bluuebunny/crossref_metadata_2025 huggingface dataset:
Integral operators and the compactness of induced representations
Authors: Busby Robert C. , Schochetman Irwin , Smith Harvey A.
Published: July 2010

Integral operators are investigated which are compositions of multiplication by bounded vector valued functions and convolutions with vector valued functions. All of the functions are based on a fixed locally compact group. Conditions are given under which certain of these operators are compact. As an application of these conditions we consider induced representations of twisted group algebras (these are generalizations of representations of groups induced from closed normal sub-groups in the sense of Mackey and include these as special cases) and we give necessary and sufficient conditions for these representations to be compact (that is, to consist entirely of compact operators).
A general class of factors of 𝐸⁴
Authors: Rubin Leonard R.
Published: July 2010

In this paper we prove that any upper semicontinuous decomposition of

E
n

E^n

which is generated by a trivial defining sequence of cubes with handles determines a factor of

E

n
+
1

E^{n + 1}

. An important corollary to this result is that every 0-dimensional point-like decomposition of

E
3

E^3

determines a factor of

E
4

E^4

. In our approach we have simplified the construction of the sequence of shrinking homeomorphisms by eliminating the necessity of shrinking sets piecewise in a collection of n-cells, the technique employed by R. H. Bing in the original result of this type.
On residually finite knot groups
Authors: Mayland E. J.
Published: July 2010

The residual finiteness of the class of groups of fibred knots, or those knot groups with finitely generated and, therefore, free commutator subgroups, has been known for some time. Using Baumslag’s results on absolutely parafree groups, this paper extends the result to twist knots (Whitehead doubles of the trivial knot) and certain other classes of nonfibred knots whose minimal spanning surface has complement with free fundamental group. As a by-product more explicit finite representations, namely cyclic extensions of certain

p
p

-groups, are obtained for these knot groups and the groups of fibred knots. Finally composites of two such knots also have residually finite groups.
Orientation-preserving mappings, a semigroup of geometric transformations, and a class of integral operators
Authors: Farias Antonio O.
Published: July 2010

A Titus transformation

T
=
⟨
α
,
v
⟩

T = \langle \alpha ,v\rangle

is a linear operator on the vector space of

C
∞

{C^\infty }

mappings from the circle into the plane given by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper T f right-parenthesis left-parenthesis t right-parenthesis equals left-parenthesis mathematical left-angle alpha comma v mathematical right-angle f right-parenthesis left-parenthesis t right-parenthesis equals f left-parenthesis t right-parenthesis plus alpha left-parenthesis t right-parenthesis det left-bracket v comma f prime left-parenthesis t right-parenthesis right-bracket v">

(
T
f
)
(
t
)
=
(
⟨
α
,
v
⟩
f
)
(
t
)
=
f
(
t
)
+
α
(
t
)
det
[
v
,

f
′

(
t
)
]
v

(Tf)(t) = (\langle \alpha ,v\rangle f)(t) = f(t) + \alpha (t)\det [v,f’(t)]v

, where

α
\alpha

is a nonnegative,

C
∞

{C^\infty }

function on the circle

S
1

{S^1}

. Let

τ
\tau

denote the semigroup generated by finite compositions of Titus transformations. A Titus mapping is the image by an element of

τ
\tau

of a degenerate curve,

α
0

v
0

{\alpha _0}{v_0}

, where

α
0

{\alpha _0}

is a

C
∞

{C^\infty }

function on

S
1

{S^1}

and

v
0

{v_0}

is fixed in the plane

R
2

{R^2}

. A

C
∞

{C^\infty }

mapping

f
:

S
1

→

R
2

f:{S^1} \to {R^2}

is called properly extendable if there is a

C
∞

{C^\infty }

mapping

F
:

D
−

→

R
2

F:{D^ - } \to {R^2}

, D the open unit disk and

D
−

{D^ - }

its closure, such that

J
F

≧
0

{J_F} \geqq 0

on

D
,

J
F

>
0

D,{J_F} > 0

near the boundary

S
1

{S^1}

of

D
−

{D^ - }

and

F

|

s
1

=
f

F{|_{{s^1}}} = f

. A

C
∞

{C^\infty }

mapping

f
:

S
1

→

R
2

f:{S^1} \to {R^2}

is called normal if it is an immersion with no triple points and all its double points are transversal. The main result of this paper can be stated: a normal mapping is extendable if and only if it is a Titus mapping. An application is made to a class of integral operators of the convolution type,

y
(
t
)
=
−

∫
0

2
π

k
(
s
)
x
(
t
−
s
)
d
s

y(t) = - \smallint _0^{2\pi } {k(s)x(t - s)ds}

. It is proved that, under certain technical conditions, such an operator is topologically equivalent to Hilbert’s transform of potential theory,

y
(
t
)
=

∫
0

2
π

cot
⁡
(
s

/

2
)
x
(
t
−
s
)
d
s

y(t) = \smallint _0^{2\pi } {\cot (s/2)x(t - s)ds}

, which gives the relation between the real and imaginary parts of the restriction to the boundary of a function holomorphic inside the unit disk.
A dominance theorem for partitioned Hermitian matrices
Authors: Merris Russell
Published: July 2010

Let

A
=
(

A

i
j

)

A = ({A_{ij}})

be a partitioned positive semidefinite hermitian matrix, where

A

i
j

{A_{ij}}

is n-square,

1
≦
i
,
j
≦
m

1 \leqq i,j \leqq m

. A class of ordered pairs of functions

(

f
1

,

f
2

)

({f_1},{f_2})

is given such that

(

f
1

(

A

i
j

)
)
−
(

f
2

(

A

i
j

)
)

({f_1}({A_{ij}})) - ({f_2}({A_{ij}}))

is positive semidefinite hermitian. Applications are given.
The connectedness of the collection of arc cluster sets
Authors: Lappan Peter
Published: July 2010

Let

f
f

be a continuous complex-valued function defined on the unit disk and let

p
p

be a boundary point of the disk. A very natural topology on the collection of all arc cluster sets of

f
f

at the point

p
p

has been investigated by Belna and Lappan [1] who proved that this collection is a compact set under certain suitable conditions. It is proved here that this collection is an arcwise connected set under the topology in question, but is not in general locally arcwise connected or even locally connected. It is also shown by example that it is generally not possible to map the real line onto the collection of arc cluster sets at

p
p

in a continuous manner.
Subharmonic functions in certain regions
Authors: Lewis John L.
Published: July 2010

In a recent paper Hellsten, Kjellberg, and Norstad considered bounded subharmonic functions u in

|

z

|

>
1

|z| > 1

which satisfy a certain inequality. They obtained an exact upper bound for the maximum modulus of u. We first show that this bound still holds when u satisfies less restrictive hypotheses. We then give an application of this result.
The gliding humps technique for 𝐹𝐾-spaces
Authors: Bennett G.
Published: July 2010

The gliging humps technique has been used by various authors to establish the existence of bounded divergent sequences in certain summability domains. The purpose of this paper is to extend these results and to obtain analogous ones for sequence spaces other than c and m. This serves to unify and improve many known results and to obtain several new ones—applications include extensions to theorems of Dawson, Lorentz-Zeller, Snyder-Wilansky and Yurimyae. Improving another result of Wilansky allows us to consider countable collections of sequence spaces—applications including the proof of a conjecture of Hill and Sledd and extensions to theorems of Berg and Brudno. A related result of Petersen is also considered and a simple proof using the Baire category theorem is given.
Operator and dual operator bases in linear topological spaces
Authors: Johnson William B.
Published: July 2010

A net

{

S
d

:
d
∈
D
}

\{ {S_d}:d \in D\}

of continuous linear projections of finite range on a Hausdorff linear topological space V is said to be a Schauder operator basis—S.O.B. —(resp. Schauder dual operator basis—S.D.O.B.) provided it is pointwise bounded and converges pointwise to the identity operator on V, and

S
e

S
d

=

S
d

{S_e}{S_d} = {S_d}

(resp.

S
d

S
e

=

S
d

{S_d}{S_e} = {S_d}

) whenever

e
≧
d

e \geqq d

. S.O.B.’s and S.D.O.B.’s are natural generalizations of finite dimensional Schauder bases of subspaces. In fact, a sequence of operators is both a S.O.B. and S.D.O.B. iff it is the sequence of partial sum operators associated with a finite dimensional Schauder basis of subspaces. We show that many duality-theory results concerning Schauder bases can be extended to S.O.B.’s or S.D.O.B.’s. In particular, a space with a S.D.O.B. is semi-reflexive if and only if the S.D.O.B. is shrinking and boundedly complete. Several results on S.O.B.’s and S.D.O.B.’s were previously unknown even in the case of Schauder bases. For example, Corollary IV.2 implies that the strong dual of an evaluable space which admits a shrinking Schauder basis is a complete barrelled space.
A sufficient condition for the lower semicontinuity of parametric integrals
Authors: Silverman Edward
Published: July 2010

We use simple convex functions and standard techniques in area theory to treat Morrey’s extension of McShane’s lower semicontinuity theorem for parametric integrals. This enables us to eliminate some technical hypotheses, simplify the proof and obtain a more general result.
Covering relations in the lattice of 𝑇₁-topologies
Authors: Larson Roland E. , Thron W. J.
Published: July 2010

A topology

T

1

{\mathcal {T}_1}

is said to cover another topology

T

2

{\mathcal {T}_2}

if

T

2

⊊

T

1

{\mathcal {T}_2} \subsetneq {\mathcal {T}_1}

and no other topology may be included between the two. In this paper, we characterize the relationship between a

T
1

{T_1}

-topology and its covers. This characterization is used to prove that the lattice of

T
1

{T_1}

-topologies is both upper and lower semimodular. We also prove that the sublattice generated by the covers of a

T
1

{T_1}

-topology is isomorphic to the Boolean lattice of all subsets of the set of covers.
Local behaviour of solutions of stochastic integral equations
Authors: Anderson William J.
Published: July 2010

Let X denote the solution process of the stochastic equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d upper X left-parenthesis t right-parenthesis equals a left-parenthesis upper X left-parenthesis t right-parenthesis right-parenthesis d t plus sigma left-parenthesis upper X left-parenthesis t right-parenthesis right-parenthesis d upper W left-parenthesis t right-parenthesis">

d
X
(
t
)
=
a
(
X
(
t
)
)
d
t
+
σ
(
X
(
t
)
)
d
W
(
t
)

dX(t) = a(X(t))dt + \sigma (X(t))dW(t)

. In this paper, conditions on

a
(
⋅
)

a( \cdot )

and

σ
(
⋅
)

\sigma ( \cdot )

are given under which the sample paths of X are differentiate at

t
=
0

t = 0

with probability one. Variations of these results are obtained leading to a new uniqueness criterion for solutions of stochastic equations. If

σ
(
⋅
)

\sigma ( \cdot )

is Hölder continuous with exponent greater than

1
2

\tfrac {1}{2}

and

a
(
⋅
)

a( \cdot )

satisfies a Lipschitz condition, it is shown that in the one-dimensional case the above equation has only one continuous solution.
A new characterization of the 𝐹 set of a rational function
Authors: Oba Marilyn K. , Pitcher Tom S.
Published: July 2010

In the early part of this century G. Julia and P. Fatou extensively studied the iteration of functions on the complex plane. More recently Hans Brolin reopened the investigation. In this paper, we are interested in the F set which is the set of points at which the family of iterates of a given rational function R is not normal and in a measure which is in some sense naturally imposed on the F set by the iterates of R. We construct a sequence of probability measures via the inverse functions of the iterates of R and almost any starting point. The measure of primary interest is the weak limit of such sequences. These weak limits are supported by F and have certain invariance properties. We establish that this weak limit measure is unique and is ergodic with respect to the transformation R on the F set for a large class of rational functions. In the course of the proof of uniqueness we develop expressions for the logarithmic potential function and for the energy integral of F. We also establish inequalities for the capacity of the F set which become equalities for the polynomial case.
A divergence theorem for Hilbert space
Authors: Goodman Victor
Published: July 2010

Let B be a real separable Banach space. A suitable linear imbedding of a real separable Hilbert space into B with dense range determines a probability measure on B which is known as abstract Wiener measure. In this paper it is shown that certain submanifolds of B carry a surface measure uniquely defined in terms of abstract Wiener measure. In addition, an identity is obtained which relates surface integrals to abstract Wiener integrals of functions associated with vector fields on regions in B. The identity is equivalent to the classical divergence theorem if the Hilbert space is finite dimensional. This identity is used to estimate the total measure of certain surfaces, and it is established that in any space B there exist regions whose boundaries have finite surface measure.
On inverse scattering for the Klein-Gordon equation
Authors: Schonbek Tomas P.
Published: July 2010

A scattering operator

S
=
S
(
V
)

S = S(V)

is set up for the Klein-Gordon equation

◻

u
=

m
2

u
(
m
>
0
)

\square u = {m^2}u(m > 0)

perturbed by a linear potential

V
=
V
(
x
)

V = V(x)

to

◻

u
=

m
2

u
+
V
u

\square u = {m^2}u + Vu

. It is found that for each

R
>
0

R > 0

there exists a constant

c
(
R
)

c(R)

(of order

R

2
−
n

{R^{2 - n}}

as

R
→
+
∞

R \to + \infty

, n = space dimension) such that if the

L
1

{L_1}

and the

L
q

{L_q}

norm of V and

V
′

V’

are bounded by

c
(
R
)
,

V
′

−
V

c(R),V’ - V

is either nonnegative or nonpositive, and

V
′

−
V

V’ - V

is of compact support having diameter

≦
R

\leqq R

, then

S
(

V
′

)
≠
S
(
V
)

S(V’) \ne S(V)

or

V
′

=
V

V’ = V

. Here

q
>
n

/

2

q > n/2

, and

c
(
R
)

c(R)

may also depend on q.
On the equivalence of multiplicity and the generalized topological degree
Authors: O’Neil T. , Thomas J. W.
Published: July 2010

In this paper we first extend the definition of the multiplicity (as defined by J. Cronin-Scanlon) of operators of the form

I
+
C
+
T

I + C + T

to operators of the form

H
+
C
+
T

H + C + T

. We then show that the generalized topological degree (as defined by F. E. Browder and W. V. Petryshyn) of operators of the form

H
+
C
+
T

H + C + T

is also defined. Finally, we show that when both the multiplicity and generalized topological degree of

H
+
C
+
T

H + C + T

are defined, they are equal.
The finiteness of 𝐼 when 𝑅[𝑋]/𝐼 is flat
Authors: Ohm Jack , Rush David E.
Published: July 2010

Let

R
R

be a commutative ring with identity, let

X
X

be an indeterminate, and let

I
I

be an ideal of the polynomial ring

R
[
X
]

R[X]

. Let

min
I

\min I

denote the set of elements of

I
I

of minimal degree and assume henceforth that

min
I

\min I

contains a regular element. Then

R
[
X
]

/

I

R[X]/I

is a flat

R
R

-module implies

I
I

is a finitely generated ideal. Under the additional hypothesis that

R
R

is quasi-local integrally closed, the stronger conclusion that

I
I

is principal holds. (An example shows that the first statement is no longer valid when

min
I

\min I

does not contain a regular element.) Let

c
(
I
)

c(I)

denote the content ideal of

I
I

, i.e.

c
(
I
)

c(I)

is the ideal of

R
R

generated by the coefficients of the elements of

I
I

. A corollary to the above theorem asserts that

R
[
X
]

/

I

R[X]/I

is a flat

R
R

-module if and only if

I
I

is an invertible ideal of

R
[
X
]

R[X]

and

c
(
I
)
=
R

c(I) = R

. Moreover, if

R
R

is quasi-local integrally closed, then the following are equivalent: (i)

R
[
X
]

/

I

R[X]/I

is a flat

R
R

-module; (ii)

R
[
X
]

/

I

R[X]/I

is a torsion free

R
R

-module and

c
(
I
)
=
R

c(I) = R

; (iii)

I
I

is principal and

c
(
I
)
=
R

c(I) = R

. Let

ξ
\xi

denote the equivalence class of

X
X

in

R
[
X
]

/

I

R[X]/I

, and let

⟨
1
,
ξ
,
⋯
,

ξ
t

⟩

\langle 1,\xi , \cdots ,{\xi ^t}\rangle

denote the

R
R

-module generated by

1
,
ξ
,
⋯
,

ξ
t

1,\xi , \cdots ,{\xi ^t}

. The following statements are also equivalent: (i)

⟨
1
,
ξ
,
⋯
,

ξ
t

⟩

\langle 1,\xi , \cdots ,{\xi ^t}\rangle

is flat for all

t
⩾
0

t \geqslant 0

; (ii)

⟨
1
,
ξ
,
⋯
,

ξ
t

⟩

\langle 1,\xi , \cdots ,{\xi ^t}\rangle

is flat for some

t
⩾
0

t \geqslant 0

for which

1
,
ξ
,
⋯
,

ξ
t

1,\xi , \cdots ,{\xi ^t}

are linearly dependent over

R
R

; (iii)

I
=
(

f
1

,
⋯
,

f
n

)
,

f
i

∈
min
I

I = ({f_1}, \cdots ,{f_n}),{f_i} \in \min I

, and

c
(
I
)
=
R

c(I) = R

; (iv)

c
(
min
I
)
=
R

c(\min I) = R

. Moreover, if

R
R

is integrally closed, these are equivalent to

R
[
X
]

/

I

R[X]/I

being a flat

R
R

-module. A certain symmetry enters in when

ξ
\xi

is regular in

R
[
ξ
]

R[\xi ]

, and in this case (i)-(iv) are also equivalent to the assertion that

R
[
ξ
]

R[\xi ]

and

R
[
1

/

ξ
]

R[1/\xi ]

are flat

R
R

-modules.
The variation of singular cycles in an algebraic family of morphisms
Authors: Roberts Joel
Published: July 2010

(1) Let

g
:

V
r

→

W
m

(
m
≧
r
)

g:{V^r} \to {W^m}(m \geqq r)

be a morphism of nonsingular varieties over an algebraically closed field. Under certain conditions, one can define a cycle

S
i

{S_i}

on

V
V

with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S u p p left-parenthesis upper S Subscript i Baseline right-parenthesis equals StartSet x vertical-bar dimension Subscript k left-parenthesis x right-parenthesis Baseline left-parenthesis normal upper Omega Subscript upper X slash upper Y Superscript 1 Baseline right-parenthesis left-parenthesis x right-parenthesis greater-than-over-equals i EndSet">

Supp
⁡
(

S
i

)
=
{
x

|

dim

k
(
x
)

(

Ω

X

/

Y

1

)
(
x
)
≧
i
}

\operatorname {Supp} ({S_i}) = \{ x|{\dim _{k(x)}}(\Omega _{X/Y}^1)(x) \geqq i\}

. The multiplicity of a component of

S
i

{S_i}

can be computed directly from local equations for

g
g

. If

V
r

⊂

P
n

{V^r} \subset {P^n}

, and if

g
:
V
→

P
m

g:V \to {P^m}

is induced by projection from a suitable linear subspace of

P
n

{P^n}

, then

S
1

{S_1}

is

c

m
−
r
+
1

(
N
⊗

O

(
−
1
)
)

{c_{m - r + 1}}(N \otimes \mathcal {O}( - 1))

, up to rational equivalence, where

N
N

is the normal bundle of

V
V

in

P
n

{P^n}

. (2) Let

f
:
X
→
S

f:X \to S

be a smooth projective morphism of noetherian schemes, where

S
S

is connected, and the fibres of

f
f

are absolutely irreducible

r
r

-dimensional varieties. For a geometric point

η
:
Spec
⁡
(
k
)
→
S

\eta :\operatorname {Spec} (k) \to S

, and a locally free sheaf

E
E

on

X
X

, let

X
η

{X_\eta }

be the corresponding geometric fibre, and

E
η

{E_\eta }

the sheaf induced on

X
η

{X_\eta }

. If

E
1

,
…
,

E
m

{E_1}, \ldots ,{E_m}

are locally free sheaves on

X
X

, and if

i
1

+
⋯
+

i
m

=
r

{i_1} + \cdots + {i_m} = r

, then the degree of the zero-cycle

c

i
1

(

E

1
η

)
⋯

c

i
m

(

E

m
η

)

{c_{{i_1}}}({E_{1\eta }}) \cdots {c_{{i_m}}}({E_{m\eta }})

is independent of the choice of

η
\eta

. (3) The results of (1) and (2) are used to study the behavior under specialization of a closed subvariety

V
′

⊂

P

2
r
−
1

V’ \subset {P^{2r - 1}}

which is the image under generic projection of a nonsingular

V
r

⊂

P
n

{V^r} \subset {P^n}

.
A characterization of 𝑀-spaces in the class of separable simplex spaces
Authors: Gleit Alan
Published: July 2010

We show that a separable simplex space is an

M
M

-space iff the arbitrary intersection of closed ideals is always an ideal.
Abelian subgroups of finite 𝑝-groups
Authors: Dancs Susan Claire
Published: July 2010

Information is obtained about the order of maximal abelian subgroups of central powers and crown products of finite

p
p

-groups. This is used to construct groups with “small” maximal abelian subgroups.
On the ideal structure of Banach algebras
Authors: Dietrich William E.
Published: July 2010

For Banach algebras

A
A

in a class which includes all group and function algebras, we show that the family of ideals of

A
A

with the same hull is typically quite large, containing ascending and descending chains of arbitrary length through any ideal in the family, and that typically a closed ideal of

A
A

whose hull meets the Šilov boundary of

A
A

cannot be countably generated algebraically.
Applications of bundle map theory
Authors: Gottlieb Daniel Henry
Published: July 2010

This paper observes that the space of principal bundle maps into the universal bundle is contractible. This fact is added to I. M. James’ Bundle map theory which is slightly generalized here. Then applications yield new results about actions on manifolds, the evaluation map, evaluation subgroups of classifying spaces of topological groups, vector bundle injections, the Wang exact sequence, and

H
H

-spaces.
Regular overrings of regular local rings
Authors: Sally Judith
Published: July 2010

The local factorization theorem of Zariski and Abhyankar characterizes all

2
2

-dimensional regular local rings which lie between a given

2
2

-dimensional regular local ring

R
R

and its quotient field as finite quadratic transforms of

R
R

. This paper shows that every regular local ring

R
R

of dimension

n
>
2

n > 2

has infinitely many minimal regular local overrings which cannot be obtained by a monoidal transform of

R
R

. These overrings are localizations of rings generated over

R
R

by certain quotients of elements of an

R
R

-sequence. Necessary and sufficient conditions are given for this type of extension of

R
R

to be regular.
A proof that 𝒞² and 𝒯² are distinct measures
Authors: Ernst Lawrence R.
Published: July 2010

We prove that there exists a nonempty family

X
X

of subsets of

R

3

{{\text {R}}^3}

such that the two-dimensional Carathéodory measure of each member of

X
X

is less than its two-dimensional

T

\mathcal {T}

measure. Every member of

X
X

is the Cartesian product of 3 copies of a suitable Cantor type subset of

R

{\text {R}}

.
Müntz-Szasz type approximation and the angular growth of lacunary integral functions
Authors: Anderson J. M.
Published: July 2010

We consider analogues of the Müntz-Szasz theorem, as in [15] and [4], for functions regular in an angle. This yields necessary and sufficient conditions for the existence of integral functions which are bounded in an angle and have gaps of a very regular nature in their power series expansion. In the case when the gaps are not so regular, similar results hold for formal power series which converge in the angle concerned.
The exceptional subset of a 𝐶₀-contraction
Authors: Herrero Domingo A.
Published: July 2010

Let

T
T

be a

C
0

{C_0}

-operator acting on a (complex separable) Hilbert space

K

\mathcal {K}

; i.e.,

T
T

is a contraction on

K

\mathcal {K}

and it satisfies the equation

q
(
T
)
=
0

q(T) = 0

for some inner function

q
q

, where

q
(
T
)

q(T)

is defined in the sense of the functional calculus of B.Sz.-Nagy and C. Foiaş. Among all those inner functions

q
q

there exists a unique minimal function

p
p

defined by the conditions: (1)

p
(
T
)
=
0

p(T) = 0

; (2) if

q
(
T
)
=
0

q(T) = 0

, then

p
p

divides

q
q

. A vector

F
∈

K

F \in \mathcal {K}

is called exceptional if there exists an inner function

r
r

such that

r
(
T
)
F
=
0

r(T)F = 0

, but

p
p

does not divide

r
r

. The existence of nonexceptional vectors plays a very important role in the theory of

C
0

{C_0}

-operators. The main result of this paper says that nonexceptional vectors actually exist; moreover, the exceptional subset of a

C
0

{C_0}

-operator is a topologically small subset of

K

\mathcal {K}

.
Boundary links and an unlinking theorem
Authors: Gutiérrez M. A.
Published: July 2010

This paper gives a homotopic theoretic criterion for a higher dimensional link to be trivial.
Analytically invariant and bi-invariant subspaces
Authors: Herrero Domingo Antonio , Salinas Norberto
Published: July 2010

The purpose of this paper is to call attention to some interesting weakly closed algebras related to a bounded linear operator

T
T

acting on a Banach space

X

\mathfrak {X}

and their associated lattices of invariant subspaces, namely, the algebras generated by the polynomials and by the rational functions in

T
T

, and the commutant and the double-commutant of

T
T

. The relationship between those algebras and their lattices, as well as the ones corresponding to the operators induced by

T
T

on an invariant subspace (restriction), or on the quotient space

X

/

M

\mathfrak {X}/\mathfrak {M}

(where

M

\mathfrak {M}

is an invariant subspace of a given type) are analyzed. Several results relative to the decomposition of invariant subspaces and the topological structure of the lattices (under the “gap-between-subspaces” metric topology) are also considered.
Infinite games with imperfect information
Authors: Orkin Michael
Published: July 2010

We consider an infinite, two person zero sum game played as follows: On the

n
n

th move, players

A
,
B

A,B

select privately from fixed finite sets,

A
n

,

B
n

{A_n},{B_n}

, the result of their selections being made known before the next selection is made. After an infinite number of selections, a point in the associated sequence space,

Ω
\Omega

, is produced upon which

B
B

pays

A
A

an amount determined by a payoff function defined on

Ω
\Omega

. In this paper we extend a result of Blackwell and show that if the payoff function is the indicator function of a set in the Boolean algebra generated by the

G
δ

{G_\delta }

’s (with respect to a natural topology on

Ω
\Omega

) then the game in question has a value.
On the classification of simple antiflexible algebras
Authors: Bhandari Mahesh Chandra
Published: July 2010

In this paper, we begin a classification of simple totally antiflexible algebras (finite dimensional) over splitting fields of characteristic

≠
2
,
3

\ne 2,3

. For such an algebra

A
A

let

P
P

be the largest associative ideal in

A
+

{A^ + }

and let

N
N

be the radical of

P
P

. We say that

A
A

is of type

(
m
,
n
)

(m,n)

if

N
N

is nilpotent of class

m
m

with

dim
⁡
A
=
n

\dim A = n

. Define

N
i

=

N

i
−
1

⋅
N
,

N
1

=
N

{N_i} = {N_{i - 1}} \cdot N,{N_1} = N

, then

A
A

is said to be of type

(
m
,
n
,

d
1

,

d
2

,
⋯
,

d
q

)

(m,n,{d_1},{d_2}, \cdots ,{d_q})

if

A
A

is of type

(
m
,
n
)
,
dim
⁡
(

N
i

−

N

i
−
1

)
=

d
i

(m,n),\dim ({N_i} - {N_{i - 1}}) = {d_i}

for

1
⩽
i
⩽
q

1 \leqslant i \leqslant q

and

dim
⁡
(

N
i

−

N

i
+
1

)
=
1

\dim ({N_i} - {N_{i + 1}}) = 1

for

q
>
i
>
m

q > i > m

. We then determine all nodal simple totally antiflexible algebras of types

(
n
,
n
)
,
(
n
−
k
,
n
,
k
+
1
)
,
(
n
−
2
,
n
)

(n,n),(n - k,n,k + 1),(n - 2,n)

(over fields of characteristic

≠
2
,
3

\ne 2,3

) and of type (3, 6) (over the field of complex numbers). We also give preliminary results for nodal simple totally antiflexible algebras of type

(
n
−
k
,
n
,
k
,
2
)

(n - k,n,k,2)

and of type

(
m
,
n
,

d
1

,
⋯
,

d
q

)

(m,n,{d_1}, \cdots ,{d_q})

in general with

m
>
2

m > 2

(the case

m
=
2

m = 2

has been classified by D. J. Rodabaugh).
On trigonometric series associated with separable, translation invariant subspaces of 𝐿^{∞}(𝐺)
Authors: Blei Ron C.
Published: July 2010

G
G

denotes a compact abelian group, and

Γ
\Gamma

denotes its dual. Our main result is that every non-Sidon set

E
⊂
Γ

E \subset \Gamma

contains a non-Sidon set

F
F

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript upper F Superscript normal infinity Baseline left-parenthesis upper G right-parenthesis equals circled-plus 1 Subscript i equals 1 Superscript normal infinity Baseline upper C Subscript upper F Sub Subscript i Baseline left-parenthesis upper G right-parenthesis">

L
F
∞

(
G
)
=

⊕
l

1

i
=
1

∞

C

F
i

(
G
)

L_F^\infty (G) = { \oplus _l}1_{i = 1}^\infty {C_{{F_i}}}(G)

, where the

F
i

{F_i}

’s are finite, mutually disjoint, and

∪

i
=
1

∞

F
i

=
F

\cup _{i = 1}^\infty {F_i} = F

.
Wall manifolds with involution
Authors: Rowlett R. J.
Published: July 2010

Consider smooth manifolds

W
W

with involution

t
t

and a Wall structure described by a map

f
:
W
→

S
1

f:W \to {S^1}

such that

f
t
=
f

ft = f

. For such objects we define cobordism theories

W

∗
I

{\text {W}}_\ast ^I

(in case

W
W

is closed,

t
t

unrestricted),

W

∗
F

{\text {W}}_ \ast ^F

(for

W
W

closed,

t
t

fixed-point free), and

W

∗

rel

{\text {W}}_ \ast ^{{\text {rel}}}

(

W
W

with boundary,

t
t

free on

W
W

). We prove that there is an exact sequence
\[

0
→

W

∗
I

→

W

∗

rel

→

W

∗
F

→
0.

0 \to {\text {W}}_ \ast ^I \to {\text {W}}_ \ast ^{{\text {rel}}} \to {\text {W}}_ \ast ^F \to 0.

\]
As a corollary,

W

∗
I

{\text {W}}_ \ast ^I

imbeds in the cobordism of unoriented manifolds with involution. We also describe how

W

∗
I

{\text {W}}_ \ast ^I

determines the

2
2

-torsion in the cobordism of oriented manifolds with involution.
Close-to-convex multivalent functions with respect to weakly starlike functions
Authors: Styer David
Published: July 2010

It is the object of this article to define close-to-convex multivalent functions in terms of weakly starlike multivalent functions. Six classes are defined, and shown to be equal. These generalize the class of close-to-convex functions developed by Livingston in the article,

p
p

-valent close-to-convex functions, Trans. Amer. Math. Soc. 115 (1965), 161-179.
Simple groups of order 2^{𝑎}3^{𝑏}5^{𝑐}7^{𝑑}𝑝
Authors: Alex Leo J.
Published: July 2010

Let

PSL

(
n
,
q
)

{\operatorname {PSL}}(n,q)

denote the projective special linear group of degree

n
n

over

GF

(
q
)

{\text {GF}}(q)

, the field with

q
q

elements. The following theorem is proved. Theorem. Let

G
G

be a simple group of order

2
a

3
b

5
c

7
d

p
,
a
>
0
,
p

{2^a}{3^b}{5^c}{7^d}p,a > 0,p

an odd prime. If the index of a Sylow

p
p

-subgroup of

G
G

in its normalizer is two, then

G
G

is isomorphic to one of the groups,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P upper S upper L left-parenthesis 2 comma 5 right-parenthesis comma upper P upper S upper L left-parenthesis 2 comma 7 right-parenthesis comma upper P upper S upper L left-parenthesis 2 comma 9 right-parenthesis comma upper P upper S upper L left-parenthesis left-parenthesis 2 comma 8 right-parenthesis comma upper P upper S upper L left-parenthesis 2 comma 16 right-parenthesis comma upper P upper S upper L left-parenthesis 2 comma 25 right-parenthesis comma upper P upper S upper L left-parenthesis 2 comma 27 right-parenthesis comma upper P upper S upper L left-parenthesis 2 comma 81 right-parenthesis">

PSL

(
2
,
5
)
,

PSL

(
2
,
7
)
,

PSL

(
2
,
9
)
,

PSL

(
(
2
,
8
)
,

PSL

(
2
,
16
)
,

PSL

(
2
,
25
)
,

PSL

(
2
,
27
)
,

PSL

(
2
,
81
)

{\operatorname {PSL}}(2,5),{\operatorname {PSL}}(2,7),{\operatorname {PSL}}(2,9),{\operatorname {PSL}}((2,8),{\operatorname {PSL}}(2,16),{\operatorname {PSL}}(2,25),{\operatorname {PSL}}(2,27),{\operatorname {PSL}}(2,81)

, and

PSL

(
3
,
4
)

{\operatorname {PSL}}(3,4)

.
Almost recursively enumerable sets
Authors: Berry John W.
Published: July 2010

An injective function on N, the nonnegative integers, taking values in N, is called almost recursive (abbreviated a.r.) if its inverse has a partial recursive extension. The range of an a.r. function f is called an almost recursively enumerable set in general; an almost recursive set if in addition f is strictly increasing. These are natural generalizations of regressive and retraceable sets respectively. We show that an infinite set is almost recursively enumerable iff it is point decomposable in the sense of McLaughlin. This leads us to new characterizations of certain classes of immune sets. Finally, in contrast to the regressive case, we show that a.r. functions and sets are rather badly behaved with respect to recursive equivalence.
On the finitely generated subgroups of an amalgamated product of two groups
Authors: Burns R. G.
Published: July 2010

Sufficient conditions are found for the free product

G
G

of two groups

A
A

and

B
B

with an amalgamated subgroup

U
U

to have the properties (1) that the intersection of each pair of finitely generated subgroups of

G
G

is again finitely generated, and (2) that every finitely generated subgroup containing a nontrivial subnormal subgroup of

G
G

has finite index in

G
G

. The known results that Fuchsian groups and free products (under the obvious conditions on the factors) have properties (1) and (2) follow as instances of the main result.
Symmetrization of distributions and its application. II. Liouville type problem in convolution equations
Authors: Chen Kuang-ho
Published: July 2010

The symmetrization of distributions corresponding to a bounded

n
−
1

n - 1

dimensional

C
∞

{C^\infty }

-submanifold of a

C
∞

{C^\infty }

-manifold is constructed. This device reduces the consideration of distributions in

R
n

{R^n}

to the one of distributions in

R
1

{R^1}

, i.e. the symmetrized distributions. Using the relation between the inverse Fourier transform of a symmetrized distribution and the one of the original (nonsymmetrized) distribution, we determine the rate of decay at infinity of solutions to a general convolution equation necessary to assure uniqueness. Using a result in the division problem for distributions, we achieve the following result: If

u
∈
C
(

R
n

)

u \in C({R^n})

is a solution of the convolution equation

S
∗
u
=
f
,
f
∈

D

(

R
n

)

S \ast u = f,f \in \mathcal {D}({R^n})

, with some suitable

S
∈

E

′

(

R
n

)

S \in \mathcal {E}’({R^n})

, then

u
∈

D

(

R
n

)

u \in \mathcal {D}({R^n})

, provided

u
u

decays sufficiently fast at infinity.
Algebraic extensions of difference fields
Authors: Evanovich Peter
Published: July 2010

An inversive difference field

K

\mathcal {K}

is a field K together with a finite number of automorphisms of K. This paper studies inversive extensions of inversive difference fields whose underlying field extensions are separable algebraic. The principal tool in our investigations is a Galois theory, first developed by A. E. Babbitt, Jr. for finite dimensional extensions of ordinary difference fields and extended in this work to partial difference field extensions whose underlying field extensions are infinite dimensional Galois. It is shown that if

L

\mathcal {L}

is a finitely generated separable algebraic inversive extension of an inversive partial difference field

K

\mathcal {K}

and the automorphisms of

K

\mathcal {K}

commute on the underlying field of

K

\mathcal {K}

then every inversive subextension of

L

/

K

\mathcal {L}/\mathcal {K}

is finitely generated. For ordinary difference fields the paper makes a study of the structure of benign extensions, the group of difference automorphisms of a difference field extension, and two types of extensions which play a significant role in the study of difference algebra: monadic extensions (difference field extensions

L

/

K

\mathcal {L}/\mathcal {K}

having at most one difference isomorphism into any extension of

K

\mathcal {K}

) and incompatible extensions (extensions

L

/

K

,

M

/

K

\mathcal {L}/\mathcal {K},\mathcal {M}/\mathcal {K}

having no difference field compositum).
On the zeros of power series with Hadamard gaps-distribution in sectors
Authors: Chang I Lok
Published: July 2010

We give a sufficient condition for a power series with Hadamard gaps to assume every complex value infinitely often in every sector of the unit disk.
On fibering of cobordism classes
Authors: Stong R. E.
Published: July 2010

This paper studies the problem posed by Conner and Floyd of finding which cobordism classes are represented by the total space of a fibering with prescribed base or fiber.
On embedding set functions into covariance functions
Authors: Allen G. D.
Published: July 2010

We consider any continuous hermitian kernel

M
(
Δ
,

Δ
′

)

M(\Delta ,\Delta ’)

on

P

×

P

\mathcal {P} \times \mathcal {P}

where

P

\mathcal {P}

is the prering of intervals of [0,1]. Conditions on M are given to find an interval covariance function

K
(
Δ
,

Δ
′

)

K(\Delta ,\Delta ’)

so that

K
(
Δ
,

Δ
′

)
=
M
(
Δ
,

Δ
′

)

K(\Delta ,\Delta ’) = M(\Delta ,\Delta ’)

for all nonoverlapping

Δ
\Delta

and

Δ
′

\Delta ’

in

P

\mathcal {P}

. The problem is solved by first treating finite hermitian matrices A and finding a positive definite matrix B so that

b

i
j

=

a

i
j

,
i
≠
j

{b_{ij}} = {a_{ij}},i \ne j

, so that tr B is minimized. Using natural correspondence between interval covariance functions and stochastic processes, a decomposition theorem is derived for stochastic processes of bounded quadratic variation into an orthogonal process and a process having minimal quadratic variation.
On the character of Weil’s representation
Authors: Howe Roger E.
Published: July 2010

The importance of certain representations of symplectic groups, usually called Weil representations, for the general problem of finding representations of certain group extensions is made explicit. Some properties of the character of Weil’s representation for a finite symplectic group are given and discussed, again in the context of finding representations of group extensions. As a by-product, the structure of anisotropic tori in symplectic groups is given.
Comparison of eigenvalues for linear differential equations of order 2𝑛
Authors: Travis Curtis C.
Published: July 2010

An abstract eigenvalue comparison theorem is proven for

u
0

{u_0}

-positive linear operators in a Banach space equippped with a cone of “nonnegative” elements. This result is then applied to certain linear differential equations of order 2n in order to obtain eigenvalue comparison theorems of an “integral type."
Closed countably generated structures in 𝐶(𝑋)
Authors: Roth B.
Published: July 2010

Let

C
(
X
)

C(X)

be the space of continuous real or complex valued functions on a compact space X with the sup norm topology. In the present paper, the subalgebras, vector lattices, and vector lattice ideals of

C
(
X
)

C(X)

which are closed and countably generated are characterized.
Higher derivations and field extensions
Authors: Davis R. L.
Published: July 2010

Let

K
K

be a field having prime characteristic

p
p

. The following conditions on a subfield

k
k

of

K
K

are equivalent: (i)

∩
n

K

p
n

(
k
)
=
k

{ \cap _n}{K^{{p^n}}}(k) = k

and

K

/

k

K/k

is separable. (ii)

k
k

is the field of constants of an infinite higher derivation defined in

K
K

. (iii)

k
k

is the field of constants of a set of infinite higher derivations defined in

K
K

. If

K

/

k

K/k

is separably generated and

k
k

is algebraically closed in

K
K

, then

k
k

is the field of constants of an infinite higher derivation in

K
K

. If

K

/

k

K/k

is finitely generated then

k
k

is the field of constants of an infinite higher derivation in

K
K

if and only if

K

/

k

K/k

is regular.
On Lagrangian groups
Authors: Humphreys J. F. , Johnson D. L.
Published: July 2010

We study the class

L

\mathcal {L}

of Lagrangian groups, that is, of finite groups

G
G

possessing a subgroup of index

n
n

for each factor

n
n

of

|

G

|

|G|

. These groups and their analogues were considered by McLain in [4] and the object of the present work is to extend the results in this article. We study the classes

(
G
)
=
{
H

|

G
×
H
∈

L

}

(G) = \{ H|G \times H \in \mathcal {L}\}

and also the closure of

L

\mathcal {L}

under wreath products. We also consider the two classes

X

\mathfrak {X}

and

Y

\mathfrak {Y}

introduced in [2] and [4] respectively.
Alternating Chebyshev approximation
Authors: Dunham Charles B.
Published: July 2010

An approximating family is called alternating if a best Chebyshev approximation is characterized by its error curve having a certain number of alternations. The convergence properties of such families are studied. A sufficient condition for the limit of best approximation on subsets to converge uniformly to the best approximation is given: it is shown that this is often (but not always) a necessary condition. A sufficient condition for the Chebyshev operator to be continuous is given: it is shown that this is often (but not always) a necessary condition.
On the Arens products and certain Banach algebras
Authors: Wong Pak Ken
Published: July 2010

In this paper, we study several problems in Banach algebras concerned with the Arens products.
Elementary properties of free groups
Authors: Sacerdote George S.
Published: July 2010

In this paper we show that several classes of elementary properties (properties definable by sentences of a first order logic) of groups hold for all nonabelian free groups. These results are obtained by examining special embeddings of these groups into one another which preserve the properties in question.
Wreath products and representations of degree one or two
Authors: Bateman J. M. , Phillips Richard E. , Sonneborn L. M.
Published: July 2010

S

2

{\mathcal {S}_2}

denotes all groups

G
G

that possess an ascending invariant series whose factors are one- or two-generated Abelian groups. We are interested in the ptoblem (1): For which nontrivial groups

A
A

and

B
B

is

A
A

wr

B
B

in

S

2

?

{\mathcal {S}_2}?

(1) has been completely solved by D. Parker in the case where

A
A

and

B
B

are finite of odd order. Parker’s results are partially extended here to cover groups of even order. Our answer to (1) is complete in the case where

A
A

is a finite

2
2

-group: If

A
A

is a finite

2
2

-group,

A
A

wr

B
B

is in

S

2

{\mathcal {S}_2}

iff

B
B

is finite and

B

/

O
2

(
B
)

B/{O_2}(B)

is isomorphic to a subgroup of a dihedral group of an elementary

3
3

-group. If

A
A

is not a

2
2

-group, we offer only necessary conditions on

B
B

. Problem (1) is closely related to Problem (2): If

F
F

is a prime field or the integers, which finite groups

B
B

have all their irreducible representations over

F
F

of degrees one or two? It is shown that all finite

B
B

which satisfy (2) are

S

2

{\mathcal {S}_2}

groups; in particular all such

B
B

are solvable.
On the convergence of best uniform deviations
Authors: Poreda S. J.
Published: July 2010

If a function f is continuous on a closed Jordan curve

Γ
\Gamma

and meromorphic inside

Γ
\Gamma

, then the polynomials of best uniform approximation to f on

Γ
\Gamma

converge interior to

Γ
\Gamma

. Furthermore, the limit function can in each case be explicitly determined in terms of the mapping function for the interior of

Γ
\Gamma

. Applications and generalizations of this result are also given.
Interpolation between consecutive conjugate points of an 𝑛th order linear differential equation
Authors: Gustafson G. B.
Published: July 2010

The interpolation problem

x

(
n
)

+

P

n
−
1

x

(
n
−
1
)

+
⋯
+

P
0

x
=
0

{x^{(n)}} + {P_{n - 1}}{x^{(n - 1)}} + \cdots + {P_0}x = 0

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript left-parenthesis i right-parenthesis Baseline left-parenthesis t Subscript j Baseline right-parenthesis equals 0 comma i equals 0 comma midline-horizontal-ellipsis comma k Subscript j Baseline minus 1 comma j equals 0 comma midline-horizontal-ellipsis comma m">

x

(
i
)

(

t
j

)
=
0
,
i
=
0
,
⋯
,

k
j

−
1
,
j
=
0
,
⋯
,
m

{x^{(i)}}({t_j}) = 0,i = 0, \cdots ,{k_j} - 1,j = 0, \cdots ,m

, is studied on the conjugate interval

[
a
,

η
1

(
a
)
]

[a,{\eta _1}(a)]

. The main result is that there exists an essentially unique nontrivial solution of the problem almost everywhere, provided

k
1

+
⋯
+

k
m

≥
n

{k_1} + \cdots + {k_m} \geq n

, and cer tain other inequalities are satisfied, with

a
=

t
0

>

t
1

>
⋯
>

t
m

=

η
1

(
a
)

a = {t_0} > {t_1} > \cdots > {t_m} = {\eta _1}(a)

. In particular, this paper corrects the results of Azbelev and Caljuk (Mat. Sb. 51 (93) (1960), 475-486; English transl., Amer. Math. Soc. Transl. (2) 42 (1964), 233-245) on third order equations, and shows that their results are correct almost everywhere.
Primary ideals in rings of analytic functions
Authors: Williams R. Douglas
Published: July 2010

Let A be the ring of all analytic functions on a connected, noncompact Riemann surface. We use the valuation theory of the ring A as developed by N. L. Alling to analyze the structure of the primary ideals of A. We characterize the upper and lower primary ideals of A and prove that every nonprime primary ideal of A is either an upper or a lower primary ideal. In addition we give some necessary and sufficient conditions for certain ideals of A to be intersections of primary ideals.
Perfect mappings and certain interior images of 𝑀-spaces
Authors: Worrell J. M. , Wicke H. H.
Published: July 2010

The main theorems of this paper show that certain conditions (called

λ
c

,

λ
b

,

β
c

{\lambda _c},{\lambda _b},{\beta _c}

, and

β
b

{\beta _b}

) are invariant, in the presence of

T
0

{T_0}

-regularity, under the application of closed continuous peripherally compact mappings. Interest in these conditions lies in the fact that they may be used to characterize certain regular

T
0

{T_0}

open continuous images of some classes of

M
M

-spaces in the sense of K. Morita, and in the fact that they are preserved by open continuous mappings with certain appropriate additional conditions. For example, the authors have shown that a regular

T
0

{T_0}

-space is an open continuous image of a paracompact Čech complete space if and only if the space satisfies condition

λ
b

{\lambda _b}

[Pacific J. Math. 37 (1971), 265-275]. Moreover, in the same paper it is shown that if a completely regular

T
0

{T_0}

-space satisfies condition

λ
b

{\lambda _b}

then any

T
0

{T_0}

completely regular open continuous image of it also satisfies

λ
b

{\lambda _b}

. These results together with the results of the present paper and certain known results lead to the following theorem: The smallest subclass of the class of regular

T
0

{T_0}

-spaces which contains all paracompact Čech complete spaces and which is closed with respect both to the application of perfect mappings and to the application of open continuous mappings preserving

T
0

{T_0}

-regularity is the subclass satisfying condition

λ
b

{\lambda _b}

. Similar results are obtained for the regular

T
0

{T_0}

-spaces satisfying

λ
c

,

β
b

{\lambda _c},{\beta _b}

, and

β
c

{\beta _c}

. The other classes of

M
M

-spaces involved are the regular

T
0

{T_0}

complete

M
M

-spaces (i.e., spaces which are quasi-perfect preimages of complete metric spaces),

T
2

{T_2}

paracompact

M
M

-spaces, and regular

T
0

M

{T_0}M

-spaces. In the last two cases besides the inferiority of the mappings the notion of uniform

λ
\lambda

-completeness, which generalizes compactness of a mapping, enters. (For details see General Topology and Appl. 1 (1971), 85-100.) The proofs are accomplished through the use of two basic lemmas on closed continuous mappings satisfying certain additional conditions.
Exit properties of stochastic processes with stationary independent increments
Authors: Millar P. W.
Published: July 2010

Let

{

X
t

,
t
≥
0
}

\{ {X_t},t \geq 0\}

be a real stochastic process with stationary independent increments. For

x
>
0

x > 0

, define the exit time

T
x

{T_x}

from the interval

(
−
∞
,
x
]

( - \infty ,x]

by

T
x

=
inf
{
t
>
0
:

X
t

>
x
}

{T_x} = \inf \{ t > 0:{X_t} > x\}

. A reasonably complete solution is given to the problem of deciding precisely when

P
0

{

X

T
x

=
x
}
>
0

{P^0}\{ {X_{{T_x}}} = x\} > 0

and precisely when

P
0

{

X

T
x

=
x
}
=
0

{P^0}\{ {X_{{T_x}}} = x\} = 0

. The solution is given in terms of parameters appearing in the Lévy formula for the characteristic function of

X
t

{X_t}

. A few applications of this result are discussed.