bluuebunny/crossref_metadata_2025 huggingface dataset:
Centralisers of 𝐶^{∞} diffeomorphisms
Authors: Anderson Boyd
Published: July 2010

It is shown that if F is a hyperbolic contraction of

R
n

{R^n}

, coordinates may be chosen so that not only is F a polynomial mapping, but so is any diffeomorphism which commutes with F. This implies an identity principle for diffeomorphisms

G
1

{G_1}

and

G
2

{G_2}

commuting with an arbitrary Morse-Smale diffeomorphism F of a compact manifold M: if

G
1

,

G
2

∈
Z
(
F
)

{G_1},{G_2} \in Z(F)

, then

G
1

=

G
2

{G_1} = {G_2}

on an open subset of

M
⇒

G
1

≡

G
2

M \Rightarrow {G_1} \equiv {G_2}

on M. Finally it is shown that under a certain linearisability condition at the saddles of F,

Z
(
F
)

Z(F)

is in fact a Lie group in its induced topology.
On almost bounded functions
Authors: Miniowitz Ruth
Published: July 2010

New results are presented with regard to the “almost bounded functions” introduced by Goodman [2], including a theorem which contains a proof of Goodman’s conjecture for a particular case.
An application of functional operator models to dissipative scattering theory
Authors: Bondy Daniel Alan
Published: July 2010

An abstract framework for dissipative scattering theory is developed and then applied to two systems previously considered by P. Lax and R. S. Phillips. Results relating the poles and zeroes of the scattering matrix to the spectra of the infinitesimal generators A (which generates the semigroup formed by mapping initial data into solution data at time t) and B (which generates a “local” semigroup) are proven. In particular these results are shown to follow from the fact that the characteristic function of A (appropriately defined) and the scattering matrix combine to form the characteristic function of B.
Some examples in shape theory using the theory of compact connected abelian topological groups
Authors: Keesling James
Published: July 2010

In previous papers the author has studied the shape of compact connected abelian topological groups. This study has led to a number of theorems and examples in shape theory. In this paper a theorem is proved concerning the Čech homology of compact connected abelian topological groups. This theorem together with the author’s previous results are then used to study the movability of general compact Hausdorff spaces. In the theory of shape for compact metric spaces, a number of significant theorems have been proved for movable compact metric spaces. Among these are a theorem of Hurewicz type due to K. Kuperberg, a Whitehead type theorem due to Moszyńska, and a theorem concerning the exactness of the Čech homology sequence for movable compact metric pairs due to Overton. In this paper examples are constructed which show that these theorems do not generalize to arbitrary movable compact Hausdorff spaces without additional assumptions.
Classification of simply connected four-dimensional 𝑅𝑅-manifolds
Authors: Tsagas Gr. , Ledger A.
Published: July 2010

Let (M, g) be a Riemannian manifold. We assume that there is a mapping

s
:
M
→
I
(
M
)

s:M \to I(M)

, where

I
(
M
)

I(M)

is the group of isometries of (M, g), such that

s
x

=
s
(
x
)
,
∀
x
∈
M

{s_x} = s(x),\forall x \in M

, has x as a fixed isolated point, then (M, g) is called a Riemannian s-manifold. If the tensor field S on M defined by the relation

S
x

=

(
d

s
x

)
x

,
∀
x
∈
M

{S_x} = {(d{s_x})_x},\forall x \in M

, is differentiable and invariant by each isometry

s
x

{s_x}

, then the manifold (M, g) is called a regularly s-symmetric Riemannian manifold. The aim of the present paper is to classify simply connected four-dimensional regularly s-symmetric Riemannian manifolds.
Weierstrass normal forms and invariants of elliptic surfaces
Authors: Kas Arnold
Published: July 2010

Let

π
:
S
→
B

\pi :S \to B

be an elliptic surface with a section

σ
:
B
→
S

\sigma :B \to S

. Let

L

−
1

→
B

{L^{ - 1}} \to B

be the normal bundle of

σ
(
B
)

\sigma (B)

in S, and let

W
=
P
(

L

⊗
2

⊕

L

⊗
3

⊕
1
)

W = P({L^{ \otimes 2}} \oplus {L^{ \otimes 3}} \oplus 1)

be a

P

2

{{\mathbf {P}}^2}

-bundle over B. Let

S
∗

{S^\ast }

be the surface obtained from S by contracting those components of fibres of S which do not intersect

σ
(
B
)

\sigma (B)

. Then

S
∗

{S^\ast }

may be imbedded in W and defined by a “Weierstrass equation":
\[

y
2

z
=

x
3

−

g
2

x

z
2

−

g
3

z
3

{y^2}z = {x^3} - {g_2}x{z^2} - {g_3}{z^3}

\]
where

g
2

∈

H
0

(
B
,

O

(

L

⊗
4

)
)

{g_2} \in {H^0}(B,\mathcal {O}({L^{ \otimes 4}}))

and

g
3

∈

H
0

(
B
,

O

(

L

⊗
6

)
)

{g_3} \in {H^0}(B,\mathcal {O}({L^{ \otimes 6}}))

. The only singularities (if any) of

S
∗

{S^\ast }

are rational double points. The triples

(
L
,

g
2

,

g
3

)

(L,{g_2},{g_3})

form a set of invariants for elliptic surfaces with sections, and a complete set of invariants is given by

{
(
L
,

g
2

,

g
3

)
}

/

G

\{ (L,{g_2},{g_3})\} /G

where

G
≅

C

∗

×

Aut

(
B
)

G \cong {{\mathbf {C}}^\ast } \times {\operatorname {Aut}}\;(B)

.
Homomorphisms of integral domains of characteristic zero
Authors: Fried E. , Sichler J.
Published: July 2010

Every category of universal algebras is isomorphic to a full subcategory of the category of all integral domains of characteristic zero and all their 1-preserving homomorphisms. Consequently, every monoid is isomorphic to the monoid of all 1-preserving endomorphisms of an integral domain of characteristic zero.
End extensions, conservative extensions, and the Rudin-Frolík ordering
Authors: Blass Andreas
Published: July 2010

The ordering of ultrafilters on the natural numbers defined by “E-prod N is an end extension of D-prod N,” the ordering defined by “E-prod N is a conservative extension of D-prod N,” and the Rudin-Frolik ordering are proved to be distinct if the continuum hypothesis holds. These three orderings are also characterized in terms of (not necessarily internal) ultrafilters in the Boolean algebra of internal sets of natural numbers in a nonstandard universe.
Stability in Witt rings
Authors: Craven Thomas C.
Published: July 2010

An abstract Witt ring R is defined to be a certain quotient of an integral group ring for a group of exponent 2. The ring R has a unique maximal ideal M containing 2. A variety of results are obtained concerning n-stability, the condition that

M

n
+
1

=
2

M
n

{M^{n + 1}} = 2{M^n}

, especially its relationship to the ring of continuous functions from the space of minimal prime ideals of R to the integers. For finite groups, a characterization of integral group rings is obtained in terms of n-stability. For Witt rings of formally real fields, conditions equivalent to n-stability are given in terms of the real places defined on the field.
Intersection properties of balls and subspaces in Banach spaces
Authors: Lima Ȧsvald
Published: July 2010

We study intersection properties of balls in Banach spaces using a new technique. With this technique we give new and simple proofs of some results of Lindenstrauss and others, characterizing Banach spaces with

L
1

(
μ
)

{L_1}(\mu )

dual spaces by intersection properties of balls, and we solve some open problems in the isometric theory of Banach spaces. We also give new proofs of some results of Alfsen and Effros characterizing M-ideals by intersection properties of balls, and we improve some of their results. In the last section we apply these results on function algebras, G-spaces and order unit spaces and we give new and simple proofs for some representation theorems for those Banach spaces with

L
1

(
μ
)

{L_1}(\mu )

dual spaces whose unit ball contains extreme points.
An 𝐿¹-space for Boolean algebras and semireflexivity of spaces 𝐿^{∞}(𝑋,Σ,𝜇)
Authors: Sentilles Dennis
Published: July 2010

In this paper we suggest a measure free construction of

L
1

{L^1}

-spaces using Boolean algebras and strict topologies and initiate a duality theory of

(

L
∞

,

L
1

)

({L^\infty },{L^1})

like that of the duality of continuous functions and Baire measures, showing that the Boolean context yields a formal link between uniform tightness, uniform

σ
\sigma

-additivity and uniform integrability.
On Lie algebras of vector fields
Authors: Koriyama Akira , Maeda Yoshiaki , Omori Hideki
Published: July 2010

This paper has two purposes. The first is a generalization of the theorem of Pursell-Shanks [10]. Our generalization goes by assuming the existence of a nontrivial core of a Lie algebra. However, it seems to be a necessary condition for the theorems of Pursell-Shanks type. The second is the classification of cores under the assumption that the core itself is infinitesimally transitive at every point. As naturally expected, we have the nonelliptic, primitive infinite-dimensional Lie algebras.
A comparison of various definitions of contractive mappings
Authors: Rhoades B. E.
Published: July 2010

A number of authors have defined contractive type mappings on a complete metric space X which are generalizations of the well-known Banach contraction, and which have the property that each such mapping has a unique fixed point. The fixed point can always be found by using Picard iteration, beginning with some initial choice

x
0

∈
X

{x_0} \in X

. In this paper we compare this multitude of definitions. X denotes a complete metric space with distance function d, and f a function mapping X into itself.
The structure of local integral orthogonal groups
Authors: James D. G.
Published: July 2010

Let M be a lattice on a regular quadratic space over a nondyadic local field. The normal subgroups of the integral orthogonal group

O
(
M
)

O(M)

are determined.
Derivatives of entire functions and a question of Pólya
Authors: Hellerstein Simon , Williamson Jack
Published: July 2010

An old question of Pólya asks whether an entire function f which has, along with each of its derivatives, only real zeros must be of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis equals z Superscript m Baseline e Superscript minus a z squared plus b z plus c Baseline product Underscript n Endscripts left-parenthesis 1 minus StartFraction z Over z Subscript n Baseline EndFraction right-parenthesis e Superscript z slash z Super Subscript n">

f
(
z
)
=

z
m

e

−
a

z
2

+
b
z
+
c

∏
n

(

1
−

z

z
n

)

e

z

/

z
n

f(z) = {z^m}{e^{ - a{z^2} + bz + c}}\prod \limits _n {\left ( {1 - \frac {z}{{{z_n}}}} \right )} {e^{z/{z_n}}}

\]
where

a
⩾
0
,
b

a \geqslant 0,b

and the

z
n

{z_n}

are real, and

Σ
n

z
n

−
2

>
∞

{\Sigma _n}z_n^{ - 2} > \infty

. This note answers this question (essentially in the affirmative) if f is of finite order; indeed, it is established that if

f
,

f
′

f,f’

, and

f
f

have only real zeros (f of finite order), then either f has the above form or f has one of the forms
\[

f
(
z
)
=
a

e

b
z

,

f
(
z
)
=
a
(

e

i
c
z

−

e

i
d

)

f(z) = a{e^{bz}},\quad f(z) = a({e^{icz}} - {e^{id}})

\]
where a, b, c, and d are constants, b complex, c and d real.
Three-dimensional manifolds with finitely generated fundamental groups
Authors: Messer Robert
Published: July 2010

Recent results of G. P. Scott and T. W. Tucker indicate that a three-dimensional manifold with a finitely generated fundamental group is, in various senses, close to being compact. In this paper the structure of such a manifold M is described in terms of a certain compact, incompressible submanifold of M. This result is used to show that the product of M with the real line is essentially the interior of a compact 4-manifold. Finally, when M is

P
2

{P^2}

-irreducible, a necessary and sufficient condition is given for M to be homeomorphic to the complement of a closed subset of the boundary of a compact 3-manifold.
The periodic points of Morse-Smale endomorphisms of the circle
Authors: Block Louis
Published: July 2010

Let

M
S
(

S
1

)

MS({S^1})

denote the set of continuously differentiable maps of the circle with finite nonwandering set, which satisfy certain generic properties. For

f
∈
M
S
(

S
1

)

f \in MS({S^1})

let

P
(
f
)

P(f)

denote the set of positive integers which occur as the period of some periodic point of f. It is shown that for

f
∈
M
S
(

S
1

)

f \in MS({S^1})

there are integers

m
⩾
1

m \geqslant 1

and

n
⩾
0

n \geqslant 0

such that

P
(
f
)
=
{
m
,
2
m
,
4
m
,
…
,

2
n

m
}

P(f) = \{ m,2m,4m, \ldots ,{2^n}m\}

. Conversely, if m and n are integers,

m
⩾
1
,
n
⩾
0

m \geqslant 1,n \geqslant 0

, there is a map

f
∈
M
S
(

S
1

)

f \in MS({S^1})

with

P
(
f
)
=
{
m
,
2
m
,
4
m
,
…
,

2
n

m
}

P(f) = \{ m,2m,4m, \ldots ,{2^n}m\}

.
Hall-Higman type theorems. III
Authors: Berger T. R.
Published: July 2010

This paper continues the investigations of this series. Suppose that

G
=
A
N
S

G = ANS

where S and NS are normal subgroups of G. Suppose that

(

|

A

|

,

|

N
S

|

)
=
1

(|A|,|NS|) = 1

, S is extraspecial, and

S

/

Z
(
S
)

S/Z(S)

is a faithful minimal module for the subgroup AN of G. Assume that k is a field of characteristic prime to

|

G

|

|G|

and V is a faithful irreducible

k

[
G
]

{\mathbf {k}}[G]

-module. The structure of G is discussed in the minimal situation where N is cyclic, A is nilpotent, and

V

|

A

V{|_A}

does not have a regular A-direct summand.
Decomposition spaces having arbitrarily small neighborhoods with 2-sphere boundaries
Authors: Woodruff Edythe P.
Published: July 2010

Let G be an u.s.c. decomposition of

S
3

{S^3}

. Let H denote the set of nondegenerate elements and P be the natural projection of

S
3

{S^3}

onto

S
3

/

G

{S^3}/G

. Suppose that each point in the decomposition space has arbitrarily small neighborhoods with 2-sphere boundaries which miss

P
(
H
)

P(H)

. We prove in this paper that this condition implies that

S
3

/

G

{S^3}/G

is homeomorphic to

S
3

{S^3}

. This answers a question asked by Armentrout [1, p. 15]. Actually, the hypothesis concerning neighborhoods with 2-sphere boundaries is necessary only for the points of

P
(
H
)

P(H)

.
Study of the permanent conjecture and some of its generalizations. II
Authors: Rothaus O. S.
Published: July 2010

In this paper we investigate in a more systematic manner some of the topics initiated in part I of the paper with same title [5]. More specifically, we study in greater detail the properties of the function

E
(
y
)

E(y)

defined in [5] attached to convex polytopes, whose properties in the special case of the space of doubly stochastic matrices are connected with the permanent conjecture. Some close links with Perron-Frobenius theory are developed, and we obtain as a by-product of our study what is, I believe, a new expression for the maximum eigenvalue of a nonnegative matrix, which leads to some new estimates of the same. A final section of the paper investigates some purely algebraic properties of

E
(
y
)

E(y)

, and we obtain some very interesting information connecting a doubly stochastic matrix and its transversals. In order to keep this paper as self-contained as possible, facts used here drawn from part I are stated with as much explicit detail as possible.
Chern-Simons invariants of reductive homogeneous spaces
Authors: Donnelly Harold
Published: July 2010

The geometric characteristic classes of Chern-Simons are computed for certain connections on the canonical bundle and tangent bundle over a reductive homogeneous space. This includes globally symmetric spaces with the Levi-Civita connection of any bi-invariant metric.
Strange billiard tables
Authors: Halpern Benjamin
Published: July 2010

A billiard table is any compact convex body T in the plane bounded by a continuously differentiable curve

∂
T

\partial T

. An idealized billiard ball is a point which moves at unit speed in a straight line except when it hits the boundary

∂
T

\partial T

where it rebounds making the angle of incidence equal to the angle of reflection. A rather surprising phenomenon can happen on such a table.
Homological algebra and set theory
Authors: Eklof Paul C.
Published: July 2010

Assuming the Axiom of Constructibility, necessary and sufficient conditions are given for the vanishing of

Ext

Λ
1

{\operatorname {Ext}}_\Lambda ^1

for rings

Λ
\Lambda

of global dimension 1. Using Martin’s Axiom, the necessity of these conditions is shown not to be a theorem of ZFC. Applications are given to abelian group theory, including a partial solution (assuming

V

=

L

{\text {V}} = {\text {L}}

) to a problem of Baer on the splitting of abelian groups. Some independence results in abelian group theory are also proved.
Matrix representation of simple halfrings
Authors: Stone H. E.
Published: July 2010

The structure of halfrings which are strong direct sums of minimal subtractive right ideals is studied. A class of right simple hemirings which contains both division hemirings and differential subsemirings of division rings is introduced and studied extensively as a tool in this investigation. A matrix representation is obtained for a class of halfrings which properly includes differential subsemirings of simple Artinian rings.
On the asymptotic distribution of closed geodesics on compact Riemann surfaces
Authors: Randol Burton
Published: July 2010

The set of lengths of closed geodesics on a compact Riemann surface is related to the Selberg zeta function in a manner which is evocative of the relationship between the rational primes and the Riemann zeta function. In this paper, this connection is developed to derive results about the asymptotic distribution of these lengths.
On the structure of principal ideals of operators
Authors: Allen G. D. , Shen L. C.
Published: July 2010

This paper considers various types of principal ideals generated by single compact operators on a separable Hilbert space. In particular, necessary and sufficient condtions that a principal ideal be normable are given. Relations between principal ideals and duals of Lorentz and Orlicz spaces are also given. All conditions are expressed using the singular numbers of the operator.
Bounded point evaluations and smoothness properties of functions in 𝑅^{𝑝}(𝑋)
Authors: Wolf Edwin
Published: July 2010

Let X be a compact subset of the complex plane C. We denote by

R
0

(
X
)

{R_0}(X)

the algebra consisting of the (restrictions to X of) rational functions with poles off X. Let m denote 2-dimensional Lebesgue measure. For

p
⩾
1

p \geqslant 1

, let

L
p

(
X
)
=

L
p

(
X
,
d
m
)

{L^p}(X) = {L^p}(X,dm)

. The closure of

R
0

(
X
)

{R_0}(X)

in

L
p

(
X
)

{L^p}(X)

will be denoted by

R
p

(
X
)

{R^p}(X)

. Whenever p and q both appear, we assume that

1

/

p
+
1

/

q
=
1

1/p + 1/q = 1

. If x is a point in X which admits a bounded point evaluation on

R
p

(
X
)

{R^p}(X)

, then the map which sends f to

f
(
x
)

f(x)

for all

f
∈

R
0

(
X
)

f \in {R_0}(X)

extends to a continuous linear functional on

R
p

(
X
)

{R^p}(X)

. The value of this linear functional at any

f
∈

R
p

(
X
)

f \in {R^p}(X)

is denoted by

f
(
x
)

f(x)

. We examine the smoothness properties of functions in

R
p

(
X
)

{R^p}(X)

at those points which admit bounded point evaluations. For

p
>
2

p > 2

we prove in Part I a theorem that generalizes the “approximate Taylor theorem” that James Wang proved for

R
(
X
)

R(X)

. In Part II we generalize a theorem of Hedberg about the convergence of a certain capacity series at a point which admits a bounded point evaluation. Using this result, we study the density of the set X at such a point.
Analytic left algebraic groups. II
Authors: Magid Andy R.
Published: July 2010

An analytic left algebraic group is a complex analytic group carrying a structure of affine algebraic variety such that left translations by fixed elements are morphisms. The core of such a group is the (algebraic) subgroup of all elements such that right translation by them is a morphism. It is shown that the core determines the left algebraic structure, and this is used to determine when left algebraic structures are conjugate by inner automorphisms.
A global theorem for singularities of maps between oriented 2-manifolds
Authors: Quine J. R.
Published: July 2010

Let M and N be smooth compact oriented connected 2-mani-folds. Suppose

f
:
M
→
N

f:M \to N

is smooth and every point

p
∈
M

p \in M

is either a fold point, cusp point, or regular point of f i.e., f is excellent in the sense of Whitney. Let

M
+

{M^ + }

be the closure of the set of regular points at which f preserves orientation and M the closure of the set of regular points at which f reverses orientation. Let

p
1

,
…
,

p
n

{p_1}, \ldots ,{p_n}

be the cusp points and

μ
(

p
k

)

\mu ({p_k})

the local degree at the cusp point

p
k

{p_k}

. We prove the following:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi left-parenthesis upper M right-parenthesis minus 2 chi left-parenthesis upper M Superscript minus Baseline right-parenthesis plus sigma-summation mu left-parenthesis p Subscript k Baseline right-parenthesis equals left-parenthesis degree f right-parenthesis chi left-parenthesis upper N right-parenthesis">

χ
(
M
)
−
2
χ
(

M
−

)
+
∑
μ
(

p
k

)
=
(
deg
⁡
f
)
χ
(
N
)

\chi (M) - 2\chi ({M^ - }) + \sum \mu ({p_k}) = (\deg f)\chi (N)

\]
where

χ
\chi

is the Euler characteristic and deg is the topological degree. We show that it is a generalization of the Riemann-Hurwitz formula of complex analysis and give some examples.
A Gross measure property
Authors: Ernst Lawrence R.
Published: July 2010

We prove that there exists a subset E of

[
0
,
1
]
×

R

2

[0,1] \times {{\mathbf {R}}^2}

such that the 2-dimensional Gross measure of E is 0, while the 1-dimensional Gross measure of

{
z
:
(
y
,
z
)
∈
E
}

\{ z:(y,z) \in E\}

is positive for all

y
∈
[
0
,
1
]

y \in [0,1]

. It is known that for Hausdorff measures no set exists satisfying these conditions.
The homotopy continuation method: numerically implementable topological procedures
Authors: Alexander J. C. , Yorke James A.
Published: July 2010

The homotopy continuation method involves numerically finding the solution of a problem by starting from the solution of a known problem and continuing the solution as the known problem is homotoped to the given problem. The process is axiomatized and an algebraic topological condition is given that guarantees the method will work. A number of examples are presented that involve fixed points, zeroes of maps, singularities of vector fields, and bifurcation. As an adjunct, proofs using differential rather than algebraic techniques are given for the Borsuk-Ulam Theorem and the Rabinowitz Bifurcation Theorem.
Classifying open principal fibrations
Authors: Edwards David A. , Hastings Harold M.
Published: July 2010

Let G be a compact metric group. We shall construct classifying spaces for open principal G-fibrations over compact metric spaces.
Pullback de Rham cohomology of the free path fibration
Authors: Chen Kuo-Tsai
Published: July 2010

Let M and N be smooth manifolds and let

B
¯

(
A
)

\bar B (A)

be the reduced bar construction on the de Rham complex

Λ
(
M
)

\Lambda (M)

or a suitable subcomplex A of M. For every smooth map

f
:
N
→
M
×
M

f:N \to M \times M

, the tensor product

Λ
(
N
)
⊗

B
¯

(
A
)

\Lambda (N) \otimes \bar B(A)

, equipped with a suitable differential, will yield the correct cohomology for the pullback of the free path fibration

P
(
M
)
→
M
×
M

P(M) \to M \times M

via the smooth map F. Moreover,

Λ
(
N
)
⊗

B
¯

(
A
)

\Lambda (N) \otimes \bar B(A)

can be taken as a de Rham subcomplex of the pullback space.
Representation theory of algebras stably equivalent to an hereditary Artin algebra
Authors: Platzeck María Inés
Published: July 2010

Two artin algebras are stably equivalent if their categories of finitely generated modules modulo projectives are equivalent. The author studies the representation theory of algebras stably equivalent to hereditary algebras, using the notions of almost split sequences and irreducible morphisms. This gives a new unified approach to the theories developed for hereditary and radical square zero algebras by Gabriel, Gelfand, Bernstein, Ponomarev, Dlab, Ringel and Müller, as well as other algebras not covered previously. The techniques are purely module theoretical and do not depend on representations of diagrams. They are similar to those used by M. Auslander and the author to study hereditary algebras.
The commutant of an analytic Toeplitz operator
Authors: Cowen Carl C.
Published: July 2010

For a function f in

H
∞

{H^\infty }

of the unit disk, the operator on

H
2

{H^2}

of multiplication by f will be denoted by

T
f

{T_f}

and its commutant by

{

T
f

}
′

\{ {T_f}\} ’

. For a finite Blaschke product B, a representation of an operator in

{

T
B

}
′

{\{ {T_B}\}’}

as a function on the Riemann surface of

B

−
1

∘
B

{B^{ - 1}} \circ B

motivates work on more general functions. A theorem is proved which gives conditions on a family

F

\mathcal {F}

of

H
∞

{H^\infty }

functions which imply that there is a function h such that

{

T
h

}
′

=

∩

f
∈

F

{

T
f

}
′

\{ {T_h}\} ’ = { \cap _{f \in \mathcal {F}}}\{ {T_f}\} ’

. As a special case of this theorem, we find that if the inner factor of

f
−
f
(
c
)

f - f(c)

is a finite Blaschke product for some c in the disk, then there is a finite Blaschke product B with

{

T
f

}
′

=
{

T
B

}
′

\{ {T_f}\} ’ = \{ {T_B}\} ’

. Necessary and sufficient conditions are given for an operator to commute with

T
f

{T_f}

when f is a covering map (in the sense of Riemann surfaces). If f and g are in

H
∞

{H^\infty }

and

f
=
h
∘
g

f = h \circ g

, then

{

T
f

}
′

⊃
{

T
g

}
′

\{ {T_f}\} ’ \supset \{ {T_g}\} ’

. This paper introduces a class of functions, the

H
2

{H^2}

-ancestral functions, for which the converse is true. If f and g are

H
2

{H^2}

-ancestral functions, then

{

T
f

}
′

≠
{

T
g

}
′

\{ {T_f}\} ’ \ne \{ {T_g}\} ’

unless

f
=
h
∘
g

f = h \circ g

where h is univalent. It is shown that inner functions and covering maps are

H
2

{H^2}

-ancestral functions, although these do not exhaust the class. Two theorems are proved, each giving conditions on a function f which imply that

T
f

{T_f}

does not commute with nonzero compact operators. It follows from one of these results that if f is an

H
2

{H^2}

-ancestral function, then

T
f

{T_f}

does not commute with any nonzero compact operators.
Growth hyperspaces of Peano continua
Authors: Curtis D. W.
Published: July 2010

For X a nondegenerate Peano continuum, let

2
X

{2^X}

be the hyperspace of all nonempty closed subsets of X, topologized with the Hausdorff metric. It is known that

2
X

{2^X}

is homeomorphic to the Hilbert cube. A nonempty closed subspace

G

\mathcal {G}

of

2
X

{2^X}

is called a growth hyperspace provided it satisfies the following condition: if

A
∈

G

A \in \mathcal {G}

, and

B
∈

2
X

B \in {2^X}

such that

B
⊃
A

B \supset A

and each component of B meets A, then also

B
∈

G

B \in \mathcal {G}

. The class of growth hyperspaces includes many previously considered subspaces of

2
X

{2^X}

. It is shown that if X contains no free arcs, and

G

\mathcal {G}

is a nontrivial growth hyperspace, then

G

∖
{
X
}

\mathcal {G}\backslash \{ X\}

is a Hilbert cube manifold. A corollary characterizes those growth hyperspaces which are homeomorphic to the Hilbert cube. Analogous results are obtained for growth hyperspaces with respect to the hyperspace

cc

(
X
)

{\text {cc}}(X)

of closed convex subsets of a convex n-cell X.
Hermitian Lie algebras and metaplectic representations. I
Authors: Sternberg Shlomo , Wolf Joseph A.
Published: July 2010

A notion of “hermitian Lie algebra” is introduced which relates ordinary and graded Lie algebra structures. In the case of real-symplectic and arbitrary-signature-unitary Lie algebras, it leads to an analysis of the minimal dimensional coadjoint orbits, and then to the metaplectic representations and their restrictions to unitary groups of arbitrary signature and parabolic subgroups of these unitary groups.
Isotopy groups
Authors: Larmore Lawrence L.
Published: July 2010

For any mapping

f
:
V
→
M

f:V \to M

(not necessarily an embedding), where V and M are differentiable manifolds without boundary of dimensions k and n, respectively, V compact, let

[
V
⊂
M

]
f

=

π
1

(

M
V

,
E
,
f
)

{[V \subset M]_f} = {\pi _1}({M^V},E,f)

, i.e., the set of isotopy classes of embeddings with a specific homotopy to f (E = space of embeddings). The purpose of this paper is to enumerate

[
V
⊂
M

]
f

{[V \subset M]_f}

. For example, if

k
⩾
3
,
n
=
2
k

k \geqslant 3,n = 2k

, and M is simply connected,

[

S
k

⊂
M

]
f

{[{S^k} \subset M]_f}

corresponds to

π
2

M

{\pi _2}M

or

π
2

M
⊗

Z
2

{\pi _2}M \otimes {Z_2}

, depending on whether k is odd or even. In the metastable range, i.e.,

3
(
k
+
1
)
>
2
n

3(k + 1) > 2n

, a natural Abelian affine structure on

[
V
⊂
M

]
f

{[V \subset M]_f}

is defined: if, further, f is an embedding

[
V
⊂
M

]
f

{[V \subset M]_f}

is then an Abelian group. The set of isotopy classes of embeddings homotopic to f is the set of orbits of the obvious left action of

π
1

(

M
V

,
f
)

{\pi _1}({M^V},f)

on

[
V
⊂
M

]
f

{[V \subset M]_f}

. A spectral sequence is constructed converging to a theory

H
∗

(
f
)

{H^\ast }(f)

. If

3
(
k
+
1
)
>
2
n
,

H
0

(
f
)
≅

[
V
⊂
M

]
f

3(k + 1) > 2n, {H^0}(f) \cong {[V \subset M]_f}

provided the latter is nonempty. A single obstruction

Γ
(
f
)
∈

H
1

(
f
)

\Gamma (f) \in {H^1}(f)

is also defined, which must be zero if f is homotopic to an embedding; this condition is also sufficient if

3
(
k
+
1
)
⩽
2
n

3(k + 1) \leqslant 2n

. The

E
2

{E_2}

terms are cohomology groups of the reduced deleted product of V with coefficients in sheaves which are not even locally trivial.

[

S
k

⊂
M

]
f

{[{S^k} \subset M]_f}

is specifically computed in terms of generators and relations if

n
=
2
k
,
k
⩾
3

n = 2k, k \geqslant 3

(Theorem 6.0.2).
Tensor products for 𝑆𝐿(2,𝑘)
Authors: Martin Robert P.
Published: July 2010

Let G be

SL

(
2
,
k
)

{\text {SL}}(2,k)

where k is a locally compact, nondiscrete, totally disconnected topological field whose residual characteristic is not 2,

π
σ

{\pi _\sigma }

, be a principal series representation of G, and

π
∈

G
^

\pi \in \hat G

be arbitrary. We determine the decomposition of

π
σ

⊗
π

{\pi _\sigma } \otimes \pi

into irreducibles by reducing this problem to decomposing the restriction of each

T
∈

G
^

T \in \hat G

to a minimal parabolic subgroup B of G and decomposing certain tensor products of irreducibles of B.
Central twisted group algebras
Authors: Smith Harvey A.
Published: July 2010

A twisted group algebra

L
1

(
A
,
G
;
T
,
α
)

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

is central iff T is trivial and A commutative. (Group algebras of central extension of G are such.) We show that if

H
2

(
G
)

{H^2}(G)

is discrete any central

L
1

(
A
,
G
;
α
)

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

is a direct sum of closed ideals

L
1

(

A
i

,
G
;

α
i

)

{L^1}({A_i},G;{\alpha _i})

having as duals fibre bundles over the duals of closed ideals

A
i

{A_i}

in A, with fibres projective duals of G, and principal

G
∧

{G^\wedge }

bundles (where

G
∧

{G^\wedge }

denotes the group of characters of G) satisfying the conditions which define characteristic bundles for G abelian. (If G is compact

H
2

(
G
)

{H^2}(G)

is always discrete, the direct sum is countable, and the bundles are locally trivial.) Applications are made to the duals of central extensions of groups and in particular to duals of “central” groups. For G commutative,

H
2

(
G
)

{H^2}(G)

discrete, and A a

C
∗

{C^\ast }

-algebra with identity, all central twisted group algebras

L
1

(
A
,
G
;
α
)

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

(and their duals) are classified in purely algebraic terms involving

H
2

(
G
)

{H^2}(G)

, the group G, and the first Čech cohomology group of the dual of A. This result allows us, in principle, to construct all the central

L
1

(
A
,
G
;
α
)

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

and their duals where A is a

C
∗

{C^\ast }

-algebra with identity and G a compact commutative group.
Groups of PL Λ-homology spheres
Authors: Anderson Gerald A.
Published: July 2010

Let

∧

=

Z

k

\wedge \, = \,{{\textbf {Z}}_k}\,

be a subring of Q. The group of

H
K

{H_K}

-cobordism classes of closed PL n-manifolds with the

∧
\wedge

-homology of

S
n

{S^n}

is computed for

n
≥
4

n \ge 4

(modulo K-torsion). The simply connected version is also computed.
Haar measure for measure groupoids
Authors: Hahn Peter
Published: July 2010

It is proved that Mackey’s measure groupoids possess an analogue of Haar measure for locally compact groups; and many properties of the group Haar measure generalize. Existence of Haar measure for groupoids permits solution of a question raised by Ramsay. Ergodic groupoids with finite Haar measure are characterized.
The regular representations of measure groupoids
Authors: Hahn Peter
Published: July 2010

Techniques are developed to study the regular representation and

σ
\sigma

-regular representations of measure groupoids. Convolution, involution, a modular Hilbert algebra, and local and global versions of the regular representation are defined. The associated von Neumann algebras, each uniquely determined by the groupoid and the cocycle

σ
\sigma

, provide a generalization of the group-measure space construction. When the groupoid is principal and ergodic, these algebras are factors. Necessary and sufficient conditions for the

σ
\sigma

-regular representations of a principal ergodic groupoid to be of type I, II, or III are given, as well as a description of the flow of weights; these are independent of

σ
\sigma

. To treat nonergodic groupoids, an ergodic decomposition theorem is provided.
Frattini subalgebras of finitely generated soluble Lie algebras
Authors: Amayo Ralph K.
Published: July 2010

This paper is motivated by a recent one of Stewart and Towers [8] investigating Lie algebras with “good Frattini structure” (definition below). One consequence of our investigations is to prove that any finitely generated metanilpotent Lie algebra has good Frattini structure, thereby answering a question of Stewart and Towers and providing a complete Lie theoretic analogue of the corresponding group theoretic result of Phillip Hall. It will also be shown that in prime characteristic, finitely generated nilpotent-by-finite-dimensional Lie algebras have good Frattini structure.
The cohomology of the symmetric groups
Authors: Mann Benjamin Michael
Published: July 2010

Let

S

n

{{\mathcal {S}}_n}

be the symmetric group on n letters and SG the limit of the sets of degree +1 homotopy equivalences of the

n
−
1

n - 1

sphere. Let p be an odd prime. The main results of this paper are the calculations of

H

∗

(

S

n

,

Z

/

p
)

{H^{\ast }}({\mathcal {S}_n},\,Z/p)

and

H

∗

(
S
G
,
Z

/

p
)

{H^{\ast }}(SG,Z/p)

as algebras, determination of the action of the Steenrod algebra,

a

(
p
)

\mathcal {a}(p)

, on

H

∗

(

S

n

,

Z

/

p
)

{H^{\ast }}({\mathcal {S}_n},\,Z/p)

and

H

∗

(
S
G
,
Z

/

p
)

{H^{\ast }}(SG,Z/p)

and integral analysis of

H

∗

(

S

n

,

Z
,

p
)

{H^{\ast }}({\mathcal {S}_n},\,Z,\,p)

and

H

∗

(
S
G
,

Z
,

p
)

{H^{\ast }}(SG,\,Z,\,p)

.
Amenable pairs of groups and ergodic actions and the associated von Neumann algebras
Authors: Zimmer Robert J.
Published: July 2010

If X and Y are ergodic G-spaces, where G is a locally compact group, and X is an extension of Y, we study a notion of amenability for the pair

(
X
,
Y
)

(X,Y)

. This simultaneously generalizes and expands upon previous work of the author concerning the notion of amenability in ergodic theory based upon fixed point properties of affine cocycles, and the work of Eymard on the conditional fixed point property for groups. We study the relations between this concept of amenability, properties of the von Neumann algebras associated to the actions by the Murray-von Neumann construction, and the existence of relatively invariant measures and conditional invariant means.
Existence theorems for Pareto optimization; multivalued and Banach space valued functionals
Authors: Cesari L. , Suryanarayana M. B.
Published: July 2010

Existence theorems are obtained for optimization problems where the cost functional takes values in an ordered Banach space. The order is defined in terms of a closed convex cone in the Banach space; and in this connection, several relevant properties of cones are studied and they are shown to coincide in the finite dimensional case. The notion of a weak (Pareto) extremum of a subset of an ordered Banach space is then introduced. Existence theorems are proved for extrema for Mayer type as well as Lagrange type problems-in a manner analogous to and including those with scalar valued cost. The side conditions are in the form of general operator equations on a class of measurable functions defined on a finite measure space. Needed closure and lower closure theorems are proved. Also, several analytic criteria for lower closure are provided. Before the appendix, several illustrative examples are given. In the appendix, a criterion (different from the one used in main text) is given and proved, for the Pareto optimality of an element.
A 𝑞-analog of restricted growth functions, Dobinski’s equality, and Charlier polynomials
Authors: Milne Stephen C.
Published: July 2010

We apply finite operator techniques due to G. C. Rota to a combinatorial identity, which counts a collection of generalized restricted growth functions in two ways, and obtain a q-analog of Charlier polynomials and Dobinski’s equality for the number of partitions of an n-set. Our methods afford a unified proof of certain identities in the combinatorics of finite dimensional vector spaces over

GF

(
q
)

{\text {GF}}(q)

.
Symmetric duality for structured convex programs
Authors: McLinden L.
Published: July 2010

A fully symmetric duality model is presented which subsumes the classical treatments given by Duffin (1956), Eisenberg (1961) and Cottle (1963) for linear, homogeneous and quadratic convex programming. Moreover, a wide variety of other special objective functional structures, including homogeneity of any nonzero degree, is handled with equal ease. The model is valid in spaces of arbitrary dimension and treats explicitly systems of both nonnegativity and linear inequality constraints, where the partial orderings may correspond to nonpolyhedral convex cones. The approach is based on augmenting the Fenchel-Rockafellar duality model (1951, 1967) with cone structure to handle constraint systems of the type mentioned. The many results and insights from Rockafellar’s general perturbational duality theory can thus be brought to bear, particularly on sensitivity analysis and the interpretation of dual variables. Considerable attention is devoted to analysis of suboptimizations occurring in the model, and the model is shown to be the projection of another model.
Projective modules for finite Chevalley groups
Authors: Ballard John W.
Published: July 2010

The purpose of this paper is to obtain character formulas for certain indecomposable projective modules for a finite Chevalley group. It is shown that these modules are also modules for the corresponding semisimple algebraic group.
Chern classes of certain representations of symmetric groups
Authors: Evens Leonard , Kahn Daniel S.
Published: July 2010

A formula is derived for the Chern classes of the representation id

∫

ξ
:
P
∫

H
→

U

p
n

\int {\xi :P\int {H \to {U_{pn}}} }

where P is cyclic of order P and

ξ
:
H
→

U
n

\xi :H \to {U_n}

is a fintie dimensional unitary representation of the group H. The formula is applied to the problem of calculating the Chern classes of the “natural” representations

π
j

:

S

j

→

U
j

{\pi _j}:{\mathcal {S}_j} \to {U_j}

of symmetric groups by permutation matrices.
Quantitative Korovkin theorems for positive linear operators on 𝐿_{𝑝}-spaces
Authors: Berens H. , DeVore R.
Published: July 2010

Let

(

L
n

)

({L_n})

be a sequence of positive linear operators on

L
p

(
Ω
)

{L_p}(\Omega )

,

1
⩽
p
>
∞

1 \leqslant p > \infty

or

C
(
Ω
)

C(\Omega )

with

Ω
⊆

R
m

\Omega \subseteq {R^m}

. For suitable

Ω
\Omega

, the functions

(

φ
i

)

i
=
0

m
+
1

({\varphi _i})_{i = 0}^{m + 1}

given by

φ
0

(
x
)
≡
1

{\varphi _0}(x) \equiv 1

,

φ
i

(
x
)
≡

x
i

{\varphi _i}(x) \equiv {x_i}

,

1
⩽
i
⩽
m

1 \leqslant i \leqslant m

,and

φ

m
+
1

(
x
)
≡

|
x
|

2

{\varphi _{m + 1}}(x) \equiv {\left | x \right |^2}

form a test set for

L
p

(
Ω
)

{L_p}(\Omega )

. That is, if

L
n

(

φ
i

)

{L_n}({\varphi _i})

converges to

φ
i

{\varphi _i}

in

‖
⋅
‖

p

{\left \| \cdot \right \|_p}

for each

i
=
0
,
1
,
…
,
m
+
1

i = 0,1, \ldots ,m + 1

, then

L
n

(
f
)

{L_n}(f)

converges to f in

‖
⋅
‖

p

{\left \| \cdot \right \|_p}

for each

f
∈

L
p

(
Ω
)

f \in {L_p}(\Omega )

. We give here quantitative versions of this result. Namely, we estimate

‖

f
−

L
n

f

‖

p

{\left \| {f - {L_n}f} \right \|_p}

in terms of the error

‖

φ
i

−

L
n

φ
i

‖

p

{\left \| {{\varphi _i} - {L_n}{\varphi _i}} \right \|_p}

,

0
⩽
i
⩽
m
+
1

0 \leqslant i \leqslant m + 1

,and the smoothness of the function f.
On complete hypersurfaces of nonnegative sectional curvatures and constant 𝑚th mean curvature
Authors: Hartman Philip
Published: July 2010

The main result is that if

M
=

M
n

M = {M^n}

is a complete Riemann manifold of nonnegative sectional curvature and

X
:

M
→

R

n
+
1

X:\,M \to {R^{n + 1}}

is an isometric immersion such that

X
(
M
)

X(M)

has a positive constant mth mean curvature, then

X
(
M
)

X(M)

is the product of a Euclidean space

R

n
−
d

{R^{n - d}}

and a d-dimensional sphere,

m
⩽
d
⩽
n

m \leqslant d \leqslant n

.
The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators. I
Authors: Weiss Gary
Published: July 2010

We prove the following statements about bounded linear operators on a separable, complex Hilbert space: (1) Every normal operator N that is similar to a Hilbert-Schmidt perturbation of a diagonal operator D is unitarily equivalent to a Hilbert-Schmidt perturbation of D; (2) For every normal operator N, diagonal operator D and bounded operator X, the Hilbert-Schmidt norms (finite or infinite) of

N
X

−

X
D

NX\, - \,XD

and

N

∗

X

−

X

D

∗

{N^{\ast }}X\, - \,X{D^{\ast }}

are equal; (3) If

N
X

−

X
N

NX\, - \,XN

and

N

∗

X

−

X

N

∗

{N^{\ast }}X\, - \,X{N^{\ast }}

are Hilbert-Schmidt operators, then their Hilbert-Schmidt norms are equal; (4) If X is a Hilbert-Schmidt operator and N is a normal operator so that

N
X

−

X
N

NX\, - \,XN

is a trace class operator, then Trace

(

N
X

−

X
N

)

=

0

\left ( {NX\, - \,XN} \right )\, = \,0

; (5) For every normal operator N that is a Hilbert-Schmidt perturbation of a diagonal operator, and every bounded operator X, the Hilbert-Schmidt norms (finite or infinite) of

N
X

−

X
N

NX\, - \,XN

and

N

∗

X

−

X

N

∗

{N^{\ast }}X\, - \,X{N^{\ast }}

are equal. The main technique employs the use of a new concept which we call ’generating functions for matrices’.
Geometric convexity. III. Embedding
Authors: Cantwell John , Kay David C.
Published: July 2010

The straight line spaces of dimension three or higher which were considered by the first author in previous papers are shown to be isomorphic with a strongly open convex subset of a real vector space. To achieve this result we consider the classical descriptive geometry studied in various papers and textbooks by Pasch, Hilbert, Veblen, Whitehead, Coxeter, Robinson, and others, with the significant difference that the geometry considered here is not restricted to 3 dimensions. Our main theorem (which is well known in dimension 3) is that any such geometry is isomorphic to a strongly open convex subset of a real vector space whose “chords” play the role of lines.
Liftings and the construction of stochastic processes
Authors: Cohn Donald L.
Published: July 2010

It is shown that if the continuum hypothesis holds, then the use of liftings to construct modifications of stochastic processes can replace measurable processes with nonmeasurable ones. The use of liftings to choose the paths, rather than the random variables, of a stochastic process is investigated.
Even triangulations of 𝑆³ and the coloring of graphs
Authors: Goodman Jacob Eli , Onishi Hironori
Published: July 2010

A simple necessary and sufficient condition is given for the vertices of a graph, planar or not, to be properly four-colorable. This criterion involves the notion of an “even” triangulation of

S
3

{S^3}

and generalizes, in a natural way, a corresponding criterion for the three-colorability of planar graphs.
Some exponential moments of sums of independent random variables
Authors: Kuelbs J.
Published: July 2010

If

{

X
n

}

\{ {X_n}\}

is a sequence of vector valued random variables,

{

a
n

}

\{ {a_n}\}

a sequence of positive constants, and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M equals sup Underscript n greater-than-or-slanted-equals 1 Endscripts double-vertical-bar left-parenthesis upper X 1 plus midline-horizontal-ellipsis plus upper X Subscript n Baseline right-parenthesis slash a Subscript n Baseline double-vertical-bar">

M
=

sup

n
⩾
1

‖

(

X
1

+
⋯
+

X
n

)

/

a
n

‖

M = {\sup _{n \geqslant 1}}\left \| {({X_1} + \cdots + {X_n})/{a_n}} \right \|

, we examine when

E
(
Φ
(
M
)
)
>
∞

E(\Phi (M)) > \infty

under various conditions on

Φ
,
{

X
n

}

\Phi ,\{ {X_n}\}

, and

{

a
n

}

\{ {a_n}\}

. These integrability results easily apply to empirical distribution functions.
Leaf prescriptions for closed 3-manifolds
Authors: Cantwell John , Conlon Lawrence
Published: July 2010

Our basic question is: What open, orientable surfaces of finite type occur as leaves with polynomial growth in what closed 3-manifolds? This question is motivated by other work of the authors. It is proven that every such surface so occurs for suitable

C
∞

{C^\infty }

foliations of suitable closed 3-manifolds and for suitable

C
1

{C^1}

foliations of all closed 3-manifolds. If the surface has no isolated nonplanar ends it also occurs for suitable

C
∞

{C^\infty }

foliations of all closed 3-manifolds. Finally, a large class of surfaces with isolated nonplanar ends occurs in suitable

C
∞

{C^\infty }

foliations of all closed, orientable 3-manifolds that are not rational homology spheres.
Weak cuts of combinatorial geometries
Authors: Nguyen Hien Q.
Published: July 2010

A weak cut of a Combinatorial Geometry G is a generalization of a modular cut, corresponding to the family of the new dependent sets in a weak map image of G. The use of weak cuts allows the construction of all weak images of G, an important result being that, to any family

M

{\mathcal {M}}

of independent sets of G, is associated a unique weak cut

C

{\mathcal {C}}

containing

M

{\mathcal {M}}

. In practice, the flats of the weak image defined by

C

{\mathcal {C}}

can be constructed directly. The weak cuts corresponding to known weak maps, such as truncation, projection, elementary quotient, are determined. The notion of weak cut is particularly useful in the study of erections. Given a geometry F and a weak image G, an F-erection of G is an erection of G which is a weak image of F. The main results are that the set of all F-erections of G is a lattice with the weak map order, and that the free F-erection can be constructed explicitly. Finally, a problem involving higher order erection is solved.
Adjacent connected sums and torus actions
Authors: McGavran Dennis
Published: July 2010

Let M and N be closed, compact manifolds of dimension m and let X be a closed manifold of dimension

n
>
m

n > m

with embeddings of

X

×

D

m
−
n

X\, \times \,{D^{m - n}}

into M and N. Suppose the interior of

X

×

D

m
−
n

X\, \times \,{D^{m - n}}

is removed from M and N and the resulting manifolds are attached via a homeomorphism

f
:

X
×

S

m
−
n
−
1

→

X

×

S

m
−
n
−
1

f:\,X \times \,{S^{m - n - 1}}\, \to \,X\, \times \,{S^{m - n - 1}}

. Let this homeomorphism be of the form

f
(
x
,

t
)

=

(
x
,

F
(
x
)
(
t
)
)

f(x,\,t)\, = \,(x,\,F(x)(t))

where

F
:

X
→

S
O
(
m
−
n
)

F:\,X \to \,SO(m - n)

. The resulting manifold, written as

M

#
X

N

M\,{\# _X}\,N

, is called the adjacent connected sum of M and N along X. In this paper definitions and examples are given and the examples are then used to classify actions of the torus

T
n

{T^n}

on closed, compact, connected, simply connected

(
n

+

2
)

(n\, + \,2)

-manifolds,

n
⩾

4

n \geqslant \,4

.