bluuebunny/crossref_metadata_2025_split huggingface dataset:
Individual Variation in Hepatic Aldehyde Oxidase Activity
Authors: Al‐Salmy H. S. 
Published: July 2002

AbstractAldehyde oxidase (AO) is a molybdo‐flavo enzyme expressed predominantly in the liver, lung, and kidney. AO plays a major role in oxidation of aldehydes, as well as oxidation of various N‐heterocyclic compounds of pharmacological and toxicological importance including antiviral (famciclovir), antimalarial (quinine), antitumour (methotrexate), and nicotine. The aim of this study was to investigate cytosolic aldehyde oxidase activity in human liver.Cytosolic AO was characterised using both the metabolism of N‐[(2‐dimethylamino)ethyl] acridine‐4‐carboxamide (DACA) and benzaldehyde to form DACA‐9(10H)‐acridone (quantified by HPLC with fluorescence detection) and benzoic acid (quantified spectrophotometrically). Thirteen livers (10 female, 3 male) were examined. The intrinsic clearance (Vmax/Km) of DACA varied 18‐fold (0.03‐0.50 ml/min/mg). Vmax ranged from 0.20‐3.10 nmol/ min/mg, and Km ranged from 3.5‐14.2 μM. In the same specimens, the intrinsic clearance for benzaldehyde varied 5‐fold (0.40‐1.8 ml/min/mg). Vmax ranged from 3.60‐12.6 nmol/min/mg and Km ranged from 3.6‐14.6 μM. Furthermore, there were no differences in AO activity between male and female human livers, nor was there any relationship to age of donor (range 29‐73 years), smoking status, or disease status. In conclusion, our results showed that there are variations in AO activity in human liver. These variations in aldehyde oxidase activity might reflect individual variations or they might be due to AO stability during processing and storage.
Fixed point and selection theorems in hyperconvex spaces
Authors: Khamsi M. , Kirk W. , Martinez Yañez Carlos 
Published: July 2002

It is shown that a set-valued mappingT∗T^{\ast }of a hyperconvex metric spaceMMwhich takes values in the space of nonempty externally hyperconvex subsets ofMMalways has a lipschitzian single valued selectionTTwhich satisfiesd(T(x),T(y))≤dH(T∗(x),T∗(y))d(T(x),T(y))\leq d_{H}(T^{\ast }(x),T^{\ast }(y))for allx,y∈Mx,y\in M. (HeredHd_{H}denotes the usual Hausdorff distance.) This fact is used to show that the space of all boundedλ\lambda-lipschitzian self-mappings ofMMis itself hyperconvex. Several related results are also obtained.
On a polynomial inequality of Kolmogoroff’s type
Authors: Bojanov B. , Varma A. 
Published: July 2002

We prove an inequality of the form
\[

‖

f

(
j
)

‖
2

≤
A
‖

f

(
m
)

‖
2

+
B
‖
f

‖
2

\|f^{(j)}\|^2\leq A\|f^{(m)}\|^2+B\|f\|^2

\]
for polynomials of degree

n
n

and any fixed

0
>
j
>
m
≤
n

0>j>m\leq n

. Here

‖
⋅
‖

\|\cdot \|

is the

L
2

L_2

-norm on

(
−
∞
,
∞
)

(-\infty ,\infty )

with a weight

e

−

t
2

e^{-t^2}

. The coefficients

A
A

and

B
B

are given explicitly and depend on

j
,
m

j,m

and

n
n

only. The equality is attained for the Hermite orthogonal polynomials

H
n

(
t
)

H_n(t)

.
Multiple path-valued conditional Yeh-Wiener integrals
Authors: Park Chull , Skoug David 
Published: July 2002

In this paper we establish various results involving parallel line-valued conditional Yeh-Wiener integrals of the type

E
(
F
(
x
)

|

x
(

s
j

,
⋅
)
=

η
j

(
⋅
)

E(F(x)|x(s_j,\boldsymbol {\cdot })=\eta _j(\boldsymbol {\cdot })

,

j
=
1
,
…
,
n
)

j=1,\dotsc ,n)

where

0
>

s
1

>
⋯
>

s
n

0>s_1>\cdots >s_n

. We then develop a formula for converting these multiple path-valued conditional Yeh-Wiener integrals into ordinary Yeh-Wiener integrals. Next, conditional Yeh-Wiener integrals for functionals

F
F

of the form
\[

F
(
x
)
=
exp
⁡

{

∫
0
S

∫
0
T

ϕ
(
s
,
t
,
x
(
s
,
t
)
)

d
t

d
s
}

F(x)=\exp \left \{\int _0^S\int _0^T\phi (s,t,x(s,t))\,dt\,ds\right \}

\]
are evaluated by solving an appropriate Wiener integral equation. Finally, a Cameron-Martin translation theorem is obtained for these multiple path-valued conditional Yeh-Wiener integrals.
Some remarks on the operator of Foias and Williams
Authors: Petrovic Srdjan 
Published: July 2002

In this paper we study the Foias-Williams operator

T
(

H

g

)
=

(

S

∗

a
m
p
;

H

g

0

a
m
p
;
S

)

\begin{equation*}T(H_{g})=\begin {pmatrix}S^{*} & H_{g} \ 0 & S \end{pmatrix} \end{equation*}

where

g
∈

L

∞

g\in L^{\infty }

, and

H

g

H_{g}

is a Hankel operator with symbol

g
g

. We exhibit a relationship between the similarity of

T
(

H

g

)

T(H_{g})

to a contraction and the rate of decay of

{

|

g

n

|

}

n
=
0

∞

\{|g_{n}|\}_{n=0}^{\infty }

, the absolute values of the Fourier coefficients of the symbol

g
g

.
Asymptotic analysis of Daubechies polynomials
Authors: Shen Jianhong , Strang Gilbert 
Published: July 2002

To study wavelets and filter banks of high order, we begin with the zeros ofBp(y){\mathbf {B}}_{p}(y). This is the binomial series for(1−y)−p(1-y)^{-p}, truncated afterppterms. Its zeros give thep−1p-1zeros of the Daubechies filter inside the unit circle, byz+z−1=2−4yz+z^{-1} = 2-4y. The filter hasppadditional zeros atz=−1z = -1, and this construction makes it orthogonal and maximally flat. The dilation equation leads to orthogonal wavelets withppvanishing moments. Symmetric biorthogonal wavelets (generally better in image compression) come similarly from a subset of the zeros ofBp(y){\mathbf {B}}_{p}(y). We study the asymptotic behavior of these zeros. Matlab shows a remarkable plot forp=70p = 70. The zeros approach a limiting curve|4y(1−y)|=1|4y(1-y)| = 1in the complex plane, which is the circle|z−z−1|=2|z-z^{-1}| = 2. All zeros have|y|≤1/2|y| \le 1/2, and the rightmost zeros approachy=1/2y = 1/2(corresponding toz=±iz= \pm i) with speedp−1/2p^{- 1/2}. The curve|4y(1−y)|=(4πp)1/2p|1−2y|1/p|4y(1-y)| = {(4 \pi p)}^{{1}/{2p}} \, |1-2y|^{ 1/p}gives a very accurate approximation for finitepp. The wide dynamic range in the coefficients ofBp(y){\mathbf {B}}_{p}(y)makes the zeros difficult to compute for largepp. Rescalingyyby44allows us to reachp=80p = 80by standard codes.
Isomorphisms of row and column finite matrix rings
Authors: Haefner J. , del Río A. , Simón J. 
Published: July 2002

This paper investigates the ring-theoretic similarities and the categorical dissimilarities between the ring

R
F
M
(
R
)

RFM(R)

of row finite matrices and the ring

R
C
F
M
(
R
)

RCFM(R)

of row and column finite matrices. For example, we prove that two rings

R
R

and

S
S

are Morita equivalent if and only if the rings

R
C
F
M
(
R
)

RCFM(R)

and

R
C
F
M
(
S
)

RCFM(S)

are isomorphic. This resembles the result of V. P. Camillo (1984) for

R
F
M
(
R
)

RFM(R)

. We also show that the Picard groups of

R
F
M
(
R
)

RFM(R)

and

R
C
F
M
(
R
)

RCFM(R)

are isomorphic, even though the rings

R
F
M
(
R
)

RFM(R)

and

R
C
F
M
(
R
)

RCFM(R)

are never Morita equivalent.
Averages of operators and their positivity
Authors: Nagisa Masaru , Wada Shuhei 
Published: July 2002

Let

T
T

be a bounded linear operator on a Hilbert space. We prove that

T
T

is positive, if there exists a positive integer

N
N

such that
\[

‖
I
−

1

N
+
1

∑

i
=
k

k
+
N

T

i

‖
,
‖
I
−

1

N
+
2

∑

i
=
k

k
+
N
+
1

T

i

‖
≤
1

\|I- {\frac {1 }{N+1 }}\sum \limits _{i=k}^{k+N} T^{i} \|, \|I- {\frac {1 }{N+2 }}\sum \limits _{i=k}^{k+N+1} T^{i} \| \le 1

\]
for any non-negative integer

k
k

. For several commuting operators, we can extend this result and get the similar statement.
A weak-type inequality of subharmonic functions
Authors: Choi Changsun 
Published: July 2002

We prove the weak-type inequality

λ
μ
(
u
+

|

v

|

≥
λ
)
≤
(
α
+
2
)

∫

∂
D

u

d
μ

\lambda \mu (u+|v|\ge \lambda )\le (\alpha +2) \int _{\partial D}u\,d\mu

,

λ
>
0

\lambda >0

, between a non-negative subharmonic function

u
u

and an

H

\mathbb {H}

-valued smooth function

v
v

, defined on an open set containing the closure of a bounded domain

D
D

in a Euclidean space

R

n

\mathbb R^n

, satisfying

|

v
(
0
)

|

≤
u
(
0
)

|v(0)|\le u(0)

,

|

∇
v

|

≤

|

∇
u

|

|\nabla v|\le |\nabla u|

and

|

Δ
v

|

≤
α
Δ
u

|\Delta v|\le \alpha \Delta u

, where

α
≥
0

\alpha \ge 0

is a constant. Here

μ
\mu

is the harmonic measure on

∂
D

\partial D

with respect to 0. This inequality extends Burkholder’s inequality in which

α
=
1

\alpha =1

and

H

=

R

ν

\mathbb {H}=\mathbb {R}^\nu

, a Euclidean space.
A note on Kamenev type theorems for second order matrix differential systems
Authors: Meng Fanwei , Wang Jizhong , Zheng Zhaowen 
Published: July 2002

Some oscillation criteria are given for the second order matrix differential system

Y
+
Q
(
t
)
Y
=
0

Y+Q(t) Y=0

, where

Y
Y

and

Q
Q

are

n
×
n

n\times n

real continuous matrix functions with

Q
(
t
)

Q(t)

symmetric,

t
∈
[

t
0

,
∞
)

t\in [t_0,\infty )

. These results improve oscillation criteria recently discovered by Erbe, Kong and Ruan by using a generalized Riccati transformation

V
(
t
)
=
a
(
t
)
{

Y
′

(
t
)

Y

−
1

(
t
)
+
f
(
t
)
I
}

V(t)=a(t)\{Y’(t) Y^{-1}(t) +f(t)I\}

, where

I
I

is the

n
×
n

n\times n

identity matrix,

f
∈

C
1

f\in C^1

is a given function on

[

t
0

,
∞
)

[t_0,\infty )

and

a
(
t
)
=
exp
⁡
{
−
2

∫
t

f
(
s
)

d
s
}

a(t)=\exp \{-2 \int ^t f(s)\,ds\}

.
The intersection of three spheres in a sphere and a new application of the Sato-Levine invariant
Authors: Ogasa Eiji 
Published: July 2002

Take transverse immersions

f
:

S

1

4

⨿

f:S^{4}_{1}\amalg

S

2

4

⨿

S^{4}_{2}\amalg

S

3

4

↬

S

6

S^{4}_{3}\looparrowright S^{6}

such that (1)

f
|

S

i

4

f\vert S^{4}_{i}

is an embedding, (2)

f
(

S

i

4

)
∩
f
(

S

j

4

)
≠
∅

f(S^{4}_{i})\cap f(S^{4}_{j})\neq \varnothing

and

f
(

S

i

4

)
∩
f
(

S

j

4

)

f(S^{4}_{i})\cap f(S^{4}_{j})

is connected, and (3)

f
(

S

1

4

)
∩
f
(

S

2

4

)
∩
f
(

S

3

4

)

f(S^{4}_{1})\cap f(S^{4}_{2})\cap f(S^{4}_{3})

=
∅

=\varnothing

. Then we obtain three surface-links

L

i

L_{i}

= (

f

−
1

(
f
(

S

i

4

)
∩
f
(

S

j

4

)
)

f^{-1}(f(S^{4}_{i})\cap f(S^{4}_{j}))

,

f

−
1

(
f
(

S

i

4

)
∩
f
(

S

k

4

)
)

f^{-1}(f(S^{4}_{i})\cap f(S^{4}_{k}))

) in

S

i

4

S^{4}_{i}

, where

(
i
,
j
,
k
)

(i,j,k)

=(1,2,3), (2,3,1), (3,1,2). We prove that, we have the equality

β
(

L

1

)
+

\beta (L_{1})+

β
(

L

2

)
+

\beta (L_{2})+

β
(

L

3

)
=
0
,

\beta (L_{3})=0,

where

β
(

L

i

)

\beta (L_{i})

is the Sato-Levine invariant of

L

i

L_{i}

, if all

L

i

L_{i}

are semi-boundary links.
High order moments of character sums
Authors: Cochrane Todd , Zheng Zhiyong 
Published: July 2002

We establish the upper bound

1

p
−
1

∑

χ
≠

χ

o

|

∑

x
=
a
+
1

a
+
B

χ
(
x
)

|

2
k

≪

ϵ
,
k

p

k
−
1
+
ϵ

+

B

k

p

ϵ

,

\begin{equation*}\frac {1}{p-1} \sum _{\chi \ne \chi _{o}}\big | \sum _{x=a+1}^{a+B} \chi (x) \big |^{2k} \ll _{\epsilon ,k} p^{k-1 +\epsilon } + B^{k} p^{\epsilon }, \end{equation*}

with

p
p

a prime and

k
k

any positive integer, the sum being over all nonprincipal multiplicative characters

(
mod

p
)

\pmod p

.
An application of Schauder’s fixed point theorem with respect to higher order BVPs
Authors: Wong Fu-Hsiang 
Published: July 2002

We shall provide conditions on the functionf(t,u1,⋯,un−1)f(t,u_{1},\cdots , u_{n-1}). The higher order boundary value problem({BVP}){(E) u(n)(t)+f(t,u(t),u(1)(t),⋯,u(n−2)(t))=0 for t∈(0,1) and n≥2, (BC) {u(i)(0)=0, 0≤i≤n−3, αu(n−2)(0)−βu(n−1)(0)=0, γu(n−2)(1)+δu(n−1)(1)=0\begin{equation*}\begin {cases}(E)~~ u^{(n)}(t)+ f(t, u(t),u^{(1)}(t),\cdots ,u^{(n-2)}(t))=0~~~~~\mathrm {~for~}~~~~~t\in (0,1)~~~~\mathrm {and}~~~~~~n\ge 2,\ (BC)~~ \begin {cases}u^{(i)}(0)=0,~~~~~0\le i \le n-3,\ \alpha u^{(n-2)}(0)-\beta u^{(n-1)}(0)=0,\ \gamma u^{(n-2)}(1)+\delta u^{(n-1)}(1)=0\end{cases} \end{cases} \tag {{BVP}}\end{equation*}has at least one solution.
A central limit theorem for Markov chains and applications to hypergroups
Authors: Gallardo Léonard 
Published: July 2002

Let

(

X
n

)

(X_n)

be a homogeneous Markov chain on an unbounded Borel subset of

R

\mathbb {R}

with a drift function

d
d

which tends to a limit

m
1

m_1

at infinity. Under a very simple hypothesis on the chain we prove that

n

−
1

/

2

(

X
n

−

∑

k
=
1

n

d
(

X

k
−
1

)
)

\displaystyle n^{-1/2} (X_n - \sum ^ n_{k=1} d(X_{k-1}))

converges in distribution to a normal law

N
(
0
,

σ
2

)

N (0, \sigma ^2)

where the variance

σ
2

\sigma ^2

depends on the asymptotic behaviour of

(

X
n

)

(X_n)

. When

d
−

m
1

d - m_1

goes to zero quickly enough and

m
1

≠
0

m_1 \neq 0

, the random centering may be replaced by

n

m
1

.

n m_1.

These results are applied to the case of random walks on some hypergroups.
Degenerations for modules over representation-finite algebras
Authors: Zwara Grzegorz 
Published: July 2002

Let

A
A

be a representation-finite algebra. We show that a finite dimensional

A
A

-module

M
M

degenerates to another

A
A

-module

N
N

if and only if the inequalities

dim

K

⁡
H
o

m

A

(
M
,
X
)
≤

dim

K

⁡
H
o

m

A

(
N
,
X
)

\dim _{K} Hom_{A}(M,X)\leq \dim _{K} Hom_{A}(N,X)

hold for all

A
A

-modules

X
X

. We prove also that if

Ext

A

1

⁡
(
X
,
X
)
=
0

\operatorname {Ext}_{A}^{1}(X,X)=0

for any indecomposable

A
A

-module

X
X

, then any degeneration of

A
A

-modules is given by a chain of short exact sequences.
Remarkable asymmetric random walks
Authors: Mattner L. 
Published: July 2002

There exists an asymmetric probability measure

P
P

on the real line with

P

∗
n

(
]
0
,
∞
[
)
+
(
1

/

2
)

P

∗
n

(
{
0
}
)
=
1

/

2

P^{\ast n} (]0,\infty [) + (1/2) P^{\ast n} (\{0\}) = 1/2

for every

n
∈

N

n \in \mathbf {N}

.

P
P

can be chosen absolutely continuous and

P
P

can be chosen to be concentrated on the integers. In both cases,

P
P

can be chosen to have moments of every order, but

P
P

cannot be determined by its moments.
Asymptotic regularity of Daubechies’ scaling functions
Authors: Lau Ka-Sing , Sun Qiyu 
Published: July 2002

Let

ϕ
N

\phi _N

,

N
≥
1

N\ge 1

, be Daubechies’ scaling function with symbol

(

1
+

e

−
i
ξ

2

)

N

Q
N

(
ξ
)

\big ({1+e^{-i\xi }\over 2}\big )^N Q_N(\xi )

, and let

s
p

(

ϕ
N

)
,
0
>
p
≤
∞

s_p(\phi _N),0>p\le \infty

, be the corresponding

L
p

L^p

Sobolev exponent. In this paper, we make a sharp estimation of

s
p

(

ϕ
N

)

s_p(\phi _N)

, and we prove that there exists a constant

C
C

independent of

N
N

such that
\[

N
−

ln
⁡

|

Q
N

(
2
π

/

3
)

|

ln
⁡
2

−

C
N

≤

s
p

(

ϕ
N

)
≤
N
−

ln
⁡

|

Q
N

(
2
π

/

3
)

|

ln
⁡
2

.

N-{\ln |Q_N(2\pi /3)|\over \ln 2}-{C\over N}\le s_p(\phi _N)\le N-{\ln |Q_N(2\pi /3)|\over \ln 2}.

\]
This answers a question of Cohen and Daubeschies (Rev. Mat. Iberoamericana, 12(1996), 527-591) positively.
Geometry of a crossed product
Authors: Nikolaev Igor 
Published: July 2002

We introduce a continuous dimension functionα:∙→R\alpha : \bullet \to \mathbb {R}on the Grothendieck groupK0K_0over the crossed productC∗C^*-algebraC(X)⋊ϕZC(X)\rtimes _{\phi }\mathbb {Z}. The functionα\alphahas an elegant geometry: on every minimal flowϕt\phi ^tit takes the value of the “rotation number" ofϕt\phi ^t; such a problem was posed in 1936 by A. Weil.
Semi-classical limit for random walks
Authors: Porod Ursula , Zelditch Steve 
Published: July 2002

Let

(
G
,
μ
)

(G, \mu )

be a discrete symmetric random walk on a compact Lie group

G
G

with step distribution

μ
\mu

and let

T

μ

T_{\mu }

be the associated transition operator on

L
2

(
G
)

L^2(G)

. The irreducibles

V

ρ

V_{\rho }

of the left regular representation of

G
G

on

L
2

(
G
)

L^2(G)

are finite dimensional invariant subspaces for

T

μ

T_{\mu }

and the spectrum of

T

μ

T_{\mu }

is the union of the sub-spectra

σ
(

T

μ

↿

V

ρ

)

\sigma (T_{\mu }\upharpoonleft _{V_{\rho }})

on the irreducibles, which consist of real eigenvalues

{

λ

ρ
1

,
.
.
.
,

λ

ρ
dim
⁡

V

ρ

}

\{ \lambda _{\rho 1},...,\lambda _{\rho \dim V_{\rho }}\}

. Our main result is an asymptotic expansion for the spectral measures
\[

m

ρ

μ

(
λ
)
:=

1

dim
⁡

V

ρ

∑

j
=
1

dim
⁡

V

ρ

δ
(
λ
−

λ

ρ
j

)

m_{\rho }^{\mu }(\lambda ) := \frac {1}{\dim V_{\rho }} \sum _{j=1}^{\dim V_{\rho }} \delta (\lambda - \lambda _{\rho j})

\]
along rays of representations in a positive Weyl chamber

t

+
∗

\mathbf {t}^*_+

, i.e. for sequences of representations

k
ρ

k \rho

,

k
∈

N

k\in \mathbb {N}

with

k
→
∞

k\rightarrow \infty

. As a corollary we obtain some estimates on the spectral radius of the random walk. We also analyse the fine structure of the spectrum for certain random walks on

U
(
n
)

U(n)

(for which

T

μ

T_{\mu }

is essentially a direct sum of Harper operators).
Homology manifold bordism
Authors: Johnston Heather , Ranicki Andrew 
Published: July 2002

The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact

A
N
R

ANR

homology manifolds of dimension

≥
6

\geq 6

is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston.

First, we establish homology manifold transversality for submanifolds of dimension

≥
7

\geq 7

: if

f
:
M
→
P

f:M \to P

is a map from an

m
m

-dimensional homology manifold

M
M

to a space

P
P

, and

Q
⊂
P

Q \subset P

is a subspace with a topological

q
q

-block bundle neighborhood, and

m
−
q
≥
7

m-q \geq 7

, then

f
f

is homology manifold

s
s

-cobordant to a map which is transverse to

Q
Q

, with

f

−
1

(
Q
)
⊂
M

f^{-1}(Q) \subset M

an

(
m
−
q
)

(m-q)

-dimensional homology submanifold.

Second, we obtain a codimension

q
q

splitting obstruction

s
Q

(
f
)
∈
L

S

m
−
q

(
Φ
)

s_Q(f) \in LS_{m-q}(\Phi )

in the Wall

L
S

LS

-group for a simple homotopy equivalence

f
:
M
→
P

f:M \to P

from an

m
m

-dimensional homology manifold

M
M

to an

m
m

-dimensional Poincaré space

P
P

with a codimension

q
q

Poincaré subspace

Q
⊂
P

Q \subset P

with a topological normal bundle, such that

s
Q

(
f
)
=
0

s_Q(f)=0

if (and for

m
−
q
≥
7

m-q \geq 7

only if)

f
f

splits at

Q
Q

up to homology manifold

s
s

-cobordism.

Third, we obtain the multiplicative structure of the homology manifold bordism groups

Ω
∗
H

≅

Ω
∗

T
O
P

[

L
0

(

Z

)
]

\Omega ^H_*\cong \Omega ^{TOP}_*[L_0(\mathbb Z)]

.
An estimate for weighted Hilbert transform via square functions
Authors: Petermichl S. , Pott S. 
Published: July 2002

We show that the norm of the Hilbert transform as an operator on the weighted space

L
2

(
w
)

L^2(w)

is bounded by a constant multiple of the

3

/

2

3/2

power of the

A
2

A_2

constant of

w
w

, in other words by

c

sup
I

(
⟨
ω

⟩
I

⟨

ω

−
1

⟩
I

)

3

/

2

c\, \sup _I (\langle \omega \rangle _I \langle \omega ^{-1} \rangle _I)^{3/2}

. We also give a short proof for sharp upper and lower bounds for the dyadic square function.
A problem of Foldes and Puri on the Wiener process
Authors: Shi Z. 
Published: July 2002

Let

W
W

be a real-valued Wiener process starting from 0, and

τ
(
t
)

\tau (t)

be the right-continuous inverse process of its local time at 0. Földes and Puri [3] raise the problem of studying the almost sure asymptotic behavior of

X
(
t
)
=

∫
0

τ
(
t
)

1

{

|

W
(
u
)

|

≤
α
t
}

d
u

X(t)=\int _0^{\tau (t)} \mathbb {1}_{\{|W(u)| \le \alpha t\}}\,du

as

t
t

tends to infinity, i.e. they ask: how long does

W
W

stay in a tube before “crossing very much” a given level? In this note, both limsup and liminf laws of the iterated logarithm are provided for

X
X

.
The local dimensions of the Bernoulli convolution associated with the golden number
Authors: Hu Tian-You 
Published: July 2002

Let

X
1

,

X
2

,
…

X_1,X_2,\dotsc

be a sequence of i.i.d. random variables each taking values of 1 and

−
1

-1

with equal probability. For

1

/

2
>
ρ
>
1

1/2>\rho >1

satisfying the equation

1
−
ρ
−
⋯
−

ρ
s

=
0

1-\rho -\dotsb -\rho ^s=0

, let

μ
\mu

be the probability measure induced by

S
=

∑

i
=
1

∞

ρ
i

X
i

S=\sum _{i=1}^\infty \rho ^iX_i

. For any

x
x

in the range of

S
S

, let
\[

d
(
μ
,
x
)
=

lim

r
→

0
+

log
⁡
μ
(
[
x
−
r
,
x
+
r
]
)

/

log
⁡
r

d(\mu ,x)=\lim _{r\to 0^+}\log \mu ([x-r,x+r])/\log r

\]
be the local dimension of

μ
\mu

at

x
x

whenever the limit exists. We prove that
\[

α
∗

=
−

log
⁡
2

log
⁡
ρ

and

α
∗

=
−

log
⁡
δ

s
log
⁡
ρ

−

log
⁡
2

log
⁡
ρ

,

\alpha ^*=-\frac {\log 2}{\log \rho }\quad \text {and}\quad \alpha _*=-\frac {\log \delta }{s\log \rho }-\frac {\log 2}{\log \rho },

\]
where

δ
=
(

5

−
1
)

/

2

\delta =(\sqrt {5}-1)/2

, are respectively the maximum and minimum values of the local dimensions. If

s
=
2

s=2

, then

ρ
\rho

is the golden number, and the approximate numerical values are

α
∗

≈
1.4404

\alpha ^*\approx 1.4404

and

α
∗

≈
0.9404

\alpha _*\approx 0.9404

.
Nonexistence of global solutions of a nonlinear hyperbolic system
Authors: Deng Keng 
Published: July 2002

Consider the initial value problem

u

t
t

=
Δ
u
+
|
v

|

p

,

a
m
p
;

v

t
t

=
Δ
v
+
|
u

|

q

,

a
m
p
;
x
∈

R

n

,

a
m
p
;
t
>
0
,

u
(
x
,
0
)
=
f
(
x
)
,

a
m
p
;
v
(
x
,
0
)
=
h
(
x
)
,

a
m
p
;

a
m
p
;

u

t

(
x
,
0
)
=
g
(
x
)
,

a
m
p
;

v

t

(
x
,
0
)
=
k
(
x
)
,

\begin{equation*} \begin {array}{llll} u_{tt} = \Delta u+\vert v\vert ^{p}, & v_{tt} = \Delta v +\vert u\vert ^{q}, &x\in \mathbb {R}^{n},&t>0,\\ u(x,0)=f(x),&v(x,0)=h(x),&{}&{}\\ u_{t}(x,0) = g(x), &v_{t}(x,0) = k(x), \end{array} \end{equation*}

with

1
≤
n
≤
3

1\le n\le 3

and

p
,
q
>
0

p,q>0

. We show that there exists a bound

B
(
n
)
(
≤
∞
)

B(n) (\le \infty )

such that if

1
>
p
q
>
B
(
n
)

1>pq>B(n)

all nontrivial solutions with compact support blow up in finite time.
Some ramifications of a theorem of Boas and Pollard concerning the completion of a set of functions in 𝐿²
Authors: Kazarian K. , Zink Robert 
Published: July 2002

About fifty years ago, R. P. Boas and Harry Pollard proved that an orthonormal system that is completable by the adjunction of a finite number of functions also can be completed by multiplying the elements of the given system by a fixed, bounded, nonnegative measurable function. In subsequent years, several variations and extensions of this theorem have been given by a number of other investigators, and this program is continued here. A mildly surprising corollary of one of the results is that the trigonometric and Walsh systems can be multiplicatively transformed into quasibases for

L

1

[
0
,
1
]

L^{1}[0,1]

.