bluuebunny/crossref_metadata_2025 huggingface dataset:
Teichmüller inequalities without coefficient normalization
Authors: Obrock Arthur E.
Published: July 2010

Teichmüller’s relation between the coefficients of extremal schlicht functions and quadratic differentials is extended. The coefficient normalization hypothesis in his theorem is dropped with the result that the new coefficient relations become more complex. This completes the partial result in this direction which is contained in Jenkins’ General Coefficient Theorem. A modification of the version of the length-area method used by Teichmüller and Jenkins is introduced in our proof.
Gelfand theory of pseudo differential operators and hypoelliptic operators
Authors: Taylor Michael E.
Published: July 2010

This paper investigates an algebra

A

\mathfrak {A}

of pseudo differential operators generated by functions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a left-parenthesis x right-parenthesis element-of upper C Superscript normal infinity Baseline left-parenthesis upper R Superscript n Baseline right-parenthesis intersection upper L Superscript normal infinity Baseline left-parenthesis upper R Superscript n Baseline right-parenthesis">

a
(
x
)
∈

C
∞

(

R
n

)
∩

L
∞

(

R
n

)

a(x) \in {C^\infty }({R^n}) \cap {L^\infty }({R^n})

such that

D
α

a
(
x
)
→
0

{D^\alpha }a(x) \to 0

as

|

x

|

→
∞

|x| \to \infty

, if

|

α

|

≧
1

|\alpha | \geqq 1

, and by operators

q
(
D
)

Q

−
1

/

2

q(D){Q^{ - 1/2}}

where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q left-parenthesis upper D right-parenthesis greater-than upper P left-parenthesis upper D right-parenthesis comma upper Q equals upper I plus upper P left-parenthesis upper D right-parenthesis Superscript asterisk Baseline upper P left-parenthesis upper D right-parenthesis">

q
(
D
)
>
P
(
D
)
,
Q
=
I
+
P

(
D

)
∗

P
(
D
)

q(D) > P(D),Q = I + P{(D)^ \ast }P(D)

, and

P
(
D
)

P(D)

is hypoelliptic. It is proved that such an algebra has compact commutants, and the maximal ideal space of the commutative

C
∗

{C^ \ast }

algebra

A

/

J

\mathfrak {A}/J

is investigated, where

J
J

consists of the elements of

A

\mathfrak {A}

which are compact. This gives a necessary and sufficient condition for a differential operator

q
(
x
,
D
)
:

B

2

,

P
~

→

L
2

q(x,D):{\mathfrak {B}_2}_{,\tilde P} \to {L^2}

to be Fredholm. (Here and in the rest of this paragraph we assume that the coefficients of all operators under consideration satisfy the conditions given on

a
(
x
)

a(x)

in the first sentence.) It is also proved that if

p
(
x
,
D
)

p(x,D)

is a formally selfadjoint operator on

R
n

{R^n}

which has the same strength as

P
(
D
)

P(D)

uniformly on

R
n

{R^n}

, then

p
(
x
,
D
)

p(x,D)

is selfadjoint, with domain

B

2
,

P
~

(

R
n

)

{\mathfrak {B}_{2,\tilde P}}({R^n})

, and semibounded, if

n
≧
2

n \geqq 2

. From this a Gårding type inequality for uniformly strongly formally hypoelliptic operators and a global regularity theorem for uniformly formally hypoelliptic operators are derived. The familiar local regularity theorem is also rederived. It is also proved that a hypoelliptic operator

p
(
x
,
D
)

p(x,D)

of constant strength is formally hypoelliptic, in the sense that for any

x
0

{x_0}

, the constant coefficients operator

p
(

x
0

,
D
)

p({x_0},D)

is hypoelliptic.
The powers of a maximal ideal in a Banach algebra and analytic structure
Authors: Read T. T.
Published: July 2010

Sufficient conditions are given for the existence of an analytic variety at an element

ϕ
\phi

of the spectrum of a commutative Banach algebra with identity. An associated graded algebra first considered by S. J. Sidney is used to determine the dimension of the analytic variety in terms of the closed powers of the maximal ideal which is the kernel of

ϕ
\phi

.
An optimization problem for unitary and orthogonal representations of finite groups
Authors: Djoković D. Ž. , Blake I. F.
Published: July 2010

Let

G
→

GL

(
V
)

G \to {\text {GL}}(V)

be a faithful orthogonal representation of a finite group G acting in an Euclidean space V. For a unit vector x we choose

g
≠
1

g \ne 1

in G so that

|

g
x
−
x

|

|gx - x|

is minimal and put

δ
(
x
)
=

|

g
x
−
x

|

\delta (x) = |gx - x|

. We study the class of vectors x which maximize

δ
(
x
)

\delta (x)

and have the additional property that

|

g
x
−
x

|

|gx - x|

depends only on the conjugacy class of

g
∈
G

g \in G

. For some special types of representations we are able to characterize completely this class of vectors.
Current valued measures and Geöcze area
Authors: Gariepy Ronald
Published: July 2010

If f is a continuous mapping of finite Geöcze area from a polyhedral region

X
⊂

R
k

X \subset {R^k}

into

R
n

,
2
≦
k
≦
n

{R^n},2 \leqq k \leqq n

, then, under suitable hypotheses, one can associate with f, by means of the Cesari-Weierstrass integral, a current valued measure T over the middle space of f. In particular, if either

k
=
2

k = 2

or the

k
+
1

k + 1

-dimensional Hausdorff measure of

f
(
X
)

f(X)

is zero, then T is essentially the same as a current valued measure defined by H. Federer and hence serves to describe the tangential properties of f and the multiplicities with which f assumes its values. Further, the total variation of T is equal to the Geöcze area of f.
Cobordism Massey products
Authors: Alexander J. C.
Published: July 2010

The structure of Massey products is introduced into the bordism ring

Ω
S

{\Omega ^S}

of manifolds with structure S and machinery is developed to investigate it. The product is changed to one in homotopy via the Pontrjagin-Thom map and methods for computation via the Adams spectral sequence are developed. To illustrate the methods, some products in

Ω

S
U

{\Omega ^{SU}}

and

Ω

S
p

{\Omega ^{Sp}}

are computed.
Morita theorems for functor categories
Authors: Newell D. C.
Published: July 2010

We generalize the Morita theorems to certain functor categories using properties of adjoint functors.
On subgroups of 𝑀₂₄. II. The maximal subgroups of 𝑀₂₄
Authors: Choi Chang
Published: July 2010

In this paper we effect a systematic study of transitive subgroups of

M

24

{M_{24}}

, obtaining 5 transitive maximal subgroups of

M

24

{M_{24}}

of which one is primitive and four imprimitive. These results, along with the results of the paper, On subgroups of

M

24

{M_{24}}

. I, enable us to enumerate all the maximal subgroups of

M

24

{M_{24}}

. There are, up to conjugacy, nine of them. The complete list includes one more in addition to those listed by J. A. Todd in his recent work on

M

24

{M_{24}}

. The two works were done independently employing completely different methods.
Two Hilbert spaces in which polynomials are not dense
Authors: Newman D. J. , Wohlgelernter D. K.
Published: July 2010

Let

S
S

be the Hilbert 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 double-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} = \iint {|f(z){|^2}}dm(z)

, where

m
m

is a positive measure defined on the Borel sets of the complex plane. Two Hilbert spaces are constructed in which polynomials are not dense. In the second example, our space is one which contains all exponentials and yet in which the exponentials are not complete. This is a somewhat surprising result since the exponentials are always complete on the real line.
A Sturmian theorem for first order partial differential equations
Authors: Wong Pui Kei
Published: July 2010

A pair of first order partial differential equations is considered. The system is transformed into a single nonlinear scalar equation of the Riccati type from which some Wirtinger type integral inequalities for functions of several variables are derived. A comparison theorem for two such pairs of first order equations is then proved using the Wirtinger inequalities.
Absolute Tauberian constants for Cesàro means
Authors: Sherif Soraya
Published: July 2010

This paper is concerned with introducing two inequalities of the form

∑

n
=
0

∞

|

τ
n

−

a
n

|

≦
K
A

\sum \nolimits _{n = 0}^\infty {|{\tau _n}} - {a_n}| \leqq KA

and

∑

n
=
0

∞

|

τ
n

−

a
n

|

≦

K
′

B

\sum \nolimits _{n = 0}^\infty {|{\tau _n}} - {a_n}| \leqq K’B

, where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau Subscript n Baseline equals upper C Subscript n Superscript left-parenthesis k right-parenthesis Baseline minus upper C Subscript n minus 1 Superscript left-parenthesis k right-parenthesis Baseline comma upper C Subscript n Superscript left-parenthesis k right-parenthesis">

τ
n

=

C
n

(
k
)

−

C

n
−
1

(
k
)

,

C
n

(
k
)

{\tau _n} = C_n^{(k)} - C_{n - 1}^{(k)},C_n^{(k)}

denote the Cesàro transform of order

k
,
K

k,K

and

K
′

K’

are absolute Tauberian constants,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts vertical-bar normal upper Delta left-parenthesis n a Subscript n Baseline right-parenthesis StartAbsoluteValue greater-than normal infinity comma upper B equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts vertical-bar normal upper Delta left-parenthesis left-parenthesis 1 slash n right-parenthesis sigma-summation Underscript v equals 1 Overscript n minus 1 Endscripts v a Subscript v Baseline right-parenthesis EndAbsoluteValue greater-than normal infinity">

A
=

∑

n
=
0

∞

|

Δ
(
n

a
n

)

|

>
∞
,
B
=

∑

n
=
0

∞

|

Δ
(
(
1

/

n
)

∑

v
=
1

n
−
1

v

a
v

)

|

>
∞

A = \sum \nolimits _{n = 0}^\infty {|\Delta (n{a_n}} )| > \infty ,B = \sum \nolimits _{n = 0}^\infty {|\Delta ((1/n)\sum \nolimits _{v = 1}^{n - 1} {v{a_v}} } )| > \infty

and

Δ

u
k

=

u
k

−

u

k
+
1

\Delta {u_k} = {u_k} - {u_{k + 1}}

. The constants

K
,

K
′

K,K’

will be determined.
Bounded continuous functions on a completely regular space
Authors: Sentilles F. Dennis
Published: July 2010

Three locally convex topologies on

C
(
X
)

C(X)

are introduced and developed, and in particular shown to coincide with the strict topology on locally compact

X
X

and yield dual spaces consisting of tight,

τ
\tau

-additive and

σ
\sigma

-additive functionals respectively for completely regular

X
X

.
Some remarks on quasi-analytic vectors
Authors: Chernoff Paul R.
Published: July 2010

Recently a number of authors have developed conditions of a generalized quasi-analytic nature which imply essential selfadjointness for semibounded symmetric operators in Hilbert space. We give a unified derivation of these results by reducing them to the basic theorems of Nelson and Nussbaum. In addition we present an extension of Nussbaum’s quasi-analytic vector theorem to the setting of semigroups in Banach spaces.
A geometry for 𝐸₇
Authors: Faulkner John R.
Published: July 2010

A geometry is defined by the 56-dimensional representation

M

\mathfrak {M}

of a Lie algebra of type

E
7

{E_7}

. Every collineation is shown to be induced by a semisimilarity of

M

\mathfrak {M}

, and the image of the automorphism group of

M

\mathfrak {M}

in the collineation group is shown to be simple.
Local to global theorems in the theory of Hurewicz fibrations
Authors: Arnold James
Published: July 2010

This paper is concerned with the problem of showing a local fibration is a fibration. There are two kinds of local to global theorems proven. The first type of theorem considers local fibrations where local is in terms of closed covers of the base (e.g. the set of closed simplices of a polyhedron, the cones of a suspension). The second type of theorem deals with local in terms of open covers of the total space.
Slicing theorems for 𝑛-spheres in Euclidean (𝑛+1)-space
Authors: Daverman Robert J.
Published: July 2010

This paper describes conditions on the intersection of an n-sphere

Σ
\Sigma

in Euclidean

(
n
+
1
)

(n + 1)

-space

E

n
+
1

{E^{n + 1}}

with the horizontal hyperplanes of

E

n
+
1

{E^{n + 1}}

sufficient to determine that the sphere be nicely embedded. The results generally are pointed towards showing that the complement of

Σ
\Sigma

is 1-ULC (uniformly locally 1-connected) rather than towards establishing the stronger property that

Σ
\Sigma

is locally flat. For instance, the main theorem indicates that

E

n
+
1

−
Σ

{E^{n + 1}} - \Sigma

is 1-ULC provided each non-degenerate intersection of

Σ
\Sigma

and a horizontal hyperplane be an

(
n
−
1
)

(n - 1)

-sphere bicollared both in that hyperplane and in

Σ
\Sigma

itself

(
n
≠
4
)

(n \ne 4)

.
Polars and their applications in directed interpolation groups
Authors: Glass A. M. W.
Published: July 2010

In the study of l-groups, as in many other branches of mathematics, use is made of the concept of “orthogonal elements". The purpose of this paper is to show that this concept can be extended to directed, interpolation groups and that most of the theorems in l-groups concerning polars hold in the more general setting of directed, interpolation groups. As consequences, generalisations of Holland’s and Lorenzen’s theorems are obtained and a result on o-simple abelian, directed, interpolation groups.
One-parameter inverse semigroups
Authors: Eberhart Carl , Selden John
Published: July 2010

This is the second in a projected series of three papers, the aim of which is the complete description of the closure of any one-parameter inverse semigroup in a locally compact topological inverse semigroup. In it we characterize all one-parameter inverse semigroups. In order to accomplish this, we construct the free one-parameter inverse semigroups and then describe their congruences.
Automorphism groups of bounded domains in Banach spaces
Authors: Greenfield Stephen J. , Wallach Nolan R.
Published: July 2010

We prove a weak Schwarz lemma in Banach space and use it to show that in Hilbert space a Siegel domain of type II is not necessarily biholomorphic to a bounded domain. We use a strong Schwarz lemma of L. Harris to find the full group of automorphisms of the infinite dimensional versions of the Cartan domains of type I. We then show that all domains of type I are holomorphically inequivalent, and are different from k-fold products of unit balls

(
k
≧
2
)

(k \geqq 2)

. Other generalizations and comments are given.
The asymptotic behavior of a Fourier transform and the localization property for eigenfunction expansions for some partial differential operators
Authors: Randol Burton
Published: July 2010

The asymptotic behavior of a certain Fourier transform is investigated, and the result is applied to obtain a localization theorem for elliptic operators on the torus.
Semigroups satisfying variable identities. II
Authors: Putcha Mohan S. , Weissglass Julian
Published: July 2010

The concept of a semigroup satisfying an identity

x
y
=
f
(
x
,
y
)

xy = f(x,y)

is generalized by considering identities in

n
n

-variables and letting the identity depend on the variables. The property of satisfying a “variable identity” is studied. Semigroups satisfying certain types of identities are characterized in terms of unions and semilattices of groups.
Correspondence between Lie algebra invariant subspaces and Lie group invariant subspaces of representations of Lie groups
Authors: Zeitlin Joel
Published: July 2010

Let G be a Lie group with Lie algebra

g

\mathfrak {g}

and

B

=

u

(

g

)

\mathfrak {B} = \mathfrak {u}(\mathfrak {g})

, the universal enveloping algebra of

g

\mathfrak {g}

; also let U be a representation of G on H, a Hilbert space, with dU the corresponding infinitesimal representation of

g

\mathfrak {g}

and

B

\mathfrak {B}

. For G semisimple Harish-Chandra has proved a theorem which gives a one-one correspondence between

d
U
(

g

)

dU(\mathfrak {g})

invariant subspaces and

U
(
G
)

U(G)

invariant subspaces for certain representations U. This paper considers this theorem for more general Lie groups. A lemma is proved giving such a correspondence without reference to some of the concepts peculiar to semisimple groups used by Harish-Chandra. In particular, the notion of compactly finitely transforming vectors is supplanted by the notion of

Δ
f

{\Delta _f}

, the

Δ
\Delta

finitely transforming vectors, for

Δ
∈

B

\Delta \in \mathfrak {B}

. The lemma coupled with results of R. Goodman and others immediately yields a generalization to Lie groups with large compact subgroup. The applicability of the lemma, which rests on the condition

g

Δ
f

⊆

Δ
f

\mathfrak {g}{\Delta _f} \subseteq {\Delta _f}

, is then studied for nilpotent groups. The condition is seen to hold for all quasisimple representations, that is representations possessing a central character, of nilpotent groups of class

≦
2

\leqq 2

. However, this condition fails, under fairly general conditions, for

g

=

N
4

\mathfrak {g} = {N_4}

, the 4-dimensional class 3 Lie algebra.

N
4

{N_4}

is shown to be a subalgebra of all class 3

g

\mathfrak {g}

and the condition is seen to fail for all

g

\mathfrak {g}

which project onto an algebra where the condition fails. The result is then extended to cover all

g

\mathfrak {g}

of class 3 with general dimension 1. Finally, it is conjectured that

g

Δ
f

⊆

Δ
f

\mathfrak {g}{\Delta _f} \subseteq {\Delta _f}

for all quasisimple representations if and only if class

g

≦
2

\mathfrak {g} \leqq 2

.
Smooth extensions in infinite dimensional Banach spaces
Authors: Renz Peter
Published: July 2010

If

B
B

is

l
p

(
ω
)

{l_p}(\omega )

or

c
0

(
ω
)

{c_0}(\omega )

we show

B
B

has the following extension property. Any homeomorphism from a compact subset

M
M

of

B
B

into

B
B

may be extended to a homeomorphism of

B
B

onto

B
B

which is a

C
∞

{C^\infty }

diffeomorphism on

B
∖
M

B\backslash M

to its image in

B
B

. This is done by writing

B
B

as a direct sum of closed subspaces

B
1

{B_1}

and

B
2

{B_2}

both isomorphically isometric to

B
B

so that the natural projection of

K
K

into

B
1

{B_1}

along

B
2

{B_2}

is one-to-one (see H. H. Corson, contribution in Symposium on infinite dimensional topology, Ann. of Math. Studies (to appear)). With

K
,
B
,

B
1

K,B,{B_1}

and

B
2

{B_2}

as above a homeomorphism of

B
B

onto itself is constructed which leaves the

B
1

{B_1}

-coordinates of points in

B
B

unchanged, carries

K
K

into

B
1

{B_1}

and is a

C
∞

{C^\infty }

diffeomorphic map on

B
∖
K

B\backslash K

. From these results the extension theorem may be proved by standard methods.
Zeros of partial sums and remainders of power series
Authors: Buckholtz J. D. , Shaw J. K.
Published: July 2010

For a power series

f
(
z
)
=

Σ

k
=
0

∞

a
k

z
k

f(z) = \Sigma _{k = 0}^\infty {a_k}{z^k}

let

s
n

(
f
)

{s_n}(f)

denote the maximum modulus of the zeros of the nth partial sum of f and let

r
n

(
f
)

{r_n}(f)

denote the smallest modulus of a zero of the nth normalized remainder

Σ

k
=
n

∞

a
k

z

k
−
n

\Sigma _{k = n}^\infty {a_k}{z^{k - n}}

. The present paper investigates the relationships between the growth of the analytic function f and the behavior of the sequences

{

s
n

(
f
)
}

\{ {s_n}(f)\}

and

{

r
n

(
f
)
}

\{ {r_n}(f)\}

. The principal growth measure used is that of R-type: if

R
=
{

R
n

}

R = \{ {R_n}\}

is a nondecreasing sequence of positive numbers such that

lim
(

R

n
+
1

/

R
n

)
=
1

\lim ({R_{n + 1}}/{R_n}) = 1

, then the R-type of f is
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau Subscript upper R Baseline left-parenthesis f right-parenthesis equals limit sup StartAbsoluteValue a Subscript n Baseline upper R 1 upper R 2 midline-horizontal-ellipsis upper R Subscript n Baseline EndAbsoluteValue Superscript 1 slash n">

τ
R

(
f
)
=
lim
sup

|

a
n

R
1

R
2

⋯

R
n

|

1

/

n

{\tau _R}(f) = \lim \sup |{a_n}{R_1}{R_2} \cdots {R_n}{|^{1/n}}

. We prove that there is a constant P such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau Subscript upper R Baseline left-parenthesis f right-parenthesis limit inf left-parenthesis s Subscript n Baseline left-parenthesis f right-parenthesis slash upper R Subscript n Baseline right-parenthesis less-than-over-equals upper P and tau Subscript upper R Baseline left-parenthesis f right-parenthesis limit sup left-parenthesis r Subscript n Baseline left-parenthesis f right-parenthesis slash upper R Subscript n Baseline right-parenthesis greater-than-over-equals left-parenthesis 1 slash upper P right-parenthesis">

τ
R

(
f
)
lim
inf
(

s
n

(
f
)

/

R
n

)
≦
P

and

τ
R

(
f
)
lim
sup
(

r
n

(
f
)

/

R
n

)
≧
(
1

/

P
)

{\tau _R}(f)\lim \inf ({s_n}(f)/{R_n}) \leqq P\quad {\text {and}}\quad {\tau _R}(f)\lim \sup ({r_n}(f)/{R_n}) \geqq (1/P)

\]
for functions f of positive finite R-type. The constant P cannot be replaced by a smaller number in either inequality; P is called the power series constant.
A continuity theorem for Fuchsian groups
Authors: Wong C. K.
Published: July 2010

On a given Riemann surface, fix a discrete (finite or infinite) sequence of points

{

P
k

}
,
k
=
1
,
2
,
3
,
…
,

\{ {P_k}\} ,k = 1,2,3, \ldots ,

and associate to each

P
k

{P_k}

an “integer”

ν
k

{\nu _k}

(which may be

1
,
2
,
3
,
…
,

or

∞
)

1,2,3, \ldots ,{\text {or}}\;\infty )

. This sequence of points and “integers” is called a “signature” on the Riemann surface. With only a few exceptions, a Riemann surface with signature can always be represented by a Fuchsian group. We investigate here the dependence of the group on the number

ν
k

{\nu _k}

. More precisely, keeping the points

P
k

{P_k}

fixed, we vary the numbers

ν
k

{\nu _k}

in such a way that the signature tends to a limit signature. We shall prove that the corresponding representing Fuchsian group converges to the Fuchsian group which corresponds to the limit signature.
On the H. Lewy extension phenomenon
Authors: Nirenberg Ricardo
Published: July 2010

We prove local approximation and extension theorems for

C
∞

{C^\infty }

submanifolds

M
M

of

C
n

{C^n}

(CR submanifolds). Under some conditions on

M
M

, any smooth solution of the induced Cauchy-Riemann equations can be extended holomorphically to bigger (and sometimes open) sets.
The space of homeomorphisms on a compact two-manifold is an absolute neighborhood retract
Authors: Luke R. , Mason W. K.
Published: July 2010

The theorem mentioned in the title is proved.
On the evaluation of Brewer’s character sums
Authors: Giudici Reinaldo E. , Muskat Joseph B. , Robinson Stanley F.
Published: July 2010

A decade ago in this journal B. W. Brewer defined a sequence of polynomials

V
n

(
x
,
1
)

{V_n}(x,1)

and for

n
=
4

n = 4

and 5 evaluated
\[

∑

x
=
1

p

χ

(

V
n

(
x
,
1
)
)

,

\sum \limits _{x = 1}^p {{}_\chi ({V_n}(x,1))},

\]

χ
\chi

the nonprincipal quadratic character of the prime

p
p

, in closed form. A. L Whiteman derived these results by means of cyclotomy. Brewer subsequently defined

V
n

(
x
,
Q
)

{V_n}(x,Q)

. This paper applies cyclotomy to the more general polynomials and provides evaluations for several more values of

n
n

. Relevant quadratic decompositions of primes are studied.
Endomorphism rings of reduced torsion-free modules over complete discrete valuation rings
Authors: Liebert Wolfgang
Published: July 2010

The purpose of this paper is to find necessary and sufficient conditions that an abstract ring be isomorphic to the ring of all endomorphisms of a reduced torsion-free module over a (possibly noncommutative) complete discrete valuation ring.
A characterization of badly approximable functions
Authors: Poreda S. J.
Published: July 2010

Complex valued functions, continuous on a closed Jordan curve in the plane and having the property that all of their polynomials of best uniform approximation on that curve are identically zero are characterized in terms of their mapping properties on that curve.
Hypersurfaces of nonnegative curvature in a Hilbert space
Authors: Jonker Leo
Published: July 2010

In this paper we prove the following generalizations of known theorems about hypersurfaces in

R

n

{{\mathbf {R}}^n}

: Let

M
M

be a hypersurface in a Hilbert space. (1) If on

M
M

the sectional curvature

K
(
σ
)

K(\sigma )

is nonnegative for every

2
2

-plane section

σ
\sigma

and if

K
(
σ
)
>
0

K(\sigma ) > 0

for at least one

σ
\sigma

, then

M
M

is the boundary of a convex body. (2) If

K
(
σ
)
=
0

K(\sigma ) = 0

for all

σ
\sigma

, then

M
M

is a hypercylinder. The main tool in these theorems is Smale’s infinite dimensional Sard’s theorem.
A sheaf-theoretic duality theory for cylindric algebras
Authors: Comer Stephen D.
Published: July 2010

Stone’s duality between Boolean algebras and Boolean spaces is extended to a dual equivalence between the category of all

α
\alpha

-dimensional cylindric algebras and a certain category of sheaves of such algebras. The dual spaces of important types of algebras are characterized and applications are given to the study of direct and subdirect decompositions of cylindric algebras.
An algebra of distributions on an open interval
Authors: Shultz Harris S.
Published: July 2010

Let

(
a
,
b
)

(a,b)

be any open subinterval of the reals which contains the origin and let

B

\mathfrak {B}

denote the family of all distributions on

(
a
,
b
)

(a,b)

which are regular in some interval

(
∈
,
0
)

( \in ,0)

, where

∈>
0

\in > 0

. Then

B

\mathfrak {B}

is a commutative algebra: Multiplication is defined so that, when restricted to those distributions on

(
a
,
b
)

(a,b)

whose supports are contained in

[
0
,
b
)

[0,b)

, it is ordinary convolution. Also,

B

\mathfrak {B}

can be injected into an algebra of operators; this family of operators is a sequentially complete locally convex space. Since it preserves multiplication, this injection serves as a generalization (there are no growth restrictions) of the two-sided Laplace transformation.
The equivariant Plateau problem and interior regularity
Authors: Lawson H. Blaine
Published: July 2010

Let

M
⊂

R

n

M \subset {{\text {R}}^n}

be a compact submanifold of Euclidean space which is invariant by a compact group

G
⊂
S
O
(
n
)

G \subset SO(n)

. When

dim
⁡
(
M
)
=
n
−
2

\dim (M) = n - 2

, it is shown that there always exists a solution to the Plateau problem for

M
M

which is invariant by

G
G

and, furthermore, that uniqueness of this solution among

G
G

-invariant currents implies uniqueness in general. This result motivates the subsequent study of the Plateau problem for

M
M

within the class of

G
G

-invariant integral currents. It is shown that this equivariant problem reduces to the study of a corresponding Plateau problem in the orbit space

R

/

G

{\text {R}}/G

where, for “big” groups, questions of uniqueness and regularity are simplified. The method is then applied to prove that for a constellation of explicit manifolds

M
M

, the cone

C
(
M
)
=
{
t
x
;
x
∈
M

C(M) = \{ tx;x \in M

and

0
⩽
t
⩽
1
}

0 \leqslant t \leqslant 1\}

is the unique solution to the Plateau problem for

M
M

, (Thus, there is no hope for general interior regularity of solutions in codimension one.) These manifolds include the original examples of type

S
n

×

S
n

⊂

R

2
n
+
2

,
n
⩾
3

{S^n} \times {S^n} \subset {{\text {R}}^{2n + 2}},n \geqslant 3

, due to Bombieri, DeGiorgi, Giusti and Simons. They also include a new example in

R

8

{{\text {R}}^8}

and examples in

R

n

{{\text {R}}^n}

for

n
⩾
10

n \geqslant 10

with any prescribed Betti number nonzero.
Actions of groups of order 𝑝𝑞
Authors: Lazarov Connor
Published: July 2010

We study the bordism group of stably complex

G
G

-manifolds in the case where

G
G

is a metacyclic group of order

p
q

pq

and

p
p

and

q
q

are distinct primes. This bordism group is a module over the complex bordism ring and we compute the projective dimension of this module. We develop some techniques necessary for the study of this module in case

G
G

is a more general metacyclic group.
A class of representations of the full linear group. II
Authors: Pierce Stephen
Published: July 2010

Let

V
V

be an

n
n

-dimensional vector space over complex numbers

C
C

. Let

W
W

be the

m
m

th tensor product of

V
V

. If

T
∈

Hom
C

(
V
,
V
)

T \in {\operatorname {Hom} _C}(V,V)

, let

⊗
m

T
∈

Hom
C

(
W
,
W
)

{ \otimes ^m}T \in {\operatorname {Hom} _C}(W,W)

be the

m
m

th tensor product of

T
T

. The homomorphism

T
→

⊗
m

T

T \to { \otimes ^m}T

is a representation of the full linear group

G

L

n

(
C
)

{\text {G}}{{\text {L}}_n}(C)

. If

H
H

is a subgroup of the symmetric group

S
m

{S_m}

, and

χ
\chi

a linear character on

H
H

, let

V
χ
m

(
G
)

V_\chi ^m(G)

be the subspace of

W
W

consisting of all tensors symmetric with respect to

H
H

and

χ
\chi

. Then

V
χ
m

(
H
)

V_\chi ^m(H)

is invariant under

⊗
m

T

{ \otimes ^m}T

. Let

K
(
T
)

K(T)

be the restriction of

⊗
m

T

{ \otimes ^m}T

to

V
χ
m

(
H
)

V_\chi ^m(H)

. For

n
n

large compared with

m
m

and for

H
H

transitive, we determine all cases when the representation

T
→
K
(
T
)

T \to K(T)

is irreducible.
𝐶*-algebras generated by Fourier-Stieltjes transforms
Authors: Dunkl Charles F. , Ramirez Donald E.
Published: July 2010

For G a locally compact group and

G
^

\hat G

its dual, let

M

d

(

G
^

)

{\mathcal {M}_d}(\hat G)

be the

C
∗

{C^ \ast }

-algebra generated by the Fourier-Stieltjes transforms of the discrete measures on G. We show that the canonical trace on

M

d

(

G
^

)

{\mathcal {M}_d}(\hat G)

is faithful if and only if G is amenable as a discrete group. We further show that if G is nondiscrete and amenable as a discrete group, then the only measures in

M

d

(

G
^

)

{\mathcal {M}_d}(\hat G)

are the discrete measures, and also the sup and lim sup norms are identical on

M

d

(

G
^

)

{\mathcal {M}_d}(\hat G)

. These results are extensions of classical theorems on almost periodic functions on locally compact abelian groups.
Localizations of HNP rings
Authors: Kuzmanovich James
Published: July 2010

In this paper it is shown that every hereditary Noetherian prime ring is the intersection of a hereditary Noetherian prime ring having no invertible ideals with a bounded hereditary Noetherian prime ring in which every nonzero two-sided ideal contains an invertible two-sided ideal. Further, it is shown that this intersection corresponds to a decomposition of torsion modules over such a ring; if

R
R

is an HNP ring with enough invertible ideals, then this decomposition coincides with that of Eisenbud and Robson. If

M
M

is a maximal invertible ideal of

R
R

where

R
R

is as above, then an overring of

R
R

is constructed which is a localization of

R
R

at

M
M

in a “classical sense"; that is, it is a ring of quotients with respect to a multiplicatively closed set of regular elements satisfying the Ore conditions. The localizations are shown to have nonzero radical and are also shown to satisfy a globalization theorem. These localizations are generalizations of ones constructed by A. V. Jategaonkar for HNP rings with enough invertible ideals.
Inductive limits of finite dimensional 𝐶*-algebras
Authors: Bratteli Ola
Published: July 2010

Inductive limits of ascending sequences of finite dimensional

C
∗

{C^ \ast }

-algebras are studied. The ideals of such algebras are classified, and a necessary and sufficient condition for isomorphism of two such algebras is obtained. The results of Powers concerning factor states and representations of UHF-algebras are generalized to this case. A study of the current algebra of the canonical anticommutation relations is then being made.
On the genus of a group
Authors: White Arthur T.
Published: July 2010

The genus of a group is defined to be the minimum genus for any Cayley color graph of the group. All finite planar groups have been determined, but little is known about the genus of finite nonplanar groups. In this paper two families of toroidal groups are presented; the genus is calculated for certain abelian groups; and upper bounds are given for the genera of the symmetric and alternating groups and for some hamiltonian groups.
Spaces of set-valued functions
Authors: O’Steen David N.
Published: July 2010

If

X
X

and

Y
Y

are topological spaces, the set of all continuous functions from

X
X

into

C
Y

CY

, the space of nonempty, compact subsets of

Y
Y

with the finite topology, contains a copy (with singleton sets substituted for points) of

Y
X

{Y^X}

, the continuous point-valued functions from

X
X

into

Y
Y

. It is shown that

Y
X

{Y^X}

is homeomorphic to this copy contained in

(
C
Y

)
X

{(CY)^X}

(where all function spaces are assumed to have the compact-open topology) and that, if

X
X

or

Y
Y

is

T
2

,

(
C
Y

)
X

{T_2},{(CY)^X}

is homoemorphic to a subspace of

(
C
Y

)

C
X

{(CY)^{CX}}

. Further, if

Y
Y

is

T
2

{T_2}

, then these images of

Y
X

{Y^X}

and

(
C
Y

)
X

{(CY)^X}

are closed in

(
C
Y

)
X

{(CY)^X}

and

(
C
Y

)

C
X

{(CY)^{CX}}

respectively. Finally, it is shown that, under certain conditions, some elements of

X
Y

{X^Y}

may be considered as elements of

(
C
Y

)
X

{(CY)^X}

and that the induced

1
1

-

1
1

function between the subspaces is open.
Extremal length and harmonic functions on Riemann surfaces
Authors: Minda Carl David
Published: July 2010

Expressions for several conformally invariant pseudometrics on a Riemann surface

R
R

are given in terms of three new forms of reduced extremal distance. The pseudometrics are defined by means of various subclasses of the set of all harmonic functions on

R
R

having finite Dirichlet integral. The reduced extremal distance between two points is defined on

R
R

, on the Alexandroff one-point compactification of

R
R

and on the Kerékjártó-Stoïlow compactification of

R
R

. These reduced extremal distances are computed in terms of harmonic functions having specified singularities and boundary behavior. The key to establishing this connection with harmonic functions is a general theorem dealing with extremal length on a compact bordered Riemann surface and its extensions to noncompact bordered surfaces. These results are used to obtain new tests for degeneracy in the classification theory of Riemann surfaces. Finally, some of the results are illustrated for a hyperbolic simply connected Riemann surface.
Approximation of strictly singular and strictly cosingular operators using nonstandard analysis
Authors: Brace J. W. , Kneece R. Royce
Published: July 2010

The strictly singular operators and the strictly cosingular operators are characterized by the manner in which they can be approximated by continuous linear operators of finite-dimensional range. We make use of linear convergence structures to obtain each class as limit points of the operators with finite-dimensional range. The construction of a nonstandard model makes it possible to replace convergence structures by topologies. Our nonstandard models are called nonstandard locally convex spaces.
Classes of automorphisms of free groups of infinite rank
Authors: Cohen Robert
Published: July 2010

This paper is concerned with finding classes of automorphisms of an infinitely generated free group F which can be generated by “elementary” Nielsen transformations. Two different notions of “elementary” Nielsen transformations are explored. One leads to a classification of the automorphisms generated by these transformations. The other notion leads to the subgroup B of

Aut

(
F
)

{\operatorname {Aut}}(F)

consisting of the “bounded length” automorphisms of F. We prove that the class of “bounded 3-length” automorphisms

B
3

{B_3}

and the class of “elementary simultaneous” Nielsen transformations generate the same subgroup of

Aut

(
F
)

{\operatorname {Aut}}(F)

. We show that for the class T of automorphisms of “2 occurring generators", the groups generated by

T
∩
B

T \cap B

and the “elementary simultaneous” Nielsen transformations are identical. These results lead to the conjecture that B is generated by the “elementary simultaneous Nielsen transformations". A study is also made of the subgroup S of the “triangular automorphisms” of

F
∞

{F_\infty }

, the free group on a countably infinite set of free generators. It is found that a “triangular automorphism” may be factored into “splitting automorphisms” of

F
∞

{F_\infty }

, which may be viewed as the “elementary” automorphisms of S.
Analytic wave front sets for solutions of linear differential equations of principal type
Authors: Andersson Karl Gustav
Published: July 2010

The propagation of analyticity for solutions u of

P
(
x
,
D
)
u
=
f

P(x,D)u = f

is studied, in terms of wave front sets, for a large class of differential operators

P
=
P
(
x
,
D
)

P = P(x,D)

of principal type. In view of a theorem by L. Hörmander [9], the results obtained imply rather precise results about the surjectivity of the mapping

P
:

C
∞

(
Ω
)
→

C
∞

(
Ω
)

P:{C^\infty }(\Omega ) \to {C^\infty }(\Omega )

.
The Picard group of noncommutative rings, in particular of orders
Authors: Fröhlich A.
Published: July 2010

The structure of the Picard group of not necessarily commutative rings, and specifically of orders, and its relation to the automorphism group are studied, mainly with arithmetic applications in mind.
Solutions of partial differential equations with support on leaves of associated foliations
Authors: Zachmanoglou E. C.
Published: July 2010

Suppose that the linear partial differential operator

P
(
x
,
D
)

P(x,D)

has analytic coefficients and that it can be written in the form

P
(
x
,
D
)
=
R
(
x
,
D
)
S
(
x
,
D
)

P(x,D) = R(x,D)S(x,D)

where

S
(
x
,
D
)

S(x,D)

is a polynomial in the homogeneous first order operators

A
1

(
x
,
D
)
,
⋯
,

A
r

(
x
,
D
)

{A_1}(x,D), \cdots ,{A_r}(x,D)

. Then in a neighborhood of any point

x
0

{x^0}

at which the principal part of

S
(
x
,
D
)

S(x,D)

does not vanish identically, there is a solution of

P
(
x
,
D
)
u
=
0

P(x,D)u = 0

with support the leaf through

x
0

{x^0}

of the foliation induced by the Lie algebra generated by

A
1

(
x
,
D
)
,
⋯
,

A
r

(
x
,
D
)

{A_1}(x,D), \cdots ,{A_r}(x,D)

. This result yields necessary conditions for hypoellipticity and uniqueness in the Cauchy problem. An application to second order degenerate elliptic operators is also given.
Two-norm spaces and decompositions of Banach spaces. II
Authors: Subramanian P. K. , Rothman S.
Published: July 2010

Let

X
X

be a Banach space,

Y
Y

a closed subspace of

X
∗

{X^\ast }

. One says

X
X

is

Y
Y

-reflexive if the canonical imbedding of

X
X

onto

Y
∗

{Y^\ast }

is an isometry and

Y
Y

-pseudo reflexive if it is a linear isomorphism onto. If

X
X

has a basis and

Y
Y

is the closed linear span of the corresponding biorthogonal functionals, necessary and sufficient conditions for

X
X

to be

Y
Y

-pseudo reflexive are due to I. Singer. To every

B
B

-space

X
X

with a decomposition we associate a canonical two-norm space

X
s

{X_s}

and show that the properties of

X
s

{X_s}

, in particular its

γ
\gamma

-completion, may be exploited to give different proofs of Singer’s results and, in particular, to extend them to

B
B

-spaces with decompositions. This technique is then applied to a study of direct sum of

B
B

-spaces with respect to a BK space. Necessary and sufficient conditions for such a space to be reflexive are obtained.
Closed hulls in infinite symmetric groups
Authors: Haimo Franklin
Published: July 2010

Let

Sym
⁡
M

\operatorname {Sym} M

be the symmetric group of an infinite set

M
M

. What is the smallest subgroup of

Sym
⁡
M

\operatorname {Sym} M

containing a given element if the subgroup is subject to the further condition that it is also the automorphism group of some finitary algebra on

M
M

? The structures of such closed hulls are related to the disjoint-cycle decompositions of the given elements. If the closed hull is not just the cyclic subgroup on the given element then it is nonminimal as a closed hull and is represented as a subdirect product of finite cyclic groups as well as by a quotient group of a group of infinite sequences. We determine the conditions under which it has a nontrivial primary component for a given prime

p
p

and show that such components must be bounded abelian groups.
Some integral inequalities with applications to the imbedding of Sobolev spaces defined over irregular domains
Authors: Adams R. A.
Published: July 2010

This paper examines the possibility of extending the Sobolev Imbedding Theorem to certain classes of domains which fail to have the “cone property” normally required for that theorem. It is shown that no extension is possible for certain types of domains (e.g. those with exponentially sharp cusps or which are unbounded and have finite volume), while extensions are obtained for other types (domains with less sharp cusps). These results are developed via certain integral inequalities which generalize inequalities due to Hardy and to Sobolev, and are of some interest in their own right. The paper is divided into two parts. Part I establishes the integral inequalities; Part II deals with extensions of the imbedding theorem. Further introductory information may be found in the first section of each part.
𝑃-commutative Banach *-algebras
Authors: Tiller Wayne
Published: July 2010

Let

A
A

be a complex

∗

^ \ast

-algebra. If

f
f

is a positive functional on

A
A

, let

I
f

=
{
x
∈
A
:
f
(

x
∗

x
)
=
0
}

{I_f} = \{ x \in A:f(x^ \ast x) = 0\}

be the corresponding left ideal of

A
A

. Set

P
=
∩

I
f

P = \cap {I_f}

, where the intersection is over all positive functionals on

A
A

. Then

A
A

is called

P
P

-commutative if

x
y
−
y
x
∈
P

xy - yx \in P

for all

x
,
y
∈
A

x,y \in A

. Every commutative

∗

^ \ast

-algebra is

P
P

-commutative and examples are given of noncommutative

∗

^ \ast

-algebras which are

P
P

-commutative. Many results are obtained for

P
P

-commutative Banach

∗

^ \ast

-algebras which extend results known for commutative Banach

∗

^ \ast

-algebras. Among them are the following: If

A
2

=
A

{A^2} = A

, then every positive functional on

A
A

is continuous. If

A
A

has an approximate identity, then a nonzero positive functional on

A
A

is a pure state if and only if it is multiplicative. If

A
A

is symmetric, then the spectral radius in

A
A

is a continuous algebra seminorm.
Probabilistic recursive functions
Authors: Mann Irwin
Published: July 2010

The underlying question considered in this paper is whether or not the purposeful introduction of random elements, effectively governed by a probability distribution, into a calculation may lead to constructions of number-theoretic functions that are not available by deterministic means. A methodology for treating this question is developed, using an effective mapping of the space of infinite sequences over a finite alphabet into itself. The distribution characterizing the random elements, under the mapping, induces a new distribution. The property of a distribution being recursive is defined. The fundamental theorem states that recursive distributions induce only recursive distributions. A function calculated by any probabilistic means is called

ψ
\psi

-calculable. For a class of such calculations, these functions are recursive. Relative to Church’s thesis, this leads to an extension of that thesis: Every

ψ
\psi

-effectively calculable function is recursive. In further development, a partial order on distributions is defined through the concept of “inducing.” It is seen that a recursive atom-free distribution induces any recursive distribution. Also, there exist distributions that induce, but are not induced by, any recursive distribution. Some open questions are mentioned.
Reductions of ideals in commutative rings
Authors: Hays James H.
Published: July 2010

All rings considered in this paper are commutative, associative, and have an identity. If A and B are ideals in a ring, then B is a reduction of A if

B
⊆
A

B \subseteq A

and if

B

A
n

=

A

n
+
1

B{A^n} = {A^{n + 1}}

for some positive integer n. An ideal is basic if it has no reductions. These definitions were considered in local rings by Northcott and Rees; this paper considers them in more general rings. Basic ideals in Noetherian rings are characterized to the extent that they are characterized in local rings. It is shown that elements of the principal class generate a basic ideal in a Noetherian ring. Prüfer domains do not have the basic ideal property, that is, there may exist ideals which are not basic; however, a characterization of Prüfer domains can be given in terms of basic ideals. A domain is Prüfer if and only if every finitely generated ideal is basic.
Euclidean 𝑛-space modulo an (𝑛-1)-cell
Authors: Bryant J. L.
Published: July 2010

This paper, together with another paper by the author titled similarly, provides a complete answer to a conjecture raised by Andrews and Curtis: if D is a k-cell topologically embedded in euclidean n-space

E
n

{E^n}

, then

E
n

/

D
×

E
1

{E^n}/D \times {E^1}

is homeomorphic to

E

n
+
1

{E^{n + 1}}

. Although there is at present only one case outstanding (

n
⩾
4

n \geqslant 4

and

k
=
n
−
1

k = n - 1

), the proof we give here works whenever

n
⩾
4

n \geqslant 4

. We resolve this conjecture (for

n
⩾
4

n \geqslant 4

) by proving a stronger result: if

Y
×

E
1

≈

E

n
+
1

Y \times {E^1} \approx {E^{n + 1}}

and if D is a k-cell in Y, then

Y

/

D
×

E
1

≈

E

n
+
1

Y/D \times {E^1} \approx {E^{n + 1}}

. This theorem was proved by Glaser for

k
⩽
n
−
2

k \leqslant n - 2

and has as a corollary: if K is a collapsible polyhedron topologically embedded in

E
n

{E^n}

, then

E
n

/

K
×

E
1

≈

E

n
+
1

{E^n}/K \times {E^1} \approx {E^{n + 1}}

. Our method of proof uses radial engulfing and a well-known procedure devised by Bing.
On the classification of metabelian Lie algebras
Authors: Gauger Michael A.
Published: July 2010

The classification of 2-step nilpotent Lie algebras is attacked by a generator-relation method. The main results are in low dimensions or a small number of relations.
Deforming cohomology classes
Authors: Wavrik John J.
Published: July 2010

Let

π
:
X
→
S

\pi :X \to S

be a flat proper morphism of analytic spaces.

π
\pi

may be thought of as providing a family of compact analytic spaces,

X
s

{X_s}

, parametrized by the space

S
S

. Let

F

\mathcal {F}

be a coherent sheaf on

X
X

flat over

S
S

.

F

\mathcal {F}

may be thought of as a family of coherent sheaves,

F

s

{\mathcal {F}_s}

, on the family of spaces

X
s

{X_s}

. Let

o
∈
S

o \in S

be a fixed point,

ξ
o

∈
H
q
(

X
o

,

F

o

)

{\xi _o} \in Hq({X_o},{\mathcal {F}_o})

. In this paper, we consider the problem of extending

ξ
o

{\xi _o}

to a cohomology class

ξ
∈
H
q
(

π

−
1

(
U
)
,

F

)

\xi \in Hq({\pi ^{ - 1}}(U),\mathcal {F})

where

U
U

is some neighborhood of

o
o

in

S
S

. Extension problems of this type were first considered by P. A. Griffiths who obtained some results in the case in which the morphism

π
\pi

is simple and the sheaf

F

\mathcal {F}

is locally free. We obtain generalizations of these results without the restrictions. Among the applications of these results is a necessary and sufficient condition for the existence of a space of moduli for a compact manifold. This application was discussed in an earlier paper by the author. We use the Grauert “direct image” theorem, the theory of Stein compacta, and a generalization of a result of M. Artin on solutions of analytic equations to reduce the problem to an algebraic problem. In §2 we discuss obstructions to deforming

ξ
o

{\xi _o}

; in §3 we show that if no obstructions exist,

ξ
o

{\xi _o}

may be extended; in §4 we give a useful criterion for no obstructions; and in §5 we discuss some examples.