bluuebunny/crossref_metadata_2025 huggingface dataset:
Mixing 𝑘-fold independent processes of zero entropy
Authors: Flaminio L.
Published: June 2010

For each fixed

k
k

we give a simple construction of a stochastic process

(

X
n

)

({X_n})

which is mixing, has zero entropy, and such that any

k
k

-tuple of the

X
n

{X_n}

are independent.
Quasi-heredity of algebras and their factor algebras
Authors: Xi Chang Chang
Published: June 2010

Let

A
A

be a finite-dimensional algebra over an algebraically closed field and denote by

N
N

the Jacobson radical of

A
A

. If there is an integer

i
⩾
2

i \geqslant 2

such that

A

/

N
i

A/{N^i}

is quasi-hereditary, then

A
A

is quasi-hereditary.
Subspaces and graphs
Authors: Chung Kin Yan
Published: June 2010

Subspaces sufficiently near an arbitrary (fixed) subspace of a Hilbert space are shown to be in one-to-one correspondence with operators defined on the given subspace. Specifically, the nearby subspaces can be regarded as the graphs of these operators. This is applied to explicitly define a

C
∞

{C^\infty }

-atlas of charts for the set of subspaces.
Value sets of polynomials over finite fields
Authors: Wan Da Qing , Shiue Peter Jau-Shyong , Chen Ching Shyang
Published: June 2010

Let

F

q

{{\mathbf {F}}_q}

be the finite field of

q
q

elements, and let

V
f

{V_f}

be the number of values taken by a polynomial

f
(
x
)

f(x)

over

F

q

{{\mathbf {F}}_q}

. We establish a lower bound and an upper bound of

V
f

{V_f}

in terms of certain invariants of

f
(
x
)

f(x)

. These bounds improve and generalize some of the previously known bounds of

V
f

{V_f}

. In particular, the classical Hermite-Dickson criterion is improved. Our bounds also give a new proof of a recent theorem of Evans, Greene, and Niederreiter. Finally, we give some examples which show that our bounds are sharp.
Weakly continuous functions on Banach spaces not containing 𝑙₁
Authors: Gutiérrez Joaquín M.
Published: June 2010

Banach spaces not containing

l
1

{l_1}

are characterized in terms of continuous and holomorphic functions and polynomials which are weakly sequentially continuous and weakly continuous on bounded subsets. An application to (bounded linear) operators is also given.
Weakly compact homomorphisms from group algebras
Authors: Johnson B. E.
Published: June 2010

A locally compact group

G
G

has the weakly compact homomorphism property if every weakly compact homomorphism from the group algebra

L
1

(
G
)

{L^1}(G)

into another Banach algebra

B

\mathfrak {B}

has finite-dimensional range. It has been shown that compact and abelian groups have this property. We extend this to a large class of groups including all solvable groups.
Rough isometries and Dirichlet finite harmonic functions on graphs
Authors: Soardi Paolo M.
Published: June 2010

Suppose that

G
1

{G_1}

and

G
2

{G_2}

are roughly isometric connected graphs of bounded degree. If

G
1

{G_1}

has no nonconstant Dirichlet finite harmonic functions, then neither has

G
2

{G_2}

.
On approximately convex functions
Authors: Ng C. T. , Nikodem K.
Published: June 2010

The Bernstein-Doetsch theorem on midconvex functions is extended to approximately midconvex functions and to approximately Wright convex functions.
The unicellularity of contractions of class 𝐶₀
Authors: Zou Cheng Zu
Published: June 2010

In this paper, we shall generalize the unicellularity of operators on finite-dimensional spaces to that of the contraction of class

C
0

{C_0}

on Hilbert spaces. We prove: (1) Each nilpotent operator on Hilbert space is Banach reducible (Theorem 3). (2) A contraction

T
T

of class

C
0

{C_0}

on Hilbert space is unicellular if and only if

T
T

has one-point spectrum and every invariant subspace for

T
T

is cyclic (Theorem 6). (3) A contraction

T
T

of class

C
0

{C_0}

on Hilbert space is unicellular if and only if

T
T

has one-point spectrum and all invariant subspaces of

T
T

are hyperinvariant subspaces of

T
T

(Theorem 8).
The 2-character table does not determine a group
Authors: Johnson Kenneth W. , Sehgal Surinder K.
Published: June 2010

Frobenius had defined the group determinant of a group

G
G

which is a polynomial in

n
=

|

G

|

n = |G|

variables. Formanek and Sibley have shown that the group determinant determines the group. Hoehnke and Johnson show that the

3
3

-characters (a part of the group determinant) determine the group. In this paper it is shown that the

2
2

-characters do not determine the group. If we start with a group

G
G

of a certain type then a group

H
H

with the same

2
2

-character table must form a Brauer pair with

G
G

. A complete description of such an

H
H

is available in Comm. Algebra 9 (1981), 627-640.
Finite loop space with maximal tori have finite Weyl groups
Authors: Smith Larry
Published: June 2010

A finite loop space

X
X

is said to have a maximal torus if there is a map

f
:
B
T
→
B
X

f:BT \to BX

where

T
T

is a torus such that

rank
⁡
(
T
)
=
rank
⁡
(
X
)

\operatorname {rank} (T) = \operatorname {rank} (X)

and the homotopy fibre of

f
f

has the homotopy type of a finite complex. The Weyl group

W
f

{W_f}

of

f
f

is the set of homotopy classes

w
:
B
T
→
B
T

w:BT \to BT

such that
\[

B
T

→

w

B
T

f
↘

↙
f

B
X

\begin {array}{*{20}{c}} {BT\xrightarrow {w}BT} \\ {f \searrow \quad \swarrow f} \\ {BX} \\ \end {array}

\]
homotopy commutes. In this note we prove that

W
f

{W_f}

is always finite.
Lyapunov characteristic exponents are nonnegative
Authors: Przytycki Feliks
Published: June 2010

We prove that, for an arbitrary rational map

f
f

on the Riemann sphere and an arbitrary probability invariant measure on the Julia set, Lyapunov characteristic exponents are nonnegative a.e. In particular

log
⁡

|

f
′

|

\log |f’|

is integrable. An analogous theorem is proved for smooth maps of an interval with all critical points being nonflat. This allows us to fill a gap in the proof of Denker and Urbański’s theorem that there exists a probability conformal measure on the Julia set with exponent equal to the supremum of the Hausdorff dimensions of probability invariant measures with positive entropy.
Additive derivations of nest algebras
Authors: Han De Guang
Published: June 2010

In this paper we prove that every additive derivation of a nest algebra acting on an infinite-dimensional Hilbert space is inner. This extends the relative result for linear derivations of nest algebras.
Integrability of superharmonic functions and subharmonic functions
Authors: Aikawa Hiroaki
Published: June 2010

We apply the coarea formula to obtain integrability of superharmonic functions and nonintegrability of subharmonic functions. The results involve the Green function. For a certain domain, say Lipschitz domain, we estimate the Green function and restate the results in terms of the distance from the boundary.
How far can one move from a potential peak with small initial speed?
Authors: Barone Netto Ângelo , Zampieri Gaetano
Published: June 2010

We consider a natural Lagrangian system and show that from a pointq0{q_0}inn-space, where the potential energyVhas a (weak) maximum, one can go near the boundary of any compact ball whereV(q)≤V(q0)V(q) \leq V({q_0})with (arbitrarily small) nonvanishing initial speeds. The result holds true for sets which areC2{C^2}-diffeomorphic to a compact ball. This property is found as a simple consequence of the Hopf-Rinow theorem and of a theorem of Gordon. As a corollary we deduce a well-known local result, namely, a ’converse’ of the Lagrange-Dirichlet theorem, thus obtained via geometric arguments.
Borel maps with the “point of continuity property” and completely Borel additive families in some nonmetrizable spaces
Authors: Holický Petr
Published: June 2010

Under the axiom that no measurable cardinal exists it is proved that "

(
F
∩
G

)
σ

{(F \cap G)_\sigma }

-measurable" maps of a hereditarily Baire and Čech analytic (e.g., compact) space into a metric space has the point of continuity property. A result on completely Borel-additive families in Čech analytic spaces is the crucial part of the proof.
Spectral theorem for unbounded strongly continuous groups on a Hilbert space
Authors: Boyadzhiev Khristo , deLaubenfels Ralph
Published: June 2010

Suppose

B
B

is a closed, densely defined linear operator on a Hilbert space and

a
>
0

a > 0

. Denote

{
z

|

|

Im
⁡
(
z
)

|

>
b
}

\{ z||\operatorname {Im} (z)| > b\}

by

H
b

{H_b}

. We show that

B
B

has an

H
∞

(

H
b

)

{H^\infty }({H_b})

functional calculus, for all

b
>
a

b > a

, if and only if

i
B

iB

generates a strongly continuous group of operators of exponential type

a
a

. We obtain specific upper bounds on

|

|

f
(
B
)

|

|

||f(B)||

, in terms of

sup
{

e

−
b

|

t

|

|

|

e

i
t
B

|

|

|

t
∈

R

}

\sup \{ {e^{ - b|t|}}||{e^{itB}}||\;|t \in {\mathbf {R}}\}

. Corollaries include the spectral theorem for closed operators on a Hilbert space and a generalization of a result due to McIntosh relating imaginary powers and

H
∞

{H^\infty }

functional calculi.
Howe/Kirillov theory for 𝑝-adic symmetric spaces
Authors: Hakim Jeffrey
Published: June 2010

The method of coadjoint orbits is adapted to sufficiently small compact subgroups of a pair

(
G
,

G
+

)

(G,{G_ + })

, where G is a p-adic group and

G
+

{G_ + }

is the subgroup of fixed points of an involution. The techniques developed here have been used to prove local integrability of certain distributions which are fundamental in the harmonic analysis of

G

/

G
+

G/{G_ + }

.
Moments of the lifetime of conditioned Brownian motion in cones
Authors: Davis Burgess , Zhang Biao
Published: June 2010

Let

τ
\tau

be the time it takes standard d-dimensional Brownian motion, started at a point inside a cone

Γ
\Gamma

in

R

d

{\mathbb {R}^d}

which has aperture angle

θ
\theta

, to leave the cone. Burkholder has determined the smallest p, denoted

p
(
θ
,
d
)

p(\theta ,d)

, such that

E

τ
p

=
∞

E{\tau ^p} = \infty

. We show that if

y
∈
∂
Γ

y \in \partial \Gamma

then the smallest p, such that

E
(

τ
p

|

B
τ

=
y
)
=
∞

E({\tau ^p}|{B_\tau } = y) = \infty

, is

p
=
2
p
(
θ
,
d
)
+
(
d
−
2
)

/

2

p = 2p(\theta ,d) + (d - 2)/2

.
Extreme points of the unit ball of the space of Lipschitz functions
Authors: Farmer Jeff D.
Published: June 2010

We consider the space of all Lipschitz functions on a metric space with bounded Lipschitz norm, and give an intrinsic characterization of the extreme points of the unit ball. We briefly discuss some examples of extreme Lipschitz functions, and apply the result to show that if the norm of a Banach space is Gateaux differentiable then extreme functions on any one-dimensional subspace may be canonically extended to extreme functions on the whole space.
On the Virasoro algebra as reduced Poisson submanifold of a Kac-Moody algebra on the circle
Authors: Marí Beffa Gloria
Published: June 2010

In this paper we show how the Lie-Poisson structure on the dual of the Virasoro algebra can be obtained through a standard Poisson reduction process performed on a Kac-Moody algebra on the circle. We use the geometrical idea of the process to establish some relation between transverse structures on both Poisson manifolds.
Classifying PL 5-manifolds up to regular genus seven
Authors: Casali Maria Rita , Gagliardi Carlo
Published: June 2010

In the present paper, we show that the only closed orientable

5
5

-manifolds of regular genus less or equal than seven are the

5
5

-sphere

S

5

{\mathbb {S}^5}

and the connected sums of

m
m

copies of

S

1

×

S

4

{\mathbb {S}^1} \times {\mathbb {S}^4}

, with

m
⩽
7

m \leqslant 7

. As a consequence, the genus of

S

3

×

S

2

{\mathbb {S}^3} \times {\mathbb {S}^2}

is proved to be eight. This suggests a possible approach to the (

3
3

-dimensional) Poincaré Conjecture, via the well-known classification of simply connected

5
5

-manifolds, obtained by Smale and Barden.
Maximal moment inequalities for stochastic processes
Authors: Móricz F.
Published: June 2010

Let

X

a
,
b

{X_{a,b}}

be nonnegative random variables with the property that

X

a
,
b

⩽

X

a
,
c

+

X

c
,
b

{X_{a,b}} \leqslant {X_{a,c}} + {X_{c,b}}

for all

0
⩽
a
>
c
>
b
⩽
T

0 \leqslant a > c > b \leqslant T

, where

T
>
0

T > 0

is fixed. We define

M

a
,
b

:=
sup
{

X

a
,
c

:
a
>
c
⩽
b
}

{M_{a,b}}: = \sup \{ {X_{a,c}}:a > c \leqslant b\}

and establish bounds for

E

M

a
,
b

γ

EM_{a,b}^\gamma

and

E
exp
⁡
(
λ

M

a
,
b

)

E\exp (\lambda {M_{a,b}})

in terms of assumed bounds for

E

X

a
,
b

γ

EX_{a,b}^\gamma

and

E
exp
⁡
(
λ

X

a
,
b

)

E\exp (\lambda {X_{a,b}})

, respectively, where

γ
⩾
1

\gamma \geqslant 1

and

λ
\lambda

runs through an interval

(

λ
0

,
∞
)

({\lambda _0},\infty )

with fixed

λ
0

⩾
0

{\lambda _0} \geqslant 0

. These bounds explicitly involve a nonnegative function

g
(
a
,
b
)

g(a,b)

assumed to be quasi-superadditive with an index

Q
Q

, which means that

g
(
a
,
c
)
+
g
(
c
,
b
)
⩽
Q
g
(
a
,
b
)

g(a,c) + g(c,b) \leqslant Qg(a,b)

for all

0
⩽
a
>
c
>
b
⩽
T

0 \leqslant a > c > b \leqslant T

, where

1
⩽
Q
>
2

1 \leqslant Q > 2

is fixed. Maximal inequalities obtained in this way can be applied to stochastic processes exhibiting long-range dependence. Among others, these applications may include certain self-similar processes such as fractional Brownian motion, stochastic processes occurring in linear time-series models, etc.
An explicit family of curves with trivial automorphism groups
Authors: Turbek Peter
Published: June 2010

It is well known that a generic compact Riemann surface of genus greater than two admits only the identity automorphism; however, examples of such Riemann surfaces with their defining algebraic equations have not appeared in the literature. In this paper we give the defining equations of a doubly infinite, two-parameter family of projective curves (Riemann surfaces if defined over the complex numbers), whose members admit only the identity automorphism.
Dense extremally disconnected subspaces
Authors: Dow A. , van Mill J.
Published: June 2010

We prove that every compact Basically Disconnected space of

π
\pi

-weight

ω
1

{\omega _1}

has a dense Extremally Disconnected subspace. In Boolean algebraic terms: every

σ
\sigma

-complete Boolean algebra B with density

ω
1

{\omega _1}

carries an ultrafilter which generates an ultrafilter in the completion of B. The statement that every compact Basically Disconnected space of weight

c

\mathfrak {c}

has a dense Extremally Disconnected subspace is shown to be equivalent to CH.
A linear recurrence system
Authors: Blasius A.
Published: June 2010

We look at a triangular system of n equations and reinvestigate a related function introduced by Chen and Kuck. Our main contribution is to provide a new proof of a result which forms the basis of their work.
On the dimension of certain totally disconnected spaces
Authors: Oversteegen Lex G. , Tymchatyn E. D.
Published: June 2010

It is well known that there exist separable, metrizable, totally disconnected spaces of all dimensions. In this note we introduce the notion of an almost 0-dimensional space and prove that every such space is a totally disconnected subspace of an R-tree and, hence, at most 1-dimensional. As applications we prove that the spaces of homeomorphisms of the universal Menger continua are 1-dimensional and that hereditarily locally connected spaces have dimension at most two.
The difference between permutation polynomials over finite fields
Authors: Cohen Stephen D. , Mullen Gary L. , Shiue Peter Jau-Shyong
Published: June 2010

Recently S. D. Cohen resolved a conjecture of Chowla and Zassenhaus (1968) in the affirmative by showing that, if

f
(
x
)

f(x)

and

g
(
x
)

g(x)

are integral polynomials of degree

n
≥
2

n \geq 2

and p is a prime exceeding

(

n
2

−
3
n
+
4

)
2

{({n^2} - 3n + 4)^2}

for which f and g are both permutation polynomials of the finite field

F
p

{F_p}

, then their difference

h
=
f
−
g

h = f - g

cannot be such that

h
(
x
)
=
c
x

h(x) = cx

for some integer c not divisible by p. In this note we provide a significant generalization by proving that, if h is not a constant in

F
p

{F_p}

and t is the degree of h, then

t
≥
3
n

/

5

t \geq 3n/5

and, provided

t
≤
n
−
3

t \leq n - 3

, t and n are not relatively prime. In a sense this measures the isolation of permutation polynomials of the same degree over large finite prime fields.
Strong incompactness for some nonperfect rings
Authors: Trlifaj Jan
Published: June 2010

Answering a question of Eklof and Mekler, for each regular uncountable cardinal

κ
\kappa

, we construct a non-left-perfect ring

R
κ

{R_\kappa }

and a non-projective strongly

κ
\kappa

-free left ideal

I
κ

{I_\kappa }

such that

gen

(

I
κ

)
=
κ

{\text {gen}}({I_\kappa }) = \kappa

. Moreover, if

κ
>

ℵ
1

\kappa > {\aleph _1}

, then

I
κ

{I_\kappa }

is not

κ
\kappa

-free. As consequences, we obtain results concerning incompactness spectra of non-perfect rings.
On the integrality of some Galois representations
Authors: Gross Robert
Published: June 2010

We find an appropriate topology to put on K, the fraction field of the Iwasawa algebra

Λ
=

Z

p

[
[
T
]
]

\Lambda = {{\mathbf {Z}}_p}[[T]]

, so that compact subgroups of

K
×

{K^ \times }

are in fact contained in

Λ
×

{\Lambda ^ \times }

. This ensures that Galois representations to

K
×

{K^ \times }

have image in

Λ
×

{\Lambda ^ \times }

.
A Chevalley-Warning approach to 𝑝-adic estimates of character sums
Authors: Wan Da Qing
Published: June 2010

The elementary Chevalley-Warning congruence method is applied to obtain several p-adic estimates of character sums over finite fields.
Axiomatization and undecidability results for metrizable betweenness relations
Authors: Mendris Robert , Zlatoš Pavol
Published: June 2010

Let d be a metric on a nonempty set A. The ternary betweenness relation

T
d

{T_d}

induced by d on A is defined by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript d Baseline left-parenthesis x comma y comma z right-parenthesis left right double arrow d left-parenthesis x comma y right-parenthesis plus d left-parenthesis y comma z right-parenthesis equals d left-parenthesis x comma z right-parenthesis">

T
d

(
x
,
y
,
z
)
⇔
d
(
x
,
y
)
+
d
(
y
,
z
)
=
d
(
x
,
z
)

{T_d}(x,y,z) \Leftrightarrow d(x,y) + d(y,z) = d(x,z)

\]
for

x
,
y
,
z
∈
A

x,y,z \in A

. Allowing the range of d to vary over some "reasonable" ordered additive algebraic structures (not just the real numbers), we will prove that the class

M

\mathcal {M}

of all metrizable ternary structures, i.e., the class of all structures

(
A
,

T
d

)

(A,{T_d})

, where d is some metric on A, is an elementary class which can be axiomatized by a set of universal Horn sentences. Further, using an algorithm of linear programming, we will show that the first-order theory of

M

\mathcal {M}

is recursively axiomatizable and its universal part is decidable. On the other hand, the theory of

M

\mathcal {M}

is not finitely axiomatizable and the theory of finite members of

M

\mathcal {M}

is hereditarily undecidable.
Some counterexamples to the regularity of Monge-Ampère equations
Authors: Wang Xu Jia
Published: June 2010

We present examples to show that the solution u of the Monge-Ampère equation

det
(

D
2

u
)
=
f
(
x
)

\det ({D^2}u) = f(x)

, with

u
=
0

u = 0

on the boundary, may not lie in

W

2
,
p

{W^{2,p}}

or in

C

1
,
α

{C^{1,\alpha }}

for noncontinuous and positive

f
(
x
)

f(x)

and for continuous and nonnegative

f
(
x
)

f(x)

.
Representing projective sets as unions of Borel sets
Authors: Becker Howard
Published: June 2010

We consider a method of representing projective sets by a particular type of union of Borel sets, assuming AD. We prove a generalization of the theorem that a set is

∑
2
1

\sum _2^1

iff it is the union of

ω
1

{\omega _1}

Borel sets.
On the compactness of strongly continuous semigroups and cosine functions of operators
Authors: Henríquez Hernán R.
Published: June 2010

In this note we relate two notions of compactness for strongly continuous semigroups of linear operators and cosine functions of linear operators. Specifically, if T denotes a strongly continuous semigroup of linear operators defined on a Banach space X, we will show that T is compact if and only if the set

{
(
T
(
∙
)
x
:
x
∈
X
,

‖
x
‖

≤
1
}

\{ (T( \bullet )x:x \in X,\left \| x \right \| \leq 1\}

is relatively compact in any space

L
p

(
[
0
,
a
]
)
;
X
)

{L^p}([0,a]);X)

for

1
≤
p
>
∞

1 \leq p > \infty

and

a
>
0

a > 0

. We establish similar results for

(
T
(
t
)
−
I

)
n

,
n
∈

N

{(T(t) - I)^n},n \in {\mathbf {N}}

, and for cosine and sine functions of operators.
Invariant theory for a parabolic subgroup of 𝑆𝐿(𝑛+1,𝑅)
Authors: Gover A. Rod
Published: June 2010

For a certain maximal parabolic P of

SL

(
n
+
1
,

R

)

{\text {SL}}(n + 1,\mathbb {R})

, the complete invariant theory is presented for a class of P-representation modules. These modules arise naturally from the geometry of

P

n

{\mathbb {P}^n}

. In particular, a means of listing all the exceptional invariants is described. This is a model problem for some parabolic invariant theory problems posed by Fefferman.
A new simple proof of the Gel′fand-Mazur-Kaplansky theorem
Authors: Cabrera García Miguel , Rodríguez-Palacios Ángel
Published: June 2010

We provide an almost purely algebraic proof of Kaplansky’s refinement of the Gelfand-Mazur theorem asserting that the reals, complex, and quaternions are the only associative normed real algebras with no nonzero topological divisors of zero.
A geometric approach to an equation of J. D’Alembert
Authors: Pràstaro A. , Rassias Th. M.
Published: June 2010

By using a geometric framework of PDE’s we prove that the set of solutions of the D’Alembert equation

(
∗
)
(

∂
2

log
⁡
f

∂
x
∂
y

)
=
0

( \ast )(\frac {{{\partial ^2}\log f}}{{\partial x\partial y}}) = 0

is larger than the set of smooth functions of two variables

f
(
x
,
y
)

f(x,y)

of the form

(
∗
∗
)
f
(
x
,
y
)
=
h
(
x
)
∙
g
(
y
)

( \ast \ast )f(x,y) = h(x) \bullet g(y)

. This agrees with a previous counterexample by Th. M. Rassias given to a statement by C. M. Stéphanos. More precisely, we have the following result. Theorem. The set of 2-dimensional integral manifolds of PDE

(
∗
)

( \ast )

properly contains the ones representable by graphs of 2-jet-derivatives of functions

f
(
x
,
y
)

f(x,y)

expressed in the form

(
∗
∗
)

( \ast \ast )

. A generalization of this result to functions of more than two variables is sketched also by considering the equation

(

∂
n

log
⁡
f

∂

x
1

⋯
∂

x
n

)
=
0

(\frac {{{\partial ^n}\log f}}{{\partial {x_1} \cdots \partial {x_n}}}) = 0

.
On tree ideals
Authors: Goldstern Martin , Repický Miroslav , Shelah Saharon , Spinas Otmar
Published: June 2010

Let

l
0

{l^0}

and

m
0

{m^0}

be the ideals associated with Laver and Miller forcing, respectively. We show that

a
d
d

(

l
0

)
>

c
o
v

(

l
0

)

{\mathbf {add}}({l^0}) > {\mathbf {cov}}({l^0})

and

a
d
d

(

m
0

)
>

c
o
v

(

m
0

)

{\mathbf {add}}({m^0}) > {\mathbf {cov}}({m^0})

are consistent. We also show that both Laver and Miller forcing collapse the continuum to a cardinal

≤

h

\leq \mathfrak {h}

.
On Hilbert spaces with unital multiplication
Authors: Zalar Borut
Published: June 2010

We give a new simplified proof of two theorems of Froelich, Ingelstam, and Smiley. Our approach enables us also to generalize both of them. In the second section we prove a related theorem which requires different methods for its proof.
On Dahlberg’s Lusin area integral theorem
Authors: Mitrea Marius
Published: June 2010

We give new proofs to the Lusin area integral theorem of Dahlberg. Our techniques rely on the theory of elliptic boundary value problems on nonsmooth domains and are shown to extend to other important cases, including systems of equations.
The Hausdorff dimension of graphs of density continuous functions. II
Authors: Buczolich Zoltán , Ostaszewski Krzysztof
Published: June 2010

In this paper we complete the proof of the fact that the Hausdorff dimensions of graphs of density continuous functions vary continuously between one and two. This result was announced in our previous paper, but the proof there contained a gap and the construction given there should also be slightly modified. This correction is done in this paper.
On the multiple points of certain meromorphic functions
Authors: Langley J. K.
Published: June 2010

We show that if f is transcendental and meromorphic in the plane and

T
(
r
,
f
)
=
o

(
log
⁡
r

)
2

T(r,f) = o{(\log r)^2}

, then f has infinitely many critical values. This is sharp. Further, we apply a result of Eremenko to show that if f is meromorphic of finite lower order in the plane and

N
(
r
,
1

/

f
f
)
=
o
(
T
(
r
,

f
′

/

f
)
)

N(r,1/ff) = o(T(r,f’ /f))

, then

f
(
z
)
=
exp
⁡
(
a
z
+
b
)

f(z) = \exp (az + b)

or

f
(
z
)
=

(
a
z
+
b

)

−
n

f(z) = {(az + b)^{ - n}}

with a and b constants and n a positive integer.
On spectral properties of perturbed operators
Authors: Nair M. Thamban
Published: June 2010

Farid (1991) has given an estimate for the norm of a perturbation V required to obtain an eigenvector for the perturbed operator

T
+
V

T + V

within a given ball centered at a given eigenvector of the unperturbed (closed linear) operator T. A similar result is derived from a more general result of the author (1989) which also guarantees that the corresponding eigenvalue is simple and also that the eigenpair is the limit of a sequence obtained in an iterative manner.
Commutation of variation and dual projection
Authors: Neal David
Published: June 2010

For a raw process of integrable variation V, taking values in a Banach space E having the Radon-Nikodyn property, the variation of the predictable (optional) dual projection is the predictable (optional) dual projection of the variation. An analogous result holds for the associated stochastic measures. The result is applied to the stochastic integral of a real, optional process H with respect to V when V is adapted.
A congruence for primes
Authors: Sun Zhi Wei
Published: June 2010

With the help of the Pell sequence we obtain the following new congruence for odd primes:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript k equals 1 Overscript left-parenthesis p minus 1 right-parenthesis slash 2 Endscripts StartFraction 1 Over k dot 2 Superscript k Baseline EndFraction identical-to sigma-summation Underscript k equals 1 Overscript left-bracket 3 p slash 4 right-bracket Endscripts StartFraction left-parenthesis negative 1 right-parenthesis Superscript k minus 1 Baseline Over k EndFraction left-parenthesis mod p right-parenthesis period">

∑

k
=
1

(
p
−
1
)

/

2

1

k
⋅

2
k

≡

∑

k
=
1

[
3
p

/

4
]

(
−
1
)

k
−
1

k

(
mod

p
)
.

\sum \limits _{k = 1}^{(p - 1)/2} {\frac {1}{{k \cdot {2^k}}} \equiv \sum \limits _{k = 1}^{[3p/4]} {\;\frac {{{{( - 1)}^{k - 1}}}}{k}} \quad \pmod p.}

\]
Uniform rational approximation
Authors: Yang Li Ming
Published: June 2010

Let K be a compact subset of the complex plane

C

\mathbb {C}

, and let

P
(
K
)

P(K)

and

R
(
K
)

R(K)

be the closures in

C
(
K
)

C(K)

of polynomials and rational functions with poles off K, respectively. Suppose that

R
(
K
)
≠
C
(
K
)
,
λ

R(K) \ne C(K),\lambda

is a nonpeak point for

R
(
K
)

R(K)

, and g is continuous on

C

\mathbb {C}

and

C
1

{C^1}

in a neighborhood of

λ
\lambda

. Then

P
(
K
)
g
+
R
(
K
)

P(K)g + R(K)

is not dense in

C
(
K
)

C(K)

. In fact, our proof shows that there are a lot of smooth functions which are not in the closure of

P
(
K
)
g
+
R
(
K
)

P(K)g + R(K)

.
A Paley-Wiener theorem for frames
Authors: Christensen Ole
Published: June 2010

We prove a stability theorem for frames. Our result is a generalization of a classical result of Paley and Wiener about Riesz bases; it is also related to the Perturbation Theorem of Kato.
Surfaces in 𝑅⁴ with constant affine Gauss maps
Authors: Vrancken Luc , Wang Chang Ping
Published: June 2010

In this paper we study the affine Gauss maps of nondegenerate surfaces in

R

4

{\mathbb {R}^4}

with respect to the Burstin-Mayer normalization (1927) and with respect to the normalization obtained by Nomizu and the first author in 1993. We determine up to affine transformations all surfaces in

R

4

{\mathbb {R}^4}

with constant affine Gauss maps.
Omitted values in domains of normality
Authors: Bergweiler Walter , Rohde Steffen
Published: June 2010

It is proved that if U and V are connected components of the Fatou set of an entire function f and if

f
(
U
)
⊂
V

f(U) \subset V

, then

V
∖
f
(
U
)

V\backslash f(U)

contains at most one point.
Star-shaped complexes and Ehrhart polynomials
Authors: Hibi Takayuki
Published: June 2010

We study Ehrhart polynomials of star-shaped triangulations of balls by means of Cohen-Macaulay rings and canonical modules.
Composition operators between Hardy and weighted Bergman spaces on convex domains in 𝐶^{𝑁}
Authors: MacCluer Barbara D. , Mercer Peter R.
Published: June 2010

Suppose

Ω
\Omega

is a bounded, strongly convex domain in

C

N

{{\mathbf {C}}^N}

with smooth boundary and

ϕ
:
Ω
→
Ω

\phi :\Omega \to \Omega

is an arbitrary holomorphic map. While in general the composition operator

C
ϕ

{C_\phi }

need not map the Hardy space

H
p

(
Ω
)

{H^p}(\Omega )

into itself when

N
>
1

N > 1

, our main theorem shows that

C
ϕ

{C_\phi }

does map

H
p

(
Ω
)

{H^p}(\Omega )

boundedly into a certain weighted Bergman space on

Ω
\Omega

, where the weight function depends on the dimension N. We also consider properties of

C
ϕ

{C_\phi }

on

H
p

(
Ω
)

{H^p}(\Omega )

when

ϕ
(
Ω
)

\phi (\Omega )

is contained in an approach region in

Ω
\Omega

.
Some complete Ricci-flat Kähler metrics in 𝐶𝑃²
Authors: Hsieh Chun Chung
Published: June 2010

In this article, we give an explicit construction of complete Ricci-flat Kahler metrics in

C

P
2

∖

\mathbb {C}{P^2}\backslash

{three complex lines in general position}.
Completeness of compact Lorentz manifolds admitting a timelike conformal Killing vector field
Authors: Romero Alfonso , Sánchez Miguel
Published: June 2010

It is proved that every compact Lorentz manifold admitting a timelike conformal Killing vector field is geodesically complete. So, a recent result by Kamishima in J. Differential Geometry [37 (1993), 569-601] is widely extended.
Existence and nonexistence of positive eigenfunctions for the 𝑝-Laplacian
Authors: Binding P. A. , Huang Y. X.
Published: June 2010

We study the relation between (i) the principal eigencurve, i.e., the graph ofμ1{\mu _1}satisfying<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk right-parenthesis EndLabel minus d i v left-parenthesis StartAbsoluteValue nabla u EndAbsoluteValue Superscript p minus 2 Baseline nabla u right-parenthesis plus left-parenthesis q left-parenthesis x right-parenthesis minus lamda w left-parenthesis x right-parenthesis right-parenthesis StartAbsoluteValue u EndAbsoluteValue Superscript p minus 2 Baseline u equals mu 1 left-parenthesis lamda right-parenthesis StartAbsoluteValue u EndAbsoluteValue Superscript p minus 2 Baseline mu EndLayout">(∗)−div(|∇u|p−2∇u)+(q(x)−λw(x))|u|p−2u=μ1(λ)|u|p−2μ\begin{equation}\tag {$ \ast $} - {\operatorname {div}}(|\nabla u{|^{p - 2}}\nabla u) + (q(x) - \lambda w(x))|u{|^{p - 2}}u = {\mu _1}(\lambda )|u{|^{p - 2}}\mu \end{equation}on a smooth bounded domainΩ\OmegainRN{{\mathbf {R}}^N}withp>1p > 1, and (ii) existence and nonexistence of positive solutions of(∗)( \ast )withμ1(λ)=0{\mu _1}(\lambda ) = 0. Eigencurve arguments are used extensively.
A short note on the full Jacobi group
Authors: Choie YoungJu
Published: June 2010

The full Jacobi group has been defined to study Jacobi forms. The full Jacobi group has properties similar to the modular group. In this note we investigate properties of the full Jacobi group to get generators and relations.
Characterizations of naturally submodular graphs: a polynomially solvable class of the TSP
Authors: Herer Yale T. , Penn Michal
Published: June 2010

Let

G
=
(
V
,
E
)

G = (V,E)

be a graph and

w
:
E
→

R
+

w:E \to {R^ + }

be a length function. Given

S
⊆
V

S \subseteq V

, a Steiner tour is a cycle passing at least once through each vertex of S. In this paper we investigate naturally submodular graphs: graphs for which the length function of the Steiner tours is submodular. We provide two characterizations of naturally submodular graphs, an

O
(
n
)

O(n)

time algorithm for identifying such graphs, and an

O
(
n
)

O(n)

time algorithm for solving the Steiner traveling salesman problem on such graphs.
On a theorem of Hartman and Wintner
Authors: Millar P. W.
Published: June 2010

An elementary stochastic process argument is given for a measuretheoretic result of Hartman-Wintner, which asserts (under a natural condition) that an infinitely divisible measure has no atoms iff its corresponding Lévy measure has infinite mass.
Noetherian subrings of power series rings
Authors: Wan Da Qing
Published: June 2010

Let R be a commutative noetherian ring with unit. It is shown that certain subrings contained between the polynomial ring

R
[
X
]

R[X]

and the power series ring

R
[
X
]
[
[
Y
]
]

R[X][[Y]]

are also noetherian. These subrings naturally arise from studying p-adic analytic variation of zeta functions over finite fields.
A Dvoretsky theorem for polynomials
Authors: Dineen Seán
Published: June 2010

We lift upper and lower estimates from linear functionals to n-homogeneous polynomials and using this result show that

l
∞

{l_\infty }

is finitely represented in the space of n-homogeneous polynomials,

n
≥
2

n \geq 2

, on any infinite-dimensional Banach space. Refinements are also given.
On 𝑝-hyponormal contractions
Authors: Duggal B. P.
Published: June 2010

The contraction A on a Hilbert space H is said to be p-hyponormal,

0
>
p
>
1

0 > p > 1

, if

(

A
∗

A

)
p

≥

(
A

A
∗

)
p

{({A^ \ast }A)^p} \geq {(A{A^ \ast })^p}

. Let A be an invertible p-hyponormal contraction. It is shown that A has

C

.0

{C_{.0}}

completely nonunitary part. Now let H be separable. If A is pure and the defect operator

D
A

=

(
1
−

A
∗

A

)

1

/

2

{D_A} = {(1 - {A^ \ast }A)^{1/2}}

is of Hilbert-Schmidt class, then

A
∈

C

10

A \in {C_{10}}

. Let

B
∗

{B^ \ast }

be a contraction such that

B
∗

{B^\ast }

has

C

.0

{C_{.0}}

completely nonunitary part,

D

B
∗

{D_{{B^ \ast }}}

is of Hilbert-Schmidt class, and

B
∗

{B^ \ast }

satisfies the property that if the restriction of

B
∗

{B^ \ast }

to an invariant subspace is normal, then the subspace reduces

B
∗

{B^ \ast }

. It is shown that if

A
X
=
X
B

AX = XB

for some quasi-affinity X, then A and B are unitarily equivalent normal contractions.
Cohen-Macaulayness of blow-ups of homogeneous weak 𝑑-sequences
Authors: Johnson Mark R. , Raghavan K. N.
Published: June 2010

Let R be a homogeneous Cohen-Macaulay algebra over a field, and let I be an ideal generated by a homogeneous weak d-sequence. We show, under reasonable conditions on the sequence, that the graded ring

gr
M

R
[
I
t
]

{\text {gr}_M}R[It]

of the Rees algebra

R
[
I
t
]
=

⊕

i
≥
0

I
i

R[It] = { \oplus _{i \geq 0}}{I^i}

is Cohen-Macaulay. In particular we obtain the Cohen-Macaulayness of the blow-up ring

R
[
I
t
]

R[It]

.
Real rank of tensor products of 𝐶*-algebras
Authors: Kodaka Kazunori , Osaka Hiroyuki
Published: June 2010

We study the real rank of tensor products of

C
∗

{C^ \ast }

-algebras. From the dimension theory:

dim
⁡
(
X
×
Y
)
≤
dim
⁡
X
+
dim
⁡
Y

\dim (X \times Y) \leq \dim X + \dim Y

, it is naturally hoped that

R
R
(
A
⊗
B
)
≤
R
R
(
A
)
+
R
R
(
B
)

RR(A \otimes B) \leq RR(A) + RR(B)

. We then prove that it is false generally. Moreover, we point out that (FS)-property for

C
∗

{C^ \ast }

-algebras is not stable under taking tensor products.
6-dimensional manifolds without totally algebraic homology
Authors: Teichner Peter
Published: June 2010

We construct 6-dimensional manifolds for which not all codimension 2 homology classes (with

Z

/

2

\mathbb {Z}/2

-coefficients) are realized by algebraic subvarieties in any real algebraic structure on the manifold. It was known that such examples exist in dimension 11 and higher, and that dimension 6 is the best possible. We also give an elementary algebraic topological proof of a connection between codimension 2 submanifolds and vector bundles which was previously proven only by algebraic geometrical methods.
Entire solutions of the abstract Cauchy problem in a Hilbert space
Authors: deLaubenfels Ralph , Yao Fuyuan
Published: June 2010

We show that, whenever the linear operator A is symmetric and densely defined, on a Hilbert space, then the abstract Cauchy problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction d Over d z EndFraction u left-parenthesis z right-parenthesis equals upper A Superscript asterisk Baseline left-parenthesis u left-parenthesis z right-parenthesis right-parenthesis left-parenthesis z element-of bold upper C right-parenthesis comma u left-parenthesis 0 right-parenthesis equals x">

d

d
z

u
(
z
)
=

A
∗

(
u
(
z
)
)

(
z
∈

C

)
,

u
(
0
)
=
x

\frac {d}{{dz}}u(z) = {A^ \ast }(u(z))\quad (z \in {\mathbf {C}}),\qquad u(0) = x

\]
has an entire solution, for all initial data x in the image of

e

−

A
¯

A
∗

{e^{ - \bar A{A^ \ast }}}

, which is a dense set.
Another generalization of Anderson’s theorem
Authors: Du Hong Ke
Published: June 2010

In this paper, we prove that if A and B are normal operators on a Hilbert space H, then, for every operator S satisfying
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper S upper B equals upper S comma double-vertical-bar upper A upper X upper B minus upper X plus upper S double-vertical-bar greater-than-or-equal-to double-vertical-bar upper A double-vertical-bar Superscript negative 1 Baseline double-vertical-bar upper B double-vertical-bar Superscript negative 1 Baseline double-vertical-bar upper S double-vertical-bar">

A
S
B
=
S
,

‖

A
X
B
−
X
+
S

‖

≥

‖
A
‖

−
1

‖
B
‖

−
1

‖
S
‖

ASB = S, \left \| {AXB - X + S} \right \| \geq {\left \| A \right \|^{ - 1}}{\left \| B \right \|^{ - 1}}\left \| S \right \|

for all operators

X
∈
B
(
H
)

X \in B(H)

, and that if A and B are contractions, then, for every operator S satisfying

A
S
B
=
S

ASB = S

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Superscript asterisk Baseline upper S upper B Superscript asterisk Baseline equals upper S comma double-vertical-bar upper A upper X upper B minus upper X plus upper S double-vertical-bar greater-than-or-equal-to double-vertical-bar upper S double-vertical-bar">

A
∗

S

B
∗

=
S
,

‖

A
X
B
−
X
+
S

‖

≥

‖
S
‖

{A^ \ast }S{B^ \ast } = S,\left \| {AXB - X + S} \right \| \geq \left \| S \right \|

for all operators

X
∈
B
(
H
)

X \in B(H)

, where

B
(
H
)

B(H)

denotes the set of all bounded linear operators on H.
Locally compact groups which have the weakly compact homomorphism property
Authors: Runde Volker
Published: June 2010

A locally compact group G is WCHP if every weakly compact homomorphism from

L
1

(
G
)

{L^1}(G)

into a Banach algebra has finite-dimensional range, and is

WCHP
+

{\text {WCHP}^ + }

if every extension of G by an abelian group is WCHP. We verify the

WCHP
+

{\text {WCHP}^ + }

property for certain locally compact groups, including all Moore groups and all connected groups.
On some modular equations of degree 5
Authors: Shen Li-Chien
Published: June 2010

The main purpose of this paper is to examine certain well-known modular equations of degree 5 from a very intuitive angle. Unlike the classical treatment of this subject, which requires parametrization of certain key quantities, we will give direct proofs to these equations by recasting them as theta function identities which are quite interesting in their own right.
𝑛-Laplacian in ℋ¹_{}𝓁ℴ𝒸 does not lead to regularity
Authors: Firoozye Nikan B.
Published: June 2010

It is well known that in two space dimensions, if a solution to Poisson’s equation has right-hand side in

H

loc

1

\mathcal {H}_{{\text {loc}}}^1

, then this solution is actually continuous. The corresponding result for n-Laplacians is shown to be false for

n
≥
3

n \geq 3

; we construct two examples with right-hand sides in

H

loc

1

(

ℜ
n

)

\mathcal {H}_{{\text {loc}}}^1({\Re ^n})

such that the corresponding solutions to the n-Laplacian are unbounded in the first case, and bounded but discontinuous in the second.
The general local form of an analytic mapping into the set of idempotent elements of a Banach algebra
Authors: Labrousse J.-Ph.
Published: June 2010

This paper gives a general formula which describes any analytical mapping of a suitably small open neighborhood U in

C

\mathbb {C}

into the set of idempotent elements of any complex Banach algebra B and an application of this formula to the case when B is a Calkin algebra.
Unique integrability of continuous 𝑘-plane fields
Authors: McSwiggen Patrick D.
Published: June 2010

It is shown that unique integrability of a continuous k-plane field is a generic property for plane field satisfying a "topological bracket condition".
Aperiodic tilings in higher dimensions
Authors: Radin Charles
Published: June 2010

We show that in dimensions

d
≥
3

d \geq 3

, aperiodic tilings can naturally avoid more symmetries than just translations.
On compact Kaehler submanifolds in 𝐶𝑃^{𝑛+𝑝} with nonnegative sectional curvature
Authors: Shen Yi Bing
Published: June 2010

A complete classification for nonnegatively curved compact Kaehler submanifolds

M
n

{M^n}

in

C

P

n
+
p

\mathbb {C}{P^{n + p}}

with

p
>
n

p > n

is given, so that a conjecture of K. Ogiue is resolved partially.
Tri-quotient maps are preserved by infinite products
Authors: Uspenskij V. V.
Published: June 2010

The theorem stated in the title answers a question of E. Michael (1977).
A note on 𝐺-invariant forms
Authors: Gagola Stephen M.
Published: June 2010

If G is a finite group, the reduction

mod

p

\bmod \; p

of a module supporting a nondegenerate G-invariant form need not itself support such a form. However, under a suitable hypothesis on the splitting field (quadratric closure) and a carefully chosen lattice within the module (for reduction

mod

p

\bmod \; p

), this will always be the case. The argument given is elementary and self-contained.
Nilpotency of derivations in prime rings
Authors: Jensen David W.
Published: June 2010

In 1957, E. C. Posner proved that if

λ
\lambda

and

δ
\delta

are derivations of a prime ring R, characteristic

R
≠
2

R \ne 2

, then

λ
δ
=
0

\lambda \delta = 0

implies either

λ
=
0

\lambda = 0

or

δ
=
0

\delta = 0

. We extend this well-known result by showing that, without any characteristic restriction,

λ

δ
m

=
0

\lambda {\delta ^m} = 0

implies either

λ
=
0

\lambda = 0

or

δ

4
m
−
1

=
0

{\delta ^{4m - 1}} = 0

. We also prove that

λ
n

δ
=
0

{\lambda ^n}\delta = 0

implies either

δ
2

=
0

{\delta ^2} = 0

or

λ

12
n
−
9

=
0

{\lambda ^{12n - 9}} = 0

. In the case where

λ
n

δ
m

=
0

{\lambda ^n}{\delta ^m} = 0

, we show that if

λ
\lambda

and

δ
\delta

commute, then at least one of the derivations must be nilpotent.
Optimal intervals of stability of a forced oscillator
Authors: Alonso Jose Miguel
Published: June 2010

Consider the differential equation of a nonlinear oscillator with linear friction and a T-periodic external force. We find optimal bounds on the derivative of the restoring force and on the period T in order to obtain a unique T-periodic solution that is asymptotically stable.
Compactifications with discrete remainders
Authors: Hatzenbuhler James P. , Mattson Don A.
Published: June 2010

Conditions are obtained which characterize when a space has a Hausdorff compactification with a discrete remainder. A characterization is also given for when the minimal perfect compactification of a 0-space has a discrete remainder. It is shown that a metric space has a compactification with a discrete remainder if and only if it is rimcompact. In general, however, for a space to have a compactification with a discrete remainder, it is not necessary that the space be rimcompact.
Wedderburn decompositions of commutative Banach algebras
Authors: Solovej Michel
Published: June 2010

We prove that if A is a commutative Banach algebra with

rad
⁡

(
A

)
2

=
0

\operatorname {rad}{(A)^2} = 0

and

A

/

rad
⁡
(
A
)
=
C
(
[
0
,
1
]
)

A/\operatorname {rad}(A) = C([0,1])

for the unit interval [0, 1], then A has a strong Wedderburn decomposition.
On the Carleson measure characterization of BMO functions on the unit sphere
Authors: Jevtić Miroljub
Published: June 2010

A higher dimensional version of the well-known Carleson measure characterization of BMO functions is given.
There are knots whose tunnel numbers go down under connected sum
Authors: Morimoto Kanji
Published: June 2010

In this paper, we show that there are infinitely many tunnel number two knots K such that the tunnel number of

K
#

K
′

K\# K’

is equal to two again for any 2-bridge knot

K
′

K’

.
A note on cohomological dimension of approximate movable spaces
Authors: Watanabe Tadashi
Published: June 2010

We show that any approximate movable compact metric space X satisfies the equality

dim
⁡
X
=

dim

Z

X

\dim X = {\dim _\mathbb {Z}}X

without finite dimensional condition. Thus there is no approximate movable compact metric space X with

dim
⁡
X
=
∞

\dim X = \infty

and

dim

Z

X
>
∞

{\dim _\mathbb {Z}}X > \infty

. Since ANRs and some generalized ANRs are approximate movable, they satisfy the above equality.
Some analytical properties of continuous stationary Markov transition functions
Authors: Kendall David G.
Published: June 2010

A systematic treatment of Markov processes with Euclidean state-spaces has recently been presented by Doob [1], the restriction on the nature of the state-space being associated with the very illuminating probabilistic method which he uses throughout. At about the same time a new step was taken by Kolmogorov [4] who established for countable state-spaces the existence and finiteness of the derivative of the transition-function

p

i
j

(
t
)

{p_{ij}}(t)

at

t
=
0
+

t = 0 +

when

i
≠
j

i \ne j

. In this paper some of Doob’s and Kolmogorov’s results are combined and shown to be valid (when suitably formulated) for an arbitrary state-space. For the sake of a generality which proves useful in the discussion of existence theorems the transition-function

P
t

(
x
,

A
)

{P_t}(x,\;A)

is not assumed to be “honest"; i.e., if

X
X

is the state-space then it is supposed that

P
t

(
x
,

X
)
≦
1

{P_t}(x,\;X) \leqq 1

.