bluuebunny/crossref_metadata_2025 huggingface dataset:
Flat core properties associated to the 𝑝-Laplace operator
Authors: Kamin Shoshana , Véron Laurent
Published: June 2010

We study the formation of a flat hat pattern in the profile of the positive solution of an equation of the type:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon normal upper Delta Subscript p Baseline u minus u Superscript p minus 1 Baseline left-parenthesis 1 minus u right-parenthesis Superscript theta Baseline equals 0 left-parenthesis 0 greater-than theta greater-than p minus 1 right-parenthesis">

ε

Δ
p

u
−

u

p
−
1

(
1
−
u

)
θ

=
0

(
0
>
θ
>
p
−
1
)

\varepsilon {\Delta _p}u - {u^{p - 1}}{(1 - u)^\theta } = 0\;(0 > \theta > p - 1)

in a bounded domain

Ω
\Omega

. When

ε
\varepsilon

tends to

0
+

{0^ + }

, the growth of the zone where

u
=

u
ε

u = {u_\varepsilon }

takes the value

1
1

in

Ω
\Omega

is studied.
Strong measure zero and strongly meager sets
Authors: Carlson Timothy J.
Published: June 2010

We consider conjectures made by Prikry and Galvin concerning strong measure zero and strongly meager sets of real numbers.
The Cayley graphs of Coxeter and Artin groups
Authors: Droms Carl , Servatius Herman
Published: June 2010

We obtain a complete classification of the Coxeter and Artin groups whose Cayley graphs with respect to the standard presentations are planar. We also classify those whose Cayley graphs have planar embeddings in which the vertices have no accumulation points.
On immersions of 𝑘-connected 𝑛-manifolds
Authors: Li B. H. , Mahowald M. E.
Published: June 2010

In this note we classify up to regular homotopy the classes of immersions of

k
k

-connected closed differentiable

n
n

-manifolds in

R

2
n
−
k

{\mathbb {R}^{2n - k}}

.
Complemented subspaces and amenability: a counterexample
Authors: Takahashi Yuji
Published: June 2010

We give an example of a left amenable discrete semigroup

S
S

such that

l
∞

(
S
)

{l^\infty }(S)

has weak

∗

^{{\ast }}

-closed selfadjoint left translation invariant subalgebras that are weak

∗

^{{\ast }}

-complemented but not invariantly complemented in

l
∞

(
S
)

{l^\infty }(S)

. This resolves negatively a problem raised by Lau.
Invariant subspaces of Toeplitz operators with piecewise continuous symbols
Authors: Peller Vladimir V.
Published: June 2010

Sufficient conditions are found for the existence of nontrivial invariant subspaces for Toeplitz operators with piecewise continuous symbols. The results are obtained by estimating the norm of the resolvents.
Independent events and independent experiments
Authors: Baryshnikov Yuliy M. , Eisenberg Bennett
Published: June 2010

It is shown that independent events in a probability space with equally likely outcomes are isomorphic to events coming from independent experiments each with equally likely outcomes.
Polar decompositions of bounded linear functionals on operator subalgebras
Authors: Kusuda Masaharu
Published: June 2010

Let

M
M

be a von Neumann algebra and let

φ
\varphi

be a normal linear functional on a strongly closed

C

∗

{C^{\ast }}

-subalgebra

N
N

of

M
M

. Denote by

F

φ

{\mathcal {F}_\varphi }

the set of normal linear functionals

ψ
\psi

on

M
M

extending

φ
\varphi

with

|

|

ψ

|

|

=

|

|

φ

|

|

||\psi || = ||\varphi ||

. It is shown that there exists a partial isometry

v
v

in

N
N

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi equals StartAbsoluteValue phi EndAbsoluteValue left-parenthesis v dot right-parenthesis comma StartAbsoluteValue phi EndAbsoluteValue equals phi left-parenthesis v Superscript asterisk Baseline dot right-parenthesis comma StartAbsoluteValue EndAbsoluteValue phi StartAbsoluteValue EndAbsoluteValue equals StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue phi EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue">

φ
=

|

φ

|

(
v
⋅
)
,

|

φ

|

=
φ
(

v

∗

⋅
)
,

|

|

φ

|

|

=

|

|

|

φ

|

|

|

\varphi = |\varphi |(v \cdot ),\qquad |\varphi | = \varphi ({v^{\ast }} \cdot ),\qquad ||\varphi || = |||\varphi |||

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi equals StartAbsoluteValue psi EndAbsoluteValue left-parenthesis v dot right-parenthesis comma StartAbsoluteValue psi EndAbsoluteValue equals psi left-parenthesis v Superscript asterisk Baseline dot right-parenthesis comma StartAbsoluteValue EndAbsoluteValue psi StartAbsoluteValue EndAbsoluteValue equals StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue psi EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue">

ψ
=

|

ψ

|

(
v
⋅
)
,

|

ψ

|

=
ψ
(

v

∗

⋅
)
,

|

|

ψ

|

|

=

|

|

|

ψ

|

|

|

\psi = |\psi |(v\cdot ),\qquad |\psi | = \psi ({v^{\ast }}\cdot ),\qquad ||\psi || = |||\psi |||

\]
for all

ψ
\psi

in

F

φ

{\mathcal {F}_\varphi }

, where

|

φ

|

|\varphi |

and

|

ψ

|

|\psi |

denote the absolute values of

φ
\varphi

and

ψ
\psi

respectively. Let

A
A

be a

C

∗

{C^{\ast }}

-algebra and let

B
B

be a

C

∗

{C^{\ast }}

-subalgebra of

A
A

. As a consequence of this result, we obtain that every state on

B
B

has a unique state extension to

A
A

if and only if every bounded linear functional on

B
B

has a unique norm-preserving extension to a bounded linear functional on

A
A

.
Some applications of the Euler-Jacobi formula to differential equations
Authors: Cima Anna , Gasull Armengol , Mañosas Francesc
Published: June 2010

The Euler-Jacobi formula gives an algebraic relation between the critical points of a vector field and their indices. Using this formula we obtain an upper bound for the number of centers that a planar polynomial differential equation can have and study the distribution of the critical points for planar quadratic and cubic differential equations.
Piecewise linear discontinuous double coverings of the circle
Authors: Galeeva Roza , Tresser Charles
Published: June 2010

In his study of a particular Lorenz-like semiflow, S. F. Kennedy introduced a two-parameter family of endomorphisms of the circle with two marked points. These are piecewise affine double coverings of the circle with a pair of discontinuities, which all have topological entropy

log
⁡
2

\log 2

. We answer the question Kennedy raised about when two such maps are topologically conjugate.
Taylor theorem for planar curves
Authors: Rababah Abedallah
Published: June 2010

We describe an approximation method for planar curves that significantly improves the standard rate obtained by local Taylor approximations. The method exploits the freedom in the choice of the parametrization and achieves the order

4
m

/

3

4m/3

where

m
m

is the degree of the approximating polynomial parametrization. Moreover, we show for a particular set of curves that the optimal rate

2
m

2m

is possible.
On the Diophantine equations 𝑑₁𝑥²+2^{2𝑚}𝑑₂=𝑦ⁿ and 𝑑₁𝑥²+𝑑₂=4𝑦ⁿ
Authors: Maohua Le
Published: June 2010

Let

d
1

,

d
2

{d_1},\;{d_2}

be coprime positive integers, which are squarefree, and let

h
h

denote the class number of the imaginary quadratic field

Q

(

−

d
1

d
2

)

\mathbb {Q}(\sqrt { - {d_1}{d_2}} )

. Let

m
,

n

m,\;n

be integers such that

m
⩾
0
,

n
>
1

m \geqslant 0,\;n > 1

, and

gcd
(
n
,
2
h
)
=
1

\gcd (n,2h) = 1

. In this paper we prove that if

n
⩾
8.5
⋅

10
6

n \geqslant 8.5 \cdot {10^6}

, then the equations

d
1

x
2

+

2

2
m

d
2

=

y
n

(
2
∤
y
)

{d_1}{x^2} + {2^{2m}}{d_2} = {y^n}(2\nmid y)

and

d
1

x
2

+

d
2

=
4

y
n

{d_1}{x^2} + {d_2} = 4{y^n}

have no positive integer solutions

(
x
,
y
)

(x,y)

with

gcd
(
x
,
y
)
=
1

\gcd (x,y) = 1

.
Constructing bases for radicals and nilradicals of Lie algebras
Authors: Merrin Stephen
Published: June 2010

The radical and nilradical of a finite-dimensional Lie algebra

L
L

are well defined unique subspaces of

L
L

. Nevertheless, we show the impossibility of ever finding a general algorithm that will construct finite bases for radicals (or nilradicals) of arbitrary finite-dimensional Lie algebras. Our approach involves an investigation of the relationship between radicals of associative algebras and radicals of Lie algebras. Building on a result of Richman in the constructive theory of associative algebras, we prove that bases for radicals and nilradicals of finite-dimensional Lie algebras over a discrete field

F
F

can always be constructed if and only if

F
F

satisfies Seidenberg’s condition P. A special case is that if we restrict ourselves to fields of characteristic zero, we can indeed always construct bases for radicals. Our proofs are entirely constructive (i.e., do not use the general law of excluded middle).
Kmax and stable rank of enveloping algebras
Authors: Tintera George
Published: June 2010

We calculate upper bounds for the stable rank of enveloping algebras of all finite-dimensional nonabelian nilpotent and some solvable Lie algebras using Kmax and Stafford’s generalization of Bass’s Stable Range Theorem. We find that the stable rank of these algebras is less than their commutative counterparts, polynomial rings.
On the minimal invariant subspaces of the hyperbolic composition operator
Authors: Matache Valentin
Published: June 2010

The composition operator induced by a hyperbolic Möbius transform

ϕ
\phi

on the classical Hardy space

H
2

{H^2}

is considered. It is known that the invariant subspace problem for Hilbert space operators is equivalent to the fact that all the minimal invariant subspaces of this operator are one- dimensional. In connection with that we try to decide by the properties of a given function

u
u

in

H
2

{H^2}

if the corresponding cyclic subspace is minimal or not. The main result is the following. If the radial limit of

u
u

is continuously extendable at one of the fixed points of

ϕ
\phi

and its value at the point is nonzero, then the cyclic subspace generated by

u
u

is minimal if and only if

u
u

is constant.
Radó’s theorem for locally solvable vector fields
Authors: Hounie Jorge , Tavares Joaquim
Published: June 2010

We extend the classical theorem of Radó, valid for the Cauchy-Riemann operator, to locally solvable first-order operators with smooth coefficients in

R

N

{\mathbb {R}^N}

.
Quadratic and quasi-quadratic functionals
Authors: Šemrl Peter
Published: June 2010

In this note we show how Jordan

∗

{\ast }

-derivations arise as a "measure" of the representability of quasi-quadratic functionals by sesquilinear ones. Our main result can be considered as an extension of the Jordan-von Neumann characterization of pre-Hilbert space.
On generalized maximal functions
Authors: Schröder Bernd S. W.
Published: June 2010

In this paper we study the question of under what circumstances the quantity
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue sup Underscript t greater-than normal infinity comma a element-of double-struck upper R Endscripts StartAbsoluteValue integral Subscript 0 Superscript t Baseline f left-parenthesis a comma upper M Subscript s Baseline right-parenthesis d upper M Subscript s Baseline EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue Subscript p">

|

|

sup

t
>
∞
,

a
∈

R

|

∫
0
t

f
(
a
,

M
s

)

d

M
s

|

|

|

p

||{\sup _{t > \infty ,\,a \in \mathbb {R}}}|\int _0^t f (a,{M_s})\,d{M_s}|\;|{|_p}

is comparable to

|

|

M
∞

∗

|

|

p

||M_\infty ^{\ast }|{|_p}

, where

M
t

{M_t}

is a continuous martingale and

f
f

is a bounded Borel-measurable function.
On mappings with integrable dilatation
Authors: Iwaniec Tadeusz , Šverák Vladimír
Published: June 2010

A factorization of Stoilow’s type has been obtained for mappings in

R

2

{\mathbb {R}^2}

with integrable dilatation.
Continuous proper holomorphic maps into bounded domains
Authors: Dor Avner
Published: June 2010

A continuous proper holomorphic map is constructed from the unit ball of

C

N

{\mathbb {C}^N}

to a smooth bounded domain in

C

M

(
2
⩽
N
⩽
M
−
1
)

{\mathbb {C}^M}(2 \leqslant N \leqslant M - 1)

. The construction is done at a strongly convex corner of the target domain. At each stage the map is pushed farther into the boundary in a direction almost tangent to the boundary at a close vicinity. The close point property is employed, along with suitable peak functions, to obtain a minimal codimension. It appears to be close to the most general construction that can be made by summation of peak functions.
Automatic continuity of homomorphisms in topological algebras
Authors: Bhatt S. J.
Published: June 2010

A homomorphism from a locally convex

Q
Q

-algebra to a uniform topological algebra is continuous. A one-to-one homomorphism from a regular complete spectrally bounded uniform topological algebra onto a dense subalgebra of a semisimple locally

m
m

-convex

Q
Q

-algebra is open. Examples are discussed to show that none of the assumptions in these results can be omitted.
The Euler characteristic of a generic wavefront in a 3-manifold
Authors: Izumiya S. , Marar W. L.
Published: June 2010

We give a relation between Euler characteristics of a generic closed Legendrian surface and its wavefront.
Sobolev and quasiconformal extension domains
Authors: Ghamsari Manouchehr
Published: June 2010

A domain

D
⊂

R

n

D \subset {\mathbb {R}^n}

has the quasiconformal extension property if each quasiconformal self-map of

D
D

extends to a quasiconformal self-map of

R

n

;

D

{\mathbb {R}^n};\;D

has the Sobolev extension property if there is a bounded linear operator
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda colon upper W Superscript 1 comma n Baseline left-parenthesis upper D right-parenthesis right-arrow upper W Superscript 1 comma n Baseline left-parenthesis double-struck upper R Superscript n Baseline right-parenthesis">

Λ
:

W

1
,
n

(
D
)
→

W

1
,
n

(

R

n

)

\Lambda :{W^{1,n}}(D) \to {W^{1,n}}({\mathbb {R}^n})

. We consider the relation between the above extension properties for

n
⩾
3

n \geqslant 3

. We show that for domains quasiconformally equivalent to a ball the quasiconformal extension property implies the Sobolev extension property. We show that this is not true in general. Next the Sobolev extension property does not imply the extension property for quasiconformal maps which extend as homeomorphisms. Finally if

G
⊂

R

2

G \subset {\mathbb {R}^2}

and if

D
=
G
×

R

D = G \times \mathbb {R}

is quasiconformally equivalent to a ball, then

D
D

has the quasiconformal extension property if and only if

D
D

is a quasiball.
Some infinite series identities
Authors: Ford Kevin B.
Published: June 2010

Certain infinite series are shown to satisfy simple identities between the square of the sum of the series and the sum of the squares of the terms of the series. The main tool is Ramanujan’s

1

ψ
1

{}_1{\psi _1}

summation formula.
A class of Riesz sets
Authors: Li Daniel
Published: June 2010

Let

G
G

be a metrizable compact abelian group. A subset

Λ
\Lambda

in the dual group is said to be ergodic if every

f
∈

L
∞

(
G
)

f \in {L^\infty }(G)

whose spectrum lies in a translate of

Λ
\Lambda

has a unique invariant mean. It is shown that such a set is a Riesz set.
Statistical limit points
Authors: Fridy J. A.
Published: June 2010

Following the concept of a statistically convergent sequence

x
x

, we define a statistical limit point of

x
x

as a number

λ
\lambda

that is the limit of a subsequence

{

x

k
(
j
)

}

\{ {x_{k(j)}}\}

of

x
x

such that the set

{
k
(
j
)
:
j
∈

N

}

\{ k(j):j \in \mathbb {N}\}

does not have density zero. Similarly, a statistical cluster point of

x
x

is a number

γ
\gamma

such that for every

ε
>
0

\varepsilon > 0

the set

{
k
∈

N

:

|

x
k

−
γ

|

>
ε
}

\{ k \in \mathbb {N}:|{x_k} - \gamma | > \varepsilon \}

does not have density zero. These concepts, which are not equivalent, are compared to the usual concept of limit point of a sequence. Statistical analogues of limit point results are obtained. For example, if

x
x

is a bounded sequence then

x
x

has a statistical cluster point but not necessarily a statistical limit point. Also, if the set

M
:=
{
k
∈

N

:

x
k

>

x

k
+
1

}

M: = \{ k \in \mathbb {N}:{x_k} > {x_{k + 1}}\}

has density one and

x
x

is bounded on

M
M

, then

x
x

is statistically convergent.
Primeness of twisted knots
Authors: Motegi Kimihiko
Published: June 2010

Let

V
V

be a standardly embedded solid torus in

S
3

{S^3}

with a meridian-preferred longitude pair

(
μ
,
λ
)

(\mu ,\lambda )

and

K
K

a knot contained in

V
V

. We assume that

K
K

is unknotted in

S
3

{S^3}

. Let

f
n

{f_n}

be an orientation-preserving homeomorphism of

V
V

which sends

λ
\lambda

to

λ
+
n
μ

\lambda + n\mu

. Then we get a twisted knot

K
n

=

f
n

(
K
)

{K_n} = {f_n}(K)

in

S
3

{S^3}

. Primeness of twisted knots is discussed and we prove: A twisted knot

K
n

{K_n}

is prime if

|

n

|

>
5

|n| > 5

. Moreover,

{

K
n

}

n
∈
Z

{\{ {K_n}\} _{n \in Z}}

contains at most five composite knots.
Associative and Jordan shift algebras
Authors: Loos Ottmar , Neher Erhard
Published: June 2010

Let

R
R

be the shift algebra, i.e., the associative algebra presented by generators

u
,
v

u,v

and the relation

u
v
=
1

uv = 1

. As N. Jacobson showed,

R
R

contains an infinite family of matrix units. In this paper, we describe the Jordan algebra

R
+

{R^ + }

and its unital special universal envelope by generators and relations. Moreover, we give a presentation for the Jordan triple system defined on

R
R

by

P
x

y
=
x

y

∗

x

{P_x}y = x{y^{\ast }}x

where

∗

^{\ast }

is the involution on

R
R

with

u

∗

=
v

{u^{\ast }} = v

. As a consequence, we prove the existence of an infinite rectangular grid in a Jordan triple system

V
V

containing tripotents

c
c

and

d
d

with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V 2 left-parenthesis c right-parenthesis equals upper V 2 left-parenthesis d right-parenthesis circled-plus left-parenthesis upper V 2 left-parenthesis c right-parenthesis intersection upper V 1 left-parenthesis d right-parenthesis right-parenthesis">

V
2

(
c
)
=

V
2

(
d
)
⊕
(

V
2

(
c
)
∩

V
1

(
d
)
)

{V_2}(c) = {V_2}(d) \oplus ({V_2}(c) \cap {V_1}(d))

and

V
2

(
c
)
∩

V
1

(
d
)
≠
0

{V_2}(c) \cap {V_1}(d) \ne 0

.
Inverse theorem for best polynomial approximation in 𝐿_{𝑝},0<𝑝<1
Authors: Ditzian Z. , Jiang D. , Leviatan D.
Published: June 2010

A direct theorem for best polynomial approximation of a function in

L
p

[
−
1
,
1
]
,

0
>
p
>
1

{L_p}[ - 1,1],\;0 > p > 1

, has recently been established. Here we present a matching inverse theorem. In particular, we obtain as a corollary the equivalence for

0
>
α
>
k

0 > \alpha > k

between

E
n

(
f

)
p

=
O
(

n

−
α

)

{E_n}{(f)_p} = O({n^{ - \alpha }})

and

ω
φ
k

(
f
,
t

)
p

=
O
(

t
α

)

\omega _\varphi ^k{(f,t)_p} = O({t^\alpha })

. The present result complements the known direct and inverse theorem for best polynomial approximation in

L
p

[
−
1
,
1
]
,

1
⩽
p
⩽
∞

{L_p}[ - 1,1],\;1 \leqslant p \leqslant \infty

. Analogous results for approximating periodic functions by trigonometric polynomials in

L
p

[
−
π
,
π
]
,
0
>
p
⩽
∞

{L_p}[ - \pi ,\pi ],0 > p \leqslant \infty

, are known.
Permutation properties of the polynomials 1+𝑥+⋯+𝑥^{𝑘} over a finite field
Authors: Matthews Rex
Published: June 2010

It is shown that a polynomial of the shape

1
+
x
+
⋯
+

x
k

1 + x + \cdots + {x^k}

is a permutation polynomial over a finite field

F

q

{\mathbb {F}_q}

of odd characteristic

p
p

if and only if

k
≡
1
mod
p
(
q
−
1
)

k \equiv 1\bmod p(q - 1)

.
A suspension theorem for continuous trace 𝐶*-algebras
Authors: Dădărlat Marius
Published: June 2010

Let

B

\mathcal {B}

be a stable continuous trace

C

∗

{C^{\ast }}

-algebra with spectrum

Y
Y

. We prove that the natural suspension map
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript asterisk Baseline colon left-bracket upper C 0 left-parenthesis upper X right-parenthesis comma script upper B right-bracket right-arrow left-bracket upper C 0 left-parenthesis upper X right-parenthesis circled-times upper C 0 left-parenthesis bold upper R right-parenthesis comma script upper B circled-times upper C 0 left-parenthesis bold upper R right-parenthesis right-bracket">

S

∗

:
[

C
0

(
X
)
,

B

]
→
[

C
0

(
X
)
⊗

C
0

(

R

)
,

B

⊗

C
0

(

R

)
]

{S_{\ast }}:[{C_0}(X),\mathcal {B}] \to [{C_0}(X) \otimes {C_0}({\mathbf {R}}),\mathcal {B} \otimes {C_0}({\mathbf {R}})]

is a bijection, provided that both

X
X

and

Y
Y

are locally compact connected spaces whose one-point compactifications have the homotopy type of a finite CW-complex and

X
X

is noncompact. This is used to compute the second homotopy group of

B

\mathcal {B}

in terms of

K
K

-theory. That is,

[

C
0

(

R

2

)
,

B

]
=

K
0

(

B

0

)

[{C_0}({{\mathbf {R}}^2}),\mathcal {B}] = {K_0}({\mathcal {B}_0})

, where

B

0

{\mathcal {B}_0}

is a maximal ideal of

B

\mathcal {B}

if

Y
Y

is compact, and

B

0

=

B

{\mathcal {B}_0} = \mathcal {B}

if

Y
Y

is noncompact.
Constructing 𝑈𝑉^{𝑘}-maps between spheres
Authors: Ferry Steven C.
Published: June 2010

The purpose of this note is to give a quick proof of an extremely counterintuitive theorem of Bestvina, Walsh, and Wilson. The theorem says, for example, that the degree

2
2

map

d
2

:

S
3

→

S
3

{d_2}:{S^3} \to {S^3}

is homotopic to a map such that

p

−
1

(
x
)

{p^{ - 1}}(x)

is connected for each

x
∈

S
3

x \in {S^3}

.
Linear monotone operators and weighted BMO
Authors: Lai Qinsheng
Published: June 2010

In this paper, the linear monotone operators, which include the well-known Hardy operator and Riemann-Liouville fractional integrals, are introduced. A necessary and sufficient condition for them to be bounded from a Banach function space into a weighted BMO is given, and their compactness in some particular cases is studied. Meanwhile, the embedding properties concerning the weighted BMO are investigated.
A nonspectral dense Banach subalgebra of the irrational rotation algebra
Authors: Schweitzer Larry B.
Published: June 2010

We give an example of a dense, simple, unital Banach subalgebra

A
A

of the irrational rotation

C

∗

{C^{\ast }}

-algebra

B
B

, such that

A
A

is not a spectral subalgebra of

B
B

. This answers a question posed by T. W. Palmer (Spectral algebras, Rocky Mountain J. Math. 22 (1992), 293-328).
An invariant Riemann type integral defined by figures
Authors: Novikov Alexei , Pfeffer Washek F.
Published: June 2010

We show that a multidimensional generalized Riemann integral defined by means of rectangular figures is already invariant with respect to lipeomorphic changes of coordinates.
The sufficient condition for a convex body to enclose another in 𝑅⁴
Authors: Zhou Jia Zu
Published: June 2010

We follow Hadwiger and Ren’s ideas to estimate the kinematic measure of a convex body

D
1

{D_1}

with

C
2

{C^2}

-boundary

∂

D
1

\partial {D_1}

moving inside another convex body

D
0

{D_0}

with the same kind of boundary

∂

D
0

\partial {D_0}

under the isometry group G in

R

4

{\mathbb {R}^4}

. By using Chern and Yen’s kinematic fundamental formula, C-S. Chen’s kinematic formula for the total square mean curvature

∫

∂

D
0

∩
g
∂

D
1

H
2

d
v

{\smallint _{\partial {D_0} \cap g\partial {D_1}}}{H^2}dv

, and some well-known results about the curvatures of the 2-dimensional intersection submanifold

∂

D
0

∩
g
∂

D
1

\partial {D_0} \cap g\partial {D_1}

, we obtain a sufficient condition to guarantee that one convex body can enclose another in

R

4

{\mathbb {R}^4}

.
Multipliers with closed range on regular commutative Banach algebras
Authors: Aiena Pietro , Laursen Kjeld B.
Published: June 2010

Conditions equivalent with closure of the range of a multiplier T, defined on a commutative semisimple Banach algebra A, are studied. A main result is that if A is regular then

T
2

A

{T^2}A

is closed if and only if T is a product of an idempotent and an invertible. This has as a consequence that if A is also Tauberian then a multiplier with closed range is injective if and only if it is surjective. Several aspects of Fredholm theory and Kato theory are covered. Applications to group algebras are included.
The eigenvalue gap for one-dimensional convex potentials
Authors: Lavine Richard
Published: June 2010

For Schrödinger operators on an interval with convex potentials, the gap between the two lowest eigenvalues is minimized when the potential is constant.
Generalizations of Deodhar’s 𝛼-localization functor
Authors: Cox Ben
Published: June 2010

In this paper we generalize the result of Deodhar (see Invent. Math. 57 (1980), 101-118) on

α
\alpha

-localization functors. Namely, we show that localization with respect to a larger family of left denominator sets "intertwines" with tensoring by finite-dimensional representations. In the language of the author’s previous work, localization with respect to such a left denominator set produces a new example of an

F

\mathfrak {F}

-functor and an

F

\mathfrak {F}

-category.
A characterisation of absolutely continuous measures on topological semigroups
Authors: Dzinotyiweyi Heneri A. M.
Published: June 2010

Let S be a completely regular topological semigroup and

μ
\mu

a bounded regular Borel measure on S. For a very large class of noncompact semigroups S, we show that the map

x
→

μ
∗

x
¯

x \to {\mu ^ \ast }\bar x

of S into the space of bounded regular Borel measures on S is norm-continuous if and only if

μ
0

f

{\mu _0}f

is a left uniformly continuous function on S, for all bounded continuous functions f on S. Here the function

μ
0

f

{\mu _0}f

is given by
\[

μ
0

f
(
x
)
:=
∫

f
(
y
x
)
d
μ
(
y
)

on

S
.

{\mu _0}f(x): = \int {f(yx)d\mu (y)\quad {\text {on}}\;S.}

\]
An elementary proof of Benedicks’s and Carleson’s estimates of harmonic measure of linear sets
Authors: Sodin Mikhail
Published: June 2010

Benedicks and Carleson proved sophisticated estimates of harmonic measure of linear sets which have found applications in approximation theory and harmonic analysis. We give an elementary proof of these estimates. This proof allows us to relax the assumptions and strengthen estimates.
Kähler-Einstein surfaces with nonpositive bisectional curvature
Authors: Zheng Fangyang
Published: June 2010

In this note we show that, for a Kähler-Einstein surface M with negative Ricci curvature and nonpositive bisectional curvature, if the cotangent bundle of M is not quasi-ample then M is a quotient of the bidisc.
Every low Boolean algebra is isomorphic to a recursive one
Authors: Downey Rod , Jockusch Carl G.
Published: June 2010

It is shown that every (countable) Boolean algebra with a presentation of low Turing degree is isomorphic to a recursive Boolean algebra. This contrasts with a result of Feiner (1967) that there is a Boolean algebra with a presentation of degree

≤

0
′

\leq 0’

which is not isomorphic to a recursive Boolean algebra. It is also shown that for each n there is a finitely axiomatizable theory

T
n

{T_n}

such that every

low
n

{\text {low}_n}

model of

T
n

{T_n}

is isomorphic to a recursive structure but there is a

low

n
+
1

{\text {low}_{n + 1}}

model of

T
n

{T_n}

which is not isomorphic to any recursive structure. In addition, we show that

n
+
2

n + 2

is the Turing ordinal of the same theory

T
n

{T_n}

, where, very roughly, the Turing ordinal of a theory describes the number of jumps needed to recover nontrivial information from models of the theory. These are the first known examples of theories with Turing ordinal

α
\alpha

for

3
≤
α
>
ω

3 \leq \alpha > \omega

.
Invariant subspaces: continuous stability implies smooth stability
Authors: Noakes Lyle , Chung Kin Yan
Published: June 2010

The stability of an invariant subspace is defined in the terms of the existence of a certain type of function. The imposition of further conditions on this function leads to different forms of stability. Of these, the equivalence of continuous and smooth stability is proved; two proofs are offered for comparison.
Growth property for the minimal surface equation in unbounded domains
Authors: Hwang Jenn-Fang
Published: June 2010

Here we prove that if u satisfies the minimal surface equation in an unbounded domain

Ω
\Omega

which is properly contained in a half plane, then the growth rate of u is of the same order as the shape of

Ω
\Omega

and

u

|

∂
Ω

u{|_{\partial \Omega }}

.
Representing characteristic homology classes of 𝑚𝐂𝐏²#𝐧\overline{𝐂}𝐏²
Authors: Guo Jian Han , Gan Dan Yan
Published: June 2010

We prove the following theorems. Theorem 1. If

m
,
n
≥
1
,
x
∈

H
2

(
m
C

P
2

#
n

C
P

¯

2

)

m,n \geq 1,x \in {H_2}(mC{P^2}\# n{\overline {CP} ^2})

is a characteristic homology class with

x
2

=
16
l
+
m
−
n
>
0

{x^2} = 16l + m - n > 0

and (1)

m
>
3
l
+
1

m > 3l + 1

provided

l
≥
0

l \geq 0

, or (2)

m
>
−
19
l
+
1

m > - 19l + 1

provided

l
>
0

l > 0

. Suppose that the 11/8-conjecture is true. Then x cannot be represented by a smoothly embedded 2-sphere. Theorem 2. Let

m
,
n
≥
4
l
>
0
,
x
∈

H
2

(
m
C

P
2

#
n

C
P

¯

2

)

m,n \geq 4l > 0,x \in {H_2}(mC{P^2}\# n{\overline {CP} ^2})

be a primitive characteristic homology class with

x
2

=
±
16
l
+
m
−
n

{x^2} = \pm 16l + m - n

. Then x can be represented by a smoothly embedded 2-sphere.
Every uncountable abelian group admits a nonnormal group topology
Authors: Trigos-Arrieta F. Javier
Published: June 2010

If G is a locally compact Abelian group, let

G

+

{{\mathbf {G}}^ + }

denote the underlying group of G equipped with the weakest topology that makes all the continuous characters of G continuous. Thus defined,

G

+

{{\mathbf {G}}^ + }

is a totally bounded topological group. We prove: Theorem.

G

+

{{\mathbf {G}}^ + }

is normal if and only if G is

σ
\sigma

-compact. When G is discrete, this theorem answers in the negative a question posed in 1990 by E. van Douwen, and it partially solves a problem posed in 1945 by A. Markov.
Siegel’s theorem for complex function fields
Authors: Voloch José Felipe
Published: June 2010

We give a short proof of the finiteness of the set of integral points on an affine algebraic curve of genus at least one, defined over a function field of characteristic zero.
On the theory of the Kontorovich-Lebedev transformation on distributions
Authors: Yakubovich Semen B. , Fisher Brian
Published: June 2010

A new proof of the representation theorem for the Kontorovich-Lebedev transformation on distributions is given.
Riemannian metrics with large 𝜆₁
Authors: Colbois B. , Dodziuk J.
Published: June 2010

We show that every compact smooth manifold of three or more dimensions carries a Riemannian metric of volume one and arbitrarily large first eigenvalue of the Laplacian.
An operator-valued Yeh-Wiener integral and a Kac-Feynman Wiener integral equation
Authors: Park Chull , Skoug David
Published: June 2010

Let

C
[
0
,
T
]

C[0,T]

denote Wiener space, i.e., the space of all continuous functions

η
(
t
)

\eta (t)

on

[
0
,
T
]

[0,T]

such that

η
(
0
)
=
0

\eta (0) = 0

. For

Q
=
[
0
,
S
]
×
[
0
,
T
]

Q = [0,S] \times [0,T]

let

C
(
Q
)

C(Q)

denote Yeh-Wiener space, i.e., the space of all

R

\mathbb {R}

-valued continuous functions

x
(
s
,
t
)

x(s,t)

on

Q
Q

such that

x
(
0
,
t
)
=
x
(
s
,
0
)
=
0

x(0,t) = x(s,0) = 0

for all

(
s
,
t
)

(s,t)

in

Q
Q

. For

h
∈

L
2

(
Q
)

h \in {L_2}(Q)

let

Z
(
x
;
s
,
t
)

Z(x;s,t)

be the Gaussian process defined by the stochastic integral
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z left-parenthesis x semicolon s comma t right-parenthesis equals integral Subscript 0 Superscript t Baseline integral Subscript 0 Superscript s Baseline h left-parenthesis u comma v right-parenthesis d x left-parenthesis u comma v right-parenthesis period">

Z
(
x
;
s
,
t
)
=

∫
0
t

∫
0
s

h
(
u
,
v
)
d
x
(
u
,
v
)
.

Z(x;s,t) = \int _0^t {\int _0^s {h(u,v)dx(u,v).} }

\]
Then for appropriate functionals

F
F

and

ψ
\psi

we show that the operator-valued function space integral
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper I Subscript lamda Superscript h Baseline left-parenthesis upper F right-parenthesis psi right-parenthesis left-parenthesis eta left-parenthesis dot right-parenthesis right-parenthesis equals upper E Subscript x Baseline left-bracket upper F left-parenthesis lamda Superscript negative 1 slash 2 Baseline upper Z left-parenthesis x semicolon dot comma dot right-parenthesis plus eta left-parenthesis dot right-parenthesis right-parenthesis psi left-parenthesis lamda Superscript negative 1 slash 2 Baseline upper Z left-parenthesis x semicolon upper S comma dot right-parenthesis plus eta left-parenthesis dot right-parenthesis right-parenthesis right-bracket">

(

I
λ
h

(
F
)
ψ
)
(
η
(
⋅
)
)
=

E
x

[
F
(

λ

−
1

/

2

Z
(
x
;
⋅
,
⋅
)
+
η
(
⋅
)
)
ψ
(

λ

−
1

/

2

Z
(
x
;
S
,
⋅
)
+
η
(
⋅
)
)
]

(I_\lambda ^h(F)\psi )(\eta ( \cdot )) = {E_x}[F({\lambda ^{ - 1/2}}Z(x; \cdot , \cdot ) + \eta ( \cdot ))\psi ({\lambda ^{ - 1/2}}Z(x;S, \cdot ) + \eta ( \cdot ))]

\]
is the unique solution of a Kac-Feynman Wiener integral equation. We also use this integral equation to evaluate various Yeh-Wiener integrals.
A sharp estimate on the Bergman kernel of a pseudoconvex domain
Authors: Fu Siqi
Published: June 2010

In this note we obtain a sharp estimate of the Bergman kernels near

C

2

{\mathcal {C}^2}

boundary points of pseudoconvex domains by induction on the dimension and a theorem of Ohsawa-Takegoshi.
Upper bounds for the derivative of exponential sums
Authors: Borwein Peter , Erdélyi Tamás
Published: June 2010

The equality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript p Endscripts StartFraction StartAbsoluteValue p prime left-parenthesis a right-parenthesis EndAbsoluteValue Over double-vertical-bar p double-vertical-bar Subscript left-bracket a comma b right-bracket Baseline EndFraction equals StartFraction 2 n squared Over b minus a EndFraction">

sup
p

|

p
′

(
a
)

|

‖
p
‖

[
a
,
b
]

=

2

n
2

b
−
a

\sup \limits _p \frac {{|p’(a)|}}{{{{\left \| p \right \|}_{[a,b]}}}} = \frac {{2{n^2}}}{{b - a}}

\]
is shown, where the supremum is taken for all exponential sums p of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p left-parenthesis t right-parenthesis equals a 0 plus sigma-summation Underscript j equals 1 Overscript n Endscripts a Subscript j Baseline e Superscript lamda Super Subscript j Superscript t Baseline comma a Subscript j Baseline element-of bold upper R comma">

p
(
t
)
=

a
0

+

∑

j
=
1

n

a
j

e

λ
j

t

,

a
j

∈

R

,

p(t) = {a_0} + \sum \limits _{j = 1}^n {{a_j}{e^{{\lambda _j}t}},\quad {a_j} \in {\mathbf {R}},}

\]
with nonnegative exponents

λ
j

{\lambda _j}

. The inequalities
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar p prime double-vertical-bar Subscript left-bracket a plus delta comma b minus delta right-bracket Baseline less-than-or-equal-to 4 left-parenthesis n plus 2 right-parenthesis cubed delta Superscript negative 1 Baseline double-vertical-bar p double-vertical-bar Subscript left-bracket a comma b right-bracket">

‖

p
′

‖

[
a
+
δ
,
b
−
δ
]

≤
4

(
n
+
2

)
3

δ

−
1

‖
p
‖

[
a
,
b
]

{\left \| {p’} \right \|_{[a + \delta ,b - \delta ]}} \leq 4{(n + 2)^3}{\delta ^{ - 1}}{\left \| p \right \|_{[a,b]}}

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar p prime double-vertical-bar Subscript left-bracket a plus delta comma b minus delta right-bracket Baseline less-than-or-equal-to 4 StartRoot 2 EndRoot left-parenthesis n plus 2 right-parenthesis cubed delta Superscript negative 3 slash 2 Baseline double-vertical-bar p double-vertical-bar Subscript upper L 2 left-bracket a comma b right-bracket">

‖

p
′

‖

[
a
+
δ
,
b
−
δ
]

≤
4

2

(
n
+
2

)
3

δ

−
3

/

2

‖
p
‖

L
2

[
a
,
b
]

{\left \| {p’} \right \|_{[a + \delta ,b - \delta ]}} \leq 4\sqrt 2 {(n + 2)^3}{\delta ^{ - 3/2}}{\left \| p \right \|_{{L_2}[a,b]}}

\]
are also proved for all exponential sums of the above form with arbitrary real exponents. These results improve inequalities of Lorentz and Schmidt and partially answer a question of Lorentz.
Uniqueness and nonuniqueness of the positive Cauchy problem for the heat equation on Riemannian manifolds
Authors: Murata Minoru
Published: June 2010

We investigate a uniqueness problem of whether a nonnegative solution of the heat equation on a noncompact Riemannian manifold is uniquely determined by its initial data. A sufficient condition for the uniqueness (resp. nonuniqueness) is given in terms of nonintegrability (resp. integrability) at infinity of

−
1

- 1

times a negative function by which the Ricci (resp. sectional) curvature of the manifold is bounded from below (resp. above) at infinity. For a class of manifolds, these sufficient conditions yield a simple criterion for the uniqueness.
A simple proof of a remarkable continued fraction identity
Authors: Anderson P. G. , Brown T. C. , Shiue P. J.-S.
Published: June 2010

We give a simple proof of a generalization of the equality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts StartFraction 1 Over 2 Superscript left-bracket n slash tau right-bracket Baseline EndFraction equals left-bracket 0 comma 2 Superscript 0 Baseline comma 2 Superscript 1 Baseline comma 2 Superscript 1 Baseline comma 2 squared comma 2 cubed comma 2 Superscript 5 Baseline comma ellipsis right-bracket comma">

∑

n
=
1

∞

1

2

[
n

/

τ
]

=
[
0
,

2
0

,

2
1

,

2
1

,

2
2

,

2
3

,

2
5

,
…
]
,

\sum \limits _{n = 1}^\infty {\frac {1}{{{2^{[n/\tau ]}}}} = [0,{2^0},{2^1},{2^1},{2^2},{2^3},{2^5}, \ldots ],}

\]
where

τ
=
(
1
+

5

)

/

2

\tau = (1 + \sqrt 5 )/2

and the exponents of the partial quotients are the Fibonacci numbers, and some closely related results.
Pitt’s inequality and the uncertainty principle
Authors: Beckner William
Published: June 2010

The "uncertainty principle" is formulated using logarithmic estimates obtained from a sharp form of Pitt’s inequality. The qualitative nature of this result underlies the relations connecting entropy, the Hardy-Littlewood-Sobolev inequality, and the logarithmic Sobolev inequality.
Dilations on invertible spaces
Authors: Nunnally Ellard
Published: June 2010

This paper primarily concerns certain groups of homeomorphisms which are associated in a natural way with a variety of spaces, which satisfy a set of axiomatic conditions put forth in §1. Let us suppose that

X
X

is a space of the type in question and that

G
G

is an appropriate group of homeomorphisms of

X
X

onto itself. In §2 we demonstrate the existence of a nonvoid subcollection

D

\mathcal {D}

, the “topological dilations,” of

G
G

which is characterized in Theorem 1 in the following fashion: suppose

f
∈

D

f \in \mathcal {D}

and

g
∈
G

g \in G

, then

g
∈

D

g \in \mathcal {D}

if and only if

f
f

is a

G
G

-conjugate of

g
g

, that is if and only if there exists an element

h
h

of

G
G

such that

f
=
h
g

h

−
1

f = hg{h^{ - 1}}

. We proceed then to show in §3 that if

f
f

and

g
g

are nonidentity elements of

G
G

, then we may find

δ
,
r
∈
G

\delta ,r \in G

such that the product

(
r
g

r

−
1

)
(
δ
f

δ

−
1

)
∈

D

(rg{r^{ - 1}})(\delta f{\delta ^{ - 1}}) \in \mathcal {D}

. We then combine this fact with the characterization of

D

\mathcal {D}

mentioned above to conclude that each element of

D

\mathcal {D}

is a “universal” element of

G
G

in the sense that if

d
∈

D

d \in \mathcal {D}

, then any element

g
g

of

G
G

may be represented as the product of two

G
G

-conjugates of

d
d

. Furthermore we conclude that if

g
g

is not the identity element of

G
G

, then

g
g

can be represented as the product of three

G
G

-conjugates of any nonidentity element of

G
G

. Finally, we apply the conclusions to groups of homeomorphisms of certain spaces: for example spheres, cells, the Cantor set, etc.
The formal linearization of a semisimple Lie algebra of vector fields about a singular point
Authors: Hermann Robert
Published: June 2010

A classical theorem by Poincaré gives conditions that a nonlinear ordinary differential equation
\[

d
x

/

d
t
=
A
(
x
)
,

dx/dt = A(x),

\]
with

A
(
0
)
=
0

A(0) = 0

in n variables

x
=
(

x
1

,
…
,

x
n

)

x = ({x_1}, \ldots ,{x_n})

can be reduced to a linear form
\[

d

x
′

d
t

=

∂
A

∂
x

(
0
)

x
′

\frac {{dx’}}{{dt}} = \frac {{\partial A}}{{\partial x}}(0)x’

\]
by a change of variables

x
′

=
f
(
x
)

x’ = f(x)

. A generalization is given for a finite set of such differential equations, which form a semisimple Lie algebra.
Strongly separable pairings of rings
Authors: Cunningham Robert S.
Published: June 2010

The theory of adjoint functors has been used by Morita to develop a theory of Frobenius and quasi-Frobenius extensions subsuming the work of Kasch, Müller, Nakayama, and others. We use adjoint functors to define a pairing of the two rings and develop a theory of relative projective and injective modules for pairings generalizing that of Hochschild for extensions. The main purpose of this paper is to define “strongly separable pairings” generalizing strongly separable (i.e. finitely generated projective separable) algebras. We show that such pairings have very close connections to category equivalences, so that it is natural to investigate those properties shared by two rings which admit a strongly separable pairing. We show that most “categorical” properties are so shared.
A density theorem with an application to gap power series
Authors: Binmore K. G.
Published: June 2010

Let N be a set of positive integers and let
\[

F
(
z
)
=
∑

A
n

z
n

F(z) = \sum {{A_n}{z^n}}

\]
be an entire function for which

A
n

=
0
(
n
∉
N
)

{A_n} = 0(n \notin N)

. It is reasonable to expect that, if D denotes the density of the set N in some sense, then

F
(
z
)

F(z)

will behave somewhat similarly in every angle of opening greater than

2
π
D

2\pi D

. For functions of finite order, the appropriate density seems to be the Pólya maximum density

P

\mathcal {P}

. In this paper we introduce a new density

D

\mathcal {D}

which is perhaps the appropriate density for the consideration of functions of unrestricted growth. It is shown that, if

|

I

|

>
2
π

D

|I| > 2\pi \mathcal {D}

, then
\[

log
⁡
M
(
r
)
∼
log
⁡
M
(
r
,
I
)

\log M(r) \sim \log M(r,I)

\]
outside a small exceptional set. Here

M
(
r
)

M(r)

denotes the maximum modulus of

F
(
z
)

F(z)

on the circle

|

z

|

=
r

|z| = r

and

M
(
r
,
I
)

M(r,I)

that of

F
(
r

e

i
θ

)

F(r{e^{i\theta }})

for values of

θ
\theta

in the closed interval I. The method used is closely connected with the question of approximating to functions on an interval by means of linear combinations of the exponentials

e

i
x
n

(
n
∈
N
)

{e^{ixn}}(n \in N)

.
On Wiener process sample paths
Authors: Foschini G. J. , Mueller R. K.
Published: June 2010

Let

{

X
t

(
ω
)
}

\{ {X_t}(\omega )\}

represent a version of the Wiener process having almost surely continuous sample paths on

(
−
∞
,
∞
)

( - \infty ,\infty )

that vanish at zero. We present a theorem concerning the local nature of the sample paths. Almost surely the local behavior at each t is of one of seven varieties thus inducing a partition of

(
−
∞
,
∞
)

( - \infty ,\infty )

into seven disjoint Borel sets of the second class. The process

{

X
t

(
ω
)
}

\{ {X_t}(\omega )\}

can be modified so that almost surely the sample paths are everywhere locally recurrent.
An elementary theory of the category of topological spaces
Authors: Schlomiuk Dana I.
Published: June 2010

An elementary system of axioms was given by F. W. Lawvere for the category of sets and mappings. The purpose of this paper is to provide a finite number of elementary axioms for the category of topological spaces and continuous mappings and to prove that any model of these axioms is equivalent to “the category of topological spaces” constructed over some model of Lawvere’s axioms. Furthermore, we prove that any complete category, model of the given axioms is equivalent to the category of topological spaces.
Identities involving the coefficients of a class of Dirichlet series. IV
Authors: Berndt Bruce C.
Published: June 2010

We consider a class of Dirichlet series satisfying a functional equation with gamma factors. We define a generalized Dirichlet series that is analogous to the generalized zeta-function of Riemann. An analytic continuation for these generalized series is derived, and a few simple properties are established. Secondly, we prove a theorem on the Abel summation of Dirichlet series that satisfy Hecke’s functional equation.
Two-sided semisimple maximal quotient rings
Authors: Cateforis Vasily C.
Published: June 2010

Let R be an associative ring with singular right ideal zero and finite right Goldie dimension; F. L. Sandomierski has shown that the (R. E. Johnson) maximal right quotient ring Q of R is then semisimple (artinian). In this paper necessary and sufficient conditions are sought that Q be also a left (necessarily the maximal) quotient ring of R. Flatness of Q as a right R-module is shown to be such a condition. The condition that R have singular left ideal zero and finite left Goldie dimension, though necessary, is shown to be not sufficient in general. Conditions of two-sidedness of Q are also obtained in terms of the homogeneous components (simple subrings) of Q and the subrings of R, they induce.
The subgroups of a free product of two groups with an amalgamated subgroup
Authors: Karrass A. , Solitar D.
Published: June 2010

We prove that all subgroups

H
H

of a free product

G
G

of two groups

A
,
B

A,B

with an amalgamated subgroup

U
U

are obtained by two constructions from the intersection of

H
H

and certain conjugates of

A
,
B

A,B

, and

U
U

. The constructions are those of a tree product, a special kind of generalized free product, and of a Higman-Neumann-Neumann group. The particular conjugates of

A
,
B

A,B

, and

U
U

involved are given by double coset representatives in a compatible regular extended Schreier system for

G
G

modulo

H
H

. The structure of subgroups indecomposable with respect to amalgamated product, and of subgroups satisfying a nontrivial law is specified. Let

A
A

and

B
B

have the property

P
P

and

U
U

have the property

Q
Q

. Then it is proved that

G
G

has the property

P
P

in the following cases:

P
P

means every f.g. (finitely generated) subgroup is finitely presented, and

Q
Q

means every subgroup is f.g.;

P
P

means the intersection of two f.g. subgroups is f.g., and

Q
Q

means finite;

P
P

means locally indicable, and

Q
Q

means cyclic. It is also proved that if

N
N

is a f.g. normal subgroup of

G
G

not contained in

U
U

, then

N
U

NU

has finite index in

G
G

.
Quadratic functionals of second order
Authors: Leighton Walter
Published: June 2010

In this paper we study the minimizing of the general second-order quadratic functional (1.3) in a class of admissible functions

y
(
x
)

y(x)

with fixed endpoint conditions on

y
(
x
)

y(x)

and its derivative at

x
=
a

x = a

and at

x
=
b

x = b

. Necessary conditions and sufficient conditions are obtained. These lead, in turn, to natural extensions of the Sturm comparison theorem to fourth-order selfadjoint equations. These extensions include and are more general than previously stated comparison theorems. Finally, it is found that the present variational theory provides an orderly approach to second-order Wirtinger-like inequalities, and the results include as special cases many published results of this type.