bluuebunny/crossref_metadata_2025 huggingface dataset:
The homological dimensions of symmetric algebras
Authors: Carrig James E.
Published: July 2010

Let D be a Dedekind domain and M a rank-one torsion-free D-module. An analysis of

A
=

S
D

(
M
)

A = {S_D}(M)

, the symmetric algebra of M, yields the following information: Theorem. (1) Tor-dim

A
⩽
2

a
n
d

=
1

i
f
f

M
=
K

A \leqslant 2\;and\; = 1\;iff\;M = K

, the quotient field of D; (2) A is coherent; (3) Global

dim
⁡
A
=
2

\dim A = 2

. For higher rank modules coherence is not assured and only rough estimates of the dimensions are found. On the other hand, if

S
D

(
M
)

{S_D}(M)

is a domain of global dimension two, then M has rank one but the dimension of D may be two. If D is local of dimension two then

M
=
K

M = K

.
Spectral theory for contraction semigroups on Hilbert space
Authors: Gearhart Larry
Published: July 2010

In this paper we determine the relationship between the spectra of a continuous contraction semigroup on Hilbert space and properties of the resolvent of its infinitesimal generator. The methods rely heavily on dilation theory. In particular, we reduce the general problem to the case that the cogenerator of the semigroup has a characteristic function with unitary boundary values. We then complete the analysis by generalizing the scalar result of J. W. Moeller on compressions of the translation semigroup to the case of infinite multiplicity.
On the number of real zeros of a random trigonometric polynomial
Authors: Sambandham M.
Published: July 2010

For the random trigonometric polynomial
\[

∑

n
=
1

N

g
n

(
t
)
cos
⁡
n
θ
,

\sum \limits _{n = 1}^N {{g_n}(t)\cos n\theta ,}

\]
where

g
n

(
t
)
,
0
⩽
t
⩽
1

{g_n}(t),0 \leqslant t \leqslant 1

, are dependent normal random variables with mean zero, variance one and joint density function
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper M EndAbsoluteValue Superscript 1 slash 2 Baseline left-parenthesis 2 pi right-parenthesis Superscript negative upper N slash 2 Baseline exp left-bracket minus left-parenthesis 1 slash 2 right-parenthesis a overbar prime upper M a overbar right-bracket">

|

M

|

1

/

2

(
2
π

)

−
N

/

2

exp
⁡
[
−
(
1

/

2
)

a
¯

′

M

a
¯

]

|M{|^{1/2}}{(2\pi )^{ - N/2}}\exp [ - (1/2)\bar a’M\bar a]

\]
where

M

−
1

{M^{ - 1}}

is the moment matrix with

ρ

i
j

=
ρ
,
0
>
ρ
>
1
,
i
≠
j
,
i
,
j
=
1
,
2
,
…
,
N

{\rho _{ij}} = \rho ,0 > \rho > 1,i \ne j,i,j = 1,2, \ldots ,N

and

a
¯

\bar a

is the column vector, we estimate the probable number of zeros.
Mesures invariantes sur les hypergroupes
Authors: Spector R.
Published: July 2010

The notion of an invariant or relatively invariant measure on a hypergroup is studied. The main result is that every commutative hypergroup carries an invariant measure.
On a degenerate principal series of representations of 𝑈(2,2)
Authors: Hua Yang
Published: July 2010

A degenerate principal series of representations

T
(
ρ
,
m
;
⋅
)
,
(
ρ
,
m
)
∈

R

×

Z

T(\rho ,m; \cdot ),(\rho ,m) \in {\mathbf {R}} \times {\mathbf {Z}}

, of

U
(
2
,
2
)

U(2,2)

, is realized on the Hilbert space of all square integrable functions on the space X of

2
×
2

2 \times 2

Hermitian matrices. Using Fourier analysis, gamma functions, and Mellin analysis, we spectrally analyze the operator equation

A
T
(
ρ
,
m
;
g
)
=
T
(
ρ
,
m
;
g
)
A

AT(\rho ,m;g) = T(\rho ,m;g)A

for all

g
∈

G

=
U
(
2
,
2
)

g \in \mathfrak {G} = U(2,2)

on an invariant subspace of

L
2

(
X
)

{L^2}(X)

, and obtain the first main result: For

ρ
≠
0

\rho \ne 0

or m odd,

T
(
ρ
,
m
;
⋅
)

T(\rho ,m; \cdot )

is irreducible. Then we define certain integral transforms on

L
2

(
X
)

{L^2}(X)

the analytic continuation of which leads to the second main result:

T
(
0
,
2
n
;
⋅
)

T(0,2n; \cdot )

is reducible.
Invariant measures and equilibrium states for some mappings which expand distances
Authors: Walters Peter
Published: July 2010

For a certain collection of transformations T we define a Perron-Frobenius operator and prove a convergence theorem for the powers of the operator along the lines of the theorem D. Ruelle proved in his investigation of the equilibrium states of one-dimensional lattice systems. We use the convergence theorem to study the existence and ergodic properties of equilibrium states for T and also to study the problem of invariant measures for T. Examples of the transformations T considered are expanding maps, transformations arising from f-expansions and shift systems.
The Riemann hypothesis for Selberg’s zeta-function and the asymptotic behavior of eigenvalues of the Laplace operator
Authors: Randol Burton
Published: July 2010

Much of that part of the theory of the Riemann zeta-function based on the Riemann hypothesis carries over to zeta-functions of Selberg’s type, and in this way one can get asymptotic information about various eigenvalue problems. The methods are illustrated in the case of a compact Riemann surface.
Desarguesian Klingenberg planes
Authors: Bacon P. Y.
Published: July 2010

Klingenberg planes are generalizations of Hjelmslev planes. If R is a local ring, one can construct a projective Klingenberg plane

V

(
R
)

{\textbf {V}}(R)

and a derived affine Klingenberg plane

A

(
R
)

{\textbf {A}}(R)

from R. If V is a projective Klingenberg plane, if

R
1

,

R
2

{R_1},\,{R_2}

and

R
3

{R_3}

are local rings, if

s
1

,

s
2

{s_1},\,{s_2}

and

s
3

{s_3}

are the sides of a nondegenerate triangle in V, and if each of the derived affine Klingenberg planes

a

(

V
,

s
i

)

\mathcal {a}\left ( {V,\,{s_i}} \right )

is isomorphic to

A

(

R
i

)
,

{\textbf {A}}({R_i}),\,

,

i

=

1
,

2
,

3

i\, = \,1,\,2,\,3

, then the rings

R
1

,

R
2

{R_1},\,{R_2}

and

R
3

{R_3}

are isomorphic, and V is isomorphic to

V

(

R
1

)
;

{\textbf {V}}({R_1});

; also, if g is a line of V, then the derived affine Klingenberg plane

a

(

V
,

g

)

\mathcal {a}({V,\,g})

is isomorphic to

A

(

R
1

)

\textbf {A}({R_1})

. Examples are given of projective Klingenberg planes V, each of which has the following two properties: (1) V is not isomorphic to

V

(
R
)

{\textbf {V}}(R)

for any local ring R; and (2) there is a flag

(
B
,

b
)

(B,\,b)

of V, and a local ring S such that each derived affine Klingenberg plane

a

(

V
,

m

)

\mathcal {a}({V,\,m})

is isomorphic to

A

(
S
)

{\textbf {A}}(S)

whenever

m

=

b

m\, = \,b

, or m is a line through B which is not neighbor to b.
Nearnesses, proximities, and 𝑇₁-compactifications
Authors: Reed Ellen E.
Published: July 2010

Gagrat, Naimpally, and Thron together have shown that separated Lodato proximities yield

T
1

{T_1}

-compactifications, and conversely. This correspondence is not

1
−
1

1 - 1

, since nonequivalent compactifications can induce the same proximity. Herrlich has shown that if the concept of proximity is replaced by that of nearness then all principal (or strict)

T
1

{T_1}

-extensions can be accounted for. (In general there are many nearnesses compatible with a given proximity.) In this paper we obtain a 1-1 correspondence between principal

T
1

{T_1}

-extensions and cluster-generated nearnesses. This specializes to a 1-1 match between principal

T
1

{T_1}

-compactifications and contigual nearnesses. These results are utilized to obtain a 1-1 correspondence between Lodato proximities and a subclass of

T
1

{T_1}

-compactifications. Each proximity has a largest compatible nearness, which is contigual. The extension induced by this nearness is the construction of Gagrat and Naimpally and is characterized by the property that the dual of each clan converges. Hence we obtain a 1-1 match between Lodato proximities and clan-complete principal

T
1

{T_1}

-compactifications. When restricted to EF-proximities, this correspondence yields the usual map between

T
2

{T_2}

-compactifications and EF-proximities.
Conditionally compact semitopological one-parameter inverse semigroups of partial isometries
Authors: Bertman M. O.
Published: July 2010

The algebraic structure of one-parameter inverse semigroups has been completely described. Furthermore, if B is the bicyclic semigroup and if B is contained in any semitopological semigroup, the relative topology on B is discrete. We show that if F is an inverse semigroup generated by an element and its inverse, and F is contained in a compact semitopological semigroup, then the relative topology is discrete; in fact, if F is any one-parameter inverse semigroup contained in a compact semitopological semigroup, then the multiplication on F is jointly continuous if and only if the inversion is continuous on F, and we describe

F
¯

\bar F

in that case. We also show that if

{

J
t

}

\{ {J_t}\}

is a one-parameter semigroup of bounded linear operators on a (separable) Hilbert space, then

{

J
t

}
∪
{

J
t
∗

}

\{ {J_t}\} \cup \{ J_t^\ast \}

generates a one-parameter inverse semigroup T with

J
t

−
1

=

J
t
∗

J_t^{ - 1} = J_t^\ast

if and only if

{

J
t

}

\{ {J_t}\}

is a one-parameter semigroup of partial isometries, and we describe the weak operator closure of T in that case.
The traces of holomorphic functions on real submanifolds
Authors: Harris Gary Alvin
Published: July 2010

Suppose M is a real-analytic submanifold of complex Euclidean n = space and consider the following question: Given a real-analytic function f defined on M, is f the restriction to M of an ambient holomorphic function? If M is a C.R. submanifold the question has been answered completely. Namely, f is the trace of a holomorphic function if and only if f is a C.R. function. The more general situation in which M need not be a C.R. submanifold is discussed in this paper. A complete answer is obtained in case the dimension of M is larger than or equal to n and M is generic in some neighborhood of each point off its C.R. singularities. The solution is of infinite order and follows from a consideration of the following problem: Given a holomorphic function f and a holomorphic mapping

Φ
\Phi

, when does there exist a holomorphic mapping F such that

f
=
F
∘
Φ

f = F \circ \Phi

?
Uniqueness in the Cauchy problem for parabolic equations
Authors: Hayne Roger M.
Published: July 2010

In a classical paper S. Täcklind (Nova Acta Soc. Sci. Upsaliensis (4) 10 (1936), 1-57) closed the uniqueness question for the Cauchy problem for the heat equation with a general growth hypothesis which was both necessary and sufficient. Täcklind’s proof of the sufficiency involved an ingenious bootstrapping comparison technique employing the maximum principle and a comparison function constructed from the Green’s function for a half cylinder. G. N. Zolotarev (Izv. Vysš. Učebn. Zaved. Matematika 2 (1958), 118-135) has extended this result using essentially the same technique to show that Täcklind’s uniqueness condition remains sufficient for a general second order parabolic equation provided the coefficients are regular enough to permit the existence and estimation of a Green’s function. We have now shown, using a new approach which replaces the construction based upon a Green’s function by an appropriate comparison solution of the maximizing equation (C. Pucci, Ann. Mat. Pura Appl. 72 (1966), 141-170), that Täcklind’s condition is sufficient without any regularity conditions on the coefficients.
The BP homology of 𝐻-spaces
Authors: Kane Richard
Published: July 2010

The BP homology of 1-connected H-spaces whose loop space is torsion free is studied. It is shown that localizing in a suitable manner kills all torsion in

B

P

∗

(
X
)

B{P_{\ast }}(X)

. Similar results are then obtained for the bordism of X. Finally the K-theory of X is shown to have no p torsion.
Limiting distributions for branching random fields
Authors: Fleischman Joseph
Published: July 2010

In this paper we derive limiting distributions for branching Brownian motion. The cases considered are where the state space is (1) the line and (2) the plane where (a) initially there’s but one particle and (b) where there’s initially a random number of independent particles. In all cases, the branching process is critical and we obtain results for the growth of selectively neutral mutant types. We use moment generating functions to derive these results.
Weak uniqueness sets for discrete groups
Authors: Bożejko Marek , Pytlik Tadeusz
Published: July 2010

For discrete groups we introduce a new class of sets, called weak uniqueness sets, which for abelian groups contains the class of sets of uniqueness. Considered is the problem of determining groups for which every finite set is a weak uniqueness set. Some examples are given.
On the growth of the integral means of subharmonic functions of order less than one
Authors: Abi-Khuzam Faruk F.
Published: July 2010

Let u be a subharmonic function of order

λ
(
0
>
λ
>
1
)

\lambda (0 > \lambda > 1)

, and let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m Subscript s Baseline left-parenthesis r comma u right-parenthesis equals left-brace left-parenthesis 1 slash 2 pi right-parenthesis integral Subscript negative pi Superscript pi Baseline StartAbsoluteValue u left-parenthesis r e Superscript i theta Baseline right-parenthesis EndAbsoluteValue Superscript s Baseline d theta right-brace Superscript 1 slash s">

m
s

(
r
,
u
)

=

{

(
1

/

2
π
)

∫

−
π

π

|

u
(
r

e

i
θ

)

|

s

d
θ

}

1

/

s

{m_s}(r,u)\, = \,{\left \{ {(1/2\pi )\int _{ - \pi }^\pi {{{\left | {u(r{e^{i\theta }})} \right |}^s}} d\theta } \right \}^{1/s}}

. We compare the growth of

m
s

(
r
,
u
)

{m_s}(r,u)

with that of the Riesz mass of u as measured by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N left-parenthesis r comma u right-parenthesis equals left-parenthesis 1 slash 2 pi right-parenthesis integral Subscript negative pi Superscript pi Baseline u left-parenthesis r e Superscript i theta Baseline right-parenthesis d theta">

N

(
r
,
u
)

=
(
1

/

2
π
)

∫

−
π

π

u
(
r

e

i
θ

)
d
θ

N\,(r,u)\, = (1/2\pi )\int _{ - \pi }^\pi {u(r{e^{i\theta }})d\theta }

. A typical result of this paper states that the following inequality is sharp:
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk right-parenthesis EndLabel limit inf Underscript x right-arrow normal infinity Endscripts StartFraction m Subscript s Baseline left-parenthesis r comma u right-parenthesis Over upper N left-parenthesis r comma u right-parenthesis EndFraction less-than-or-slanted-equals m Subscript s Baseline left-parenthesis psi Subscript lamda Baseline right-parenthesis EndLayout">

(

∗

)

lim

inf

x
→
∞

m

s

(
r
,

u
)

N

(
r
,

u
)

⩽

m

s

(

ψ

λ

)

\begin{equation} \tag {$\ast $} \underset {x\to \infty }{\lim \,\inf }\,\,\frac {{{m}_{s}}\left ( r,\,u \right )}{N\left ( r,\,u \right )}\,\leqslant \,{{m}_{s}}\left ( {{\psi }_{\lambda }} \right )\end{equation}

where

ψ
λ

(
θ
)

=

(
π
λ

/

sin

λ
)
cos

λ
θ

\psi _\lambda (\theta )\, = \,(\pi \lambda /\sin \,\lambda )\cos \,\lambda \theta

. The case

s

=

1

s\, = \,1

is due to Edrei and Fuchs, the case

s

=

2

s\, = \,2

is due to Miles and Shea and the case

s

=

∞

s\, = \,\infty

is due to Valiron.
Periodic solutions for a differential equation in Banach space
Authors: Lightbourne James H.
Published: July 2010

Suppose X is a Banach space,

Ω
⊂
X

\Omega \subset X

is closed and convex, and

A
:
[
0
,
∞
)
×
Ω
→
X

A:[0,\infty ) \times \Omega \to X

is continuous. Then if
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript h right-arrow 0 Endscripts StartAbsoluteValue x plus h upper A left-parenthesis t comma x right-parenthesis semicolon normal upper Omega EndAbsoluteValue slash h equals 0 for all left-parenthesis t comma x right-parenthesis element-of left-bracket 0 comma normal infinity right-parenthesis times normal upper Omega comma">

lim

h
→
0

|

x
+
h
A
(
t
,
x
)
;
Ω

|

/

h
=
0

for

all

(
t
,
x
)
∈
[
0
,
∞
)
×
Ω
,

\lim \limits _{h \to 0} |x + hA(t,x);\Omega |/h = 0\quad {\text {for}}\;{\text {all}}\;(t,x) \in [0,\infty ) \times \Omega ,

\]
there exist approximate solutions to the initial value problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis upper I upper V upper P right-parenthesis EndLabel u prime left-parenthesis t right-parenthesis equals upper A left-parenthesis t comma u left-parenthesis t right-parenthesis right-parenthesis comma u left-parenthesis 0 right-parenthesis equals x element-of normal upper Omega period EndLayout">

(

I
V
P

)

u
′

(
t
)
=
A
(
t
,
u
(
t
)
)
,

u
(
0
)
=
x
∈
Ω
.

\begin{equation}\tag {$IVP$}u’(t) = A(t,u(t)),\quad u(0) = x \in \Omega .\end{equation}

In the case that

A
(
t
,
x
)
=
B
(
t
,
x
)
+
C
(
t
,
x
)

A(t,x) = B(t,x) + C(t,x)

, where B satisfies a dissipative condition and C is compact, we obtain a growth estimate on the measure of noncompactness of trajectories for a class of approximate solutions. This estimate is employed to obtain existence of periodic solutions to (IVP).
A Hopf global bifurcation theorem for retarded functional differential equations
Authors: Nussbaum Roger D.
Published: July 2010

We prove a result concerning the global nature of the set of periodic solutions of certain retarded functional differential equations. Our main theorem is an analogue, for retarded F.D.E.’s, of a result by J. Alexander and J. Yorke for ordinary differential equations.
The cohomology of semisimple Lie algebras with coefficients in a Verma module
Authors: Williams Floyd L.
Published: July 2010

The structure of the cohomology of a complex semisimple Lie algebra with coefficients in an arbitrary Verma module is completely determined. Because the Verma modules are infinite-dimensional, the cohomology need not vanish (as it does for nontrivial finite-dimensional modules). The methods presented exploit the homological machinery of Cartan-Eilenberg [3]. The results of [3], when applied to the universal enveloping algebra of a semisimple Lie algebra and when coupled with key results of Kostant [12], Hochschild-Serre [9], yield the basic structure theorem-Theorem 4.19. Our results show, incidently, that an assertion of H. Kimura, Theorem 2 of [13] is false. A counterexample is presented in §6.
Unitary representations of Lie groups with cocompact radical and applications
Authors: Pukanszky L.
Published: July 2010

The paper gives a necessary and sufficient condition in order that a connected and simply connnected Lie group with cocompact radical be of type I. This result is then applied to a characterization of Lie groups, all irreducible unitary representations of which are completely continuous.
Hypoellipticity on the Heisenberg group-representation-theoretic criteria
Authors: Rockland Charles
Published: July 2010

A representation-theoretic characterization is given for hypoellipticity of homogeneous (with respect to dilations) left-invariant differential operators P on the Heisenberg group

H
n

{H_n}

; it is the precise analogue for

H
n

{H_n}

of the statement for

R

n

{{\mathbf {R}}^n}

that a homogeneous constant-coefficient differential operator is hypoelliptic if and only if it is elliptic. Under these representation-theoretic conditions a parametrix is constructed for P by means of the Plancherel formula. However, these conditions involve all the irreducible representations of

H
n

{H_n}

, whereas only the generic, infinite-dimensional representations occur in the Plancherel formula. A simple class of examples is discussed, namely

P
=

Σ

i
=
1

n

X
i

2
m

+

Y
i

2
m

P = \Sigma _{i = 1}^nX_i^{2m} + Y_i^{2m}

, where

X
i

,

Y
i

,
i
=
1
,
…
,
n

{X_i},{Y_i},i = 1, \ldots ,n

, and Z generate the Lie algebra of

H
n

{H_n}

via the commutation relations

[

X
i

,

Y
j

]
=

δ

i
j

Z

[{X_i},{Y_j}] = {\delta _{ij}}Z

, and where m is a positive integer. In the course of the proof a connection is made between homogeneous left-invariant operators on

H
n

{H_n}

and a class of degenerate-elliptic operators on

R

n
+
1

{{\mathbf {R}}^{n + 1}}

studied by Grušin. This connection is examined in the context of localization in enveloping algebras.
Distances between two-state Markov processes attainable by Markov joinings
Authors: Ellis Martin H.
Published: July 2010

The function which assigns to each pair of two-state Markov processes the set of partition distances between them attainable by a Markov process on their joint atoms is computed. It is found that the infimum of these distances, the “Markov distance” between the pair, fails to satisfy the Triangle Inequality, hence fails to be a metric; thus in some cases the

d
¯

\overline d

-distance between two two-state Markov processes cannot be attained by a Markov process on their joint atoms.
Sets of divergence on the group 2^{𝜔}
Authors: Harris David C. , Wade William R.
Published: July 2010

We show that there exist uncountable sets of divergence for

C
(

2
ω

)

C({2^\omega })

. We also show that a necessary and sufficient condition that a set E be a set of divergence for

L
p

(

2
ω

)
,
1
>
p
>
∞

{L^p}({2^\omega }),1 > p > \infty

, is that E be of Haar measure zero.
Ultrapowers and local properties of Banach spaces
Authors: Stern Jacques
Published: July 2010

The present paper is an approach to the local theory of Banach spaces via the ultrapower construction. It includes a detailed study of ultrapowers and their dual spaces as well as a definition of a new notion, the notion of a u-extension of a Banach space. All these tools are used to give a unified definition of many classes of Banach spaces characterized by local properties (such as the

L

p

{\mathcal {L}_p}

-spaces). Many examples are given; also, as an application, it is proved that any

L

p

{\mathcal {L}_p}

-space,

1
>
p
>
∞

1 > p > \infty

, has an ultrapower which is isomorphic to an

L
p

{L_p}

-space.
Reductions of residuals are finite
Authors: Hindley R.
Published: July 2010

An important theorem of the

λ
β
K

\lambda \beta K

-calculus which has not been fully appreciated up to now is D. E. Schroer’s finiteness theorem (1963), which states that all reductions of residuals are finite. The present paper gives a new proof of this theorem and extends it from

λ
β

\lambda \beta

-reduction to

λ
β
η

\lambda \beta \eta

-reduction and reductions with certain extra operators added, for example the pairing, iteration and recursion operators. Combinatory weak reduction, with or without extra operators, is also included.
Partition theorems and ultrafilters
Authors: Baumgartner James E. , Taylor Alan D.
Published: July 2010

We introduce a class of ultrafilters on

ω
\omega

called k-arrow ultrafilters and characterized by the partition relation

U
→

(
U
,
k

)
2

U \to {(U,k)^2}

. These are studied in conjunction with P-points, Q-points, weakly Ramsey and Ramsey ultrafilters.
Interpretation of the 𝑝-adic log gamma function and Euler constants using the Bernoulli measure
Authors: Koblitz Neal
Published: July 2010

A regularized version of J. Diamond’sp-adic log gamma function and hisp-adic Euler constants are represented as integrals using B. Mazur’sp-adic Bernoulli measure.
The order and symbol of a distribution
Authors: Weinstein Alan
Published: July 2010

A definition is given, for an arbitrary distribution g on a manifold X, of the order and symbol of g at a point

(

χ
,
ξ

)

({\chi ,\xi })

of the cotangent bundle

T

∗

X

T^{\ast }X

. If

X
=

R

n

X = \textbf {R}^n

, the order of g at

(

0
,
ξ

)

({0,\xi })

is the growth order as

τ
→
∞

\tau \to \infty

of the distributions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g Superscript tau Baseline left-parenthesis x right-parenthesis equals e Superscript minus i StartRoot tau EndRoot mathematical left-angle x comma xi mathematical right-angle Baseline g left-parenthesis x slash StartRoot tau EndRoot right-parenthesis">

g
τ

(
x
)
=

e

−
i

τ

⟨
x
,
ξ
⟩

g

(

x

/

τ

)

{g^\tau }(x) = {e^{ - i\sqrt \tau \langle x,\xi \rangle }}g\left ( {x /\sqrt \tau } \right )

; if the order is less than or equal to N, the N-symbol of g is the family

g
τ

{g^\tau }

modulo

O
(

τ

N
−
1

/

2

)

O({{\tau ^{N - 1/2}}})

. It is shown that the order and symbol behave in a simple way when g is acted upon by a pseudo-differential operator. If g is a Fourier integral distribution, suitable identifications can be made so that the symbol defined here agrees with the bundle-valued symbol defined by Hörmander.
Solution of the nonlinear problem 𝐴𝑢=𝑁(𝑢) in a Banach space
Authors: Schechter Martin , Shapiro Jack , Snow Morris
Published: July 2010

We solve a nonlinear problem

A
u

=

N
(
u
)

Au\, = \,N(u)

where A is semi-Fredholm and N is a nonlinear compact operator.
Ultrafilter invariants in topological spaces
Authors: Saks Victor
Published: July 2010

Let

m

⩾

ℵ
0

\mathfrak {m}\, \geqslant \,{\aleph _0}

and

X

=

∏

i

∈

I

X
i

X\, = \,\prod \nolimits _{i\, \in \,I} {{X_i}}

. Then X is

[

ℵ
0

,

m

]

[{\aleph _0},\,\mathfrak {m}]

-compact if and only if

∏

i

∈

J

X
i

\prod \nolimits _{i\, \in \,J} {{X_i}}

is

[

ℵ
0

,

m

]

[{\aleph _0},\,\mathfrak {m}]

-compact for all

J

⊂

I

J\, \subset \,I

with

|

J

|

⩽

2

2

m

|J|\, \leqslant \,{2^{{2^\mathfrak {m}}}}

. Let

m

≥

ℵ
0

\mathfrak {m}\, \geq \,{\aleph _0}

,

(

x
ξ

:

ξ

>

m

)

({x_\xi }:\,\xi \, > \,\mathfrak {m})

a net in X,

p

∈

X

p\, \in \,X

, and

D

∈

β
(

m

)

\mathcal {D}\, \in \,\beta (\mathfrak {m})

. Then

p

=

D

−

lim

ξ
>

m

x
ξ

p\, = \,\mathcal {D}\, - \,{\lim _{\xi > \mathfrak {m}}}\,{x_\xi }

if

{
ξ

>

m

:

x
ξ

∈

U
}

∈

D

\{ \xi \, > \,\mathfrak {m}:\,{x_\xi }\, \in \,U \} \, \in \, \mathcal {D}

for every neighborhood U of p. Every topological space is characterized by its

D

\mathcal {D}

-limits. Several topological properties are described using ultrafilter invariants, including compactness and perfect maps. If X is a Hausdorff space and D is a discrete space equipotent with a dense subset of X, then X is a continuous perfect image of a subspace of

β
D

\beta D

which contains D if and only if X is regular.
Characterizations of amenable groups
Authors: Emerson William R.
Published: July 2010

Generalizing a construction of Banach from 1923 we obtain new criteria for the amenability of a locally compact group G. The relationship of these new criteria to known characterizations is then investigated, and in particular a formally strenghthened version of the von Neumann/Dixmier condition for amenability is established.
3-pseudomanifolds with preassigned links
Authors: Altshuler Amos
Published: July 2010

A 3-pseudomanifold is a finite connected simplicial 3-complex

K

\mathcal {K}

such that every triangle in

K

\mathcal {K}

belongs to precisely two 3-simplices of

K

\mathcal {K}

, the link of every edge in

K

\mathcal {K}

is a circuit, and the link of every vertex in

K

\mathcal {K}

is a closed 2-manifold. It is proved that for every finite set

∑
\sum

of closed 2-manifolds, there exists a 3-pseudomanifold

K

\mathcal {K}

such that the link of every vertex in

K

\mathcal {K}

is homeomorphic to some

S

∈

∑

S\, \in \,\sum

, and every

S

∈

∑

S\, \in \,\sum

is homeomorphic to the link of some vertex in

K

\mathcal {K}

.
Structural stability for flows on the torus with a cross-cap
Authors: Gutiérrez Carlos
Published: July 2010

Let

X

r

(

M
~

)
,
r
≥
1

{{\mathcal {X}}^r}({\tilde M}), r \geq 1

, denote the space of

C
r

{C^r}

-vector fields on the torus with a cross-cap

M
~

\tilde M

. We show that the Morse-Smale vector fields of

X

r

(

M
~

)

{{\mathcal {X}}^r}({\tilde M})

are dense on it. We also give a simple proof that a

C
0

{C^0}

-flow on the Klein bottle cannot support a nontrivial

ω
\omega

-recurrent trajectory.
Centers of hypergroups
Authors: Ross Kenneth A.
Published: July 2010

This paper initiates a study of Z-hypergroups, that is, commutative topological hypergroups K such that

K

/

Z

K/Z

is compact where Z denotes the maximum subgroup (equivalently, the center) of K. The character hypergroup

K
∧

{K^\wedge }

is studied and its connection with the locally compact abelian group

Z
∧

{Z^\wedge }

is given. Each Z-group is shown to correspond in a natural way to a Z-hypergroup. It is observed that the dual of a Z-group is itself a hypergroup. The basic orthogonality relations on Z-groups due to S. Grosser and M. Moskowitz are shown to hold for most Z-hypergroups. Some results on measure algebras of compact hypergroups due to C. F. Dunkl are extended to a class of noncompact hypergroups.
Asymptotic formulas for Toeplitz determinants
Authors: Basor Estelle
Published: July 2010

The object of this paper is to find an asymptotic formula for determinants of finite dimensional Toeplitz operators generated by a class of functions with singularities. The formula is a generalization of the Strong Szegö Limit Theorem.
Pure weak mixing
Authors: Ellis R. , Glasner S.
Published: July 2010

We show that for the group of integers Z, the maximal common factor of the maximal weakly mixing minimal flows coincides with the universal purely weakly mixing flow. We show that the maximal weakly mixing minimal flows are not all isomorphic to each other. None of the maximal w.m. flows are regular.
The existence of combinatorial formulae for characteristic classes
Authors: Levitt Norman , Rourke Colin
Published: July 2010

Given a characteristic class on a locally ordered combinatorial manifold M there exists a cocycle which represents the class on M and is locally defined, i.e. its value on

σ
∈
M

\sigma \in M

depends only on the ordered star

st

(
σ
,
M
)

{\text {st}}(\sigma ,M)

. For rational classes the dependence on order disappears. There is also a locally defined cycle which carries the dual homology class.
The signature of symplectic manifolds
Authors: Jones Leslie P.
Published: July 2010

The motivation for this work was a calculation by Oshanin of the image of the signature homomorphism from the special unitary cobordism ring into the integers. Here we compute this image for symplectic cobordism. This is accomplished by proving two divisibility theorems and then giving examples to show the theorems are the best possible.
Alternators of a right alternative algebra
Authors: Hentzel Irvin Roy
Published: July 2010

We show that in any right alternative algebra, the additive span of the alternators is nearly an ideal. We give an easy test to use to determine if a given set of additional identities will imply that the span of the alternators is an ideal. We apply our technique to the class of right alternative algebras satisfying the condition

(
a
,
a
,
b
)
=
λ
[
a
,
[
a
,
b
]
]

(a,a,b) = \lambda [a,[a,b]]

. We show that any semiprime algebra over a field of characteristic

≠
2

\ne 2

,

≠
3

\ne 3

which satisfies the right alternative law and the above identity with

λ
≠
0

\lambda \ne 0

is a subdirect sum of (associative and commutative) integral domains.
The ninety-one types of isogonal tilings in the plane
Authors: Grünbaum Branko , Shephard G. C.
Published: July 2010

A tiling of the plane by closed topological disks of isogonal if its symmetries act transitively on the vertices of the tiling. Two isogonal tilings are of the same type provided the symmetries of the tiling relate in the same way every vertex in each to its set of neighbors. Isogonal tilings were considered in 1916 by A. V. Šubnikov and by others since then, without obtaining a complete classification. The isogonal tilings are vaguely dual to the isohedral (tile transitive) tilings, but the duality is not strict. In contrast to the existence of 81 isohedral types of planar tilings we prove the following result: There exist 91 types of isogonal tilings of the plane in which each tile has at least three neighbors.
Classification of circle actions on 4-manifolds
Authors: Fintushel Ronald
Published: July 2010

This article studies locally smooth

S
1

{S^1}

-actions on closed oriented 4-manifolds in terms of the orbit space, orbit type data, and the characteristic class of the action which lies in

H
1

(

M

∗

,

S

∗

)

{H_1}({M^{\ast }},{S^{\ast }})

where

M

∗

{M^{\ast }}

is the orbit space and

S

∗

{S^{\ast }}

is the image of a certain collection of singular orbits. It is proved that such actions are determined by their weighted orbit spaces and are in 1-1 correspondence with “legally-weighted” 3-manifolds. The information contained in the weighted orbit space is used to give a presentation of the fundamental group of the 4-manifold, and in certain cases the quadratic form is computed.
Vector valued eigenfunctions of ergodic transformations
Authors: Flytzanis E.
Published: July 2010

We study the solutions X, T, of the eigenoperator equation
\[

X
(
h
(
⋅
)
)

=

T
X
(
⋅
)

a

.e

.

X(h( \cdot ))\,= \,TX( \cdot )\,{\text {a}}{\text {.e}}{\text {.}}

\]
, where h is a measurable transformation in a

σ
\sigma

-finite measure space

(
S
,
Σ
,
m
)

(S,\Sigma ,m)

, T is a bounded linear operator in a separable Hilbert space H and

X
:
S
→
H

X:S \to H

is Borel measurable. We solve the equation for some classes of measure preserving transformations. For the general case we obtain necessary conditions concerning the eigenoperators, in terms of operators induced by h in the scalar function spaces over the measure space. Finally we investigate integrability properties of the eigenfunctions.
Functions operating on positive definite matrices and a theorem of Schoenberg
Authors: Christensen Jens Peter Reus , Ressel Paul
Published: July 2010

We prove that the set of all functions

f
:

[
−
1
,

1
]
→
[
−
1
,

1
]

f:\,[ - 1,\,1] \to [ - 1,\,1]

operating on real positive definite matrices and normalized such that

f
(
1
)

=

1

f(1)\, = \,1

, is a Bauer simplex, and we identify its extreme points. As an application we obtain Schoenberg’s theorem characterising positive definite kernels on the infinite dimensional Hilbert sphere.
Absolute continuity in the dual of a Banach algebra
Authors: Berman Stephen Jay
Published: July 2010

If A is a Banach algebra, G is in the dual space

A

∗

{A^{\ast }}

, and I is a closed ideal in A, then let

‖
G
‖

I

∗

{\left \| G \right \|_{{I^{\ast }}}}

denote the norm of the restriction of G to I. We define a relation

≪
\ll

in

A

∗

{A^{\ast }}

as follows:

G
≪
L

G \ll L

if for every

ε
>
0

\varepsilon > 0

there exists a

δ
>
0

\delta > 0

such that if I is a closed ideal in A and

‖
L
‖

I

∗

>
δ

{\left \| L \right \|_{{I^{\ast }}}} > \delta

then

‖
G
‖

I

∗

>
ε

{\left \| G \right \|_{{I^{\ast }}}} > \varepsilon

. We explore this relation (which coincides with absolute continuity of measures when A is the algebra of continuous functions on a compact space) and related concepts in the context of several Banach algebras, particularly the algebra

C
1

[
0
,
1
]

{C^1}[0,1]

of differentiable functions and the algebra of continuous functions on the disc with holomorphic extensions to the interior. We also consider generalizations to noncommutative algebras and Banach modules.
Degrees of irreducible characters of (𝐵,𝑁)-pairs of types 𝐸₆ and 𝐸₇
Authors: Surowski David B.
Published: July 2010

Let G be a finite (B, N)-pair whose Coxeter system is of type

E
6

{E_6}

or

E
7

{E_7}

. Let

1
B
G

1_B^G

be the permutation character of the action of G on the left cosets of the Borel subgroup B in G. In this paper we give the character degrees of the irreducible constituents of

1
B
G

1_B^G

.
Application of the sector condition to the classification of sub-Markovian semigroups
Authors: Silverstein Martin L.
Published: July 2010

Let

p

t

{p_{t}}

,

t
>
0

t > 0

, be a strongly continuous submarkovian semigroup on a real Hilbert space

L
2

(
X
,
m
)

{L^2}(X, m)

. The measure m is assumed to be excessive and the

L
2

{L^2}

generator A is assumed to satisfy an estimate (the sector condition) which permits the application of Dirichlet spaces (not necessarily symmetric). Other submarkovian semigroups

P
t
∼

P_t^ \sim

with the same local generator and cogenerator and relative to which m is again excessive are classified in terms of generators for processes which live on a suitable boundary.
A noncommutative probability theory
Authors: Gudder S. P. , Hudson R. L.
Published: July 2010

A noncommutative probability theory is developed in which no boundedness, finiteness, or “tracial” conditions are imposed. The underlying structure of the theory is a “probability algebra”

(

a

,
ω
)

(\mathcal {a},\omega )

where

a

\mathcal {a}

is a *-algebra and

ω
\omega

is a faithful state on

a

\mathcal {a}

. Conditional expectations and coarse-graining are discussed. The bounded and unbounded commutants are considered and commutation theorems are proved. Two classes of probability algebras, which we call closable and symmetric probability algebras are shown to have important regularity properties. The canonical algebra of quantum mechanics is considered in some detail and a strong commutation theorem is proven for this case. Moreover, in this case, isotropic normal states, KMS states, and stable states are defined and characterized.
Explosions in completely unstable flows. I. Preventing explosions
Authors: Nitecki Zbigniew
Published: July 2010

Several conditions are equivalent to the property that a flow (on an open manifold) and its

C
0

{C^0}

perturbations have only wandering points. These conditions are: (i) there exists a strong Liapunov function; (ii) there are no generalized recurrent points in the sense of Auslander; (iii) there are no chain recurrent points, in the sense of Conley; (iv) there exists a fine sequence of filtrations; (v) relative to some metric; the flow is the gradient flow of a function without critical points. We establish these equivalences, and consider a few questions related to structural stability when all orbits wander.
Explosions in completely unstable flows. II. Some examples
Authors: Nitecki Zbigniew
Published: July 2010

A dynamical system with all points wandering is “explosive” if someC0{C^0}perturbation has nonwandering points. It is known that the plane admits no explosive cascades or flows; in this paper, examples are constructed to show that all open manifolds exceptR1{R^1}andR2{R^2}admit explosive flows (and hence cascades).
Conformality and semiconformality of a function holomorphic in the disk
Authors: Yamashita Shinji
Published: July 2010

Conformality and semiconformality at a boundary point, of a function f nonconstant and holomorphic in

|
z
|

>
1

\left | z \right | > 1

are local properties. Therefore one would suspect the requirement of such global conditions on f as f is univalent in

|
z
|

>
1

\left | z \right | > 1

, or f is a member of a larger class which contains all univalent functions in

|
z
|

>
1

\left | z \right | > 1

. We shall prove some extensions and new results without any assumption on f, or with a local assumption on f at most. Our methods are, for the most part, different from the ones in the classical cases. One of the main tools is Theorem 8 on the angular limits of the real part of a holomorphic function and its derivative.
Dolbeault homotopy theory
Authors: Neisendorfer Joseph , Taylor Laurence
Published: July 2010

For complex manifolds, we define “complex homotopy groups” in terms of the Dolbeault complex. Many theorems of classical homotopy theory are reflected in the properties of complex homotopy groups. Analytic fibre bundles yield long exact sequences of complex homotopy groups and various Hurewicz theorems relate complex homotopy groups to the Dolbeault cohomology. In a more analytic vein, the classical Fröhlicher spectral sequence has a complex homotopy analogue. We compute these complex homotopy invariants for such examples as Calabi-Eckmann manifolds, Stein manifolds, and complete intersections.
The real and rational cohomology of differential fibre bundles
Authors: Wolf Joel
Published: July 2010

Consider a differential fibre bundle (E,

π
\pi

, X,

G

/

G
H

H

{G \left / {\vphantom {G H}} \right . H}

, G). Under certain reasonable hypotheses, the cohomology of the total space E is computed in terms of the cohomology of the base space X and algebraic invariants of the imbedding of H into G.
Hyperbolicity and cycles
Authors: Franke J. E. , Selgrade J. F.
Published: July 2010

In this paper cycle points are defined without the assumption of Axiom A. The closure of the set of cycle points

C

\mathcal {C}

being quasi-hyperbolic is shown to be equivalent to Axiom A plus no cycles. Also we give a sufficient condition for

C

\mathcal {C}

to equal the chain recurrent set. In proving these theorems, a spectral decomposition for quasi-hyperbolic invariant sets is used.
An integral Riemann-Roch formula for induced representations of finite groups
Authors: Evens Leonard , Kahn Daniel S.
Published: July 2010

Let H be a subgroup of the finite group G,

ξ
\xi

a finite dimensional complex representation of H and

ρ
\rho

the induced representation of G. If

s
k

(
ρ
)
∈

H

2
k

(
G
,

Z

)

{s_k}(\rho ) \in {H^{2k}}(G,\textbf {Z})

,

k
⩾
1

k \geqslant 1

denote the characteristic classes bearing the same relation to power sums that Chern classes bear to elementary symmetric functions, then we prove the following,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper N With bar left-parenthesis k right-parenthesis left-parenthesis s Subscript k Baseline left-parenthesis rho right-parenthesis minus upper T r Subscript upper H right-arrow upper G Baseline left-parenthesis s Subscript k Baseline left-parenthesis xi right-parenthesis right-parenthesis right-parenthesis equals 0 comma">

N
¯

(
k
)
(

s
k

(
ρ
)
−

T

r

H
→
G

(

s
k

(
ξ
)
)

)
=
0
,

\begin{equation} \bar N (k)( {{s_k}(\rho ) - {\text {T}}{{\text {r}}_{H \to G}}({s_k}(\xi ))}) = 0, \end{equation}

where

N
¯

(
k
)
=

∏

p

|

N
(
k
)

p

prime

p

\begin{equation} \bar N(k) = {\prod _{\begin {array}{*{20}{c}} {p|N(k)} \\ {p{\text {prime}}} \\ \end{array}}}p \end{equation}

and

N
(
k
)
=

(

∏

p

prime

p

[
k

/

p
−
1
]

)

/

k
!
.

\begin{equation} N(k) = \left ( {\begin {array}{*{20}{c}} {\prod \limits _{p{\text {prime}}} {{p^{[k/p - 1]}}}} \end{array} } \right )/k!. \end{equation}

(Tr denotes transfer.) Moreover,

N
¯

(
k
)

\bar N (k)

is the least integer with this property. This settles a question originally raised in a paper of Knopfmacher in which it was conjectured that the required bound was N(k).
Harmonic functions and mass cancellation
Authors: Baxter J. R.
Published: July 2010

If a function on an open set in

R

n

{\textbf {R}^n}

has the mean value property for one ball at each point of the domain, the function will be said to possess the restricted mean value property. (The ordinary or unrestricted mean value property requires that the mean value property hold for every ball in the domain.) We specify the single ball at each point x by its radius

δ
(
x
)

\delta (x)

, a function of x. Under appropriate conditions on

δ
\delta

and the function, the restricted mean value property implies that the function is harmonic, giving a converse to the mean value theorem (see references). In the present paper a converse to the mean value theorem is proved, in which the function

δ
\delta

is well behaved, but the function is only required to be nonnegative. A converse theorem for more general means than averages over balls is also obtained. These results extend theorems of D. Heath, W. Veech, and the author (see references). Some connections are also pointed out between converse mean value theorems and mass cancellation.
A new approach to the limit theory of recurrent Markov chains
Authors: Athreya K. B. , Ney P.
Published: July 2010

Let

{

X
n

;

n
⩾
0
}

\{ {X_n};\,n \geqslant 0\}

be a Harris-recurrent Markov chain on a general state space. It is shown that there is a sequence of random times

{

N
i

;

i
⩾
1
}

\{ {N_i};\,i \geqslant 1\}

such that

{

X

N
i

;

 

i
⩾
1
}

\{ {X_{{N_i}}};{\text { }}i \geqslant 1\}

are independent and identically distributed. This idea is used to show that

{

X
n

}

\{ {X_n}\}

is equivalent to a process having a recurrence point, and to develop a regenerative scheme which leads to simple proofs of the ergodic theorem, existence and uniqueness of stationary measures.
Contractible 3-manifolds of finite genus at infinity
Authors: Brown E. M.
Published: July 2010

A class of contractible open 3-manifolds is defined. It is shown that all contractible open 3-manifolds which can be written as a union of cubes with a bounded number of handles are in this class. It is shown that a proper map between manifolds of this class which induces an isomorphism of proper fundamental groups (e.g. a proper homotopy equivalence) is proper homotopic to a homeomorphism. A naturality condition for homomorphisms of proper fundamental groups is developed. It is shown that a natural homomorphism between the proper fundamental groups of these manifolds is induced by a proper map.
Multidimensional quality control problems and quasivariational inequalities
Authors: Anderson Robert F. , Friedman Avner
Published: July 2010

A machine can manufacture any one of n m-dimensional Brownian motions with drift

λ
j

{\lambda _j}

,

P
x

λ
j

P_x^{{\lambda _j}}

, defined on the space of all paths

x

(
t
)

∈

C

(

[

0
,

∞

)

;

R
m

)

x\left ( t \right )\, \in \,C\left ( {\left [ {0,\,\infty } \right );\,{R^m}} \right )

. It is given that the product is a random evolution dictated by a Markov process

θ

(
t
)

\theta \left ( t \right )

with n states, and that the product is

P
x

λ
j

P_x^{{\lambda _j}}

when

θ

(
t
)

=

j
,

1

⩽

j

⩽

n

\theta \left ( t \right )\, = \,j,\,1\, \leqslant \,j\, \leqslant \,n

. One observes the

σ
\sigma

-fields of

x

(
t
)

x\left ( t \right )

, but not of

θ

(
t
)

\theta \left ( t \right )

. With each product

P
x

λ
j

P_x^{{\lambda _j}}

there is associated a cost

c
j

{c_j}

. One inspects

θ
\theta

at a sequence of times (each inspection entails a certain cost) and stops production when the state

θ

=

n

\theta \, = \,n

is reached. The problem is to find an optimal sequence of inspections. This problem is reduced to solving a certain elliptic quasi variational inequality. The latter problem is actually solved in a rather general case.
On the free boundary of a quasivariational inequality arising in a problem of quality control
Authors: Friedman Avner
Published: July 2010

In some recent work in stochastic optimization with partial observation occurring in quality control problems, Anderson and Friedman [1], [2] have shown that the optimal cost can be determined as a solution of the quasi variational inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row upper M w left-parenthesis p right-parenthesis plus f left-parenthesis p right-parenthesis greater-than-or-equal-to 0 comma w left-parenthesis p right-parenthesis less-than-or-equal-to psi left-parenthesis p semicolon w right-parenthesis comma left-parenthesis upper M w left-parenthesis p right-parenthesis plus f left-parenthesis p right-parenthesis right-parenthesis left-parenthesis w left-parenthesis p right-parenthesis minus psi left-parenthesis p semicolon w right-parenthesis right-parenthesis equals 0 EndLayout">

M
w

(
p
)

+

f

(
p
)

≥

0
,

w

(
p
)

≤

ψ

(

p
;

w

)

,

(

M
w

(
p
)

+

f

(
p
)

)

(

w

(
p
)

−

ψ

(

p
;

w

)

)

=

0

\begin {gathered} Mw\left ( p \right )\, + \,f\left ( p \right )\, \geq \,0,\,w\left ( p \right )\, \leq \,\psi \left ( {p;\,w} \right ), \left ( {Mw\left ( p \right )\, + \,f\left ( p \right )} \right )\left ( {w\left ( p \right )\, - \,\psi \left ( {p;\,w} \right )} \right )\, = \,0 \end {gathered}

\]
in the simplex

p
i

>

0

{p_i}\, > \,0

,

∑

i

=

1

n

p
i

=

1

\sum \nolimits _{i\, = \,1}^n {{p_i}\, = \,1}

. Here f,

ψ
\psi

are given functions of p,

ψ
\psi

is a functional of w, and M is a given elliptic operator degenerating on the boundary. This system has a unique solution when M does not degenerate in the interior of the simplex. The aim of this paper is to study the free boundary, that is, the boundary of the set where

w

(
p
)

>

ψ

(

p
;

w

)

w\left ( p \right )\, > \,\psi \left ( {p;\,w} \right )

.