bluuebunny/crossref_metadata_2025 huggingface dataset:
Generalizations of l’Hôpital’s rule
Authors: Lee Cheng Ming
Published: June 2010

An essential limit, similar to the concept of essential bounded functions, is defined and briefly discussed. Using the essential limit, l’Hôpital’s rule is generalized to include the following theorem as a special case. Theorem. Let F, G be real-valued functions defined on the open interval (a, b). Suppose that the approximate derivatives

F

ap

′

(
x
)

{F’_{{\text {ap}}}}(x)

and

G

ap

′

(
x
)

{G’_{{\text {ap}}}}(x)

exist finitely,

G

ap

′

(
x
)
>
0

{G’_{{\text {ap}}}}(x) > 0

for almost all x in (a, b), and the extreme approximate derivates of both F and G are finite nearly everywhere in (a, b). Then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript x right-arrow a plus Endscripts left-bracket upper F left-parenthesis x right-parenthesis slash upper G left-parenthesis x right-parenthesis right-bracket equals ess limit Underscript x right-arrow a plus Endscripts left-bracket upper F prime Subscript ap Baseline left-parenthesis x right-parenthesis slash upper G prime Subscript ap Baseline left-parenthesis x right-parenthesis right-bracket">

lim

x
→
a
+

[
F
(
x
)

/

G
(
x
)
]
=

ess

lim

x
→
a
+

[

F

ap

′

(
x
)

/

G

ap

′

(
x
)
]

{\lim _{x \to a + }}[F(x)/G(x)] = {\text {ess}}\;{\lim _{x \to a + }}[{F’_{{\text {ap}}}}(x)/{G’_{{\text {ap}}}}(x)]

provided that the essential limit in the right-hand side exists and that

lim

x
→
a
+

F
(
x
)
=

lim

x
→
a
+

G
(
x
)
=
0

{\lim _{x \to a + }}F(x) = {\lim _{x \to a + }}G(x) = 0

or

lim

x
→
a
+

G
(
x
)
=
−
∞

{\lim _{x \to a + }}G(x) = - \infty

.
Nonexistence of global solutions and bifurcation analysis for a boundary-value problem of parabolic type
Authors: Pao C. V.
Published: June 2010

The aim of this paper is to present a bifurcation analysis on the existence of the nonexistence of a global solution for a semilinear parabolic equation and to characterize the local stability and the instability of the corresponding steady-state solutions. The bifurcation result can be described either by a parameter

λ
\lambda

for a fixed spatial domain

Ω
\Omega

or by varying

Ω
\Omega

for a fixed

λ
\lambda

. The stability analysis gives a result which can be used to determine the stability or instability problem when the system possesses nonintersecting multiple steady-state solutions.
A note on generators of semigroups
Authors: Tsui Sze Kai
Published: June 2010

The generator T of a norm-continuous semigroup of identity preserving positive linear mappings on a

C
∗

{C^\ast }

-algebra

A

\mathfrak {A}

is characterized as one that satisfies
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T left-parenthesis u Superscript asterisk Baseline u right-parenthesis greater-than-or-slanted-equals u Superscript asterisk Baseline upper T left-parenthesis u right-parenthesis plus upper T left-parenthesis u Superscript asterisk Baseline right-parenthesis u">

T
(

u
∗

u
)
⩾

u
∗

T
(
u
)
+
T
(

u
∗

)
u

T({u^\ast }u) \geqslant {u^\ast }T(u) + T({u^\ast })u

for all unitary elements u in

A

\mathfrak {A}

.
Upper and lower Fredholm spectra. I
Authors: Buoni John J. , Harte Robin , Wickstead Tony
Published: June 2010

Joint upper and lower Fredholm spectra are defined for n-tuples of bounded linear operators, and the upper Fredholm spectrum is represented both as the simultaneous eigenvalues and as the simultaneous approximate eigenvalues of an n-tuple of operators obtained by a Berberian-Quigley construction.
Some explicitly invertible Toeplitz operators on the quarter-plane
Authors: Radlow James
Published: June 2010

A 2-variable Toeplitz operator T is invertible only under special conditions on its symbol t(z) = t(zx, zj). Here we make three additions to the known list of such special conditions and we construct explicit inverses in each of the three cases. The results are obtained by the use of a 2-variable Wiener-Hopf factorization of the symbol.
Locally free actions and Stiefel-Whitney numbers. II
Authors: Stong R. E. , Winkelnkemper H. E.
Published: June 2010

This paper determines the possible bordism classes of manifolds with a locally free G action for G one of

S
1

×

S
1

,

(

S
1

)
4

{S^1} \times {S^1},{({S^1})^4}

or

S
3

{S^3}

and gets partial information for

(

S
1

)
3

{({S^1})^3}

.
A vector field with one zero on 𝐺/𝑃
Authors: Akyildiz Ersan
Published: June 2010

We will give a

G
a

{G_a}

-action on G/P with only one fixed point, where G is a connected linear algebraic group defined over an algebraically closed field k of characteristic zero and P is a parabolic subgroup of G.
A family of Lie algebras not extendible to a family of Lie groups
Authors: Coppersmith Don
Published: June 2010

We produce an analytic family of finite-dimensional Lie algebras, parameterized by a smooth Hausdorff manifold, which does not correspond to the Lie algebra of any Hausdorff (separable) family of Lie groups. This answers a question of Douady and Lazard.
Conditions for generating a nonvanishing bounded analytic function
Authors: Mantel J. H.
Published: June 2010

B. A. Taylor and L. A. Rubel have posed the problem of finding necessary and sufficient conditions on a set of given functions

f
1

,

f
2

,
…
,

f
n

{f_1},{f_2}, \ldots ,{f_n}

in

H
∞

{H^\infty }

such that there exist functions

g
1

,

g
2

,
…
,

g
n

{g_1},{g_2}, \ldots ,{g_n}

in

H
∞

{H^\infty }

with

Σ

i
=
1

n

f
i

g
i

≠
0

\Sigma _{i = 1}^n{f_i}{g_i} \ne 0

in the open unit disc. L. A. Rubel has conjectured that a necessary and sufficient condition is that there exist a harmonic minorant of

log
⁡
[

Σ

i
=
1

n

|

f
i

|

]

\log [\Sigma _{i = 1}^n|{f_i}|]

in the open unit disc. The major result of this paper proves that the conjecture is true if one of the given functions

f
1

,

f
2

,
…
,

f
n

{f_1},{f_2}, \ldots ,{f_n}

has a zero set which is an interpolation set for

H
∞

{H^\infty }

.
A theorem on infinite products of eigenvalues of Sturm-Liouville type operators
Authors: Levit S. , Smilansky U.
Published: June 2010

Infinite products of ratios of eigenvalues of Sturm-Liouville operators are expressed in a closed form in terms of corresponding solutions of initial-value problems.
Extension of invariant linear functionals
Authors: Fan Ky
Published: June 2010

In a recent paper [4] we considered a semigroup of linear contractions or a group of linear isometries in a normed vector space, and obtained sufficient conditions for the orbits to lie on parallel closed hyperplanes. In the present note, we take a more general viewpoint and shall prove some theorems on extension of continuous linear functionals which are invariant under a group or a semigroup of continuous linear maps on a locally convex topological vector space. These more general results include those in [4] as direct consequences.
The numerical range and the essential numerical range
Authors: Williams J. P.
Published: June 2010

A simple proof is given of Lancaster’s theorem that the convex hull of the numerical and essential numerical ranges of a Hilbert space operator is the closure of the numerical range.
Schur multipliers of some finite nilpotent groups
Authors: Jackson David A.
Published: June 2010

Let B denote the Burnside group,

B
(

p
α

,
d
)

B({p^\alpha },d)

and let

G
=
B

/

B
k

G = B/{B_k}

where p is a prime and

1
>
k
>
p

1 > k > p

. We show that the Schur multiplier,

M
(
G
)

M(G)

, is a direct power of

Ψ
(
k
,
d
)

\Psi (k,d)

cyclic groups, each having order

p
α

{p^\alpha }

, where

Ψ
(
k
,
d
)
=

k

−
1

Σ

n

|

k

μ
(
k

/

n
)

d
n

\Psi (k,d) = {k^{ - 1}}{\Sigma _{n|k}}\mu (k/n){d^n}

. (This is Witt’s formula for the rank of

F
k

/

F

k
+
1

{F_k}/{F_{k + 1}}

where F is free on d generators.) In addition we can show that

M
(
B
(
3
,
d
)
)

M(B(3,d))

is elementary abelian of exponent 3 and rank
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 left-parenthesis Subscript 2 Baseline Superscript d Baseline right-parenthesis plus 4 left-parenthesis Subscript 3 Baseline Superscript d Baseline right-parenthesis plus 3 left-parenthesis Subscript 4 Baseline Superscript d Baseline right-parenthesis">

2

(
2
d

)
+
4

(
3
d

)
+
3

(
4
d

)

2(_2^d) + 4(_3^d) + 3(_4^d)

.
The automorphism group of 2_{𝐹₄}(2)’
Authors: Weiss Richard
Published: June 2010

We give a new proof that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A u t left-parenthesis Superscript 2 Baseline upper F 4 left-parenthesis 2 right-parenthesis Superscript prime Baseline right-parenthesis approximately-equals upper A u t left-parenthesis Superscript 2 Baseline upper F 4 left-parenthesis 2 right-parenthesis right-parenthesis approximately-equals squared upper F 4 left-parenthesis 2 right-parenthesis">

Aut

(
2

F
4

(
2

)
′

)
≅

Aut

(
2

F
4

(
2
)
)
≅

2

F
4

(
2
)

{\operatorname {Aut}}{(^2}{F_4}(2)’) \cong {\operatorname {Aut}}{(^2}{F_4}(2)) \cong {\;^2}{F_4}(2)

(see [4]).
𝐶*-algebras isomorphic after tensoring
Authors: Plastiras Joan
Published: June 2010

It is always true that whenever

A

\mathfrak {A}

and

B

\mathfrak {B}

are isomorphic

C
∗

{C^\ast }

-algebras then

M

2

⊗

A

{\mathfrak {M}_2} \otimes \mathfrak {A}

and

M

2

⊗

B

{\mathfrak {M}_2} \otimes \mathfrak {B}

are also isomorphic, and the converse holds for many standard examples. In this note we present two

C
∗

{C^\ast }

-algebras

A

\mathfrak {A}

and

B

\mathfrak {B}

such that

M

2

⊗

A

{\mathfrak {M}_2} \otimes \mathfrak {A}

and

M

2

⊗

B

{\mathfrak {M}_2} \otimes \mathfrak {B}

are isomorphic whereas

A

\mathfrak {A}

and

B

\mathfrak {B}

are not.
A short short proof of the Cartwright-Littlewood theorem
Authors: Brown Morton
Published: June 2010

Each orientation preserving homeomorphism of the plane that is invariant on a nonseparating bounded continuum has a fixed point on the continuum.
A normal form theorem for lattices completely generated by a subset
Authors: Grätzer George , Kelly David
Published: June 2010

For an

m

\mathfrak {m}

-complete lattice L (

m

\mathfrak {m}

is an infinite regular cardinal) and subset X of L that

m

\mathfrak {m}

-generates L, we prove a Normal Form Theorem for elements of L expressed as polynomials over X. This generalizes a theorem of B. Jónsson in which such a representation is found for the lattice L freely

m

\mathfrak {m}

-generated by a poset X. We also apply this result to free

m

\mathfrak {m}

-products of

m

\mathfrak {m}

-complete lattices.
A measure theoretic variant of Blumberg’s theorem
Authors: Brown Jack B.
Published: June 2010

It is the purpose of this note to present a measure theoretic variant of Blumberg’s theorem about continuous restrictions of arbitrary real functions.
An isoperimetric inequality involving conformal mapping
Authors: Acker Andrew
Published: June 2010

An isoperimetric inequality due to L. Bieberbach is generalized to the case of weighted areas.
Group-quotients with positive sectional curvatures
Authors: Geroch Robert
Published: June 2010

Let H be a closed subgroup of compact Lie group G. A necessary and sufficient condition is obtained for the existence of a left-invariant Riemannian metric on G such that the subduced metric on the quotient H G has strictly positive sectional curvatures.
The Ising model limit of 𝜙⁴ lattice fields
Authors: Rosen Jay
Published: June 2010

We show that the

λ
→
∞

\lambda \to \infty

limit of

λ

ϕ
4

\lambda {\phi ^4}

lattice fields is an Ising model.
Dilations on involution semigroups
Authors: Szafraniec F. H.
Published: June 2010

We present an equivalent form of the boundedness condition involved in the Sz.-Nagy general dilation theorem and, as a consequence, we prove a dilation theorem for a product of commuting dilatable operator functions on involution semigroups. Also we show that the Bram-Halmos criterion of subnormality can be directly deduced from the proposed boundedness condition.
Fixed point theorems for mappings of nonexpansive type
Authors: Lim Teck-Cheong
Published: June 2010

Constructive fixed point theorems for single-valued and compact-valued nonexpansive mappings, which map a closed convex subset C of a Banach space X into X and send the boundary of C relative to X into C, are given. Mappings for which the method of asymptotic center applies are also considered.
The lower central series of the group ⟨𝑥,𝑦: 𝑥^{𝑝}=1⟩
Authors: Labute John P.
Published: June 2010

In this paper we determine the Lie algebra associated to the lower central series of the group

⟨
x
,
y
:

x
p

=
1
⟩

\langle x,y:{x^p} = 1\rangle

, p a prime.
The range of a vector measure has the Banach-Saks property
Authors: Anantharaman R.
Published: June 2010

The above result of Diestel and Seifert is proved using the Banach-Saks theorem for

L
p

(
λ
)
,
1
>
p
>
∞

{L^p}(\lambda ),1 > p > \infty

.
On the geometric interpretation of seminormality
Authors: Davis Edward D.
Published: June 2010

We give a new characterization of the property of seminormality for 1-dimensional local rings; and with the help of this result we develop extensions of the theorems of Salmon and Bombieri which interpret this property in terms of singularities of algebraic curves.
On interpolating functions with minimal norm
Authors: Stray A. , Øyma K. O.
Published: June 2010

Let

H
∞

{H^\infty }

denote the Banach algebra of bounded analytic functions in the unit disc

{
z
:

|

z

|

>
1
}

\{ z:|z| > 1\}

. If f is an extreme point in the unit ball of

H
∞

{H^\infty }

, there is always a Blaschke product B, whose zeros form an interpolating sequence tending to one point of the unit circle, such that

‖

f
+
B
h

‖

∞

>
1

{\left \| {f + Bh} \right \|_\infty } > 1

if

h
∈

H
∞

h \in {H^\infty }

and

h
≠
0

h \ne 0

. An application of this result to the theory of best approximation is given.
Lacunary series and exponential moments
Authors: Kuelbs J. , Woyczyński W. A.
Published: June 2010

In the present paper we prove the boundedness of second exponential moments for the suprema of norms of partial sums of random series in Banach spaces, Sidon lacunary series on a compact Abelian group with coefficients in a Banach space, and for random lacunary series with similar coefficients. Exponential moments related to the law of iterated logarithm for lacunary series are also proved to be finite. Our results, even in case of standard trigonometric series with real coefficients, strengthen the classical theorem of Salem and Zygmund.
A simple Noetherian ring not Morita equivalent to a domain
Authors: Stafford J. T.
Published: June 2010

An example of Zalesskii and Neroslavskii is used to produce an example of a simple ring that is not Morita equivalent to a domain.
Piecewise linear embeddings of balls into contractible manifolds
Authors: Millett Kenneth C.
Published: June 2010

The simplicial space of proper piecewise linear embeddings of k-balls into n-dimensional contractible manifolds which are equal to a fixed embedding along the boundary is contractible if

n
−
k
=
3

n - k = 3

.
A ternary function for distributivity and permutability of an equivalence lattice
Authors: Korec Ivan
Published: June 2010

The main result of the paper is Theorem 1. Let A be a countable set and L be a complete sublattice of the equivalence lattice on A. The following are equivalent (i) L is a distributive lattice of permutable equivalence relations. (ii) There is an algebra with congruence lattice L among the fundamental operations of which is a ternary function f with the property
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis 1 right-parenthesis EndLabel f left-parenthesis a comma b comma b right-parenthesis equals f left-parenthesis a comma b comma a right-parenthesis equals f left-parenthesis b comma b comma a right-parenthesis equals a EndLayout">

(

1

)

f
(
a
,
b
,
b
)
=
f
(
a
,
b
,
a
)
=
f
(
b
,
b
,
a
)
=
a

\begin{equation}\tag {$1$} \quad f(a,b,b) = f(a,b,a) = f(b,b,a) = a\end{equation}

for all

a
,
b
∈
A

a, b \in A

.
Regularity, normality and multifunctions
Authors: Joseph James E.
Published: June 2010

In this note we give several characterizations of regularity and normality in terms of weakly upper-semicontinuous multifunctions.
Metric characterizations of dimension for separable metric spaces
Authors: Janos Ludvik , Martin Harold
Published: June 2010

A subset B of a metric space (X, d) is called a d-bisector set iff there are distinct points x and y in X with

B
=
{
z
:
d
(
x
,
z
)
=
d
(
y
,
z
)
}

B = \{ z:d(x,z) = d(y,z)\}

. It is shown that if X is a separable metrizable space, then

dim
⁡
(
X
)
⩽
n

\dim (X) \leqslant n

iff X has an admissible metric d for which

dim
⁡
(
B
)
⩽
n
−
1

\dim (B) \leqslant n - 1

whenever B is a d-bisector set. For separable metrizable spaces, another characterization of n-dimensionality is given as well as a metric dependent characterization of zero dimensionality.
A generalized Kleene-Moschovakis theorem
Authors: Harrington Leo , Kirousis Lefteris , Schlipf John
Published: June 2010

Moschovakis generalized a theorem of Kleene to prove that if

X

\mathfrak {X}

is a collection of subsets of any acceptable structure

M

\mathfrak {M}

such that

(

M

,

X

)
⊨

Δ
1
1

(\mathfrak {M},\mathfrak {X}) \vDash \Delta _1^1

comprehension, every hyperelementary subset of

M

\mathfrak {M}

is in

X

\mathfrak {X}

. We prove an analogous result for arbitrary

M

\mathfrak {M}

. We also get analogous results for

M

\mathfrak {M}

with an extra quantifier Q.
Integration by substitution
Authors: Goodman Gerald S.
Published: June 2010

The author shows how N-functions provide a natural setting in which to establish the change of variables formula for Lebesgue or Denjoy/Perron integrals. By abolishing the need to pass to the limit under the integral sign, the validity of the classical formula is significantly extended, yielding new results even in the case of Lebesgue integrals.
Unique balayage in Fourier transforms on compact abelian groups
Authors: Shapiro George S.
Published: June 2010

Let K be a compact subset of the compact abelian group G and let

Λ
\Lambda

be a subset of the dual group

Γ
\Gamma

. Unique balayage is said to be possible for

(
K
,
Λ
)

(K,\Lambda )

if, for every

μ
\mu

in

M
(
G
)

M(G)

, there is a unique

ν
\nu

in

M
(
K
)

M(K)

whose Fourier transform,

ν
^

\hat \nu

, agrees on

Λ
\Lambda

with

μ
^

\hat \mu

. We prove that in order that there be any K with unique balayage possible for

(
K
,
Λ
)
,
Λ

(K,\Lambda ),\Lambda

must belong to the coset ring of

Γ
\Gamma

. The converse of this statement is false. Some examples are given for the case where G is the circle group.
Multipliers on compact groups
Authors: Inglis Ian
Published: June 2010

We give some sufficient conditions for a function on a compact totally disconnected abelian group to be an

L
p

{L^p}

Fourier multiplier.
The Fredholm radius of a bundle of closed linear operators
Authors: Liebetrau E.-O.
Published: June 2010

Given a bundle of linear operators

T
−
λ
S

T - \lambda S

, where T is closed and S is bounded, a sequence

{

δ
m

(
T
:
S
)
}

\{ {\delta _m}(T:S)\}

of extended real numbers is defined. If T is a Fredholm operator, the limit

lim

δ
m

(
T
:
S

)

1

/

m

{\lim \delta _m}{(T:S)^{1/m}}

exists and is equal to the supremum of all

r
>
0

r > 0

such that

T
−
λ
S

T - \lambda S

is a Fredholm operator for

|

λ

|

>
r

|\lambda | > r

.
Markov property of extremal local fields
Authors: Dang-Ngoc N. , Royer G.
Published: June 2010

We show that extremal local field on

(
E
,

E

)
T

{(E,\mathcal {E})^T}

, with

T
=

Z

T = {\mathbf {Z}}

or R and

(
E
,

E

)

(E,\mathcal {E})

standard, possesses the Markov property. This result generalizes that of F. Spitzer in the case

T
=

Z

T = {\mathbf {Z}}

, E countable and a result of G. Royer and M. Yor on extremal measures associated to certain diffusion processes.
Intervals of continua which are Hilbert cubes
Authors: Eberhart Carl
Published: June 2010

If P is a subcontinuum of a metric continuum X, then by the interval of continua

C

(
P
,
X
)

\mathcal {C}(P,X)

we mean the space of all subcontinua of X which contain P (with the Hausdorff metric). We show that

C

(
P
,
X
)

\mathcal {C}(P,X)

is often homeomorphic with the Hilbert cube.
Proper regular semigroups
Authors: Masat F. E.
Published: June 2010

In a recent paper, D. B. McAlister gave several characterizations of proper inverse semigroups. In this paper, the concept of proper is extended to the class of regular semigroups. This is done by requiring that the set of idempotents of the semigroup coincides with the kernel of the minimum group congruence on the semigroup. A theorem is presented which contains several characterizations of proper regular semigroups, and the related result of McAlister then follows as a corollary. The paper concludes with some open questions and examples.
Boundary representations and tensor products of 𝐶*-algebras
Authors: Hopenwasser Alan
Published: June 2010

Let A and B be unital, generating linear subspaces of

C
∗

{C^ \ast }

-algebras

A

\mathcal {A}

and

B

\mathcal {B}

, respectively. If either

A

\mathcal {A}

or

B

\mathcal {B}

is a GCR algebra, then the set of boundary representations for

A
⊗
B

A \otimes B

can be identified with the Cartesian product of the boundary representations for A with the boundary representations for B.
On the area of harmonic surfaces
Authors: Beeson Michael
Published: June 2010

We prove that if the sequence

S
n

{S_n}

of harmonic surfaces converges to the harmonic surface S, and if the boundary of

S
n

{S_n}

is a rectifiable Jordan curve

C
n

{C_n}

, whose length is uniformly bounded by L, then the area

A
(

S
n

)

A({S_n})

converges to

A
(
S
)

A(S)

. This solves an old problem, several special cases of which have been solved in the literature.
Sets of natural numbers with no minimal asymptotic bases
Authors: Erdős Paul , Nathanson Melvyn B.
Published: June 2010

The set A of natural numbers is an asymptotic basis for S if the sets S and 2A eventually coincide. An asymptotic basis A for S is minimal if no proper subset of A is a basis for S. Sets S are constructed which possess infinitely many asymptotic bases, but no minimal asymptotic basis.
Symmetry for finite dimensional Hopf algebras
Authors: Humphreys J. E.
Published: June 2010

This note refines criteria given by R. G. Larson and M. E. Sweedler for a finite dimensional Hopf algebra to be a symmetric algebra, with applications to restricted universal enveloping algebras and to certain finite dimensional subalgebras of the hyperalgebra of a semisimple algebraic group in characteristic p.
Representable monoids
Authors: Taft Earl J.
Published: June 2010

A representable monoid is one with enough representative functions to separate points. It is shown that the monoid algebra of a representable monoid is a proper algebra. In particular, the group algebra of a residually-finite group is a proper algebra. It is also shown that the free product of two representable monoids is again representable.
Differentiability via directional derivatives
Authors: Lau Ka Sing , Weil Clifford E.
Published: June 2010

Let F be a continuous function from an open subset D of a separable Banach space X into a Banach space Y. We show that if there is a dense

G
δ

{G_\delta }

subset A of D and a

G
δ

{G_\delta }

subset H of X whose closure has nonempty interior, such that for each

a
∈
A

a \in A

and each

x
∈
H

x \in H

the directional derivative

D
x

F
(
a
)

{D_x}F(a)

of F at a in the direction x exists, then F is Gâteaux differentiable on a dense

G
δ

{G_\delta }

subset of D. If X is replaced by

R
n

{R^n}

, then we need only assume that the n first order partial derivatives exist at each

a
∈
A

a \in A

to conclude that F is Frechet differentiable on a dense,

G
δ

{G_\delta }

subset of D.
Approximation of induced automorphisms and special automorphisms
Authors: Whitman P. N.
Published: June 2010

A class of measure-preserving invertible point transformations which admit approximations is defined. If T is an automorphism which admits an approximation, conditions are given such that an induced automorphism and a special automorphism over T again admit an approximation.
Remainders in Hausdorff compactifications
Authors: Chandler Richard E. , Tzung Fu-Chien
Published: June 2010

We generalize a theorem of Steiner and Steiner and use it to obtain new results concerning remainders for completely regular spaces. Both the locally compact and the nonlocally compact cases are considered.
Witt classes of integral representations of an abelian 2-group
Authors: Gibbs David E.
Published: June 2010

In this paper the Witt groups of integral representations of an abelian 2-group

π
,

W
0

(
π
;
Z
)

\pi ,{W_0}(\pi ;Z)

and

W
2

(
π
;
Z
)

{W_2}(\pi ;Z)

are calculated. Invariants are listed which completely determine

W
0

(

Z
4

;
Z
)

{W_0}({Z_4};Z)

and

W
2

(

Z
4

;
Z
)

{W_2}({Z_4};Z)

and can be extended to the case

π
=

Z

2
k

\pi = {Z_{{2^k}}}

. If

π
\pi

is an elementary abelian 2-group, it is shown that

W
2

(
π
;
Z
)
=
0

{W_2}(\pi ;Z) = 0

and

W
0

(
π
;
Z
[

1
2

]
)

{W_0}(\pi ;Z[\tfrac {1}{2}])

is ring isomorphic to the group ring

W
(
Z
[

1
2

]
)
(

Hom

(
π
,

Z
2

)
)

W(Z[\tfrac {1}{2}])({\operatorname {Hom}}(\pi ,{Z_2}))

.
Unter- und Oberfunktionen und Differentialungleichungen zweiter Ordnung in normierten Räumen
Authors: Lemmert Roland
Published: June 2010

An invariance theorem for a cone in a normed linear space with respect to solutions of general elliptic inequalities is obtained, which includes and generalizes monotonicity theorems for systems of second order ordinary, parabolic and elliptic differential inequalities.
On A. Hurwitz’ method in isoperimetric inequalities
Authors: Chavel Isaac
Published: June 2010

We show that if M is complete simply connected with nonpositive sectional curvatures,

Ω
\Omega

a minimal submanifold of M with connected suitably oriented boundary

Γ
\Gamma

then

λ

1

/

2

V

/

A
⩽

(
n
−
1

)

1

/

2

/

n

{\lambda ^{1/2}}V/A \leqslant {(n - 1)^{1/2}}/n

where V is the volume of

Ω
\Omega

, A the volume of

Γ
,
λ

\Gamma ,\lambda

the first nonzero eigenvalue of the Laplacian of

Γ
\Gamma

, and

n
(
⩾
2
)

n( \geqslant 2)

is the dimension of

Ω
\Omega

.
Invariants and asymptotic behavior of solutions of a conservation law
Authors: Liu Tai Ping
Published: June 2010

We study the asymptotic behavior of solutions of the initial value problem for a conservation law

u
t

+
f

(
u

)
x

=
0

{u_t} + f{(u)_x} = 0

. It is assumed that the initial data agrees with the Riemann data for

|

x

|

|x|

large. We show that the solution approaches that of the corresponding Riemann problem at algebraic rates.
The Banach-Mazur distance between the trace classes 𝑐ⁿ_{𝑝}
Authors: Tomczak-Jaegermann Nicole
Published: June 2010

The Banach-Mazur distance between

l
2
n

⊗
^

l
2
m

l_2^n \hat {\otimes } l_2^m

and

l
2
n

⊗
^

^

l
2
m

l_2^n \hat {\hat {\otimes }} l_2^m

is shown to be of the order

min
(
n
,
m
)

\sqrt {\min (n,m)}

. Our proof yields that the distance between the trace classes

c
p
n

c_p^n

and

c
q
n

c_q^n

is of the same order as

d
(

l
p
n

,

l
q
n

)

d(l_p^n,l_q^n)

.
On the successive derived sets of the Pisot numbers
Authors: Boyd David W.
Published: June 2010

Let S denote the set of Pisot (or Pisot-Vijayaraghavan) numbers and let

S

(
k
)

{S^{(k)}}

be the kth derived set of S. It is shown that

k

1

/

2

⩽
min

S

(
k
)

{k^{1/2}} \leqslant \min {S^{(k)}}

and that

lim
sup
(
min

S

(
k
)

/

k
)
>
1

\lim \sup (\min {S^{(k)}}/k) > 1

. The lower bound improves the estimate

k

1

/

4

⩽
min

S

(
k
)

{k^{1/4}} \leqslant \min {S^{(k)}}

of Dufresnoy and Pisot, while the upper bound improves the obvious estimate

min

S

(
k
)

⩽
k
+
1

\min {S^{(k)}} \leqslant k + 1

.
Recursive determination of the sum-of-divisors function
Authors: Ewell John A.
Published: June 2010

A recursive scheme for determination of the sum-of-divisors function is presented. As all of the formulas involve triangular numbers, the scheme is therefore compared for efficiency with another known recursive triangular-number formula for this function.
Some remarks on the sequences of exhaustive measures and uniform boundedness of measures
Authors: Khurana Surjit Singh
Published: June 2010

Nikodym-type theorems and uniform boundedness theorems for measures on Boolean algebra with interpolation property are derived from corresponding results for

σ
\sigma

-algebras.
On the existence of maximal and minimal solutions for parabolic partial differential equations
Authors: Bebernes J. W. , Schmitt K.
Published: June 2010

The existence of maximal and minimal solutions for initial-boundary value problems and the Cauchy initial value problem associated with

L
u
=
f
(
x
,
t
,
u
,
∇
u
)

Lu = f(x,t,u,\nabla u)

where L is a second order uniformly parabolic differential operator is obtained by constructing maximal and minimal solutions from all possible lower and all possible upper solutions, respectively. This approach allows f to be highly nonlinear, i.e., f locally Hölder continuous with almost quadratic growth in

|

∇
u

|

|\nabla u|

.
On the commutative properties of boundary integral operators
Authors: Roach G. F.
Published: June 2010

A discussion of the interior Dirichlet and Neumann problems of classical potential theory can be given in terms of the symmeterisers of certain related integral operators. Recent developments in the theory and application of integral equations of the first kind have made this approach towards the solution of boundary value problems a more attractive proposition. However for problems more general than those arising in potential theory a greater knowledge of associated spectral properties is required together with a realisation that much of the symmetry occurring in potential problems will be lost and that attention must be directed instead towards commutativity relations. This is demonstrated by considering boundary value problems associated with the Helmholtz equation.
Paracompactness, metacompactness, and semi-open covers
Authors: Junnila Heikki J. K.
Published: June 2010

Paracompactness and metacompactness are characterized in terms of locally finite and point-finite semi-open refinements of open covers. It follows from one of these characterizations that a continuous image of a paracompact space under a pseudo-open and compact mapping is metacompact.
Equivalence and slice theory for symplectic forms on closed manifolds
Authors: Swanson R. C. , Chicone C. C.
Published: June 2010

In this paper, a study is made of the pullback action of the diffeomorphism group on the totality of symplectic forms on a compact manifold. For this action, the orbit is shown to be a smooth (Banach) manifold consisting of a denumerable union of submanifolds, each lying in a fixed cohomology class. In addition, a precise characterization is given of those symplectic manifolds for which there is a local factorization of the pullback action in the sense of a transverse “slice” of closed 2-forms, invariant under the group of symplectic diffeomorphisms.
On the SQ-universality of HNN groups
Authors: Fine Benjamin , Tretkoff Marvin
Published: June 2010

A group G is SQ-universal if every countable group can be embedded in a quotient of G. In this paper we give conditions under which HNN groups are SQ-universal. These are then applied to showing the SQ-universality of certain classes of Kleinian groups (discontinuous subgroups of

P
S

L
2

(

C

)

PS{L_2}({\mathbf {C}})

. Finally some connections are mentioned between SQ-universality and the theory of ends.
Principal congruences of pseudocomplemented semilattices and congruence extension property
Authors: Sankappanavar H. P.
Published: June 2010

Principal congruences of pseudocomplemented semilattices are characterized and shown to be definable. This characterization is then applied to give a new proof of the fact that the variety of pseudocomplemented semilattices has the congruence extension property.
A relaxed Picard iteration process for set-valued operators of the monotone type
Authors: Dunn J. C.
Published: June 2010

The fixed points

x
¯

\bar x

of set-valued operators,

T
:
X
→

2
X

T:X \to {2^X}

, satisfying a condition of the monotonicity type on convex subsets X of a Hilbert space are approximated by a relaxation process,

x

n
+
1

=

x
n

+

ω
n

(
T

x
n

−

x
n

)

{x_{n + 1}} = {x_n} + {\omega _n}(T{x_n} - {x_n})

, in which

T
~

\tilde T

is a single-valued branch of T and the relaxation parameter

ω
n

∈
[
0
,
1
]

{\omega _n} \in [0,1]

is made to depend in a certain way on the prior history of the process. If

T
~

\tilde T

is bounded on bounded subsets of X, then

‖

x
n

−

x
¯

‖

\left \| {{x_n} - \bar x} \right \|

converges to 0 like

O
(

n

−
1

/

2

)

O({n^{ - 1/2}})

. If

T
~

\tilde T

is also continuous at

x
¯

\bar x

and if

x
¯

=

T
~

x
¯

\bar x = \tilde T\bar x

, then

‖

x
n

−

x
¯

‖

=
o
(

n

−
1

/

2

)

\left \| {{x_n} - \bar x} \right \| = o({n^{ - 1/2}})

. If

T
~

\tilde T

satisfies a condition of the Lipschitz type at

x
¯

\bar x

, then

‖

x
n

−

x
¯

‖

=
O
(

μ

n

/

2

)

\left \| {{x_n} - \bar x} \right \| = O({\mu ^{n/2}})

for some

μ
∈
[
0
,
1
)

\mu \in [0,1)

.
A dilation theorem for ⋅-preserving maps of 𝒞*-algebras
Authors: Gardner L. Terrell
Published: June 2010

A linear map of a (unital)

C

∗

{\mathcal {C}^\ast }

-algebra into a

C

∗

{\mathcal {C}^\ast }

-algebra, which preserves the absolute value, is analyzed as the composition of a (unital)

∗

^\ast

-homomorphism into a usually larger target algebra, followed by a suitable multiplication

x
↦
x
b
=

b

1

/

2

x

b

1

/

2

x \mapsto xb = {b^{1/2}}x{b^{1/2}}

.
Invariant subspaces of von Neumann algebras. II
Authors: Peligrad Costel
Published: June 2010

It is shown that every parareductive operator algebra

A
⊂
B
(
H
)

A \subset B(H)

(as defined below) is a von Neumann algebra. For the proof of this result, some new properties of paraclosed operators are obtained. Finally, a sufficient condition that a reductive algebra be a von Neumann algebra is given.
The Haar functions almost diagonalize multiplication by 𝑥
Authors: Anderson Joel
Published: June 2010

It is shown that if A is multiplication by x on

L
2

(
0
,
1
)

{L^2}(0,1)

, then the matrix for A given by the Haar functions has the form diagonal plus Hilbert-Schmidt.
Free completely distributive lattices
Authors: Markowsky George
Published: June 2010

We show that the usual construction of the free distributive lattice on n generators generalizes to an arbitrary quantity of generators and actually yields a free completely distributive lattice. Furthermore, for an infinite number of generators the cardinality of the corresponding free completely distributive lattice is exactly that of the power set of the power set of the set of generators.
Logconcavity of the cooling of a convex body
Authors: Deslauriers Gilles , Dubuc Serge
Published: June 2010

The solution

u
(
x
,
t
)

u(x,t)

of the heat equation is logconcave in the space variable x whenever the initial temperature

u
0

(
x
)

{u_0}(x)

of the convex body is logconcave.
Martingale representation in uniformly convex spaces
Authors: Landers D. , Rogge L.
Published: June 2010

In this paper we define the concept of a martingale in a uniformly convex Banach space and show that each bounded martingale is convergent and can be represented as a sequence of nearest point projections onto closed convex sets of one element of the Banach space. This result contains classical representation theorems for usual martingales and yields applications in the theory of nearest point projections in

L
p

{L_p}

-spaces.
Locally compact groups which are amenable as discrete groups
Authors: Chou Ching
Published: June 2010

In the courses of their independent studies of the existence of invariant means which are not topologically invariant, E. Granirer and W. Rudin have considered several properties of a locally compact group G which are satisfied if G is amenable as a discrete group. By applying a result of J. Rosenblatt (together with ideas of Granirer and Rudin) we show some of these properties on G imply that G is amenable as a discrete group.
A theorem for prime rings
Authors: Richoux Anthony
Published: June 2010

Let n be a positive integer and let R be a prime ring either of characteristic zero or of characteristic

⩾
n

\geqslant n

. Then for any

a
1

,

a
2

,
…
,

a

n
+
1

∈
R

{a_1},{a_2}, \ldots ,{a_{n + 1}} \in R

, if

a
1

x

a
2

x
⋯

a
n

x

a

n
+
1

=
0

{a_1}x{a_2}x \cdots {a_n}x{a_{n + 1}} = 0

for all

x
∈
R

x \in R

. Then

a
i

=
0

{a_i} = 0

for some

1
⩽
i
⩽
n
+
1

1 \leqslant i \leqslant n + 1

.
Note on simple integral extension domains and maximal chains of prime ideals
Authors: Ratliff L. J.
Published: June 2010

It is shown that if R is a semi-local (Noetherian) domain, then there exists a simple integral extension domain

R
[
e
]

R[e]

of R such that there exists a maximal chain of prime ideals of length n in some integral extension domain of R if and only if there exists a maximal chain of prime ideals of length n in

R
[
e
]

R[e]

. An interesting existence theorem on a certain type of height one prime ideals in

R
[
X
]

R[X]

follows.
Absolute Abel summability and capacity
Authors: Welland G. V.
Published: June 2010

Precise limits on the size of exceptional sets for which functions in the Lebesgue class,

L

α
p

\mathcal {L}_\alpha ^p

, can fail to be absolutely Abel summable are given in terms of Bessel capacity.
On a class of finite-dimensional Lie algebras
Authors: Amayo Ralph K.
Published: June 2010

Over fields of prime characteristic the centre of the universal enveloping algebra of a finite-dimensional Lie algebra contains a finitely generated polynomial algebra over which the universal envelope is a finitely generated module. This result, which is due to Curtis, is crucial in certain investigations of finitely generated soluble Lie algebras and motivates the introduction of the class Max-cu, which will be called the class of Curtis algebras, consisting of Lie algebras whose universal envelopes have the property described above. It has been an open question whether the class Max-cu consists of finite-dimensional Lie algebras. This paper gives an affirmative answer to the question.
Universally Lusin-measurable and Baire-1 projections
Authors: Saab Elias
Published: June 2010

It is obvious that a dual Banach space

E
∗

{E^\ast }

is reflexive if and only if the natural projection P from

E

∗
∗
∗

{E^{ \ast \ast \ast }}

to

E
∗

{E^\ast }

is

weak
∗

{\text {weak}^\ast }

to weak continuous. In this paper it is proved that the next best condition on P, namely that P is

weak
∗

{\text {weak}^\ast }

to weak universally Lusin-measurable is necessary and sufficient for

E
∗

{E^\ast }

to have the Radon-Nikodým property. In addition we prove that if E is any Banach space that is complemented in its second dual by a

weak
∗

{\text {weak}^\ast }

to weak Baire-1 projection, then E has the Radon-Nikodým property. We also prove that if E is a Banach space that is complemented in its second dual

E

∗
∗

{E^{ \ast \ast }}

by a projection

P
:

E

∗
∗

→
E

P:{E^{\ast \ast }} \to E

with

F
=

P

−
1

(
0
)

F = {P^{ - 1}}(0)

weakly K-analytic; then saying that

E

∗
∗

{E^{ \ast \ast }}

has the Radon-Nikodým property is equivalent to saying that P is

weak
∗

{\text {weak}^\ast }

to weak universally Lusin-measurable.
Kaehler manifolds of positive curvature operator
Authors: Ogiue Koichi , Tachibana Shun-ichi
Published: June 2010

An n-dimensional compact Kaehler manifold of positive curvature operator is real cohomologically equivalent to

P
n

(
C
)

{P_n}(C)

.
A note on the Borel formula
Authors: Dotzel Ronald M.
Published: June 2010

A new proof of the Borel formula is obtained for

G
=

(

Z
p

)
r

G = {({Z_p})^r}

actions on spaces X having

H
i

(
X
;

Z
p

)
=
0
,
i
≠
n

{H_i}(X;{Z_p}) = 0,i \ne n

(some n) and

H
n

(
X
;

Z
p

)
=

Z
p

⊕

{H_n}(X;{Z_p}) = {Z_p} \oplus

Free

Z
p

G

{Z_p}G

module. Each

X
H

{X^H}

must be a

Z
p

{Z_p}

-homology

n
(
H
)

n(H)

-sphere and then

n
−
n
(
G
)
=
Σ
(
n
(
H
)
−
(
G
)
)

n - n(G) = \Sigma (n(H) - (G))

, sum running over corank 1 subgroups. A discussion of examples follows.
On autonomous control systems on certain manifolds
Authors: Liang Chao Chu
Published: June 2010

Let

M
n

{M^n}

be a compact

C
∞

{C^\infty }

manifold,

n
⩾
4

n \geqslant 4

, admitting a vector field with every orbit a circle. Then there exists a completely controllable set S consisting of two nonsingular

C
∞

{C^\infty }

vectors X and Y such that every orbit of X is a circle.
Triangles in arrangements of lines. II
Authors: Purdy George
Published: June 2010

We show that given n lines in the real projective plane, no

n
−
1

n - 1

of which are concurrent, the number

p
3

{p_3}

of triangular regions formed is at most

2
5

n
(
n
−
1
)

\tfrac {2}{5}n(n - 1)

, equality being possible. We also show that if

n
⩾
6

n \geqslant 6

then

p
3

⩽

7

18

n
(
n
−
1
)
+

1
3

{p_3} \leqslant \tfrac {7}{{18}}n(n - 1) + \tfrac {1}{3}

. Grünbaum has conjectured

p
3

⩽

1
3

n
(
n
−
1
)

{p_3} \leqslant \tfrac {1}{3}n(n - 1)

.
The spectrum of the Laplacian of Kähler manifolds
Authors: Chen Bang-Yen , Vanhecke Lieven
Published: June 2010

We strengthen some results on the spectrum of the Laplacian for 0- and 1-forms [8], [9] on Kähler manifolds and give some new results for the 2-forms.
A linearly ordered ring whose theory admits elimination of quantifiers is a real closed field
Authors: van den Dries L. P. D.
Published: June 2010

Linearly ordered rings and preordered fields whose first order theory admits quantifier elimination are shown to be real closed fields.
A strong convergence theorem for stationary Gaussian sequences in a Hilbert space
Authors: Deo Chandrakant M.
Published: June 2010

An a.s. functional LIL is proved for stationary Gaussian sequences taking values in a Hilbert space.
A class of 4-manifolds which have 2-spines
Authors: Trace Bruce
Published: June 2010

In this note we establish the following. Theorem. Let

M
4

{M^4}

denote a piecewise linear or smooth compact 4-manifold-with-boundary having a handle decomposition consisting of 0-, 2- and 3-handles. Then

M
4

{M^4}

has a 2-dimensional spine if and only if

H
3

(

M
4

)
=
0

{H_3}({M^4}) = 0

.
On an analog of Lagrange’s theorem for commutative Hopf algebras
Authors: Radford David E.
Published: June 2010

We show that the Hopf algebra A over any field k representing the affine group scheme

SL

(
2
,
)

{\text {SL}}(2,)

is not a free B-module for some sub-Hopf algebra B of A. In particular k can be algebraically closed, or of characteristic 0 in which case A is also cosemisimple.
Elementary functional equations on partially ordered linear algebra
Authors: Dai Taen Yu
Published: June 2010

Suppose that a Dedekind

σ
\sigma

-complete partially ordered linear algebra A admits a function with familiar functional and order properties as possessed by the real-valued exponential, logarithm, root or power function on the real line. Then A can be characterized as an algebra of almost finite extended-real-valued continuous functions defined on some proper topological space.
Stability of stationary points and cohomology of groups
Authors: Hirsch Morris W.
Published: June 2010

A stationary point of a group action is called stable if nearby actions have nearby stationary points. Sufficient conditions for stability are found for certain linear actions. In special cases necessary and sufficient conditions are determined.
Derivations on continuous functions
Authors: Watatani Yasuo
Published: June 2010

We shall give a simple proof of Sakai’s characterization on derivations of continuous functions.
A two-point boundary value problem with jumping nonlinearities
Authors: Castro B. Alfonso
Published: June 2010

We prove that a certain two point BVP with jumping nonlinearities has a solution. Our result generalizes that of [2]. We use variational methods which permit giving a minimax characterization of the solution. Our proof exposes the similarities between the variational behavior of this problem and that of other semilinear problems with noninvertible linear part (see [5]).
Local ergodic theorems for noncommuting semigroups
Authors: McGrath S. A.
Published: June 2010

Let

(
X
,
μ
)

(X,\mu )

be a

σ
\sigma

-finite measure space and

L
p

(
μ
)
,
1
⩽
p
⩽
∞

{L_p}(\mu ),1 \leqslant p \leqslant \infty

, the usual Banach spaces of complex-valued functions. For

k
=
1
,
2
,
…
,
n

k = 1,2, \ldots ,n

, let

{

T
k

(
t
)
:
t
⩾
0
}

\{ {T_k}(t):t \geqslant 0\}

be a strongly continuous semigroup of Dunford-Schwartz operators. If
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper R Subscript n minus 1 Baseline equals StartSet g colon integral Underscript StartAbsoluteValue g EndAbsoluteValue greater-than t Endscripts StartAbsoluteValue g slash t EndAbsoluteValue left-parenthesis log StartAbsoluteValue g slash t EndAbsoluteValue right-parenthesis Superscript n minus 1 Baseline d mu greater-than normal infinity for all t greater-than 0 EndSet comma">

f
∈

R

n
−
1

=

{

g
:

∫

|

g

|

>
t

|

g

/

t

|

(
log
⁡

|

g

/

t

|

)

n
−
1

d
μ
>
∞

for

all

t
>
0

}

,

f \in {R_{n - 1}} = \left \{ {g:\int _{|g| > t} {|g/t|{{(\log |g/t|)}^{n - 1}}d\mu > \infty {\text {for}}\;{\text {all}}\;t > 0} } \right \},

\]
then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over alpha 1 alpha 2 midline-horizontal-ellipsis alpha Subscript n Baseline EndFraction integral Subscript 0 Superscript alpha Subscript n Baseline midline-horizontal-ellipsis integral Subscript 0 Superscript alpha Subscript i Baseline upper T Subscript n Baseline left-parenthesis t Subscript n Baseline right-parenthesis midline-horizontal-ellipsis upper T 1 left-parenthesis t 1 right-parenthesis f left-parenthesis x right-parenthesis d t 1 midline-horizontal-ellipsis d t Subscript n Baseline right-arrow f left-parenthesis x right-parenthesis">

1

α
1

α
2

⋯

α
n

∫
0

α
n

⋯

∫
0

α
i

T
n

(

t
n

)
⋯

T
1

(

t
1

)
f
(
x
)
d

t
1

⋯
d

t
n

→
f
(
x
)

\frac {1}{{{\alpha _1}{\alpha _2} \cdots {\alpha _n}}}\int _0^{{\alpha _n}} \cdots \int _0^{{\alpha _i}} {{T_n}({t_n}) \cdots {T_1}({t_1})f(x)d{t_1} \cdots d{t_n} \to f(x)}

\]

μ
\mu

-a.e. as

α
1

↘
0
,
…
,

α
n

↘
0

{\alpha _1} \searrow 0, \ldots ,{\alpha _n} \searrow 0

independently. If

f
∈

L
p

(
μ
)
,
1
>
p
>
∞

f \in {L_p}(\mu ),1 > p > \infty

, then the limit exists in norm as well as pointwise.
The length of a curve in a space of curvature ≤𝐾
Authors: Dekster B. V.
Published: June 2010

Let M be a compact ball in a Riemannian manifold with sectional curvatures

⩽
K

\leqslant K

. Suppose its radius

R
0

{R_0}

is less than the injectivity radius at the center of M and

R
0

>
π

/

2

K

{R_0} > \pi /2\sqrt K

if

K
>
0

K > 0

. Denote by

M
0

{M_0}

a circle of radius

R
0

{R_0}

in the plane of constant curvature K and by

κ
\kappa

the curvature of

∂

M
0

\partial {M_0}

. Then any curve in M with curvature

⩽
χ
>
κ

\leqslant \chi > \kappa

is not longer than a circular arc of curvature

χ
\chi

in

M
0

{M_0}

whose ends are opposite points of

∂

M
0

\partial {M_0}

. Any curve in M with total curvature not exceeding some

τ
>
0

\tau > 0

(

τ
=
π

/

2

\tau = \pi /2

if

K
⩽

κ
2

K \leqslant {\kappa ^2}

) is not longer than the longest curve in

M
0

{M_0}

with the same total curvature whose tangent vector rotates in a permanent direction.
Real minimal hypersurfaces in a complex projective space
Authors: Kon Masahiro
Published: June 2010

A characterization of the geodesic minimal hypersphere in a complex projective space is given.
The Hanf number of 𝐿_{𝜔₁𝜔₁}
Authors: Väänänen Jouko
Published: June 2010

A model of set theory is constructed in which the Hanf number of

L

ω
1

ω
1

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

is below the first weakly compact cardinal. This answers a question of J. Silver.
Shrinking decompositions of 𝐸ⁿ with countably many 1-dimensional, star-like equivalent nondegenerate elements
Authors: Lay Terry L.
Published: June 2010

It is shown that an upper semicontinuous decomposition of

E
n

(
n
⩾
1
)

{E^n}(n \geqslant 1)

with countably many 1-dimensional, star-like equivalent nondegenerate elements is shrinkable.