bluuebunny/crossref_metadata_2025 huggingface dataset:
On the Friedrichs operator
Authors: Lin Peng , Rochberg Richard
Published: June 2010

Let

Ω
\Omega

be a simply connected domain in

C

1

{\mathbb {C}^1}

with the area measure dA. Let

P
¯

Ω

{\bar P_\Omega }

be the orthogonal projection from

L
2

(
Ω
,
d
A
)

{L^2}(\Omega ,dA)

onto the closed subspace of antiholomorphic functions in

L
2

(
Ω
,
d
A
)

{L^2}(\Omega ,dA)

. The Friedrichs operator

T
¯

Ω

{\bar T_\Omega }

associated to

Ω
\Omega

is the operator from the Bergman space

L
a
2

(
Ω
)

L_a^2(\Omega )

into

L
2

(
Ω
,
d
A
)

{L^2}(\Omega ,dA)

defined by

T
¯

Ω

f
=

P
¯

Ω

f

{\bar T_\Omega }f = {\bar P_\Omega }f

. In this note, some smoothness conditions on the boundary of

Ω
\Omega

are given such that the Friedrichs operator

T
¯

Ω

{\bar T_\Omega }

belongs to the Schatten classes

S
p

{S_p}

.
A measure with a large set of tangent measures
Authors: O’Neil Toby
Published: June 2010

There exists a Borel regular, finite, non-zero measure

μ
\mu

on

R

d

{\mathbb {R}^d}

such that for

μ
\mu

-a.e. x the set of tangent measures of

μ
\mu

at x consists of all non-zero, Borel regular, locally finite measures on

R

d

{\mathbb {R}^d}

.
On the existence of flat planes in spaces of nonpositive curvature
Authors: Bridson Martin R.
Published: June 2010

Let X be a proper 1-connected geodesic metric space which is non-positively curved in the sense that it satisfies Gromov’s

CAT

(
0
)

{\text {CAT}}(0)

condition globally. If X is cocompact, then either it is

δ
\delta

-hyperbolic, for some

δ
>
0

\delta > 0

, or else it contains an isometrically embedded copy of the Euclidean plane; these conditions are mutually exclusive. It follows that if the fundamental group of a compact non-positively curved polyhedron K is not word-hyperbolic, then the universal cover of K contains a flat plane.
Weak convergence and weak compactness in abstract 𝑀 spaces
Authors: Chen Shu Tao , Sun Hui Ying
Published: June 2010

This paper presents some properties of bounded linear functionals on

σ
\sigma

complete abstract M spaces, from which some criteria for weak convergence and weak compactness in such spaces are obtained.
On the torsion part of 𝐶^{[𝑛]} with respect to the action of a derivation
Authors: Coomes Brian
Published: June 2010

We show that the torsion part of

C

[
n
]

{\mathbb {C}^{[n]}}

with respect to the action of a derivation is algebraically closed in

C

[
n
]

{\mathbb {C}^{[n]}}

if the flow associated with the derivation is analytic on

C

×

C

n

\mathbb {C} \times {\mathbb {C}^n}

. We also present a connection between this result and Keller’s Jacobian conjecture.
Knotted symmetric graphs
Authors: Livingston Charles
Published: June 2010

For a knotted graph in

S
3

{S^3}

we define the vertex constant group, a quotient of the fundamental group of the complement. For planar graphs the group is cyclic. For graphs with periodic symmetry the group is related to the fundamental group of the branched cover of

S
3

{S^3}

branched over knots contained in the quotient of the graph under the symmetry. These tools are used to demonstrate that a large family of knotted graphs are not planar.
Conformally homeomorphic Lorentz surfaces need not be conformally diffeomorphic
Authors: Smyth Robert W. , Weinstein Tilla
Published: June 2010

A Lorentz surfaceL\mathcal {L}is an ordered pair (S, [h]) whereSis an orientedC∞{C^\infty }2-manifold and [h] the set of allC∞{C^\infty }metrics conformally equivalent to a fixedC∞{C^\infty }Lorentzian metrichonS. (Thus Lorentz surfaces are the indefinite metric analogs of Riemann surfaces.) This paper describes subsets of the Minkowski 2-plane which are conformally homeomorphic, but not evenC1{C^1}conformally diffeomorphic. It also describes subsets of the Minkowski 2-plane which areCj{C^j}but notCj+1{C^{j + 1}}conformally diffeomorphic for any fixedj=1,2,…j = 1,2, \ldots. Finally, the paper describes a Lorentz surface conformally homeomorphic to a subset of the Minkowski 2-plane, but notC1{C^1}conformally diffeomorphic to any subset of the Minkowski 2-plane.
Points joined by three shortest paths on convex surfaces
Authors: Zamfirescu Tudor
Published: June 2010

LetSbe a convex surface andx∈Sx \in S. It is shown here that the set of all points ofSjoined withxby at least three shortest paths can be dense inS. It is proven that, in fact, in the sense of Baire categories most convex surfaces have this property, for anyx. Moreover, on most convex surfaces, for most of their points, there is just one farthest point (in the intrinsic metric), and precisely three shortest paths lead to that point.
Number of equilibrium states of piecewise monotonic maps of the interval
Authors: Buzzi Jérôme
Published: June 2010

We prove a bound of the form suggested by S. Newhouse for the number of measures with maximal entropy for a piecewise monotonic map with N monotonicity intervals:

4
(
N
−
1
)

4(N - 1)

. More generally we consider a potential

ϕ
\phi

of bounded distortion. If

sup
ϕ
>
P
(
f
,
ϕ
)

\sup \phi > P(f,\phi )

, we give an explicit bound in terms of N and of the pressure.
Operators with finite chain length and the ergodic theorem
Authors: Laursen K. B. , Mbekhta M.
Published: June 2010

With a technical assumption (E-k), which is a relaxed version of the condition

T
n

/

n
→
0
,
n
→
∞

{T^n}/n \to 0,n \to \infty

, where T is a bounded linear operator on a Banach space, we prove a generalized uniform ergodic theorem which shows, inter alias, the equivalence of the finite chain length condition
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X equals left-parenthesis upper I minus upper T right-parenthesis Superscript k Baseline upper X circled-plus kernel left-parenthesis upper I minus upper T right-parenthesis Superscript k Baseline right-parenthesis">

(
X
=

(
I
−
T

)
k

X
⊕
ker
⁡

(
I
−
T

)
k

)

(X = {(I - T)^k}X \oplus \ker {(I - T)^k})

, of closedness of

(
I
−
T

)
k

X

{(I - T)^k}X

, and of quasi-Fredholmness of

I
−
T

I - T

. One consequence, still assuming (E-k), is that

I
−
T

I - T

is semi-Fredholm if and only if

I
−
T

I - T

is Riesz-Schauder. Other consequences are: a uniform ergodic theorem and conditions for ergodicity for certain classes of multipliers on commutative semisimple Banach algebras.
Amenability and uniformly continuous functionals on the algebras 𝐴_{𝑝}(𝐺) for discrete groups
Authors: Xu Guangwu
Published: June 2010

In this paper, it is shown that for every discrete group G and

1
>
p
>
∞
,

A
p

(
G
)
⋅
P

M
p

(
G
)

1 > p > \infty ,{A_p}(G) \cdot P{M_p}(G)

is closed in

P

M
p

(
G
)

P{M_p}(G)

if and only if G is amenable.
Solution of the Baire order problem of Mauldin
Authors: Balcerzak Marek , Rogowska Dorota
Published: June 2010

Let X be an uncountable Polish space, and let I be a proper

σ
\sigma

-ideal of subsets of X such that

{
x
}
∈
I

\{ x\} \in I

for each

x
∈
X

x \in X

. Denote by

B
α

(
I
)
,
α
≤

ω
1

{B_\alpha }(I),\alpha \leq {\omega _1}

, the Baire system generated by the family of functions

f
:
X
→

R

f:X \to \mathbb {R}

continuous I almost everywhere. We prove that if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r left-parenthesis upper I right-parenthesis equals min left-brace right-brace colon less-than-or-equal-to less-than-or-equal-to alpha omega 1 colon equals equals of BB plus plus alpha 1 left-parenthesis right-parenthesis upper I of BB alpha left-parenthesis right-parenthesis upper I">

r
(
I
)
=
min
{
α
≤

ω
1

:

B

α
+
1

(
I
)
=

B
α

(
I
)
}

r(I) = \min \{ \alpha \leq {\omega _1}:{B_{\alpha + 1}}(I) = {B_\alpha }(I)\}

, then either

r
(
I
)
=
1

r(I) = 1

or

r
(
I
)
=

ω
1

r(I) = {\omega _1}

. This answers the problem raised by R. D. Mauldin in 1973.
The radical of a modular alternative loop algebra
Authors: Goodaire Edgar G.
Published: June 2010

IfGis a group of order2n{2^n}andFis a field of characteristic 2, it is well known that the augmentation ideal of the group algebraFGis nilpotent. In this paper, we extend this result to alternative loop algebras.
Periodicity and indecomposability
Authors: Ingram W. T.
Published: June 2010

In this paper we characterize the existence of periodic points of odd period greater than one for unimodal mappings of an interval onto itself. The interesting juxtaposition of this condition with the occurrence in inverse limits of the well-known Brouwer-Janiszewski-Knaster continuum is explored. Also obtained is a characterization of indecomposability of certain inverse limits using a single unimodal bonding map.
Fixed points via “biased maps”
Authors: Jungck G. , Pathak H. K.
Published: June 2010

A generalization of compatible maps called "biased maps" is introduced and used to prove fixed point theorems for Meir-Keeler type contractions involving four maps. Extensions of known results are thereby obtained. In particular, a theorem by Kang and Rhoades is generalized.
Free amalgams of rank two
Authors: Rosset Shmuel
Published: June 2010

For every

n
>
1

n > 1

and

m
>
1

m > 1

we construct an amalgamated free product

U

∗
W

V

U{ \ast _W}V

which is free of rank 2, while U is (free) of rank n , V is of rank m, and W is, necessarily, of rank

n
+
m
−
2

n + m - 2

.
Theorem of Kuratowski-Suslin for measurable mappings
Authors: Wiśniewski Andrzej
Published: June 2010

The purpose of this paper is to describe these Borel mappings on a separable complete metric space X which transform every measurable set (with respect to some measure

μ
\mu

on X) onto a measurable one. It is shown that a one-to-one Borel mapping f on X fulfills the above property if and only if the measure

μ
\mu

is absolutely continuous with respect to the measure

μ
f

{\mu _f}

(an image of

μ
\mu

under the mapping f). Our results are a generalization of the classical results of Suslin and Kuratowski.
Countably additive homomorphisms between von Neumann algebras
Authors: Bunce L. J. , Hamhalter J.
Published: June 2010

Let M and N be von Neumann algebras where M has no abelian direct summand. A

∗
\ast

-homomorphism

π
:
M
→
N

\pi :M \to N

is said to be countably additive if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi left-parenthesis sigma-summation Underscript 1 Overscript normal infinity Endscripts e Subscript n Baseline right-parenthesis equals sigma-summation Underscript 1 Overscript normal infinity Endscripts pi left-parenthesis e Subscript n Baseline right-parenthesis">

π
(

∑
1
∞

e
n

)
=

∑
1
∞

π
(

e
n

)

\pi (\sum \nolimits _1^\infty {{e_n}) = \sum \nolimits _1^\infty {\pi ({e_n})} }

, for every sequence

(

e
n

)

({e_n})

of orthogonal projections in M. We prove that a

∗
\ast

-homomorphism

π
:
M
→
N

\pi :M \to N

is countably additive if and only if

π
(
e
∨
f
)
=
π
(
e
)
∨
π
(
f
)

\pi (e \vee f) = \pi (e) \vee \pi (f)

for every pair of projections e and f of M. A corollary is that if, in addition, M has no Type

I

2

{{\text {I}}_2}

direct summands, then every lattice morphism from the projections of M into the projections of N is a

σ
\sigma

-lattice morphism.
Nonnegative solutions of the radial Laplacian with nonlinearity that changes sign
Authors: Cac N. P. , Fink A. M. , Gatica J. A.
Published: June 2010

We find a solution to the radial Laplacian equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y plus StartFraction upper N minus 1 Over x EndFraction y Superscript prime Baseline plus lamda a left-parenthesis x right-parenthesis f left-parenthesis y right-parenthesis equals 0 comma y prime left-parenthesis 0 right-parenthesis equals y left-parenthesis 1 right-parenthesis equals 0">

y
+

N
−
1

x

y
′

+
λ
a
(
x
)
f
(
y
)
=
0
,

y
′

(
0
)
=
y
(
1
)
=
0

y + \frac {{N - 1}}{x}y’ + \lambda a(x)f(y) = 0,y’ (0) = y(1) = 0

when a may change sign and is "sufficiently positive". The function f is qualitatively like

e
y

{e^y}

, and we conclude solutions for

0
≤
λ
≤

λ
0

0 \leq \lambda \leq {\lambda _0}

.
The minimal error conjugate gradient method is a regularization method
Authors: Hanke Martin
Published: June 2010

The regularizing properties of the conjugate gradient iteration, applied to the normal equation of a linear ill-posed problem, were established by Nemirovskii in 1986. A seemingly more attractive variant of this algorithm is the minimal error method suggested by King. The present paper analyzes the regularizing properties of the minimal error method. It is shown that the discrepancy principle is no regularizing stopping rule for the minimal error method. Instead, a different stopping rule is suggested which leads to order-optimal convergence rates.
Compact elements and smallest faithful representation of 𝐶*-algebras
Authors: Hauser Raphael A. , Matzinger Heinrich
Published: June 2010

Let A be a

C
∗

{C^ \ast }

-algebra which either allows a faithful separable representation or is postliminal. We prove that A then admits a smallest faithful representation if and only if the ideal of compact elements is an essential ideal in A.
Weak maximality condition and polycyclic groups
Authors: Kim Y. K. , Rhemtulla A. H.
Published: June 2010

A group G is called strongly restrained if there exists an integer n such that

⟨

x

(
y
)

⟩

\langle {x^{(y)}}\rangle

can be generated by n elements for all x, y in G. We show that a group G is polycyclic-by-finite if and only if G is a finitely generated strongly restrained group in which every nontrivial finitely generated subgroup has a nontrivial finite quotient. This provides a general setting for various results in soluble and residually finite groups that have appeared recently.
Constant mean curvature discs with bounded area
Authors: López Rafael , Montiel Sebastián
Published: June 2010

It has been long conjectured that the two spherical caps are then only discs in the Euclidean three-space

R

3

{\mathbb {R}^3}

with non-zero constant mean curvature spanning a round circle. In this work, we prove that it is true when the area of such a disc is less than or equal to that of the big spherical cap.
The Tate conjecture for 𝑡-motives
Authors: Taguchi Yuichiro
Published: June 2010

A version of the Tate conjecture is proved for

φ
\varphi

-modules of "t-motive type".
Topological classification of irreducible plane curve singularities in terms of Weierstrass polynomials
Authors: Kang Chunghyuk
Published: June 2010

Let

f
(
z
,
y
)

f(z,y)

be analytically irreducible at 0 and

f
(
0
)
=
0

f(0) = 0

. Then the plane curve singularity defined by f has the same topological type as the curve defined by

f

k
+
1

{f_{k + 1}}

for some

k
≥
0

k \geq 0

where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f 1 equals z Superscript a Baseline plus y Superscript b Baseline comma f 2 equals f 1 Superscript n 21 Baseline plus y Superscript m 11 Baseline z Superscript m 12 Baseline comma f 3 equals f 2 Superscript n 31 Baseline plus f 1 Superscript n 22 Baseline y Superscript m 21 Baseline z Superscript m 22 Baseline comma ellipsis">

f
1

=

z
a

+

y
b

,

f
2

=

f
1

n

21

+

y

m

11

z

m

12

,

f
3

=

f
2

n

31

+

f
1

n

22

y

m

21

z

m

22

,
…

{f_1} = {z^a} + {y^b},{f_2} = {f_1}^{{n_{21}}} + {y^{{m_{11}}}}{z^{{m_{12}}}},{f_3} = {f_2}^{{n_{31}}} + {f_1}^{{n_{22}}}{y^{{m_{21}}}}{z^{{m_{22}}}}, \ldots

are defined by induction on k with distinct numerical conditions topologically invariant. Moreover, we give an easy alternate proof of Zariski’s topological classification theorem of irreducible plane curve singularities.
On residually finite-dimensional 𝐶*-algebras
Authors: Archbold R. J.
Published: June 2010

Exel and Loring have listed several conditions that are equivalent to the residual finite-dimensionality of a

C
∗

{C^ \ast }

-algebra. We review and extend this list.
Congruence lattices of small planar lattices
Authors: Grätzer G. , Lakser H. , Schmidt E. T.
Published: June 2010

For a finite distributive lattice D with n join-irreducible elements, we construct a finite (planar) lattice L with

O
(

n
2

)

O({n^2})

elements such that the congruence lattice of L is isomorphic to D. This improves on an early result of R. P. Dilworth (around 1940) and G. Grätzer and E. T. Schmidt (1962) constructing such a (nonplanar) lattice L with

O
(

2

2
n

)

O({2^{2n}})

elements, and on a recent construction of G. Grätzer and H. Lakser which yields a finite (planar) lattice L with

O
(

n
3

)

O({n^3})

elements.
On a fixed point theorem of Kirk
Authors: Morales Claudio H. , Mutangadura Simba A.
Published: June 2010

Let X be a reflexive Banach space, D an open and bounded subset of X, and

T
:

D
¯

→
X

T:\bar D \to X

a continuous mapping which is locally pseudocontractive on D. Suppose there exists an element

z
∈
D

z \in D

such that

‖

z
−
T
z

‖

>

‖

x
−
T
x

‖

\left \| {z - Tz} \right \| > \left \| {x - Tx} \right \|

for all x on the boundary of D. Then under the so-called condition (S), T has a fixed point in D. Although this result was proved earlier by Kirk, we show here a much easier approach.
Commutants for some classes of Hausdorff matrices
Authors: Rhoades B. E.
Published: June 2010

Let

Γ
\Gamma

denote the algebra of all bounded infinite matrices on c, the space of convergent sequences,

Δ
\Delta

the subalgebra of

Γ
\Gamma

consisting of lower triangular matrices. It is well known that, if H is any Hausdorff matrix with distinct diagonal entries, then the commutant of H in

Δ
\Delta

contains only Hausdorff matrices. In previous work the author has shown that a necessary condition for the commutant of a Hausdorff matrix H to be the same in

Γ
\Gamma

and

Δ
\Delta

is that H have distinct diagonal entries, but that the condition is not sufficient. In this paper it is shown that certain Hausdorff matrices, with distinct diagonal entries, have the same commutants in

Γ
\Gamma

and

Δ
\Delta

.
A geometrical characterization of algebraic varieties of 𝐶²
Authors: Aupetit Bernard
Published: June 2010

We prove that a closed subset

V
≠

C

2

V \ne {\mathbb {C}^2}

is an algebraic variety if all its horizontal sections and its vertical sections are finite or a complex line and if

C

2

∖
V

{\mathbb {C}^2}\backslash V

is an open set of holomorphy (equivalently pseudoconvex). This result has important consequences in the theory of the socle for Jordan-Banach algebras.
Generalizations of 𝑀-groups
Authors: Berkovich Yakov
Published: June 2010

In this note we prove some generalizations of Taketa’s theorem on solvability of M-groups.
Quasi-isometries of hyperbolic space are almost isometries
Authors: Cooper Daryl
Published: June 2010

In this paper we show that for

n
≥
3

n \geq 3

a quasi-isometry of hyperbolic n-space

H

n

{\mathbb {H}^n}

to itself is almost an isometry, in the sense that the image of most points on a sphere of radius r are close to a sphere of the same radius. To be more precise, the result is that given

K
>
1

K > 1

and

ϵ
>
0

\epsilon > 0

there is a

δ
>
0

\delta > 0

such that the image of any sphere of any radius r under any K-quasi-isometry lies within a distance of

δ
\delta

of another sphere of radius r, except for the image of a proportion

ϵ
\epsilon

of the source sphere. We illustrate our result with a quasi-isometry of

H

n

{\mathbb {H}^n}

for which the image of a sphere is the analog of an ellipsoid in Euclidean space. There is no corresponding result when

n
=
2

n = 2

. This failure is illustrated by lifting to the universal cover a surface diffeomorphism which is not isotopic to an isometry.
Completion theorem for cohomological dimensions
Authors: Olszewski Wojciech
Published: June 2010

We prove that for every separable metrizable space X with

dim
G

X
≤
n

{\dim _G}X \leq n

, there exists a metrizable completion Y of X with

dim
G

Y
≤
n

{\dim _G}Y \leq n

provided that G is either a countable group or a torsion group, and with

dim
G

Y
≤
n
+
1

{\dim _G}Y \leq n + 1

if G is an arbitrary group.
Absolutely continuous invariant measures for a class of affine interval exchange maps
Authors: Coelho Zaqueu , Lopes Artur , da Rocha Luiz Fernando
Published: June 2010

We consider a class

A

\mathcal {A}

of affine interval exchange maps of the interval and we analyse several ergodic properties of the elements of this class, among them the existence of absolutely continuous invariant probability measures. The maps of this class are parametrised by two values a and b, where

a
,
b
∈
(
0
,
1
)

a,b \in (0,1)

. There is a renormalization map T defined from

A

\mathcal {A}

to itself producing an attractor given by the set

R

\mathcal {R}

of pure rotations, i.e. the set of (a, b) such that

b
=
1
−
a

b = 1 - a

. The density of the absolutely continuous invariant probability and the rotation number of the elements of the class

A

\mathcal {A}

are explicitly calculated. We also show how the continued fraction expansion of this rotation number can be obtained from the renormalization map.
Some new analytical techniques and their application to irregular cases for the third order ordinary linear boundary-value problem
Authors: Stanley Nathaniel R.
Published: June 2010

1. For the operator

T
3
−

(
D
)

T_3^ - (D)

defined by

−

d
3

/

d

x
3

- {d^3}/d{x^3}

and a triple of boundary conditions irregular in the sense of Birkhoff, the reduction of this triple to canonical forms is implicit in the reduction made for a more general third order operator (Theorem 1.2). 2. A new technique is developed for calculating the Green’s function for the nth order ordinary linear boundary-value problem (Theorem 2.4), and is applied to

T
3
−

T_3^ -

; a necessary and sufficient condition is given for the identification of degenerate sets of boundary conditions for

T
3
−

T_3^ -

(Theorem 2.6). 3. A new technique is developed for calculating asymptotic expansions for large zeros of exponential sums, and the form of the expansion, which includes a logarithmic asymptotic series, is established by induction (Theorem 3.1); expansions for the cube roots of the eigenvalues of

T
3
−

T_3^ -

then follow as special cases. 4. A theorem of Dunford and Schwartz (Theorem 4.0) giving a sufficient condition for completeness of eigenfunctions in terms of growth of the norm of the resolvent operator, is applied to prove that, with a possible exception, the eigenfunctions of

T
3
−

T_3^ -

span

L
2

(
0
,
1
)

{L_2}(0,1)

(Theorem 4.5).
Extension methods in cardinal arithmetic
Authors: Ellentuck Erik
Published: June 2010

Functions (relations) defined on the nonnegative integers are extended to the cardinal numbers by the method of Myhill (Nerode) respectively. We obtain various results relating these extensions and conclude with an analysis of AE Horn sentences interpreted in the cardinal numbers. Let

A

\mathfrak {A}

be the sentence

(
∀

x
1

)
⋯
(
∀

x
n

)
(
∃
!
y
)

b

(\forall {x_1}) \cdots (\forall {x_n})(\exists !y)\mathfrak {b}

where quantifiers are restricted to the Dedekind cardinals and

b

\mathfrak {b}

is an equation built up from functors for cardinal addition, multiplication, and integer constants. One of our principal results is that

A

\mathfrak {A}

is a theorem of set theory (with the axiom of choice replaced by the axiom of choice for sets of finite sets) if and only if we can prove that the uniquely determined Skolem function for

A

\mathfrak {A}

extends an almost combinatorial function.
Singly generated homogeneous 𝐹-algebras
Authors: Carpenter Ronn
Published: June 2010

With each point

m
m

in the spectrum of a singly generated

F
F

-algebra we associate an algebra

A
m

{A_m}

of germs of functions. It is shown that if

A
m

{A_m}

is isomorphic to the algebra of germs of analytic functions of a single complex variable, then the spectrum of

A
A

contains an analytic disc about

m
m

. The algebra

A
A

is called homogeneous if the algebras

A
m

{A_m}

are all isomorphic. If

A
A

is homogeneous and none of the algebras

A
m

{A_m}

have zero divisors, we show that

A
A

is the direct sum of its radical and either an algebra of analytic functions or countably many copies of the complex numbers. If

A
A

is a uniform algebra which is homogeneous, then it is shown that

A
A

is either the algebra of analytic functions on an open subset of the complex numbers or the algebra of all continuous functions on its spectrum.
Estimates for the number of real-valued continuous functions
Authors: Comfort W. W. , Hager Anthony W.
Published: June 2010

It is a familiar fact that

|

C
(
X
)

|

≦

2

δ
X

|C(X)| \leqq {2^{\delta X}}

, where

|

C
(
X
)

|

|C(X)|

is the cardinal number of the set of real-valued continuous functions on the infinite topological space

X
X

, and

δ
X

\delta X

is the least cardinal of a dense subset of

X
X

. While for metrizable spaces equality obtains, for some familiar spaces—e.g., the one-point compactification of the discrete space of cardinal

2
ℵ
0

2\aleph 0

—the inequality can be strict, and the problem of more delicate estimates arises. It is hard to conceive of a general upper bound for

|

C
(
X
)

|

|C(X)|

which does not involve a cardinal property of

X
X

as an exponent, and therefore we consider exponential combinations of certain natural cardinal numbers associated with

X
X

. Among the numbers are

w
X

wX

, the least cardinal of an open basis, and

w
c
X

wcX

, the least

m

\mathfrak {m}

for which each open cover of

X
X

has a subfamily with

m

\mathfrak {m}

or fewer elements whose union is dense. We show that

|

C
(
X
)

|

≦

(
w
X

)

w
c
X

|C(X)| \leqq {(wX)^{wcX}}

, and that this estimate is best possible among the numbers in question. (In particular,

(
w
X

)

w
c
X

≦

2

δ
X

{(wX)^{wcX}} \leqq {2^{\delta X}}

always holds.) In fact, it is only with the use of a version of the generalized continuum hypothesis that we succeed in finding an

X
X

for which

|

C
(
X
)

|

>

(
w
X

)

w
c
X

|C(X)| > {(wX)^{wcX}}

.
The local limit theorem and some related aspects of super-critical branching processes
Authors: Athreya Krishna B. , Ney Peter
Published: June 2010

Let

{

Z
n

:
n
=
0
,
1
,
2
,
…
}

\{ {Z_n}:n = 0,1,2, \ldots \}

be a Galton-Watson branching process with offspring p.g.f.

f
(
s
)
=

Σ
0
∞

p
j

s
j

f(s) = \Sigma _0^\infty {p_j}{s^j}

. Assume (i)

1
>
m
=

f
′

(
1
−
)
=

Σ
1
∞

j

p
j

>
∞

1 > m = f’(1 - ) = \Sigma _1^\infty j{p_j} > \infty

, (ii)

Σ
1
∞

j
2

p
j

>
∞

\Sigma _1^\infty {j^2}{p_j} > \infty

and (iii)

γ
0

=

f
′

(
q
)
>
0

{\gamma _0} = f’(q) > 0

, where

q
q

is the extinction probability of the process. Let

w
(
x
)

w(x)

denote the density function of

W
W

, the almost sure limit of

Z
n

m

−
n

{Z_n}{m^{ - n}}

with

Z
0

=
1
,

w

(
i
)

(
x
)

{Z_0} = 1,{w^{(i)}}(x)

the

i
i

-fold convolution of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w left-parenthesis x right-parenthesis comma upper P Subscript n Baseline left-parenthesis i comma j right-parenthesis equals upper P left-parenthesis upper Z Subscript n Baseline equals j vertical-bar upper Z 0 equals i right-parenthesis comma delta 0 equals left-parenthesis log gamma 0 Superscript negative 1 Baseline right-parenthesis left-parenthesis log m right-parenthesis Superscript negative 1">

w
(
x
)
,

P
n

(
i
,
j
)
=
P
(

Z
n

=
j

|

Z
0

=
i
)
,

δ
0

=
(
log
⁡

γ
0

−
1

)

(
log
⁡
m

)

−
1

w(x),{P_n}(i,j) = P({Z_n} = j|{Z_0} = i),{\delta _0} = (\log \gamma _0^{ - 1}){(\log m)^{ - 1}}

and

β
0

=

m

δ
0

/

(
3
+

δ
0

)

{\beta _0} = {m^{{\delta _0}/(3 + {\delta _0})}}

. Then for any

0
>
β
>

β
0

0 > \beta > {\beta _0}

and

i
i

we can find a constant

C
=
C
(
i
,
β
)

C = C(i,\beta )

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue m Superscript n Baseline upper P Subscript n Baseline left-parenthesis i comma j right-parenthesis minus w Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis m Superscript negative n Baseline j right-parenthesis EndAbsoluteValue less-than-over-equals upper C left-bracket beta 0 Superscript negative n Baseline left-parenthesis m Superscript negative n Baseline j right-parenthesis Superscript negative 1 Baseline plus beta Superscript negative n Baseline right-bracket">

|

m
n

P
n

(
i
,
j
)
−

w

(
i
)

(

m

−
n

j
)

|

≦
C
[

β
0

−
n

(

m

−
n

j

)

−
1

+

β

−
n

]

|{m^n}{P_n}(i,j) - {w^{(i)}}({m^{ - n}}j)| \leqq C[\beta _0^{ - n}{({m^{ - n}}j)^{ - 1}} + {\beta ^{ - n}}]

\]
for all

j
≧
1

j \geqq 1

. Applications to the boundary theory of the associated space time process are also discussed.
Strong renewal theorems with infinite mean
Authors: Erickson K. Bruce
Published: June 2010

Let F be a nonarithmetic probability distribution on

(
0
,
∞
)

(0,\infty )

and suppose

1
−
F
(
t
)

1 - F(t)

is regularly varying at

∞
\infty

with exponent

α
,
0
>
α
≦
1

\alpha ,0 > \alpha \leqq 1

. Let

U
(
t
)
=
Σ

F

n
∗

(
t
)

U(t) = \Sigma {F^{{n^ \ast }}}(t)

be the renewal function. In this paper we first derive various asymptotic expressions for the quantity

U
(
t
+
h
)
−
U
(
t
)

U(t + h) - U(t)

as

t
→
∞
,
h
>
0

t \to \infty ,h > 0

fixed. Next we derive asymptotic relations for the convolution

U
∗

z
(
t
)
,
t
→
∞

{U^ \ast }z(t),t \to \infty

, for a large class of integrable functions z. All of these asymptotic relations are expressed in terms of the truncated mean function

m
(
t
)
=

∫
0
t

[
1
−
F
(
x
)
]
d
x

m(t) = \smallint _0^t[1 - F(x)]dx

, t large, and appear as the natural extension of the classical strong renewal theorem for distributions with finite mean. Finally in the last sections of the paper we apply the special case

α
=
1

\alpha = 1

to derive some limit theorems for the distributions of certain waiting times associated with a renewal process.
The spectrum of partial differential operators on 𝐿^{𝑝}(𝑅ⁿ)
Authors: Iha Franklin T. , Schubert C. F.
Published: June 2010

The purpose of this paper is to prove that if the polynomial

P
(
ξ
)

P(\xi )

associated with a partial differential operator

P
P

on

L
p

(

R
n

)

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

, with constant coefficients, has the growth property,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper P left-parenthesis xi right-parenthesis EndAbsoluteValue Superscript negative 1 Baseline equals upper O left-parenthesis StartAbsoluteValue xi EndAbsoluteValue Superscript negative r Baseline right-parenthesis comma StartAbsoluteValue xi EndAbsoluteValue right-arrow normal infinity">

|

P
(
ξ
)

|

−
1

=
O
(

|

ξ

|

−
r

)
,

|

ξ

|

→
∞

|P(\xi ){|^{ - 1}} = O(|\xi {|^{ - r}}),|\xi | \to \infty

for some

r
>
0

r > 0

, then the spectrum of

P
P

is either the whole complex plane or it is the numerical range of

P
(
ξ
)

P(\xi )

; and if

P
(
ξ
)

P(\xi )

has some additional property (all the coefficients of

P
(
ξ
)

P(\xi )

being real, for example), then the spectrum of

P
P

is the numerical range for those

p
p

sufficiently close to 2. It is further shown that the growth property alone is not sufficient to ensure that the spectrum of

P
P

is the numerical range of

P
(
ξ
)

P(\xi )

.
Logarithmic convexity, first order differential inequalities and some applications
Authors: Levine Howard Allen
Published: June 2010

Let, for

t
∈
[
0
,
T
)
(
T
>
∞
)
,
D
(
t
)

t \in [0,T)(T > \infty ),D(t)

be a dense linear subspace of a Hilbert space

H
H

, and let

M
(
t
)

M(t)

and

N
(
t
)

N(t)

be linear operators (possibly unbounded) mapping

D
(
t
)

D(t)

into

H
H

. Let

f
:
[
0
,
T
)
×
H
→
H

f:[0,T) \times H \to H

. We give sufficient conditions on

M
,
N

M,N

and

f
f

in order to insure uniqueness and stability of solutions to
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 right-parenthesis upper M left-parenthesis t right-parenthesis d u slash d t equals upper N left-parenthesis t right-parenthesis u plus f left-parenthesis t comma u right-parenthesis comma u left-parenthesis 0 right-parenthesis given period">

(
1
)

M
(
t
)
d
u

/

d
t
=
N
(
t
)
u
+
f
(
t
,
u
)
,

u
(
0
)

given
.

(1)\quad M(t)du/dt = N(t)u + f(t,u),\quad u(0)\;\text {given}.

\]
This problem is not in general well posed in the sense of Hadamard. We cite some examples of (1) from the literature. We also give some examples of the problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 2 right-parenthesis upper M left-parenthesis t right-parenthesis StartFraction d squared u Over d t squared EndFraction equals upper N left-parenthesis t right-parenthesis u plus f left-parenthesis t comma u comma StartFraction d u Over d t EndFraction right-parenthesis comma u left-parenthesis 0 right-parenthesis comma StartFraction d u Over d u EndFraction left-parenthesis 0 right-parenthesis prescribed comma">

(
2
)

M
(
t
)

d
2

u

d

t
2

=
N
(
t
)
u
+
f

(

t
,
u
,

d
u

d
t

)

,

u
(
0
)
,

d
u

d
u

(
0
)

prescribed
,

(2)\quad M(t)\frac {{{d^2}u}}{{d{t^2}}} = N(t)u + f\left ( {t,u,\frac {{du}}{{dt}}} \right ),\quad u(0),\frac {{du}}{{du}}(0)\;\text {prescribed},

\]
for which questions of uniqueness and stability were discussed in a previous paper.
On 2-groups with no normal abelian subgroups of rank 3, and their occurrence as Sylow 2-subgroups of finite simple groups
Authors: MacWilliams Anne R.
Published: June 2010

We prove that in a finite

2
2

-group with no normal Abelian subgroup of rank

≧
3

\geqq 3

, every subgroup can be generated by four elements. This result is then used to determine which

2
2

-groups

T
T

with no normal Abelian subgroup of rank

≧
3

\geqq 3

can occur as

S
2

{S_2}

’s of finite simple groups

G
G

, under certain assumptions on the embedding of

T
T

in

G
G

.
The existence of solutions of abstract partial difference polynomials.
Authors: Bentsen Irving
Published: June 2010

A partial difference (p.d.) ring is a commutative ring

R
R

together with a (finite) set of isomorphisms (called transforming operators) of

R
R

into

R
R

which commute under composition. It is shown here that (contrary to the ordinary theory [R. M. Cohn, Difference algebra]) there exist nontrivial algebraically irreducible abstract p.d. polynomials having no solution and p.d. fields having no algebraically closed p.d. overfield. If

F
F

is a p.d. field with two transforming operators, then the existence of a p.d. overfield of

F
F

whose underlying field is an algebraic closure of that of

F
F

is a necessary and sufficient condition for every nontrivial algebraically irreducible abstract p.d. polynomial

P
P

in the p.d. polynomial ring
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-brace y Superscript left-parenthesis 1 right-parenthesis Baseline comma y Superscript left-parenthesis 2 right-parenthesis Baseline comma ellipsis comma y Superscript left-parenthesis n right-parenthesis Baseline right-brace">

F
{

y

(
1
)

,

y

(
2
)

,
…
,

y

(
n
)

}

F\{ {y^{(1)}},{y^{(2)}}, \ldots ,{y^{(n)}}\}

to have a solution

η
\eta

(in some p.d. overfield of

F
F

) such that:

η
\eta

has

n
−
1

n - 1

transformal parameters,

η
\eta

is not a proper specialization over

F
F

of any other solution of

P
P

, and, if

Q
Q

is a p.d. polynomial whose indeterminates appear effectively in

P
P

and

Q
Q

is annulled by

η
\eta

, then

Q
Q

is a multiple of

P
.
P

P.P

has at most finitely many isomorphically distinct such solutions. Necessity holds if

F
F

has finitely many transforming operators.
Locally noetherian commutative rings
Authors: Heinzer William , Ohm Jack
Published: June 2010

This paper centers around the theorem that a commutative ring

R
R

is noetherian if every

R
P

,
P

{R_P},P

prime, is noetherian and every finitely generated ideal of

R
R

has only finitely many weak-Bourbaki associated primes. A more precise local version of this theorem is also given, and examples are presented to show that the assumption on the weak-Bourbaki primes cannot be deleted nor replaced by the assumption that Spec

(
R
)

(R)

is noetherian. Moreover, an alternative statement of the theorem using a variation of the weak-Bourbaki associated primes is investigated. The proof of the theorem involves a knowledge of the behavior of associated primes of an ideal under quotient ring extension, and the paper concludes with some remarks on this behavior in the more general setting of flat ring extensions.
Bounded holomorphic functions of several complex variables. I
Authors: Kim Dong Sie
Published: June 2010

A domain of bounded holomorphy in a complex analytic manifold is a maximal domain for which every bounded holomorphic function has a bounded analytic continuation. In this paper, we show that this is a local property: if, for each boundary point of a domain, there exists a bounded holomorphic function which cannot be continued to any neighborhood of the point, then there exists a single bounded holomorphic function which cannot be continued to any neighborhood of the boundary points.
Necessary conditions for stability of diffeomorphisms
Authors: Franks John
Published: June 2010

S. Smale has recently given sufficient conditions for a diffeomorphism to be

Ω
\Omega

-stable and conjectured the converse of his theorem. The purpose of this paper is to give some limited results in the direction of that converse. We prove that an

Ω
\Omega

-stable diffeomorphism

f
f

has only hyperbolic periodic points and moreover that if

p
p

is a periodic point of period

k
k

then the

k
k

th roots of the eigenvalues of

d

f
p
k

df_p^k

are bounded away from the unit circle. Other results concern the necessity of transversal intersection of stable and unstable manifolds for an

Ω
\Omega

-stable diffeomorphism.
Generating and cogenerating structures
Authors: Beachy John A.
Published: June 2010

A functor

T
:

A

→

B

T:\mathcal {A} \to \mathcal {B}

acts faithfully on the right of a class of objects

A

′

\mathcal {A}’

of

A

\mathcal {A}

if it distinguishes morphisms out of objects of

A

′

\mathcal {A}’

(that is,

A
′

∈

A

′

,
X
∈

A

,
f
,
g
∈

A

(

A
′

,
X
)

A’ \in \mathcal {A}’,X \in \mathcal {A},f,g \in \mathcal {A}(A’,X)

and

f
≠
g

f \ne g

implies

T
(
f
)
≠
T
(
g
)
)

T(f) \ne T(g))

. We define a full subcategory

R

F

(
T
)

\mathcal {R}\mathcal {F}(T)

such that

T
T

acts faithfully on the right of the objects of

R

F

(
T
)

\mathcal {R}\mathcal {F}(T)

. An object

U
∈

A

U \in \mathcal {A}

is a generator if

H
U

:

A

→

E

n
s

{H^U}:\mathcal {A} \to \mathcal {E}ns

is faithful, and if

H
U

{H^U}

is not faithful, we may still consider

R

F

(

H
U

)

\mathcal {R}\mathcal {F}({H^U})

. This gives rise to the notion of a generating structure. Cogenerating structures are defined dually, and various canonical generating and cogenerating structures are defined for the category of

R
R

-modules. Relationships between these can be used in the homological classification of rings.
Monotonicity, convexity and symmetric derivates
Authors: Weil Clifford E.
Published: June 2010

If the first lower symmetric derivate of a continuous function is nonnegative, then it is nondecreasing. If the second lower symmetric derivate of a continuous function is nonnegative, then it is convex. In this paper it is shown that if continuity is replaced by Baire one, Darboux in each of these, then the resulting statements are true.
Closed 3-manifolds with no periodic maps
Authors: Raymond Frank , Tollefson Jeffrey L.
Published: June 2010

Examples of closed, orientable, aspherical 3-manifolds are constructed on which every action of a finite group is trivial.
The spectral theory of posets and its applications to 𝐶*-algebras
Authors: Dooley A. H.
Published: June 2010

This paper uses methods from the spectral theory of partially ordered sets to clarify and extend some recent results concerning approximately finite-dimensional

C
∗

{C^\ast }

-algebras. An extremely explicit description is obtained of the Jacobson topology on the primitive ideal space, and it is shown that this topology has a basis of quasi-compact open sets. In addition, the main results of [4] are proved using only elementary means.
𝑍_{𝑝}actions on symplectic manifolds
Authors: Rowlett R. J.
Published: June 2010

A bordism classification is studied for periodic maps of prime period p preserving a symplectic structure on a smooth manifold. In sharp contrast to the corresponding oriented bordism, this theory contains nontrivial ptorsion even when p is odd. Calculation gives an upper limit on the size of this p-torsion.
Docility at infinity and compactifications of ANR’s
Authors: Sher R. B.
Published: June 2010

Various conditions of contractibility and extensibility at

∞
\infty

for locally compact metric spaces are studied. These are shown to be equivalent if the space under consideration is an absolute neighborhood retract (ANR) and an ANR satisfying them is called docile at

∞
\infty

. Docility at

∞
\infty

is invariant under proper homotopy domination. The ANR X is docile at

∞
\infty

if and only if FX (the Freudenthal compactification of X) is an ANR and

F
X
−
X

FX - X

is unstable in FX; the inclusion of X into FX is a homotopy equivalence.
A minimax formula for dual 𝐵*-algebras
Authors: Wong Pak Ken
Published: June 2010

Let A be a dual

B
∗

{B^\ast }

-algebra. We give a minimax formula for the positive elements in A. By using this formula and some of its consequent results, we introduce and study the symmetric norms and symmetrically-normed ideals in A.
The degree of approximation for generalized polynomials with integral coefficients
Authors: von Golitschek M.
Published: June 2010

The classcal Müntz theorem and the so-called Jackson-Müntz theorems concern uniform approximation on [0, 1] by polynomials whose exponents are taken from an increasing sequence of positive real numbers

Λ
\Lambda

. Under mild restrictions on the exponents, the degree of approximation for

Λ
\Lambda

-polynomials with real coefficients is compared with the corresponding degree of approximation when the coefficients are taken from the integers.
Lie algebras of type 𝐵𝐶₁
Authors: Allison B. N.
Published: June 2010

Let L be a central simple Lie algebra of type

B

C
1

B{C_1}

with highest root space of dimension greater than one over a field of characteristic zero. It is shown that either L is isomorphic to the simple Lie algebra associated with a skew hermitian form of index one or L can be constructed from the tensor product of two composition algebras. This result is obtained by completing the description (begun in [3]) of the corresponding class of ternary algebras.
The decay of solutions of the two dimensional wave equation in the exterior of a straight strip
Authors: Wolfe Peter
Published: June 2010

We study an initial boundary value problem for the wave equation in the exterior of a straight strip. We assume the initial data has compact support and that the solution vanishes on the strip. We then show that at any point in space the solution is

O
(
1

/

t
)

O(1/t)

as

t
→
∞

t \to \infty

. This is the same rate of decay as obtains for the solution of the initial boundary value problem posed in the exterior of a smooth star shaped region. Our method is to use a Laplace transform. This reduces the problem to a consideration of a boundary value problem for the Helmholtz equation. We derive estimates for the solution of the Helmholtz equation for both high and low frequencies which enable us to obtain our results by estimating the Laplace inversion integral asymptotically.
An example where topological entropy is continuous
Authors: Block Louis
Published: June 2010

Let ent denote topological entropy, and let

C
r

(

S
1

,

S
1

)

{C^r}({S^1},{S^1})

denote the space of continuous functions of the circle to itself having r continuous derivatives with the

C
r

{C^r}

(uniform) topology. Let

f
0

{f_0}

denote a particular

C
2

{C^2}

map of the circle (

f
0

{f_0}

is the first bifurcation point one comes to in a bifurcation from a full three shift to a map with finite nonwandering set). The main results of this paper are the following: Theorem A. The map ent:

C
0

(

S
1

,

S
1

)
→
R
∪
{
∞
}

{C^0}({S^1},{S^1}) \to R \cup \{ \infty \}

is lower-semicontinuous at

f
0

{f_0}

. Theorem B. The map ent:

C
2

(

S
1

,

S
1

)
→
R

{C^2}({S^1},{S^1}) \to R

is continuous at

f
0

{f_0}

. In proving these two theorems several general results on entropy of mappings of the circle are proved.
Lattices of convex sets
Authors: Bennett Mary Katherine
Published: June 2010

If V is a vector space over an ordered division ring, C a convex subset of V and L the lattice of convex subsets of C, then we call L a convexity lattice. We give necessary and sufficient conditions for an abstract lattice to be a convexity lattice in the finite dimensional case.
Weak convergence of the area of nonparametric 𝐿₁ surfaces
Authors: Michener Kim E.
Published: June 2010

The main purpose of this work is to obtain an analogue to a theorem of L. C. Young on the behavior of the nonparametric surface area of continuous functions. The analogue is for

L
1

{L^1}

functions of generalized bounded variation. By considering arbitrary Borel vector measures and kernels other than the area kernel, results concerning the weak behavior of measures induced by a class of sublinear functionals are obtained.
Ergodic equivalence relations, cohomology, and von Neumann algebras. II
Authors: Feldman Jacob , Moore Calvin C.
Published: June 2010

Let R be a Borel equivalence relation with countable equivalence classes, on the standard Borel space

(
X
,

A

,
μ
)

(X,\mathcal {A},\mu )

. Let

σ
\sigma

be a 2-cohomology class on R with values in the torus

T

\mathbb {T}

. We construct a factor von Neumann algebra

M

(
R
,
σ
)

{\mathbf {M}}(R,\sigma )

, generalizing the group-measure space construction of Murray and von Neumann [1] and previous generalizations by W. Krieger [1] and G. Zeller-Meier [1]. Very roughly,

M

(
R
,
σ
)

{\mathbf {M}}(R,\sigma )

is a sort of twisted matrix algebra whose elements are matrices

(

a

x
,
y

)

({a_{x,y}})

, where

(
x
,
y
)
∈
R

(x,y) \in R

. The main result, Theorem 1, is the axiomatization of such factors; any factor M with a regular MASA subalgebra A, and possessing a conditional expectation from M onto A, and isomorphic to

M

(
R
,
σ
)

{\mathbf {M}}(R,\sigma )

in such a manner that A becomes the “diagonal matrices";

(
R
,
σ
)

(R,\sigma )

is uniquely determined by M and A. A number of results are proved, linking invariants and automorphisms of the system (M, A) with those of

(
R
,
σ
)

(R,\sigma )

. These generalize results of Singer [1] and of Connes [1]. Finally, several results are given which contain fragmentary information about what happens with a single M but two different subalgebras

A

1

,

A

2

{{\mathbf {A}}_1},{{\mathbf {A}}_2}

.
The periodic behavior of Morse-Smale diffeomorphisms on compact surfaces
Authors: Narasimhan Carolyn C.
Published: June 2010

Necessary and sufficient conditions are given for the existence of Morse-Smale diffeomorphisms homotopic to the identity with prescribed periodic characteristics on any compact 2-manifold.
Ramsey’s theorem for spaces
Authors: Spencer Joel H.
Published: June 2010

A short proof is given of the following known result. For all k, r, t there exists n so that if the t-spaces of an n-space are r-colored there exists a k-space all of whose t-spaces are the same color. Here t-space refers initially to a t-dimensional affine space over a fixed finite field. The result is also shown for a more general notion of t-space.
A construction of uncountably many weak von Neumann transformations
Authors: David Karl
Published: June 2010

We define weak von Neumann transformations and discuss some of their properties, using several examples of countable classes of these transformations. Then we construct an uncountable class by the cutting-and-stacking method. We show that each member of this class is ergodic and has zero entropy.
Two approaches to supermanifolds
Authors: Batchelor Marjorie
Published: June 2010

The problem of supplying an analogue of a manifold whose sheaf of functions contains anticommuting elements has been approached in two ways. Either one extends the sheaf of functions formally, as in the category of graded manifolds [3], [8], or one mimicks the usual definition of a manifold, having replaced Euclidean space with a suitable product of the odd and even parts of an exterior algebra as in the category of supermanifolds [6]. This paper establishes the equivalence of the category of supermanifolds with the category of graded manifolds.