bluuebunny/crossref_metadata_2025 huggingface dataset:
Weak compactness in locally convex spaces
Authors: Tacon D. G.
Published: July 2010

The notion of weak compactness plays a central role in the theory of locally convex topological vector spaces. However, in the statement of many theorems, completeness of the space, or at least quasi-completeness of the space in the Mackey topology is an important assumption. In this paper we extend the concept of weak compactness in a general way and obtain a number of useful particular cases. If we replace weak compactness by these generalized notions we can drop the completeness assumption from the statement of many theorems; for example, we generalize the classical theorems of Eberlein and Kreĭn. We then consider generalizations of semireflexivity and reflexivity and characterize these properties in terms of our previous ideas as well as in terms of known concepts. In most of the proofs we use techniques of nonstandard analysis.
Boundary representations on 𝐶*-algebras with matrix units
Authors: Hopenwasser Alan
Published: July 2010

Let

A

\mathcal {A}

be a

C
∗

{C^\ast }

-algebra with unit, let

S

\mathcal {S}

be a linear subspace of

A

⊗

M
n

\mathcal {A} \otimes {M_n}

which contains the natural set of matrix units and which generates

A

\mathcal {A}

as a

C
∗

{C^\ast }

-algebra. Let

J

\mathcal {J}

be the subset of

A

\mathcal {A}

consisting of entries of matrices in

S

\mathcal {S}

. Then the boundary representations of

A

⊗

M
n

\mathcal {A} \otimes {M_n}

relative to

S

\mathcal {S}

are parametrized by the boundary representations of

A

\mathcal {A}

relative to

J

\mathcal {J}

. Also, a nontrivial example is given of a subalgebra of a

C
∗

{C^\ast }

-algebra which possesses exactly one boundary representation.
A nonlinear optimal control minimization technique
Authors: Rupp Russell D.
Published: July 2010

Hestenes’ method of multipliers is applied to a nonlinear optimal control problem. This requires that a differentially constrained problem be embedded in a family of unconstrained problems so as to preserve standard sufficiency criteria. Given an initial estimate of the Lagrange multipliers, a convergent sequence of arcs is generated. They are minimizing with respect to members of the above family, and their limit is the solution to the differentially constrained problem.
Properties of fixed-point sets of nonexpansive mappings in Banach spaces
Authors: Bruck Ronald E.
Published: July 2010

Let C be a closed convex subset of the Banach space X. A subset F of C is called a nonexpansive retract of C if either

F
=
∅

F = \emptyset

or there exists a retraction of C onto F which is a nonexpansive mapping. The main theorem of this paper is that if

T
:
C
→
C

T:C \to C

is nonexpansive and satisfies a conditional fixed point property, then the fixed-point set of T is a nonexpansive retract of C. This result is used to generalize a theorem of Belluce and Kirk on the existence of a common fixed point of a finite family of commuting nonexpansive mappings.
Concerning the shapes of finite-dimensional compacta
Authors: Geoghegan Ross , Summerhill R. Richard
Published: July 2010

It is shown that two “tamely” embedded compacta of dimension

≤
k

\leq k

lying in

E
n

(
n
≥
2
k
+
2
)

{E^n}(n \geq 2k + 2)

have the same (Borsuk) shape if and only if their complements are homeomorphic. In particular, two k-dimensional closed submanifolds of

E

2
k
+
2

{E^{2k + 2}}

have the same homotopy type if and only if their complements are homeomorphic.
Decomposable braids and linkages
Authors: Levinson H.
Published: July 2010

An n-braid is called k-decomposable if and only if the removal of k arbitrary strands results in a trivial

(
n
−
k
)

(n - k)

-braid. k-decomposable n-linkages are similarly defined. All k-decomposable n-braids are generated by an explicit geometric process, and so are all k-decomposable n-linkages. The latter are not always closures of k-decomposable n-braids. Many examples are given.
The structure of Dedekind cardinals
Authors: Ellentuck Erik
Published: July 2010

Semantic criteria are given for provability in set theory without the axiom of choice of positive sentences about the Dedekind cardinals. These criteria suggest that Dedekind cardinals (as well as general cardinals) have an internal structure.
On the congruence lattice characterization theorem
Authors: Lampe William A.
Published: July 2010

A simplified proof is given for the theorem characterizing the congruence lattice of a universal algebra.
A Wedderburn theorem for alternative algebras with identity over commutative rings
Authors: Brown W. C.
Published: July 2010

In this paper, we study alternative algebras

Λ

{\mathbf {\Lambda }}

over a commutative, associative ring R with identity. When

Λ

{\mathbf {\Lambda }}

is finitely generated as an R-module, we define the radical J of

Λ

{\mathbf {\Lambda }}

. We show that matrix units and split Cayley algebras can be lifted from

Λ

/

J

{\mathbf {\Lambda }}/J

to

Λ

{\mathbf {\Lambda }}

when R is a Hensel ring. We also prove the following Wedderburn theorem: Let

Λ

{\mathbf {\Lambda }}

be an alternative algebra over a complete local ring R of equal characteristic. Suppose

Λ

{\mathbf {\Lambda }}

is finitely generated as an R-module, and

Λ

/

J

{\mathbf {\Lambda }}/J

is separable over

R
¯

\bar R

(

R
¯

\bar R

the residue class field of R). Then there exists an

R
¯

\bar R

-subalgebra S of

Λ

{\mathbf {\Lambda }}

such that

S
+
J
=

Λ

S + J = {\mathbf {\Lambda }}

and

S
∩
J
=
0

S \cap J = 0

.
On the Green’s function for the biharmonic equation in an infinite wedge
Authors: Seif Joseph B.
Published: July 2010

The Green’s function for the biharmonic equation in an infinite angular wedge is considered. The main result is that if the angle a is less than

a
1

≅
0.812
π

{a_1} \cong 0.812\pi

, then the Green’s function does not remain positive; in fact it oscillates an infinite number of times near zero and near

∞
\infty

. The method uses a number of transformations of the problem including the Fourier transform. The inversion of the Fourier transform is accomplished by means of the calculus of residues and depends on the zeros of a certain transcendental function. The distribution of these zeros in the complex plane gives rise to the determination of the angle

a
1

{a_1}

. A general expression for the asymptotic behavior of the solution near zero and near infinity is obtained. This result has the physical interpretation that if a thin elastic plate is deflected downward at a point, the resulting shape taken by the plate will have ripples which protrude above the initial plane of the plate.
Irreducible representations of the 𝐶*-algebra generated by a quasinormal operator
Authors: Bunce John W.
Published: July 2010

For A a quasinormal operator on Hilbert space, we determine the irreducible representations of

C
∗

(
A
)

{C^ \ast }(A)

, the

C
∗

{C^ \ast }

-algebra generated by A and the identity. We also explicitly describe the topology on the space of unitary equivalence classes of irreducible representations of

C
∗

(
A
)

{C^ \ast }(A)

and calculate the regularized transform of

C
∗

(
A
)

{C^ \ast }(A)

, thus exhibiting an isomorphic copy of

C
∗

(
A
)

{C^ \ast }(A)

.
Involutions on 𝑆¹×𝑆² and other 3-manifolds
Authors: Tollefson Jeffrey L.
Published: July 2010

This paper exploits the following observation concerning involutions on nonreducible 3-manifolds: If the dimension of the fixed point set of a PL involution is less than or equal to one then there exists a pair of disjoint 2-spheres that do not bound 3-cells and whose union is invariant under the given involution. The classification of all PL involutions of

S
1

×

S
2

{S^1} \times {S^2}

is obtained. In particular,

S
1

×

S
2

{S^1} \times {S^2}

admits exactly thirteen distinct PL involutions (up to conjugation). It follows that there is a unique PL involution of the solid torus

S
1

×

D
2

{S^1} \times {D^2}

with 1-dimensional fixed point set. Furthermore, there are just four fixed point free

Z

2
k

{Z_{2k}}

-actions and just one fixed point free

Z

2
k
+
1

{Z_{2k + 1}}

-action on

S
1

×

S
2

{S^1} \times {S^2}

for each positive integer k (again, up to conjugation). The above observation is also used to obtain a general description of compact, irreducible 3-manifolds that admit two-sided embeddings of the projective plane.
Isomorphism of simple Lie algebras
Authors: Allison B. N.
Published: July 2010

Let

L

\mathcal {L}

and

L

′

\mathcal {L}’

be finite dimensional simple Lie algebras over a field of characteristic zero. A necessary and sufficient condition is given for

L

\mathcal {L}

and

L

′

\mathcal {L}’

to be isomorphic. The anisotropic kernel of

L

\mathcal {L}

is also studied. In particular, a result about this kernel in the rank one reduced case is proved. This result is then used to prove a conjugacy theorem for the simple summands of the anisotropic kernel in the general reduced case. The results and methods of this paper are rational in the sense that they involve no extension of the base field.
Some stable results on the cohomology of the classical infinite-dimensional Lie algebras
Authors: Guillemin Victor , Shnider Steven
Published: July 2010

In this paper we compute the cohomology of various classical infinite-dimensional Lie algebras generalizing results of Gel’fand-Fuks for the Lie algebra of all formal power series vector fields.
Finite- and infinite-dimensional representation of linear semisimple groups
Authors: Lepowsky James , Wallach Nolan R.
Published: July 2010

Every representation in the nonunitary principal series of a noncompact connected real semisimple linear Lie group G with maximal compact subgroup K is shown to have a K-finite cyclic vector. This is used to give a new proof of Harish-Chandra’s theorem that every member of the nonunitary principal series has a (finite) composition series. The methods of proof are based on finite-dimensional G-modules, concerning which some new results are derived. Further related results on infinite-dimensional representations are also obtained.
On the singular boundary value problem for elliptic equations
Authors: Hayashida Kazunari
Published: July 2010

The operator

L

\mathcal {L}

is elliptic and of second order in a domain

Ω
\Omega

in

R
N

{R^N}

. We consider the following boundary value problem:

L

u
=
f

\mathcal {L}u = f

in

Ω
\Omega

and

B

u
=
0

\mathcal {B}u = 0

on

∂
Ω

\partial \Omega

, where

B

=
a
d

/

d
n
+
β

\mathcal {B} = ad/dn + \beta

(d/dn is the conormal derivative on

∂
Ω

\partial \Omega

). The coefficient

α
\alpha

is assumed to be nonnegative. However,

α
\alpha

may vanish partly on

∂
Ω

\partial \Omega

. Then the regularity of the weak solutions for the above problem is shown by the variational method.
Restricting a Schauder basis to a set of positive measure
Authors: Shirey James
Published: July 2010

Let

{

f
n

}

\{ {f_n}\}

be an orthonormal system of functions on [0, 1] containing a subsystem

{

f

n
k

}

\{ {f_{{n_k}}}\}

for which (a)

f

n
k

→
0

{f_{{n_k}}} \to 0

weakly in

L
2

{L_2}

, and (b) given

E
⊂
[
0
,
1
]

E \subset [0,1]

,

|

E

|

>
0

|E| > 0

,

Lim

Inf

∫
E

|

f

n
k

(
x
)

|

d
x
>
0

{\operatorname {Lim}}\;{\operatorname {Inf}}{\smallint _E}|{f_{{n_k}}}(x)|dx > 0

. There then exists a subsystem

{

g
n

}

\{ {g_n}\}

of

{

f
n

}

\{ {f_n}\}

such that for any set E as above, the linear span of

{

g
n

}

\{ {g_n}\}

in

L
1

(
E
)

{L_1}(E)

is not dense. For every set E as above, there is an element of

L
p

(
E
)

{L_p}(E)

,

1
>
p
>
∞

1 > p > \infty

, whose Walsh series expansion converges conditionally and an element of

L
1

(
E
)

{L_1}(E)

whose Haar series expansion converges conditionally.
Limit properties of Poisson kernels of tube domains
Authors: Dickson Lawrence J.
Published: July 2010

If certain local boundary conditions hold near

P
∈
∂
Γ

P \in \partial \Gamma

, the Poisson kernel belonging to a proper cone

Γ
⊂

R

n

\Gamma \subset {{\mathbf {R}}^n}

converges to a tight

C
0
∗

C_0^\ast

limit as its parameter converges admissibly to P in

Γ
\Gamma

. This limit can be identified with a lower-dimensional Poisson kernel. The result always works for polytopic and “rounded” cones; for these, a result on the decrease at infinity is obtained which in fact implies convergence almost everywhere in the appropriate sense of the Poisson integral to certain of its boundary values.
Fourier analysis on linear metric spaces
Authors: Kuelbs J.
Published: July 2010

Probability measures on a real complete linear metric space

E
E

are studied via their Fourier transform on

E
′

E’

provided

E
E

has the approximation property and possesses a real positive definite continuous function

Φ
(
x
)

\Phi (x)

such that

|

|

x

|

|

>
ϵ

||x|| > \epsilon

implies

Φ
(
0
)
−
Φ
(
x
)
>
c
(
ϵ
)

\Phi (0) - \Phi (x) > c(\epsilon )

where

c
(
ϵ
)
>
0

c(\epsilon ) > 0

. In this setting we obtain conditions on the Fourier transforms of a family of tight Borel probabilities which yield tightness of the family of measures. This then is applied to obtain necessary and sufficient conditions for a complex valued function on

E
′

E’

to be the Fourier transform of a tight Borel probability on

E
E

. An extension of the Levy continuity theorem as given by

L

{\text {L}}

. Gross for a separable Hilbert space is obtained for such metric spaces. We also prove that various Orlicz-type spaces are in the class of spaces to which our results apply. Finally we apply our results to certain Orlicz-type sequence spaces and obtain conditions sufficient for tightness of a family of probability measures in terms of uniform convergence of the Fourier transforms on large subsets of the dual. We also obtain a more explicit form of Bochner’s theorem for these sequence spaces. The class of sequence spaces studied contains the

l
p

{l_p}

spaces

(
0
>
p
⩽
2
)

(0 > p \leqslant 2)

and hence these results apply to separable Hilbert space.
Multilinear identities of the matrix ring
Authors: Leron Uri
Published: July 2010

Let V be a vector space over a field F of zero characteristic, which is acted upon by the symmetric group. Systems of generators for V are constructed, which have special symmetry and skew symmetry properties. This is applied to prove that every multilinear polynomial identity of degree

2
n
+
1

2n + 1

which holds in the matrix ring

F
n

(
n
>
2
)

{F_n}(n > 2)

is a consequence of the standard identity

s

2
n

{s_{2n}}

. The notions of rigid and semirigid sequences of matrices are defined and treated.
Appell polynomial expansions and biorthogonal expansions in Banach spaces
Authors: Buckholtz J. D.
Published: July 2010

Let

{

p
k

}
0
∞

\{ {p_k}\} _0^\infty

denote the sequence of Appell polynomials generated by an analytic function

ϕ
\phi

with the property that the power series for

θ
=
1

/

ϕ

\theta = 1/\phi

has a larger radius of convergence than the power series for

ϕ
\phi

. The expansion and uniqueness properties of

{

p
k

}

\{ {p_k}\}

are determined completely. In particular, it is shown that the only convergent

{

p
k

}

\{ {p_k}\}

expansions are basic series, and that there are no nontrivial representations of 0. An underlying Banach space structure of these expansions is also studied.
Hermitian vector bundles and value distribution for Schubert cycles
Authors: Cowen Michael J.
Published: July 2010

R. Bott and S. S. Chern used the theory of characteristic differential forms of a holomorphic hermitian vector bundle to study the distribution of zeroes of a holomorphic section. In this paper their methods are extended to study how often a holomorphic mapping into a Grassmann manifold hits Schubert cycles of fixed type.
The Cauchy problem for Douglis-Nirenberg elliptic systems of partial differential equations
Authors: Kramer Richard J.
Published: July 2010

Several partial answers are given to the question: Suppose U is a solution of the Douglis-Nirenberg elliptic system

L
U
=
F

LU = F

where F is analytic and L has analytic coefficients. If

U
=
0

U = 0

in some appropriate sense on a hyperplane (or any analytic hypersurface) must U vanish identically? One answer follows from introducing a so-called formal Cauchy problem for Douglis-Nirenberg elliptic systems and establishing existence and uniqueness theorems. A second Cauchy problem, in some sense a more natural one, is discussed for an important subclass of the Douglis-Nirenberg elliptic systems. The results in this case give a second partial answer to the original question. The methods of proof employed are largely algebraic. The systems are reduced to systems to which the Cauchy-Kowalewski theorem applies.
Pseudo-differential estimates for linear parabolic operators
Authors: Ellis David
Published: July 2010

In recent papers, S. Kaplan and D. Ellis have used singular integral operator theory, multilinear interpolation and forms of the classical “energy inequality” to obtain results for linear parabolic operators. For higher order linear parabolic operators the local estimates were globalized by a Gårding type partition of unity. In the present paper it is shown how the theory of pseudo-differential operators can be used to study linear parabolic operators without recourse to multilinear interpolation. We also prove that the Gårding type partition of unity is square summable in the Sobolev type spaces

H
S

{H^S}

and

K

r
,
S

{\mathcal {K}^{r,S}}

.
Localization, homology and a construction of Adams
Authors: Deleanu Aristide , Hilton Peter
Published: July 2010

In recent papers, the authors have developed the technique of using Kan extensions to obtain extensions of homology and cohomology theories from smaller to larger categories of topological spaces. In the present paper, it is shown that the conditions imposed there to guarantee that the Kan extension of a cohomology theory is again a cohomology theory in fact also imply that the Kan extension commutes with stabilization. A construction, due to Adams, for completing a space with respect to a homology theory by using categories of fractions is generalized to triangulated categories, and it is shown that, for any family of primes P, the Adams completion of a space X with respect to the homology theory

H
~

∗

(
−
;

Z

P

)

{\tilde H_ \ast }( - ;{{\mathbf {Z}}_P})

is the localization of X at P in the sense of Sullivan. Using this, the Kan extension of the restriction of a homology theory to the category of spaces having P-torsion homotopy groups is determined.
Lattice points and Lie groups. I
Authors: Cahn Robert S.
Published: July 2010

Assume that G is a compact semisimple Lie group and

G

\mathfrak {G}

its associated Lie algebra. It is shown that the number of irreducible representations of G of dimension less than or equal to n is asymptotic to

k

n

a

/

b

k{n^{a/b}}

, where a = the rank of

G

\mathfrak {G}

and b = the number of positive roots of

G

\mathfrak {G}

.
Absolute convergence of series of Fourier coefficients
Authors: McLaughlin James R.
Published: July 2010

In this article the author unifies and generalizes practically all known sufficiency results for absolute convergence of series of Fourier coefficients that are given in terms of the integrated modulus of continuity, best approximation, or bounded pth variation. This is done for the trigonometric, Walsh, Haar, Franklin, and related systems as well as general orthonormal systems. Many of the original proofs of previous results relied upon special properties of the trigonometric, Haar, and other systems and were done independently of one another. Also, several authors have proved results which at the time they believed to be generalizations of past results, but are, in fact, corollaries of them. The present author will expose underlying principles and illustrate their usefulness.
Schur multipliers of finite simple groups of Lie type
Authors: Griess Robert L.
Published: July 2010

This paper presents results on Schur multipliers of finite groups of Lie type. Specifically, let p denote the characteristic of the finite field over which such a group is defined. We determine the p-part of the multiplier of the Chevalley groups

G
2

(
4
)
,

G
2

(
3
)

{G_2}(4),{G_2}(3)

and

F
4

(
2
)

{F_4}(2)

the Steinberg variations; the Ree groups of type

F
4

{F_4}

and the Tits simple group

2

F
4

(
2

)
′

^2{F_4}(2)’

.
On the integrals of Perron type
Authors: Bullen P. S. , Lee C. M.
Published: July 2010

An abstract derivate system is defined axiomatically, and then a naturally corresponding Perron integral theory is developed, unifying all the existing integral theories of Perron type of first order. A new scale of approximately mean-continuous integrals and a new scale of symmetric Cesàro-Perron integrals are obtained as examples of the general theory. Also, the MZ-integral and the SCP-integral are proved to be equivalent.
Analytic functions characterized by their means on an arc
Authors: Ching Chin Hung , Chui Charles K.
Published: July 2010

It is known that a function f, holomorphic in the open unit disc U with

C

1
+
ε

{C^{1 + \varepsilon }}

boundary data for some

ε
>
0

\varepsilon > 0

, is uniquely determined by its arithmetic means over equally spaced points on

∂
U

\partial U

. By using different techniques, we weaken the hypothesis

C

1
+
ε

(
∂
U
)

{C^{1 + \varepsilon }}(\partial U)

to functions with

L
p

{L^p}

derivatives,

1
>
p
≤
∞

1 > p \leq \infty

. We also prove that a function is determined by its averages over an arc K if f is holomorphic in a neighborhood of

U
¯

\bar U

, and that this result is false for some functions f in

A
∩

C
∞

(

U
¯

)

A \cap {C^\infty }(\bar U)

. On the other hand, we can capture a

A
∩

C
2

(

U
¯

)

A \cap {C^2}(\bar U)

function from its means and shifted means on K.
Slicing and intersection theory for chains modulo 𝜈 associated with real analytic varieties
Authors: Hardt Robert M.
Published: July 2010

In a real analytic manifold a k dimensional (real) analytic chain is a locally finite sum of integral multiples of chains given by integration over certain k dimensional analytic submanifolds (or strata) of some k dimensional real analytic variety. In this paper, for any integer

ν
≥
2

\nu \geq 2

, the concepts and results of [6] on the continuity of slicing and the intersection theory for analytic chains are fully generalized to the modulo

ν
\nu

congruence classes of such chains.
On the isotropic group of a homogeneous polynomial
Authors: Ho Siu Ming
Published: July 2010

Let G be the linear group leaving a homogeneous polynomial of degree k fixed. The author shows that either the polynomial is a polynomial in fewer than the assigned number of variables or that the

(
k
−
1
)

(k - 1)

st prolongation of G is 0. The author also shows that this result is optimal.
On the uniform convergence of quasiconformal mappings
Authors: Palka Bruce
Published: July 2010

Let D be a domain in extended Euclidean n-space with “smooth” boundary and let

{

f
j

}

\{ {f_j}\}

be a sequence of K-quasiconformal mappings of D into

R
n

{R^n}

which converges uniformly on compact sets in D to a quasiconformal mapping. This paper considers the question: When does the sequence

{

f
j

}

\{ {f_j}\}

converge uniformly on all of D? Geometric conditions on the domains

f
j

(
D
)

{f_j}(D)

are given which are sufficient and, in many cases, necessary for uniform convergence. The particular case where D is the unit ball in

R
n

{R^n}

is examined to obtain analogues to classical convergence theorems for conformal mappings in the plane.
Inversion formulae for the probability measures on Banach spaces
Authors: Gharagoz Hamedani G. , Mandrekar V.
Published: July 2010

Let

B
B

be a real separable Banach space, and let

μ
\mu

be a probability measure on

B

(
B
)

\mathcal {B}(B)

, the Borel sets of

B
B

. The characteristic functional (Fourier transform)

ϕ
\phi

of

μ
\mu

, defined by

ϕ
(
y
)
=

∫
B

exp
⁡
{
i
(
y
,
x
)
}
d
μ
(
x
)

\phi (y) = \int _B {\exp \{ i(y,x)\} d\mu (x)\;}

for

y
∈

B
∗

y \in {B^\ast }

(the topological dual of

B
B

), uniquely determines

μ
\mu

. In order to determine

μ
\mu

on

B

(
B
)

\mathcal {B}(B)

, it suffices to obtain the value of

∫
B

G
(
s
)
d
μ
(
s
)

\int _B {G(s)d\mu (s)}

for every real-valued bounded continuous function

G
G

on

B
B

. Hence an inversion formula for

μ
\mu

in terms of

ϕ
\phi

is obtained if one can uniquely determine the value of

∫
B

G
(
s
)
d
μ
(
s
)

\int _B {G(s)d\mu (s)}

for all real-valued bounded continuous functions

G
G

on

B
B

in terms of

ϕ
\phi

and

G
G

. The main efforts of this paper will be to prove such inversion formulae of various types. For the Orlicz space

E
α

{E_\alpha }

of real sequences we establish inversion formulae (Main Theorem II) which properly generalize the work of L. Gross and derive as a corollary the extension of the Main Theorem of L. Gross to

E
α

{E_\alpha }

spaces (Corollary 2.2.12). In Part I we prove a theorem (Main Theorem I) which is Banach space generalization of the Main Theorem of L. Gross by reinterpreting his necessary and sufficient conditions in terms of convergence of Gaussian measures. Finally, in Part III we assume our Banach space to have a shrinking Schauder basis to prove inversion formulae (Main Theorem III) which express the measure directly in terms of

ϕ
\phi

and

G
G

without the use of extension of

ϕ
\phi

as required in the Main Theorems I and II. Furthermore this can be achieved without using the Lévy Continuity Theorem and we hope that one can use this theorem to obtain a direct proof of the Lévy Continuity Theorem.
Solvable groups having system normalizers of prime order
Authors: Seitz Gary M.
Published: July 2010

Let G be a solvable group having system normalizer D of prime order. If G has all Sylow groups abelian then we prove that

l
(
G
)
=
l
(

C
G

(
D
)
)
+
2

l(G) = l({C_G}(D)) + 2

, provided

l
(
G
)
≥
3

l(G) \geq 3

(here

l
(
H
)

l(H)

denotes the nilpotent length of the solvable group H). We conjecture that the above result is true without the condition on abelian Sylow subgroups. Other special cases of the conjecture are handled.
The Cauchy problem for degenerate parabolic equations with discontinuous drift
Authors: Conway Edward D.
Published: July 2010

The coefficient of the gradient is allowed to be discontinuous but is assumed to satisfy a “one-sided” Lipschitz condition. This condition insures the pathwise uniqueness of the underlying Markov process which in turn yields the existence of a unique stable generalized solution of the parabolic equation. If the data is Lipschitz continuous, then so is the solution.
Local and asymptotic approximations of nonlinear operators by (𝑘₁,…𝑘_{𝑁})-homogeneous operators
Authors: Moore R. H. , Nashed M. Z.
Published: July 2010

Notions of local and asymptotic approximations of a nonlinear mapping F between normed linear spaces by a sum of N

k
i

{k_i}

-homogeneous operators are defined and investigated. It is shown that the approximating operators inherit from F properties related to compactness and measures of noncompactness. Nets of equi-approximable operators with collectively compact (or bounded) approximates, which arise in approximate solutions of integral and operator equations, are studied with particular reference to pointwise (or weak convergence) properties. As a by-product, the well-known result that the Fréchet (or asymptotic) derivative of a compact operator is compact is generalized in several directions and to families of operators.
A free boundary problem connected with the optimal stopping problem for diffusion processes
Authors: Kotlow Daniel B.
Published: July 2010

This paper deals with a free boundary problem for a parabolic equation in one space variable which arises from the problem of selecting an optimal stopping strategy for the diffusion process connected with the equation. It is shown that a solution of the free boundary problem yields the solution of a minimum problem concerning supersolutions of the parabolic equation as well as the solution of the optimal stopping problem. Theorems regarding the existence, uniqueness, regularity, and approach to the steady state of solutions of the free boundary problem are established.
Some highly homogeneous groups
Authors: Dornhoff Larry
Published: July 2010

We study finite rank 4 permutation groups which are solvable and imprimitive, with a regular normal subgroup. This means determining those groups N with a solvable automorphism group A which has only three orbits on the nonidentity elements of N.
Nilpotent-by-finite groups with isomorphic finite quotients
Authors: Pickel P. F.
Published: July 2010

Let

F

(
G
)

\mathcal {F}(G)

denote the set of isomorphism classes of finite homomorphic images of a group G. We say that groups G and H have isomorphic finite quotients if

F

(
G
)
=

F

(
H
)

\mathcal {F}(G) = \mathcal {F}(H)

. Let

H

\mathcal {H}

denote the class of finite extensions of finitely generated nilpotent groups. In this paper we show that if G is in

H

\mathcal {H}

, then the groups H in

H

\mathcal {H}

for which

F

(
G
)
=

F

(
H
)

\mathcal {F}(G) = \mathcal {F}(H)

lie in only finitely many isomorphism classes.
Finite groups with nicely supplemented Sylow normalizers
Authors: Perin David
Published: July 2010

This paper considers finite groups G whose Sylow normalizers are supplemented by groups D having a cyclic Hall

2
′

2’

-subgroup. G is solvable and all odd order composition factors of G are cyclic. If

S
∈

Syl
2

(
D
)

S \in {\text {Syl}_2}(D)

is cyclic, dihedral, semidihedral, or generalized quaternion, then G is almost super-solvable.
Moment and BV-functions on commutative semigroups
Authors: Maserick P. H.
Published: July 2010

A general notion of variation of functions on an arbitrary commutative semigroup with identity is introduced. The concept includes Hausdorff’s for the additive semigroup of nonnegative integers as well as the more recent notions introduced for semilattices. An abstract moment problem is formulated and solved.
Subgroups of free products with amalgamated subgroups: A topological approach
Authors: Chipman J. C.
Published: July 2010

The structure of an arbitrary subgroup of the limit of a group system is shown to be itself the limit of a group system, the elements of which can be described in terms of subgroups of the original group system.
A topology for a lattice-ordered group
Authors: Redfield R. H.
Published: July 2010

Let G be an arbitrary lattice-ordered group. We define a topology on G, called the

J

\mathcal {J}

-topology, which is a group and lattice topology for G and which is preserved by cardinal products. The

J

\mathcal {J}

-topology is the interval topology on totally ordered groups and is discrete if and only if G is a lexico-sum of lexico-extensions of the integers. We derive necessary and sufficient conditions for the

J

\mathcal {J}

-topology to be Hausdorff, and we investigate

J

\mathcal {J}

-topology convergence.
Galois theory for fields 𝐾/𝑘 finitely generated
Authors: Heerema Nickolas , Deveney James
Published: July 2010

Let K be a field of characteristic

p
≠
0

p \ne 0

. A subgroup G of the group

H
t

(
K
)

{H^t}(K)

of rank t higher derivations

(
t
≤
∞
)

(t \leq \infty )

is Galois if G is the group of all d in

H
t

(
K
)

{H^t}(K)

having a given subfield h in its field of constants where K is finitely generated over h. We prove: G is Galois if and only if it is the closed group (in the higher derivation topology) generated over K by a finite, abelian, independent normal iterative set F of higher derivations or equivalently, if and only if it is a closed group generated by a normal subset possessing a dual basis. If

t
>
∞

t > \infty

the higher derivation topology is discrete. M. Sweedler has shown that, in this case, h is a Galois subfield if and only if

K

/

h

K/h

is finite modular and purely inseparable. Also, the characterization of Galois groups for

t
>
∞

t > \infty

is closely related to the Galois theory announced by Gerstenhaber and and Zaromp. In the case

t
=
∞

t = \infty

, a subfield h is Galois if and only if

K

/

h

K/h

is regular. Among the applications made are the following: (1)

∩
n

h
(

K

p
n

)

{ \cap _n}h({K^{{p^n}}})

is the separable algebraic closure of h in K, and (2) if

K

/

h

K/h

is algebraically closed,

K

/

h

K/h

is regular if and only if

K

/

h
(

K

p
n

)

K/h({K^{{p^n}}})

is modular for

n
>
0

n > 0

.
Moufang loops of small order. I
Authors: Chein Orin
Published: July 2010

The main result of this paper is the determination of all nonassociative Moufang loops of orders≤31\leq 31. Combinatorial type methods are used to consider a number of cases which lead to the discovery of 13 loops of the type in question and prove that there can be no others. All of the loops found are isomorphic to all of their loop isotopes, are solvable, and satisfy both Lagrange’s theorem and Sylow’s main theorem. In addition to finding the loops referred to above, we prove that Moufang loops of ordersp,p2{p^2},p3{p^3}orpq(forpandqprime) must be groups. Finally, a method is found for constructing nonassociative Moufang loops as extensions of nonabelian groups by the cyclic group of order 2.
𝐵-convexity and reflexivity in Banach spaces
Authors: Brown Dean R.
Published: July 2010

A proof of James that uniformly nonsquare spaces are reflexive is extended in part to B-convex spaces. A condition sufficient for non-B-convexity and related conditions equivalent to non-B-convexity are given. The following theorem is proved: A Banach space is B-convex if each subspace with basis is B-convex.
Analytic equivalence in the disk algebra
Authors: Warren Hugh E.
Published: July 2010

The notion of analytically equivalent domains can be extended from the complex plane to commutative Banach algebras with identity. In

C
(
X
)

C(X)

a domain equivalent to the unit ball must have a boundary that is in a certain sense continuous. This paper shows that in the disk algebra “continuous” must be replaced with “analytic.” These results set limits in the classical Riemann mapping theorem on how smoothly the mapping can respond to changes in the domain being mapped.
Selfadjoint algebras of unbounded operators. II
Authors: Powers Robert T.
Published: July 2010

Unbounded selfadjoint representations of

∗

^\ast

-algebras are studied. It is shown that a selfadjoint representation of the enveloping algebra of a Lie algebra can be exponentiated to give a strongly continuous unitary representation of the simply connected Lie group if and only if the representation preserves a certain order structure. This result follows from a generalization of a theorem of Arveson concerning the extensions of completely positive maps of

C
∗

{C^ \ast }

-algebras. Also with the aid of this generalization of Arveson’s theorem it is shown that an operator

π
(
A
)

¯

\overline {\pi (A)}

is affiliated with the commutant

π
(

A

)
′

\pi (\mathcal {A})’

of a selfadjoint representation

π
\pi

of a

∗

^\ast

-algebra

A

\mathcal {A}

, with

A
=

A
∗

∈

A

A = {A^ \ast } \in \mathcal {A}

, if and only if

π
\pi

preserves a certain order structure associated with A and

A

\mathcal {A}

. This result is then applied to obtain a characterization of standard representations of commutative

∗

^\ast

-algebras in terms of an order structure.
Equivariant method for periodic maps
Authors: Huang Wu Hsiung
Published: July 2010

The notion of coherency with submanifolds for a Morse function on a manifold is introduced and discussed in a general way. A Morse inequality for a given periodic transformation which compares the invariants called qth Euler numbers on fixed point set and the invariants called qth Lefschetz numbers of the transformations is thus obtained. This gives a fixed point theorem in terms of qth Lefschetz number for arbitrary q.
A combinatorial approach to the diagonal 𝑁-representability problem
Authors: Yoseloff Mark Laurance
Published: July 2010

The problem considered is that of the diagonal N-representability of a pth-order reduced density matrix,

p
≥
2

p \geq 2

, for a system of N identical fermions or bosons. A finite number M of allowable single particle states is assumed. The problem is divided into three cases, namely: Case I.

M
=
N
+
p

M = N + p

; Case II.

M
>
N
+
p

M > N + p

; Case III.

M
>
N
+
p

M > N + p

. Using the theory of polyhedral convex cones, a complete set of necessary and sufficient conditions is first found for Case I. This solution is then employed to find such conditions for Case II. For Case III, two algorithms are developed to generate solutions for the problem, and examples of the usage of these algorithms are given.
Countable unions of totally projective groups
Authors: Hill Paul
Published: July 2010

Let the p-primary abelian group G be the set-theoretic union of a countable collection of isotype subgroups

H
n

{H_n}

of countable length. We prove that if

H
n

{H_n}

is totally projective for each n, then G must be totally projective. In particular, an ascending sequence of isotype and totally projective subgroups of countable length leads to a totally projective group. The result generalizes and complements a number of theorems appearing in various articles in the recent literature. Several applications of the main result are presented.
Square integrable representations of semisimple Lie groups
Authors: Tirao Juan A.
Published: July 2010

Let D be a bounded symmetric domain. Let G be the universal covering group of the identity component

A
0

(
D
)

{A_0}(D)

of the group of all holomorphic diffeomorphisms of D onto itself. In this case, any G-homogeneous vector bundle

E
→
D

E \to D

admits a natural structure of G-homogeneous holomorphic vector bundles. The vector bundle

E
→
D

E \to D

must be holomorphically trivial, since D is a Stein manifold. We exhibit explicitly a holomorphic trivialization of

E
→
D

E \to D

by defining a map

Φ
:
G
→

GL

(
V
)

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

(V being the fiber of the vector bundle) which extends the classical “universal factor of automorphy” for the action of

A
0

(
D
)

{A_0}(D)

on D. Then, we study the space H of all square integrable holomorphic sections of

E
→
D

E \to D

. The natural action of G on H defines a unitary irreducible representation of G. The representations obtained in this way are square integrable over

G

/

Z

G/Z

(Z denotes the center of G) in the sense that the absolute values of their matrix coefficients are in

L
2

(
G

/

Z
)

{L_2}(G/Z)

.
Koebe sequences of arcs and normal meromorphic functions
Authors: Dragosh Stephen
Published: July 2010

Let f be a normal meromorphic function in the unit disk. An estimate for the growth of the modulus of f on a Koebe sequence of arcs is obtained; the estimate is in terms of the order of normality of f. An immediate consequence of the estimate is the following theorem due to F. Bagemihl and W. Seidel: A nonconstant normal meromorphic function has no Koebe values. Another consequence is that each level set of a nonconstant normal meromorphic function cannot contain a Koebe sequence of arcs provided the order of normality of f is less than a certain positive constant

C
∗

{C^\ast }

.
Conditions for a TVS to be homeomorphic with its countable product
Authors: Terry Wesley E.
Published: July 2010

C. Bessaga has given conditions for a Banach space to be homeomorphic with its countable product. In this paper, we extend and generalize these results to complete metric topological vector spaces by using infinite dimensional techniques.
Jordan algebras and connections on homogeneous spaces
Authors: Sagle Arthur A.
Published: July 2010

We use the correspondence between G-invariant connections on a reductive homogeneous space

G

/

H

G/H

and certain nonassociative algebras to explicitly compute the pseudo-Riemannian connections in terms of a Jordan algebra J of endomorphisms. It is shown that if G and H are semisimple Lie groups, then J is a semisimple Jordan algebra. Also a general method for computing examples of J is given.
van Kampen’s theorem for 𝑛-stage covers
Authors: Chipman J. C.
Published: July 2010

A version of van Kampen’s theorem is obtained for covers whose members do not share a common point and whose pairwise intersection need not be connected.
A new characterization of tame 2-spheres in 𝐸³
Authors: Weill Lawrence R.
Published: July 2010

It is shown in Theorem 1 that a 2-sphere S in

E
3

{E^3}

is tame from

A
=

Int

S

A = {\text {Int}}\;S

if and only if for each compact set

F
⊂
A

F \subset A

there exists a 2-sphere

S
′

S’

with complementary domains

A
′

=

Int

S
′

,

B
′

=

Ext

S
′

A’ = {\text {Int}}\;S’,B’ = {\text {Ext}}\;S’

, such that

F
⊂

A
′

⊂

A
′

¯

⊂
A

F \subset A’ \subset \overline {A’} \subset A

and for each

x
∈

S
′

x \in S’

there exists a path in

B
′

¯

\overline {B’}

of diameter less than

ρ
(
F
,
S
)

\rho (F,S)

which runs from x to a point

y
∈
S

y \in S

. Furthermore, the theorem holds when A is replaced by B,

A
′

A’

by

B
′

,

B
′

B’,B’

by

A
′

A’

, and Int by Ext. Two applications of this characterization are given. Theorem 2 states that a 2-sphere is tame from the complementary domain C if for arbitrarily small

ε
>
0

\varepsilon > 0

, S has a metric

ε
\varepsilon

-envelope in C which is a 2-sphere. Theorem 3 answers affirmatively the following question: Is a 2-sphere

S
⊂

E
3

S \subset {E^3}

tame in

E
3

{E^3}

if there exists an

ε
>
0

\varepsilon > 0

such that if a,

b
∈
S

b \in S

satisfy

ρ
(
a
,
b
)
>
ε

\rho (a,b) > \varepsilon

, then there exists a path in S of spherical diameter

ρ
(
a
,
b
)

\rho (a,b)

which connects a and b?
The concordance diffeomorphism group of real projective space
Authors: Wells Robert
Published: July 2010

Let

P
r

{P_r}

be r-dimensional real projective space with r odd, and let

π
0

Diff
+

:

P
r

{\pi _0}{\text {Diff}^ + }:{P_r}

be the group of orientation preserving diffeomorphisms

P
r

→

P
r

{P_r} \to {P_r}

factored by the normal subgroup of those concordant (= pseudoisotopic) to the identity. The main theorem of this paper is that for

r
≡
11
mod
16

r \equiv 11 \bmod 16

the group

π
0

Diff
+

:

P
r

{\pi _0}{\text {Diff}^ + }:{P_r}

is isomorphic to the homotopy group

π

r
+
1
+
k

(

P
∞

/

P

k
−
1

)

{\pi _{r + 1 + k}}({P_\infty }/{P_{k - 1}})

, where

k
=
d

2
L

−
r
−
1

k = d{2^L} - r - 1

with

L
≥
φ
(
(
r
+
1
)

/

2
)

L \geq \varphi ((r + 1)/2)

and

d

2
L

≥
r
+
1

d{2^L} \geq r + 1

. The function

φ
\varphi

is denned by

φ
(
l
)
=
{
i

|

0
>
i
≤
l
,
i
≡
0
,
1
,
2
,
4
mod
(
8
)
}

\varphi (l) = \{ i|0 > i \leq l,i \equiv 0,1,2,4 \bmod (8)\}

. The theorem is proved by introducing a cobordism version of the mapping torus construction; this mapping torus construction is a homomorphism

t
:

π
0

Diff
+

:

P
r

→

Ω

r
+
1

(
v
)

t:{\pi _0}{\text {Diff}^ + }:{P_r} \to {\Omega _{r + 1}}(v)

for

r
≡
11
mod
16

r \equiv 11 \bmod 16

and

Ω

r
+
1

(
v
)

{\Omega _{r + 1}}(v)

a suitable Lashof cobordism group. It is shown that t is an isomorphism onto the torsion subgroup

Ω

r
+
1

(
v
)

{\Omega _{r + 1}}(v)

, and that this subgroup is isomorphic to

π

r
+
1
+
k

(

P
∞

/

P

k
−
1

)

{\pi _{r + 1 + k}}({P_\infty }/{P_{k - 1}})

as above. Then one reads off from Mahowald’s tables of

π

n
+
m

(

P
∞

/

P

m
−
1

)

{\pi _{n + m}}({P_\infty }/{P_{m - 1}})

that

π
0

Diff
+

:

P

11

=

Z
2

{\pi _0}{\text {Diff}^ + }:{P_{11}} = {Z_2}

and

π
0

Diff
+

:

P

27

=
6

Z
2

{\pi _0}{\text {Diff}^ + }:{P_{27}} = 6{Z_2}

.
Countable box products of ordinals
Authors: Rudin Mary Ellen
Published: July 2010

The countable box product of ordinals is examined in the paper for normality and paracompactness. The continuum hypothesis is used to prove that the box product of countably many

σ
\sigma

-compact ordinals is paracompact and that the box product of another class of ordinals is normal. A third class trivially has a nonnormal product.
Singular perturbations for systems of linear partial differential equations
Authors: Livne A. , Schuss Z.
Published: July 2010

We consider the system of linear partial differential equations

ε

A

i
j

u

i
j

ε

+

B
i

u
i
ε

+
C

u
ε

=
f

\varepsilon {A^{ij}}u_{ij}^\varepsilon + {B^i}u_i^\varepsilon + C{u^\varepsilon } = f

where

A

i
j

,

B
i

{A^{ij}},{B^i}

are symmetric

m
×
m

m \times m

matrices and — C is a sufficiently large positive definite matrix. We prove that under suitable conditions

‖

u
ε

−
u

‖

L
2

≤
c
√
ε

‖
f
‖

H
1

{\left \| {{u^\varepsilon } - u} \right \|_{{L^2}}} \leq c\surd \varepsilon {\left \| f \right \|_{{H^1}}}

where u is the solution of a suitable boundary value problem for the system

B
i

u
i

+
C
u
=
f

{B^i}{u_i} + Cu = f

.
Monotone decompositions of continua not separated by any subcontinua
Authors: Vought Eldon J.
Published: July 2010

Let M be a compact, metric continuum that is separated by no subcontinuum. If such a continuum has a monotone, upper semicontinuous decomposition, the elements of which have void interior and for which the quotient space is a simple closed curve, then it is said to be of type

A

′

{\text {A}}’

. It is proved that a bounded plane continuum is of type

A’

{\text {A’}}

if and only if M contains no indecomposable subcontinuum with nonvoid interior. In

E
3

{E^3}

this condition is not sufficient and an example is given to illustrate this. However, it is shown that if M is hereditarily decomposable then M is of type

A

′

{\text {A}}’

. Next, a condition is given that characterizes continua of type

A’

{\text {A’}}

. Also the structure of the elements in the decomposition of a continuum of type

A’

{\text {A’}}

is discussed and the decomposition is shown to be unique. Finally, some consequences of these results and some remarks are given.
Extensions of normal immersions of 𝑆¹ into 𝑅²
Authors: Marx Morris L.
Published: July 2010

Suppose that

f
:

S
1

→

R
2

f:{S^1} \to {R^2}

is an immersion, i.e., a

C
1

{C^1}

map such that

f
′

f’

is never zero. We call f normal if there are only finitely many self-intersections and these are transverse double points. A normal immersion f can be topologically determined by a finite number of combinatorial invariants. Using these invariants it is possible to give considerable information about extensions of f to

D
2

{D^2}

. In this paper we give a new set of invariants, inspired by the work of S. Blank, to solve several problems concerning the existence of certain kinds of extensions. The problems solved are as follows: (1) When does f have a light open extension

F
:

D
2

→

R
2

F:{D^2} \to {R^2}

? (Recall that light means

F

−
1

(
y
)

{F^{ - 1}}(y)

is totally disconnected for all y and open means F maps open sets of the interior of

D
2

{D^2}

to open sets of

R
2

{R^2}

.) Because of the work of Stoïlow, the question is equivalent to the following: when does there exist a homeomorphism

h
:

S
1

→

S
1

h:{S^1} \to {S^1}

, such that fh has an analytic extension to

D
2

{D^2}

? (2) Suppose that

F
:

D
2

→

R
2

F:{D^2} \to {R^2}

is light, open, sense preserving, and, at each point of

S
1

{S^1}

, F is a local homeomorphism. At each point of the interior of

D
2

{D^2}

, F is locally topologically equivalent to the power mapping

z
m

{z^m}

on

D
2

,
m
≥
1

{D^2},m \geq 1

. Points where

m
>
1

m > 1

are called branch points and

m
−
1

m - 1

is the multiplicity of the point. There are only a finite number of branch points. The problem is to discover the minimum number of branch points of any properly interior extension of f. Also we can ask what multiplicities can arise for extensions of a given f. (3) Given a normal f, find the maximum number of properly interior extensions of f that are pairwise inequivalent. Since each immersion of the disk is equivalent to a local homeomorphism, the problem of immersion extensions is a special case of this. It is Blank’s solution of the immersion problem that prompted this paper.
Local norm convergence of states on the zero time Bose fields
Authors: Bratteli Ola
Published: July 2010

For a sequence of vector states on the Boson Fock space which are norm convergent on the Newton-Wigner local algebras, conditions are given which guarantee norm convergence on the relativistic local algebras also. These conditions are verified for the cutoff physical vacuum states of the

P

(
ϕ

)
2

P{(\phi )_2}

field theory, and yield a simplification of the proof of the locally normal property of the physical vacuum in that theory.