bluuebunny/crossref_metadata_2025 huggingface dataset:
The index of harmonic foliations on spheres
Authors: Kamber Franz W. , Tondeur Philippe
Published: June 2010

For foliations on a compact oriented manifold there is a natural energy functional, defined with respect to a Riemannian metric. Harmonic Riemannian foliations are then the critical foliations for this functional under an appropriate class of special variations. The index of the title is the index of the Hessian of the energy functional at a critical, i.e., harmonic foliation. It is a finite number. In this note it is shown that for a harmonic Riemannian foliation

F

\mathcal {F}

of codimension

q
q

on the

n
n

-sphere (

n
>
2

n > 2

) this index is greater or equal to

q
+
1

q + 1

. Thus

F

\mathcal {F}

is unstable. Moreover the given bound is best possible.
An embedding characterization of almost compact spaces
Authors: Salbany Sergio
Published: June 2010

We characterize almost compact and almost realcompact spaces in terms of their situation in the product

(
J
,
u

)
C

{(J,u)^C}

. In the characterization of almost compactness

J
J

is the two point set or the unit interval; in the characterization of almost realcompactness

J
J

is the set of nonnegative integers or the nonnegative reals.

u
u

is the upper topology on the real line restricted to

J
J

.
Regular functions of restricted growth and their zeros in tangential regions
Authors: Linden C. N.
Published: June 2010

For a given function

k
k

, positive, continuous, nondecreasing and unbounded on

[
0
,
1
)

[0,1)

, let

A

(
k
)

{A^{(k)}}

denote the class of functions regular in the unit disc for which log

|

f
(
z
)

|

>
k
(

|

z

|

)

|f(z)| > k(|z|)

when

|

z

|

>
1

|z| > 1

. Hayman and Korenblum have shown that a necessary and sufficient condition for the sets of positive zeros of all functions in

A

(
k
)

{A^{(k)}}

to be Blaschke is that
\[

∫
0
1

(
k
(
t
)

/

(
1
−
t
)
)

d
t

\int _0^1 {\sqrt {(k(t)/(1 - t))\,dt} }

\]
is finite. It is shown that the imposition of a further regularity condition on the growth of

k
k

ensures that in some tangential region the zero set of each function in

A

(
k
)

{A^{(k)}}

is also Blaschke.
On homogeneous polynomials on a complex ball
Authors: Ryll J. , Wojtaszczyk P.
Published: June 2010

We prove that there exist

n
n

-homogeneous polynomials

p
n

{p_n}

on a complex

d
d

-dimensional ball such that

‖

p
n

‖

∞

=
1

{\left \| {{p_n}} \right \|_\infty } = 1

and

‖

p
n

‖

2

⩾

π

2

−
d

{\left \| {{p_n}} \right \|_2} \geqslant \sqrt \pi {2^{- d}}

. This enables us to answer some questions about

H
p

{H_p}

and Bloch spaces on a complex ball. We also investigate interpolation by

n
n

-homogeneous polynomials on a

2
2

-dimensional complex ball.
On wave fronts propagation in multicomponent media
Authors: Freĭdlin M. I.
Published: June 2010

The behavior as

t
→
∞

t \to \infty

of solutions of some parabolic systems of differential equations of the Kolmogorov-Petrovskii-Piskunov type is investigated. The present approach uses the Kac-Feynman formula and estimates on large deviations.
Applications of variational inequalities to the existence theorem on quadrature domains
Authors: Sakai Makoto
Published: June 2010

In this paper we shall study quadrature domains for the class of subharmonic functions. By using the theory of variational inequalities, we shall give a new proof of the existence and uniqueness theorem. As an application, we deal with Hele-Shaw flows with a free boundary and show that their two weak solutions, one of which was defined by the author using quadrature domains and the other was defined by Gustafsson [3] using variational inequalities, are identical with each other.
Cartesian-closed coreflective subcategories of uniform spaces
Authors: Rice M. D. , Tashjian G. J.
Published: June 2010

This paper characterizes the coreflective subcategories

C

\mathcal {C}

of uniform spaces for which a natural function space structure generates the exponential law

X

Y
⊗
Z

=

(

X
Y

)
Z

{X^{Y \otimes Z}} = {({X^Y})^Z}

on

C

\mathcal {C}

. Such categories are cartesian-closed. Specifically, we show that

C

\mathcal {C}

is cartesian-closed in this way if and only if

C

\mathcal {C}

is inductively generated by a finitely productive family of locally fine spaces. The results divide naturally into two cases: those subcategories containing the unit interval are generated by precompact spaces, while the subcategories not containing the unit interval are generated by spaces which admit an infinite cardinal. These results may be used to derive the characterizations of cartesian-closed coreflective subcategories of Tychonoff spaces found in [10].
Primarité de 𝐿^{𝑝}(𝑋)
Authors: Capon Michèle
Published: June 2010

Soit

X
X

un espace de Banach à base symétrique. Nous étudions les opérateurs de

L
p

(
X
)

{L^p}(X)

dans lui-même, en leur associant une "matrice" d’opérateurs de

L
p

{L^p}

. Cette technique nous permet de démontrer que pour tout

p
p

réel dans

[
1
,
+
∞
[
,

l’espace

 

L
p

(
X
)

[1, + \infty [,{\text {l’espace}}\ L^p(X)

est primaire.
Neighborhoods of algebraic sets
Authors: Durfee Alan H.
Published: June 2010

In differential topology, a smooth submanifold in a manifold has a tubular neighborhood, and in piecewise-linear topology, a subcomplex of a simplicial complex has a regular neighborhood. The purpose of this paper is to develop a similar theory for algebraic and semialgebraic sets. The neighborhoods will be defined as level sets of polynomial or semialgebraic functions.
The sharp form of Oleĭnik’s entropy condition in several space variables
Authors: Hoff David
Published: June 2010

We investigate the conditions under which the Volpert-Kruzkov solution of a single conservation law in several space variables with flux

F
F

will satisfy the simplified entropy condition

div

F
′

(
u
)
⩽
1

/

t

\operatorname {div}\,F’(u) \leqslant 1/t

, and when this condition guarantees uniqueness for given

L
∞

{L^\infty }

Cauchy data. We show that, when

F
F

is

C
1

{C^1}

, our condition guarantees uniqueness iff

F
F

is isotropic, and that, for such

F
F

, the Volpert-Kruzkov solution always satisfies our condition.
Axioms for Stiefel-Whitney homology classes of some singular spaces
Authors: Veljan Darko
Published: June 2010

A system of axioms for the Stiefel-Whitney classes of certain type of singular spaces is established. The main examples of these singular spaces are Euler manifolds

mod

2

\text {mod}\, 2

and homology manifolds

mod

2

\text {mod}\, 2

. As a consequence, it is shown that on homology manifolds

mod

2

\text {mod}\, 2

the generalized Stiefel conjecture holds.
Borel functions of bounded class
Authors: Fremlin D. H. , Hansell R. W. , Junnila H. J. K.
Published: June 2010

Let

X
X

and

Y
Y

be metric spaces and

f
:
X
→
Y

f:X \to Y

a Borel measurable function. Does

f
f

have to be of bounded class, i.e. are the sets

f

−
1

[
H
]

{f^{ - 1}}[ H ]

, for open

H
⊆
Y

H \subseteq Y

, of bounded Baire class in

X
?

X?

This is an old problem of A. H. Stone. Positive answers have been given under a variety of extra hypotheses and special axioms. Here we show that (i) unless something similar to a measurable cardinal exists, then

f
f

is of bounded class and (ii) if

f
f

is actually a Borel isomorphism, then

f

(

and

 

f

−
1

)

f\,({\text {and}}\ {f^{ - 1}})

are of bounded class.
Analytic functionals with unbounded carriers and mean periodic functions
Authors: Meril Alex
Published: June 2010

We study certain ideals in some spaces of analytic functionals with unbounded carriers introduced by T. Kawaï, M. Morimoto and J. W. de Roever. Using Banach algebra methods, we show an example of space without spectral synthesis. Using Hörmander’s

L
2

{L^2}

estimates, we prove a spectral synthesis theorem for mean periodic functions.
A new bound for the Steiner ratio
Authors: Du D. Z. , Hwang F. K.
Published: June 2010

Let

V
V

denote a given set of

n
n

points in the euclidean plane. A Steiner minimal tree for

V
V

is the shortest network (clearly, it has to be a tree) interconnecting

V
V

. Junctions of the network which are not in

V
V

are called Steiner points (those in

V
V

will be called regular points). A shortest tree interconnecting

V
V

without using any Steiner points is called a minimal tree. Let

σ
(
V
)

\sigma (V)

and

μ
(
V
)

\mu (V)

denote the lengths of a Steiner minimal tree and a minimal tree, respectively. Define

ρ
\rho

to be the greatest lower bound for the ratio

σ
(
V
)

/

μ
(
V
)

\sigma (V)/\mu (V)

over all

V
V

. We prove

ρ
>
.8

\rho > .8

.
Earthquakes and tessellations of Teichmüller space
Authors: Waterman Peter , Wolpert Scott
Published: June 2010

A brief exposition of the Teichmüller and earthquake theory is given for the once punctured torus. Using these ideas, computer plots, of the tessellation of Teichmüller space by the mapping class group and of the earthquake paths leaving a point, were generated. The algorithms for generating the plots are discussed.
Existence of infinitely many solutions for a forward backward heat equation
Authors: Höllig Klaus
Published: June 2010

Let

ϕ
\phi

be a piecewise linear function which satisfies the condition

s
ϕ
(
s
)
⩾
c

s
2

,
c
>
0
,
s
∈

R

s\phi (s) \geqslant c{s^2},c > 0,s \in {\mathbf {R}}

, and which is monotone decreasing on an interval

(
a
,
b
)
⊂

R

+

(a,b) \subset {{\mathbf {R}}_ + }

. It is shown that for

f
∈

C
2

[
0
,
1
]

f \in {C^2}[0,1]

, with

max

f
′

>
a

\max f^\prime > a

, there exists a

T
>
0

T > 0

such that the initial boundary value problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t Baseline equals phi left-parenthesis u Subscript x Baseline right-parenthesis Subscript x Baseline comma u Subscript x Baseline left-parenthesis 0 comma t right-parenthesis equals u Subscript x Baseline left-parenthesis 1 comma t right-parenthesis equals 0 comma u left-parenthesis dot comma 0 right-parenthesis equals f comma">

u
t

=
ϕ

(

u
x

)
x

,

u
x

(
0
,
t
)
=

u
x

(
1
,
t
)
=
0
,

u
(
⋅
,
0
)
=
f
,

{u_t} = \phi \,{({u_x})_x},\qquad {u_x}(0,t) = {u_x}(1,t) = 0,\qquad u( \cdot ,0) = f,

\]
has infinitely many solutions

u
u

satisfying

∥

u

∥

α

,
∥

u
x

∥

∞

,
∥

u
t

∥

2

⩽
c
(
f
,
ϕ
)

\parallel \;u\;{\parallel _{\alpha }},\parallel \;{u_x}{\parallel _{\infty }},\parallel \;{u_t}{\parallel _{2}} \leqslant c(f,\phi )

on

[
0
,
1
]
×
[
0
,
T
]

[0,1] \times [0,T]

.
An application of asymptotic techniques to certain problems of spectral and scattering theory of Stark-like Hamiltonians
Authors: Ben-Artzi Matania
Published: June 2010

Let

L
0

=
−
Δ
+
V
(

x
1

)
,
L
=

L
0

+

V
p

(
x
)

{L_0} = - \Delta + V({x_1}),L = {L_0} + {V_p}(x)

be selfadjoint in

L
2

(

R
n

)

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

. Here

V
,

V
p

V,{V_p}

are real functions,

V
(

x
1

)

V({x_1})

depends only on the first coordinate. Existence of the wave-operators
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Subscript plus-or-minus Baseline left-parenthesis upper L comma upper L 0 right-parenthesis equals s hyphen limit Underscript t right-arrow plus-or-minus normal infinity Endscripts exp left-parenthesis i t upper L right-parenthesis exp left-parenthesis minus i t upper L 0 right-parenthesis">

W
±

(
L
,

L
0

)
=
s
-

lim

t
→
±
∞

exp
⁡
(
i
t
L
)
exp
⁡
(
−
i
t

L
0

)

{W_ \pm }\,(L,{L_0}) = s \text {-} {\lim _{t \to \pm \infty }}\,\exp (itL)\exp ( - it{L_0})

is proved, using the stationary phase method. For this, an asymptotic technique is applied to the study of

−

d
2

/

d

t
2

+
V
(
t
)

-{d^2}/d{t^2} + V(t)

in

L
2

(
R
)

{L^2}(R)

. Its absolute continuity is proved as well as a suitable eigenfunction expansion.

V
V

is a "Stark-like" potential. In particular, the cases

V
(

x
1

)
=
(
−
sgn
⁡

x
1

)

|

x
1

|

α

,
0
>
α
⩽
2

V({x_1}) = ( - \operatorname {sgn}{x_1})|{x_1}\,{|^\alpha },0 > \alpha \leqslant 2

, are included.

V
p

{V_p}

may be taken as the sum of an

L
2

{L^2}

-function and a function satisfying growth conditions in the

+

x
1

+ {x_1}

direction.

V
p

(
x
)
=

|

x

|

−
1

{V_p}(x) = |x|^{ - 1}

is included.
Group actions on aspherical 𝐴_{𝑘}(𝑁)-manifolds
Authors: Ku Hsü Tung , Ku Mei Chin
Published: June 2010

By an aspherical

A
k

(
N
)

{A_k}(N)

-manifold, we mean a compact connected manifold

M
M

together with a map

f
f

from

M
M

into an aspherical complex

N
N

such that

f

∗

:

H
k

(
N
;
Q
)
→

H
k

(
M
;
Q
)

{f^{\ast }}: H^k(N;Q)\to H^k(M;Q)

is nontrivial. In this paper we shall show that if

S
1

{S^1}

acts effectively and smoothly on a smooth aspherical

A
k

(
N
)

{A_k}(N)

-manifold,

k
>
1

k > 1

,

N
N

a closed oriented Riemannian

k
k

-manifold, with strictly negative curvature, and the

K
K

-degree

K
(
f
)
≠
0

K(f) \ne 0

, then the fixed point set

F
F

is not empty, and at least one component of

F
=

∪

j

F
j

F = { \cup _{j}}{F_j}

is an aspherical

A
k

(
N
)

{A_k}(N)

-manifold. Moreover,

Sign

(
f
)
=

Σ
j

Sign

(
f

|

F
j

)

{\operatorname {Sign}}(f) = {\Sigma _j}\,{\operatorname {Sign}}(f|{F_j})

. We also study the degree of symmetry and semisimple degree of symmetry of aspherical

A
k

(
N
)

{A_k}(N)

-manifolds.
A restriction theorem for semisimple Lie groups of rank one
Authors: Tirao Juan A.
Published: June 2010

Let

g

R

=

f

R

+

p

R

{\mathfrak {g}_{\mathbf {R}}} = {\mathfrak {f}_{\mathbf {R}}} + {\mathfrak {p}_{\mathbf {R}}}

be a Cartan decomposition of a real semisimple Lie algebra

g

R

{\mathfrak {g}_{\mathbf {R}}}

and let

g

=

f

+

p

\mathfrak {g} = \mathfrak {f} + \mathfrak {p}

be the corresponding complexification. Also let

a

R

{\mathfrak {a}_{\mathbf {R}}}

be a maximal abelian subspace of

p

R

{\mathfrak {p}_{\mathbf {R}}}

and let

a

\mathfrak {a}

be the complex subspace of

p

\mathfrak {p}

generated by

a

R

{\mathfrak {a}_{\mathbf {R}}}

. We assume

dim
⁡

a

R

=
1

\dim {\mathfrak {a}_{\mathbf {R}}} = 1

. Now let

G
G

be the adjoint group of

g

\mathfrak {g}

and let

K
K

be the analytic subgroup of

G
G

with Lie algebra

ad

g

(

f

)

{\text {ad}}_\mathfrak {g}(\mathfrak {f})

. If

S
′

(

g

)

S^\prime (\mathfrak {g})

denotes the ring of all polynomial functions on

g

\mathfrak {g}

then clearly

S
′

(

g

)

S^\prime (\mathfrak {g})

is a

G
G

-module and a fortiori a

K
K

-module. In this paper, we determine the image of the subring

S
′

(

g

)
K

S^\prime {(\mathfrak {g})^K}

of

K
K

-invariants in

S
′

(

g

)

S^\prime (\mathfrak {g})

under the restriction map

f
↦
f

|

f

+

a

(
f
∈

S
′

(

g

)
K

)

f \mapsto f{|_{\mathfrak {f} + \mathfrak {a}}}(f \in S^\prime {(\mathfrak {g})^K})

.
Invariant subspaces on Riemann surfaces of Parreau-Widom type
Authors: Hayashi Mikihiro
Published: June 2010

In this paper we generalize Beurling’s invariant subspace theorem to the Hardy classes on a Riemann surface with infinite handles. The problem is to classify all closed (

weak

∗

\text {weak}^{\ast }

closed, if

p
=
∞

p = \infty

)

H
∞

(
d
χ
)

{H^\infty }(d\chi )

-submodules, say

m

\mathfrak {m}

, of

L
p

(
d
χ
)

{L^p}(d\chi )

,

1
⩽
p
⩽
∞

1 \leqslant p \leqslant \infty

, where

d
χ

d\chi

is the harmonic measure on the Martin boundary of a Riemann surface

R
R

, and

H
∞

(
d
χ
)

{H^\infty }(d\chi )

is the set of boundary functions of all bounded analytic functions on

R
R

. Our main result is stated roughly as follows. Let

R
R

be of Parreau-Widom type, that is, the space

H
∞

(
R
,
γ
)

{H^\infty }(R,\gamma )

of bounded analytic sections contains a nonzero element for every complex flat line bundle

γ
∈
π

(
R

)

∗

\gamma \in \pi {(R)^{\ast }}

. We may assume, without loss of generality, that the Green’s function of

R
R

vanishes at the infinity. Set
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m Superscript normal infinity Baseline left-parenthesis gamma right-parenthesis equals sup left-brace StartAbsoluteValue f left-parenthesis bold upper O right-parenthesis EndAbsoluteValue colon f element-of upper H Superscript normal infinity Baseline left-parenthesis upper R comma gamma right-parenthesis comma StartAbsoluteValue f EndAbsoluteValue less-than-or-slanted-equals 1 right-brace">

m
∞

(
γ
)
=
sup
{

|

f
(

O

)

|

:
f
∈

H
∞

(
R
,
γ
)
,

|

f

|

⩽
1
}

{m^\infty }(\gamma ) = \sup \{ |f({\mathbf {O}})|:f \in {H^\infty }(R,\gamma ),|f| \leqslant 1\}

for a fixed point

O

{\mathbf {O}}

of

R
R

. Then, a necessary and sufficient condition in order that every such an

m

\mathfrak {m}

takes either the form

m

=

C
E

L
p

(
d
χ
)

\mathfrak {m} = {C_E}{L^p}(d\chi )

, where

C
E

{C_E}

is the characteristic function of a set

E
E

, or the form

m

=
q

H
p

(
d
χ
,
γ
)

\mathfrak {m} = q{H^p}(d\chi ,\gamma )

, where

|

q

|

=
1

|q| = 1

a.e. and

γ
\gamma

is some element of

π

(
R

)

∗

\pi {(R)^{\ast }}

is that

m
∞

(
γ
)

{m^\infty }(\gamma )

is continuous for the variable

γ
∈
π

(
R

)

∗

\gamma \in \pi {(R)^{\ast }}

.
Some research problems about algebraic differential equations
Authors: Rubel Lee A.
Published: June 2010

Twenty-four new research problems are posed, and their background and partial solutions are sketched. Many of these problems are in the (somewhat unexpected) area of interaction between algebraic differential equations, topology, and mathematical logic.
The word problem for lattice-ordered groups
Authors: Glass A. M. W. , Gurevich Yuri
Published: June 2010

Theorem. There is a finitely generated one relator lattice-ordered group with insoluble (group) word problem.
Maximal abelian subalgebras of von Neumann algebras and representations of equivalence relations
Authors: Sutherland Colin E.
Published: June 2010

We associate to each pair

(

M

,

A

)

(\mathcal {M},\mathcal {A})

, (with

M

\mathcal {M}

a von Neumann algebra, and

A

\mathcal {A}

a maximal abelian subalgebra) a representation

α
\alpha

of the Takesaki equivalence relation

R

(

M

,

A

)

\mathcal {R}\,(\mathcal {M},\mathcal {A})

of

(

M

,

A

)

(\mathcal {M},\mathcal {A})

as automorphisms of a

I

∞

{{\text {I}}_\infty }

factor. Conversely each such representation

α
\alpha

of

R

\mathcal {R}

on

(
X
,
μ
)

(X,\mu )

as automorphisms of

B

(

H

)

\mathcal {B}\,(\mathcal {H})

determines a von Neumann algebra-abelian subalgebra pair

S
′

(

R

,
α
)
=
(

N

,

B

)

S’\,(\mathcal {R},\alpha ) = (\mathcal {N},\mathcal {B})

where

N

\mathcal {N}

is the commutant of the algebra of "self-intertwiners" for

α
\alpha

and

B

=

L
∞

(
X
,
μ
)
⊗
1

\mathcal {B} = {L^\infty }(X,\mu ) \otimes 1

on

L
2

(
X
,
μ
)
⊗

H

{L^2}(X,\mu ) \otimes \mathcal {H}

. The main concern is the assignments
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis script upper M comma script upper A right-parenthesis right-arrow script upper T left-parenthesis script upper M comma script upper A right-parenthesis equals left-parenthesis script upper R left-parenthesis script upper M comma script upper A right-parenthesis comma alpha right-parenthesis">

(

M

,

A

)
→

T

(

M

,

A

)
=
(

R

(

M

,

A

)
,
α
)

(\mathcal {M},\mathcal {A}) \to \mathcal {T}\;(\mathcal {M},\mathcal {A}) = (\mathcal {R}\,(\mathcal {M},\mathcal {A}),\alpha )

and

(

R

,
α
)
→

S
′

(

R

,
α
)

(\mathcal {R},\alpha ) \to S’(\mathcal {R},\alpha )

, and in particular, the extent to which they are inverse to each other—this occurs if

R

\mathcal {R}

is countable nonsingular and

α
\alpha

is (conjugation by) a projective square-integrable representation (cf. [8]), or if

A

\mathcal {A}

is a Cartan subalgebra (cf. [5]), among other cases. A partial dictionary between the representations

(

R

,
α
)

(\mathcal {R},\alpha )

and pairs

(

M

,

A

)

(\mathcal {M},\mathcal {A})

is given—thus if

R

\mathcal {R}

is countable nonsingular and

α
\alpha

is what we term replete,

S
′

(

R

,
α
)

S’(\mathcal {R},\alpha )

is injective whenever

R

\mathcal {R}

is amenable, and a complete Galois theory generalizing that for crossed products by discrete groups is available. We also show how to construct various pathological examples such as a singular maximal abelian subalgebra

A

⊆

M

\mathcal {A} \subseteq \mathcal {M}

for which the Takesaki equivalence relation

R

(

M

,

A

)

\mathcal {R}\,(\mathcal {M},\mathcal {A})

is nontrivial.
The initial trace of a solution of the porous medium equation
Authors: Aronson D. G. , Caffarelli L. A.
Published: June 2010

Let

u
=
u
(
x
,
t
)

u = u(x,t)

be a continuous weak solution of the porous medium equation in

R

d

×
(
0
,
T
)

{{\mathbf {R}}^d} \times (0,T)

for some

T
>
0

T > 0

. We show that corresponding to

u
u

there is a unique nonnegative Borel measure

ρ
\rho

on

R

d

{{\mathbf {R}}^d}

which is the initial trace of

u
u

. Moreover, we show that the initial trace

ρ
\rho

must belong to a certain growth class. Roughly speaking, this growth restriction shows that there are no solutions of the porous medium equation whose pressure grows, on average, more rapidly then

|

x

|

2

|x{|^2}

as

|

x

|

→
∞

|x| \to \infty

.
Linearized stability of extreme shock profiles in systems of conservation laws with viscosity
Authors: Pego Robert L.
Published: June 2010

For a genuinely nonlinear hyperbolic system of conservation laws with added artificial viscosity,

u
t

+
f

(
u

)
x

=
ε

u

x
x

{u_t} + f{(u)_x} = \varepsilon {u_{xx}}

, we prove that traveling wave profiles for small amplitude extreme shocks (the slowest and fastest) are linearly stable to perturbations in initial data chosen from certain spaces with weighted norm; i.e., we show that the spectrum of the linearized equation lies strictly in the left-half plane, except for a simple eigenvalue at the origin (due to phase translations of the profile). The weight

e

c
x

{e^{cx}}

is used in components transverse to the profile, where, for an extreme shock, the linearized equation is dominated by unidirectional convection.
Lower semicontinuity, existence and regularity theorems for elliptic variational integrals of multiple valued functions
Authors: Mattila Pertti
Published: June 2010

Let

A
A

be an open set in

R

m

{{\mathbf {R}}^m}

with compact smooth boundary, and let

Q

{\mathbf {Q}}

be the space of unordered

Q
Q

tuples of points of

R

n

{{\mathbf {R}}^n}

. F. J. Almgren, Jr. has developed a theory for functions

f
:
A
→

Q

f:A \to {\mathbf {Q}}

and used them to prove regularity theorems for area minimizing integral currents. In particular, he has defined in a natural way the space

Y

2

(
A
,

Q

)

{\mathcal {Y}_2}(A,{\mathbf {Q}})

of functions

f
:
A
→

Q

f:A \to {\mathbf {Q}}

with square summable distributional partial derivatives and the Dirichlet integral

Dir
⁡
(
f
;
A
)

\operatorname {Dir}(f;A)

of such functions. In this paper we study more general constant coefficient quadratic integrals

G

(
f
;
A
)

{\mathbf {G}}(f;A)

which are

Q
Q

elliptic in the sense that there is

c
>
0

c > 0

such that

G

(
f
;
A
)
⩾
c

Dir
⁡
(
f
;
A
)

{\mathbf {G}}(f;A) \geqslant c\,\operatorname {Dir}(f;A)

for

f
∈

Y

2

(
A
;

Q

)

f \in {\mathcal {Y}_2}(A;{\mathbf {Q}})

with zero boundary values. We prove a lower semicontinuity theorem which leads to the existence of a

G

{\mathbf {G}}

minimizing function with given reasonable boundary values. In the case

m
=
2

m = 2

we also show that such a function is Hölder continuous and regular on an open dense set. In the case

m
⩾
3

m \geqslant 3

the regularity problem remains open.
Invariant theory and the lambda algebra
Authors: Singer William M.
Published: June 2010

Let

A
A

be the Steenrod algebra over the field

F
2

{F_2}

. In this paper we construct for any left

A
A

-module

M
M

a chain complex whose homology groups are isomorphic to the groups

Tor
s
A

⁡
(

F
2

,
M
)

\operatorname {Tor}_s^A({F_2},M)

. This chain complex in homological degree

s
s

is built from a ring of invariants associated with the action of the linear group

G

L
s

(

F
2

)

G{L_s}({F_2})

on a certain algebra of Laurent series. Thus, the homology of the Steenrod algebra (and so the Adams spectral sequence for spheres) is seen to have a close relationship to invariant theory. A key observation in our work is that the Adem relations can be described in terms of the invariant theory of

G

L
2

(

F
2

)

G{L_2}({F_2})

. Our chain complex is not new: it turns out to be isomorphic to the one constructed by Kan and his coworkers from the dual of the lambda algebra. Thus, one effect of our work is to give an invariant-theoretic interpretation of the lambda algebra. As a consequence we find that the dual of lambda supports an action of the Steenrod algebra that commutes with the differential. The differential itself appears as a kind of "residue map". We are also able to describe the coalgebra structure of the dual of lambda using our invariant-theoretic language.
Systems of conservation laws with invariant submanifolds
Authors: Temple Blake
Published: June 2010

Systems of conservation laws with coinciding shock and rarefaction curves arise in the study of oil reservoir simulation, multicomponent chromatography, as well as in the study of nonlinear motion in elastic strings. Here we characterize this phenomenon by deriving necessary and sufficient conditions on the geometry of a wave curve in order that the shock wave curve coincide with its associated rarefaction wave curve for a system of conservation laws. This coincidence is the one dimensional case of a submanifold of the state variables being invariant for the system of equations, and the necessary and sufficient conditions are derived for invariant submanifolds of arbitrary dimension. In the case of

2
×
2

2 \times 2

systems we derive explicit formulas for the class of flux functions that give rise to the coupled nonlinear conservation laws for which the shock and rarefaction wave curves coincide.
Uniqueness of torsion free connection on some invariant structures on Lie groups
Authors: Boyom Michel Nguiffo , Giraud Georges
Published: June 2010

Let

G

\mathcal {G}

be a connected Lie group with Lie algebra

g

\mathfrak {g}

. Let

Int
⁡
(

g

)

\operatorname {Int}(\mathfrak {g})

be the group of inner automorphisms of

g

\mathfrak {g}

. The group

G

\mathcal {G}

is naturally equipped with

Int
⁡
(

g

)

\operatorname {Int}(\mathfrak {g})

-reductions of the bundle of linear frames on

G

\mathcal {G}

. We investigate for what kind of Lie group the

0
0

-connection of E. Cartan is the unique torsion free connection adapted to any of those

Int
⁡
(

g

)

\operatorname {Int}(\mathfrak {g})

-reductions.
On some subalgebras of a von Neumann algebra crossed product
Authors: Solel Baruch
Published: June 2010

We study conditions for a nonselfadjoint subalgebra of a von Neumann crossed product

L

\mathcal {L}

to be an algebra of analytic operators with respect to a flow on

L

\mathcal {L}

. We restrict ourselves to the case where

L

\mathcal {L}

is constructed from a finite von Neumann algebra

M
M

with a trace preserving

∗

^{\ast }

-automorphism

α
\alpha

that acts ergodically on the center of

M
M

.
Approximate subdifferentials and applications. I. The finite-dimensional theory
Authors: Ioffe A. D.
Published: June 2010

We introduce and study a new class of subdifferentials associated with arbitrary functions. Among the questions considered are: connection with other derivative-like objects (e.g. derivatives, convex subdifferentials, generalized gradients of Clarke and derivate containers of Warga), calculus of approximate subdifferentials and applications to analysis of set-valued maps and to optimization. It turns out that approximate subdifferentials are minimal (as sets) among other conceivable subdifferentials satisfying some natural requirements. This shows that certain results involving approximate subdifferentials are the best possible and, at the same time, marks certain limitations of nonsmooth analysis. Another important property of approximate subdifferentials is that, being essentially nonconvex, they admit a rich calculus that covers the calculus of convex subdifferentials and leads to more precise and sometimes new results for generalized gradients of Clarke.
Generalized Hua-operators and parabolic subgroups. The cases of 𝑆𝐿(𝑛,𝐶) and 𝑆𝐿(𝑛,𝑅)
Authors: Johnson Kenneth D.
Published: June 2010

Suppose

G
=

SL

(
n
,

C

)

G = {\text {SL}}(n,{\mathbf {C}})

or

SL

(
n
,

R

)

{\text {SL}}(n,{\mathbf {R}})

and

K
K

is a maximal compact subgroup of

G
G

. If

P
P

is any parabolic subgroup of

G
G

, we determine a system of differential equations on

G

/

K

G/K

with the property that any function on

G

/

K

G/K

satisfies these differential equations if and only if it is the Poisson integral of a hyperfunction on

G

/

P

G/P

.
Supercompactness of compactifications and hyperspaces
Authors: Bell Murray G.
Published: June 2010

We prove a theorem which implies that if

γ
ω

\gamma \omega

is a supercompact compactification of the countable discrete space

ω
\omega

then

γ
ω
−
ω

\gamma \omega - \omega

is separable. This improves an earlier result of the author’s that such a

γ
ω

\gamma \omega

must have

γ
ω
−
ω

ccc

\gamma \omega - \omega \;{\text {ccc}}

. We prove a theorem which implies that the hyperspace of closed subsets of

2

ω
2

{2^{\omega _2}}

is not a continuous image of a supercompact space. This improves an earlier result of

L

{\text {L}}

. Šapiro that the hyperspace of closed subsets of

2

ω
2

{2^{\omega _2}}

is not dyadic.
Endomorphisms of the cohomology of complex Grassmannians
Authors: Hoffman Michael
Published: June 2010

For any complex Grassmann manifold

G
G

, we classify all endomorphisms of the rational cohomology ring of

G
G

which are nonzero on dimension

2
2

. Some applications of this result are given.
Invariant densities for random maps of the interval
Authors: Pelikan S.
Published: June 2010

A random map is a discrete time process in which one of a number of functions is selected at random and applied. Here we study random maps of

[
0
,
1
]

[0,1]

which represent dynamical systems on the square

[
0
,
1
]
×
[
0
,
1
]

[0,1] \times [0,1]

. Sufficient conditions for a random map to have an absolutely continuous invariant measure are given, and the number of ergodic components of a random map is discussed.
A property of complete minimal surfaces
Authors: Hasanis Thomas , Koutroufiotis Dimitri
Published: June 2010

If

M
M

is a complete minimal surface in

R
n

{R^n}

, we denote by

W
W

the set of points in

R
n

{R^n}

that do not lie on any tangent plane of

M
M

. By taking a point in

W
W

as origin, the position vector of

M
M

determines a global unit normal vector field

e
e

to

M
M

. We prove that if

e
e

is a minimal section, then

M
M

is a plane. In particular, the set of tangent planes of a nonflat complete minimal surface in

R
3

{R^3}

covers all

R
3

{R^3}

. We also prove a similar result for a complete minimal surface

M
M

in

S
3

{S^3}

, and deduce from it that if the spherical image of

M
M

lies in a closed hemisphere, then

M
M

is a great

S
2

{S^2}

.
The number of factorizations of numbers less than 𝑥 into divisors greater than 𝑦
Authors: Hensley Douglas
Published: June 2010

Let

A
(
x
,

y
)

A(x,\,y)

be the number in the title. There is a function

h
:
[
0
,

∞
)
→
[
0
,

2
]

h:[0,\,\infty ) \to [0,\,2]

, decreasing and convex, with

h
(
0
)
=
2

h(0) = 2

and

lim

r
→
∞

⁡
h
(
r
)
=
0

\operatorname {lim}_{r \to \infty }h(r) = 0

, such that if

r
=
log
⁡
y

/

log

x

r = \operatorname {log} y/\sqrt {\operatorname {log} \,x}

then as

x
→
∞

x \to \infty

with

r
r

fixed,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis x comma y right-parenthesis equals StartFraction upper C left-parenthesis r right-parenthesis x exp left-parenthesis h left-parenthesis r right-parenthesis StartRoot log x EndRoot right-parenthesis Over left-parenthesis log x right-parenthesis Superscript 3 slash 4 Baseline EndFraction left-parenthesis 1 plus upper O left-parenthesis log x right-parenthesis Superscript negative 1 slash 4 Baseline right-parenthesis">

A
(
x
,
y
)
=

C
(
r
)
x

exp
⁡

(

h
(
r
)

log

x

)

(
log

x
)

3

/

4

(

1
+
O

(
log

x
)

−
1

/

4

)

A(x,y) = \frac {{C(r)x\,\operatorname {exp} \left ( {h(r)\sqrt {\operatorname {log} \,x} } \right )}}{{{{(\operatorname {log} \,x)}^{3/4}}}}\left ( {1 + O{{(\operatorname {log} \,x)}^{ - 1/4}}} \right )

\]
. The estimate is uniform on intervals

0
>
r
≤

R
0

0 > r \leq {R_0}

. As corollaries we have for

log

y
=
θ

(
log

x

)

1

/

4

\operatorname {log} \,y = \theta {(\operatorname {log} \,x)^{1/4}}

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript x right-arrow normal infinity Endscripts StartFraction upper A left-parenthesis x comma y right-parenthesis Over upper A left-parenthesis x comma 1 right-parenthesis slash y EndFraction equals e Superscript theta squared slash 2">

lim

x
→
∞

A
(
x
,

y
)

A
(
x
,

1
)

/

y

=

e

θ
2

/

2

\lim \limits _{x \to \infty } \,\frac {{A(x,\,y)}} {{A(x,\,1)/y}} = {e^{{\theta ^2}/2}}

\]
,and if

log

y
=
o

(
log

x

)

1

/

4

\operatorname {log} \,y = o {(\operatorname {log} \,x)^{1/4}}

then

A
(
x
,

y
)
≈
A
(
x
,

1
)

/

y

A(x,\,y) \approx A(x,\,1)/y

.
On specializations of curves. I
Authors: Nobile A.
Published: June 2010

The following is proved: Given a family of projective reduced curves

X
→
T

X \to T

(

T
T

irreducible), if

X
t

{X_t}

(the general curve) is integral and

X
0

{X_0}

is a special curve (having irreducible components

X
1

,
…
,

X
r

{X_1}, \ldots ,{X_r}

), then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript i equals 1 Overscript r Endscripts g Subscript i Baseline left-parenthesis upper X Subscript i Baseline right-parenthesis less-than-or-slanted-equals g left-parenthesis upper X Subscript t Baseline right-parenthesis">

∑

i
=
1

r

g
i

(

X
i

)
⩽
g
(

X
t

)

\sum \nolimits _{i = 1}^r {{g_i}({X_i}) \leqslant g({X_t})}

, where

g
(
Z
)
=

g(Z) =

geometric genus of

Z
Z

. Conversely, if

A
A

is a reduced plane projective curve, of degree

n
n

with irreducible components

X
1

,
…
,

X
r

{X_1}, \ldots ,{X_r}

, and

g
g

satisfies
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript i equals 1 Overscript r Endscripts g Subscript i Baseline left-parenthesis upper X Subscript i Baseline right-parenthesis less-than-or-slanted-equals g less-than-or-slanted-equals one half left-parenthesis n minus 1 right-parenthesis left-parenthesis n minus 2 right-parenthesis">

∑

i
=
1

r

g
i

(

X
i

)
⩽
g
⩽

1
2

(
n
−
1
)
(
n
−
2
)

\sum \nolimits _{i = 1}^r {{g_i}({X_i}) \leqslant g \leqslant \frac {1} {2}(n - 1)(n - 2)}

, then a family of plane curves

X
→
T

X \to T

(with

T
T

integral) exists, where for some

t
0

∈
T
,

X

t
0

=
Z

{t_0} \in T,{X_{{t_0}}} = Z

and for

t
t

generic,

X
t

{X_t}

is integral and has only nodes as singularities. Results of this type appear in an old paper by G. Albanese, but the exposition is rather obscure.
Structural stability of equivariant vector fields on two-manifolds
Authors: dos Reis G. L.
Published: June 2010

A class of vector fields on two-dimensional manifolds equivariant under the action of a compact Lie group is defined. Properties of openness, structural ability, and density are proved.
On the arithmetic and homology of algebras of linear type
Authors: Herzog J. , Simis A. , Vasconcelos W. V.
Published: June 2010

Three modifications of the symmetric algebra of a module are introduced and their arithmetical and homological properties studied. Emphasis is placed on converting syzygetic properties of the modules into ideal theoretic properties of the algebras, e.g. Cohen-Macaulayness, factoriality. The main tools are certain Fitting ideals of the module and an extension to modules of a complex of not necessarily free modules that we have used in studying blowing-up rings.
Filters and the weak almost periodic compactification of a discrete semigroup
Authors: Berglund John F. , Hindman Neil
Published: June 2010

The weak almost periodic compactification of a semigroup is a compact semitopological semigroup with certain universal properties relative to the original semigroup. It is not, in general, a topological compactification. In this paper an internal construction of the weak almost periodic compactification of a discrete semigroup is constructed as a space of filters, and it is shown that for discrete semigroups, the compactification is usually topological. Other results obtained on the way to the main one include descriptions of weak almost periodic functions on closed subsemigroups of topological groups, conditions for functions on the additive natural numbers or on the integers to be weak almost periodic, and an example to show that the weak almost periodic compactification of the natural numbers is not the closure of the natural numbers in the weak almost periodic compactification of the integers.
Actions of finite groups on homotopy 3-spheres
Authors: Feighn M. E.
Published: June 2010

It is conjectured that the action of a finite group of diffeomorphisms of the

3
3

-sphere is equivariantly diffeomorphic to a linear action. This conjecture is verified if both of the following conditions hold: (i) Each isotropy subgroup is dihedral or cyclic. (ii) There is a point with cyclic isotropy subgroup of order not

1
,
2
,
3

1,2,3

or

5
5

.
Complexity of subcases of Presburger arithmetic
Authors: Scarpellini Bruno
Published: June 2010

We consider formula subclasses of Presburger arithmetic which have a simple structure in one sense or the other and investigate their computational complexity. We also prove some results on the lower bounds of lengths of formulas which are related to questions on quantifier elimination.
Moments of balanced measures on Julia sets
Authors: Barnsley M. F. , Harrington A. N.
Published: June 2010

By a theorem of S. Demko there exists a balanced measure on the Julia set of an arbitrary nonlinear rational transformation on the Riemann sphere. It is proved here that if the transformation admits an attractive or indifferent cycle, then there is a point with respect to which all the moments of a balanced measure exist; moreover, these moments can be calculated exactly. An explicit balanced measure is exhibited in an example where the Julia set is the whole sphere and for which the moments, with respect to any point, do not all exist.
Universal families for conull FK spaces
Authors: Snyder A. K.
Published: June 2010

This paper considers the problem of determining a useful family of sequence spaces which is universal for conull

FK
\text {FK}

spaces in the following sense: An

FK
\text {FK}

space is conull if and only if it contains a member of the family. In the equivalent context of weak wedge spaces, an appropriate family of subspaces of boundedness domains

m
A

{m_A}

of matrices is shown to be universal. Most useful is the fact that the members of this family exhibit unconditional sectional convergence. The latter phenomenon is known for wedge spaces. Another family of spaces which is universal for conull spaces among semiconservative spaces is provided. The spaces are designed to simplify gliding humps arguments. Improvements are thereby obtained for some pseudoconull type theorems of Bennett and Kalton. Finally, it is shown that conull spaces must contain pseudoconull

BK
\text {BK}

algebras.
Twistor CR manifolds and three-dimensional conformal geometry
Authors: LeBrun Claude R.
Published: June 2010

A

CR
\text {CR}

(i.e. partially complex)

5
5

-manifold is contructed as a sphere bundle over an arbitrary

3
3

-manifold with conformal metric. This so-called twistor

C
R

CR

manifold is show to capture completely the original geometry, and necessary and sufficient conditions are given for an abstract

CR
\text {CR}

manifold to arise via the construction. The above correspondence is then used to prove that a twistor

CR
\text {CR}

manifold is locally imbeddable as a real hypersurface in

C

3

{{\mathbf {C}}^3}

only if it is real-analytic with respect to a suitable atlas.
On positive solutions of some pairs of differential equations
Authors: Dancer E. N.
Published: June 2010

In this paper, we discuss the existence of solutions, with both components positive, of a Dirichlet problem for a coupled pair of partial differential equations. The main result is proved by using degree theory in cones. We also discuss the asymptotic behaviour of solutions as a parameter tends to zero.
Ramsey games
Authors: Hajnal A. , Nagy Zs.
Published: June 2010

The paper deals with game-theoretic versions of the partition relations

α
→
(
β

)
2

>
τ

\alpha \to (\beta )_2^{ > \tau }

and

α
→
(
β

)
2
τ

\alpha \to (\beta )_2^\tau

introduced in [2]. The main results are summarized in the Introduction.
Necessary and sufficient conditions for oscillations of higher order delay differential equations
Authors: Ladas G. , Sficas Y. G. , Stavroulakis I. P.
Published: June 2010

Consider the

n

th

n{\text {th}}

order delay differential equation (1)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis t right-parenthesis plus left-parenthesis negative 1 right-parenthesis Superscript n plus 1 Baseline sigma-summation Underscript i equals 0 Overscript k Endscripts p Subscript i Baseline x left-parenthesis t minus tau Subscript i Baseline right-parenthesis equals 0 comma t greater-than-or-equal-to t 0 comma">

x

(
n
)

(
t
)
+

(
−
1

)

n
+
1

∑

i
=
0

k

p
i

x
(
t
−

τ
i

)
=
0
,

t
≥

t
0

,

{x^{(n)}}(t) + {( - 1)^{n + 1}}\sum \limits _{i = 0}^k {{p_i}x(t - {\tau _i}) = 0, \qquad t \geq {t_0}},

\]
where the coefficients and the delays are constants such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 equals tau 0 greater-than tau 1 greater-than midline-horizontal-ellipsis greater-than tau Subscript k Baseline semicolon p 0 greater-than-or-equal-to 0 comma p Subscript i Baseline greater-than 0 comma i equals 1 comma 2 comma ellipsis comma k semicolon k greater-than-or-equal-to 1">

0
=

τ
0

>

τ

1

>
⋯
>

τ
k

;

p
0

≥
0
,

p
i

>
0
,
i
=
1
,
2
,
…
,
k
;
k
≥
1

0 = {\tau _0} > {\tau _{1}}\, > \cdots > {\tau _k};{p_0}\, \geq 0,{p_i} > 0,i = 1,2,\ldots ,k;k \geq 1

and

n
≥
1

n \geq 1

. The characteristic equation of (1) is (2)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Superscript n Baseline plus left-parenthesis negative 1 right-parenthesis Superscript n plus 1 Baseline sigma-summation Underscript i equals 0 Overscript k Endscripts p Subscript i Baseline e Superscript minus lamda tau Super Subscript i Superscript Baseline equals 0 period">

λ
n

+

(
−
1

)

n
+
1

∑

i
=
0

k

p
i

e

−
λ

τ
i

=
0

.

{\lambda ^n} + {( - 1)^{n + 1}}\;\sum \limits _{i = 0}^k {{p_i}{e^{ - \lambda {\tau _i}}} = 0}.

\]
We prove the following theorem. Theorem. (i) For

n
n

odd every solution of (1) oscillates if and only if (2) has no real roots. (ii) For

n
n

even every bounded solution of (1) oscillates if and only if (2) has no real roots in

(
−
∞
,
0
]

( - \infty ,0]

. The above results have straightforward extensions for advanced differential equations.
Factorizing the polynomial of a code
Authors: Hansel G. , Perrin D. , Reutenauer C.
Published: June 2010

We give an extension and a simplified presentation of a theorem of Schützenberger. This theorem describes the factorization of the commutative polynomial associated with a finite maximal code. It is the deepest result known so far in the theory of (variable-length) codes.
Riesz decompositions in Markov process theory
Authors: Getoor R. K. , Glover J.
Published: June 2010

Riesz decompositions of excessive measures and excessive functions are obtained by probabilistic methods without regularity assumptions. The decomposition of excessive measures is given for Borel right processes. The results for excessive functions are formulated within the framework of weak duality. These results extend and generalize the pioneering work of Hunt in this area.
The theory of 𝐺^{∞}-supermanifolds
Authors: Boyer Charles P. , Gitler Samuel
Published: June 2010

A theory of supermanifolds is developed in which a supermanifold is an ordinary manifold associated with a certain integrable second order

G
G

-structure. A structure theorem is proved showing that every

G
∞

{G^\infty }

-supermanifold has a complete distributive lattice of foliations with flat affine leaves. Furthermore, an existence and uniqueness theorem for local flows of

G
∞

{G^\infty }

vector fields is proved.
Compressed algebras: Artin algebras having given socle degrees and maximal length
Authors: Iarrobino Anthony
Published: June 2010

J. Emsalem and the author showed in [18] that a general polynomial

f
f

of degree

j
j

in the ring

R

=
k
[

y
1

,
…
,

y
r

]

\mathcal {R} = k[ {{y_1},\ldots ,{y_r}} ]

has

(

j
+
r
−
1

r
−
1

)

\left ( {\begin {array}{*{20}{c}} {j + r - 1} \\ {r - 1} \\ \end {array} } \right )

linearly independent partial derivates of order

i
i

, for

i
=
0
,
1
,
…
,
t
=
[

j

/

2

]

i = 0,1,\ldots ,t = [ {j/2} ]

. Here we generalize the proof to show that the various partial derivates of

s
s

polynomials of specified degrees are as independent as possible, given the room available. Using this result, we construct and describe the varieties

G
(
E
)

G(E)

and

Z
(
E
)

Z(E)

parametrizing the graded and nongraded compressed algebra quotients

A
=
R

/

I

A = R/I

of the power series ring

R
=
k
[
[

x
1

,
…
,

x
r

]
]

R = k[[{x_1},\ldots ,{x_r}]]

, having given socle type

E
E

. These algebras are Artin algebras having maximal length

dim
⁡

k

A

\dim {_{k}}A

possible, given the embedding degree

r
r

and given the socle-type sequence

E
=
(

e
1

,
…
,

e
s

)

E = ({e_1},\ldots ,{e_s})

, where

e
i

{e_i}

is the number of generators of the dual module

A
^

\hat A

of

A
A

, having degree

i
i

. The variety

Z
(
E
)

Z(E)

is locally closed, irreducible, and is a bundle over

G
(
E
)

G(E)

, fibred by affine spaces

A

N

{{\mathbf {A}}^N}

whose dimension is known. We consider the compressed algebras a new class of interesting algebras and a source of examples. Many of them are nonsmoothable—have no deformation to

(
k
+
⋯
+
k
)

(k + \cdots + k)

—for dimension reasons. For some choices of the sequence

E
,

D

E,{\text {D}}

. Buchsbaum,

D

{\text {D}}

. Eisenbud and the author have shown that the graded compressed algebras of socle-type

E
E

have almost linear minimal resolutions over

R
R

, with ranks and degrees determined by

E
E

. Other examples have given type

e
=

dim
k

(

socle

A
)

e = {\dim _k}\;({\text {socle}}\;A)

and are defined by an ideal

I
I

with certain given numbers of generators in

R
=
k
[
[

x
1

,
…

,

x
r

]
]

R = k[[{x_1},\ldots \;,{x_r}]]

. An analogous construction of thin algebras

A
=
R

/

(

f
1

,
…
,

f
s

)

A = R/({f_1},\ldots ,{f_s})

of minimal length given the initial degrees of

f
1

,
…
,

f
s

{f_1},\ldots ,{f_s}

is compared to the compressed algebras. When

r
=
2

r = 2

, the thin algebras are characterized and parametrized, but in general when

r
>
3

r > 3

, even their length is unknown. Although

k
=

C

k = {\mathbf {C}}

through most of the paper, the results extend to characteristic

p
p

.
On certain Boolean algebras 𝒫(𝜔)/ℐ
Authors: Just Winfried , Krawczyk Adam
Published: June 2010

We consider possible isomorphisms between algebras of the form

P

(
ω
)

/

I

\mathcal {P}(\omega )/I

, assuming

C
H

{\rm {CH}}

. In particular, the solution of a problem of Erdös and Ulam is given. We include some remarks on the completeness of such algebras.
On Kummer’s twenty-four solutions of the hypergeometric differential equation
Authors: Dwork B.
Published: June 2010

The

p
p

-adic analyticity of the Boyarsky matrix associated with the hypergeometric function

F
(
a
,
b
;
c
;
x
)

F(a,b;c;x)

has been investigated in an earlier article. The transformation of this matrix under translation of

(
a
,
b
,
c
)

(a, b, c)

by

Z

3

{{\bf {Z}}^3}

was determined at that time. This article gives the transformation of this matrix under the extended Kummer group. The

p
p

-adic implications of quadratic and higher-degree transformations remain open.