bluuebunny/crossref_metadata_2025 huggingface dataset:
Linearization and mappings onto pseudocircle domains
Authors: Haas Andrew
Published: June 2010

We demonstrate the existence of linearizations for groups of conformal and anticonformal homeomorphisms of Riemann surfaces. The finitely generated groups acting on plane domains are classified in terms of specific linearizations. This extends Maskit’s work in the directly conformal case. As an application we prove that there exist conformal representations of finite genus open Riemann surfaces for which accessible boundary points are either isolated or lie on circular arcs of pseudocircular boundary components. In many cases these are actually circle domains. Along the way we extend the applicability of Carathéodory’s boundary correspondence theorem for prime ends.
Mean convergence of Lagrange interpolation. III
Authors: Nevai Paul
Published: June 2010

Necessary and sufficient conditions are found for weighted mean convergence of Lagrange and quasi-Lagrange interpolation based at the zeros of generalized Jacobi polynomials.
Brauer factor sets and simple algebras
Authors: Rowen Louis H.
Published: June 2010

It is shown that the Brauer factor set

(

c

i
j
k

)

({c_{ijk}})

of a finite-dimensional division algebra of odd degree

n
n

can be chosen such that

c

i
j
i

=

c

i
i
j

=

c

j
i
i

=
1

{c_{iji}} = {c_{iij}} = {c_{jii}} = 1

for all

i
,
j

i,j

and

c

i
j
k

=

c

k
j
i

−
1

{c_{ijk}} = c_{kji}^{ - 1}

. This implies at once the existence of an element

a
≠
0

a \ne 0

with

tr

(
a
)
=

tr

(

a
2

)
=
0

{\text {tr}}(a) = {\text {tr}}({a^2}) = 0

; the coefficients of

x

n
−
1

{x^{n - 1}}

and

x

n
−
2

{x^{n - 2}}

in the characteristic polynomial of

a
a

are thus

0
0

. Also one gets a generic division algebra of degree

n
n

whose center has transcendence degree

n
+
(
n
−
1
)
(
n
−
2
)

/

2

n + (n - 1)(n - 2)/2

, as well as a new (simpler) algebra of generic matrices. Equations are given to determine the cyclicity of these algebras, but they may not be tractable.
Dichotomies and asymptotic behaviour for linear differential systems
Authors: Muldowney James S.
Published: June 2010

Sufficient conditions that a system of differential equations

x
′

=
A
(
t
)
x

x’ = A(t)x

have a dichotomy usually require that the matrix

A
(
t
)

A(t)

be bounded or at least that some restriction be placed on the rate of growth or decay of solutions. Here three sets of necessary and sufficient conditions for a dichotomy which do not impose such a restriction are given in terms of Liapunov functions. Each of the theorems gives practical criteria for a dichotomy including the extension to unbounded matrices of criteria which depend on a concept of diagonal dominance for

A
(
t
)

A(t)

. An asymptotic analysis is also given for subspaces of the solution set by means of the associated compound equations.
On pairs of recursively enumerable degrees
Authors: Ambos-Spies Klaus
Published: June 2010

Lachlan and Yates proved that some, but not all, pairs of incomparable recursively enumerable (r.e.) degrees have an infimum. We answer some questions which arose from this situation. We show that not every nonzero incomplete r.e. degree is half of a pair of incomparable r.e. degrees which have an infimum, whereas every such degree is half of a pair without infimum. Further, we prove that every nonzero r.e. degree can be split into a pair of r.e. degrees which have no infimum, and every interval of r.e. degrees contains such a pair of degrees.
Subordination-preserving integral operators
Authors: Miller Sanford S. , Mocanu Petru T. , Reade Maxwell O.
Published: June 2010

Let

β
\beta

and

γ
\gamma

be complex numbers and let

H
H

be the space of functions regular in the unit disc. Subordination of functions

f
f

,

g
∈
H

g \in H

is denoted by

f
≺
g

f \prec g

. Let

K
⊂
H

K \subset H

and let the operator

A
:
K
→
H

A:K \to H

be defined by

F
=
A
(
f
)

F = A(f)

, where
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis z right-parenthesis equals left-bracket StartFraction 1 Over z Superscript gamma Baseline EndFraction integral Subscript 0 Superscript z Baseline f Superscript beta Baseline left-parenthesis t right-parenthesis t Superscript gamma minus 1 Baseline d t right-bracket Superscript 1 slash beta Baseline period">

F
(
z
)
=

[

1

z
γ

∫
0
z

f
β

(
t
)

t

γ
−
1

d
t

]

1

/

β

.

F(z) = {\left [ {\frac {1} {{{z^\gamma }}}\int _0^z {{f^\beta }(t){t^{\gamma - 1}}dt} } \right ]^{1/\beta }}.

\]
The authors determine conditions under which
\[

f
≺
g
⇒
A
(
f
)
≺
A
(
g
)
,

f \prec g \Rightarrow A(f) \prec A(g),

\]
and then use this result to obtain new distortion theorems for some classes of regular functions.
Points of continuity for semigroup actions
Authors: Lawson Jimmie D.
Published: June 2010

The purpose of this paper is to provide a more unified approach to questions involving the existence of points of joint continuity in separately continuous semigroup actions by deriving a small number of general principles which suffice to deduce previously derived results and generalizations thereof. The first major result gives sufficient conditions for a point to be a point of joint continuity in a general setting of "migrants", a useful symmetric generalization of semigroup actions. Results concerning actions of semigroups with group-like properties follow. In the latter part of the paper the notion of a subordinate point is introduced and joint continuity at subordinate points for various settings is proved. Finally, these results are applied to linear actions on locally convex spaces.
Fixed point sets of metric and nonmetric spaces
Authors: Martin John R. , Weiss William
Published: June 2010

A space

X
X

is said to have the complete invariance property

(
CIP
)

(\text {CIP})

if every nonempty closed subset of

X
X

is the fixed point set of some self-mapping of

X
X

. It is shown that connected subgroups of the plane and compact groups need not have

CIP
\text {CIP}

, and

CIP
\text {CIP}

need not be preserved by self-products of Peano continua, nonmetric manifolds or

0
0

-dimensional spaces. Sufficient conditions are given for an infinite product of spaces to have

CIP
\text {CIP}

. In particular, an uncountable product of real lines, circles or two-point spaces has

CIP
\text {CIP}

. Examples are given which contrast the behavior of

CIP
\text {CIP}

in the nonmetric and metric cases, and examples of spaces are given where the existence of

CIP
\text {CIP}

is neither provable nor refutable with the usual axioms of set theory.
Proper holomorphic mappings that must be rational
Authors: Bell Steven
Published: June 2010

Suppose

f
:

D
1

→

D
2

f:{D_1} \to {D_2}

is a proper holomorphic mapping between bounded domains in

C

n

{{\mathbf {C}}^n}

. We shall prove that under certain circumstances

f
f

must be a rational mapping, i.e., that the

n
n

component functions

f
i

{f_i}

of

f
f

are rational functions.
Diffusion approximation and computation of the critical size
Authors: Bardos C. , Santos R. , Sentis R.
Published: June 2010

This paper is devoted to the mathematical definition of the extrapolation length which appears in the diffusion approximation. To obtain this result, we describe the spectral properties of the transport equation and we show how the diffusion approximation is related to the computation of the critical size. The paper also contains some simple numerical examples and some new results for the Milne problem.
The strong conclusion of the F. and M. Riesz theorem on groups
Authors: Glicksberg I.
Published: June 2010

LetSSbe a closed proper generating subsemigroup of the dualΓ\Gammaof a locally compact abelian groupGG. Then there are Haar singular measures onGGorthogonal toSSunlessG=R×ΔG = {\mathbf {R}} \times \DeltaorT×Δ{\mathbf {T}} \times \DeltawithΔ\Deltadiscrete, and then allμ\muorthogonal toSSare Haar absolutely continuous.
On Skolem’s exponential functions below 2^{2^{𝑋}}
Authors: van den Dries Lou , Levitz Hilbert
Published: June 2010

A result of Ehrenfeucht implies that the smallest class of number-theoretic functions

f
:

N

→

N

f:{\mathbf {N}} \to {\mathbf {N}}

containing the constants

0
,
1
,
2
,
…

0,1,2, \ldots

, the identity function

X
X

, and closed under addition, multiplication and

f
→

f
X

f \to {f^X}

, is well-ordered by the relation of eventual dominance. We show that its order type is

ω

ω
ω

{\omega ^{{\omega ^\omega }}}

, and that for any two nonzero functions

f
,
g

f,g

in the class the quotient

f
(
n
)

/

g
(
n
)

f(n)/g(n)

tends to a limit in

E
+

∪
{
0
,
∞
}

{E^ + } \cup \{ 0,\infty \}

as

n
→
∞

n \to \infty

, where

E
+

{E^ + }

is the smallest set of positive real numbers containing

1
1

and closed under addition, multiplication and under the operations

x
→

x

−
1

,
x
→

e
x

x \to {x^{ - 1}},x \to {e^x}

.
Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in 𝑅⁴
Authors: Dunbar Steven R.
Published: June 2010

We establish the existence of traveling wave solutions for a reaction-diffusion system based on the Lotka-Volterra differential equation model of a predator and prey interaction. The waves are of transition front type, analogous to the solutions discussed by Fisher and Kolmogorov et al. for a scalar reaction-diffusion equation. The waves discussed here are not necessarily monotone. There is a speed

c
∗

>
0

{c^\ast } > 0

such that for

c
>

c
∗

c > {c^\ast }

there is a traveling wave moving with speed

c
c

. The proof uses a shooting argument based on the nonequivalence of a simply connected region and a nonsimply connected region together with a Liapunov function to guarantee the existence of the traveling wave solution. The traveling wave solution is equivalent to a heteroclinic orbit in

4
4

-dimensional phase space.
The radiance obstruction and parallel forms on affine manifolds
Authors: Goldman William , Hirsch Morris W.
Published: June 2010

A manifold

M
M

is affine if it is endowed with a distinguished atlas whose coordinate changes are locally affine. When they are locally linear

M
M

is called radiant. The obstruction to radiance is a one-dimensional class

c
M

{c_M}

with coefficients in the flat tangent bundle of

M
M

. Exterior powers of

c
M

{c_M}

give information on the existence of parallel forms on

M
M

, especially parallel volume forms. As applications, various kinds of restrictions are found on the holonomy and topology of compact affine manifolds.
Asymptotic periodicity of the iterates of Markov operators
Authors: Lasota A. , Li T.-Y. , Yorke J. A.
Published: June 2010

We say

P
:

L
1

→

L
1

P:{L^1} \to {L^1}

is a Markov operator if (i)

P
f
≥
0

Pf \geq 0

for

f
≥
0

f \geq 0

and (ii)

‖
P
f
‖
=
‖
f
‖

\| Pf\| = \| f\|

if

f
≥
0

f \geq 0

. It is shown that any Markov operator

P
P

has certain spectral decomposition if, for any

f
∈

L
1

f \in {L^1}

with

f
≥
0

f \geq 0

and

‖
f
‖
=
1

\| f\| = 1

,

P
n

f
→

F

{P^n}f \to \mathcal {F}

when

n
→
∞

n \to \infty

, where

F

\mathcal {F}

is a strongly compact subset of

L
1

{L^1}

. It follows from this decomposition that

P
n

f

{P^n}f

is asymptotically periodic for any

f
∈

L
1

f \in {L^1}

.
An idempotent completion functor in homotopy theory
Authors: Hastings Harold M.
Published: June 2010

We observe that Artin-Mazur style

R
R

-completions (

R
R

is a commutative ring with identity) induce analogous idempotent completions on the weak prohomotopy category pro-Ho(Top). Because Ho(Top) is a subcategory of pro-Ho(Top) and pro-Ho(Top) is closely related to the topologized homotopy category of J. F. Adams and D. Sullivan, our construction represents the Sullivan completions as homotopy limits of idempotent functors. In addition, we show that the Sullivan completion is idempotent on those spaces (in analogy with the Bousfield and Kan

R
∞

{R_\infty }

-completion on

R
R

-good spaces) for which its cohomology with coefficients in

R
R

agrees with that of our Artin-Mazur style completion. Finally, we rigidify the Artin-Mazur completion to obtain an idempotent Artin-Mazur completion on a category of generalized prospaces which preserves fibration and suitably defined cofibration sequences. (Our previous results on idempotency and factorization lift to the rigid completion.) Our results answer questions of Adams, Sullivan, and, later, A. Deleanu.
Complete linear systems on rational surfaces
Authors: Harbourne Brian
Published: June 2010

We determine the dimension, fixed components and base points of complete linear systems on blowings-up of

P

2

{{\mathbf {P}}^2}

having irreducible anticanonical divisor.
a.c. groups: extensions, maximal subgroups, and automorphisms
Authors: Hickin Kenneth
Published: June 2010

In

§
1

\S 1

we extend the results of [

3

{\mathbf {3}}

] on centralizers to r.e. subgroups and show, e.g., that every a.c. group has an

∞
−
ω

\infty - \omega

-equivalent subgroup of the same power which is embedded maximally in itself; and we pursue a natural typology of maximal subgroups.

§
2

\S 2

shows that if

A
A

is a countable group of automorphisms of a countable a.c. group

G
G

such that

A
⊃
Inn

G

A \supset \operatorname {Inn}\;G

, then there exists

τ
∈
Aut

G

\tau \in \operatorname {Aut}\;G

such that the

HNN

{\text {HNN}}

extension (

A
,
τ
:

τ

−
1

g
τ
=
τ
(
g
)

A,\tau :{\tau ^{ - 1}}g\tau = \tau (g)

for all

g
∈
Inn

G

g \in \operatorname {Inn}\;G

) is a subgroup of

Aut

G

\operatorname {Aut}\;G

. We show in

§
3

\S 3

that every a.c. group with a countable skeleton has a proper extension to an a.c. group having any skeleton that contains the original one and any f.g. group which contains the countable a.c. group equivalent to the original one as an r.e. subset. This uses Ziegler’s construction [

7

{\mathbf {7}}

]. Finally, in

§
4

\S 4

, also using Ziegler’s construction we show that there exists an a.c. group

A
A

of any power and having any countable skeleton which has a free subgroup

M
M

such that for all

x
∈
A
−
M

x \in A - M

and

y
∈
A

y \in A

there exist free generators

a
,
b
,
c
∈
M

a,b,c \in M

such that

y
=

(
a
x

)
b

(
a
x

)
c

y = {(ax)^b}{(ax)^c}

.
The dual Weierstrass-Laguerre transform
Authors: Haimo Deborah Tepper
Published: June 2010

An inversion algorithm is derived for the dual Weierstrass-Laguerre transform
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript normal infinity Baseline g Subscript alpha Baseline left-parenthesis x comma y semicolon 1 right-parenthesis phi left-parenthesis y right-parenthesis y Superscript alpha Baseline e Superscript negative y slash left-parenthesis alpha plus 1 right-parenthesis d y">

∫
0
∞

g
α

(
x
,
y
;
1
)
φ
(
y
)

y
α

e

−
y

/

(
α
+
1
)
d
y

\int _0^\infty {{g_\alpha }(x,y;1)\varphi (y){y^\alpha }{e^{ - y}}/(\alpha + 1)dy}

, where the function

g
α

(
x
,
y
,
t
)

{g_\alpha }(x,y,t)

is associated with the source solution of the Laguerre differential heat equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x u Subscript x x Baseline left-parenthesis x comma t right-parenthesis equals left-parenthesis alpha plus 1 minus x right-parenthesis u Subscript x Baseline left-parenthesis x comma t right-parenthesis equals u Subscript t Baseline left-parenthesis x comma t right-parenthesis">

x

u

x
x

(
x
,
t
)
=
(
α
+
1
−
x
)

u
x

(
x
,
t
)
=

u
t

(
x
,
t
)

x{u_{xx}}(x,t) = (\alpha + 1 - x){u_x}(x,t) = {u_t}(x,t)

. Correspondingly, sufficient conditions are established for a function to be represented by a Weierstrass-Laguerre Stieltjes transform

∫
0
∞

g
α

(
x
,
y
;
1
)

d
β
(
y
)

\int _0^\infty {{g_\alpha }(x,y;1)\;d\beta (y)}

of a nondecreasing function

β
\beta

.
The determinant of the Eisenstein matrix and Hilbert class fields
Authors: Efrat I. , Sarnak P.
Published: June 2010

We compute the determinant of the Eisenstein matrix associated to the Hilbert-Blumenthal modular group

PSL
2

(

O

k

)

{\text {PSL}_2}({\mathcal {O}_k})

, and express it in terms of the zeta function of the Hilbert class field of

K
K

.
A note on automorphic forms of weight one and weight three
Authors: Stiller Peter F.
Published: June 2010

In this paper the author develops an interesting relationship between classical automorphic forms of weights one and three, and the solutions of certain second order differential equations related to elliptic (modular) surfaces. In particular for a cusp form of weight three, it is shown that the special values of the associated Dirichlet series can be determined from the periods of an inhomogeneous differential equation, or what is the same thing, the monodromy of an associated third order differential equation. Explicit examples are provided for principal congruence subgroups

Γ
(
N
)

\Gamma (N)

with

N
≡
0

mod

4

N \equiv 0\,\operatorname {mod} \,4

.
Subgraphs of random graphs
Authors: Fremlin D. H. , Talagrand M.
Published: June 2010

Let

Δ
⊆

[
ω

]
2

\Delta \subseteq {[\omega ]^2}

be an undirected graph on

ω
\omega

, and let

u
∈
[
0
,

1
]

u \in [0,\,1]

. Following P. Erdös and A. Hajnal, we write

(
ω
,

2
,

u
)
⇒
Δ

(\omega ,\,2,\,u) \Rightarrow \Delta

to mean: whenever

E
1

⊆
[
0
,

1
]

{E_1} \subseteq [0,\,1]

is a measurable set of Lebesgue measure at least

u
u

for every

I
∈

[
ω

]
2

I \in {[\omega ]^2}

, then there is some

t
∈
[
0
,

1
]

t \in [0,\,1]

such that

Δ
\Delta

appears in the graph

Γ
t

=
{
I
:

t
∈

E
I

}

{\Gamma _t} = \{ I:\,t \in {E_I}\}

in the sense that there is a strictly increasing function

f
:

ω
→
ω

f:\,\omega \to \omega

such that

{
f
(
i
)
,

f
(
j
)
}
∈

Γ
t

\{ f(i),\,f(j)\} \in {\Gamma _t}

whenever

{
i
,

j
}
∈
Δ

\{ i,\,j\} \in \Delta

. We give an algorithm for determining when

(
ω
,

2
,

u
)
⇒
Δ

(\omega ,\,2,\,u) \Rightarrow \Delta

for finite

Δ
\Delta

, and we show that for infinite

Δ
,

(
ω
,

2
,

u
)
⇒
Δ

\Delta ,\,(\omega ,\,2,\,u) \Rightarrow \Delta

if there is a

υ
>
u

\upsilon > u

such that

(
ω
,

2
,

υ
)
⇒

Δ
′

(\omega ,\,2,\,\upsilon ) \Rightarrow {\Delta ^\prime }

for every finite

Δ
′

⊆
Δ

\Delta ’ \subseteq \Delta

. Our results depend on a new condition, expressed in terms of measures on

β
ω

\beta \omega

, sufficient to imply that

Δ
\Delta

appears in

Γ
\Gamma

(Theorem 2F), and enable us to identify the extreme points of some convex sets of measures (Theorem 5H).
Convergence of conditional expectations and strong laws of large numbers for multivalued random variables
Authors: Hiai Fumio
Published: June 2010

Fatou’s lemmas and Lebesgue’s convergence theorems are established for multivalued conditional expectations of random variables having values in the closed subsets of a separable Banach space. Strong laws of large numbers are also given for such multivalued random variables.
Arbitrarily large continuous algebras on one generator
Authors: Adámek Jiří , Koubek Václav , Nelson Evelyn , Reiterman Jan
Published: June 2010

Generation of order-continuous algebras is investigated for various concepts of continuity. For the continuity understood as the preservation of joins of countably-directed sets, arbitrarily large infinitary continuous algebras on one generator are constructed.
Quasiconformal and bi-Lipschitz homeomorphisms, uniform domains and the quasihyperbolic metric
Authors: Martin Gaven J.
Published: June 2010

Let

D
D

be a proper subdomain of

R
n

{R^n}

and

k
D

{k_D}

the quasihyperbolic metric defined by the conformal metric tensor

d

s
¯

2

=
dist
⁡

(
x
,
∂
D

)

−
2

d

s
2

d{\overline s ^2} = \operatorname {dist} {(x,\partial D)^{ - 2}}d{s^2}

. The geodesics for this and related metrics are shown, by purely geometric methods, to exist and have Lipschitz continuous first derivatives. This is sharp for

k
D

{k_D}

; we also obtain sharp estimates for the euclidean curvature of such geodesics. We then use these results to prove a general decomposition theorem for uniform domains in

R
n

{R^n}

, in terms of embeddings of bi-Lipschitz balls. We also construct a counterexample to the higher dimensional analogue of the decomposition theorem of Gehring and Osgood.
Algebraic and etale 𝐾-theory
Authors: Dwyer William G. , Friedlander Eric M.
Published: June 2010

We define etale

K
K

-theory, interpret various conjectures of Quillen and Lichtenbaum in terms of a map from algebraic

K
K

-theory to etale

K
K

-theory, and then prove that this map is surjective in many cases of interest.
Strange attractors of uniform flows
Authors: Kan Ittai
Published: June 2010

Consider orbitally stable attractors of those flows on the open solid torus

D
2

×

S
1

{D^2} \times {S^1}

which have uniform velocity in the

S
1

{S^1}

direction (uniform flows). It is found that any such attractor is the frontier of a strictly nested sequence of positively invariant open solid tori. Necessary and sufficient conditions related to these tori are derived for an arbitrary set to be an orbitally stable attractor. When the cross-section of an orbitally stable attractor is a Cantor set, the first return map is found to be conjugate to an irrational rotation on a certain compact abelian group. New examples are constructed of orbitally stable attractors of uniform

C
∞

{C^\infty }

flows whose cross-sections have uncountably many components (one of these attractors has positive

3
3

-dimensional Lebesgue measure).
Reflexivity and order properties of scalar-type spectral operators in locally convex spaces
Authors: Dodds P. G. , de Pagter B. , Ricker W.
Published: June 2010

One of the principal results of the paper is that each scalar-type spectral operator in the quasicomplete locally convex space

X
X

is reflexive. The paper also studies in detail the relation between the theory of equicontinuous spectral measures in locally convex spaces and the order properties of equicontinuous Bade complete Boolean algebras of projections.
Long time asymptotics of the Korteweg-de Vries equation
Authors: Venakides Stephanos
Published: June 2010

We study the long time evolution of the solution to the Kortewegde Vries equation with initial data

υ
(
x
)

\upsilon (x)

which satisfy
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript x right-arrow negative normal infinity Endscripts upsilon left-parenthesis x right-parenthesis equals negative 1 comma limit Underscript x right-arrow plus normal infinity Endscripts upsilon left-parenthesis x right-parenthesis equals 0">

lim

x
→
−
∞

υ
(
x
)
=
−
1
,

lim

x
→
+
∞

υ
(
x
)
=
0

\lim \limits _{x \to - \infty } \upsilon (x) = - 1,\qquad \lim \limits _{x \to + \infty } \upsilon (x) = 0

\]
We show that as

t
→
∞

t \to \infty

the step emits a wavetrain of solitons which asymptotically have twice the amplitude of the initial step. We derive a lower bound of the number of solitons separated at time

t
t

for

t
t

large.
Best rational approximations of entire functions whose Maclaurin series coefficients decrease rapidly and smoothly
Authors: Levin A. L. , Lubinsky D. S.
Published: June 2010

Let

f
=

Σ

j
=
0

∞

a
j

z
j

f = \Sigma _{j = 0}^\infty {a_j}{z^j}

be an entire function which satisfies
\[

|

a

j
−
1

a

j
+
1

/

a
j
2

|

⩽

ρ
2

,

j
=
1
,
2
,
3
,
…
,

|{a_{j - 1}}a{ _{j + 1}}/a_j^2| \leqslant {\rho ^2},\qquad j = 1,2,3, \ldots ,

\]
where

0
>
ρ
>

ρ
0

0 > \rho > {\rho _0}

and

ρ
0

=
0.4559
…

{\rho _0} = 0.4559 \ldots

is the positive root of the equation

2

Σ

j
=
1

∞

ρ

j
2

=
1

2\Sigma _{j = 1}^\infty {\rho ^{{j^2}}} = 1

. Let

r
>
0

r > 0

be fixed. Let

W

L
M

{W_{LM}}

denote the rational function of type

(
L
,
M
)

(L,M)

of best approximation to

f
f

in the uniform norm on

|

z

|

⩽
r

|z| \leqslant r

. We show that for any sequence of nonnegative integers

{

M
L

}

L
=
1

∞

\{ {M_L}\} _{L = 1}^\infty

that satisfies

M
L

⩽
10
L
,

L
=
1
,
2
,
3
,
…

{M_L} \leqslant 10L,\,L = 1,2,3, \ldots

, the rational approximations

W

L

M
L

{W_{L{M_L}}}

converge to

f
f

throughout

C

{\mathbf {C}}

as

L
→
∞

L \to \infty

. In particular, convergence takes place for the diagonal sequence and for the row sequences of the Walsh array for

f
f

.
Conditions suffisantes de résolubilité locale pour des opérateurs invariants à gauche sur des groupes nilpotents. II
Authors: Lévy-Bruhl Pierre
Published: June 2010

On donne des conditions suffisantes de résolubilité locale pour des opérateurs différentiels invariants à gauche sur certains groupes de Lie nilpotents gradués. Ces conditions portent sur l’image de l’opérateur par certaines représentations unitaires irréductibles du groupe.
A new construction of noncrossed product algebras
Authors: Jacob Bill , Wadsworth Adrian R.
Published: June 2010

New examples of noncrossed product division algebras are obtained, using methods different from all previous noncrossed product constructions. The examples are division algebras over intersections ofpp-Henselian valued fields, and they have Schur indexpm{p^m}and exponentpn{p^n}for any prime numberppand any integersm⩾n⩾2(n⩾3ifp=2)m \geqslant n \geqslant 2\;(n \geqslant 3\;{\text {if}}\;p = 2). The basic tools used in the construction are valuation theory and Galois cohomology; no generic methods are applied and there is no p.i. theory. Along the way, local-global principles are proved for central simple algebras over intersections ofpp-Henselian valued fields.
Extensions of Verma modules
Authors: Carlin Kevin J.
Published: June 2010

A spectral sequence is introduced which computes extensions in category

O

\mathcal {O}

in terms of derived functors associated to coherent translation functors. This is applied to the problem of computing extensions of one Verma module by another when the highest weights are integral and regular. Some results are obtained which are consistent with the Gabber-Joseph conjecture. The main result is that the highest-degree nonzero extension is one-dimensional. The spectral sequence is also applied to the Kazhdan-Lusztig conjecture and related to the work of Vogan in this area.
On type of metric spaces
Authors: Bourgain J. , Milman V. , Wolfson H.
Published: June 2010

Families of finite metric spaces are investigated. A notion of metric type is introduced and it is shown that for Banach spaces it is consistent with the standard notion of type. A theorem parallel to the Maurey-Pisier Theorem in Local Theory is proved. Embeddings of

l
p

{l_p}

-cubes into the

l
1

{l_1}

-cube (Hamming cube) are discussed.
Volume of mixed bodies
Authors: Lutwak Erwin
Published: June 2010

By using inequalities obtained for the volume of mixed bodies and the Petty Projection Inequality, (sharp) isoperimetric inequalities are derived for the projection measures (Quermassintegrale) of a convex body. These projection measure inequalities, which involve mixed projection bodies (zonoids), are shown to be strengthened versions of the classical inequalities between the projection measures of a convex body. The inequality obtained for the volume of mixed bodies is also used to derive a form of the Brunn-Minkowski inequality involving mixed bodies. As an application, inequalities are given between the projection measures of convex bodies and the mixed projection integrals of the bodies.
Left separated spaces with point-countable bases
Authors: Fleissner William G.
Published: June 2010

Theorem 2.2 lists properties equivalent to left separated spaces in the class of

T
1

{T_1}

with point-countable bases, with examples preventing plausible additions to this list. For example,

X
X

is left iff

X
X

is

σ
\sigma

-weakly separated or

X
X

has a closure preserving cover by countable closed sets, but

X
X

is left separated does not imply that

X
X

is

σ
\sigma

-discrete. Theorem 2.2 is used to show that the following reflection property holds after properly collapsing a supercompact cardinal to

ω
2

{\omega _2}

: If

X
X

is a not

σ
\sigma

-discrete metric space, then

X
X

has a not

σ
\sigma

-discrete subspace of cardinality less than

ω
2

{\omega _2}

. Similar reflection properties are shown true in some models and false in others.
The complex equilibrium measure of a symmetric convex set in 𝑅ⁿ
Authors: Bedford Eric , Taylor B. A.
Published: June 2010

We give a formula for the measure on a convex symmetric set

K
K

in

R

n

{{\mathbf {R}}^n}

which is the Monge-Ampere operator applied to the extremal plurisubharmonic function

L
K

{L_K}

for the convex set. The measure is concentrated on the set

K
K

and is absolutely continuous with respect to Lebesgue measure with a density which behaves at the boundary like the reciprocal of the square root of the distance to the boundary. The precise asymptotic formula for

x
∈
K

x \in K

near a boundary point

x
0

{x_0}

of

K
K

is shown to be of the form

c
(

x
0

)

/

[

dist

(
x
,

∂
K
)

]

−
1

/

2

c({x_0})/{[{\operatorname {dist}}(x,\,\partial K)]^{ - 1/2}}

, where the constant

c
(

x
0

)

c({x_0})

depends both on the curvature of

K
K

at

x
0

{x_0}

and on the global structure of

K
K

.
Brownian motions of ellipsoids
Authors: Norris J. R. , Rogers L. C. G. , Williams David
Published: June 2010

The object of this paper is to provide an elementary treatment (involving no differential geometry) of Brownian motions of ellipsoids, and, in particular, of some remarkable results first obtained by Dynkin. The canonical right-invariant Brownian motion

G
=
{
G
(
t
)
}

G = \{ G(t)\}

on

GL

(
n
)

{\text {GL}}(n)

induces processes

X
=
G

G
T

X = G{G^T}

and

Y
=

G
T

G

Y = {G^T}G

on the space of positive-definite symmetric matrices. The motion of the common eigenvalues of

X
X

and

Y
Y

is analysed. It is further shown that the orthonormal frame of eigenvectors of

X
X

ultimately behaves like Brownian motion on

O

(
n
)

{\text {O}}(n)

, while that of

Y
Y

converges to a limiting value. The

Y
Y

process is that studied by Dynkin and Orihara. From a naive standpoint, the

X
X

process would seem to provide a more natural model.
Deformations of complete minimal surfaces
Authors: Rosenberg Harold
Published: June 2010

A notion of deformation is defined and studied for complete minimal surfaces in

R
3

{R^3}

and

R
3

/

G
,
G

{R^3}/G,G

a group of translations. The catenoid, Enneper’s surface, and the surface of Meeks-Jorge, modelled on a

3
3

-punctured sphere, are shown to be isolated. Minimal surfaces of total curvature

4
π

4\pi

in

R
3

/

Z

{R^3}/Z

and

R
3

/

Z
2

{R^3}/{Z^2}

are studied. It is proved that the helicoid and Scherk’s surface are isolated under periodic perturbations.
Nonlinear oblique boundary value problems for nonlinear elliptic equations
Authors: Lieberman Gary M. , Trudinger Neil S.
Published: June 2010

We consider the nonlinear oblique derivative boundary value problem for quasilinear and fully nonlinear uniformly elliptic partial differential equations of second order. The elliptic operators satisfy natural structure conditions as introduced by Trudinger in the study of the Dirichlet problem while for the boundary operators we formulate general structure conditions which embrace previously considered special cases such as the capillarity condition. The resultant existence theorems include previous work such as that of Lieberman on quasilinear equations and Lions and Trudinger on Neumann boundary conditions.
Generic dynamics and monotone complete 𝐶*-algebras
Authors: Sullivan Dennis , Weiss B. , Wright J. D. Maitland
Published: June 2010

Let

R
R

be any ergodic, countable generic equivalence relation on a perfect Polish space

X
X

. It follows from the main theorem of

§
1

\S 1

that, modulo a meagre subset of

X
,
R

X,R

may be identified with the relation of orbit equivalence ensuing from a canonical action of

Z

{\mathbf {Z}}

. Answering a longstanding problem of Kaplansky, Takenouchi and Dyer independently gave cross-product constructions of Type III

A

W
∗

A{W^\ast }

-factors which were not von Neumann algebras. As a specialization of a much more general result, obtained in

§
3

\S 3

, we show that the Dyer factor is isomorphic to the Takenouchi factor.
Real hypersurfaces and complex submanifolds in complex projective space
Authors: Kimura Makoto
Published: June 2010

Let

M
M

be a real hypersurface in

P
n

(

C

)

{P^n}({\mathbf {C}})

be the complex structure and

ξ
\xi

denote a unit normal vector field on

M
M

. We show that

M
M

is (an open subset of) a homogeneous hypersurface if and only if

M
M

has constant principal curvatures and

J
ξ

J\xi

is principal. We also obtain a characterization of certain complex submanifolds in a complex projective space. Specifically,

P
m

(

C

)

{P^m}({\mathbf {C}})

(totally geodesic),
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q Superscript n Baseline comma upper P Superscript 1 Baseline left-parenthesis bold upper C right-parenthesis times upper P Superscript n Baseline left-parenthesis bold upper C right-parenthesis comma upper S upper U left-parenthesis 5 right-parenthesis slash upper S left-parenthesis upper U left-parenthesis 2 right-parenthesis times upper U left-parenthesis 3 right-parenthesis right-parenthesis">

Q
n

,

P
1

(

C

)
×

P
n

(

C

)
,
S
U
(
5
)

/

S
(
U
(
2
)
×
U
(
3
)
)

{Q^n},{P^1}({\mathbf {C}}) \times {P^n}({\mathbf {C}}),SU(5)/S(U(2) \times U(3))

and

S
O
(
10
)

/

U
(
5
)

SO(10)/U(5)

are the only complex submanifolds whose principal curvatures are constant in the sense that they depend neither on the point of the submanifold nor on the normal vector.
Orbits of the pseudocircle
Authors: Kennedy Judy , Rogers James T.
Published: June 2010

The following theorem is proved. Theorem. The pseudocircle has uncountably many orbits under the action of its homeomorphism group. Each orbit is the union of uncountably many composants.
BMO rational approximation and one-dimensional Hausdorff content
Authors: Verdera Joan
Published: June 2010

Let

X
⊂

C

X \subset {\mathbf {C}}

be compact and let

f
∈
VMO
⁡
(

C

)

f \in \operatorname {VMO} ({\mathbf {C}})

. We give necessary and sufficient conditions on

f
f

and

X
X

for

f

|

X

{f_{|X}}

to be the limit of a sequence of rational functions without poles on

X
X

in the norm of

BMO
⁡
(
X
)

\operatorname {BMO} (X)

, the space of functions of bounded mean oscillation on

X
X

. We also characterize those compact

X
⊂

C

X \subset {\mathbf {C}}

with the property that the restriction to

X
X

of each function in

VMO
⁡
(

C

)

\operatorname {VMO} ({\mathbf {C}})

, which is holomorphic on

X
˚

\mathring {X}

, is the limit of a sequence of rational functions with poles off

X
X

. Our conditions involve the notion of one-dimensional Hausdorff content. As an application, a result related to the inner boundary conjecture is proven.
The Radon transform on 𝑆𝐿(2,𝑅)/𝑆𝑂(2,𝑅)
Authors: Wallace D. I. , Yamaguchi Ryuji
Published: June 2010

Let

G
G

be

S
L
(
2
,

R

)

SL(2,{\mathbf {R}})

.

G
G

acts on the upper half-plane

H

\mathcal {H}

by the Möbius transformation, providing

H

\mathcal {H}

with the Riemannian metric structure along with the Laplacian,

Δ
\Delta

. We study the integral transform along each geodesic.

G
G

acts on

P

\mathcal {P}

, the space of all geodesics, in a natural way, providing

P

\mathcal {P}

with its invariant measure and its own Laplacian. (

P

\mathcal {P}

actually is a coset space of

G
G

.) Therefore the above transform can be viewed as a map from a suitable function space on

H

\mathcal {H}

to a suitable function space on

P

\mathcal {P}

. We prove a number of properties of this transform, including the intertwining properties with its Laplacians and its relation to the Fourier transforms.
Analytic perturbation of the Taylor spectrum
Authors: Slodkowski Zbigniew
Published: June 2010

Let

T
1

(
z
)
,
…
,

T
m

(
z
)

{T_1}(z), \ldots ,{T_m}(z)

,

z
∈
G
⊂

C

k

z \in G \subset {{\mathbf {C}}^k}

, be analytic families of bounded operators in a complex Banach space

X
X

, such that for each

z
∈
G

z \in G

the operators

T
i

(
z
)

{T_i}(z)

and

T
j

(
z
)

{T_j}(z)

,

i
,
j
=
1
,
…
,
n

i,j = 1, \ldots ,n

, commute. Main result: If

K
(
z
)

K(z)

denotes the Taylor spectrum of the tuple

(

T
1

(
z
)
,
…
,

T
m

(
z
)
)

({T_1}(z), \ldots ,{T_m}(z))

, then the set-valued function

K
:
G
→

2

C

m

K:G \to {2^{{\mathbf {C}}m}}

is analytic. Analyticity of such set-valued functions is defined here by a simultaneous local maximum property of

k
k

-tuples of complex polynomials on the graph of

K
K

.
Regularity results for an elliptic-parabolic free boundary problem
Authors: Bertsch M. , Hulshof J.
Published: June 2010

We study an elliptic-parabolic free boundary problem in one space dimension. We give several regularity results for both the weak solution and the free boundary. In particular conditions are given which ensure that the free boundary is a

C
1

{C^1}

-curve.
A global theory of internal solitary waves in two-fluid systems
Authors: Amick C. J. , Turner R. E. L.
Published: June 2010

The problem analyzed is that of two-dimensional wave motion in a heterogeneous, inviscid fluid confined between two rigid horizontal planes and subject to gravity

g
g

. It is assumed that a fluid of constant density

ρ
+

{\rho _ + }

lies above a fluid of constant density

ρ
−

>

ρ
+

>
0

{\rho _ - } > {\rho _ + } > 0

and that the system is nondiffusive. Progressing solitary waves, viewed in a moving coordinate system, can be described by a pair

(
λ
,
w
)

(\lambda ,w)

, where the constant

λ
=
g

/

c
2

\lambda = g/{c^2}

,

c
c

being the wave speed, and where

w
(
x
,
η
)
+
η

w(x,\eta ) + \eta

is the height at a horizontal position

x
x

of the streamline which has height

η
\eta

at

x
=
±
∞

x = \pm \infty

. It is shown that among the nontrivial solutions of a quasilinear elliptic eigenvalue problem for

(
λ
,
w
)

(\lambda ,w)

is an unbounded connected set in

R

×
(

H
0
1

∩

C

0
,
1

)

{\mathbf {R}} \times (H_0^1 \cap {C^{0,1}})

. Various properties of the solution are shown, and the behavior of large amplitude solutions is analyzed, leading to the alternative that internal surges must occur or streamlines with vertical tangents must occur.
Vector bundles on complex projective spaces and systems of partial differential equations. I
Authors: Stiller Peter F.
Published: June 2010

This paper establishes and investigates a relationship between the space of solutions of a system of constant coefficient partial differential equations and the cohomology (

H
1

{H^1}

in particular) of an associated vector bundle/reflexive sheaf on complex projective space. Using results of Grothendieck and Shatz on vector bundles over projective one-space, the case of partial differential equations in two variables is completely analyzed. The final section applies results about vector bundles on higher-dimensional projective spaces to the case of three or more variables.
Representations of crossed products by coactions and principal bundles
Authors: Landstad M. B. , Phillips J. , Raeburn I. , Sutherland C. E.
Published: June 2010

The main purpose of this paper is to establish a covariant representation theory for coactions of locally compact groups on

C

∗

{C^{\ast }}

-algebras (including a notion of exterior equivalence), to show how these results extend the usual notions for actions of groups on

C

∗

{C^{\ast }}

-algebras, and to apply these ideas to classes of coactions termed pointwise unitary and locally unitary to obtain a complete realization of the isomorphism theory of locally trivial principal

G
G

-bundles in this context. We are also able to obtain all (Cartan) principal

G
G

-bundles in this context, but their isomorphism theory remains elusive.
Regressive partition relations for infinite cardinals
Authors: Hajnal András , Kanamori Akihiro , Shelah Saharon
Published: June 2010

The regressive partition relation, which turns out to be important in incompleteness phenomena, is completely characterized in the transfinite case. This work is related to Schmerl

[

S

]

\left [ {\mathbf {S}} \right ]

, whose characterizations we complete.
Growth properties of functions in Hardy fields
Authors: Rosenlicht Maxwell
Published: June 2010

This paper continues the author’s earlier work on the notion of rank in a Hardy field. Further results are given on functions in Hardy fields of finite rank, including extensions of Hardy’s results on the rates of growth of his logarithmico-exponential functions.
On the behavior near the crest of waves of extreme form
Authors: Amick C. J. , Fraenkel L. E.
Published: June 2010

The angle

ϕ
\phi

which the free boundary of an extreme wave makes with the horizontal is the solution of a singular, nonlinear integral equation that does not fit (as far as we know) into the theory of compact operators on Banach spaces. It has been proved only recently that solutions exist and that (as Stokes suggested in 1880) these solutions represent waves with sharp crests of included angle

2
π

/

3

2\pi /3

. In this paper we use the integral equation, known properties of solutions and the technique of the Mellin transform to obtain the asymptotic expansion
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis asterisk right-parenthesis phi left-parenthesis s right-parenthesis equals StartFraction pi Over 6 EndFraction plus sigma-summation Underscript n equals 1 Overscript k Endscripts a Subscript n Baseline s Superscript mu Super Subscript n Superscript Baseline plus o left-parenthesis s Superscript mu Super Subscript k Superscript Baseline right-parenthesis as s down-arrow 0">

(
∗
)

ϕ
(
s
)
=

π
6

+

∑

n
=
1

k

a
n

s

μ
n

+
o
(

s

μ
k

)

as

s
↓
0

( * )\qquad \phi (s) = \frac {\pi } {6} + \sum \limits _{n = 1}^k {{a_n}{s^{{\mu _n}}} + o({s^{{\mu _k}}})} \quad {\text {as}}\,s \downarrow 0

\]
, to arbitrary order; the coordinate

s
s

is related to distance from the crest as measured by the velocity potential rather than by length. The first few (and probably all) of the exponents

μ
n

{\mu _n}

are transcendental numbers. We are unable to evaluate the coefficients

a
n

{a_n}

explicitly, but define some in terms of global properties of

ϕ
\phi

, and the others in terms of earlier coefficients. It is proved in [8] that

a
1

>
0

{a_1} > 0

, and follows here that

a
2

>
0

{a_2} > 0

. The derivation of (*) includes an assumption about a question in number theory; if that assumption should be false, logarithmic terms would enter the series at very large values of

n
n

.
On embedding of group rings of residually torsion free nilpotent groups into skew fields
Authors: Eizenbud A. , Lichtman A. I.
Published: June 2010

It is proven that the group ring of an amalgamated free product of residually torsion free nilpotent groups is a domain and can be embedded in a skew field. This is a generalization of J. Lewin’s theorem, proven for the case of free groups. Our proof is based on the study of the Malcev-Neumann power series ring

K

⟨
G
⟩

K\left \langle G \right \rangle

of a residually torsion free nilpotent group

G
G

. It is shown that its subfield

D
D

, generated by the group ring

K
G

KG

, does not depend on the order of

G
G

for many kinds of orders and the study of

D
D

can be reduced in some sense to the case when

G
G

is nilpotent.
The homology and higher representations of the automorphism group of a Riemann surface
Authors: Broughton S. A.
Published: June 2010

The representations of the automorphism group of a compact Riemann surface on the first homology group and the spaces of

q
q

-differentials are decomposed into irreducibles. As an application it is shown that

M

24

{M_{24}}

is not a Hurwitz group.
Two-dimensional nonlinear boundary value problems for elliptic equations
Authors: Lieberman Gary M.
Published: June 2010

Boundary regularity of solutions of the fully nonlinear boundary value problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis x comma u comma upper D u comma upper D squared u right-parenthesis equals 0 in normal upper Omega comma upper G left-parenthesis x comma u comma upper D u right-parenthesis equals 0 on partial-differential normal upper Omega">

F
(
x
,
u
,
D
u
,

D
2

u
)
=
0

in

Ω
,

G
(
x
,
u
,
D
u
)
=
0

on

∂
Ω

F(x,u,Du,{D^2}u) = 0\quad {\text {in}}\;\Omega ,\qquad G(x,u,Du) = 0\quad {\text {on}}\;\partial \Omega

\]
is discussed for two-dimensional domains

Ω
\Omega

. The function

F
F

is assumed uniformly elliptic and

G
G

is assumed to depend (in a nonvacuous manner) on

D
u

Du

. Continuity estimates are proved for first and second derivatives of

u
u

under weak hypotheses for smoothness of

F
F

,

G
G

, and

Ω
\Omega

.
Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval
Authors: Block Louis , Coven Ethan M.
Published: June 2010

We say that a continuous map

f
f

of a compact interval to itself is linear Markov if it is piecewise linear, and the set of all

f
k

(
x
)

{f^k}(x)

, where

k
⩾
0

k \geqslant 0

and

x
x

is an endpoint of a linear piece, is finite. We provide an effective classification, up to topological conjugacy, for linear Markov maps and an effective procedure for determining whether such a map is transitive. We also consider expanding Markov maps, partly to motivate the proof of the more complicated linear Markov case.
On weakly countably determined Banach spaces
Authors: Mercourakis Sophocles
Published: June 2010

For a topological space

X
X

, let

C
1

(
X
)

{C_1}(X)

denote the Banach space of all bounded functions

f
:
X
→

R

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

such that for every

ε
>
0

\varepsilon > 0

the set

{
x
∈
X
:

|

f
(
x
)

|

⩾
ε
}

\{ x \in X:|f(x)| \geqslant \varepsilon \}

is closed and discrete in

X
X

, endowed with the supremum norm. The main theorem is the following: Let

L
L

be a weakly countably determined subset of a Banach space; then there exist a subset

Σ
′

\Sigma ’

of the Baire space

Σ
\Sigma

, a compact space

K
K

, and a bounded linear one-to-one operator

T
:
C
(
L
)
→

C
1

(

Σ
′

×
K
)

T:C(L) \to {C_1}(\Sigma ’ \times K)

that is pointwise to pointwise continuous. In the case where

L
L

is weakly analytic,

Σ
′

\Sigma ’

can be replaced by

Σ
\Sigma

. This theorem is connected with the basic result of Amir-Lindenstrauss on WCG Banach spaces and has corresponding consequences such as: the representation of Gulko (resp. Talagrand) compact spaces as pointwise compact subsets of

C
1

(

Σ
′

×
K
)

{C_1}(\Sigma ’ \times K)

(resp.

C
1

(
Σ
×
K
)

{C_1}(\Sigma \times K)

) (a compact space

Ω
\Omega

is called Gulko or Talagrand compact if

C
(
Ω
)

C(\Omega )

is WCD or a weakly

K
K

-analytic Banach space); the characterization of WCD (resp. weakly

K
K

-analytic) Banach spaces

E
E

, using one-to-one operators from

E

∗

{E^{\ast }}

into

C
1

(

Σ
′

×
K
)

{C_1}(\Sigma ’ \times K)

(resp.

C
1

(
Σ
×
K
)

{C_1}(\Sigma \times K)

); and the existence of equivalent "good" norms on

E
E

and

E

∗

{E^{\ast }}

simultaneously.
The binary matroids with no 4-wheel minor
Authors: Oxley James G.
Published: June 2010

The cycle matroids of wheels are the fundamental building blocks for the class of binary matroids. Brylawski has shown that a binary matroid has no minor isomorphic to the rank-3 wheel

M
(

W

3

)

M({\mathcal {W}_3})

if and only if it is a series-parallel network. In this paper we characterize the binary matroids with no minor isomorphic to

M
(

W

4

)

M({\mathcal {W}_4})

. This characterization is used to solve the critical problem for this class of matroids and to extend results of Kung and Walton and Welsh for related classes of binary matroids.
A regularity result for viscosity solutions of Hamilton-Jacobi equations in one space dimension
Authors: Jensen R. , Souganidis P. E.
Published: June 2010

Viscosity solutions of Hamilton-Jacobi equations need only to be continuous. Here we prove that, in the special case of a one-dimensional stationary problem, under quite general assumptions, Lipschitz continuous viscosity solutions have right and left derivatives at every point. Moreover, these derivatives have some kind of continuity properties.