bluuebunny/crossref_metadata_2025 huggingface dataset:
Permutations and presentations
Authors: Cholak Peter , Downey Rod
Published: July 2010

We say that an automorphismΦ\PhiofE∗{\mathcal {E}^ \ast }(the lattice of recursively enumerable sets modulo the finite sets) isinducedby a permutationpiff for alle,Φ(We)=∗p(We)\Phi ({W_e}){ = ^ \ast }p({W_e}). A permutationhis called apresentationofΦ\Phiiff for alle,Φ(We)=∗Wh(e)\Phi ({W_e}){ = ^\ast }{W_{h(e)}}. In this paper, we will explore the degree-theoretic connections between these two notions. Using a new proof of the well-known fact that every automorphism is induced by a permutationp, we show that such apcan be found recursively inh⊕∅h \oplus \emptyset, wherehis a presentation ofΦ\Phi. The main result of the paper is to show that there is an effective automorphism ofE∗{\mathcal {E}^ \ast }which is not induced by aΔ2{\Delta _2}-permutation.
A note on discriminantal arrangements
Authors: Falk Michael
Published: July 2010

Let

A

0

{\mathcal {A}_0}

be a fixed affine arrangement of n hyperplanes in general position in

K

k

{{\mathbf {K}}^k}

. Let

U
(
n
,
k
)

U(n,k)

denote the set of general position arrangements whose elements are parallel translates of the hyperplanes of

A

0

{\mathcal {A}_0}

. Then

U
(
n
,
k
)

U(n,k)

is the complement of a central arrangement

B

(
n
,
k
)

\mathcal {B}(n,k)

. These are the well-known discriminantal arrangements introduced by Y. I. Manin and V. V. Schechtman. In this note we give an explicit description of

B

(
n
,
k
)

\mathcal {B}(n,k)

in terms of the original arrangement

A

0

{\mathcal {A}_0}

. In terms of the underlying matroids,

B

(
n
,
k
)

\mathcal {B}(n,k)

realizes an adjoint of the dual of the matroid associated with

A

0

{\mathcal {A}_0}

. Using this description we show that, contrary to the conventional wisdom, neither the intersection lattice of

B

(
n
,
k
)

\mathcal {B}(n,k)

nor the topology of

U
(
n
,
k
)

U(n,k)

is independent of the original arrangement

A

0

{\mathcal {A}_0}

.
Baire paradoxical decompositions need at least six pieces
Authors: Wehrung Friedrich
Published: July 2010

We show that in certain cases paradoxical decompositions of compact metric spaces using sets (or even [0, 1 ]-valued functions) with the property of Baire modulo meager sets need more pieces than paradoxical decompositions with unrestricted pieces. In particular, any Baire paradoxical decomposition of the sphere

S
2

{S^2}

using isometries needs at least six pieces.
All cluster points of countable sets in supercompact spaces are the limits of nontrivial sequences
Authors: Yang Zhong Qiang
Published: July 2010

A space is called supercompact if it has an open subbase such that every cover consisting of elements of the subbase has a subcover consisting of two elements. In this paper we prove that, in a continuous image of a closed

G
δ

{G_\delta }

-set of a supercompact space, a point is a cluster point of a countable set if and only if it is the limit of a nontrivial sequence. As corollaries, we answer questions asked by J. van Mill et al.
Ricci tensors of real hypersurfaces in a complex projective space
Authors: Maeda Sadahiro
Published: July 2010

This paper gives a classification of real hypersurfaces in a complex projective space under assumptions that the structure vector

ξ
\xi

is principal, the focal map has constant rank, and

∇
ξ

S
=
0

{\nabla _\xi }S = 0

, where S is the Ricci tensor of the real hypersurface.
Geometric properties of some subspaces of 𝑉𝑁(𝐺)
Authors: Belanger Alain , Forrest Brian E.
Published: July 2010

Let G be a locally compact group. We show that any one of the spaces
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U upper C upper B left-parenthesis ModifyingAbove upper G With caret right-parenthesis comma upper W upper A upper P left-parenthesis ModifyingAbove upper G With caret right-parenthesis comma upper A upper P left-parenthesis ModifyingAbove upper G With caret right-parenthesis">

U
C
B
(

G
^

)
,
W
A
P
(

G
^

)
,
A
P
(

G
^

)

UCB(\hat G),WAP(\hat G),AP(\hat G)

, and

C
δ
∗

(
G
)

C_\delta ^ \ast (G)

is Asplund if and only if the group G is finite. We also show that any one of the spaces

V
N
(
G
)
,
U
C
B
(

G
^

)

VN(G),UCB(\hat G)

, and

C
δ
∗

(
G
)

C_\delta ^\ast (G)

has the DPP if and only if the group G has an abelian subgroup of finite index.
A note on endomorphisms of irrational rotation 𝐶*-algebras
Authors: Kodaka Kazunori
Published: July 2010

Let

θ
\theta

be an irrational number and

A
θ

{A_\theta }

the corresponding irrational rotation

C

∗

{{\text {C}}^ \ast }

-algebra. In the present note we will show that proper endomorphisms of

A
θ

{A_\theta }

of a certain sort do not occur if

θ
\theta

is not quadratic.
Pathwise connectivity of the spatial numerical range
Authors: Kuliyev Tofik Y.
Published: July 2010

In this paper we prove that the spatial numerical range of a given operator on a separable Banach space is pathwise connected.
Bias and variance reduction in estimation of model dimension
Authors: Loh Wei-Yin , Zheng Xiaodong
Published: July 2010

The problem of estimating the number of regressors to include in a linear regression model is considered. Estimators based on the final prediction error and Akaike’s criterion frequently have large positive bias. Shrinkage correction factors and bootstrapping are used to produce new estimators with reduced bias. The asymptotic bias and mean-squared errors of these estimators are derived analytically. Finite-sample estimates are obtained by simulation.
On quasi-continuous rings
Authors: Nicholson W. K. , Yousif M. F.
Published: July 2010

A well-known result of Utumi asserts that a two-sided continuous two-sided artinian ring is quasi-Frobenius. In this paper we extend Utumi’s result to quasi-continuous rings.
Solutions of three-term relations in several variables
Authors: Xu Yuan
Published: July 2010

A system of multivariate orthogonal polynomials satisfies a matrix equation which plays the role of a three-term relation of the orthogonal polynomial of one variable. However, unlike the case of one variable, there does not exist a second solution of this matrix equation that is linearly independent to the orthogonal polynomials. In particular, there is no analogy of the associated polynomials in several variables.
Approximation method and equilibria of abstract economies
Authors: Tan Kok-Keong , Yuan Xian-Zhi
Published: July 2010

By applying an equilibrium existence theorem for a qualitative game due to Ding and Tan and by employing an "approximation" method used by Tulcea, we obtain an equilibrium existence theorem for an abstract economy (generalized game) in which the constraint correspondences are not assumed to have open graphs nor open lower sections (which are generally assumed in the literature). Our result generalizes the corresponding results of Shafer and Sonnenschein (1975), Borglin and Keiding (1976), Yannelis and Prabhakar (1983), Tulcea (1986), and Chang (1990) in several ways.
Hardy inequalities and imbeddings in domains generalizing 𝐶^{0,𝜆} domains
Authors: Wannebo Andreas
Published: July 2010

Sufficient conditions for Hardy inequalities in a domain are studied together with related imbedding inequalities. The weights are nonnegative monotone functions of the distance to the boundary, and the domains are a generalization of

C

0
,
λ

{C^{0,\lambda }}

domains, obtained by replacing the power function in the definition with a general nondecreasing one.
A Leray-Schauder type theorem for approximable maps
Authors: Ben-El-Mechaiekh H. , Idzik A.
Published: July 2010

We present an elementary proof of a Leray-Schauder type theorem for approximable set-valued maps. Our theorem generalizes many results for convex as well as nonconvex maps. Our argument is not based on a homotopy invariance property but, quite surprisingly, on a matching theorem of Ky Fan on closed covers of convex sets.
On the modular equations of degree 3
Authors: Shen Li-Chien
Published: July 2010

In this paper, we use the twelve Jacobian elliptic functions to derive a collection of 24 identities which are essential in the study of the modular equations of degree 3.
On the number of generators of modules over polynomial rings
Authors: Li Hongnian
Published: July 2010

In this paper we prove the following Theorem. Let

B
=
A
[

X
1

,
⋅
,

X
n

]

B = A[{X_1}, \cdot ,{X_n}]

, where A is a universally catenary equidimensional ring. Let M be a finitely generated B-module of rank r. Denote by d the Krull dimension of A, by

μ
(
M
)

\mu (M)

the minimal number of generators of M, and by

I
M

{I_M}

the (radical) ideal which defines the set of primes of B at which M is not locally free. Assume that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu left-parenthesis upper M slash upper I Subscript upper M Baseline upper M right-parenthesis less-than-or-equal-to eta a n d eta greater-than-or-equal-to max left-brace right-brace comma plus plus dr comma plus plus plus slash slash of dimdimBIMr 1 comma">

μ
(
M

/

I
M

M
)
≤
η

a
n
d

η
≥
max
{
d
+
r
,
dim
⁡
B

/

I
M

+
r
+
1
}
,

\mu (M/{I_M}M) \leq \eta \;and\;\eta \geq \max \{ d + r,\dim B/{I_M} + r + 1\} ,

\]
where

η
\eta

is a positive integer. Then

μ
(
M
)
≤
η

\mu (M) \leq \eta

. This improves a result of R. G. López, On the number of generators of modules over polynomial affine rings, Math. Z. 208 (1991), 11-21.
Hecke algebra on homogeneous trees and relations with Toeplitz and Hankel operators
Authors: Wysoczański Janusz
Published: July 2010

We consider the Hecke algebra on homogeneous trees. We prove that it is a maximal abelian subalgebra of some operator algebras if the degree of the tree is greater than 2. There we show the influence of geometry of the tree on that fact. If the degree is 2 (for example, in the case of integers) then we show that operators which commute with the Hecke algebra can be uniquely represented as a sum of Hankel and Toeplitz matrices.
Existence of 𝜋₁-negligible embeddings in 4-manifolds. A correction to Theorem 10.5 of Freedmann and Quinn
Authors: Stong Richard
Published: July 2010

The purpose of this note is to provide a correction to the existence part of Theorems 10.3 and 10.5 of Topology of

4
4

-manifolds, Princeton Univ. Press, Princeton, NJ, 1990, which analyze when one can find a connected sum decomposition of a

4
4

-manifold or a

π
1

{\pi _1}

-negligible embedding in a

4
4

-manifold respectively. In particular this gives a correction to the definition of the

4
4

-dimensional Kervaire-Milnor invariant. We also define this invariant in a slightly more general context.
On upper semicontinuity of duality mappings
Authors: Contreras Manuel D. , Payá Rafael
Published: July 2010

We give new sufficient conditions for a Banach space to be an Asplund (or reflexive) space in terms of certain upper semicontinuity of the duality mapping.
Spatiality of isomorphisms between certain reflexive algebras
Authors: Lambrou M. S. , Longstaff W. E.
Published: July 2010

Two subspaces M and N of a Hilbert space H are in generalized generic position if

M
∩
N
=

M
⊥

∩

N
⊥

=
(
0
)

M \cap N = {M^ \bot } \cap {N^ \bot } = (0)

and

dim
⁡
(

M
⊥

∩
N
)
=
dim
⁡
(
M
∩

N
⊥

)

\dim ({M^ \bot } \cap N) = \dim (M \cap {N^ \bot })

. If H is separable and both the pairs

{

M
1

,

N
1

}

\{ {M_1},{N_1}\}

and

{

M
2

,

N
2

}

\{ {M_2},{N_2}\}

are in generalized generic position, then every algebraic isomorphism

φ
:

Alg

{

M
1

,

N
1

}
→

Alg

{

M
2

,

N
2

}

\varphi :{\operatorname {Alg}}\{ {M_1},{N_1}\} \to {\operatorname {Alg}}\{ {M_2},{N_2}\}

is spatially induced, that is, there exists an invertible operator

T
0

∈

B

(
H
)

{T_0} \in \mathcal {B}(H)

such that

φ
(
B
)
=

T
0

B

T
0

−
1

\varphi (B) = {T_0}BT_0^{ - 1}

, for every

B
∈

Alg

{

M
1

,

N
1

}

B \in {\operatorname {Alg}}\{ {M_1},{N_1}\}

. The proof of this uses the following result: If H is separable,

M

⊆
H

\mathcal {M} \subseteq H

is a proper operator range in H, and the operator

T
∈

B

(
H
)

T \in \mathcal {B}(H)

has the property that, for every

W
∈

B

(
H
)

W \in \mathcal {B}(H)

leaving

M

\mathcal {M}

invariant, the range of

W
T
−
T
W

WT - TW

is included in

M

\mathcal {M}

, then the range of

T
−
λ

T - \lambda

is included in

M

\mathcal {M}

, for some unique scalar

λ
\lambda

.
Homogeneous Hamiltonian 𝐺-bundles
Authors: Lashof R.
Published: July 2010

The Kirillov, Kostant, Souriau, et al. theorem on homogeneous HamiltonianG-manifolds is generalized to (closed) HamiltonianG-manifolds with isotropy groups of (fixed rank) maximal rank.
A note concerning a theorem of Cramér
Authors: Mitrović Slobodanka
Published: July 2010

In this paper we consider H. Cramer’s Theorem 5.2 from Structural and statistical problems for a class of stochastic processes, Princeton Univ. Press, Princeton, NJ, 1971, and show that the proof of this theorem is not valid.
The existence of bounded infinite 𝐷Tr-orbits
Authors: Liu Shi Ping , Schulz Rainer
Published: July 2010

We construct an indecomposable module over a symmetric algebra whose DTr-ovbil is infinite and bounded. This yields a counterexample to a conjecture which states that the number of modules in an Auslander-Reiten component having the same length is finite.
Values of 𝐿-functions at the critical point
Authors: Mai Liem
Published: July 2010

For a discriminant D of a binary quadratic form, we study the average value of

L
(
s
,

ε
D

)

L(s,{\varepsilon _D})

at the critical point

1
2

\frac {1}{2}

where

ε
D

{\varepsilon _D}

is defined by W. Kohnen and D. Zagier:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon Subscript upper D Baseline left-parenthesis n right-parenthesis equals sigma-summation Underscript StartLayout 1st Row g greater-than 0 2nd Row g StartAbsoluteValue delta comma g squared EndAbsoluteValue n 3rd Row left-parenthesis delta slash g comma n slash g squared right-parenthesis equals 1 EndLayout Endscripts left-parenthesis StartFraction upper D 0 Over g Superscript negative 2 Baseline n EndFraction right-parenthesis g">

ε
D

(
n
)
=

∑

g
>
0

g

|

δ
,

g
2

|

n

(
δ

/

g
,
n

/

g
2

)
=
1

(

D
0

g

−
2

n

)

g

{\varepsilon _D}(n) = \sum \limits _{\begin {array}{*{20}{c}} {g > 0} \\ {g|\delta ,{g^2}|n} \\ {(\delta /g,n/{g^2}) = 1} \\ \end {array} } {\left ( {\frac {{{D_0}}}{{{g^{ - 2}}n}}} \right )} \,g

\]
for

n
∈

N

n \in \mathbb {N}

and

D
=

D
0

δ
2

,

D
0

D = {D_0}{\delta ^2},{D_0}

a fundamental discriminant and

δ
∈

N

\delta \in \mathbb {N}

. When

D
=

D
0

,
L
(
s
,

ε

D
0

)

D = {D_0},L(s,{\varepsilon _{{D_0}}})

is the Dirichlet series

L
(
s
,
(

D
0

⋅

)
)

L(s,(\frac {{{D_0}}}{ \cdot }))

. We derive an asymptotic formula for

∑
D

L
(

1
2

,

ε
D

)

\sum \nolimits _D {L(\frac {1}{2},{\varepsilon _D})}

, where the sum runs over all discriminants

D
∈
(
0
,
Y
]

D \in (0,Y]

or

[
−
Y
,
0
)

[ - Y,0)

.
Bootstrap sample size in nonregular cases
Authors: Shao Jun
Published: July 2010

We study the bootstrap estimator of the sampling distribution of a given statistic in some nonregular cases where the given statistic is nonsmooth or not-so-smooth. It is found that the ordinary bootstrap, based on a bootstrap sample of the same size as the original data set, produces an inconsistent bootstrap estimator. On the other hand, when we draw a bootstrap sample of a smaller size with the ratio of the size of the bootstrap sample over the size of the original data set tending to zero, the resulting bootstrap estimator is consistent. Examples of these nonregular cases are given, including the cases of functions of means with null first-order derivatives, differentiable statistical functionals with null influence function, nondifferentiable functions of means, and estimators based on some test results.
A connection between weak regularity and the simplicity of prime factor rings
Authors: Birkenmeier Gary F. , Kim Jin-Yong , Park Jae Keol
Published: July 2010

In this paper, we show that a reduced ring R is weakly regular (i.e.,

I
2

=
I

{I^2} = I

for each one-sided ideal I of R) if and only if every prime ideal is maximal. This result extends several well-known results. Moreover, we provide examples which indicate that further generalization of this result is limited.
Finding a boundary for a Menger manifold
Authors: Chigogidze A.
Published: July 2010

We give a characterization of k-dimensional

(
k
≥
1
)

(k \geq 1)

Menger manifolds admitting boundaries in the sense of Chapman and Siebenmann.
The structure of Johns rings
Authors: Faith Carl , Menal Pere
Published: July 2010

In this paper we continue our study of right Johns rings, that is, right Noetherian rings in which every right ideal is an annihilator. Specifically we study strongly right Johns rings, or rings such that every

n
×
n

n \times n

matrix ring

R
n

{R_n}

is right Johns. The main theorem (Theorem 1.1) characterizes them as the left

FP

{\text {FP}}

-injective right Noetherian rings, a result that shows that not all Johns rings are strong. (This first was observed by Rutter for Artinian Johns rings; see Theorem 1.2.) Another characterization is that all finitely generated right

R
R

-modules are Noetherian and torsionless, that is, embedded in a product of copies of

R
R

. A corollary to this is that a strongly right Johns ring

R
R

is preserved by any group ring

R
G

RG

of a finite group (Theorem 2.1). A strongly right Johns ring is right

F
P
F

FPF

(Theorem 4.2).
Almost periodic homeomorphisms and 𝑝-adic transformation groups on compact 3-manifolds
Authors: Lee Joo S.
Published: July 2010

In this paper we prove that regularly almost periodic is equivalent to nearly periodic for homeomorphisms on compact metric spaces and give an example to show that the above is false without the compactness assumption. We also prove that the following statement is equivalent to the Hilbert-Smith conjecture on compact 3-manifolds

M
3

{M^3}

: If h is almost periodic on

M
3

{M^3}

, with h = identity on

∂

M
3

\partial {M^3}

, then h = identity on

M
3

{M^3}

.
Two-generator cable knots are tunnel one
Authors: Bleiler Steven A.
Published: July 2010

A two-generator cable knot exterior is a genus two handlebody with a single two-handle attached.
Finite groups containing many involutions
Authors: Mann Avinoam
Published: July 2010

If the ratio of the number of involutions of a finite group to the group order is bounded below, the group is bounded by abelian by bounded.
Idempotents in matrix rings
Authors: Barnett Christopher , Camillo Victor
Published: July 2010

Let R be a commutative, von Neumann regular ring and

M
n

(
R
)

{M_n}(R)

the ring of

n
×
n

n \times n

matrices over R. What are the idempotents in

M
n

(
R
)

{M_n}(R)

? Our motivation is to think of R as the sort of ring that occurs in functional analysis, for example a ring of measurable functions. We show how to uniquely write down all idempotents in

M
n

(
R
)

{M_n}(R)

in terms of arbitrary parameters. The main theorem is stated in language to appeal to an audience wider than algebraists, but in a remark, we give a more refined statement for specialists.
Free actions of zero-dimensional compact groups on Menger manifolds
Authors: Sakai Katsuro
Published: July 2010

It is proved that every separable zero-dimensional compact group acts freely on any Menger manifold M. In case M is compact, this result was proved by Dranishnikov. Here is provided an alternative short proof.
A relation between Nörlund and Abel summabilities
Authors: Sarıgöl Mehmet A.
Published: July 2010

This paper gives a necessary and sufficient condition in order that every series summable by a Nörlund method with

p
n

=
o
(

P
n

)

{p_n} = o({P_n})

as

n
→
∞

n \to \infty

should be summable by Abel’s method to the same sum and so extends two theorems of Zygmund.
Every 3-manifold with boundary embeds in 𝑇𝑟𝑖𝑜𝑑×𝑇𝑟𝑖𝑜𝑑×𝐼
Authors: Li Zhong Mou
Published: July 2010

Let M be a compact, connected 3-manifold with nonempty boundary. Then M embeds in

T
×
T
×
I

T \times T \times I

, where T is a triod and

I
=
[
0
,
1
]

I = [0,1]

.
A constraint on the existence of simple torsion free Lie modules
Authors: Britten Daniel , Lemire Frank , Tarokh Vahid
Published: July 2010

For any simple Lie algebra L with Cartan subalgebra H the classification of all simple H-diagonalizable L-modules having a finite-dimensional weight space is known to depend on determining the simple torsion-free L-modules of finite degree. It is further known that the only simple Lie algebras which admit simple torsion-free modules of finite degree are those of types

A
n

{A_n}

and

C
n

{C_n}

. For the case of

A
n

{A_n}

we show that there are no simple torsion-free

A
n

{A_n}

-modules of degree k for

n
≥
4

n \geq 4

and

2
≤
k
≤
n
−
2

2 \leq k \leq n - 2

. We conclude with some examples showing that there exist simple torsion-free

A
n

{A_n}

-modules of degrees

1
,
n
−
1

1,n - 1

, and n.
Putnam’s inequality for 𝑝-hyponormal operators
Authors: Chō Muneo , Itoh Masuo
Published: July 2010

The purpose of this paper is to show the following: Let

0
>
p
>

1
2

0 > p > \frac {1}{2}

. If T is a p-hyponormal operator on a Hilbert space, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar left-parenthesis upper T Superscript asterisk Baseline upper T right-parenthesis Superscript p Baseline minus left-parenthesis upper T upper T Superscript asterisk Baseline right-parenthesis Superscript p Baseline double-vertical-bar less-than-or-equal-to StartFraction p Over pi EndFraction double-integral Underscript sigma left-parenthesis upper T right-parenthesis Endscripts rho Superscript 2 p minus 1 Baseline d rho d theta period">

‖

(

T
∗

T
)

p

−

(
T

T
∗

)

p

‖

≤

p
π

∬

σ
(
T
)

ρ

2
p
−
1

d
ρ
d
θ

.

\left \| {{{({T^ \ast }T)}^p} - {{(T{T^ \ast })}^p}} \right \| \leq \frac {p}{\pi }\iint _{\sigma (T)} {{\rho ^{2p - 1}}d\rho d\theta }.

\]
That is, Putnam’s inequality holds for p-hyponormal operators.
On characterizing derivatives
Authors: Preiss D. , Tartaglia M.
Published: July 2010

We show that the set of derivatives of real functions of a real variable can be characterized by prescribing, for each subset of the real line, the set of its possible inverse images.
Spectrally bounded generalized inner derivations
Authors: Curto Raúl , Mathieu Martin
Published: July 2010

We characterize the generalized inner derivations on a unital Banach algebra which are spectrally bounded. In particular, a simplified argument for the recent result due to Brešar that every spectrally bounded inner derivation maps into the radical is given.
Embedding theorems for residually Černikov 𝐶𝐶-groups
Authors: González M. , Otal J.
Published: July 2010

Embedding theorems for residually Černikov CC-groups are obtained, extending the corresponding results on FC-groups and improving some previous results on CC-groups.
Semidirect products of 𝐼-𝐸 groups
Authors: Lyons Carter G. , Peterson Gary L.
Published: July 2010

An I-E group is a group G in which the endomorphism near-ring generated by the inner automorphisms of G equals the endomorphism near-ring generated by the endomorphisms of G. In this paper we obtain a result characterizing when a semidirect product of I-E groups of relatively prime orders is an I-E group. We then use this result to show that a semidirect product of cyclic groups of relatively prime orders is an I-E group.
Nonsingular modules and 𝑅-homogeneous maps
Authors: Albrecht Ulrich , Hausen Jutta
Published: July 2010

A non-singular R-module M is a ray for the class of all non-singular modules if every R-homogeneous map from M into a non-singular module is additive. Every essential extension of a non-singular locally cyclic module is a ray. We investigate the structure of rays, and determine those semi-prime Goldie-rings for which all non-singular modules are rays and those rings for which the only rays are essential extensions of locally cyclic modules.
Convex functions and Schwarz derivatives
Authors: Fejzić Hajrudin
Published: July 2010

If the lower Schwarz derivate of a continuous function is nonnegative, then it is convex. The main result in this paper is that if the lower Schwarz derivate of a measurable function f is nonnegative, then there is a dense open set with f convex on each component.
Uniform asymptotic stability via Liapunov-Razumikhin technique
Authors: Liu James H.
Published: July 2010

The Liapunov-Razumikhin technique is applied to obtain the uniform asymptotic stability for linear integrodifferential equations in Hilbert spaces,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x prime left-parenthesis t right-parenthesis equals upper A left-bracket x left-parenthesis t right-parenthesis plus integral Subscript number-sign Superscript t Baseline upper F left-parenthesis t minus s right-parenthesis x left-parenthesis s right-parenthesis d s right-bracket comma t greater-than-or-equal-to t 0 greater-than-or-equal-to 0 left-parenthesis number-sign equals 0 or negative normal infinity right-parenthesis comma">

x
′

(
t
)
=
A

[

x
(
t
)
+

∫
#
t

F
(
t
−
s
)
x
(
s
)

d
s

]

,

t
≥

t
0

≥
0
(
#
=
0

or

−
∞
)
,

x’(t) = A\left [ {x(t) + \int _\# ^t {F(t - s)x(s)\,ds} } \right ],\quad t \geq {t_0} \geq 0(\# = 0\;{\text {or}} - \infty ),

\]
which occur in viscoelasticity and in heat conduction for materials with memory.
Rang réel de certaines extensions
Authors: Elhage Hassan Nawfal
Published: July 2010

On montre pour toutes

C
∗

{C^ \ast }

-algebras A et J l’égalité entre les rangs réels de A et de A/J si J est essentiel dans A et isomorphe à l’algèbre des opérateurs compacts. En particulier, le rang réel de l’algèbre de Toeplitz

T

\mathcal {T}

est égal à 1.
The Lavrentiev phenomenon and the obstacle problem for the Dirichlet integral
Authors: Kilpeläinen T. , Lindqvist P.
Published: July 2010

It is shown that the Lavrentiev phenomenon can occur for the Dirichlet integral, when the obstacle problem is considered.
On a rectangle inclusion problem
Authors: Pawlikowski Janusz
Published: July 2010

We show that if every set of reals of size

2

ℵ
0

{2^{{\aleph _0}}}

contains a meager-to-one continuous image of a set that cannot be covered by less than

2

ℵ
0

{2^{{\aleph _0}}}

meager sets, then there exists a null (Lebesgue measure zero) subset of the plane

R

×

R

\mathbb {R} \times \mathbb {R}

that meets every nonnull rectangle

X
×
Y

X \times Y

. The antecedent is satisfied, e.g., if

ω
2

{\omega _2}

Cohen reals are added to a model of the continuum hypothesis.
On the number of Galois 𝑝-extensions of a local field
Authors: Yamagishi Masakazu
Published: July 2010

Let p be a prime, k a finite extension of the p-adic field

Q

p

{\mathbb {Q}_p}

, and G a finite p-group. Let

v
(
k
,
G
)

v(k,G)

denote the number of non-isomorphic Galois extensions of k whose Galois groups are isomorphic to G. When k does not contain a primitive p-th root of unity, I. R. Šafarevič gave an explicit formula for

v
(
k
,
G
)

v(k,G)

. In this note, we treat the case when k contains a primitive p-th root of unity. After giving a general formula for

v
(
k
,
G
)

v(k,G)

(Theorem 1), we calculate

v
(
k
,
G
)

v(k,G)

explicitly for some special p-groups (Theorem 2.2).
Every Wijsman topology relative to a Polish space is Polish
Authors: Costantini C.
Published: July 2010

Generalizing a result of G. Beer and a result of E. Effros, we show that if (X, d) is a separable and completely metrizable metric space, then the hyperspace of X endowed with the Wijsman topology is separable and completely metrizable.
Identities for 𝑞-ultraspherical polynomials and Jacobi functions
Authors: Koelink H. T.
Published: July 2010

A q-analogue of a result by Badertscher and Koornwinder [Canad. J. Math. 44 (1992), 750-773] relating the action of a Hahn polynomial of differential operator argument on ultraspherical polynomials to an ultraspherical polynomial of shifted order and degree is derived. The q-analogue involves q-Hahn polynomials, continuous q-ultraspherical polynomials, and a shift operator. Another limit as q tends to 1 yields an identity for Jacobi functions. Combination with another result of Badertscher and Koornwinder gives a curious formula for Jacobi functions.
Stable rank of 𝐻^{∞} in multiply connected domains
Authors: Tolokonnikov V.
Published: July 2010

The stable rank of the algebra

H
∞

(
G
)

{H^\infty }(G)

of bounded analytic functions in every finitely connected open Riemann surface is equal to one. The same is true for some infinitely connected plain domains (Behrens domains). The proof is based on the Treil theorem, which considered the case of

H
∞

{H^\infty }

in the unit disk.
The stability of positive semigroups on 𝐿_{𝑝} spaces
Authors: Weis Lutz
Published: July 2010

For a positive semigroup

T
t

{T_t}

on

L
p

(
Ω
,
m
)

{L_p}(\Omega ,m)

with generator A, the growth bound of

(

T
t

)

({T_t})

equals the spectral bound of A. In particular, if

s
(
A
)
>
0

s(A) > 0

, the mild solutions of the Cauchy problem

u
′

=
A
u

u’ = Au

are asymptotically stable.
A note on the construction of a certain class of Kleinian groups
Authors: Bianconi Ricardo
Published: July 2010

We prove that if

{

S
1

,

S
1

′

,
…
,

S
n

,

S
n
′

}

\{ {S_1},{S_1}’, \ldots ,{S_n},S_n’\}

is a collection of distinct spheres in

R

m

{\mathbb {R}^m}

with common exterior, and

g
1

,
…
,

g
n

{g_1}, \ldots ,{g_n}

are Möbius transformations such that for each i,

S
i

{S_i}

is the isometric sphere of

g
i

{g_i}

and

S
i
′

S_i’

is the isometric sphere of

g
i

−
1

g_i^{ - 1}

and such that

g
i

{g_i}

maps points of contact of

S
i

{S_i}

, to points of contact of

S
i
′

S_i’

, then the group G generated by the

g
i

{g_i}

’s is Kleinian.
Asymptotics of reproducing kernels on a plane domain
Authors: Engliš Miroslav
Published: July 2010

Let

Ω
\Omega

. be a plane domain of hyperbolic type,

|

d
z

|

/

w
(
z
)

|dz|/w(z)

the Poincaré metric on

Ω
\Omega

, and

K

Ω
,
q

(
x
,

y
¯

)

{K_{\Omega ,q}}(x,\bar y)

the reproducing kernel for the Hilbert space

A

q
2

(
Ω
)

\mathcal {A}_q^2(\Omega )

of all holomorphic functions on

Ω
\Omega

square-integrable with respect to the measure

w

(
z

)

2
q
−
2

|

d
z
∧
d

z
¯

|

w{(z)^{2q - 2}}|dz \wedge d\bar z|

. It is proved that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript q right-arrow plus normal infinity Endscripts StartFraction upper K Subscript normal upper Omega comma q Baseline left-parenthesis z comma z overbar right-parenthesis w left-parenthesis z right-parenthesis Superscript 2 q Baseline Over 2 q EndFraction equals StartFraction 1 Over pi EndFraction period">

lim

q
→
+
∞

K

Ω
,
q

(
z
,

z
¯

)
w

(
z
)

2
q

2
q

=

1
π

.

\lim \limits _{q \to + \infty } \frac {{{K_{\Omega ,q}}(z,\bar z)w{{(z)}^{2q}}}}{{2q}} = \frac {1}{\pi }.

\]
A Kronecker theorem for higher order Hankel forms
Authors: Rochberg Richard
Published: July 2010

A classical theorem of Kronecker describes those Hankel forms that are of finite rank. Here an analogous characterization is given for the higher order (higher rank) Hankel forms introduced by Janson and Peetre. The methods apply to spaces of holomorphic functions in which the polynomials are dense.
On principal sections of a pair of forms
Authors: Cheng Che-Man
Published: July 2010

Let H and C be

n
×
n

n \times n

Hermitian matrices with C positive definite. Let

H
(

i
1

,
…
,

i
r

)

H({i_1}, \ldots ,{i_r})

denote the submatrix of H formed by deleting the rows and columns

i
1

,
…
,

i
r

{i_1}, \ldots ,{i_r}

, of H. In this paper, with

r
1

+
⋯
+

r
k

≤
n

{r_1} + \cdots + {r_k} \leq n

, we study the roots of the determinantal equation

det
(
λ
C
−
H
)
=
0

\det (\lambda C - H) = 0

and those of
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="det left-parenthesis left-parenthesis lamda upper C minus upper H right-parenthesis left-parenthesis r 1 plus midline-horizontal-ellipsis plus r Subscript i minus 1 Baseline plus 1 comma ellipsis comma r 1 plus midline-horizontal-ellipsis plus r Subscript i Baseline right-parenthesis right-parenthesis equals 0">

det
(
(
λ
C
−
H
)
(

r
1

+
⋯
+

r

i
−
1

+
1
,
…
,

r
1

+
⋯
+

r
i

)
)
=
0

\det ((\lambda C - H)({r_1} + \cdots + {r_{i - 1}} + 1, \ldots ,{r_1} + \cdots + {r_i})) = 0

\]
for

i
=
1
,
…
,
k

i = 1, \ldots ,k

.
A strong law for 𝐵-valued arrays
Authors: Li De Li , Rao M. Bhaskara , Tomkins R. J.
Published: July 2010

Let

(
B
,

‖
∙
‖

)

(B,\left \| \bullet \right \|)

be a real separable Banach space and

{

X

n
,
k

;
n
≥
1
,
1
≤
k
≤
n
}

\{ {X_{n,k}};n \geq 1,1 \leq k \leq n\}

a triangular array of iid B-valued random variables. Set

S
(
n
)
=

∑

k
=
1

n

X

n
,
k

,
n
≥
1

S(n) = \sum \nolimits _{k = 1}^n {{X_{n,k}},n \geq 1}

, and

Log

t
=
log
⁡
max
{
e
,
t
}
,
t
∈
ℜ

{\operatorname {Log}}\,t = \log \max \{ e,t\} ,t \in \Re

. In this paper, we characterize the limit behavior of

S
(
n
)

/

2
n

Log

n

,
n
≥
1

S(n)/\sqrt {2n\,{\operatorname {Log}}\,n} ,n \geq 1

. As an application of our result, we resolve an open problem posed by Hu and Weber (1992). The case of row-wise independent arrays is also dealt with.
Metric flows in space forms of nonpositive curvature
Authors: Basmajian Ara , Walschap Gerard
Published: July 2010

We characterize those space forms of nonpositive curvature that admit one-dimensional Riemannian foliations. The hyperbolic ones are essentially the trivial line bundles over the flat ones. In particular, any such space admits a flat metric.
A generalized translation theorem for free homeomorphisms of surfaces
Authors: Guillou Lucien
Published: July 2010

Let h be a free homeomorphism with a finite set of fixed points on a compact surface. Then h is, on the complement of the fixed point set, everywhere ’semi-locally’ conjugate to a translation. This generalizes the Brouwer plane translation theorem.
On the existence of periodic solutions for nonconvex-valued differential inclusions in 𝐑^{𝐍}
Authors: Hu Shou Chuan , Papageorgiou Nikolaos S.
Published: July 2010

In this paper we investigate the existence of periodic solutions for differential inclusions with nonconvex-valued orientor field. Using a tangential condition and directionally continuous selectors, we establish the existence of periodic trajectories.
Subsets close to invariant subsets for group actions
Authors: Brailovsky Leonid , Pasechnik Dmitrii V. , Praeger Cheryl E.
Published: July 2010

Let G be a group acting on a set

Ω
\Omega

and k a non-negative integer. A subset (finite or infinite)

A
⊆
Ω

A \subseteq \Omega

is called k-quasi-invariant if

|

A
g

∖
A

|

≤
k

|{A^g}\backslash A| \leq k

for every

g
∈
G

g \in G

. It is shown that if A is k-quasi-invariant for

k
≥
1

k \geq 1

, then there exists an invariant subset

Γ
⊆
Ω

\Gamma \subseteq \Omega

such that

|

A
△
Γ

|

>
2
e
k

⌈

(
ln
⁡
2
k
)

⌉

|A\vartriangle \Gamma | > 2ek\left \lceil {(\ln 2k)} \right \rceil

. Information about G-orbit intersections with A is obtained. In particular, the number m of G-orbits which have non-empty intersection with A, but are not contained in A, is at most

2
k
−
1

2k - 1

. Certain other bounds on

|

A
△
Γ

|

|A\vartriangle \Gamma |

, in terms of both m and k, are also obtained.
Fixed point theorems for arc-preserving mappings of uniquely arcwise-connected continua
Authors: Fugate J. B. , Mohler Lee
Published: July 2010

Let X be a uniquely arcwise-connected continuum and

f
:
X
→
X

f:X \to X

a map which is arc-preserving (i.e., the image of each arc is an arc or a point). We prove that f has a fixed point.
Projective normal flatness and Hilbert functions
Authors: Orbanz U. , Robbiano L.
Published: July 2010

Projective normal flatness of a local ring

R
R

along an ideal

I
I

is defined to be the flatness of the morphism on the exceptional divisor induced by blowing up

R
R

with center

I
I

. It is shown that most of the important properties of normal flatness have an analogue for projective normal flatness. In particular, we study the local Hilbert function in connection with projective normal flatness. If

R

/

I

R/I

is regular and

R
R

projectively normally flat along

I
I

, then we obtain the same inequality for the local Hilbert functions under blowing up as in the permissible case.
Path derivatives: a unified view of certain generalized derivatives
Authors: Bruckner A. M. , O’Malley R. J. , Thomson B. S.
Published: July 2010

A collection

E
=
{

E
x

:
x
∈
R
}

E = \{ {E_x}:x \in R\}

is a system of paths if each set

E
x

{E_x}

has

x
x

as a point of accumulation. For such a system

E
E

the derivative

F
E
′

(
x
)

F_E’(x)

of a function

F
F

at a point

x
x

is just the usual derivative at

x
x

relative to the set

E
x

{E_x}

. The goal of this paper is the investigation of properties that

F
F

and its derivative

F
E
′

F_E’

must have under certain natural assumptions about the collection

E
E

. In particular, it is shown that most of the familiar properties of approximate derivatives and approximately differentiable functions follow in this setting from three conditions on the collection

E
E

relating to the "thickness" of the sets

E
x

{E_x}

and the way in which the sets intersect.
A nonlinear integral equation occurring in a singular free boundary problem
Authors: Höllig Klaus , Nohel John A.
Published: July 2010

We study the Cauchy problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row u Subscript t Baseline equals phi left-parenthesis u Subscript x Baseline right-parenthesis Subscript x Baseline comma left-parenthesis x comma t right-parenthesis element-of bold upper R times bold upper R Subscript plus Baseline comma 2nd Row u left-parenthesis dot comma 0 right-parenthesis equals f EndLayout">

{

u
t

=
ϕ

(

u
x

)
x

,

(
x
,
t
)
∈

R

×

R

+

,

u
(
⋅
,
0
)
=
f

\left \{ \begin {gathered} {u_t} = \phi {({u_x})_x},\qquad (x,t) \in {\mathbf {R}} \times {{\mathbf {R}}_ + }, \hfill \\ u( \cdot ,0) = f \hfill \\ \end {gathered} \right .

\]
with the piecewise linear constitutive function

ϕ
(
ξ
)
=

ξ
+

=
max
(
0
,
ξ
)

\phi (\xi ) = {\xi _ + } = \max (0,\xi )

and with smooth initial data

f
f

which satisfy

x

f
′

(
x
)
⩾
0

xf’(x) \geqslant 0

,

x
∈

R

x \in {\mathbf {R}}

, and

f
(
0
)
>
0

f(0) > 0

. We prove that free boundary

s
s

, given by

u
x

(
s

(
t

)
+

,
t
)
=
0

{u_x}(s{(t)^ + },t) = 0

, is of the form
\[

s
(
t
)
=
−
κ

t

+
o

(

t

)

,

t
→

0
+

,

s(t) = - \kappa \sqrt t + o\left ( {\sqrt t } \right ),\qquad t \to {0^ + },

\]
where the constant

κ
=
0.9034
…

\kappa = 0.9034 \ldots

is the (numerical) solution of a particular nonlinear equation. Moreover, we show that for any

α
∈
(
0
,
1

/

2
)

\alpha \in (0,1/2)

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue StartFraction d squared Over d t squared EndFraction f left-parenthesis s left-parenthesis t right-parenthesis right-parenthesis EndAbsoluteValue equals upper O left-parenthesis t Superscript alpha minus 1 Baseline right-parenthesis comma t right-arrow 0 Superscript plus Baseline period">

|

d
2

d

t
2

f
(
s
(
t
)
)

|

=
O
(

t

α
−
1

)
,

t
→

0
+

.

\left | {\frac {{{d^2}}} {{d{t^2}}}f(s(t))} \right | = O({t^{\alpha - 1}}),\qquad t \to {0^ + }.

\]
The proof involves the analysis of a nonlinear singular integral equation.
An integral inequality with applications
Authors: Leckband M. A.
Published: July 2010

Using a technical integral inequality, J. Moser proved a sharp result on exponential integrability of a certain space of Sobolev functions. In this paper, we show that the integral inequality holds in a general setting using nonincreasing functions and a certain class of convex functions. We then apply the integral inequality to extend the above result by J. Moser to other spaces of Sobolev functions. A second application is given generalizing some different results by M. Jodeit.
Where the continuous functions without unilateral derivatives are typical
Authors: Malý Jan
Published: July 2010

An alternative proof of the existence of a Besicovitch function (i.e. a continuous function which has nowhere a unilateral derivative) is presented. The method consists in showing the residuality of Besicovitch functions in special subspaces of the Banach space of all continuous functions on

[
0
,
1
]

[0,1]

and yields Besicovitch functions with additional properties of Morse or Hölder type. A way how to obtain functions with a similar behavior on normed linear spaces is briefly mentioned.
A Cayley-Dickson process for a class of structurable algebras
Authors: Allison B. N. , Faulkner J. R.
Published: July 2010

In this paper, we study the class of all simple structurable algebras with the property that the space of skew-hermitian elements has dimension

1
1

. These algebras with involution have arisen in the study of Lie algebra constructions. The reduced algebras are isotopic to

2
×
2

2 \times 2

matrix algebras. We study a Cayley-Dickson process for rationally constructing some algebras in the class including division algebras and nonreduced nondivision algebras. An important special case of the process endows the direct sum of two copies of a

28
28

-dimensional degree

4
4

central simple Jordan algebra

B

\mathcal {B}

with the structure of an algebra with involution. In preparatory work, we obtain a procedure for giving the space

B

0

{\mathcal {B}_0}

of trace zero elements of any such Jordan algebra

B

\mathcal {B}

the structure of a

27
27

-dimensional exceptional Jordan algebra.
Applications of uniform convexity of noncommutative 𝐿^{𝑝}-spaces
Authors: Kosaki Hideki
Published: July 2010

We consider noncommutative

L
p

{L^p}

-spaces,

1
>
p
>
∞

1 > p > \infty

, associated with a von Neumann algebra, which is not necessarily semifinite, and obtain some consequences of their uniform convexity. Among other results, we obtain (i) the norm continuity of the "absolute value part" map from each

L
p

{L^p}

-space onto its positive part; (ii) a certain continuity result on Radon-Nikodym derivatives in the context of positive cones introduced by H. Araki; and (iii) the necessary and sufficient condition for certain

L
p

{L^p}

-norm inequalities to become equalities. Some dominated convergence theorems for a probability gage are also considered.
Decomposability of Radon measures
Authors: Gardner R. J. , Pfeffer W. F.
Published: July 2010

A topological space is called metacompact or metalindelöf if each open cover has a point-finite or point-countable refinement, respectively. It is well known that each Radon measure is expressible as a sum of Radon measures supported on a disjoint family of compact sets, called a concassage. If the unions of measurable subsets of the members of a concassage are also measurable, the Radon measure is called decomposable. We show that Radon measures in a metacompact space are always saturated, and therefore decomposable whenever they are complete. The previous statement is undecidable in ZFC if "metacompact" is replaced by "metalindelöf". The proofs are based on structure theorems for a concassage of a Radon measure. These theorems also show that the union of a concassage of a Radon measure in a metacompact space is a Borel set, which is paracompact in the subspace topology whenever the main space is regular.
Suspension spectra and homology equivalences
Authors: Kuhn Nicholas J.
Published: July 2010

Let

f
:

Σ
∞

X
→

Σ
∞

Y

f:{\Sigma ^\infty }X \to {\Sigma ^\infty }Y

be a stable map between two connected spaces, and let

E

∗

{E_{\ast }}

be a generalized homology theory. We show that if

E

∗

(
f
)

{E_{\ast }}(f)

is an isomorphism then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript asterisk Baseline left-parenthesis normal upper Omega Superscript normal infinity Baseline f right-parenthesis colon upper E Subscript asterisk Baseline left-parenthesis upper Q upper X right-parenthesis right-arrow upper E Subscript asterisk Baseline left-parenthesis upper Q upper Y right-parenthesis">

E

∗

(

Ω
∞

f
)
:

E

∗

(
Q
X
)
→

E

∗

(
Q
Y
)

{E_{\ast }}({\Omega ^\infty }f):{E_{\ast }}(QX) \to {E_{\ast }}(QY)

is a monomorphism, but possibly not an epimorphism. Applications of this theorem include results of Miller and Snaith concerning the

K
K

-theory of the Kahn-Priddy map.
The quotient semilattice of the recursively enumerable degrees modulo the cappable degrees
Authors: Schwarz Steven
Published: July 2010

In this paper, we investigate the quotient semilattice

R
_

/

M
_

\underline R /\underline M

of the r.e. degrees modulo the cappable degrees. We first prove the

R
_

/

M
_

\underline {R} /\underline {M}

counterpart of the Friedberg-Muchnik theorem. We then show that minimal elements and minimal pairs are not present in

R
_

/

M
_

\underline R /\underline M

. We end with a proof of the

R
_

/

M
_

\underline {R} /\underline {M}

counterpart to Sack’s splitting theorem.
New inequalities for polynomials
Authors: Frappier C. , Rahman Q. I. , Ruscheweyh St.
Published: July 2010

Using a recently developed method to determine bound-preserving convolution operators in the unit disk, we derive various refinements and generalizations of the well-known inequalities of S. Bernstein and M. Riesz for polynomials. Many of these results take into account the size of one or more of the coefficients of the polynomial in question. Other results of similar nature are obtained from a new interpolation formula.
Coordinatization in superstable theories. I. Stationary types
Authors: Buechler Steven
Published: July 2010

Suppose

T
T

is superstable and

P
P

is a complete type over some finite set with

U
(
p
)
=
α
+
1

U(p) = \alpha + 1

for some

α
\alpha

. We show how to associate with

p
p

an incidence geometry which measures the complexity of the family of extensions of

p
p

of rank

α
\alpha

. When

p
p

is stationary we give a characterization of the possible incidence geometries. As an application we prove Theorem. Suppose

M
M

is superstable and has only one

1
1

-type

p
∈
S
(
∅
)

p \in S(\emptyset )

. Further suppose

p
p

is stationary with

U
(
p
)
=
α
+
1

U(p) = \alpha + 1

for some

α
\alpha

. Then one of the following holds: (i) There is an equivalence relation

E
⊂

M
2

E \subset {M^2}

with infinitely many infinite classes definable over

∅
\emptyset

. (ii)

M
M

is the algebraic closure of a set of Morley rank

1
1

. In particular,

M
M

is

ℵ
0

{\aleph _0}

-stable of finite rank.
Polar coordinates induced by actions of compact Lie groups
Authors: Dadok Jiri
Published: July 2010

Let

G
G

be a connected Lie subgroup of the real orthogonal group

O
(
n
)

O(n)

. For the action of

G
G

on

R

n

{{\mathbf {R}}^n}

, we construct linear subspaces

a

\mathfrak {a}

that intersect all orbits. We determine for which

G
G

there exists such an

a

\mathfrak {a}

meeting all the

G
G

-orbits orthogonally; groups that act transitively on spheres are obvious examples. With few exceptions all possible

G
G

arise as the isotropy subgroups of Riemannian symmetric spaces.
A bijective proof of Stanley’s shuffling theorem
Authors: Goulden I. P.
Published: July 2010

For two permutations

σ
\sigma

and

ω
\omega

on disjoint sets of integers, consider forming a permutation on the combined sets by "shuffling"

σ
\sigma

and

ω
\omega

(i.e.,

σ
\sigma

and

ω
\omega

appear as subsequences). Stanley [10], by considering

P
P

-partitions and a

q
q

-analogue of Saalschutz’s

3

F
2

_3{F_2}

summation, obtained the generating function for shuffles of

σ
\sigma

and

ω
\omega

with a given number of falls (an element larger than its successor) with respect to greater index (sum of positions of falls). It is a product of two

q
q

-binomial coefficients and depends only on remarkably simple parameters, namely the lengths, numbers of falls and greater indexes of

σ
\sigma

and

ω
\omega

. A combinatorial proof of this result is obtained by finding bijections for lattice path representations of shuffles which reduce

σ
\sigma

and

ω
\omega

to canonical permutations, for which a direct evaluation of the generating function is given.