bluuebunny/crossref_metadata_2025 huggingface dataset:
Remarks on some integrals and series involving the Stirling numbers and 𝜁(𝑛)
Authors: Shen Li-Chien
Published: June 2012

From the perspective of the well-known identity
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 2 Baseline upper F 1 left-parenthesis a comma b semicolon c semicolon 1 right-parenthesis equals StartFraction normal upper Gamma left-parenthesis c right-parenthesis normal upper Gamma left-parenthesis c minus a minus b right-parenthesis Over normal upper Gamma left-parenthesis c minus a right-parenthesis normal upper Gamma left-parenthesis c minus b right-parenthesis EndFraction comma">

2

F
1

(
a
,
b
;
c
;
1
)
=

Γ
(
c
)
Γ
(
c
−
a
−
b
)

Γ
(
c
−
a
)
Γ
(
c
−
b
)

,

{}_2{F_1}(a,b;c;1) = \frac {{\Gamma (c)\Gamma (c - a - b)}} {{\Gamma (c - a)\Gamma (c - b)}},

\]
we clarify the connections between the Stirling numbers

s
k
n

s_k^n

and the Riemann zeta function

ζ
(
n
)

\zeta (n)

. As a consequence, certain series and integrals can be evaluated in terms of

ζ
(
n
)

\zeta (n)

and

s
k
n

s_k^n

.
When does unique local support ensure convexity?
Authors: Young Donald Francis
Published: June 2012

A basic theorem of convex analysis states that a real-valued function on an open interval of the real line is convex and differentiable if at each point of its domain there exists a unique supporting line. In this paper we show that the same conclusion can be drawn under the weaker hypothesis that there exists a unique locally supporting line at each point. We also show by counterexample that convexity cannot be concluded under analogous circumstances for

f
:
S
→

R

f:S \to \mathbb {R}

, where

S
⊂

R

n

S \subset {\mathbb {R}^n}

is open and convex, if

n
>
1

n > 1

.
On the coefficient groups of equivariant 𝐾-theory
Authors: Yang Yimin
Published: June 2012

We calculated the coefficient groups of equivariant

K
K

-theory for any cyclic group, and we proved that, for any compact Lie group, the coefficient groups can only have

2
2

-torsion. If there is any

2
2

-torsion, it is detected by

2
2

-primary finite subgroups of

G
G

. The rationalization of the coefficient groups then can be easily calculated.
Examples of 𝐵(𝐷,𝜆)-refinable and weak \overline𝜃-refinable spaces
Authors: Fast Stephen H. , Smith J. C.
Published: June 2012

In 1980, J. C. Smith asked for examples which would demonstrate the relationships between the properties

B
(
D
,
λ
)

B(D,\lambda )

-refinability,

B
(
D
,

ω
0

)

B(D,{\omega _0})

-refinability, and weak

θ
¯

\bar \theta

-refinability. This paper gives such examples in the class of

T
4

{T_4}

spaces.
Hausdorff measure and level sets of typical continuous mappings in Euclidean spaces
Authors: Kirchheim Bernd
Published: June 2012

We determine the Hausdorff dimension of level sets and of sets of points of multiplicity for mappings in a residual subset of the space of all continuous mappings from

R

n

{\mathbb {R}^n}

to

R

m

{\mathbb {R}^m}

.
Sub-self-similar sets
Authors: Falconer K. J.
Published: June 2012

A compact set

E
⊆

R

n

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

is called sub-self-similar if

E
⊆

⋃

i
=
1

m

S
i

(
E
)

E \subseteq \bigcup \nolimits _{i = 1}^m {{S_i}(E)}

, where the

S
i

{S_i}

are similarity transfunctions. We consider various examples and constructions of such sets and obtain formulae for their Hausdorff and box dimensions, generalising those for self-similar sets.
On the Littlewood-Paley-Stein 𝑔-function
Authors: Meda Stefano
Published: June 2012

We consider semigroups

(

T
t

)

({T_t})

, which are contractive on

L
p

(
M
)

{L^p}(M)

for all

p
∈
[
q
,

q
′

]

p \in [q,q’]

and

q
∈
[
1
,
2
)

q \in [1,2)

. We give an example (on symmetric spaces of the noncompact type) which shows that the Littlewood-Paley-Stein

g
g

-function associated to the infinitesimal generator of

(

T
t

)

({T_t})

may be unbounded on

L
q

(
M
)

{L^q}(M)

and on

L

q
′

(
M
)

{L^{q’}}(M)

. We prove that variants of the

g
g

-function are bounded on these Lebesgue spaces.
Holomorphic martingales and interpolation between Hardy spaces: the complex method
Authors: Müller P. F. X.
Published: June 2012

A probabilistic proof is given to identify the complex interpolation space of

H
1

(

T

)

{H^1}(\mathbb {T})

and

H
∞

(

T

)

{H^\infty }(\mathbb {T})

as

H
p

(

T

)

{H^p}(\mathbb {T})

.
Irreducible semigroups of functionally positive nilpotent operators
Authors: Zhong Yong
Published: June 2012

For each irrational number

θ
∈
(
0
,
1
)

\theta \in (0,1)

, we construct a semigroup

S

θ

{\mathcal {S}_\theta }

of nilpotent operators on

S

2

(
[
0
,
1
]
)

{\mathcal {S}^2}([0,1])

that are also partial isometries and positive in the sense that the operator maps nonnegative functions to nonnegative functions. We prove that each semigroup

S

θ

{\mathcal {S}_\theta }

is discrete in the norm topology and hence norm-closed and that the weak closure of

S

θ

{\mathcal {S}_\theta }

is independent of

S

θ

{\mathcal {S}_\theta }

. We show that each semigroup

S

θ

{\mathcal {S}_\theta }

has no nontrivial invariant subspaces.
Applications of simplicial 𝑀-sets to proper and strong shape theories
Authors: Hernández Paricio L. J.
Published: June 2012

In this paper we have tried to reduce the classical classification problems for spaces and maps of the proper category and of the strong shape category to similar problems in the homotopy category of simplicial sets or in the homotopy category of simplicial

M
M

-sets, which

M
M

is the monoid of proper selfmaps of the discrete space

N

\mathbb {N}

of nonnegative integers. Given a prospace (prosimplicial set)

Y
Y

, we have constructed a simplicial set

P

¯

R

Y

{\overline {\mathcal {P}} ^R}Y

such that the Hurewicz homotopy groups of

P

¯

R

Y

{\overline {\mathcal {P}} ^R}Y

are the Grossman homotopy groups of

Y
Y

. For the case of the end prospace

Y
=
ε
X

Y = \varepsilon X

of a space

X
X

, we obtain Brown’s proper homotopy groups; and for the Vietoris prospace

Y
=
V
X

Y = VX

(introduced by Porter) of a compact metrisable space

X
X

, we have Quigley’s inward groups. The simplicial subset

P

¯

R

Y

{\overline {\mathcal {P}} ^R}Y

of a tower

Y
Y

contains, as a simplicial subset, the homotopy limit

lim
R

Y

{\lim ^R}Y

. The inclusion

lim
R

Y
→

P

¯

R

Y

{\lim ^R}Y \to {\overline {\mathcal {P}} ^R}Y

induces many relations between the homotopy and (co)homology invariants of the prospace

Y
Y

. Using the functor

P

¯

R

{\overline {\mathcal {P}} ^R}

we prove Whitehead theorems for proper homotopy, prohomotopy, and strong shape theories as a particular case of the standard Whitehead theorem. The algebraic condition is given in terms of Brown’s proper groups, Grossman’s homotopy groups and Quigley’s inward groups, respectively. In all these cases an equivalent cohomological condition can be given by taking twisted coefficients. The "singular" homology groups of

P

¯

R

Y

{\overline {\mathcal {P}} ^R}Y

provide homology theories for the Brown, Grossman and Quigley homotopy groups that satisfy Hurewicz theorems in the corresponding settings. However, there are other homology theories for the homotopy groups above satisfying other Hurewicz theorems. We also analyse the notion of

P

¯

\overline {\mathcal {P}}

-movable prospace. For a

P

¯

\overline {\mathcal {P}}

-movable tower we prove easily (without

lim
1

{\lim ^1}

functors) that the strong homotopy groups agree with the Čech homotopy groups and the Grossman homotopy groups are determined by the Čech (or strong) groups by the formula

G

π
q

=

P

¯

π
ˇ

q

^G{\pi _q} = \overline {\mathcal {P}} \check {\pi }_q

. This implies that the algebraic condition of the Whitehead theorem can be given in terms of strong (Čech) groups when the condition of

P

¯

\overline {\mathcal {P}}

-movability is included. We also study homology theories for the strong (Steenrod) homotopy groups which satisfy Hurewicz theorems but in general do not agree with the corresponding Steenrod-Sitnikov homology theories.
Attractors in inhomogeneous conservation laws and parabolic regularizations
Authors: Fan Hai Tao , Hale Jack K.
Published: June 2012

The asymptotic behavior of inhomogeneous conservation laws is considered. The attractor of the equation is characterized. The relationship between attractors of the equation and that of its parabolic regularization is studied.
Localization and genus in group theory
Authors: Peschke G.
Published: June 2012

We provide a unifying category theoretical framework to discuss various kinds of local global phenomena. Specializing to localization of groups at sets of primes

P
P

, we identify a large class of groups for which localization supports a passage from local information to global information. Local global principles for groups in this class are established and used to calculate certain homomorphism sets as well as splittings of epimorphisms and monomorphisms from local data.
Some inequalities of algebraic polynomials with nonnegative coefficients
Authors: Chen Weiyu
Published: June 2012

Let

S
n

{S_n}

be the collection of all algebraic polynomials of degree

⩽
n

\leqslant n

with nonnegative coefficients. In this paper we discuss the extremal problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript p Subscript n Baseline left-parenthesis x right-parenthesis element-of upper S Subscript n Endscripts StartFraction integral Subscript a Superscript b Baseline left-parenthesis p prime Subscript n Baseline left-parenthesis x right-parenthesis right-parenthesis squared omega left-parenthesis x right-parenthesis d x Over integral Subscript a Superscript b Baseline p Subscript n Superscript 2 Baseline left-parenthesis x right-parenthesis omega left-parenthesis x right-parenthesis d x EndFraction">

sup

p
n

(
x
)
∈

S
n

∫
a
b

(

p
′

n

(
x
)
)

2

ω
(
x
)
d
x

∫
a
b

p
n
2

(
x
)
ω
(
x
)
d
x

\sup \limits _{{p_n}(x) \in {S_n}} \frac {{\int \limits _a^b {{{({{p’}_n}(x))}^2}\omega (x)dx} }} {{\int \limits _a^b {p_n^2(x)\omega (x)dx} }}

\]
where

ω
(
x
)

\omega (x)

is a positive and integrable function. This problem is solved completely in the cases
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis i right-parenthesis left-bracket a comma b right-bracket equals left-bracket negative 1 comma 1 right-bracket comma omega left-parenthesis x right-parenthesis equals left-parenthesis 1 minus x squared right-parenthesis Superscript alpha Baseline comma alpha greater-than negative 1 semicolon">

(

i

)
[
a
,
b
]
=
[
−
1
,
1
]
,
ω
(
x
)
=

(
1
−

x
2

)
α

,
α
>
−
1
;

({\text {i}})[a,b] = [ - 1,1],\omega (x) = {(1 - {x^2})^\alpha },\alpha > - 1;

\]

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis ii right-parenthesis left-bracket a comma b right-parenthesis equals left-bracket 0 comma normal infinity right-parenthesis comma omega left-parenthesis x right-parenthesis equals x Superscript alpha Baseline e Superscript negative x Baseline comma alpha greater-than negative 1 semicolon">

(

ii

)
[
a
,
b
)
=
[
0
,
∞
)
,
ω
(
x
)
=

x
α

e

−
x

,
α
>
−
1
;

({\text {ii}})[a,b) = [0,\infty ),\omega (x) = {x^\alpha }{e^{ - x}},\alpha > - 1;

\]

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis iii right-parenthesis left-parenthesis a comma b right-parenthesis equals left-parenthesis negative normal infinity comma normal infinity right-parenthesis comma omega left-parenthesis x right-parenthesis equals e Superscript minus alpha x squared Baseline comma alpha greater-than 0 period">

(

iii

)
(
a
,
b
)
=
(
−
∞
,
∞
)
,
ω
(
x
)
=

e

−
α

x
2

,
α
>
0.

({\text {iii}})(a,b) = ( - \infty ,\infty ),\omega (x) = {e^{ - \alpha {x^2}}},\alpha > 0.

\]
The second case was solved by Varma for some values of

α
\alpha

and by Milovanović completely. We provide a new proof here in this case.
Affine transformations and analytic capacities
Authors: Dowling Thomas , O’Farrell Anthony G.
Published: June 2012

Analytic capacities are set functions defined on the plane which may be used in the study of removable singularities, boundary smoothness and approximation of analytic functions belonging to some function space. The symmetric concrete Banach spaces form a class of function spaces that includes most spaces usually studied. The Beurling transform is a certain singular integral operator that has proved useful in analytic function theory. It is shown that the analytic capacity associated to each Beurling-invariant symmetric concrete Banach space behaves reasonably under affine transformation of the plane. It is not known how general analytic capacities behave under affine maps.
Tate cohomology of periodic 𝐾-theory with reality is trivial
Authors: Fajstrup Lisbeth
Published: June 2012

We calculate the

R
O
(

Z

/

2
)

RO(\mathbb {Z}/2)

-graded spectrum for Atiyah’s periodic

K
K

-theory with reality and the Tate cohomology associated to it. The latter is shown to be trivial.
Compact composition operators on the Bloch space
Authors: Madigan Kevin , Matheson Alec
Published: June 2012

Necessary and sufficient conditions are given for a composition operator

C
ϕ

f
=
f

o

ϕ

{C_\phi }f = f{\text {o}}\phi

to be compact on the Bloch space

B

\mathcal {B}

and on the little Bloch space

B

0

{\mathcal {B}_0}

. Weakly compact composition operators on

B

0

{\mathcal {B}_0}

are shown to be compact. If

ϕ
∈

B

0

\phi \in {\mathcal {B}_0}

is a conformal mapping of the unit disk

D

\mathbb {D}

into itself whose image

ϕ
(

D

)

\phi (\mathbb {D})

approaches the unit circle

T

\mathbb {T}

only in a finite number of nontangential cusps, then

C
ϕ

{C_\phi }

is compact on

B

0

{\mathcal {B}_0}

. On the other hand if there is a point of

T

∩
ϕ
(

D

)

\mathbb {T} \cap \phi (\mathbb {D})

at which

ϕ
(

D

)

\phi (\mathbb {D})

does not have a cusp, then

C
ϕ

{C_\phi }

is not compact.
Uniformly ergodic multioperators
Authors: Mbekhta M. , Vasilescu F.-H.
Published: June 2012

A version of the uniform ergodic theorem valid for commuting multioperators is given.
An access theorem for analytic functions
Authors: Ortel Marvin
Published: June 2012

Suppose that

M

\mathcal {M}

is an analytic manifold,

m
0

∈

M

,
f
:

M

→

R

{m_0} \in \mathcal {M},f:\mathcal {M} \to \mathbb {R}

, and

f
f

is analytic. Then at least one of the following three statements is true: (1)

m
0

{m_0}

is a local maximum of

f
f

. (2) There is a continuous path

σ
:
[
0
,
1
]
→

M

\sigma :[0,1] \to \mathcal {M}

such that

σ
(
0
)
=

m
0

\sigma (0) = {m_0}

,

f
∘
σ

f \circ \sigma

is strictly increasing on

[
0
,
1
]

[0,1]

, and

σ
(
1
)

\sigma (1)

is a local maximum of

f
f

. (3) There is a continuous path

σ
:
[
0
,
1
)
→

M

\sigma :[0,1) \to \mathcal {M}

with these properties:

σ
(
0
)
=

m
0

;
f
∘
σ

\sigma (0) = {m_0};f \circ \sigma

is strictly increasing on

[
0
,
1
)

[0,1)

; whenever

K
K

is a compact subset of

M

\mathcal {M}

, there is a corresponding number

d
(
K
)
∈
[
0
,
1
)

d(K) \in [0,1)

such that

σ
(
t
)
∉
K

\sigma (t) \notin K

for all

t
∈
[
d
(
K
)
,
1
)

t \in [d(K),1)

.
Circle actions on rational homology manifolds and deformations of rational homotopy types
Authors: Raussen Martin
Published: June 2012

The aim of this paper is to follow up the program set in [LR85, Rau92], i.e., to show the existence of nontrivial group actions ("symmetries") on certain classes of manifolds. More specifically, given a manifold

X
X

with submanifold

F
F

, I would like to construct nontrivial actions of cyclic groups on

X
X

with

F
F

as fixed point set. Of course, this is not always possible, and a list of necessary conditions for the existence of an action of the circle group

T
=

S
1

T = {S^1}

on

X
X

with fixed point set

F
F

was established in [Rau92]. In this paper, I assume that the rational homotopy types of

F
F

and

X
X

are related by a deformation in the sense of [A1178] between their (Sullivan) models as graded differential algebras (cf. [Sul77, Hal83]). Under certain additional assumptions, it is then possible to construct a rational homotopy description of a

T
T

-action on the complement

X
∖
F

X\backslash F

that fits together with a given

T
T

-bundle action on the normal bundle of

F
F

in

X
X

. In a subsequent paper [Rau94], I plan to show how to realize this

T
T

-action on an actual manifold

Y
Y

rationally homotopy equivalent to

X
X

with fixed point set

F
F

and how to "propagate" all but finitely many of the restricted cyclic group actions to

X
X

itself.
Homology operations on a new infinite loop space
Authors: Totaro Burt
Published: June 2012

Boyer et al. [1] defined a new infinite loop space structure on the space

M
0

=

∏

n
⩾
1

K
(

Z

,
2
n
)

{M_0} = {\prod _{n \geqslant 1}}K({\mathbf {Z}},2n)

such that the total Chern class map

B
U
→

M
0

BU \to {M_0}

is an infinite loop map. This is a sort of Riemann-Roch theorem without denominators: for example, it implies Fulton-MacPherson’s theorem that the Chern classes of the direct image of a vector bundle

E
E

under a given finite covering map are determined by the rank and Chern classes of

E
E

. We compute the Dyer-Lashof operations on the homology of

M
0

{M_0}

. They provide a new explanation for Kochman’s calculation of the operations on the homology of

B
U

BU

, and they suggest a possible characterization of the infinite loop structure on

M
0

{M_0}

.
Curvature conditions on Riemannian manifolds with Brownian harmonicity properties
Authors: Hughes H. R.
Published: June 2012

The time and place that Brownian motion on a Riemannian manifold first exits a normal ball of radius

ε
\varepsilon

is considered and a general procedure is given for computing asymptotic expansions, as

ε
\varepsilon

decreases to zero, for joint moments of the first exit time and place random variables. It is proven that asymptotic versions of exit time and place distribution properties that hold on harmonic spaces are equivalent to certain curvature conditions for harmonic spaces. In particular, it is proven that an asymptotic mean value condition involving first exit place is equivalent to certain levels of curvature conditions for harmonic spaces depending on the order of the asymptotics. Also, it is proven that an asymptotic uncorrelated condition for first exit time and place is equivalent to weaker curvature conditions at corresponding orders of asymptotics.
Diophantine approximation in 𝑅ⁿ
Authors: Vulakh L. Ya.
Published: June 2012

A modification of the Ford geometric approach to the problem of approximation of irrational real numbers by rational fractions is developed. It is applied to find an upper bound for the Hurwitz constant for a discrete group acting in a hyperbolic space. A generalized Khinchine’s approximation theorem is also given.
Weighted boundary limits of the generalized Kobayashi-Royden metrics on weakly pseudoconvex domains
Authors: Yu Ji Ye
Published: June 2012

The purpose of this paper is to study the existence of weighted boundary limits of the generalized Kobayashi-Royden metrics on weakly pseudoconvex domains in

C

n

{\mathbb {C}^n}

and to explore the connections between the limits and the Levi invariants. The main result extends Graham’s result on strongly pseudoconvex domains to a large class of weakly pseudoconvex domains.
Stable range one for rings with many idempotents
Authors: Camillo Victor P. , Yu Hua-Ping
Published: June 2012

An associative ring

R
R

is said to have stable range

1
1

if for any

a
a

,

b
∈
R

b \in R

satisfying

a
R
+
b
R
=
R

aR + bR = R

, there exists

y
∈
R

y \in R

such that

a
+
b
y

a + by

by is a unit. The purpose of this note is to prove the following facts. Theorem

3
3

: An exchange ring

R
R

has stable range

1
1

if and only if every regular element of

R
R

is unit-regular. Theorem

5
5

: If

R
R

is a strongly

π
\pi

-regular ring with the property that all powers of every regular element are regular, then

R
R

has stable range

1
1

. The latter generalizes a recent result of Goodearl and Menal [

5
5

].
On a singular quasilinear anisotropic elliptic boundary value problem
Authors: Choi Y. S. , Lazer A. C. , McKenna P. J.
Published: June 2012

We consider the problem
\[

u
a

u

x
x

+

u
b

u

y
y

+
p
(

x

)
=
0

{u^a}{u_{xx}} + {u^b}{u_{yy}} + p({\mathbf {x}}) = 0

\]
with

a
⩾
0

a \geqslant 0

,

b
⩾
0

b \geqslant 0

, on a smooth convex bounded region in

R

2

{{\mathbf {R}}^2}

with Dirichlet boundary conditions. We show that if the positive function

p
p

is uniformly bounded away from zero, then the problem has a classical solution.
Matrix variate 𝜃-generalized normal distribution
Authors: Gupta A. K. , Varga T.
Published: June 2012

In this paper, the matrix variate

θ
\theta

-generalized normal distribution is introduced. Then its properties are studied. In particular, it is proved that this distribution has maximal entropy in a certain class of distributions.
The Toeplitz theorem and its applications to approximation theory and linear PDEs
Authors: Jia Rong Qing
Published: June 2012

We take an algebraic approach to the problem of approximation by dilated shifts of basis functions. Given a finite collection

Φ
\Phi

of compactly supported functions in

L
p

(

R

s

)

(
1
⩽
p
⩽
∞
)

{L_p}({\mathbb {R}^s})\quad (1 \leqslant p \leqslant \infty )

, we consider the shift-invariant space

S
S

generated by

Φ
\Phi

and the family

(

S
h

:
h
>
0
)

({S^h}:h > 0)

, where

S
h

{S^h}

is the

h
h

-dilate of

S
S

. We prove that

(

S
h

:
h
>
0
)

({S^h}:h > 0)

provides

L
p

{L_p}

-approximation order

r
r

only if

S
S

contains all the polynomials of total degree less than

r
r

. In particular, in the case where

Φ
\Phi

consists of a single function

φ
\varphi

with its moment

∫

φ
≠
0

\int {\varphi \ne 0}

, we characterize the approximation order of

(

S
h

:
h
>
0
)

({S^h}:h > 0)

by showing that the above condition on polynomial containment is also sufficient. The above results on approximation order are obtained through a careful analysis of the structure of shift-invariant spaces. It is demonstrated that a shiftinvariant space can be described by a certain system of linear partial difference equations with constant coefficients. Such a system then can be reduced to an infinite system of linear equations, whose solvability is characterized by an old theorem of Toeplitz. Thus, the Toeplitz theorem sheds light into approximation theory. It is also used to give a very simple proof for the well-known Ehrenpreis principle about the solvability of a system of linear partial differential equations with constant coefficients.
Distinct degree factorizations for polynomials over a finite field
Authors: Knopfmacher Arnold , Warlimont Richard
Published: June 2012

Let

F

~

q

[
X
]

{\widetilde {\mathbb {F}}_q}[X]

denote the multiplicative semigroup of monic polynomials in one indeterminate

X
X

, over a finite field

F

q

{\mathbb {F}_q}

. We determine for each fixed

q
q

and fixed

n
n

the probability that a polynomial of degree

n
n

in

F

q

[
X
]

{\mathbb {F}_q}[X]

has irreducible factors of distinct degrees only. These results are of relevance to various polynomial factorization algorithms.
Conditions on the logarithmic derivative of a function implying boundedness
Authors: MacGregor T. H. , Rønning F.
Published: June 2012

In this paper we investigate functions analytic and nonvanishing in the unit disk, with the property that the logarithmic derivative is contained in some domain

Ω
\Omega

. We obtain conditions on

Ω
\Omega

which imply that the functions are bounded and that their first derivatives belong to

H
p

{H^p}

for some

p
⩾
1

p \geqslant 1

. For certain domains

Ω
\Omega

the sufficient conditions that we give are also, in some sense, necessary. Examples of domains to which the results apply are given.
Π¹₁ functions are almost internal
Authors: Živaljević Boško
Published: June 2012

In Analytic mappings on hyperfinite sets [Proc. Amer. Math. Soc. 2 (1993), 587-596] Henson and Ross asked for what hyperfinite sets

S
S

and

T
T

does there exists a bijection

f
f

from

S
S

onto

T
T

whose graph is a projective subset of

S
×
T

S \times T

? In particular, when is there a

Π
1
1

\Pi _1^1

bijection from

S
S

onto

T
T

? In this paper we prove that given an internal, bounded measure

μ
\mu

, any

Π
1
1

\Pi _1^1

function is

L
(
μ
)

L(\mu )

a.e. equal to an internal function, where

L
(
μ
)

L(\mu )

is the Loeb measure associated with

μ
\mu

. It follows that if two

Π
1
1

\Pi _1^1

subsets

S
S

and

T
T

of a hyperfinite set

X
X

are

Π
1
1

\Pi _1^1

bijective, then

S
S

and

T
T

have the same measure for every uniformly distributed counting measure

μ
\mu

. When

S
S

and

T
T

are internal it turns out that any

Π
1
1

\Pi _1^1

bijection between them must already be Borel. We also prove that if a

Π
1
1

\Pi _1^1

graph in the product of two hyperfinite sets

X
X

and

Y
Y

is universal for all internal subsets of

Y
Y

, then

|

X

|

⩾

2

|

Y

|

|X| \geqslant {2^{|Y|}}

, which is a partial answer to Henson and Ross’s Problem 1.5. At the end we prove some standard results about the projections and a structure of co-proper

K
K

-analytic subsets of the product of two completely regular Hausdorff topological spaces with open vertical sections. We were able to prove the above results by revealing the structure of

Π
1
1

\Pi _1^1

subsets of the products

X
×
Y

X \times Y

of two internal sets

X
X

and

Y
Y

, all of whose

Y
Y

-sections are

Σ
1
0

(
κ
)

\Sigma _1^0(\kappa )

sets.
An optimal condition for the LIL for trigonometric series
Authors: Berkes I.
Published: June 2012

By a classical theorem (Salem-Zygmund [6], Erdős-Gàl [3]), if

(

n
k

)

({n_k})

is a sequence of positive integers satisfying

n

k
+
1

/

n
k

⩾
q
>
1

(
k
=
1
,
2
,
…
)

{n_{k + 1}}/{n_k} \geqslant q > 1\;(k = 1,2, \ldots )

then

(
cos
⁡

n
k

x
)

(\cos {n_k}x)

obeys the law of the iterated logarithm, i.e., (1)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit sup Underscript upper N right-arrow normal infinity Endscripts left-parenthesis upper N log log upper N right-parenthesis Superscript negative 1 slash 2 Baseline sigma-summation Underscript k less-than-or-slanted-equals upper N Endscripts cosine n Subscript k Baseline x equals 1 a period e period">

lim

sup

N
→
∞

(
N
log
⁡
log
⁡
N

)

−
1

/

2

∑

k
⩽
N

cos
⁡

n
k

x
=
1

a

.e

.

\lim \sup \limits _{N \to \infty } {(N\log \log N)^{ - 1/2}}\sum \limits _{k \leqslant N} {\cos {n_k}x = 1\quad {\text {a}}{\text {.e}}{\text {.}}}

\]
It is also known (Takahashi [7, 8]) that the Hadamard gap condition

n

k
+
1

/

n
k

⩾
q
>
1

{n_{k + 1}}/{n_k} \geqslant q > 1

can be essentially weakened here but the problem of finding the precise gap condition for the LIL (1) has remained open. In this paper we find, using combinatorial methods, an optimal gap condition for the upper half of the LIL, i.e., the inequality

⩽
1

\leqslant 1

in (1).
Immersions and embeddings in domains of holomorphy
Authors: Dor Avner
Published: June 2012

Let

D
1

{D_1}

be a bounded smooth strongly pseudoconvex domain in

C

N

{\mathbb {C}^N}

and let

D
2

{D_2}

be a domain of holomorphy in

C

M

(
2
⩽
N
,
5
⩽
M
,
2
N
⩽
M
)

{\mathbb {C}^M}(2 \leqslant N,5 \leqslant M,2N \leqslant M)

. There exists then a proper holomorphic immersion from

D
1

{D_1}

to

D
2

{D_2}

. Furthermore if

P
I

(

D
1

,

D
2

)

{\mathbf {PI}}({D_1},{D_2})

is the set of proper holomorphic immersions from

D
1

{D_1}

to

D
2

{D_2}

and

A
(

D
1

,

D
2

)

A({D_1},{D_2})

is the set of holomorphic maps from

D
1

{D_1}

to

D
2

{D_2}

that are continuous on the boundary, then the closure of

P
I

(

D
1

,

D
2

)

{\mathbf {PI}}({D_1},{D_2})

in the topology of uniform convergence on compacta contains

A
(

D
1

,

D
2

)

A({D_1},{D_2})

. The approximating proper maps can be made tangent to any finite order of contact at a given point. The same result was obtained for proper holomorphic maps, in one codimension, when the target domain has a plurisubharmonic exhaustion function with no saddle critical points. This includes the case where the target domain is convex. Density in a weaker sense was derived in one codimension when the critical points are contained in a compact subset of the target domain. This occurs (for example) when the target domain is bounded weakly pseudoconvex with

C
2

{C^2}

-smooth boundary. If the target domain is strongly pseudoconvex then the approximating proper holomorphic maps can also be made continuous on the boundary. A lesser degree of pseudoconvexity is required from the target domain when the codimension is larger than the minimal. A domain in

C

L

{\mathbb {C}^L}

is called "

M
M

dimensional-pseudoconvex" (where

L
⩾
M

L \geqslant M

) if it has a smooth exhaustion function

r
r

such that every point

w
w

in this domain has some

M
M

-dimensional complex affine subspace going through this point for which

r
r

, restricted to this subspace, is strictly plurisubharmonic in

w
w

. In the result mentioned above the assumption that the target domain is pseudoconvex in

C

M

(
M
⩾
2
N
,
5
)

{\mathbb {C}^M}(M \geqslant 2N,5)

can be substituted for the assumption that the domain is "

M
M

-dimensional-pseudoconvex". Similarly, the assumption that the target domain

D
2

{D_2}

is "

(
N
+
1
)

(N + 1)

-dimensional-pseudoconvex" and all the critical points of some appropriate exhaustion function are "

(
N
+
1
)

(N + 1)

-dimensional-convex" (defined in a similar manner) yields that the closure of the set of proper holomorphic maps from

D
1

{D_1}

to

D
2

{D_2}

contains

A
(

D
1

,

D
2

)

A({D_1},{D_2})

. All the results are obtained with embeddings when the Euclidean dimensions are such that

dim

C

(

D
2

)
⩾
2

dim

C

(

D
1

)
+
1

{\dim _\mathbb {C}}({D_2}) \geqslant 2{\dim _\mathbb {C}}({D_1}) + 1

. Thus, in this case, when one of the assumptions mentioned above is fulfilled, then the closure of the set of embeddings from

D
1

{D_1}

to

D
2

{D_2}

contains

A
(

D
1

,

D
2

)

A({D_1},{D_2})

.
Complete quotient Boolean algebras
Authors: Kanamori Akihiro , Shelah Saharon
Published: June 2012

For

I
I

a proper, countably complete ideal on the power set

P

(
x
)

\mathcal {P}(x)

for some set

X
X

, can the quotient Boolean algebra

P

(
X
)

/

I

\mathcal {P}(X)/I

be complete? We first show that, if the cardinality of

X
X

is at least

ω
3

{\omega _3}

, then having completeness implies the existence of an inner model with a measurable cardinal. A well-known situation that entails completeness is when the ideal

I
I

is a (nontrivial) ideal over a cardinal

κ
\kappa

which is

κ
+

{\kappa ^ + }

-saturated. The second author had established the sharp result that it is consistent by forcing to have such an ideal over

κ
=

ω
1

\kappa = {\omega _1}

relative to the existence of a Woodin cardinal. Augmenting his proof by interlacing forcings that adjoin Boolean suprema, we establish, relative to the same large cardinal hypothesis, the consistency of:

2

ω
1

=

ω
3

{2^{{\omega _1}}} = {\omega _3}

and there is an ideal ideal

I
I

over

ω
1

{\omega _1}

such that

P

(

ω
1

)

/

I

\mathcal {P}({\omega _1})/I

is complete. (The cardinality assertion implies that there is no ideal over

ω
1

{\omega _1}

which is

ω
2

{\omega _2}

-saturated, and so completeness of the Boolean algebra and saturation of the ideal has been separated.)
Conormal and piecewise smooth solutions to quasilinear wave equations
Authors: Kang Seong Joo
Published: June 2012

In this paper, we show first that if a solution

u
u

of the equation

P
2

(
t
,
x
,
u
,
D
u
,
D
)
u
=
f
(
t
,
x
,
u
,
D
u
)

{P_2}(t,x,u,Du,D)u = f(t,x,u,Du)

, where

P
2

(
t
,
x
,
u
,
D
u
,
D
)

{P_2}(t,x,u,Du,D)

is a second order strictly hyperbolic quasilinear operator, is conormal with respect to a single characteristic hypersurface

Σ
\Sigma

of

P
2

{P_2}

in the past and

Σ
\Sigma

is smooth in the past, then

Σ
\Sigma

is smooth and

u
u

is conormal with respect to

Σ
\Sigma

for all time. Second, let

Σ
0

{\Sigma _0}

and

Σ
1

{\Sigma _1}

be characteristic hypersurfaces of

P
2

{P_2}

which intersect transversally and let

Γ
=

Σ
0

∩

Σ
1

\Gamma = {\Sigma _0} \cap {\Sigma _1}

. If

Σ
0

{\Sigma _0}

and

Σ
1

{\Sigma _1}

are smooth in the past and

u
u

is conormal with repect to

{

Σ
0

,

Σ
1

}

\{ {\Sigma _0},{\Sigma _1}\}

in the past, then

Γ
\Gamma

is smooth, and

u
u

is conormal with respect to

{

Σ
0

,

Σ
1

}

\{ {\Sigma _0},{\Sigma _1}\}

locally in time outside of

Γ
\Gamma

, even though

Σ
0

{\Sigma _0}

and

Σ
1

{\Sigma _1}

are no longer necessarily smooth across

Γ
\Gamma

. Finally, we show that if

u
(
0
,
x
)

u(0,x)

and

∂
t

u
(
0
,
x
)

{\partial _t}u(0,x)

are in an appropriate Sobolev space and are piecewise smooth outside of

Γ
\Gamma

, then

u
u

is piecewise smooth locally in time outside of

Σ
0

∪

Σ
1

{\Sigma _0} \cup {\Sigma _1}

.
Hopf constructions and higher projective planes for iterated loop spaces
Authors: Kuhn Nicholas J. , Slack Michael , Williams Frank
Published: June 2012

We define a category,

H

p
n

\mathcal {H}_p^n

(for each

n
n

and

p
p

), of spaces with strong homotopy commutativity properties. These spaces have just enough structure to define the

mod
p

\bmod p

Dyer-Lashof operations for

n
n

-fold loop spaces. The category

H

p
n

\mathcal {H}_p^n

is very convenient for applications since its objects and morphisms are defined in a homotopy invariant way. We then define a functor,

P
p
n

P_p^n

, from

H

p
n

\mathcal {H}_p^n

to the homotopy category of spaces and show

P
p
n

P_p^n

to be left adjoint to the

n
n

-fold loop space functor. We then show how one can exploit this adjointness in cohomological calculations to yield new results about iterated loop spaces.
Epi-derivatives of integral functionals with applications
Authors: Loewen Philip D. , Zheng Harry H.
Published: June 2012

We study first- and second-order epi-differentiability for integral functionals defined on

L
2

[
0
,
T
]

{L^2}[0,T]

, and apply the results to obtain first- and second-order necessary conditions for optimality in free endpoint control problems.
Convergence of diagonal Padé approximants for functions analytic near 0
Authors: Lubinsky D. S.
Published: June 2012

For functions analytic in a neighbourhood of

0
0

, we show that at least for a subsequence of the diagonal Padé approximants, the point

0
0

attracts a zero proportion of the poles. The same is true for every "sufficiently dense" diagonal subsequence. Consequently these subsequences have a convergence in capacity type property, which is possibly the correct analogue of the Nuttall-Pommerenke theorem in this setting.
Involutory Hopf algebras
Authors: Passman D. S. , Quinn Declan
Published: June 2012

In 1975, Kaplansky conjectured that a finite-dimensional semisimple Hopf algebra is necessarily involutory. Twelve years later, Larson and Radford proved the conjecture in characterisitic

0
0

and obtained significant partial results in positive characteristics. The goal of this paper is to offer an efficient proof of these results using rather minimal prerequisites, no "harpoons", and gratifyingly few "hits".
Stochastic control problems and spherical functions on symmetric spaces
Authors: Duncan T. E. , Upmeier H.
Published: June 2012

A family of explicitly solvable stochastic control problems is formulated and solved in noncompact symmetric spaces. The symmetric spaces include all of the classical spaces and four of the exceptional spaces. The stochastic control problems are the control of Brownian motion in these symmetric spaces by a drift vector field. For each symmetric space a family of stochastic control problems is formulated by using spherical functions in the cost functionals. These spherical functions are explicitly described and are polynomials in suitable coordinates. A generalization to abstract root systems is given.
Generalizations of Browder’s degree theory
Authors: Hu Shou Chuan , Papageorgiou Nikolaos S.
Published: June 2012

The starting point of this paper is the recent important work of F. E. Browder, who extended degree theory to operators of monotone type. The degree function of Browder is generalized to maps of the form

T
+
f
+
G

T + f + G

, where

T
T

is maximal monotone,

f
f

is of class

(
S

)
+

{(S)_ + }

bounded, and

G
(
⋅
)

G( \cdot )

is an u.s.c. compact multifunction. It is also generalized to maps of the form

f
+

N
G

f + {N_G}

, with

f
f

of class

(
S

)
+

{(S)_ + }

and

N
G

{N_G}

the Nemitsky operator of a multifunction

G
(
x
,
r
)

G(x,r)

satisfying various types of sign conditions. Some examples are also included to illustrate the abstract results.
Hadamard convexity and multiplicity and location of zeros
Authors: Abi-Khuzam Faruk F.
Published: June 2012

We consider certain questions arising from a paper of Hayman concerning quantitative versions of the Hadamard three-circle theorem for entire functions. If

b
(
r
)

b(r)

denotes the second derivative of

log
⁡
M
(
r
)

\log M(r)

with respect to

log
⁡
r

\log r

, the principal contributions of this work are (i) a characterization of those entire

f
f

with nonnegative Maclaurin coefficients for which

lim
sup
b
(
r
)
=

1
4

\lim \sup b(r) = \frac {1} {4}

and (ii) some exploration of the relationship between multiple zeros of

f
f

and the growth of

b
(
r
)

b(r)

.
The regionally proximal relation
Authors: Auslander Joseph , Ellis David B. , Ellis Robert
Published: June 2012

Sufficient conditions for the regionally proximal relation

Q
(
X
)

Q(X)

of a minimal flow to be an equivalence relation are obtained in terms of the group

G

(
X
)

\mathcal {G}(X)

of the flow and various groups which depend only on the acting group

T
T

.
Variational formulas on Lipschitz domains
Authors: Elcrat Alan R. , Miller Kenneth G.
Published: June 2012

A rigorous treatment is given of variational formulas for solutions of certain Dirichlet problems for the Laplace operator on Lipschitz domains under interior variations. In particular we extend well-known variational formulas for the torsional rigidity and for capacity from the class of

C
1

{C^1}

domains to the class of Lipschitz domains. A motivation for this work comes from the use of variational methods in the study of Prandtl-Batchelor flows in fluid mechanics.
Existence of extremal metrics on compact almost homogeneous Kähler manifolds with two ends
Authors: Guan Daniel
Published: June 2012

In this note we prove the existence and the uniqueness of extremal metrics in every Kähler class of any compact almost homogeneous Kähler manifold with two ends by considering the scalar curvature equations, those manifolds might not be projective. We also prove that there are extremal metrics in some Kähler classes of a completion of the multicanonical line bundle of a Kähler-Einstein manifold of positive Ricci curvature.
Left annihilators characterized by GPIs
Authors: Lee Tsiu Kwen
Published: June 2012

Let

R
R

be a semiprime ring with extended centroid

C
C

,

U
U

the right Utumi quotient ring of

R
R

,

S
S

a subring of

U
U

containing

R
R

and

ρ
1

{\rho _1}

,

ρ
2

{\rho _2}

two right ideals of

R
R

. In the paper we show that

l
S

(

ρ
1

)
=

l
S

(

ρ
2

)

{l_S}({\rho _1}) = {l_S}({\rho _2})

if and only if

ρ
1

{\rho _1}

and

ρ
2

{\rho _2}

satisfy the same generalized polynomial identities (GPIs) with coefficients in

S
C

SC

, where

l
S

(

ρ
i

)

{l_S}({\rho _i})

denotes the left annihilator of

ρ
i

{\rho _i}

in

S
S

. As a consequence of the result, if

ρ
\rho

is a right ideal of

R
R

such that

l
R

(
ρ
)
=
0

{l_R}(\rho ) = 0

, then

ρ
\rho

and

U
U

satisfy the same GPIs with coefficients in the two-sided Utumi quotient ring of

R
R

.
On the relational basis of Cayley’s theorem and of similar representations for algebras
Authors: Sedaghat Hassan
Published: June 2012

Considering a binary operation as a ternary relation permits certain sections of the latter (which are functions) to be used in representing an abstract semigroup as a family of the self-maps of its underlying set under function composition. The idea is thus seen to be entirely similar to the way that the sections of a partial ordering under set inclusion represent the (abstract) partially ordered set. An extension of this procedure yields a uniform set of representation theorems for a large class of associative algebras.
Construction of homomorphisms of 𝑀-continuous lattices
Authors: Xu Xiao Quan
Published: June 2012

We present a direct approach to constructing homomorphisms of

M
M

-continuous lattices, a generalization of continuous lattices, into the unit interval, and show that an

M
M

-continuous lattice has sufficiently many homomorphisms into the unit interval to separate the points.
The large deviation principle for a general class of queueing systems. I
Authors: Dupuis Paul , Ellis Richard S.
Published: June 2012

We prove the existence of a rate function and the validity of the large deviation principle for a general class of jump Markov processes that model queueing systems. A key step in the proof is a local large deviation principle for tubes centered at a class of piecewise linear, continuous paths mapping [0,1] into

[
0
,
1
]

[0,1]

. In order to prove certain large deviation limits, we represent the large deviation probabilities as the minimal cost functions of associated stochastic optimal control problems and use a subadditivity—type argument. We give a characterization of the rate function that can be used either to evaluate it explicitly in the cases where this is possible or to compute it numerically in the cases where an explicit evaluation is not possible.
Generalized (𝑡,𝑠)-sequences, Kronecker-type sequences, and Diophantine approximations of formal Laurent series
Authors: Larcher Gerhard , Niederreiter Harald
Published: June 2012

The theory of

(
t
,
s
)

(t,s)

-sequences leads to powerful constructions of low-discrepancy sequences in an

s
s

-dimensional unit cube. We generalize this theory in order to cover arbitrary sequences constructed by the digital method and, in particular, the Kronecker-type sequences introduced by the second author. We define diophantine approximation constants for formal Laurent series over finite fields and show their connection with the distribution properties of Kronecker-type sequences. The main results include probabilistic theorems on the distribution of sequences constructed by the digital method and on the diophantine approximation character of

s
s

-tuples of formal Laurent series over finite fields.
Partial extensions of Attouch’s theorem with applications to proto-derivatives of subgradient mappings
Authors: Levy A. B. , Poliquin R. , Thibault L.
Published: June 2012

Attouch’s Theorem, which gives on a reflexive Banach space the equivalence between the Mosco epi-convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients, has many important applications in convex optimization. In particular, generalized derivatives have been defined in terms of the epi-convergence or graph convergence of certain difference quotient mappings, and Attouch’s Theorem has been used to relate these various generalized derivatives. These relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems. We prove in a Hilbert space several "partial extensions" of Attouch’s Theorem to functions more general than convex; these functions are called primal-lower-nice. Furthermore, we use our extensions to derive a relationship between the second-order epi-derivatives of primal-lower-nice functions and the proto-derivative of their associated subgradient mappings.
The relative Burnside module and the stable maps between classifying spaces of compact Lie groups
Authors: Minami Norihiko
Published: June 2012

Tom Dieck’s Burnside ring of compact Lie groups is generalized to the relative case: For anyG▹NG \triangleright N, a compact Lie group and its normal subgroupA(G▹N)A(G \triangleright N)is defined to be an appropriate set of the equivalence classes of compactGG-ENR’s with freeNN-action, in such a way that<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi colon upper A left-parenthesis upper G white right pointing small triangle upper N right-parenthesis asymptotically-equals pi Subscript upper G slash upper N Superscript 0 Baseline left-parenthesis upper S Superscript 0 Baseline semicolon upper B left-parenthesis upper N comma upper G right-parenthesis Subscript plus Baseline right-parenthesis">ψ:A(G▹N)≃πG/N0(S0;B(N,G)+)\psi :A(G \triangleright N) \simeq \pi _{G/N}^0({S^0};B{(N,G)_ + }), whereB(N,G)B(N,G)is the classifying space of principal(N,G)(N,G)-bundle. Under the "product" situation, i.e.G=F×K,N=K,A(F×K▹K)G = F \times K,\;N = K,\;A(F \times K \triangleright K)is also denoted byA(F,K)A(F,K), as it turns out to be the usualA(F,K)A(F,K)when bothFFandKKare finite. Then a couple of applications are given to the study of stable maps between classifying spaces of compact Lie groups: a conceptual proof of Feshbach’s double coset formula, and a density theorem on the map<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha Subscript p Superscript logical-and Baseline colon upper A left-parenthesis upper L comma upper H right-parenthesis Subscript p Superscript logical-and Baseline right-arrow left-brace upper B upper L Subscript plus comma Baseline upper B upper H Subscript plus Baseline right-brace Subscript p Superscript logical-and">αp∧:A(L,H)p∧→{BL+,BH+}p∧\alpha _p^ \wedge :A(L,H)_p^ \wedge \to \{ B{L_{ + ,}}B{H_ + }\} _p^ \wedgefor any compact Lie groupsL,KL,\;Kwhenppis odd. (Some restriction is applied toLLwhenp=2p = 2.) This latter result may be regarded as the pushout of Feshbach’s density theorem and the theorem of May-Snaith-Zelewski, over the celebrated Carlsson solution of Segal’s Burnside ring conjecture.
An analogue of the Jacobson-Morozov theorem for Lie algebras of reductive groups of good characteristics
Authors: Premet Alexander
Published: June 2012

Let

g

\mathfrak {g}

be the Lie algebra of a connected reductive group

G
G

over an algebraically closed field of characteristic

p
>
0

p > 0

. Suppose that

G

(
1
)

{G^{(1)}}

is simply connected and

p
p

is good for the root system of

G
G

. Given a one-dimensional torus

λ
⊂
G

\lambda \subset G

let

g

(
λ
,
1
)

\mathfrak {g}(\lambda ,1)

denote the weight component of

Ad(

λ

)

{\text {Ad(}}\lambda {\text {)}}

corresponding to weight

i
∈
X
(
λ
)
≅

Z

i \in X(\lambda ) \cong \mathbb {Z}

. It is proved in the paper that, for any nonzero nilpotent element

e
∈

g

e \in \mathfrak {g}

, there is a one-dimentional torus

λ
e

⊂
G

{\lambda _e} \subset G

such that

e
∈

g

(

λ
e

,
2
)

e \in \mathfrak {g}({\lambda _e},2)

and

Ker

ad

e
⊆

⊕

i
⩾
0

g

(

λ
e

,
i
)

{\text {Ker}}{\text {ad}}e \subseteq { \oplus _{i \geqslant 0}}\mathfrak {g}({\lambda _e},i)

.
Schur’s partition theorem, companions, refinements and generalizations
Authors: Alladi Krishnaswami , Gordon Basil
Published: June 2012

Schur’s partition theorem asserts the equality

S
(
n
)
=

S
1

(
n
)

S(n) = {S_1}(n)

, where

S
(
n
)

S(n)

is the number of partitions of

n
n

into distinct parts

≡
1
,
2
(

mod

3
)

\equiv 1,2(\mod 3)

and

S
1

(
n
)

{S_1}(n)

is the number of partitions of

n
n

into parts with minimal difference

3
3

and no consecutive multiples of

3
3

. Using a computer search Andrews found a companion result

S
(
n
)
=

S
2

(
n
)

S(n) = {S_2}(n)

, where

S
2

(
n
)

{S_2}(n)

is the number of partitions of

n
n

whose parts

e
i

{e_i}

satisfy

e
i

−

e

i
+
1

⩾
3
,
2
o
r
5

{e_i} - {e_{i + 1}} \geqslant 3,2or5

according as

e
i

≡
1
,
2

{e_i} \equiv 1,2

or

(
mod
3
)

(\bmod 3)

. By means of a new technique called the method of weighted words, a combinatorial as well as a generating function proof of both these theorems are given simultaneously. It is shown that

S
1

(
n
)

{S_1}(n)

and

S
2

(
n
)

{S_2}(n)

are only two of six companion partition functions

S
j

(
n
)
,
j
=
1
,
2
,
…
6

{S_j}(n),j = 1,2, \ldots 6

, all equal to

S
(
n
)

S(n)

. A three parameter refinement and generalization of these results is obtained.