bluuebunny/crossref_metadata_2025 huggingface dataset:
New criteria for freeness in abelian groups
Authors: Hill Paul
Published: July 2010

A new criterion is established for an abelian group to be free. The criterion applies to an ascending chain of free subgroups. The result is used to construct groups that are almost free but not free. In particular, we construct examples that show that the class of free abelian groups is not definable in the logical language

L

∞
κ

{L_{\infty \kappa }}

if

κ
≤

ℵ
2

\kappa \leq {\aleph _2}

. In doing so, we take advantage of a recent theorem of P. Eklof.
Complete topologies on spaces of Baire measure
Authors: Kirk R. B.
Published: July 2010

Let X be a completely regular Hausdorff space, let L be the linear space of all finite linear combinations of the point measures on X and let

M
σ

{M_\sigma }

denote the space of Baire measures on X. The following is proved: If

M
σ

{M_\sigma }

is endowed with the topology of uniform convergence on the uniformly bounded, equicontinuous subsets of

C
b

(
X
)

{C^b}(X)

, then

M
σ

{M_\sigma }

is a complete locally convex space in which L is dense and whose dual is

C
b

(
X
)

{C^b}(X)

, provided there are no measurable cardinals. A complete description of the situation in the presence of measurable cardinals is also given. Let

M
C

{M_C}

be the subspace of

M
σ

{M_\sigma }

consisting of those measures which have compact support in the realcompactification of X. The following result is proved: If

M
C

{M_C}

is endowed with the topology of uniform convergence on the pointwise bounded and equicontinuous subsets of

C
(
X
)

C(X)

, then

M
C

{M_C}

is a complete locally convex space in which L is dense and whose dual is

C
(
X
)

C(X)

, provided there are no measurable cardinals. Again the situation if measurable cardinals exist is described completely. Let M denote the Banach dual of

C
b

(
X
)

{C^b}(X)

. The following is proved: If M is endowed with the topology of uniform convergence on the norm compact subsets of

C
b

(
X
)

{C^b}(X)

, then M is a complete locally convex space in chich L is dense. It is also proved that

M
σ

{M_\sigma }

is metrizable if and only if X is discrete and that the metrizability of either

M
C

{M_C}

or M is equivalent to X being finite. Finally the following is proved: If

M
C

{M_C}

has the Mackey topology for the pair

(

M
C

,
C
(
X
)
)

({M_C},C(X))

, then

M
C

{M_C}

is complete and L is dense in

M
C

{M_C}

.
Higher dimensional generalizations of the Bloch constant and their lower bounds
Authors: Hahn Kyong T.
Published: July 2010

A higher dimensional generalization of the classical Bloch theorem depends in an essential way on the “boundedness” of the family of holomorphic mappings considered. In this paper the author considers two types of such “bounded” families and obtains explicit lower bounds of the generalized Bloch constants of these families on the hyperball in the space

C

n

{{\mathbf {C}}^n}

in terms of universal constants which characterize the families.
Smoothness of certain metric projections on Hilbert space
Authors: Holmes Richard B.
Published: July 2010

A study is made of differential properties of the distance function and the metric projection defined by a closed convex subset of Hilbert space. The former mapping is also considered within the context of more general Banach spaces.
On arbitrary sequences of isomorphisms in 𝑅^{𝑚}→𝑅^{𝑚}
Authors: Pugh Charles C.
Published: July 2010

In this paper a new, clean proof of an algebraic theorem needed in ordinary differential equations is presented. The theorem involves the existence and uniqueness of a “complete splitting” for some subsequence of an arbitrary sequence of isomorphisms of Euclidean m-space. In the positive-definite case, a complete splitting is a limit condition on eigenspaces and eigenvalues.
A characterization of the invariant measures for an infinite particle system with interactions
Authors: Liggett Thomas M.
Published: July 2010

Let

p
(
x
,
y
)

p(x,y)

be the transition function for a symmetric, irreducible, transient Markov chain on the countable set S. Let

η
t

{\eta _t}

be the infinite particle system on S with the simple exclusion interaction and one-particle motion determined by p. A characterization is obtained of all the invariant measures for

η
t

{\eta _t}

in terms of the bounded functions on S which are harmonic with respect to

p
(
x
,
y
)

p(x,y)

. Ergodic theorems are proved concerning the convergence of the system to an invariant measure.
On sequences containing at most 3 pairwise coprime integers
Authors: Choi S. L. G.
Published: July 2010

It has been conjectured by Erdös that the largest number of natural numbers not exceeding n from which one cannot select

k
+
1

k + 1

pairwise coprime integers, where

k
≥
1

k \geq 1

and

n
≥

p
k

n \geq {p_k}

, with

p
k

{p_k}

denoting the kth prime, is equal to the number of natural numbers not exceeding n which are multiples of at least one of the first k primes. It is known that the conjecture holds for k = 1 and 2. In this paper we establish the truth of the conjecture for k = 3.
Decomposition theories for abelian categories
Authors: Fisher Joe W. , Wolff Harvey
Published: July 2010

Both the classical approach to decomposition theories and Fisher’s technique of constructing decomposition theories from radical functions are extended to and exploited in the context of abelian categories. These two different approaches to decomposition theories for abelian categories intertwine in one theorem from which flows necessary and sufficient conditions for the existence of the tertiary, primary, and Bourbaki’s

P

\mathcal {P}

-primary decomposition theories.
Extension of Loewner’s capacity theorem
Authors: Näkki Raimo
Published: July 2010

Analogues of a well-known theorem of Loewner concerning conformal capacity of a space ring are given in the case of an arbitrary domain.
The periods of Eichler integrals for Kleinian groups
Authors: Sato Hiroki
Published: July 2010

We shall give period relations and inequalities for Eichler integrals for Kleinian groups

Γ
\Gamma

which have simply connected components of of the region of discontinuity of

Γ
\Gamma

. These are a generalization of those for abelian integrals. By using the period inequality, we shall give an alternate proof of a result of Kra.
Mielnik’s probability spaces and characterization of inner product spaces
Authors: Stanojevic C. V.
Published: July 2010

A characterization of inner product spaces is given in terms of Mielnik’s probability function. The generalized parallelogram law is related to the functional equation

f
+
f
∘
g
=
1

f + f \circ g = 1

.
Fixed point structures
Authors: Muenzenberger T. B. , Smithson R. E.
Published: July 2010

A fixed point structure is a triple

(
X
,

P

,

F

)

(X,\mathcal {P},\mathcal {F})

where X is a set,

P

\mathcal {P}

a collection of subsets of X, and

F

\mathcal {F}

a family of multifunctions on X into itself together with a set of axioms which insure that each member of

F

\mathcal {F}

has a fixed point. A fixed point structure for noncontinuous multifunctions on semitrees is established that encompasses fixed point theorems of Wallace-Ward and Young-Smithson as well as new fixed point theorems for partially ordered sets and closed stars in real vector spaces. Also two other fixed point structures are presented that subsume fixed point theorems of Tarski-Ward-Smithson on semilattices and, more generally, partially ordered sets. Also the Davis-Ward converse to this last fixed point theorem is obtained.
Products of decompositions of 𝐸ⁿ
Authors: Smith Brian J.
Published: July 2010

In this paper we give a sufficient condition for the existence of a homeomorphism

h
:

E
m

/

G
×

E
n

/

H
→

E

m
+
n

h:{E^m}/G \times {E^n}/H \to {E^{m + n}}

, where G and H are u.s.c. decompositions of Euclidean space. This condition is then shown to hold for a wide class of examples in which the decomposition spaces

E
m

/

G

{E^m}/G

and

E
n

/

H

{E^n}/H

may fail to be Euclidean.
On unaveraged convergence of positive operators in Lebesgue space
Authors: Fong H. , Sucheston L.
Published: July 2010

Let T be a power-bounded positive conservative operator on

L
1

{L_1}

of a

σ
\sigma

-finite measure space. Let e be a bounded positive function invariant under the operator adjoint to T. Theorem. (1)

∫

|

T
n

f

|

⋅

e
→
0

\smallint |{T^n}f| \cdot \;e \to 0

implies (2)

∫

|

T
n

f

|

→
0

\smallint |{T^n}f| \to 0

. If T and all its powers are ergodic, and T satisfies an abstract Harris condition, then(l) holds by the Jamison-Orey theorem for all integrable f with

∫
f
⋅
e
=
0

\smallint f \cdot e = 0

, and hence also (2) holds for such f. A new proof of the Jamison-Orey theorem is given, via the ’filling scheme’. For discrete measure spaces this is due to Donald Ornstein, Proc. Amer. Math. Soc. 22 (1969), 549-551. If T is power-bounded, conservative and ergodic, and

0
>

f
0

=
T

f
0

0 > {f_0} = T{f_0}

, then

f
0

⋅

e
∈

L
1

{f_0}\; \cdot \;e \in {L_1}

implies

f
0

∈

L
1

{f_0} \in {L_1}

, hence (2) implies that

T
n

f

{T^n}f

converges for each

f
∈

L
1

f \in {L_1}

. Theorem. Let T be a positive conservative contraction on

L
1

{L_1}

; then the class of functions

{
f
−
T
f
,
f
∈

L
1
+

}

\{ f - Tf,f \in L_1^ + \}

is dense in the class of functions

{
f
−
T
f
,
f
∈

L
1

}

\{ f - Tf,f \in {L_1}\}

.
On the asymptotic behavior of nonlinear wave equations
Authors: Glassey Robert T.
Published: July 2010

Positive energy solutions of the Cauchy problem for the equation

◻

u
=

m
2

u
+
F
(
u
)

\square u = {m^2}u + F(u)

are considered. With

G
(
u
)
=

∫
0
u

F
(
s
)
d
s

G(u) = \smallint _0^uF(s)ds

, it is proven that

G
(
u
)

G(u)

must be nonnegative in order for uniform decay and the existence of asymptotic “free” solutions to hold. When

G
(
u
)

G(u)

is nonnegative and satisfies a growth restriction at infinity, the kinetic and potential energies (with m = 0) are shown to be asymptotically equal. In case

F
(
u
)

F(u)

has the form

|

u

|

p
−
1

u

|u{|^{p - 1}}u

, scattering theory is shown to be impossible if

1
>
p
≤
1
+
2

n

−
1

(
n
≥
2
)

1 > p \leq 1 + 2{n^{ - 1}}\;(n \geq 2)

.
Stability of foliations
Authors: Levine Harold I. , Shub Michael
Published: July 2010

Let X be a compact manifold and let k be an integer. It is shown that the set of homeomorphism conjugacy classes of germs at X of foliations of codimension k and the set of homeomorphism conjugacy classes of (holonomy) representations of

∏
1

(
X
)

{\prod _1}(X)

in the group of germs at 0 of 0-fixed self-diffeomorphisms of

R

k

{{\text {R}}^k}

are homeomorphic when given appropriate topologies. Stable foliation germs and stable holonomy representations correspond under this homeomorphism. It is shown that there are no stable foliation germs at a toral leaf if the dimension of the torus is greater than one.
Visitations of ruled sums
Authors: Baum Leonard E. , Stratton H. H.
Published: July 2010

Let

{

X
i

}

\{ {X_i}\}

be a sequence of independent identically distributed random variables and for

D
⊆

I
+

D \subseteq {I^ + }

let

S
D

=

Σ

i
∈
D

X
i

{S_D} = {\Sigma _{i \in D}}{X_i}

. A rule

(

)

(\;)

is a mapping

I
+

→

2

I
+

:
∀
n

|

(
n
)

|

=
n

{I^ + } \to {2^{{I^ + }}}:\forall n|(n)| = n

and

S

(

)

=
{

S

(
n
)

}

{S_{(\;)}} = \{ {S_{(n)}}\}

is its associated ruled sum. Ruled sums generalize ordinary sums

S
n

{S_n}

. Indeed, all a.e. results for

S
n

{S_n}

can be investigated for

S

(
n
)

{S_{(n)}}

frequently requiring different methods and sometimes yielding different conclusions. In a previous paper we studied strong laws of large numbers and the law of the iterated logarithm. In this paper we study infinite visitation. Under suitable hypotheses on the basic distribution function F of the

X
i

{X_i}

we show that, for all rules

(

)
,

S

(
n
)

(\;),{S_{(n)}}

visits each integer infinitely often a.e. in the lattice case (or has all points of the real line as accumulation points in the nonlattice case). In fact we obtain a “rate of visitation.” There follows extensions of the Pólya theorem on encounters in the plane and 3-space from random walks to these ruled sums. Some equivalence relations and partial orderings on rules are defined. For normal variables this leads to an extension of the previously mentioned result for ruled sums of the type of the iterated logarithm law.
Jacobson structure theory for Hestenes ternary rings
Authors: Stephenson Robert Allan
Published: July 2010

The principal results are an extension of the density theorem to Hestenes ternary rings and a characterization of primitive ternary rings.
Coterminal families and the strong Markov property
Authors: Pittenger A. O. , Shih C. T.
Published: July 2010

Let

E
Δ

{E_\Delta }

be a compact metric space and assume that a strong Markov process X is defined on

E
Δ

{E_\Delta }

. Under the assumption that X has right continuous paths with left limits, it is shown that a version of the strong Markov property extends to coterminal families, a class of random times which can be visualized as last exit times before t from a fixed subset of

E
Δ

{E_\Delta }

. Since the random times are not Markov times, the conditioning

σ
\sigma

-field and the new conditional probabilities must be defined. If X is also assumed to be nearly quasileft continuous, i.e. branching points are permitted, two different conditionings are possible—one on the “past” of the random time and one on the “past plus present"—and two different conditional probabilities must be defined.
Functions automorphic on large domains
Authors: James David A.
Published: July 2010

For a discontinuous group

Γ
⊂

SL

(
2
,
R
)

\Gamma \subset {\text {SL}}(2,R)

, Poincaré produced a corresponding nonconstant automorphic form, meromorphic on the open upper half plane

Π
+

{\Pi ^ + }

. When the domain of meromorphicity grows larger than

Π
+

{\Pi ^ + }

, the type of group which can support an automorphic form is restricted, and the corresponding forms are generally quite simple. A complete analysis of this phenomenon is presented, with examples which show results are best possible.
Topological entropy for noncompact sets
Authors: Bowen Rufus
Published: July 2010

For

f
:
X
→
X

f:X \to X

continuous and

Y
⊂
X

Y \subset X

a topological entropy

h
(
f
,
Y
)

h(f,Y)

is defined. For X compact one obtains results generalizing known theorems about entropy for compact Y and about Hausdorff dimension for certain

Y
⊂
X
=

S
1

Y \subset X = {S^1}

. A notion of entropy-conjugacy is proposed for homeomorphisms.
Heegaard splittings of homology 3-spheres
Authors: Neumann Dean A.
Published: July 2010

We investigate properties of Heegaard splittings of closed

3
3

-manifolds which are known for simply-connected manifolds and which might provide the basis for a general test for simple-connectivity. Our results are negative: each property considered is shown to hold in a wider class of manifolds.
Completely seminormal operators with boundary eigenvalues
Authors: Clancey Kevin
Published: July 2010

For

f
∈

L
2

(
E
)

f \in {L^2}(E)

we consider the singular integral operator
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript upper E Baseline f left-parenthesis s right-parenthesis equals s f left-parenthesis s right-parenthesis plus pi Superscript negative 1 Baseline integral Underscript upper E Endscripts f left-parenthesis t right-parenthesis left-parenthesis t minus s right-parenthesis Superscript negative 1 Baseline d t">

T
E

f
(
s
)
=
s
f
(
s
)
+

π

−
1

∫
E

f
(
t
)

(
t
−
s

)

−
1

d
t

{T_E}f(s) = sf(s) + {\pi ^{ - 1}}{\smallint _E}f(t){(t - s)^{ - 1}}dt

. These singular integral operators are a special case of operators acting on a Hilbert space with one dimensional self-commutator. We discover generalized eigenfunctions of the equation

T
E

f
=
0

{T_E}f = 0

and, for

p
>
2

p > 2

, we will give an

L
p

(
E
)

{L^p}(E)

solution of the equation

T
E

f
=

χ
E

{T_E}f = {\chi _E}

. The main result of the paper is an example of a nonzero

L
2

(
E
)

{L^2}(E)

solution of

T
E

f
=
0

{T_E}f = 0

, with

λ
=
0

\lambda = 0

a boundary point of the spectrum of

T
E

{T_E}

.
Bernstein approximation problem for differentiable functions and quasi-analytic weights
Authors: Zapata Guido
Published: July 2010

The Bernstein problem for differentiable functions is considered. Sufficient conditions in order that a decreasing family of weights be fundamental are given. Some of these conditions are also related to the concept of quasi-analytic weight.
On the Picard-Lefschetz transformation for algebraic manifolds acquiring general singularities
Authors: Landman Alan
Published: July 2010

We consider a holomorphic family

{

V
t

}

t
∈
D

{\{ {V_t}\} _{t \in D}}

of projective algebraic varieties

V
t

{V_t}

parametrized by the unit disc

D
=
{
t
∈

C

:

|

t

|

>
1
}

D = \{ t \in {\mathbf {C}}:|t| > 1\}

and where

V
t

{V_t}

is smooth for

t
≠
0

t \ne 0

but

V
0

{V_0}

may have arbitrary singularities. Displacement of cycles around a path

t
=

t
0

e

i
θ

(
0
⩽
θ
⩽
2
π
)

t = {t_0}{e^{i\theta }}(0 \leqslant \theta \leqslant 2\pi )

leads to the Picard-Lefschetz transformation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T colon upper H Subscript asterisk Baseline left-parenthesis upper V Subscript t 0 Baseline comma bold upper Z right-parenthesis right-arrow upper H Subscript asterisk Baseline left-parenthesis upper V Subscript t 0 Baseline comma bold upper Z right-parenthesis">

T
:

H
∗

(

V

t
0

,

Z

)
→

H
∗

(

V

t
0

,

Z

)

T:{H_\ast }({V_{{t_0}}},{\mathbf {Z}}) \to {H_\ast }({V_{{t_0}}},{\mathbf {Z}})

on the homology of a smooth

V

t
0

{V_{t0}}

. We prove that the eigenvalues of

T
T

are roots of unity and obtain an estimate on the elementary divisors of

T
T

. Moreover, we give a global inductive procedure for calculating

T
T

in specific examples, several of which are worked out to illustrate the method.
Inner product modules over 𝐵*-algebras
Authors: Paschke William L.
Published: July 2010

This paper is an investigation of right modules over a

B
∗

{B^\ast }

-algebra B which posses a B-valued “inner product” respecting the module action. Elementary properties of these objects, including their normability and a characterization of the bounded module maps between two such, are established at the beginning of the exposition. The case in which B is a

W
∗

{W^\ast }

-algebra is of especial interest, since in this setting one finds an abundance of inner product modules which satisfy an analog of the self-duality property of Hilbert space. It is shown that such self-dual modules have important properties in common with both Hilbert spaces and

W
∗

{W^\ast }

-algebras. The extension of an inner product module over B by a

B
∗

{B^\ast }

-algebra A containing B as a

∗

^\ast

-subalgebra is treated briefly. An application of some of the theory described above to the representation and analysis of completely positive maps is given.
A Stong-Hattori spectral sequence
Authors: Johnson David Copeland
Published: July 2010

Let

G
∗

(

)

{G_ \ast }(\;)

be the Adams summand of connective K-theory localized at the prime p. Let

B

P
∗

(

)

B{P_\ast }(\;)

be Brown-Peterson homology for that prime. A spectral sequence is constructed with

E
2

{E^2}

term determined by

G
∗

(
X
)

{G_ \ast }(X)

and whose

E
∞

{E^\infty }

terms give the quotients of a filtration of

B

P
∗

(
X
)

B{P_ \ast }(X)

where X is a connected spectrum. A torsion property of the differentials implies the Stong-Hattori theorem.
Generalized Dedekind eta-functions and generalized Dedekind sums
Authors: Berndt Bruce C.
Published: July 2010

A transformation formula under modular substitutions is derived for a very large class of generalized Eisenstein series. The result also gives a transformation formula for generalized Dedekind eta-functions. Various types of Dedekind sums arise, and reciprocity laws are established.
Convolution equations and harmonic analysis in spaces of entire functions
Authors: Dickson D. G.
Published: July 2010

If H is the topological space of functions analytic in the simply connected open set

Ω
\Omega

of the plane with the topology of compact convergence, its dual may be identified with the space E of functions of exponential type whose Borel transforms have their singularities in

Ω
\Omega

. For f in H and

ϕ
\phi

in E,

(
f
∗
ϕ
)
(
z
)
≡

⟨

f
,

ϕ
z

⟩

(f \ast \phi )(z) \equiv \left \langle {f,{\phi _z}} \right \rangle

where

ϕ
z

{\phi _z}

is the z-translate of

ϕ
\phi

. If

f

≢

0

f{\nequiv }0

in any component of

Ω
,
f
∗
ϕ
=
0

\Omega ,f \ast \phi = 0

if and only if

ϕ
\phi

is a finite linear combination of monomial-exponentials

z
p

exp
⁡
(
ω
z
)

{z^p} \exp (\omega z)

where

ω
\omega

is a zero of f in

Ω
\Omega

of order at least

p
+
1

p + 1

. For such f and

ψ
\psi

in E,

f
∗
ϕ
=
ψ

f \ast \phi = \psi

is solved explicitly for

ϕ
\phi

. If E is assigned its strong dual topology and

τ
(
ϕ
)

\tau (\phi )

is the closed linear span in E of the translates of

ϕ
\phi

, then

τ
(
ϕ
)

\tau (\phi )

is a finite direct sum of closed subspaces spanned by monomial-exponentials. Each closed translation invariant subspace of E is the kernel of a convolution mapping

ϕ
→
f
∗
ϕ

\phi \to f \ast \phi

; there is a one-to-one correspondence between such subspaces and the closed ideals of H with the correspondence that of annihilators.
Groups of diffeomorphisms and their subgroups
Authors: Omori Hideki
Published: July 2010

This paper has two purposes. The first is to prove the existence of a normal coordinate with respect to a connection defined on the group of diffeomorphisms of a closed manifold, relating to an elliptic complex. The second is to prove a Frobenius theorem with respect to a right invariant distribution defined on the group of diffeomorphisms of a closed manifold, relating to an elliptic complex. Consequently, the group of all volume preserving diffeomorphisms and the group of all symplectic diffeomorphisms are Fréchet Lie groups.
A completely mitotic nonrecursive r.e. degree
Authors: Ladner Richard E.
Published: July 2010

A nonrecursive r.e. degree d is constructed that has the property that every r.e. set of degree d is mitotic. The degree d has several other interesting properties including the property that any two r.e. sets of degree d are weak truth table equivalent.
Multiplicities of second order linear recurrences
Authors: Alter Ronald , Kubota K. K.
Published: July 2010

A second order linear recurrence is a sequence

{

a
n

}

\{ {a_n}\}

of integers satisfying a

a

n
+
2

=
M

a

n
+
1

−
N

a
n

{a_{n + 2}} = M{a_{n + 1}} - N{a_n}

where N and M are fixed integers and at least one

a
n

{a_n}

is nonzero. If k is an integer, then the number

m
(
k
)

m(k)

of solutions of

a
n

=
k

{a_n} = k

is at most 3 (respectively 4) if

M
2

−
4
N
>
0

{M^2} - 4N > 0

and there is an odd prime

q
≠
3

q \ne 3

(respectively q = 3) such that

q

|

M

q|M

and

q
∤
k
N

q\nmid kN

. Further

M
=

sup
k

integer

m
(
k
)

M = {\sup _k}{\;_{{\text {integer}}}}m(k)

is either infinite or

≤
5

\leq 5

provided that either (i)

(
M
,
N
)
=
1

(M,N) = 1

or (ii)

6
∤
N

6\nmid N

.
Structure theory for equational classes generated by quasi-primal algebras
Authors: Quackenbush Robert W.
Published: July 2010

Quasi-primal algebras (which include finite simple polyadic and cylindric algebras) were introduced by A. S. Pixley. In this paper equational classes generated hy quasi-primal algebras are investigated with respect to the following concepts: the congruence extension property, the amalgamation property and the amalgamation class, weak injectives and weak injective hulls, the standard semigroup of operators. A brief discussion of monadic algebras is included to illustrate the results of the paper.
Rank 𝑟 solutions to the matrix equation 𝑋𝐴𝑋^{𝑇}=𝐶,𝐴 alternate, over 𝐺𝐹(2^{𝑦})
Authors: Buckhiester Philip G.
Published: July 2010

Let

GF

(
q
)

{\text {GF}}(q)

denote a finite field of characteristic two. Let

V
n

{V_n}

denote an n-dimensional vector space over

GF

(
q
)

{\text {GF}}(q)

. An

n
×
n

n \times n

symmetric matrix A over

GF

(
q
)

{\text {GF}}(q)

is said to be an alternate matrix if A has zero diagonal. Let A be an

n
×
n

n \times n

alternate matrix over

GF

(
q
)

{\text {GF}}(q)

and let C be an

s
×
s

s \times s

symmetric matrix over

GF

(
q
)

{\text {GF}}(q)

. By using Albert’s canonical forms for symmetric matrices over fields of characteristic two, the number

N
(
A
,
C
,
n
,
s
,
r
)

N(A,C,n,s,r)

of

s
×
n

s \times n

matrices X of rank r over

GF

(
q
)

{\text {GF}}(q)

such that

X
A

X
T

=
C

XA{X^T} = C

is determined. A symmetric bilinear form on

V
n

×

V
n

{V_n} \times {V_n}

is said to be alternating if

f
(
x
,
x
)
=
0

f(x,x) = 0

, for each x in

V
n

{V_n}

. Let f be such a bilinear form. A basis

(

x
1

,
…
,

x
ρ

,

y
1

,
…
,

y
ρ

)
,
n
=
2
ρ

({x_1}, \ldots ,{x_\rho },{y_1}, \ldots ,{y_\rho }),n = 2\rho

, for

V
n

{V_n}

is said to be a symplectic basis for

V
n

{V_n}

if

f
(

x
i

,

x
j

)
=
f
(

y
i

,

y
j

)
=
0

f({x_i},{x_j}) = f({y_i},{y_j}) = 0

and

f
(

x
i

,

y
j

)
=

δ

i
j

f({x_i},{y_j}) = {\delta _{ij}}

, for each i,

j
=
1
,
2
,
…
,
ρ

j = 1,2, \ldots ,\rho

. In determining the number

N
(
A
,
C
,
n
,
s
,
r
)

N(A,C,n,s,r)

, it is shown that a symplectic basis for any subspace of

V
n

{V_n}

, can be extended to a symplectic basis for

V
n

{V_n}

. Furthermore, the number of ways to make such an extension is determined.
Some combinatorial principles
Authors: Ketonen Jussi
Published: July 2010

We extend some large cardinal axioms of Jensen to weakly inaccessible cardinals. Related problems regarding the saturatedness of certain filters are also studied.
𝐾₁ of a curve of genus zero
Authors: Roberts Leslie G.
Published: July 2010

We determine the structure of the vector bundles on a curve of genus zero and calculate the “universal determinant”

K
1

{K_1}

of such a curve.
Counting commutators
Authors: Miech R. J.
Published: July 2010

Let G be a group generated by x and y,

G
2

{G_2}

be the commutator subgroup of G, and

G
1

{G_1}

be the group generated by y and

G
2

{G_2}

. This paper contains explicit expansions of

y

x
m

{y^{{x^m}}}

modulo [

G
2

,

G
2

,

G
2

{G_2},{G_2},{G_2}

] and

(
x
y

)
m

{(xy)^m}

modulo [

G
1

,

G
1

,

G
1

{G_1},{G_1},{G_1}

]. The motivation for these results stem from the p-groups of maximal class, for a large number of these groups have

[

G
1

,

G
1

,

G
1

]
=
1

[{G_1},{G_1},{G_1}] = 1

.
Uniformly distributed sequences in locally compact groups. I
Authors: Benzinger Leonora
Published: July 2010

We investigate the notion of uniformly distributed sequences in locally compact groups. Our main result is the following: A locally compact group G possesses a uniformly distributed sequence if and only if it possesses a sequence whose homomorphic images are dense in each of the compact quotients of G.
On proper homotopy theory for noncompact 3-manifolds
Authors: Brown E. M. , Tucker T. W.
Published: July 2010

Proper homotopy groups analogous to the usual homotopy groups are defined. They are used to prove, modulo the Poincaré conjecture, that a noncompact 3-manifold having the proper homotopy type of a closed product

F
×
[
0
,
1
]

F \times [0,1]

or a half-open product

F
×
[
0
,
1
)

F \times [0,1)

where F is a 2-manifold is actually homeomorphic to

F
×
[
0
,
1
]

F \times [0,1]

or

F
×
[
0
,
1
)

F \times [0,1)

, respectively. By defining a concept for noncompact manifolds similar to boundary-irreducibility, a well-known result of Waldhausen concerning homotopy and homeomorphism type of compact 3-manifolds is extended to the noncompact case.
Weakly almost periodic and uniformly continuous functionals on the Fourier algebra of any locally compact group
Authors: Granirer Edmond E.
Published: July 2010

We define for any locally compact group G, the space of bounded uniformly continuous functionals on Ĝ,

U
C
B
(

G
^

)

UCB(\hat G)

, in the context of P. Eymard [Bull. Soc. Math. France 92 (1964), 181-236. MR 37 #4208] (for notations see next section). For

u
∈
A
(
G
)

u \in A(G)

let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Superscript up-tack Baseline equals left-brace phi element-of upper V upper N left-parenthesis upper G right-parenthesis semicolon phi left-bracket upper A left-parenthesis upper G right-parenthesis u right-bracket equals 0 right-brace">

u
⊥

=
{
ϕ
∈
V
N
(
G
)
;
ϕ
[
A
(
G
)
u
]
=
0
}

{u^ \bot } = \{ \phi \in VN(G);\phi [A(G)u] = 0\}

. Theorem. If for some norm separable subspace

X
⊂
V
N
(
G
)

X \subset VN(G)

and some positive definite

0
≠
u
∈
A
(
G
)
,
U
C
B
(

G
^

)
⊂

0 \ne u \in A(G),UCB(\hat G) \subset

norm closure

[
W
(

G
^

)
+
X
+

u
⊥

]

[W(\hat G) + X + {u^ \bot }]

then G is discrete. If G is discrete then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U upper C upper B left-parenthesis ModifyingAbove upper G With caret right-parenthesis subset-of upper A upper P left-parenthesis ModifyingAbove upper G With caret right-parenthesis subset-of upper W left-parenthesis ModifyingAbove upper G With caret right-parenthesis">

U
C
B
(

G
^

)
⊂
A
P
(

G
^

)
⊂
W
(

G
^

)

UCB(\hat G) \subset AP(\hat G) \subset W(\hat G)

.
Egoroff properties and the order topology in Riesz spaces
Authors: Dodds Theresa K. Y. Chow
Published: July 2010

In this paper we prove that, for a Riesz space L, the order closure of each subset of L coincides with its pseudo order closure if and only if the order closure of each convex subset of L coincides with its pseudo order closure; moreover, each of these statements is equivalent to the strong Egoroff property. For Archimedean Riesz spaces, similar results hold for the relative uniform topology.
Symmetric norm ideals and relative conjugate ideals
Authors: Salinas Norberto
Published: July 2010

In this paper some aspects of the algebraic structure of the ring of all bounded linear operators on an infinite dimensional separable complex Hilbert space are discussed. In particular, a comparison criterion for maximal and minimal norm ideals is established. Also, a general notion of the conjugate of an ideal relative to another ideal is studied and some questions concerning joins and intersections of ideals are solved.
On the structure of the set of solutions of equations involving 𝐴-proper mappings
Authors: Fitzpatrick P. M.
Published: July 2010

Let X and Y be Banach spaces having complete projection schemes (say, for example, they have Schauder bases). We consider various properties of mappings

T
:
D
⊂
X
→
Y

T:D \subset X \to Y

which are either Approximation-proper (A-proper) or the uniform limit of such mappings. In §1 general properties, including those of the generalized topological degree, of such mappings are discussed. In §2 we give sufficient conditions in order that the solutions of an equation involving a nonlinear mapping be a continuum. The conditions amount to requiring that the generalized topological degree not vanish, and that the mapping involved be the uniform limit of well structured mappings. We devote §3 to proving a result connecting the topological degree of an A-proper Fréchet differentiable mapping to the degree of its derivative. Finally, in §4, various Lipschitz-like conditions are discussed in an A-proper framework, and constructive fixed point and surjectivity results are obtained.
Classical quotient rings
Authors: Shock Robert C.
Published: July 2010

Throughout R is a ring with right singular ideal

Z
(
R
)

Z(R)

. A right ideal K of R is rationally closed if

x

−
1

K
=
{
y
∈
R
:
x
y
∈
K
}

{x^{ - 1}}K = \{ y \in R:xy \in K\}

is not a dense right ideal for all

x
∈
R
−
K

x \in R - K

. A ring R is a Cl-ring if R is (Goldie) right finite dimensional,

R

/

Z
(
R
)

R/Z(R)

is semiprime,

Z
(
R
)

Z(R)

is rationally closed, and

Z
(
R
)

Z(R)

contains no closed uniform right ideals. These rings include the quasi-Frobenius rings as well as the semiprime Goldie rings. The commutative Cl-rings have Cl-classical quotient rings. The injective ones are congenerator rings. In what follows, R is a Cl-ring. A dense right ideal of R contains a right nonzero divisor. If R satisfies the minimum condition on rationally closed right ideals then R has a classical Artinian quotient ring. The complete right quotient ring Q (also called the Johnson-Utumi maximal quotient ring) of R is a Cl-ring. If R has the additional property that bR is dense whenever b is a right nonzero divisor, then Q is classical. If Q is injective, then Q is classical.
Hereditary 𝑄𝐼-rings
Authors: Boyle Ann K.
Published: July 2010

We consider in this paper rings in which every quasi-injective right R-module is injective. These rings will be referred to as right QI-rings. For a hereditary ring, this is equivalent to the condition that R be noetherian and a right V-ring. We also consider rings in which proper cyclic right R-modules are injective. These are right QI-rings which are either semisimple or right hereditary, right Ore domains in which indecomposable injective right R-modules are either simple or isomorphic to the injective hull of

R
R

{R_R}

.
A matrix representation for associative algebras. I
Authors: Lewin Jacques
Published: July 2010

Let F be a mixed free algebra on a set X over the field K. Let U, V be two ideals of F, and

{
δ
(
x
)
,
(
x
∈
X
)
}

\{ \delta (x),(x \in X)\}

a basis for a free

(
F

/

U
,
F

/

V
)

(F/U,F/V)

-bimodule T. Then the map

x
→
(

x
+
V

a
m
p
;
0

δ
(
x
)

a
m
p
;

x
+
U

)

x \to (\begin {array}{*{20}{c}} {x + V} & 0 \\ {\delta (x)} & {x + U} \\ \end {array} )

induces an injective homomorphism

F

/

U
V
→
(

F

/

V

a
m
p
;
0

T

a
m
p
;

F

/

U

)

F/UV \to (\begin {array}{*{20}{c}} {F/V} & 0 \\ T & {F/U} \\ \end {array} )

. If

F

/

U

F/U

and

F

/

V

F/V

are embeddable in matrices over a commutative algebra, so is

F

/

U
V

F/UV

. Some special cases are investigated and it is shown that a PI algebra with nilpotent radical satisfies all identities of some full matrix algebra.
A divergent weighted orthonormal series of broken line Franklin functions
Authors: Reed Coke S.
Published: July 2010

The purpose of this paper is to define a differentiable function F and an inner product on the space of continuous functions on [0,1] in such a way that the Fourier expansion of F obtained by orthonormalizing the broken line Franklin functions according to this inner product is divergent.
Symmetric integro-differential-boundary problems
Authors: Zimmerberg Hyman J.
Published: July 2010

Necessary and sufficient conditions for a linear vector integro-differential-boundary problem to be symmetric (selfadjoint) are developed, and then applied to obtain canonical forms of such symmetric problems. Moreover, the formulation of the integro-boundary conditions herein yields a simplification of one of the conditions for selfadjointness of a differential-boundary operator previously announced.
Asymptotic solutions of linear Volterra integral equations with singular kernels
Authors: Wong J. S. W. , Wong R.
Published: July 2010

Volterra integral equations of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u prime left-parenthesis t right-parenthesis equals minus integral Subscript 0 Superscript t Baseline a left-parenthesis t minus tau right-parenthesis u left-parenthesis tau right-parenthesis d tau comma u left-parenthesis 0 right-parenthesis equals 1">

u
′

(
t
)
=
−

∫
0
t

a
(
t
−
τ
)
u
(
τ
)
d
τ
,
u
(
0
)
=
1

u’(t) = - \smallint _0^ta(t - \tau )u(\tau )d\tau ,u(0) = 1

, are considered, where

a
(
t
)
∈
C
(
0
,
∞
)
∩

L
1

(
0
,
1
)

a(t) \in C(0,\infty ) \cap {L_1}(0,1)

. Explicit asymptotic forms are obtained for the solutions, when the kernels

a
(
t
)

a(t)

have a specific asymptotic representation.
Linear operators and vector measures
Authors: Brooks J. K. , Lewis P. W.
Published: July 2010

Compact and weakly compact operators on function spaces are studied. Those operators are characterized by properties of finitely additive set functions whose existence is guaranteed by Riesz representation theorems.
Indecomposable polytopes
Authors: Meyer Walter
Published: July 2010

The space of summands (with respect to vector addition) of a convex polytope in n dimensions is studied. This space is shown to be isomorphic to a convex pointed cone in Euclidean space. The extreme rays of this cone correspond to similarity classes of indecomposable polytopes. The decomposition of a polytope is described and a bound is given for the number of indecomposable summands needed. A means of determining indecomposability from the equations of the bounding hyperplanes is given.
Representation of functions as limits of martingales
Authors: Lamb Charles W.
Published: July 2010

In this paper we show that if

(
Ω
,

F

,
P
)

(\Omega ,\mathcal {F},P)

is a probability space and if

{

F

n

}

n
≥
1

{\{ \mathcal {F}{_n}\} _{n \geq 1}}

is an increasing sequence of sub-

σ
\sigma

-fields of

F

\mathcal {F}

which satisfy an additional condition, then every real valued,

F

∞

{\mathcal {F}_\infty }

-measurable function f can be written as the a.e. limit of a martingale

{

f
n

,

F

n

}

n
≥
1

{\{ {f_n},{\mathcal {F}_n}\} _{n \geq 1}}

. The case where f takes values in the extended real line is also studied. A construction is given of a “universal” martingale

{

f
n

,

F

n

}

n
≥
1

{\{ {f_n},{\mathcal {F}_n}\} _{n \geq 1}}

such that any

F

∞

{\mathcal {F}_\infty }

-measurable function is the a.e. limit of a suitably chosen subsequence.
Weighted norm inequalities for fractional integrals
Authors: Muckenhoupt Benjamin , Wheeden Richard
Published: July 2010

The principal problem considered is the determination of all nonnegative functions,

V
(
x
)

V(x)

, such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper T Subscript gamma Baseline f left-parenthesis x right-parenthesis upper V left-parenthesis x right-parenthesis double-vertical-bar Subscript q Baseline less-than-or-equal-to upper C double-vertical-bar f left-parenthesis x right-parenthesis upper V left-parenthesis x right-parenthesis double-vertical-bar Subscript p">

‖

T
γ

f
(
x
)
V
(
x
)
‖

q

≤
C

‖
f
(
x
)
V
(
x
)
‖

p

\left \|{T_\gamma }f(x)V(x)\right \|_q \leq C\left \|f(x)V(x)\right \|_p

where the functions are defined on

R
n

,
0
>
γ
>
n
,
1
>
p
>
n

/

γ
,
1

/

q
=
1

/

p
−
γ

/

n

{R^n},0 > \gamma > n,1 > p > n/\gamma ,1/q = 1/p - \gamma /n

, C is a constant independent of f and

T
γ

f
(
x
)
=
∫
f
(
x
−
y
)

|

y

|

γ
−
n

d
y

{T_\gamma }f(x) = \smallint f(x - y)|y{|^{\gamma - n}}dy

. The main result is that

V
(
x
)

V(x)

is such a function if and only if
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis StartFraction 1 Over StartAbsoluteValue upper Q EndAbsoluteValue EndFraction integral Underscript upper Q Endscripts left-bracket upper V left-parenthesis x right-parenthesis right-bracket Superscript q Baseline d x right-parenthesis Superscript 1 slash q Baseline left-parenthesis StartFraction 1 Over StartAbsoluteValue upper Q EndAbsoluteValue EndFraction integral Underscript upper Q Endscripts left-bracket upper V left-parenthesis x right-parenthesis right-bracket Superscript minus p prime Baseline d x right-parenthesis Superscript 1 slash p prime Baseline less-than-or-equal-to upper K">

(

1

|

Q

|

∫
Q

[
V
(
x
)
]

q

d
x

)

1

/

q

(

1

|

Q

|

∫
Q

[
V
(
x
)
]

−

p
′

d
x

)

1

/

p
′

≤
K

{\left ( {\frac {1}{{|Q|}}\int _Q {{{[V(x)]}^q}dx} } \right )^{1/q}}{\left ( {\frac {1}{{|Q|}}\int _Q {{{[V(x)]}^{ - p’}}dx} } \right )^{1/p’}} \leq K

\]
where Q is any n dimensional cube,

|

Q

|

|Q|

denotes the measure of Q,

p
′

=
p

/

(
p
−
1
)

p’ = p/(p - 1)

and K is a constant independent of Q. Substitute results for the cases

p
=
1

p = 1

and

q
=
∞

q = \infty

and a weighted version of the Sobolev imbedding theorem are also proved.
Differentiability of solutions to hyperbolic initial-boundary value problems
Authors: Rauch Jeffrey B. , Massey Frank J.
Published: July 2010

This paper establishes conditions for the differentiability of solutions to mixed problems for first order hyperbolic systems of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis partial-differential slash partial-differential t minus sigma-summation upper A Subscript j Baseline partial-differential slash partial-differential x Subscript j Baseline minus upper B right-parenthesis u equals upper F">

(
∂

/

∂
t
−
∑

A
j

∂

/

∂

x
j

−
B
)
u
=
F

(\partial /\partial t - \sum {A_j}\partial /\partial {x_j} - B)u = F

on

[
0
,
T
]
×
Ω
,
M
u
=
g

[0,T] \times \Omega ,Mu = g

on

[
0
,
T
]
×
∂
Ω
,
u
(
0
,
x
)
=
f
(
x
)
,
x
∈
Ω

[0,T] \times \partial \Omega ,u(0,x) = f(x),x \in \Omega

. Assuming that

L

2

{\mathcal {L}_2}

a priori inequalities are known for this equation, it is shown that if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F element-of upper H Superscript s Baseline left-parenthesis left-bracket 0 comma upper T right-bracket times normal upper Omega right-parenthesis comma g element-of upper H Superscript s plus 1 slash 2 Baseline left-parenthesis left-bracket 0 comma upper T right-bracket times partial-differential normal upper Omega right-parenthesis comma f element-of upper H Superscript s Baseline left-parenthesis normal upper Omega right-parenthesis">

F
∈

H
s

(
[
0
,
T
]
×
Ω
)
,
g
∈

H

s
+
1

/

2

(
[
0
,
T
]
×
∂
Ω
)
,
f
∈

H
s

(
Ω
)

F \in {H^s}([0,T] \times \Omega ),g \in {H^{s + 1/2}}([0,T] \times \partial \Omega ),f \in {H^s}(\Omega )

satisfy the natural compatibility conditions associated with this equation, then the solution is of class

C
p

{C^p}

from [0, T] to

H

s
−
p

(
Ω
)
,
0
≤
p
≤
s

{H^{s - p}}(\Omega ),0 \leq p \leq s

. These results are applied to mixed problems with distribution initial data and to quasi-linear mixed problems.
𝜆 connected plane continua
Authors: Hagopian Charles L.
Published: July 2010

A continuum M is said to be

λ

{\mathbf {\lambda }}

connected if any two distinct points of M can be joined by a hereditarily decomposable continuum in M. Recently this generalization of arcwise connectivity has been related to fixed point ptoblems in the plane. In particular, it is known that every

λ

{\mathbf {\lambda }}

connected nonseparating plane continuum has the fixed point property. The importance of arcwise connectivity is, to a considerable extent, due to the fact that it is a continuous invariant. To show that

λ

{\mathbf {\lambda }}

connectivity has a similar feature is the primary purpose of this paper. Here it is proved that if M is a

λ

{\mathbf {\lambda }}

connected continuum and f is a continuous function of M into the plane, then

f
(
M
)

f(M)

is

λ

{\mathbf {\lambda }}

connected. It is also proved that every semiaposyndetic plane continuum is

λ

{\mathbf {\lambda }}

connected.
Wreath products and existentially complete solvable groups
Authors: Saracino D.
Published: July 2010

It is known that the theory of abelian groups has a model companion but that the theory of groups does not. We show that for any fixed

n
≥
2

n \geq 2

the theory of groups solvable of length

≤
n

\leq n

has no model companion. For the metabelian case

(
n
=
2
)

(n = 2)

we prove the stronger result that the classes of finitely generic, infinitely generic, and existentially complete metabelian groups are all distinct. We also give some algebraic results on existentially complete metabelian groups.
Waring’s problem for twenty-two biquadrates
Authors: Thomas Henry E.
Published: July 2010

That every natural number is the sum of at most twenty-two biquadrates is proven by ascent from machine results on sums of six fourth powers.
Symplectic homogeneous spaces
Authors: Chu Bon Yao
Published: July 2010

It is proved in this paper that for a given simply connected Lie group G with Lie algebra

g

\mathfrak {g}

, every left-invariant closed 2-form induces a symplectic homogeneous space. This fact generalizes the results in [7] and [12] that if

H
1

(

g

)
=

H
2

(

g

)
=
0

{H^1}(\mathfrak {g}) = {H^2}(\mathfrak {g}) = 0

, then the most general symplectic homogeneous space covers an orbit in the dual of the Lie algebra

g

\mathfrak {g}

. A one-to-one correspondence can be established between the orbit space of equivalent classes of 2-cocycles of a given Lie algebra and the set of equivalent classes of simply connected symplectic homogeneous spaces of the Lie group. Lie groups with left-invariant symplectic structure cannot be semisimple; hence such groups of dimension four have to be solvable, and connected unimodular groups with left-invariant symplectic structure are solvable [4].
Regularly varying functions and convolutions with real kernels
Authors: Jordan G. S.
Published: July 2010

Let

ϕ
\phi

be a positive, measurable function and k a real-valued function on

(
0
,
∞
)
,
k
∈

L
1

(
d
t

/

t
)

(0,\infty ),k \in {L^1}(dt/t)

. We give conditions on

ϕ
\phi

and k sufficient to deduce the regular variation of

ϕ
\phi

from the assumption that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha equals limit Underscript x right-arrow normal infinity Endscripts StartFraction 1 Over phi left-parenthesis x right-parenthesis EndFraction integral Subscript 0 Superscript normal infinity Baseline phi left-parenthesis t right-parenthesis k left-parenthesis StartFraction x Over 2 EndFraction right-parenthesis StartFraction d t Over t EndFraction exits left-parenthesis alpha not-equals 0 comma normal infinity right-parenthesis period">

α
=

lim

x
→
∞

1

ϕ
(
x
)

∫
0
∞

ϕ
(
t
)
k

(

x
2

)

d
t

t

exits

(
α
≠
0
,
∞
)
.

\alpha = \lim \limits _{x \to \infty } \frac {1}{{\phi (x)}}\int _0^\infty {\phi (t)k\left ( {\frac {x}{2}} \right )} \;\frac {{dt}}{t}\;{\text {exits}}\;(\alpha \ne 0,\infty ).

\]
The general theorems extend in certain ways results of other authors and yield a new theorem on the relation between the radial growth and zero-distribution of those entire functions which are canonical products of nonintegral order with negative zeros.
An extension of Weyl’s lemma to infinite dimensions
Authors: Elson Constance M.
Published: July 2010

A theory of distributions analogous to Schwartz distribution theory is formulated for separable Banach spaces, using abstract Wiener space techniques. A distribution T is harmonic on an open set U if for any test function f on U,

T
(
Δ
f
)
=
0

T(\Delta f) = 0

, where

Δ
f

\Delta f

fis the generalized Laplacian of f. We prove that a harmonic distribution on U can be represented as a unique measure on any subset of U which is a positive distance from

U
C

{U^C}

. In the case where the space is finite dimensional, it follows from Weyl’s lemma that the measure is in fact represented by a

C
∞

{C^\infty }

function. This functional representation cannot be expected in infinite dimensions, but it is shown that the measure has smoothness properties analogous to infinite differentiability of functions.
Primitive elements and one relation algebras
Authors: Aust Catherine
Published: July 2010

Let F be a free algebra in a variety V. An element p of F is called primitive if it is contained in some free generating set for F. In 1936, J. H. C. Whitehead proved that a group with generators

g
1

,
…
,

g
n

{g_1}, \ldots ,{g_n}

and one relation

r
=
1

r = 1

is free if and only if the relator r is primitive in the free group on

g
1

,
…
,

g
n

{g_1}, \ldots ,{g_n}

. In tnis paper, tne question of whether there is an analogous theorem for other varieties is considered. A necessary and sufficient condition that a finitely generated, one relation algebra be free is proved for any Schreier variety of nonassociative linear algebras and for any variety defined by balanced identities. An identity

u
(

x
1

,
…
,

x
n

)
=
v
(

x
1

,
…
,

x
n

)

u({x_1}, \ldots ,{x_n}) = v({x_1}, \ldots ,{x_n})

is called balanced if each of u and v has the same length and number of occurrences of each

x
i

{x_i}

. General sufficiency conditions that a finitely generated, one relation algebra be free are given, and all of the known results analogous to the Whitehead theorem are shown to be equivalent to a general necessary condition. Also an algebraic proof of Whitehead’s theorem is outlined to suggest the line of argument for other varieties.