bluuebunny/crossref_metadata_2025 huggingface dataset:
On Hilbert class fields in characteristic 𝑝>0 and their 𝐿-functions
Authors: Turner Stuart
Published: June 2010

Let k be a global field of characteristic

p
>
0

p > 0

with field of constants

F

q

{{\mathbf {F}}_q}

. Let

k
¯

\bar k

be an algebraic closure of k. In this note we study the subfields of

k
¯

\bar k

which are maximal unramified abelian extensions of k with field of constants

F

q

{{\mathbf {F}}_q}

. Each of these fields may be regarded as an analogue of the Hilbert class field of algebraic number theory [1, p. 79]. In §1 we recall the construction of these class fields and in §2 we show that if k has genus one, they are all

F

q

{{\mathbf {F}}_q}

-isomorphic. In §3 we show that this is not necessarily the case if the genus of k is greater than one. The argument there is based on an observation about the L-functions of the fields.
Nonoscillation criteria for elliptic equations in conical domains
Authors: Allegretto W.
Published: June 2010

Nonoscillation criteria are established for a class of higher order elliptic equations defined in conical domains.
Binary operations in the set of solutions of a partial difference equation
Authors: Zeilberger Doron
Published: June 2010

Let

P

\mathcal {P}

be a partial difference operator with constant coefficients in n independent (discrete) variables, and let

S

P

=
{
f
:

Z
n

→

C

;

P

f
=
0
}

{\mathcal {S}_\mathcal {P}} = \{ f:{Z^n} \to {\mathbf {C}};\mathcal {P}f = 0\}

. We introduce a certain class of binary operations

S

P

×

S

P

→

S

P

{\mathcal {S}_\mathcal {P}} \times {\mathcal {S}_\mathcal {P}} \to {\mathcal {S}_\mathcal {P}}

generalizing a binary operation introduced by Duffin and Rohrer.
On commutants of reductive algebras
Authors: Fong Che Kao
Published: June 2010

It is proved, for certain operators T, if

A

\mathcal {A}

is a reductive algebra and

T
∈

A

′

T \in \mathcal {A}’

, then

T
∗

∈

A

′

{T^ \ast } \in \mathcal {A}’

.
A characterization of certain compact flows
Authors: Knight Ronald A.
Published: June 2010

Compact flows on certain 2-manifolds are characterized in terms of the bilateral stability properties of the orbits.
On the nonexistence of a projection from functions of 𝑥 to functions of 𝑥ⁿ
Authors: Milman P.
Published: June 2010

The subspace
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi Superscript asterisk Baseline upper C Superscript normal infinity Baseline left-parenthesis bold upper R Superscript 1 Baseline right-parenthesis subset-of upper C Superscript normal infinity Baseline left-parenthesis bold upper R Superscript 1 Baseline right-parenthesis">

ϕ
∗

C
∞

(

R

1

)
⊂

C
∞

(

R

1

)

{\phi ^ \ast }{C^\infty }({{\mathbf {R}}^1}) \subset {C^\infty }({{\mathbf {R}}^1})

of all

C
∞

{C^\infty }

functions of

ϕ
(
x
)
=

x
n

,
n
=
1
,
2
,
3
,
…

\phi (x) = {x^n},n = 1,2,3, \ldots

, is a closed subspace of

C
∞

(

R

1

)

{C^\infty }({{\mathbf {R}}^1})

by Glaeser’s Composition Theorem. We prove that for

n
>
2

n > 2

there does not exist a linear continuous projection

π
\pi

from

C
∞

(

R

1

)

{C^\infty }({{\mathbf {R}}^1})

onto

ϕ
∗

C
∞

(

R

1

)

{\phi ^ \ast }{C^\infty }({{\mathbf {R}}^1})

.
The monotone class theorem in infinitary logic
Authors: Keisler H. Jerome
Published: June 2010

A monotone formula in the infinitary logic

L

ω
1

ω

{L_{{\omega _1}\omega }}

is a formula which is built up from finite formulas using only quantifiers and monotone countable conjunctions and disjunctions. The monotone class theorem from measure theory is used to show that every formula of

L

ω
1

ω

{L_{{\omega _1}\omega }}

is logically equivalent to a monotone formula (the monotone normal form theorem). The proof is effectivized in order to obtain similar normal form theorems for admissible logics

L
A

{L_A}

.
Recurrences for the sum of divisors
Authors: Ewell John A.
Published: June 2010

The author presents two recursive determinations of the sum of positive divisors of a given positive integer. Each recurrence is then discussed with regard to economy of computation, and in this light is compared with the well-known recurrence of Niven and Zuckerman. As far as methods of proof are concerned, everything is accomplished within the algebra of formal power series.
A remarkable class of continued fractions
Authors: Adams William W. , Davison J. L.
Published: June 2010

For any irrational number

α
\alpha

and integer

a
>
1

a > 1

, the continued fraction of

(
a
−
1
)

∑

r
=
1

∞

1

/

a

[
r
α
]

(a - 1)\sum _{r = 1}^\infty 1/{a^{[r\alpha ]}}

is computed explicitly in terms of the continued fraction of

α
\alpha

.
Mackey’s theorem for nonunitary representations
Authors: Kutzko P. C.
Published: June 2010

We present an analog to the Intertwining Number Theorem of Mackey in a setting which arises naturally in the study of the representation theory of p-adic linear groups.
A complete Kähler metric of positive curvature on 𝐶ⁿ
Authors: Klembeck Paul F.
Published: June 2010

A complete Kähler metric of positive curvature on

C

n

{{\mathbf {C}}^n}

is constructed and its importance is discussed.
A disconjugacy criterion for linear scalar differential operators
Authors: Muldowney James S.
Published: June 2010

It is shown that if a linear scalar differential operator is not disconjugate on an interval then each member of a certain family of first order vector differential equations has an oscillatory solution. Thus any condition which guarantees the nonoscillation of a member of the family is a disconjugacy criterion for the scalar operator. The form of the vector systems is convenient for the use of nonoscillation conditions developed by Nehari, Schwarz and Friedland.
A best possible extension of the Hausdorff-Young theorem
Authors: Young Robert M.
Published: June 2010

The purpose of this note is to show that a recent result of A. M. Sedleckiĭ on nonharmonic Fourier series in

L
p

(
−
π
,
π
)

{L^p}( - \pi ,\pi )

has as a simple consequence a “best possible” extension of the classical Hausdorff-Young theorem.
Quasi-complementors and complementors on certain Banach algebras
Authors: Wong Pak Ken
Published: June 2010

We study quasi-complementors and complementors on annihilator Banach algebras. We give conditions for a quasi-complementor to be a complementor.
Relationships between continuum neighborhoods in inverse limit spaces and separations in inverse limit sequences
Authors: Davis Harvey S.
Published: June 2010

The main result of this paper is the following theorem. Let

{

X
α

,

f

α
β

,
α
,
β
∈
I
}

\{ {X_\alpha },{f_{\alpha \beta }},\alpha ,\beta \in I\}

be an inverse system of compact Hausdorff spaces and continuous onto maps with inverse limit X. Let

p
∈
X

p \in X

and A be closed in X. There exists a continuum neighborhood of p disjoint from A if and only if there exists

α
∈
I

\alpha \in I

and disjoint sets U and V open in

X
α

{X_\alpha }

, neighborhoods respectively of

p
α

{p_\alpha }

and

A
α

{A_\alpha }

such that for all

β
⩾
α
,

f

α
β

−
1

(
U
)

\beta \geqslant \alpha ,f_{\alpha \beta }^{ - 1}(U)

lies in a single component of

X
β

−

f

α
β

−
1

(
V
)

{X_\beta } - f_{\alpha \beta }^{ - 1}(V)

. This is Theorem B of the text.
𝜎-locally finite maps
Authors: Michael E.
Published: June 2010

A map

f
:
X
→
Y

f:X \to Y

is called

σ
\sigma

-locally finite if every

σ
\sigma

-locally finite cover

A

\mathcal {A}

of X has a refinement

B

\mathcal {B}

such that

f
(

B

)

f(\mathcal {B})

is

σ
\sigma

-locally finite. The principal purpose of this paper is to provide proofs of some results on these maps which were announced by the author in a previous note.
Orientation-reversing periodic PL maps of lens spaces
Authors: Kim Paik Kee
Published: June 2010

We complete the classification of all orientation-reversing PL maps of period

4
k
(
k
>
1
)

4k(k > 1)

on lens spaces. All

Z
4

{Z_4}

-actions on the projective 3-space are also classified.
A new expression for umbral operators and power series inversion
Authors: Garsia A. M. , Joni S. A.
Published: June 2010

It is shown here the compositional inversion of formal power series has a natural formulation in terms of umbral operators. This fact is used to derive a new proof of the Lagrange inversion theorem by operator theoretic methods.
A note on two congruences on a groupoid
Authors: Nirmala Kumari Amma K.
Published: June 2010

Let S be a groupoid and

θ
p

,

θ
m

{\theta _p},{\theta _m}

the congruences on S defined as follows:

x

θ
p

y

(
x

θ
m

y
)

x{\theta _p}y\;(x{\theta _m}y)

iff every prime (minimal prime) ideal of S containing x contains y and vice versa. It is proved that

θ
p

{\theta _p}

is the smallest congruence on S for which the quotient is a semilattice. It is also shown that

S

/

θ
m

S/{\theta _m}

is a disjunction semilattice if S has 0 and is a Boolean algebra if S is intraregular and closed for pseudocomplements. Some connections between the ideals of S and those of the quotients are established. Congruences similar to

θ
p

{\theta _p}

and

θ
m

{\theta _m}

are defined on a lattice using lattice-ideals; quotients under these are distributive lattices.
A Loewner approach to a coefficient inequality for bounded univalent functions
Authors: De Temple Duane W. , Jenkins James A.
Published: June 2010

The Loewner theory is used to obtain the sharp upper bound for the functional

Re
⁡
{

e

2
i
θ

(

a
3

−

a
2
2

)
+
4
σ

e

i
θ

a
2

}

\operatorname {Re} \{ {e^{2i\theta }}({a_3} - a_2^2) + 4\sigma {e^{i\theta }}{a_2}\}

over the class of univalent functions

f
(
z
)
=
b
(
z
+

a
2

z
2

+

a
3

z
3

+
…
)

f(z) = b(z + {a_2}{z^2} + {a_3}{z^3} + \ldots )

which map the unit disc into itself;

θ
∈

R

,
σ
∈
[
0
,
1
]

\theta \in {\mathbf {R}},\sigma \in [0,1]

and

b
∈
(
0
,
1
]

b \in (0,1]

are fixed parameters.
Asymptotic behavior of solutions of a fourth order nonlinear differential equation
Authors: Taylor W. E.
Published: June 2010

In this paper, the asymptotic properties of solutions of a certain fourth order differential equation are considered. Sufficient conditions for oscillation are also given.
Extensions of fixed point theorems of Rhoades and Ćirić
Authors: Pal T. K. , Maiti M.
Published: June 2010

In this paper fixed point theorems have been established for the mappings which are contractive over two consecutive elements of an orbit. The results, thus established, extend the theorems of B. E. Rhoades and Lj. B. Ćirić.
The Turán-Kubilius inequality
Authors: Elliott P. D. T. A.
Published: June 2010

The author gives a short proof of the Turán-Kubilius inequality with an improved constant.
Distinct sums over subsets
Authors: Hanson F. , Steele J. M. , Stenger F.
Published: June 2010

A finite set of integers with distinct subset sums has a precisely bounded Dirichlet series.
On the intersection of free factors of a free group
Authors: Burns R. G. , Chau T. C. , Solitar D.
Published: June 2010

An example is given of a descending sequence of free factors of a free group F whose intersection is not a free factor of F.
Conditions for commutative semigroups to have nontrivial homomorphisms into nonnegative (positive) reals
Authors: Kobayashi Yuji
Published: June 2010

Let S be a commutative cancellative semigroup of finite rank (the free rank of the quotient group of S is finite). We give a necessary and sufficient condition for S to have a nontrivial homomorphism into the additive semigroup of nonnegative (positive) reals. We also give a counter-example in the case of infinite rank.
A mapping theorem for logarithmic and integration-by-parts operators
Authors: Appling William D. L.
Published: June 2010

Suppose U is a set, F is a field of subsets of U,

p

A
B

{\mathfrak {p}_{AB}}

is the set of all bounded real-valued finitely additive functions defined on F, and W is a collection of functions from F into

exp
⁡
(

R

)

\exp ({\mathbf {R}})

, closed under multiplication, each element of which has range union bounded and bounded away from 0. Let

P

\mathcal {P}

denote the set to which T belongs iff T is a function from W into

p

A
B

{\mathfrak {p}_{AB}}

such that if each of

α
\alpha

and

β
\beta

is in W and V is in F, then the following integrals exist and the following “integration-by-parts” equation holds:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper V Endscripts alpha left-parenthesis upper I right-parenthesis upper T left-parenthesis beta right-parenthesis left-parenthesis upper I right-parenthesis plus integral Underscript upper V Endscripts beta left-parenthesis upper I right-parenthesis upper T left-parenthesis alpha right-parenthesis left-parenthesis upper I right-parenthesis equals upper T left-parenthesis alpha beta right-parenthesis left-parenthesis upper V right-parenthesis period">

∫
V

α
(
I
)
T
(
β
)
(
I
)
+

∫
V

β
(
I
)
T
(
α
)
(
I
)
=
T
(
α
β
)
(
V
)
.

\int _V \alpha (I)T(\beta )(I) + \int _V {\beta (I)T(\alpha )(I) = T(\alpha \beta )(V).}

\]
Let

L

\mathfrak {L}

denote the set to which S belongs iff S is a function from W into

p

A
B

{\mathfrak {p}_{AB}}

such that if each of

α
\alpha

and

β
\beta

is in W, then the integral

∫
U

α
(
I
)
S
(
β
)
(
I
)

\smallint _U {\alpha (I)S(\beta )(I)}

exists and the following “logarithmic” equation holds:

S
(
α
β
)
=
S
(
α
)
+
S
(
β
)

S(\alpha \beta ) = S(\alpha ) + S(\beta )

. It is shown that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace left-parenthesis upper T comma upper S right-parenthesis colon upper T in script upper P comma upper S equals left-brace left-parenthesis alpha comma integral left-parenthesis 1 slash alpha right-parenthesis upper T left-parenthesis alpha right-parenthesis right-parenthesis colon alpha in upper W right-brace right-brace">

{
(
T
,
S
)
:
T

in

P

,

S
=
{
(
α
,

∫

(
1

/

α
)
T
(
α
)
)
:

α

in

W
}
}

\{ (T,S):T\;{\text {in}}\;\mathcal {P},\;S = \{ (\alpha ,\;\smallint {(1/\alpha )T(\alpha )):\;\alpha \;{\text {in}}\;W\} \} }

is a one-one mapping from

P

\mathcal {P}

onto

L

\mathfrak {L}

.
On the diameters of compact Riemann surfaces
Authors: Chang Fwu Ranq
Published: June 2010

We derive an inequality relating the diameter and the length of a simple closed geodesic on a compact Riemann surface.
𝐻^{∞}(𝑅)+𝐴𝑃 is closed
Authors: Power Stephen
Published: June 2010

Let

H
∞

(
R
)

{H^\infty }(R)

be the space of functions on the real line R which are boundary functions of functions bounded and analytic in the upper half-plane and let AP denote the space of uniformly almost periodic functions on R. We show that

H
∞

(
R
)
+
A
P

{H^\infty }(R) + AP

is closed and is not an algebra.
Growth conditions and similarity orbits
Authors: Ghatage P.
Published: June 2010

It is proved that an operator T which acts on a separable Hilbert space and whose resolvent satisfies a first-order growth condition inside and outside the unit circle, can be approximated in norm by operators which are similar to a single unitary operator. In particular the ’Markus-operator’ can be approximated by similarities of a diagonal unitary operator and is a compact perturbation of the same.
Conditions for restricted translation operators to belong to 𝑆_{𝑝}
Authors: Dym Harry , Shapiro Isidor
Published: June 2010

Let B be a Blaschke product, let P be the orthogonal projection of

L
2

(

R
1

)

{L^2}({R^1})

onto the left invariant subspace

H
2

⊖
B

H
2

{H^2} \ominus B{H^2}

and let

E
t

:
f
(
γ
)
→

e

−
i
γ
t

f
(
γ
)

{E_t}:f(\gamma ) \to {e^{ - i\gamma t}}f(\gamma )

. Conditions are given on the roots of B for

P

E
t

P

P{E_t}P

to belong to

S
p

,
0
>
p
>
∞

{S_p}, 0 > p > \infty

.
Locally convex hypersurfaces of negatively curved spaces
Authors: Alexander S.
Published: June 2010

A well-known theorem due to Hadamard states that if the second fundamental form of a compact immersed hypersurface M of Euclidean space

E
n

(
n
⩾
3
)

{E^n}(n \geqslant 3)

is positive definite, then M is imbedded as the boundary of a convex body. There have been important generalizations of this theorem concerning hypersurfaces of

E
n

,

H
n

{E^n},{H^n}

, and

S
n

{S^n}

, but there seem to be no versions for hypersurfaces of spaces of variable curvature, and no proofs which generalize to these spaces. Our main result is that Hadamard’s theorem holds in any complete, simply connected Riemannian manifold of nonpositive sectional curvature.
A class of two-bridge knots with property-𝑃
Authors: Mayland E. J.
Published: June 2010

A knot k has property-P provided no simply connected manifold results from performing a nontrivial elementary surgery along k. We establish property-P for certain families of two-bridge knots generalizing twist knots (Whitehead doubles of the trivial knot).
A note on the central limit theorem for square-integrable processes
Authors: Hahn Marjorie G.
Published: June 2010

A method is given for constructing sample-continuous processes which do not satisfy the central limit theorem in

C
[
0
,
1
]

C[0,1]

. Let

{
X
(
t
)
:
t
∈
[
0
,
1
]
}

\{ X(t):t \in [0,1]\}

be a stochastic process. Using our method we characterize all possible nonnegative functions f for which the condition
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis upper X left-parenthesis t right-parenthesis minus upper X left-parenthesis s right-parenthesis right-parenthesis squared less-than-or-slanted-equals f left-parenthesis StartAbsoluteValue t minus s EndAbsoluteValue right-parenthesis">

E
(
X
(
t
)
−
X
(
s
)

)
2

⩽
f
(

|

t
−
s

|

)

E( X(t) - X(s) )^2 \leqslant f( | t - s | )

\]
alone is sufficient to imply that

X
(
t
)

X(t)

satisfies the central limit theorem in

C
[
0
,
1
]

C[0,1]

.
Extensions of totally projective groups
Authors: Linton Ronald , Megibben Charles
Published: June 2010

It is an unpublished observation of L. Fuchs and E. A. Walker that if L is a fully invariant subgroup of the totally projective p-group G, then both L and G/L are totally projective. In this note we treat the more difficult converse question.
A nonspectral Birkhoff-regular differential operator
Authors: Walker Philip W.
Published: June 2010

It is shown by example that Birkhoff-regular differential operators need not be spectral.
Quasisimilarity does not preserve the hyperlattice
Authors: Herrero Domingo A.
Published: June 2010

Two quasisimilar nilpotent Hilbert space operators of order three can have nonisomorphic hyperinvariant subspace lattices.
Multiple integrals of Lipschitz functions in the calculus of variations
Authors: Clarke Frank H.
Published: June 2010

We consider a multiple integral problem in the calculus of variations in which the integrand is locally Lipschitz but not differentiable, and in which minimization takes place over a Sobolev space. Using a minimax theorem, we derive an analogue of the classical Euler condition for optimality, couched in terms of “generalized gradients". We proceed to indicate how these results may be applied to deduce existence and smoothness properties of solutions to certain Poisson equations.
On Scholz’s reciprocity law
Authors: Williams Kenneth S.
Published: June 2010

An elementary proof is given of a reciprocity law proved by Scholz using class-field theory.
The completion of Prüfer domains
Authors: Močkoř Jiří
Published: June 2010

The completion of Prüfer and almost Dedekind domains in topologies

sup
{

T

w

:
w
∈
Ω
}

\sup \{ {\mathcal {T}_w}:w \in \Omega \}

, where

T

w

{\mathcal {T}_w}

is a topology induced by a valuation

w
∈
Ω

w \in \Omega

, are characterized in terms of Manis valuations.
Representations of solvable Lie algebras. IV. An elementary proof of the (𝑈/𝑃)_{𝐸}-structure theorem
Authors: McConnell J. C.
Published: June 2010

In this paper we give a shorter and much more elementary proof of a theorem which describes the structure of certain localisations of the enveloping algebra of a completely solvable Lie algebra. Such a localisation is shown to be a twisted group algebra where the group is free abelian of finite rank and the coefficient ring is a polynomial extension of a Weyl algebra.
Coefficients of inverses of meromorphic univalent functions
Authors: Schober Glenn
Published: June 2010

Sharp bounds are obtained for the coefficients

B

2
n
−
1

,
1
⩽
n
⩽
7

{B_{2n - 1}},1 \leqslant n \leqslant 7

, of the inverses of normalized meromorphic univalent functions. For the remaining values of n, sharp bounds are given in case initial segments of even coefficients are zero.
On the holomorphic curvature of some intrinsic metrics
Authors: Wong B.
Published: June 2010

If G is a hyperbolic manifold in the sense of Kobayashi and the differential Kobayashi metric

K
G

{K_G}

is of class

C
2

{C^2}

, then the holomorphic curvature of

K
G

{K_G}

is greater than or equal to

−
4

- 4

. If G is Carathéodory-hyperbolic and the differential Carathéodory metric

C
G

{C_G}

is of class

C
2

{C^2}

, then the holomorphic curvature of

C
G

{C_G}

is less than or equal to

−
4

- 4

. With this result we obtain an intrinsic characterization of the unit ball.
Operators in the commutant of a reductive algebra
Authors: Moore Robert L.
Published: June 2010

Let

A

\mathcal {A}

be a reductive algebra. It is shown that there is a subspace

M

\mathcal {M}

that reduces

A

\mathcal {A}

and such that the commutant of

A

|

M

\mathcal {A}|\mathcal {M}

is selfadjoint and the commutant of

A

|

M

⊥

\mathcal {A}|{\mathcal {M}^ \bot }

consists of hyporeductive operators. It is then shown that under a variety of conditions, if an operator T is in

A

′

\mathcal {A}’

, then

T
∗

{T^ \ast }

is in

A

′

\mathcal {A}’

.
A remark on directional contractions
Authors: Kirk W. A. , Ray William O.
Published: June 2010

Let X be a Banach space and D a convex subset of X. A mapping

T
:
D
→
X

T:D \to X

is called a directional contraction if there exists a constant

α
∈
(
0
,
1
)

\alpha \in (0,1)

such that corresponding to each

x
,
y
∈
D

x,y \in D

there exists

ε
=
ε
(
x
,
y
)
∈
(
0
,
1
]

\varepsilon = \varepsilon (x,y) \in (0,1]

for which
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper T left-parenthesis x plus epsilon left-parenthesis y minus x right-parenthesis right-parenthesis minus upper T left-parenthesis x right-parenthesis double-vertical-bar less-than-or-slanted-equals alpha epsilon double-vertical-bar x minus y double-vertical-bar">

‖

T
(
x
+
ε
(
y
−
x
)
)
−
T
(
x
)

‖

⩽
α
ε

‖

x
−
y

‖

\left \| {T(x + \varepsilon (y - x)) - T(x)} \right \| \leqslant \alpha \varepsilon \left \| {x - y} \right \|

. Tests for lipschitzianness are obtained which yield the fact that if a closed mapping is a directional contraction, then it must be a global contraction, and sufficient conditions are given under which a nonclosed directional contraction

T
:
D
→
D

T:D \to D

always has a fixed point.
On the first countability of hyperspaces
Authors: Mizokami Takemi
Published: June 2010

In this paper, the heredity of first countability of a space X on the hyperspace

C

(
X
)

\mathcal {C}(X)

of nonempty compact subsets of X with the finite topology is investigated and some variations are given.
𝑚𝑜𝑑 𝑝 Wu formulas for the Steenrod algebra and the Dyer-Lashof algebra
Authors: Shay P. Brian
Published: June 2010

Formulas for the polynomials

P

i

(

c
j

)

{\mathcal {P}^i}({c_j})

and

Q
i

(

a
j

)

{Q^i}({a_j})

in

H
∗

(
B
U
;

Z
p

)

{H^\ast }(BU;{Z_p})

and

H
∗

(
B
U
;

Z
p

)

{H_\ast }(BU;{Z_p})

, analogous to Wu’s formulas for

S

q
i

(

w
j

)

S{q^i}({w_j})

, are given.
Generators for 𝐺 bordism
Authors: Beem R. Paul
Published: June 2010

We show that a certain collection of G manifolds generates

N
∗

(
G
)

{N_\ast }(G)

as an algebra over

N
∗

{N_\ast }

, the unoriented bordism ring, where G is finite abelian and of order not divisible by four.
On the modulus of boundary values of holomorphic functions
Authors: Range R. Michael
Published: June 2010

A differential geometric method is introduced to study the modulus of boundary values of holomorphic functions on smoothly bounded pseudoconvex domains D in

C

n

,
n
⩾
2

{{\mathbf {C}}^n},n \geqslant 2

. It is shown that functions in

A
(
D
)

A(D)

are determined up to a constant factor by their modulus on an open subset of the Shilov boundary. For the case of

H
∞

(
D
)

{H^\infty }(D)

, it is shown that inner functions which satisfy a certain local condition are constant.
Asymptotic decay of oscillatory solutions of second order differential equations with forcing term
Authors: Kusano Takaŝi , Onose Hiroshi
Published: June 2010

The ordinary differential equation

(
p
(
t
)

y
′

)
′

+
q
(
t
)
f
(
y
)
=
r
(
t
)

(p(t)y’)’ + q(t)f(y) = r(t)

and its companion functional differential equation are considered. Sufficient conditions are given which ensure that all oscillatory solutions tend to zero as

t
→
∞

t \to \infty

.
On James’ quasi-reflexive Banach space
Authors: Casazza P. G. , Lin Bor Luh , Lohman R. H.
Published: June 2010

In the James’ space J, there exist complemented reflexive subspaces which are not uniformly convexifiable and there are uncountably many mutually nonequivalent unconditional basic sequences in J each of which spans a complemented subspace of J. If

{

y
n

}

\{ {y_n}\}

is a block basic sequence with constant coefficients of the unit vector basis of J, then its closed linear span

[

y
n

]

[{y_n}]

is complemented in J and the space

[

y
n

]

[{y_n}]

is either isomorphic to J or to

(

Σ

n
=
1

∞

J

t
n

)

l
2

{(\Sigma _{n = 1}^\infty {J_{{t_n}}})_{{l_2}}}

for some

{

t
n

}

\{ {t_n}\}

where

J
n

=
[

e
1

,

e
2

,
…
,

e
n

]

{J_n} = [{e_1},{e_2}, \ldots ,{e_n}]

.
Nearest normal approximation for certain operators
Authors: Phillips John
Published: June 2010

In this paper we show that binormal operators have nearest normal approximants. In fact, we exhibit nearest normals to such operators and, as a corollary, show that the hermitian part of a binormal operator with real spectrum is a nearest normal. We obtain further corollaries on nearest normal approximation to operators which are square roots of normal operators and then apply these results to perturbations of operator algebras.
On the twenty-ninth question of Allen Shields
Authors: Stampfli Joseph G.
Published: June 2010

Not every hyponormal weighted shift is the sum of a subnormal weighted shift (in the same basis) and a trace class operator.
Landau-Kolmogorov inequalities for semigroups and groups
Authors: Certain Melinda W. , Kurtz Thomas G.
Published: June 2010

An elementary functional analytic argument is given showing how inequalities of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar f Superscript left-parenthesis k right-parenthesis Baseline double-vertical-bar Superscript n Baseline less-than-or-slanted-equals upper K Subscript n comma k Baseline double-vertical-bar f double-vertical-bar Superscript n minus k Baseline double-vertical-bar f Superscript left-parenthesis n right-parenthesis Baseline double-vertical-bar Superscript k">

‖

f

(
k
)

‖

n

⩽

K

n
,
k

‖
f
‖

n
−
k

‖

f

(
n
)

‖

k

{\left \| {{f^{(k)}}} \right \|^n} \leqslant {K_{n,k}}{\left \| f \right \|^{n - k}}{\left \| {{f^{(n)}}} \right \|^k}

, where f is a real, n-times differentiable function and

‖
⋅
‖

\left \| \cdot \right \|

denotes the sup norm on

(
0
,
∞
)

(0,\infty )

(or

(
−
∞
,
∞
)

( - \infty ,\infty )

), yield corresponding inequalities,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper A Superscript k Baseline x EndAbsoluteValue Superscript n Baseline less-than-or-slanted-equals upper K Subscript n comma k Baseline StartAbsoluteValue x EndAbsoluteValue Superscript n minus k Baseline StartAbsoluteValue upper A Superscript n Baseline x EndAbsoluteValue Superscript k">

|

A
k

x

|

n

⩽

K

n
,
k

|

x

|

n
−
k

|

A
n

x

|

k

|{A^k}x{|^n} \leqslant {K_{n,k}}|x{|^{n - k}}|{A^n}x{|^k}

, for generators of linear contraction semigroups (or groups) on arbitrary Banach spaces with norm

|

⋅

|

| \cdot |

. Since Landau, Kolmogorov, Schoenberg and Cavaretta have established the function inequalities with the best possible constants, this argument gives the generator inequalities with the best possible constants for general Banach spaces extending work of Kallman and Rota, Hille and others. Questions concerning the best possible constants for specific Banach spaces remain open.
A characterization of metric completeness
Authors: Weston J. D.
Published: June 2010

A proof is given of a theorem, relevant to fixed-point theory, which implies that a metric space (X, d) is complete if and only if, for each continuous function

h
:
X
→

R

h:X \to {\mathbf {R}}

bounded below on X, there is a point

x
0

{x_0}

such that

h
(

x
0

)
−
h
(
x
)
>
d
(

x
0

,
x
)

h({x_0}) - h(x) > d({x_0},x)

for every other point x.
𝐶(𝑋) is not necessarily a retract of 2^{𝑋}
Authors: Goodykoontz Jack T.
Published: June 2010

Let X be a continuum and

2
X

(
C
(
X
)
)

{2^X}(C(X))

denote the hyperspace of closed subsets (subcontinua) of X. An example is given of a continuum X such that

C
(
X
)

C(X)

is not a retract of

2
X

{2^X}

.
On the existence of 𝜔-points
Authors: Szymański Andrzej
Published: June 2010

Without additional axioms to ZFC it is shown that each point of a dense-in-itself Hausdorff space in which nonempty

G
δ

{G_\delta }

’s have nonempty interiors is an

ω
\omega

-point.
An algebraic classification of some links of codimension two
Authors: Liang Chao Chu
Published: June 2010

For

q
⩾
2

q \geqslant 2

, J. Levine proved that two simple

(
2
q
−
1
)

(2q - 1)

-knots are isotopic if and only if their Seifert matrices are equivalent. In this paper, we will prove the analogue of Levine’s result for simple boundary

(
2
q
−
1
)

(2q - 1)

-links; we will show that: “For

q
⩾
3

q \geqslant 3

, two simple boundary

(
2
q
−
1
)

(2q - 1)

-links are isotopic if and only if their Seifert matrices are l-equivalent (defined by some algebraic moves)."
When is the free product of lattices complete?
Authors: Grätzer George , Kelly David
Published: June 2010

Yu. I. Sorkin proved that, up to isomorphism, there are three finite lattices that can be represented as a free product of two lattices. In this note we prove that, up to isomorphism, there are five complete lattices that can be so represented.
A property of finite 𝑝-groups
Authors: Kondo Shoichi
Published: June 2010

Let

R
(
G
)

R(G)

denote the character ring of a finite group G and let

Λ
\Lambda

be a commutative ring with identity. In this paper we show that if

G
≠
{
1
}

G \ne \{ 1\}

, then

Λ

⊗
Z

R
(
G
)

\Lambda { \otimes _Z}R(G)

has only one maximal ideal if and only if G is a p-group and

Λ
\Lambda

has only one maximal ideal m such that

Λ

/

m

\Lambda /\mathfrak {m}

is of characteristic p.
A note on Sylow intersections
Authors: Ish-Shalom Ariel
Published: June 2010

Let p be a prime number, G a finite group whose order is divisible by p, and S a Sylow p-subgroup of G. We say that G satisfies

(
∗
)

( \ast )

if there exists

g
∈
G

g \in G

such that

S
∩

S
g

=

O
p

(
G
)

S \cap {S^g} = {O_p}(G)

. There are examples of primes p and finite groups G that do not satisfy

(
∗
)

( \ast )

. In this note we discuss several related properties satisfied in general, as well as giving sufficient conditions for G to satisfy

(
∗
)

( \ast )

.
Successive derivatives of entire functions
Authors: Hellerstein Simon , Williamson Jack
Published: June 2010

We show that if f is a real entire function which has, along with each of its derivatives, only real nonpositive zeros, then either

f
(
z
)
=
c

e

σ
z

,
c

f(z) = c{e^{\sigma z}},c

and

σ
\sigma

real constants, or
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis equals c z Superscript m Baseline e Superscript sigma z Baseline product Underscript n Endscripts left-parenthesis 1 plus StartFraction z Over StartAbsoluteValue z Subscript n Baseline EndAbsoluteValue EndFraction right-parenthesis">

f
(
z
)
=
c

z
m

e

σ
z

∏
n

(

1
+

z

|

z
n

|

)

f(z) = c{z^m}{e^{\sigma z}}\prod \limits _n {\left ( {1 + \frac {z}{{|{z_n}|}}} \right )}

\]
where

σ
⩾
0

\sigma \geqslant 0

and

∑
n

|

z
n

|

−
1

>
∞

\sum \nolimits _n {|{z_n}{|^{ - 1}} > \infty }

.
A note on comparison theorems
Authors: Abramovich Shoshana
Published: June 2010

Comparison theorems for solutions, for focal points and conjugate points of second order linear differential equations are given in this paper.
The Banach-Stone property and the weak Banach-Stone property in three-dimensional spaces
Authors: Cambern Michael
Published: June 2010

Let X and Y be compact Hausdorff spaces, E a Banach space, and

C
(
X
,
E
)

C(X,E)

the space of continuous functions on X to E. E has the weak Banach-Stone property if, whenever

C
(
X
,
E
)

C(X,E)

and

C
(
Y
,
E
)

C(Y,E)

are isometric, then X and Y are homeomorphic. E has the Banach-Stone property if the descriptive as well as the topological conclusions of the Banach-Stone theorem for scalar functions remain valid in the case of isometries of

C
(
X
,
E
)

C(X,E)

onto

C
(
Y
,
E
)

C(Y,E)

. These two properties were first studied by M. Jerison, and it we later shown that every space E found by Jerison to have the weak Banach-Stone property actually has the Banach-Stone property, thus raising the question of whether the two properties are distinct. Here we characterize all three-dimensional spaces with the weak Banach-Stone property, and, in so doing, show the properties to be distinct.
On contractions satisfying 𝐴𝑙𝑔𝑇={𝑇}’
Authors: Wu Pei Yuan
Published: June 2010

For a bounded linear operator T on a Hilbert space let

{
T

}
′

\{ T\} ’

and

Alg

T

{\operatorname {Alg}}\;T

denote the commutant, the double commutant and the weakly closed algebra generated by T and 1, respectively. Assume that T is a completely nonunitary contraction with a scalar-valued characteristic function

ψ
(
λ
)

\psi (\lambda )

. In this note we prove the equivalence of the following conditions: (i)

|

ψ
(

e

i
t

)

|

=
1

|\psi ({e^{it}})| = 1

on a set of positive Lebesgue measure; (ii)

Alg

T
=
{
T

}
′

{\operatorname {Alg}}\;T = \{ T\} ’

; (iii) every invariant subspace for T is hyperinvariant. This generalizes the well-known fact that compressions of the shift satisfy

Alg

T
=
{
T

}
′

{\operatorname {Alg}}\;T = \{T\}’

.
On asymptotic properties of several classes of operators
Authors: Campbell Stephen L. , Gellar Ralph
Published: June 2010

Let

p
(
T
,

T
∗

)

p(T,{T^ \ast })

be a polynomial in T and

T
∗

{T^ \ast }

where T is a bounded linear operator on a separable Hilbert space. Let

Δ
=
{
T

|

p
(
T
,

T
∗

)
=
0
}

\Delta = \{ T|p(T,{T^ \ast }) = 0\}

. Then

Δ
\Delta

is said to be asymptotic for p if for every

K
>
0

K > 0

, there exists an

ε
0

>
0

{\varepsilon _0} > 0

and function

δ
(
ε
,
K
)
,

lim

ε
→
0

δ
(
ε
,
K
)
=
0

\delta (\varepsilon ,K),{\lim _{\varepsilon \to 0}}\delta (\varepsilon ,K) = 0

, such that if

ε
>

ε
0

,

‖
T
‖

⩽
K

\varepsilon > {\varepsilon _0},\left \| T \right \| \leqslant K

, and

‖

p
(
T
,

T
∗

)

‖

>
ε

\left \| {p(T,{T^ \ast })} \right \| > \varepsilon

, then there exists

T
^

∈
Δ

\hat T \in \Delta

such that

‖

T
−

T
^

‖

>
δ
(
ε
,
K
)

\left \| {T - \hat T} \right \| > \delta (\varepsilon ,K)

. It is observed that the hermitian, unitary, and isometric operators are asymptotic for the obvious polynomials. It is known that the normals are not asymptotic for

p
(
T
,

T
∗

)
=

T
∗

T
−
T

T
∗

p(T,{T^ \ast }) = {T^ \ast }T - T{T^ \ast }

. An example gives several negative results including one that says the quasinormals are not asymptotic for

p
(
T
,

T
∗

)
=
T

T
∗

T
−

T
∗

T
2

p(T,{T^\ast }) = T{T^\ast }T - {T^\ast }{T^2}

. It is shown that if p is any polynomial in just one of T or

T
∗

{T^ \ast }

, then

Δ
\Delta

is asymptotic for p.
The curvatures of some skew fundamental forms
Authors: Milnor Tilla Klotz
Published: June 2010

Fix a unit normal vector field on a surface

C
4

{C^4}

-immersed in a Riemannian 3-manifold of constant sectional curvature. Suppose H and K are mean and Gauss curvatures respectively, and that

H
′

=

H
2

−
K

H’ = \sqrt {{H^2} - K}

. Wherever

H
′

≠
0

H’ \ne 0

, define I’, II’ and III’ by

H
′

I
′

=
II
−
H
I
,

H
′

II
′

=
H
II
−
K
I

H’\text {I}’ = \text {II} - H\text {I}, H’\text {II}’ = H\text {II} - K\text {I}

and

III
′

=
H

II
′

−
K

I
′

\text {III}’ = H\text {II}’ - K\text {I}’

, where I and II are the first and second fundamental forms. For constants

α
,
β

\alpha ,\beta

, and

γ
\gamma

, let

Λ
′

=
α

I

′

+
β

II

′

+
γ

III

′

\Lambda ’ = \alpha {\text {I}}’ + \beta {\text {II}}’ + \gamma {\text {III}}’

. Wherever

H
′

≠
0

H’ \ne 0

and

Λ
′

\Lambda ’

is nondegenerate, the curvature of this (not necessarily Riemannian) metric

Λ
′

\Lambda ’

is computed in terms of

K
(

I

′

)

K({\text {I}}’)

and more familiar quantities on the surface. Some discussion of

K
(

I

′

)

K({\text {I}}’)

is also included.
The class of compact* spaces is productive and closed hereditary
Authors: Govaerts W.
Published: June 2010

W. W. Comfort defined compact

∗

^ \ast

spaces as completely regular Hausdorff spaces X such that each maximal ideal in the ring

C
∗

(
X
,

R

)

{C^ \ast }(X,{\mathbf {R}})

of bounded continuous real-valued functions on X is fixed. He showed that, independently of the axiom of choice, the class of compact

∗

^ \ast

spaces is productive and closed hereditary. We give a short new proof of this.
Convolution powers of singular-symmetric measures. II
Authors: Izuchi Keiji
Published: June 2010

Let G be an infinite compact abelian group such that its dual group contains an infinite independent subset.

L

(
G
)

\mathfrak {L}(G)

denotes the sum of all radicals of group algebras contained in the measure algebra on G. Then, for a positive integer k, there is a measure

μ
\mu

on G such that

μ
n

{\mu ^n}

is singular-symmetric for

1
⩽
n
⩽
k

1 \leqslant n \leqslant k

and

μ
n

∈

L

(
G
)

{\mu ^n} \in \mathfrak {L}(G)

for

n
>
k

n > k

.
A type of nearest point set in a complete 𝑙-group
Authors: Keisler Michael
Published: June 2010

A theorem by W. D. L. Appling (Riv. Mat. Univ. Parma, (3) 2 (1973), 251-276) demonstrates that a C-set is a nearest point set in

ba

(
S
,
Σ
)

{\text {ba}}(S,\Sigma )

with respect to the variation norm. This paper demonstrates an analogous result for a generalized form of C-set in a complete l-group with distance with respect to the norm being replaced by a stronger property definable in a complete l-group (distance between elements x and y of a complete l-group G is taken to be

|

x
−
y

|

|x - y|

). The result is then shown to be a characterization of sets possessing the stronger property in the case of a complete vector lattice, but not a characterization in the case of a complete l-group.
An incidence theorem for a class of continuous real functions
Authors: Baston V. J. , Bostock F. A.
Published: June 2010

Sufficient conditions are obtained for a real function to have a line cutting its graph in exactly two points. Examples are given to show that none of these conditions may be omitted.
Global solvability of an abstract complex
Authors: Cardoso Fernando , Hounie Jorge
Published: June 2010

In a recent paper F. Treves studied a model of complexes of pseudodifferential operators in an open set of

R
n

{R^n}

, establishing necessary and sufficient conditions for its semiglobal solvability. In the present paper, the authors give necessary and sufficient conditions for the global solvability of an analogous complex defined on an orientable, compact smooth manifold without boundary.
Extension of invariant linear functionals: a sequel to Fan’s paper
Authors: Lau Anthony Toming
Published: June 2010

We establish the relationship between certain invariant extension properties for linear functional of K. Fan when a semigroup S acts on a locally convex topological vector space and the existence of a left-invariant mean on the space of almost periodic and weakly almost periodic functions on S.
Locally equational completeness of rings and semigroups
Authors: Chiaro M. R.
Published: June 2010

Hu has developed locally equational classes as a generalization of equational classes. It is shown here that the lattice of equational classes of rings is a proper sublattice of the lattice of locally equational classes of rings but the locally equationally complete rings are precisely the equationally complete rings. Using the equational and locally equational class operators the locally equationally complete classes of semigroups are shown to be those that are equationally complete.
Stiefel-Whitney homology classes of quasi-regular cell complexes
Authors: Goldstein Richard , Turner Edward C.
Published: June 2010

A quasi-regular cell complex is defined and shown to have a reasonable barycentric subdivision. In this setting, Whitney’s theorem that the k-skeleton of the barycentric subdivision of a triangulated n-manifold is dual to the

(
n
−
k
)

(n - k)

th Stiefel-Whitney cohomology class is proven, and applied to projective spaces, lens spaces and surfaces.
Characterizations of strictly monotone sets
Authors: Stanek Jean Chan
Published: June 2010

One natural generalization of convex sets is the concept of strictly monotone sets. In this paper, Tietze’s characterization theorem on convex sets is generalized to strictly monotone sets. The main result is that a closed connected set in the Euclidean plane

E
2

{E_2}

is strictly monotone if and only if it is locally strictly monotone. This theorem is shown to hold in

E
n

{E_n}

with additional hypotheses.
A Helly type theorem on the sphere
Authors: Katchalski Meir
Published: June 2010

This paper establishes a Helly type theorem for convex sets in

S
n

{S_n}

, the n-dimensional unit sphere.
A rather classless model
Authors: Kaufmann Matt
Published: June 2010

Using

♢

ω
1

{\diamondsuit _{{\omega _1}}}

, a model

N

\mathfrak {N}

of Peano arithmetic is constructed which has a unique extension

(

N

,
χ
)

(\mathfrak {N},\chi )

to a model of

Δ
1
1

\Delta _1^1

-PA, Peano arithmetic with

Δ
1
1

\Delta _1^1

-comprehension.
A note on 𝑘-critically 𝑛-connected graphs
Authors: Entringer R. C. , Slater Peter J.
Published: June 2010

A graph G is said to be

(

n
∗

,
k
)

({n^\ast },k)

-connected if it has connectivity n and every set of k vertices is contained in an n-cutset. It is shown that an

(

n
∗

,
k
)

({n^\ast },k)

-connected graph G contains an n-cutset C such that G — C has a component with at most

n

/

(
k
+
1
)

n/(k + 1)

vertices, thereby generalizing a result of Chartrand, Kaugars and Lick. It is conjectured, however, that

n

/

(
k
+
1
)

n/(k + 1)

can be replaced with

n

/

2
k

n/2k

and this is shown to be best possible.
Cartesian closed topological hulls
Authors: Herrlich H. , Nel L. D.
Published: June 2010

It is shown in this paper that if a concrete category

A

\mathfrak {A}

admits embedding as a full finitely productive subcategory of a cartesian closed topological (CCT) category, then

A

\mathfrak {A}

admits such embedding into a smallest CCT category, its CCT hull. This hull is characterized internally by means of density properties and externally by means of a universal property. The problem is posed of whether every topological category has a CCT hull.
Inequalities between intrinsic metrics
Authors: Burbea Jacob
Published: June 2010

We introduce the “mth order Carathéodory-Reiffen metric,” the “mth order Bergman metric” and the “mth order modified Bergman metric” on M. Here M is a complex manifold which is ample in a suitable sense. These “metrics” are defined on

T
(
M
)

T(M)

and they are intrinsic. They arise as solutions of maximum problems. The first orders of these “metrics” (except for the modified Bergman metric) are the corresponding familiar metrics. All these metrics are biholomorphically invariant. We establish a chain of inequalities between them. This generalizes an earlier result of Hahn, proved by different methods.
Inverse limits and absolutes of 𝐻-closed spaces
Authors: Vinson T. O. , Dickman R. F.
Published: June 2010

The inverse limit of an inverse system of nonempty (respectively, minimal Hausdorff) H-closed spaces with continuous open surjective bonding maps is shown to be a nonempty H-closed (respectively, minimal Hausdorff) space. This result is used to resolve a conjecture, in the affirmative, of S. W. Willard: Every H-closed space is the continuous and open image of a minimal Hausdorff space.
Reflexivity of topological groups
Authors: Nickolas Peter
Published: June 2010

It is shown that under mild conditions the path-component of the identity in the dual group

G

∅
^

G\hat \emptyset

of an Abelian topological group G is precisely the union of the one-parameter subgroups of

G

∅
^

G\hat \emptyset

. This yields several corollaries, including a necessary condition for certain groups to be reflexive (to satisfy the Pontrjagin duality theorem), and a negative answer to a question of N. Noble.
A characteristic property of the sphere
Authors: Koufogiorgos Themis , Hasanis Thomas
Published: June 2010

On an ovaloid S with Gaussian curvature

K
>
0

K > 0

in Euclidean three-space

E
3

{E^3}

, the second fundamental form defines a nondegenerate Riemannian metric with curvature

K

II

{K_{{\text {II}}}}

. It is shown that S is a sphere if

K

II

=
c

H
s

K
r

{K_{{\text {II}}}} = c{H^s}{K^r}

, where c, s and r are constants, H is the mean curvature of S and

0
⩽
s
⩽
1

0 \leqslant s \leqslant 1

.
The index of a holomorphic mapping and the index theorem
Authors: Ishihara Tôru
Published: June 2010

The index theorem for a harmonic mapping of riemannian manifolds is given. Let

f
:
M
→
N

f:M \to N

be a holomorphic mapping of Kaehler manifolds. Then it is shown that the index of f is zero and that a Jacobi field along f is a holomorphic section of the bundle

f
∗

T
(
N
)

{f^ \ast }T(N)

induced by f.
On the support of symmetric infinitely divisible and stable probability measures on LCTVS
Authors: Rajput Balram S.
Published: June 2010

It is shown that the topological support (supp.) of a

τ
\tau

-regular, symmetric, infinitely divisible (resp. stable of any index

α
∈
(
0
,
2
)

\alpha \in (0,2)

) probability measure on a Hausdorff LCTVS E is a subgroup (resp. a subspace) of E. The part regarding the support of a stable probability measure of this theorem completes a result of A. De-Acosta [Ann. of Probability 3 (1975), 865-875], who proved a similar result for

α
∈
(
1
,
2
)

\alpha \in (1,2)

, and the author [Proc. Amer. Math. Soc. 63 (1977), 306-312], who proved it for

α
∈
[
1
,
2
)

\alpha \in [1,2)

. Further, it provides a complete affirmative solution to the question, raised by J. Kuelbs and V. Mandrekar [Studia Math. 50 (1974), 149-162], of whether the supp. of a symmetric stable probability measure of index

α
∈
(
0
,
1
]

\alpha \in (0,1]

on a separable Hilbert space H is a subspace of H.
A note on flat algebras
Authors: Nobile Augusto
Published: June 2010

The following results are proved. If

f
:
A
→
B

f:A \to B

is a homomorphism of rings, with B noetherian, I is an ideal of B contained in the Jacobson radical, and

B

/

I
n

B/{I^n}

is A-flat, for all n, then f is flat. If, using similar notations and assumptions, I is generated by a regular sequence, then the flatness of B/I implies the flatness of f. A simple geometric application is given.
On the absence of necessary conditions for linear evolution operators
Authors: Goldstein Jerome A.
Published: June 2010

There exist selfadjoint operators

A
(
t
)
(
t
⩾
0
)

A(t)(t \geqslant 0)

, whose resolvents depend smoothly on t, such that the initial value (Schrödinger) problem

d
u
(
t
)

/

d
t
=
i
A
(
t
)
u
(
t
)
,
u
(
0
)
=
f

du(t)/dt = iA(t)u(t),u(0) = f

has a unique

(

C
∞

)

({C^\infty })

solution u for each f in a dense set in the underlying Hilbert space, with u depending continuously on f, and yet the intersection over

t
⩾
0

t \geqslant 0

of the domain of

A
(
t
)

A(t)

is {0}.
Generalized Hewitt-Savage theorems for strictly stationary processes
Authors: Isaac Richard
Published: June 2010

Generalizations of the Hewitt-Savage zero-one law are proved for strictly stationary processes. This takes the form of statements concerning inclusion and equality relationships among certain sigma-fields related to the process.