bluuebunny/crossref_metadata_2025 huggingface dataset:
A method of lines for a nonlinear abstract functional evolution equation
Authors: Kartsatos A. G. , Parrott M. E.
Published: June 2010

Let

X
X

be a real Banach space with

X
∗

{X^\ast }

uniformly convex. A method of lines is introduced and developed for the abstract functional problem (E)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u prime left-parenthesis t right-parenthesis plus upper A left-parenthesis t right-parenthesis u left-parenthesis t right-parenthesis equals upper G left-parenthesis t comma u Subscript t Baseline right-parenthesis comma u 0 equals phi comma t element-of left-bracket 0 comma upper T right-bracket period">

u
′
(
t
)
+
A
(
t
)
u
(
t
)
=
G
(
t
,

u
t

)
,

u
0

=
ϕ
,

t
∈
[
0
,
T
]
.

u\prime (t) + A(t)u(t) = G(t,{u_t}), \quad {u_0} = \phi , \quad t \in [0,T].

\]
The operators

A
(
t
)
:
D
⊂
X
→
X

A(t):D \subset X \to X

are

m
m

-accretive and

G
(
t
,
ϕ
)

G(t,\phi )

is a global Lipschitzian-like function in its two variables. Further conditions are given for the convergence of the method to a strong solution of (E). Recent results for perturbed abstract ordinary equations are substantially improved. The method applies also to large classes of functional parabolic problems as well as problems of integral perturbations. The method is straightforward because it avoids the introduction of the operators

A
^

(
t
)

\hat A(t)

and the corresponding use of nonlinear evolution operator theory.
Stability of the travelling wave solution of the FitzHugh-Nagumo system
Authors: Jones Christopher K. R. T.
Published: June 2010

Travelling wave solutions for the FitzHugh-Nagumo equations have been proved to exist, by various authors, close to a certain singular limit of the equations. In this paper it is proved that these waves are stable relative to the full system of partial differential equations; that is, initial values near (in the sup norm) to the travelling wave lead to solutions that decay to some translate of the wave in time. The technique used is the linearised stability criterion; the framework for its use in this context has been given by Evans [6-9]. The search for the spectrum leads to systems of linear ordinary differential equations. The proof uses dynamical systems arguments to analyse these close to the singular limit.
The transformation of vector-functions, scaling and bifurcation
Authors: Magnus R. J.
Published: June 2010

Various known methods for studying the bifurcation of zeros of a Banach-space-valued mapping are unified under a single idea, akin to using a coordinate transformation to obtain a simple form of the function under consideration. The general nature of the hypotheses permits the dropping of the pervasive "Fredholm condition" of bifurcation theory.
Functions of Φ-bounded variation and Riemann-Stieltjes integration
Authors: Schramm Michael
Published: June 2010

A notion of generalized bounded variation is introduced which simultaneously generalizes many of those previously examined. It is shown that the class of functions arising from this definition is a Banach space with a suitable norm. Appropriate variation functions are defined and examined, and an analogue of Helly’s theorem is estabished. The significance of this class to convergence of Fourier series is briefly discussed. A result concerning Riemann-Stieltjes integrals of functions of this class is proved.
𝐵𝑀𝑂(𝜌) and Carleson measures
Authors: Smith Wayne Stewart
Published: June 2010

This paper concerns certain generalizations of

BMO

{\text {BMO}}

, the space of functions of bounded mean oscillation. Let

ρ
\rho

be a positive nondecreasing function on

(
0
,
∞
)

(0,\infty )

with

ρ
(
0
+
)
=
0

\rho (0 + ) = 0

. A locally integrable function on

R

m

{{\mathbf {R}}^m}

is said to belong to

BMO

(
ρ
)

{\text {BMO}}(\rho )

if its mean oscillation over any cube

Q
Q

is

O
(
ρ
(
l
(
Q
)
)
)

O(\rho (l(Q)))

, where

l
(
Q
)

l(Q)

is the edge length of

Q
Q

. Carleson measures are known to be closely related to

BMO

{\text {BMO}}

. Generalizations of these measures are shown to be similarly related to the spaces

BMO

(
ρ
)

{\text {BMO}}(\rho )

. For a cube

Q
Q

in

R

m

,

|

Q

|

{{\mathbf {R}}^m},\;|Q|

denotes its volume and

R
(
Q
)

R(Q)

is the set
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace left-parenthesis x comma y right-parenthesis element-of bold upper R Subscript plus Superscript m plus 1 Baseline colon x element-of upper Q comma 0 greater-than y greater-than l left-parenthesis upper Q right-parenthesis right-brace">

{
(
x
,
y
)
∈

R

+

m
+
1

:
x
∈
Q
,

0
>
y
>
l
(
Q
)
}

\{ (x,y) \in {\mathbf {R}}_ + ^{m + 1}:x \in Q,\;0 > y > l(Q)\}

. A measure

μ
\mu

on

R

+

m
+
1

{\mathbf {R}}_ + ^{m + 1}

is called a

ρ
\rho

-Carleson measure if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue mu EndAbsoluteValue left-parenthesis upper R left-parenthesis upper Q right-parenthesis right-parenthesis equals upper O left-parenthesis rho left-parenthesis l left-parenthesis upper Q right-parenthesis right-parenthesis StartAbsoluteValue upper Q EndAbsoluteValue right-parenthesis">

|

μ

|

(
R
(
Q
)
)
=
O
(
ρ
(
l
(
Q
)
)

|

Q

|

)

|\mu |(R(Q)) = O(\rho (l(Q))|Q|)

, for all cubes

Q
Q

. L. Carleson proved that a compactly supported function in

BMO

{\text {BMO}}

can be represented as the sum of a bounded function and the balyage, or sweep, of some Carleson measure. A generalization of this theorem involving

BMO

(
ρ
)

{\text {BMO}}(\rho )

and

ρ
\rho

-Carleson measures is proved for a broad class of growth functions, and this is used to represent

BMO

(
ρ
)

{\text {BMO}}(\rho )

as a dual space. The proof of the theorem is based on a proof of J. Garnett and P. Jones of Carleson’s theorem. Another characterization of

BMO

(
ρ
)

{\text {BMO}}(\rho )

using

ρ
\rho

-Carleson measures is a corollary. This result generalizes a characterization of

BMO

{\text {BMO}}

due to C. Fefferman. Finally, an atomic decomposition of the predual of

BMO

(
ρ
)

{\text {BMO}}(\rho )

is given.
Of planar Eulerian graphs and permutations
Authors: Moran Gadi
Published: June 2010

Infinite planar Eulerian graphs are used to show that for

v
>
0

v > 0

the covering number of the infinite simple group

H
v

=
S

/

S
v

{H_v} = S/{S^v}

is two. Here

S
S

denotes the group of all permutations of a set of cardinality

ℵ
v

,

S
v

{\aleph _v},{S^v}

denotes its subgroup consisting of the permutations moving less than

ℵ
v

{\aleph _v}

elements, and the covering number of a (simple) group

G
G

is the smallest positive integer

n
n

satisfying

C
n

=
G

{C^n} = G

for every nonunit conjugacy class

C
C

in

G
G

.
On the model equations which describe nonlinear wave motions in a rotating fluid
Authors: Kim Jong Uhn
Published: June 2010

This paper concerns mathematical aspects of the two model equations describing nonlinear wave motions in a rotating fluid. We establish local existence of solutions and show that singularities occur in a finite time under certain hypotheses. We also show that these equations admit nonconstant travelling wave solutions.
Abstract theory of abelian operator algebras: an application of forcing
Authors: Jech Thomas J.
Published: June 2010

The abstract abelian operator theory is developed from a general standpoint, using the method of forcing and Boolean-valued models.
On the structure of weakly compact subsets of Hilbert spaces and applications to the geometry of Banach spaces
Authors: Argyros S. , Farmaki V.
Published: June 2010

A characterization of weakly compact subsets of a Hilbert space, when they are considered as subsets of

B
B

-spaces with an unconditional basis, is given. We apply this result to renorm a class of reflexive

B
B

-spaces by defining a norm uniformly convex in every direction. We also prove certain results related to the factorization of operators. Finally, we investigate the structure of weakly compact subsets of

L
1

(
μ
)

{L^1}(\mu )

.
Convexity conditions and intersections with smooth functions
Authors: Agronsky S. , Bruckner A. M. , Laczkovich M. , Preiss D.
Published: June 2010

A continuous function that agrees with each member of a family

F

\mathcal {F}

of smooth functions in a small set must itself possess certain desirable properties. We study situations that arise when

F

\mathcal {F}

consists of the family of polynomials of degree at most

n
n

, as well as certain larger families and when the small sets of agreement are finite. The conclusions of our theorems involve convexity conditions. For example, if a continuous function

f
f

agrees with each polynomial of degree at most

n
n

in only a finite set, then

f
f

is

(
n
+
1
)

(n + 1)

-convex or

(
n
+
1
)

(n + 1)

-concave on some interval. We consider also certain variants of this theorem, provide examples to show that certain improvements are not possible and present some applications of our results.
A geometric approach to the linear Penrose transform
Authors: Penkov I. B.
Published: June 2010

We show that under a certain cohomological condition the theorem of Witten, Isenberg, Yasskin and Green about the inverse Penrose transform of a (non-self-dual) connection

∇
\nabla

(together with Manin’s description of its curvature

F
∇

{F_\nabla }

) is true in a quite general situation. We then present a (multidimensional) version of the Penrose transform of a vector bundle in the language of jets. This gives a coordinate-free interpretation of certain results of Henkin and Manin, coding a number of classical field equations in terms of obstructions to infinitesimal extension of cohomology classes.
The de Branges theorem on univalent functions
Authors: FitzGerald Carl H. , Pommerenke Ch.
Published: June 2010

We present a simplified version of the de Branges proof of the Lebedev-Milin conjecture which implies the Robertson and Bieberbach conjectures. As an application of an analysis of the technique, it is shown that the method could not be used directly to prove the Bieberbach conjecture.
A note on a question of Bass
Authors: Dutta Sankar P.
Published: June 2010

In this paper we study the following question raised by Bass: Is a local ring with a finitely generated module of finite injective dimension Cohen-Macaulay? We prove that the answer is in the affirmative when a certain local cohomology of the ring is either decomposible or cyclic. We also study the above question in some special cases and some of its implications.
A quasi-invariance theorem for measures on Banach spaces
Authors: Bell Denis
Published: June 2010

We show that for a measure

γ
\gamma

on a Banach space directional differentiability implies quasi-translation invariance. This result is shown to imply the Cameron-Martin theorem. A second application is given in which

γ
\gamma

is the image of a Gaussian measure under a suitably regular map.
The cuspidal group and special values of 𝐿-functions
Authors: Stevens Glenn
Published: June 2010

The structure of the cuspidal divisor class group is investigated by relating this structure to arithmetic properties of special values of

L
L

-functions of weight two Eisenstein series. A new proof of a theorem of Kubert (Proposition 3.1) concerning the group of modular units is derived as a consequence of the method. The key lemma is a nonvanishing result (Theorem 2.1) for values of the “

L
L

-function” attached to a one-dimensional cohomology class over the relevant-congruence subgroup. Proposition 4.7 provides data regarding Eisenstein series and associated subgroups of the cuspidal divisor class group which the author hopes will simplify future calculations in the cuspidal group.
Duality theorems in deformation theory
Authors: Goldschmidt Hubert
Published: June 2010

We give a unified treatment of the construction of the Calabi sequence, which is a resolution of the sheaf of Killing vector fields on a Riemannian manifold of constant curvature, and of the resolution of the sheaf of conformal Killing vector fields on a conformally flat Riemannian manifold of dimension

⩾
3

\geqslant 3

introduced in [7]. We also explain why the latter resolution is selfadjoint and associate to certain geometric structures selfadjoint resolutions of their infinitesimal automorphisms.
Radial limits of 𝑀-subharmonic functions
Authors: Ullrich David
Published: June 2010

"

M
M

-subharmonic" functions are defined in the unit ball of

C

n

{{\mathbf {C}}^n}

. Their basic properties are developed, leading to the following generalization of a theorem of Littlewood: An

M
M

-subharmonic function such that its restrictions to spheres centered at the origin are bounded in

L
1

{L^1}

must have radial limits almost everywhere on the unit sphere.
Joint continuity of division of smooth functions. II. The distance to a Whitney stratified set from a transversal submanifold
Authors: Mostow Mark Alan
Published: June 2010

Let

S
S

be a closed set in

R

m

{{\mathbf {R}}^m}

, and let a

C
1

{C^1}

Whitney stratification of

S
S

be given. (Actually, only Whitney (a)-regularity is needed.) Let

f
:

R

n

→

R

m

f:{{\mathbf {R}}^n} \to {{\mathbf {R}}^m}

be a

C
1

{C^1}

map transversal to all the strata. Assume that the image of

f
f

intersects

S
S

. Then for each compact set

K
K

in

R

n

{{\mathbf {R}}^n}

, the Euclidean distances

ρ
(
x
,

f

−
1

(
S
)
)

\rho (x,{f^{ - 1}}(S))

and

ρ
(
f
(
x
)
,
S
)

\rho (f(x),S)

, for

x
x

in

K
K

, are bounded by constant multiples of each other. The bounding constants can be chosen to work for all maps

g
g

which are close enough to

f
f

in a

C
1

{C^1}

sense on a neighborhood of

K
K

. This result is used in part I (written jointly with S. Shnider) to prove a result on the joint continuity of the division of smooth functions [MS].
Approximation theorems for Nash mappings and Nash manifolds
Authors: Shiota Masahiro
Published: June 2010

Let

0
≤
r
>
∞

0 \leq r > \infty

. A Nash function on

R

n

{\mathbf {R}^n}

is a

C
r

{C^r}

function whose graph is semialgebraic. It is shown that a

C
r

{C^r}

Nash function is approximated by a

C
ω

{C^\omega }

Nash one in a strong topology defined in the same way as the usual topology on the space

S

\mathcal {S}

of rapidly decreasing

C
∞

{C^\infty }

functions. A

C
r

{C^r}

Nash manifold in

R

n

{\mathbf {R}^n}

is a semialgebraic

C
r

{C^r}

manifold. We also prove that a

C
r

{C^r}

Nash manifold for

r
≥
1

r \ge 1

is approximated by a

C
ω

{C^\omega }

Nash manifold, from which we can classify all

C
r

{C^r}

Nash manifolds by

C
r

{C^r}

Nash diffeomorphisms.
Solvability of differential equations with linear coefficients of real type
Authors: Felix Rainer
Published: June 2010

Let

L
L

be the infinitesimal generator associated with a flow on a manifold

M
M

. Regarding

L
L

as an operator on a space of testfunctions we deal with the question if

L
L

has closed range. (Questions of this kind are investigated in [4, 1, 2].) We provide conditions under which

L
+
μ
1
:

S

(
M
)
→

S

(
M
)

L + \mu 1:\mathcal {S}(M) \to \mathcal {S}(M)

,

μ
∈

C

\mu \in {\mathbf {C}}

, has closed range, where

M
=

R

n

×
K

M = {{\mathbf {R}}^n} \times K

,

K
K

being a compact manifold; here

S

(
M
)

\mathcal {S}(M)

is the Schwartz space of rapidly decreasing smooth functions. As a consequence we show that the differential operator

Σ

i
,
j

a

i
j

x
j

(
∂

/

∂

x
i

)
+
b

{\Sigma _{i,j}}{a_{ij}}{x_j}(\partial /\partial {x_i}) + b

defines a surjective mapping of the space

S

(

R

n

)

\mathcal {S}({{\mathbf {R}}^n})

of tempered distributions onto itself provided that all eigenvalues of the matrix

(

a

i
j

)

({a_{ij}})

are real. (In the case of imaginary eigenvalues this is not true in general [3].)
A solution of Warner’s 3rd problem for representations of holomorphic type
Authors: Williams Floyd L.
Published: June 2010

In response to one of ten problems posed by G. Warner, we assign (to the extent that it is possible) a geometric or cohomological interpretation— in the sense of Langlands—to the multiplicty in

L
2

(
Γ
∖
G
)

{L^2}(\Gamma \backslash G)

of an irreducible unitary representation

π
\pi

of a semisimple Lie group

G
G

, where

Γ
\Gamma

is a discrete subgroup of

G
G

, in the case when

π
\pi

has a highest weight.
A Martin boundary in the plane
Authors: Salisbury Thomas S.
Published: June 2010

Let

E
E

be an open connected subset of Euclidean space, with a Green function, and let

λ
\lambda

be harmonic measure on the Martin boundary

Δ
\Delta

of

E
E

. We will show that, except for a

λ
⊗
λ

\lambda \otimes \lambda

-null set of

(
x
,
y
)
∈

Δ
2

(x,y) \in {\Delta ^2}

,

x
x

is an entrance point for Brownian motion conditioned to leave

E
E

at

y
y

. R. S. Martin gave examples in dimension

3
3

or higher, for which there exist minimal accessible Martin boundary points

x
≠
y

x \ne y

for which this condition fails. We will give a similar example in dimension

2
2

.
Equilibrium points of nonatomic games over a Banach space
Authors: Khan M. Ali
Published: June 2010

Schmeidler’s results on the equilibrium points of nonatomic games with strategy sets in Euclidean

n
n

-space are generalized to nonatomic games with strategy sets in a Banach space. Our results also extend previous work of the author which assumed the Banach space to be separable and its dual to possess the Radon-Nikodým property. Our proofs use recent results in functional analysis.
Weak limits of stopped diffusions
Authors: Baxter J. R. , Chacon R. V. , Jain N. C.
Published: June 2010

We consider the following homogenization problem: Let Brownian motion in

R

d

{{\mathbf {R}}^d}

,

d
⩾
3

d \geqslant 3

, be killed on the surface of many small absorbing bodies (standard diffusion equation with Dirichlet boundary conditions). We investigate the limit as the number of bodies approaches infinity and the size of the bodies approaches

0
0

. By taking a weak limit of stopping times we replace a convergence problem on the state space by an identification of the limit on the sample space. This technique then gives results without smoothness assumptions which were previously necessary.
Bounds on the dimension of variations of Hodge structure
Authors: Carlson James A.
Published: June 2010

We derive upper bounds on the dimension of a variation of Hodge structure of weight two and show that these bounds are sharp. Using them we exhibit maximal geometric variations of Hodge structure. Analogous results for higher weight are obtained in the presence of a nondegeneracy hypothesis, and variations coming from hypersurfaces are shown to be nondegenerate. Maximal geometric variations of higher weight are also constructed.
On linking double lines
Authors: Migliore Juan
Published: June 2010

A double line is a nonreduced locally Cohen-Macaulay scheme of degree two supported on a line in projective three-space. The heart of this work is to compute the associated Hartshorne-Rao module for such a curve. We can then say exactly when two such curves are in the same liaison class and in fact when they are directly linked. In particular, we find that

C
C

is only self-linked in characteristic two.
On coupled multiparameter nonlinear elliptic systems
Authors: Cantrell Robert Stephen
Published: June 2010

This paper considers the system of nonlinear Dirichlet boundary value problems
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartBinomialOrMatrix upper L u left-parenthesis x right-parenthesis equals lamda f left-parenthesis u left-parenthesis x right-parenthesis comma v left-parenthesis x right-parenthesis right-parenthesis Choose upper L v left-parenthesis x right-parenthesis equals mu g left-parenthesis u left-parenthesis x right-parenthesis comma v left-parenthesis x right-parenthesis right-parenthesis EndBinomialOrMatrix comma x element-of normal upper Omega comma">

{

L
u
(
x
)
=
λ
f
(
u
(
x
)
,
v
(
x
)
)

L
v
(
x
)
=
μ
g
(
u
(
x
)
,
v
(
x
)
)

}

,

x
∈
Ω
,

\left \{ \begin {gathered} Lu(x) = \lambda f(u(x),v(x)) \hfill \\ Lv(x) = \mu g(u(x),v(x)) \hfill \\ \end {gathered} \right \},\qquad x \in \Omega ,

\]
a bounded domain in

R

n

{{\mathbf {R}}^n}

. Here

L
L

is a strongly, uniformly elliptic linear partial differential operator,

λ
\lambda

,

μ
\mu

are real parameters, and

f
f

,

g
:

R

2

→
R

g:{{\mathbf {R}}^2} \to R

are smooth with
\[

f
(
0
,
0
)
=
0
=
g
(
0
,
0
)
.

f(0,0) = 0 = g(0,0).

\]
A detailed analysis of the solution set to the system is given from the point of view of several parameter bifurcation theory.
Contributions from conjugacy classes of regular elliptic elements in Hermitian modular groups to the dimension formula of Hermitian modular cusp forms
Authors: Eie Min King
Published: June 2010

The dimension of the vector space of hermitian modular cusp forms on the hermitian upper half plane can be obtained from the Selberg trace formula; in this paper we shall compute the contributions from conjugacy classes of regular elliptic elements in hermitian modular groups by constructing an orthonomal basis in a certain Hilbert space of holomorphic functions. A generalization of the main Theorem can be applied to the dimension formula of cusp forms of

S
U
(
p
,

q
)

SU(p,\,q)

. A similar theorem was given for the case of regular elliptic elements of

Sp

(
n
,

Z

)

{\text {Sp}}(n,\,{\mathbf {Z}})

in [5] via a different method.
Total stability of sets for nonautonomous differential systems
Authors: Athanassov Zhivko S.
Published: June 2010

The principal purpose of this paper is to present sufficient conditions for total stability, or stability under constantly acting perturbations, of sets of a sufficiently general kind for nonautonomous ordinary differential equations. To do this, two Liapunov-like functions with specific properties are used. The obtained results include and considerably improve the classical results on total stability of isolated equilibrium points. Applications are presented to study the stability of nonautonomous Lurie-type nonlinear equations.
The spectrum (𝑃∧𝐵𝑃⟨2⟩)_{-∞}
Authors: Davis Donald M. , Johnson David C. , Klippenstein John , Mahowald Mark , Wegmann Steven
Published: June 2010

The spectrum

(
P
∧

BP

⟨

2

⟩

)

−
∞

{(P \wedge {\text {BP}}\langle {\text {2}}\rangle )_{ - \infty }}

is defined to be the homotopy inverse limit of spectra

P

−
k

∧

BP

⟨

2

⟩

{P_{ - k}} \wedge {\text {BP}}\langle {\text {2}}\rangle

, where

P

−
k

{P_{ - k}}

is closely related to stunted real projective spaces, and

BP

⟨

2

⟩

{\text {BP}}\langle {\text {2}}\rangle

is formed from the Brown-Peterson spectrum. It is proved that this spectrum is equivalent to the infinite product of odd suspensions of the

2
2

-adic completion of the spectrum of connective

K
K

-theory. An odd-primary analogue is also proved.
Insufficiency of Torres’ conditions for two-component classical links
Authors: Platt M. L.
Published: June 2010

Torres has given necessary conditions for a polynomial to be the Alexander polynomial of a two component link. For certain links, additional conditions are necessary. Hillman gave one example for linking number

6
6

. Here we give examples for all other linking numbers except

0
,
±
1

0, \pm 1

, and

±
2

\pm 2

.
Harmonic analysis on Grassmannian bundles
Authors: Strichartz Robert S.
Published: June 2010

The harmonic analysis of the Grassmannian bundle of

k
k

-dimensional affine subspaces of

R

n

{{\mathbf {R}}^n}

, as a homogeneous space of the Euclidean motion group, is given explicitly. This is used to obtain the diagonalization of various generalizations of the Radon transform between such bundles. In abstract form, the same technique gives the Plancherel formula for any unitary representation of a semidirect product

G
×
V

G \times V

(

V
V

a normal abelian subgroup) induced from an irreducible unitary representation of a subgroup of the form

H
×
W

H \times W

.
On the depth of the symmetric algebra
Authors: Herzog J. , Rossi M. E. , Valla G.
Published: June 2010

Let

(
R
,

m

)

(R,\mathfrak {m})

be a local ring. Assume that

R
=
A

/

I

R = A/I

, where

(
A
,

n

)

(A,\mathfrak {n})

is a regular local ring and

I
⊆

n

2

I \subseteq {\mathfrak {n}^2}

is an ideal. The depth of the symmetric algebra

S
(

m

)

S(\mathfrak {m})

of

m

\mathfrak {m}

over

R
R

is computed in terms of the depth of the associated graded module

gr

n

(
I
)

{\text {gr}_\mathfrak {n}}(I)

and the so-called "strong socle condition." Explicit results are obtained, for instance, if

I
I

is generated by a super-regular sequence, if

I
I

has a linear resolution or if

I
I

has projective dimension one.
The dual of the Bergman space 𝐴¹ in symmetric Siegel domains of type 𝐼𝐼
Authors: Békollé David
Published: June 2010

An affirmative answer is given to the following conjecture of R. Coifman and R. Rochberg: in any symmetric Siegel domain of type II, the dual of the Bergman space

A
1

{A^1}

coincides with the Bloch space of holomorphic functions and can be realized as the Bergman projection of

L
∞

{L^\infty }

.
Distant future and analytic measures
Authors: Tanaka Jun-ichi
Published: June 2010

Using a representation of analytic measures in terms of a flow built under a function, it is shown that a positive measure is the total variation measure of an analytic measure if and only if the distant future is the zero subspace. This settles a problem posed by Forelli in connection with his generalization of F. and M. Riesz theorems. We also provide another version of Helson’s existence theorem.
Explosion problems for symmetric diffusion processes
Authors: Ichihara Kanji
Published: June 2010

We discuss the explosion problem for a symmetric diffusion process. Hasminskii’s idea cannot be applied to this case. Instead, the theory of Dirichlet forms is employed to obtain criteria for conservativeness and explosion of the process. The fundamental criteria are given in terms of the

α
\alpha

-equilibrium potentials and

α
\alpha

-capacities of the unit ball centered at the origin. They are applied to obtain sufficient conditions on the coefficients of the infinitesimal generator for conservativeness and explosion.
The variation of the de Rham zeta function
Authors: Rosenberg Steven
Published: June 2010

Special values of the zeta function

ζ
(
s
)

\zeta (s)

for the Laplacian on forms

Δ
\Delta

on a compact Riemannian manifold are known to have geometric significance. We compute the variation of these special values with respect to the variation of the metric and write down the Euler-Lagrange equation for conformal variations. The invariant metric on a locally symmetric space is shown to be critical for every local Lagrangian. We also compute the variation of

ζ
′

(
0
)

\zeta ’(0)

, or equivalently of det

Δ
\Delta

. Finally, flat manifolds are characterized by flatness at a point and a condition on the amplitudes of the eigenforms of

Δ
\Delta

.
Concavity of solutions of the porous medium equation
Authors: Bénilan Philippe , Vázquez Juan Luis
Published: June 2010

We consider the problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper P right-parenthesis StartLayout Enlarged left-brace 1st Row u Subscript t Baseline equals left-parenthesis u Superscript m Baseline right-parenthesis Subscript x x Baseline comma 2nd Row u left-parenthesis x comma 0 right-parenthesis equals u 0 left-parenthesis x right-parenthesis EndLayout StartLayout 1st Row with x element-of bold upper R comma t greater-than 0 2nd Row for x element-of bold upper R EndLayout">

(

P

)

{

u
t

=

(

u
m

)

x
x

,

u
(
x
,
0
)
=

u
0

(
x
)

with

x
∈

R

,

t
>
0

for

x
∈

R

\left ( {\text {P}} \right )\quad \quad \left \{ {\begin {array}{*{20}{c}} {{u_t} = {{({u^m})}_{xx}},} \\ {u(x,0) = {u_0}(x)} \\ \end {array} } \right .\quad \begin {array}{*{20}{c}} {{\text {with}}\,x \in {\mathbf {R}},\,t > 0} \\ {{\text {for}}\,x \in {\mathbf {R}}} \\ \end {array}

\]
where

m
>
1

m > 1

and

u
0

{u_0}

is a continuous, nonnegative function that vanishes outside an interval

(
a
,

b
)

(a,\,b)

and such that

(

u
0

m
−
1

)
≤
−
C
≤
0

(u_0^{m - 1}) \leq - C \leq 0

in

(
a
,

b
)

(a,\,b)

. Using a Trotter-Kato formula we show that the solution conserves the concavity in time: for every

t
>
0
,

u
(
x
,
t
)

t > 0,\,u(x,t)

vanishes outside an interval
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega left-parenthesis t right-parenthesis equals left-parenthesis Subscript zeta Baseline 1 Baseline left-parenthesis t right-parenthesis comma Subscript zeta Baseline 2 Baseline left-parenthesis t right-parenthesis right-parenthesis">

Ω
(
t
)
=
(

ζ
1

(
t
)
,

ζ
2

(
t
)
)

\Omega (t) = ({}_{\zeta 1}(t),\,{}_{\zeta 2}(t))

and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis u Superscript m minus 1 Baseline right-parenthesis Subscript x x Baseline less-than-or-equal-to minus StartFraction upper C Over 1 plus upper C left-parenthesis m left-parenthesis m plus 1 right-parenthesis slash left-parenthesis m minus 1 right-parenthesis right-parenthesis t EndFraction">

(

u

m
−
1

)

x
x

≤
−

C

1
+
C
(
m
(
m
+
1
)

/

(
m
−
1
)
)
t

{({u^{m - 1}})_{xx}} \leq - \frac {C} {{1 + C(m(m + 1)/(m - 1))t}}

\]
in

Ω
(
t
)

\Omega (t)

. Consequently the interfaces

x

=

ζ
i

(
t
)

x{ = _{\zeta i}}(t)

,

i
=
1
,

2

i = 1,\,2

, are concave curves. These results also give precise information about the large time behavior of solutions and interfaces.
Category and group rings in homotopy theory
Authors: Ralph William J.
Published: June 2010

It frequently arises in algebraic topology that a function

β
:
G
→
H

\beta :G \to H

, between two groups, is not a homomorphism. We show that in many standard situations

β
\beta

induces a group homomorphism

β
¯

:

Z

(
G
)

/

A

d

→
H

\overline \beta :{\mathbf {Z}}(G)/{\mathcal {A}^d} \to H

, where

A

d

{\mathcal {A}^d}

is a power of the augumentation ideal in the group ring

Z

(
G
)

{\mathbf {Z}}(G)

. A typical example is

β
:
[
X
,

Y
]
→
[

S
2

X
,

S
2

Y
]

\beta :[X,\,Y] \to [{S^2}X,\,{S^2}Y]

where

Y
Y

is some

H
H

-group, in which case

d
d

can be taken to be

1
+

cat

X

1 + {\text {cat}}\,X

.
Geometric theory of extremals in optimal control problems. I. The fold and Maxwell case
Authors: Kupka I.
Published: June 2010

The behavior of the extremal curves in optimal control theory is much more complex than that of their namesakes in the classical calculus of variations. Here we analyze the simplest instances of singular behavior of these extremals. Among others, in sharp contrast to the classical case, a

C
0

{C^0}

-limit of extremals may not be an extremal. In the simplest cases (elliptic fold and Maxwell points) of this occurrence, the limits are trajectories of a new vector field. A special case of this field showed up in some work of V. I. Arnold. Results related to ours have been obtained in low dimension by I. Ekeland.
The dimension of closed sets in the Stone-Čech compactification
Authors: Keesling James
Published: June 2010

In this paper properties of compacta

K
K

in

β
X
∖
X

\beta X\backslash X

are studied for Lindelöf spaces

X
X

. If

dim

K
=
∞

{\operatorname {dim}}\,K = \infty

, then there is a mapping

f
:
K
→

T
c

f:K \to {T^c}

such that

f
f

is onto and every mapping homotopic to

f
f

is onto. This implies that there is an essential family for

K
K

consisting of

c
c

disjoint pairs of closed sets. It also implies that if

K
=
∪

{

K
α

|

α
>
c

}

K = \cup \left \{ {{K_\alpha }|\alpha > c} \right \}

with each

K
α

{K_\alpha }

closed, then there is a

β
\beta

such that

dim

K
β

=
∞

{\operatorname {dim}}\,{K_\beta } = \infty

. Assume

K
K

is a compactum in

β
X
∖
X

\beta X\backslash X

as above. Then if

dim

K
=
n

{\operatorname {dim}}\,K = n

, there is a closed set

K
′

K’

in

K
K

such that

dim

K
′

=
n

{\operatorname {dim}}\,K’ = n

and such that every nonempty

G
δ

{G_\delta }

-set in

K
′

K’

contains an

n
n

-dimensional compactum. This holds for

n
n

finite or infinite. If

dim

K
=
n

{\operatorname {dim}}\,K = n

and

K
=
∪

{

K
α

|

α
>

ω
1

}

K = \cup \left \{ {{K_\alpha }|\alpha > {\omega _1}} \right \}

with each

K
α

{K_\alpha }

closed, then there must be a

β
\beta

such that

dim

K
β

=
n

{\operatorname {dim}}\,{K_\beta } = n

.
Balanced subgroups of finite rank completely decomposable abelian groups
Authors: Nongxa Loyiso G.
Published: June 2010

It is proved that, if a finite rank completely decomposable group has extractable typeset of cardinality at most 5, all its balanced subgroups are also completely decomposable. Balanced Butler groups with extractable typeset of size at most 3 are almost completely decomposable and decompose into rank 1 and/or rank 3 indecomposable summands. We also construct an indecomposable balanced Butler group whose extractable typeset is of size 4 which fails to be almost completely decomposable.
Quadratic geometry of numbers
Authors: Schlickewei Hans Peter , Schmidt Wolfgang M.
Published: June 2010

We give upper bounds for zeros of quadratic forms. For example we prove that for any nondegenerate quadratic form

F

(

x
1

,
…
,

x
n

)

\mathfrak {F}({x_1}, \ldots ,\,{x_n})

with rational integer coefficients which vanishes on a

d
d

-dimensional rational subspace

(
d
>
0
)

(d > 0)

there exist sublattices

Γ
0

,

Γ
1

,

…

,

Γ

n
−
d

{\Gamma _0},\,{\Gamma _1},\, \ldots \,,{\Gamma _{n - d}}

of

Z

n

{\mathbf {Z}^n}

of rank

d
d

, on which

F

\mathfrak {F}

vanishes, with the following properties:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rank left-parenthesis normal upper Gamma 0 intersection normal upper Gamma Subscript i Baseline right-parenthesis equals d minus 1 comma rank left-parenthesis normal upper Gamma 0 union normal upper Gamma 1 union midline-horizontal-ellipsis union normal upper Gamma Subscript n minus d Baseline right-parenthesis equals n">

rank

(

Γ
0

∩

Γ
i

)
=
d
−
1
,

rank

(

Γ
0

∪

Γ
1

∪
⋯
∪

Γ

n
−
d

)
=
n

{\text {rank}}({\Gamma _0} \cap {\Gamma _i}) = d - 1,\quad {\text {rank}}({\Gamma _0} \cup {\Gamma _1} \cup \cdots \cup {\Gamma _{n - d}}) = n

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis det normal upper Gamma 0 right-parenthesis Superscript n minus d Baseline det normal upper Gamma 1 midline-horizontal-ellipsis det normal upper Gamma Subscript n minus d Baseline much-less-than upper F Superscript left-parenthesis n minus d right-parenthesis squared">

(
det

Γ
0

)

n
−
d

det

Γ
1

⋯
det

Γ

n
−
d

≪

F

(
n
−
d
)

2

{(\det \,{\Gamma _0})^{n - d}}\det \,{\Gamma _1} \cdots \det \,{\Gamma _{n - d}} \ll {F^{{{(n - d)}^2}}}

\]
, where

F
F

is the maximum modulus of the coefficients of

F

\mathfrak {F}

.
Toeplitz operators on the Segal-Bargmann space
Authors: Berger C. A. , Coburn L. A.
Published: June 2010

In this paper, we give a complete characterization of those functions on

2
n

2n

-dimensional Euclidean space for which the Berezin-Toeplitz quantizations admit a symbol calculus modulo the compact operators. The functions in question are characterized by a condition of “small oscillation at infinity” .
𝐾-theory and right ideal class groups for HNP rings
Authors: Hodges Timothy J.
Published: June 2010

Let

R
R

be an hereditary Noetherian prime ring, let

S
S

be a "Dedekind closure" of

R
R

and let

T

\mathcal {T}

be the category of finitely generated

S
S

-torsion

R
R

-modules. It is shown that for all

i
≥
0

i \geq 0

, there is an exact sequence
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 right-arrow upper K Subscript i Baseline left-parenthesis script upper T right-parenthesis right-arrow upper K Subscript i Baseline left-parenthesis upper R right-parenthesis right-arrow upper K Subscript i Baseline left-parenthesis upper S right-parenthesis right-arrow 0">

0
→

K
i

(

T

)
→

K
i

(
R
)
→

K
i

(
S
)
→
0

0 \to {K_i}(\mathcal {T}) \to {K_i}(R) \to {K_i}(S) \to 0

. If

i
=
0

i = 0

, or

R
R

has finitely many idempotent ideals then this sequence splits. A notion of "right ideal class group" is then introduced for hereditary Noetherian prime rings which generalizes the standard definition of class group for hereditary orders over Dedekind domains. It is shown that there is a decomposition

K
0

(
R
)
≅

Cl

(
R
)
⊕
F

{K_0}(R) \cong {\text {Cl}}(R) \oplus F

where

F
F

is a free abelian group whose rank depends on the number of idempotent maximal ideals of

R
R

. Moreover there is a natural isomorphism

Cl

(
R
)
≅

Cl

(
S
)

{\text {Cl}}(R) \cong {\text {Cl}}(S)

and this decomposition corresponds closely to the splitting of the above exact sequence for

K
0

{K_0}

.
Holomorphic mappings on 𝑙₁
Authors: Ryan Raymond A.
Published: June 2010

We describe the holomorphic mappings of bounded type, and the arbitrary holomorphic mappings from the complex Banach space

l
1

{l_1}

into a complex Banach space

X
X

. It is shown that these mappings have monomial expansions and the growth of the norms of the coefficients is characterized in each case. This characterization is used to give new descriptions of the compact open topology and the Nachbin ported topology on the space

H

(

l
1

;
X
)

\mathcal {H}({l_1};X)

of holomorphic mappings, and to prove a lifting property for holomorphic mappings on

l
1

{l_1}

. We also show that the monomials form an equicontinuous unconditional Schauder basis for the space

(

H

(

l
1

)
,

τ
0

)

(\mathcal {H}({l_1}),{\tau _0})

of holomorphic functions on

l
1

{l_1}

with the topology of uniform convergence on compact sets.
A new proof that Teichmüller space is a cell
Authors: Fischer A. E. , Tromba A. J.
Published: June 2010

A new proof is given, using the energy of a harmonic map, that Teichmüller space is a cell.
Homological stability for 𝑂_{𝑛,𝑛} over a local ring
Authors: Betley Stanisław
Published: June 2010

Let

R
R

be a local ring,

V

2
n

{V^{2n}}

a free module over

R
R

of rank

2
n

2n

and

q
q

a bilinear form on

V

2
n

{V^{2n}}

which has in some basis the matrix

|

0

a
m
p
;
1

1

a
m
p
;
0

|

\left | {\begin {array}{*{20}{c}} 0 & 1 \\ 1 & 0 \\ \end {array} } \right |\,

. Let

O

n
,
n

{O_{n,n}}

be the group of automorphisms of

V

2
n

{V^{2n}}

which preserve

q
q

. We prove the following theorem: if

n
n

is big enough with respect to

k
k

then the inclusion homomorphism

i
:

O

n
,
n

→

O

n
+
1
,
n
+
1

i:{O_{n,n}} \to {O_{n + 1,n + 1}}

induces an isomorphism
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i Subscript asterisk Baseline colon upper H Subscript k Baseline left-parenthesis upper O Subscript n comma n Baseline semicolon upper Z right-parenthesis right-arrow upper H Subscript k Baseline left-parenthesis upper O Subscript n plus 1 comma n plus 1 Baseline semicolon upper Z right-parenthesis">

i

∗

:

H
k

(

O

n
,
n

;

Z
)
→

H
k

(

O

n
+
1
,
n
+
1

;
Z
)

{i_{\ast }}:{H_k}({O_{n,n}};\,Z) \to {H_k}({O_{n + 1,n + 1}};Z)

.
Singularly perturbed Dirichlet problems with subquadratic nonlinearities
Authors: DeSanti Albert J.
Published: June 2010

Boundary and interior layer theory is provided for a class of singularly perturbed Dirichlet problems with subquadratic nonlinearities in the derivative terms. The results obtained generalize and extend well-known results on the semilinear problem.
Taming wild extensions with Hopf algebras
Authors: Childs Lindsay N.
Published: June 2010

Let

K
⊂
L

K \subset L

be a Galois extension of number fields with abelian Galois group

G
G

and rings of integers

R
⊂
S

R \subset S

, and let

A

\mathcal {A}

be the order of

S
S

in

K
G

KG

. If

A

\mathcal {A}

is a Hopf

R
R

-algebra with operations induced from

K
G

KG

, then

S
S

is locally isomorphic to

A

\mathcal {A}

as

A

\mathcal {A}

-module. Criteria are found for

A

\mathcal {A}

to be a Hopf algebra when

K
=

Q

K = {\mathbf {Q}}

or when

L

/

K

L/K

is a Kummer extension of prime degree. In the latter case we also obtain a complete classification of orders over

R
R

in

L
L

which are tame or Galois

H
H

-extensions,

H
H

a Hopf order in

K
G

KG

, using a generalization of the discriminant.
Ideals of holomorphic functions with 𝐶^{∞} boundary values on a pseudoconvex domain
Authors: Bierstone Edward , Milman Pierre D.
Published: June 2010

We give natural sufficient conditions for the solution of several problems concerning division in the space

A

∞

(
Ω
)

{\mathcal {A}^\infty }(\Omega )

of holomorphic functions with

C

∞

{\mathcal {C}^\infty }

boundary values on a pseudoconvex domain

Ω
\Omega

with smooth boundary. The sufficient conditions come from upper semicontinuity with respect to the analytic Zariski topology of a local invariant of coherent analytic sheaves (the "invariant diagram of initial exponents"), and apply to division in the space of

C

∞

{\mathcal {C}^\infty }

Whitney functions on an arbitrary closed set. Our theorem on division in

A

∞

(
Ω
)

{\mathcal {A}^\infty }(\Omega )

follows using Kohn’s theorem on global regularity in the

∂
¯

\bar \partial

-Neumann problem.
All infinite groups are Galois groups over any field
Authors: Dugas Manfred , Göbel Rüdiger
Published: June 2010

Let

G
G

be an arbitrary monoid with

1
1

and right cancellation, and

K
K

be a given field. We will construct extension fields

F
⊇
K

F \supseteq K

with endomorphism monoid End

F
F

isomorphic to

G
G

modulo Frobenius homomorphisms. If

G
G

is a group, then Aut

F
=
G

F = G

. Let

F
G

{F^G}

denote the fixed elements of

F
F

under the action of

G
G

. In the case that

G
G

is an infinite group, also

F
G

=
K

{F^G} = K

and

G
G

is the Galois group of

F
F

over

K
K

. If

G
G

is an arbitrary group, and

G
=
1

G = 1

, respectively, this answers an open problem (R. Baer 1967, E. Fried, C. U. Jensen, J. Thompson) and if

G
G

is infinite, the result is an infinite analogue of the still unsolved Hilbert-Noether conjecture inverting Galois theory. Observe that our extensions

K
⊂
F

K \subset F

are not algebraic. We also suggest to consider the case

K
=

C

K = {\mathbf {C}}

and

G
=
{
1
}

G = \{ 1\}

.
Existence domains of holomorphic functions of restricted growth
Authors: Jarnicki M. , Pflug P.
Published: June 2010

The paper presents a large class of domains

G
G

of holomorphy in

C

n

{{\mathbf {C}}^n}

such that, for any

N
>
0

N > 0

, there exists a nonextendable holomorphic function

f
f

on

G
G

with

|

f

|

δ
G
N

|f|\delta _G^N

bounded where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta Subscript upper G Baseline left-parenthesis z right-parenthesis colon equals min left-parenthesis left-parenthesis 1 plus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis Superscript negative 1 slash 2 Baseline comma d i s t left-parenthesis z comma partial-differential upper G right-parenthesis right-parenthesis">

δ
G

(
z
)
:=
min
(

(
1
+

|

z

|

2

)

−
1

/

2

,

dist
⁡
(
z
,

∂
G
)
)

{\delta _G}(z): = \min ({(1 + |z{|^2})^{ - 1 / 2}},\,\operatorname {dist} (z,\,\partial G))

. Any fat Reinhardt domain of holomorphy belongs to this class. On the other hand we characterize those Reinhardt domains of holomorphy which are existence domains of bounded holomorphic functions.
Dense morphisms in commutative Banach algebras
Authors: Corach Gustavo , Suárez Fernando Daniel
Published: June 2010

Using a new notion of stability we compute exactly the stable rank of the polydisc algebra, extend Oka’s extension theorem to

n
n

-tuples of functions without common zeros and give an estimation for a question raised by Swan concerning the stable rank of a dense subalgebra of a given Banach algebra.
Chaotic maps with rational zeta function
Authors: Nusse H. E.
Published: June 2010

Fix a nontrivial interval

X
⊂

R

X \subset {\mathbf {R}}

and let

f
∈

C
1

(
X
,

X
)

f \in {C^1}(X,\,X)

be a chaotic mapping. We denote by

A
∞

(
f
)

{A_\infty }(f)

the set of points whose orbits do not converge to a (one-sided) asymptotically stable periodic orbit of

f
f

or to a subset of the absorbing boundary of

X
X

for

f
f

. A. We assume that

f
f

satisfies the following conditions: (1) the set of asymptotically stable periodic points for

f
f

is compact (an empty set is allowed), and (2)

A

∞

(
f
)

A{\,_\infty }(f)\,

is compact,

f
f

is expanding on

A
∞

(
f
)

{A_\infty }(f)

. Then we can associate a matrix

A
f

{A_f}

with entries either zero or one to the mapping

f
f

such that the number of periodic points for

f
f

with period

n
n

is equal to the trace of the matrix

[

A
f

]

n

{\left [ {{A_f}} \right ]^n}

; furthermore the zeta function of

f
f

is rational having the eigenvalues of

A
f

{A_f}

as poles. B. We assume that

f
∈

C
3

(
X
,

X
)

f \in {C^3}(X,\,X)

such that: (1) the Schwarzian derivative of

f
f

is negative, and (2) the closure of

A
∞

(
f
)

{A_\infty }(f)

is compact and

f
′

(
x
)
≠
0

f’ (x) \ne 0

for all

x
x

in the closure of

A
∞

(
f
)

{A_\infty }(f)

. Then we obtain the same result as in A.
On the initial-boundary value problem for a Bingham fluid in a three-dimensional domain
Authors: Kim Jong Uhn
Published: June 2010

The initial-boundary value problem associated with the motion of a Bingham fluid is considered. The existence and uniqueness of strong solution is proved under a certain assumption on the data. It is also shown that the solution exists globally in time when the data are small and that the solution converges to a periodic solution if the external force is time-periodic.
Symmetry breaking for a class of semilinear elliptic problems
Authors: Ramaswamy Mythily , Srikanth P. N.
Published: June 2010

We study positive solutions of the Dirichlet problem for

−
Δ
u
=

u
p

−
λ

- \Delta u = {u^p} - \lambda

,

p
>
1

p > 1

,

λ
>
0

\lambda > 0

, on the unit ball

Ω
\Omega

. We show that there exists a positive solution

(

u
0

,

λ
0

)

({u_0},\,{\lambda _0})

of this problem which satisfies in addition

∂

u
0

/

∂
n
=
0

\partial {u_0}/\partial n = 0

on

∂
Ω

\partial \Omega

. We prove also that at

(

u
0

,

λ
0

)

({u_0},\,{\lambda _0})

, the symmetry breaks, i.e. asymmetric solutions bifurcate from the positive radial solutions.
A lifting theorem and uniform algebras
Authors: Nakazi Takahiko , Yamamoto Takanori
Published: June 2010

In this paper we discuss the possible generalizations of a lifting theorem of a

2
×
2

2 \times 2

matrix to uniform algebras. These have applications to Hankel operators, weighted norm inequalities for conjugation operators and Toeplitz operators on uniform algebras. For example, the Helson-Szegö theorems for general uniform algebras follow.
𝑘-dimensional regularity classifications for 𝑠-fractals
Authors: Martín Miguel Ángel , Mattila Pertti
Published: June 2010

We study subsets

E
E

of

R

n

{{\mathbf {R}}^n}

which are

H
s

{H^s}

measurable and have

0
>

H
s

(
E
)
>
∞

0 > {H^s}(E) > \infty

, where

H
s

{H^s}

is the

s
s

-dimensional Hausdorff measure. Given an integer

k
k

,

s
⩽
k
⩽
n

s \leqslant k \leqslant n

, we consider six (

s
s

,

k
k

) regularity definitions for

E
E

in terms of

k
k

-dimensional subspaces or surfaces of

R

n

{{\mathbf {R}}^n}

. If

s
=
k

s = k

, they all agree with the (

H
k

{H^k}

,

k
k

) rectifiability in the sense of Federer, but in the case

s
>
k

s > k

we show that only two of them are equivalent. We also study sets with positive lower density, and projection properties in connection with these regularity definitions.
There is no exactly 𝑘-to-1 function from any continuum onto [0,1], or any dendrite, with only finitely many discontinuities
Authors: Heath Jo W.
Published: June 2010

Katsuura and Kellum recently proved [8] that any (exactly)

k
k

-to

1
1

function from

[
0
,

1
]

[0,\,1]

onto

[
0
,

1
]

[0,\,1]

must have infinitely many discontinuities, and they asked if the theorem remains true if the domain is any (compact metric) continuum. The result in this paper, that any (exactly)

k
k

-to-

1
1

function from a continuum onto any dendrite has finitely many discontinuities, answers their question in the affirmative.