bluuebunny/crossref_metadata_2025 huggingface dataset:
On going-down for simple overrings. III
Authors: Dobbs David E. , Papick Ira J.
Published: June 2010

Theorem 1. Let

R
R

be an integral domain with quotient field

K
K

. The following three conditions are equivalent: (a)

R
⊂
R
[
u
]

R \subset R[u]

satisfies going-down

(
G
D
)

(GD)

for each

u
u

in

K
K

; (b)

R
⊂
V

R \subset V

satisfies

G
D

GD

for each valuation overring

V
V

of

R
R

; (c)

R
⊂
S

R \subset S

satisfies

G
D

GD

for each domain

S
S

containing

R
R

. If (d) is the condition obtained by restricting the domains

S
S

in

(

c)

({\text {c)}}

to be overrings of

R
R

, then

(

a)

⇔

(d)

({\text {a)}} \Leftrightarrow {\text {(d)}}

has been proved in case

R
R

is Krull or integrally closed finite-conductor (e.g., pseudo-Bézout) or Noetherian. Theorem 2. Let

R
⊂
T

R \subset T

be domains such that either

Spec
⁡
(
R
)

\operatorname {Spec} (R)

or

Spec
⁡
(
T
)

\operatorname {Spec} (T)

, as a poset under inclusion, is a tree. If

R
⊂
R
[
u
,
v
]

R \subset R[u,v]

satisfies

G
D

GD

for each

u
u

and

v
v

in

T
T

, then

R
⊂
T

R \subset T

satisfies

G
D

GD

.
The prime radical in alternative rings
Authors: Rich Michael
Published: June 2010

The characterization by J. Levitzki of the prime radical of an associative ring

R
R

as the set of strongly nilpotent elements of

R
R

is adapted here to apply to a wide class of nonassociative rings. As a consequence it is shown that the prime radical is a hereditary radical for the class of alternative rings and that the prime radical of an alternative ring coincides with the prime radical of its attached Jordan ring.
𝑝-Helson sets
Authors: Oberlin Daniel M.
Published: June 2010

Every infinite compact abelian group

G
G

contains a subset which is a set of interpolation for each space

m
p

(
G
)
(
1
>
p
>
2
)

{m_p}(G)(1 > p > 2)

, but which is not Helson.
Compact Lie groups which act on Euclidean space without fixed points
Authors: Edmonds Allan L. , Lee Ronnie
Published: June 2010

It is shown that a compact Lie group

G
G

with identity component

G
0

{G_0}

acts without fixed points on euclidean space if and only if

G
0

{G_0}

is nonabelian or

G

/

G
0

G/{G_0}

is not of prime power order, which completes earlier work of P. E. Conner and E. E. Floyd, Conner and D. Montgomery, and W.-C. Hsiang and W.-Y. Hsiang.
Any unitary principal series representation of (𝐺𝐿)_{𝑛} over a 𝑝-adic field is irreducible
Authors: Howe Roger , Silberger Allan
Published: June 2010

This paper proves that for the group

GL
n

{\operatorname {GL} _n}

over the

p
p

-adics every unitary principal series representation is irreducible.
The free topological group on a simply connected space
Authors: Hardy J. P. L. , Morris Sidney A.
Published: June 2010

It is shown that the free

k
k

-group on a simply connected locally equiconnected space is simply connected. This result is then used to verify, for a large class of groups, the conjecture of Ordman that

π
1

(
G

∗

H
)
=

π
1

(
G
)
×

π
1

(
H
)

{\pi _1}(G{\ast }H) = {\pi _1}(G) \times {\pi _1}(H)

, where

G

∗

H

G{\ast }H

is the free product of

G
G

and

H
H

.
Bounds for solutions to ordinary differential equations applied to a singular Cauchy problem
Authors: Walker W. J.
Published: June 2010

The Cauchy problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript t t Baseline minus t Superscript 2 plus epsilon Baseline u Subscript x x Baseline minus u Subscript y Baseline equals 0 comma epsilon greater-than 0 comma u left-parenthesis x comma y comma 0 right-parenthesis equals alpha left-parenthesis x comma y right-parenthesis comma u Subscript t Baseline left-parenthesis x comma y comma 0 right-parenthesis equals gamma left-parenthesis x comma y right-parenthesis">

u

t
t

−

t

2
+
ε

u

x
x

−

u
y

=
0
,
ε
>
0
,
u
(
x
,
y
,
0
)
=
α
(
x
,
y
)
,

u
t

(
x
,
y
,
0
)
=
γ
(
x
,
y
)

{u_{tt}} - {t^{2 + \varepsilon }}{u_{xx}} - {u_y} = 0,\varepsilon > 0,u(x,y,0) = \alpha (x,y),{u_t}(x,y,0) = \gamma (x,y)

, is shown to be unstable by demonstrating that there exists a sequence of solutions which increase indefinitely on a sequence of neighbourhoods of

t
=
0

t = 0

which shrink to zero, while at the same time the initial data is tending to zero. The equation

u

t
t

−

t

2
+
ε

u

x
x

−

u

y
y

−

u
y

=
0

{u_{tt}} - {t^{2 + \varepsilon }}{u_{xx}} - {u_{yy}} - {u_y} = 0

is investigated with the same initial data and in this case it is shown that the sequence of solutions remains bounded on a neighbourhood of

t
=
0

t = 0

which suggests but does not prove that the Cauchy problem for this equation is well posed. The latter result is a consequence of bounds obtained on a neighbourhood of

t
=
0

t = 0

for complex-valued solutions of the ordinary differential equation
\[

y
+
(
a
(
t
)
+
i
b
(
t
)
)
y
=
0.

y + (a(t) + ib(t))y = 0.

\]
Two remarks on global solutions of ordinary differential equations in the real line
Authors: Vidossich Giovanni
Published: June 2010

We prove two theorems concerning global solutions of initial and terminal value problems: a criteria for the existence of maximal and minimal solutions, as well as the connectedness of the set of solutions without any assumption but the Carathéodory hypotheses.
𝐶𝑅 submanifolds of a Kaehler manifold. I
Authors: Bejancu Aurel
Published: June 2010

The differential geometry of CR submanifolds of a Kaehler manifold is studied. Theorems about totally geodesic CR submanifolds and totally umbilical CR submanifolds are given.
The volume of a slightly curved submanifold in a convex region
Authors: Dekster B. V.
Published: June 2010

Let T be a compact convex region in an n-dimensional Riemannian space,

k
s

{k_s}

be the minimum sectional curvature in T, and

κ
>
0

\kappa > 0

be the minimum normal curvature of the boundary of T. Denote by

P
ν

(
ξ
)

{P^\nu }(\xi )

a v-dimensional sphere, plane or hyperbolic plane of curvature

ξ
\xi

. We assume that

k
s

{k_s}

, k are such that on

P
2

(

k
s

)

{P^2}({k_s})

there exists a circumference of curvature k. Let

R
0

=

R
0

(
κ
,

k
s

)

{R_0} = {R_0}(\kappa ,{k_s})

be its radius. Now, let Q be a convex (in interior sense) m-dimensional surface in T whose normal curvatures with respect to any normal are not greater than x satisfying

0
⩽
χ
>
κ

0 \leqslant \chi > \kappa

. Denote by

L
χ

{L_\chi }

the length of a circular arc of curvature x in

P
2

(

k
s

)

{P^2}({k_s})

with the distance

2

R
0

2{R_0}

between its ends. We prove that the volume of Q does not exceed the volume of a ball in

P
m

(

k
s

−
(
n
−
m
)

χ
2

)

{P^m}({k_s} - (n - m){\chi ^2})

of radius

1
2

L
χ

\tfrac {1}{2}{L_\chi }

. These volumes are equal when T is a ball in

P
n

(

k
s

)

{P^n}({k_s})

and Q is its m-dimensional diameter.
On the Floquet problem for quasiperiodic systems
Authors: Murdock J. A.
Published: June 2010

The problem of reducing

x
˙

=
A
(
t
)
x

\dot x = A(t)x

, where

A
(
t
)

A(t)

is a quasiperiodic

n
×
n

n \times n

matrix, to a system with constant coefficients is studied by means of an associated linear partial differential equation. A necessary and sufficient condition is given which, while not always computable, appears to provide a new approach to this classical problem.
A note on automorphism groups of algebraic number fields
Authors: Fried M.
Published: June 2010

For any finite group G the paper gives an explicit and simple construction of (not necessarily Galois) algebraic extensions of Q having their full automorphism group equal to G.
The Hilbert transform of Schwartz distributions
Authors: Pandey J. N.
Published: June 2010

Let

D

\mathcal {D}

be the Schwartz space of infinitely differentiable complex-valued functions defined on the real line with compact supports equipped with the usual topology. Assume

H
(

D

)

H(\mathcal {D})

to be the space of

C
∞

{C^\infty }

functions defined on the real line whose every element is the Hilbert transform of an element of

D

\mathcal {D}

. We equip the space

H
(

D

)

H(\mathcal {D})

with an appropriate topology and show that the classical Hilbert transformation

H
H

, defined by

H
f
=
P

∫

−
∞

∞

f
(
t
)

/

(
t
−
x
)
d
t

Hf = P\int _{ - \infty }^\infty {f(t)/(t - x)dt}

, is a homeomorphism from

D

\mathcal {D}

onto

H
(

D

)

H(\mathcal {D})

. The Hilbert transform

H
f

Hf

of

f
∈

D

′

f \in \mathcal {D}’

is then defined to be an element of

H
′

(

D

)

H’(\mathcal {D})

given by the relation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper H f comma phi mathematical right-angle equals mathematical left-angle f comma minus upper H phi mathematical right-angle for-all phi element-of upper H left-parenthesis script upper D right-parenthesis period">

⟨

H
f
,
φ

⟩

=

⟨

f
,
−
H
φ

⟩

∀
φ
∈
H
(

D

)
.

\left \langle {Hf,\varphi } \right \rangle = \left \langle {f, - H\varphi } \right \rangle \forall \varphi \in H(\mathcal {D}).

\]
It then follows that -

−

H
2

f

/

π
2

=
f
∀
f
∈

D

′

- {H^2}f/{\pi ^2} = f\forall f \in \mathcal {D}’

. Applications of our results in solving some singular integral equations are also discussed.
A spectral characterization of universally weakly inner automorphisms of separable 𝐶*-algebras
Authors: Wright Steve
Published: June 2010

Let

A
A

be a separable

C
∗

{C^ * }

-algebra,

α
\alpha

a

∗
*

-automorphism of

A
A

. The following theorem is proven:

α
\alpha

is weakly inner in every faithful, nondegenerate representation of

A
A

if and only if

α
\alpha

fixes each closed, two-sided ideal of

A
A

and the Borchers spectrum of each quotient automorphism vanishes.
Linear extensions of convex subsets
Authors: Fujihira Hideyuki
Published: June 2010

We give conditions for linear extensions of closed convex subsets of topological vector spaces to be closed.
Order of magnitude of the concentration function
Authors: Hall Peter
Published: June 2010

Suppose a sum of independent random variables, when scaled in a suitable way, is stochastically compact. It is proved that the precise order of magnitude of the concentration function of the sum equals the inverse of the scale factor.
Homogeneous circlelike continua
Authors: Lewis Wayne
Published: June 2010

We extend earlier work of Bing, Jones, Hagopian, and Rogers to give a complete classification of nondegenerate homogeneous circle-like continua as pseudo-arcs, solenoids, or solenoids of pseudo-arcs. by showing that solenoids of pseudo-arcs are unique.
Extremal values of continuants
Authors: Ramharter G.
Published: June 2010

The following question was posed by C. A. Nicol: Given an arbitrary set

B
B

of positive integers, find the extremal denominators of regular continued fractions with partial denominators from

B
B

, each element occurring a given number of times. Partial solutions have been given by T. S. Motzkin and E. G. Straus, and later by T. W. Cusick. We derive the general solutions from a purely combinatorial theorem about the set of permutations of a vector with components from an arbitrary linearly ordered set. We also consider certain halfregular continued fractions. Here the maximizing arrangements have to be described in terms of an algorithmic procedure, as their combinatorial structure is exceptionally complicated. Its investigation leads to a connection with the well-known Markov spectrum. Finally we obtain an asymptotic formula for the ratio of extremal continuants and some sharp (essentially analytic) inequalities concerning cyclic continuants.
Sets of uniqueness for a certain class ℳ_{𝜖} on the dyadic group
Authors: Yoneda Kaoru
Published: June 2010

For each sequence

ε
=

{

ε
n

}

\varepsilon = \left \{ {{\varepsilon _n}} \right \}

of real numbers which satisfies

lim

inf

n
→
∞

ε

2

n
+
1

/

ε

2
n

>
0

\lim {\inf _{n \to \infty }}{\varepsilon _{{2^{n + 1}}}}/{\varepsilon _{{2^n}}} > 0

and

ε
n

↓
0

{\varepsilon _n} \downarrow 0

as

n
→
∞

n \to \infty

, let

M

ε

{\mathfrak {M}_\varepsilon }

denote the set of all Walsh series

μ
∼

∑

k
=
0

∞

μ
^

(
k
)

w
k

(
x
)

\mu \sim \sum \nolimits _{k = 0}^\infty {\hat \mu (k){w_k}(x)}

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript k equals 0 Overscript normal infinity Endscripts epsilon Subscript k Baseline StartAbsoluteValue ModifyingAbove mu With caret left-parenthesis k right-parenthesis EndAbsoluteValue squared greater-than normal infinity">

∑

k
=
0

∞

ε
k

|

μ
^

(
k
)

|

2

>
∞

\sum \nolimits _{k = 0}^\infty {{\varepsilon _k}{{\left | {\hat \mu (k)} \right |}^2} > \infty }

. We give a necessary and sufficient condition for a subset of the dyadic group to be a set of uniqueness for

M

ε

{\mathfrak {M}_\varepsilon }

.
A Hilbert space limit for the iterated hyperspace functor
Authors: Toruńczyk H. , West J.
Published: June 2010

Let

X
X

be a nondegenerate metric Peano continuum and let

P
(
X
)

P(X)

be the hyperspace of closed, nonvoid subsets of

X
X

equipped with the Hausdorff metric. Then the inclusion of

X
X

into

P
(
X
)

P(X)

as the single element sets is an isometry and we have a direct system

X
→
P
(
X
)
→
P
(
P
(
X
)
)
→
⋯

X \to P(X) \to P(P(X)) \to \cdots

of isometric inclusions. Let

X
′

X’

be the metric direct limit and

X
∗

{X^*}

be its completion. We prove that the pair

(

X
∗

,

X
′

)

({X^*},X’)

is homeomorphic to the pair

(

l
2

,

l
σ
2

)

({l^2},l_\sigma ^2)

, where

l
σ
2

l_\sigma ^2

is the linear span in

l
2

{l^2}

of the Hilbert cube.
Negligible sets of Radon measures
Authors: Prinz P.
Published: June 2010

Let

m
m

be a Radon measure on a Hausdorff topological space

X
X

. Corresponding to three kinds of outer measures, three kinds of

m
m

-negligible sets are considered. The main theorem states that in a metacompact space

X
X

each locally

m
m

-negligible set is

m
m

-negligible.
Sharp estimates for Lebesgue constants
Authors: Carenini Marco , Soardi Paolo M.
Published: June 2010

Suppose

C
⊂

R
N

C \subset {R^N}

is a closed convex bounded body containing 0 in its interior. If

∂
C

\partial C

is sufficiently smooth with strictly positive Gauss curvature at each point, then, denoting by

L

r
,
C

{L_{r,C}}

the Lebesgue constant relative to

C
C

, there exists a constant

A
>
0

A > 0

such that

L

r
,
C

⩾
A

r

(
N
−
1
)

/

2

{L_{r,C}} \geqslant A{r^{(N - 1)/2}}

for

r
r

sufficiently large. This complements the known result that there exists a constant

B
B

such that

L

r
,
C

⩽
B

r

(
N
−
1
)

/

2

{L_{r,C}} \leqslant B{r^{(N - 1)/2}}

for

r
r

sufficiently large.
Embedding cosmic spaces in Lusin spaces
Authors: Bešlagić Amer
Published: June 2010

We show that every regular cosmic space can be embedded in a Lusin space. This answers a question posed by J. P. R. Christensen.
Cardinalities of first countable 𝑅-closed spaces
Authors: Dow Alan , Porter Jack
Published: June 2010

It is now well known that first countable compact Hausdorff spaces are either countable or have cardinality

c
c

. The situation for first countable

H
H

-closed spaces is that they have cardinality less than or equal

c
c

, and it is at least consistent that they may have cardinality

ℵ
1

>
c

{\aleph _1} > c

. We show that the situation is quite different for first countable

R
R

-closed spaces. We begin by constructing an example which has cardinality

ℵ
1

{\aleph _1}

. Let

λ
0

{\lambda _0}

be the smallest cardinal greater than

c
c

which is not a successor. For each cardinal

κ
\kappa

with

c
⩽
κ
⩽

λ
0

c \leqslant \kappa \leqslant {\lambda _0}

we construct a first countable

R
R

-closed space of cardinality

κ
\kappa

. We also construct a first countable

R
R

-closed space of cardinality

λ
0
ω

\lambda _0^\omega

. This seems to indicate that there is no reasonable upper bound to the cardinalities of

R
R

-closed spaces as a function of their character.
Embeddings in 𝐺(1,3)
Authors: Papantonopoulou A.
Published: June 2010

We give a description of nonsingular surfaces of degree

d
⩽
8

d \leqslant 8

in the Grassmannian

G
(
1
,
3
)

G(1,3)

embedded in

P

5

{{\mathbf {P}}^5}

.
Three convex sets
Authors: Talagrand Michel
Published: June 2010

We construct a Choquet simplex

K
K

such that there is a universally measurable affine function

f
f

on

K
K

, which satisfies the barycentric calculus, and is zero on the set of extreme points, but is not identically zero. We also construct a closed convex bounded set of a Banach space without extreme points, but such that each point is the barycenter of a maximal measure. Finally, we construct a closed bounded set

L
L

of

l
1

(

R

)

{l^1}({\mathbf {R}})

and a maximal measure on

L
L

which is supported by a weak Baire set which contains no extreme points.
Equivalence of the classical theorems of Schottky, Landau, Picard and hyperbolicity
Authors: Hahn Kyong T.
Published: June 2010

Modifying the classical theorems of Schottky and Landau, the author obtains the converses of these theorems. More precisely, the author defines the notions of Schottky, Landau and Picard properties and proves that a plane domain

D
D

satisfies any of these properties if and only if

C

∖
D

{\mathbf {C}}\backslash D

contains at least two points. The method of proofs is completely elementary and uses only some basic properties of the Kobayashi metric.
Minimax and variational inequalities for compact spaces
Authors: McClendon J. F.
Published: June 2010

The minimax inequality

min
x

sup
y

f
(
x
,
y
)
⩽

sup
x

f
(
x
,
x
)

{\min _x}{\sup _y}f(x,y) \leqslant {\sup _x}f(x,x)

,proved by K. Fan for convex spaces, is proved here for certain contractible and acyclic spaces. Some variational inequality and fixed point theorems are deduced.
Cobordism and the nonfinite homotopy type of some diffeomorphism groups
Authors: Chess Daniel S.
Published: June 2010

Unoriented cobordism, a geometric construction, and a theorem of Browder on finite

H
H

-spaces are used to give new examples of manifolds whose diffeomorphism groups have identity component of nonfinite homotopy type.
Gel′fand-Kirillov dimension of rings of formal differential operators on affine varieties
Authors: Smith S. P.
Published: June 2010

Let

A
A

be the coordinate ring of a smooth affine algebraic variety defined over a field

k
k

. Let

D
D

be the module of

k
k

-linear derivations on

A
A

and form

A
[
D
]

A[D]

, the ring of differential operators on

A
A

, as follows: consider

A
A

and

D
D

as subspaces of

End
k

A

{\operatorname {End}_k}A

(

A
A

acting by left multiplication on itself), and define

A
[
D
]

A[D]

to be the subalgebra generated by

A
A

and

D
D

. Let

rk
⁡
D

\operatorname {rk} D

denote the torsion-free rank of

D
D

(that is,

rk
⁡
D
=

dim
F

F

⊗
A

D

\operatorname {rk}D = {\dim _F}F{ \otimes _A}D

where

F
F

is the quotient field of

A
A

). The ring

A
[
D
]

A[D]

is a finitely generated

k
k

-algebra so its Gelfand-Kirillov dimension

GK

(
A
[
D
]
)

{\text {GK}}(A[D])

may be defined. The following is proved. Theorem.

GK

(
A
[
D
]
)
=

tr de

g

k

A
+
rk
⁡
D
=
2

 tr de

g

k

A

{\text {GK}}(A[D]) = {\text {tr de}}{{\text {g}}_k}A + \operatorname {rk} D = 2{\text { tr de}}{{\text {g}}_k}A

. Actually we work in a more general setting than that just described, and although a more general result is obtained, this is the most natural and important application of the main theorem.
The equivalence of zero span and zero semispan
Authors: Davis James Francis
Published: June 2010

In this paper we introduce the idea of the symmetric span of a continuum, and show that continua with zero symmetric span are in class

W
W

. Continua with zero span have zero symmetric span, but the converse does not hold. We also show that if every subcontinuum of the continuum

M
M

is in class

W
W

then the span of

M
M

and the semispan of

M
M

agree. These results are then applied to show that the properties of having zero span and of having zero semispan are equivalent.
Embedding phenomena based upon decomposition theory: locally spherical but wild codimension one spheres
Authors: Daverman Robert J.
Published: June 2010

For

n
⩾
7

n \geqslant 7

we describe an

(
n
−
1
)

(n - 1)

-sphere

Σ
\Sigma

wildly embedded in the

n
n

-sphere yet every point of

Σ
\Sigma

has arbitrarily small neighborhoods bounded by flat

(
n
−
1
)

(n - 1)

-spheres, each intersecting

Σ
\Sigma

in an

(
n
−
2
)

(n - 2)

-sphere. Not only do these examples for large

n
n

run counter to what can occur when

n
=
3

n = 3

, they also illustrate the sharpness of high-dimensional taming theorems developed by Cannon and Harrold and Seebeck. Furthermore, despite their wildness, they have mapping cylinder neighborhoods, which both run counter to what is possible when

n
=
3

n = 3

and also partially illustrate the sharpness of another high-dimensional taming theorem due to Bryant and Lacher.
The Gel′fand subalgebra of real or non-Archimedean valued continuous functions
Authors: Domínguez Jesús M.
Published: June 2010

Let

L
L

be either the field of real numbers or a nonarchimedean rank-one valued field. For topological space

T
T

we study the Gelfand subalgebra

C
0

(
T
,
L
)

{C_0}(T,L)

of the algebra of all

L
L

-valued continuous functions

C
(
T
,
L
)

C(T,L)

. The main result is that if

T
T

is a paracompact locally compact Hausdorff space, which is ultraregular if

L
L

is nonarchimedean, then for

f
∈
C
(
T
,
L
)

f \in C(T,L)

the following statements are equivalent: (1) There exists a compact set

K
⊂
T

K \subset T

such that

f
(
T
−
K
)

f(T - K)

is finite, (2)

f
f

has finite range on every discrete closed subset of

T
T

, and (3)

f
∈

C
0

(
T
,
L
)

f \in {C_0}(T,L)

.
Final sets for operators on real entire functions of order one, normal type
Authors: Prather C. L.
Published: June 2010

Let

f
f

be a real entire function of order one, normal type that is bounded on the real axis and

L
=
φ

(
D
)

L = \varphi \left ( D \right )

,

D
=

(

d

/

d
z

)

D = \left ( {d / dz} \right )

with

φ

(
ω
)

\varphi \left ( \omega \right )

a Laguerre-Pólya function satisfying

φ

(
0
)

=
0

\varphi \left ( 0 \right ) = 0

. Then the final set of

f
f

with respect to

L
L

is contained in the real axis as either a discrete subset or the whole axis.
On divergent lacunary trigonometric series
Authors: Ramsey L. Thomas
Published: June 2010

Let

S
=

{

λ
n

}

n
=
1

∞

S = \left \{ {{\lambda _n}} \right \}_{n = 1}^\infty

be a sequence of positive real numbers such that

λ

n
+
1

/

λ
n

⩾
q
>
1

{\lambda _{n + 1}}/{\lambda _n} \geqslant q > 1

for all

n
n

. If

q
>
8

q > 8

then any divergent series with frequencies in

S
S

has its real part diverging (uniformly) to

+
∞

+ \infty

on a set of positive logarithmic capacity. It is necessary that

q
>
2

q > 2

. A new sufficient condition for the generalized capacity of a set to be positive is developed and then applied in the proof.
Cell-like mappings between CS sets
Authors: Henderson James P.
Published: June 2010

A CS set is a certain type of stratified metric space. The structure of a CS set which is a generalized

n
n

-manifold is determined. A class of cell-like mappings between CS sets is defined, and it is shown that such a cell-like mapping with manifold image

M
n

{M^n}

,

n
≠
4

n \ne 4

, is approximable by homeomorphisms.
The set of zeroes of an “almost polynomial” function
Authors: Yomdin Y.
Published: June 2010

Let

f
f

be a smooth function on the unit

n
n

-dimensional ball, with the

C
0

{C^0}

-norm, equal to one. We prove that if for some

k
⩾
2

k \geqslant 2

, the norm of the

k
k

th derivative of

f
f

is bounded by

2

−
k
−
1

{2^{ - k - 1}}

, then the set of zeroes

Y
Y

of

f
f

is similar to that of a polynomial of degree

k
−
1

k - 1

. In particular,

Y
Y

is contained in a countable union of smooth hypersurfaces; "many" straight lines cross

Y
Y

in not more than

k
−
1

k - 1

points, and the

n
−
1

n - 1

-volume of

Y
Y

is bounded by a constant, depending only on

n
n

and

k
k

.
Sturmian theory and disconjugacy of second order systems
Authors: Dannan Fozi M.
Published: June 2010

A generalization of the Sturm Comparison Theorem is given to second order linear systems. In addition, an analogue to Sternberg disconjugacy criterion for nonselfadjoint second order linear systems is given.
A topological space without a complete quasi-uniformity
Authors: Künzi Hans-Peter A. , Fletcher Peter
Published: June 2010

We show that an example of Burke and van Douwen has no complete quasi-uniformity. Moreover, we show that it is almost finitely-fully normal but not almost

ℵ
0

{\aleph _0}

-fully normal.
Criteria for metrisability
Authors: Collins P. J. , Roscoe A. W.
Published: June 2010

A simple condition on the local bases of a first countable space is shown to imply metrisability, and some new and some well-known metrisation theorems are deduced. Weakening the condition gives new classes of spaces distinct from the class of metrisable spaces.
Totally Zippin 𝑝-groups
Authors: Megibben Charles
Published: June 2010

If

G
G

is a

p
p

-group of limit length

λ
\lambda

, then it satisfies the

λ
\lambda

-Zippin property provided that whenever

A

/

p
λ

A
≅
G
≅
B

/

p
λ

B

A/{p^\lambda }A \cong G \cong B/{p^\lambda }B

, every isomorphism between

p
λ

A

{p^\lambda }A

and

p
λ

B

{p^\lambda }B

extends to an isomorphism between

A
A

and

B
B

. We show that if

G
G

is almost balanced in a totally projective group, then

G
G

does satisfy the

λ
\lambda

-Zippin property. This leads to the existence of a great variety of

G
G

’s that are totally Zippin in the sense that

G

/

p
α

G

G/{p^\alpha }G

satisfies the

α
\alpha

-Zippin property for all limit ordinals

α
⩽
λ
=

length of 

G

\alpha \leqslant \lambda = {\text {length of }}G

. Hence totally Zippin

p
p

-groups need not be

S
S

-groups, although those of countable length turn out to be direct sums of countable groups.
Extreme points of subordination families with univalent majorants
Authors: Hallenbeck David J.
Published: June 2010

Let

s
(
F
)

s(F)

denote the set of functions subordinate to a univalent function

F
F

in

Δ
\Delta

the unit disc. Let

B
0

{B_0}

denote the set of functions

ϕ
(
z
)

\phi (z)

analytic in

Δ
\Delta

satisfying

ϕ
(
z
)

|

>
1

\phi (z)\left | { > 1} \right .

and

ϕ
(
0
)
=
0

\phi (0) = 0

. We prove the following results: If

f
=
F
∘
ϕ

f = F \circ \phi

is an extreme point of

s
(
F
)

s(F)

and

F
(
Δ
)

F(\Delta )

is a Jordan domain, then

ϕ
\phi

is an extreme point of

B
0

{B_0}

.
Approximate unitary equivalence of power partial isometries
Authors: Davidson Kenneth R.
Published: June 2010

Every power partial isometry (p.p.i.) in the Calkin algebra lifts to a p.p.i. in

B

(

H

)

B\left ( \mathcal {H} \right )

.
Additivity of measure implies dominating reals
Authors: Miller Arnold W.
Published: June 2010

We show that additivity of measure

A

(
m
)

A\left ( m \right )

, the union of less than continuum many measure zero sets has measure zero) implies that every family

F
⊆

ω
ω

F \subseteq {\omega ^\omega }

of cardinality less than continuum is eventually dominated (this is the property

D
D

). This yields as a corollary from known results that

A

(
m
)

+
B

(
c
)

→
A

(
c
)

A\left ( m \right ) + B\left ( c \right ) \to A\left ( c \right )

.

A

(
c
)

A\left ( c \right )

is the property that the union of less than continuum many first category sets has first category and

B

(
c
)

B\left ( c \right )

is the property that the real line is not the union of less than continuum many first category sets. Also, a new property of measure and category is introduced, the covering property,

C

(
m
)

C\left ( m \right )

, which says that for any family of measure zero (first category) sets of cardinality less than the continuum there is some measure zero (first category) set not covered by any member of the family. By dualizing the proof that

A

(
m
)

→
D

A\left ( m \right ) \to D

we show that

w
D
→
C

(
m
)

wD \to C\left ( m \right )

. The weak dominating property,

w
D

wD

, says that no small family contained in

ω
ω

{\omega ^\omega }

dominates every element of

ω
ω

{\omega ^\omega }

.
A note on Ščepin’s theorem
Authors: Tsuda Kōichi
Published: June 2010

We shall generalize a theorem of Ščepin as follows. Theorem. Every finite dimensional

ANR

(

P

1

)

{\text {ANR}}({\mathcal {P}_1})

is metrizable, where

P

1

{\mathcal {P}_1}

denotes the class of all

σ
\sigma

-locally compact, paracompact

p
p

-spaces.
Roots of invertibly weighted shifts with finite defect
Authors: Keough Gerard E.
Published: June 2010

Let

T
T

be a unilateral invertibly weighted shift; i.e.,

T
T

maps a square-summable vector sequence

{

x

0
,

x
1

,
…

}

\left \{ {{x_{0,}}{x_1}, \ldots } \right \}

from a Hilbert space

H
H

to the sequence

{

0
,

A
0

x
0

,

A
1

x

1
,

…

}

\left \{ {0,{A_0}{x_0},{A_1}{x_{1,}} \ldots } \right \}

, where

{

A
n

}

\left \{ {{A_n}} \right \}

is a uniformly bounded sequence of invertible operators on

H
H

. If

S
0

{S_0}

is the identity operator on

H
H

, and

S
n

=

A

n
−
1

A

n
−
2

⋯

A
0

{S_n} = {A_{n - 1}}{A_{n - 2}} \cdots {A_0}

for

n
⩾
1

n \geqslant 1

, then

T
T

is unitarily equivalent to multiplication by the variable

Z
Z

on the space

H
2

(
T
)

{H^2}\left ( T \right )

consisting of formal series

∑

x
n

Z
n

\sum {x_n}{Z^n}

having coefficients

x
n

∈
H

{x_n} \in H

which satisfy

∑

‖

S
n

x
n

‖

2

>
+
∞

\sum {\left \| {{S_n}{x_n}} \right \|^2} > + \infty

. The commutant of this multiplication consists of formal series

∑

F
n

Z
n

\sum {F_n}{Z^n}

which define bounded operators on

H
2

(
T
)

{H^2}\left ( T \right )

—where each

F
n

{F_n}

is an operator on

H
H

, and the action of such a series on an element of

H
2

(
T
)

{H^2}\left ( T \right )

is given by the Cauchy product of the two series. Using these characterizations, it is shown that if

H
H

has finite dimension

m
⩾
2

m \geqslant 2

, then

T
T

has an

n
n

th root only if

n
n

divides

m
m

. Examples are given of shifts

T
T

with (a)

m
=
2

m = 2

, but

T
T

has no square root, and (b)

m
=
4

m = 4

,

T
T

has a square root, but no fourth root.
On spaces of maps between complex projective spaces
Authors: Møller Jesper Michael
Published: June 2010

When

1
⩽
m
⩽
n

1 \leqslant m \leqslant n

, the space

M
(

P
m

,

P
n

)

M({P^m},{P^n})

of maps of complex projective

m
m

-space

P
m

{P^m}

into complex projective

n
n

-space

P
n

{P^n}

has a countably infinite number of components enumerated by degrees of maps in

H
2

(

P
m

;

Z

)

{H^2}({P^m};{\mathbf {Z}})

. By calculating their

(
2
n
−
2
m
+
1
)

(2n - 2m + 1)

-dimensional integral homology group we show that two components of

M
(

P
m

,

P
n

)

M({P^m},{P^n})

are homotopy equivalent if and only if their associated degrees have the same absolute value.
Groups of permutation projective dimension two
Authors: Arnold James E.
Published: June 2010

In previous papers we developed a version of homological algebra for

Z
[
G
]

Z[G]

modules (

G
G

a finite group) using summands of permutation modules in place of projective modules. The resulting theory is more discriminating than the usual homological algebra. For example, an f.g.

Z
Z

-torsion free

Z
[
G
]

Z[G]

module is either projective or has infinite projective dimension, whereas for

G
G

cyclic, all f.g.

Z
[
G
]

Z[G]

modules have permutation projective dimension one. In this paper we apply results of Endo and Miyata on permutation projective modules to characterize groups of dimension two.
On the rationality of the variety of smooth rational space curves with fixed degree and normal bundle
Authors: Ballico Edoardo
Published: June 2010

Let

S
~

n
,
a

{\tilde S_{n,a}}

be the variety of smooth, rational curves of degree

n
n

in

P

3

{{\mathbf {P}}_3}

whose normal bundle has a factor of degree

2
n
−
1
+
a

2n - 1 + a

and a factor of degree

2
n
−
1
−
a

2n - 1 - a

. In this paper we prove that

S
~

n
,
a

{\tilde S_{n,a}}

is rational if

n
−
a

n - a

is even and

a
>
0

a > 0

.
Central zero divisors in group algebras
Authors: Smith Martha K.
Published: June 2010

A central element of the complex group algebra

C

G

{\mathbf {C}}G

which is a zero divisor in the

W
∗

{W^ * }

group algebra

W
(
G
)

W(G)

is also a zero divisor in

C

G

{\mathbf {C}}G

. As a corollary, if

K
K

is a field of characteristic zero,

G
G

is a group,

A
A

is an abelian normal subgroup of

G
G

, and

R
R

is the Ore localization of

K
G

KG

obtained by inverting all nonzero elements of

K
A

KA

, then all matrix rings over

R
R

are directly finite and

R
R

has the invariant basis property.
Indecomposability of ideals in group rings
Authors: Parmenter M. M.
Published: June 2010

Let

H
H

be a subgroup of

G
G

and let

I
I

be the (two-sided) ideal of

Z

G

{\mathbf {Z}}G

generated by

ω
(

Z

H
)

\omega ({\mathbf {Z}}H)

. In this note, we show that

I
I

is indecomposable as an ideal in

Z

G

{\mathbf {Z}}G

. This extends a result of Linnell [1] and simplifies his argument somewhat.
Invertibility in nest algebras
Authors: Feintuch Avraham , Lambert Alan
Published: June 2010

Let

F

\mathcal {F}

denote a complete nest of subspaces of a complex Hilbert space

H

\mathcal {H}

, and let

C

\mathcal {C}

denote the nest algebra defined by

F

\mathcal {F}

. Let

K

\mathcal {K}

denote the ideal of compact operators on

H

\mathcal {H}

. If

F

\mathcal {F}

has no infinite-dimensional gaps then

T
∈

C

T \in \mathcal {C}

is invertible in

C

\mathcal {C}

if and only if it is invertible in

C

+

K

\mathcal {C} + \mathcal {K}

. An example is given of a nest with an infinite gap for which there exists an operator in

C

\mathcal {C}

which is invertible in

C

+

K

\mathcal {C} + \mathcal {K}

but not in

C

\mathcal {C}

.
Complex foliations generated by (1,1)-forms
Authors: Klimek M.
Published: June 2010

Complex foliations generated by

(
1
,
1
)

(1,1)

-forms are studied in order to describe geometric properties of complex partial differential equations of Monge-Ampère type.
Compact 3-manifolds with infinitely-generated groups of self-homotopy-equivalences
Authors: McCullough Darryl
Published: June 2010

Examples are constructed of compact

3
3

-manifolds with boundary whose groups of self-homotopy-equivalences are not finitely-generated.
On some continued fraction identities of Srinivasa Ramanujan
Authors: Bhargava S. , Adiga Chandrashekar
Published: June 2010

The main purpose of this note is to state and prove, in a simple, unified manner, several

q
q

-continued fraction expansions found in Ramanujan’s "lost" notebook. This is related to some recent works of G. E. Andrews and M. D. Hirschhorn.
On certain support points of the class 𝑆
Authors: Brickman Louis , Ruscheweyh Stephan
Published: June 2010

A support point of

S
S

maps the unit disk onto the complement of an analytic arc going to

∞
\infty

. We study the case where this arc has an analytic continuation through its finite endpoint and back to

∞
\infty

. As an application we find that only under severe restrictions can a Bazilevič function be a support point of

S
S

.
On a problem of Hellerstein, Shen and Williamson
Authors: Hinkkanen A. , Rossi J.
Published: June 2010

Suppose that

f
f

is a nonentire transcendental meromorphic function, real on the real axis, such that

f
f

and

f
′

f’

have only real zeros and poles, and

f
′

f’

omits a nonzero value. Confirming a conjecture of Hellerstein, Shen and Williamson, it is shown that then

f
f

is essentially

f

(
z
)

=
tan
⁡
z
−
B
z
−
C

f\left ( z \right ) = \tan z - Bz - C

for suitable values of

B
B

and

C
C

.
Borel measurable images of Polish spaces
Authors: Levi Sandro , Maitra Ashok
Published: June 2010

We determine the Borel class of the image of a Polish space under a Borel measurable function of class

η
\eta

which maps open sets in the Polish space to Borel sets of additive class

ξ
\xi

in the range, under a mild restriction on the inverse images of singleton sets. Our computation is based on a selection theorem proved in this article.
A wave equation with a possibly jumping nonlinearity
Authors: Ward J. R.
Published: June 2010

Existence of a doubly periodic solution to a forced semilinear wave equation is established. The nonlinearity may "jump" across any finite number of eigenvalues of finite multiplicity.
A characterization of absolutely 𝐶*-smooth continua
Authors: Proctor C. Wayne
Published: June 2010

A continuum

X
X

is proven to be absolutely

C
∗

{C^*}

-smooth if and only if each compactification

Y
Y

of the half line

[
0
,
∞
)

[0,\infty )

with remainder

X
X

has the property that the space of all subcontinua of

Y
Y

is a compactification of the space of all subcontinua of

[
0
,
∞
)

[0,\infty )

.
PMEA and first countable, countably paracompact spaces
Authors: Burke Dennis K.
Published: June 2010

The Product Measure Extension Axiom is used to prove a "

θ
\theta

-expandable" type property for first countable, countably paracompact (countably metacompact) spaces. Among other results, it follows (under PMEA) that countably paracompact Moore spaces are metrizable, and first countable, countably paracompact Hausdorff spaces are strongly collectionwise normal with respect to compact sets.
A Wiener type theorem for Dirichlet series
Authors: Goodman Arthur , Newman D. J.
Published: June 2010

A famous theorem of Wiener states the conditions under which the reciprocal of a function with an absolutely convergent Fourier series also has an absolutely convergent Fourier series. We offer an elementary proof of the fact, first proven in [2], that if

F
(
s
)

F(s)

has an absolutely convergent Dirichlet series then

1

/

F
(
s
)

1/F(s)

has an absolutely convergent Dirichlet series if and only if

|

F
(
s
)

|

\left | {F(s)} \right |

is bounded away from zero in the closed right half-plane.
Convex functions of bounded type
Authors: Goodman A. W.
Published: June 2010

We introduce a new class of normalized functions univalent and convex in the unit disk. These are called convex of bounded type and the set is denoted by

C
V
(

R
1

,

R
2

)

CV({R_1},{R_2})

. For this set we find the Koebe domain, a coefficient bound, and a bound for

|

f
(
z
)

|

\left | {f(z)} \right |

. We also mention a few of the many questions that can be asked about this new class of univalent functions.
Markov’s inequality for polynomials with real zeros
Authors: Borwein Peter
Published: June 2010

Markov’s inequality asserts that

|

|

p
n
′

|

|

⩽

n
2

|

|

p
n

|

|

||{p’_n}|| \leqslant {n^2}||{p_n}||

for any polynomial

p
n

{p_n}

of degree

n
n

. (We denote the supremum norm on

[
−
1
,
1
]

[ - 1,1]

by

|

|

.

|

|

||.||

.) In the case that

p
n

{p_n}

has all real roots, none of which lie in

[
−
1
,
1
]

[ - 1,1]

, Erdös has shown that

|

|

p
n
′

|

|

⩽
e
n

|

|

p
n

|

|

/

2

||{p’_n}|| \leqslant en||{p_n}||/2

. We show that if

p
n

{p_n}

has

n
−
k

n - k

real roots, none of which lie in

[
−
1
,
1
]

[ - 1,1]

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue p Subscript n Superscript prime Baseline less-than-or-slanted-equals c n left-parenthesis k plus 1 right-parenthesis StartAbsoluteValue EndAbsoluteValue p Subscript n Baseline StartAbsoluteValue EndAbsoluteValue">

|

|

p
n
′

⩽
c
n
(
k
+
1
)

|

|

p
n

|

|

||{p’_n} \leqslant cn(k + 1)||{p_n}||

, where

c
c

is independent of

n
n

and

k
k

. This extension of Markov’s and Erdös’ inequalities was conjectured by Szabados.
A necessary and sufficient condition for a connected amenable group to have polynomial growth
Authors: Ganesan S.
Published: June 2010

It is shown that a connected amenable group

G
G

has polynomial growth if, and only if, given any open subsemigroup

S
S

of

G
G

and a compact set

K
K

in

G
G

there exists an

s
s

in

S
S

such that

K
s

⊂
S

{K_s} \subset S

.
Strongly 𝜋-regular matrix semigroups
Authors: Okniński Jan
Published: June 2010

We prove that if

S
S

is a strongly

π
\pi

-regular multiplicative sub-semigroup of the matrix algebra

M
n

(
K
)

{M_n}(K)

,

K
K

being a field, then there exists a chain of ideals

S
1

◃
⋯
◃

S
t

=
S

{S_1} \triangleleft \cdots \triangleleft {S_t} = S

such that

t
≤

2

n
+
1

t \leq {2^{n + 1}}

and any Rees factor semigroup

S
i

/

S

i
−
1

{S_i}/{S_{i - 1}}

is either completely

0
0

-simple or nilpotent of index not exceeding

∏

j
=
0

n

(
j
n

)

\prod _{j = 0}^n(_j^n)

. This sharpens the main result of [4], in particular solving Problem 3.9 from [3].
Pairs where 2^{𝑎}-2^{𝑏} divides 𝑛^{𝑎}-𝑛^{𝑏} for all 𝑛
Authors: Sun Qi , Zhang Ming Zhi
Published: June 2010

Recently Selfridge asked for what

a
a

and

b
b

does

2
a

−

2
b

{2^a} - {2^b}

divide

n
a

−

n
b

{n^a} - {n^b}

for all

n
n

? In this paper, the authors prove that there exist only fourteen such pairs

(
a
,
b
)

(a,b)

when

0
≤
b
>
a

0 \leq b > a

.
On recursive trees with a unique infinite branch
Authors: Clote Peter
Published: June 2010

In this paper we analyze the Turing degree of an infinite branch in a recursive tree

T
⊆

ω

>
ω

T \subseteq {\omega ^{ > \omega }}

and its relation to the well-founded part of the tree. It is, of course, not surprising that the two notions are related, but it is of a certain technical interest (in terms of the coding procedure used) to establish the exact interrelation. An interpretation of our result in terms of a Cantor-Bendixson derivative operation on trees

T
⊆

ω

>
ω

T \subseteq {\omega ^{ > \omega }}

is given.
A nonstandard functional approach to Fubini’s theorem
Authors: Loeb Peter A.
Published: June 2010

In this note we use a functional approach to the integral to obtain a special case of the Keisler-Fubini theorem; the general case can be obtained with a similar proof. An immediate application is the standard Fubini theorem for products of Radon measures. Similar methods give the Weil formula for quotient groups of compact Abelian groups.
A short proof of the Birkhoff-Smale theorem
Authors: Mrowka T.
Published: June 2010

A short proof of the Birkhoff-Smale theorem on homoclinic points of diffeomorphisms is given. The proof is elementary in that only the local properties of a hyperbolic fixed point are assumed.
Symmetric determinants and Jordan norm similarities in characteristic 2
Authors: Waterhouse William C.
Published: June 2010

We first determine all linear changes of variable formally preserving symmetric determinants in characteristic 2; there are just slightly more of them than in other characteristics. We then restate this result in terms of affine group schemes. This allows us to apply descent theory, and thereby we prove a theorem on norm similarities of Jordan algebras in the one case left open by Jacobson.
On the combinatorial properties of Blackwell spaces
Authors: Jasiński Jakub
Published: June 2010

Under

MA + 

¬

CH

{\text {MA + }}\neg {\text {CH}}

(Martin’s Axiom and negation of the Continuum Hypothesis) we prove that the intersection of a Blackwell space with the analytic set and the Cartesian product of a Blackwell space and a Borel set do not need to be Blackwell spaces.
The solution sets of extremal problems in 𝐻¹
Authors: Hayashi Eric
Published: June 2010

Let

u
u

be an essentially bounded function on the unit circle

T
T

. Let

S
u

{S_u}

denote the subset of the unit sphere of

H
1

{H^1}

on which the functional
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F right-arrow from bar integral Subscript 0 Superscript 2 pi Baseline ModifyingAbove u With bar left-parenthesis e Superscript i t Baseline right-parenthesis upper F left-parenthesis e Superscript i t Baseline right-parenthesis d t slash 2 pi">

F
↦

∫
0

2
π

u
¯

(

e

i
t

)
F
(

e

i
t

)
d
t

/

2
π

F \mapsto \smallint _0^{2\pi }\bar u({e^{it}})F({e^{it}})dt/2\pi

attains its norm. A complete description of

S
u

{S_u}

is given in terms of an inner function

b
0

{b_0}

and an outer fun tion

g
0

{g_0}

in

H
2

{H^2}

for which

g
0
2

g_0^2

is an exposed point in the unit ball of

H
1

{H^1}

. An explicit description is given for the kernel of an arbitrary Toeplitz operator on

H
2

{H^2}

. The exposed points in

H
1

{H^1}

are characterized; an example is given of a strong outer function in

H
1

{H^1}

which is not exposed.
Pinching theorems of Simons type for complete minimal submanifolds in the sphere
Authors: Yanglian Pan
Published: June 2010

In this paper the author proves some pinching theorems of Simons type for complete minimal submanifolds in the sphere, which generalize the relative results by Simons and Yau.
Preprojective partitions and the determinant of the Hom matrix
Authors: Igusa K. , Todorov G.
Published: June 2010

If

Λ
\Lambda

is an artin algebra and

Λ
\Lambda

is the set of isomorphism classes of indecomposable finitely generated

Λ
\Lambda

-modules, then there is a partition

ind
⁡
Λ
=

∪

i
⩾
0

P
_

_

i

\operatorname {ind}\Lambda = { \cup _{i \geqslant 0}}{\underline {\underline P} _i}

, called the preprojective partition. We give an algorithm for computing this partition, which is given only in terms of numerical properties of the Auslander-Reiten quiver of

Λ
\Lambda

. If

Λ
\Lambda

is of finite representation type, then there are two essentially different proofs that the matrix
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H o m equals left-parenthesis lengt h Subscript End left-parenthesis upper N right-parenthesis slash rEnd left-parenthesis upper N right-parenthesis Baseline upper H o m Subscript normal upper Lamda Baseline left-parenthesis upper M comma upper N right-parenthesis right-parenthesis">

Hom
=
(

lengt

h

End(

N
)

/

rEnd(

N

)

Hom

Λ

⁡
(
M
,
N
)
)

\operatorname {Hom} = ({\text {lengt}}{{\text {h}}_{{\text {End(}}N)/{\text {rEnd(}}N{\text {)}}}}\operatorname {Hom}_{\Lambda }(M,N))

, where

M
,
n
∈
ind
⁡
Λ

M,n \in \operatorname {ind}\Lambda

has determinant

+
1
[

I
T
1

,

W
1

,

Z
1

]

+ 1[{\mathbf {IT1}},{\mathbf {W1}},{\mathbf {Z1}}]

. We show that the paths between the Hom matrix and the identity matrix in

GL

n

⁡
(

R

)

\operatorname {GL}_{n}(\mathbf {R})

given by these two proofs are homotopic.
Convergence of Fourier series in discrete crossed products of von Neumann algebras
Authors: Mercer Richard
Published: June 2010

The convergence of the generalized Fourier series

Σ
π
(
x
(
g
)
)
u
(
g
)

\Sigma \pi (x(g))u(g)

is considered in the crossed product of a von Neumann algebra by a discrete group. An example from classical theory shows that this series does not converge in any of the usual topologies. It is proven that this series does converge in a topology introduced by Bures which is well suited to a crossed product situation. As an elementary application, we answer the question: In what topology is an infinite matrix (representing a bounded operator) the sum of its diagonals?
The strong limit of von Neumann subalgebras with conditional expectations
Authors: Tsukada Makoto
Published: June 2010

The strong lower limit and the weak upper limit of a net of von Neumann subalgebras on which the conditional expectations exist with respect to a fixed faithful normal state are defined. The limits coincide if and only if the corresponding conditional expectations converge strongly.
Equivariant 𝐾-theory and representations of Hecke algebras
Authors: Lusztig George
Published: June 2010

We construct some representations of the Hecke algebra of an affine Weyl group using equivariant

K
K

-theory and state a conjecture on some

q
q

-analogs of the Springer representations.
A note on infinite-dimensional spaces defined by topological games
Authors: Hattori Yasunao
Published: June 2010

We get the following result: For a totally normal space

X
,
X

X,X

is a strongly countable dimensional space if and only if

X
X

is an

F

(

d
i
m

)

{\mathbf {F}}({\mathbf {dim}})

-like space. This result answers a question of R. Telgársky.