \ldots ,{x_t}]

.
A construction of simple principal right ideal domains
Authors: Cohn P. M.
Published: June 2010

It is shown how simple principal ideal domains can be obtained from any principal right ideal domain by localization. When no localization is needed one can, under favourable conditions, obtain a simple principal right (but not left) ideal domain, and an easy example is given.
Linear homeomorphisms of some classical families of univalent functions
Authors: Hartmann Frederick W.
Published: June 2010

The extreme points of the closed convex hull of some classical families of univalent functions analytic on the open unit disk, e.g. the convex, K, and starlike, St, have recently been characterized. These characterizations are used to determine an explicit representation for the class of linear homeomorphisms of the extreme points of the closed convex hulls of K and St and thus of the hulls themselves. With the aid of these representations it is shown that every linear homeomorphism of K or St is a rotation, i.e. convolution with

{
exp
⁡
(
i
n
θ
)
:
n
=
0
,
1
,
…
}

\{ \exp (in \theta ):n = 0,1, \ldots \}

. In the way of a positive result: if

P

\mathcal {P}

is the convex set of analytic functions with positive real part and

f
(
0
)
=
1

f(0) = 1

and

L

\mathcal {L}

is a linear homeomorphism of

P

\mathcal {P}

, then

L

(
S
t
)
⊂
S
t

\mathcal {L}(St) \subset St

, but

L

(
K
)
⊄
K

\mathcal {L}(K) \not \subset K

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

Let

f
(
z
)

f(z)

be an entire function of order 1, type

τ
\tau

having no zero in

Im

z
>
0

\operatorname {Im} \;z > 0

. If

h
f

(
−
π

/

2
)
=
τ
,

h
f

(
π

/

2
)
⩽
0

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

then it is known that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript negative normal infinity greater-than x greater-than normal infinity Endscripts StartAbsoluteValue f prime left-parenthesis x right-parenthesis EndAbsoluteValue greater-than-or-slanted-equals left-parenthesis tau slash 2 right-parenthesis sup Underscript negative normal infinity greater-than x greater-than normal infinity Endscripts StartAbsoluteValue f left-parenthesis x right-parenthesis EndAbsoluteValue">

sup

−
∞
>
x
>
∞

|

f
′

(
x
)

|

⩾
(
τ

/

2
)

sup

−
∞
>
x
>
∞

|

f
(
x
)

|

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

. In this paper we consider the case when

f
(
z
)

f(z)

has no zero in

Im

z
>
k
,
k
⩽
0

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

and obtain a sharp result.
Some counterexamples concerning intrinsic distances
Authors: Barth Theodore J.
Published: June 2010

Examples of the following are given: (i) complex spaces whose Carathéodory distances are not inner; (ii) a taut complex space that is not complete hyperbolic; (iii) a complex space that is complete hyperbolic

mod

Δ

\bmod \; \Delta

but not taut

mod

Δ

\bmod \; \Delta

; (iv) a bounded pseudoconvex domain that is not taut.
Reproducing kernels for 𝑞-Jacobi polynomials
Authors: Al-Salam Waleed A. , Ismail Mourad E. H.
Published: June 2010

We derive a family of reproducing kernels for the q-Jacobi polynomials
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi Subscript n Superscript left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis x right-parenthesis equals Subscript 2 Baseline normal upper Phi 1 left-parenthesis q Superscript negative n Baseline comma q Superscript n minus 1 plus beta Baseline semicolon q Superscript alpha Baseline semicolon q comma q x right-parenthesis">

Φ
n

(
α
,
β
)

(
x
)

=
2

Φ
1

(

q

−
n

,

q

n
−
1
+
β

;

q
α

;
q
,
q
x
)

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

. This is achieved by proving that the polynomials

Φ
n

(
α
,
β
)

(
x
)

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

satisfy a discrete Fredholm integral equation of the second kind with a positive symmetric kernel, then applying Mercer’s theorem.
On points at which a set is cone-shaped
Authors: Edelstein M. , Keener L. , O’Brien R.
Published: June 2010

A set

S

\mathcal {S}

in a normed linear space X is said to be cone-shaped at

x
∈
X

x \in X

if there is a closed half-space that has x in its bounding hyperplane and contains

{
y
∈

S

:
[
x
,
y
]
⊂
S
}

\{ y \in \mathcal {S}:[x,y] \subset S\}

. The point x is called a cone point. In this paper it is shown that if X has an equivalent uniformly convex and uniformly smooth norm and if

S

\mathcal {S}

is a closed bounded subset with the finite visibility property for cone points (i.e., for every finite set F of cone points of S there is a point

z
∈
S

z \in S

such that

[
z
,
y
]
⊂

S

[z,y] \subset \mathcal {S}

for all

y
∈
F

y \in F

), then S is starshaped.
Unitary parts of generalized Toeplitz operators
Authors: Butz Jeffrey R.
Published: June 2010

For a generalized Toeplitz operator acting on a Hilbert space with unilateral shift, necessary and sufficient conditions are obtained for the existence of a nontrivial unitary part, and an explicit description of this unitary part is given.
On the extreme points of the interval between two operators
Authors: Narcowich Francis J.
Published: June 2010

Given that A, B are operators on a complex Hilbert space, and that

B
−
A

B - A

is nonnegative, the interval between A and B consists of every operator, G, such that both

B
−
G

B - G

and

G
−
A

G - A

are nonnegative. The extreme points of such an interval are exhibited and the interval is shown to be the closure of the convex hull of these extreme points in the weak-operator topology.
Open and uniformly open relations
Authors: Mah P. , Naimpally S. A.
Published: June 2010

It is shown that if

(
X
,
δ
)

(X,\delta )

is an Efremovič proximity space, Y is a topological space,

R
⊂
X
×
Y

R \subset X \times Y

is an injective relation, then R is open if and only if R is weakly open, nearly open and

R
[
X
]

R[X]

is open in Y. An analogous result is proved when X is a uniform space and Y a Morita uniform space: (i) if R is uniformly open, then R is weakly open and uniformly nearly open; (ii) if R is weakly open, and uniformly nearly open, then R is uniformly open on X to

R
[
X
]

R[X]

. These results include, as special cases, results of Kelley, Pettis and Weston.
An estimate of the density at the boundary of an integral current modulo 𝑣
Authors: Paur Sandra O.
Published: June 2010

An inequality is obtained which bounds the density at

z
∈

R

n

z \in {{\mathbf {R}}^n}

of the boundary of a

k
+
1

k + 1

dimensional integral current modulo

ν

(
S

)
ν

\nu \;{(S)^\nu }

by the density of

(
S

)
ν

{(S)^\nu }

at z. Also, the concept of boundary tangent developed in [3] is shown to be in agreement with Federer’s concept of a measuretheoretic exterior normal if

ν
=
0

\nu = 0

and S is obtained by integration over an

L

n

{\mathfrak {L}^n}

measurable subset of

R

n

{{\mathbf {R}}^n}

.
Continuous homomorphisms are differentiable
Authors: Holmes J. P.
Published: June 2010

Suppose X is a Banach space, D is an open set of X containing 0, and V is a continuously differentiable function from DXD to X satisfying

V
(
0
,
x
)
=
V
(
x
,
0
)
=
x

V(0,x) = V(x,0) = x

for each x in D. If T is a continuous function from

[
0
,
1
]

[0,1]

into D satisfying

T
(
0
)
=
0

T(0) = 0

and

V
(
T
(
s
)
,
T
(
t
)
)
=
T
(
s
+
t
)

V(T(s),T(t)) = T(s + t)

whenever each of s, t, and

s
+
t

s + t

is in

[
0
,
1
]

[0,1]

then T is continuously differentiable on

[
0
,
1
]

[0,1]

.
An 𝑛log𝑛 lower bound on synchronous combinational complexity
Authors: Harper L. H.
Published: June 2010

Synchronous combinational machines are combinational machines such that the length of all paths from inputs to a logic element are the same. In this paper is is shown that any Boolean function of n variables satisfying certain subfunction conditions (which are satisfied by “almost all” such functions) must have synchronous combinational complexity at least

n
log
⁡
n

n\log n

.
Left and right invariance in an integral domain
Authors: Beauregard Raymond A.
Published: June 2010

A ring is said to be right (left) invariant if each of its right (left) ideals is twosided. In this paper we resolve the conjecture: Every right invariant integral domain which satisfies the left Ore (multiple) condition is left invariant. A proof is given for the class of LCM domains satisfying a finiteness condition. An example is given to show that the LCM hypothesis cannot be dropped. A second example shows that the conjecture fails even in a Bezout domain which does not have the finiteness condition. The problem of right versus left boundedness is also considered.
A superposition theorem for bounded continuous functions
Authors: Demko Stephen
Published: June 2010

It is shown that there exist

2
n
+
1

2n + 1

real valued, continuous functions

ϕ
0

,
…
,

ϕ

2
n

{\phi _0}, \ldots ,{\phi _{2n}}

defined on

R

n

{{\mathbf {R}}^n}

such that every bounded real valued continuous function on

R

n

{{\mathbf {R}}^n}

is expressible in the form

Σ

i
=
0

2
n

g
∘

ϕ
i

\Sigma _{i = 0}^{2n}g \circ {\phi _i}

for some

g
∈
C
(

R

)

g \in C({\mathbf {R}})

. Extensions to some unbounded functions are also made.
A boundary value problem for 𝐻^{∞}(𝐷)
Authors: Cahill Rotraut Goubau
Published: June 2010

Let

W
=

⋃

n
=
1

∞

W
n

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

be an

F
σ

{F_\sigma }

subset of the unit circle of measure 0 and let

{

q
n

}
,
n
⩾
1

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

, be a decreasing sequence with

q
1

⩽
1

{q_1} \leqslant 1

and

lim
n

q
n

=
0

{\lim _n}{q_n} = 0

. There exists an H in

H
∞

(
D
)

{H^\infty }(D)

of norm

q
1

{q_1}

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

q
1

{q_1}

on

W
1

{W_1}

and

q

n
+
1

{q_{n + 1}}

on

W

n
+
1

∖

⋃

k
=
1

n

W
k

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

. If W is simultaneously a

G
δ

{G_\delta }

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

W
=
{

e

i

w
n

}
,
n
⩾
1

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

, there is such an H of norm 1 for which

lim

r
→
1

H
(
r

e

i

w
n

)
=
1

/

n

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

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

{

S
λ

}
,
λ
⩾
1

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

, real, for which the function h, defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h left-parenthesis x right-parenthesis equals a Subscript n Baseline plus left-bracket left-parenthesis inf left-brace right-brace vertical-bar lamda vertical-bar element-of element-of xS right-parenthesis minus n right-bracket left-parenthesis a Subscript n plus 1 Baseline minus a Subscript n Baseline right-parenthesis comma a Subscript n Baseline equals minus ln q Subscript n Baseline">

h
(
x
)
=

a
n

+
[
(
inf
{
λ

|

x
∈
S
}
)
−
n
]
(

a

n
+
1

−

a
n

)
,

a
n

=
−
ln

q
n

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

, is such that the function
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H left-parenthesis w right-parenthesis equals exp left-parenthesis negative 1 slash 2 pi right-parenthesis integral left-bracket left-parenthesis w plus e Superscript i u Baseline right-parenthesis slash left-parenthesis e Superscript i u Baseline minus w right-parenthesis right-bracket h left-parenthesis u right-parenthesis">

H
(
w
)
=
exp
⁡
(
−
1

/

2
π
)
∫
[
(
w
+

e

i
u

)

/

(

e

i
u

−
w
)
]
h
(
u
)

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

du has the required properties. Some of the techniques used are similar to those developed in an earlier paper [1].
Cluster sets on open Riemann surfaces
Authors: Niimura Mikio
Published: June 2010

Generalizations of theorems of Gross-Iversen type on exceptional values are given for analytic mappings on open Riemann surfaces.
The hyperbolicity of the complement of 2𝑛+1 hyperplanes in general position in 𝑃_{𝑛} and related results
Authors: Green Mark L.
Published: June 2010

Using a modified version of a technique of R. Brody, a simple proof is found that the complement of 2n + 1 hyperplanes in general position in

P

n

{{\mathbf {P}}_n}

is complete hyperbolic and hyperbolically embedded in

P

n

{{\mathbf {P}}_n}

. In fact, a more general result is obtained showing that a suitable Picard theorem is sufficient to imply hyperbolicity in a large class of algebro-geometric situations.
Essentially Hermitian operators on 𝑙₁ are compact perturbations of Hermitians
Authors: Legg David , Ward Joseph
Published: June 2010

In this paper, we present a solution to one case of a problem of F. F. Bonsall; namely, that every essentially Hermitian operator on

l
1

{l_1}

is a compact perturbation of a Hermitian operator.
New topological extension properties
Authors: Terada Toshiji
Published: June 2010

A topological property

P

\mathcal {P}

is called an extension property if

P

\mathcal {P}

is closed-hereditary and productive. In this paper new topological extension properties are introduced, which to some extent fill up a gap between complete regularity and compactness.
A new proof of Caristi’s fixed point theorem
Authors: Siegel Jerrold
Published: June 2010

An elementary proof of Caristi’s generalization of the contraction theorem is presented.
Isomorphisms of sums of Boolean algebras
Authors: Trnková Věra , Koubek Václav
Published: June 2010

We prove that any Boolean algebra is a homomorphic image of a Boolean algebra B isomorphic to

B
+
B
+
B

B + B + B

but not to

B
+
B

B + B

.
Simple continued fractions and special relativity
Authors: Hensley Douglas
Published: June 2010

Let

E
0

,

E
1

,
…
,

E
n

{E_0},{E_1}, \ldots ,{E_n}

be inertial frames of reference in a one dimensional relativistic universe where the speed of light is

c
=

k

c = \sqrt k

, k some natural number. For

n
⩾
1

n \geqslant 1

let

E
n

{E_n}

have velocity 1 with respect to

E

n
−
1

{E_{n - 1}}

. Let

x
n

{x_n}

denote the velocity of

E
n

{E_n}

with respect to

E
0

{E_0}

. Then only if

k
=
2
,
3

k = 2,3

or 5 will

x
n

{x_n}

be a simple continued fraction convergent of

k

\sqrt k

infinitely often.
On Weitzenböck’s theorem in positive characteristic
Authors: Fauntleroy A.
Published: June 2010

Let k be an algebraically closed field and let

f
:

G
a

→

GL

(
V
)

f:{G_a} \to {\text {GL}}(V)

be a finite-dimensional k-rational representation of the additive group

G
a

{G_a}

. If the subspace of

G
a

{G_a}

-fixed points in V is a hyperplane, then the ring of

G
a

{G_a}

-invariant polynomial functions on V is finitely generated over k. This result is an analog of a classical theorem of Weitzenböck, a modern proof of which has been given by C. S. Seshadri.
Groups with an abelian Sylow subgroup
Authors: Ballard John W.
Published: June 2010

The purpose of this note is to obtain a modular refinement of Brauer’s induction theorem for groups with an abelian Sylow subgroup.
Generalizations of the Vitali-Hahn-Saks theorem on vector measures
Authors: Oberle Richard A.
Published: June 2010

In this paper an extension of the Brook’s control measure existence theorem for families of vector measures is established. This result is applied to pointwise convergent sequences of finitely additive vector measures to obtain a generalization of the Vitali-Hahn-Saks theorem.
Equipartition of energy for a class of second order equations
Authors: Costa David G.
Published: June 2010

We consider the Cauchy problem for a class of second order equations of the form

(
d

/

d
t
−

A
2

)
(
d

/

d
t
−

A
1

)
u
(
t
)
=
0

(d/dt - {A_2})(d/dt - {A_1})u(t) = 0

in a Hilbert space H. A d’Alembert type solution formula is presented and we give a suitable definition of energy. Also, we derive a necessary and sufficient condition for the asymptotic equipartition of energy (Kinetic and Potential) to hold. These results generalize corresponding results for the abstract wave equation

(

d
2

/

d

t
2

+

A
2

)
u
(
t
)
=
0

({d^2}/d{t^2} + {A^2})u(t) = 0

.
Unitary parts of contractive Hankel matrices
Authors: Butz Jeffrey R.
Published: June 2010

For a Hankel matrix

H
=
(

c

j
+
k

)

H = ({c_{j + k}})

which is a contraction, necessary and sufficient conditions are obtained for the existence of a nontrivial unitary part, and an explicit description of this unitary part is given.
Zero-extreme points and the generalized convex kernel
Authors: Sparks Arthur G.
Published: June 2010

Let S be a compact simply connected set in the plane. Let

K
(
n
)

K(n)

denote the generalized convex kernel of S of order n, bd S the boundary of S,

E
(
0
,
S
)

E(0,S)

the set of 0-extreme points of S, and for

x
∈
S

x \in S

, let

K
(
n
,
x
)

K(n,x)

denote the nth order convex kernel of x in S. It is known that

K
(
n
)
=
∩
{
K
(
n
,
x
)

|

x
∈

bd

S
}

K(n) = \cap \{ K(n,x)|x \in {\text {bd}}\;S\}

and in certain cases, it is known that

K
(
1
)
=
∩
{
K
(
1
,
x
)

|

x
∈
E
(
0
,
S
)
}

K(1) = \cap \{ K(1,x)|x \in E(0,S)\}

. The main result of this paper extends the above mentioned results for certain sets. It is shown that

K
(
n
)
=
∩
{
K
(
n
,
x
)

|

x
∈
E
(
0
,
S
)
}

K(n) = \cap \{ K(n,x)|x \in E(0,S)\}

for certain compact simply connected sets S in the plane. In the process of obtaining this result, a characterization of the boundary is also obtained.
Torus actions on cohomological complex projective spaces
Authors: Wang Kai
Published: June 2010

In this note, we will compute the Pontrjagin classes of those topological manifolds which have the same rational cohomology algebra as

C

P
n

C{P^n}

and which admit effective topological

T
n

{T^n}

actions.
On Anosov diffeomorphisms on the plane
Authors: Mendes P.
Published: June 2010

A diffeomorphism on the plane holding an integrable Anosov splitting is called an Anosov diffeomorphism on the plane. In this paper the author proves that Anosov diffeomorphisms on the plane are structurally stable.
On the innovation theorem
Authors: Lin T. F.
Published: June 2010

Let

z
(
t
)
,
0
⩽
t
⩽
T

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

, be the signal process and

y
(
t
)
=

∫
0
t

z
(
r
)

d
r
+
w
(
t
)

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

be the observation process where

w
(
t
)

w(t)

is a process of independent increments. It is shown that, under certain conditions, the innovation process
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v left-parenthesis t right-parenthesis equals y left-parenthesis t right-parenthesis minus integral Subscript 0 Superscript t Baseline upper E left-parenthesis z left-parenthesis r right-parenthesis vertical-bar y left-parenthesis u right-parenthesis comma 0 less-than-or-slanted-equals u less-than-or-slanted-equals r right-parenthesis d r">

v
(
t
)
=
y
(
t
)
−

∫
0
t

E
(
z
(
r
)

|

y
(
u
)
,
0
⩽
u
⩽
r
)

d
r

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

, has the same probability law as

w
(
t
)

w(t)

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

Assosymmetric rings are ones which satisfy the law
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x comma y comma z right-parenthesis equals left-parenthesis upper P left-parenthesis x right-parenthesis comma upper P left-parenthesis y right-parenthesis comma upper P left-parenthesis z right-parenthesis right-parenthesis">

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

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

for each permutation P of x, y, z. Let A be an assosymmetric ring having characteristic different from 2 or 3. We show that if A is solvable then A is nilpotent. Also, if each subring generated by a single element is nilpotent, and if A has D.C.C. on right ideals, then A is nilpotent. We also give an example showing that the Wedderburn Principal Theorem fails for assosymmetirc rings.
Abstract reflections and Coxeter groups
Authors: Goldschmidt David M.
Published: June 2010

A notion of “abstract reflection” is introduced and used to characterize the Coxeter groups. Several known results on Coxeter groups are obtained as easy corollaries.
On nonoscillatory linear differential equations of second order
Authors: Hartman Philip
Published: June 2010

For the equation

y
+
q
(
t
)
y
=
0

y + q(t)y = 0

, we derive necessary conditions for nonoscillation in terms of “general means” of

Q
(
t
)
=

∫
0
t

q

d
s

Q(t) = \smallint _0^tq\,ds

.
A class of singular neutral-differential systems
Authors: Fitzpatrick W. J. , Grimm L. J.
Published: June 2010

Noetherian operator theory is used to prove an existence theorem for a singular functional-differential system. An analogue of the standard existence and uniqueness results at an ordinary point follows as a corollary.
A remark on the direct method of the calculus of variations
Authors: Penot J. P.
Published: June 2010

This note deals with the problem of minimizing a real-valued function f on a weakly closed subset of a reflexive Banach space. We use a mild monotonicity assumption introduced by P. Hess [11] on the derivative f’ of f to get the weak lower semicontinuity of f. We show that one can dispense with any continuity assumption on f’, so that we get a true generalization of F. E. Browder’s results [4]. The relevance of the monotonicity property to the calculus of variations is shown by an example.
On 𝜇-spaces and 𝑘_{𝑅}-spaces
Authors: Blasco J. L.
Published: June 2010

In this paper it is proved that when X is a

k
R

{k_R}

-space then

μ
X

\mu X

(the smallest subspace of

β
X

\beta X

containing X with the property that each of its bounded closed subsets is compact) also is a

k
R

{k_R}

-space; an example is given of a

k
R

{k_R}

-space X such that its Hewitt realcompactification,

υ
X

\upsilon X

, is not a

k
R

{k_R}

-space. We show with an example that there is a non-

k
R

{k_R}

-space X such that

υ
X

\upsilon X

and

μ
X

\mu X

are

k
R

{k_R}

-spaces. Also we answer negatively a question posed by Buchwalter: Is

μ
X

\mu X

the union of the closures in

υ
X

\upsilon X

of the bounded subsets of X? Finally, without using the continuum hypothesis, we give an example of a locally compact space X of cardinality

ℵ
1

{\aleph _1}

such that

υ
X

\upsilon X

is not a k-space.
Counting group elements of order 𝑝 modulo 𝑝²
Authors: Herzog Marcel
Published: June 2010

Let G be a finite group of order divisible by the prime p. It is shown that the number of elements of G of order p is congruent to

−
1

- 1

modulo

p
2

{p^2}

, unless a Sylow p-subgroup of G is cyclic, generalized quaternion, dihedral or quasidihedral.
A unique continuation theorem involving a degenerate parabolic operator
Authors: Lair Alan V.
Published: June 2010

We consider the degenerate parabolic operator

L
[
u
]
=
γ
B
[
u
]
−

u
t

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

on a domain

D
=
Ω
×
(
0
,
T
]

D = \Omega \times (0,T]

where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B left-bracket u right-bracket equals normal upper Sigma Subscript i comma j equals 1 Superscript n Baseline left-parenthesis a Subscript i j Baseline left-parenthesis x right-parenthesis u Subscript x Sub Subscript j Subscript Baseline right-parenthesis Subscript x Sub Subscript i">

B
[
u
]
=

Σ

i
,
j
=
1

n

(

a

i
j

(
x
)

u

x
j

)

x
i

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

and

γ
\gamma

is an arbitrary complex number. Classically,

γ
=
1

\gamma = 1

and the real-valued matrix

(

a

i
j

)

({a_{ij}})

is positive definite. We assume

(

a

i
j

)

({a_{ij}})

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

|

L
[
u
]

|

⩽
c

|

u

|

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

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

Ω
\Omega

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

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

h

n
k

=

(

n
−
k

n
+
α

)

Δ

n
−
k

ν
k

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

, where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta nu Subscript n Baseline equals nu Subscript n Baseline minus nu Subscript n plus 1 Baseline comma nu Subscript n Baseline equals integral Subscript 0 Superscript 1 Baseline t Superscript n plus alpha Baseline d beta left-parenthesis t right-parenthesis">

Δ

ν
n

=

ν
n

−

ν

n
+
1

,

ν
n

=

∫
0
1

t

n
+
α

d
β
(
t
)

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

for some

β
(
t
)

\beta (t)

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

α
⩾
0

\alpha \geqslant 0

. The matrix

H
α

{H_\alpha }

generated by

ν
n

=

(
n
+
α
+
1

)

−
1

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

is shown to be a subnormal operator on

l
2

{l^2}

if

α
\alpha

is a nonnegative integer.
Decomposable tensors as a quadratic variety
Authors: Grone Robert
Published: June 2010

Let

V
i

{V_i}

be a finite dimensional vector space over a field F for each

i
=
1
,
2
,
…
,
m

i = 1,2, \ldots ,m

, and let z be a tensor in

V
1

⊗
⋯
⊗

V
m

{V_1} \otimes \cdots \otimes {V_m}

. In this paper a set of homogeneous quadratic polynomials in the coordinates of z is exhibited for which the associated variety is the set of decomposable tensors. In addition, a question concerning the maximal tensor rank in such a situation is answered, and an application to other symmetry classes of tensors is cited.
An unbaireable stratifiable space
Authors: van Douwen Eric K.
Published: June 2010

There is a stratifiable space which cannot be densely embedded in a stratifiable Baire space, in fact not even in a regular Baire

σ
\sigma

-space. Every regular Baire

σ
\sigma

-space, hence every stratifiable Baire space, has a dense metrizable

G
δ

{G_\delta }

-subspace.
On the normability of the intersection of 𝐿_{𝑝} spaces
Authors: Bell Wayne C.
Published: June 2010

The set

L
ω

=

⋂

p
=
1

∞

L
p

[
0
,
1
]

{L_\omega } = \bigcap \nolimits _{p = 1}^\infty {{L_p}[0,1]}

is not equal to

L
∞

[
0
,
1
]

{L_\infty }[0,1]

since

L
ω

{L_\omega }

contains the function

−
ln
⁡
x

- \ln x

. Using the theory of

L
p

{L_p}

spaces for finitely additive set functions developed by Leader [9] we will prove several necessary and sufficient conditions for the normability of a generalization of

L
ω

{L_\omega }

. These include the equality and finite dimensionality of all the

L
p

{L_p}

spaces,

p
⩾
1

p \geqslant 1

.