bluuebunny/crossref_metadata_2025 huggingface dataset:
Higher algebraic 𝐾-theories
Authors: Anderson D. , Karoubi M. , Wagoner J.
Published: July 2010

A homotopy fibration is established relating the Volodin or BN-pair definition of algebraic K-theory to the theory defined by Quillen. In [2] we outlined the construction of natural homomorphisms
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript asterisk Superscript upper Q Baseline right-arrow upper K Subscript asterisk Superscript upper B upper N Baseline right-arrow upper K Subscript asterisk Superscript upper V Baseline right-arrow upper K Subscript asterisk Superscript upper K upper V">

K
∗
Q

→

K
∗

B
N

→

K
∗
V

→

K
∗

K
V

K_ \ast ^Q \to K_ \ast ^{BN} \to K_ \ast ^V \to K_ \ast ^{KV}

\]
between higher algebraic K-theories

K
∗
Q

K_ \ast ^Q

of [10] and [11],

K
∗

B
N

K_ \ast ^{BN}

of [17],

K
∗
V

K_ \ast ^V

of [16], and

K
∗

K
V

K_ \ast ^{KV}

of [7] and [8]. This was one of the steps in proving the various definitions of higher K-theory are equivalent. It turns out they all agree-including the theory

K
∗
S

K_ \ast ^S

of [14], [5], and [8]-provided one restricts to the category of regular rings when using

K
∗

K
V

K_\ast ^{KV}

. See [1], [2], [5], [8] and [18]. The purpose of this paper is to prove the following theorem, announced in [2], which yields the construction of

K
∗
Q

→

K
∗

B
N

K_ \ast ^Q \to K_ \ast ^{BN}

. Theorem. For any associative ring with identity A
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L Superscript upper B upper N Baseline left-parenthesis upper A right-parenthesis right-arrow upper B left-brace upper U Subscript upper F Baseline right-brace Superscript plus Baseline right-arrow upper B upper G upper L left-parenthesis upper A right-parenthesis Superscript plus">

G

L

B
N

(
A
)
→
B

{

U
F

}
+

→
B
G
L

(
A

)
+

G{L^{BN}}(A) \to B{\{ {U_F}\} ^ + } \to BGL{(A)^ + }

\]
is a homotopy fibration. For the reader’s convenience and because the presentation of the BN-pair K-theory

K
∗

B
N

K_ \ast ^{BN}

used here is slightly different from that of [17], we shall briefly recall the definition of

G

L

B
N

G{L^{BN}}

and

B
{

U
F

}

B\{ {U_F}\}

in the first section.
Alternator and associator ideal algebras
Authors: Hentzel Irvin Roy , Piacentini Cattaneo Giulia Maria , Floyd Denis
Published: July 2010

If I is the ideal generated by all associators,

(
a
,
b
,
c
)
=
(
a
b
)
c
−
a
(
b
c
)

(a,b,c) = (ab)c - a(bc)

, it is well known that in any nonassociative algebra

R
,
I
⊆
(
R
,
R
,
R
)
+
R
(
R
,
R
,
R
)

R,I \subseteq (R,R,R) + R(R,R,R)

. We examine nonassociative algebras where

I
⊆
(
R
,
R
,
R
)

I \subseteq (R,R,R)

. Such algebras include

(
−
1
,
1
)

( - 1,1)

algebras, Lie algebras, and, as we show, a large number of associator dependent algebras. An alternator is an associator of the type (a, a, b), (a, b, a), (b, a, a). We next study algebras where the additive span of all alternators is an ideal. These include all algebras where

I
=
(
R
,
R
,
R
)

I = (R,R,R)

as well as alternative algebras. The last section deals with prime, right alternative, alternator ideal algebras satisfying an identity of the form

[
x
,
(
x
,
x
,
a
)
]
=
γ
(
x
,
x
,
[
x
,
a
]
)

[x,(x,x,a)] = \gamma (x,x,[x,a])

for fixed

γ
\gamma

. With two exceptions, if this algebra has an idempotent e such that

(
e
,
e
,
R
)
=
0

(e,e,R) = 0

, then the algebra is alternative. All our work deals with algebras with an identity element over a field of characteristic prime to 6. All our containment relations are given by identities.
Composition series and intertwining operators for the spherical principal series. I
Authors: Johnson Kenneth D. , Wallach Nolan R.
Published: July 2010

Let G be a connected semisimple Lie group with finite center and let K be a maximal compact subgroup. Let

π
\pi

be a not necessarily unitary principal series representation of G on the Hilbert space

H
π

{H^\pi }

. If

X
π

{X^\pi }

denotes the space of K-finite vectors of

H
π

,
π

{H^\pi },\pi

induces a representation

π
0

{\pi _0}

of

U
(
g
)

U(g)

, the enveloping algebra of the Lie algebra of G, on

X
π

{X^\pi }

. In this paper, we determine when

π
0

{\pi _0}

is irreducible, and if

π
0

{\pi _0}

is not irreducible we determine the composition series of

X
π

{X^\pi }

and the structure of the induced representations on the subquotients. Explicit computation of the intertwining operators for the different principal series representations are obtained and their relationship to polynomials defined by B. Kostant are obtained.
On everywhere-defined integrals
Authors: Dubins Lester E.
Published: July 2010

Hardly any finite integrals can be defined for all real-valued functions. In contrast, if infinity is admitted as a possible value for the integral, then every finite integral can be extended to all real-valued functions.
Representation theorems for nonlinear disjointly additive functionals and operators on Sobolev spaces
Authors: Marcus Moshe , Mizel Victor J.
Published: July 2010

An abstract characterization is obtained for a class of nonlinear differential operators defined on the subspace

S
=

W
˚

k
p

[
a
,
b
]

S = {\mathring {W}}_k^p[a,b]

of the kth order Sobolev space

W
k
p

[
a
,
b
]
,
1
⩽
k
,
1
⩽
p
⩽
∞

W_k^p[a,b], 1 \leqslant k, 1 \leqslant p \leqslant \infty

. It is shown that every mapping

T
:
S
→

L
1

[
a
,
b
]

T:S \to {L^1}[a,b]

which is local, continuous and

D
k

{D^k}

-disjointly additive has the form

(
T
u
)
(
t
)
=
H
(
t
,

D
k

u
(
t
)
)

(Tu)(t) = H(t,{D^k}u(t))

, where

H
:
[
a
,
b
]
×
R
→
R

H:[a,b] \times R \to R

is a function obeying Carathéodory conditions as well as

(
∗
)
H
(
⋅
,
0
)
=
0

(\ast )H( \cdot ,0) = 0

. Here

D
k

{D^k}

-disjoint additivity means

T
(
u
+
v
)
=
T
u
+
T
v

T(u + v) = Tu + Tv

whenever

(

D
k

u
)
(

D
k

v
)
=
0

({D^k}u)({D^k}v) = 0

. Likewise, every real functional N on S which is continuous and

D
k

{D^k}

-disjointly additive has the form

N
(
u
)
=
∫
T
u

N(u) = \smallint Tu

, with T as above. Liapunov’s theorem on vector measures plays a crucial role, and the analysis suggests new questions about such measures. Likewise, a new type of Radon-Nikodým theorem is employed in an essential way.
Knot modules. I
Authors: Levine Jerome
Published: July 2010

For a differentiable knot, i.e. an imbedding

S
n

⊂

S

n
+
2

{S^n} \subset {S^{n + 2}}

, one can associate a sequence of modules

{

A
q

}

\{ {A_q}\}

over the ring

Z
[
t
,

t

−
1

]

Z[t,{t^{ - 1}}]

, which are the source of many classical knot invariants. If X is the complement of the knot, and

X
~

→
X

\tilde X \to X

the canonical infinite cyclic covering, then

A
q

=

H
q

(

X
~

)

{A_q} = {H_q}(\tilde X)

. In this work a complete algebraic characterization of these modules is given, except for the Z-torsion submodule of

A
1

{A_1}

.
Restrictions of convex subsets of 𝐶(𝑋)
Authors: Hag Per
Published: July 2010

The main result of this paper is a theorem giving a measure-theoretic condition which is necessary and sufficient for a closed convex subset S of

C
(
X
)

C(X)

to have the so-called bounded extension property with respect to a closed subset F of X. This theorem generalizes well-known results on closed subspaces by Bishop, Gamelin and Semadeni.
The lattice of patterns induced by a positive cone of functions
Authors: Hartfiel D. J. , Maxson C. J.
Published: July 2010

This paper contains a study of a special type of lattice which arises by considering the supports of functions in a positive cone of functions. It is shown that many known combinatorial structures provide examples of such a lattice. The basic problem addressed in the paper is that of determining the structure of this lattice.
Holomorphic continuation of smooth functions over Levi-flat hypersurfaces
Authors: Bedford Eric
Published: July 2010

Here we consider the singularities of a Levi-flat real hypersurface S in C that lie in an analytic variety of codimension 2. It is shown that, from the geometric point of view, there are two kinds of singularities, and the type of singularity determines whether S bounds a domain of holomorphy of type

A
∞

{A^\infty }

.
On analytically invariant subspaces and spectra
Authors: Herrero Domingo A.
Published: July 2010

Let T be a bounded linear operator from a complex Banach space

X

\mathfrak {X}

into itself. Let

A

T

{\mathcal {A}_T}

and

A

T
a

\mathcal {A}_T^a

denote the weak closure of the polynomials and the rational functions (with poles outside the spectrum

σ
(
T
)

\sigma (T)

of T) in T, respectively. The lattice

Lat

A

T
a

{\operatorname {Lat}}\;\mathcal {A}_T^a

of (closed) invariant subspaces of

A

T
a

\mathcal {A}_T^a

is a very particular subset of the invariant subspace lattice

Lat

A

T

=

Lat

T

{\operatorname {Lat}}\;{\mathcal {A}_T} = {\operatorname {Lat}}\;T

of T. It is shown that: (1) If the resolvent set of T has finitely many components, then

Lat

A

T
a

{\operatorname {Lat}}\;\mathcal {A}_T^a

is a clopen (i.e., closed and open) sublattice of

Lat

T

{\operatorname {Lat}}\;T

, with respect to the “gap topology” between subspaces. (2) If

M

1

,

M

2

∈

Lat

T
,

M

1

∩

M

2

∈

Lat

A

T
a

{\mathfrak {M}_1},{\mathfrak {M}_2} \in {\operatorname {Lat}}\;T,{\mathfrak {M}_1} \cap {\mathfrak {M}_2} \in {\operatorname {Lat}}\;\mathcal {A}_T^a

and

M

1

+

M

2

{\mathfrak {M}_1} + {\mathfrak {M}_2}

is closed in

X

\mathfrak {X}

and belongs to

Lat

A

T
a

{\operatorname {Lat}}\;\mathcal {A}_T^a

, then

M

1

{\mathfrak {M}_1}

and

M

2

{\mathfrak {M}_2}

also belong to

Lat

A

T
a

{\operatorname {Lat}}\;\mathcal {A}_T^a

. (3) If

M

∈

Lat

T
,
R

\mathfrak {M} \in {\operatorname {Lat}}\;T,R

is the restriction of T to

M

\mathfrak {M}

and

T
¯

\bar T

is the operator induced by T on the quotient space

X

/

M

\mathfrak {X}/\mathfrak {M}

, then

σ
(
T
)
⊂
σ
(
R
)
∪
σ
(

T
¯

)

\sigma (T) \subset \sigma (R) \cup \sigma (\bar T)

. Moreover,

σ
(
T
)
=
σ
(
R
)
∪
σ
(

T
¯

)

\sigma (T) = \sigma (R) \cup \sigma (\bar T)

if and only if

M

∈

Lat

A

T
a

\mathfrak {M} \in {\operatorname {Lat}}\;\mathcal {A}_T^a

. The results also include an analysis of the semi-Fredholm index of R and

T
¯

\bar T

at a point

λ
∈
σ
(
R
)
∪
σ
(

T
¯

)
∖
σ
(
T
)

\lambda \in \sigma (R) \cup \sigma (\bar T)\backslash \sigma (T)

and extensions of the results to algebras between

A

T

{\mathcal {A}_T}

and

A

T
a

\mathcal {A}_T^a

.
Some properties of families of convex cones
Authors: Katchalski Meir
Published: July 2010

The purpose of this paper is to study properties of finite families of convex cones in n-dimensional Euclidean space

R
n

{R^n}

, whose members all have the origin as a common apex. Of special interest are such families of convex cones in

R
n

{R^n}

which have the following property: Each member of the family is of dimension n, the intersection of any two members is at least

(
n
−
1
)

(n - 1)

-dimensional, ..., the intersection of any n members is at least 1-dimensional and the intersection of all the members is the origin.
The 𝑝-adic log gamma function and 𝑝-adic Euler constants
Authors: Diamond Jack
Published: July 2010

We define

G
p

{G_p}

, a p-adic analog of the classical log gamma function and show it satisfies relations similar to the standard formulas for log gamma. We also define p-adic Euler constants and use them to obtain results on

G

p
′

G{’_p}

and on the logarithmic derivative of Morita’s

Γ
p

{\Gamma _p}

.
A bound on the rank of purely simple systems
Authors: Okoh Frank
Published: July 2010

A pair of complex vector spaces (V, W) is called a system if and only if there is a C-bilinear map from

C

2

×
V

{{\mathbf {C}}^2} \times V

to W. The category of systems contains subcategories equivalent to the category of modules over the ring of complex polynomials. Many concepts in the latter generalize to the category of systems. In this paper the pure projective systems are characterized and a bound on the rank of purely simple systems is obtained.
Horocycle flows on certain surfaces without conjugate points
Authors: Eberlein Patrick
Published: July 2010

We study the topological but not ergodic properties of the horocycle flow

{

h
t

}

\{ {h_t}\}

in the unit tangent bundle SM of a complete two dimensional Riemannian manifold M without conjugate points that satisfies the “uniform Visibility” axiom. This axiom is implied by the curvature condition

K
⩽
c
>
0

K \leqslant c > 0

but is weaker so that regions of positive curvature may occur. Compactness is not assumed. The method is to relate the horocycle flow to the geodesic flow for which there exist useful techniques of study. The nonwandering set

Ω
h

⊆
S
M

{\Omega _h} \subseteq SM

for

{

h
t

}

\{ {h_t}\}

is classified into four types depending upon the fundamental group of M. The extremes that

Ω
h

{\Omega _h}

be a minimal set for

{

h
t

}

\{ {h_t}\}

and that

Ω
h

{\Omega _h}

admit periodic orbits are related to the existence or nonexistence of compact “totally convex” sets in M. Periodic points are dense in

Ω
h

{\Omega _h}

if they exist at all. The only compact minimal sets in

Ω
h

{\Omega _h}

are periodic orbits if M is noncompact The flow

{

h
t

}

\{ {h_t}\}

is minimal in SM if and only if M is compact. In general

{

h
t

}

\{ {h_t}\}

is topologically transitive in

Ω
h

{\Omega _h}

and the vectors in

Ω
h

{\Omega _h}

with dense orbits are classified. If the fundamental group of M is finitely generated and

Ω
h

=
S
M

{\Omega _h} = SM

then

{

h
t

}

\{ {h_t}\}

is topologically mixing in SM.
Almost split sequences for group algebras of finite representation type
Authors: Reiten Idun
Published: July 2010

Let k be an algebraically closed field of characteristic p and G a finite group such that p divides the order of G. We compute all almost split sequences over kG when kG is of finite representation type, or more generally, for a finite dimensional k-algebra

Λ
\Lambda

given by a Brauer tree. We apply this to show that if

Λ
\Lambda

and

Λ
′

\Lambda ’

are stably equivalent k-algebras given by Brauer trees, then they have the same number of simple modules.
Fréchet differentiable functionals and support points for families of analytic functions
Authors: Cochrane Paul , MacGregor Thomas H.
Published: July 2010

Given a closed subset of the family

S
∗

(
α
)

{S^\ast }(\alpha )

of functions starlike of order

α
\alpha

of a particular form, a continuous Fréchet differentiable functional, J, is constructed with this collection as the solution set to the extremal problem

max
Re
⁡
J
(
f
)

\max \operatorname {Re} J(f)

over

S
∗

(
α
)

{S^\ast }(\alpha )

. Similar results are proved for families which can be put into one-to-one correspondence with

S
∗

(
α
)

{S^\ast }(\alpha )

. The support points of

S
∗

(
α
)

{S^\ast }(\alpha )

and

K
(
α
)

K(\alpha )

, the functions convex of order

α
\alpha

, are completely characterized and shown to coincide with the extreme points of their respective convex hulls. Given any finite collection of support points of

S
∗

(
α
)

{S^\ast }(\alpha )

(or

K
(
α
)

K(\alpha )

), a continuous linear functional, J, is constructed with this collection as the solution set to the extremal problem

max
Re
⁡
J
(
f
)

\max \operatorname {Re} J(f)

over

S
∗

(
α
)

{S^\ast }(\alpha )

(or

K
(
α
)

K(\alpha )

).
On the rationality of divisors and meromorphic functions
Authors: Tung Chia Chi
Published: July 2010

Let E be a holomorphic vector bundle over a connected complex manifold X and D a divisor on E. Let

A
(
D
)

A(D)

be the set of all

x
∈
X

x \in X

for which

(

supp

D
)
∩

E
x

({\text {supp}}\;D) \cap {E_x}

is a proper algebraic set in

E
x

{E_x}

. The purpose of this paper is to prove that the following conditions are equivalent: (i)

A
(
D
)

A(D)

has positive measure in X; (ii) D extends to a unique divisor on the projective completion Ē of E; (iii) D is locally given by the divisors of rational meromorphic functions defined over open sets in X. Similar results for meromorphic functions are derived. The proof requires an extension theorem for analytic set: Assume E is a holomorphic vector bundle over a pure p-dimensional complex space X and S an analytic set in E of pure codimension 1. Then the closure S of S in E is analytic if and only if

S
∩

E
x

S \cap {E_x}

is a proper algebraic set for all x in a set of positive 2p-measure in every branch of X.
Global structural stability of a saddle node bifurcation
Authors: Robinson Clark
Published: July 2010

S. Newhouse, J. Palis, and F. Takens have recently proved the global structural stability of a one parameter unfolding of a saddle node when the nonwandering set is finite and transversality conditions are satisfied. (The diffeomorphism is Morse-Smale except for the saddle node.) Using their local unfolding of a saddle node and our method of compatible families of unstable disks (instead of the more restrictive method of compatible systems of unstable tubular families), we are able to extend one of their results to the case where the nonwandering set is infinite. We assume that a saddle node is introduced away from the rest of the nonwandering set which is hyperbolic (Axiom A), and that a (strong) transversality condition is satisfied.
Perturbation of translation invariant positivity preserving semigroups on 𝐿²(𝑅^{𝑁})
Authors: Herbst Ira W. , Sloan Alan D.
Published: July 2010

The theory of singular local perturbations of translation invariant positivity preserving semigroups on

L
2

(

R

N

,

d
N

x
)

{L^2}({{\mathbf {R}}^N},{d^N}x)

is developed. A powerful approximation theorem is proved which allows the treatment of a very general class of singular perturbations. Estimates on the local singularities of the kernels of the semigroups,

e

−
t
H

{e^{ - tH}}

, are given. Eigenfunction expansions are derived. The local singularities of the eigenfunction and generalized eigenfunctions are discussed. Results are illustrated with examples involving singular perturbations of —

Δ
\Delta

.
Generic properties of eigenfunctions of elliptic partial differential operators
Authors: Albert Jeffrey H.
Published: July 2010

The problem considered here is that of describing generically the zeros, critical points and critical values of eigenfunctions of elliptic partial differential operators. We consider operators of the form

L
+
ρ

L + \rho

, where L is a fixed, second-order, selfadjoint,

C
∞

{C^\infty }

linear elliptic partial differential operator on a compact manifold (without boundary) and

ρ
\rho

is a

C
∞

{C^\infty }

function. It is shown that, for almost all

ρ
\rho

, i.e. for a residual set, the eigenvalues of

L
+
ρ

L + \rho

are simple and the eigenfunctions have the following properties: (1) they are Morse functions; (2) distinct critical points have distinct critical values; (3) 0 is not a critical value.
Continuous maps of the interval with finite nonwandering set
Authors: Block Louis
Published: July 2010

Let f be a continuous map of a closed interval into itself, and let

Ω
(
f
)

\Omega (f)

denote the nonwandering set of f. It is shown that if

Ω
(
f
)

\Omega (f)

is finite, then

Ω
(
f
)

\Omega (f)

is the set of periodic points of f. Also, an example is given of a continuous map g, of a compact, connected, metrizable, one-dimensional space, for which

Ω
(
g
)

\Omega (g)

consists of exactly two points, one of which is not periodic.
Operators with small self-commutators
Authors: Del Valle J. W.
Published: July 2010

Let A be a bounded operator on a Hilbert space H. The self-commutator of A, denoted [A], is

A
∗

A
−
A

A
∗

{A^\ast }A - A{A^\ast }

. An operator is of commutator rank n if the rank of [A] is n. In this paper operators of commutator rank one are studied. Two particular subclasses are investigated in detail. First, completely nonnormal operators of commutator rank one for which

A
∗

A

{A^\ast }A

and

A

A
∗

A{A^\ast }

commute are completely characterized. They are shown to be special types of simple weighted shifts. Next, operators of commutator rank one for which

{

A
n

e

}

n
=
0

∞

\{ {A^n}e\} _{n = 0}^\infty

is an orthogonal sequence (where e is a generator of the range of [A]) are characterized as a type of weighted operator shift.
On the Seifert manifold of a 2-knot
Authors: Gutierrez M. A.
Published: July 2010

From geometric facts about embeddings

S
2

→

S
4

{S^2} \to {S^4}

we study the relationship between the smallest number of normal generators (weight) of a group and its preabelian presentations.
Processes with independent increments on a Lie group
Authors: Feinsilver Philip
Published: July 2010

The Lévy-Khinchin representation for processes with independent increments is extended to processes taking values in a Lie group. The basis of the proof is to approximate continuous time processes by Markov chains. The processes involved are handled by the technique, developed by Stroock and Varadhan, of characterizing Markov processes by associated martingales.
On the growth of solutions of algebraic differential equations
Authors: Bank Steven B.
Published: July 2010

In this paper we determine estimates for the growth of both real-valued and complex-valued solutions of algebraic differential equations on an interval

(

x
0

,
+
∞
)

({x_0}, + \infty )

. One of the main results of the paper (Theorem 3) confirms E. Borel’s conjecture on the growth of real-valued solutions for a broad class of solutions of second-order algebraic differential equations. The conjecture had previously been shown to be false for third-order equations.
Absolutely area minimizing singular cones of arbitrary codimension
Authors: Bindschadler David
Published: July 2010

The examples of area minimizing singular cones of codimension one discovered by Bombieri, DeGiorgi and Guisti are generalized to arbitrary codimension, thus filling a dimensional gap. Previously the only nontrivial examples of singular area minimizing integral currents of codimension other than one were obtained from holomorphic varieties and hence of even codimension. Specifically, let S be the N-fold Cartesian product of p-dimensional spheres and C be the cone over S. We prove that for p sufficiently large, C is absolutely area minimizing. It follows from the technique used that C restricted to the ball of radius

N

1

/

2

{N^{1/2}}

is the unique solution to the oriented Plateau problem with boundary S.
Diffeomorphisms almost regularly homotopic to the identity
Authors: Wells Robert
Published: July 2010

Let

f
:
M
→
M

f:M \to M

be a self-map of a closed smooth

n
n

-manifold. Does there exist a diffeomorphism

φ
:
M
→
M

\varphi :M \to M

homotopic to

f
f

? Define

φ
\varphi

to be almost regularly homotopic to the identity if

φ

|

M
−

p
t

\varphi |M - \mathrm {pt}

. is regularly homotopic to the inclusion

M
−

p
t

.
⊂
M

M - \mathrm {pt}. \subset M

. Let

ψ
:
M
→
M
∨
M

\psi :M \to M \vee M

be the result of collapsing the boundary of a smooth

n
n

-cell in

M
M

, and let

M
∨
M

→

Δ
′

M

M \vee M \stackrel {\Delta ’}{\to } M

be the codiagonal. For

ξ
∈

π
n

(
M
)

\xi \in {\pi _n}(M)

define

τ
(
ξ
)

\tau (\xi )

to be the composition
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M right-arrow Overscript psi Endscripts upper M logical-or upper M right-arrow Overscript 1 logical-or xi Endscripts upper M logical-or upper M right-arrow Overscript normal upper Delta Superscript prime Baseline Endscripts upper M period">

M

→

ψ

M
∨
M

→

1
∨
ξ

M
∨
M

→

Δ
′

M
.

M \stackrel {\psi }{\rightarrow } M \vee M \stackrel {1 \vee \xi }{\rightarrow } M \vee M \stackrel {\Delta ’}{\rightarrow } M.

\]
Theorem. If

M
M

is

2
2

-connected,

s
s

-parallelizable, and

n
=
2
l
>
5

n = 2l > 5

with

l

≢

0
mod
(
4
)

l \nequiv 0 \bmod (4)

, then

τ
(
ξ
)

\tau (\xi )

contains a diffeomorphism almost regularly homotopic to the identity iff

ξ
\xi

is in the kernel of the stabilization map

π
n

(
M
)
→

π
n
s

(
M
)

{\pi _n}(M) \to \pi _n^s(M)

.
Erickson’s conjecture on the rate of escape of 𝑑-dimensional random walk
Authors: Kesten Harry
Published: July 2010

We prove a strengthened form of a conjecture of Erickson to the effect that any genuinely d-dimensional random walk

S
n

,
d
⩾
3

{S_n},d \geqslant 3

, goes to infinity at least as fast as a simple random walk or Brownian motion in dimension d. More precisely, if

S
n
∗

S_n^\ast

is a simple random walk and

B
t

{B_t}

, a Brownian motion in dimension d, and

ψ
:
[
1
,
∞
)
→
(
0
,
∞
)

\psi :[1,\infty ) \to (0,\infty )

a function for which

t

−
1

/

2

ψ
(
t
)
↓
0

{t^{ - 1/2}}\psi (t) \downarrow 0

, then

ψ

(
n

)

−
1

|

S
n
∗

|

→
∞

\psi {(n)^{ - 1}}|S_n^\ast | \to \infty

w.p.l, or equivalently,

ψ

(
t

)

−
1

|

B
t

|

→
∞

\psi {(t)^{ - 1}}|{B_t}| \to \infty

w.p.l, iff

∫
1
∞

ψ

(
t

)

d
−
2

t

−
d

/

2

>
∞

\smallint _1^\infty \psi {(t)^{d - 2}}{t^{ - d/2}} > \infty

; if this is the case, then also

ψ

(
n

)

−
1

|

S
n

|

→
∞

\psi {(n)^{ - 1}}|{S_n}| \to \infty

w.p.l for any random walk Sn of dimension d.
Hyperarithmetically encodable sets
Authors: Solovay Robert M.
Published: July 2010

We say that a set of integers, A, is hyperarithmetically (recursively) encodable, if every infinite set of integers X contains an infinite subset Y in which A is hyperarithmetical (recursive). We show that the recursively encodable sets are precisely the hyperarithmetic sets. Let

σ
\sigma

be the closure ordinal of a universal

Σ
1
1

\Sigma _1^1

inductive definition. Then A is hyperarithmetically encodable iff it is constructible before stage

σ
\sigma

. We also prove an effective version of the Galvin-Prikry results that open sets, and more generally Borel sets, are Ramsey, and in the case of open sets prove that our improvement is optimal.
The Diophantine problem for polynomial rings and fields of rational functions
Authors: Denef J.
Published: July 2010

We prove that the diophantine problem for a ring of polynomials over an integral domain of characteristic zero or for a field of rational functions over a formally real field is unsolvable.
Standard and normal reductions
Authors: Hindley R.
Published: July 2010

Curry and Feys’ original standardization proof for

λ
β

\lambda \beta

-reduction is analyzed and generalized to

λ
β
η

\lambda \beta \eta

-reductions with extra operators. There seem to be two slightly different definitions of ’standard reduction’ in current use, without any awareness that they are different; it is proved that although these definitions turn out to be equivalent for

λ
β

\lambda \beta

-reduction, they become different for

λ
β
η

\lambda \beta \eta

and for reductions involving extra operators, for example the recursion operator. Normal reductions are also studied, and it is shown that the basic normal-reduction theorem stays true when fairly simple operators like Church’s

δ
\delta

and Curry’s iterator Z are added, but fails for more complicated ones like the recursion operator R. Finally, a table is given summarizing the results, and showing how far the main theorems on

λ
β

\lambda \beta

-reductions extend to reductions with various extra operators.
A characterization of upper-embeddable graphs
Authors: Jungerman Mark
Published: July 2010

It is proved that a pseudograph G is upper-embeddable if and only if it has a spanning tree T such that G - T has at most one component with an odd number of edges. This result is then used to show that all 4-edge connected graphs are upper-embeddable.
On approximation by shifts and a theorem of Wiener
Authors: Zalik R. A.
Published: July 2010

We study the completeness in

L
2

(
R
)

{L_2}(R)

of sequences of the form

{
f
(

c
n

−
t
)
}

\{ f({c_n} - t)\}

, where

{

c
n

}

\{ {c_n}\}

is a sequence of distinct real numbers. A Müntztype theorem is proved, valid for a large class of functions and, in particular, for

f
(
t
)
=
exp
⁡
(
−

t
2

)

f(t) = \exp ( - {t^2})

.
Infinitesimal calculus on locally convex spaces. I. Fundamentals
Authors: Stroyan K. D.
Published: July 2010

Differential calculus on nonnormed locally convex spaces suffers from technical difficulties (and the subsequent plethora of different definitions) partly because the families of multilinear maps over the spaces do not inherit a suitable topology. In this note we give the elementary ingredients of a strong differentiation based on Abraham Robinson’s theory of infinitesimals. Though nontopologizable, our theory is a natural generalization of standard infinitesimal calculus (finite dimensional or Banach space), see Robinson [1966], Keisler [1976], or Stroyan and Luxemburg [1976]. It is simpler than many recent developments, e.g., Yamamuro [1974] and Keller [1974]. The technical improvement of our approach should lead to advances in a variety of subjects.
𝑅-separable coordinates for three-dimensional complex Riemannian spaces
Authors: Boyer C. P. , Kalnins E. G. , Miller Willard
Published: July 2010

We classify allR-separable coordinate systems for the equations<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma Subscript i comma j equals 1 Superscript 3 Baseline g Superscript negative 1 slash 2 Baseline partial-differential Subscript j Baseline left-parenthesis g Superscript 1 slash 2 Baseline g Superscript i j Baseline partial-differential Subscript i Baseline psi right-parenthesis equals 0">Σi,j=13g−1/2∂j(g1/2gij∂iψ)=0\Sigma _{i,j = 1}^3\,{g^{ - 1/2}}{\partial _j}({g^{1/2}}{g^{ij}}{\partial _i}\psi ) = 0andΣi,j=13gij∂iW∂jW=0\Sigma _{i,j\, = \,1}^3 {{g^{ij}}{\partial _i}W{\partial _j}W\, = \,0}with special emphasis on nonorthogonal coordinates, and give a group-theoretic interpretation of the results. We show that for flat space the two equations separate in exactly the same coordinate systems.
The theorem of Torelli for singular curves
Authors: Jambois Thomas
Published: July 2010

Let C be a compact (singular) curve embedded in a surface. Then C carries a canonical sheaf

Ω
\Omega

which is locally free of rank 1. Moreover, C has a generalized Jacobian J which fits in an exact sequence

(

∗

)

0
→
F
→
J
→
A
→
0

\begin{equation}\tag {$\ast $} 0 \to F \to J \to A \to 0\end{equation}

of algebraic groups such that A is an abelian variety and

F
=

(

C

∗

)
r

×

C

s

F = {({{\mathbf {C}}^\ast })^r} \times {{\mathbf {C}}^s}

. Let

C
_

\underline {C}

be the set of nonsingular points of C and let

θ
\theta

= Zariski-closure of the image of

(

C
_

)

(
g
−
1
)

(\underline {C})^{(g - 1)}

in J. Then: Theorem. If C is irreducible and sections of

Ω
\Omega

map C onto X in

P

g
−
1

{P^{g - 1}}

then the isomorphism class of J together with the translation class of the divisor

θ
\theta

on J determine the isomorphism class of X. As a corollary, if

ψ
:
C
→
X

\psi :C \to X

is an isomorphism (in which case we call C nonhyperelliptic) the above data determine the isomorphism class of C. I do not know if this remains true when C is hyperelliptic. It should be noted that the linear equivalence class of

θ
\theta

is not enough to determine X. The principal idea of the proof is that of Andreotti, that is, to recover the curve as the dual of the branch locus of the Gauss map from

θ
\theta

to

P

g
−
1

{P^{g - 1}}

; however our arguments are usually analytic. The organization of this paper is as follows: In §1 we prove a stronger than usual version of Abel’s theorem for Riemann surfaces and in §2 we extend this theorem to apply to singular curves. In succeeding sections we construct the generalized Jacobian as a complex Lie group J and embed J in an analytic fibre bundle over A with projective spaces as fibre. This we use to endow J with the structure of an algebraic group. §7 contains a miscellany of facts about branch loci and dual varieties, and in §8 the main theorems are stated and proved. We should mention here that the variations on Abel’s theorem proved in this paper (1.2.4 and 3.0.1) were proved by Severi, at least in the special case corresponding to ordinary double points [12].
Square integrable representations and a Plancherel theorem for parabolic subgroups
Authors: Keene Frederick W.
Published: July 2010

Let G be a semisimple Lie group with Iwasawa decomposition

G

=

K
A
N

G\, = \,KAN

. Let

g

0

=

f

0

+

a

+

n

{\mathfrak {g}_0} \,=\, {\mathcal {f}_0} \,+\, \mathfrak {a} \,+\, \mathfrak {n}

be the corresponding decomposition of the Lie algebra of G. Then the nilpotent subgroup N has square integrable representations if and only if the reduced restricted root system is of type

A
1

{A_1}

or

A
2

{A_2}

. The Plancherel measure for N can be found explicitly in these cases. We then prove the Plancherel theorem in the

A
1

{A_1}

case for the solvable subgroup NA by combining Mackey’s “Little Group” method with an idea due to C. C. Moore: we find an operator D, defined on the

C
∞

{C^\infty }

functions on NA with compact support, such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis e right-parenthesis equals integral Underscript left-parenthesis upper N upper A right-parenthesis Superscript logical-and Baseline Endscripts trace left-parenthesis upper D pi left-parenthesis phi right-parenthesis right-parenthesis d mu left-parenthesis pi right-parenthesis">

ϕ
(
e
)
=

∫

(
N
A
)

∧

tr

(
D
π
(
ϕ
)
)
d
μ
(
π
)

\phi (e) = \int _{{{(NA)}^\wedge }} {{\text {tr}}} (D\pi (\phi ))d\mu (\pi )

\]
where

(
N
A

)
∧

{(NA)^ \wedge }

is the unitary dual, e is the identity, and

μ
\mu

is the Plancherel measure for NA, and D is an unbounded selfadjoint operator. In the

A
1

{A_1}

case, D involves fractional powers of the Laplace operator and hence is not a differential operator.
Singularly perturbed boundary value problems with angular limiting solutions
Authors: Howes F. A.
Published: July 2010

A basic result of Haber and Levinson which describes the behavior of solutions of

ε
y
=
f
(
t
,
y
,

y
′

)

\varepsilon y = f(t,y,y’)

,

a
>
t
>
b

a > t > b

,

y
(
a
,
ε
)

y(a,\varepsilon )

,

y
(
b
,
ε
)

y(b,\varepsilon )

, prescribed, in the presence of a reduced solution with corners is modified to treat related classes of problems. Under various stability assumptions, solutions are shown to remain, for small

ε

>

0

\varepsilon \, > \,0

, in a o(l)-neighborhood of an angular reduced solution with the possible exception of narrow layers near the boundaries in some cases. Each aspect of the theory developed here is illustrated by several examples.
Branch point structure of covering maps onto nonorientable surfaces
Authors: Ezell Cloyd L.
Published: July 2010

Let

f
:
M

→

N

f:M\, \to \,N

be a degree n branched cover onto a compact, connected nonorientable surface with branch points

y
1

,

y
2

,

…
,

y
m

{y_1},\,{y_2},\, \ldots ,\,{y_m}

in N, and let the multiplicities at points in

f

−
1

(

y
i

)

{f^{ - 1}}({y_i})

be

μ

i
1

,

μ

i
2

,

…
,

μ

i

k
i

{\mu _{i1}},\,{\mu _{i2}},\, \ldots ,\,{\mu _{i{k_i}}}

. The branching array of f, designated by B, is the following array of numbers:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row mu 11 comma mu 12 comma ellipsis comma mu Subscript 1 k 1 Baseline mu 21 comma mu 22 comma ellipsis comma mu Subscript 2 k 2 Baseline vertical-ellipsis mu Subscript m Baseline 1 Baseline comma mu Subscript m Baseline 2 Baseline comma ellipsis comma mu Subscript m k Sub Subscript m Subscript Baseline EndLayout">

μ

11

,

μ

12

,

…
,

μ

1

k
1

μ

21

,

μ

22

,
…
,

μ

2

k
2

⋮

μ

m
1

,

μ

m
2

,

…
,

μ

m

k
m

\begin {gathered} {\mu _{11}},\,{\mu _{12}},\, \ldots ,\,{\mu _{1{k_1}}} {\mu _{21}},\,{\mu _{22\,}}, \ldots ,\,{\mu _{2{k_2}}} \,\,\,\,\,\,\,\,\,\,\,\, \vdots {\mu _{m1}},\,{\mu _{m2}},\, \ldots ,\,{\mu _{m{k_m}}} \end {gathered}

\]
We show that the numbers in the branching array must always satisfy the following conditions: (1)
\[

∑

{

μ

i
j

+

1

|

j

=

1
,

2
,

…
,

k
i

}

=

n

\sum {\{ {\mu _{ij}} \,+ \,1|j \,=\, 1,\,2,\, \ldots ,\,{k_i}\} \,=\, n}

\]
, (2)

∑

{

μ

i
j

|

i

=

1
,

2
,
…
,
m
;
j

=

1
,

2
,
…
,

k
i

}

\sum {\{ {\mu _{ij}}|i\,=\, 1,\,2, \ldots ,m;j\,=\, 1,\,2, \ldots ,{k_i}\} }

is even. Furthermore, if B is any array of numbers satisfying these conditions, and if N is not the projective plane, then there is a branched cover onto N with B as its branching array.
Invariant differential equations on certain semisimple Lie groups
Authors: Rouvière F.
Published: July 2010

If G is a semisimple Lie group with one conjugacy class of Cartan subalgebras (e.g. a complex semisimple Lie group), a bi-invariant differential equation on G can be reduced by means of the Radon transform to one on the subgroup MA. In particular, all polynomials of the Casimir operator have a central fundamental solution, and are solvable in

C
∞

(
G
)

{C^\infty }(G)

; but, for G complex, the “imaginary” Casimir operator is not.
The isotopy problem for Jordan matrix algebras
Authors: Petersson Holger P.
Published: July 2010

Conditions are given which are necessary and sufficient for two members of a certain class of Jordan matrix algebras to be isotopic. The main ingredient of these conditions is McCrimmon’s notion of isotopy for alternative algebras.
Hamiltonian systems in a neighborhood of a saddle point
Authors: Barbu Viorel
Published: July 2010

The behavior of Hamiltonian differential systems associated with a concave convex function H in a Hilbert space is studied by variational methods. It is shown that under quite general conditions on the function H the system behaves in a neighborhood of a minimax saddle point of H much like as in the classical theory of ordinary differential systems. The results extend previous work of R. T. Rockafellar.
Analytic equations and singularities of plane curves
Authors: Wavrik John J.
Published: July 2010

Theorems (Artin, Wavrik) exist which show that sufficiently good approximate (power series) solutions to a system of analytic equations may be approximated by convergent solutions. This paper considers the problem of explicity determining the order,

β
\beta

, to which an approximate solution must solve the system of equations. The paper deals with the case of one equation,

f
(
x
,
y
)
=
0

f(x,y) = 0

, in two variables. It is shown how

β
\beta

depends on the singularities of the curve

f
(
x
,
y
)
=
0

f(x,y) = 0

. A method for obtaining the minimal

β
\beta

is given. A rapid way of finding

β
\beta

using the Newton Polygon for f applies in special cases.
Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity
Authors: Hartman Philip
Published: July 2010

We find analogues of known results onnth order linear differential equations fornth order linear difference equations. These include the concept of disconjugacy, Pólya’s criterion for disconjugacy, Frobenius factorizations, generalized Sturm theorems, existence and properties of principal solutions, signs of Green’s functions, and completely monotone families of solutions of equations depending on a parameter.
Quality control for Markov chains and free boundary problems
Authors: Anderson Robert F. , Friedman Avner
Published: July 2010

A machine can manufacture any one of n Markov chains

P
x

λ
j

(

1

≤

j

≤

n

)

P_x^{{\lambda _j}}\,\left ( {1\, \leq \,j\, \leq \,n} \right )

; the

P
x

λ
j

P_x^{{\lambda _j}}

are defined on the space of all sequences

x

=

{

x

(
m
)

}

(

1

≤

m

≤

∞

)

x\, = \,\left \{ {x\left ( m \right )} \right \}\,\left ( {1\, \leq \,m\, \leq \,\infty } \right )

and are absolutely continuous (in finite times) with respect to one another. It is assumed that chains

P
x

λ
j

P_x^{{\lambda _j}}

evolve in a random way, dictated by a Markov chain

θ

(
m
)

\theta \left ( m \right )

with n states, so that when

θ

(
m
)

=

j

\theta \left ( m \right )\, = \,j

the machine is producing

P
x

λ
j

P_x^{{\lambda _j}}

. One observes the

σ
\sigma

-fields of

x

(
m
)

x\left ( m \right )

in order to determine when to inspect

θ

(
m
)

\theta \left ( m \right )

. With each product

P
x

λ
j

P_x^{{\lambda _j}}

there is associated a cost

c
j

{c_j}

. One inspects

θ
\theta

at a sequence of times (each inspection entails a certain cost) and stops production when the state

θ

=

n

\theta \, = \,n

is reached. The problem is to find an optimal sequence of inspections. This problem is reduced, in this paper, to solving a certain free boundary problem. In case

n

=

2

n\, = \,2

the latter problem is solved.
Quantization and projective representations of solvable Lie groups
Authors: Moscovici Henri , Verona Andrei
Published: July 2010

Kostant’s quantization procedure is applied for constructing irreducible projective representations of a solvable Lie group from symplectic homogeneous spaces on which the group acts. When specialized to a certain class of such groups, including the exponential ones, the technique exposed in the present paper provides a complete parametrization of all irreducible projective representations.
The spectral theory of distributive continuous lattices
Authors: Hofmann Karl H. , Lawson Jimmie D.
Published: July 2010

In this paper various properties of the spectrum (i.e. the set of prime elements endowed with the hull-kernel topology) of a distributive continuous lattice are developed. It is shown that the spectrum is always a locally quasicompact sober space and conversely that the lattice of open sets of a locally quasicompact sober space is a continuous lattice. Algebraic lattices are a special subclass of continuous lattices and the special properties of their spectra are treated. The concept of the patch topology is extended from algebraic lattices to continuous lattices, and necessary and sufficient conditions for its compactness are given.
The heat equation on a compact Lie group
Authors: Fegan H. D.
Published: July 2010

Recently there has been much work related to Macdonald’s

η
\eta

-function identities. In the present paper the aim is to give another proof of these identities using analytical methods. This is done by using the heat equation to obtain Kostant’s form of the identities. The basic idea of the proof is to look at subgroups of the Lie group which are isomorphic to the group

S
U
(
2
)

SU(2)

. When this has been done the problem has essentially been reduced to that for the group

S
U
(
2
)

SU(2)

, which is a classical result.
Amalgamation and inverse and regular semigroups
Authors: Hall T. E.
Published: July 2010

A method for proving the embeddability of semigroup amalgams is introduced. After providing necessary and sufficient conditions in terms of representations for the weak embeddability of a semigroup amalgam, it successfully deals with the embedding of inverse semigroup amalgams into inverse semigroups and the embedding of an amalgam of regular semigroups whose core is full in each member.
A characterization and sum decomposition for operator ideals
Authors: Blass Andreas , Weiss Gary
Published: July 2010

Let

L
(
H
)

L(H)

be the ring of bounded operators on a separable Hubert space. Assuming the continuum hypothesis, we prove that in

L
(
H
)

L(H)

every two-sided ideal that contains an operator of infinite rank is the sum of two smaller two-sided ideals. The proof involves a new combinatorial description of ideals of

L
(
H
)

L(H)

. This description is also used to deduce some related results about decompositions of ideals. Finally, we discuss the possibility of proving our main theorem under weaker assumptions than the continuum hypothesis and the impossibility of proving it without the axiom of choice.
The fixed-point construction in equivariant bordism
Authors: Rowlett Russell J.
Published: July 2010

Consider the bordism

Ω

∗

(
G
)

{\Omega _ {\ast }}(G)

of smooth G-actions. If K is a subgroup of G, with normalizer NK, there is a standard

N
K

/

K

NK/K

-action on

Ω

∗

(
K
)

{\Omega _ {\ast }}(K)

(All, Proper). If M has a smooth G-action, a tubular neighborhood of the fixed set of K in M represents an element of

Ω

∗

(
K
)

(

All, Proper

)

N
K

/

K

{\Omega _ {\ast }}(K){({\text {All, Proper}})^{NK/K}}

. One thus obtains the “fixed point homomorphism”

ϕ
\phi

carrying

Ω

∗

(
G
)

{\Omega _ {\ast }}(G)

to the sum of the

Ω

∗

(
K
)

(

All, Proper

)

N
K

/

K

{\Omega _ {\ast }}(K){({\text {All, Proper}})^{NK/K}}

, summed over conjugacy classes of subgroups K. Let P be the collection of primes not dividing the order of G. We show that the P-localization of

ϕ
\phi

is an isomorphism, and give several applications.
The immersion conjecture for 𝑅𝑃^{8𝑙+7} is false
Authors: Davis Donald M. , Mahowald Mark
Published: July 2010

Let

α
(
n
)

\alpha (n)

denote the number of l’s in the binary expansion of n. It is proved that if

n
≡
7

n \equiv 7

(8),

α
(
n
)
=
6

\alpha (n) = 6

, and

n
≠
63

n \ne 63

, then

R

P
n

{\mathbf {R}}{P^n}

can be immersed in

R

2
n
−
14

{{\mathbf {R}}^{2n - 14}}

. This provides the first counterexample to the well-known conjecture that the best immersion is in

R

2
n
−
2
α
(
n
)
+
1

{{\mathbf {R}}^{2n - 2\alpha (n) + 1}}

(when

α
(
n
)
≡
1

\alpha (n) \equiv 1

or

2
mod
4

2 \bmod 4

). The method of proof is obstruction theory.
On the construction of branched coverings of low-dimensional manifolds
Authors: Berstein Israel , Edmonds Allan L.
Published: July 2010

Several general results are proved concerning the existence and uniqueness of various branched coverings of manifolds in dimensions 2 and 3. The results are applied to give a rather complete account as to which 3-manifolds are branched coverings of

S
3

{S^3}

,

S
2

×

S
1

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

,

P
2

×

S
1

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

, or the nontrivial

S
3

{S^3}

-bundle over

S
1

{S^1}

, and which degrees can be achieved in each case. In particular, it is shown that any closed nonorientable 3-manifold is a branched covering of

P
2

×

S
1

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

of degree which can be chosen to be at most 6 and with branch set a simple closed curve. This result is applied to show that a closed nonorientable 3-manifold admits an open book decomposition which is induced from such a decomposition of

P
2

×

S
1

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

.
Examples of noncatenary rings
Authors: Heitmann Raymond C.
Published: July 2010

A technique is developed for constructing a new family of noetherian integral domains. To each domain, there naturally corresponds its poset (partially ordered set) of prime ideals. The resulting family of posets has the following property: every finite poset is isomorphic to a saturated subset of some poset in the family. In the process, it is determined when certain power series may be adjoined to noetherian rings without destroying the noetherian property.
Differential algebraic Lie algebras
Authors: Cassidy Phyllis Joan
Published: July 2010

A class of infinite-dimensional Lie algebras over the field

K

\mathcal {K}

of constants of a universal differential field

U

\mathcal {U}

is studied. The simplest case, defined by homogeneous linear differential equations, is analyzed in detail, and those with underlying set

U

×

U

\mathcal {U}\, \times \,\mathcal {U}

are classified.
Isotoping mappings to open mappings
Authors: Walsh John J.
Published: July 2010

Let f be a quasi-monotone mapping from a compact, connected manifold

M
m

(
m

⩾

3
)

{M^m}\,(m\, \geqslant \,3)

onto a space Y; then there is an open mapping g from M onto Y such that, for each

y

∈

Y
,

g

−
1

(
y
)

y\, \in \,Y,\,{g^{ - 1}}(y)

is not a point and

g

−
1

(
y
)

{g^{ - 1}}(y)

and

f

−
1

(
y
)

{f^{ - 1}}(y)

are equivalently embedded in M (in particular,

g

−
1

(
y
)

{g^{ - 1}}(y)

and

f

−
1

(
y
)

{f^{ - 1}}(y)

have the same shape). Applying the result with f equal to the identity mapping on M yields a continuous decomposition of M into cellular sets each of which is not a point.
Space curves that point almost everywhere
Authors: Wilker J. B.
Published: July 2010

We construct a simple, closed, continuously differentiable curve

r
:

[
0
,

1
]

→

E
d

(
d

⩾

3
)

r:\,[0,\,1]\, \to \,{E^d}\,(d\, \geqslant \,3)

whose tangent vector never points twice in the same direction of

S

d

−

1

{S^{d\, - \,1}}

yet sweeps out a set of directions equal to almost all of

S

d

−

1

{S^{d\, - \,1}}

.