bluuebunny/crossref_metadata_2025_split huggingface dataset:
Asymptotics for the heat equation in the exterior of a shrinking compact set in the plane via Brownian hitting times
Authors: Pinsky Ross 
Published: July 2002

Let

D

r

=
{
x
∈

R

2

:

|

x

|

≤
r
}

D_{r}=\{x\in R^{2}:|x|\le r\}

and let

γ
\gamma

be a continuous, nonincreasing function on

[
0
,
∞
)

[0,\infty )

satisfying

lim

t
→
∞

γ
(
t
)
=
0

\lim _{t\to \infty }\gamma (t)=0

. Consider the heat equation in the exterior of a time-dependent shrinking disk in the plane:

a
m
p
;

u

t

=

1
2

Δ
u
,

x
∈

R

2

−

D

γ
(
t
)

,

t
>
0
,

a
m
p
;
u
(
x
,
0
)
=
0
,

x
∈

R

2

−

D

γ
(
t
)

,

a
m
p
;
u
(
x
,
t
)
=
1
,

x
∈

D

γ
(
t
)

,
t
>
0.

\begin{equation*} \begin {split} &u_{t}=\frac {1}{2}\Delta u, \quad x\in R^{2}-D_{\gamma (t)}, \quad t>0,\\ &u(x,0)=0,\quad x\in R^{2}-D_{\gamma (t)},\\ &u(x,t)=1,\quad x\in D_{\gamma (t)}, t>0. \end{split} \end{equation*}

If there exist constants

0
>

c

1

>

c

2

0>c_{1}>c_{2}

and a constant

k
>
0

k>0

such that

c

1

t

−
k

≤
γ
(
t
)
≤

c

2

t

−
k

c_{1}t^{-k}\le \gamma (t)\le c_{2}t^{-k}

, for sufficiently large

t
t

, then

lim

t
→
∞

u
(
x
,
t
)
=

1

1
+
2
k

\lim _{t\to \infty }u(x,t)=\frac {1}{1+2k}

. The same result is also shown to hold when

D

γ
(
t
)

D_{\gamma (t)}

is replaced by

L

γ
(
t
)

L_{\gamma (t)}

, where

L

r

=
{
(

x

1

,
0
)
∈

R

2

:

|

x

1

|

≤
r
}

L_{r}=\{(x_{1},0)\in R^{2}:|x_{1}|\le r\}

. Also, a discrepancy is noted between the asymptotics for the above forward heat equation and the corresponding backward one. The method used is probabilistic.
From 𝐾(𝑛+1)_{*}(𝑋) to 𝐾(𝑛)_{*}(𝑋)
Authors: Minami Norihiko 
Published: July 2002

Let

X
X

be a space of finite type. Set

q
=
2
(
p
−
1
)

q=2(p-1)

as usual, and define the mod

q
q

support of

K
(
n

)
∗

(
X
)

K(n)^*(X)

by

S
(
X
,
K
(
n
)
)
=
{
m
∈

Z

/

q

Z

∣

⨁

d
≡
m
mod
q

K
(
n

)
d

≠
0
}

S(X,K(n)) = \{ m \in \mathbb {Z}/q\mathbb {Z} \mid \bigoplus _{d \equiv m \bmod q}\ K(n)^d \neq 0 \}

for

n
>
0.

n>0.

Call

K
(
n

)
∗

(
X
)

K(n)^*(X)

sparse if there is no

m
∈

Z

/

q

Z

m \in \mathbb {Z}/q\mathbb {Z}

with

m
,
m
+
1
∈
S
(
X
,
K
(
n
)
)
.

m, m+1 \in S(X,K(n)).

Then we show the relation

S
(
X
,
K
(
n
)
)
⫅
S
(
X
,
K
(
n
+
1
)
)

S(X,K(n)) \subseteqq S(X,K(n+1))

for any finite type space

X
X

with

K
(
n
+
1

)
∗

(
X
)

K(n+1)^*(X)

being sparse.

As a special case, we have

K
(
n
+
1

)

o
d
d

(
X
)
=
0
⟹
K
(
n

)

o
d
d

(
X
)
=
0
,

K(n+1)^{odd}(X) = 0 \Longrightarrow K(n)^{odd}(X) = 0,

and the main theorem of Ravenel, Wilson and Yagita is also generalized in terms of the mod

q
q

support.
Critical points of the area functional of a complex closed curve on the manifold of Kähler metrics
Authors: Castorena Abel 
Published: July 2002

We consider a compact complex manifold

M
M

of dimension

n
n

that admits Kähler metrics and we assume that

C
↪
M

C\hookrightarrow M

is a closed complex curve. We denote by

K
C

(
1
)

\mathcal {KC}(1)

the space of classes of Kähler forms

[
ω
]
∈

H

1
,
1

(
M
,

R

)

[\omega ]\in H^{1,1}(M,\mathbb {R})

that define Kähler metrics of volume 1 on

M
M

and define

A

C

:

K
C

(
1
)
→

R

\mathbf {A}_{C}:\mathcal {KC}(1)\to \mathbb {R}

by

A

C

(
[
ω
]
)
=

∫

C

ω
=
area of
C
in the induced metric by
ω

\mathbf {A}_{C}([\omega ])=\int _{C} \omega =\text {area of }C\text { in the induced metric by }\omega

. We show how the Riemann-Hodge bilinear relations imply that any critical point of

A

C

\mathbf {A}_{C}

is the strict global minimum and we give conditions under which there is such a critical point

[
ω
]

[\omega ]

: A positive multiple of

[
ω

]

n
−
1

∈

H

2
n
−
2

(
M
,

R

)

[\omega ]^{n-1}\in H^{2n-2}(M,\mathbb {R})

is the Poincaré dual of the homology class of

C
C

. Applying this to the Abel-Jacobi map of a curve into its Jacobian,

C
↪
J
(
C
)

C\hookrightarrow J(C)

, we obtain that the Theta metric minimizes the area of

C
C

within all Kähler metrics of volume 1 on

J
(
C
)

J(C)

.
Best possibility of the Furuta inequality
Authors: Tanahashi Kôtarô 
Published: July 2002

Let

0
≤
p
,
q
,
r
∈

R

,
p
+
2
r
≤
(
1
+
2
r
)
q

0\le p,q,r\in \Bbb R, p+2r\le (1+2r)q

, and

1
≤
q

1\le q

. Furuta (1987) proved that if bounded linear operators

A
,
B
∈
B
(
H
)

A,B\in B(H)

on a Hilbert space

H
H

(
dim
⁡
(
H
)
≥
2
)

(\dim (H)\ge 2)

satisfy

0
≤
B
≤
A

0\le B\le A

, then

(

A
r

B
p

A
r

)

1

/

q

≤

A

(
p
+
2
r
)

/

q

(A^r B^p A^r)^{1/q} \le A^{(p+2r)/q}

. In this paper, we prove that the range

p
+
2
r
≤
(
1
+
2
r
)
q

p+2r\le (1+2r)q

and

1
≤
q

1\le q

is best possible with respect to the Furuta inequality, that is, if

(
1
+
2
r
)
q
>
p
+
2
r

(1+2r) q>p+2r

or

0
>
q
>
1

0>q>1

, then there exist

A
,
B
∈
B
(

R

2

)

A,B\in B(\Bbb R^2)

which satisfy

0
≤
B
≤
A

0\le B\le A

but

(

A
r

B
p

A
r

)

1

/

q

≰

A

(
p
+
2
r
)

/

q

(A^r B^p A^r)^{1/q}\nleq A^{(p+2r)/q}

.
𝐴-differentiability and 𝐴-analyticity
Authors: Gadea P. , Masqué J. 
Published: July 2002

Let

A
A

be a finite-dimensional commutative algebra over

R

\mathbb {R}

and let

C

A

r

(
U
)

C_{A}^{r}(U)

,

C

ω

(
U
,
A
)

C^{\omega }(U,A)

and

O

A

(
U
)

\mathcal { O}_{A}(U)

be the ring of

A
A

-differentiable functions of class

C

r

,

0
≤
r
≤
∞

C^{r},\,0 \leq r \leq \infty

, the ring of real analytic mappings with values in

A
A

and the ring of

A
A

-analytic functions, respectively, defined on an open subset

U
U

of

A

n

A^{n}

. We prove two basic results concerning

A
A

-differentiability and

A
A

-analyticity:

1

s
t

1^{st}

)

O

A

(
U
)
=

C

A

∞

(
U
)
⋂

C

ω

(
U
,
A
)

\mathcal { O}_{A}(U) = C^{\infty }_{A}(U) \bigcap C^{\omega }(U,A)

,

2

n
d

2^{nd}

)

O

A

(
U
)
=

C

A

∞

(
U
)

\mathcal { O}_{A}(U) = C^{\infty }_{A}(U)

if and only if

A
A

is defined over

C

\mathbb {C}

.
Invariant points of maps between Grassmannians
Authors: Mukherjea Kalyan , Sankaran Parameswaran 
Published: July 2002

We show that for a very large class of integers

l
>
k
>
n

l>k>n

and any map

f
:

G
k

(

R

n

)
→

G
l

(

R

n

)

f\colon G_k(\mathbb R^n)\to G_l(\mathbb R^n)

between Grassmannians, there is some

k
k

-plane of

R

n

\mathbb R^n

which is mapped into a subspace of itself.
The distance from the Apostol spectrum
Authors: Kordula V. , Müller V. 
Published: July 2002

If

T
T

is an s-regular operator in a Banach space (i.e.

T
T

has closed range and

N
(
T
)
⊂

R

∞

(
T
)

N(T)\subset R^{\infty }(T)

) and

γ
(
T
)

\gamma (T)

is the Kato reduced minimum modulus, then

lim

n
→
∞

γ
(

T

n

)

1

/

n

=
sup
{
r
:
T
−
λ

i
s
s
-
r
e
g
u
l
a
r
f
o
r

|

λ

|

>
r
}
.

\begin{equation*}\lim _{n\to \infty }\gamma (T^{n})^{1/n}=\sup \{r: T-\lambda { \operatorname {is s-regular for }}|\lambda |>r\}. \end{equation*}
Derivations with Engel conditions on multilinear polynomials
Authors: Lee Pjek-Hwee , Lee Tsiu-Kwen 
Published: July 2002

Let

R
R

be a prime algebra over a commutative ring

K
K

with unity and let

f
(

X

1

,
…
,

X

n

)

f(X_{1}, \ldots , X_{n})

be a multilinear polynomial over

K
K

. Suppose that

d
d

is a nonzero derivation on

R
R

such that for all

r

1

,
…
,

r

n

r_{1}, \ldots , r_{n}

in some nonzero ideal

I
I

of

R
R

,

[

d

(

f
(

r

1

,
…
,

r

n

)

)

,
f
(

r

1

,
…
,

r

n

)

]

k

=
0

\Big [ d\big ( f(r_{1}, \ldots , r_{n})\big ), f(r_{1}, \ldots , r_{n}) \Big ]_{k} = 0

with

k
k

fixed. Then

f
(

X

1

,
…
,

X

n

)

f(X_{1}, \ldots , X_{n})

is central–valued on

R
R

except when char

R
=
2

R=2

and

R
R

satisfies the standard identity

s

4

s_{4}

in 4 variables.
Radial symmetry of large solutions of nonlinear elliptic equations
Authors: Taliaferro Steven 
Published: July 2002

We give conditions under which all

C
2

C^2

solutions of the problem

a
m
p
;
Δ
u
=
f
(

|

x

|

,
u
)
,

x
∈

R

n

,

a
m
p
;

lim

|

x

|

→
∞

u
(
x
)
=
∞

\begin{align*} &\Delta u = f(|x|,u),\qquad x\in {\mathbb {R}}^n,\ &\lim _{|x|\to \infty } u(x) = \infty \end{align*}

are radial. We assume

f
(

|

x

|

,
u
)

f(|x|,u)

is positive when

|

x

|

|x|

and

u
u

are both large and positive. Since this problem with

f
(

|

x

|

,
u
)
=
u

f(|x|,u) = u

has non-radial solutions, we rule out this possibility by requiring that

f
(

|

x

|

,
u
)

f(|x|,u)

grow superlinearly in

u
u

when

|

x

|

|x|

and

u
u

are both large and positive. However we make no assumptions on the rate of growth of solutions.
The classical Banach spaces ℓ_{𝑣𝑎𝑟𝑝ℎ𝑖}/ℎ_{𝜑}
Authors: Granero Antonio , Hudzik Henryk 
Published: July 2002

In this paper we study some structural and geometric properties of the quotient Banach spaces

ℓ

φ

(
I
)

/

h

φ

(

S

)

\ell _{\varphi }(I)/ h_{\varphi }(\mathcal {S})

, where

I
I

is an arbitrary set,

φ
\varphi

is an Orlicz function,

ℓ

φ

(
I
)

\ell _{\varphi }(I)

is the corresponding Orlicz space on

I
I

and

h

φ

(

S

)
=
{
x
∈

ℓ

φ

(
I
)
:
∀
λ
>
0
,
∃
s
∈

S

such that

I

φ

(

x
−
s

λ

)
>
∞
}

h_{\varphi }(\mathcal {S}) =\{x\in \ell _{\varphi }(I) :\forall \lambda >0, \exists s\in \mathcal {S}\text { such that } I_{\varphi } (\frac {x-s}{\lambda })>\infty \}

,

S

\mathcal {S}

being the ideal of elements with finite support. The results we obtain here extend and complete the ones obtained by Leonard and Whitfield (Rocky Mountain J. Math. 13 (1983), 531–539). We show that

ℓ

φ

(
I
)

/

h

φ

(

S

)

\ell _{\varphi }(I) / h_{\varphi }(\mathcal {S})

is not a dual space, that

E
x
t
(

B

ℓ

φ

(
I
)

/

h

φ

(

S

)

)
=
∅

Ext(B_{ \ell _{\varphi }(I)/ h_{\varphi }(\mathcal {S}) })=\emptyset

, if

φ
(
t
)
>
0

\varphi (t)>0

for every

t
>
0

t>0

, that

S

ℓ

φ

(
I
)

/

h

φ

(

S

)

S_{ \ell _{\varphi }(I)/ h_{\varphi }(\mathcal {S})}

has no smooth points, that it cannot be renormed equivalently with a strictly convex or smooth norm, that

ℓ

φ

(
I
)

/

h

φ

(

S

)

\ell _{ \varphi }(I)/h_{ \varphi }(\mathcal {S})

is a Grothendieck space, etc.
An inequality for polyhedra and ideal triangulations of cusped hyperbolic 3-manifolds
Authors: Wada Masaaki , Yamashita Yasushi , Yoshida Han 
Published: July 2002

It is not known whether every noncompact hyperbolic 3-manifold of finite volume admits a decomposition into ideal tetrahedra. We give a partial solution to this problem: Let

M
M

be a hyperbolic 3-manifold obtained by identifying the faces of

n
n

convex ideal polyhedra

P

1

,
…
,

P

n

P_{1},\dots ,P_{n}

. If the faces of

P

1

,
…
,

P

n
−
1

P_{1},\dots ,P_{n-1}

are glued to

P

n

P_{n}

, then

M
M

can be decomposed into ideal tetrahedra by subdividing the

P

i

P_{i}

’s.
Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption
Authors: Haiyun Zhou , Yuting Jia 
Published: July 2002

In the present paper, the following result is shown: Let

X
X

be a real Banach space with a uniformly convex dual

X
∗

X^*

, and let

K
K

be a nonempty closed convex and bounded subset of

X
X

. Assume that

T
:

K
→
K

T:\,K\rightarrow K

is a continuous strong pseudocontraction. Let

{

α
n

}

n
=
1

∞

\{\alpha _n\}^{\infty }_{n=1}

and

{

β
n

}

n
=
1

∞

\{\beta _n\}^{\infty }_{n=1}

be two real sequences satisfying (i)

0
>

α
n

,

β
n

>
1

0>\alpha _n,\,\beta _n>1

for all

n
≥
1

n\ge 1

; (ii)

∑

n
=
1

∞

α
n

=
∞

\sum _{n=1}^{\infty }\alpha _n=\infty

; and (iii)

α
n

→
0
,

β
n

→
0

\alpha _n \rightarrow 0,\, \beta _n \rightarrow 0

as

n
→
∞
.

n\rightarrow \infty .

Then the Ishikawa iterative sequence

{

x
n

}

n
=
1

∞

\{x_n\}_{n=1}^{\infty }

generated by

(
I
)

{

x
1

∈
K
,

x

n
+
1

=
(
1
−

α
n

)

x
n

+

α
n

T

y
n

,

y
n

=
(
1
−

β
n

)

x
n

+

β
n

T

x
n

,

n
≥
1
,

\begin{equation*} \mathrm {(I)} \quad \left \{ \begin {array}{l} x_1\in K,\\ x_{n+1}=(1-\alpha _n)x_n+\alpha _nTy_n,\\ y_n=(1-\beta _n)x_n+\beta _nTx_n,\,n\geq 1, \end{array} \right . \end{equation*}

converges strongly to the unique fixed point of

T
T

.
On simultaneous extension of continuous partial functions
Authors: Künzi Hans-Peter , Shapiro Leonid 
Published: July 2002

For a metric space

X
X

let

C

v
c

(
X
)

\mathcal {C}_{vc}(X)

(that is, the set of all graphs of real-valued continuous functions with a compact domain in

X
X

) be equipped with the Hausdorff metric induced by the hyperspace of nonempty closed subsets of

X
×

R

.

X\times {\mathbf {R}}.

It is shown that there exists a continuous mapping

Φ
:

C

v
c

(
X
)
→

C

b

(
X
)

\Phi :\mathcal {C}_{vc}(X)\rightarrow \mathcal {C}_b(X)

satisfying the following conditions: (i)

Φ
(
f
)
|
dom
⁡
f
=
f

\Phi (f)\vert \operatorname {dom}f= f

for all partial functions

f
.

f.

(ii) For every nonempty compact subset

K
K

of

X
,

X,

Φ
|

C

b

(
K
)
:

C

b

(
K
)
→

C

b

(
X
)

\Phi \vert \mathcal {C}_b(K):\mathcal {C}_b(K) \rightarrow \mathcal {C}_b(X)

is a linear positive operator such that

Φ
(

1
K

)
=

1
X

\Phi (1_K)=1_X

.
Group algebras whose units satisfy a group identity. II
Authors: Passman D. 
Published: July 2002

Let

K
[
G
]

K[G]

be the group algebra of a torsion group

G
G

over an infinite field

K
K

, and let

U
=
U
(
G
)

U=U(G)

denote its group of units. A recent paper of A. Giambruno, S. K. Sehgal, and A. Valenti proved that if

U
U

satisfies a group identity, then

K
[
G
]

K[G]

satisfies a polynomial identity, thereby confirming a conjecture of Brian Hartley. Here we add a footnote to their result by showing that the commutator subgroup

G
′

G’

of

G
G

must have bounded period. Indeed, this additional fact enables us to obtain necessary and sufficient conditions for

U
(
G
)

U(G)

to satisfy an identity.
On self-adjointness of a Schrödinger operator on differential forms
Authors: Braverman Maxim 
Published: July 2002

Let

M
M

be a complete Riemannian manifold and let

Ω

∙

(
M
)

\Omega ^{\bullet }(M)

denote the space of differential forms on

M
M

. Let

d
:

Ω

∙

(
M
)
→

Ω

∙
+
1

(
M
)

d:\Omega ^{\bullet }(M)\to \Omega ^{\bullet +1}(M)

be the exterior differential operator and let

Δ
=
d

d

∗

+

d

∗

d

\Delta =dd^{*}+d^{*}d

be the Laplacian. We establish a sufficient condition for the Schrödinger operator

H
=
Δ
+
V
(
x
)

H=\Delta +V(x)

(where the potential

V
(
x
)
:

Ω

∙

(
M
)
→

Ω

∙

(
M
)

V(x):\Omega ^{\bullet }(M)\to \Omega ^{\bullet }(M)

is a zero order differential operator) to be self-adjoint. Our result generalizes a theorem by I. Oleinik about self-adjointness of a Schrödinger operator which acts on the space of scalar valued functions.
Sufficient conditions for the oscillation of forced super-linear second order differential equations with oscillatory potential
Authors: Nasr A. 
Published: July 2002

In the case of oscillatory potentials, we give sufficient conditions for the oscillation of the forced super-linear equation
\[

x
(
t
)
+
a
(
t
)

|

x
(
t
)

|

ν

sgn
⁡
x
(
t
)
=
g
(
t
)
.

x(t)+a(t)|x(t)|^{\nu }\operatorname {sgn} x(t)=g(t).

\]
This answers a question raised by J. S. W. Wong.
Eigenvalue pinching theorems on compact symmetric spaces
Authors: Suzuki Yuuichi , Urakawa Hajime 
Published: July 2002

We prove two first eigenvalue pinching theorems for Riemannian symmetric spaces (Theorems 1 and 2). As their application, we answer negatively a question raised by Elworthy and Rosenberg, who proposed to show that for every compact simple Lie group

G
G

with a bi-invariant Riemannian metric

h
h

on

G
G

with respect to

−

1
2

B

-\frac {1}{2} \,B

,

B
B

being the Killing form of the Lie algebra

g

\mathfrak {g}

, the first eigenvalue

λ

1

(
h
)

\lambda _{1}(h)

would satisfy

∑

j
=
1

2

∑

ℓ
=
3

n

|

[

v

j

,

v

ℓ

]

|

2

>
n
(
2

λ

1

(
h
)
−
1
)
,

\begin{equation*}\sum _{j=1}^{2}\sum _{\ell =3}^{n} |[v_{j},v_{\ell }]|^{2}>n(2\lambda _{1}(h)-1),\end{equation*}

for all orthonormal bases

{

v

j

}

j
=
1

n

\{v_{j}\}_{j=1}^{n}

of tangent spaces of

G
G

(cf. Corollary 3). This problem arose in an attempt to give a spectral geometric proof that

π

2

(
G
)
=
0

\pi _{2}(G)=0

for a Lie group

G
G

.
Products of similar matrices
Authors: Witte Dave 
Published: July 2002

Let

A
A

and

B
B

be

n
×
n

n \times n

matrices of determinant

1
1

over a field

K
K

, with

n
>
2

n >2

or

|

K

|

>
3

|K|>3

. We show that if

A
A

is not a scalar matrix, then

B
B

is a product of matrices similar to

A
A

. Analogously, we conjecture that if

a
a

and

b
b

are elements of a semisimple algebraic group

G
G

over a field of characteristic zero, and if there is no normal subgroup of

G
G

containing

a
a

but not

b
b

, then

b
b

is a product of conjugates of

a
a

. The conjecture is verified for orthogonal groups and symplectic groups, and for all semisimple groups over local fields. Thus, in a connected, semisimple Lie group with finite center, the only conjugation-invariant subsemigroups are the normal subgroups.
Higher order Turán inequalities
Authors: Dimitrov Dimitar 
Published: July 2002

The celebrated Turán inequalities

P

n

2

(
x
)
−

P

n
−
1

(
x
)

P

n
+
1

(
x
)
≥
0

P_{n}^{2}(x) - P_{n-1}(x) P_{n+1}(x) \geq 0

,

x
∈
[
−
1
,
1
]

x \in [-1,1]

,

n
≥
1

n \geq 1

, where

P

n

(
x
)

P_{n}(x)

denotes the Legendre polynomial of degree

n
n

, are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities

γ

n

2

−

γ

n
−
1

γ

n
+
1

≥
0

\gamma _{n}^{2} - \gamma _{n-1} \gamma _{n+1} \geq 0

,

n
≥
1

n \geq 1

, which hold for the Maclaurin coefficients of the real entire function

ψ
\psi

in the Laguerre-Pólya class,

ψ
(
x
)
=

∑

n
=
0

∞

γ

n

x

n

/

n
!

\psi (x) = \sum _{n=0}^{\infty } \gamma _{n} x^{n}/n!

.
Noncomplete linear systems on elliptic curves and Abelian varieties: Addendum to a paper by Ch. Birkenhake
Authors: Ballico E. 
Published: July 2002

Here we give a result on the postulation (i.e. the 2-normality) of nonlinearly normal embeddings of Abelian varieties. This result improves some of the results proved in a recent paper by Ch. Birkenhake.
Positive solutions of nonlinear elliptic equations in the Euclidean plane
Authors: Ufuktepe U. , Zhao Z. 
Published: July 2002

In the present paper, we study the existence of solutions to the problem
\[

{

Δ
u
+
f
(
x
,
u
)
=
0

a
m
p
;

in

D

u
>
0

a
m
p
;

in

D

u
=
0

a
m
p
;

on

∂
D

\begin {cases} \Delta u+f(x,u)=0 & \text {in $D$}\\ u>0&\text {in $D$}\\ u=0 & \text {on $\partial D$} \end {cases}

\]
where

D
D

is an unbounded domain in

R

2

\mathbb {R}^2

with a compact nonempty boundary

∂
D

\partial D

consisting of finitely many Jordan curves. The goal is to prove an existence theorem for the above problem in a general setting by using Brownian path integration and potential theory.
Solution of a functional equation arising from utility that is both separable and additive
Authors: Aczél János , Ger Roman , Járai Antal 
Published: July 2002

The problem of determining all utility measures over binary gambles that are both separable and additive leads to the functional equation

f
(
v
)
=
f
(
v
w
)
+
f
[
v
Q
(
w
)
]
,

v
,
v
Q
(
w
)
∈
[
0
,
k
)
,

w
∈
[
0
,
1
]

.

\begin{equation*}f(v) = f(vw) + f[vQ(w)], \qquad v, vQ(w) \in [0,k),\quad w \in [0,1]\,. \end{equation*}

The following conditions are more or less natural to the problem:

f
f

strictly increasing,

Q
Q

strictly decreasing; both map their domains onto intervals (

f
f

onto a

[
0
,
K
)

[0,K)

,

Q
Q

onto

[
0
,
1
]

[0,1]

); thus both are continuous,

k
>
1

k>1

,

f
(
0
)
=
0

f(0)=0

,

f
(
1
)
=
1

f(1)=1

,

Q
(
1
)
=
0

Q(1)=0

,

Q
(
0
)
=
1

Q(0)=1

. We determine, however, the general solution without any of these conditions (except

f
:
[
0
,
k
)
→

R

+

:=
[
0
,
∞
)

f: [0,k) \to \mathbb {R}_{+}:= [0, \infty )

,

Q
:
[
0
,
1
]
→

R

+

Q:[0,1]\to \mathbb {R}_{+}

, both into). If we exclude two trivial solutions, then we get as general solution

f
(
v
)
=
α

v

β

f(v) = \alpha v^{\beta }

(

β
>
0

\beta > 0

,

α
>
0

\alpha > 0

;

α
=
1

\alpha = 1

for

f
(
1
)
=
1

f(1) = 1

), which satisfies all the above conditions. The paper concludes with a remark on the case where the equation is satisfied only almost everywhere.
A counterexample to the Fredholm alternative for the 𝑝-Laplacian
Authors: Drábek Pavel , Takáč Peter 
Published: July 2002

The following nonhomogeneous Dirichlet boundary value problem for the one-dimensional

p
p

-Laplacian with

1
>
p
>
∞

1 > p > \infty

is considered:

(*)

−
(

|

u
′

|

p
−
2

u
′

)
′

−
λ

|

u

|

p
−
2

u
=
f
(
x
)

for

0
>
x
>
T
;

u
(
0
)
=
u
(
T
)
=
0
,

\begin{equation*} - (|u’|^{p-2} u’)’ - \lambda |u|^{p-2} u = f(x) \;\mbox { for } 0 > x > T ; \quad u(0) = u(T) = 0 , \tag {*}\end{equation*}

where

f
≡
1
+
h

f\equiv 1 + h

with

h
∈

L
∞

(
0
,
T
)

h\in L^\infty (0,T)

small enough. Solvability properties of Problem (*) with respect to the spectral parameter

λ
∈

R

\lambda \in \mathbb {R}

are investigated. We focus our attention on some fundamental differences between the cases

p
≠
2

p\neq 2

and

p
=
2

p=2

. For

p
≠
2

p\neq 2

we give a counterexample to the classical Fredholm alternative (which is valid for the linear case

p
=
2

p=2

).
Completeness in the set of wavelets
Authors: Garrigós Gustavo , Speegle Darrin 
Published: July 2002

We study the completeness properties of the set of wavelets in

L

2

(

R

)

L^{2}(\mathbb {R})

. It is well-known that this set is not closed in the unit ball of

L

2

(

R

)

L^{2}(\mathbb {R})

. However, if one considers the metric inherited as a subspace (in the Fourier transform side) of

L

2

(

R

,
d
ξ
)
∩

L

2

(

R

∗

,

d
ξ

|

ξ

|

)

L^{2}(\mathbb {R},d\xi ) \cap L^{2}(\mathbb {R}_{*},{\frac {{d\xi }}{{|\xi |}}})

, we do obtain a complete metric space.
Splittings of Banach spaces induced by Clifford algebras
Authors: Carothers N. , Dilworth S. , Sobecki David 
Published: July 2002

Let

H
H

be an infinite-dimensional Hilbert space of density character

m

\mathfrak {m}

. By representing

H
H

as a module over an appropriate Clifford algebra, it is proved that

H
H

possesses a family

{

A

α

}

α
∈

m

\{A_{\alpha }\}_{\alpha \in \mathfrak {m}}

of proper closed nonzero subspaces such that

d
(

S

A

α

,

S

A

β

)
=
d
(

S

A

α

⊥

,

S

A

β

)
=
d
(

S

A

α

⊥

,

S

A

β

⊥

)
=

2
−

2

(
α
≠
β
)
.

\begin{equation*}d(S_{A_{\alpha }},S_{A_{\beta }})=d(S_{A^{\perp }_{\alpha }},S_{A_{\beta }}) =d(S_{A^{\perp }_{\alpha }},S_{A^{\perp }_{\beta }})=\sqrt {2-\sqrt 2}\qquad (\alpha \ne \beta ).\end{equation*}

Analogous results are proved for

L

p

L_{p}

spaces and for

c

0

(
X
)

c_{0}(X)

and

ℓ

p

(
X
)

\ell _{p}(X)

(

1
≤
p
≤
∞

1 \le p \le \infty

) when

X
X

is an arbitrary nonzero Banach space.
The super order dual of an ordered vector space and the Riesz–Kantorovich formula
Authors: Aliprantis Charalambos , Tourky Rabee 
Published: July 2002

A classical theorem of F. Riesz and L. V. Kantorovich asserts that if

L
L

is a vector lattice and

f
f

and

g
g

are order bounded linear functionals on

L
L

, then their supremum (least upper bound)

f
∨
g

f\lor g

exists in

L
∼

L^\sim

and for each

x
∈

L
+

x\in L_+

it satisfies the so-called Riesz–Kantorovich formula:
\[

[

f
∨
g

]

(
x
)
=
sup

{

f
(
y
)
+
g
(
z
)
:

y
,
z
∈

L
+

a
n
d

y
+
z
=
x

}

.

\bigl [f\lor g\bigr ](x)=\sup \bigl \{f(y)+g(z)\colon \ y,z\in L_+\ \,\mathrm {and}\ \, y+z=x\bigr \}\,.

\]
Related to the Riesz–Kantorovich formula is the following long-standing problem: If the supremum of two order bounded linear functionals

f
f

and

g
g

on an ordered vector space exists, does it then satisfy the Riesz–Kantorovich formula?

In this paper, we introduce an extension of the order dual of an ordered vector space and provide some answers to this long-standing problem. The ideas regarding the Riesz–Kantorovich formula owe their origins to the study of the fundamental theorems of welfare economics and the existence of competitive equilibrium. The techniques introduced here show that the existence of decentralizing prices for efficient allocations is closely related to the above-mentioned problem and to the properties of the Riesz–Kantorovich formula.