bluuebunny/crossref_metadata_2025 huggingface dataset:
Duality theories for metabelian Lie algebras. II
Authors: Gauger Michael A.
Published: July 2010

In this paper I have replaced one of the axioms given in my Duality theory for metabelian Lie algebras (Trans. Amer. Math. Soc. 187 (1974), 89-102) concerning duality theories by a considerably more natural assumption which yields identical results—a uniqueness theorem.
Asymptotic values of modulus 1 of functions in the unit ball of 𝐻^{∞}
Authors: Leung Kar Koi
Published: July 2010

The main purpose of this paper is to prove a theorem concerning a necessary and sufficient condition for an inner function to have a limiting value of modulus 1 along an arc inside the unit disc, terminating at a point of the unit circle.
𝜔-cohesive sets
Authors: Ryan Barbara F.
Published: July 2010

We define and investigate

ω
\omega

-cohesiveness, a strong notion of indecomposability for subsets of the integers and their isols. This notion says, for example, that if

X
X

is the isol of an

ω
\omega

-cohesive set then, for any integer

n
n

implies that, for some integer

k
,
⋅
(

X
−
k

n

)
≤
Y

k, \cdot (\begin {array}{*{20}{c}} {X - k} \\ n \\ \end {array} ) \leq Y

or

Z
Z

. From this it follows that if

f
(
x
)
∈

T
1

f(x) \in {T_1}

, the collection of almost recursive combinatorial polynomials, then the predecessors of

f
Λ

(
X
)

{f_\Lambda }(X)

are limited to isols

g
Λ

(
X
)

{g_\Lambda }(X)

where

g
(
X
)
∈

T
1

g(X) \in {T_1}

. We show existence of

ω
\omega

-cohesive sets. And we show that the isol of an

ω
\omega

-cohesive set is an

n
n

-order indecomposable isol as defined by Manaster. This gives an alternate proof to one half of Ellentuck’s theorem showing a simple algebraic difference between the isols and cosimple isols. In the last section we study functions of several variables when applied to isols of

ω
\omega

-cohesive sets.
Quasi-equivalence classes of normal representations for a separable 𝐶*-algebra
Authors: Halpern Herbert
Published: July 2010

It is shown that the set of quasi-equivalence classes of normal representations of a separable

C
∗

{C^\ast }

-algebra is a Borel subset of the quasi-dual with the Mackey Borel structure and forms a standard Borel space in the induced Borel structure. It is also shown that the set of factor states which induce normal representations forms a Borel set of the space of factor states with the

w
∗

{w^\ast }

-topology and that this set has a Borel transversal.
Surgery on a curve in a solid torus
Authors: Neuzil J. P.
Published: July 2010

We consider the following surgery question: If a regular neighborhood of a polyhedral knot in a solid torus is removed and then sewn back differently, what manifold results? We consider two classes of knots, torus knots and what we call doubly twisted knots. We obtain some related results on surgery on knots in

S
3

{S^3}

.
𝑃𝐿 involutions of 𝑆¹×𝑆¹×𝑆¹
Authors: Kwun Kyung Whan , Tollefson Jeffrey L.
Published: July 2010

We prove that the

3
3

-dimensional torus

S
1

×

S
1

×

S
1

{S^1} \times {S^1} \times {S^1}

admits exactly nine nonequivalent PL involutions. With the exception of the four fixed point free ones, the involutions may be distinguished by their fixed point sets: (1) eight points, (2) two simple closed curves, (3) four simple closed curves, (4) one torus, (5) two tori.
A classification theorem for abelian 𝑝-groups
Authors: Warfield R. B.
Published: July 2010

A new class of Abelian

p
p

-groups, called

S
S

-groups, is studied, and the groups in this class are classified in terms of cardinal invariants. The class of

S
S

-groups includes Nunke’s totally projective

p
p

-groups. The invariants consist of the Ulm invariants (which Hill has shown can be used to classify the totally projective groups) together with a new sequence of invariants indexed by limit ordinals which are not cofinal with

ω
\omega

. The paper includes a fairly complete discussion of dense isotype subgroups of totally projective

p
p

-groups, including necessary and sufficient conditions for two of them to be congruent under the action of an automorphism of the group. It also includes an extension of Ulm’s theorem to a class of mixed modules over a discrete valuation ring.
Nonlinear techniques for linear oscillation problems
Authors: Nehari Zeev
Published: July 2010

It is shown that for differential equations of the form

y

(
n
)

+
p
y
=
0

{y^{(n)}} + py = 0

there exist associated sets of systems of nonlinear equations which play a role similar to that of the ordinary Riccati equation in the case

n
=
2

n = 2

. In particular, the existence of continuous solutions of the nonlinear system is equivalent to the absence of certain types of oscillatory solutions of the linear equation. If

p
p

is of constant sign, the coefficients of the “Riccati systems” are all nonnegative, and the resulting positivity and monotonicity properties make it possible to obtain explicit oscillation criteria for the original equation.
One-dimensional polyhedral irregular sets of homomorphisms of 3-manifolds
Authors: Husch L. S. , Row W. H.
Published: July 2010

Examples are given to show that there exist homeomorphisms of open

3
3

-manifolds whose sets of irregular points are wildly embedded one-dimensional polyhedra. The main result of the paper is that a one-dimensional polyhedral set of irregular points can fail to be locally tame on, at most, a discrete subset of the set of points of order greater than one. Necessary and sufficient conditions are given so that the set of irregular points is locally tame at each point.
The center of an order with finite global dimension
Authors: Ramras Mark
Published: July 2010

Let

Λ
\Lambda

be a quasi-local ring of global dimension

n
>
∞

n > \infty

. Assume that its center

R
R

is a noetherian domain, that

Λ
\Lambda

is finitely generated torsion-free as an

R
R

-module, and that

R
R

is an

R
R

-direct summand of

Λ
\Lambda

. Then

R
R

is integrally closed in its quotient field

K
K

and Macauley of dimension

n
n

. Furthermore, when

n
=
2
,
Λ

n = 2,\Lambda

is a maximal

R
R

-order in the central simple

K
K

-algebra

Λ

⊗
R

K

\Lambda { \otimes _R}K

. This extends an earlier result of the author, in which

R
R

was assumed to have global dimension 2. Examples are given to show that in the above situation

R
R

can have infinite global dimension.
On smoothing compact measure spaces by multiplication
Authors: Maharam Dorothy
Published: July 2010

Let

μ
\mu

be a regular Borel probability measure on a compact Hausdorff space

S
S

, and let

m
m

be Lebesgue measure on the unit interval

I
I

. It is proved that the measure-theoretic product

(
S
,
μ
)
×
(

I
w

,

m
w

)

(S,\mu ) \times ({I^w},{m^w})

, where

w
w

is a large enough cardinal and

m
w

{m^w}

denotes product Lebesgue measure, is “pseudo-isometric” to

(

I
w

,

m
w

)

({I^w},{m^w})

. Here a pseudo-isometry

ϕ
\phi

is a point-isometry except that, instead of

ϕ
(
A
)

\phi (A)

being measurable for every measurable

A
A

, it is required only that

A
A

differ by a null set from a set with measurable image. If instead

μ
\mu

is a Baire probability measure and

S
S

is a Baire subset of

I
w

{I^w}

, then

(
S
,
μ
)
×
(

I
w

,

m
w

)

(S,\mu ) \times ({I^w},{m^w})

is point-isometric to

(

I
w

,

m
w

)

({I^w},{m^w})

. Finally it is shown that (roughly speaking) continuous maps can be “smoothed” into projection maps (to within pseudo-isometries) by multiplication by suitable projection maps.
Primitive ideals of twisted group algebras
Authors: Britton Otha L.
Published: July 2010

E. Effros and F. Hahn have conjectured that if

(
G
,
Z
)

(G,Z)

is a second countable locally compact transformation group, with

G
G

amenable, then every primitive ideal of the associated

C
∗

{C^\ast }

-algebra arises as the kernel of an irreducible representation induced from a stability subgroup. Results of Effros and Hahn concerning this conjecture are extended to include the twisted group algebra

L
1

(
G
,
A
;
T
,
α
)

{L^1}(G,A;T,\alpha )

, where

A
A

is a separable type I

C
∗

{C^\ast }

-algebra.
The similarity orbit of a normal operator
Authors: Fialkow L. A.
Published: July 2010

If

N
N

is a bounded normal operator on a separable Hilbert space

H

\mathcal {H}

, let

S

(
N
)

\mathcal {S}(N)

denote the similarity orbit of

N
N

in

L
(

H

)

L(\mathcal {H})

and let

S

k

(
N
)

{\mathcal {S}_k}(N)

denote the set of all compact perturbations of elements of

S

(
N
)

\mathcal {S}(N)

. It is proved that

S

(
N
)
(

S

K

(
N
)
)

\mathcal {S}(N)({\mathcal {S}_K}(N))

is norm closed in

L
(

H

)

L(\mathcal {H})

if and only if the spectrum (essential spectrum) of

N
N

is finite. If the essential spectrum of

N
N

is infinite and

M
M

is a normal operator whose spectrum is connected and contains that of

N
N

, then

M
M

is in the closure of

S

(
N
)

\mathcal {S}(N)

. If the spectrum of

N
N

is connected, this result characterizes the normal elements of the closure of

S

(
N
)

\mathcal {S}(N)

. A normal operator is similar to a nonquasidiagonal operator if and only if its essential spectrum contains more than two points.
The spinor genus of quaternion orders
Authors: Nipp Gordon L.
Published: July 2010

Let D be a global domain whose quotient field F does not have characteristic 2, let

A

\mathfrak {A}

be a quaternion algebra over F, and let

D

\mathfrak {D}

be an order on

A

\mathfrak {A}

over D. A right

D

\mathfrak {D}

-module M which is simultaneously a lattice on

A

\mathfrak {A}

over D is said to be right

D

\mathfrak {D}

-generic if there exists

α
∈

A

,
N
(
α
)
≠
0

\alpha \in \mathfrak {A},N(\alpha ) \ne 0

, such that

α

−
1

M
∈

gen

D

{\alpha ^{ - 1}}M \in {\operatorname {gen}}\;\mathfrak {D}

. Our main result is that every right

D

\mathfrak {D}

-generic module is cyclic if and only if every class in the spinor genus of

D

\mathfrak {D}

represents a unit in D. One consequence is that

D

\mathfrak {D}

is in a spinor genus of one class if and only if

D

\mathfrak {D}

-generic modules are cyclic and

D

\mathfrak {D}

represents every unit represented by its spinor genus. In addition, it is shown that a necessary and sufficient condition that an integral ternary lattice L be in a spinor genus of one class is that every right

D

L

{\mathfrak {D}_L}

-generic pair be equivalent to a two-sided

D

L

{\mathfrak {D}_L}

-generic pair, where

D

L

{\mathfrak {D}_L}

is the quaternion order associated with L.
On the Harish-Chandra homomorphism
Authors: Lepowsky J.
Published: July 2010

Using the Iwasawa decomposition

g

=

k

⊕

a

⊕

n

\mathfrak {g} = \mathfrak {k} \oplus \mathfrak {a} \oplus \mathfrak {n}

of a real semisimple Lie algebra

g

\mathfrak {g}

, Harish-Chandra has defined a now-classical homomorphism from the centralizer of

k

\mathfrak {k}

in the universal enveloping algebra of

g

\mathfrak {g}

into the enveloping algebra

A

\mathcal {A}

of

a

\mathfrak {a}

. He proved, using analysis, that its image is the space of Weyl group invariants in

A

\mathcal {A}

. Here the weaker fact that the image is contained in this space of invariants is proved “purely algebraically". In fact, this proof is carried out in the general setting of semisimple symmetric Lie algebras over arbitrary fields of characteristic zero, so that Harish-Chandra’s result is generalized. Related results are also obtained.
Projective limits in harmonic analysis
Authors: Greene William A.
Published: July 2010

A treatment of induced transformations of measures and measurable functions is presented. Given a diagram

φ
:
G
→
H

\varphi :G \to H

in the category of locally compact groups and continuous proper surjective group homomorphisms, functors are produced which on objects are given by

G
→

L
2

(
G
)
,

L
1

(
G
)

G \to {L^2}(G),{L^1}(G)

,

M
(
G
)
,
W
(
G
)

M(G),W(G)

, denoting, resp., the

L
2

{L^2}

-space,

L
1

{L^1}

-algebra, measure algebra, and von Neu mann algebra generated by left regular representation of

L
1

{L^1}

on

L
2

{L^2}

. All functors but but the second are shown to preserve projective limits; by example, the second is shown not to do so. The category of Hilbert spaces and linear transformations of norm

⩽
1

\leqslant 1

is shown to have projective limits; some propositions on such limits are given. Also given is a type and factor characterization of projective limits in the category of

W
∗

{W^ \ast }

-algebras and surjective normal

∗
\ast

-algebra homomorphisms.
Comparison theorems for bounded solutions of \triangle𝑢=𝑃𝑢
Authors: Glasner Moses
Published: July 2010

Let

P
P

and

Q
Q

be

C
1

{C^1}

densities on a hyperbolic Riemann surface

R
R

. A characterization of isomorphisms between the spaces of bounded solutions of

Δ
u
=
P
u

\Delta u = Pu

and

Δ
u
=
Q
u

\Delta u = Qu

on

R
R

in terms of the Wiener harmonic boundary is given.
Sheaves of 𝐻-spaces and sheaf cohomology
Authors: Parks James M.
Published: July 2010

The concept of a sheaf of

H
H

-spaces is introduced and, using the Čech technique, a cohomology theory is defined in which the cohomology “groups” are

H
H

-spaces. The corresponding axioms of Cartan [3] for this theory are verified and other properties of the theory are investigated.
Polynomials related to the Bessel functions
Authors: Howard F. T.
Published: July 2010

In this paper we examine the polynomials

W
n

(
a
)

{W_n}(a)

defined by means of
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus 4 e Superscript x a Baseline left-bracket x left-parenthesis e Superscript x Baseline minus 1 right-parenthesis minus 2 left-parenthesis e Superscript x Baseline plus 1 right-parenthesis right-bracket Superscript negative 1 Baseline equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts upper W Subscript n Baseline left-parenthesis a right-parenthesis x Superscript n Baseline slash n factorial period">

−
4

e

x
a

[
x
(

e
x

−
1
)
−
2
(

e
x

+
1
)

]

−
1

=

∑

n
=
0

∞

W
n

(
a
)

x
n

/

n
!

.

- 4{e^{xa}}{[x({e^x} - 1) - 2({e^x} + 1)]^{ - 1}} = \sum \limits _{n = 0}^\infty {{W_n}(a){x^n}/n!} .

\]
These polynomials are closely related to the zeros of the Bessel function of the first kind of index —3/2, and they are in some ways analogous to the Bernoulli and Euler polynomials. This analogy is discussed, and the real and complex roots of

W
n

(
a
)

{W_n}(a)

are investigated. We show that if

n
n

is even then

W
n

(
a
)
>
0

{W_n}(a) > 0

for all

a
a

, and if

n
n

is odd then

W
n

(
a
)

{W_n}(a)

has only the one real root

a
=
1

/

2

a = 1/2

. Also we find upper and lower bounds for all

b
b

such that

W
n

(
a
+
b
i
)
=
0

{W_n}(a + bi) = 0

. The problem of multiple roots is discussed and we show that if

n
≡
0
,
1
,
5
,
8

n \equiv 0,1,5,8

or 9

(
mod

12
)

(\bmod \; 12)

, then

W
n

(
a
)

{W_n}(a)

has no multiple roots. Finally, if

n
≡
0
,
1
,
2
,
5
,
6

n \equiv 0,1,2,5,6

or 8

(
mod

12
)

(\bmod \; 12)

, then

W
n

(
a
)

{W_n}(a)

has no factor of the form

a
2

+
c
a
+
d

{a^2} + ca + d

where

c
c

and

(
mod

12
)

(\bmod \; 12)

are integers.
On the inverse problem of Galois theory
Authors: Kovacic J.
Published: July 2010

Let

k
k

be a field,

F
F

a finite subfield and

G
G

a connected solvable algebraic matric group defined over

F
F

. Conditions on

G
G

and

k
k

are given which ensure the existence of a Galois extension of

k
k

with group isomorphic to the

F
F

-rational points of

G
G

.
A Galois theory for a class of inseparable field extensions
Authors: Davis R. L.
Published: July 2010

The structure of the group of rank n higher derivations in a field K is discussed and a characterization of its Galois subgroups is given. This yields a Galois type correspondence between these subgroups and the subfields over which K is purely inseparable, finite dimensional and modular.
On the two sheeted coverings of conics by elliptic curves
Authors: MacRae R. E.
Published: July 2010

Let K be the field of algebraic functions on an elliptic curve that can be described by an equation of the form

y
2

=
f
(
x
)

{y^2} = f(x)

where

f
(
x
)

f(x)

is a quartic polynomial over a field k. Moreover, assume that the Riemann surface for K contains no points rational over k. When k is the field of real numbers it is well known that K may also be expressed as a quadratic extension of a function field

L
=
k
(
u
,
v
)

L = k(u,v)

of algebraic functions on a conic whose Riemann surface also contains no points rational over k. We extend this result to p-adic ground fields k. Moreover, we describe the various subfields of index two and genus zero (conic subfields) in terms of the k-rational points on the Jacobian of K. This is done for arbitrary ground fields. In particular, the embedding of the projective class group of K (over k) is seen to describe exactly those conic subfields that possess k-rational points.
On semisimple commutative semigroups
Authors: Arendt B. D.
Published: July 2010

This paper presents an application of radical theory to the structure of commutative semigroups via their semilattice decomposition. Maximal group congruences and semisimplicity are characterized for certain classes of commutative semigroups and

N
N

-semigroups.
On bounded elements of linear algebraic groups
Authors: Sit Kwan-Yuk Claire
Published: July 2010

Let

F
F

be a local field of characteristic zero and

G 

{\text {G }}

a connected algebraic group defined over

F
F

. Let

G
G

be the locally compact group of

F
F

-rational points. One characterizes the group

B
(
G
)

B(G)

of

g
∈
G

g \in G

whose conjugacy class is relatively compact. For instance, if

G

{\text {G}}

is

F
F

-split or reductive without anisotropic factors then

B
(
G
)

B(G)

is the center of

G
G

. If

H
H

is a closed subgroup of

G
G

such that

G

/

H

G/H

has finite volume, then the centralizer of

H
H

in

G
G

is contained in

B
(
G
)

B(G)

. If, moreover,

H
H

is the centralizer of some

x
∈
G

x \in G

then

G

/

H

G/H

is compact.
Uniqueness and 𝛼-capacity on the group 2^{𝜔}
Authors: Wade William R.
Published: July 2010

We introduce a class of Walsh series

J

α
+

\mathcal {J}_\alpha ^ +

for each

0
>
α
>
1

0 > \alpha > 1

and show that a necessary and sufficient condition that a closed set

E
⊆

2
ω

E \subseteq {2^\omega }

be a set of uniqueness for

J

α
+

\mathcal {J}_\alpha ^ +

is that the

α
\alpha

-capacity of

E
E

be zero.
A family of countably compact 𝑃_{∗}-hypergroups
Authors: Dunkl Charles F. , Ramirez Donald E.
Published: July 2010

An infinite compact group is necessarily uncountable, by the Baire category theorem. A compact

P
∗

{P_\ast }

-hypergroup, in which the product of two points is a probability measure, is much like a compact group, having an everywhere supported invariant measure, an orthogonal system of characters which span the continuous functions in the uniform topology, and a multiplicative semigroup of positive-definite functions. It is remarkable that a compact

P
∗

{P_\ast }

-hypergroup can be countably infinite. In this paper a family of such hypergroups, which include the algebra of measures on the

p
p

-adic integers which are invariant under the action of the units (for

p
=
2
,
3
,
5
,
⋯
)

p = 2,3,5, \cdots )

) is presented. This is an example of the symmetrization technique. It is possible to give a nice characterization of the Fourier algebra in terms of a bounded-variation condition, which shows that the usual Banach algebra questions about the Fourier algebra, such as spectral synthesis, and Helson sets have easily determinable answers. Helson sets are finite, each closed set is a set of synthesis, the maximal ideal space is exactly the underlying hypergroup, and the functions that operate are exactly the Lip 1 functions.
𝐵𝑉-functions, positive-definite functions and moment problems
Authors: Maserick P. H.
Published: July 2010

Let S be a commutative semigroup with identity 1 and involution. A complex valued function f on S is defined to be positive definite if

Π
j

Δ
j

f
(
1
)
⩾
0

{\Pi _j}{\Delta _j}f(1) \geqslant 0

where the

Δ
j

{\Delta _j}

’s belong to a certain class of linear sums of shift operators. For discrete groups the positive definite functions defined herein are shown to be the classically defined positive definite functions. An integral representation theorem is proved and necessary and sufficient conditions for a function to be the difference of two positive-definite functions, i.e. a BV-function, are given. Moreover the BV-function defined herein agrees with those previously defined for semilattices, with respect to the identity involution. Connections between the positive-definite functions and completely monotonic functions are discussed along with applications to moment problems.
Uniqueness of commuting compact approximations
Authors: Holmes Richard B. , Scranton Bruce E. , Ward Joseph D.
Published: July 2010

Let

H
H

be an infinite dimensional complex Hilbert space, and let

B

(
H
)

\mathcal {B}(H)

(resp.

C

(
H
)

\mathcal {C}(H)

) be the algebra of all bounded (resp. compact) linear operators on

H
H

. It is well known that every

T
∈

B

(
H
)

T \in \mathcal {B}(H)

has a best approximation from the subspace

C

(
H
)

\mathcal {C}(H)

. The purpose of this paper is to study the uniqueness problem concerning the best approximation of a bounded linear operator by compact operators. Our criterion for selecting a unique representative from the set of best approximants is that the representative should commute with

T
T

. In particular, many familiar operators are shown to have zero as a unique commuting best approximant.
On imbedding finite-dimensional metric spaces
Authors: Lipscomb Stephen Leon
Published: July 2010

The classical imbedding theorem in dimension theory gives a nice topological characterization of separable metric spaces of finite covering dimension. The longstanding problem of obtaining an analogous theorem for the nonseparable case is solved.
On the 2-realizability of 2-types
Authors: Dyer Micheal N.
Published: July 2010

A

2
2

-type is a triple

(
π
,

π
2

,
k
)

(\pi ,{\pi _2},k)

, where

π
\pi

is a group,

π
2

{\pi _2}

a

π
\pi

-module and

k
∈

H
3

(
π
,

π
2

)

k \in {H^3}(\pi ,{\pi _2})

. The following question is studied: When is a

2
2

-type

(
π
,

π
2

,
k
)

(\pi ,{\pi _2},k)

realizable by

2
2

-dimensional CW-complex

X
X

such that the

2
2

-type

(

π
1

X
,

π
2

X
,
k
(
X
)
)

({\pi _1}X,{\pi _2}X,k(X))

is equivalent to

(
π
,

π
2

,
k
)

(\pi ,{\pi _2},k)

? A long list of necessary conditions is given (2.2). One necessary and sufficient condition (3.1) is proved, provided

π
\pi

has the property that stably free, finitely generated

π
\pi

-modules are free. “Stable”

2
2

-realizability is characterized (4.1) in terms of the Wall invariant of [15]. Finally, techniques of [5] are used to extend C. T. C. Wall’s Theorem F of [15] to a space

X
X

which is dominated by a finite CW-complex of dimension 2, provided

π
1

X

{\pi _1}X

is finite cyclic. Under these conditions

X
X

has the homotopy type of a finite

2
2

-complex if and only if the Wall invariant vanishes.
Converses to the Ω-stability and invariant lamination theorems
Authors: Gottlieb Allan
Published: July 2010

In 1967 Smale proved that for diffeomorphisms on closed smooth manifolds, Axiom

A

{\text {A}}

and no cycles are sufficient conditions for

Ω
\Omega

-stability and asserted the analogous theorem for vectorfields. Pugh and Shub have supplied a proof of the latter. Since then a major problem in dynamical systems has been Smale’s conjecture that Axiom

A

{\text {A}}

(resp.

A’

{\text {A’}}

) and no cycles are necessary as well as sufficient for

Ω
\Omega

-stability of diffeomorphisms (resp. vectorfields). Franks and Guckenheimer have worked on the diffeomorphism problem by strengthening the definition of

Ω
\Omega

-stable diffeomorphisms. In this paper it will be shown that an analogous strengthening of

Ω
\Omega

-stable vectorfields forces Smale’s conditions to be necessary. The major result of this paper is the following THEOREM. If

(
Λ
,
L
)

(\Lambda ,L)

is a compact laminated set,

N
N

is a normal bundle to the lamination, and

f
f

is an absolutely and differentiably

L
L

-stable diffeomorphism of a closed smooth manifold then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis id hyphen f Subscript number-sign Baseline overbar right-parenthesis colon upper C Superscript 0 Baseline left-parenthesis upper N right-parenthesis right-arrow upper C Superscript 0 Baseline left-parenthesis upper N right-parenthesis">

(

id - 

f
#

¯

)
:

C
0

(
N
)
→

C
0

(
N
)

({\text {id - }}\overline {{f_\# }} ):{C^0}(N) \to {C^0}(N)

is surjective. If the lamination is just a compact submanifold, the theorem is already new. When applied to flows, this theorem gives the above result on

Ω
\Omega

-stable vectorfields.
On a varietal structure of algebras
Authors: Giri R. D.
Published: July 2010

Shafaat introduced two successive generalisations of the variety of algebras: namely the semivariety and the quasivariety. We study a slightly more generalised concept which we call a pseudovariety.
Topological dynamics and 𝐶*-algebras
Authors: Green William L.
Published: July 2010

If

G
G

is a group of automorphisms of a

C
∗

{C^ \ast }

-algebra

A
A

with identity, then

G
G

acts in a natural way as a transformation group on the state space

S
(
A
)

S(A)

of

A
A

. Moreover, this action is uniformly almost periodic if and only if

G
G

has compact pointwise closure in the space of all maps of

A
A

into

A
A

. Consideration of the enveloping semigroup of

(
S
(
A
)
,
G
)

(S(A),G)

shows that, in this case, this pointwise closure

G
¯

\bar G

is a compact topological group consisting of automorphisms of

A
A

. The Haar measure on

G
¯

\bar G

is used to define an analogue of the canonical center-valued trace on a finite von Neumann algebra. If

A
A

possesses a sufficiently large group

G
0

{G_0}

of inner automorphisms such that

(
S
(
A
)
,

G
0

)

(S(A),{G_0})

is uniformly almost periodic, then

A
A

is a central

C
∗

{C^ \ast }

-algebra. The notion of a uniquely ergodic system is applied to give necessary and sufficient conditions that an approximately finite dimensional

C
∗

{C^ \ast }

-algebra possess exactly one finite trace.
On a Galois theory for inseparable field extensions
Authors: Mordeson John N.
Published: July 2010

Heerema has developed a Galois theory for fields L of characteristic

p
≠
0

p \ne 0

in which the Galois subfields K are those for which

L

/

K

L/K

is normal, modular and, for some nonnegative integer

e
,
K
(

L

p

e
+
1

)

/

K

e,K({L^{{p^{e + 1}}}})/K

is separable. The related automorphism groups G are subgroups of a particular group A of automorphisms on

L
[
x
]

/

x

p
e

+
1

L
[
x
]

L[x]/{x^{{p^e} + 1}}L[x]

where x is an indeterminate over L. For

H
⊆
G

H \subseteq G

Galois subgroups of A, we give a necessary and sufficient condition for H to be G-invariant. An extension of a result of the classical Galois theory is also given as is a necessary and sufficient condition for every intermediate field of

L

/

K

L/K

to be Galois where K is a Galois subfield of L.
A decomposition for certain real semisimple Lie groups
Authors: Michelson H. Lee
Published: July 2010

For a class of real semisimple Lie groups, including those for which G and K have the same rank, Kostant introduced the decomposition

G
=
K

N
0

K

G = K{N_0}K

, where

N
0

{N_0}

is a certain abelian subgroup of N, and conjectured that the Jacobian of the decomposition with respect to Haar measure, as well as the spherical functions, would be polynomial in the canonical coordinates of

N
0

{N_0}

. We compute here the Jacobian, which turns out to be polynomial precisely when the equality of ranks is satisfied. We also compute those spherical functions which restrict to polynomials on

N
0

{N_0}

.
Weakly smooth continua
Authors: Lum Lewis
Published: July 2010

We define and investigate a class of continua called weakly smooth. Smooth dendroids, weakly smooth dendroids, generalized trees, and smooth continua are all examples of weakly smooth continua. We generalize characterizations of the above mentioned examples to weakly smooth continua. In particular, we characterize them as compact Hausdorff spaces which admit a quasi order satisfying certain properties.
Extensions of maps as fibrations and cofibrations
Authors: Quinn Frank
Published: July 2010

Suppose

f
:
X
→
Y

f:X \to Y

is a map of 1-connected spaces. In the “stable” range, roughly where the connectivity of Y exceeds the homology, or homotopy, dimension of X, it is well known that f can be extended as a cofibration

C
→
X
→
Y

C \to X \to Y

, or respectively a fibration

X
→
Y
→
B

X \to Y \to B

. A criterion is given for the existence of such extensions in a less restrictive “metastable” range. A main result is that if f is at least 2-connected and 2 con

Y
≥
dim
⁡
Y
−
1
,
dim
⁡
X

Y \geq \dim Y - 1,\dim X

, then f extends as a cofibration if and only if the map

(
1
×
f
)
Δ
:
X
→
(
X
×
Y
)

/

X

(1 \times f)\Delta :X \to (X \times Y)/X

factors through f.
The algebraic EHP sequence
Authors: Singer William M.
Published: July 2010

Let

A
A

be the dual of the

mod
−
2

\bmod - 2

Steenrod algebra. If

M
,
N

M,N

, are graded unstable

A
A

-comodules, one can define and compute the derived functors

Coext
A

(
M
,
N
)

{\text {Coext} _A}(M,N)

using unstable injective resolutions of

N
N

. Bousfield and Curtis have shown that these unstable Coext groups can be fit into a long exact “EHP sequence", an algebraic analogue of the EHP sequence of homotopy theory. Our object in the present paper is to study the relationship between the

E
,
H

E,H

, and

P
P

homomorphisms and the composition pairing
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Coext Subscript upper A Baseline left-parenthesis upper N comma upper R right-parenthesis circled-times Coext Subscript upper A Baseline left-parenthesis upper M comma upper N right-parenthesis right-arrow Coext Subscript upper A Baseline left-parenthesis upper M comma upper R right-parenthesis">

Coext
A

(
N
,
R
)
⊗

Coext
A

(
M
,
N
)
→

Coext
A

(
M
,
R
)

{\text {Coext} _A}(N,R) \otimes {\text {Coext} _A}(M,N) \to {\text {Coext} _A}(M,R)

. Among our results is a formula that measures the failure of the composition product to commute.
Conditions for the existence of contractions in the category of algebraic spaces
Authors: Mazur Joseph
Published: July 2010

Artin’s conditions for the existence of a new contraction of an algebraic subspace are changed. The new conditions are more applicable in special cases, especially when the subspace has a conormal bundle.
Duality theory for locally compact groups with precompact conjugacy classes. I. The character space
Authors: Sund Terje
Published: July 2010

Let G be a locally compact group, and let

X

(
G
)

\mathcal {X}(G)

consist of the nonzero extreme points of the set of continuous, G-invariant, positive definite functions f on G such that

f
(
e
)
≤
1

f(e) \leq 1

.

X

(
G
)

\mathcal {X}(G)

is called the character space, and is given the topology of uniform convergence on compacta. The purpose of the present paper is to extend the main results from the duality theory of abelian groups and [Z] groups to the class of

[
F
C

]
−

{[FC]^ - }

groups (i.e., groups with precompact conjugacy classes), letting

X

(
G
)

\mathcal {X}(G)

play the role of the character group in the abelian theory. Some of our theorems are only proved for the class

[
F
D

]
−

(
⊂

[
F
C

]
−

)

{[FD]^ - }\;( \subset {[FC]^ - })

. If

G
∈

[
F
C

]
−

G \in {[FC]^ - }

then

X

(
G
)
≈

X

(
H
)

\mathcal {X}(G) \approx \mathcal {X}(H)

where H is a certain

[
F
I
A

]
−

{[FIA]^ - }

quotient group. Hence there is no loss of generality to study character spaces of

[
F
I
A

]
−

{[FIA]^ - }

groups.
Holomorphic functions on nuclear spaces
Authors: Boland Philip J.
Published: July 2010

The space

H

(
E
)

\mathcal {H}(E)

of holomorphic functions on a quasi-complete nuclear space is investigated. If

H

(
E
)

\mathcal {H}(E)

is endowed with the compact open topology, it is shown that

H

(
E
)

\mathcal {H}(E)

is nuclear if and only if

E
′

E’

(continuous dual of

E
E

) is nuclear. If

E
E

is a

D

F
N

\mathcal {D}FN

(dual of a Fréchet nuclear space) and

F
F

is a closed subspace of

E
E

, then the restriction mapping

H

(
E
)
→

H

(
F
)

\mathcal {H}(E) \to \mathcal {H}(F)

is a surjective strict morphism.
Fiber preserving equivalence
Authors: Miller Richard T.
Published: July 2010

We give a theory of fibered regular neighborhoods based on a remarkable property of simplicial fibered projections. All the usual properties of regular neighborhoods are retained. Using Millett’s fibered general position, together with the regular neighborhoods, we prove THEOREM. The simplicial space of codimension 4 PL embeddings of a complex into a PL manifold is locally contractible at each point of the space of topological embeddings.
Immersions of complex hypersurfaces
Authors: Samsky Stanley R.
Published: July 2010

The varieties
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V Superscript n Baseline left-parenthesis d right-parenthesis equals left-brace left-bracket z 0 comma ellipsis comma z Subscript n Baseline right-bracket element-of upper C upper P Superscript n Baseline colon z 0 Superscript d Baseline plus midline-horizontal-ellipsis plus z Subscript n Superscript d Baseline equals 0 comma d greater-than 0 right-brace">

V
n

(
d
)
=
{
[

z
0

,
…
,

z
n

]
∈
C

P
n

:

z
0
d

+
⋯
+

z
n
d

=
0
,
d
>
0
}

{V^n}(d) = \{ [{z_0}, \ldots ,{z_n}] \in C{P^n}:z_0^d + \cdots + z_n^d = 0,d > 0\}

form a class of manifolds containing the complex projective spaces. Maps from

V
n

(
d
)

{V^n}(d)

to

V
k

(
e
)

{V^k}(e)

are partially characterized by a “degree". We prove some nonimmersion results which are phrased in terms of this degree, and which generalize the results of S. Feder [4] on complex projective spaces.
Extension of Fourier 𝐿^{𝑝}—𝐿^{𝑞} multipliers
Authors: Cowling Michael G.
Published: July 2010

By

M
p
q

(
Γ
)

M_p^q(\Gamma )

we denote the space of Fourier

L
p

−

L
q

{L^p} - {L^q}

multipliers on the LCA group

Γ
\Gamma

. K. de Leeuw [4] (for

Γ
=

R
a

\Gamma = {R^a}

), N. Lohoué [16] and S. Saeki [19] have shown that if

Γ
0

{\Gamma _0}

is a closed subgroup of

Γ
\Gamma

, and

ϕ
\phi

is a continuous function in

M
p
p

(
Γ
)

M_p^p(\Gamma )

, then the restriction

ϕ
0

{\phi _0}

of

ϕ
\phi

to

Γ
0

{\Gamma _0}

is in

M
p
p

(

Γ
0

)

M_p^p({\Gamma _0})

, and

‖

ϕ
0

‖

M
p
p

⩽

‖
ϕ
‖

M
p
p

{\left \| {{\phi _0}} \right \|_{M_p^p}} \leqslant {\left \| \phi \right \|_{M_p^p}}

. We answer here a natural question arising from this result: we show that every continuous function

ψ
\psi

in

M
p
p

(
Γ
)

M_p^p(\Gamma )

is the restriction to

Γ
0

{\Gamma _0}

of a continuous

M
p
p

(
Γ
)

M_p^p(\Gamma )

function whose norm is the same as that of

ψ
\psi

. A Figà-Talamanca and G. I. Gaudry [8] proved this with the extra condition that

Γ
0

{\Gamma _0}

be discrete: our technique develops their ideas. An extension theorem for

M
p
q

(

Γ
0

)

M_p^q({\Gamma _0})

is obtained: this complements work of Gaudry [11] on restrictions of

M
p
q

(
Γ
)

M_p^q(\Gamma )

-functions to

Γ
0

{\Gamma _0}

.
A characterization of manifolds
Authors: McAuley Louis F.
Published: July 2010

The purpose of this paper is (1) to give a proof of one general theorem characterizing certain manifolds and (2) to illustrate a technique which should be useful in proving various theorems analogous to the one proved here. Theorem. Suppose that

f
:
X
⇒
[
0
,
1
]

f:X \Rightarrow [0,1]

, where

X
X

is a compactum, and that

f
f

has the properties: (1) for

0
⩽
x
>
1

/

2
,

f

−
1

(
x
)
=

S
n

≅

M
0

0 \leqslant x > 1/2,{f^{ - 1}}(x) = {S^n} \cong {M_0}

, (2)

f

−
1

(
1

/

2
)
≅

S
n

{f^{ - 1}}(1/2) \cong {S^n}

with a tame (or flat)

k
k

-sphere

S
k

{S^k}

shrunk to a point, (3) for

1

/

2
>
x
⩽
1
,

f

−
1

(
x
)
≅

1/2 > x \leqslant 1,{f^{ - 1}}(x) \cong

a compact connected

n
n

-manifold

M
1

≅

S

n
−
(
k
+
1
)

×

S

k
+
1

{M_1} \cong {S^{n - (k + 1)}} \times {S^{k + 1}}

(a spherical modification of

M
0

{M_0}

of type

k
k

), and (4) there is a continuum

C
C

in

X
X

such that (letting

C
x

=

f

−
1

(
x
)
∩
C

{C_x} = {f^{ - 1}}(x) \cap C

) (a)

0
⩽
x
>
1

/

2
,

C
x

≅

S
k

0 \leqslant x > 1/2,{C_x} \cong {S^k}

, (b)

C

1

/

2

=
{
p
}

{C_{1/2}} = \{ p\}

a point, (c) for

1

/

2
>
x
⩽
1

1/2 > x \leqslant 1

, and (d) each of

f

|

(
X
−
C
)
,
f

|

f

−
1

[
0
,
1

/

2
)

f|(X - C),f|{f^{ - 1}}[0,1/2)

, and

f

|

f

−
1

(
1

/

2
,
1
]

f|{f^{ - 1}}(1/2,1]

is completely regular. Then

X
X

is homeomorphic to a differentiable

(
n
+
1
)

(n + 1)

-manifold

M
M

whose boundary is the disjoint union of

M
¯

0

{\bar M_0}

and

M
¯

1

{\bar M_1}

where

M
i

=

M
¯

i

,
i
=
0
,
1

{M_i} = {\bar M_i},i = 0,1

.
Normal structure of the one-point stabilizer of a doubly-transitive permutation group. I
Authors: O’Nan Michael E.
Published: July 2010

Let G be a doubly-transitive permutation group on a finite set X and x a point of X. Let

N
x

{N^x}

be a normal subgroup of

G
x

{G_x}

, the subgroup fixing x, such that

N
x

{N^x}

is a T.I. set and not semiregular on

X
−
x

X - x

. Then,

P
S
L
(
n
,
q
)
⊆
G
⊆
P
Γ
L
(
n
,
q
)

PSL(n,q) \subseteq G \subseteq P\Gamma L(n,q)

. Geometrical consequences of this result are also obtained.
On the construction of the Bockstein spectral sequence
Authors: Siegel Jerrold
Published: July 2010

The Bockstein spectral sequence is developed from a direct limit construction. This is shown to clarify its relation to certain associated structures, in particular the divided power operations. Finally, the direct limit construction is used to study the problem of enumerating the Bockstein spectral sequences over a given simple

R
R

-module.
Commutative regular rings with integral closure
Authors: Lipshitz L.
Published: July 2010

First order conditions are given which are necessary for a commutative regular ring to have a prime integrally closed extension. If the ring is countable these conditions are also sufficient.
On differentials of the first kind and theta constants for certain congruence subgroups
Authors: Crisalli A. J.
Published: July 2010

Let

Γ
(
8
)

\Gamma (8)

denote the principal congruence subgroup of level 8 and let

Γ
(
16
,
32
)

\Gamma (16,32)

denote the subgroup of

Γ
(
16
)

\Gamma (16)

satisfying

c
d
≡
a
b
≡
0

(
mod
32
)

cd \equiv ab \equiv 0\;(\bmod 32)

. We are dealing only with the elliptic modular case. Consider the spaces of cusp forms of weight 2 (differentials of the first kind) with respect to these groups. It is proved that these spaces are generated by certain monomials of theta constants of degree 4.
On the Calkin algebra and the covering homotopy property
Authors: Conway John B.
Published: July 2010

Let

H

\mathcal {H}

be a separable Hilbert space,

B

(

H

)

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

the bounded operators on

H

,

K

\mathcal {H},\mathcal {K}

the ideal of compact operators, and

π
\pi

the natural map from

B

(

H

)

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

onto the Calkin algebra

B

(

H

)

/

K

\mathcal {B}(\mathcal {H})/\mathcal {K}

. Suppose X is a compact metric space and

Φ
:
C
(
X
)
×
[
0
,
1
]
→

B

(

H

)

/

K

\Phi :C(X) \times [0,1] \to \mathcal {B}(\mathcal {H})/\mathcal {K}

is a continuous function such that

Φ
(
⋅
,
t
)

\Phi ( \cdot ,t)

is a

∗
\ast

-isomorphism for each t and such that there is a

∗
\ast

-isomorphism

ψ
:
C
(
X
)
→

B

(

H

)

\psi :C(X) \to \mathcal {B}(\mathcal {H})

with

π
ψ
(
⋅
)
=
Φ
(
⋅
,
0
)

\pi \psi ( \cdot ) = \Phi ( \cdot ,0)

. It is shown in this paper that if X is a simple Jordan curve, a simple closed Jordan curve, or a totally disconnected metric space then there is a continuous map

Ψ
:
C
(
X
)
×
[
0
,
1
]
→

B

(

H

)

\Psi :C(X) \times [0,1] \to \mathcal {B}(\mathcal {H})

such that

π
Ψ
=
Φ

\pi \Psi = \Phi

and

Ψ
(
⋅
,
0
)
=
ψ
(
⋅
)

\Psi ( \cdot ,0) = \psi ( \cdot )

. Furthermore if X is the disjoint union of two spaces that both have this property, then X itself has this property.
Cone complexes and PL transversality
Authors: McCrory Clint
Published: July 2010

A definition of PL transversality is given, using the orderreversing duality on partially ordered sets. David Stone’s theory of stratified polyhedra is thereby simplified; in particular, the symmetry of blocktransversality is proved. Also, polyhedra satisfying Poincaré duality are characterized geometrically.
Rings of continuous functions with values in a topological field
Authors: Bachman George , Beckenstein Edward , Narici Lawrence , Warner Seth
Published: July 2010

Let

F
F

be a complete topological field. We undertake a study of the ring

C
(
X
,
F
)

C(X,F)

of all continuous

F
F

-valued functions on a topological space

X
X

whose topology is determined by

C
(
X
,
F
)

C(X,F)

, in that it is the weakest making each function in

C
(
X
,
F
)

C(X,F)

continuous, and of the ring

C
∗

(
X
,
F
)

{C^\ast }(X,F)

of all continuous

F
F

-valued functions with relatively compact range, where the topology of

X
X

is similarly determined by

C
∗

(
X
,
F
)

{C^\ast }(X,F)

. The theory of uniform structures permits a rapid construction of the appropriate generalizations of the Hewitt realcompactification of

X
X

in the former case and of the Stone-Čech compactification of

X
X

in the latter. Most attention is given to the case where

F
F

and

X
X

are ultraregular; in this case we determine conditions on

F
F

that permit a development parallel to the classical theory where

F
F

is the real number field. One example of such conditions is that the cardinality of

F
F

be nonmeasurable and that the topology of

F
F

be given by an ultrametric or a valuation. Measure-theoretic interpretations are given, and a nonarchimedean analogue of Nachbin and Shirota’s theorem concerning the bornologicity of

C
(
X
)

C(X)

is obtained.
𝑎*-closures of lattice-ordered groups
Authors: Bleier Roger , Conrad Paul
Published: July 2010

A convex

l
l

-subgroup of an

l
l

-group

G
G

is closed if it contains the join of each of its subsets that has a join in

G
G

. An extension of

G
G

which preserves the lattice of closed convex

l
l

-subgroups of

G
G

is called an

a
∗

{a^ \ast }

-extension of

G
G

. In this paper we consider

a
∗

{a^ \ast }

-extensions and

a
∗

{a^ \ast }

-closures of

G
G

.
The oscillation of an operator on 𝐿^{𝑝}
Authors: Barnes George R. , Whitley Robert
Published: July 2010

We introduce and discuss the oscillation of an operator T mapping

L
p

(
S
,
Σ
,
μ
)

{L^p}(S,\Sigma ,\mu )

into a Banach space. We establish results relating the oscillation, a “local norm", to the norm of the operator. Also using the oscillation we define a generalization of the Fredholm operators T with index

κ
(
T
)
>
∞

\kappa (T) > \infty

and a corresponding perturbation class which contains the compact operators.
A canonical form for symmetric and skew-symmetric extended symplectic modular matrices with applications to Riemann surface theory
Authors: Zarrow Robert
Published: July 2010

The (extended) symplectic modular group

(

∧
n

)

Γ
n

({ \wedge _n}){\Gamma _n}

is the set of all

2
n
×
2
n

2n \times 2n

integer matrices

M
M

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper M upper J Superscript t Baseline upper M equals plus-or-minus upper J right-parenthesis comma upper M upper J Superscript t Baseline upper M equals upper J comma upper J equals Start 2 By 2 Matrix 1st Row 1st Column 0 2nd Column a m p semicolon upper I 2nd Row 1st Column negative upper I 2nd Column a m p semicolon 0 EndMatrix comma">

(
M

J
t

M
=
±
J
)
,
M

J
t

M
=
J
,
J
=

[

0

a
m
p
;
I

−
I

a
m
p
;
0

]

,

(M{J^t}M = \pm J),M{J^t}M = J,J = \left [ {\begin {array}{*{20}{c}} 0 & I \\ { - I} & 0 \\ \end {array} } \right ],

\]

I
I

being the

n
×
n

n \times n

identity matrix. Let

S
n

=
{
M
∈

∧
n

−

Γ
n

|

M
=
−

t

M
}

{S_n} = \{ M \in { \wedge _n} - {\Gamma _n}|M = - {}^tM\}

and

T
n

=
{
M
∈

∧
n

−

Γ
n

|

M
=

t

M
}

{T_n} = \{ M \in { \wedge _n} - {\Gamma _n}|M = {}^tM\}

. We say

M
∼
N

M \sim N

if there exists

K
∈

Γ
n

K \in {\Gamma _n}

such that

M
=
K
N

t

K

M = KN{}^tK

. This defines an equivalence relation on each of these sets separately and we obtain a canonical form for this equivalence. We use this canonical form to study two types of Riemann surfaces which are conformally equivalent to their conjugates and obtain characterizations of their period matrices. We also obtain characterizations of the symplectic matrices which the conformal equivalence induces on the first homology group. One type of surface dealt with is the symmetric Riemann surfaces, i.e. those surfaces which have a conjugate holomorphic self-map of order 2. The other type of surface studied we we call pseudo-symmetric surfaces. These are the hyperelliptic surfaces with the property that the sheet interchange is the square of a conjugate holomorphic automorphism.
On angular momentum Helmholtz theorems and cohomology of Lie algebras
Authors: Stetkaer Henrik
Published: July 2010

Helmholtz’ 2nd theorem (that every vector field on

R

3

{{\mathbf {R}}^3}

with vanishing curl is gradient of a function) can be viewed as a statement about the group of translations of

R

3

{{\mathbf {R}}^3}

. We prove similar theorems for other Lie transformation groups, in particular for semidirect products of abelian and compact semisimple groups. Using Hodge theory we also obtain results analogous to the 1st Helmholtz theorem, but only for compact Lie transformation groups.
Ruelle’s operator theorem and 𝑔-measures
Authors: Walters Peter
Published: July 2010

We use g-measures to give a proof of a convergence theorem of Ruelle. The method of proof is used to gain information about the ergodic properties of equilibrium states for subshifts of finite type.
Equisingular deformations of Puiseux expansions
Authors: Nobile Augusto
Published: July 2010

Parametrizations of a deformation (over a complete local ring) of an irreducible algebroid curve are studied. With these parametrizations another definition of equisingularity (equivalent to the known ones) is given, by using their characteristic numbers. These methods are also used in the complex analytic case. The following applications of these techniques are given: a proof of the existence of a versal equisingular deformation in the complex analytic case; a proof that an equisingular formal family of branches is determined by its

ν
\nu

-truncation (

ν
\nu

large enough, depending only on the characteristic of the special fiber).
Large models of countable height
Authors: Friedman Harvey
Published: July 2010

Eery countable transitive model

M
M

of ZF (without choice) has an ordinal preserving extension satisfying ZF, of power

⊐

M
∩
O
n

{ \sqsupset _{M \cap On}}

. An application to infinitary logic is given.
The residue calculus in several complex variables
Authors: Gordon Gerald Leonard
Published: July 2010

Let W be a complex manifold and V an analytic variety. Then homology classes in

W
−
V

W - V

which bound in V, called the geometric residues, are studied. In fact, a long exact sequence analogous to the Thom-Gysin sequence for nonsingular V is formed by a geometric construction. A geometric interpretation of the Leray spectral sequence of the inclusion of

W
−
V
⊂
V

W - V \subset V

is also given. If the complex codimension of V is one, then one shows that each cohomology class of

W
−
V

W - V

can be represented by a differential form of the type

θ
∧
λ
+
η

\theta \wedge \lambda + \eta

where

λ
\lambda

is the kernel associated to V and

θ

|

V

\theta |V

is the Poincaré residue of this class.