bluuebunny/crossref_metadata_2025 huggingface dataset:
Trotter’s product formula for semigroups generated by quasilinear elliptic operators
Authors: Watanabe Michiaki
Published: June 2010

Trotter’s product formula is given for nonlinear semigroups in

L
1

(

R
N

)

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

generated by quasilinear operators of the form

Δ
ϕ

\Delta \phi

, where

ϕ
\phi

is a suitable function: formally
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="exp left-parenthesis t normal upper Delta phi right-parenthesis u equals limit Underscript h down-arrow 0 Endscripts left-brace exp left-parenthesis h normal upper Delta phi 1 right-parenthesis midline-horizontal-ellipsis exp left-parenthesis h normal upper Delta phi Subscript k Baseline right-parenthesis right-brace Superscript left-bracket t slash h right-bracket Baseline u">

exp
⁡
(
t
Δ
ϕ
)
u
=

lim

h
↓
0

{

exp
⁡
(
h
Δ

ϕ
1

)
⋯
exp
⁡
(
h
Δ

ϕ
k

)

}

[
t

/

h
]

u

\exp (t\Delta \phi )u = {\lim _{h \downarrow 0}}{\left \{ {\exp (h\Delta {\phi _1}) \cdots \exp (h\Delta {\phi _k})} \right \}^{[t/h]}}u

, where

ϕ
=

ϕ
1

+
⋯
+

ϕ
k

\phi = {\phi _1} + \cdots + {\phi _k}

. The proof is carried out by a new method for construction of a semigroup with generator

Δ
ϕ

\Delta \phi

in

L
1

(

R
N

)

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

.
Lower estimates for zeros of stochastic Sturm-Liouville problems
Authors: Kreith Kurt
Published: June 2010

Letting

T
(
ω
)

T(\omega )

denote the first zero of a class of stochastic Sturm-Liouville initial value problems, a lower bound is established for the expected value of the random variable

T
T

.
Normal curvatures and Euler classes for polyhedral surfaces in 4-space
Authors: Banchoff Thomas F.
Published: June 2010

Using the approach of singularities of projections into lower dimensional spaces it is possible to define nonintrinsic local curvature quantities at each vertex of a polyhedral surface immersed in

4
4

-space which add up to the normal Euler number of the immersion. Related uniqueness results for lattice polyhedra have been established by B. Yusin.
Jet-detectable extrema
Authors: Netto Angelo Barone
Published: June 2010

Given a real polynomial

f
f

with

f
(
0
)
=
0

f(0) = 0

, we define a finite family of germs of smooth one-variable functions that can be used to test if

f
f

has an extreme value at 0. A generalization is obtained for functions that have

k
k

-jets in a suitable sense.
Derivatives of Bernstein polynomials and smoothness
Authors: Ditzian Z.
Published: June 2010

Equivalence relations between the asymptotic behaviour of derivatives of Bernstein polynomials and the smoothness of the function they approximate are given. This is achieved with an a priori condition that the function is of class

Lip
⁡
β

\operatorname {Lip}\beta

with some small

β
>
0

\beta > 0

. The a priori condition is dropped when a similar equivalence relation using the Katorovich operator is proved.
On the elliptic equation Δ𝑢=𝜑(𝑥)𝑢^{𝛾} in 𝑅²
Authors: Kawano Nichiro , Kusano Takaŝi , Naito Manabu
Published: June 2010

The equation

(
∗
)
Δ
u
=
ϕ
(
x
)

u
γ

( * )\Delta u = \phi (x){u^\gamma }

is considered in

R

2

{{\mathbf {R}}^2}

, where

γ
≠
1

\gamma \ne 1

and

ϕ
(
x
)
⩾
0

\phi (x) \geqslant 0

is locally Hölder continuous. Sufficient conditions are obtained for

(
∗
)

( * )

to possess infinitely many positive solutions which are defined throughout

R
2

{R^2}

and have logarithmic growth as

|

x

|

→
∞

|x| \to \infty

. An extension of the main result to the higher-dimensional case is also attempted.
A sharp inequality for the 𝑝-center of gravity of a random variable
Authors: Cox David C.
Published: June 2010

Let

X
X

be a real-valued random variable. For

p
>
1

p > 1

, define the

p
p

-center of gravity of

X
X

,

C
p

(
X
)

{C_p}(X)

, as the unique number

c
c

which minimizes

‖

X
−
c

‖

p

{\left \| {X - c} \right \|_p}

. This paper exhibits a more-or-less explicit expression for the best constant

γ
=

γ
p

\gamma = {\gamma _p}

in the inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper E left-parenthesis upper X right-parenthesis minus upper C Subscript p Baseline left-parenthesis upper X right-parenthesis EndAbsoluteValue less-than-or-slanted-equals gamma double-vertical-bar upper X minus upper C Subscript p Baseline left-parenthesis upper X right-parenthesis double-vertical-bar Subscript p">

|

E
(
X
)
−

C
p

(
X
)

|

⩽
γ

‖

X
−

C
p

(
X
)

‖

p

\left | {E(X) - {C_p}(X)} \right | \leqslant \gamma {\left \| {X - {C_p}(X)} \right \|_p}

, and presents asymptotic formulas for

γ
p

{\gamma _p}

as

p
→
1
,
2

p \to 1,2

and

+
∞

+ \infty

, respectively. The definition of

C
p

(
X
)

{C_p}(X)

is extended to variables taking values in an arbitrary Hilbert space

H
H

, and it is shown that

γ
p

{\gamma _p}

is not increased by this extension.
A note on skew-Hopf fibrations
Authors: Gage Michael E.
Published: June 2010

Each great circle fibration of the unit

3
3

-sphere in

4
4

-space can be identified with a subset of the Grassmann manifold of oriented

2
2

-planes in

4
4

-space by associating each great circle fiber with the

2
2

-plane it lies in. This Grassmann manifold can be identified with the space

S
2

×

S
2

{S^2} \times {S^2}

. H. Gluck and F. Warner, in Great circle fibrations of the three sphere, Duke Math. J. 50 (1983), 107-132, have shown that the subsets of this Grassmann manifold which correspond to great circle fibrations can be interpreted as the graphs of distance decreasing maps from

S
2

{S^2}

and

S
2

{S^2}

and that Hopf fibrations correspond to constant maps. This note characterizes explicitly the maps which correspond to "skew-Hopf" fibrations: those fibrations of the

3
3

-sphere obtained from Hopf fibrations by applying a linear transformation of

4
4

-space followed by projection of the fibers back to the unit

3
3

-sphere.
Some remarks on homeomorphisms of compact Lie groups
Authors: Bagchi S. C. , Sitaram A.
Published: June 2010

It is proved that if

G
G

is a compact connected semisimple Lie group and

H
H

a compact group of homeomorphisms of

G
G

containing all left and right translations of

G
G

, then there exists a positive integer

k
k

such that for any

τ
∈
H

\tau \in H

,

τ
k

{\tau ^k}

is, modulo a translation, an inner automorphism.
Martin compactifications and quasiconformal mappings
Authors: Segawa Shigeo , Tada Toshimasa
Published: June 2010

It is shown that there exists a quasiconformal mapping

T
T

of a Riemann surface

R
1

{R_1}

onto another

R
2

{R_2}

such that

T
T

cannot be extended to a homeomorphism of the Martin compactification

R
1
∗

R_1^*

of

R
1

{R_1}

onto that

R
2
∗

R_2^*

of

R
2

{R_2}

.
Some topological properties of the 1-set contractions
Authors: Domínguez Benavides Tomás
Published: June 2010

Let

C
C

be a bounded closed convex subset of a Banach space

X
X

. It is shown that, in the category sense, almost all

1
1

-set-contractions

f
:
C
→
C

f:C \to C

are condensing. To know how the condensing mappings are scattered in the set

Σ
1

(
C
)

{\Sigma _1}(C)

of

1
1

-set-contractions it is proved that the set of noncondensing mappings is dense in

Σ
1

(
C
)

{\Sigma _1}(C)

.
Applications of the spaces of homogeneous polynomials to some problems on the ball algebra
Authors: Bourgain J.
Published: June 2010

Denote by

B
2

{B_2}

the unit ball in

C

2

{{\mathbf {C}}^2}

. The existence is shown of a uniformly bounded orthonormal basis in

H
2

(

B
2

)

{H^2}({B_2})

, by constructing such systems in the spaces of homogeneous polynomials. In the second part of the paper, those spaces of homogeneous polynomials are exploited to disprove the existence of generalized analytic projections, the so-called

(

i
p

−

π
p

)

({i_p} - {\pi _p})

property, for the ball algebra.
Reductive algebras containing a direct sum of the unilateral shift and a certain other operator are selfadjoint
Authors: Ansari Mohamad A.
Published: June 2010

We give a partial solution of the reductive algebra problem to prove that: a reductive algebra containing the direct sum of a unilateral shift of finite multiplicity and a finite-dimensional completely nonunitary contraction is a von Neumann algebra.
Relations between travelling wave solutions of quasilinear parabolic equations
Authors: Engler Hans
Published: June 2010

A transformation between travelling wave solutions of

v
t

=

v

x
x

+
f

(
v
)

{v_t} = {v_{xx}} + f\left ( v \right )

and

u
t

=

(

D

(
u
)

u
x

)

x

+
g

(
u
)

{u_t} = {\left ( {D\left ( u \right ){u_x}} \right )_x} + g\left ( u \right )

is given.
Continuous selections for weakly Hausdorff lower semicontinuous multifunctions
Authors: De Blasi F. S. , Myjak J.
Published: June 2010

The existence of continuous selections is proved for a class of weakly Hausdorff lower semicontinuous multifunctions.
Selfadjointness of ∗-representations generated by positive linear functionals
Authors: Inoue A.
Published: June 2010

The first purpose of this paper is to prove that

π
τ

{\pi _\tau }

is selfadjoint when

π
ϕ

{\pi _\phi }

is selfadjoint and

π
ψ

{\pi _\psi }

is bounded, where

τ
\tau

is the sum of positive linear functionals

ϕ
,
ψ

\phi ,\psi

on a

∗
*

-algebra

A

\mathcal {A}

and

π
τ

,

π
ϕ

{\pi _\tau },{\pi _\phi }

and

π
ψ

{\pi _\psi }

are

∗
*

-representations generated by

τ
,
ϕ

\tau ,\phi

and

ψ
\psi

, respectively. The second purpose is to prove that

π
ϕ

{\pi _\phi }

is standard, where

ϕ
\phi

is a positive linear functional on

A

\mathcal {A}

such that there exists a net

{

ϕ
α

}

\left \{ {{\phi _\alpha }} \right \}

of positive linear functionals on

A

\mathcal {A}

satisfying

ϕ
α

≦
ϕ
,

π

ϕ
α

{\phi _\alpha } \leqq \phi ,{\pi _{{\phi _\alpha }}}

is bounded for all

α
\alpha

and

lim

α

ϕ
α

(
x
)

=
ϕ

(
x
)

\lim _{\alpha }{\phi _\alpha }\left ( x \right ) = \phi \left ( x \right )

for each

x
∈

A

x \in \mathcal {A}

.
Powers of transitive bases of measure and category
Authors: Pawlikowski Janusz
Published: June 2010

We prove that on the real line the minimal cardinality of a base of measure zero sets equals the minimal cardinality of their transitive base. Next we show that it is relatively consistent that the minimal cardinality of a base of meager sets is greater than the minimal cardinality of their transitive base. We also prove that it is relatively consistent that the transitive additivity of measure zero sets is greater than the ordinary additivity and that the same is true about meager sets.
Discrete cells properties in the boundary set setting
Authors: Bowers Philip L.
Published: June 2010

Let

X
X

be the complement of a

σ
−
Z

\sigma - Z

-set in a locally compact separable ANR. It is proved that

X
X

satisfies the discrete

n
n

-cells property for each nonnegative integer

n
n

if and only if

X
X

satisfies the discrete approximation property. As a consequence, Hilbert space manifolds that arise as complements of boundary sets in Hilbert cube manifolds are characterized in terms of their homological structure coupled with a minimal amount of general positioning.
Ring of endomorphisms of a finite length module
Authors: Gupta R. N. , Singh Surjeet
Published: June 2010

An example of a uniserial module

M
R

{M_R}

of composition length 2, such that

S
=
End
⁡
(

M
R

)

S = \operatorname {End}({M_R})

acting on the left is not right artinian, is given. An elementary proof of a known result, that the ring of endomorphism of a finite length quasi-injective module

M
R

{M_R}

acting on the left is left artinian, is also given.
Best monotone approximation in 𝐿₁[0,1]
Authors: Huotari Robert , Legg David
Published: June 2010

If

f
f

is a bounded Lebesgue measurable function on [0,1] and

1
>
p
>
∞

1 > p > \infty

, let

f
p

{f_p}

denote the best

L
p

{L_p}

-approximation to

f
f

by nondecreasing functions. It is shown that

f
p

{f_p}

converges almost everywhere as

p
p

decreases to one to a best

L
1

{L_1}

-approximation to

f
f

by nondecreasing functions. The set of best

L
1

{L_1}

-approximations to

f
f

by nondecreasing functions is shown to include its supremum and infimum.
Viscosity solutions of Hamilton-Jacobi equations at the boundary
Authors: Crandall Michael G. , Newcomb Richard
Published: June 2010

When considering classical solutions of boundary value problems for nonlinear first-order scalar partial differential equations, one knows that there are parts of the boundary of the region under consideration where one cannot specify data and would not expect to require data in order to prove uniqueness. Of course, classical solutions of such problems rarely exist in the large owing to the crossing of characteristics. The theory of a sort of generalized solution—called "viscosity solutions"—for which good existence and uniqueness theorems are valid has been developed over the last few years. In this note we give some results concerning parts of the boundary on which one need not know (prescribe) viscosity solutions to be able to prove comparison (and hence uniqueness) results. In this context, this amounts to identifying boundary points with the property that solutions in the interior which are continuous up to the boundary are also viscosity solutions at the boundary point. Examples indicating the sharpness of the results are given.
Conformal mapping of regions bounded by curvilinear polygons
Authors: Rodin Burton
Published: June 2010

The main result is the proof that, under a natural notion of conformal equivalence, any curvilinear polygon is conformally equivalent to a curvilinear polygon whose inside region is starlike. The method of proof is a new type of continuity argument which, in distinction to the classical continuity method, yields existence without requiring uniqueness.
A note on simple duality
Authors: Davis James F. , Löffler Peter
Published: June 2010

Certain spaces satisfying Poincaré duality are shown to have the homotopy type of simple Poincaré complexes. The motivation for and applications of this result are the study of free actions of finite groups on manifolds.
Functions with a dense set of proper local maximum points
Authors: Villani Alfonso
Published: June 2010

Let

X
X

be any metric space. The existence of continuous real functions on

X
X

, with a dense set of proper local maximum points, is shown. Indeed, given any

σ
\sigma

-discrete set

S
⊂
X

S \subset X

, the set of all

f
∈
C
(
X
)

f \in C(X)

, which assume a proper local maximum at each point of

S
S

, is a dense subset of

C
(
X
)

C(X)

. This implies, for a perfect metric space

X
X

, the density in

C
(
X
,
Y
)

C(X,Y)

of "nowhere constant" continuous functions from

X
X

to a normed space

Y
Y

. In this way, two questions raised in [2] are solved.
Translation properties of sets of positive upper density
Authors: Bergelson Vitaly , Weiss Benjamin
Published: June 2010

Generalizing a result of Raimi we show that there exists a set

E
⊂

N

E \subset {\mathbf {N}}

such that if

A
⊂

N

A \subset {\mathbf {N}}

is a set with positive upper density, then there exists a number

k
∈

N

k \in {\mathbf {N}}

such that

d
∗

(
(
A
+
k
)
∩
E
)
>
0

{d^ * }((A + k) \cap E) > 0

and

d
∗

(
(
A
+
k
)
∩

E
c

)
>
0

{d^ * }((A + k) \cap {E^c}) > 0

. Some extensions and further results are also obtained.
A common fixed-point theorem in reflexive locally uniformly convex Banach spaces
Authors: Edelstein Michael , Kiang Mo Tak
Published: June 2010

Let

X
X

be a reflexive locally uniformly convex Banach space and

G
G

an ultimately nonexpansive commutative semigroup of continuous self-maps of

X
X

. If there exists a point

x
x

in

X
X

recurrent under

G
G

such that

G
(
x
)

G(x)

is bounded, then

G
G

has a common fixed point in

co

¯

(
G
(
x
)
)

\overline {{\text {co}}} (G(x))

. If

X
X

is a Hilbert space then there is exactly one such point in

co

¯

(
G
(
x
)
)

\overline {{\text {co}}} (G(x))

.
Almost Euclidean quotient spaces of subspaces of a finite-dimensional normed space
Authors: Milman V. D.
Published: June 2010

The main result of this article is Theorem 1 which states that a quotient space

Y
,
dim
⁡
Y
=
k

Y,\dim Y = k

, of a subspace of any finite dimensional normed space

X
X

, may be chosen to be

d
d

-isomorphic to a euclidean space even for

k
=
[
λ
n
]

k = [\lambda n]

for any fixed

λ
>
1

\lambda > 1

(and

d
d

depending on

λ
\lambda

only).
Structural instability of 𝑒𝑥𝑝(𝑧)
Authors: Devaney Robert L.
Published: June 2010

The entire function

exp
⁡

(
z
)

\operatorname {exp}\left ( z \right )

has a Julia set equal to the whole plane. We show that there are complex

λ
\lambda

’s near 1 such that

λ

e
z

\lambda {e^z}

has an attracting periodic orbit. Hence

e
z

{e^z}

is not structurally stable.
On a theorem of Cohen and Montgomery
Authors: Van den Bergh Michel
Published: June 2010

In a recent paper, Cohen and Montgomery proved a conjecture of Bergman concerning the relation between the Jacobson radical and the graded Jacobson radical of a ring graded by a finite group. In their proof they made use of the theory of Hopf algebras. In this note we give a short and elementary proof of the Bergman conjecture.
A priori bounds for positive solutions of a semilinear elliptic equation
Authors: Cosner Chris , Schmitt Klaus
Published: June 2010

We consider the semilinear elliptic equation

−
Δ
u
=
f
(
u
)

- \Delta u = f(u)

,

x
∈
Ω

x \in \Omega

, subject to zero Dirichlet boundary conditions, where

Ω
⊂

R

n

\Omega \subset {{\mathbf {R}}^n}

is a bounded domain with smooth boundary and the nonlinearity

f
f

assumes both positive and negative values. Under the assumption that

Ω
\Omega

satisfies certain symmetry conditions we establish two results providing lower bounds on the

C
0

(

Ω
¯

)

{C^0}(\overline \Omega )

norm of positive solutions. The bounds derived are the same one obtains in dimension

n
=
1

n = 1

.
On two conjectures concerning the partial sums of the harmonic series
Authors: Zemyan Stephen M.
Published: June 2010

Let

S
n

{S_n}

denote the

n
n

th partial sum of the harmonic series. For a given positive integer

k
>
1

k > 1

, there exists a unique integer

n
k

{n_k}

such that

S

n
k

−
1

>
k
>

S

n
k

{S_{{n_k} - 1}} > k > {S_{{n_k}}}

. It has been conjectured that

n
k

{n_k}

is equal to the integer nearest

e

k
−
y

{e^{k - y}}

, where

γ
\gamma

is Euler’s constant. We provide an estimate on

n
k

{n_k}

which suggests that this conjecture may have to be modified. We also propose a conjecture concerning the amount by which

S

n
k

−
1

{S_{{n_k} - 1}}

and

S

n
k

{S_{{n_k}}}

differ from

k
k

.
𝐻¹ subordination and extreme points
Authors: Abu-Muhanna Yusuf
Published: June 2010

Suppose that

F
F

is an element of

H
1

{H^1}

(Hardy class of order 1 over the unit disc). Let

s

(
F
)

{\text {s}}(F)

denote the set of functions subordinate to

F
F

. We show that if

ϕ
\phi

is inner and

ϕ
(
0
)
=
0

\phi (0) = 0

; then

F
∘
ϕ

F \circ \phi

is an extreme point of the closed convex hull of

s

(
F
)

{\text {s}}(F)

.
The Forelli problem concerning ideals in the disk algebra 𝐴(𝐷)
Authors: Mortini Raymond
Published: June 2010

Let

Z
(
f
)

Z(f)

be the zero set of a function

f
∈
A
(

D

)

f \in A({\mathbf {D}})

and

Z
(
I
)
=

∩

f
∈
I

Z
(
f
)

Z(I) = { \cap _{f \in I}}Z(f)

the zero set of an ideal

I
I

in

A
(

D

)

A({\mathbf {D}})

. It is shown that in the disk algebra

A
(

D

)

A({\mathbf {D}})

every finitely generated ideal

I
I

has the weak Forelli property, i.e. there exists a function

f
∈
I

f \in I

such that

Z
(
f
)
∩
T
=
Z
(
I
)
∩
T

Z(f) \cap T = Z(I) \cap T

, where

T
T

is the boundary of the unit circle

D

{\mathbf {D}}

. On the other hand, there exists a finitely generated ideal

I
I

in

A
(

D

)

A({\mathbf {D}})

such that

Z
(
f
)
≠
Z
(
I
)

Z(f) \ne Z(I)

for each choice of

f
∈
I

f \in I

. This provides us with a negative answer to a problem of F. Forelli [1].
The first coefficient of the Conway polynomial
Authors: Hoste Jim
Published: June 2010

A formula is given for the first coefficient of the Conway polynomial of a link in terms of its linking numbers. A graphical interpretation of this formula is also given.
The algebraic independence of certain Liouville continued fractions
Authors: Adams William W.
Published: June 2010

This work uses some simple Liouville type arguments to extend some recent work of Bundschuh and of Laohakosol and Ubolsri on algebraic independence. The results are stronger and are not restricted to just two numbers. We then use the results to give a new and simple proof of Bundschuh’s result concerning the algebraic independence of certain numbers whose

g
g

-adic and continued fraction expansions are both known.
Numerical radius-attaining operators on 𝐶(𝐾)
Authors: Cardassi Carmen Silvia
Published: June 2010

Using a construction due to Johnson and Wolfe, we show that the numerical radius-attaining operators from

C
(
K
)

C(K)

into itself are dense in the space of all operators, where

K
K

is a compact Hausdorff space.
For any 𝑋, the product 𝑋×𝑌 has remote points for some 𝑌
Authors: Peters Thomas J.
Published: June 2010

Any space with a

σ
\sigma

-locally finite

π
\pi

-base will be called a

σ
−
π

\sigma - \pi

space. The work of Chae and Smith can be extended to show that every nonpseudocompact

σ
−
π

\sigma - \pi

space has remote points.

2

^{2}

Sufficient conditions for a product to be a

σ
−
π

\sigma - \pi

space are developed. It is shown that, for each space, if

α
\alpha

is a cardinal with the discrete topology, where

α
\alpha

is not less than

π
\pi

-weight of

X
X

, then

X
×

α
ω

X \times {\alpha ^\omega }

has remote points. Cardinal function criteria are developed for the existence of

σ
−
π

\sigma - \pi

spaces. An example is given of a pathological product which is a

σ
−
π

\sigma - \pi

space even though none of its finite partial products is a

σ
−
π

\sigma - \pi

space.
On the existence of Green’s function in Riemannian manifolds
Authors: Fernández José L.
Published: June 2010

This note provides a sufficient condition of geometric character for the existence of Green’s function in an arbitrary complete Riemannian manifold.
Comparison theorems for second order differential systems
Authors: Kim W. J.
Published: June 2010

Comparison theorems are proved for second order linear differential systems of the form

(

R
i

y
′

)
′

+

P
i

y
=
0

({R_i}y’)’ + {P_i}y = 0

, where

R
i

{R_i}

and

P
i

{P_i}

are continuous

n
×
n

n \times n

matrices and

R
i

{R_i}

is invertible,

i
=
1
,
2

i = 1,2

.
A remark about viscosity solutions of Hamilton-Jacobi equations at the boundary
Authors: Souganidis P. E.
Published: June 2010

We consider viscosity solutions of first-order partial differential equations of Hamilton-Jacobi type in bounded domains. We give criteria which identify boundary points at which the equation is automatically satisfied in the viscosity sense, if it holds in the interior. These complement some recent results of M. G. Crandall and R. Newcomb [3].
Ground states and small perturbations for 𝐶*-dynamical systems
Authors: Ikunishi Akio
Published: June 2010

Existence of ground states for

C
∗

{C^*}

-dynamical systems is stable under perturbations with relative bound

>
1

> 1

.
Comparison theorem for conjugate points of systems of 𝑛th order nonselfadjoint differential equations
Authors: Tomastik E. C.
Published: June 2010

A comparison theorem for conjugate points of the two systems of linear differential equations

x

(
n
)

−

(
−
1

)

n
−
k

p
(
t
)
x
=
0

{x^{(n)}} - {( - 1)^{n - k}}p(t)x = 0

and

y

(
n
)

−

(
−
1

)

n
−
k

q
(
t
)
y
=
0

{y^{(n)}} - {( - 1)^{n - k}}q(t)y = 0

, where

p
(
t
)

p(t)

and

q
(
t
)

q(t)

are

m
×
m

m \times m

matrices of continuous functions, is given. It is assumed that

q
(
t
)

q(t)

is positive with respect to a certain cone but no positivity conditions of any kind are imposed on

p
(
t
)

p(t)

. No selfadjointness conditions are assumed; however, the results are new even in the selfadjoint case.
A partial integration formula for product integrals of unbounded operator-valued functions
Authors: Hughes Rhonda J.
Published: June 2010

The partial integration formula for product integrals
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="product Underscript y Overscript x Endscripts e Superscript left-parenthesis upper A left-parenthesis s right-parenthesis plus upper B left-parenthesis s right-parenthesis right-parenthesis d s Baseline equals product Underscript y Overscript x Endscripts e Superscript upper A left-parenthesis s right-parenthesis d s Baseline product Underscript y Overscript x Endscripts exp left-parenthesis left-parenthesis product Underscript s Overscript y Endscripts e Superscript upper A left-parenthesis u right-parenthesis d u Baseline upper B left-parenthesis s right-parenthesis product Underscript y Overscript s Endscripts e Superscript upper A left-parenthesis u right-parenthesis d u Baseline right-parenthesis d s right-parenthesis comma">

∏
y
x

e

(
A
(
s
)
+
B
(
s
)
)
d
s

=

∏
y
x

e

A
(
s
)
d
s

∏
y
x

exp

(

(

∏
s
y

e

A
(
u
)
d
u

B
(
s
)

∏
y
s

e

A
(
u
)
d
u

)

d
s

)

,

\prod \limits _y^x {{e^{(A(s) + B(s))ds}} = } \prod \limits _y^x {{e^{A(s)ds}}} \prod \limits _y^x {\exp } \left ( {\left ( {\prod \limits _s^y {{e^{A(u)du}}} B(s)\prod \limits _y^s {{e^{A(u)du}}} } \right )ds} \right ),

\]
of which the Trotter product formula is a consequence, is established for a wide class of unbounded operator-valued functions

A
(
s
)
,
B
(
s
)

A(s),B(s)

.
Invariant subspaces for algebras of subnormal operators
Authors: Thomson James E.
Published: June 2010

Every rationally cyclic subnormal operator has a hyperinvariant subspace.
The maximum modulus principle for CR functions
Authors: Iordan Andrei
Published: June 2010

Let

M
M

be a CR submanifold of

C

n

{{\mathbf {C}}^n}

without extreme points. Then, the modulus of any CR function on

M
M

cannot have a strong local maximum at any point of

M
M

.
Antisymmetry and the direct integral decomposition of unstarred operator algebras
Authors: Szymański Wacław
Published: June 2010

It is shown that the direct integral decomposition of a non-self-adjoint operator algebra

A

{\mathcal A}

has the diagonal

A

∩

A

∗

{\mathcal A} \cap {{\mathcal A}^ * }

of this algebra as the algebra of diagonalizable operators if and only if almost all direct integrands of

A

{\mathcal A}

are antisymmetric algebras. By using the antisymmetric decomposition a direct integral model of a commutative, reflexive algebra is obtained.
𝐾-divisibility and a theorem of Lorentz and Shimogaki
Authors: Bennett Colin , Sharpley Robert
Published: June 2010

The Brudnyi-Krugljak theorem on the

K
K

-divisibility of Gagliardo couples is derived by elementary means from earlier results of Lorentz-Shimogaki on equimeasurable rearrangements of measurable functions. A slightly stronger form of Calderón’s theorem describing the Hardy-Littlewood-Pólya relation in terms of substochastic operators (which itself generalizes the classical Hardy-Littlewood-Pólya result for substochastic matrices) is obtained.
A trivial link with no linear unlinking
Authors: Knoblauch Tom
Published: June 2010

Not every relative link (finite collection of disjoint polygonal spanning arcs of a cube) that is trivial allows a linear unlinking.
A nonlinear ergodic theorem for a reversible semigroup of nonexpansive mappings in a Hilbert space
Authors: Takahashi Wataru
Published: June 2010

Let

C
C

be a nonempty closed convex subset of a Hilbert space,

S
S

a right reversible semitopological semigroup,

S

=
{

T
t

:
t
∈
S
}

\mathcal {S} = \{ {T_t}:t \in S\}

a continuous representation of

S
S

as nonexpansive mappings on a closed convex subset

C
C

into

C
C

, and

F
(

S

)

F(\mathcal {S})

the set of common fixed points of mappings

T
t

,

t
∈
S

{T_t},\;t \in S

. Then we deal with the existence of a nonexpansive retraction

P
P

of

C
C

onto

F
(

S

)

F(\mathcal {S})

such that

P

T
t

=

T
t

P
=
P

P{T_t} = {T_t}P = P

for each

t
∈
S

t \in S

and

P
x

{P_x}

is contained in the closure of the convex hull of

{

T
t

x
:
t
∈
S
}

\{ {T_t}x:t \in S\}

for each

x
∈
C

x \in C

.
Second order differential equations with transcendental coefficients
Authors: Rossi John
Published: June 2010

Let

w
1

{w_1}

and

w
2

{w_2}

be two linearly independent solutions to

w
+
A
w
=
0

w + Aw = 0

, where

A
A

is a transcendental entire function of order

ρ
(
A
)
>
1

\rho (A) > 1

. We show that the exponent of convergence

λ
(
E
)

\lambda (E)

of the zeros of

E
=

w
1

w
2

E = {w_1}{w_2}

is either infinite or satisfies

ρ

(
A

)

−
1

+
λ

(
E

)

−
1

≤
2

\rho {(A)^{ - 1}} + \lambda {(E)^{ - 1}} \leq 2

. For

ρ
(
A
)
=

1
2

\rho (A) = \tfrac {1}{2}

, this answers a question of Bank.
Complete metrics conformal to the hyperbolic disc
Authors: Bland J. , Kalka Morris
Published: June 2010

In this paper we study complete metrics conformal to the hyperbolic disc. We show that any smooth function

K
K

bounded between two negative constants is the curvature of such a metric. We also show that if

K
≥
0

K \geq 0

near the boundary,

K
K

cannot be the curvature of such a metric.
An answer to a question of M. Newman on matrix completion
Authors: Vaserstein L. N.
Published: June 2010

Let

R
R

be a principal ideal ring,

A
A

a symmetric

t
t

-by-

t
t

matrix over

R
R

,

B
B

a

t
t

-by-

(
n
−
t
)

(n - t)

matrix over

R
R

such that the

t
t

-by-

n
n

matrix

(
A
,
B
)

(A,B)

is primitive. Newman [2] proved that

(
A
,
B
)

(A,B)

may be completed (as the first

t
t

rows) to a symmetric

n
n

-by-

n
n

matrix of determinant 1, provided that

1
≤
t
≤
n

/

3

1 \leq t \leq n/3

. He showed that the result is false, in general, if

t
=
n

/

2

t = n/2

, and he asked to determine all values of

t
t

such that

1
≤
t
≤
n

1 \leq t \leq n

and the result holds. It is shown here that these values are exactly

t
t

satisfying

1
≤
t
≤
n

/

2

1 \leq t \leq n/2

. Moreover, the result is proved for a larger (than the principal ideal rings) class of commutative rings, namely, for the rings satisfying the second stable range condition of Bass [1]. Also, it is observed that Theorems 2 and 3 of [2, p. 40] proved there for principal ideal rings are true for this larger class of rings, as well as the basic result of [2, p. 39].
Higher order singularities of morphisms to projective space
Authors: Hansen Johan P.
Published: June 2010

This paper proves existence theorems for higher order singularities of a finite morphism to

P

m

{{\mathbf {P}}^m}

and deduces a result on simple connectivity of varieties admitting a finite morphism of bounded singularity. The singularities are obtained by successive degeneration of double points of

f
f

. Our main tool is R. Schwarzenberger’s notion of generalized secant sheaves and the connectedness theorem obtained by W. Fulton and the author.
Approximation on disks
Authors: de Paepe P. J.
Published: June 2010

It is shown that if the functions

F
F

and

G
G

are defined in a neighborhood of the origin in the complex plane and are in a certain sense like

z
m

{z^m}

and

z

−
n

{z^{ - n}}

with

gcd
(
m
,
n
)
=
1

\gcd (m,n) = 1

, then on sufficiently small closed disks

D
D

around 0 every continuous function on

D
D

can be uniformly approximated by polynomials in

F
F

and

G
G

.
Exact sequences of spectra and duality
Authors: Kuhn Nicholas J.
Published: June 2010

We note that duality arguments show that, under certain circumstances, an exact sequence of spectra remains exact after smashing with a finite complex. Among various applications, we show that the validity for

S
0

{S^0}

of G. W. Whitehead’s symmetric products conjecture implies that the conjecture is "almost" true for all finite complexes.
Counterexample to the Lu Qi-Keng conjecture
Authors: Boas Harold P.
Published: June 2010

There exists a smooth bounded strongly pseudoconvex domain in

C

2

{{\mathbf {C}}^2}

, smoothly equivalent to the ball, whose Bergman kernel function has zeroes.
On a theorem of Hardy and Littlewood on the polydisc
Authors: Kim Hong Oh
Published: June 2010

We prove the polydisc version of the theorem of Hardy and Littlewood on the fractional integral: If

0
>
α
>
1

/

p

0 > \alpha > 1/p

and if

f
∈

H
p

f \in {H^p}

, then

I
α

f
∈

H
q

{I^\alpha }f \in {H^q}

with

q
=
p

/

(
1
−
α
p
)

q = p/(1 - \alpha p)

where

I
α

f

{I^\alpha }f

is the fractional integral of

f
f

of order

α
\alpha

.
Linear perturbations of a nonoscillatory second order equation
Authors: Trench William F.
Published: June 2010

It is shown that the equation

(
r
(
t
)

x
′

)
′

+
g
(
t
)
x
=
0

(r(t)x’)’ + g(t)x = 0

has solutions which behave asymptotically like those of a nonoscillatory equation

(
r
(
t
)

y
′

)
′

+
f
(
t
)
y
=
0

(r(t)y’)’ + f(t)y = 0

, provided that a certain integral involving

f
−
g

f - g

converges (perhaps conditionally) and satisfies a second condition which has to do with its order of convergence. The result improves upon a theorem of Hartman and Wintner.
Kinematic formulas for Weyl’s curvature invariants of submanifolds in complex projective space
Authors: Ishihara Tōru
Published: June 2010

It is shown in [5] that Weyl’s curvature invariants

k

2
p

(
M
)

{k_{2p}}(M)

can be expressed by

γ
q

∧

F

m
−
q

[
M
]

{\gamma _q} \wedge {F^{m - q}}[M]

, where

M
M

is a

2
m

2m

-dimensional Kähler submanifold with compact closure in a space of constant holomorphic curvature,

γ
q

{\gamma _q}

is the

q
q

th Chern form of

M
M

and

F
F

is the Kähler form of

M
M

. In this paper, we shall show that each

γ
q

∧

F

m
−
q

[
M
]

{\gamma _q} \wedge {F^{m - q}}[M]

is expressible in terms of

F
F

and

k

2
p

(
M
)

{k_{2p}}(M)

. Using this result, we get kinematic formulas for

k

2
p

(
M
)

{k_{2p}}(M)

from Shifrin’s [8] kinematic formulas for Chern classes.
The Thompson-Wielandt theorem
Authors: Fan Paul S.
Published: June 2010

In this short paper we give a completely elementary argument for the Thompson-Wielandt theorem and note its significance for the Goldschmidt-Sims conjecture.
On the algebra of bounded holomorphic martingales
Authors: Arai Hitoshi
Published: June 2010

The properties of the algebra

H
∞

{H^\infty }

of bounded holomorphic martingales are further studied following the work of N. Th. Varopoulos [11]. The purpose of this paper is to discuss the behavior of weak* closed algebras

B
B

with

H
∞

≠
⊂

B
≠
⊂

L
∞

{H^\infty }_ \ne ^ \subset B_ \ne ^ \subset {L^\infty }

and the support sets of

H
∞

{H^\infty }

-functions. We also give some results on the factorization theorem for

H
∞

{H^\infty }

and extreme points of the unit ball of

H
∞

{H^\infty }

.
A note on uniform operators
Authors: Wang Hsiao Lan
Published: June 2010

An operator is uniform if its restriction to any infinite-dimensional invariant subspace is unitarily equivalent to itself. We show that a uniform operator having a proper infinite-dimensional invariant subspace resembles an analytic Toeplitz operator in the way that the weakly closed algebra generated by it and the identity operator is isomorphic to a subalgebra of the Calkin algebra; furthermore, this algebra contains no nonscalar operator which is quasi-similar to a normal operator.
Average radial limits in weighted Hardy spaces
Authors: Matheson Alec , Ullrich David C.
Published: June 2010

Weighted Hardy spaces are defined in the unit disc by specifying the rate of growth of

p
p

th means near the boundary. Although a function in one of these spaces need have no radial limits, it is shown that in certain of these spaces "average radial limits" exist over an interval on the boundary. An integral representation in terms of these average radial limits is given, with an application to the question of existence of (pointwise) radial limits.
A simplified proof of Heinz inequality and scrutiny of its equality
Authors: Furuta Takayuki
Published: June 2010

An operator means a bounded linear operator on a Hilbert space

H
H

. We give a simplified proof of the following inequality:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper I Subscript 1 Baseline right-parenthesis StartAbsoluteValue left-parenthesis upper T x comma y right-parenthesis EndAbsoluteValue squared less-than-or-equal-to left-parenthesis StartAbsoluteValue upper T EndAbsoluteValue Superscript 2 alpha Baseline x comma x right-parenthesis left-parenthesis StartAbsoluteValue upper T Superscript asterisk Baseline EndAbsoluteValue Superscript 2 left-parenthesis 1 minus alpha right-parenthesis Baseline y comma y right-parenthesis">

(

I

1

)

|

(
T
x
,
y
)

|

2

≤
(

|

T

|

2
α

x
,
x
)
(

|

T
∗

|

2
(
1
−
α
)

y
,
y
)

({{\text {I}}_1})\quad |(Tx,y){|^2} \leq (|T{|^{2\alpha }}x,x)(|{T^*}{|^{2(1 - \alpha )}}y,y)

\]
for any operator

T
T

and for any

x
,
y
∈
H

x, y \in H

and for any real number

α
\alpha

with

0
≤
α
≤
1

0 \leq \alpha \leq 1

. In case

0
>
α
>
1

0 > \alpha > 1

, the equality in

(

I

1

)

({{\text {I}}_1})

holds iff

|

T

|

2
α

x

|T{|^{2\alpha }}x

and

T
∗

y

{T^*}y

are linearly dependent iff

T
x

Tx

and

|

T
∗

|

2
(
1
−
α
)

y

|{T^*}{|^{2(1 - \alpha )}}y

are linearly dependent.

(

I

1

)

({{\text {I}}_1})

is equivalent to
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper I Subscript 2 Baseline right-parenthesis StartAbsoluteValue left-parenthesis upper T x comma y right-parenthesis EndAbsoluteValue less-than-or-equal-to StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue upper T EndAbsoluteValue Superscript alpha Baseline x StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue upper T Superscript asterisk Baseline EndAbsoluteValue Superscript 1 minus alpha Baseline y StartAbsoluteValue EndAbsoluteValue comma">

(

I

2

)

|

(
T
x
,
y
)

|

≤

|

|

|

T

|

α

x

|

|

|

|

|

T
∗

|

1
−
α

y

|

|

,

({{\text {I}}_2})\quad |(Tx,y)| \leq ||\;|T{|^\alpha }x||\;||\;|{T^*}{|^{1 - \alpha }}y||,

\]
, so one might believe that the equality in

(

I

1

)

({{\text {I}}_1})

or

(

I

2

)

({{\text {I}}_2})

would hold iff

|

T

|

2
α

x

|T{|^{2\alpha }}x

and

|

T
∗

|

2
(
1
−
α
)

y

|{T^*}{|^{2(1 - \alpha )}}y

are linearly dependent or iff

|

T

|

α

x

|T{|^\alpha }x

and

|

T
∗

|

1
−
α

y

|{T^*}{|^{1 - \alpha }}y

are linearly dependent, but we can give counterexamples to these mistakes. By this fact, the form of

(

I

1

)

({{\text {I}}_1})

is more convenient than

(

I

2

)

({{\text {I}}_2})

in order to remind us of the case when the equality in

(

I

1

)

({{\text {I}}_1})

or

(

I

2

)

({{\text {I}}_2})

holds.
On binomial units of certain cubic field
Authors: Ozeki Kiyota
Published: June 2010

Scarowsky has conjectured about a binomial unit in a cubic field. We discuss a relation between binomial units and a diophantine equation.
On completeness of the products of harmonic functions
Authors: Ramm A. G.
Published: June 2010

Let

L
L

be a partial differential operator in

R
n

{R^n}

with constant coefficients. We prove that, under some assumption on

L
L

, the set of products of the elements of the null-space of

L
L

forms a complete set in

L
p

(
D
)

{L^p}(D)

,

p
⩾
1

p \geqslant 1

, where

D
D

is any bounded domain. In particular, the products of harmonic functions form a complete set in

L
p

(
D
)

{L^p}(D)

,

p
⩾
1

p \geqslant 1

.
Structural theorems for periodic ultradistributions
Authors: Pilipović Stevan
Published: June 2010

Structural theorems for periodic ultradistributions of Roumieu and Beurling types are given.
A complex space whose spectrum is not locally compact anywhere
Authors: Hayes Sandra , Vigué Jean-Pierre
Published: June 2010

An example of a two-dimensional complex space is given with the property that the continuous spectrum of the global holomorphic functions is not locally compact at any point.
Fibrations of locally symmetric spaces and the failure of the Jordan-Hölder property
Authors: Johnson F. E. A.
Published: June 2010

We construct examples of locally symmetric spaces which fibre, with both base space and fibre also locally symmetric, in two distinct ways so that neither fibre nor base of one is homeomorphic to either fibre or base of the other. The simplest example is a complex surface which fibres over a complex curve in two essentially distinct ways. In consequence, the Jordan-Hölder property fails for large classes of irreducible discrete subgroups of Lie groups.
Examples of topological groups homeomorphic to 𝑙^{𝑓}₂
Authors: Dobrowolski Tadeusz
Published: June 2010

We prove that the following spaces are homeomorphic to

l
2
f

l_2^f

: (1) the group of piecewise continuous autotransformations of

[
−
1
,
1
]

[ - 1,1]

preserving Lebesgue measure, and (2) certain subgroups obtained as group spans of linearly independent arcs in linear spaces. These are consequences of our discussion of the problem whether

σ
\sigma

-fd-compact locally contractible metric groups must be either finite-dimensional or locally homeomorphic to

l
2
f

l_2^f

.
A weakly infinite-dimensional space whose product with the irrationals is strongly infinite-dimensional
Authors: Pol Elżbieta
Published: June 2010

We give an example of a weakly infinite-dimensional space

X
X

such that the product

X
×
B

X \times B

of

X
X

and a subspace

B
B

of the irrationals is strongly infinite-dimensional; under the assumption of the Continuum Hypothesis,

B
B

can be the irrationals. This example answers a question of Addis and Gresham [AG].
Harnack’s inequality for Schrödinger operators and the continuity of solutions
Authors: Chiarenza F. , Fabes E. , Garofalo N.
Published: June 2010

We prove a uniform Harnack inequality for nonnegative solutions of

A
u
=
V
u

Au = Vu

, where

A
A

is a second order elliptic operator in divergence form, and

V
V

belongs to the Stummel class of potentials. As a consequence we obtain the continuity of a general weak solution. These results extend the previous work of Aizenman and Simon for

Δ
u
=
V
u

\Delta u = Vu

.
A typical property of Baire 1 Darboux functions
Authors: Evans Michael J. , Humke Paul D.
Published: June 2010

It is well known that a real-valued, bounded, Baire class one function of a real variable is the derivative of its indefinite integral at every point except possibly those in a set which is both of measure zero and of first category. In the present paper, a bounded, Darboux, Baire class one function is constructed to have the property that its indefinite integral fails to be differentiable at non-

σ
\sigma

-porous set of points. Such functions are then shown to be "typical" in the sense of category in several standard function spaces.
On the volumes of images of holomorphic mappings in 𝐶ⁿ
Authors: Alexander H.
Published: June 2010

Let

f
f

be a holomorphic map from the unit ball in complex

n
n

-space to complex

n
n

-space. We establish a lower bound for the volume, taken without multiplicity, of the image of

f
f

. The estimate is in terms of the boundary values of

f
f

. This generalizes some known results in one complex variable. The proof uses the methods of uniform algebras.
Simple examples of nonrealizable CR hypersurfaces
Authors: Jacobowitz Howard
Published: June 2010

A new proof is provided of a nonrealizability result due to Hill, Penrose, and Sparling. This result is then generalized to higher dimensions: Each

∂
b

{\partial _b}

-cohomology class in

H

0
,
1

(
M
)

{H^{0,1}}(M)

can be used to define a nonrealizable CR structure on

M
×

R

2

M \times {{\mathbf {R}}^2}

.
Two-bridge knots with unknotting number one
Authors: Kanenobu Taizo , Murakami Hitoshi
Published: June 2010

We determine all two-bridge knots with unknotting number one. In fact we prove that a two-bridge knot has unknotting number one iff there exist positive integers

p
p

,

m
m

, and

n
n

such that

(
m
,
n
)
=
1

(m,n) = 1

and

2
m
n
=
p
±
1

2mn = p \pm 1

, and it is equivalent to

S
(
p
,
2

n
2

)

S(p,2{n^2})

in Schubert’s notation. It is also shown that it can be expressed as

C
(
a
,

a
1

,

a
2

,
…
,

a
k

,
±
2
,
−

a
k

,
…
,
−

a
2

,
−

a
1

)

C(a,{a_1},{a_2}, \ldots ,{a_k}, \pm 2, - {a_k}, \ldots , - {a_2}, - {a_1})

using Conway’s notation.
Maps of the interval with every point chain recurrent
Authors: Block Louis , Coven Ethan M.
Published: June 2010

We show that if

f
f

is a continuous map of a compact interval to itself and every point is chain recurrent, then either

f
2

{f^2}

is the identity map or

f
2

{f^2}

is turbulent.
The word problem for free lattice-ordered groups (and some other free algebras)
Authors: Evans Trevor
Published: June 2010

It is known that in the lattice of all varieties of

l
l

-groups, there is a unique maximal proper subvariety. Based on this fact we give a simple algorithm for deciding the word problem in free

l
l

-groups. Some other applications are also given.
A note on the automorphism groups of Lie module triple systems
Authors: Hopkins Nora C.
Published: June 2010

Using representation theory and well-known facts about automorphism groups of reductive Lie algebras, the automorphism group of a basic Lie module triple system

(
M
,
{
,
,
}
,

L

,
b
,
ϕ
)

(M,\{ ,,\} ,\mathcal {L},b,\phi )

over an algebraically closed field of characteristic zero is related to the automorphism group of the Lie algebra

L

\mathcal {L}

, the automorphism group of the standard embedding

S

\mathcal {S}

of

(
M
,
{
,
,
}
)

(M,\{ ,,\} )

, and the automorphism group of the split null extension

S

\mathcal {S}

of

L

\mathcal {L}

by

M
M

.
Four algebraic spectral properties
Authors: Call Frederick W.
Published: June 2010

For a commutative ring, we examine the four concepts of flat surjections, exact torsion functors, induced closed affine subschemes, and type I BSP torsion theories. We show they are equivalent, they correspond to closed subsets of Spec that are also stable under generalization, and thus are spectral in nature.
General position theorems for generalized manifolds
Authors: Bryant J. L.
Published: June 2010

It is an open question as to whether a generalized

n
n

-manifold,

n
⩾
5

n \geqslant 5

, that satisfies the disjoint disks property is a topological manifold. In this paper it is shown that any such space

X
X

satisfies general position properties for maps of polyhedra into

X
X

.
Relationship between the meet and join operators in the lattice of group topologies
Authors: Clark Bradd , Schneider Victor
Published: June 2010

Let

L
(
G
)

L(G)

be the lattice of all topologies on the group

G
G

which make

G
G

into a topological group. If

τ
1

{\tau _1}

and

τ
2

{\tau _2}

are Hausdorff group topologies and

τ
1

∨

τ
2

{\tau _1} \vee {\tau _2}

is the discrete topology, then

τ
1

∧

τ
2

{\tau _1} \wedge {\tau _2}

is a Hausdorff topology. If

τ
1

{\tau _1}

and

τ
2

{\tau _2}

are locally compact Hausdorff group topologies, then

τ
1

∨

τ
2

{\tau _1} \vee {\tau _2}

is locally compact if and only if

τ
1

∧

τ
2

{\tau _1} \wedge {\tau _2}

is Hausdorff.
A short proof of Cauchy’s polygonal number theorem
Authors: Nathanson Melvyn B.
Published: June 2010

This paper presents a simple proof that every nonnegative integer is the sum of

m
+
2

m + 2

polygonal numbers of order

m
+
2

m + 2

.
Pseudo-Riemannian metric singularities and the extendability of parallel transport
Authors: Kossowski Marek
Published: June 2010

We are given a

C
∞

C\infty

immersion

i
:
N
→
(
M
,
⟨

⟩
)

i:N \to (M,\langle \;\rangle )

, and

p
∈
N

p \in N

is a point where

N
p
⊥

∩

T
p

N

N_p^ \bot \cap {T_p}N

is one-dimensional. We have shown that there is a tensor

I

I

p

:

T
p

N
×

T
p

N
×

Rad
p

→

R

{\text {I}}{{\text {I}}_p}:{T_p}N \times {T_p}N \times \operatorname {Rad}_p \to {\mathbf {R}}

intrinsic to

(
N
,

i
∗

⟨

⟩
)

(N,{i^*}\langle \;\rangle )

which determines an extrinsic feature of the immersion. The purpose of this paper is to show that II controls the following two intrinsic properties. First, II determines which pairs of vector fields

X
X

,

Y
Y

on

N
N

have the property that intrinsic covariant derivative

∇
x

Y

{\nabla _x}Y

extends smoothly to all of

N
N

. Second, given a curve in

N
N

containing

p
,

I

I

p

p,{\text {I}}{{\text {I}}_p}

determines which parallel vector fields along the curve extend smoothly through

p
p

. As an application we locally characterize product and flat metric singularities.