bluuebunny/crossref_metadata_2025 huggingface dataset:
Almost locally tame 2-manifolds in a 3-manifold
Authors: Rosen Harvey
Published: July 2010

Several conditions are given which together imply that a 2-manifold M in a 3-manifold is locally tame from one of its complementary domains, U, at all except possibly one point. One of these conditions is that certain arbitrarily small simple closed curves on M can be collared from U. Another condition is that there exists a certain sequence

M
1

,

M
2

,
…

{M_1},{M_2}, \ldots

of 2-manifolds in U converging to M with the property that each unknotted, sufficiently small simple closed curve on each

M
i

{M_i}

is nullhomologous on

M
i

{M_i}

. Moreover, if each of these simple closed curves bounds a disk on a member of the sequence, then it is shown that M is tame from

U
(
M
≠

S
2

)

U(M \ne {S^2})

. As a result, if U is the complementary domain of a torus in

S
3

{S^3}

that is wild from U at just one point, then U is not homeomorphic to the complement of a tame knot in

S
3

{S^3}

.
Expanding gravitational systems
Authors: Saari Donald G.
Published: July 2010

In this paper we obtain a classification of motion for Newtonian gravitational systems as time approaches infinity. The basic assumption is that the motion survives long enough to be studied, i.e., the solution exists in the interval

(
0
,
∞
)

(0,\infty )

. From this classification it is possible to obtain a sketch of the evolving Newtonian universe.
How changing 𝐷[[𝑥]] changes its quotient field
Authors: Sheldon Philip B.
Published: July 2010

Let

D
[
[
x
]
]

D[[x]]

be the ring of formal power series over the commutative integral domain

D
D

. It is shown that changing

D
[
[
x
]
]

D[[x]]

to

D
[
[
x

/

a
]
]

D[[x/a]]

changes (i.e. increases) the quotient field by an infinite transcendence degree over the original field whenever

∩

i
=
1

∞

a
i

D
=
0

\cap _{i = 1}^\infty {a^i}D = 0

. From this it follows that if

D
1

{D_1}

and

D
2

{D_2}

are two distinct rings between the integers and the rational numbers, with

D
1

{D_1}

contained in

D
2

{D_2}

, then the change in the ring of coefficients from

D
1

[
[
x
]
]

{D_1}[[x]]

to

D

2

[
[
x
]
]

D_{2}[[x]]

again yields a change in the quotient fields by an infinite transcendence degree. More generally, it is shown that

D
D

is completely integrally closed iff any increase in the ring of coefficients yields an increase in the quotient field of

D
[
[
x
]
]

D[[x]]

. Moreover,

D
D

is a one-dimensional Prüfer domain iff any change in the ring of coefficients from one overring of

D
D

to another overring of

D
D

yields a change in the quotient field of the respective power series rings. Finally it is shown that many of the domain properties of interest are really properties of their divisibility groups, and some examples are constructed by first constructing the required divisibility groups.
Identities involving the coefficients of a class of Dirichlet series. V, VI
Authors: Berndt Bruce C.
Published: July 2010

In 1949 Chowla and Selberg gave a very useful formula for the Epstein zeta-function associated with a positive definite binary quadratic form. Several generalizations of this formula are given here. The method of proof is new and is based on a theorem that we formerly proved for “generalized” Dirichlet series. An easy proof of Kronecker’s second limit formula is also given.
Comparison and oscillation theorems for matrix differential inequalitites
Authors: Noussair E. S.
Published: July 2010

Strong comparison theorems of Sturm’s type are established for systems of second order quasilinear elliptic partial differential equations. The technique used leads to new oscillation and nonoscillation criteria for such systems. Some criteria are deduced from a comparison theorem, and others are derived by a direct variational method. Some of our results constitute extensions of known theorems to nonselfadjoint quasilinear systems.
A pointwise convergence theorem for sequences of continuous functions.
Authors: Schrader K.
Published: July 2010

Let

{

f
k

}

\{ {f_k}\}

be a sequence of continuous real valued functions defined on an interval

I
I

and

N
N

a fixed nonnegative integer such that if

f
k

(
x
)
=

f
i

(
x
)

{f_k}(x) = {f_i}(x)

for more than

N
N

distinct values of

x
∈
I

x \in I

then

f

k

(
x
)
≡

f
i

(
x
)

{f_{k}}(x) \equiv {f_i}(x)

for

x
∈
I

x \in I

. It follows that there is a subsequence

{

g
j

}

\{ {g_j}\}

of

{

f
k

}

\{ {f_k}\}

such that for each

x
x

the subsequence

{

g
j

(
x
)
}

\{ {g_j}(x)\}

is eventually monotone. Thus

lim

j
→
+
∞

g
j

(
x
)
=
f
(
x
)

{\lim _{j \to + \infty }}{g_j}(x) = f(x)

exists for all

x
x

, where

f
f

is an extended real valued function. If

|

f
k

(
x
)

|

|{f_k}(x)|

is bounded for each

x
∈
I

x \in I

then

lim

j
→
+
∞

g
j

(
x
)
=
f
(
x
)

{\lim _{j \to + \infty }}{g_j}(x) = f(x)

exists as a finite limit for all

x
∈
I

x \in I

. For

N
=
0

N = 0

this reduces to picking a monotone subsequence from a sequence of continuous functions whose graphs are pairwise disjoint.
Commutators, 𝐶^{𝑘}-classification, and similarity of operators
Authors: Kantorovitz Shmuel
Published: July 2010

We generalize the results of our recent paper, The

C
k

{C^k}

-classification of certain operators in

L
p

{L_p}

. II, to the abstract setting of a pair of operators satisfying the commutation relation

[
M
,
N
]
=

N
2

[M,N] = {N^2}

.
Real orthogonal representations of algebraic groups
Authors: Grosshans Frank
Published: July 2010

The purpose of this paper is to determine explicitly, nondegenerate real symmetric bilinear forms invariant under a real absolutely irreducible representation of a real semisimple algebraic group,

G
G

. If

G
G

is split, we construct an extension

G
∗

{G^ \ast }

containing

G
G

and those outer automorphisms of

G
G

fixing the highest weight of the representation. The representation is then extended to

G
∗

{G^ \ast }

and the form is described in terms of the character of this extension. The case of a nonsplit algebraic group is then reduced to the above. The corresponding problem for representations by matrices over the real quaternion division algebra is also considered using similar methods.
Rings of invariant polynomials for a class of Lie algebras
Authors: Takiff S. J.
Published: July 2010

Let

G
G

be a group and let

π
:
G
→
G
L
(
V
)

\pi :G \to GL(V)

be a finite-dimensional representation of

G
G

. Then for

g
∈
G
,
π
(
g
)

g \in G,\pi (g)

induces an automorphism of the symmetric algebra

S
(
V
)

S(V)

of

V
V

. We let

I
(
G
,
V
,
π
)

I(G,V,\pi )

be the subring of

S
(
V
)

S(V)

consisting of elements invariant under this induced action. If

G
G

is a connected complex semisimple Lie group with Lie algebra

L
L

and if Ad is the adjoint representation of

G
G

on

L
L

, then Chevalley has shown that

I
(
G
,
L
,
Ad
)

I(G,L,\text {Ad} )

is generated by a finite set of algebraically independent elements. However, relatively little is known for nonsemisimple Lie groups. In this paper the author exhibits and investigates a class of nonsemisimple Lie groups

G
G

with Lie algebra

L
L

for which

I
(
G
,
L
,
Ad
)

I(G,L,\text {Ad} )

is also generated by a finite set of algebraically independent elements.
Uniqueness of solutions of the Dirichlet and Neumann problems for hyperbolic equations
Authors: Young Eutiquio C.
Published: July 2010

Conditions for uniqueness of solutions of the Dirichlet and Neumann problems are obtained for a singular hyperbolic equation involving a real parameter.
A generalization of the strict topology
Authors: Giles Robin
Published: July 2010

The strict topology

β
\beta

on the space

C
(
X
)

C(X)

of bounded real-valued continuous functions on a topological space X was defined, for locally compact X, by Buck (Michigan Math. J. 5 (1958), 95-104). Among other things he showed that (a)

C
(
X
)

C(X)

is

β
\beta

-complete, (b) the dual of

C
(
X
)

C(X)

under the strict topology is the space of all finite signed regular Borel measures on X, and (c) a Stone-Weierstrass theorem holds for

β
\beta

-closed subalgebras of

C
(
X
)

C(X)

. In this paper the definition of the strict topology is generalized to cover the case of an arbitrary topological space and these results are established under the following conditions on X: for (a) X is a k-space; for (b) X is completely regular; for (c) X is unrestricted.
Existence and duality theorems for convex problems of Bolza
Authors: Rockafellar R. T.
Published: July 2010

The theory of conjugate convex functions is applied to a fundamental class of “convex” problems in the calculus of variations and optimal control. This class has many special properties which have not previously been exploited and for which the standard methods of approach are inadequate. Duality theorems are established which yield new results on the existence of optimal arcs, as well as necessary and sufficient conditions for optimality. These results have some relevance also to the study of “nonconvex” problems.
Degree of symmetry of a homotopy real projective space
Authors: Ku H. T. , Mann L. N. , Sicks J. L. , Su J. C.
Published: July 2010

The degree of symmetry

N
(
M
)

N(M)

of a compact connected differentiable manifold M is the maximum of the dimensions of the compact Lie groups which can act differentiably and effectively on it. It is well known that

N
(
M
)
≦
dim

S
O
(
m
+
1
)

N(M) \leqq \dim \; SO(m + 1)

, for an m-dimensional manifold, and that equality holds only for the standard m-sphere and the standard real projective m-space. W. Y. Hsiang has shown that for a high dimensional exotic m-sphere M,

N
(
M
)
>

m
2

/

8
+
1
>

(

1
4

)

dim
⁡
S
O
(
m
+
1
)

N(M) > {m^2}/8 + 1 > \left ( {\frac {1}{4}} \right )\dim SO(m + 1)

, and that

N
(
M
)
=

m
2

/

8
+
7

/

8

N(M) = {m^2}/8 + 7/8

for some exotic m-spheres. It is shown here that the same results are true for exotic real projective spaces.
Oscillation properties of two term linear differential equations
Authors: Bogar G. A.
Published: July 2010

The two term differential equations

L
n

[
y
]
+
p
y
=
0

{L_n}[y] + py = 0

, where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 0 left-bracket y right-bracket equals y comma upper L Subscript i Baseline left-bracket y right-bracket equals left-parenthesis rho Subscript i Baseline left-parenthesis t right-parenthesis upper L Subscript i Baseline left-bracket y left-parenthesis t right-parenthesis right-bracket right-parenthesis prime">

L
0

[
y
]
=
y
,

L
i

[
y
]
=
(

ρ
i

(
t
)

L
i

[
y
(
t
)
]

)
′

{L_0}[y] = y,{L_i}[y] = ({\rho _i}(t){L_i}[y(t)])’

, were recently studied by Z. Nehari. In this paper we give integral conditions which assure the integrability of

ρ
1

−
1

(
t
)
p
(
t
)

\rho _1^{ - 1}(t)p(t)

on

[
a
,
∞
)

[a,\infty )

when

L
n

[
y
]

{L_n}[y]

is disconjugate. By changing the integral conditions slightly we then prove that the equation has n linearly independent oscillatory solutions.
Well distributed sequences of integers
Authors: Veech William A.
Published: July 2010

Niven’s notion of a uniformly distributed sequence of integers is generalized to well distribution, and two classes of integer sequences are studied in terms of this generalization.
Conjugacy separability of the groups of hose knots
Authors: Stebe Peter F.
Published: July 2010

Let

G
G

be a group. An element

g
g

of

G
G

is c.d. in

G
G

if and only if, given any element

h
h

of

G
G

, either

h
h

is conjugate to

g
g

or there is a homomorphism

ξ
\xi

from

G
G

onto a finite group such that

ξ
(
g
)

\xi (g)

is not conjugate to

ξ
(
h
)

\xi (h)

. Following A. Mostowski, a group is conjugacy separable or c.s. if and only if every element of the group is c.d. In this paper we show that the groups of hose knots are c.s.
Singular integrals and fractional powers of operators
Authors: Fisher Michael J.
Published: July 2010

Recently R. Wheeden studied a class of singular integral operators, the hypersingular integrals, as operators from

L
p
α

(
H
)

L_p^\alpha (H)

to

L
p

(
H
)
;

L
p
α

(
H
)

{L_p}(H);L_p^\alpha (H)

is the range of the

α
\alpha

th order Bessel potential operator acting on

L
p

(
H
)

{L_p}(H)

with the inherited norm. The purposes of the present paper are to extend the known results on hypersingular integrals to complex indices, to extend these results to operators defined over a real separable Hilbert space, and to use Komatsu’s theory of fractional powers of operators to show that the hypersingular integral operator

G
α

{G^\alpha }

is

∫
H

(
−

A
y

)
α

f

d
μ
(
y
)

{\smallint _H}{( - {A_y})^\alpha }f\,d\mu (y)

when

Im
⁡
(
α
)
≠
0

\operatorname {Im}(\alpha ) \ne 0

or when

ℜ
(
α
)

\Re (\alpha )

is not a positive integer where

A
y

g

{A_y}g

is the derivative of g in the direction y. The case where

Im
⁡
(
α
)
=
0

\operatorname {Im} (\alpha ) = 0

and

ℜ
(
α
)

\Re (\alpha )

is a positive integer is treated in a sequel to the present paper.
Weighted norm inequalities for singular and fractional integrals
Authors: Muckenhoupt Benjamin , Wheeden Richard L.
Published: July 2010

Inequalities of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar StartAbsoluteValue x EndAbsoluteValue Superscript alpha Baseline upper T f double-vertical-bar Subscript q Baseline less-than-over-equals upper C double-vertical-bar StartAbsoluteValue x EndAbsoluteValue Superscript alpha Baseline f double-vertical-bar Subscript p">

‖

|
x
|

α

T
f

‖

q

≦
C

‖

|
x
|

α

f

‖

p

{\left \| {{{\left | x \right |}^\alpha }Tf} \right \|_q} \leqq C{\left \| {{{\left | x \right |}^\alpha }f} \right \|_p}

are proved for certain well-known integral transforms, T, in

E
n

{E^n}

. The transforms considered include Calderón-Zygmund singular integrals, singular integrals with variable kernel, fractional integrals and fractional integrals with variable kernel.
The space of all self-homeomorphisms of a two-cell which fix the cell’s boundary is an absolute retract
Authors: Mason W. K.
Published: July 2010

The theorem mentioned in the title is proved. A corollary of the title theorem is: any homeomorphism between two compact subsets of the function space mentioned in the title can be extended to a homeomorphism of the function space onto itself.
The two-piece property and tight 𝑛-manifolds-with-boundary in 𝐸ⁿ
Authors: Banchoff Thomas F.
Published: July 2010

The two-piece property for a set A is a generalization of convexity which reduces to the condition of minimal total absolute curvature if A is a compact 2-manifold. We show that a connected compact 2-manifold-with-boundary in

E
2

{E^2}

has the TPP if and only if each component of the boundary has the TPP. The analogue of this result is not true in higher dimensions without additional conditions, and we introduce a stronger notion called k-tightness and show that an

(
n
+
1
)

(n + 1)

-manifold-with-boundary

M

n
+
1

{M^{n + 1}}

embedded in

E

n
+
1

{E^{n + 1}}

is 0- and

(
n
−
1
)

(n - 1)

-tight if and only if its boundary is 0- and

(
n
−
1
)

(n - 1)

-tight.
Estimates for best approximation to rational functions
Authors: Poreda S. J.
Published: July 2010

Estimates for the deviation of certain rational functions and their polynomials of best uniform approximation on various sets are given. As a result, in some cases these deviation and polynomials are explicitly calculated. For example, the polynomials of best uniform approximation to the function

(
α
z
+
β
)

/

(
z
−
a
)
(
1
−

a
¯

z
)
,

|

a

|

≠
1

(\alpha z + \beta )/(z - a)(1 - \bar az),|a| \ne 1

, on the unit circle are given.
The addition theorem for the entropy of transformations of 𝐺-spaces
Authors: Thomas R. K.
Published: July 2010

For a measure-preserving transformation

T
T

which is a skew-product of a measure-preserving transformation

S
S

and a topological group endomorphism

σ
\sigma

, it is shown that the entropy

h
h

satisfies the following “addition theorem":

h
(
T
)
=
h
(
S
)
+
h
(
σ
)

h(T) = h(S) + h(\sigma )

.
On branch loci in Teichmüller space
Authors: Harvey W. J.
Published: July 2010

The branch locus of the ramified covering of the space of moduli of Fuchsian groups with fixed presentation by the corresponding Teichmüller space is decomposed into a union of Teichmüller spaces, each characterised by a description of the action of the conformal self-mappings admitted by the underlying Riemann surfaces. Equivalence classes of subloci under the action of the modular group are studied, and counted in certain simple cases. One may compute as a result the number of conjugacy classes of elements of prime order in the mapping class group of closed surfaces.
Complex Fourier analysis on a nilpotent Lie group
Authors: Goodman Roe
Published: July 2010

Let

G
G

be a simply-connected nilpotent Lie group, with complexification

G
c

{G_c}

. The functions on

G
G

which are analytic vectors for the left regular representation of

G
G

on

L
2

(
G
)

{L_2}(G)

are determined in this paper, via a dual characterization in terms of their analytic continuation to

G
c

{G_c}

, and by properties of their

L
2

{L_2}

Fourier transforms. The analytic continuation of these functions is shown to be given by the Fourier inversion formula. An explicit construction is given for a dense space of entire vectors for the left regular representation. In the case

G
=
R

G = R

this furnishes a group-theoretic setting for results of Paley and Wiener concerning functions holomorphic in a strip.
Nonresidually finite one-relator groups
Authors: Meskin Stephen
Published: July 2010

The study of one-relator groups includes the connections between group properties and the form of the relator. In this paper we discuss conditions on the form

u

−
1

v
l

u

v
m

{u^{ - 1}}{v^l}u{v^m}

which force the corresponding one-relator groups to be nonresidually finite, i.e. the intersection of the normal subgroups of finite index to be nontrivial. Moreover we show that these forms can be detected amongst the words of a free group.
Entropy-expansive maps
Authors: Bowen Rufus
Published: July 2010

Let

f
:
X
→
X

f:X \to X

be a uniformly continuous map of a metric space. f is called h-expansive if there is an

ε
>
0

\varepsilon > 0

so that the set
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi Subscript epsilon Baseline left-parenthesis x right-parenthesis equals left-brace y colon d left-parenthesis f Superscript n Baseline left-parenthesis x right-parenthesis comma f Superscript n Baseline left-parenthesis y right-parenthesis right-parenthesis less-than-over-equals epsilon">

Φ
ε

(
x
)
=
{
y
:
d
(

f
n

(
x
)
,

f
n

(
y
)
)
≦
ε

{\Phi _\varepsilon }(x) = \{ y:d({f^n}(x),{f^n}(y)) \leqq \varepsilon

for all

n
≧
0

n \geqq 0

} has zero topological entropy for each

x
∈
X

x \in X

. For X compact, the topological entropy of such an f is equal to its estimate using

ε
:
h
(
f
)
=
h
(
f
,
ε
)

\varepsilon :h(f) = h(f,\varepsilon )

. If X is compact finite dimensional and

μ
\mu

an invariant Borel measure, then

h
μ

(
f
)
=

h
μ

(
f
,
A
)

{h_\mu }(f) = {h_\mu }(f,A)

for any finite measurable partition A of X into sets of diameter at most

ε
\varepsilon

. A number of examples are given. No diffeomorphism of a compact manifold is known to be not h-expansive.
Weakly wandering vectors and weakly independent partitions
Authors: Krengel Ulrich
Published: July 2010

We first characterize continuous spectrum and purely discrete spectrum of an isometry U of a Hilbert space geometrically by the existence of a spanning system, resp. by the absence, of vectors with infinitely many orthogonal images under powers of U. We then characterize weak mixing and discrete spectrum of an invertible measure preserving transformation of a probability space in terms of the null sets of the space. Finally for two-fold weakly mixing transformations the result on isometries is strengthened by proving the density of the set of partitions with infinitely many mutually independent images in the set of all finite partitions.
𝐺₀ of a graded ring
Authors: Roberts Leslie G.
Published: July 2010

We consider the Grothendieck group

G
0

{G_0}

of various graded rings, including

G
0

(

A
n
r

)

{G_0}(A_n^r)

where A is a commutative noetherian ring, and

A
n
r

A_n^r

is the A-subalgebra of the polynomial ring

A
[

X
0

,
…
,

X
n

]

A[{X_0}, \ldots ,{X_n}]

generated by monomials of degree r. If A is regular, then

G
0

(

A
n
r

)

{G_0}(A_n^r)

has a ring structure. The ideal class groups of these rings are also considered.
Temperatures in several variables: Kernel functions, representations, and parabolic boundary values
Authors: Kemper John T.
Published: July 2010

This work develops the notion of a kernel function for the heat equation in certain regions of

n
+
1

n + 1

-dimensional Euclidean space and applies that notion to the study of the boundary behavior of nonnegative temperatures. The regions in question are bounded between spacelike hyperplanes and satisfy a parabolic Lipschitz condition at points on the lateral boundary. Kernel functions (normalized, nonnegative temperatures which vanish on the parabolic boundary except at a single point) are shown to exist uniquely. A representation theorem for nonnegative temperatures is obtained and used to establish the existence of finite parabolic limits at the boundary (except for a set of heat-related measure zero).
Hyperbolic limit sets
Authors: Newhouse Sheldon E.
Published: July 2010

Many known results for diffeomorphisms satisfying Axiom A are shown to be true with weaker assumptions. It is proved that if the negative limit set

L
−

(
f
)

{L^ - }(f)

of a diffeomorphism f is hyperbolic, then the periodic points of f are dense in

L
−

(
f
)

{L^ - }(f)

. A spectral decomposition theorem and a filtration theorem for such diffeomorphisms are obtained and used to prove that if

L
−

(
f
)

{L^ - }(f)

is hyperbolic and has no cycles, then f satisfies Axiom A, and hence is

Ω
\Omega

-stable. Examples are given where

L
−

(
f
)

{L^ - }(f)

is hyperbolic, there are cycles, and f fails to satisfy Axiom A.
Stochastic integral representation of multiplicative operator functionals of a Wiener process
Authors: Pinsky Mark A.
Published: July 2010

Let M be a multiplicative operator functional of (X, L) where X is a d-dimensional Wiener process and L is a separable Hilbert space. Sufficient conditions are given in order that M be equivalent to a solution of the linear Itô equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M left-parenthesis t right-parenthesis equals upper I plus sigma-summation Underscript j equals 1 Overscript d Endscripts integral Subscript 0 Superscript t Baseline upper M left-parenthesis s right-parenthesis upper B Subscript j Baseline left-parenthesis x left-parenthesis s right-parenthesis right-parenthesis d x Subscript j Baseline left-parenthesis s right-parenthesis plus integral Subscript 0 Superscript t Baseline upper M left-parenthesis s right-parenthesis upper B 0 left-parenthesis x left-parenthesis s right-parenthesis right-parenthesis d s comma">

M
(
t
)
=
I
+

∑

j
=
1

d

∫
0
t

M
(
s
)

B
j

(
x
(
s
)
)
d

x
j

(
s
)
+

∫
0
t

M
(
s
)

B
0

(
x
(
s
)
)
d
s
,

M(t) = I + \sum \limits _{j = 1}^d {\int _0^t {M(s){B_j}} } (x(s))d{x_j}(s) + \int _0^t {M(s){B_0}(x(s))ds,}

\]
where

B
0

,
…
,

B
d

{B_0}, \ldots ,{B_d}

are bounded operator functions on

R
d

{R^d}

. The conditions require that the equation

T
(
t
)
f
=
E
[
M
(
t
)
f
(
x
(
t
)
)
]

T(t)f = E[M(t)f(x(t))]

define a semigroup on

L
2

(

R
d

)

{L^2}({R^d})

whose infinitesimal generator has a domain which contains all linear functions of the coordinates

(

x
1

,
…
,

x
d

)

({x_1}, \ldots ,{x_d})

. The proof of this result depends on an a priori representation of the semigroup

T
(
t
)

T(t)

in terms of the Wiener semigroup and a first order matrix operator. A second result characterizes solutions of the above Itô equation with

B
0

=
0

{B_0} = 0

. A sufficient condition that M belong to this class is that

E
[
M
(
t
)
]

E[M(t)]

be the identity operator on L and that

M
(
t
)

M(t)

be invertible for each

t
>
0

t > 0

. The proof of this result uses the martingale stochastic integral of H. Kunita and S. Watanabe.
Noetherian intersections of integral domains
Authors: Heinzer William , Ohm Jack
Published: July 2010

Let

D
>
R

D > R

be integral domains having the same quotient field K and suppose that there exists a family

{

V
i

}

i
∈
I

{\{ {V_i}\} _{i \in I}}

of 1-dim quasi-local domains having quotient field K such that

D
=
R
∩
{

V
i

|

i
∈
I
}

D = R \cap \{ {V_i}|i \in I\}

. The goal of this paper is to find conditions on R and the

V
i

{V_i}

in order for D to be noetherian and, conversely, conditions on D in order for R and the

V
i

{V_i}

to be noetherian. An important motivating case is when the set

{

V
i

}

\{ {V_i}\}

consists of a single element V and V is a valuation ring. It is shown, for example, in this case that (i) if V is centered on a finitely generated ideal of D, then V is noetherian and (ii) if V is centered on a maximal ideal of D, then D is noetherian if and only if R and V are noetherian.
Inverse limits, entropy and weak isomorphism for discrete dynamical systems
Authors: Brown James R.
Published: July 2010

A categorical approach is taken to the study of a single measure-preserving transformation of a finite measure space and to inverse systems and inverse limits of such transformations. The questions of existence and uniqueness of inverse limits are settled. Sinaǐ’s theorem on generators is recast and slightly extended to say that entropy respects inverse limits, and various known results about entropy are obtained as immediate corollaries, e.g. systems with quasi-discrete or quasi-periodic spectrum have zero entropy. The inverse limit

Φ
\Phi

of an inverse system

{

Φ
α

:
α
∈
J
}

\{ {\Phi _\alpha }:\alpha \in J\}

of dynamical systems is (1) ergodic, (2) weakly mixing, (3) mixing (of any order) iff each

Φ
α

{\Phi _\alpha }

has the same property. Finally, inverse limits are used to lift a weak isomorphism of dynamical systems

Φ
1

{\Phi _1}

and

Φ
2

{\Phi _2}

to an isomorphism of systems

Φ
^

1

{\hat \Phi _1}

and

Φ
^

2

{\hat \Phi _2}

with the same entropy.
Skew products of dynamical systems
Authors: Kin Eijun
Published: July 2010

In 1950-1951, H. Anzai introduced a method of skew products of dynamical systems in connection with isomorphism problems in ergodic theory. There is a problem to give a necessary and sufficient condition under which an ergodic skew product dynamical system has pure point spectrum. For the special case, translations on the torus, he gave a partial answer for this question. However, this problem has been open in the general case. In the present paper, we generalize the notion of skew products proposed by Anzai and give a complete answer for this problem.
Asymptotic formulae for the eigenvalues of a two-parameter ordinary differential equation of the second order
Authors: Faierman M.
Published: July 2010

We consider a two-point boundary value problem associated with an ordinary differential equation defined over the unit interval and containing the two parameters

λ
\lambda

and

μ
\mu

. If for each real

μ
\mu

we denote the

m
m

th eigenvalue of our system by

λ
m

(
μ
)

{\lambda _m}(\mu )

, then it is known that

λ
m

(
μ
)

{\lambda _m}(\mu )

is real analytic in

−
∞
>
μ
>
∞

- \infty > \mu > \infty

. In this paper we concern ourselves with the asymptotic development of

λ
m

(
μ
)

{\lambda _m}(\mu )

as

μ
→
∞

\mu \to \infty

, and indeed obtain such a development to an accuracy determined by the coefficients of our differential equation. With suitable conditions on the coefficients of our differential equation, the asymptotic formula for

λ
m

(
μ
)

{\lambda _m}(\mu )

may be further developed using the methods of this paper. These results may be modified so as to apply to

λ
m

(
μ
)

{\lambda _m}(\mu )

as

μ
→
−
∞

\mu \to - \infty

if the coefficients of our differential equation are also suitably modified.
Abstract evolution equations and the mixed problem for symmetric hyperbolic systems
Authors: Massey Frank J.
Published: July 2010

In this paper we show that Kato’s theory of linear evolution equations may be applied to the mixed problem for first order symmetric hyperbolic systems of partial differential equations.
Nested sequences of local uniform spaces
Authors: Williams James
Published: July 2010

A locally uniform space is a pair of sets

(
X
,

V

)
,

V

(X,\mathcal {V}),\mathcal {V}

being a filter on

X
×
X

X \times X

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="for-all upper U element-of script upper V comma normal upper Delta left-parenthesis upper X right-parenthesis equals StartSet mathematical left-angle x comma x mathematical right-angle colon x element-of upper X EndSet subset-of-or-equal-to upper U comma upper U Superscript negative 1 Baseline equals StartSet mathematical left-angle y comma x mathematical right-angle colon mathematical left-angle x comma y mathematical right-angle element-of upper U EndSet element-of script upper V">

∀
U
∈

V

,
Δ
(
X
)
=
{
⟨
x
,
x
⟩
:
x
∈
X
}
⊆
U
,

U

−
1

=
{
⟨
y
,
x
⟩
:
⟨
x
,
y
⟩
∈
U
}
∈

V

\forall U \in \mathcal {V},\Delta (X) = \{ \langle x,x\rangle :x \in X\} \subseteq U,{U^{ - 1}} = \{ \langle y,x\rangle :\langle x,y\rangle \in U\} \in \mathcal {V}

, and

∀
x
∈
X
,
∃
V
∈

V

:
(
V
∘
V
)
[
x
]
⊆
U
[
x
]

\forall x \in X,\exists V \in \mathcal {V}:(V \circ V)[x] \subseteq U[x]

. We shall say that a sequence

{
(

X
n

,

V

n

)
:
n
∈
ω
}

\{ ({X_n},{\mathcal {V}_n}):n \in \omega \}

is nested iff

∀
n
∈
ω
,

X
n

⊆

X

n
+
1

\forall n \in \omega ,{X_n} \subseteq {X_{n + 1}}

and

V

n
+
1

|

X
n

=

V

n

{\mathcal {V}_{n + 1}}|{X_n} = {\mathcal {V}_n}

. By a limit for a nested sequence

{
(

X
n

,

V

n

)
:
n
∈
ω
}

\{ ({X_n},{\mathcal {V}_n}):n \in \omega \}

, we shall mean any locally uniform space

(
X
,

V

)

(X,\mathcal {V})

such that

X
=
∪
{

X
n

:
n
∈
ω
}

X = \cup \{ {X_n}:n \in \omega \}

and

∀
n
∈
ω
,

V

|

X
n

=

V

n

\forall n \in \omega ,\mathcal {V}|{X_n} = {\mathcal {V}_n}

. Our first task will be to consider when a nested sequence of locally uniform spaces has a limit; in order to do this, we shall introduce a weak generalization of pseudo-metric functions. We shall also show that, in contrast to locally uniform spaces, each nested sequence of uniform spaces has a limit. With each locally uniform space one can associate a regular relative topology in the obvious fashion. E. Hewitt and J. Novak have constructed regular spaces of the type on which every real-valued continuous function is constant; we shall use our results about limits to give a relatively simple general construction for producing locally uniform spaces which have this type of relative topology. The construction may be done in such a way that the spaces produced have several pleasant topological properties.
The fix-points and factorization of meromorphic functions
Authors: Gross Fred , Yang Chung-chun
Published: July 2010

In this paper, we use the Nevanlinna theory of meromorphic functions and a result of Goldstein to generalize some known results in factorization and fixpoints of entire functions. Specifically, we prove (1) If

f
f

and

g
g

are nonlinear entire functions such that

f
(
g
)

f(g)

is transcendental and of finite order, then

f
(
g
)

f(g)

has infinitely many fix-points. (2) If

f
f

is a polynomial of degree

≧
3

\geqq 3

, and

g
g

is an arbitrary transcendental meromorphic function, then

f
(
g
)

f(g)

must have infinitely many fix-points. (3) Let

p
(
z
)
,
q
(
z
)

p(z),q(z)

be any nonconstant polynomials, at least one of which is not

c
c

-even, and let

a
a

and

b
b

be any constants with

a
a

or

b
≠
0

b \ne 0

. Then

h
(
z
)
=
q
(
z
)
exp
⁡
(
a

z
2

+
b
z
)
+
p
(
z
)

h(z) = q(z)\exp (a{z^2} + bz) + p(z)

is prime.
The envelope of holomorphy of Riemann domains over a countable product of complex planes
Authors: Matos Mário C.
Published: July 2010

This paper deals with the problem of constructing envelopes of holomorphy for Riemann domains over a locally convex space. When this locally convex space is a countable product of complex planes the existence of the envelope of holomorphy is proved and the domains of holomorphy are characterized.
Solvable fundamental groups of compact 3-manifolds
Authors: Evans Benny , Moser Louise
Published: July 2010

A classification is given for groups which can occur as the fundamental group of some compact

3
3

-manifold. In most cases we are able to determine the topological structure of a compact

3
3

-manifold whose fundamental group is known to be solvable. Using the results obtained for solvable groups, we are able to extend some known results concerning nilpotent groups of closed

3
3

-manifolds to the more general class of compact

3
3

-manifolds. In the final section it is shown that each nonfinitely generated abelian group which occurs as a subgroup of the fundamental group of a

3
3

-manifold is a subgroup of the additive group of rationals.
Groups with finite dimensional irreducible representations
Authors: Moore Calvin C.
Published: July 2010

It will be shown that a locally compact group has a finite bound for the dimensions of its irreducible unitary representations if and only if it has a closed abelian subgroup of finite index. It will further be shown that a locally compact group has all of its irreducible representations of finite dimension if and only if it is a projective limit of Lie groups with the same property, and finally that a Lie group has this property if and only if it has a closed subgroup H of finite index such that H modulo its center is compact.
Holomorphic maps into complex projective space omitting hyperplanes
Authors: Green Mark L.
Published: July 2010

Using methods akin to those of E. Borel and R. Nevanlinna, a generalization of Picard’s theorem to several variables is proved by reduction to a lemma on linear relations among exponentials of entire functions. More specifically, it is shown that a holomorphic map from

C

m

{{\mathbf {C}}^m}

to

P

n

{{\mathbf {P}}_n}

omitting

n
+
2

n + 2

distinct hyperplanes has image lying in a hyperplane. If the map omits

n
+
2

n + 2

or more hyperplanes in general position, the image will lie in a linear subspace of low dimension, being forced to be constant if the map omits

2
n
+
1

2n + 1

hyperplanes in general position. The limits found for the dimension of the image are shown to be sharp.
Asymptotic behavior of linear integrodifferential systems
Authors: Barbu Viorel , Grossman Stanley I.
Published: July 2010

We consider the system
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper L right-parenthesis y prime left-parenthesis t right-parenthesis equals upper A y left-parenthesis t right-parenthesis plus integral Subscript negative normal infinity Superscript t Baseline upper B left-parenthesis t minus s right-parenthesis y left-parenthesis s right-parenthesis d s comma y left-parenthesis t right-parenthesis equals f left-parenthesis t right-parenthesis comma t less-than-or-slanted-equals 0">

(

L)

y
′

(
t
)
=
A
y
(
t
)
+

∫

−
∞

t

B
(
t
−
s
)
y
(
s
)
d
s
,
y
(
t
)
=
f
(
t
)
,
t
⩽
0

({\text {L)}}y’(t) = Ay(t) + \int _{ - \infty }^t {B(t - s)y(s)ds,y(t) = f(t),t \leqslant 0}

where

y
(
t
)

y(t)

is an

n
n

-vector and

A
A

and

B
(
t
)

B(t)

are

n
×
n

n \times n

matrices. System

(

L)

({\text {L)}}

generates a semigroup given by

T
t

f
(
s
)
=
y
(
t
+
s
;
f
)

{T_t}f(s) = y(t + s;f)

for

f
f

bounded, continuous and having a finite limit at

−
∞

- \infty

. Under hypotheses concerning the roots of

det
(
λ
I
−
A
−

B
^

(
λ
)
)

\det (\lambda I - A - \hat B(\lambda ))

, where

B
^

(
λ
)

\hat B(\lambda )

is the Laplace transform, various results about the asymptotic behavior of

y
(
t
)

y(t)

are derived, generally after invoking the Hille-Yosida theorem. Two typical results are Theorem 1. If

B
(
t
)
∈

L
1

[
0
,
∞
)

B(t) \in {L^1}[0,\infty )

and

(
λ
I
−
A
−

B
^

(
λ
)

)

−
1

{(\lambda I - A - \hat B(\lambda ))^{ - 1}}

exists for

Re
⁡
λ
>
0

\operatorname {Re} \lambda > 0

, then for every

ϵ
>
0

\epsilon > 0

, there is an

M

ϵ

{M_{\epsilon }}

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper T Subscript t Baseline f StartAbsoluteValue EndAbsoluteValue less-than-or-slanted-equals upper M Subscript epsilon Baseline e Superscript epsilon t Baseline StartAbsoluteValue EndAbsoluteValue f StartAbsoluteValue EndAbsoluteValue">

|

|

T
t

f

|

|

⩽

M

ϵ

e

ϵ
t

|

|

f

|

|

||{T_t}f|| \leqslant {M_{\epsilon }}{e^{\epsilon t}}||f||

. Theorem 2. If

(
λ
I
−
A
−

B
^

(
λ
)

)

−
1

{(\lambda I - A - \hat B(\lambda ))^{ - 1}}

exists for

Re
⁡
λ
>
−
α
(
α
>
0
)

\operatorname {Re} \lambda > - \alpha (\alpha > 0)

and if

B
(
t
)

e

α
t

∈

L
1

[
0
,
∞
)

B(t){e^{\alpha t}} \in {L^1}[0,\infty )

, then the solution to

(

L)

({\text {L)}}

is exponentially asymptotically stable.
The tangential Cauchy-Riemann complex on spheres
Authors: Folland G. B.
Published: July 2010

This paper investigates the

∂
¯

b

{\overline \partial _b}

complex of Kohn and Rossi on the unit sphere in complex

n
n

-space (considered as the boundary of the unit ball). The methods are Fourier-analytic, exploiting the fact that the unitary group

U
(
n
)

U(n)

acts homogeneously on the complex. We decompose the spaces of sections into irreducible components under the action of

U
(
n
)

U(n)

and compute the action of

∂
¯

b

{\overline \partial _b}

on each irreducible piece. We then display the connection between the

∂
¯

b

{\overline \partial _b}

complex and the Dolbeault complexes of certain line bundles on complex projective space. Precise global regularity theorems for

∂
¯

b

{\overline \partial _b}

are proved, including a Sobolev-type estimate for norms related to

∂
¯

b

{\overline \partial _b}

. Finally, we solve the

∂
¯

\overline \partial

-Neumann problem on the unit ball and obtain a proof by explicit calculations of the noncoercive nature of this problem.
Automorphisms of a free associative algebra of rank 2. II
Authors: Czerniakiewicz Anastasia J.
Published: July 2010

Let

R
R

be a commutative domain with 1.

R
⟨
x
,
y
⟩

R\langle x,y\rangle

stands for the free associative algebra of rank 2 over

R
;
R
[

x
~

,

y
~

]

R;R[\tilde x,\tilde y]

is the polynomial algebra over

R
R

in the commuting indeterminates

x
~

\tilde x

and

y
~

\tilde y

. We prove that the map
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Ab colon upper A u t left-parenthesis upper R mathematical left-angle x comma y mathematical right-angle right-parenthesis right-arrow upper A u t left-parenthesis upper R left-bracket x overTilde comma y overTilde right-bracket right-parenthesis">

Ab
:
Aut
⁡
(
R
⟨
x
,
y
⟩
)
→
Aut
⁡
(
R
[

x
~

,

y
~

]
)

\text {Ab}: \operatorname {Aut} (R\langle x,y\rangle ) \to \operatorname {Aut} (R[\tilde x,\tilde y])

induced by the abelianization functor is a monomorphism. As a corollary to this statement and a theorem of Jung [5], Nagata [7] and van der Kulk [8]* that describes the automorphisms of

F
[

x
~

,

y
~

]

F[\tilde x,\tilde y]

(

F
F

a field) we are able to conclude that every automorphism of

F
⟨
x
,
y
⟩

F\langle x,y\rangle

is tame (i.e. a product of elementary automorphisms).
On bounded oscillation and asymptotic expansion of conformal strip mappings
Authors: Obrock Arthur E.
Published: July 2010

Relations between the boundary parameters

ϕ
−

,

ϕ
+

{\phi _ - },{\phi _ + }

of a strip

S
=
{

ϕ
−

(
x
)
>
y
>

ϕ
+

(
x
)
}

S = \{ {\phi _ - }(x) > y > {\phi _ + }(x)\}

and the values

f
(
x
)

f(x)

of its canonical conformal mapping onto a horizontal strip

H
=
{

|

υ

|

>
1
}

H = \{ |\upsilon | > 1\}

are studied. Bounded oscillation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis max Underscript y Endscripts upper R e f left-parenthesis x plus i y right-parenthesis minus min Underscript y Endscripts upper R e f left-parenthesis x plus i y right-parenthesis equals omega left-parenthesis x right-parenthesis equals upper O left-parenthesis 1 right-parenthesis right-parenthesis">

(

max
y

Re
⁡
f
(
x
+
i
y
)
−

min
y

Re
⁡
f
(
x
+
i
y
)
=
ω
(
x
)
=
O
(
1
)
)

({\max _y}\operatorname {Re} f(x + iy) - {\min _y}\operatorname {Re} f(x + iy) = \omega (x) = O(1))

is characterized in terms of

ϕ
−

,

ϕ
+

{\phi _ - },{\phi _ + }

. A formal series expansion

υ
=
∑

y
m

a

m
,
n

(
x
)

\upsilon = \sum {y^m}{a_{m,n}}(x)

is derived for the solution to the Dirichlet problem on

S
S

and its partial sums are used to obtain formulas for the asymptotic expansion of

f
f

in terms of

ϕ
+

,

ϕ
−

{\phi _ + },{\phi _ - }

.
Mean Cesàro summability of Laguerre and Hermite series
Authors: Poiani Eileen L.
Published: July 2010

The primary purpose of this paper is to prove inequalities of the type
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue sigma Subscript n Baseline left-parenthesis f comma x right-parenthesis upper W left-parenthesis x right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript p Baseline less-than-or-slanted-equals upper C StartAbsoluteValue EndAbsoluteValue f left-parenthesis x right-parenthesis upper W left-parenthesis x right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript p">

|

|

σ
n

(
f
,
x
)
W
(
x
)

|

|

p

⩽
C

|

|

f
(
x
)
W
(
x
)

|

|

p

||{\sigma _n}(f,x)W(x)|{|_p} \leqslant C||f(x)W(x)|{|_p}

where

σ
n

(
f
,
x
)

{\sigma _n}(f,x)

is the

n
n

th

(
C
,
1
)

(C,1)

mean of the Laguerre or Hermite series of

f
,
W
(
x
)

f, W(x)

is a suitable weight function of particular form,

C
C

is a constant independent of

f
(
x
)

f(x)

and

n
n

, and the norm is taken over

(
0
,
∞
)

(0,\infty )

in the Laguerre case and

(
−
∞
,
∞
)

( - \infty ,\infty )

in the Hermite case for

1
⩽
p
⩽
∞

1 \leqslant p \leqslant \infty

. Both necessary and sufficient conditions for these inequalities to remain valid are determined. For

p
>
∞

p > \infty

and

f
(
x
)
W
(
x
)
∈

L
p

f(x)W(x) \in {L^p}

, mean summability results showing that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow normal infinity Endscripts StartAbsoluteValue EndAbsoluteValue left-bracket sigma Subscript n Baseline left-parenthesis f comma x right-parenthesis minus f left-parenthesis x right-parenthesis right-bracket upper W left-parenthesis x right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript p Baseline equals 0">

lim

n
→
∞

|

|

[

σ
n

(
f
,
x
)
−
f
(
x
)
]
W
(
x
)

|

|

p

=
0

\lim \nolimits _{n \to \infty } ||[{\sigma _n}(f,x) - f(x)]W(x)|{|_p} = 0

are derived by use of the appropriate density theorems. Detailed proofs are presented for the Laguerre expansions, and the analogous results for Hermite series follow as corollaries.
Some asymptotic fixed point theorems
Authors: Nussbaum Roger D.
Published: July 2010

By an asymptotic fixed point theorem we mean a theorem in functional analysis in which the existence of fixed points of a map

f
f

is established with the aid of assumptions on the iterates

f
n

{f^n}

of

f
f

. We prove below some new theorems of this type, and we obtain as corollaries results of F. E. Browder, G. Darbo, R. L. Frum-Ketkov, W. A. Horn and others. We also state a number of conjectures about fixed point theorems at the end of the paper.
Radon-Nikodym theorems for vector valued measures
Authors: Kupka Joseph
Published: July 2010

Let

μ
\mu

be a nonnegative measure, and let

m
m

be a measure having values in a real or complex vector space

V
V

. This paper presents a comprehensive treatment of the question: When is

m
m

the indefinite integral with respect to

μ
\mu

of a

V
V

valued function

f
?

f?

Previous results are generalized, and two new types of Radon-Nikodym derivative, the “type

ρ
\rho

” function and the “strongly

Γ
\Gamma

integrable” function, are introduced. A derivative of type

ρ
\rho

may be obtained in every previous Radon-Nikodym theorem known to the author, and a preliminary result is presented which gives necessary and sufficient conditions for the measure

m
m

to be the indefinite integral of a type

ρ
\rho

function. The treatment is elementary throughout, and in particular will include the first elementary proof of the Radon-Nikodym theorem of Phillips.
Differentiable structures on function spaces
Authors: Krikorian Nishan
Published: July 2010

A

C
s

{C^s}

differentiable manifold structure is constructed for spaces of maps from a compact

C
r

{C^r}

manifold

M
M

to a

C

r
+
s

{C^{r + s}}

manifold

N
N

. The method (1) is inspired by Douady; (2) does not require any additional structure on

N
N

(such as sprays); (3) includes the case when

N
N

is an analytic manifold and concludes that the mapping space is also an analytic manifold; (4) can be used to treat all the classical mapping spaces (

C
r

{C^r}

functions,

C
r

{C^r}

functions with Hölder conditions, and Sobolev functions). Several interesting aspects of these manifolds are investigated such as their tangent spaces, their behavior with respect to functions, and realizations of Lie group structures on them. Differentiable structures are also exhibited for spaces of compact maps with noncompact domain.
Cyclic vectors and irreducibility for principal series representations. II
Authors: Wallach Nolan R.
Published: July 2010

This paper is a continuation of the author’s paper Cyclic vectors and irreducibility for principal series representations. In this paper the nonunitary principal series is studied. Using a theorem of Kostant, a sufficient condition is found for irreducibility of nonunitary principal series representations.
On the semisimplicity of group rings of solvable groups
Authors: Hampton C. R. , Passman D. S.
Published: July 2010

Let

K
[
G
]

K[G]

denote the group ring of

G
G

over the field

K
K

of characteristic

p
>
0

p > 0

. An interesting unsolved problem is to find necessary and sufficient conditions on

G
G

for

K
[
G
]

K[G]

to be semisimple. Even the special case in which

G
G

is assumed to be a solvable group is still open. In this paper we prove a number of theorems which may be of use in this special case.
Transverse cellular mappings of polyhedra
Authors: Akin Ethan
Published: July 2010

We generalize Marshall Cohen’s notion of transverse cellular map to the polyhedral category. They are described by the following: Proposition. Let

f
:
K
→
L

f:K \to L

be a proper simplicial map of locally finite simplicial complexes. The following are equivalent: (1) The dual cells of the map are all cones. (2) The dual cells of the map are homogeneously collapsible in

K
K

. (3) The inclusion of

L
L

into the mapping cylinder of

f
f

is collared. (4) The mapping cylinder triad

(

C
f

,
K
,
L
)

({C_f},K,L)

is homeomorphic to the product triad

(
K
×
I
;
K
×
1
,
K
×
0
)

(K \times I;K \times 1,K \times 0)

rel

K
=
K
×
1

K = K \times 1

. Condition (2) is slightly weaker than

f

−
1

{f^{ - 1}}

(point) is homogeneously collapsible in

K
K

. Condition (4) when stated more precisely implies

f
f

is homotopic to a homeomorphism. Furthermore, the homeomorphism so defined is unique up to concordance. The two major applications are first, to develop the proper theory of “attaching one polyhedron to another by a map of a subpolyhedron of the former into the latter". Second, we classify when two maps from

X
X

to

Y
Y

have homeomorphic mapping cylinder triads. This property turns out to be equivalent to the equivalence relation generated by the relation

f
∼
g

f \sim g

, where

f
,
g
:
X
→
Y

f,g:X \to Y

means

f
=
g
r

f = gr

for

r
:
X
→
X

r:X \to X

some transverse cellular map.