bluuebunny/crossref_metadata_2025 huggingface dataset:
A character-theoretic approach to embeddings of rooted maps in an orientable surface of given genus
Authors: Jackson D. M. , Visentin T. I.
Published: June 2012

The group algebra of the symmetric group and properties of the irreducible characters are used to derive combinatorial properties of embeddings of rooted maps in orientable surfaces of arbitrary genus. In particular, we show that there exists, for each genus, a correspondence between the set of rooted quadrangulations and a set of rooted maps of all lower genera with a distinguished subset of vertices.
Cardinal conditions for strong Fubini theorems
Authors: Shipman Joseph
Published: June 2012

If

κ
1

,

κ
2

,
…
,

κ
n

{\kappa _1},{\kappa _2}, \ldots ,{\kappa _n}

are cardinals with

κ
1

{\kappa _1}

the cardinality of a nonmeasurable set, and for

i
=
2
,
3
,
…
,
n

i = 2,3, \ldots ,n

κ
i

{\kappa _i}

is the cardinality of a set of reals which is not the union of

κ

i
−
1

{\kappa _{i - 1}}

measure-

0
0

sets, then for any nonnegative function

f
:

R

n

→

R

f:{{\mathbf {R}}^n} \to {\mathbf {R}}

all of the iterated integrals
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript sigma Baseline equals double-integral midline-horizontal-ellipsis integral f left-parenthesis x 1 comma x 2 comma ellipsis comma x Subscript n Baseline right-parenthesis d x Subscript sigma left-parenthesis 1 right-parenthesis Baseline d x Subscript sigma left-parenthesis 2 right-parenthesis Baseline midline-horizontal-ellipsis d x Subscript sigma left-parenthesis n right-parenthesis Baseline comma sigma element-of upper S Subscript n Baseline">

I
σ

=
∬
⋯
∫

f
(

x
1

,

x
2

,
…
,

x
n

)
d

x

σ
(
1
)

d

x

σ
(
2
)

⋯
d

x

σ
(
n
)

,

σ
∈

S
n

{I_\sigma } = \iint \cdots \int {f({x_1},{x_2}, \ldots ,{x_n})d{x_{\sigma (1)}}d{x_{\sigma (2)}} \cdots d{x_{\sigma (n)}},\quad \sigma \in {S_n}}

\]
, which exist are equal. If all

n
!

n!

of the integrals exist, then the weaker condition of the case

n
=
2

n = 2

implies they are equal. These cardinal conditions are consistent with and independent of ZFC, and follow from the existence of a real-valued measure on the continuum. Other necessary conditions and sufficient conditions for the existence and equality of iterated integrals are also treated.
The generalized Lusternik-Schnirelmann category of a product space
Authors: Clapp Mónica , Puppe Dieter
Published: June 2012

We continue to study the notions of

A

\mathcal {A}

-category and strong

A

\mathcal {A}

-category which we introduced in [2]. We give a characterization of them in terms of homotopy colimits and then use it to prove some product theorems in this context.
Dualizing complexes of affine semigroup rings
Authors: Schäfer Uwe , Schenzel Peter
Published: June 2012

For an affine semigroup ring we construct the dualizing complex in terms of the semigroup and the homology of the face lattice of the polyhedral cone spanned by the semigroup. As a consequence there are characterizations of locally Cohen-Macaulay rings, Buchsbaum rings, and Cohen-Macaulay rings as well as Serre’s condition

S

l

{\mathcal {S}_l}

.
Weak type (1,1) estimates for heat kernel maximal functions on Lie groups
Authors: Cowling Michael , Gaudry Garth , Giulini Saverio , Mauceri Giancarlo
Published: June 2012

For a Lie group

G
G

with left-invariant Haar measure and associated Lebesgue spaces

L
p

(
G
)

{L^p}(G)

, we consider the heat kernels

{

p
t

}

t
>
0

{\{ {p_t}\} _{t > 0}}

arising from a right-invariant Laplacian

Δ
\Delta

on

G
G

: that is,

u
(
t
,
⋅
)
=

p
t

∗

f

u(t, \cdot ) = {p_t}{\ast }f

solves the heat equation

(
∂

/

∂
t
−
Δ
)
u
=
0

(\partial /\partial t - \Delta )u = 0

with initial condition

u
(
0
,
⋅
)
=
f
(
⋅
)

u(0, \cdot ) = f( \cdot )

. We establish weak-type

(
1
,
1
)

(1,1)

estimates for the maximal operator

M

(

M

f
=

sup

t
>
0

|

p
t

∗

f

|

)

\mathcal {M}(\mathcal {M}\;f = {\sup _{t > 0}}|{p_t}{\ast }f|)

and for related Hardy-Littlewood maximal operators in a variety of contexts, namely for groups of polynomial growth and for a number of classes of Iwasawa

A
N

AN

groups. We also study the "local" maximal operator

M

0

(

M

0

f
=

sup

0
>
t
>
1

|

p
t

∗

f

|

)

{\mathcal {M}_0}({\mathcal {M}_0}f = {\sup _{0 > t > 1}}|{p_t}{\ast }f|)

and related Hardy-Littlewood operators for all Lie groups.
Theorems of Hardy and Paley for vector-valued analytic functions and related classes of Banach spaces
Authors: Blasco O. , Pełczyński A.
Published: June 2012

We investigate the classes of Banach spaces where analogues of the classical Hardy inequality and the Paley gap theorem hold for vector-valued functions. We show that the vector-valued Paley theorem is valid for a large class of Banach spaces (necessarily of cotype

2
2

) which includes all Banach lattices of cotype

2
2

, all Banach spaces whose dual is of type

2
2

and also the preduals of

C
∗

{C^ * }

-algebras. For the trace class

S
1

{S_1}

and the dual of the algebra of all bounded operators on a Hilbert space a stronger result holds; namely, the vector-valued analogue of the Fefferman theorem on multipliers from

H
1

{H^1}

into

l
1

{l^1}

; in particular for the latter spaces the vector-valued Hardy inequality holds. This inequality is also true for every Banach space of type

>
1

> 1

(Bourgain).
Indecomposable Cohen-Macaulay modules and their multiplicities
Authors: Popescu Dorin
Published: June 2012

The main aim of this paper is to find a large class of rings for which there are indecomposable maximal Cohen-Macaulay modules of arbitrary high multiplicity (or rank in the case of domains).
Locally flat 2-knots in 𝑆²×𝑆² with the same fundamental group
Authors: Sato Yoshihisa
Published: June 2012

We consider a locally flat

2
2

-sphere in

S
2

×

S
2

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

representing a primitive homology class

ξ
\xi

, which is referred to as a

2
2

-knot in

S
2

×

S
2

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

representing

ξ
\xi

. Then for any given primitive class

ξ
\xi

, there exists a

2
2

-knot in

S
2

×

S
2

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

representing

ξ
\xi

with simply-connected complement. In this paper, we consider the classification of

2
2

-knots in

S
2

×

S
2

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

whose complements have a fixed fundamental group. We show that if the complement of a

2
2

-knot

S
S

in

S
2

×

S
2

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

is simply connected, then the ambient isotopy type of

S
S

is determined. In the case of nontrivial

π
1

{\pi _1}

, however, we show that the ambient isotopy type of a

2
2

-knot in

S
2

×

S
2

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

with nontrivial

π
1

{\pi _1}

is not always determined by

π
1

{\pi _1}

.
On collectionwise normality of locally compact, normal spaces
Authors: Balogh Zoltán T.
Published: June 2012

We prove that by adjoining supercompact many Cohen or random reals to a model of ZFC set theory, in the resulting model, every normal locally compact space is collectionwise normal. In the same models, countably paracompact, locally compact

T
3

{T_3}

-spaces are expandable. Local compactness in the above theorems can be weakened to being of point-countable type, a condition that is implied by both Čech-completeness and first countability.
On the topology and geometric construction of oriented matroids and convex polytopes
Authors: Richter Jürgen , Sturmfels Bernd
Published: June 2012

This paper develops new combinatorial and geometric techniques for studying the topology of the real semialgebraic variety

R

(
M
)

\mathcal {R}(M)

of all realizations of an oriented matroid

M
M

. We focus our attention on point configurations in general position, and as the main result we prove that the realization space of every uniform rank

3
3

oriented matroid with up to eight points is contractible. For these special classes our theorem implies the isotopy property which states the spaces

R

(
M
)

\mathcal {R}(M)

are path-connected. We further apply our methods to several related problems on convex polytopes and line arrangements. A geometric construction and the isotopy property are obtained for a large class of neighborly polytopes. We improve a result of M. Las Vergnas by constructing a smallest counterexample to a conjecture of G. Ringel, and, finally, we discuss the solution to a problem of R. Cordovil and P. Duchet on the realizability of cyclic matroid polytopes.
On the braid index of alternating links
Authors: Murasugi Kunio
Published: June 2012

We show that, at least for an alternating fibered link or

2
2

-bridge link

L
L

, there is an exact formula which expresses the braid index

b

(
L
)

{\mathbf {b}}(L)

of

L
L

as a function of the

2
2

-variable generalization

P
L

(
l
,
m
)

{P_L}(l,m)

of the Jones polynomial.
The holomorphic discrete series of an affine symmetric space and representations with reproducing kernels
Authors: Ólafsson G. , Ørsted B.
Published: June 2012

Consider a semisimple connected Lie group

G
G

with an affine symmetric space

X
X

. We study abstractly the intertwining operators from the discrete series of

X
X

into representations with reproducing kernel and, in particular, into the discrete series of

G
G

; each such is given by a convolution with an analytic function. For

X
X

of Hermitian type, we consider the holomorphic discrete series of

X
X

and here derive very explicit formulas for the intertwining operators. As a corollary we get a multiplicity one result for the series in question.
Representations of knot groups in 𝑆𝑈(2)
Authors: Klassen Eric Paul
Published: June 2012

This paper is a study of the structure of the space

R
(
K
)

R(K)

of representations of classical knot groups into

SU

(
2
)

{\text {SU}}(2)

. Let

R
^

(
K
)

\hat R(K)

equal the set of conjugacy classes of irreducible representations. In

§
I

\S I

, we interpret the relations in a presentation of the knot group in terms of the geometry of

SU

(
2
)

{\text {SU}}(2)

; using this technique we calculate

R
^

(
K
)

\hat R(K)

for

K
K

equal to the torus knots, twist knots, and the Whitehead link. We also determine a formula for the number of binary dihedral representations of an arbitrary knot group. We prove, using techniques introduced by Culler and Shalen, that if the dimension of

R
^

(
K
)

\hat R(K)

is greater than

1
1

, then the complement in

S
3

{S^3}

of a tubular neighborhood of

K
K

contains closed, nonboundary parallel, incompressible surfaces. We also show how, for certain nonprime and doubled knots,

R
^

(
K
)

\hat R(K)

has dimension greater than one. In

§
I
I

\S II

, we calculate the Zariski tangent space,

T
ρ

(
R
(
K
)
)

{T_\rho }(R(K))

, for an arbitrary knot

K
K

, at a reducible representation

ρ
\rho

, using a technique due to Weil. We prove that for all but a finite number of the reducible representations,

dim
⁡

T
ρ

(
R
(
K
)
)
=
3

\dim {T_\rho }(R(K))= 3

. These nonexceptional representations possess neighborhoods in

R
(
K
)

R(K)

containing only reducible representations. At the exceptional representations, which correspond to real roots of the Alexander polynomial,

dim
⁡

T
ρ

(
R
(
K
)
)
=
3
+
2
k

\dim {T_\rho }(R(K)) = 3 + 2k

for a positive integer

k
k

. In those examples analyzed in this paper, these exceptional representations can be expressed as limits of arcs of irreducible representations. We also give an interpretation of these "extra" tangent vectors as representations in the group of Euclidean isometries of the plane.
A transitive homeomorphism on the pseudoarc which is semiconjugate to the tent map
Authors: Kennedy Judy
Published: June 2012

A powerful theorem and construction of Wayne Lewis are used to build two homeomorphisms on the pseudoarc, each of which is semiconjugate to the tent map on the unit interval. The first homeomorphism is transitive, thus answering a question of Marcy Barge as to whether such homeomorphisms exist. The second homeomorphism admits wandering points. Also, it is proven that any homeomorphism on the pseudoarc that is semiconjugate to the tent map and is irreducible with respect to the semiconjugacy must either be transitive or admit wandering points.
Invariant arcs, Whitney levels, and Kelley continua
Authors: van de Vel M.
Published: June 2012

As an application of convexity in spaces of arcs, three results of a somewhat different nature have been obtained. The first one gives some simple conditions under which an arc of a semilattice is mapped back into itself by an order-preserving function. The second result states that certain Whitney levels are absolute retracts. Finally, Kelley continua are characterized by what we call approximating coselections.
The cohomology of certain function spaces
Authors: Bendersky Martin , Gitler Sam
Published: June 2012

We study a spectral sequence converging to the cohomology of the configuration space of

n
n

ordered points in a manifold. A chain complex is constructed with homology equal to the

E
2

{E_2}

term. If the field is the rationals and the manifold is formal then the spectral sequence is shown to collapse. The results are applied to compute the Anderson spectral sequence converging to the cohomology of a function space.
Maxwell’s equations in a periodic structure
Authors: Chen Xinfu , Friedman Avner
Published: June 2012

Consider a diffraction of a beam of particles in

R

3

{\mathbb {R}^3}

when the dielectric coefficient is a constant

ε
1

{\varepsilon _1}

above a surface

S
S

and a constant

ε
2

{\varepsilon _2}

below a surface

S
S

, and the magnetic permeability is constant throughout

R

3

{\mathbb {R}^3}

.

S
S

is assumed to be periodic in the

x
1

{x_1}

direction and of the form

x
1

=

f
1

(
s
)
,

x
3

=

f
3

(
s
)
,

x
2

{x_1} = {f_1}(s),\,{x_3} = {f_3}(s),\,{x_2}

arbitrary. We prove that there exists a unique solution to the time-harmonic Maxwell equations in

R

3

{\mathbb {R}^3}

having the form of refracted waves for

x
3

≪
1

{x_3} \ll 1

and of transmitted waves for

−

x
3

≫
1

- {x_3} \gg 1

if and only if there exists a unique solution to a certain system of two coupled Fredholm equations. Thus, in particular, for all the

ε
\varepsilon

’s, except for a discrete number, there exists a unique solution to the Maxwell equations.
Monogenic differential calculus
Authors: Sommen F.
Published: June 2012

In this paper we study differential forms satisfying a Dirac type equation and taking values in a Clifford algebra. For them we establish a Cauchy representation formula and we compute winding numbers for pairs of nonintersecting cycles in

R

m

{\mathbb {R}^m}

as residues of special differential forms. Next we prove that the cohomology spaces for the complex of monogenic differential forms split as direct sums of de Rham cohomology spaces. We also study duals of spaces of monogenic differential forms, leading to a general residue theory in Euclidean space. Our theory includes the one established in our paper [11] and is strongly related to certain differential forms introduced by Habetha in [4].
Stable rank and approximation theorems in 𝐻^{∞}
Authors: Laroco Leonardo A.
Published: June 2012

It is conjectured that for

H
∞

{H^\infty }

the Bass stable rank

(
bsr
)

(\text {bsr})

is

1
1

and the topological stable rank

(
tsr
)

(\text {tsr})

is

2
2

.

bsr

(

H
∞

)
=
1

{\text {bsr}}({H^\infty })= 1

if and only if for every

(

f
1

,

f
2

)

∈

H
∞

×

H
∞

({f_1},{f_2})\; \in {H^\infty } \times {H^\infty }

which is a corona pair (i.e., there exist

g

1

{g_{1}}

,

g
2

∈

H
∞

{g_2} \in {H^\infty }

such that

f
1

g
1

+

f
2

g
2

=
1

{f_1}{g_1} + {f_2}{g_2}= 1

) there exists a

g
∈

H
∞

g \in {H^\infty }

such that

f
1

+

f
2

g
∈

(

H
∞

)

−
1

{f_1} + {f_2}g \in {({H^\infty })^{ - 1}}

, the invertibles in

H
∞

{H^\infty }

; however, it suffices to consider corona pairs

(

f
1

,

f
2

)

({f_1},{f_2})

where

f
1

{f_1}

is a Blaschke product. It is also shown that there exists a

g
∈

H
∞

g \in {H^\infty }

such that

f
1

+

f
2

g
∈
exp
⁡
(

H
∞

)

{f_1} + {f_2}g \in \exp ({H^\infty })

if and only if

log
⁡

f
1

\log {f_1}

can be boundedly, analytically defined on

{

z
∈

D

:

|

f
2

(
z
)

|

>
δ

}

\{ {z \in \mathbb {D}:|\,{f_2}(z)| > \delta } \}

, for some

δ
>
0

\delta > 0

.

tsr

(

H
∞

)
=
2

{\text {tsr}}({H^\infty })= 2

if and only if the corona pairs are uniformly dense in

H
∞

×

H
∞

{H^\infty } \times {H^\infty }

; however, it suffices to show that the corona pairs are uniformly dense in pairs of Blaschke products. This condition would be satisfied if the interpolating Blaschke products were uniformly dense in the Blaschke products. For

b
b

an inner function,

K
=

H
2

⊖
b

H
2

K= {H^2} \ominus b\,{H^2}

is an

H
∞

{H^\infty }

-module via the compressed Toeplitz operators

C
f

=

P
K

T
f

|

K

{C_f}= {P_K}{T_f}{|_K}

, for

f
∈

H
∞

f \in {H^\infty }

, where

T
f

{T_f}

is the Toeplitz operator

T
f

g
=
f

g

{T_f}g= f\,g

, for

g
∈

H
2

g \in {H^2}

. Some stable rank questions can be recast as lifting questions: for

(

f
i

)
1
n

⊂

H
∞

({f_i})_1^n \subset {H^\infty }

, there exist

(

g
i

)
1
n

({g_i})_1^n

,

(

h
i

)
1
n

⊂

H
∞

({h_i})_1^n \subset {H^\infty }

such that

∑

i
=
1

n

(

f
i

+
b

g
i

)

h
i

=
1

\sum \nolimits _{i = 1}^n {({f_i} + b\,{g_i}){h_i}= 1}

if and only if the compressed Toeplitz operators

(

C

f
i

)
1
n

({C_{{f_i}}})_1^n

may be lifted to Toeplitz operators

(

T

F
i

)
1
n

({T_{{F_i}}})_1^n

which generate

B
(

H
2

)

B({H^2})

as an ideal.
Fixed set systems of equivariant infinite loop spaces
Authors: Costenoble Steven R. , Waner Stefan
Published: June 2012

We develop machinery enabling us to show that suitable

G
G

-spaces, including the equivariant version of

B
F

BF

, are equivariant infinite loop spaces. This involves a "recognition principle" for systems of spaces which behave formally like the system of fixed sets of a

G
G

-space; that is, we give a necessary and sufficient condition for such a system to be equivalent to the fixed set system of an equivariant infinite loop space. The advantage of using the language of fixed set systems is that one can frequently replace the system of fixed sets of an actual

G
G

-space by an equivalent formal system which is considerably simpler, and which admits the requisite geometry necessary for delooping. We also apply this machinery to construct equivariant Eilenberg-Mac Lane spaces corresponding to Mackey functors.
Pureté, rigidité, et morphismes entiers
Authors: Picavet Gabriel
Published: June 2012

Bousfield and Kan have shown that a ring morphism with domain

Z

{\mathbf {Z}}

is rigid; we say that a ring morphism is rigid if it admits a factorization by an epimorphism, followed by a pure morphism. A ring

A
A

is said to be rigid if every morphism with domain

A
A

is a rigid one. Our principal results are: the rigid domains are the Prüferian rings

A
A

, with

Dim
⁡
(
A
)
≤
1

\operatorname {Dim} (A) \leq 1

, and the Noetherian rigid rings are the Z.P.I. rings. The quasi-compact open sets of an affine rigid scheme, having as underlying ring a domain or a Noetherian ring, are affine and schematically dense if they contain the assassin of the ring. Every injective integral ring morphism with rigid domain is a pure morphism. We give two criteria of purity for integral injective morphisms. As a consequence of these results we obtain the following properties: if

A
A

is a normal ring, containing the field of rationals, or is a regular ring, containing a field, every injective integral morphism with domain

A
A

is a pure one. For a reduced ring, we define the category of reduced modules and show that any injective integral morphism is pure with respect to the category of the reduced modules.
Coefficient ideals
Authors: Shah Kishor
Published: June 2012

Let

R
R

be a

d
d

-dimensional Noetherian quasi-unmixed local ring with maximal ideal

M
M

and an

M
M

-primary ideal

I
I

with integral closure

I
¯

\overline I

. We prove that there exist unique largest ideals

I
k

{I_k}

for

1
≤
k
≤
d

1 \leq k \leq d

lying between

I
I

and

I
¯

\overline I

such that the first

k
+
1

k + 1

Hilbert coefficients of

I
I

and

I
k

{I_k}

coincide. These coefficient ideals clarify some classical results related to

I
¯

\overline I

. We determine their structure and immediately apply the structure theorem to study the associated primes of the associated graded ring of

I
I

.
The Schwartz space of a general semisimple Lie group. II. Wave packets associated to Schwartz functions
Authors: Herb Rebecca A.
Published: June 2012

Let

G
G

be a connected semisimple Lie group. If

G
G

has finite center, Harish-Chandra used Eisenstein integrals to construct Schwartz class wave packets of matrix coefficients and showed that every

K
K

-finite function in the Schwartz space is a finite sum of such wave packets. This paper is the second in a series which generalizes these results of Harish-Chandra to include the case that

G
G

has infinite center. In this paper, the Plancherel theorem is used to decompose

K
K

-compact Schwartz class functions (those with

K
K

-types in a compact set) as finite sums of wave packets. A new feature of the infinite center case is that the individual wave packets occurring in the decomposition of a Schwartz class function need not be Schwartz class. These wave packets are studied to obtain necessary conditions for a wave packet of Eisenstein integrals to occur in the decomposition of a Schwartz class function. Applied to the case that

f
f

itself is a single wave packet, the results of this paper yield a complete characterization of Schwartz class wave packets.
Hyperholomorphic functions and second order partial differential equations in 𝑅ⁿ
Authors: Yeh R. Z.
Published: June 2012

Hyperholomorphic functions in

R
n

{R^n}

with

n
≥
2

n \geq 2

are introduced, extending the hitherto considered hyperholomorphic functions in

R
2

{R^2}

. A Taylor formula is derived, and with it a unique representation theorem is proved for hyperholomorphic functions that are real analytic at the origin. Hyperanalyticity is seen to be generally a consequence of hyperholomorphy and real analyticity combined. Results for hyperholomorphic functions are applied to gradients of solutions of second order homogeneous partial differential equations with constant coefficients. Polynomial solutions of such a second order equation are obtained by a matrix algorithm. These polynomials are modified and combined to form polynomial bases for real analytic solutions. It is calculated that in such a basis there are

(
m
+
n
−
3
)
!
(
2
m
+
n
−
2
)

/

m
!
(
n
−
2
)
!

(m + n - 3)!(2m + n - 2)/m!(n - 2)!

homogeneous polynomials of degree

m
m

.
Cardinal representations for closures and preclosures
Authors: Galvin F. , Milner E. C. , Pouzet M.
Published: June 2012

A cardinal representation of a preclosure

φ
\varphi

on a set

E
E

is a family

A

⊆

P

(
E
)

\mathcal {A} \subseteq \mathcal {P}(E)

such that for any set

X
⊆
∪

A

,
φ
(
X
)
=
E

X \subseteq \cup \mathcal {A},\varphi (X) = E

holds if and only if

|

X
∩
A

|

=

|

A

|

|X \cap A|= |A|

for every

A
∈

A

A \in \mathcal {A}

. We show, for example (Theorem 2.3) that any topological closure has such a representation, but there are closures which have no cardinal representation (Theorem 11.2). We prove that, if

k
k

is finite and a closure has no independent set of size

k
+
1

k + 1

, then it has a cardinal representation,

A

\mathcal {A}

, of size

|

A

|

≤
k

|\mathcal {A}| \leq k

(Theorem 2.4). This result is used to give a new proof of a theorem of D. Duffus and M. Pouzet [4] about gaps in a lattice of finite breadth. We do not know if a closure which has no infinite independent set necessarily has a cardinal representation, but we do prove this is so for the special case of a closure on a countable set (Theorem 2.5). Even in this special case, nothing can be said about the size of the cardinal representation; however, if the closure is algebraic, then there is a finite cardinal representation (Theorem 2.6). These results do not hold for preclosures in general, but if a preclosure on a countable set has no independent set of size

k
+
1

k + 1

(

k
k

finite), then it has a cardinal representation

A

\mathcal {A}

of size

|

A

|

≤
k

|\mathcal {A}| \leq k

(Theorem 2.7).
Commuting fully invariant congruences on free completely regular semigroups
Authors: Pastijn F.
Published: June 2012

We show that "almost all" fully invariant congruences on every free completely regular semigroup commute. From this it is shown that the lattice of completely regular semigroup varieties is arguesian.
Quantitative stability of variational systems. I. The epigraphical distance
Authors: Attouch Hédy , Wets Roger J.-B.
Published: June 2012

This paper proposes a global measure for the distance between the elements of a variational system (parametrized families of optimization problems).
On Hermite-Fejér interpolation in a Jordan domain
Authors: Chui Charles K. , Shen Xie Chang
Published: June 2012

The Hermite-Fejér interpolation problem on a Jordan domain is studied. Under certain mild conditions on the smoothness of the boundary curve, we give both uniform and

L
p

{L^p}

,

0
>
p
>
∞

0 > p > \infty

, estimates on the rate of convergence. Our estimates are sharp even for the unit disk setting.
Genericity of nontrivial 𝐻-superrecurrent 𝐻-cocycles
Authors: Dajani Karma
Published: June 2012

We prove that most

H

{\text {H}}

-cocycles for a nonsingular ergodic transformation of type

II

I

λ

{\text {II}}{{\text {I}}_\lambda }

,

0
>
λ
>
1

0 > \lambda > 1

, are

H

{\text {H}}

-superrecurrent. This is done by showing that the set of nontrivial

H

{\text {H}}

-superrecurrent

H

{\text {H}}

-cocycles form a dense

G
δ

{G_\delta }

set with respect to the topology of convergence in measure.
Change of variable results for 𝐴_{𝑝}- and reverse Hölder 𝑅𝐻ᵣ-classes
Authors: Johnson R. , Neugebauer C. J.
Published: June 2012

We study conditions under which the map
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript h comma gamma Baseline w left-parenthesis x right-parenthesis equals w left-parenthesis h left-parenthesis x right-parenthesis right-parenthesis StartAbsoluteValue h prime left-parenthesis x right-parenthesis EndAbsoluteValue Superscript gamma">

T

h
,
γ

w
(
x
)
=
w
(
h
(
x
)
)

|

h
′

(
x
)

|

γ

{T_{h,\gamma }}w(x)= w(h(x))|h’(x){|^\gamma }

\]
maps the Muckenhoupt class

A
p

{A_p}

into

A
q

{A_q}

and the reverse Hölder class

R

H

r
1

R{H_{{r_1}}}

into

R

H

r
2

R{H_{{r_2}}}

.
Area integral estimates for the biharmonic operator in Lipschitz domains
Authors: Pipher Jill , Verchota Gregory
Published: June 2012

Let

D
⊆

R

n

D \subseteq {{\mathbf {R}}^n}

be a Lipschitz domain and let

u
u

be a function biharmonic in

D
D

, i.e.,

Δ
Δ
u
=
0

\Delta \Delta u= 0

in

D
D

. We prove that the nontangential maximal function and the square function of the gradient of

u
u

have equivalent

L
p

(
d
μ
)

{L^p}(d\mu )

norms, where

d
μ
∈

A
∞

(
d
σ
)

d\mu \in {A^\infty }\,(d\sigma )

and

d
σ

d\sigma

is surface measure on

∂
D

\partial D

.
Alexander duality and Hurewicz fibrations
Authors: Ferry Steven C.
Published: June 2012

We explore conditions under which the restriction of the projection map

p
:

S
n

×
B
→
B

p:{S^n} \times B \to B

to an open subset

U
⊂

S
n

×
B

U \subset S^n \times B

is a Hurewicz fibration. As a consequence, we exhibit Hurewicz fibrations

p
:
E
→
I

p:E \to I

such that: (i)

p
:
E
→
I

p:E \to I

is not a locally trivial bundle, (ii)

p

−
1

(
t
)

p^{ - 1}(t)

is an open

n
n

-manifold for each

t
t

, and (iii)

p

∘

proj:E

×

R
1

→
I

p\; \circ \;{\text {proj:E}} \times {R^1} \to I

is a locally trivial bundle. The fibers in our examples are distinguished by having nonisomorphic fundamental groups at infinity. We also show that when the fibers of a Hurewicz fibration with open

n
n

-manifold fibers have finitely generated

(
n
−
1
)

st

(n - 1){\text {st}}

homology, then all fibers have the same finite number of ends. This last shows that the punctured torus and the thrice punctured two-sphere cannot both be fibers of a Hurewicz fibration

p
:
E
→
I

p:E \to I

with open

2
2

-manifold fibers.
Generalizations of Picard’s theorem for Riemann surfaces
Authors: Järvi Pentti
Published: June 2012

Let

D
D

be a plane domain,

E
⊂
D

E \subset D

a compact set of capacity zero, and

f
f

a holomorphic mapping of

D
∖
E

D\backslash E

into a hyperbolic Riemann surface

W
W

. Then there is a Riemann surface

W
′

W’

containing

W
W

such that

f
f

extends to a holomorphic mapping of

D
D

into

W
′

W’

. The same conclusion holds if hyperbolicity is replaced by the assumption that the genus of

W
W

be at least two. Furthermore, there is quite a general class of sets of positive capacity which are removable in the above sense for holomorphic mappings into Riemann surfaces of positive genus, except for tori.
Unit groups and class numbers of real cyclic octic fields
Authors: Shen Yuan Yuan
Published: June 2012

The generating polynomials of D. Shanks’ simplest quadratic and cubic fields and M.-N. Gras’ simplest quartic and sextic fields can be obtained by working in the group

P
G

L

2

(

Q

)

{\mathbf {PG}}{{\mathbf {L}}_2}({\mathbf {Q}})

. Following this procedure and working in the group

P
G

L

2

(

Q

(

2

)
)

{\mathbf {PG}}{{\mathbf {L}}_2}({\mathbf {Q}}(\sqrt 2))

, we obtain a family of octic polynomials and hence a family of real cyclic octic fields. We find a system of independent units which is close to being a system of fundamental units in the sense that the index has a uniform upper bound. To do this, we use a group theoretic argument along with a method similar to one used by T. W. Cusick to find a lower bound for the regulator and hence an upper bound for the index. Via Brauer-Siegel’s theorem, we can estimate how large the class numbers of our octic fields are. After working out the first three examples in

§
5

\S 5

, we make a conjecture that the index is

8
8

. We succeed in getting a system of fundamental units for the quartic subfield. For the octic field we obtain a set of units which we conjecture to be fundamental. Finally, there is a very natural way to generalize the octic polynomials to get a family of real

2
n

{2^n}

-tic number fields. However, to select a subfamily so that the fields become Galois over

Q

{\mathbf {Q}}

is not easy and still a lot of work on these remains to be done.
Microlocal Holmgren’s theorem for a class of hypo-analytic structures
Authors: Berhanu S.
Published: June 2012

A microlocal version of Holmgren’s Theorem is proved for a certain class of the hypo-analytic structures of Baouendi, Chang, and Treves.
Automorphisms and twisted forms of generalized Witt Lie algebras
Authors: Waterhouse William C.
Published: June 2012

We prove that the automorphisms of the generalized Witt Lie algebras

W
(
m
,

n

)

W(m,{\mathbf {n}})

over arbitrary commutative rings of characteristic

p
≥
3

p \geq 3

all come from automorphisms of the algebras on which they are defined as derivations. By descent theory, this result then implies that if a Lie algebra over a field becomes isomorphic to

W
(
m
,

n

)

W(m,{\mathbf {n}})

over the algebraic closure, it is a derivation algebra of the type studied long ago by Ree. Furthermore, all isomorphisms of those derivation algebras are induced by isomorphisms of their underlying associative algebras.
Action on Grassmannians associated with a field extension
Authors: Rabau Patrick
Published: June 2012

We examine the action of the general linear group

GL

L

(
V
)

{\text {GL}}_L(V)

on the set of all

K
K

-subspaces of

V
V

, where

L

/

K

L/K

is a finite field extension and

V
V

is a finite-dimensional vector space over

L
L

. The orbits are completely classified in the case of quadratic and cubic extensions; for infinite fields, the number of orbits is shown to be infinite if the degree of the extension is at least four. As an application we obtain

q
q

-analogues of tranformation and evaluation formulas for hypergeometric functions due to Gessel and Stanton.
Remarks on forced equations of the double pendulum type
Authors: Tarantello Gabriella
Published: June 2012

Motivated by the double pendulum equation we consider Lagrangian systems with potential

V
=
V
(
t
,
q
)

V = V(t,q)

periodic in each of the variables

t
,
q
=
(

q
1

,
…
,

q
N

)

t,q = ({q_1}, \ldots ,{q_N})

. We study periodic solutions for the corresponding equation of motion subject to a periodic force

f
=
f
(
t
)

f = f(t)

. If

f
f

has mean value zero, the corresponding variational problem admits a

Z

N

{{\mathbf {Z}}^N}

symmetry which yields

N
+
1

N + 1

distinct periodic solutions (see [9]). Here we consider the case where the average of

f
f

, though bounded, is no longer required to be zero. We show how this situation becomes more delicate, and in general it is only possible to claim no more than two periodic solutions.
𝑘-cobordism for links in 𝑆³
Authors: Cochran Tim D.
Published: June 2012

We give an explicit finite set of (based) links which generates, under connected sum, the

k
k

-cobordism classes of links. We show that the union of these generating sets,

2
≤
k
>
∞

2 \leq k > \infty

, is not a generating set for

ω
\omega

-cobordism classes or even

∞
\infty

-cobordism classes. For

2
2

-component links in

S
3

{S^3}

we define

(
2
,
k
)

(2,k)

-corbordism and show that the concordance invariants

β
i

,
i
∈

Z

+

{\beta ^i},i \in {\mathbb {Z}^+}

, previously defined by the author, are invariants under

(
2
,
i
+
1
)

(2,i + 1)

-cobordism. Moreover we show that the

(
2
,
k
)

(2,k)

-cobordism classes of links (with linking number 0) is a free abelian group of rank

k
−
1

k - 1

, detected precisely by

β
1

×
⋯
×

β

k
−
1

{\beta ^1} \times \cdots \times {\beta ^{k - 1}}

. We write down a basis. The union of these bases

(
2
≤
k
>
∞
)

(2 \leq k > \infty )

is not a generating set for

(
2
,
∞
)

(2,\infty )

or

(
2
,
ω
)

(2,\omega )

-cobordism classes. However, we can show that

∏

i
=
1

∞

β
i

(

)

\prod _{i = 1}^\infty {\beta ^i}(\;)

is an isomorphism from the group of

(
2
,
∞
)

(2,\infty )

-cobordism classes to the subgroup

R

⊂

∏

i
=
1

∞

Z

\mathcal {R} \subset \prod _{i = 1}^\infty \mathbb {Z}

of linearly recurrent sequences, so a basis exists by work of T. Jin.
On lifting Hecke eigenforms
Authors: Walling Lynne H.
Published: June 2012

A classical Hilbert modular form

f
∈

M

k

(

Γ
0

(

N

,

I

)
,

χ

N

)

f \in {\mathcal {M}_k}({\Gamma _0}(\mathcal {N},\mathcal {I}),{\chi _\mathcal {N}})

cannot be an eigenform for the full Hecke algebra. We develop a means of lifting a classical form to a modular form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F element-of circled-plus script upper M Subscript k Baseline left-parenthesis normal upper Gamma 0 left-parenthesis script upper N comma script upper I Subscript lamda Baseline right-parenthesis comma chi Subscript script upper N Baseline right-parenthesis">

F
∈

⊕
λ

M

k

(

Γ
0

(

N

,

I

λ

)
,

χ

N

)

F \in { \oplus _\lambda }{\mathcal {M}_k}({\Gamma _0}(\mathcal {N},{\mathcal {I}_\lambda }),{\chi _\mathcal {N}})

which is an eigenform for the full Hecke algebra. Using this lift, we develop the newform theory for a space of cusp forms

S

k

(

Γ
0

(

N

,

I

)
,

χ

N

)

{\mathcal {S}_k}({\Gamma _0}(\mathcal {N},\mathcal {I}),{\chi _\mathcal {N}})

; we also use theta series to construct eigenforms for the full Hecke algebra.
Action on Grassmannians associated with commutative semisimple algebras
Authors: Kim Dae San , Rabau Patrick
Published: June 2012

Let

A
A

be a finite-dimensional commutative semisimple algebra over a field

k
k

and let

V
V

be a finitely generated faithful

A
A

-module. We study the action of the general linear group

GL

A

(
V
)

{\text {GL}}_A(V)

on the set of all

k
k

-subspaces of

V
V

and show that, if the field

k
k

is infinite, there are infinitely many orbits as soon as

A
A

has dimension at least four. If

A
A

has dimension two or three, the number of orbits is finite and independent of the field; in each such case we completely classify the orbits by means of a certain number of integer parameters and determine the structure of the quotient poset obtained from the action of

GL

A

(
V
)

{\text {GL}}_A(V)

on the poset of

k
k

-subspaces of

V
V

.
The transverse homoclinic dynamics and their bifurcations at nonhyperbolic fixed points
Authors: Deng Bo
Published: June 2012

The complete description of the dynamics of diffeomorphisms in a neighborhood of a transverse homoclinic orbit to a hyperbolic fixed point is obtained. It is topologically conjugate to a non-Bernoulli shift called

{

∑
,
σ

}

\{ {\sum ,\sigma } \}

. We also obtain a more or less complete picture, referred to as the net weaving bifurcation, when the fixed point of such a system is undergoing the generic saddle-node bifurcation. The idea of homotopy conjugacy is naturally introduced to show that systems whose fixed points undergo the pitchfork, transcritical, periodic doubling, and Hopf bifurcations are all homotopically conjugate to our shift dynamics

{

∑
,
σ

}

\{ {\sum ,\sigma } \}

in a neighborhood of a transverse homoclinic orbit. These bifurcations are also examined in the context of the spectral decomposition with respect to the maximal indecomposable nonwandering sets.
Locally constant cohomology
Authors: Spanier E.
Published: June 2012

In this paper we study locally constant cohomology theories on a space

X
X

. We prove that for cohomology theories on a category of paracompact spaces the homotopy axiom of Eilenberg-Steenrod is a consequence of the other Eilenberg-Steenrod axioms together with continuity and either additivity or weak additivity. We also prove that if

H
H

is a cohomology theory on the space of a simplicial complex

K
K

which is locally constant on every open simplex of

K
K

there is a spectral sequence converging to

H
(

|

K

|

)

H(|K|)

whose

E
2

{E_2}

-term is the usual simplicial cohomology of

K
K

with coefficients in various stacks on

K
K

defined by

H
H

. This generalizes some known spectral sequences.
Symmetric local algebras with 5-dimensional center
Authors: Chlebowitz M. , Külshammer B.
Published: June 2012

We prove that a symmetric split local algebra whose center is

5
5

-dimensional has dimension

5
5

or

8
8

. This implies that the defect groups of a block of a finite group containing exactly five irreducible Frobenius characters and exactly one irreducible Brauer character have order

5
5

or are nonabelian of order

8
8

.
Characterization for the solvability of nonlinear partial differential equations
Authors: Rosinger Elemer E.
Published: June 2012

Within the nonlinear theory of generalized functions introduced earlier by the author a number of existence and regularity results have been obtained. One of them has been the first global version of the Cauchy-Kovalevskaia theorem, which proves the existence of generalized solutions on the whole of the domain of analyticity of arbitrary analytic nonlinear

PDEs
\text {PDEs}

. These generalized solutions are analytic everywhere, except for closed, nowhere dense subsets which can be chosen to have zero Lebesgue measure. This paper gives a certain extension of that result by establishing an algebraic necessary and sufficient condition for the existence of generalized solutions for arbitrary polynomial nonlinear

PDEs
\text {PDEs}

with continuous coefficients. This algebraic characterization, given by the so-called neutrix or off diagonal condition, is proved to be equivalent to certain densely vanishing conditions, useful in the study of the solutions of general nonlinear

PDEs
\text {PDEs}

.
Stable splittings of the dual spectrum of the classifying space of a compact Lie group
Authors: Lee Chun-Nip
Published: June 2012

For a compact Lie group

G
G

, there is a map from the

G
G

-equivariant fixed point spectrum of the zero sphere to the dual spectrum of the classifying space of

G
,
D
B

G
+

G, DB{G_ + }

. When

G
G

is finite, the affirmative solution to Segal’s conjecture states that this map is an equivalence upon appropriate completion of the source. In the case of a compact Lie group, we obtain splitting results of

D
B

G
+

DB{G_ + }

via this map upon taking

p
p

-adic completions.
Symmetry of knots and cyclic surgery
Authors: Wang Shi Cheng , Zhou Qing
Published: June 2012

If a nontorus knot

K
K

admits a symmetry which is not a strong inversion, then there exists no nontrivial cyclic surgery on

K
K

. No surgery on a symmetric knot can produce a fake lens space or a

3
3

-manifold

M
M

with

|

π
1

(
M
)

|

=
2

|{\pi _1}(M)|= 2

. This generalizes the result of Culler-Gordon-Luecke-Shalen-Bleiler-Scharlemann and supports the conjecture that no nontrivial surgery on a nontrivial knot yields a

3
3

-manifold

M
M

with

|

π
1

(
M
)

|

>
5

|{\pi _1}(M)| > 5

.
Superharmonic functions on foliations
Authors: Adams S. R.
Published: June 2012

We use techniques from geometric analysis to prove that any positive, leafwise superharmonic, measurable function on a Riemannian measurable foliation with transverse invariant measure, finite total volume and complete leaves is, in fact, constant on a.e. leaf.
Grothendieck groups of quotient singularities
Authors: Marcos Eduardo do Nascimento
Published: June 2012

Given a quotient singularity

R
=

S
G

R = {S^G}

where

S
=

C

[
[

x
1

,
…
,

x
n

]
]

S = {\mathbf {C}}[[{x_1}, \ldots ,{x_n}]]

is the formal power series ring in

n
n

-variables over the complex numbers

C

{\mathbf {C}}

, there is an epimorphism of Grothendieck groups

ψ
:

G
0

(
S
[
G
]
)
→

G
0

(
R
)

\psi :{G_0}(S[G]) \to {G_0}(R)

, where

S
[
G
]

S[G]

is the skew group ring and

ψ
\psi

is induced by the fixed point functor. The Grothendieck group of

S
[
G
]

S[G]

carries a natural structure of a ring, isomorphic to

G
0

(

C

[
G
]
)

{G_0}({\mathbf {C}}[G])

. We show how the structure of

G
0

(
R
)

{G_0}(R)

is related to the structure of the ramification locus of

V
V

over

V

/

G

V/G

, and the action of

G
G

on it. The first connection is given by showing that

Ker

ψ

\operatorname {Ker}\;\psi

is the ideal generated by

[

C

]

[{\mathbf {C}}]

if and only if

G
G

acts freely on

V
V

. That this is sufficient has been proved by Auslander and Reiten in [4]. To prove the necessity we show the following: Let

U
U

be an integrally closed domain and

T
T

the integral closure of

U
U

in a finite Galois extension of the field of quotients of

U
U

with Galois group

G
G

. Suppose that

|

G

|

|G|

is invertible in

U
U

, the inclusion of

U
U

in

T
T

is unramified at height one prime ideals and

T
T

is regular. Then

G
0

(
T
[
G
]
)
≅
Z

{G_0}(T[G]) \cong Z

if and only if

U
U

is regular. We analyze the situation

V
=

V
1

∐

C

[
G
]

V
2

V = {V_1}{\coprod } _{\mathbf {C}[G]}{V_2}

where

G
G

acts freely on

V
1

,

V
1

≠
0

{V_1},{V_1} \ne 0

. We prove that for a quotient singularity

R
,

G
0

(
R
)
≅

G
0

(
R
[
[
t
]
]
)

R,{G_0}(R) \cong {G_0}(R[[t]])

. We also study the structure of

G
0

(
R
)

{G_0}(R)

for some cases with

dim
⁡
R
=
3

\dim R = 3

.
Hyperfinite transversal theory
Authors: Živaljević Boško
Published: June 2012

A measure theoretic version of a well-known P. Hall’s theorem, about the existence of a system of distinct representatives of a finite family of finite sets, has been proved for the case of the Loeb space of an internal, uniformly distributed, hyperfinite counting space. We first prove Hall’s theorem for

Π
1
0

(
κ
)

\Pi _1^0(\kappa )

graphs after which we develop the version of discrete Transversal Theory. We then prove a new version of Hall’s theorem in the case of

Σ
1
0

(
κ
)

\Sigma _1^0(\kappa )

monotone graphs and give an example of a

Σ
1
0

\Sigma _1^0

graph which satisfies Hall’s condition and which does not possess an internal a.e. matching.
Central limit theorems for sums of Wick products of stationary sequences
Authors: Avram Florin , Fox Robert
Published: June 2012

We show, by the method of cumulants, that checking whether the central limit theorem for sums of Wick powers of a stationary sequence holds can be reduced to the study of an associated graph problem (see Corollary 1). We obtain thus central limit theorems under various integrability conditions on the cumulant spectral functions (Theorems 2, 3).
Adams’ cobar equivalence
Authors: Félix Yves , Halperin Stephen , Thomas Jean-Claude
Published: June 2012

Let

F
F

be the homotopy fibre of a continuous map

Y

→

ω

X

Y\xrightarrow {\omega }X

, with

X
X

simply connected. We modify and extend a construction of Adams to obtain equivalences of DGA’s and DGA modules,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega upper C Subscript asterisk Baseline left-parenthesis upper X right-parenthesis ModifyingAbove right-arrow With asymptotically-equals upper C upper U Subscript asterisk Baseline left-parenthesis normal upper Omega upper X right-parenthesis comma">

Ω

C

∗

(
X
)

→

≃

C

U

∗

(
Ω
X
)
,

\Omega {C_{\ast }}(X)\xrightarrow { \simeq }C{U_{\ast }}(\Omega X),

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega left-parenthesis upper C Subscript asterisk Superscript omega Baseline left-parenthesis upper Y right-parenthesis semicolon upper C Subscript asterisk Baseline left-parenthesis upper X right-parenthesis right-parenthesis ModifyingAbove right-arrow With asymptotically-equals upper C upper U Subscript asterisk Baseline left-parenthesis upper F right-parenthesis comma">

Ω
(

C

∗

ω

(
Y
)
;

C

∗

(
X
)
)

→

≃

C

U

∗

(
F
)
,

\Omega (C_{\ast }^\omega (Y);{C_{\ast }}(X))\xrightarrow { \simeq }C{U_{\ast }}(F),

\]
where on the left-hand side

Ω
(
−
)

\Omega ( - )

denotes the cobar construction. Our equivalences are natural in

X
X

and

ω
\omega

. Using this result we show how to read off the algebra

H

∗

(
Ω
X
;
R
)

{H_{\ast }}(\Omega X;R)

and the

H

∗

(
Ω
X
;
R
)

{H_{\ast }}(\Omega X;R)

module,

H

∗

(
F
;
R
)

{H_{\ast }}(F;R)

, from free models for the singular cochain algebras

C

S

∗

(
X
)

C{S^{\ast }}(X)

and

C

S

∗

(
Y
)

C{S^{\ast }}(Y)

; here we assume

R
R

is a principal ideal domain and

X
X

and

Y
Y

are of finite

R
R

type.
A quasiregular analogue of a theorem of Hardy and Littlewood
Authors: Nolder Craig A.
Published: June 2012

Suppose that

f
f

is analytic in the unit disk. A theorem of Hardy and Littlewood relates the Hölder continuity of

f
f

over the unit disk to the growth of the derivative. We prove here a quasiregular analogue of this result in certain domains in

n
n

-dimensional space. We replace values of the derivative with a local integral average. In the process we generalize a result on the continuity of quasiconformal mappings due to Nakki and Palka. We also present another proof of the relationship between the growth of the derivative and quasiregular mappings in

BMO
\text {BMO}

.
An open collar theorem for 4-manifolds
Authors: Guilbault Craig R.
Published: June 2012

Let

M
4

{M^4}

be an open

4
4

-manifold with boundary. Conditions are given under which

M
4

{M^4}

is homeomorphic to

∂
M
×
[
0
,
1
)

\partial M \times [0,1)

. Applications include a

4
4

-dimensional weak

h
h

-cobordism theorem and a classification of weakly flat embeddings of

2
2

-spheres in

S
4

{S^4}

. Specific examples of

(
n
−
2
)

(n - 2)

-spheres embedded in

S
n

{S^n}

(including

n
=
4

n = 4

) are also discussed.
Trace functions in the ring of fractions of polycyclic group rings
Authors: Lichtman A. I.
Published: June 2012

Let

K
G

KG

be the group ring of a polycyclic-by-finite group

G
G

over a field

K
K

of characteristic zero,

R
R

be the Goldie ring of fractions of

K
G

KG

,

S
S

be an arbitrary subring of

R

n
×
n

{R_{n \times n}}

. We prove that the intersection of the commutator subring

[
S
,
S
]

[S,S]

with the center

Z
(
S
)

Z(S)

is nilpotent. This implies the existence of a nontrivial trace function in

R

n
×
n

{R_{n \times n}}

.