bluuebunny/crossref_metadata_2025 huggingface dataset:
Topological equivalence of gradient vectorfields
Authors: Shafer Douglas S.
Published: June 2010

This paper is a study of the behavior of the topological equivalence class of the planar gradient vectorfield

X

=

grad
g

V

X\, = \,{\operatorname {grad} _g}\,V

, in a neighborhood of a degenerate singularity of V, as g varies over all Riemannian metrics. It is shown that under simple restrictions on V the topological equivalence class of X is determined by its first nonvanishing jet, and that only finitely many equivalence classes occur (for fixed V). In this case, when the degree of the first nonvanishing jet of V is less than five, necessary and sufficient conditions for change in equivalence class are given, both in terms of the coefficients of the homogeneous part of V and geometrically in terms of its level curves. A catalogue of possible phase portraits, up to topological equivalence, is included. Necessary conditions are given for change in higher degree.
The number of groups of a given genus
Authors: Tucker T. W.
Published: June 2010

It is shown that the number of groups with a given genus greater than one is finite. The proof depends heavily on V. K. Proulx’s classification of groups of genus one. The key observation is that as the number of vertices of a graph imbedded on a given surface increases, the average face size of the imbedding approaches the average face size of a toroidal imbedding. The result appears to be related to Hurwitz’s theorem bounding the order of a group of conformal automorphisms on a Riemann surface of genus g.
Basic sequences in non-Schwartz Fréchet spaces
Authors: Bellenot Steven F.
Published: June 2010

Obliquely normalized basic sequences are defined and used to characterize non-Schwartz-Fréchet spaces. It follows that each non-Schwartz-Fréchet space E has a non-Schwartz subspace with a basis and a quotient which is not Montel (which has a normalized basis if E is separable). Stronger results are given when more is known about E, for example, if E is a subspace of a Fréchet

l
p

{l_p}

-Köthe sequence space, then E has the Banach space

l
p

{l_p}

as a quotient and E has a subspace isomorphic to a non-Schwartz

l
p

{l_p}

-Köthe sequence space. Examples of Fréchet-Montel spaces which are not subspaces of any Fréchet space with an unconditional basis are given. The question of the existence of conditional basic sequences in non-Schwartz-Fréchet spaces is reduced to questions about Banach spaces with symmetric bases. Nonstandard analysis is used in some of the proofs and a new nonstandard characterization of Schwartz spaces is given.
Cosmoi of internal categories
Authors: Street Ross
Published: June 2010

An internal full subcategory of a cartesian closed category

A

\mathcal {A}

, is shown to give rise to a structure on the 2-category

C
a
t
(

A

)

Cat(\mathcal {A})

of categories in

A

\mathcal {A}

which introduces the notion of size into the analysis of categories in

A

\mathcal {A}

and allows proofs by transcendental arguments. The relationship to the currently popular study of locally internal categories is examined. Internal full subcategories of locally presentable categories (in the sense of Gabriel-Ulmer) are studied in detail. An algorithm is developed for their construction and this is applied to the categories of double categories, triple categories, and so on.
Scattering theory and polynomials orthogonal on the real line
Authors: Geronimo J. S. , Case K. M.
Published: June 2010

The techniques of scattering theory are used to study polynomials orthogonal on a segment of the real line. Instead of applying these techniques to the usual three-term recurrence formula, we derive a set of two two-term recurrence formulas satisfied by these polynomials. One of the advantages of these new recurrence formulas is that the Jost function is related, in the limit as

n

→

∞

n\, \to \,\infty

, to the solution of one of the recurrence formulas with the boundary conditions given at

n

=

0

n\, = \,0

. In this paper we investigate the properties of the Jost function and the spectral function assuming the coefficients in the recurrence formulas converge at a particular rate.
Twisted Lubin-Tate formal group laws, ramified Witt vectors and (ramified) Artin-Hasse exponentials
Authors: Hazewinkel Michiel
Published: June 2010

For any ring

R
R

let

Λ
(
R
)

\Lambda (R)

denote the multiplicative group of power series of the form

1
+

a
1

t
+
⋯

1 + {a_1}t + \cdots

with coefficients in

R
R

. The Artin-Hasse exponential mappings are homomorphisms
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Subscript p comma normal infinity Baseline left-parenthesis k right-parenthesis right-arrow normal upper Lamda left-parenthesis upper W Subscript p comma normal infinity Baseline left-parenthesis k right-parenthesis right-parenthesis">

W

p
,
∞

(
k
)
→
Λ
(

W

p
,
∞

(
k
)
)

W_{p, \infty } (k) \to \Lambda ({W_{p, \infty }}(k))

, which satisfy certain additional properties. Somewhat reformulated, the Artin-Hasse exponentials turn out to be special cases of a functorial ring homomorphism
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E colon upper W Subscript p comma normal infinity Baseline left-parenthesis minus right-parenthesis right-arrow upper W Subscript p comma normal infinity Baseline left-parenthesis upper W Subscript p comma normal infinity Baseline left-parenthesis minus right-parenthesis right-parenthesis">

E
:

W

p
,
∞

(
−
)
→

W

p
,
∞

(

W

p
,
∞

(
−
)
)

E: {W_{p, \infty }}( - ) \to {W_{p,\infty }}({W_{p,\infty }}( - ))

, where

W

p
,
∞

{W_{p,\infty }}

is the functor of infinite-length Witt vectors associated to the prime

p
p

. In this paper we present ramified versions of both

W

p
,
∞

(
−
)

{W_{p,\infty }}( - )

and

E
E

, with

W

p
,
∞

(
−
)

{W_{p,\infty }}( - )

replaced by a functor

W

q
,
∞

F

(
−
)

W_{q,\infty }^F( - )

, which is essentially the functor of

q
q

-typical curves in a (twisted) Lubin-Tate formal group law over

A
A

, where

A
A

is a discrete valuation ring that admits a Frobenius-like endomorphism

σ
\sigma

(we require

σ
(
a
)
≡

a
q

mod

m

\sigma (a) \equiv {a^q} \bmod \mathfrak {m}

for all

a
∈
A

a \in A

, where

m

\mathfrak {m}

is the maximal idea of

A
A

). These ramified-Witt-vector functors

W

q
,
∞

F

(
−
)

W_{q,\infty }^F( - )

do indeed have the property that, if

k
=
A

/

m

k = A/\mathfrak {m}

is perfect,

A
A

is complete, and

l

/

k

l/k

is a finite extension of

k
k

, then

W

q
,
∞

F

(
l
)

W_{q,\infty }^F(l)

is the ring of integers of the unique unramified extension

L

/

K

L/K

covering

l

/

k

l/k

.
Convergence and Cauchy structures on lattice ordered groups
Authors: Ball Richard N.
Published: June 2010

This paper employs the machinery of convergence and Cauchy structures in the task of obtaining completion results for lattice ordered groups. §§1 and 2 concern l-convergence and l-Cauchy structures in general. §4 takes up the order convergence structure; the resulting completion is shown to be the Dedekind-MacNeille completion. §5 concerns the polar convergence structure; the corresponding completion has the property of lateral completeness, among others. A simple theory of subset types routinizes the adjoining of suprema in §3. This procedure, nevertheless, is shown to be sufficiently general to prove the existence and uniqueness of both the Dedekind-MacNeille completion in §4 and the lateral completion in §5. A proof of the existence and uniqueness of a proper class of similar completions comes free. The principal new hull obtained by the techniques of adjoining suprema is the type

Y

\mathcal {Y}

hull, strictly larger than the lateral completion in general.
Riemann surfaces and bounded holomorphic functions
Authors: Pranger Walter
Published: June 2010

The principal result of this article asserts the equivalence of the following four conditions on a hyperbolic Riemann surface X: (a) the following set

z

|

|

f
(
z
)

|

⩽

sup

|

f

|

z|\,|f(z)|\, \leqslant \,{\text {sup}}\,|f|

on K for every bounded holomorphic section f of

ξ
\xi

is compact for every unitary vector bundle

ξ
\xi

and every compact set K; (b) every unitary line bundle has nontrivial bounded holomorphic sections and the condition in (a) holds for

ξ

=

i
d

\xi \, = \,{i_d}

; (c) every unitary line bundle has nontrivial bounded holomorphic sections and X is regular for potential theory; (d) every unitary line bundle has nontrivial bounded holomorphic sections and X is its own B-envelope of holomorphy. If X is a subset of C, these are also equivalent to the following: (e) for every unitary line bundle

ξ
\xi

the bounded holomorphic sections are dense in the holomorphic sections.
Central Fourier-Stieltjes transforms with an isolated value
Authors: Armstrong Alan
Published: June 2010

Let

μ
\mu

be a central Borel measure on a compact, connected group G. If 0 is isolated in the range of

μ
^

{\hat \mu }

, then there exists a closed, normal subgroup H of G such that

π
H

μ

{\pi _H}\mu

, the restriction of

μ
\mu

to the cosets of H, is the convolution of an invertible measure with a nonzero idempotent measure. This result extends I. Glicksberg’s result for LCA groups. An example is given which shows that this result is false in general for disconnected groups.
Locally free affine group actions
Authors: Plante J. F.
Published: June 2010

Differentiable actions by the nonabelian 2-dimensional Lie group on compact manifolds are considered. When the action is locally free and the orbits have codimension one it is shown that there are at most finitely many minimal sets each containing a countably infinite number of cylindrical orbits. Examples are given to show that various codimension, differentiability, and minimality restrictions are necessary.
Branched extensions of curves in compact surfaces
Authors: Ezell Cloyd L.
Published: June 2010

A polymersion is a map

F
:

M

→

N

F:\,M\, \to \,N

where M and N are compact surfaces, orientable or nonorientable, M a surface with boundary, where (a) At each interior point of M, there is an integer

n

⩾

1

n\, \geqslant \,1

such that F is topologically equivalent to the complex map

z
n

{z^n}

in a neighborhood about the point. (b) At each point x in the boundary of M,

δ
M

\delta M

, there is a neighborhood U containing x such that U is homeomorphic to F(U). A normal polymersion is one where

F
(
δ
M
)

F(\delta M)

is a normal set of curves in N. We are concerned with establishing a combinatorial representation for normal polymersions which map to arbitrary compact surfaces.
On the topology of simply connected algebraic surfaces
Authors: Mandelbaum Richard , Moishezon Boris
Published: June 2010

Suppose x is a smooth simply-connected compact 4-manifold. Let

p

=

C

P
2

p\, = \,{\textbf {C}}{P^2}

and

Q

=

−

C

P
2

Q\, = \, - {\textbf {C}}{P^2}

be the complex projective plane with orientation opposite to the usual. We shall say that X is completely decomposable if there exist integers a, b such that X is diffeomorphic to

a
P

\# 

b
Q

aP\,{\text {\# }}\,bQ

. By a result of Wall [W1] there always exists an integer k such that

X

#

(
k

+

1
)
P

#
k
Q

X\,\# \,(k\, + \,1)P\,\# kQ

is completely decomposable. If

X

#

P

X\,\# \,P

is completely decomposable we shall say that X is almost completely decomposable. In [MM] we demonstrated that any nonsingular hypersurface of

C

P
3

{\textbf {C}}{P^3}

is almost completely decomposable. In this paper we generalize this result in two directions as follows: Theorem 3.5. Suppose W is a simply-connected nonsingular complex projective 3-fold. Then there exists an integer

m
0

⩾

1

{m_0}\, \geqslant \,1

such that any hypersurface section

V
m

{V_m}

of W of degree

m

⩾

m
0

m\, \geqslant \,{m_0}

which is nonsingular will be almost completely decomposable. Theorem 5.3. Let V be a nonsingular complex algebraic surface which is a complete intersection. Then V is almost completely decomposable.
Univalence criteria in multiply-connected domains
Authors: Osgood Brad G.
Published: June 2010

Theorems due to Nehari and Ahlfors give sufficient conditions for the univalence of an analytic function in relation to the growth of its Schwarzian derivative. Nehari’s theorem is for the unit disc and was generalized by Ahlfors to any simply-connected domain bounded by a quasiconformal circle. In both cases the growth is measured in terms of the hyperbolic metric of the domain. In this paper we prove a corresponding theorem for finitely-connected domains bounded by points and quasiconformal circles. Metrics other than the hyperbolic metric are also considered and similar results are obtained.
A stable converse to the Vietoris-Smale theorem with applications to shape theory
Authors: Ferry Steve
Published: June 2010

Our main result says that if

f
:

X

→

Y

f:\,X\, \to \,Y

is a map between finite polyhedra which has k-connected homotopy fiber, then there is an n such that

f

×

id:

X

×

I
n

→
Y

f\, \times \,{\text {id:}}\,X\, \times \,{I^n} \to Y

is homotopic to a map with k-connected point-inverses. This result is applied to give an algebraic characterization of compacta shape equivalent to locally n-connected compacta. We also show that a

U

V
1

U{V^1}

compactum can be “improved” within its shape class until its homotopy theory and strong shape theory are the same with respect to finite dimensional polyhedra.
Countable ultrahomogeneous undirected graphs
Authors: Lachlan A. H. , Woodrow Robert E.
Published: June 2010

Let

G

=

⟨

V
G

,

E
G

⟩

G\, = \,\left \langle {{V_G},\,{E_G}} \right \rangle

be an undirected graph. The complementary graph

G
~

\tilde G

is

⟨

V
G

,

E

G
~

⟩

\left \langle {{V_G},\,{E_{\tilde G}}} \right \rangle

where

(

V
1

,

V
2

)

∈

E

G
~

({V_1},\,{V_2})\, \in \,{E_{\tilde G}}

iff

V
1

≠

V
2

{V_1}\, \ne \,{V_2}

and

(

V
1

,

V
2

)

∉

E
G

({V_1},\,{V_2})\, \notin \,{E_G}

. Let

K
(
n
)

K(n)

be the complete undirected graph on n vertices and let E be the graph [ill] i.e.

⟨

{
a
,

b
,

c
}
,

{
(
b
,

c
)
,

(
c
,

b
)
}

⟩

\left \langle {\{ a,\,b,\,c\} ,\,\{ (b,\,c),\,(c,\,b)\} } \right \rangle

. G is ultrahomogeneous just in case every isomorphism of subgraph of smaller cardinality can be lifted to an automorphism of G. Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper D equals StartSet upper K left-parenthesis n right-parenthesis colon n element-of omega EndSet union StartSet upper E comma upper E overTilde EndSet union StartSet ModifyingAbove upper K With tilde left-parenthesis n right-parenthesis colon n element-of omega EndSet">

D

=

{
K
(
n
)
:

n

∈

ω
}

∪

{
E
,

E
~

}

∪

{

K
~

(
n
)
:

n

∈

ω
}

\mathcal {D}\, = \,\{ K(n):\,n\, \in \,\omega \} \, \cup \,\{ E,\,\tilde E\} \, \cup \,\{ \tilde K(n):\,n\, \in \,\omega \}

. Theorem: Let

G
1

{G_1}

,

G
2

{G_2}

be two countable (infinite) ultrahomogeneous graphs such that for each

H

∈

D

H

H\, \in \,\mathcal {D}\,H

can be embedded in

G
1

{G_1}

, just in case it can be embedded in

G
2

{G_2}

. Then

G
1

≅

G
2

{G_1}\, \cong \,{G_2}

. Corollary: There are a countable number of countable ultrahomogeneous (undirected) graphs.
Preservation of closure in a locally convex space. I
Authors: Brodsky I.
Published: June 2010

This paper is concerned with the lifting of the closures of sets. If H is a topological vector space, G a subspace and A closed in G for the induced topology, under what conditions on A in G is it true that the closure of A is preserved in H, i.e., A is closed in H? In this paper a fundamental lifting proposition is proved. ’Preservation of closure’ will prove to be a fruitful technique in obtaining some interesting results in the theory of locally convex spaces. Using this technique, we will first show when closure is equivalent to completeness. Then we will prove a generalization to locally convex spaces of the classical Heine-Borel Theorem for Euclidean n-space. Generalizing a result of Petunin, we will also give some necessary and sufficient conditions on semireflexivity. Finally, we will give a necessary and sufficient condition for the sum of two closed subspaces to be closed.
Nonexistence of nontrivial 𝑐𝑚”-harmonic 1-forms on a complete foliated Riemannian manifold
Authors: Kitahara Haruo
Published: June 2010

We study the nonexistence of nontrivial

◻

\square

-harmonic 1-forms on a complete foliated riemannian manifold with positive definite Ricci curvature. It is well known that the harmonic 1-form on a compact and orientable riemannian manifold with positive definite Ricci curvature is trivial. Our main theorem is an extension of this fact in the complete foliated riemannian case.
Weak subordination and stable classes of meromorphic functions
Authors: Stephenson Kenneth
Published: June 2010

This paper introduces the notion of weak subordination: If

F
F

and

G
G

are meromorphic in the unit disc

U

\mathcal {U}

, then

F
F

is weakly subordinate to

G
G

, written

F
>
G

F > G

, provided there exist analytic functions

ϕ
\phi

and

ω
:

U

→

U

\omega :\mathcal {U} \to \mathcal {U}

, with

ϕ
\phi

an inner function, so that

F
∘
ϕ
=
G
∘
ω

F \circ \phi = G \circ \omega

. A class

X

\mathcal {X}

of meromorphic functions is termed stable if

F

≺

w

G

F \stackrel {w}{\prec } G

and

G
∈

X

⇒
F
∈

X

G \in \mathcal {X} \Rightarrow F \in \mathcal {X}

. The motivation is recent work of Burkholder which relates the growth of a function with its range and boundary values. Assume

F
F

and

G
G

are meromorphic and

G
G

has nontangential limits, a.e. Assume further that

F
(

U

)
∩
G
(

U

)
≠
∅

F(\mathcal {U}) \cap G(\mathcal {U}) \ne \emptyset

and

G
(

e

i
θ

)
∉
F
(

U

)

G({e^{i\theta }}) \notin F(\mathcal {U})

, a.e. This is denoted by

F
>
G

F > G

. Burkholder proved for several classes

X

\mathcal {X}

that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis left-parenthesis asterisk right-parenthesis right-parenthesis EndLabel upper F greater-than upper G and upper G element-of script upper X right double arrow upper F element-of script upper X period EndLayout">

(

(
∗
)

)

F
>
G

and

G
∈

X

⇒
F
∈

X

.

\begin{equation}\tag {$(\ast )$}F > G \qquad {\text {and}}\quad G \in \mathcal {X} \Rightarrow F \in \mathcal {X}.\end{equation}

The main result of this paper is the Theorem:

F
>
G
⇒
F

≺
w

G

F > G \Rightarrow F{ \prec ^w}G

. In particular, implication (*) holds for all stable classes

X

\mathcal {X}

. The paper goes on to study various stable classes, which include BMOA,

H
p

{H^p}

,

0
>
p
⩽
∞

0 > p \leqslant \infty

,

N

∗

{N_{\ast }}

, the space of functions of bounded characteristic, and the

M
Φ

{M^\Phi }

spaces introduced by Burkholder. VMOA and the Bloch functions are examples of classes which are not stable.
A maximal function characterization of 𝐻^{𝑝} on the space of homogeneous type
Authors: Uchiyama Akihito
Published: June 2010

Let

ψ
0

(
x
)

∈

S
(

R
n

)

{\psi _0}(x)\, \in \,{\mathcal {S(}}{R^n}{\text {)}}

and let

∫

R
n

ψ
0

(
y
)

d
y

≠

0

\int _{{R^n}} {{\psi _0}(y)\,dy\, \ne \,0}

.For

f

∈

S

′

R
n

f\, \in \,{\mathcal {S}’}{R^n}

,

x

∈

R
n

x\, \in \,{R^n}

and

M

⩾

0

M\, \geqslant \,0

, let
\[

f
+

(
x
)

=

sup

t

>

0

|

f

∗

ψ

0
t

(
x
)

|

{f^ + }(x)\, = \,\sup \limits _{t\, > \,0} \,\left | {f\,{\ast }\,{\psi _{0t}}(x)} \right |

\]
and let

f

∗

M

(
x
)

=

sup
{

|

f

∗

ψ
t

(
x
)

|

:

t

>

0

{f^{{\ast }M}}(x)\, = \,\sup \{ \left | {f\,{\ast }\,{\psi _t}(x)} \right |:\,t\, > \,0

,

ψ
(
y
)

∈

S
(

R
n

)

\psi (y)\, \in \,{\mathcal {S(}}{R^n})

,

supp

ψ

⊂

{
y

∈

R
n

:

|
y
|

>

1
}

\operatorname {supp} \,\psi \, \subset \,\{ y\, \in \,{R^n}:\,\left | y \right |\, > \,1\}

,

‖

D
α

ψ

‖

L
∞

⩽

1

{\left \| {{D^\alpha }\psi } \right \|_{{L^\infty }}}\, \leqslant \,1

for any multi-index

α

=

(

α
1

,

…
,

α
n

)

\alpha \, = \,({\alpha _1},\, \ldots ,\,{\alpha _n})

such that

Σ

i
=
1

n

α
i

⩽

M
}

\Sigma _{i = 1}^n\,{\alpha _i}\, \leqslant \,M\}

where

ψ
t

(
y
)

=

t

−
n

ψ
(
y

/

t
)

{\psi _t}(y)\, = \,{t^{ - n}}\psi (y/t)

. Fefferman-Stein [11] showed Theorem A. Let

p

>

0

p\, > \,0

. Then there exists

M
(
p
,

n
)

M(p,\,n)

, depending only on p and n, such that if

M

⩾

M
(
p
,

n
)

M\, \geqslant \,M(p,\,n)

, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c double-vertical-bar f Superscript plus Baseline double-vertical-bar upper L Superscript p Baseline less-than-or-slanted-equals double-vertical-bar f Superscript asterisk upper M Baseline double-vertical-bar upper L Superscript p Baseline less-than-or-slanted-equals bold upper C double-vertical-bar f Superscript plus Baseline double-vertical-bar upper L Superscript p">

c

‖

f
+

‖

L
p

⩽

‖

f

∗

M

‖

L
p

⩽

C

‖

f
+

‖

L
p

c\left \| {{f^ + }} \right \|{L^p}\, \leqslant \,\left \| {{f^{{\ast }M}}} \right \|{L^p}\, \leqslant \,{\textbf {C}}\left \| {{f^ + }} \right \|{L^p}

\]
for any

f

∈

S

′

(

R
n

)

f\, \in \,{\mathcal {S}’}({R^n})

, where c and C are positive constants depending only on

ψ
0

{\psi _0}

, p, M and n. We investigate this on the space of homogeneous type with certain assumptions.
A partition theorem for the infinite subtrees of a tree
Authors: Milliken Keith R.
Published: June 2010

We prove a generalization for infinite trees of Silver’s partition theorem. This theorem implies a version for trees of the Nash-Williams partition theorem.
Fine convergence and admissible convergence for symmetric spaces of rank one
Authors: Korányi Adam , Taylor J. C.
Published: June 2010

The connections between fine convergence in the sense of potential theory and admissible convergence to the boundary for quotients of eigenfunctions of the Laplace-Beltrami operator are investigated. This leads to a version of the local Fatou theorem on symmetric spaces of rank one which is considerably stronger than previously known results. The appendix establishes the relationship between harmonic measures on the intersection of the Martin boundaries of a domain and a subdomain.
Invariance of solutions to invariant nonparametric variational problems
Authors: Brothers John E.
Published: June 2010

Let

f
f

be a weak solution to the Euler-Lagrange equation of a convex nonparametric variational integral in a bounded open subset

D
D

of

R

n

{{\mathbf {R}}^n}

. Assume the boundary

B
B

of

D
D

to be rectifiable. Let

D
D

be a compact connected Lie group of diffeomorphisms of a neighborhood of

D
∪
B

D \cup B

which leave

D
D

invariant and assume the variational integral to be

G
G

-invariant. Conditions are formulated which imply that if

f
f

is continuous on

D
∪
B

D \cup B

and

f
∘
g

|

B
=
f

|

B

f \circ g|B = f|B

for

g
∈
G

g \in G

then

f
∘
g
=
f

f \circ g = f

for every

g
∈
G

g \in G

. If the integrand

L
L

is strictly convex then

f
f

can be shown to have a local uniqueness property which implies invariance. In case

L
L

is not strictly convex the graph

T
f

{T_f}

of

f
f

in

R

n

×

R

{{\mathbf {R}}^n} \times {\mathbf {R}}

is interpreted as the solution to an invariant parametric variational problem, and invariance of

T
f

{T_f}

, hence of

f
f

, follows from previous results of the author. For this purpose a characterization is obtained of those nonparametric integrands on

R

n

{{\mathbf {R}}^n}

which correspond to a convex positive even parametric integrand on

R

n

×

R

{{\mathbf {R}}^n} \times {\mathbf {R}}

in the same way that the nonparametric area integrand corresponds to the parametric area integrand.
The cohomology algebras of finite-dimensional Hopf algebras
Authors: Wilkerson Clarence
Published: June 2010

The cohomology algebra of a finite dimensional graded connected cocommutative biassociative Hopf algebra over a field

K
K

is shown to be a finitely generated

K
K

-algebra. Counterexamples to the analogue of a result of Quillen (that nonnilpotent cohomology classes should have nonzero restriction to some abelian sub-Hopf algebra) are constructed, but an elementary proof of the validity of this "detection principle" for the special case of finite sub-Hopf algebras of the

mod
⁡
2

\operatorname {mod} 2

Steenrod algebra is given. As an application, an explicit formula for the Krull dimension of the cohomology algebras of the finite skeletons of the

mod
⁡
2

\operatorname {mod} 2

Steenrod algebra is given.
Integral geometric properties of capacities
Authors: Mattila Pertti
Published: June 2010

Let

m
m

and

n
n

be positive integers,

0
>
m
>
n

0 > m > n

, and

C
K

{C_K}

and

C
H

{C_H}

the usual potential-theoretic capacities on

R
n

{R^n}

corresponding to lower semicontinuous kernels

K
K

and

H
H

on

R
n

×

R
n

{R^n} \times {R^n}

with

H
(
x
,
y
)
=
K
(
x
,
y
)

|

x
−
y

|

n
−
m

⩾
1

H(x,y) = K(x,y){\left | {x - y} \right |^{n - m}} \geqslant 1

for

|

x
−
y

|

⩽
1

\left | {x - y} \right | \leqslant 1

. We consider relations between the capacities

C
K

(
E
)

{C_K}(E)

and

C
H

(
E
∩
A
)

{C_H}(E \cap A)

when

E
⊂

R
n

E \subset {R^n}

and

A
A

varies over the

m
m

-dimensional affine subspaces of

R
n

{R^n}

. For example, we prove that if

E
E

is compact,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript upper K Baseline left-parenthesis upper E right-parenthesis less-than-or-slanted-equals c integral upper C Subscript upper H Baseline left-parenthesis upper E intersection upper A right-parenthesis d lamda Subscript n comma m Baseline upper A">

C
K

(
E
)
⩽
c
∫

C
H

(
E
∩
A
)
d

λ

n
,
m

A

{C_K}(E) \leqslant c\smallint {C_H}(E \cap A)d{\lambda _{n,m}}A

where

λ

n
,
m

{\lambda _{n,m}}

is a rigidly invariant measure and

c
c

is a positive constant depending only on

n
n

and

m
m

.
Manifolds of nonanalyticity of solutions of certain linear PDEs
Authors: Zachmanoglou E. C.
Published: June 2010

It is shown that if

P
P

is a linear partial differential operator with analytic coefficients, and if

M
M

is an analytic manifold of codimension

3
3

which is partially characteristic with respect to

P
P

and satisfies certain additional conditions, then one can find, in a neighborhood of any point of

M
M

, solutions of the equation

P
u
=
0

Pu = 0

which are flat or singular precisely on

M
M

.
A relative Nash theorem
Authors: Akbulut Selman , King Henry C.
Published: June 2010

We prove that if

M
M

is a closed smooth manifold and

M
i

{M_i}

,

i
=
1
,
…
,
k

i = 1, \ldots ,k

, are transversally intersecting closed smooth submanifolds of

M
M

, then there exist a nonsingular algebraic set

Z
Z

and nonsingular algebraic subsets

Z
i

{Z_i}

,

i
=
1
,
…
,
k

i = 1, \ldots ,k

, of

Z
Z

such that

(
M
;

M
1

,
…
,

M
k

)

(M;{M_1}, \ldots ,{M_k})

is diffeomorphic to

(
Z
;

Z
1

,
…
,

Z
k

)

(Z;{Z_1}, \ldots ,{Z_k})

. We discuss a generalization and the consequences of this result.
Probability and interpolation
Authors: Lorentz G. G. , Lorentz R. A.
Published: June 2010

An

m
×
n

m \times n

matrix

E
E

with

n
n

ones and

(
m
−
1
)
n

(m - 1)n

zeros, which satisfies the Pólya condition, may be regular and singular for Birkhoff interpolation. We prove that for random distributed ones,

E
E

is singular with probability that converges to one if

m
m

,

n
→
∞

n \to \infty

. Previously, this was known only if

m
⩾
(
1
+
δ
)
n

/

log
⁡
n

m \geqslant (1 + \delta )n/\log n

. For constant

m
m

and

n
→
∞

n \to \infty

, the probability is asymptotically at least

1
2

\tfrac {1} {2}

.
Structure theory for a class of grade four Gorenstein ideals
Authors: Kustin Andrew , Miller Matthew
Published: June 2010

An ideal

I
I

in a commutative noetherian ring

R
R

is a Gorenstein ideal of

grade
⁡
g

\operatorname {grade} g

if

pd
R

(
R

/

I
)
=
grade

I
=
g

{\operatorname {pd} _R}(R / I) = \operatorname {grade} \,I = g

and the canonical module

Ext
R
g

⁡
(
R

/

I
,

R
)

\operatorname {Ext} _R^g(R / I,\,R)

is cyclic. Serre showed that if

g
=
2

g = 2

then

I
I

is a complete intersection, and Buchsbaum and Eisenbud proved a structure theorem for the case

g
=
3

g = 3

. We present generic resolutions for a class of Gorenstein ideals of

grade
⁡
4

\operatorname {grade} 4

, and we illustrate the structure of the resolution with various specializations. Among these examples there are Gorenstein ideals of

grade

4

\operatorname {grade} \,4

in

k
[
[
x
,

y
,

z
,

v
]
]

k[[x,\,y,\,z,\,v]]

that are

n
n

-generated for any odd integer

n
⩾
7

n \geqslant 7

. We construct other examples from almost complete intersections of

grade

3

\operatorname {grade} \,3

and their canonical modules. In the generic case the ideals are shown to be normal primes. Finally, we conclude by giving an explicit associative algebra structure for the resolutions. It is this algebra structure that we use to classify the different Gorenstein ideals of

grade

4

\operatorname {grade} \,4

, and which may be the key to a complete structure theorem.
Differentiability of the metric projection in Hilbert space
Authors: Fitzpatrick Simon , Phelps R. R.
Published: June 2010

A study is made of differentiability of the metric projection

P
P

onto a closed convex subset

K
K

of a Hilbert space

H
H

. When

K
K

has nonempty interior, the Gateaux or Fréchet smoothness of its boundary can be related with some precision to Gateaux or Fréchet differentiability properties of

P
P

. For instance, combining results in

§
3

\S 3

with earlier work of R. D. Holmes shows that

K
K

has a

C
2

{C^2}

boundary if and only if

P
P

is

C
1

{C^1}

in

H
∖
K

H\backslash K

and its derivative

P
′

P’

has a certain invertibility property at each point. An example in

§
5

\S 5

shows that if the

C
2

{C^2}

condition is relaxed even slightly then

P
P

can be nondifferentiable (Fréchet) in

H
∖
K

H\backslash K

.
Universal bundles for deformations of asymmetric structures
Authors: Dippolito Paul R.
Published: June 2010

Riemannian parallel transport of frames and Cartan connections can be considered as special cases of a type of structure in which the connection form on the principal bundle can transform by an arbitrary extension of the adjoint representation of the fiber group; there always exist universal deformations for asymmetric objects in any of these geometries.
Siegel domains and representations of Jordan algebras
Authors: Rothaus O. S.
Published: June 2010

In the analysis of infinitesimal automorphisms of arbitrary Siegel domains, a certain class of nonsemisimple Jordan algebras occurs. The description of all the infinitesimal automorphisms of the domain may be based on a study of representations of the associated Jordan algebra satisfying a certain "strange identity". In this paper, all the possibilities for the Jordan algebra and representations satisfying the identity are given.
Balanced Howell rotations of the twin prime power type
Authors: Du Ding Zhu , Hwang F. K.
Published: June 2010

We prove by construction that a balanced Howell rotation for

n
n

players always exists if

n
=

p
r

q
s

n = {p^r}{q^s}

where

p
p

and

q
≠
3

q \ne 3

are primes and

q
s

=

p
r

+
2

{q^s} = {p^r} + 2

. This generalizes a much weaker previous result. The construction uses properties of a Galois domain which is a direct sum of two Galois fields.
On the construction of relative genus fields
Authors: Cornell Gary
Published: June 2010

We show how to construct the relative genus field in many cases. This is then applied to constructing fields with interesting class groups.
Saturation properties of ideals in generic extensions. II
Authors: Baumgartner James E. , Taylor Alan D.
Published: June 2010

The general type of problem considered here is the following. Suppose

I
I

is a countably complete ideal on

ω
1

{\omega _1}

satisfying some fairly strong saturation requirement (e.g.

I
I

is precipitous or

ω
2

{\omega _2}

-saturated), and suppose that

P
P

is a partial ordering satisfying some kind of chain condition requirement (e.g.

P
P

has the c.c.c. or forcing with

P
P

preserves

ω
1

{\omega _1}

). Does it follow that forcing with

P
P

preserves the saturation property of

I
I

? In this context we consider not only precipitous and

ω
2

{\omega _2}

-saturated ideals, but we also introduce and study a class of ideals that are characterized by a property lying strictly between these two notions. Some generalized versions of Chang’s conjecture and Kurepa’s hypothesis also arise naturally from these considerations.
Theta-characteristics on algebraic curves
Authors: Harris Joe
Published: June 2010

The theory of theta-characteristics is developed algebraically, so that it may be applied to possibly singular and/or reducible algebraic curves. The configuration of theta-characteristics on a curve is described in terms of its singularities, with applications to the geometry of plane quartic curves and the problem of Appolonius. Some results on Gorenstein local rings are appended.
Quaternionic Kaehler manifolds
Authors: Whitt Lee
Published: June 2010

The topological classification of

4
4

- and

8
8

- (real) dimensional compact quaternionic Kaehler manifolds is given. There is only the torus in dimension 4. In dimension 8, there are 12 homeomorphism classes; representatives are given explicitly.
On the Schwarz reflection principle
Authors: Hwang J. S.
Published: June 2010

Recently, we have solved a long outstanding problem of A. J. Lohwater (1953) by showing that if

f
(
z
)

f(z)

is meromorphic in

|

z

|

>
1

|z|> 1

whose radial limits have modulus 1 for almost all points on an arc

A
A

of

|

z

|

>
1

|z|> 1

, and if

P
P

is a singular point of

f
(
z
)

f(z)

on

A
A

, then every value of modulus 1 which is not in the range of

f
(
z
)

f(z)

at

P
P

is an asymptotic value of

f
(
z
)

f(z)

at some point of each subarc of

A
A

containing the point

P
P

. Lohwater proved this theorem for functions of bounded characteristic and he made a comment that his method is not, in general, applicable to functions of unbounded characteristic. In this paper, we shall present an alternative proof of the above theorem based on the very method of Lohwater.
Unipotent characters of the even orthogonal groups over a finite field
Authors: Lusztig George
Published: June 2010

The characters of unipotent representations of a simple algebraic group over

F
q

F_q

of type

≠

D
n

\ne D_n

on any regular semisimple element are explicitly known for large

q
q

. This paper deals with the remaining case: type

D
n

D_n

.
A nonshrinkable decomposition of 𝑆ⁿ involving a null sequence of cellular arcs
Authors: Daverman R. J. , Walsh J. J.
Published: June 2010

This paper presents a decomposition

G
G

of

S
n

(
n
≥
3
)

S^n(n\ge 3)

into points and a null sequence of cellular arcs such that

S
n

/

G

S^n/G

is not a manifold; furthermore, the union of the nondegenerate elements from

G
G

lies in a

2
2

-cell in

S
n

S^n

and the image in

S
n

/

G

S^n/G

of this union has

0
0

-dimensional closure. Examples of nonshrinkable decompositions with a null sequence of cellular arcs have been constructed in the case

n
=
3

n=3

by D. S. Gillman and J. M. Martin and by R. H. Bing and M. Starbird. We construct another example in this dimension, for which all the arcs lie in the boundary of a crumpled cube

C
C

, and then produce higher dimensional examples by spinning

C
C

.
Rational LS category and its applications
Authors: Félix Yves , Halperin Stephen
Published: June 2010

Let

S
S

be a

1
1

-connected CW-complex of finite type and put

ca

t

0

(
S
)
=

{\text {ca}}{{\text {t}}_0}(S) =

Lusternik-Schnirelmann category of the localization

S

Q

{S_{\mathbf {Q}}}

. This invariant is characterized in terms of the minimal model of

S
S

. It is shown that if

ϕ
:
S
→
T

\phi :S \to T

is injective on

π
∗

⊗

Q

{\pi _ \ast } \otimes {\mathbf {Q}}

then

ca

t

0

(
S
)
⩽

ca

t

0

(
T
)

{\text {ca}}{{\text {t}}_0}(S) \leqslant {\text {ca}}{{\text {t}}_0}(T)

, and this result is strengthened when

ϕ
\phi

is the fibre inclusion of a fibration. It is also shown that if

H
∗

(
S
;

Q

)
>
∞

{H^ \ast }(S;{\mathbf {Q}}) > \infty

then either

π
∗

(
S
)
⊗

Q

>
∞

{\pi _ \ast }(S) \otimes {\mathbf {Q}} > \infty

or the groups

π
k

(
S
)
⊗

Q

{\pi _k}(S) \otimes {\mathbf {Q}}

grow exponentially with

k
k

.
Trace-like functions on rings with no nilpotent elements
Authors: Cohen M. , Montgomery Susan
Published: June 2010

Let

R
R

be a ring with no nilpotent elements, with extended center

C
C

, and let

E
E

be the set of idempotents in

C
C

. Our first main result is that for any finite group

G
G

acting as automorphisms of

R
R

, there exist a finite set

L
⊆
E

L \subseteq E

and an

R
G

{R^G}

-bimodule homomorphism

τ
:
R
→

(
R
L

)
G

\tau :R \to {(RL)^G}

such that

τ
(
R
)

\tau (R)

is an essential ideal of

(
R
E

)
G

{(RE)^G}

. This theorem is applied to show the following: if

R
R

is a Noetherian, affine

P
I

PI

-algebra (with no nilpotent elements) over the commutative Noetherian ring

A
A

, and

G
G

is a finite group of

A
A

-automorphisms of

R
R

such that

R
G

{R^G}

is Noetherian, then

R
G

{R^G}

is affine over

A
A

.
The Catalan equation over function fields
Authors: Silverman Joseph H.
Published: June 2010

Let

K
K

be the function field of a projective variety. Fix

a
,
b
,
c
∈

K
∗

a,b,c \in {K^ \ast }

. We show that if

max
{
m
,
n
}

\max \{ m,n\}

is sufficiently large, then the Catalan equation

a

x
m

+
b

y
n

=
c

a{x^m} + b{y^n} = c

has no nonconstant solutions

x
,
y
∈
K

x,y \in K

.
Global solvability on compact Heisenberg manifolds
Authors: Richardson Leonard F.
Published: June 2010

We apply the methods of primary and irreducible Fourier series on compact nilmanifolds to determine the ranges of all first order invariant operators on the compact Heisenberg manifolds. We show that the sums of primary solutions behave better on these manifolds than on any multidimensional torus.
Local analyticity in weighted 𝐿¹-spaces and applications to stability problems for Volterra equations
Authors: Jordan G. S. , Staffans Olof J. , Wheeler Robert L.
Published: June 2010

We study the qualitative properties of the solutions of linear convolution equations such as

x
′

+

x

∗

μ
=
f
,

x
+
a
∗
x
=
f

x’ + x ^{\ast } \mu = f,\,x + a \ast x = f

and

x
∗
μ
=
f

x \ast \mu = f

. We are especially concerned with finding conditions which ensure that these equations have resolvents which belong to, or are determined up to a term belonging to, certain weighted

L
1

{L^1}

-spaces. Our results are obtained as consequences of more general Banach algebra results on functions that are locally analytic with respect to the elements of a weighted

L
1

{L^1}

-space. In particular, we derive a proposition of Wiener-Lévy type for weighted

L
1

{L^1}

-spaces which underlies all subsequent results. Our method applies equally well to equations more general than those mentioned above. We unify and sharpen the results of several recent papers on the asymptotic behavior of Volterra convolution equations of the types mentioned above, and indicate how many of them can be extended to the Fredholm case. In addition, we give necessary and sufficient conditions on the perturbation term

f
f

for the existence of bounded or integrable solutions

x
x

in some critical cases when the corresponding limit equations have nontrivial solutions.
Disproof of a coefficient conjecture for meromorphic univalent functions
Authors: Tsao Anna
Published: June 2010

Let

Σ
\Sigma

denote the class of functions

g
(
z
)
=
z
+

b
0

+

b
1

z

−
1

+
⋯

g(z) = z + {b_0} + {b_1}{z^{ - 1}} + \cdots

analytic and univalent in

|

z

|

>
1

|z|> 1

except for a simple pole at

∞
\infty

. A well-known conjecture asserts that

|

b
n

|

≤
2

/

(
n
+
1
)

(
n
=
1
,
2
,
…
)

|{b_n}|\, \leq 2/(n + 1)\qquad (n = 1,2, \ldots )

with equality for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g left-parenthesis z right-parenthesis equals left-parenthesis 1 plus z Superscript n plus 1 Baseline right-parenthesis Superscript 2 slash left-parenthesis n plus 1 right-parenthesis Baseline slash z equals z plus 2 z Superscript negative n Baseline slash left-parenthesis n plus 1 right-parenthesis plus midline-horizontal-ellipsis">

g
(
z
)
=

(
1
+

z

n
+
1

)

2

/

(
n
+
1
)

/

z
=
z
+
2

z

−
n

/

(
n
+
1
)
+
⋯

g(z) = {(1 + {z^{n + 1}})^{2/(n + 1)}}/z = z + 2{z^{ - n}}/(n + 1) + \cdots

. Although the conjecture is true for

n
=
1
,
2

n=1,2

and certain subclasses of the class

Σ
\Sigma

, the general conjecture is known to be false for all odd

n
≥
3

n\ge 3

and

n
=
4

n=4

. In

§
2

\S 2

, we generalize a variational method of Goluzin and develop second-variational techniques. This enables us in

§
3

\S 3

to construct explicit counterexamples to the conjecture for all

n
>
4

n > 4

. In fact, the conjectured extremal function does not even provide a local maximum for

Re

{

b
n

}

{\text {Re}}\{ {b_n}\}

,

n
>
4

n > 4

.
Dynamical systems and extensions of states on 𝐶*-algebras
Authors: Dang-Ngoc Nghiem
Published: June 2010

Let

(
A
,
G
,
τ
)

(A,G,\tau )

be a noncommutative dynamical system, i.e.

A
A

is a

C

∗

{C^{\ast } }

-algebra,

G
G

a topological group and

τ
\tau

an action of

G
G

on

A
A

by

∗

^{\ast }

-automorphisms, and let

(

M
α

)

({M_\alpha })

be an

M
M

-net on

G
G

. We characterize the set of

a
a

in

A
A

such that

M
α

a

{M_\alpha }a

converges in norm. We show that this set is intimately related to the problem of extensions of pure states of R. V. Kadison and I. M. Singer: if

B
B

is a maximal abelian subalgebra of

A
A

, we can associate a dynamical system

(
A
,
G
,
τ
)

(A,G,\tau )

such that

M
α

a

{M_\alpha }a

converges in norm if and only if all extensions to

A
A

, of a homomorphism of

B
B

, coincide on

a
a

. This result allows us to construct different examples of a

C

∗

{C^{\ast } }

-algebra

A
A

with maximal abelian subalgebra

B
B

(isomorphic to

C
(

R

/

Z

)

C({\mathbf {R}}/{\mathbf {Z}})

or

L
∞

[
0
,
1
]
)

{L^\infty }[0,1])

for which the property of unique pure state extension of homomorphisms is or is not verified.
Some canonical cohomology classes on groups of volume preserving diffeomorphisms
Authors: McDuff Dusa
Published: June 2010

We discuss some canonical cohomology classes on the space

B
¯

D

i
f

f

ω
0

c

M

\bar B\mathcal {D}iff_{\omega 0}^cM

, where

D

i
f

f

ω
0

c

M

\mathcal {D}iff_{\omega 0}^cM

is the identity component of the group of compactly supported diffeomorphisms of the manifold

M
M

which preserve the volume form

ω
\omega

. We first look at some classes

c
k

(
M
)
,
1
⩽
k
⩽
n
=

dim

M

{c_k}(M),1 \leqslant k \leqslant n = {\text {dim}}\,M

, which are defined for all

M
M

, and show that the top class
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c Subscript n Baseline left-parenthesis upper M right-parenthesis element-of upper H Superscript n Baseline left-parenthesis upper B overbar script upper D i f f Subscript omega Baseline 0 Superscript c Baseline upper M semicolon bold upper R right-parenthesis">

c
n

(
M
)
∈

H
n

(

B
¯

D

i
f

f

ω
0

c

M
;

R

)

{c_n}(M) \in \,{H^n}(\bar B\mathcal {D}iff_{\omega 0}^cM;{\mathbf {R}})

is nonzero for

M
=

S
n

,
n

M = {S^n},n

odd, and is zero for

M
=

S
n

,
n

M = {S^n},n

even. When

H
c
i

(
M
;

R

) = 0

H_c^i(M;{\mathbf {R}}{\text {) = 0}}

for

0
⩽
i
>
n

0 \leqslant i > n

, the classes

c
k

(
M
)

{c_k}(M)

all vanish and a secondary class
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s left-parenthesis upper M right-parenthesis element-of upper H Superscript n minus 1 Baseline left-parenthesis upper B overbar script upper D i f f Subscript omega Baseline 0 Superscript c Baseline upper M semicolon bold upper R right-parenthesis">

s
(
M
)
∈

H

n
−
1

(

B
¯

D

i
f

f

ω
0

c

M
;

R

)

s(M) \in \,{H^{n - 1}}(\bar B\mathcal {D}iff_{\omega 0}^cM; {\mathbf {R}})

may be defined. This is trivially zero when

n
n

is odd, and is twice the Calabi invariant for symplectic manifolds when

n
=
2

n = 2

. We prove that

s
(

R

n

)
≠
0

s({{\mathbf {R}}^n}) \ne 0

when

n
n

is even by showing that it is one of a set of nonzero classes which were defined by Hurder in [7].
Orthogonal geodesic and minimal distributions
Authors: Bivens Irl
Published: June 2010

Let

F

\mathfrak {F}

be a smooth distribution on a Riemannian manifold

M
M

with

H

\mathfrak {H}

the orthogonal distribution. We say that

F

\mathfrak {F}

is geodesic provided

F

\mathfrak {F}

is integrable with leaves which are totally geodesic submanifolds of

M
M

. The notion of minimality of a submanifold of

M
M

may be defined in terms of a criterion involving any orthonormal frame field tangent to the given submanifold. If this criterion is satisfied by any orthonormal frame field tangent to

H

\mathfrak {H}

then we say

H

\mathfrak {H}

is minimal. Suppose that

F

\mathfrak {F}

and

H

\mathfrak {H}

are orthogonal geodesic and minimal distributions on a submanifold of Euclidean space. Then each leaf of

F

\mathfrak {F}

is also a submanifold of Euclidean space with mean curvature normal vector field

η
\eta

. We show that the integral of

|

η

|

2

|\eta {|^2}

over

M
M

is bounded below by an intrinsic constant and give necessary and sufficient conditions for equality to hold. We study the relationships between the geometry of

M
M

and the integrability of

H

\mathfrak {H}

. For example, if

F

\mathfrak {F}

and

H

\mathfrak {H}

are orthogonal geodesic and minimal distributions on a space of nonnegative sectional curvature then

H

\mathfrak {H}

is integrable iff

F

\mathfrak {F}

and

H

\mathfrak {H}

are parallel distributions. Similarly if

M
n

{M^n}

has constant negative sectional curvature and dim

H

=
2
>
n

\mathfrak {H} = 2 > n

then

H

\mathfrak {H}

is not integrable. If

F

\mathfrak {F}

is geodesic and

H

\mathfrak {H}

is integrable then we characterize the local structure of the Riemannian metric in the case that the leaves of

H

\mathfrak {H}

are flat submanifolds of

M
M

with parallel second fundamental form.
Spectral families of projections, semigroups, and differential operators
Authors: Benzinger Harold , Berkson Earl , Gillespie T. A.
Published: June 2010

This paper presents new developments in abstract spectral theory suitable for treating classical differential and translation operators. The methods are specifically geared to conditional convergence such as arises in Fourier expansions and in Fourier inversion in general. The underlying notions are spectral family of projections and well-bounded operator, due to D. R. Smart and J. R. Ringrose. The theory of well-bounded operators is considerably expanded by the introduction of a class of operators with a suitable polar decomposition. These operators, called polar operators, have a canonical polar decomposition, are free from restrictions on their spectra (in contrast to well-bounded operators), and lend themselves to semigroup considerations. In particular, a generalization to arbitrary Banach spaces of Stone’s theorem for unitary groups is obtained. The functional calculus for well-bounded operators with spectra in a nonclosed arc is used to study closed, densely defined operators with a well-bounded resolvent. Such an operatorLLis represented as an integral with respect to the spectral family of its resolvent, and a sufficient condition is given for(−L)(- L)to generate a strongly continuous semigroup. This approach is applied to a large class of ordinary differential operators. It is shown that this class contains significant subclasses of operators which have a polar resolvent or generate strongly continuous semigroups. Some of the latter consist of polar operators up to perturbation by a semigroup continuous in the uniform operator topology.
An algebraic approach to Grothendieck’s residue symbol
Authors: Hopkins Glenn
Published: June 2010

A certain map—the "residue map"—is defined and its properties are investigated. The impetus for the definition and study of this map is a definition by A. Grothendieck of a homomorphism, the "residue symbol", which has been found applicable in several areas, including the duality theory of algebraic varieties.