bluuebunny/crossref_metadata_2025 huggingface dataset:
Nonlinear approximation in uniformly smooth Banach spaces
Authors: Rozema Edward R. , Smith Philip W.
Published: July 2010

John R. Rice [Approximation of functions. Vol. II, Addison-Wesley, New York, 1969] investigated best approximation from a nonlinear manifold in a finite dimensional, smooth, and rotund space. The authors define the curvature of a manifold by comparing the manifold with the unit ball of the space and suitably define the “folding” of a manifold. Rice’s Theorem 11 extends as follows: Theorem. Let X be a uniformly smooth Banach space, and

F
:

R
n

→
X

F:{R^n} \to X

be a homeomorphism onto

M
=
F
(

R
n

)

M = F({R^n})

. Suppose

∇
F
(
a
)

\nabla F(a)

exists for each a in X,

∇
F

\nabla F

is continuous as a function of a, and

∇
F
(
a
)
⋅

R
n

\nabla F(a) \cdot {R^n}

has dimension n. Then, if M has bounded curvature, there exists a neighborhood of M each point of which has a unique best approximation from M. A variation theorem was found and used which locates a critical point of a differentiable functional defined on a uniformly rotund space Y. [See M. S. Berger and M. S. Berger, Perspectives in nonlinearity, Benjamin, New York, 1968, p. 58ff. for a similar result when

Y
=

R
n

Y = {R^n}

.] The paper is concluded with a few remarks on Chebyshev sets.
𝑃-convexity and 𝐵-convexity in Banach spaces
Authors: Brown Dean R.
Published: July 2010

Two properties of B-convexity are shown to hold for P-convexity: (1) Under certain conditions, the direct sum of two P-convex spaces is P-convex. (2) A Banach space is P-convex if each subspace having a Schauder decomposition into finite dimensional subspaces is P-convex.
Kähler differentials and differential algebra in arbitrary characteristic
Authors: Johnson Joseph
Published: July 2010

Let L and K be differential fields with L an extension of K. It is shown how the module of Kähler differentials

Ω

L

/

K

1

\Omega _{L/K}^1

can be used to “linearize” properties of a differential field extension

L

/

K

L/K

. This is done without restriction on the characteristic p and yields a theory which for

p
≠
0

p \ne 0

is no harder than the case

p
=
0

p = 0

. As an application a new proof of the Ritt basis theorem is given.
𝑝-absolutely summing operators and the representation of operators on function spaces
Authors: Rice John William
Published: July 2010

We introduce a class of p-absolutely summing operators which we call p-extending. We show that for a logmodular function space

A
(
K
)

A(K)

, an operator

T
:
A
(
K
)
→
X

T:A(K) \to X

is p-extending if and only if there exists a probability measure

μ
\mu

on K such that T extends to an isometry

T
:

A
p

(
K
,
μ
)
→
X

T:{A^p}(K,\mu ) \to X

. We use this result to give necessary and sufficient conditions under which a bounded linear operator is isometrically equivalent to multiplication by z on a space

L
p

(
K
,
μ
)

{L^p}(K,\mu )

and certain Hardy spaces

H
p

(
K
,
μ
)

{H^p}(K,\mu )

.
One-parameter semigroups holomorphic away from zero
Authors: Certain Melinda W.
Published: July 2010

Suppose

T
T

is a one-parameter semigroup of bounded linear operators on a Banach space, strongly continuous on

[
0
,
∞
)

[0,\infty )

. It is known that

lim

sup

x
→
0

|

T
(
x
)
−
I

|

>
2

\lim {\sup _{x \to 0}}|T(x) - I| > 2

implies

T
T

is holomorphic on

(
0
,
∞
)

(0,\infty )

. Theorem I is a generalization of this as follows: Suppose

M
>
0
,
0
>
r
>
s

M > 0,0 > r > s

, and

ρ
\rho

is in (1,2). If

|

(
T
(
h
)
−
I

)
n

|

≤
M

ρ
n

|{(T(h) - I)^n}| \leq M{\rho ^n}

whenever

n
h

nh

is in

[
r
,
s
]
,
n
=
1
,
2
,
⋯
,
h
>
0

[r,s],n = 1,2, \cdots ,h > 0

, then there exists

b
>
0

b > 0

such that

T
T

is holomorphic on

[
b
,
∞
)

[b,\infty )

. Theorem II shows that, in some sense,

b
→
0

b \to 0

as

r
→
0

r \to 0

. Theorem I is an application of Theorem III: Suppose

M
>
0
,
0
>
r
>
s
,
ρ

M > 0,0 > r > s,\rho

is in (1,2), and

f
f

is continuous on

[
−
4
s
,
4
s
]

[ - 4s,4s]

. If
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue sigma-summation Underscript q equals 0 Overscript n Endscripts StartBinomialOrMatrix n Choose q EndBinomialOrMatrix left-parenthesis negative 1 right-parenthesis Superscript n minus q Baseline f left-parenthesis t plus q h right-parenthesis EndAbsoluteValue less-than-or-equal-to upper M rho Superscript n">

|

∑

q
=
0

n

(

n
q

)

(
−
1

)

n
−
q

f
(
t
+
q
h
)

|

≤
M

ρ
n

|\sum _{q = 0}^n \binom {n}{q}( - 1)^{n - q} f(t + qh)| \leq M \rho ^n

whenever

n
h

nh

is in

[
r
,
s
]

[r,s]

,

n
=
1

n = 1

,

2
2

, …,

h
>
0

h > 0

,

[
t
,
t
+
n
h
]
⊂
[
−
4
s
,
4
s
]

[t,t + nh] \subset [ - 4s,4s]

, then

f
f

has an analytic extension to an ellipse with center zero. Theorem III is a generalization of a theorem of Beurling in which the inequality on the differences is assumed for all

n
h

nh

. An example is given to show the hypothesis of Theorem I does not imply

T
T

holomorphic on

(
0
,
∞
)

(0,\infty )

.
Topological dynamics and group theory
Authors: Glasner Shmuel
Published: July 2010

We prove, using notions and techniques of topological dynamics, that a nonamenable group contains a finitely-generated subgroup of exponential growth. We also show that a group which belongs to a certain class, defined by means of topological dynamical properties, always contains a free subgroup on two generators.
A summation formula involving 𝜎(𝑁)
Authors: Nasim C.
Published: July 2010

The

L
2

{L^2}

theories are known of the summation formula involving

σ
k

(
n
)

{\sigma _k}(n)

, the sum of the kth power of divisors of n, as coefficients, for all k except

k
=
1

k = 1

. In this paper, techniques are used to overcome the extra convergence difficulty of the case

k
=
1

k = 1

, to establish a symmetric formula connecting the sums of the form

∑

σ
1

(
n
)

n

−
1

/

2

f
(
n
)

\sum {\sigma _1}(n){n^{ - 1/2}}f(n)

and

∑

σ
1

(
n
)

n

−
1

/

2

g
(
n
)

\sum {{\sigma _1}} (n){n^{ - 1/2}}g(n)

, where

f
(
x
)

f(x)

and

g
(
x
)

g(x)

are Hankel transforms of each other.
The theory of 𝑄-rings
Authors: Matlis Eben
Published: July 2010

An integral domain R with quotient field Q is defined to be a Q-ring if

Ext
R
1

⁡
(
Q
,
R
)
≅
Q

\operatorname {Ext}_R^1(Q,R) \cong Q

. It is shown that R is a Q-ring if and only if there exists an R-module A such that

Hom
R

(
A
,
R
)
=
0

{\operatorname {Hom}_R}(A,R) = 0

and

Ext
R
1

⁡
(
A
,
R
)
≅
Q

\operatorname {Ext}_R^1(A,R) \cong Q

. If A is such an R-module and

t
(
A
)

t(A)

is its torsion submodule, then it is proved that

A

/

t
(
A
)

A/t(A)

necessarily has rank one. There are only three kinds of Q-rings, namely,

Q
0

-

,

Q
1

-

{Q_0}{\text {-}},{Q_1}{\text {-}}

, or

Q
2

{Q_2}

-rings. These are described by the fact that if R is a Q-ring, then

K
=
Q

/

R

K = Q/R

can only have 0, 1, or 2 proper h-divisible submodules. If H is the completion of R in the R-topology, then R is one of the three kinds of Q-rings if and only if

H

⊗
R

Q

H{ \otimes _R}Q

is one of the three possible kinds of 2-dimensional commutative Q-algebras. Examples of all three kinds of Q-rings are produced, and the behavior of Q-rings under ring extensions is examined. General conditions are given for a ring not to be a Q-ring. As an application of the theory, necessary and sufficient conditions are found for the integral closure of a non-complete Noetherian domain to be a complete discrete valuation ring.
Some thin sets in discrete abelian groups
Authors: Blei Ron C.
Published: July 2010

Let

Γ
\Gamma

be a discrete abelian group, and

E
⊂
Γ

E \subset \Gamma

. For

F
⊂
E

F \subset E

, we say that

F
∈

P

(
E
)

F \in \mathcal {P}(E)

, if for all

Λ
\Lambda

, finite subsets of

Γ
,
0
∉
Λ
,
Λ
+
F
∩
F

\Gamma ,0 \notin \Lambda ,\Lambda + F \cap F

is finite. Having defined the Banach algebra,

A
~

(
E
)
=
c
(
E
)
∩
B
(
E
)

\tilde A(E) = c(E) \cap B(E)

, we prove the following: (i)

E
⊂
Γ

E \subset \Gamma

is a Sidon set if and only if every

F
∈

P

(
E
)

F \in \mathcal {P}(E)

is a Sidon set; (ii)

E
∈

P

(
Γ
)

E \in \mathcal {P}(\Gamma )

is a Sidon set if and only if

A
~

(
E
)
=
A
(
E
)

\tilde A(E) = A(E)

.
Modules over quadratic and quaternion rings and transformations of quadratic forms
Authors: Rice Bart
Published: July 2010

A study is made of transformations carrying certain quadratic and quaternary quadratic forms into multiples of themselves, and it is shown how these are related to the study of modules over quadratic and quaternion rings. Special automorphic transformations of n-ary quadratic forms may also exhibit a structure like those in the quadratic and quaternary cases.
Square-integrable representations and the Mackey theory
Authors: Sund Terje
Published: July 2010

It is the purpose of this paper to clarify the relationship between the square-integrable irreducible representations of a 2nd countable unimodular locally compact group G and a closed normal subgroup N using the Mackey theory relating the dual spaces

G
^

\hat G

and

N
^

\hat N

.
Disjoint meromorphic functions and nonoscillatory differential systems
Authors: Aharonov D. , Lavie M.
Published: July 2010

Conditions satisfied by disjoint meromorphic functions are obtained. These results are applied to nonoscillatory differential systems and disfocal differential equations.
Shape theory and compact connected abelian topological groups
Authors: Keesling James
Published: July 2010

Let C denote the category of compact Hausdorff spaces and continuous maps. Let

S
:
C
→
S
C

S:C \to SC

denote the functor of shape in the sense of Holsztyński from C to the shape category SC determined by the homotopy functor

H
:
C
→
H
C

H:C \to HC

from C to the homotopy category HC. Let A, B, and D denote compact connected abelian topological groups. In this paper it is shown that if G is a morphism in the shape category from A to B, then there is a unique continuous homomorphism

g
:
A
→
B

g:A \to B

such that

S
(
g
)
=
G

S(g) = G

. This theorem is used in a study of shape properties of continua which support an abelian topological group structure. The following results are shown: (1) The spaces A and B are shape equivalent if and only if

A
≃
B

A \simeq B

. (2) The space A is movable if and only if A is locally connected. (3) The space A shape dominates

B
,
S
(
A
)
≥
S
(
B
)

B,S(A) \geq S(B)

, if and only if there is a D such that

A
≃
B
×
D

A \simeq B \times D

. (4) The fundamental dimension of A is the same as the dimension of

A
,

Sd

(
A
)
=
dim
⁡
A

A,{\text {Sd}}(A) = \dim A

. In an Appendix it is shown that the Holsztyński approach to shape and the approach of Mardešić and Segal using ANR-systems are equivalent. Thus, the results apply to either theory and to the Borsuk theory in the metrizable case.
The slimmest geometric lattices
Authors: Dowling Thomas A. , Wilson Richard M.
Published: July 2010

The Whitney numbers of a finite geometric lattice L of rank r are the numbers

W
k

{W_k}

of elements of rank k and the coefficients

w
k

{w_k}

of the characteristic polynomial of L, for

0
≤
k
≤
r

0 \leq k \leq r

. We establish the following lower bounds for the

W
k

{W_k}

and the absolute values

w
k
+

=

(
−
1

)
k

w
k

w_k^ + = {( - 1)^k}{w_k}

and describe the lattices for which equality holds (nontrivially) in each case:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Subscript k Baseline greater-than-or-equal-to Start 2 By 3 Matrix 1st Row 1st Column r 2nd Column a m p semicolon minus 3rd Column a m p semicolon 2 2nd Row 1st Column k 2nd Column a m p semicolon minus 3rd Column a m p semicolon 1 EndMatrix left-parenthesis n minus r right-parenthesis plus StartBinomialOrMatrix r Choose k EndBinomialOrMatrix comma w Subscript k Superscript plus Baseline greater-than-or-equal-to Start 2 By 3 Matrix 1st Row 1st Column r 2nd Column a m p semicolon minus 3rd Column a m p semicolon 1 2nd Row 1st Column k 2nd Column a m p semicolon minus 3rd Column a m p semicolon 1 EndMatrix left-parenthesis n minus r right-parenthesis plus StartBinomialOrMatrix r Choose k EndBinomialOrMatrix comma">

W
k

≥

(

r

a
m
p
;
−

a
m
p
;
2

k

a
m
p
;
−

a
m
p
;
1

)

(
n
−
r
)
+

(

r

k

)

,

w
k
+

≥

(

r

a
m
p
;
−

a
m
p
;
1

k

a
m
p
;
−

a
m
p
;
1

)

(
n
−
r
)
+

(

r

k

)

,

{W_k} \geq \left ( {\begin {array}{*{20}{c}} r \hfill & - \hfill & 2 \hfill \\ k \hfill & - \hfill & 1 \hfill \\ \end {array} } \right )(n - r) + \left ( {\begin {array}{*{20}{c}} r \hfill \\ k \hfill \\ \end {array} } \right ),\quad w_k^ + \geq \left ( {\begin {array}{*{20}{c}} r \hfill & - \hfill & 1 \hfill \\ k \hfill & - \hfill & 1 \hfill \\ \end {array} } \right )(n - r) + \left ( {\begin {array}{*{20}{c}} r \hfill \\ k \hfill \\ \end {array} } \right ),

\]
where

n
=

W
1

n = {W_1}

is the number of points of L.
Nonattainability of a set by a diffusion process
Authors: Friedman Avner
Published: July 2010

Consider a system of n stochastic differential equations

d
ξ
=
b
(
ξ
)
d
t
+
σ
(
ξ
)
d
w

d\xi = b(\xi )dt + \sigma (\xi )dw

. Let M be a k-dimensional submanifold in

R
n

,
k
≤
n
−
1

{R^n},k \leq n - 1

. For

x
∈
M

x \in M

, denote by

d
(
x
)

d(x)

the rank of

σ

σ
∗

\sigma {\sigma ^ \ast }

restricted to the linear space of all normals to M at x. It is proved that if

d
(
x
)
≥
2

d(x) \geq 2

for all

x
∈
M

x \in M

, then

ξ
(
t
)

\xi (t)

does not hit M at finite time, given

ξ
(
0
)
∉
M

\xi (0) \notin M

, i.e., M is nonattainable. The cases

d
(
x
)
≥
1
,
d
(
x
)
≥
0

d(x) \geq 1,d(x) \geq 0

are also studied.
A nonlinear Boltzmann equation in transport theory
Authors: Pao C. V.
Published: July 2010

The method of successive approximations is used to show the existence of a unique solution to a model of a nonlinear Boltzmann equation under the homogeneous boundary and typical initial conditions. An explicit formula in terms of the prescribed functions for the calculation of an approximate solution and its error estimate are given. This formula reveals an interesting analogy between the initial-boundary value problem of the Boltzmann equation and the Cauchy problem for ordinary differential equations. Numerical results for approximate solutions of the problem can be computed by using a computer. The linear Boltzmann equation is considered as a special case and a similar formula for the calculation of approximate solutions is included.
Axisymmetric harmonic interpolation polynomials in 𝑅^{𝑁}
Authors: Marden Morris
Published: July 2010

Corresponding to a given function

F
(
x
,
ρ
)

F(x,\rho )

which is axisymnetric harmonic in an axisymmetric region

Ω
⊂

R

3

\Omega \subset {{\text {R}}^3}

and to a set of

n
+
1

n + 1

circles

C
n

{C_n}

in an axisymmetric subregion

A
⊂
Ω

A \subset \Omega

, an axisymmetric harmonic polynomial

Λ
n

(
x
,
ρ
;

C
n

)

{\Lambda _n}(x,\rho ;{C_n})

is found which on the

C
n

{C_n}

interpolates to

F
(
x
,
ρ
)

F(x,\rho )

or to its partial derivatives with respect to x. An axisymmetric subregion

B
⊂
Ω

B \subset \Omega

is found such that

Λ
n

(
x
,
ρ
;

C
n

)

{\Lambda _n}(x,\rho ;{C_n})

converges uniformly to

F
(
x
,
ρ
)

F(x,\rho )

on the closure of B. Also a

Λ
n

(
x
,
ρ
;

x
0

,

ρ
0

)

{\Lambda _n}(x,\rho ;{x_0},{\rho _0})

is determined which, together with its first n partial derivatives with respect to x, coincides with

F
(
x
,
ρ
)

F(x,\rho )

on a single circle

(

x
0

,

ρ
0

)

({x_0},{\rho _0})

in

Ω
\Omega

and converges uniformly to

F
(
x
,
ρ
)

F(x,\rho )

in a closed torus with

(

x
0

,

ρ
0

)

({x_0},{\rho _0})

as central circle.
Generalized almost periodicity in groups
Authors: Davis Henry W.
Published: July 2010

A module of almost periodic functions on a group is closed with respect to a quite general seminorm. The new space of functions is characterized in terms of the internal properties of its members. This yields new characterizations of Besicovitch and Weyl almost periodic functions in a variety of group-theoretic settings. Eberlein’s theorem that weakly almost periodic functions on the real line are Weyl almost periodic is extended to locally compact groups.
Splitting the tangent bundle
Authors: Iberkleid Wolf
Published: July 2010

We determine those unoriented cobordism classes which can be realized by a manifold whose tangent bundle splits into a sum of real (complex) line bundles.
Symmetric jump processes
Authors: Silverstein Martin L.
Published: July 2010

We use the theory of Dirichlet spaces to construct symmetric Markov processes of pure jump type and to identify the Lévy measures for these processes. Particular attention is paid to lattice and hard sphere systems which interact through speed change and exclusion.
Many-one reducibility within the Turing degrees of the hyperarithmetic sets 𝐻ₐ(𝑥)
Authors: Nelson G. C.
Published: July 2010

Spector [13] has proven that the hyperarithmetic sets

H
a

(
x
)

{H_a}(x)

and

H
b

(
x
)

{H_b}(x)

have the same Turing degree iff

|

a

|

=

|

b

|

|a| = |b|

. Y. Moschovakis has proven that the sets

H
a

(
x
)

{H_a}(x)

under many-one reducibility for

|

a

|

=
γ

|a| = \gamma

and

a
∈

O

a \in \mathcal {O}

have nontrivial reducibility properties if

γ
\gamma

is not of the form

α
+
1

\alpha + 1

or

α
+
ω

\alpha + \omega

for any ordinal a. In particular, he proves that there are chains of order type

ω
1

{\omega _1}

and incomparable many-one degrees within these Turing degrees. In Chapter II, we extend this result to show that any countable partially ordered set can be embedded in the many-one degrees within these Turing degrees. In Chapter III, we prove that if

γ
\gamma

is also not of the form

α
+

ω
2

\alpha + {\omega ^2}

for some ordinal a, then there is no minimal many-one degree of the form

H
a

(
x
)

{H_a}(x)

in this Turing degree, answering a question of Y. Moschovakis posed in [8]. In fact, we prove that given

H
a

(
x
)

{H_a}(x)

there are

H
b

(
x
)

{H_b}(x)

and

H
c

(
x
)

{H_c}(x)

both many-one reducible to

H
a

(
x
)

{H_a}(x)

with incomparable many-one degrees,

|

a

|

=

|

b

|

=

|

c

|

=
γ

|a| = |b| = |c| = \gamma

.
A Siegel formula for orthogonal groups over a function field
Authors: Haris Stephen J.
Published: July 2010

We obtain a Siegel formula for a quadratic form over a function field, by establishing the convergence of the corresponding Eisenstein-Siegel series directly, then via the Hasse principle, that of the associated Poisson formula.
Constructions in algebra
Authors: Seidenberg A.
Published: July 2010

It is shown how to construct a primary decomposition and to find the associated prime ideals of a given ideal in a polynomial ring. This is first done from a classical, and then from a strictly constructivist, point of view.
Rational points of commutator subgroups of solvable algebraic groups
Authors: Fauntleroy Amassa
Published: July 2010

Let G be a connected algebraic group defined over a field k. Denote by

G
(
k
)

G(k)

the group of k-rational points of G. Suppose that A and B are closed subgroups of G defined over k. Then

[
A
,
B
]
(
k
)

[A,B](k)

is not equal to

[
A
(
k
)
,
B
(
k
)
]

[A(k),B(k)]

in general. Here [A,B] denotes the group generated by commutators

a
b

a

−
1

b

−
1

,
a
∈
A
,
b
∈
B

ab{a^{ - 1}}{b^{ - 1}},a \in A,b \in B

. We say that a field of k of characteristic p is p-closed if given any additive polynomial

f
(
x
)

f(x)

in

k
[
x
]

k[x]

and any element c in k, there exists an element

α
\alpha

in k such that

f
(
α
)
=
c

f(\alpha ) = c

. Theorem 1. Let G be a connected solvable algebraic group defined over the p-closed field k. Let A and B be closed connected subgroups of G, which are also defined over k, and suppose A normalizes B.
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T h e n left-bracket upper A comma upper B right-bracket left-parenthesis k right-parenthesis equals left-bracket upper A left-parenthesis upper K right-parenthesis comma upper B left-parenthesis upper K right-parenthesis right-bracket">

T
h
e
n

[
A
,
B
]

(
k
)
=
[
A
(
K
)
,
B
(
K
)
]

Then\;[A,B]\;(k) = [A(K),B(K)]

. 2. If G, A and B are as above and k is only assumed to be perfect then there exists a finite extension

k
0

{k_0}

of k such that if K is the maximal p-extension of

k
0

{k_0}

, then

[
A
,
B
]
(
K
)
=
[
A
(
K
)
,
B
(
K
)
]

[A,B](K) = [A(K),B(K)]

.
Extensions of the 𝑣-integral
Authors: Edwards J. R. , Wayment S. G.
Published: July 2010

In Representations for transformations continuous in the BV norm [J. R. Edwards and S. G. Wayment, Trans. Amer. Math. Soc. 154 (1971), 251-265] the

ν
\nu

-integral is defined over intervals in

E
1

{E^1}

and is used to give a representation for transformations continuous in the BV norm. The functions f considered therein are real valued or have values in a linear normed space X, and the transformation

T
(
f
)

T(f)

is real or has values in a linear normed space Y. In this paper the

ν
\nu

-integral is extended in several directions: (1) The domain space to (a)

E
n

{E^n}

, (b) an arbitrary space S, a field

Σ
\Sigma

of subsets of S and a bounded positive finitely additive set function

μ
\mu

on

Σ
\Sigma

(in this setting the function space is replaced by the space of finitely additive set functions which are absolutely continuous with respect to

μ
\mu

); (2) the function space to (a) bounded continuous, (b)

C
c

{C_c}

, (c)

C
0

{C_0}

, (d) C with uniform convergence on compact sets; (3) range space X for the functions and Y for the transformation to topological vector spaces (not necessarily convex); (4) when X and Y are locally convex spaces, then a representation for transformations on a

C
1

{C_1}

-type space of continuously differentiable functions with values in X is given.
Asymptotic properties of 𝑈-statistics
Authors: Sproule Raymond N.
Published: July 2010

Let

r
r

be a fixed positive integer. A

U
U

-statistic

U
n

{U_n}

is an average of a symmetric measurable function of

r
r

arguments over a random sample of size

n
n

. Such a statistic may be expressed as an average of independent and identically distributed random variables plus a remainder term. We develop a Kolmogorov-like inequality for this remainder term as well as examine some of its (a.s.) convergence properties. We then relate these properties to the

U
U

-statistic. In addition, the asymptotic normality of

U
N

{U_N}

, where

N
N

is a positive integer-valued random variable, is established under certain conditions.
Local decay of solutions of conservative first order hyperbolic systems in odd dimensional space
Authors: Ralston James V.
Published: July 2010

This paper deals with symmetric hyperbolic systems,

∂
u

/

∂
t
=
L
u

\partial u/\partial t = Lu

, where L is equal to the homogeneous, constant coefficient operator

L
0

{L_0}

for

|

x

|

>
R

|x| > R

. Under the hypothesis that L has simple null bicharacteristics and these propagate to infinity, local decay of solutions and completeness of the wave operators relating solutions of

∂
u

/

∂
t
=
L
u

\partial u/\partial t = Lu

and solutions of

∂
u

/

∂
t
=

L
0

u

\partial u/\partial t = {L_0}u

are established. Results of this type for elliptic L are due to Lax and Phillips. The proof here is based, in part, on a new estimate of the regularity of the

L
2

{L^2}

-solutions of the equation

L
u
+
(
i
λ
+
ε
)
u
=
g

Lu + (i\lambda + \varepsilon )u = g

for smooth g with support in

|

x

|

≤
R

|x| \leq R

.
Approximation of analytic functions on compact sets and Bernstein’s inequality
Authors: Baouendi M. S. , Goulaouic C.
Published: July 2010

The characterization of analytic functions defined on a compact set K in

R

N

{{\mathbf {R}}_N}

by their polynomial approximation is possible if and only if K satisfies some “Bernstein type inequality", estimating any polynomial P in some neighborhood of K using the supremum of P on K. Some criterions and examples are given. Approximation by more general sets of analytic functions is also discussed.
Characterization of privileged polydomains
Authors: Siu Yum Tong
Published: July 2010

This paper gives a number of equivalent conditions for a bounded polydomain to be privileged with respect to a coherent analytic sheaf in the sense of Douady. One of the equivalent conditions is in terms of the homological codimensions of the sheaf at the boundary of the polydomain. In the case of a polydisc, this condition about homological codimensions coincides with a conjecture of Douady. The other equivalent conditions concern some weaker concepts of privilegedness and the existence of privileged sets at the boundary.
A general representation theorem for analytic solutions of first-order algebraic differential equations in sectors
Authors: Bank Steven B.
Published: July 2010

In this paper, we obtain precise asymptotic representations for a broad class of solutions of first-order algebraic differential equations whose coefficients belong to a certain type of function field.
Pseudo-boundaries and pseudo-interiors in Euclidean spaces and topological manifolds
Authors: Geoghegan Ross , Summerhill R. Richard
Published: July 2010

The negligibility theorems of infinite-dimensional topology have finite-dimensional analogues. The role of the Hilbert cube

I
ω

{I^\omega }

is played by euclidean n-space

E
n

{E^n}

, and for any nonnegative integer

k
>
n

k > n

, k-dimensional dense

F
σ

{F_\sigma }

-subsets of

E
n

{E^n}

exist which play the role of the pseudo-boundary of

I
ω

{I^\omega }

. Their complements are

(
n
−
k
−
1
)

(n - k - 1)

-dimensional dense

G
δ

{G_\delta }

pseudo-interiors of

E
n

{E^n}

. Two kinds of k-dimensional pseudo-boundaries are constructed, one from universal compacta, the other from polyhedra. All the constructions extend to topological manifolds.
Free 𝑆¹ actions and the group of diffeomorphisms
Authors: Wang Kai
Published: July 2010

Let

S
1

{S^1}

act linearly on

S

2
p
−
1

×

D

2
q

{S^{2p - 1}} \times {D^{2q}}

and

D

2
p

×

S

2
q
−
1

{D^{2p}} \times {S^{2q - 1}}

and let

f
:

S

2
p
−
1

×

S

2
q
−
1

→

S

2
p
−
1

×

S

2
q
−
1

f:{S^{2p - 1}} \times {S^{2q - 1}} \to {S^{2p - 1}} \times {S^{2q - 1}}

be an equivariant diffeomorphism. Then there is a well-defined

S
1

{S^1}

action on

S

2
p
−
1

×

D

2
q

∪
f

D

2
p

×

S

2
q
−
1

{S^{2p - 1}} \times {D^{2q}}{ \cup _f}{D^{2p}} \times {S^{2q - 1}}

. An

S
1

{S^1}

action on a homotopy sphere is decomposable if it can be obtained in this way. In this paper, we will apply surgery theory to study in detail the set of decomposable actions on homotopy spheres.
The genera of edge amalgamations of complete bigraphs
Authors: Alpert Seth R.
Published: July 2010

If G and H are graphs, then

G
∨
H

G \vee H

is defined to be a graph obtained by identifying some edge of G with some edge of H. It is shown that for all m, n, p, and q the genus

g
(

K

m
,
n

∨

K

p
,
q

)

g({K_{m,n}} \vee {K_{p,q}})

is either

g
(

K

m
,
n

)
+
g
(

K

p
,
q

)

g({K_{m,n}}) + g({K_{p,q}})

or else

g
(

K

m
,
n

)
+
g
(

K

p
,
q

)
−
1

g({K_{m,n}}) + g({K_{p,q}}) - 1

. The latter value is attained if and only if both

K

m
,
n

{K_{m,n}}

and

K

p
,
q

{K_{p,q}}

are critical in the sense that the deletion of any edge results in a graph whose genus is one less than the genus of the original graph.
Sufficient sets for some spaces of entire functions
Authors: Schneider Dennis M.
Published: July 2010

B. A. Taylor [13] has shown that the lattice points in the plane form a sufficient set for the space of entire functions of order less than two. We obtain a generalization of this result to functions of several variables and to more general spaces of entire functions. For example, we prove that if

S
⊂

C

n

S \subset {{\mathbf {C}}^n}

such that

d
(
z
,
S
)
≤
const
⁡

|

z

|

1
−
ρ

/

2

d(z,S) \leq \operatorname {const}|z{|^{1 - \rho /2}}

for all

z
∈

C

n

z \in {{\mathbf {C}}^n}

, then S is a sufficient set for the space of entire functions on

C

n

{{\mathbf {C}}^n}

of order less than

ρ
\rho

. The proof involves estimating the growth rate of an entire function from its growth rate on S. We also introduce the concept of a weakly sufficient set and obtain sufficient conditions for a set to be weakly sufficient. We prove that sufficient sets are weakly sufficient and that certain types of effective sets [8] are weakly sufficient.
Möbius transformations of the disc and one-parameter groups of isometries of 𝐻^{𝑝}
Authors: Berkson Earl , Kaufman Robert , Porta Horacio
Published: July 2010

Let

{

T
t

}

\{ {T_t}\}

be a strongly continuous one-parameter group of isometries in

H
p

(
1
⩽
p
>
∞
,
p
≠
2
)

{H^p}(1 \leqslant p > \infty ,p \ne 2)

with unbounded generator. There is a uniquely determined one-parameter group

{

ϕ
t

}
,
t
∈

R

\{ {\phi _t}\} ,t \in {\mathbf {R}}

, of Möbius transformations of the (open) disc

D
D

corresponding to

{

T
t

}

\{ {T_t}\}

by way of Forelli’s theorem. The interplay between

{

T
t

}

\{ {T_t}\}

and

{

ϕ
t

}

\{ {\phi _t}\}

is studied, and the spectral properties of the generator

A
A

of

{

T
t

}

\{ {T_t}\}

are analyzed in this context. The nature of the set

S
S

of common fixed points of the functions

ϕ
t

{\phi _t}

plays a crucial role in determining the behavior of

A
A

. The spectrum of

A
A

, which is a subset of

i

R

i{\mathbf {R}}

, can be a discrete set, a translate of

i

R

+

i{{\mathbf {R}}_ + }

or of

i

R

−

i{{\mathbf {R}}_ - }

, or all of

i

R

i{\mathbf {R}}

. If

S
S

is not a doubleton subset of the unit circle,

{

T
t

}

\{ {T_t}\}

can be extended to a holomorphic semigroup of

H
p

{H^p}

-operators, the semigroup being defined on a half-plane. The treatment of

{

T
t

}

\{ {T_t}\}

is facilitated by developing appropriate properties of one-parameter groups of Möbius transformations of

D
D

. In particular, such groups are in one-to-one correspondence (via an initial-value problem) with the nonzero polynomials

q
q

, of degree at most 2, such that

Re
[

z
¯

q
(
z
)
]
=
0

\text {Re} [\bar zq(z)] = 0

for all unimodular

z
z

. A has an explicit description (in terms of the polynomial corresponding to

{

ϕ
t

}

\{ {\phi _t}\}

) as a differential operator.
𝐻^{𝑟,}^{∞}(𝑅)- and 𝑊^{𝑟,∞}(𝑅)-splines
Authors: Smith Philip W.
Published: July 2010

Let E be a subset of R the real line and

f
:
E
→
R

f:E \to R

. Necessary and sufficient conditions are derived for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="inf left-parenthesis double-vertical-bar upper D Superscript r Baseline x double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript Baseline colon x vertical-bar Subscript upper E Baseline equals f right-parenthesis">

inf
(

‖

D
r

x
‖

L
∞

:
x

|

E

=
f
)

\inf (\left \|{D^r}x\right \|_{{L^\infty }}:x{|_E} = f)

to have a solution. When restricted to quasi-uniform partitions E, necessary and sufficient conditions are derived for the solution to be in

L
∞

{L^\infty }

. For finite partitions E it is shown that a solution to the

L
∞

{L^\infty }

infimum problem can be obtained by solving

inf
(

‖

D
r

x
‖

L
p

:
x

|

E

=
f
)

\inf (\left \|{D^r}x\right \|_{{L^p}}:x{|_E} = f)

and letting p go to infinity. In this way it was discovered that solutions to the

L
∞

{L^\infty }

problem could be chosen to be piecewise polynomial (of degree r or less). The solutions to the

L
p

{L^p}

problem are called

H

r
,
p

{H^{r,p}}

-splines and were studied extensively by Golomb in [3].
𝜔-linear vector fields on manifolds
Authors: Perrizo William
Published: July 2010

The classical study of a flow near a fixed point is generalized by composing, at each point in the manifold, the flow derivative with a parallel translation back along the flow. Circumstances under which these compositions form a one-parameter group are studied. From the point of view of the linear frame bundle, the condition is that the canonical lift commute with its horizontal part (with respect to some metric connection). The connection form applied to the lift coincides with the infinitesimal generator of the one-parameter group. Analysis of this matrix provides dynamical information about the flow. For example, if such flows are equicontinuous, they have uniformly bounded derivatives and therefore the enveloping semigroup is a Lie transformation group. Subclasses of ergodic, minimal, and weakly mixing flows with integral invariants are determined according to the eigenvalues of the matrices. Such examples as Lie algebra flows, infinitesimal affine transformations, and the geodesic flows on manifolds of constant negative curvature are examined.
A relation between 𝐾-theory and cohomology
Authors: Thomas Alan
Published: July 2010

It is well known that for X a CW-complex,

K
(
X
)

K(X)

and

H

ev

(
X
)

{H^{{\text {ev}}}}(X)

are isomorphic modulo finite groups, although the “isomorphism” is not natural. The purpose of this paper is to improve this result for X a finite CW-complex.
Conditions for the absolute continuity of two diffusions
Authors: Orey Steven
Published: July 2010

Consider two diffusion processes on the line. For each starting point x and each finite time t, consider the measures these processes induce in the space of continuous functions on [0, t]. Necessary and sufficient conditions on the generators are found for the induced measures to be mutually absolutely continuous for each x and t. If the first process is Brownian motion, the second one must be Brownian motion with drift

b
(
x
)

b(x)

, where

b
(
x
)

b(x)

is locally in

L
2

{L_2}

and satisfies a certain growth condition at

±
∞

\pm \infty

.
“Image of a Hausdorff arc” is cyclically extensible and reducible
Authors: Cornette J. L.
Published: July 2010

It is shown that a Hausdorff continuum

S
S

is the continuous image of an arc (respectively arcwise connected) if and only if each cyclic element of

S
S

is the continuous image of an arc (respectively, arcwise connected). Also, there is given an analogue to the metric space cyclic chain approximation theorem of G. T. Whyburn which applies to locally connected Hausdorff continua.
Algebras of analytic operator valued functions
Authors: Leland Kenneth O.
Published: July 2010

This paper proves and generalizes the following characterization of the algebra

A
(
K
,
K
)

A(K,K)

of complex analytic functions on open subsets of the complex plane K into K to the case of algebras of functions on a real Euclidean space E into a real Banach algebra B. Theorem. Let

F
(
K
,
K
)

F(K,K)

be the algebra of all continuous functions on open subsets of K into K, and F a subalgebra of

F
(
K
,
K
)

F(K,K)

with nonconstant elements such that

∪

f
∈
F

{ \cup _{f \in F}}

range

f
=
K
,
F

f = K,F

is closed under uniform convergence on compact sets and domain transformations of the form

z
→

z
0

+
z
σ
,
z
,

z
0

,
σ
∈
K

z \to {z_0} + z\sigma ,z,{z_0},\sigma \in K

. Then F is

F
(
K
,
K
)

F(K,K)

or

A
(
K
,
K
)

A(K,K)

or

A
¯

(
K
,
K
)
=
{

f
¯

;
f
∈
A
(
K
,
K
)
}

\bar A(K,K) = \{ \bar f;f \in A(K,K)\}

. In the general case conditions on B are studied that insure that either F contains an embedment of

F
(
R
,
R
)

F(R,R)

and thus contains quite arbitrary continuous functions or that the elements of F are analytic and F can be expressed as a direct sum of algebras

F
1

,
…
,

F
n

{F_1}, \ldots ,{F_n}

such that for

i
=
1
,
…
,
n

i = 1, \ldots ,n

, there exist complexifications

M
i

{M_i}

of E and

N
i

{N_i}

of

∪

f
∈

F
i

{\cup _{f \in {F_i}}}

range f, such that with respect to

M
i

{M_i}

and

N
i

{N_i}

the elements of

F
i

{F_i}

are complex differentiable.
Laplace operators and the 𝔥 module structure of certain cohomology groups
Authors: Williams Floyd L.
Published: July 2010

Let

n

\mathfrak {n}

be the maximal nilpotent ideal of a Borel subalgebra of a complex semisimple Lie algebra

g

\mathfrak {g}

. Under the adjoint action

n

,

g

/

n

\mathfrak {n},\mathfrak {g}/\mathfrak {n}

, and

n

′

\mathfrak {n}’

(the dual space of

n

\mathfrak {n}

) are

n

\mathfrak {n}

modules. Laplace operators for these three modules are computed by techniques which extend those introduced by B. Kostant in [6]. The kernels of these operators are then determined and, in view of the existence of a Hodge decomposition, the detailed structure of the first degree cohomology groups of

n

\mathfrak {n}

with coefficients in

n

,

g

/

n

\mathfrak {n},\mathfrak {g}/\mathfrak {n}

, and

n

′

\mathfrak {n}’

is obtained. These cohomology groups (spaces) are described, in fact, as completely reducible modules of a Cartan subalgebra

h

\mathfrak {h}

of

g

\mathfrak {g}

.
Weighted join semilattices and transversal matroids
Authors: Brualdi Richard A.
Published: July 2010

We investigate join-semilattices in which each element is assigned a nonnegative weight in a strictly increasing way. A join-subsemilattice of a Boolean lattice is weighted by cardinality, and we give a characterization of these in terms of the notion of a spread. The collection of flats with no coloops (isthmuses) of a matroid or pregeometry, partially ordered by set-theoretic inclusion, forms a join-semilattice which is weighted by rank. For transversal matroids these join-semilattices are isomorphic to join-subsemilattices of Boolean lattices. Using a previously obtained characterization of transversal matroids and results on weighted join-semilattices, we obtain another characterization of transversal matroids. The problem of constructing a transversal matroid whose join-semilattice of flats is isomorphic to a given join-subsemilattice of a Boolean lattice is then investigated.
Locally 𝑒-fine measurable spaces
Authors: Frolík Zdeněk
Published: July 2010

Hyper-Baire sets and hyper-cozero sets in a uniform space are introduced, and it is shown that for metric-fine spaces the property “every hypercozero set is a cozero set” is equivalent to several much stronger properties like being locally e-fine (defined in §1), or having locally determined precompact part (introduced in §2). The metric-fine spaces with these additional properties form a coreflective subcategory of uniform spaces; the coreflection is explicitly described. The theory is applied to measurable uniform spaces. It is shown that measurable spaces with the additional properties mentioned above are coreflective and the coreflection is explicitly described. The two coreflections are not metrically determined.
Complementation for right ideals in generalized Hilbert algebras
Authors: Phillips John
Published: July 2010

Let

A

\mathfrak {A}

be a generalized Hilbert algebra and let

J

\mathcal {J}

be a closed right ideal of

A

\mathfrak {A}

. Let

J

⊥

{\mathcal {J}^ \bot }

denote the pre-Hilbert space orthogonal complement of

J

\mathcal {J}

in

A

\mathfrak {A}

. The problem investigated in this paper is: for which algebras

A

\mathfrak {A}

is it true that

A

=

J

⊕

J

⊥

\mathfrak {A} = \mathcal {J} \oplus {\mathcal {J}^ \bot }

for every closed right ideal

J

\mathcal {J}

of

A

\mathfrak {A}

? In the case that

A

\mathfrak {A}

is achieved, a slightly stronger property is characterized and these characterizations are then used to investigate some interesting examples.
Weak compactness in the order dual of a vector lattice
Authors: Burkinshaw Owen
Published: July 2010

A sequence

{

x
n

}

\{ {x_n}\}

in a vector lattice E will be called an l’-sequence if there exists an x in E such that

Σ

k
=
1

n

|

x
k

|

≤
x

\Sigma _{k = 1}^n|{x_k}| \leq x

for all n. Denote the order dual of E by

E
b

{E^b}

. For a set

A
⊂

E
b

A \subset {E^b}

, let

‖
⋅
‖

A
∘

{\left \| \cdot \right \|_{{A^ \circ }}}

denote the Minkowski functional on E defined by its polar

A
∘

{A^ \circ }

in E. A set

A
⊂

E
b

A \subset {E^b}

will be called equi-l’-continuous on E if

lim

‖

x
n

‖

A
∘

=
0

\lim {\left \| {{x_n}} \right \|_{{A^ \circ }}} = 0

for each l’-sequence

{

x
n

}

\{ {x_n}\}

in E. The main objective of this paper will be to characterize compactness in

E
b

{E^b}

in terms of the order structure on E and

E
b

{E^b}

. In particular, the relationship of equi-l’-continuity to compactness is studied. §2 extends to

E

σ
c

{E^{\sigma c}}

the results in Kaplan [8] on vague compactness in

E
C

{E^C}

. Then this is used to study vague convergence of sequences in

E
b

{E^b}

.
Smooth complex projective space bundles and 𝐵𝑈(𝑛)
Authors: Beem R. Paul
Published: July 2010

Smooth fiberings with complex projective and Dold manifold fibers are studied and a bordism classification for even complex projective space bundles is given. The

Z
2

{Z_2}

-cohomology of

B

U
~

(
n
)

B\tilde U(n)

is computed with its Steenrod algebra action.
On homeomorphisms of infinite dimensional bundles. III
Authors: Chapman T. A. , Wong R. Y. T.
Published: July 2010

In this paper we continue the study of homeomorphisms and prove an analogue of the homeomorphism extension theorem for bundles modeled on Hilbert cube manifolds; thus we generalize previous results for Q-manifolds (Anderson-Chapman). This analogy, as in the case of manifolds, requires a consideration of proper maps and proper homotopies. The approach to the present problem is similar to that considered in our previous papers. Bear in mind several distinct difficulties occur in our setting.
A finitely additive generalization of the Fichtenholz-Lichtenstein theorem
Authors: Sinclair George Edward
Published: July 2010

Let

μ
\mu

and

ν
\nu

be bounded, finitely additive measures on algebras over sets X and Y, respectively. Conditions are determined for a bounded function

f
:
X
×
Y
→

R

f:X \times Y \to {\mathbf {R}}

, without assuming bimeasurability, so that the iterated integrals

∫
X

∫
Y

f
d
μ
d
μ

\smallint _X {\smallint _Y {fd\mu d\mu } }

and

∫
Y

∫
X

f
d
μ
d
ν

\smallint _Y {\smallint _X {fd\mu d\nu } }

exist and are equal. This result is then used to construct a product algebra and finitely additive product measure for

μ
\mu

and

ν
\nu

. Finally, a simple Fubini theorem with respect to this product algebra and product measure is established.
On centralizers of generalized uniform subgroups of locally compact groups
Authors: Sit Kwan-Yuk Law
Published: July 2010

Let

G
G

be a locally compact group and

H
H

a closed subgroup of

G
G

such that the homogeneous space

G

/

H

G/H

admits a finite invariant measure. Let

Z
G

(
H
)

{Z_G}(H)

be the centralizer of

H
H

in

G
G

. It is shown that if

G
G

is connected then

Z
G

(
H
)

{Z_G}(H)

modulo its center is compact. If

G
G

is only assumed to be locally connected it is shown that the commutator subgroup of

Z
G

(
H
)

{Z_G}(H)

has compact closure. Consequences of these results are found for special classes of groups, such as Lie groups. An example of a totally disconnected group

G
G

is given to show that the results for

Z
G

(
H
)

{Z_G}(H)

need not hold if

G
G

is not connected or locally connected.
The structure of nested spaces
Authors: Muenzenberger T. B. , Smithson R. E.
Published: July 2010

The structure of nested spaces is studied in this paper using such tools as branches, chains, partial orders, and rays in the context of semitrees. A classification scheme for various kinds of acyclic spaces is delineated in terms of semitrees. Several families of order compatible topologies for semitrees are investigated, and these families are grouped in a spectrum (inclusion chain) of topologies compatible with the semitree structure. The chain, interval, and tree topologies are scrutinized in some detail. Several topological characterizations of semitrees with certain order compatible topologies are also derived.
Blanchfield duality and simple knots
Authors: Kearton C.
Published: July 2010

The method of presentation for

n
n

-knots is used to classify simple

(
2
q
−
1
)

(2q - 1)

-knots,

q
>
3

q > 3

, in terms of the Blanchfield duality pairing. As a corollary, we characterize the homology modules and pairings which can arise from classical knots.
Weakly starlike meromorphic univalent functions
Authors: Libera Richard J. , Livingston Albert E.
Published: July 2010

A weakly starlike meromorphic univalent function is one of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis equals minus rho z g left-parenthesis z right-parenthesis left-bracket left-parenthesis z minus rho right-parenthesis left-parenthesis 1 minus rho z right-parenthesis right-bracket Superscript negative 1">

f
(
z
)
=
−
ρ
z
g
(
z
)

[
(
z
−
ρ
)
(
1
−
ρ
z
)

]

−
1

f(z) = - \rho zg(z){[(z - \rho )(1 - \rho z)]^{ - 1}}

for

0
>
ρ
>
1

0 > \rho > 1

and

g
(
z
)

g(z)

a meromorphic starlike function. The behavior of coefficients and growth of this class of functions and of a subset are studied.
Asymptotic values of modulus 1 of Blaschke products
Authors: Leung K. K. , Linden C. N.
Published: July 2010

A sufficient condition is found for each subproduct of a Blaschke product to have an asymptotic value of modulus 1 along a prescribed arc of a specified type in the unit disc. The condition obtained is found to be necessary in the case of further restrictions of the arc, and the two results give rise to a necessary and sufficient condition for the existence of

T
γ

{T_\gamma }

-limits of modulus 1 for Blaschke products.
Decomposable braids as subgroups of braid groups
Authors: Levinson H.
Published: July 2010

The group of all decomposable

3
3

-braids is the commutator subgroup of the group

I
3

{I_3}

of all

3
3

-braids which leave strand positions invariant. The group of all

2
2

-decomposable

4
4

-braids is the commutator subgroup of

I
4

{I_4}

, and the group of all decomposable

4
4

-braids is explicitly characterized as a subgroup of the second commutator subgroup of

I
4

{I_4}

.
Corrections and additions to: “On the degrees and rationality of certain characters of finite Chevalley groups” (Trans. Amer. Math. Soc. 165 (1972), 251–273) by C. T. Benson and Curtis
Authors: Curtis C. W.
Published: July 2010

Some theorems in the Benson-Curtis paper [1] were stated subject to possible exceptions in type

E
7

{E_7}

, corresponding to the two irreducible characters of the Weyl group of degree 512. An argument due to T. A. Springer(

1

^{1}

) shows that these cases actually are exceptions to the theorems, and also that there are four exceptional cases in type

E
8

{E_8}

(whose possible existence was overlooked in the original version of the paper), corresponding to the characters of the Weyl group of degree 4096.
Two model theoretic proofs of Rückert’s Nullstellensatz
Authors: Weispfenning Volker
Published: July 2010

Rückert’s Nullstellensatz for germs of analytic functions and its analogue for germs of real analytic functions are proved by a combination of nonstandard analysis with a model theoretic transfer principle. It is also shown that Rückert’s Nullstellensatz is constructive essentially relative to the Weierstrass preparation theorem.
A stability theorem for minimum edge graphs with given abstract automorphism group
Authors: McCarthy Donald J. , Quintas Louis V.
Published: July 2010

Given a finite abstract group

G

\mathcal {G}

, whenever

n
n

is sufficiently large there exist graphs with

n
n

vertices and automorphism group isomorphic to

G

\mathcal {G}

. Let

(

G

,
n
)

(\mathcal {G},n)

denote the minimum number of edges possible in such a graph. It is shown that for each

G

\mathcal {G}

there always exists a graph

M
M

such that for

n
n

sufficiently large,

e
(

G

,
n
)

e(\mathcal {G},n)

is attained by adding to

M
M

a standard maximal component asymmetric forest. A characterization of the graph

M
M

is given, a formula for

e
(

G

,
n
)

e(\mathcal {G},n)

is obtained (for large

n
n

), and the minimum edge problem is re-examined in the light of these results.
On the analytic continuation of the Minakshisundaram-Pleijel zeta function for compact Riemann surfaces
Authors: Randol Burton
Published: July 2010

A formula is derived for the Minakshisundaram-Pleijel zeta function in the half-plane Re

s
>
0

s > 0

.