bluuebunny/crossref_metadata_2025 huggingface dataset:
Plane models for Riemann surfaces admitting certain half-canonical linear series. II
Authors: Accola Robert D. M.
Published: June 2010

For

r
⩾
2

r \geqslant 2

, closed Riemann surfaces of genus

3
r
+
2

3r + 2

admitting two simple half-canonical linear series

g

3
r
+
1

r

,

h

3
r
+
1

r

g_{3r + 1}^r,h_{3r + 1}^r

are characterized by the existence of certain plane models of degree

2
r
+
3

2r + 3

where the linear series are apparent. The plane curves have

r
−
2

r - 2

3
3

-fold singularities, one

(
2
r
−
1
)

(2r - 1)

-fold singularity

Q
Q

, and two other double points (typically tacnodes) whose tangents pass through

Q
Q

. The lines through

Q
Q

cut out a

g
4
1

g_4^1

which is unique. The case where the

g
4
1

g_4^1

is the set of orbits of a noncyclic group of automorphisms of order four is characterized by the existence of

3
r
+
3

3r + 3

pairs of half-canonical linear series of dimension

r
−
1

r - 1

, where the sum of the two linear series in any pair is linearly equivalent to

g

3
r
+
1

r

+

h

3
r
+
1

r

g_{3r + 1}^r + h_{3r + 1}^r

.
Some general theorems on the cohomology of classifying spaces of compact Lie groups
Authors: Feshbach Mark
Published: June 2010

This paper is divided into two parts. The first part proves a number of general theorems on the cohomology of the classifying spaces of compact Lie groups. These theorems are proved by transfer methods, relying heavily on the double coset theorem [F

1

_{1}

]. Several of these results are well known while others are quite new. For the most part the proofs of the theorems are independent of each other and are quite short. Nevertheless they are true in great generality. Several are proven for arbitrary compact Lie groups and arbitrary cohomology theories. Perhaps the most interesting of the new results relates the cohomology of the classifying space of an arbitrary compact Lie group with that of the normalizer of a maximal torus. The second part of the paper generalizes many theorems to certain equivariant cohomology theories. Some of these theorems appear in [F

2

_{2}

].
Uniqueness of product and coproduct decompositions in rational homotopy theory
Authors: Douglas Roy , Renner Lex
Published: June 2010

Let

X
X

be a nilpotent rational homotopy type such that (1)

S
(
X
)

S(X)

, the image of the Hurewicz map has finite total rank, and (2) the basepoint map of

M
M

, a minimal algebra for

X
X

, is an element of the Zariski closure of

Aut

(
M
)

{\text {Aut}}(M)

in

End

(
M
)

{\text {End}}(M)

(i.e.

X
X

has "positive weight"). Then (A) any retract of

X
X

satisfies the two properties above, (B) any two irreducible product decompositions of

X
X

are equivalent, and (C) any two irreducible coproduct decompositions of

X
X

are equivalent.
Class groups of cyclic groups of square-free order
Authors: Matchett Andrew
Published: June 2010

Let

G
G

be a finite cyclic group of square free order. Let

Cl(

Z
G

)

{\text {Cl(}}ZG{\text {)}}

denote the projective class group of the integral group ring

Z
G

ZG

. Our main theorem describes explicitly the quotients of a certain filtration of

Cl(

Z
G

)

{\text {Cl(}}ZG{\text {)}}

. The description is in terms of class groups and unit groups of the rings of cyclotomic integers involved in

Z
G

ZG

. The proof is based on a Mayer-Vietoris sequence.
Arc-smooth continua
Authors: Fugate J. B. , Gordh G. R. , Lum Lewis
Published: June 2010

Continua admitting arc-structures and arc-smooth continua are introduced as higher dimensional analogues of dendroids and smooth dendroids, respectively. These continua include such spaces as: cones over compacta, convex continua in

l
2

{l_2}

, strongly convex metric continua, injectively metrizable continua, as well as various topological semigroups, partially ordered spaces, and hyperspaces. The arc-smooth continua are shown to coincide with the freely contractible continua and with the metric

K
K

-spaces of Stadtlander. Known characterizations of smoothness in dendroids involving closed partial orders, the set function

T
T

, radially convex metrics, continuous selections, and order preserving mappings are extended to the setting of continua with arc-structures. Various consequences of the special contractibility properties of arc-smooth continua are also obtained.
Uniqueness of invariant means for measure-preserving transformations
Authors: Rosenblatt Joseph
Published: June 2010

For some compact abelian groups

X
X

(e.g.

T
n

T^n

,

n
⩾
2

n \geqslant 2

, and

∏

n
=
1

∞

Z
2

\prod \nolimits _{n = 1}^\infty {{Z_2}}

), the group

G
G

of topological automorphisms of

X
X

has the Haar integral as the unique

G
G

-invariant mean on

L
∞

(
X
,

λ
X

)

{L_\infty }(X,{\lambda _X})

. This gives a new characterization of Lebesgue measure on the bounded Lebesgue measurable subsets

β
\beta

of

R
n

{R^n}

,

n
⩾
3

n \geqslant 3

; it is the unique normalized positive finitely-additive measure on

β
\beta

which is invariant under isometries and the transformation of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript n Baseline colon left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis right-arrow from bar left-parenthesis x 1 plus x 2 comma x 2 comma ellipsis comma x Subscript n Baseline right-parenthesis">

R
n

:
(

x
1

,
…
,

x
n

)
↦
(

x
1

+

x
2

,

x
2

,
…
,

x
n

)

{R^n}:({x_1}, \ldots ,{x_n}) \mapsto ({x_1} + {x_2},{x_2}, \ldots ,{x_n})

. Other examples of, as well as necessary and sufficient conditions for, the uniqueness of a mean on

L
∞

(
X
,
β
,
p
)

{L_\infty }(X,\beta ,p)

, which is invariant by some group of measure-preserving transformations of the probability space

(
X
,
β
,
p
)

(X,\beta ,p)

, are described.
Isomorphism theorems for octonion planes over local rings
Authors: Bix Robert
Published: June 2010

It is proved that there is a collineation between two octonion planes over local rings if and only if the underlying octonion algebras are isomorphic as rings. It is shown that every isomorphism between the little or middle projective groups of two octonion planes over local rings is induced by conjugation with a collineation or a correlation of the planes when the local rings contain

1
2

\frac {1} {2}

.
On purely inseparable algebras and P.H.D. rings
Authors: Satô Shizuka
Published: June 2010

M. E. Sweedler has considered purely inseparable algebras over rings. We define a stronger notion for purely inseparable algebras over rings and we study the fundamental properties of purely inseparable algebras. Moreover, we consider the relations between purely inseparable algebras and P.H.D. rings.
On asymptotically almost periodic solutions of a convolution equation
Authors: Staffans Olof J.
Published: June 2010

We study questions related to asymptotic almost periodicity of solutions of the linear convolution equation

(
∗
)
μ
∗
x
=
f

( \ast )\mu \ast x = f

. Here

μ
\mu

is a complex measure, and

x
x

and

f
f

are bounded functions. Basically we are interested in conditions which imply that bounded solutions of

(
∗
)

( \ast )

are asymptotically almost periodic. In particular, we show that a certain necessary condition on

f
f

for this to happen is also sufficient, thereby strengthening earlier results. We also include a result on existence of bounded solutions, and indicate a generalization to a distribution equation.
A rigid subspace of 𝐿₀
Authors: Kalton N. J. , Roberts James W.
Published: June 2010

We construct a closed infinite-dimensional subspace of

L
0

(
0
,
1
)

{L_0}(0,1)

(or

L
p

{L_p}

for

0
>
p
>
1

0 > p > 1

) which is rigid, i.e. such that every endomorphism in the space is a multiple of the identity.
When is a linear functional multiplicative?
Authors: Roitman M. , Sternfeld Y.
Published: June 2010

We prove here by elementary arguments a generalization of a theorem by Gleason, Kahane and Żelazko: If

φ
\varphi

is a linear functional on an algebra with unit

A
A

such that

φ
(
1
)
=
1

\varphi (1) = 1

and

φ
(
u
)
≠
0

\varphi (u) \ne 0

for any invertible

u
u

in

A
A

, then

φ
\varphi

is multiplicative, provided the spectrum of each element in

A
A

is bounded. We present also other conditions which may replace the assumptions on

A
A

in the theorem above.
Whitney stratified chains and cochains
Authors: Goresky R. Mark
Published: June 2010

This paper contains the technical constructions necessary for a "geometric cycle" definition of cohomology and homology in the context of Whitney stratifications. Cup and cap products are interpreted as the transverse intersection of geometric cocycles and cycles.
Singular integrals and maximal functions associated with highly monotone curves
Authors: Nestlerode W. C.
Published: June 2010

Let

γ
:
[
−
1
,
1
]
→

R

n

\gamma :[ - 1,1] \to {{\mathbf {R}}^n}

be an odd curve. Set
\[

H
γ

f
(
x
)
=

PV

∫

f
(
x
−
γ
(
t
)
)

(
d
t

/

t
)

{H_\gamma }f(x) = {\text {PV}}\int {f(x - \gamma (t))\,(dt/t)}

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript gamma Baseline f left-parenthesis x right-parenthesis equals sup h Superscript negative 1 Baseline integral Subscript 0 Superscript h Baseline StartAbsoluteValue f left-parenthesis x minus gamma left-parenthesis t right-parenthesis right-parenthesis EndAbsoluteValue d t">

M
γ

f
(
x
)
=
sup

h

−
1

∫
0
h

|

f
(
x
−
γ
(
t
)
)

|

d
t

{M_\gamma }f(x) = \sup {h^{ - 1}}\int _0^h {|f(x - \gamma (t))|\,dt}

\]
. We introduce a class of highly monotone curves in

R

n

{{\mathbf {R}}^n}

,

n
⩾
2

n \geqslant 2

, for which we prove that

H
γ

{H_\gamma }

and

M
γ

{M_\gamma }

are bounded operators on

L
2

(

R

n

)

{L^2}({{\mathbf {R}}^n})

. These results are known if

γ
\gamma

has nonzero curvature at the origin, but there are highly monotone curves which have no curvature at the origin. Related to this problem, we prove a generalization of van der Corput’s estimate of trigonometric integrals.
Smooth perturbations of a function with a smooth local time
Authors: Geman D. , Horowitz J.
Published: June 2010

A real Borel function on

[
0
,

1
]

[0,\,1]

has a local time if its occupation measure up to each time

t
t

(equivalently: its increasing, equimeasurable rearrangement on

[
0
,

t
]

[0,\,t]

) is absolutely continuous; the local time

α
t

(
x
)

{\alpha _t}(x)

is then the density. An inverse relationship exists between the smoothness of the local time in

(
t
,

x
)

(t,\,x)

and that of the original function. The sum of a function with a smooth local time and a well-behaved (e.g. absolutely continuous) function is shown to have a local time, which inherits certain significant properties from the old local time, and for which an explicit formula is given. Finally, using a probabilistic approach, examples are given of functions having local times of prescribed smoothness.
On the topological structure of even-dimensional complete intersections
Authors: Libgober A. S. , Wood J. W.
Published: June 2010

A topological connected sum decomposition into indecomposable pieces is given for complete intersections, and these pieces are described by plumbing constructions. The principal technical results are structure theorems for the intersection form on the middle dimensional homology and the submodule of spherical classes.
The integrability tensor for bundle-like foliations
Authors: Escobales Richard H.
Published: June 2010

A certain function is introduced which is useful in the study of a bundle-like foliation on a Riemannian manifold. Under the assumption that the leaves are totally geodesic, the Laplacian of this function is computed along a leaf. From this computation a sufficient condition is provided for the ambient manifold to be locally isometric to a product.
Nonorientable surfaces in some non-Haken 3-manifolds
Authors: Rubinstein J. H.
Published: June 2010

If a closed, irreducible, orientable

3
3

-manifold

M
M

does not possess any

2
2

-sided incompressible surfaces, then it can be very useful to investigate embedded one-sided surfaces in

M
M

of minimal genus. In this paper such

3
3

-manifolds

M
M

are studied which admit embeddings of the nonorientable surface

K
K

of genus

3
3

. We prove that a

3
3

-manifold

M
M

of the above type has at most

3
3

different isotopy classes of embeddings of

K
K

representing a fixed element of

H
2

(
M
,

Z
2

)

{H_2}(M,\,{Z_2})

. If

M
M

is either a binary octahedral space, an appropriate lens space or Seifert manifold, or if

M
M

has a particular type of fibered knot, then it is shown that the embedding of

K
K

in

M
M

realizing a specific homology class is unique up to isotopy.
Spectral permanence for joint spectra
Authors: Curto Raul E.
Published: June 2010

For a

C

∗

{C^{\ast }}

-subalgebra

A
A

of a

C

∗

{C^{\ast }}

-algebra

B
B

and a commuting

n
n

-tuple

a
=
(

a
1

,
…
,

a
n

)

a = ({a_1}, \ldots ,{a_n})

of elements of

A
A

, we prove that

Sp
⁡
(
a
,

A
)
=
Sp
⁡
(
a
,

B
)

\operatorname {Sp} (a,\,A) = \operatorname {Sp} (a,\,B)

, where

Sp
\operatorname {Sp}

denotes Taylor spectrum. As a consequence we prove that

0
∉
Sp
⁡
(
a
,

A
)

0 \notin \operatorname {Sp} (a,\,A)

if and only if
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove a With caret equals Start 3 By 3 Matrix 1st Row 1st Column d 1 2nd Column a m p semicolon 3rd Column a m p semicolon 2nd Row 1st Column d 2 Superscript asterisk Baseline 2nd Column a m p semicolon d 3 3rd Column a m p semicolon 3rd Row 1st Column Blank 2nd Column a m p semicolon d 4 Superscript asterisk Baseline 3rd Column a m p semicolon down-right-diagonal-ellipsis EndMatrix element-of upper L left-parenthesis upper A circled-times bold upper C Superscript 2 Super Superscript n minus 1 Superscript Baseline right-parenthesis">

a
^

=

(

d
1

a
m
p
;

a
m
p
;

d
2

∗

a
m
p
;

d
3

a
m
p
;

a
m
p
;

d
4

∗

a
m
p
;
⋱

)

∈
L

(

A
⊗

C

2

n
−
1

)

\hat a = \left ( {\begin {array}{*{20}{c}} {{d_1}} & {} & {} \\ {d_2^{\ast }} & {{d_3}} & {} \\ {} & {d_4^{\ast }} & \ddots \\ \end {array} } \right ) \in L\left ( {A \otimes {{\mathbf {C}}^{{2^{n - 1}}}}} \right )

\]
is invertible, where

d
i

{d_i}

is the

i
i

th boundary map in the Koszul complex for

A
A

. More generally, we show that

σ

δ
,
k

(
a
,

A
)
=

σ

δ
,
k

(

a
,

B

)

{\sigma _{\delta ,k}}(a,\,A) = {\sigma _{\delta ,k}}\left ( {a,\,B} \right )

and

σ

π
,
k

(
a
,

A
)
=

σ

π
,
k

(
a
,

B
)

{\sigma _{\pi ,k}}(a,\,A) = {\sigma _{\pi ,k}}(a,\,B)

(all

k
k

), where

σ

δ
,
⋅

{\sigma _{\delta ,\cdot }}

and

σ

π
,
⋅

{\sigma _{\pi ,\cdot }}

are the joint spectra considered by Z. Słodkowski.
Tauberian conditions for 𝐿¹-convergence of Fourier series
Authors: Stanojević Časlav V.
Published: June 2010

It is proved that Fourier series with asymptotically even coefficients and satisfying
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript lamda right-arrow 1 Endscripts limit sup Underscript n right-arrow normal infinity Endscripts sigma-summation Underscript j equals n Overscript left-bracket lamda n right-bracket Endscripts j Superscript p minus 1 Baseline StartAbsoluteValue normal upper Delta ModifyingAbove f With caret left-parenthesis j right-parenthesis EndAbsoluteValue Superscript p Baseline equals 0">

lim

λ
→
1

lim

sup

n
→
∞

∑

j
=
n

[
λ
n
]

j

p
−
1

|

Δ

f
^

(
j
)

|

p

=
0

{\lim _{\lambda \to 1}}\lim {\sup _{n \to \infty }}\sum _{j = n}^{[\lambda n]}{j^{p - 1}}|\Delta \hat f(j){|^p} = 0

, for some

1
>
p
⩽
2

1 > p \leqslant 2

, converge in

L
1

{L^1}

-norm if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue ModifyingAbove f With caret left-parenthesis n right-parenthesis upper E Subscript n Baseline plus ModifyingAbove f With caret left-parenthesis negative n right-parenthesis upper E Subscript negative n Baseline StartAbsoluteValue EndAbsoluteValue equals o left-parenthesis 1 right-parenthesis">

|

|

f
^

(
n
)

E
n

+

f
^

(
−
n
)

E

−
n

|

|

=
o
(
1
)

||\hat f(n){E_n} + \hat f( - n){E_{ - n}}|| = o(1)

, where

E
n

(
t
)
=

∑

k
=
0

n

e

i
k
t

{E_n}(t) = \sum _{k = 0}^n{e^{ikt}}

. Recent results of Stanojević [1], Bojanic and Stanojević [2], and Goldberg and Stanojević [3] are special cases of some corollaries to the main theorem.
Free products of 𝐶*-algebras
Authors: Avitzour Daniel
Published: June 2010

Small ("spatial") free products of

C

∗

{C^{\ast }}

-algebras are constructed. Under certain conditions they have properties similar to those proved by Paschke and Salinas for the algebras

C
r

∗

(

G
1

∗

G
2

)

C_r^{\ast }({G_1}{\ast }{G_2})

where

G
1

{G_1}

,

G
2

{G_2}

are discrete groups. The free-product analogs of noncommutative Bernoulli shifts are discussed.
Existence of Chebyshev centers, best 𝑛-nets and best compact approximants
Authors: Amir Dan , Mach Jaroslav , Saatkamp Klaus
Published: June 2010

In this paper we investigate the existence and continuity of Chebyshev centers, best

n
n

-nets and best compact sets. Some of our positive results were obtained using the concept of quasi-uniform convexity. Furthermore, several examples of nonexistence are given, e.g., a sublattice

M
M

of

C
[
0
,

1
]

C[0,\,1]

, and a bounded subset

B
⊂
M

B \subset M

is constructed which has no Chebyshev center, no best

n
n

-net and not best compact set approximant.
Some varieties containing relation algebras
Authors: Maddux Roger
Published: June 2010

Three varieties of algebras are introduced which extend the varietyRARAof relation algebras. They are obtained fromRARAby weakening the associative law for relative product, and are consequently called nonassociative, weakly-associative and semiassociative relation algebras, orNANA,WAWA, andSASA, respectively. Each of these varieties arises naturally in solving various problems concerning relation algebras. We show, for example, thatWAWAis the only one of these varieties which is closed under the formation of complex algebras of atom structures of algebras, and thatWAWAis the closure of the variety of representableRARA’s under relativization. The paper also contains a study of the elementary theories of these varieties, various representation theorems, and numerous examples.
Smoothness of the boundary values of functions bounded and holomorphic in the disk
Authors: Yamashita Shinji
Published: June 2010

The non-Euclidean counterparts of Hardy-Littlewood’s theorems on Lipschitz and mean Lipschitz functions are considered. Let

1
≤
p
>
∞

1\le p > \infty

and

0
>
α
≤
1

0 > \alpha \le 1

. For

f
f

holomorphic and bounded,

|

f

|

>
1

|f|> 1

, in

|

z

|

>
1

|z|> 1

, the condition that is necessary and sufficient for

f
f

to be continuous on

|

z

|

≤
1

|z|\le 1

with the boundary function

f
(

e

i
t

)
∈
σ

Λ
α

f({e^{it}}) \in \sigma {\Lambda _\alpha }

, the hyperbolic Lipschitz class. Furthermore, the condition that the

p
p

th mean of

f

∗

f^{\ast }

on the circle

|

z

|

=
r
>
1

|z|=r > 1

is

O
(

(
1
−
r

)

α
−
1

)

O({(1 - r)^{\alpha - 1}})

is necessary and sufficient for

f
f

to be of the hyperbolic Hardy class

H
σ

p

H_\sigma ^{p}

and for the radial limits to be of the hyperbolic mean Lipschitz class

σ

Λ
α

p

\sigma \Lambda _\alpha ^{p}

.
Spaces which look like quaternionic projective 𝑛-space
Authors: McGibbon C. A.
Published: June 2010

The projective

n
n

-spaces which correspond to the various multiplicative structures on the three sphere are studied. Necessary and sufficient conditions for a projective

n
n

-space to extend to a projective

n
+
1

n+1

-space are described. At each odd prime, an infinite family of exotic projective spaces is constructed. These exotic spaces are not homotopy equivalent, at the prime in question, to the classical quaternionic projective

n
n

-space. It is also shown that these exotic projective

n
n

-spaces do not occur as the finite skeleton of a classifying space for a group with the homotopy type of the three sphere.
On actions of adjoint type on complex Stiefel manifolds
Authors: Wang McKenzie Y.
Published: June 2010

Let

G
(
m
)

G(m)

denote

S
U

(
m
)

{\rm {SU}}(m)

or

S
p

(
m
)

{\rm {Sp}}(m)

. It is shown that when

m
≥
5

G
(
m
)

m \geq 5\,G(m)

cannot act smoothly on

W

n
,
2

W_{n,2}

, the complex Stiefel manifold of orthonormal

2
2

-frames in

C

n

\mathbf C^n

, for

n
n

odd, with connected principal isotropy type equal to the class of maximal tori in

G
(
m
)

G(m)

. This demonstrates an important difference between

W

n
,
2

W_{n,2}

,

n
n

odd, and

S

2
n
−
3

×

S

2
n
−
1

S^{2n-3}\times S^{2n-1}

in the behavior of differentiable transformation groups. Exactly the same holds for

S
O

(
m
)

{\rm {SO}}(m)

or Spin

(
m
)

(m)

if it is further assumed that a maximal

2
2

-torus of

S
O

(
m
)

{\rm {SO}}(m)

has fixed points.

2

^{2}
Branching degrees above low degrees
Authors: Fejer Peter A.
Published: June 2010

In this paper, we investigate the location of the branching degrees within the recursively enumerable (r.e.) degrees. We show that there is a branching degree below any given nonzero r.e. degree and, using a new branching degree construction and a technique of Robinson, that there is a branching degree above any given low r.e. degree. Our results extend work of Shoenfield and Soare and Lachlan on the generalized nondiamond question and show that the branching degrees form an automorphism base for the r.e. degrees.
Tilted algebras
Authors: Happel Dieter , Ringel Claus Michael
Published: June 2010

Let

A
A

be a finite dimensional hereditary algebra over a field, with

n
n

simple

A
A

-modules. An

A
A

-module

T
A

T_A

with

n
n

pairwise nonisomorphic indecomposable direct summands and satisfying

Ex

t

1

(

T
A

,

T
A

)
=
0

{\text {Ex}}{{\text {t}}^1}({T_A},\,{T_A}) = 0

is called a tilting module, and its endomorphism ring

B
B

is a tilted algebra. A tilting module defines a (usually nonhereditary) torsion theory, and the indecomposable

B
B

-modules are in one-to-one correspondence to the indecomposable

A
A

-modules which are either torsion or torsionfree. One of the main results of the paper asserts that an algebra of finite representation type with an indecomposable sincere representation is a tilted algebra provided its Auslander-Reiten quiver has no oriented cycles. In fact, tilting modules are introduced and studied for any finite dimensional algebra, generalizing recent results of Brenner and Butler.
Nonstandard construction of the stochastic integral and applications to stochastic differential equations. II
Authors: Hoover Douglas N. , Perkins Edwin
Published: June 2010

H. J. Keisler has recently used a nonstandard theory of Itô integration (due to R. M. Anderson) to construct solutions of Itô integral equations by solving an associated internal difference equation. In this paper we use the same general approach to find solutions

y
(
t
)

y(t)

of semimartingale integral equations of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y left-parenthesis t comma omega right-parenthesis equals h left-parenthesis t comma omega right-parenthesis plus integral Subscript 0 Superscript t Baseline f left-parenthesis s comma omega comma y left-parenthesis dot comma omega right-parenthesis right-parenthesis d z left-parenthesis s right-parenthesis">

y
(
t
,
ω
)
=
h
(
t
,
ω
)
+

∫
0
t

f
(
s
,
ω
,
y
(
⋅
,
ω
)
)

d
z
(
s
)

y(t,\omega ) = h(t,\omega ) + \int _0^t {f(s,\omega ,y(\cdot ,\omega ))\,dz(s)}

\]
, where

z
z

is a given semimartingale,

h
h

is a right-continuous process and

f
(
s
,
ω
,
⋅
)

f(s,\omega , \cdot )

is continuous on the space of right-continuous functions with left limits, with the topology of uniform convergence on compacts. In addition, we generalize Keisler’s continuity theorem and give necessary and sufficient conditions for an internal martingale to be

S
S

-continuous.
Tauberian 𝐿¹-convergence classes of Fourier series. I
Authors: Bray William O. , Stanojević Časlav V.
Published: June 2010

It is shown that the Stanojević [2] necessary and sufficient conditions for

L
1

{L^1}

-convergence of Fourier series of

f
∈

L
1

(
T
)

f \in {L^1}(T)

can be reduced to the classical form. A number of corollaries of a recent Tauberian theorem are obtained for the subclasses of the class of Fourier coefficients satisfying
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n Superscript alpha Baseline StartAbsoluteValue normal upper Delta ModifyingAbove f With caret left-parenthesis n right-parenthesis EndAbsoluteValue equals o left-parenthesis l right-parenthesis left-parenthesis n right-arrow normal infinity right-parenthesis">

n
α

|

Δ

f
^

(
n
)

|

=
o
(
l
)

(
n
→
∞
)

{n^\alpha }|\Delta \hat {f}(n)| = o(l)\,(n \to \infty )

for some

0
>
α
⩽

1
2

0 > \alpha \leqslant \frac {1}{2}

. For Fourier series with coefficients asymptotically even with respect to a sequence

{

l
n

}
,

l
n

=
o
(
n
)

(
n
→
∞
)

\{{l_n}\} ,{l_n} = o(n)\,(n \to \infty )

, and satisfying
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l Subscript n Superscript negative 1 slash q Baseline left-parenthesis sigma-summation Underscript k equals n Overscript n plus left-bracket n slash l Subscript n Baseline right-bracket Endscripts k Superscript p minus 1 Baseline vertical-bar normal upper Delta ModifyingAbove f With caret left-parenthesis k right-parenthesis vertical-bar Superscript p Baseline right-parenthesis Superscript 1 slash p Baseline equals o left-parenthesis 1 right-parenthesis left-parenthesis n right-arrow normal infinity right-parenthesis comma 1 slash p plus 1 slash q equals 1 comma">

l
n

−
1

/

q

(

∑

k
=
n

n
+
[
n

/

l
n

]

k

p
−
1

|

Δ

f
^

(
k
)

|

p

)

1

/

p

=
o
(
1
)

(
n
→
∞
)
,

1

/

p
+
1

/

q
=
1
,

l_n^{ - 1/q}{\left ({\sum \limits _{k = n}^{n + [n/{l_n}]} {{k^{p - 1}}|\Delta \,\hat f(k)} {|^p}} \right )^{1/p}} = o(1)\, \quad (n \to \infty ), \quad 1/p + 1/q = 1,

\]
necessary and sufficient conditions for

L
1

{L^1}

-convergence are obtained. In particular for

l
n

=
[
∥

σ
n

(
f
)
−
f

∥

−
1

]

{l_n} = [\parallel {\sigma _n}(f) - f{\parallel ^{ - 1}}]

, an important corollary is obtained which connects smoothness of

f
f

with smoothness of

{

f
^

(
n
)
}

\{\hat f(n)\}

.
Quadratic forms permitting triple composition
Authors: McCrimmon Kevin
Published: June 2010

In an algebraic investigation of isoparametric hypersurfaces, J. Dorfmeister and E. Neher encountered a nondegerate quadratic form which permitted composition with a trilinear product,

Q

(
{

x
y
z

}
)

=
Q
(
x
)
Q
(
y
)
Q
(
z
)

Q\left (\{{xyz} \} \right ) = Q(x)Q(y)Q(z)

. In this paper we give a complete description of such composition triples: they are all obtained as isotopes of permutations of standard triples

{
x
y
z
}
=
(
x
y
)
z

\{xyz \} = (xy)z

or

x
(
y
z
)

x(yz)

determined by a composition algebra, with the quadratic form

Q
Q

the usual norm form. For any fixed

Q
Q

this leads to

1
1

isotopy class in dimensions

1
1

and

2
2

,

3
3

classes in the dimension

4
4

quaternion case, and

6
6

classes in the dimension

8
8

octonion case.
On strongly indefinite functionals with applications
Authors: Hofer Helmut
Published: June 2010

Recently, in their remarkable paper Critical point theory for indefinite functionals, V. Benci and P. Rabinowitz gave a direct approach—avoiding finite-dimensional approximations—to the existence theory for critical points of indefinite functionals. In this paper we develop under weaker hypotheses a simpler but more general theory for such problems. In the second part of the paper the abstract results are applied to a class of resonance problems of the Landesman and Lazer type, and moreover they are illustrated by an application to a wave equation problem.
Weighted norm inequalities for homogeneous families of operators
Authors: Rubio de Francia José L.
Published: June 2010

If a family of operators in

R
n

{R^n}

is invariant under rotations and dilations and satisfy a certain inequality in

L
p

(

l
r

)

{L^p}({l^r})

, then it is uniformly bounded in the weighted space

L
r

(

|

x

|

n
(
r

/

p
−
1
)

d
x
)

{L^r}(|x|{^{n(r/p - 1)}}\,dx)

. This is the main consequence of a more general result for operators in homogeneous spaces. Applications are given to certain maximal operators, the Fourier transform and Bochner-Riesz multipliers.
A sphere theorem for manifolds of positive Ricci curvature
Authors: Shiohama Katsuhiro
Published: June 2010

Instead of injectivity radius, the contractibility radius is estimated for a class of complete manifolds such that

Ri

c

M

⩾
1
,

K
M

⩾
−

κ
2

{\text {Ri}}{{\text {c}}_M} \geqslant 1,{K_M} \geqslant - {\kappa ^2}

and the volume of

M
⩾

M \geqslant

the volume of the

(
π
−
ε
)

(\pi - \varepsilon )

-ball on the unit

m
m

-sphere,

m
=

dim 

M

m = {\text {dim }}M

. Then for a suitable choice of

ε
=
ε
(
m
,
k
)

\varepsilon = \varepsilon (m,k)

every

M
M

belonging to this class is homeomorphic to

S
m

{S^m}

.
CR-hypersurfaces in a space with a pseudoconformal connection
Authors: Markowitz Michael J.
Published: June 2010

In this paper we study a submanifold in a space with a pseudoconformal connection. We assume that the submanifold

M
M

is so situated that it inherits the structure of a

CR

{\text {CR}}

-hypersurface from the ambient space.

M
M

then supports two natural Cartan connections, the normal pseudoconformal connection of Cartan-Chern-Tanaka and an induced pseudoconformal connection. Analogues of the Gauss-Codazzi equations are derived and applied to determine necessary and sufficient conditions for the equivalence of these connections.
Inequalities for holomorphic functions of several complex variables
Authors: Burbea Jacob
Published: June 2010

Sharp norm-inequalities, valid for functional Hilbert spaces of holomorphic functions on the polydisk, unit ball and

C

n

{{\mathbf {C}}^n}

are established by using the notion of reproducing kernels. These inequalities extend earlier results of Saitoh and ours.
A characterization of bounded symmetric domains by curvature
Authors: D’Atri J. E. , Dotti Miatello I.
Published: June 2010

This paper will prove that a bounded homogeneous domain is symmetric if and only if, in the Bergman metric, all sectional curvatures are nonpositive.
Viscosity solutions of Hamilton-Jacobi equations
Authors: Crandall Michael G. , Lions Pierre-Louis
Published: June 2010

Problems involving Hamilton-Jacobi equations—which we take to be either of the stationary form

H
(
x
,
u
,
D
u
)
=
0

H(x,u,Du) = 0

or of the evolution form

u

t

+
H
(
x
,
t
,
u
,
D
u
)
=
0

{u_{t}} + H(x,t,u,Du) = 0

, where

D
u

Du

is the spatial gradient of

u
u

—arise in many contexts. Classical analysis of associated problems under boundary and/or initial conditions by the method of characteristics is limited to local considerations owing to the crossing of characteristics. Global analysis of these problems has been hindered by the lack of an appropriate notion of solution for which one has the desired existence and uniqueness properties. In this work a notion of solution is proposed which allows, for example, solutions to be nowhere differentiable but for which strong uniqueness theorems, stability theorems and general existence theorems, as discussed herein, are all valid.
Completeness and basis properties of complex exponentials
Authors: Redheffer Raymond M. , Young Robert M.
Published: June 2010

This paper is concerned with what might be termed the "fine structure" of the completeness and basis properties of complex exponentials. We give new criteria for two sequences to have the same excess in the sense of Paley and Wiener, a result that illuminates and supplements a well-known completeness criterion of Levinson, and new examples and counterexamples pertaining to Riesz bases.
Applications of 𝑞-Lagrange inversion to basic hypergeometric series
Authors: Gessel Ira , Stanton Dennis
Published: June 2010

A family of

q
q

-Lagrange inversion formulas is given. Special cases include quadratic and cubic transformations for basic hypergeometric series. The

q
q

-analogs of the so-called "strange evaluations" are also corollaries. Some new RogersRamanujan identities are given. A connection between the work of Rogers and Andrews, and

q
q

-Lagrange inversion is stated.
Spectra of invariant uniform and transform algebras
Authors: Glicksberg I.
Published: June 2010

For

G
G

a locally compact abelian group, any closed invariant proper subalgebra of

C
0

(
G
)

{C_0}(G)

has analytic discs in its spectrum. Related results are given for

A
(
G
)

A(G)

and

B
(
G
)

B(G)

.
Decay of Walsh series and dyadic differentiation
Authors: Wade William R.
Published: June 2010

Let

W
2

n

[
f
]

{W_2}\,n\,[f]

denote the

2
n

th

{2^n}{\text {th}}

partial sums of the Walsh-Fourier series of an integrable function

f
f

. Let

ρ
n

(
x
)

{\rho _n}(x)

represent the ratio

W
2

n
[
f
,
x
]

/

2
n

{W_2}n[f,x]/{2^n}

, for

x
∈
[
0
,
1
]

x \in [0,1]

, and let

T
(
f
)

T(f)

represent the function

(
Σ

ρ
n
2

)

1

/

2

{(\Sigma \rho _n^2)^{1/2}}

. We prove that

T
(
f
)

T(f)

belongs to

L
p

[
0
,
1
]

{L^p}[0,1]

for all

0
>
p
>
∞

0 > p > \infty

. We observe, using inequalities of Paley and Sunouchi, that the operator

f
→
T
(
f
)

f \to T(f)

arises naturally in connection with dyadic differentiation. Namely, if

f
f

is strongly dyadically differentiable (with derivative

D
˙

f

\dot Df

) and has average zero on the interval [0, 1], then the

L
p

{L^p}

norms of

f
f

and

T
(

D
˙

f
)

T(\dot Df)

are equivalent when

1
>
p
>
∞

1 > p > \infty

. We improve inequalities implicit in Sunouchi’s work for the case

p
=
1

p = 1

and indicate how they can be used to estimate the

L
1

{L^1}

norm of

T
(

D
˙

f
)

T(\dot Df)

and the dyadic

H
1

{H^1}

norm of

f
f

by means of mixed norms of certain random Walsh series. An application of these estimates establishes that if

f
f

is strongly dyadically differentiable in dyadic

H
1

{H^1}

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript 1 Baseline normal upper Sigma Subscript upper N equals 1 Superscript normal infinity Baseline vertical-bar upper W Subscript upper N Baseline left-bracket f comma x right-bracket minus sigma Subscript upper N Baseline left-bracket f comma x right-bracket slash upper N d x greater-than normal infinity">

∫
0
1

Σ

N
=
1

∞

|

W
N

[
f
,
x
]
−

σ
N

[
f
,
x
]

/

N

d
x
>
∞

\int _0^1 {\Sigma _{N = 1}^\infty |{W_N}[f,x] - {\sigma _N}[f,x]/N\,dx > \infty }

.
The optimal accuracy of difference schemes
Authors: Iserles Arieh , Strang Gilbert
Published: June 2010

We consider difference approximations to the model hyperbolic equation

u

t

=

u
x

{u_{t}} = {u_x}

which compute each new value

U
(
x
,
t
+
Δ
t
)

U(x,t + \Delta t)

as a combination of the known values

U
(
x
−
r
Δ
x
,
t
)
,
…
,
U
(
x
+
s
Δ
x
,
Δ
t
)

U(x - r\Delta x,t),\ldots ,U(x + s\Delta x,\Delta t)

. For such schemes we find the optimal order of accuracy: stability is possible for small

Δ
t

/

Δ
x

\Delta t/\Delta x

if and only if

p
⩽
min
{

r
+
s
,
2
r
+
2
,
2
s

}

p \leqslant \min \{ {r + s,2r + 2,2s} \}

. A similar bound is established for implicit methods. In this case the most accurate schemes are based on Padé approximations

P
(
z
)

/

Q
(
z
)

P(z)/Q(z)

to

z
λ

{z^\lambda }

near

z
=
1

z = 1

, and we find an expression for the difference

|

Q

|

2

−

|

P

|

2

|Q{|^2} - |P{|^2}

; this allows us to test the von Neumann condition

|

P

/

Q

|

⩽
1

|P/Q| \leqslant 1

. We also determine the number of zeros of

Q
Q

in the unit circle, which decides whether the implicit part is uniformly invertible.
Steiner minimal trees on zig-zag lines
Authors: Du D. Z. , Hwang F. K. , Weng J. F.
Published: June 2010

A Steiner minimal tree for a given set

P
P

of points in the Euclidean plane is a shortest network interconnecting

P
P

whose vertex set may include some additional points. The construction of Steiner minimal trees has been proved to be an

N
P

NP

-complete problem for general

P
P

. However, the

N
P

NP

-completeness does not exclude the possibility that Steiner trees for sets of points with special structures can be efficiently determined. In this paper we determine the Steiner mimmal trees for zig-zag lines with certain regularity properties. We also give an explicit formula for the length of such a tree.
Equilibrium states of an elastic conductor in a magnetic field: a paradigm of bifurcation theory
Authors: Wolfe Peter
Published: June 2010

In this paper we study the equilibrium states of a nonlinearly elastic conducting wire in a magnetic field. The wire is perfectly flexible and is suspended between fixed supports. The wire carries an electric current and is subjected to a constant magnetic field whose direction is parallel to the line between the supports. We solve this problem exactly and show that the set of solutions gives rise to a paradigmatic bifurcation diagram. We then carry out a study of the equations obtained by linearization about the nontrivial solutions in order to gain some insight into the stability of the various solution branches.
Solvable groups acting on the line
Authors: Plante J. F.
Published: June 2010

Actions of discrete groups on the real line are considered. When the group of homeomorphisms is solvable several sufficient conditions are given which guarantee that the group is semiconjugate to a subgroup of the affine group of the line. In the process of obtaining these results sufficient conditions are also determined for the existence of invariant (quasi-invariant) measures for abelian (solvable) groups acting on the line. It turns out, for example, that any solvable group of real analytic diffeomorphisms or a polycyclic group of homeomorphisms has a quasi-invariant measure, and that any abelian group of

C
2

{C^2}

diffeomorphisms has an invariant measure. An example is given to show that some restrictions are necessary in order to obtain such conclusions. Some applications to the study of codimension one foliations are indicated.
On some cheap control problems for diffusion processes
Authors: Menaldi José-Luis , Robin Maurice
Published: June 2010

We consider several cases of control problems for diffusion processes when the payoff functional does not depend explicitly on the control. We prove the continuity of the optimal cost function and give a characterization of this cost with a quasi-variational inequality interpreting the problem as limit of an impulse control problem when the cost of impulse tends to zero. Moreover, we show the existence of an optimal control for some particular situations.
Doubly slice knots and the Casson-Gordon invariants
Authors: Ruberman Daniel
Published: June 2010

We find knots in all dimensions which are algebraically but not geometrically doubly slice. Our new obstructions involve the Casson-Gordon invariants of the finite cyclic covers in odd dimensions and of the infinite cyclic cover in even dimensions. These same invariants provide new criteria for amphicheirality and invertibility of even-dimensional knots.
Radial limits of 𝑛-subharmonic functions in the polydisc
Authors: Nestlerode W. C. , Stoll M.
Published: June 2010

We prove a relation between a certain weighted radial limit of an

n
n

-subharmonic function in the polydisc

U
n

{U^n}

and the representing measure of its least

n
n

-harmonic majorant. We apply this result to functions in

N
(

U
n

)

N({U^n})

, the Nevalinna class of

U
n

{U^n}

. In particular, we obtain a necessary condition for a function to belong to the component of the origin in

N
(

U
n

)

N({U^n})

. These results are extensions of the work of J. H. Shapiro and A. L. Shields to

n
>
1

n > 1

.
Harmonic maps and classical surface theory in Minkowski 3-space
Authors: Milnor Tilla Klotz
Published: June 2010

Harmonic maps from a surface

S
S

with nondegenerate prescribed and induced metrics are characterized, showing that holomorphic quadratic differentials play the same role for harmonic maps from a surface with indefinite prescribed metric as they do in the Riemannian case. Moreover, holomorphic quadratic differentials are shown to arise as naturally on surfaces of constant

H
H

or

K
K

in

M
3

{M^3}

as on their counterparts in

E
3

{E^3}

. The connection between the sine-Gordon,

sinh
\sinh

-Gordon and

cosh
\cosh

-Gordon equations and harmonic maps is explained. Various local and global results are established for surfaces in

M
3

{M^3}

with constant

H
H

, or constant

K
≠
0

K \ne 0

. In particular, the Gauss map of a spacelike or timelike surface in

M
3

{M^3}

is shown to be harmonic if and only if

H
H

is constant. Also,

K
K

is shown to assume values arbitrarily close to

H
2

{H^2}

on any entire, spacelike surface in

M
3

{M^3}

with constant

H
H

, except on a hyperbolic cylinder.
Convolution theorems with weights
Authors: Kerman R. A.
Published: June 2010

Analogues of Young’s Inequality and the Convolution Theorem are shown to hold when the

L
p

{L_p}

and

L
(
p
,
q
)

L(p,q)

spaces have underlying measure defined in terms of power weights.
Analyticity on rotation invariant families of curves
Authors: Globevnik Josip
Published: June 2010

Let

G

\mathfrak {G}

be a rotation invariant family of smooth Jordan curves contained in

Δ
\Delta

, the open unit disc in

C

{\mathbf {C}}

. For each

Γ
∈

G

\Gamma \in \mathfrak {G}

let

D
Γ

{D_\Gamma }

be the simply connected domain bounded by

Γ
\Gamma

. We present various conditions which imply that if

f
f

is a continuous function on

Δ
\Delta

such that for every

Γ
∈

G

\Gamma \in \mathfrak {G}

the function

f

|

Γ

f|\Gamma

has a continuous extension to

D
Γ

¯

\overline {{D_\Gamma }}

which is analytic in

D
Γ

{D_\Gamma }

, then

f
f

is analytic in

Δ
\Delta

.
Selfadjoint representations of polynomial algebras
Authors: Inoue Atsushi , Takesue Kunimichi
Published: June 2010

In this paper we will investigate the selfadjointness of unbounded

∗

^{\ast }

-representations of the polynomial algebra. In particular, it is shown that the notion of selfadjoint representation is equivalent to that of standard representation in the case of the polynomial algebra generated by one hermitian element. Although the notion of standardness implies that of selfadjointness, the converse is not true in general. Therefore, we consider under what conditions a

∗

^{\ast }

-representation is standard.
Strong Fatou-1-points of Blaschke products
Authors: Belna C. L. , Carroll F. W. , Piranian G.
Published: June 2010

This paper shows that to every countable set

M
M

on the unit circle there corresponds a Blaschke product whose set of strong Fatou-

1
1

-points contains

M
M

. It also shows that some Blaschke products have an uncountable set of strong Fatou-

1
1

-points.
Reye congruences
Authors: Cossec François R.
Published: June 2010

This paper studies the congruences of lines which are included in two distinct quadrics of a given generic three-dimensional projective space of quadrics in

P

3

{{\mathbf {P}}^3}

.
Extensions of algebraic systems
Authors: Iskander Awad A.
Published: June 2010

There are several generalizations to universal algebras of the notion "The groupA\mathfrak {A}is an extension of the groupB\mathfrak {B}by the groupC\mathfrak {C}". In this paper we study three such generalizations and the corresponding products of classes of algebraic systems. Various results are presented. One such theorem characterizes the weakly congruence regular varieties admitting extensions of a particular sort. Another result gives, under a weak congruence permutability condition, an equational basis for the variety obtained by applying one such product to two other varieties.
An improved stability result for resonances
Authors: Ashbaugh Mark S. , Sundberg Carl
Published: June 2010

We prove stability of shape resonances for the sequence of Schrödinger equations
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis minus d squared slash d x squared plus upper U left-parenthesis x right-parenthesis plus upper W Subscript n Baseline left-parenthesis x right-parenthesis right-parenthesis psi left-parenthesis x right-parenthesis equals upper E psi left-parenthesis x right-parenthesis comma 0 less-than-or-slanted-equals x greater-than normal infinity">

(
−

d
2

/

d

x
2

+
U
(
x
)
+

W
n

(
x
)
)
ψ
(
x
)
=
E
ψ
(
x
)
,
0
⩽
x

>
∞

( - {d^2}/d{x^2} + U(x) + {W_n}(x))\psi (x) = E\psi (x),0 \leqslant x\, > \infty

, in the limit

n
→
∞

n \to \infty

where the barrier potentials

W
n

(
x
)

{W_n}(x)

are integrable, nonnegative, supported in the interval

[
1
,
a
]

(
1
>
a
>
∞
)

[1,a]\;(1 > a > \infty )

, and approach infinity pointwise a.e. for

x
∈
[
1
,
a
]

x \in [1,a]

as

n
→
∞

n \to \infty

. In the course of our investigation we prove that for suitable complex initial conditions the solution to the Riccati equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S prime left-parenthesis x right-parenthesis equals 1 minus left-parenthesis upper W Subscript n Baseline left-parenthesis x right-parenthesis minus upper E right-parenthesis left-bracket upper S left-parenthesis x right-parenthesis right-bracket squared">

S
′

(
x
)
=
1
−
(

W
n

(
x
)
−
E
)

[
S
(
x
)

]
2

S’(x) = 1 - ({W_n}(x) - E){[S(x)]^2}

goes to

0
0

as

n
→
∞

n \to \infty

uniformly on compact subsets of

[
1
,
a
]

[1,a]

. Our approach is via ordinary differential equations using outgoing wave boundary conditions to define resonances. Our stability result extends a similar result of Ashbaugh and Harrell, who use an argument based on asymptotics and the implicit function theorem to study the above problem with

λ
V
(
x
)

\lambda V(x)

replacing

W
n

(
x
)

{W_n}(x)

. Our approach is to use the Riccati equation analysis mentioned above and an application of Hurwitz’s Theorem from complex variable theory.
The free boundary of a semilinear elliptic equation
Authors: Friedman Avner , Phillips Daniel
Published: June 2010

The Dirichlet problem

Δ
u
=
λ

f
(
u
)

\Delta u = \lambda \,f(u)

in a domain

Ω
,

u
=
1

\Omega ,\,u = 1

on

∂
Ω

\partial \Omega

is considered with

f
(
t
)
=
0

f(t) = 0

if

t
≤
0
,

f
(
t
)
>
0

t \leq 0,\,f(t) > 0

if

t
>
0
,

f
(
t
)
∼

t
p

t > 0,\,f(t) \sim {t^p}

if

t
↓
0
,
0
>
p
>
1
;
f
(
t
)

t \downarrow 0,0 > p > 1;f(t)

is not monotone in general. The set

{
u
=
0
}

\{ u = 0\}

and the “free boundary”

∂
{
u
=
0
}

\partial \{ u = 0\}

are studied. Sharp asymptotic estimates are established as

λ
→
∞

\lambda \to \infty

. For suitable

f
f

, under the assumption that

Ω
\Omega

is a two-dimensional convex domain, it is shown that

{
u
=
0
}

\{ u = 0\}

is a convex set. Analogous results are established also in the case where

∂
u

/

∂
v
+
μ
(
u
−
1
)
=
0

\partial u/\partial v + \mu (u - 1) = 0

on

∂
Ω

\partial \Omega

.
The radius ratio and convexity properties in normed linear spaces
Authors: Amir D. , Franchetti C.
Published: June 2010

The supremum of the ratios of the self-radiusrA(A){r_A}(A)of a convex bounded set in a normed linear spaceXXto its absolute radiusrX(A){r_X}(A)is related to the supremum of the relative projection constants of the maximal subspaces ofXX. Necessary conditions and sufficient conditions for these suprema to be smaller than 2 are given. These conditions are selfadjoint superproperties similar toBB-convexity, superreflexivity andPP-convexity.
Positive solutions of nonlinear elliptic equations—existence and nonexistence of solutions with radial symmetry in 𝐿_{𝑝}(𝑅^{𝑁})
Authors: Toland J. F.
Published: June 2010

It is shown that when

r
r

is nonincreasing, radially symmetric, continuous and bounded below by a positive constant, the solution set of the nonlinear elliptic eigenvalue problem
\[

−
Δ
u
=
λ
u
+
r

u

1
+
σ

,

u
>
0

on

R

N

,

u
→
0

as

|x|

→
∞

- \Delta u = \lambda u + r{u^{1 + \sigma }},\qquad u > 0\qquad {\text {on}}\,{\mathbf {R}^N},\qquad u \to 0\qquad {\text {as}}\,{\text {|x|}} \to \infty

\]
, contains a continuum

C

\mathcal {C}

of nontrivial solutions which is unbounded in

R

×

L
p

(

R

N

)

\mathbf {R}\, \times \,{L_p}({\mathbf {R}^N})

for all

p
≥
1

p \geq 1

. Various estimates of the

L
p

{L_p}

norm of

u
u

are obtained which depend on the relative values of

σ
\sigma

and

p
p

, and the Pohozaev and Sobolev embedding constants.