bluuebunny/crossref_metadata_2025 huggingface dataset:
Surjective stability in dimension 0 for 𝐾₂ and related functors
Authors: Stein Michael R.
Published: July 2010

This paper continues the investigation of generators and relations for Chevalley groups over commutative rings initiated in [14]. The main result is that if A is a semilocal ring generated by its units, the groups

L
(

Φ

,
A
)

L({\mathbf {\Phi }},A)

of [14] are generated by the values of certain cocycles on

A
∗

×

A
∗

{A^\ast } \times {A^\ast }

. From this follows a surjective stability theorem for the groups

L
(

Φ

,
A
)

L({\mathbf {\Phi }},A)

, as well as the result that

L
(

Φ

,
A
)

L({\mathbf {\Phi }},A)

is the Schur multiplier of the elementary subgroup of the points in A of the universal Chevalley-Demazure group scheme with root system

Φ

{\mathbf {\Phi }}

, if

Φ

{\mathbf {\Phi }}

has large enough rank. These results are proved via a Bruhat-type decomposition for a suitably defined relative group associated to a radical ideal. These theorems generalize to semilocal rings results of Steinberg for Chevalley groups over fields, and they give an effective tool for computing Milnor’s groups

K
2

(
A
)

{K_2}(A)

when A is semilocal.
Prime ideals and sheaf representation of a pseudo symmetric ring
Authors: Shin Gooyong
Published: July 2010

Almost symmetric rings and pseudo symmetric rings are introduced. The classes of symmetric rings, of almost symmetric rings, and of pseudo symmetric rings are in a strictly increasing order. A sheaf representation is obtained for pseudo symmetric rings, similar to the cases of symmetric rings, semiprime rings, and strongly harmonic rings. Minimal prime ideals of a pseudo symmetric ring have the same characterization, due to J. Kist, as for the commutative case. A characterization is obtained for a pseudo symmetric ring with a certain right quotient ring to have compact minimal prime ideal space, extending a result due to Mewborn.
Hull subordination and extremal problems for starlike and spirallike mappings
Authors: MacGregor Thomas H.
Published: July 2010

Let

F

\mathfrak {F}

be a compact subset of the family

A

\mathcal {A}

of functions analytic in

Δ
=
{
z
:

|

z

|

>
1
}

\Delta = \{ z:\;|z| > 1\}

, and let

L

\mathcal {L}

be a continuous linear operator of order zero on

A

\mathcal {A}

. We show that if the extreme points of the closed convex hull of

F

\mathcal {F}

is the set

{

f
0

(
x
z
)
}
(

|

x

|

=
1
)

\{ {f_0}(xz)\} (|x| = 1)

, then

L

(
f
)

\mathcal {L}(f)

is hull subordinate to

L

(

f
0

)

\mathcal {L}({f_0})

in

Δ
\Delta

. This generalizes results of R. M. Robinson corresponding to families

F

\mathcal {F}

of functions that are subordinate to

(
1
+
z
)

/

(
1
−
z
)

(1 + z)/(1 - z)

or to

1

/

(
1
−
z

)
2

1/{(1 - z)^2}

. Families

F

\mathcal {F}

to which this theorem applies are discussed and we identify each such operator

L

\mathcal {L}

with a suitable sequence of complex numbers. Suppose that

Φ
\Phi

is a nonconstant entire function and that

0
>

|

z
0

|

>
1

0 > |{z_0}| > 1

. We show that the maximum of

Re
⁡
{
Φ
[
log
⁡
(
f
(

z
0

)

/

z
0

)
]
}

\operatorname {Re} \{ \Phi [\log (f({z_0})/{z_0})]\}

over the class of starlike functions of order a is attained only by the functions

f
(
z
)
=
z

/

(
1
−
x
z

)

2
−
2
α

,

|

x

|

=
1

f(z) = z/{(1 - xz)^{2 - 2\alpha }},\;|x| = 1

. A similar result is obtained for spirallike mappings. Both results generalize a theorem of G. M. Golusin corresponding to the family of starlike mappings.
Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques
Authors: Bonami Aline , Clerc Jean-Louis
Published: July 2010

Nous établissons une inégalité entre les sommes de Cesáro et la fonction maximale associées á une fonction définie sur la sphére, et nous en déduisons divers résultats de convergence en norme

L
p

{L^p}

, convergence presque partout, localisation des développements en harmoniques sphériques, ainsi qu’un théorème de multiplicateurs qui généralise le théorème classique de Marcinkiewicz sur les séries trigonométriques. La même étude est faite pour les développements suivant les polynômes ultrasphériques. Nous montrons de plus que les sommes partielles du développement en harmoniques sphériques d’une fonction de

L
p

(

Σ
n

)
,
p
≠
2

{L^p}({\Sigma _n}),p \ne 2

, ne convergent pas forcément en norme.
A fixed point theorem-free approach to weak almost periodicity
Authors: Veech William A.
Published: July 2010

In this paper we present a generalization of the Eberlein, de Leeuw and Glicksberg decomposition theorem for weakly almost periodic functions which does not rely on any fixed point theorem for its proof. A generalization of the Ryll-Nardzewski fixed point theorem is given.
Parametrizations of analytic varieties
Authors: Becker Joseph
Published: July 2010

Let V be an analytic subvariety of an open subset

Ω
\Omega

of

C

n

{{\text {C}}^n}

of pure dimension r; for any

p
∈
V

p \in V

, there exists an

n
−
r

n - r

dim plane T such that

π
T

:
V
→

C

r

{\pi _T}:V \to {{\text {C}}^r}

, the projection along T to

C

r

{{\text {C}}^r}

, is a branched covering of finite sheeting order

μ
(
V
,
p
,
T
)

\mu (V,p,T)

in some neighborhood of V about p.

π
T

{\pi _T}

is called a global parametrization of V if

π
T

{\pi _T}

has all discrete fibers, e.g.

dim
p

V
∩
(
T
+
p
)
=
0

{\dim _p}V \cap (T + p) = 0

for all

p
∈
V

p \in V

. Theorem.
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B equals StartSet left-parenthesis p comma upper T right-parenthesis element-of upper V times upper G left-parenthesis n minus r comma n right-parenthesis vertical-bar dimension Subscript p Baseline upper V intersection left-parenthesis upper T plus p right-parenthesis greater-than 0 EndSet">

B
=
{
(
p
,
T
)
∈
V
×
G
(
n
−
r
,
n
)

|

dim
p

V
∩
(
T
+
p
)
>
0
}

B = \{ (p,T) \in V \times G(n - r,n)|{\dim _p}V \cap (T + p) > 0\}

is an analytic set. If

π
2

:
V
×
G
→
G

{\pi _2}:V \times G \to G

is the natural projection, then

π
2

(
B
)

{\pi _2}(B)

is a negligible set in G. Theorem.

{
(
p
,
T
)
∈
V
×
G

|

μ
(
V
,
p
,
T
)
≥
k
}

\{ (p,T) \in V \times G|\mu (V,p,T) \geq k\}

is an analytic set. For each

p
∈
V

p \in V

, there is a least

μ
(
V
,
p
)

\mu (V,p)

and greatest

m
(
V
,
p
)

m(V,p)

sheeting multiplicity over all

T
∈
G

T \in G

. If

Ω
\Omega

is Stein, V is the locus of finitely many holomorphic functions but its ideal in

O

(
Ω
)

\mathcal {O}(\Omega )

is not necessarily finitely generated. Theorem. If

μ
(
V
,
p
)

\mu (V,p)

is bounded on V, then its ideal is finitely generated.
Discrete 𝜔-sequences of index sets
Authors: Hay Louise
Published: July 2010

We define a discrete

ω
\omega

-sequence of index sets to be a sequence

{
θ

A
n

}

n
≥
0

{\{ \theta {A_n}\} _{n \geq 0}}

, of index sets of classes of recursively enumerable sets, such that for each n,

θ

A

n
+
1

\theta {A_{n + 1}}

is an immediate successor of

θ

A
n

\theta {A_n}

in the partial order of degrees of index sets under one-one reducibility. The main result of this paper is that if S is any set to which the complete set K is not Turing-reducible, and

A
S

{A^S}

is the class of recursively enumerable subsets of S, then

θ

A
S

\theta {A^S}

is at the bottom of c discrete

ω
\omega

-sequences. It follows that every complete Turing degree contains c discrete

ω
\omega

-sequences.
Quasiconformal mappings and sets of finite perimeter
Authors: Kelly James C.
Published: July 2010

Let

D
D

be a domain in

R
n

,
n
⩾
2
,
f

{R^n},n \geqslant 2,f

a quasiconformal mapping on

D
D

. We give a definition of bounding surface of codimension one lying in

D
D

, and show that, given a system

Σ
\Sigma

of such surfaces, the image of the restriction of

f
f

to “almost every” surface is again a surface. Moreover, on these surfaces,

f
f

takes

H

n
−
1

{H^{n - 1}}

(Hausdorff

(
n
−
1
)

(n - 1)

-dimensional) null sets to

H

n
−
1

{H^{n - 1}}

null sets. “Almost every” surface is given a precise meaning via the concept of the module of a system of measures, a generalization of the concept of extremal length.
Logic and invariant theory. I. Invariant theory of projective properties
Authors: Whiteley Walter
Published: July 2010

This paper initiates a series of papers which will reexamine some problems and results of classical invariant theory, within the framework of modern first-order logic. In this paper the notion that an equation is of invariant significance for the general linear group is extended in two directions. It is extended to define invariance of an arbitrary first-order formula for a category of linear transformations between vector spaces of dimension n. These invariant formulas are characterized by equivalence to formulas of a particular syntactic form: homogeneous formulas in determinants or “brackets". The fuller category of all semilinear transformations is also introduced in order to cover all changes of coordinates in a projective space. Invariance for this category is investigated. The results are extended to cover invariant formulas with both covariant and contravariant vectors. Finally, Klein’s Erlanger Program is reexamined in the light of the extended notion of invariance as well as some possible geometric categories.
The structure of completely regular semigroups
Authors: Petrich Mario
Published: July 2010

The principal result is a construction of completely regular semigroups in terms of semilattices of Rees matrix semigroups and their translational hulls. The main body of the paper is occupied by considerations of various special cases based on the behavior of either Green’s relations or idempotents. The influence of these special cases on the construction in question is studied in considerable detail. The restrictions imposed on Green’s relations consist of the requirement that some of them be congruences, whereas the restrictions on idempotents include various covering conditions or the requirement that they form a subsemigroup.
Maximal 𝛼-r.e. sets
Authors: Lerman Manuel
Published: July 2010

Various generalizations of maximal sets from ordinary recursion theory to recursion theory on admissible ordinals are considered. A justification is given for choosing one of these definitions as superior to the rest. For all the definitions considered to be reasonable, a necessary and sufficient condition for the existence of such maximal

α
\alpha

-r.e. sets is obtained.
Elementary divisor rings and finitely presented modules
Authors: Larsen Max D. , Lewis William J. , Shores Thomas S.
Published: July 2010

Throughout, rings are commutative with unit and modules are unital. We prove that R is an elementary divisor ring if and only if every finitely presented module over R is a direct sum of cyclic modules, thus providing a converse to a theorem of Kaplansky and answering a question of Warfield. We show that every Bezout ring with a finite number of minimal prime ideals is Hermite. So, in particular, semilocal Bezout rings are Hermite answering affirmatively a question of Henriksen. We show that every semihereditary Bezout ring is Hermite. Semilocal adequate rings are characterized and a partial converse to a theorem of Henriksen is established.
Fields of fractions for group algebras of free groups
Authors: Lewin Jacques
Published: July 2010

Let KF be the group algebra over the commutative field K of the free group F. It is proved that the field generated by KF in any Mal’cev-Neumann embedding for KF is the universal field of fractions

U
(
K
F
)

U(KF)

of KF. Some consequences are noted. An example is constructed of an embedding

K
F
⊂
D

KF \subset D

into a field D with

D

≄

U
(
K
F
)

D\;\not \simeq \;U(KF)

. It is also proved that the generalized free product of two free groups can be embedded in a field.
Minimal sequences in semigroups
Authors: Putcha Mohan S.
Published: July 2010

In this paper we generalize a result of Tamura on

δ
\delta

-indecomposable semigroups. Based on this, the concept of a minimal sequence between two points, and from a point to another, is introduced. The relationship between two minimal sequences between the same points is studied. The rank of a semigroup S is defined to be the supremum of the lengths of the minimal sequences between points in S. The semirank of a semigroup S is defined to be the supremum of the lengths of the minimal sequences from a point to another in S. Rank and semirank are further studied.
Some positive trigonometric sums
Authors: Askey Richard , Steinig John
Published: July 2010

Vietoris found an interesting generalization of the classical inequality

Σ

k
=
1

n

(
sin

k
θ

/

k
)
>
0
,
0
>
θ
>
π

\Sigma _{k = 1}^n(\sin \,k\theta /k) > 0,0 > \theta > \pi

. A simplified proof is given for his inequality and his similar inequality for cosine series. Various new results which follow from these inequalities include improved estimates for the location of the zeros of a class of trigonometric polynomials and new positive sums of ultraspherical polynomials which extend an old inequality of Fejér. Both of Vietoris’ inequalities are special cases of a general problem for Jacobi polynomials, and a summary is given of known results on this problem.
Property 𝑆𝑈𝑉^{∞} and proper shape theory
Authors: Sher R. B.
Published: July 2010

A class of spaces called the

S
U

V
∞

SU{V^\infty }

spaces has arisen in the study of a possibly noncompact variant of cellularity. These spaces play a role in this new theory analogous to that of the

U

V
∞

U{V^\infty }

spaces in cellularity theory. Herein it is shown that the locally compact metric space X is an

S
U

V
∞

SU{V^\infty }

space if and only if there exists a tree T such that X and T have the same proper shape. This result is then used to classify the proper shapes of the

S
U

V
∞

SU{V^\infty }

spaces, two such being shown to have the same proper shape if and only if their end-sets are homeomorphic. Also, a possibly noncompact analog of property

U

V
n

U{V^n}

, called

S
U

V
n

SU{V^n}

, is defined and it is shown that if X is a closed connected subset of a piecewise linear n-manifold, then X is an

S
U

V
n

SU{V^n}

space if and only if X is an

S
U

V
∞

SU{V^\infty }

space. Finally, it is shown that a locally finite connected simplicial complex is an

S
U

V
∞

SU{V^\infty }

space if and only if all of its homotopy and proper homotopy groups vanish.
Periodic solutions of 𝑥”+𝑔(𝑥)+𝜇ℎ(𝑥)=0
Authors: Butler G. J. , Freedman H. I.
Published: July 2010

Necessary and sufficient conditions for

x
+
f
(
x
)
=
0

x + f(x) = 0

to admit at least one nontrivial periodic solution are given. The results are applied to

x
+
g
(
x
)
+
μ
h
(
x
)
=
0
,
x
(
0
)
=
A
,

x
′

(
0
)
=
0

x + g(x) + \mu h(x) = 0,x(0) = A,x’(0) = 0

in order to characterize those regions of the

(
μ
,
A
)

(\mu ,A)

-plane for which nontrivial periodic solutions exist. A converse theorem is given; together with some illustrative examples.
Coherent extensions and relational algebras
Authors: Bunge Marta C.
Published: July 2010

The notion of a lax adjoint to a 2-functor is introduced and some aspects of it are investigated, such as an equivalent definition and a corresponding theory of monads. This notion is weaker than the notion of a 2-adjoint (Gray) and may be obtained from the latter by weakening that of 2functor and replacing the adjointness equations by adding 2-cells satisfying coherence conditions. Lax monads are induced by and resolve into lax adjoint pairs, the latter via 2-categories of lax algebras. Lax algebras generalize the relational algebras of Barr in the sense that a relational algebra for a monad in Sets is precisely a lax algebra for the lax monad induced in Rel. Similar considerations allow us to recover the T-categories of Burroni as well. These are all examples of lax adjoints of the “normalized” sort and the universal property they satisfy can be expressed by the requirement that certain generalized Kan extensions exist and are coherent. The most important example of relational algebras, i.e., topological spaces, is analysed in this new light also with the purpose of providing a simple illustration of our somewhat involved constructions.
An induction principle for spectral and rearrangement inequalitities
Authors: Chong Kong Ming
Published: July 2010

In this paper, expressions of the form

f
≺
g

f \prec g

or

f
≺≺
g

f \prec \prec g

(where

≺
\prec

and

≺≺
\prec \prec

denote the Hardy-Littlewood-Pólya spectral order relations) are called spectral inequalities. Here a general induction principle for spectral and rearrangement inequalities involving a pair of n-tuples in

R
n

{R^n}

as well as their decreasing and increasing rearrangements is developed. This induction principle proves that such spectral or rearrangement inequalities hold iff they hold for the case when

n
=
2

n = 2

, and that, under some mild conditions, this discrete result can be generalized to include measurable functions with integrable positive parts. A similar induction principle for spectral and rearrangement inequalities involving more than two measurable functions is also established. With this induction principle, some well-known spectral or rearrangement inequalities are obtained as particular cases and additional new results given.
Ergodic measure preserving transformations with quasi-discrete spectrum
Authors: Robertson James B.
Published: July 2010

It is shown that an ergodic measure preserving transformation with quasi-discrete spectrum is conjugate to: (a) the skew-product of an ergodic measure preserving transformation with discrete spectrum and a measurable family of totally ergodic measure preserving transformations with quasi-discrete spectrum; (b) a factor of the direct product of an ergodic measure preserving transformation with discrete spectrum and a totally ergodic measure preserving transformation with quasi-discrete spectrum. Sufficient conditions are given to insure that an ergodic measure preserving transformation with quasi-discrete spectrum is conjugate to the direct product of an ergodic measure preserving transformation with discrete spectrum and a totally ergodic measure preserving transformation with quasi-discrete spectrum.
A Prüfer transformation for the equation of the vibrating beam
Authors: Banks D. O. , Kurowski G. J.
Published: July 2010

In this paper, the oscillatory properties of the eigenfunctions of an elastically constrained beam are studied. The method is as follows. The eigenfunction and its first three derivatives are considered as a four-dimensional vector,

(
u
,

u
′

,
p
u
,
(
p
u

)
′

)

(u,u’,pu,(pu)’)

. This vector is projected onto two independent planes and polar coordinates are introduced in each of these two planes. The resulting transformation is then used to study the oscillatory properties of the eigenfunctions and their derivatives in a manner analogous to the use of the Prüfer transformation in the study of second order Sturm-Liouville systems. This analysis yields, for a given set of boundary conditions, the number of zeros of each of the derivatives,

u
′

,
p
u
,
(
p
u

)
′

u’,pu,(pu)’

and the relation of these zeros to the

n
−
1

n - 1

zeros of the

n
n

th eigenfunction. The method also can be used to establish comparison theorems of a given type.
Spectrum and direct integral
Authors: Azoff Edward A.
Published: July 2010

Let

T
=

∫
Z
⊕

T
(

E

)

T = \smallint _Z^ \oplus T(\mathcal {E})

be a direct integral of Hilbert space operators, and equip the collection

C

\mathcal {C}

of compact subsets of C with the Hausdorff metric topology. Consider the [set-valued] function sp which associates with each

E

∈
Z

\mathcal {E} \in Z

the spectrum of

T
(

E

)

T(\mathcal {E})

. The main theorem of this paper states that sp is measurable. The relationship between

σ
(
T
)

\sigma (T)

and

{
σ
(
T
(

E

)
)
}

\{ \sigma (T(\mathcal {E}))\}

is also examined, and the results applied to the hyperinvariant subspace problem. In particular, it is proved that if

σ
(
T
(

E

)
)

\sigma (T(\mathcal {E}))

consists entirely of point spectrum for each

E

∈
Z

\mathcal {E} \in Z

, then either T is a scalar multiple of the identity or T has a hyperinvariant subspace; this generalizes a theorem due to T. Hoover.
Lattices of topological extensions
Authors: Mack John , Rayburn Marlon , Woods Grant
Published: July 2010

For completely regular Hausdorff spaces, we consider topological properties P which are akin to compactness in the sense of Herrlich and van der Slot and satisfy the equivalent of Mrowka’s condition (W). The algebraic structure of the family of tight extensions of X (which have P and contain no proper extension with that property) is studied. Where X has P locally but not globally, the relations between the complete lattice

P
∗

(
X
)

{P^ \ast }(X)

of those tight extensions which are above the maximal one-point extension and the topology of the P-reflection are investigated and conditions found under which

P
∗

(
X
)

{P^\ast }(X)

characterizes

γ
X
−
X

\gamma X - X

. The results include those of Magill on the lattice of compactifications of a locally compact space, and other applications are considered.
Bounded mean oscillation and regulated martingales
Authors: Herz Carl
Published: July 2010

In the martingale context, the dual Banach space to

H
1

{H_1}

is BMO in analogy with the result of Charles Fefferman [4] for the classical case. This theorem is an easy consequence of decomposition theorems for

H
1

{H_1}

-martingales which involve the notion of

L
p

{L_p}

-regulated

L
1

{L_1}

-martingales where

1
>
p
≤
∞

1 > p \leq \infty

. The strongest decomposition theorem is for

p
=
∞

p = \infty

, and this provides full information about BMO. The weaker

p
=
2

p = 2

decomposition is fundamental in the theory of martingale transforms.
Chapman-Enskog-Hilbert expansion for the Ornstein-Uhlenbeck process and the approximation of Brownian motion
Authors: Ellis Richard S.
Published: July 2010

Let

(
x
(
t
)
,
υ
(
t
)
)

(x(t),\upsilon (t))

denote the joint Ornstein-Uhlenbeck position-velocity process. Special solutions of the backward equation of this process are studied by a technique used in statistical mechanics. This leads to a new proof of the fact that, as

ε
↓
0
,
ε
x
(
t

/

ε
2

)

\varepsilon \downarrow 0,\varepsilon x(t/{\varepsilon ^2})

tends weakly to Brownian motion. The same problem is then considered for

υ
(
t
)

\upsilon (t)

belonging to a large class of diffusion processes.
Generation of analytic semigroups by strongly elliptic operators
Authors: Stewart H. Bruce
Published: July 2010

Strongly elliptic operators realized under Dirichlet boundary conditions in unbounded domains are shown to generate analytic semigroups in the topology of uniform convergence. This fact is applied to initial-boundary value problems for temporally homogeneous and temporally inhomogeneous parabolic equations.
A discontinuous intertwining operator
Authors: Sinclair Allan M.
Published: July 2010

If T and R are continuous linear operators on Banach spaces X and Y with the spectrum of R countable, we obtain necessary and sufficient conditions on the pair T, R that imply the continuity of every linear operator S from X into Y satisfying

S
T
=
R
S

ST = RS

.
Finite groups with a proper 2-generated core
Authors: Aschbacher Michael
Published: July 2010

H. Bender’s classification of finite groups with a strongly embedded subgroup is an important tool in the study of finite simple groups. This paper proves two theorems which classify finite groups containing subgroups with similar but somewhat weaker embedding properties. The first theorem, classifying the groups of the title, is useful in connection with signalizer functor theory. The second theorem classifies a certain subclass of the class of finite groups possessing a permutation representation in which some involution fixes a unique point.
The Riemann problem for general 2×2 conservation laws
Authors: Liu Tai Ping
Published: July 2010

The Riemann Problem for a system of hyperbolic conservation laws of form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 right-parenthesis StartLayout 1st Row u Subscript t Baseline plus f left-parenthesis u comma upsilon right-parenthesis Subscript x Baseline equals 0 comma 2nd Row upsilon Subscript t Baseline plus g left-parenthesis u comma upsilon right-parenthesis Subscript x Baseline equals 0 EndLayout">

(
1
)

u
t

+
f

(
u
,
υ
)

x

=
0
,

υ
t

+
g

(
u
,
υ
)

x

=
0

(1)\quad \begin {array}{*{20}{c}} {{u_t} + f{{(u,\upsilon )}_x} = 0,} \\ {{\upsilon _t} + g{{(u,\upsilon )}_x} = 0} \\ \end {array}

\]
with arbitrary initial constant states
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 2 right-parenthesis left-parenthesis u 0 left-parenthesis x right-parenthesis comma v 0 left-parenthesis x right-parenthesis right-parenthesis equals StartLayout Enlarged left-brace 1st Row left-parenthesis u Subscript l Baseline comma v Subscript l Baseline right-parenthesis comma x greater-than 0 comma 2nd Row left-parenthesis u Subscript r Baseline comma v Subscript r Baseline right-parenthesis comma x greater-than 0 comma EndLayout">

(
2
)

(

u
0

(
x
)
,

v
0

(
x
)
)
=

{

(

u
l

,

v
l

)
,

x
>
0
,

(

u
r

,

v
r

)
,

x
>
0
,

(2)\quad ({u_0}(x),{v_0}(x)) = \left \{ {\begin {array}{*{20}{c}} {({u_l},{v_l}),\quad x > 0,} \\ {({u_r},{v_r}),\quad x > 0,} \\ \end {array} } \right .

\]
is considered. We assume that

f
υ

>
0
,

g
u

>
0

{f_\upsilon } > 0,{g_u} > 0

. Let

l
i

(

r
i

)

{l_i}({r_i})

be the left (right) eigenvectors of

d
F
≡
d
(
f
,
g
)

dF \equiv d(f,g)

for eigenvalues

λ
1

>

λ
2

{\lambda _1} > {\lambda _2}

. Instead of assuming the usual convexity condition

d

λ
i

(

r
i

)
≠
0
,
i
=
1
,
2

d{\lambda _i}({r_i}) \ne 0,i = 1,2

we assume that

d

λ
i

(

r
i

)
=
0

d{\lambda _i}({r_i}) = 0

on disjoint union of

1
1

-dim manifolds in the

(
u
,
υ
)

(u,\upsilon )

plane. Oleinik’s condition (E) for single equation is extended to system (1); again call this new condition (E). Our condition (E) implies Lax’s shock inequalities and, in case

d

λ
i

(

r
i

)
≠
0

d{\lambda _i}({r_i}) \ne 0

, the two are equivalent. We then prove that there exists a unique solution to the Riemann Problem (1) and (2) in the class of shocks, rarefaction waves and contact discontinuities which satisfies condition (E).
Power residues and nonresidues in arithmetic progressions
Authors: Hudson Richard H.
Published: July 2010

Let k be an integer

≥
2

\geq 2

and p a prime such that

v
k

(
p
)
=
(
k
,
p
−
1
)
>
1

{v_k}(p) = (k,p - 1) > 1

. Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b n plus c left-parenthesis n equals 0 comma 1 comma ellipsis semicolon b greater-than-or-equal-to 2 comma 1 less-than-or-equal-to c greater-than b comma left-parenthesis b comma p right-parenthesis equals left-parenthesis c comma p right-parenthesis equals 1 right-parenthesis">

b
n
+
c
(
n
=
0
,
1
,
…
;
b
≥
2
,
1
≤
c
>
b
,
(
b
,
p
)
=
(
c
,
p
)
=
1
)

bn + c(n = 0,1, \ldots ;b \geq 2,1 \leq c > b,(b,p) = (c,p) = 1)

be an arithmetic progression. We denote the smallest kth power nonresidue in the progression

b
n
+
c

bn + c

by

g
(
p
,
k
,
b
,
c
)

g(p,k,b,c)

, the smallest quadratic residue in the progression

b
n
+
c

bn + c

by

r
2

(
p
,
b
,
c
)

{r_2}(p,b,c)

, and the nth smallest prime kth power nonresidue by

g
n

(
p
,
k
)
,
n
=
0
,
1
,
2
,
…

{g_n}(p,k),n = 0,1,2, \ldots

. If

C
(
p
)

C(p)

is the multiplicative group consisting of the residue classes

mod

p

\bmod \;p

, then the kth powers

mod

p

\bmod \;p

form a multiplicative subgroup,

C
k

(
p
)

{C_k}(p)

. Among the

v
k

(
p
)

{v_k}(p)

cosets of

C
k

(
p
)

{C_k}(p)

denote by T the coset to which c belongs (where c is the first term in the progression

b
n
+
c
)

bn + c)

, and let

h
(
p
,
k
,
b
,
c
)

h(p,k,b,c)

denote the smallest number in the progression

b
n
+
c

bn + c

which does not belong to T so that

h
(
p
,
k
,
b
,
c
)

h(p,k,b,c)

is a natural generalization of

g
(
p
,
k
,
b
,
c
)

g(p,k,b,c)

. We prove by purely elementary methods that

h
(
p
,
k
,
b
,
c
)

h(p,k,b,c)

is bounded above by

2

7

/

4

b

5

/

2

p

2

/

5

+
3

b
3

p

1

/

5

+

b
2

{2^{7/4}}{b^{5/2}}{p^{2/5}} + 3{b^3}{p^{1/5}} + {b^2}

if p is a prime for which either b or

p
−
1

p - 1

is a kth power nonresidue. The restriction on b and

p
−
1

p - 1

may be lifted if

p
>

(

g
1

(
p
,
k
)

)

7.5

p > {({g_1}(p,k))^{7.5}}

. We further obtain a similar bound for

r
2

(
p
,
b
,
c
)

{r_2}(p,b,c)

for every prime p, without exception, and we apply our results to obtain a bound of the order of

p

2

/

5

{p^{2/5}}

for the nth smallest prime kth power nonresidue of primes which are large relative to

Π

j
=
1

n
−
1

g
j

(
p
,
k
)

\Pi _{j = 1}^{n - 1}{g_j}(p,k)

.
The unitary representations of the generalized Lorentz groups
Authors: Thieleker Ernest A.
Published: July 2010

For

n
⩾
2

n \geqslant 2

, let

G
(
n
)

G(n)

denote the two-fold covering group of

SO
e

(
1
,
n
)

{\text {SO} _e}(1,n)

. In case

n
⩾
3
,
G
(
n
)

n \geqslant 3,G(n)

is isomorphic to

Spin
⁡
(
1
,
n
)

\operatorname {Spin} (1,n)

and is simply connected. In a previous paper we determined all the irreducible quasi-simple representations of these groups, up to infinitesimal equivalence. The main purpose of the present paper is to determine which of these representations are unitarizable. Thus, with the aid of some results of Harish-Chandra and Nelson we determine all the irreducible unitary representations of

G
(
n
)

G(n)

, up to unitary equivalence. One by-product of our analysis is the explicit construction of the infinitesimal equivalences, which are known to exist from our previous work, between the various subquotient representations and certain subrepresentations in the nonirreducible cases of the nonunitary principal series representations of

G
(
n
)

G(n)

.
Quasi-bounded and singular functions
Authors: Arsove Maynard , Leutwiler Heinz
Published: July 2010

A general formulation is given for the concepts of quasi-bounded and singular functions, thereby extending to a much broader class of functions the concepts initially formulated by Parreau in the harmonic case. Let

Ω
\Omega

be a bounded Euclidean region. With the underlying space taken as the class

M

\mathcal {M}

of all nonnegative functions u on

Ω
\Omega

admitting superharmonic majorants, an operator S is introduced by setting Su equal to the regularization of the infimum over

λ
≥
0

\lambda \geq 0

of the regularized reduced functions for

(
u
−
λ

)
+

{(u - \lambda )^ + }

. Quasi-bounded and singular functions are then defined as those u for which

S
u
=
0

Su = 0

and

S
u
=
u

Su = u

, respectively. A development based on properties of the operator S leads to a unified theory of quasi-bounded and singular functions, correlating earlier work of Parreau (1951), Brelot (1967), Yamashita(1968), Heins (1969), and others. It is shown, for example, that a nonnegative function u on

Ω
\Omega

is quasi-bounded if and only if there exists a nonnegative, increasing, convex function

φ
\varphi

on

[
0
,
∞
]

[0,\infty ]

such that

φ
(
x
)

/

x
→
+
∞

\varphi (x)/x \to + \infty

as

x
→
∞

x \to \infty

and

φ
∘
u

\varphi \circ u

admits a superharmonic majorant. Extensions of the theory are made to the vector lattice generated by the positive cone of functions u in

M

\mathcal {M}

satisfying

S
u
≤
u

Su \leq u

.
Interpolation between 𝐻^{𝑝} spaces: the real method
Authors: Fefferman C. , Rivière N. M. , Sagher Y.
Published: July 2010

The interpolation spaces in the Lions-Peetre method between

H
p

{H^p}

spaces,

0
>
p
>
∞

0 > p > \infty

, are calculated.
On homeomorphisms of infinite dimensional bundles. II
Authors: Chapman T. A. , Wong R. Y. T.
Published: July 2010

This paper presents some aspects of homeomorphism theory in the setting of (fibre) bundles modeled on separable Hilbert manifolds and generalizes results previously established. The main result gives a characterization of subsets of infinite deficiency in a bundle by means of their restriction to the fibres, from which we are able to prove theorems of the following types: (a) mapping replacement, (b) separation of sets, (c) negligibility of subsets, and (d) extending homeomorphisms.
Generators for 𝐴(Ω)
Authors: Sibony N. , Wermer J.
Published: July 2010

We consider a bounded domain

Ω
\Omega

in

C

n

{{\mathbf {C}}^n}

and the Banach algebra

A
(
Ω
)

A(\Omega )

of all continuous functions on

Ω
¯

\bar \Omega

which are analytic in

Ω
\Omega

. Fix

f
1

,
…
,

f
k

{f_1}, \ldots ,{f_k}

in

A
(
Ω
)

A(\Omega )

. We say they are a set of generators if

A
(
Ω
)

A(\Omega )

is the smallest closed subalgebra containing the

f
i

{f_i}

. We restrict attention to the case when

Ω
\Omega

is strictly pseudoconvex and smoothly bounded and the

f
i

{f_i}

are smooth on

Ω
¯

\bar \Omega

. In this case, Theorem 1 below gives conditions assuring that a given set

f
i

{f_i}

is a set of generators.
Bounds in piecewise linear topology
Authors: Treybig L. B.
Published: July 2010

The following types of results are obtained: Given a polyhedral

2
2

-sphere

P
P

with rectilinear triangulation

T
T

lying in the interior of a solid tetrahedron

G
G

in

E
3

{E^3}

, then there is a simplicial isotopy

f
:
G
×
[
0
,
1
]
→
G

f:G \times [0,1] \to G

taking

P
P

onto a tetrahedron so that for

t
t

in

[
0
,
1
]
,
f
(
x
,
t
)
=
x

[0,1],f(x,t) = x

on

Bd
(
G
)

\text {Bd} (G)

and

f
t

{f_t}

is affine on each element of the triangulation

S
S

of

G
G

, where card

(
S
)

(S)

is a known function of card

(
T
)

(T)

. Also, given (1)

P
P

as above, (2) polyhedral disks

D
1

{D_1}

and

D
2

{D_2}

, where

Bd
(

D
1

)
=
Bd
(

D
2

)
⊂
P

\text {Bd} ({D_1}) = \text {Bd} ({D_2}) \subset P

and

Int
⁡
(

D
1

)
∪
Int
⁡
(

D
2

)
⊂
Int
⁡
(
P
)

\operatorname {Int} ({D_1}) \cup \operatorname {Int} ({D_2}) \subset \operatorname {Int} (P)

and (3) a triangulation

T
T

of

D
1

∪

D
2

∪
P

{D_1} \cup {D_2} \cup P

, then analogous results are found for a simplicial isotopy

f
f

which is fixed on

P
P

and takes

D
1

{D_1}

onto

D
2

{D_2}

. Given

G
G

as above and a piecewise linear homeomorphism

h
:
G
→
G

h:G \to G

which is fixed on

Bd(G)

{\text {Bd(G)}}

and affine on each

r
∈
R

r \in R

, then analogous bounds are found for a simplicial isotopy

f
:
G
×
[
0
,
1
]
→
G

f:G \times [0,1] \to G

so that

f
0

(
x
)
=
x

{f_0}(x) = x

and

f
1

(
r
)
=
h
(
r
)

{f_1}(r) = h(r)

for all

r
r

in

R
R

. In the second half of this paper the normal surface and normal equation theory of Haken is briefly explained and extended slightly. Bounds are found in connection with nontrivial integer entried solutions of normal equations. Also bounds are found for the number of Simplexes used in triangulating normal surfaces associated with certain solutions of the extended normal equations.
Modular and distributive semilattices
Authors: Rhodes Joe B.
Published: July 2010

A modular semilattice is a semilattice

S
S

in which

w
≥

w \geq

implies that there exist

x
,
y
∈
S

x,y \in S

such that

x
≥
a
,
y
≥
b

x \geq a,y \geq b

and

x
∧
y
=
x
∧
w

x \wedge y = x \wedge w

. This is equivalent to modularity in a lattice and in the semilattice of ideals of the semilattice, and the condition implies the Kurosh-Ore replacement property for irreducible elements in a semilattice. The main results provide extensions of the classical characterizations of modular and distributive lattices by their sublattices: A semilattice

S
S

is modular if and only if each pair of elements of

S
S

has an upper bound in

S
S

and there is no retract of

S
S

isomorphic to the nonmodular five lattice. A semilattice is distributive if and only if it is modular and has no retract isomorphic to the nondistributive five lattice.
Weighted 𝐿² approximation of entire functions
Authors: Wohlgelernter Devora
Published: July 2010

Let

S
S

be the space of entire functions

f
(
z
)

f(z)

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue f left-parenthesis z right-parenthesis StartAbsoluteValue EndAbsoluteValue squared equals integral integral StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue squared d m left-parenthesis z right-parenthesis">

|

|

f
(
z
)

|

|

2

=
∫
∫

|

f
(
z
)

|

2

d
m
(
z
)

||f(z)|{|^2} = \smallint \smallint |f(z){|^2}dm(z)

, where

m
m

is a positive measure defined on the Borel sets of the complex plane. Write

d
m
(
z
)
=
K
(
z
)
d

A
z

=
K
(
r
,
θ
)
d
A
z

dm(z) = K(z)d{A_z} = K(r,\theta )dAz

. Theorem 1. If

ln
⁡

inf
θ

K
(
r
,
θ
)

\ln {\inf _\theta }K(r,\theta )

is asymptotic to

ln
⁡

sup
θ

K
(
r
,
θ
)

\ln {\sup _\theta } K(r,\theta )

(together with other mild restrictions) then polynomials are dense in

S
S

. Theorem 2. Let

K
(
z
)
=

e

−
ϕ
(
z
)

K(z) = {e^{ - \phi (z)}}

where

ϕ
(
z
)

\phi (z)

is a convex function of

z
z

such that all exponentials belong to

S
S

. Then polynomials are dense in

S
S

.
Symmetries of spherical harmonics
Authors: Mendes Roberto De Maria Nunes
Published: July 2010

Let

G
G

be a group of linear transformations of

R
n

{R^n}

and

H
k

(
G
)

{H_k}(G)

the vector space of spherical harmonics invariant under

G
G

. The Pálya function is the formal power series

Σ

k
≥
0

t
k

dim
⁡

H
k

(
G
)

{\Sigma _{k \geq 0}}{t^k}\dim {H_k}(G)

. In this paper, after classifying all closed subgroups of

O
(
4
)

O(4)

, we compute the Pólya functions for these groups. These functions have recently proved to be of interest in quantum mechanics and elementary particle physics.
Topological dynamics on 𝐶*-algebras
Authors: Laison Diane , Laison Gary
Published: July 2010

Dynamical properties of a group of homeomorphisms of a compact Hausdorff space

X
X

can be interpreted in terms of the commutative

C
∗

{C^\ast }

-algebra

C
(
X
)

C(X)

. We investigate a noncommutative topological dynamics extending dynamical concepts to the context of a group of automorphisms on a general

C
∗

{C^\ast }

-algebra with unit. Such concepts as minimality, almost periodicity, and point-wise almost periodicity are extended to this situation. Theorems are obtained extending commutative dynamical results and relating the noncommutative dynamics to the transformation groups induced on the state space and the weak* closure of the pure states. We show, for example, that the group acts almost periodically on the

C
∗

{C^\ast }

-algebra if and only if each of these induced transformation groups is almost periodic.
An extremal length problem on a bordered Riemann surface
Authors: Wiener Jeffrey Clayton
Published: July 2010

Partition the contours of a compact bordered Riemann surface

R
′

R’

into four disjoint closed sets

α
0

,

α
1

,

α
2

{\alpha _0},{\alpha _1},{\alpha _2}

and

γ
\gamma

with

α
0

{\alpha _0}

and

α
1

{\alpha _1}

nonempty. Let

F
F

denote the family of all locally rectifiable

1
1

-chains in

R
′

−
γ

R’ - \gamma

which join

α
0

{\alpha _0}

to

α
1

{\alpha _1}

. The extremal length problem on

R
′

R’

considers the existence of a real-valued harmonic function

u
u

on

R
′

R’

which is 0 on

α
0

,
1

{\alpha _0},1

on

α
1

{\alpha _1}

, a constant on each component

ν
k

{\nu _k}

of

α
2

{\alpha _2}

with

∫

ν
k

∗

d
u
=
0

{\smallint _{{\nu _k}}}^ \ast du = 0

and

∗

d
u
=
0

^ \ast du = 0

along

γ
\gamma

such that the extremal length of

F
F

is equal to the reciprocal of the Dirichlet integral of

u
u

, that is,

λ
(
F
)
=

D

R
′

(
u

)

−
1

\lambda (F) = {D_{R’}}{(u)^{ - 1}}

. Let

R
¯

\bar R

denote a bordered Riemann surface with a finite number of boundary components and

S
S

a compactification of

R
¯

\bar R

with the property that

∂

R
¯

⊂
S

\partial \bar R \subset S

. We consider the extremal length problem on

R
¯

\bar R

(as a subset of

S
S

) when

α
0

,

α
1

{\alpha _0},{\alpha _1}

, and

α
2

{\alpha _2}

are relatively closed subarcs of

∂

R
¯

\partial \bar R

and when

α
0

,

α
1

{\alpha _0},{\alpha _1}

and

α
2

{\alpha _2}

are closed subsets of

∂
S
=
(
S
−

R
¯

)
∪
∂

R
¯

\partial S = (S - \bar R) \cup \partial \bar R

.
Deformations of group actions
Authors: Edmonds Allan L.
Published: July 2010

Let

G
G

be a finite group and

M
M

be a compact piecewise linear (PL) manifold. Define a PL

G
G

-isotopy to be a level-preserving PL action of

G
G

on

M
×
[
0
,
1
]

M \times [0,1]

. In this paper PL

G
G

-isotopies are studied and PL

G
G

-isotopic actions (which need not be equivalent) are characterized.
The regular ring and the maximal ring of quotients of a finite Baer *-ring
Authors: Pyle Ernest S.
Published: July 2010

Necessary and sufficient conditions are obtained for extending the involution of a Baer

∗
\ast

-ring to its maximal ring of quotients. Berberian’s construction of the regular ring of a Baer

∗
\ast

-ring is generalized and this ring is identified (under suitable hypotheses) with the maximal ring of quotients.
Necessary conditions for isomorphism of Lie algebras of Block
Authors: Jacobs John B.
Published: July 2010

Two algebras of Block,

L

(
G
,
δ
,
f
)

\mathcal {L}(G,\delta ,f)

and

L

(

G
′

,

δ
′

,

f
′

)

\mathcal {L}(G’,\delta ’,f’)

, are isomorphic only if

m
(
G
)
=
m
(

G
′

)

m(G) = m(G’)

. This is not sufficient for isomorphism.
Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in 𝐶ⁿ with smooth boundary
Authors: Graham Ian
Published: July 2010

The Carathéodory and Kobayashi distance functions on a bounded domain

G
G

in

C

n

{{\mathbf {C}}^n}

have related infinitesimal forms. These are the Carathéodory and Kobayashi metrics. They are denoted by

F
(
z
,
ξ
)

F(z,\xi )

(length of the tangent vector

ξ
\xi

at the point

z
z

). They are defined in terms of holomorphic mappings, from

G
G

to the unit disk for the Carathéodory metric, and from the unit disk to

G
G

for the Kobayashi metric. We consider the boundary behavior of these metrics on strongly pseudoconvex domains in

C

n

{{\mathbf {C}}^n}

with

C
2

{C^2}

boundary.

ξ
\xi

is fixed and

z
z

is allowed to approach a boundary point

z
0

{z_0}

. The quantity

F
(
z
,
ξ
)
d
(
z
,
∂
G
)

F(z,\xi )d(z,\partial G)

is shown to have a finite limit. In addition, if

ξ
\xi

belongs to the complex tangent space to the boundary at

z
0

{z_0}

, then this first limit is zero, and

(
F
(
z
,
ξ
)

)
2

d
(
z
,
∂
G
)

{(F(z,\xi ))^2}d(z,\partial G)

has a (nontangential) limit in which the Levi form appears. We prove an approximation theorem for bounded holomorphic functions which uses peak functions in a novel way. The proof was suggested by N. Kerzman. This theorem is used here in studying the boundary behavior of the Carathéodory metric.
Convergent subsequences from sequences of functions
Authors: Thornburg James L.
Published: July 2010

Let

{

y
k

}

\{ {y_k}\}

be a sequence of functions,

y
k

∈

Π

s
∈
S

E
s

{y_k} \in {\Pi _{s \in S}}{E_s}

where

S
S

is a nonempty subset of the

l
l

-dimensional Euclidean space and

E
s

{E_s}

is an ordered vector space with positive cone

K
s

{K_s}

. If

y
k

∈

Π

s
∈
S

E
s

{y_k} \in {\Pi _{s \in S}}{E_s}

, sufficient conditions are given that

{

y
k

}

\{ {y_k}\}

have a subsequence

{

h
k

}

\{ {h_k}\}

such that for each

t
∈
S

t \in S

the sequence

{

h
k

(
t
)
}

\{ {h_k}(t)\}

is monotone for

k
k

sufficiently large, depending on

t
t

. If each

E
s

{E_s}

is an ordered topological vector space, sufficient conditions are given that

{

y
k

}

\{ {y_k}\}

has a subsequence

{

h
k

}

\{ {h_k}\}

such that for every

t
∈
S

t \in S

the sequence

{

h
k

(
t
)
}

\{ {h_k}(t)\}

is either monotone for

k
k

sufficiently large depending on

t
t

, or else the sequence

{

h
k

(
t
)
}

\{ {h_k}(t)\}

is convergent. If

E
s

=
B

{E_s} = B

for each

s
s

and

B
B

a Banach space then a definition of bounded variation is given so that if

{

y
k

}

\{ {y_k}\}

is uniformly norm bounded and the variation of the functions

y
k

{y_k}

is uniformly bounded then there is a convergent subsequence

{

h
k

}

\{ {h_k}\}

of

{

y
k

}

\{ {y_k}\}

. In the case

E
s

=
E

{E_s} = E

for each

s
∈
S

s \in S

and

E
E

is such that bounded monotone sequences converge then the given conditions imply the existence of a subsequence

{

h
k

}

\{ {h_k}\}

of

{

y
k

}

\{ {y_k}\}

which converges for each

t
∈
S

t \in S

.
Finitary imbeddings of certain generalized sample spaces
Authors: Gaudard Marie A. , Weaver Robert J.
Published: July 2010

A generalized sample space each of whose subspaces has as its logic an orthomodular poset is called an HD sample space. In this paper it is shown that any HD sample space may be imbedded in a natural way in a generalized sample space which is HD and at the same time admits a full set of dispersion free weight functions.
On the decomposition of tensor products of principal series representations for real-rank one semisimple groups
Authors: Martin Robert Paul
Published: July 2010

Let

G
G

be a connected semisimple real-rank one Lie group with finite center. It is shown that the decomposition of the tensor product of two representations from the principal series of

G
G

consists of two pieces,

T
c

{T_c}

and

T
d

{T_d}

, where

T
c

{T_c}

is a continuous direct sum with respect to Plancherel measure on

G
^

\hat G

of representations from the principal series only, occurring with explicitly determined multiplicities, and

T
d

{T_d}

is a discrete sum of representations from the discrete series of

G
G

, occurring with multiplicities which are, for the present, undetermined.
Families of holomorphic maps into Riemann surfaces
Authors: Barth Theodore J.
Published: July 2010

In analogy with the Hartogs theorem that separate analyticity of a function implies analyticity, it is shown that a separately normal family of holomorphic maps from a polydisk into a Riemann surface is a normal family. This contrasts with examples of discontinuous separately analytic maps from a bidisk into the Riemann sphere. The proof uses a theorem on pseudoconvexity of normality domains, which is proved via the following convergence criterion: a sequence

{

f
j

}

\{ {f_j}\}

of holomorphic maps from a complex manifold into a Riemann surface converges to a nonconstant holomorphic map if and only if the sequence

{

f
j

−
1

}

\{ f_j^{ - 1}\}

of set-valued maps, defined on the Riemann surface, converges to a suitable set-valued map. Extending Osgood’s theorem, it is also shown that a separately analytic map (resp. a separately normal family of holomorphic maps) from a polydisk into a hyperbolic complex space is analytic (resp. normal).
Topological extension properties
Authors: Woods R. Grant
Published: July 2010

It is known that if a topological property

P

\mathcal {P}

of Tychonoff spaces is closed-hereditary, productive, and possessed by all compact

P

\mathcal {P}

-regular spaces, then each

P

\mathcal {P}

-regular space

X
X

is a dense subspace of a space

γ

P

X

{\gamma _\mathcal {P}}X

with

P

\mathcal {P}

such that if

Y
Y

has

P

\mathcal {P}

and

f
:
X
→
Y

f:X \to Y

is continuous, then

f
f

extends continuously to

f
γ

:

γ

P

X
→
Y

{f^\gamma }:{\gamma _\mathcal {P}}X \to Y

. Such topological properties are called extension properties;

γ

P

X

{\gamma _\mathcal {P}}X

is called the maximal

P

\mathcal {P}

-extension of

X
X

. In this paper we study the relationships between pairs of extension properties and their maximal extensions. A basic tool is the concept of

P

\mathcal {P}

-pseudocompactness, which is studied in detail (a

P

\mathcal {P}

-regular space

X
X

is

P

\mathcal {P}

-pseudocompact if

γ

P

X

{\gamma _\mathcal {P}}X

is compact). A classification of extension properties is attempted, and several means of constructing extension properties are studied. A number of examples are considered in detail.