bluuebunny/crossref_metadata_2025 huggingface dataset:
A finiteness condition on regular local overrings of a local domain
Authors: Johnston Bernard
Published: June 2010

The local factorization theorem of Zariski and Abhyankar implies that between a given pair of

2
2

-dimensional regular local rings,

S
⊇
R

S \supseteq R

, having the same quotient field, every chain of regular local rings must be finite. It is shown in this paper that this property extends to every such pair of regular local rings, regardless of dimension. An example is given to show that this does not hold if "regular" is replaced by "Cohen-Macaulay," by "normal," or by "rational singularity." More generally, it is shown that the set

R

(
R
)

\mathcal {R}(R)

of

n
n

dimensional regular local rings birationally dominating a given

n
n

-dimensional local domain,

R
R

, and ordered by containment, satisfies the descending chain condition. An example is given to show that if

R
R

is regular the two examples of minimal elements of

R

(
R
)

\mathcal {R}(R)

given by J. Sally do not exhaust the set of minimal elements of

R

(
R
)

\mathcal {R}(R)

.
Polynomial invariants of graphs
Authors: Negami Seiya
Published: June 2010

We define two polynomials

f
(
G
)

f(G)

and

f

∗

(
G
)

{f^{\ast }}(G)

for a graph

G
G

by a recursive formula with respect to deformation of graphs. Analyzing their various properties, we shall discuss when two graphs have the same polynomials.
Counting cycles in permutations by group characters, with an application to a topological problem
Authors: Jackson D. M.
Published: June 2010

The character theory of the symmetric group is used to derive properties of the number of permutations, with

k
k

cycles, which are expressible as the product of a full cycle with an element of an arbitrary, but fixed, conjugacy class. For the conjugacy class of fixed point free involutions, this problem has application to the analysis of singularities in surfaces.
On the canonical element conjecture
Authors: Dutta Sankar P.
Published: June 2010

The canonical element conjecture is proved in the following two cases: (i) depth

A
=
dim
⁡
A
−
1

A = \dim A - 1

,

H
m

n
−
1

(
A
)

H_m^{n - 1}(A)

is decomposable; (ii) depth

A
=
dim
⁡
A
−
1

A = \dim A - 1

,

H
m

n
−
1

(
A

)
∨

H_m^{n - 1}{(A)^ \vee }

is cyclic. The equivalence of the C.E.C. and the improved new intersection theorem is also established.
Univalent harmonic functions
Authors: Hengartner W. , Schober G.
Published: June 2010

Several families of complex-valued, univalent, harmonic functions are studied from the point of view of geometric function theory. One class consists of mappings of a simply-connected domain onto an infinite horizontal strip with a normalization at the origin. Extreme points and support points are determined, as well as sharp estimates for Fourier coefficients and distortion theorems. Next, mappings in

|
z
|

>
1

\left | z \right | > 1

are considered that leave infinity fixed. Some coefficient estimates, distortion theorems, and covering properties are obtained. For such mappings with real boundary values, many extremal problems are solved explicitly.
A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations
Authors: Kavian O.
Published: June 2010

We consider solutions to

i

u
t

=
Δ
u
+

|
u
|

p
−
1

u

i{u_t} = \Delta u + {\left | u \right |^{p - 1}}u

,

u
(
0
)
=

u
0

u(0) = {u_0}

, where

x
x

belongs to a smooth domain

Ω
⊂

R

N

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

, and we prove that under suitable conditions on

p
p

,

N
N

and

u
0

∈

H
2

(
Ω
)
∩

H
0
1

(
Ω
)

{u_0} \in {H^2}(\Omega ) \cap H_0^1(\Omega )

,

‖

∇
u
(
t
)

‖

L
2

{\left \| {\nabla u(t)} \right \|_{{L^2}}}

blows up in finite time. The range of

p
p

’s for which blowing-up occurs depends on whether

Ω
\Omega

is starshaped or not. Examples of blowing-up under Neuman or periodic boundary conditions are given. On considère des solutions de

i

u
t

=
Δ
u
+

|
u
|

p
−
1

u

i{u_t} = \Delta u + {\left | u \right |^{p - 1}}u

,

u
(
0
)
=

u
0

u(0) = {u_0}

, où la variable d’espace

x
x

appartient à un domaine régulier

Ω
⊂

R

N

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

, et on prouve que sous des conditions adéquates sur

p
p

,

N
N

et

u
0

∈

H
2

(
Ω
)
∩

H
0
1

(
Ω
)

{u_0} \in {H^2}(\Omega ) \cap H_0^1(\Omega )

,

‖

∇
u
(
t
)

‖

L
2

{\left \| {\nabla u(t)} \right \|_{{L^2}}}

explose en temps fini. Les valeurs de

p
p

pour lesquelles l’explosion a lieu dépend de la forme de l’ouvert

Ω
\Omega

(en fait

Ω
\Omega

étoilé ou non). On donne également des exemples d’explosion sous des conditions de Neuman ou périodiques au bord.
Integration on noncompact supermanifolds
Authors: Rothstein Mitchell J.
Published: June 2010

We note that the Berezin integral, which is ill-defined for noncompact supermanifolds, is a distribution with support on the underlying manifold. This leads to the discovery of correction terms in the Berezinian transformation law and thereby eliminates the boundary ambiguities.
The Szegő projection: Sobolev estimates in regular domains
Authors: Boas Harold P.
Published: June 2010

The Szegö projection preserves global smoothness in weakly pseudoconvex domains that are regular in the sense of Diederich, Fornæss, and Catlin. It preserves local smoothness near boundary points of finite type.
A global approach to the Rankin-Selberg convolution for 𝐺𝐿(3,𝑍)
Authors: Friedberg Solomon
Published: June 2010

We discuss the Rankin-Selberg convolution on

GL
⁡
(
3
,

Z

)

\operatorname {GL} (3,{\mathbf {Z}})

in the ‘classical’ language of symmetric spaces and automorphic forms.
Periodic phenomena in the classical Adams spectral sequence
Authors: Mahowald Mark , Shick Paul
Published: June 2010

We investigate certain periodic phenomena in the classical Adams sepctral sequence which are related to the polynomial generators

ν
n

{\nu _n}

in

π

∗

(
BP
)

{\pi _{\ast }}(\operatorname {BP} )

. We define the notion of a class

a
a

in

Ext
A

(

Z

/

2
,

Z

/

2
)

{\operatorname {Ext} _A}({\mathbf {Z}}/2,{\mathbf {Z}}/2)

being

ν
n

{\nu _n}

-periodic or

ν
n

{\nu _n}

-torsion and prove that classes that are

ν
n

{\nu _n}

-torsion are also

ν
k

{\nu _k}

-torsion for all

k
k

such that

0
⩽
k
⩽
n

0 \leqslant k \leqslant n

. This allows us to define a chromatic filtration of

Ext
A

(

Z

/

2
,

Z

/

2
)

{\operatorname {Ext} _A}({\mathbf {Z}}/2,{\mathbf {Z}}/2)

paralleling the chromatic filtration of the Novikov spectral sequence

E
2

{E_2}

-term given in [13].
Stability results for a diffusion equation with functional drift approximating a chemotaxis model
Authors: Greenberg James M. , Alt Wolfgang
Published: June 2010

A hyperbolic-parabolic "chemotaxis" system modelling aggregation of motile cells by production of a diffusible chemoattractant, is approximated by a scalar diffusion equation for the cell density, where the drift term is an explicit functional of the current density profile. We prove the unique existence and, using the Hopf-Cole transformation, the local stability of an equilibrium, i.e. a steady aggregation state. We also discuss the limiting hyperbolic case of vanishing random motility with the formation of shocks describing cell clumps.
Area and Hausdorff dimension of Julia sets of entire functions
Authors: McMullen Curt
Published: June 2010

We show the Julia set of

λ
sin
⁡
(
z
)

\lambda \sin (z)

has positive area and the action of

λ
sin
⁡
(
z
)

\lambda \sin (z)

on its Julia set is not ergodic; the Julia set of

λ
exp
⁡
(
z
)

\lambda \exp (z)

has Hausdorff dimension two but in the presence of an attracting periodic cycle its area is zero.
T-degrees, jump classes, and strong reducibilities
Authors: Downey R. G. , Jockusch C. G.
Published: June 2010

It is shown that there exist r.e. degrees other than 0 and

0

′

\mathbf {0}’

which have a greatest r.e.

1
1

-degree. This solves an old question of Rogers and Jockusch. We call such degrees

1
1

-topped. We show that there exist incomplete

1
1

-topped degrees above any low r.e. degree, but also show that no nonzero low degree is

1
1

-topped. It then follows by known results that all incomplete

1
1

-topped degrees are low

2

_{2}

but not low. We also construct cappable nonzero

1
1

-topped r.e. degrees and examine the relationships between

1
1

-topped r.e. degrees and high r.e. degrees. Finally, we give an analysis of the “local” relationships of r.e. sets under various strong reducibilities. In particular, we analyze the structure of r.e.

wtt-

{\text {wtt-}}

and

tt

{\text {tt}}

-degrees within a single r.e.

T

{\text {T}}

-degree. We show, for instance, that there is an r.e. degree which contains a greatest r.e.

wtt-

{\text {wtt-}}

-degree and a least r.e.

tt

{\text {tt}}

-degree yet does not consist of a single r.e.

wtt

{\text {wtt}}

-degree. This depends on a new construction of a nonzero r.e.

T

{\text {T}}

-degree with a least

tt

{\text {tt}}

-degree, which proves to have several further applications.
Prime ideals in polycyclic crossed products
Authors: Passman D. S.
Published: June 2010

In this paper, we describe the prime ideals

P
P

in crossed products

R
∗
G

R \ast G

with

R
R

a right Noetherian ring and with

G
G

a polycyclic-by-finite group. This is achieved through a series of reductions. To start with, we may assume that

P
∩
R
=
0

P \cap R = 0

so that

R
R

is a

G
G

-prime ring. The first step uses a technique of M. Lorenz and the author to reduce to a prime ring and a subgroup of finite index in

G
G

. Next if

R
R

is prime, then we show that the prime ideals of

R
∗
G

R \ast G

disjoint from

R
R

are explicitly determined by the primes of a certain twisted group algebra of a normal subgroup of

G
G

. Finally the prime ideals in twisted group algebras of polycyclic-by-finite groups are studied by lifting the situation to ordinary group algebras where the results of J. E. Roseblade can be applied.
Nonsingular quadratic differential equations in the plane
Authors: Camacho M. I. T. , Palmeira C. F. B.
Published: June 2010

We consider the problem of determining the number of inseparable leaves of nonsingular polynomial differential equations of degree two. As a corollary of a classification theorem for the foliation defined by these equations, we prove that this number is at most 2.
Graded Lie algebras of the second kind
Authors: Chêng Jih Hsin
Published: June 2010

The associated Lie algebra of the Cartan connection for an abstract CR-hypersurface admits a gradation of the second kind. In this article, we give two ways to characterize this kind of graded Lie algebras, namely, geometric characterization in terms of symmetric spaces and algebraic characterization in terms of root systems. A complete list of this class of Lie algebras is given.
Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations
Authors: Schaaf Klaus W.
Published: June 2010

This paper is a generalization of the theory of the KPP and bistable nonlinear diffusion equations. It is shown that traveling wave solutions exist for nonlinear parabolic functional differential equations (FDEs) which behave very much like the well-known solutions of the classical KPP and bistable equations. Among the techniques used are maximum principles, sub- and supersolutions, phase plane techniques for FDEs and perturbation of linear operators.
The connectedness of the group of automorphisms of 𝐿¹(0,1)
Authors: Ghahramani F.
Published: June 2010

For each of the radical Banach algebras

L
1

(
0
,
1
)

{L^1}(0,1)

and

L
1

(
w
)

{L^1}(w)

an integral representation for the automorphisms is given. This is used to show that the groups of the automorphisms of

L
1

(
0
,
1
)

{L^1}(0,1)

and

L
1

(
w
)

{L^1}(w)

endowed with bounded strong operator topology (BSO) are arcwise connected. Also it is shown that if

|

|

|

⋅

|

|

|

p

||| \cdot ||{|_p}

denotes the norm of

B
(

L
p

(
0
,
1
)

B({L^p}(0,1)

,

L
1

(
0
,
1
)
)

{L^1}(0,1))

,

1
>
p
≤
∞

1 > p \leq \infty

, then the group of automorphisms of

L
1

(
0
,
1
)

{L^1}(0,1)

topologized by

|

|

|

⋅

|

|

|

p

||| \cdot ||{|_p}

is arcwise connected. It is shown that every automorphism

θ
\theta

of

L
1

(
0
,
1
)

{L^1}(0,1)

is of the form

θ
=

e

λ
d

lim

e

q
n

(

BSO

)

\theta = {e^{\lambda d}}{\operatorname {lim}}{e^{qn}}({\text {BSO}})

, where each

q
n

{q_n}

is a quasinilpotent derivation. It is shown that the group of principal automorphisms of

l
1

(
w
)

{l^1}(w)

under operator norm topology is arcwise connected, and every automorphism has the form

e

i
α
d

(

e

λ
d

e
D

e

−
λ
d

)
−

{e^{i\alpha d}}{({e^{\lambda d}}{e^D}{e^{ - \lambda d}})^ - }

, where

α
∈

R

\alpha \in {\mathbf {R}}

,

λ
>
0

\lambda > 0

, and

D
D

is a derivation, and where

(

e

λ
d

e
D

e

−
λ
d

)
−

{({e^{\lambda d}}{e^D}{e^{ - \lambda d}})^ - }

denotes the extension by continuity of

e

λ
d

e
D

e

−
λ
d

{e^{\lambda d}}{e^D}{e^{ - \lambda d}}

from a dense subalgebra of

l
1

(
w
)

{l^1}(w)

to

l
1

(
w
)

{l^1}(w)

.
Degrees of splittings and bases of recursively enumerable subspaces
Authors: Downey R. G. , Remmel J. B. , Welch L. V.
Published: June 2010

This paper analyzes the interrelationships between the (Turing) of r.e. bases and of r.e. splittings of r.e. vector spaces together with the relationship of the degrees of bases and the degrees of the vector spaces they generate. For an r.e. subspace

V
V

of

V
∞

{V_\infty }

, we show that

α
\alpha

is the degree of an r.e. basis of

V
V

iff

α
\alpha

is the degree of an r.e. summand of

V
V

iff

α
\alpha

is the degree and dependence degree of an r.e. summand of

V
V

. This result naturally leads to explore several questions regarding the degree theoretic properties of pairs of summands and the ways in which bases may arise.
Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data
Authors: Hoff David
Published: June 2010

We prove the global existence of weak solutions of the Cauchy problem for the Navier-Stokes equations of compressible, isentropic flow of a polytropic gas in one space dimension. The initial velocity and density are assumed to be in

L
2

{L^2}

and

L
2

∩
B
V

{L^2} \cap BV

respectively, modulo additive constants. In particular, no smallness assumptions are made about the intial data. In addition, we prove a result concerning the asymptotic decay of discontinuities in the solution when the adiabatic constant exceeds

3

/

2

3/2

.
Polynomial approximation in the mean with respect to harmonic measure on crescents
Authors: Akeroyd John
Published: June 2010

For

1
⩽
s
>
∞

1 \leqslant s > \infty

and "nice" crescents

G
G

, this paper gives a necessary condition (Theorem 2.6) and a sufficient condition (Theorem 2.5) for density of the polynomials in the generalized Hardy space

H
s

(
G
)

{H^s}(G)

. These conditions are easily tested and almost equivalent.
On the differentiability of Lipschitz-Besov functions
Authors: Dorronsoro José R.
Published: June 2010

L
r

{L^r}

and ordinary differentiability is proved for functions in the Lipschitz-Besov spaces

B
a

p
,
q

,

1
⩽
p
>
∞
,

1
⩽
q
⩽
∞
,

a
>
0

B_a^{p,q},\;1 \leqslant p > \infty ,\;1 \leqslant q \leqslant \infty ,\;a > 0

, using certain maximal operators measuring smoothness. These techniques allow also the study of lacunary directional differentiability and of tangential convergence of Poisson integrals.
Sporadic and irrelevant prime divisors
Authors: McAdam Stephen , Ratliff L. J.
Published: June 2010

Let

I
I

represent a regular ideal in a Noetherian ring

R
R

. If

W
W

is a finite set of prime ideals in

R
R

, some conditions on

W
W

are given assuring that an

I
I

can be found such that

W
W

is exactly the set of primes which are in

Ass
⁡
R

/

I

\operatorname {Ass} R/I

but not in

Ass
⁡
R

/

I
n

\operatorname {Ass} R/{I^n}

for all large

n
n

. Furthermore, if

I
I

is fixed, and if

P
P

is a prime ideal containing

I
I

, some conditions are given assuring that in the Rees ring

R

=
R
[
u
,

I
t
]
,

(
u
,

P
,

I
t
)

R

{\mathbf {R}} = R[u,\,It],\,(u,\,P,\,It){\mathbf {R}}

is a prime divisor of

u

R

u{\mathbf {R}}

.
Scalar curvatures on 𝑆²
Authors: Chen Wen Xiong , Ding Wei Yue
Published: June 2010

A theorem for the existence of solutions of the nonlinear elliptic equation

−
Δ
u
+
2
=
R
(
x
)

e
u

,

x
∈

S
2

- \Delta u + 2 = R(x){e^u},\;x \in {S^2}

, is proved by using a "mass center" analysis technique and by applying a continuous "flow" in

H
1

(

S
2

)

{H^1}({S^2})

controlled by

∇
R

\nabla R

.
Convergence in distribution of products of random matrices: a semigroup approach
Authors: Mukherjea Arunava
Published: June 2010

The problem of weak convergence of the sequence of convolution powers of a probability measure has been considered in this paper in the general context of a noncompact semigroup and in particular, in the semigroup of nonnegative and real matrices. Semigroup methods have been used to give simple proofs of some recent results of Kesten and Spitzer in nonnegative matrices. It has been also shown that these methods often lead to similar results in the more general context of real matrices.
𝑝-ranks and automorphism groups of algebraic curves
Authors: Nakajima Shōichi
Published: June 2010

Let

X
X

be an irreducible complete nonsingular curve of genus

g
g

over an algebraically closed field

k
k

of positive characteristic

p
p

. If

g
⩾
2

g \geqslant 2

, the automorphism group

Aut
⁡
(
X
)

\operatorname {Aut} (X)

of

X
X

is known to be a finite group, and moreover its order is bounded from above by a polynomial in

g
g

of degree four (Stichtenoth). In this paper we consider the

p
p

-rank

γ
\gamma

of

X
X

and investigate relations between

γ
\gamma

and

Aut
⁡
(
X
)

\operatorname {Aut} (X)

. We show that

γ
\gamma

affects the order of a Sylow

p
p

-subgroup of

Aut
⁡
(
X
)

(
§
3
)

\operatorname {Aut} (X)\;(\S 3)

and that an inequality

|

Aut
⁡
(
X
)

|

⩽
84
(
g
−
1
)
g

|\operatorname {Aut} (X)| \leqslant 84(g - 1)g

holds for an ordinary (i.e.

γ
=
g

\gamma = g

) curve

X

(
§
4
)

X\,(\S 4)

.
On weak continuity and the Hodge decomposition
Authors: Robbin Joel W. , Rogers Robert C. , Temple Blake
Published: June 2010

We address the problem of determining the weakly continuous polynomials for sequences of functions that satisfy general linear first-order differential constraints. We prove that wedge products are weakly continuous when the differential constraints are given by exterior derivatives. This is sufficient for reproducing the Div-Curl Lemma of Murat and Tartar, the null Lagrangians in the calculus of variations and the weakly continuous polynomials for Maxwell’s equations. This result was derived independently by Tartar who stated it in a recent survey article [7]. Our proof is explicit and uses the Hodge decomposition.
Quasi 𝐹-covers of Tychonoff spaces
Authors: Henriksen M. , Vermeer J. , Woods R. G.
Published: June 2010

A Tychonoff topological space is called a quasi

F
F

-space if each dense cozero-set of

X
X

is

C

∗

{C^{\ast }}

-embedded in

X
X

. In Canad. J. Math. 32 (1980), 657-685 Dashiell, Hager, and Henriksen construct the "minimal quasi

F
F

-cover"

Q
F
(
X
)

QF(X)

of a compact space

X
X

as an inverse limit space, and identify the ring

C
(
Q
F
(
X
)
)

C(QF(X))

as the order-Cauchy completion of the ring

C

∗

(
X
)

{C^{\ast }}(X)

. In On perfect irreducible preimages, Topology Proc. 9 (1984), 173-189, Vermeer constructed the minimal quasi

F
F

-cover of an arbitrary Tychonoff space. In this paper the minimal quasi

F
F

-cover of a compact space

X
X

is constructed as the space of ultrafilters on a certain sublattice of the Boolean algebra of regular closed subsets of

X
X

. The relationship between

Q
F
(
X
)

QF(X)

and

Q
F
(
β
X
)

QF(\beta X)

is studied in detail, and broad conditions under which

β
(
Q
F
(
X
)
)
=
Q
F
(
β
X
)

\beta (QF(X)) = QF(\beta X)

are obtained, together with examples of spaces for which the relationship fails. (Here

β
X

\beta X

denotes the Stone-Čech compactification of

X
X

.) The role of

Q
F
(
X
)

QF(X)

as a "projective object" in certain topological categories is investigated.
Application of group cohomology to space constructions
Authors: Igodt Paul , Lee Kyung Bai
Published: June 2010

From a short exact sequence of crossed modules

1
→
K
→
H
→

H
¯

→
1

1 \to K \to H \to \bar H \to 1

and a

2
2

-cocycle

(
ϕ
,

h
)
∈

Z
2

(
G
;

H
)

(\phi ,\,h) \in {Z^2}(G;\,H)

, a

4
4

-term cohomology exact sequence
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript a b Superscript 1 Baseline left-parenthesis upper G semicolon upper Z right-parenthesis right-arrow upper H Subscript left-parenthesis phi overbar comma h overbar right-parenthesis Superscript 1 Baseline left-parenthesis upper G semicolon upper H overbar comma upper Z overbar right-parenthesis right-arrow Overscript delta Endscripts union StartSet upper H Subscript psi Superscript 2 Baseline left-parenthesis upper G semicolon upper K right-parenthesis colon psi Subscript normal o normal u normal t Baseline equals phi Subscript normal o normal u normal t Baseline EndSet right-arrow upper H Subscript a b Superscript 2 Baseline left-parenthesis upper G semicolon upper Z right-parenthesis">

H

a
b

1

(
G
;
Z
)
→

H

(

ϕ
¯

,

h
¯

)

1

(
G
;

H
¯

,

Z
¯

)

→

δ

⋃
{

H
ψ
2

(
G
;
K
)
:

ψ

o
u
t

=

ϕ

o
u
t

}
→

H

a
b

2

(
G
;

Z
)

H_{ab}^1(G;Z) \to H_{(\bar {\phi }, \bar {h})}^1 (G; \bar {H}, \bar {Z}) \stackrel {\delta }{\to } \bigcup \{ H_\psi ^2 (G;K) : \psi _{\mathrm {out}} = \phi _{\mathrm {out}} \} \to H_{ab}^2(G;\,Z)

is obtained. Here the first and the last term are the ordinary (=abelian) cohomology groups, and

Z
Z

is the center of the crossed module

H
H

. The second term is shown to be in one-to-one correspondence with certain geometric constructions, called Seifert fiber space construction. Therefore, it follows that, if both the end terms vanish, the geometric construction exists and is unique.
A strong generalization of Helgason’s theorem
Authors: Johnson Kenneth D.
Published: June 2010

Let

G
G

be a simple Lie group with

K
A
N

KAN

an Iwasawa decomposition of

G
G

, and let

M
M

be the centralizer of

A
A

in

K
K

. Suppose

K
1

{K_1}

is a fixed, closed, normal, analytic subgroup of

K
K

, and set

P

(

K
1

)

{\mathbf {P}}({K_1})

equal to the set of all parabolic subgroups

P
P

of

G
G

which contain

M
A
N

MAN

such that

K
1

P
=
G

{K_1}P = G

and

K
1

∩
P

{K_1} \cap P

is normal in the reductive part of

P
P

. Suppose

π
:
G
→
G
L
(
V
)

\pi :G \to GL(V)

is an irreducible representation of

G
G

. Then, if

P

(

K
1

)
≠
∅

{\mathbf {P}}({K_1}) \ne \emptyset

, we obtain necessary and sufficient conditions for

V

K
1

{V^{{K_1}}}

, the space of

K
1

{K_1}

-fixed vectors, to be

≠
(
0
)

\ne (0)

. Moreover, reciprocity formulas are obtained which determine

dim
⁡

V

K
1

\dim {V^{{K_1}}}

.
Equivariant geometry and Kervaire spheres
Authors: Back Allen , Hsiang Wu-Yi
Published: June 2010

The intrinsic geometry of metrics on the Kervaire sphere which are invariant under a large transformation group (cohomogeneity one) is studied. Invariant theory is used to describe the behavior of these metrics near the singular orbits. Nice expressions for the Ricci and sectional curvatures are obtained. The nonexistence of invariant metrics of positive sectional curvature is proven, and Cheeger’s construction of metrics of positive Ricci curvature is discussed.
Universal Loeb-measurability of sets and of the standard part map with applications
Authors: Landers D. , Rogge L.
Published: June 2010

It is shown in this paper that for

K
K

-saturated models many important external sets of nonstandard analysis—such as monadic sets or the set of all near-standard points or all pre-near-standard points or all compact points—are universally Loeb-measurable, i.e. Loeb-measurable with respect to every internal content. We furthermore obtain universal Loeb-measurability of the standard part map for topological spaces which are not covered by previous results in this direction. Moreover, the standard part map can be used as a measure preserving transformation for all

τ
\tau

-smooth measures, and not only for Radon-measures as known up to now. Applications of our results lead to simple new proofs for theorems of classical measure theory. We obtain e.g. the extension of

τ
\tau

-smooth Baire-measures to

τ
\tau

-smooth Borel-measures, the decomposition theorems for

τ
\tau

-smooth Baire-measures and

τ
\tau

-smooth Borel-measures and Kakutani’s theorem for product measures.
Longer than average intervals containing no primes
Authors: Cheer A. Y. , Goldston D. A.
Published: June 2010

We present two methods for proving that there is a positive proportion of intervals which contain no primes and are longer than the average distance between consecutive primes. The first method is based on an argument of Erdös which uses a sieve upper bound for prime twins to bound the density function for gaps between primes. The second method uses known results about the first three moments for the distribution of intervals with a given number of primes. Better results are obtained by assuming that the first

n
n

moments are Poisson. The related problem of longer than average gaps between primes is also considered.
Invariant subspaces in Banach spaces of analytic functions
Authors: Richter Stefan
Published: June 2010

We study the invariant subspace structure of the operator of multiplication by

z
z

,

M
z

{M_z}

, on a class of Banach spaces of analytic functions. For operators on Hilbert spaces our class coincides with the adjoints of the operators in the Cowen-Douglas class

B

1

(

Ω
¯

)

{\mathcal {B}_1}(\overline \Omega )

. We say that an invariant subspace

M

\mathcal {M}

satisfies

cod
⁡

M

=
1

\operatorname {cod} \mathcal {M} = 1

if

z

M

z\mathcal {M}

has codimension one in

M

\mathcal {M}

. We give various conditions on invariant subspaces which imply that

cod
⁡

M

=
1

\operatorname {cod} \mathcal {M} = 1

. In particular, we give a necessary and sufficient condition on two invariant subspaces

M

\mathcal {M}

,

N

\mathcal {N}

with

cod
⁡

M

=
cod
⁡

N

=
1

\operatorname {cod} \mathcal {M} = \operatorname {cod} \mathcal {N} = 1

so that their span again satisfies

cod
⁡
(

M

∨

N

)
=
1

\operatorname {cod} (\mathcal {M} \vee \mathcal {N}) = 1

. This result will be used to show that any invariant subspace of the Bergman space

L
a
p

,

p
⩾
1

L_a^p,\,p \geqslant 1

, which is generated by functions in

L
a

2
p

L_a^{2p}

, must satisfy

cod
⁡

M

=
1

\operatorname {cod} \mathcal {M} = 1

. For an invariant subspace

M

\mathcal {M}

we then consider the operator

S
=

M
z

∗

|

M

⊥

S = M_z^{\ast }|{\mathcal {M}^ \bot }

. Under some extra assumption on the domain of holomorphy we show that the spectrum of

S
S

coincides with the approximate point spectrum iff

cod
⁡

M

=
1

\operatorname {cod} \mathcal {M} = 1

. Finally, in the last section we obtain a structure theorem for invariant subspaces with

cod
⁡

M

=
1

\operatorname {cod} \mathcal {M} = 1

. This theorem applies to Dirichlet-type spaces.
Weighted norm estimates for Sobolev spaces
Authors: Schechter Martin
Published: June 2010

We give sufficient conditions for estimates of the form\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral StartAbsoluteValue u left-parenthesis x right-parenthesis EndAbsoluteValue Superscript q Baseline d mu left-parenthesis x right-parenthesis less-than-or-slanted-equals upper C double-vertical-bar u double-vertical-bar Subscript s comma p Superscript 1 Baseline comma u element-of upper H Superscript s comma p Baseline comma">∫|u(x)|qdμ(x)⩽C‖u‖s,p1,u∈Hs,p,{\int {\left | {u(x)} \right |} ^q}d\mu (x) \leqslant C\left \| u \right \|_{s,p}^1,\qquad u \in {H^{s,p}},\]to hold, whereμ(x)\mu (x)is a measure and‖u‖s,p{\left \| u \right \|_{s,p}}is the norm of the Sobolev spaceHs,p{H^{s,p}}. Ifdμ=dxd\mu = dx, this reduces to the usual Sobolev inequality. The general form has much wider applications in both linear and nonlinear partial differential equations. An application is given in the last section.
Euler-Poincaré characteristic and higher order sectional curvature. I
Authors: Hsiung Chuan-Chih , Shiskowski Kenneth Michael
Published: June 2010

The following long-standing conjecture of H. Hopf is well known. Let

M
M

be a compact orientable Riemannian manifold of even dimension

n
⩾
2

n \geqslant 2

. If

M
M

has nonnegative sectional curvature, then the Euler-Poincaré characteristic

χ
(
M
)

\chi (M)

is nonnegative. If

M
M

has nonpositive sectional curvature, then

χ
(
M
)

\chi (M)

is nonnegative or nonpositive according as

n
≡
0

n \equiv 0

or

2
mod
4

2\bmod 4

. This conjecture for

n
=
4

n = 4

was proved first by J. W. Milnor and then by S. S. Chern by a different method. The main object of this paper is to prove this conjecture for a general

n
n

under an extra condition on higher order sectional curvature, which holds automatically for

n
=
4

n = 4

. Similar results are obtained for Kähler manifolds by using holomorphic sectional curvature, and F. Schur’s theorem about the constancy of sectional curvature on a Riemannian manifold is extended.
Convergence acceleration for generalized continued fractions
Authors: Levrie Paul , Jacobsen Lisa
Published: June 2010

The main result in this paper is the proof of convergence acceleration for a suitable modification (as defined by de Bruin and Jacobsen) in the case of an

n
n

-fraction for which the underlying recurrence relation is of Perron-Kreuser type. It is assumed that the characteristic equations for this recurrence relation have only simple roots with differing absolute values.
The ideal structure of certain nonselfadjoint operator algebras
Authors: Peters Justin
Published: June 2010

Let

(
X
,

ϕ
)

(X,\,\phi )

be a locally compact dynamical system, and

Z

+

×
ϕ

C
0

(
X
)

{{\mathbf {Z}}^ + }{ \times _\phi }\,{C_0}(X)

the norm-closed subalgebra of the crossed product

Z

×
ϕ

C
0

(
X
)

Z{ \times _\phi }{C_0}(X)

generated by the nonnegative powers of

ϕ
\phi

in case

ϕ
\phi

is a homeomorphism. If

ϕ
\phi

is just a continuous map,

Z

+

×
ϕ

C
0

{{\mathbf {Z}}^ + }{ \times _\phi }{C_0}

can still be defined by a crossed product type construction. The ideal structure of these algebras is determined in case

ϕ
\phi

acts freely. A class of strictly transitive Banach modules is described, indicating that for the nonselfadjoint operator algebras considered here, not all irreducible representations are on Hilbert space. Finally in a special case, the family of all invariant maximal right ideals is given.
Tiling the projective foliation space of a punctured surface
Authors: Mosher Lee
Published: June 2010

There is a natural way to associate, to each ideal triangulation of a punctured surface a cell decomposition of the projective foliation space of the punctured surface.
Euler spaces of analytic functions
Authors: Rovnyak James
Published: June 2010

A formula due to Euler and Legendre is used to construct finite-difference counterparts to the Dirichlet space. The spaces have integral representations and characterizations in terms of area integrals. Their reproducing kernels are logarithms of the reproducing kernels of the Newton spaces, which are counterparts to the Hardy class. A Hilbert space with reproducing kernel
\[

log
⁡
[
(
1

/

w
¯

z
)

log

1

/

(
1
−

w
¯

z
)
]

\log [(1/\overline w z)\,\log \;1/(1 - \overline w z)]

\]
is also shown to exist and to be related to Bernoulli numbers and combinatorial theory.
Bounds on the 𝐿² spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality
Authors: Lawler Gregory F. , Sokal Alan D.
Published: June 2010

We prove a general version of Cheeger’s inequality for discrete-time Markov chains and continuous-time Markovian jump processes, both reversible and nonreversible, with general state space. We also prove a version of Cheeger’s inequality for Markov chains and processes with killing. As an application, we prove

L
2

{L^2}

exponential convergence to equilibrium for random walk with inward drift on a class of countable rooted graphs.
𝑘-flat structures and exotic characteristic classes
Authors: Goldberg Lisa R.
Published: June 2010

We generalize the concept of "foliation" and define

k
k

-flat structures; these are smooth vector bundles with affine connections whose characteristic forms vanish above a certain dimension. Using semisimplicial techniques we construct a classifying space for

k
k

-flat structures, and prove a classification theorem for these structures on smooth manifolds. Techniques from rational homotopy theory are used to relate the exotic characteristic classes of foliations to the rational homotopy groups and cohomology of the classifying space.
Classifying 1-handles attached to knotted surfaces
Authors: Boyle Jeffrey
Published: June 2010

We study a method of obtaining knotted surfaces in the

4
4

-sphere

S
4

{S^4}

by attaching embedded

2
2

-dimensional

1
1

-handles to a given knot. The main result is there is a one-to-one correspondence between the

1
1

-handles that can be attached to a knot and the double cosets of the peripheral subgroup in the group of the knot. Many examples and applications are given.
Topological equivalence of flows on homogeneous spaces, and divergence of one-parameter subgroups of Lie groups
Authors: Benardete Diego
Published: June 2010

Let

Γ
\Gamma

and

Γ
′

\Gamma ’

be lattices, and

ϕ
\phi

and

ϕ
′

\phi ’

one-parameter subgroups of the connected Lie groups

G
G

and

G
′

G’

. If one of the following conditions (a), (b), or (c) hold, Theorem A states that if the induced flows on the homogeneous spaces

G

/

Γ

G/\Gamma

and

G
′

/

Γ
′

G’ /\Gamma ’

are topologically equivalent, then they are topologically equivalent by an affine map. (a)

G
G

and

G
′

G’

are one-connected and nilpotent. (b)

G
G

and

G
′

G’

are one-connected and solvable, and for all

X
X

in

L
(
G
)

L(G)

and

X
′

X’

in

L
(

G
′

)

L(G’ )

,

ad
⁡
(
x
)

\operatorname {ad} (x)

and

ad
⁡
(

X
′

)

\operatorname {ad} (X’ )

have only real eigenvalues, (c)

G
G

and

G
′

G’

are centerless and semisimple with no compact direct factor and no direct factor

H
H

isomorphic to

PSL
⁡
(
2
,

R
)

\operatorname {PSL} (2,\,R)

such that

Γ
H

\Gamma H

is closed in

G
G

. Moreover, in condition (c), the induced flow of

ϕ
\phi

on

G

/

Γ

G/\Gamma

is assumed to be ergodic. Theorem A depends on Theorem B, which concerns divergence properties of one-parameter subgroups. We say

ϕ
\phi

is isolated if and only if for any

ϕ
′

\phi ’

which recurrently approaches

ϕ
\phi

for positive and negative time,

ϕ
\phi

equals

ϕ
′

\phi ’

up to sense-preserving reparameterization. Theorem B(a) states that if

G
G

is one-connected and nilpotent, or one-connected and solvable with exp:

L
(
G
)
→
G

L(G) \to G

a diffeomrophism, then every

ϕ
\phi

of

G
G

is isolated. Let

G
G

be connected and semisimple and

ϕ
(
t
)
=
exp
⁡
(
t
X
)

\phi (t) = \exp (tX)

. Then Theorem B(b) states that

ϕ
\phi

is isolated, if

[
X
,

Y
]
=
0

[X,\,Y] = 0

and

ad
⁡
(
Y
)

\operatorname {ad} (Y)

being semisimple imply that

ad
⁡
(
Y
)

\operatorname {ad} (Y)

has some eigenvalue not pure imaginary and not zero. If

G
G

has finite center,

ϕ
\phi

is isolated if there is no compact connected subgroup in the centralizer of

ϕ
\phi

.
Flows on vector bundles and hyperbolic sets
Authors: Salamon Dietmar , Zehnder Eduard
Published: June 2010

This note deals with C. Conley’s topological approach to hyperbolic invariant sets for continuous flows. It is based on the notions of isolated invariant sets and Morse decompositions and it leads to the concept of weak hyperbolicity.
Riemannian 4-symmetric spaces
Authors: Jiménez J. A.
Published: June 2010

The main purpose of this paper is to classify the compact simply connected Riemannian

4
4

-symmetric spaces. As homogeneous manifolds, these spaces are of the form

G

/

L

G/L

where

G
G

is a connected compact semisimple Lie group with an automorphism

σ
\sigma

of order four whose fixed point set is (essentially)

L
L

. Geometrically, they can be regarded as fiber bundles over Riemannian

2
2

-symmetric spaces with totally geodesic fibers isometric to a Riemannian

2
2

-symmetric space. A detailed description of these fibrations is also given. A compact simply connected Riemannian

4
4

-symmetric space decomposes as a product

M
1

×
…
×

M
r

{M_1} \times \ldots \times {M_r}

where each irreducible factor is: (i) a Riemannian

2
2

-symmetric space, (ii) a space of the form

{
U
×
U
×
U
×
U
}

/

Δ
U

\{ U \times U \times U \times U\} /\Delta U

with

U
U

a compact simply connected simple Lie group,

Δ
U
=

\Delta U =

diagonal inclusion of

U
U

, (iii)

{
U
×
U
}

/

Δ

U
θ

\{ U \times U\} /\Delta {U^\theta }

with

U
U

as in (ii) and

U
θ

{U^\theta }

the fixed point set of an involution

θ
\theta

of

U
U

, and (iv)

U

/

K

U/K

with

U
U

as in (ii) and

K
K

the fixed point set of an automorphism of order four of

U
U

. The core of the paper is the classification of the spaces in (iv). This is accomplished by first classifying the pairs

(

g

,

σ
)

(\mathfrak {g},\,\sigma )

with

g

\mathfrak {g}

a compact simple Lie algebra and

σ
\sigma

an automorphism of order four of

g

\mathfrak {g}

. Tables are drawn listing all the possibilities for both the Lie algebras and the corresponding spaces. For

U
U

"classical," the automorphisms

σ
\sigma

are explicitly constructed using their matrix representations. The idea of duality for

2
2

-symmetric spaces is extended to

4
4

-symmetric spaces and the duals are determined. Finally, those spaces that admit invariant almost complex structures are also determined: they are the spaces whose factors belong to the class (iv) with

K
K

the centralizer of a torus.
Calibrations on 𝑅⁸
Authors: Dadok J. , Harvey R. , Morgan F.
Published: June 2010

Calibrations are a powerful tool for constructing minimal surfaces. In this paper we are concerned with

4
4

-manifolds

M
⊂

R

8

M \subset {{\mathbf {R}}^8}

. If a differential form

φ
∈

⋀
4

R

8

\varphi \in { \bigwedge ^4}{{\mathbf {R}}^8}

calibrates all tangent planes of

M
M

then

M
M

is area minimizing. For

φ
\varphi

in one of several large subspaces of

⋀
4

R

8

{ \bigwedge ^4}{{\mathbf {R}}^8}

we compute its comass, that is the maximal value of

φ
\varphi

on the Grassmannian of oriented

4
4

-planes. We then determine the set

G
(
φ
)
⊂
G
(
4
,

R

8

)

G(\varphi ) \subset G(4,\,{{\mathbf {R}}^8})

on which this maximum is attained. These are the planes calibrated by

φ
\varphi

, the possible tangent planes to a manifold calibrated by

φ
\varphi

. The families of calibrations obtained include the well-known examples: special Lagrangian, Kähler, and Cayley.
Construction of manifolds of positive scalar curvature
Authors: Carr Rodney
Published: June 2010

We prove that a regular neighborhood of a codimension

⩾
3

\geqslant 3

subcomplex of a manifold can be chosen so that the induced metric on its boundary has positive scalar curvature. A number of useful facts concerning manifolds of positive scalar curvature follow from this construction. For example, we see that any finitely presented group can appear as the fundamental group of a compact

4
4

-manifold with such a metric.
Functions that preserve the uniform distribution of sequences
Authors: Bosch William
Published: June 2010

In this paper, necessary and sufficient conditions are given for certain functions to preserve the uniform distribution of sequences. An analytic condition allows the construction of examples. An application is also given.
Odd primary periodic phenomena in the classical Adams spectral sequence
Authors: Shick Paul
Published: June 2010

We study certain periodic phenomena in the cohomology of the

mod

p

\bmod \;p

Steenrod algebra which are related to the polynomial generators

v
n

∈

π

∗

B
P

{v_n} \in {\pi _{\ast }}BP

. A chromatic resolution of the

E
2

{E_2}

term of the classical Adams spectral sequence is constructed.
Travelling wave solutions to a gradient system
Authors: Reineck James F.
Published: June 2010

Given a system of reaction-diffusion equations where the nonlinearity is derived from a potential with certain restrictions, we use the Conley index and the connection matrix to show that there is a travelling wave solution connecting the maxima of the potential.
Tangent cones to discriminant loci for families of hypersurfaces
Authors: Smith Roy , Varley Robert
Published: June 2010

A deformation of a variety with (nonisolated) hypersurface singularities, such as a projective hypersurface or a theta divisor of an abelian variety, determines a rational map of the singular locus to projective space and the resulting projective geometry of the singular locus describes how the singularities propagate in the deformation. The basic principle is that the projective model of the singular locus is dual to the tangent cone to the discriminant of the deformation. A detailed study of the method, which emerged from interpreting Andreotti-Mayer’s work on theta divisors in terms of Schlessinger’s deformation theory of singularities, is given along with examples, applications, and multiplicity formulas.
Automorphisms and isomorphisms of real Henselian fields
Authors: Brown Ron
Published: June 2010

Let

K
K

and

L
L

be ordered algebraic extensions of an ordered field

F
F

. Suppose

K
K

and

L
L

are Henselian with Archimedean real closed residue class fields. Then

K
K

and

L
L

are shown to be

F
F

-isomorphic as ordered fields if they have the same value group. Analogues to this result are proved involving orderings of higher level, unordered extensions, and, when

K
K

and

L
L

are maximal valued fields, transcendental extensions. As a corollary, generalized real closures at orderings of higher level are shown to be determined up to isomorphism by their value groups. The results on isomorphisms are applied to the computation of automorphism groups of

K
K

and to the study of the fixed fields of groups of automorphisms of

K
K

. If

K
K

is real closed and maximal with respect to its canonical valuation, then these fixed fields are shown to be exactly those real closed subfields of

K
K

which are topologically closed in

K
K

. Generalizations of this fact are proved. An example is given to illustrate several aspects of the problems considered here.
Chains on CR manifolds and Lorentz geometry
Authors: Koch Lisa K.
Published: June 2010

We show that two nearby points of a strictly pseudoconvex CR manifold are joined by a chain. The proof uses techniques of Lorentzian geometry via a correspondence of Fefferman. The arguments also apply to more general systems of chain-like curves on CR manifolds.
Conformal distortion of boundary sets
Authors: Hamilton D. H.
Published: June 2010

Conformal maps

f
f

of the disk into itself have the property that

dim
⁡

f

−
1

(
F
)
⩽
dim
⁡
F

\dim {f^{ - 1}}(F) \leqslant \dim F

for any set

F
F

on the unit circle.
Proper knot theory in open 3-manifolds
Authors: Churchyard Peter , Spring David
Published: June 2010

This paper introduces a theory of proper knots, i.e., smooth proper embeddings of

R

1

{{\mathbf {R}}^1}

into open

3
3

-manifolds. Proper knot theory is distinguished by the fact that proper isotopies of knots are not ambient in general. A uniqueness theorem for proper knots is proved in case the target manifold is the interior of a one-dimensional handlebody.
Pseudo-orbit shadowing in the family of tent maps
Authors: Coven Ethan M. , Kan Ittai , Yorke James A.
Published: June 2010

We study the family of tent maps—continuous, unimodal, piecewise linear maps of the interval with slopes

±
s

\pm s

,

2

⩽
s
⩽
2

\sqrt 2 \leqslant s \leqslant 2

. We show that tent maps have the shadowing property (every pseudo-orbit can be approximated by an actual orbit) for almost all parameters

s
s

, although they fail to have the shadowing property for an uncountable, dense set of parameters. We also show that for any tent map, every pseudo-orbit can be approximated by an actual orbit of a tent map with a perhaps slightly larger slope.
Variational principles for Hill’s spherical vortex and nearly spherical vortices
Authors: Wan Yieh Hei
Published: June 2010

In this paper, vortex rings are regarded as axisymmetric motions without swirl of an incompressible inviscid fluid in space, with vorticity confined to their finite cores. The main results of this paper are (H) Hill’s spherical vortex is a "nondegenerate" local maximum of the energy function subject to a fixed impulse, among vortex rings. (N) Norbury’s nearly spherical vortex is a "nondegenerate" local maximum of the energy function subject to a fixed impulse, and a fixed circulation. Estimates are established to overcome the discontinuity of vorticity distributions, and the singular behavior of Stoke’s stream functions near the axis of symmetry. The spectral analysis involves the use of Legendre’s functions.
Differential delay equations that have periodic solutions of long period
Authors: Chapin Steven A.
Published: June 2010

If

f
:

R

→

R

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

is a continuous odd function satisfying

x
f
(
x
)
>
0

xf(x) > 0

,

x
≠
0

x \ne 0

, and

f
(
x
)
=
o
(

x

−
2

)

f(x) = o({x^{ - 2}})

as

x
→
∞

x \to \infty

, then so-called periodic solutions of long period seem to play a prominent role in the dynamics of

(

∗

)

({\ast })

\[

x
′

(
t
)
=
−
α
f
(
x
(
t
−
1
)
)
,

α
>
0.

x’(t) = - \alpha f(x(t - 1)),\qquad \alpha > 0.

\]
In this paper we prove the existence of long-period periodic solutions of

(

∗

)

({\ast })

for a class of nonodd functions that decay "rapidly" to

0
0

at infinity and satisfy

x
f
(
x
)
⩾
0

xf(x) \geqslant 0

. These solutions have quite different qualitative features than in the odd case.
Decay rates of Fourier transforms of curves
Authors: Marshall B. P.
Published: June 2010

Let

d
μ

d\mu

be a smooth measure on a nondegenerate curve in

R

n

{{\mathbf {R}}^n}

. This paper examines the decay rate of spherical averages of its Fourier transform

d
μ

^

\widehat {d\mu }

. Thus estimates of the following form are considered:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis integral Underscript sigma-summation Underscript r Endscripts Endscripts vertical-bar ModifyingAbove d mu With caret left-parenthesis xi vertical-bar Superscript p Baseline d xi right-parenthesis Superscript 1 slash p Baseline less-than-or-slanted-equals upper C r Superscript negative sigma Baseline StartAbsoluteValue EndAbsoluteValue f StartAbsoluteValue EndAbsoluteValue">

(

∫

∑
r

|

d
μ

^

(
ξ

|

p

d
ξ

)

1

/

p

⩽
C

r

−
σ

|

|

f

|

|

{\left ( {\int _{{\sum _r}} {|\widehat {d\mu }(\xi {|^p}d\xi } } \right )^{1/p}} \leqslant C{r^{ - \sigma }}||f||

\]
where

∑
r

=
{
ξ
∈

R

n

:

|

ξ

|

=
r
}

{\sum _r} = \{ \xi \in {{\mathbf {R}}^n}:|\xi | = r\}

.
The number of solutions to linear Diophantine equations and multivariate splines
Authors: Dahmen Wolfgang , Micchelli Charles A.
Published: June 2010

In this paper we study how the number of nonnegative integer solutions of

s
s

integer linear equations in

n
⩾
s

n \geqslant s

unknowns varies as a function of the inhomogeneous terms. Aside from deriving various recurrence relations for this function, we establish some of its detailed structural properties. In particular, we show that on certain subsets of lattice points it is a polynomial. The univariate case (

s
=
1

s = 1

) yields E. T. Bell’s description of Sylvester’s denumerants. Our approach to this problem relies upon the use of polyhedral splines. As an example of this method we obtain results of R. Stanley on the problem of counting the number of magic squares.